Lecture Notes in Economics
and Mathematical Systems
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H.P. Künzi
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Fachbereich Wirtschaftswissenschaften
Fernuniversität Hagen
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Institut für Mathematische Wirtschaftsforschung (IMW)
Universität Bielefeld
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Editorial Board:
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628
Martin Albrecht
Supply Chain
Coordination Mechanisms
New Approaches for Collaborative Planning
ABC
Dr. Martin Albrecht
PAUL HARTMANN AG
martin.albrecht@hartmann.info
ISSN 0075-8442
ISBN 978-3-642-02832-8
e-ISBN 978-3-642-02833-5
DOI 10.1007/978-3-642-02833-5
Springer Heidelberg Dordrecht London New York
Library of Congress Control Number: 2009931327
c Springer-Verlag Berlin Heidelberg 2010
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For Rita and Amalia Isabel
Foreword
Inter-organizational supply chains have to coordinate their material, information,
and financial flows efficiently to be competitive. However, legally independent supply chain (SC) partners are often reluctant to share critical data such as costs or
capacity utilization, which is a prerequisite for central planning or hierarchical
planning – the planning paradigm of today’s Advanced Planning Systems (APS).
Consequently, concepts for collaborative planning are needed, considering a joint
decision making process for aligning plans of individual SC members with the aim
of achieving coordination in the light of information asymmetry.
This is the starting point and challenge of the PhD thesis of Martin Albrecht
because little is known about how to design a solution for this difficult decision
problem. Starting from an initial solution – that may be generated by upstream planning – improved solutions are looked for. This is achieved by computer-supported
negotiations, i.e., an exchange of different order proposals within the planning interval among the SC partners involved, where partners are free to accept or reject
proposals.
One challenge in this negotiation process is to find new proposals and counterproposals which have a good chance of acceptance while improving the competitive
position of a SC as a whole. Here, Albrecht devised new generic coordination
schemes for planning tasks which can be modeled either by Linear Programming
(LP) or Mixed Integer Linear Programming. For the LP case finite convergence to
the optimum has been proved.
While previous research on collaborative planning stopped with a clever coordination scheme Albrecht also considered a further, very important aspect of
negotiations: How to get the partners to tell the truth when exchanging information
and to accept a very promising solution for the supply chain as a whole. Formally
speaking, coordination mechanisms are needed where the coordination schemes can
be embedded. One of the coordination mechanisms advocated by Albrecht is the
surplus sharing by an initially agreed upon lump sum payment to one party. He has
been able to show that the corresponding mechanism results in truth-telling as a
weakly dominant strategy. The reader can expect both analytical results as well as
computational tests of collaborative planning schemes for various lot-sizing problems including some from industrial practice – and there is a lot more to be gained
vii
viii
Foreword
from reading this thesis but I will not reveal more details here. I wish this excellent
thesis a wide audience of interested and very satisfied readers and a large impact on
collaborative planning.
Hamburg, April 2009
Hartmut Stadtler
Preface
When I started my research, most known collaborative planning approaches dealing
with mathematical programming models were based on a serious oversimplification
of reality: They presumed a team setting, where parties honestly disclose information and sometimes even accept deteriorations if this benefits the supply chain as a
whole. One of the major contributions of this work has been to relax this assumption.
I have developed mechanisms, which achieve coordination despite self-interested
behavior of parties. Without wanting to relativize the importance of this contribution, I would like to point out the existence of a particular real-world team: The
people supporting me when I was writing this thesis.
First of all, I am indebted to Prof. Dr. Hartmut Stadtler. He not only set the example for my research, but also provided (sometimes incredibly) generous advice and
professional and personal support. Among many other things, he has patiently read
my papers many times and supplied several insightful suggestions at all stages of
this work. I am also grateful to Prof. Dr. Karl-Werner Hansmann for his willingness
to serve as the co-referee for this thesis.
Apart from my academic advisers, I am indebted to my colleagues and collaborating researchers. Particularly, I want to thank Carolin Püttmann for her great
teamwork in the EU-project InCoCo-S, for listening to many of my (not always fully
worked out) ideas, and for carefully proofreading the whole dissertation. Dr. Bernd
Wagner and Volker Windeck also read parts of the thesis and provided many valuable suggestions. Last, but not least, I am thankful to Prof. Dr. Heinrich Braun and
Benedikt Scheckenbach from the SAP AG for challenging discussions and for making available the real-world test data used in this work.
I also thank the Gesellschaft für Logistik und Verkehr for subsidizing the printing
of this work.
Certainly most important for this dissertation has been my family, although not
interested in supply chain management at all. My parents supported my education,
without expecting anything in return. My wife Rita not only renounced to much
shared time, but encouraged me with all her love to keep on researching until I have
(finally) been satisfied with this work.
Thank you, everybody.
Heidenheim, May 2009
Martin Albrecht
ix
Contents
Abbreviations .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . xiii
Nomenclature .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . xv
1
Introduction .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .
1.1 Motivation and Goals of This Work . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .
1.2 Methodology .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .
1.3 Outline . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .
1
1
2
2
2
Supply Chain Planning and Coordination .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 5
2.1 Supply Chain Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 5
2.1.1 Definitions and Overview .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 5
2.1.2 Master Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 8
2.2 Model Formulations for Master Planning . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 9
2.2.1 Generic Master Planning Model .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 10
2.2.2 Extension to Lot-Sizing .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 12
2.3 Decentralized Planning and Coordination .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 20
2.3.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 20
2.3.2 Decentralized Supply Chain Planning .. . . . . . . . . . . . . . .. . . . . . . . . . . 24
2.3.3 Upstream Vs. Collaborative Planning . . . . . . . . . . . . . . . .. . . . . . . . . . . 30
3
Coordination Mechanisms for Supply Chain Planning . . . . . . . .. . . . . . . . . . . 35
3.1 Symmetric Information .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 35
3.1.1 Non-cooperative Game Theory .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 36
3.1.2 Cooperative Game Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 41
3.2 One-Sided Information Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 43
3.2.1 Signaling .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 43
3.2.2 Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 45
3.3 Multilateral Information Asymmetry .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 48
3.3.1 Auctions and Their Application to Supply Chain Coordination 48
3.3.2 Mechanisms with Focus on Proposal Generation .. . .. . . . . . . . . . . 51
xi
xii
Contents
4
New Coordination Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 63
4.1 Generic Scheme for Linear Programming and Analytical Results . . . . . 64
4.1.1 Version with Iterative, Unilateral Exchange
of Cost Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 64
4.1.2 Version with One-Shot Exchange of Cost Information .. . . . . . . . 80
4.2 Scheme for Uncapacitated Dynamic Lot-Sizing
and Analytical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 82
4.3 Application to Master Planning .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 99
4.3.1 Linearization .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 99
4.3.2 Adaptation to Master Planning . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .103
4.3.3 Generic Modifications.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .111
4.3.4 Modifications for Master Planning . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .114
4.4 Customizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .120
4.4.1 Master Planning with Lot-Sizing . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .120
4.4.2 Voluntary Compliance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .122
4.4.3 Lost Sales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .123
4.4.4 Multiple Suppliers .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .126
5
New Coordination Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .129
5.1 Surplus Sharing Determined by the Informed Party . . . . . . . . .. . . . . . . . . . .130
5.2 Surplus Sharing Determined by Lump-Sum Payments . . . . . .. . . . . . . . . . .133
5.3 Surplus Sharing by a Double Auction . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .141
5.4 Comparison of Mechanisms and Discussion . . . . . . . . . . . . . . . . .. . . . . . . . . . .149
5.5 Application with Rolling Schedules . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .151
6
Computational Tests of Coordination Schemes . . . . . . . . . . . . . . . . .. . . . . . . . . . .155
6.1 General Master Planning Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .155
6.1.1 Generation of Test Instances and Performance Indicators.. . . . .155
6.1.2 Analysis of Solutions for the Generic Scheme .. . . . . .. . . . . . . . . . .162
6.1.3 Analysis of Solutions for the Modified Scheme . . . . .. . . . . . . . . . .164
6.2 Uncapacitated Lot-Sizing Problem . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .169
6.2.1 Generation of Test Instances .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .169
6.2.2 Analysis of Solutions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .171
6.3 Multi-level Capacitated Lot-Sizing Problem . . . . . . . . . . . . . . . . .. . . . . . . . . . .174
6.4 Models for Campaign Planning .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .179
6.4.1 Generation of Test Instances .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .179
6.4.2 Analysis of Solutions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .180
6.5 Real-World Supply Chain Planning Problems . . . . . . . . . . . . . . .. . . . . . . . . . .184
6.5.1 Planning Problems and Model Formulation . . . . . . . . .. . . . . . . . . . .185
6.5.2 Analysis of Solutions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .192
7
Summary and Outlook .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .197
References .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .201
Abbreviations
AGC
Average gap closure achieved by the scheme
AGS
Average gap after the application of the scheme
AGU
Average gap of the uncoordinated solution
APO
Advanced planner and optimizer
APS
Advanced planning system
arb.
Arbitrary
B
Buyer
BOM
Bill of material
B2B
Business-to-business
CLSP
Capacitated lot-sizing problem
CLSPL Capacitated lot-sizing problem with linked lot sizes
cos.
Cosinus
CSLP
Continuous setup lot-sizing problem
CU
Capacity unit
DLSP
Discrete lot-sizing and scheduling problem
EOQ
Economic order quantity
GC
Gap closure achieved by the scheme
GLSP
General lot-sizing and scheduling problem
GM
Generic master planning model
GS
Gap after the application of the scheme
H
High
IGFR
Increasing generalized failure rate
IP
Informed party
L
Lot size (driver)
L
Low (type of demand forecast)
LB
Lower bound
LP
Linear programming
MLCLSP Multi-level capacitated lot-sizing problem
MLCLSPL Multi-level capacitated lot-sizing problem with linked lot sizes
MLPLSP Multi-level proportional lot-sizing and scheduling problem
MLULSP Multi-level uncapacitated lot-sizing problem
MINLP Mixed-integer nonlinear programming
MIP
Mixed-integer programming
xiii
xiv
MU
NLP
OEM
Q
RHS
RP
PPM
S
Sched.
SMP
SNP
SOS2
spl.
T
TBO
TC
TS
U
UB
UT
Abbreviations
Monetary unit
Nonlinear programming
Original equipment manufacturer
Quantity
Right-hand side
Reporting party
Production process model
Supplier
Scheduling
Single machine processor
Supply network planning
Special ordered set of type 2
Split
Time
Time between orders
Time for solving the centralized model
Time for running the scheme
Unit
Upper bound
Unit time
Nomenclature
Indices, Sets, and Index Sets
˘E
Set of proposals already found
˘iE
Set of proposals previously generated by the scheme
˘B
Set of proposals identified by the buyer
˘S
Set of proposals identified by the supplier
˘ up
Set of proposals identified by GMupS
E
i
Set of cost changes associated with proposals for central resource use
a
Arc linking two locations
ABa
Location at the beginning of arc a
AEa
Location at the end of arc a
Br .x/ r-neighborhood of x
CS
Set of solutions identified by the scheme for the MLULSP
DS
Set of proposals with delayed supply compared to the starting proposal
ES
Set of proposals with early supply compared to the starting proposal
f
Superindex denoting the first proposal generated
i
Decentralized parties
i
Suppliers
init
Superindex denoting the initial solution
J
Set of items
j
Items or operations
JB
Set of items produced by the buyer
JD
Set of items supplied
JE
Subset of items sold to external customers
JlE
Items sold at location l
S
Set of items produced by the supplier
J
Jm
Set of items produced on resource m
L
Set of locations
l
Location, l 2 L
M
Set of resources
m
Resources (e.g., personnel, machines, production lines)
MB
Set of resources of the buyer
MS
Set of resources of the supplier
NDS Set of proposals without delayed supply compared to the starting proposal
xv
xvi
Nomenclature
NES
Set of proposals without early supply compared to the starting proposal
new
Superindex indicating a new proposal
P
Set of PPM
P
Set of parties
p
PPM, p 2 P
Pc
Set of parties for which coordinated proposals have been chosen
PS
Set of suppliers
PB
Set of supply proposals optimal for the buyer subject to any N B
Rj
Set of predecessor items of item j
S
Set of customer classes
s
Customer class, s 2 S
Sj
Set of immediate successors of item j in the BOM
st
Superindex indicating the starting solution
T
Set of periods
t
Periods
Xis
Subset of feasible solutions to CS1i
XiE
Set of proposals found so far
Subset of vertex solutions to DPi
Xivd
Xiv
Set of vertex solutions identified so far
Parameters and Random Variables
tiB
Periods between subsequent setups of the buyer
tiS
Periods between subsequent setups of the supplier
Unit penalty costs for arbitrary deviations
O ; A ; B ; C Weights
i e
Scalars
i e
Scalars
aj
Cumulated capacity requirements of an item j
ei
Number of different solutions found so far for party i
ej
Average secondary demand for item j
Proposal
PB
Proposal out of PB with the same N B and N S as in the systemwide
optimal solution
ie
Proposal e for the central resource use by party i
ij
Resource use for the j th proposal generated by party i
p
Production lead time of PPM p
aj
Transportation lead time for item j along arc a
j
Lead time for item j
Type of demand forecast
hB
Lower bound for the probability distribution of hB
b
be
lc
blcj
j
e
i
a; b
Maximum costs for backorders that are caused by a shortage in the supply
or the production of item j
Potential cost impact of backorders in the supply of item j
Cost effects of the proposal e by party i
Lower and upper bounds for S
Nomenclature
xvii
Ai
Use of the central resources by decisions xi
amj
Capacity needed on resource m for one unit of item j
amp
Capacity needed on resource m for one unit of PPM p
b
Purchase price
Bk1 ,Bk2 Prior knowledge of parties #1, #2 about the other party’s bids for proposals
k (upper bounded by bj1 ,bj2 )
Bids by parties #1, #2 for proposal k
Total amount of the central resources
Bi
Use of the decentralized resources by decisions xi
bi
Total amount of the decentralized resources
bjt
Large number, not limiting feasible lot size of item j in period t
bpt
Big number indicating the maximum production quantity of PPM p in
period t
blj0
Amount of backorders for item j at the beginning of the planning interval
bljT
Amount of backorders for item j at the end of the planning interval
blcj
Backorder costs for one unit of item j in a period
blljs
Backorder costs for one unit of item j of customer class s in a period at
location l
bsj
Batch size for item j
bsp
Batch size for PPM p
cB
Buyer’s costs of the systemwide optimal solution
csys
Overall costs in the systemwide optimum
PB
cB
Buyer’s costs for the proposal out of PB with the same N B and N S as in
the systemwide optimal solution
c0
Production costs at the supplier’s site
ci
Costs associated with decisions xi
cce n;n Costs of the solution to the centralized model for test instance n
ccor;n;i Costs after i iterations of the scheme
csys ./ Systemwide costs resulting from an implementation of CS
csys
Costs for the best solution out of CS
PB
csys
Costs of an implementation of the best proposal out of PB
cunc;n Costs of the uncoordinated solution (usually determined by upstream
planning)
cam0j Initial campaign quantity for item j
cb e
Buyer’s costs change of the previous proposal e compared to the initial
solution
conoc Average ratio between backorder and overtime costs
cp
Unit penalty costs
i
cpB
Penalty costs for supplier i
cs e
Supplier’s costs change of the previous proposal e
csie
Costs of proposal e for supplier i
csel
Costs for one unit of storage capacity increase at location l
csslj Penalty costs for one unit of stock below the required safety stock of item
j at location l
bk1 ,bk2
b0
xviii
Nomenclature
ctaj
cvp
D
d
dj0
djt
dljst
Transportation costs for one unit of item j along arc a
Variable production costs of PPM p
Demand (random variable)
Demand per unit time
Value of the j -th dimension of Ai xist xi
Primary, gross demand for item j in period t
Primary, gross demand for item j of customer class s in period t at location l
dlbejt Deviation of proposal e that is due to lost sales and relevant for the buyer
dlsejt
Deviation of proposal e due to lost sales for the supplier
cum
ejt
Cumulated secondary demand for item j in period t
F ./
Cumulated density function
f ./
Probability density function
fmt
Randomly generated factor determining capacity profiles
g max Expected surplus of the best solution identified by the scheme
g mech Expected surplus that can be realized by the mechanism
gRP .l/ Gains of the RP from coordination subject to l
h
Unit holding cost per unit time
hB
Limit for acceptable hB
hj
Holding cost for one unit of item j in a period
hS
Holding cost of the supplier
hlj
Holding cost for one unit of item j at location l in a period
hbj
Buyer’s unit costs for inventory holding of the supplied item j
ij0
Inventory of item j at the beginning of the planning interval
P rev
ijt
Inflow of item j in period t originating from earlier production periods
i clmax Maximum storage capacity at location l
i clj
Consumption of storage capacity at location l by one unit of item j
i celmax Maximum extension of storage capacity at location l
i nilj Inventory of item j at location l at the beginning of the planning interval
K
Constant of an arbitrary value (e.g., 1) used for the correct transformation
of the unit of aj
k
Parameter for surplus sharing in the sealed bid double auction
kj0
Value of the j -th dimension of kiT
kmt
Available capacity of resource m in period t
L; LO
Lump sum payment
l; lO
Markup (above the lump sum)
L
Optimal lump sum
L1 ,L2 Prior knowledge of parties #1, #2 about the leeway in general, (upper
bounded by lj1 )
1
2
Lk ,Lk Prior knowledge of parties #1, #2 about the leeway for proposals k (upper
i
L
lk
lb
bounded by lj1 )
Lump sum required by party i
Markup for proposal k
Lower bound
Nomenclature
0
xix
lbjd
Lower bound for dj0
0
lbjk
Lower bound for kj0
lmaxljs Maximum lateness for demand fulfillment of item j of customer class s at
location l
lscj
Costs for lost sales of one unit of item j in a period
lsljs
Costs for lost sales of item j of customer class s at location l
m
Number of approximation intervals
m1 ,m2 (General) markdowns chosen by parties #1, #2
m1k ,m2k Markdowns chosen by parties #1, #2 for proposal k
Mi
Vector made up of big numbers that exceed marginal cost savings resulting
from increases in central resource use
mjt
Big number, denoting the maximum cost change per unit deviation in the
supply quantities
mfjp Material flow of item j from PPM p
minlotp Minimum lot size for item p
N B;up Number of setups in the upstream planning solution
NkB
Number of buyer’s setups in the planning interval for items k 2 SJ D \ J B
NjS
Number of supplier’s setups in the planning interval for items j 2 J D
pre
nk
Number of items preceding item k
Number of the buyer’s orders within tiS ; tiSC1
oi
Overtime costs for one unit of resource m
ocm
P
Set of decentralized parties
p
Selling price
P .Q/ Purchase price dependent on the purchase quantity Q
r
Parameter denoting the ratio between N B and N S (rounded down)
r .l/
Function that maps the expected reduction of S with l
cum
rjk
Number of units of item j required to produce one unit of the (direct or
indirect) successor item k
rjk
Number of units of item j required to produce one unit of the immediate
successor item k
S
Subset of decentralized parties
S
Systemwide surplus from coordination (random variable)
s
Selling price
s
Share of the revenue generated
sk1 ,sk2 Savings by parties #1, #2 for proposal k
Si
Marginal
i i surplus from coordination for party i defined within the interval
a ; b (random variable)
S sys
Expected surplus for the whole system (random variable)
sji
Savings of party i with proposal j
Sk
Systemwide surplus for proposal k (random variable)
sc
Setup cost
scB
Setup cost of the buyer
scj
Setup cost for a lot of item j
scS
Setup cost of the supplier
xx
Nomenclature
stj
tiB
TL
TU
tiS
C
tcB
tcSC
u
u
ub
0
ubjd
0
ubjk
ut
v
v .S /
w
w
w0j
X
f;C
xnjt
Setup time for item j
Periods in which setups of the buyer occur
Time horizon in setting L
Time horizon in setting U
Periods in which setups of the supplier occur
Reservation value of the buyer
Reservation value of the supplier
Prices for central resource use
Utility vector
Upper bound
Upper bound for dj0
Upper bound for kj0
Average capacity utilization
Salvage value
Surplus from forming set S
Target for the reduction of the number of setups for the items supplied
Wholesale price
Initial setup state of item j
Random variable denoting perturbation
C
st
Node n (x-coordinate) for the linearization of f f Kjt
; XBjt xbjt
f;
st
xnjt
Node n (x-coordinate) for the linearization of f f Kjt
; XBjt xbjt
s;C
C
st
xnjt
Node n (x-coordinate) for the linearization of f s Kjt
; XBjt xbjt
s;
st
xnjt
Node n (x-coordinate) for the linearization of f s Kjt
; XBjt xbjt
Solution previously found (in step e of the scheme)
xie
xiv
Value taken by variables xi in the vertex v identified so far
xO ; xA ; xB ; xC Breakpoints
0
xtjt
Modified target supply quantity of item j in period t
e
xtjt
Amount of item j supplied in period t in the previous proposal e
e;i
xtjt
Supply quantity of item j in period t delivered by supplier i and specified
by proposal e
max
xtaj
Maximum transportation quantity of item j along arc a in a period
mi n
xtaj
Minimum transportation quantity of item j along arc a in a period
xtjt
Target for the supply quantity of item j in period t
ysjup Number of orders for item j in the proposal from upstream planning
z
Parameter indicating the ratio between T and N B (rounded down)
Variables
C
f
st
f;C
K
Weight
for
node
n
for
the
linearization
of
f
;
XB
xb
jt
njt
jt
jt f;
st
f
njt
Kjt ; XBjt xbjt
Weight for node n for the linearization of f
C
st
s;C
Weight for node n for the linearization of f s Kjt
; XBjt xbjt
njt
s
st
s;
K
Weight
for
node
n
for
the
linearization
of
f
;
XB
xb
jt
njt
jt
jt
Nomenclature
xxi
i e
Decision variables defining a linear combination of previous proposals
about the central resource use of party i
BLjt Amount of backorders for item j in period t
BLljst Amount of backorders of item j of customer class s at location l in period t
p;C
Cjt
Penalties or bonuses for greater supply of item j in period t
p;
Cjt
Penalties or bonuses for less supply of item j in period t
CBd
Costs for the decisions of the buyer’s planning domain
Costs for the decisions of the supplier’s planning domain
CSd
CMLCLSP Value of the objective function of the MLCLSP
CAMjt Campaign variable for item j in period t (quantity of the current campaign
up to period t)
CAMpt Campaign variable for PPM p in period t (quantity of the current campaign up to period t)
ls
Djt
Difference in the supply quantity of item j in period t due to lost sales
B
g
Profit of the buyer
I .Q/ (Leftover) inventory
Ijt
Inventory of item j at the end of period t
Iljt
Amount of inventory of item j at location l at the end of period t
IBjt
Inventory of the (supplied) item j at the buyer’s site in period t
ICElt Increase of storage capacity at location l in period t
ISjt
Inventory of item j at the supplier’s site in period t
ki
Prices for changes in central resource use
C
Kjt
Endogenously determined unit prices for positive deviations from the startst
ing proposal xjt
of item j in period t
Kjt
Unit prices for negative deviations of item j in period t
ls
Kjt
Penalty costs for lost sales of item j in period t
Kjagg;C Endogenously determined unit penalty costs for shifts of the supply of item
j to later periods compared to the starting supply pattern
Kjagg; Endogenously determined unit penalty costs for shifts of the supply of item
j to earlier periods compared to the starting supply pattern
LSjt Amount of lost sales of item j in period t
LSljst Amount of lost sales of item j of customer class s at location l in period t
M .Q/ Quantity sold to the market
Omt
Amount of overtime on resource m in period t
Q
Order quantity
QB
Optimal order quantity for the buyer
QSC
Optimal order quantity for the supply chain
Rjt
Integer number of full batches produced in the current campaign of item j
up to period t
Rpt
Integer number of full batches produced in the current campaign of PPM p
up to period t
Sjt
Quantity of the last batch of item j in period t which is not finished in t
Spt
Quantity of the last batch of PPM p in period t which is not finished in t
S Sljt Undershot of safety stock of item j at location l in period t
xxii
Nomenclature
Wjt
Setup state indicator variable (=1 if item j is set up at the end of period t, 0,
otherwise)
Setup state indicator variable (=1 if PPM p is set up at the end of period
t, =0 otherwise)
Production quantity of item j at the beginning of period t (i.e., of the first
campaign in t)
Production quantity of PPM p at the beginning of period t
Production quantity of item j that is not produced at the beginning of period
t (i.e., not part of the first campaign in t)
Production quantity of PPM p that is not produced at the beginning of
period t
Decision variables in the generic LP model
Production amount of item j in period t
Amount of item j delivered to the buyer in period t
Amount of item j delivered by the supplier in period t
st
Increase in the supply of item j in period t compared to xjt
Decrease in the supply of item j in period t
Transportation quantity of item j along arc a in period t
Binary setup variable (=1 if item j is produced in period t, =0 otherwise)
Binary setup variable (=1, if PPM p is produced in period t, =0 otherwise)
Setup operation indicator for resource m in period t (=1 if a setup occurs
on resource m in period t, =0 otherwise)
Indicator variable, =1 if item j is ordered in period t, =0 otherwise
Objective function value of CS1i
Binary variable (=1 if proposal i of party j is implemented, =0 otherwise).
Wpt
b
Xjt
b
Xpt
e
Xjt
e
Xpt
xi
Xjt
XBjt
XSjt
X TjtC
X Tjt
X Tajt
Yjt
Ypt
Y Imt
Y Sjt
ZCS1i
Zij
Chapter 1
Introduction
1.1 Motivation and Goals of This Work
Supply chain planning is concerned with the determination of integrated operational
plans for all functional areas and members within a supply chain. Depending on the
organizational structure of the supply chain, this task can either be considered as the
state-of-the-art or as a challenge for future supply chain excellence.
State-of-the-art is the planning in intra-organizational supply chains. This task
is supported by a broad range of procedures elaborated in the literature during the
last decades as well as modeling tools, APS (Advanced Planning Systems), which
are widely used by practitioners.1
This, however, is not the case for inter-organizational supply chains consisting
of multiple, legally independent parties. Current APS only provide interfaces for
data exchange between parties, but do not support inter-organizational collaborative
planning. In APS, an integrated planning requires a (central) entity equipped with
all relevant data and the decision authority to implement the systemwide optimal
plan. However, this approach comes with a number of downsides: The need for disclosing potentially confidential information by the decentralized parties, the conflict
of central targets with the incentive structure in decentralized organizations, and the
missing guarantee for truthful information disclosure; indeed, very few applications
of this approach have been reported so far.2
This result stands in sharp contrast to the literature, where coordination has been
widely recognized as one of the key drivers of supply chain performance in the last
10–15 years. A large number of papers evaluating the benefits from coordination
and proposing new coordination mechanisms have been produced. Unfortunately,
these mechanisms have severe limitations making it impossible to apply them to
inter-organizational supply chain planning. Among these limitations are a complete
knowledge about the others’ model data and team behavior by the participating
parties as well as the restriction on economic order quantity or newsvendor models.
1
2
See, e.g., the case studies reported by Stadtler and Kilger (2007, p. 367).
E.g., Shirodkar and Kempf (2006, p. 420).
M. Albrecht, Supply Chain Coordination Mechanisms: New Approaches
for Collaborative Planning, Lecture Notes in Economics and Mathematical Systems 628,
c Springer-Verlag Berlin Heidelberg 2010
DOI 10.1007/978-3-642-02833-5 1, 1
2
1 Introduction
In this study, we augment the existing literature by new coordination
mechanisms, which lift the major limitations as needed for a potential practical
application. These mechanisms are the first to simultaneously include several
generic features like:
The assumption of multilateral information asymmetry about other parties’
detailed data (before, during, and after the application of the mechanism)
No need for involving a third party
Self-interest of parties
Complex mathematical programming models including discrete decisions run by
the decentralized parties
In addition to a theoretical development and foundation, computational tests for
randomly generated data as well as real-world data indicate that substantial savings
can be obtained by the mechanisms proposed.
1.2 Methodology
The coordination mechanisms have to identify an improvement compared to an
initial, uncoordinated solution and to include incentives to implement the improved
solution. For that purpose, this work combines methodologies from two different
areas of economic research: Operations research and game theory.
The improvements are identified by innovative mathematical programming
models. We assume that such models are used by the parties for their decentralized
planning and develop extensions, which can be applied in an iterative manner for
the generation and identification of potentially coordinating supply proposals; the
single steps undertaken for this purpose are called a coordination scheme. The effectiveness of the schemes proposed is demonstrated by analytical and computational
results. We analytically prove the convergence of the schemes for specific model
classes, and show by computational tests that the schemes are able to substantially
mitigate the suboptimality from decentralized planning.
To determine the incentives for the decentralized parties to follow the rules of the
schemes, the mechanisms rely on concepts from the area of game theory. Strategic
(and potentially untruthful) behavior of decentralized parties is explicitly taken into
account. We build on insights and ideas from bilateral bargaining and behavioral
research to design several mechanisms that can be applied in different organizational structures. For two of these mechanisms, upper bounds for the losses due to
information asymmetry will be derived.
1.3 Outline
The thesis is organized as follows. Chapter 2 provides the basis for the mechanisms developed in this work. First, we describe the task of Master (i.e., mid-term)
1.3 Outline
3
Planning and corresponding mathematical model formulations. Second, we define
basic terms used in this work and discuss the consequences of decentralized planning. We show how the resulting planning processes can be modeled mathematically
and identify drivers for the systemwide suboptimality of unilateral targets, which
are established without coordination. Finally, we outline upstream planning, which
we assume as the standard planning procedure without coordination, as well as our
concept for collaborative planning.
Chapter 3 surveys the literature on coordination mechanisms. We have structured
this review according to the assumptions on the knowledge about the other parties’
data, i.e., we distinguish between symmetric, one-sided asymmetric, and multilateral asymmetric information. Moreover, we elaborate the basic ideas regarding the
design of the related mechanisms and provide classifications for the literature of
this area.
Chapters 4 and 5 comprise the core contributions of this work. In Chap. 4, the different coordination schemes are outlined. We begin with two versions of a generic
scheme for coordinating decentralized parties running arbitrary linear programming models (in one version, even one of the parties may run a mixed-integer
programming model). Moreover, we present a scheme for coordinating uncapacitated lot-sizing models in supply chains of one buyer and one or multiple suppliers.
For both schemes, analytical results about their convergence behavior can be derived. Apart from that, for a two-party supply chain, we present modified versions
of these schemes with improved convergence rate and improved applicability for
Master Planning problems that include discrete decisions. Amongst others, we cover
extensions of these modified versions to voluntary compliance by the supplier, the
modeling of lot-sizing and lost sales, and settings with multiple suppliers.
As a second component of the coordination mechanisms, several contractual
frameworks are outlined and analyzed in Chap. 5. Resulting are mechanisms applicable for different organizational structures and different distributions of the
bargaining power between the decentralized parties. The strategies adopted by the
decentralized parties using these mechanisms are discussed in light of behavioral
theories and analytical reasoning. As a third issue in the chapter, we outline how
the mechanisms can be adapted for rolling schedules, which are frequently used in
real-world production planning.
Computational tests are provided in Chap. 6. We examine the performance of the
schemes based on randomly generated test instances for different Master Planning
models as well as for real-world Master Planning data provided by the SAP AG. For
all problems investigated, significant improvements compared to upstream planning
can be identified after a modest number of iterations.
Finally, Chap. 7 summarizes the contributions of this work and outlines opportunities for further research.
Chapter 2
Supply Chain Planning and Coordination
The aim of this chapter is to familiarize the reader with the topic of this work,
how to coordinate mid-term planning in decentralized supply chains, i.e., supply
chains that comprise several independent, legally separated parties with their
own decision authorities. We start with a description of planning in centralized
supply chains (Sect. 2.1), where the decision authority and the knowledge of all
relevant planning data is hold by a single party. In Sect. 2.2 we provide centralized
mathematical model formulations for mid-term supply chain planning (Master Planning). Section 2.3 deals with supply chain planning in decentralized environments.
We describe the differences in the planning processes compared to centralized planning, provide reasons for the potential suboptimality of decentralized planning and
introduce coordination and, more specifically, collaborative planning as approaches
to mitigate this suboptimality.
2.1 Supply Chain Planning
2.1.1 Definitions and Overview
We begin with an abstract and often cited definition of a supply chain: A supply
chain is a “. . . network of organizations that are involved, through upstream and
downstream linkages, in the different processes and activities that produce value
in the form of products and services in the hands of the ultimate consumer.”1 As
an illustration, we depict in Fig. 2.1 a supply chain consisting of a set of vendors,
plants, distribution centers, and customers that are linked by material flows.2
From a business economics perspective, supply chains require supply chain
management, that can be defined as “the task of integrating organizational units
along a supply chain and coordinating material, information and financial flows
in order to fulfill (ultimate) customer demands with the aim of improving the
1
2
Christopher (2005, p. 17).
For similar representations, see, e.g., Shapiro (2001, p. 6) and Stadtler (2007b, p. 10).
M. Albrecht, Supply Chain Coordination Mechanisms: New Approaches
for Collaborative Planning, Lecture Notes in Economics and Mathematical Systems 628,
c Springer-Verlag Berlin Heidelberg 2010
DOI 10.1007/978-3-642-02833-5 2, 5
6
2 Supply Chain Planning and Coordination
Fig. 2.1 Sketch of a supply chain (example)
competitiveness of a supply chain as a whole.”3 One of the building blocks of supply
chain management is (advanced) supply chain planning.4 The aim of supply chain
planning is to determine an integrated plan for the whole supply chain; referring to
Fig. 2.1, such a plan comprises appropriate quantities of the raw materials procured,
of the products manufactured in the plants, of the products distributed, and of the
products sold to the customers.
Of course, this is a very complex task especially for real-world organizations,
which may comprise a large number of facilities, customers, and products. Therefore, it has been proposed in the literature5 to organize (production) planning in a
hierarchical way.6 The basic idea of hierarchical planning is the separation of decisions according to their impact, e.g., on the profitability of the supply chain. The
decisions at the upper levels, i.e., those with greater impact, are determined first
and implemented as targets for the planning of the lower levels. Further important
characteristics are the aggregation of data and decisions at the upper levels and the
provision of feedback by the lower levels.7
A common representation for the individual tasks of supply chain planning
is the Supply Chain Planning Matrix (see Fig. 2.2).8 Frequently, these planning
tasks are supported by software tools in practice. As a standard software, Advanced Planning Systems (APS) have been developed by different companies (e.g.,
SAP, Oracle9 ). Each APS contains several modules that cover in part the planning tasks stated in Fig. 2.2. The Advanced Planning Matrix provides an overview
of these modules (see Fig. 2.3).10 Long-term planning is the object of Strategic
3
Stadtler (2007b, p. 11).
See Stadtler (2007b, p. 27).
5
See, Hax and Meal (1975, p. 53) and the work of Stadtler (1988) as a comprehensive introduction
into hierarchical planning.
6
Although originally proposed for production planning, this concept equally applies to planning
in supply chains.
7
See, e.g., Stadtler (2007b, p. 32).
8
See Fleischmann et al. (2007, p. 102).
9
See SAP (2008) and Oracle (2008). The market share of APS is substantial and steadily growing;
e.g., in 2005, revenues increased by $40 million up to $741 million, see White et al. (2006, p. 2).
10
Figure 2.3 has been adapted from Meyr et al. (2007b, p. 109) with slight modifications (see
below).
4
2.1 Supply Chain Planning
7
procurement
production
long-term • materials programme
• plant location
• production system
• supplier selection
• cooperations
mid-term • personnel planning
• material requirements
planning
• contracts
short-term
• personnel planning
• ordering materials
• master production
scheduling
• capacity planning
• lot-sizing
• machine scheduling
• shop floor control
distribution
sales
• physical distribution • product programme
structure
• strategic sales
planning
• distribution planning
• mid-term
sales planning
• warehouse
replenishment
• transport planning
• short-term
sales planning
information flows
flow of goods
Fig. 2.2 Supply Chain Planning Matrix
procurement
production
distribution
long-term
Strategic Network Design
mid-term
Master Planning
Master
Planning
short-term
Purchasing
&
Material
Requirements
Planning
Production
Planning
Distribution
Planning
Scheduling
Transport
Planning
sales
Demand
Planning
Demand
Fulfillment
& ATP
Fig. 2.3 Advanced Planning Matrix
Network Design, whereas the mid-term planning tasks are covered by Purchasing &
Material Requirements Planning, Master Planning, Production Planning, Distribution Planning, and Demand Planning. Analogously, the short-term planning tasks
are tackled by Purchasing & Material Requirements Planning, Production Planning, Scheduling, Transport Planning, Demand Fulfillment & Available-To-Promise
(ATP). Note that Fig. 2.3 differs from the original Advanced Planning Matrix by the
8
2 Supply Chain Planning and Coordination
overlap of Master Planning into sales (see the shadowed area in Fig. 2.3). In our
opinion, this is more concise than the original representation since Master Planning
frequently involves sales-related decisions like backorders and lost sales.11
In the following, we will provide a more detailed description of Master Planning,
which is the main focus of this work. For an in-depth explanation of the other
modules, we refer to the textbook of Stadtler and Kilger (2007).12
2.1.2 Master Planning
Master Planning means mid-term operational decision-making carried out
simultaneously for all functional areas participating in the order fulfillment process:
Procurement, production, distribution, and sales. In this work, we focus on this
planning level since the potential financial impact for collaborating enterprises is
largest here.
Master Planning is based on monthly or weekly time buckets; hence, planning
horizons of, e.g., 12, 24, or 52 periods are used. It is important that the planning
horizon is chosen such large that effects due to seasonal demand are considered.
This requires that the planning interval comprises one seasonal cycle at least.
Table 2.1 provides an overview about the basic decisions made within Master
Planning. Most of these decisions are likewise mentioned in other descriptions of
Master Planning.13 A somewhat ambiguous role in this context plays lot-sizing,
Table 2.1 Basic decisions of Master Planning
Functional area Decision
Procurement
Quantities of raw materials to purchase
Production
Production quantities
Resource utilization
Inventory levels
Utilization of overtime
Lot-sizing
Distribution
Quantities of products to transport
Inventory levels
Sales
Quantities of products to deliver to the customers
(including backorders and lost sales)
11
See also Sect. 2.1.2. The importance of (lost) sales for integrated mid-term planning is further
supported by the practitioner-oriented literature, where mid-term planning models with the aim of
profit maximization have been proposed, see, e.g., Timpe and Kallrath (2001, p. 423) and Kallrath
(2002, p. 315) for a model that additionally includes strategic aspects.
12
See Stadtler and Kilger (2007, p. 117).
13
See Rohde and Wagner (2007, p. 160) and Günther (2006, p. 20).
2.2 Model Formulations for Master Planning
9
which is often regarded as a short-term planning task in the literature.14 Indeed,
lot-sizing is not an issue for mid-term planning in many industries.15 A noteworthy exception, however, are process industries, where lot-sizing decisions have a
substantial impact on the quality of the resulting plans. Due to large setup times
and expensive setup activities, practical mid-term planning model formulations
originating from this area usually include lot-sizing.16
The results of these decisions (e.g., the planned amount of stock or the planned
use of overtime17) constitute targets for the lower-level modules Purchasing &
Material Requirements Planning, Production Planning, and Distribution Planning.
Note that, according to the principles of hierarchical planning, the corresponding
data has to be disaggregated for this purpose.
The input data for Master Planning is deterministic. Real-world data, especially
demand forecasts, however, always comprise some uncertainty. Because of that,
mid-term planning models incorporating stochastic data have been elaborated in
the literature.18 In practice, however, stochastic decision models are rarely used for
mid-term planning. Instead, planning is done based on rolling schedules.19 This
means that only the first periods of plans (i.e., the periods before the frozen horizon)
are implemented; the rest is determined later by re-planning, which is undertaken
periodically (e.g., once a month). This approach provides the flexibility to react with
plan changes if uncertainty is revealed in future periods.
2.2 Model Formulations for Master Planning
In this section, we present mathematical model formulations for Master Planning
that are used throughout this work (among others, for the computational verification of the coordination schemes proposed). First of all, we provide a generic linear
programming (LP) model in Sect. 2.2.1. Compared to mixed-integer programming
(MIP) or nonlinear programming (NLP), the mathematical structure of LP is much
simpler, which makes approaches based on LP particularly suited for structural
analyses. However, results valid for LP do not necessarily extend to other model
classes.20 These classes include MIP models, which are required for modeling
14
As an example, see the Supply Chain Planning Matrix on p. 7 of this work.
This holds, e.g., for the (German) automotive industry, see Meyr (2004, p. 447).
16
E.g., Timpe and Kallrath (2001, p. 42) and Grunow et al. (2003, p. 109).
17
See, e.g., Rohde and Wagner (2007, p. 161).
18
E.g., Leung et al. (2007, p. 2282), Escudero et al. (1993, p. 311), and Scholl (2001, p. 295).
19
See, e.g., Fleischmann et al. (2007, p. 84).
20
As an example, consider the simplex algorithm, a common solution procedure for LP models,
which yields the optimal solution to, e.g., MIP models only in case of the total unimodularity of
the underlying matrix of coefficients, see Domschke and Drexl (2006, p. 91); for a description of
the simplex algorithm and of solution procedures for MIP models such as branch and bound, see,
e.g., Domschke and Drexl (2006, p. 21 and p. 126).
15
10
2 Supply Chain Planning and Coordination
lot-sizing decisions at the Master Planning level. Therefore, we additionally provide
model formulations accounting for lot-sizing in Sect. 2.2.2.21
2.2.1 Generic Master Planning Model
Before presenting the mathematical formulation of the generic Master Planning
model (GM), we state its underlying assumptions:
Several items are arranged in a general bill of material (BOM) and produced on
one or more specific resources. Production results in variable capacity loads.
The capacities of the resources are finite and can be extended by costly, infinitely
available overtime.
Demand is dynamic and deterministic for all items.
Unfulfilled demand can either be backlogged or lost; both actions incur additional
costs.
Inventory holding of items is possible and results in holding costs.
min
XX
hj Ijt C
j 2J t 2T
X X
ocm Omt C
m2M t 2T
X X
blcj BLjt C lscj LSjt
j 2J E t 2T
s.t. BLjt C LSjt C Ijt 1 C Xjt D djt C BLjt1 C Ijt 8j 2 J E ; t 2 T (2.1)
X
(GM)
Ijt 1 C Xjt D
rjk Xkt C Ijt 8j 2 J nJ E ; t 2 T
(2.2)
X
k2Sj
amj Xjt kmt C Omt
8m 2 M; t 2 T
(2.3)
j 2J
BLj 0 D blj0
8j 2 J E
BLj jT j D bljT 8j 2 J E
Ij 0 D ij0 8j 2 J
BLjt 0 8j 2 J E ; t 2 T [ f0g
21
(2.4)
(2.5)
(2.6)
Ijt 0
8j 2 J; t 2 T [ f0g
(2.7)
(2.8)
LSjt 0
8j 2 J E ; t 2 T
(2.9)
Omt 0 8m 2 M; t 2 T
Xjt 0 8j 2 J; t 2 T :
(2.10)
(2.11)
Note that there are further decisions and restrictions in Master Planning that rely on MIP for
their modeling. An example is the restriction that overtime can only be taken in an integer number
of shifts, which is used in our computational tests of Sect. 6.1 for analyzing the sensitivity of the
scheme regarding the presence of integer variables.
2.2 Model Formulations for Master Planning
11
Indices and Index Sets
j Items or operations (e.g., end products, intermediate products, raw
materials), j 2 J ; J E is the subset of (end) items sold to external customers
m Resources (e.g., personnel, machines, production lines), m 2 M
t Periods, t 2 T , with T D 1; : : : ; jT j
Sj Set of immediate successors of item j in the BOM
Data
amj Capacity needed on resource m for one unit of item j
blcj Backorder costs for one unit of item j in a period
bl0j Amount of backorders for item j at the beginning of the planning interval
bljT Amount of backorders for item j at the end of the planning interval
djt Primary, gross demand for item j in period t
ij0 Inventory of item j at the beginning of the planning interval
hj Holding costs for one unit of item j in a period
kmt Available capacity of resource m in period t
lscj Costs for lost sales of one unit of item j in a period
ocm Overtime costs for one unit of resource m
rjk Number of units of item j required to produce one unit of the
immediate successor item k
Variables
BLjt Amount of backorders for item j in period t
Ijt Inventory of item j at the end of period t
LSjt Amount of lost sales of item j in period t
Omt Amount of overtime on resource m in period t
Xjt Production amount of item j in period t
The objective function minimizes the costs for inventory holding, overtime,
backorders, and lost sales. Constraints (2.1) determine the quantities of the external
demand that are backlogged and lost. Constraints (2.2) ensure the fulfillment of
the secondary demand.22 Constraints (2.3) limit the capacity used for production to
the sum of normal capacity and overtime. Constraints (2.4)–(2.6) fix the amounts
of backorders and inventories at the borders of the planning interval.23 Finally,
(2.7)–(2.11) determine the nonnegativity of the decision variables.
Note that due to its generic character, this model formulation does not cover
all decisions potentially relevant for Master Planning.24 For lot-sizing, we refer to
22
For ease of exposition, we have separated items into two groups: End items with external demand
(J E ) and intermediate items used for production (J nJ E ). In a setting where an item is used for
production and has external demand, this model formulation would have to be adapted accordingly.
23
This fixation seems the most straightforward possibility for the modeling of backorders. Note
that when planning is based on rolling schedules, the modeling of maximum latenesses may be
more adequate.
24
I.e., those listed in Table 2.1 (see p. 8).
12
2 Supply Chain Planning and Coordination
the following subsection. Moreover, an exemplary modeling of storage capacities
and transportation is included in the model for the real-world planning problems
presented in Sect. 6.5.
2.2.2 Extension to Lot-Sizing
A broad variety of models incorporating lot-sizing decisions has been proposed in
the literature. We begin with some basic formulations and discuss how to integrate
them into Master Planning. Built on this, we present a Master Planning model that
is extended to production campaigns. Campaign planning is a variant of lot-sizing
with practical relevance for process industries and additional difficulties for supply
chain coordination,25 which makes this extension particularly suited for examining
the performance of the coordination schemes proposed in this work.
2.2.2.1 Basic Models
One of the first production planning problems analyzed in the literature is the
determination of the economic order quantity (EOQ),26 i.e., the optimal order
quantity of an item based on a number of restrictive assumptions. The most
important assumptions are:27
One single item is considered.
Demand is deterministic and constant.
The replenishment lead time is zero.
Inventory holding is possible and results in holding costs.
Each replenishment requires fixed ordering costs.
Then the total relevant costs per unit time can be expressed by
c .Q/ D
hQ
sc d
C
:
Q
2
Data
d Demand per unit time
h Unit holding cost per unit time
sc Cost for one replenishment order (D setup cost)
Variables
Q Order quantity
25
See Example 2.6 as an illustration of this issue.
See Harris (1913, p. 135).
27
For alternative listings of assumptions underlying the EOQ model, see, e.g., Neumann (1996,
p. 28) and Silver et al. (1998, p. 150).
26
2.2 Model Formulations for Master Planning
13
p
c .Q/ is a convex function. It takes its minimum with the EOQ Q D 2sc d= h.
Due to the restrictive assumptions, direct applications of this model are rather rare
in practice.28 In spite of that, this model has proved useful as a basis for analyzing
lot-sizing decisions in broader contexts, which include the potential cost impact of
deviations of lot sizes from the EOQ29 and supply chain coordination mechanisms.30
The term lot-sizing also means the determination of optimal order quantities,
but – in contrast to the EOQ – without the limitation to constant demand.31 The most
basic lot-sizing model is the uncapacitated dynamic single-item lot-sizing model
developed by Wagner and Within (1958).32 Since the applicability of this model is
again rather limited, we state a more relevant extension to several items and several levels of the BOM, the MLULSP (D Multi-Level Uncapacitated Lot-Sizing
Problem).33
XX
XX
min
hj Ijt C
scj Yjt
(2.12)
j 2J t 2T
j 2J t 2T
s.t. (2.2), (2.6), (2.8), (2.11)
(MLULSP)
Ijt 1 C Xjt D djt C Ijt
8j 2 J E ; t 2 T
Xjt bjt Yjt 8j 2 J; t 2 T
Yjt 2 f0; 1g 8j 2 J; t 2 T :
(2.13)
(2.14)
(2.15)
Data
bjt Large number, not limiting feasible lot size of item j in period t,
PjT j
e.g., bjt D Dt
dj for j 2 J E ; for j 2 J nJ E , bjt can be calculated
P
recursively by bjt D k2Sj rjk bkt
hj Holding cost for one unit of item j in a period
scj Setup cost for a lot of item j
Variables
Yjt Binary setup variable (D 1, if item j is produced in period t, D 0 otherwise)
The objective function (2.12) minimizes the sum of inventory holding and setup
costs. Constraints (2.2), (2.6), (2.8), and (2.11) are taken from GM. Constraints
(2.13) ensure together with (2.2) the fulfillment of external and secondary demand,
respectively. Setup constraints (2.14) enforce variables Yjt to 1 if a lot of item j is
produced in period t. Constraints (2.15) define variables Yjt as binary.
28
Note that this only holds for this model in its pure form presented above. For some extensions
(e.g., to a multi-level BOM), real-world applications have been reported, see, e.g., Muckstadt and
Roundy (1993, p. 61) for an automotive manufacturer and Stadtler (1992, p. 217) for a light alloy
foundry.
29
See, e.g., Zangwill (1987, p. 1209) and Stadtler (2007a, p. 407).
30
See Chap. 3 for examples.
31
See, e.g., Silver et al. (1998, p. 198). Note that the optimal quantities for single orders usually
differ from each other in case of dynamic demand.
32
See Wagner and Within (1958, p. 89).
33
See, e.g., Domschke et al. (1997, p. 154).
14
2 Supply Chain Planning and Coordination
Combining the MLULSP and GM, we obtain a Multi-Level Capacitated
Lot-Sizing Problem (MLCLSP) with backorders and lost sales as a generic Master
Planning model that includes decisions related to lot-sizing. This model differs from
the original MLCLSP developed by Billington et al. (1983)34 by the negligence of
penalties for undertime and by the inclusion of backorders and lost sales.
min CMLCLSP D
(MLCLSP)
X X
m2M t 2T
XX
hj Ijt C
j 2J t 2T
ocm Omt C
X X
XX
scj Yjt C
j 2J t 2T
blcj BLjt C
j 2J E t 2T
X X
lscj LSjt
j 2J E t 2T
s.t. (2.1)–(2.11), (2.14), (2.15):
Variables
CMLCLSP Value of the objective function of the MLCLSP
2.2.2.2 Extension to Campaign Planning
Campaign planning is a variant of lot-sizing, which raises additional challenges for
an efficient mathematical modeling and is of great importance in process industries.35 Analogously to a production lot, a campaign means the production of several
units of items without performing any additional setup operation.36 The sizes of
these units usually cannot be chosen continuously; due to technical restrictions, e.g.,
fixed tank or reaction volumes, whole batches, i.e., prespecified amounts of items,
have to be produced. Hence, a campaign length corresponds to an integer number
of batch sizes.37
For the modeling of campaign planning, the MLCLSP has to be altered in two respects. First, of course, we have to assure that only complete batches are produced.
Second, the MLCLSP contains a representation defect, which affects the applicability of this model for campaign planning. The MLCLSP comprises the restrictive
assumption that, whenever an item is produced in a period, a setup has to be performed for this item. The setup is required irrespectively whether the resource has
already been set up for this item at the end of the preceding period, i.e., the setup
state could have been preserved. This representation defect can affect the optimality
of the resulting production plans in general.38 For campaign planning, this effect is
significantly aggravated, particularly if the production of a single batch requires a
considerable share of the available capacity.
34
See Billington et al. (1983, p. 6).
For real-world applications and case studies for campaign planning in process industries, see,
e.g., Brandenburg and Tölle (2008), Grunow et al. (2002, p. 281), and Rajaram and Karmarkar
(2004, p. 253).
36
See, e.g., Suerie (2005c, p. 2).
37
See Kallrath (2005, p. 341).
38
For an extensive discussion of this issue, see Suerie (2005c, p. 34).
35
2.2 Model Formulations for Master Planning
15
Table 2.2 Comparison of lot-sizing models with the preservation of setup states
DLSP CSLP PLSP CLSPL
GLSP
Maximum number of items per period 1
1
2
Arbitrary Arbitrary
Sequence-dependent setups allowed?
No
No
No
No
Yes
In the literature, several model formulations have been developed that overcome
the above mentioned representation defect by allowing lot sizes that overlap period
boundaries.39 These models differ by their scope and their computational complexity. Table 2.2 provides a comparison of five basic models with this property,
the DLSP (Discrete Lot-sizing and Scheduling Problem),40 the CSLP (Continuous
Setup Lot-sizing Problem),41 the PLSP (Proportional Lot-sizing and Scheduling
Problem),42 the GLSP (General Lot-sizing and Scheduling Problem),43 and the
CLSPL (Capacitated Lot-Sizing Problem with Linked lot sizes).44
The scopes of these models differ by the maximum number of items that can be
produced per time period45 and by the question whether sequence-dependent setups
can be modeled.46 We choose the CLSPL as the basis for evaluating the coordination
schemes proposed. The main reason for this is that the CLSPL allows the production
of an arbitrary number of items per period. In anticipation to Sect. 2.3, we want to
point out that coordination becomes most relevant if several items are ordered and
if decentralized parties have little leeway for adapting their production plans (e.g.,
due to tight capacities and elevated costs for shortages). In such situation, parties
would hardly confine themselves to models which artificially restrict the production
to one or two items per period. Potential modest increases in inventory holding and
setup costs with the CLSPL are of secondary relevance then. Such increases may
be caused by the greater computational complexity of the CLSPL, which usually
results in larger optimality gaps compared to the DLSP, CSLP, and PLSP, provided
that a limit on the solution time is applied.47 Sequence-dependent setups, in turn,
39
For comprehensive surveys of these models see, e.g., Drexl and Kimms (1997, p. 221) and Jans
and Degraeve (2008, p. 1619).
40
See Fleischmann (1990, p. 338).
41
See Karmarkar and Schrage (1985, p. 328).
42
See Drexl and Haase (1995, p. 75).
43
See Fleischmann and Meyr (1997, p. 12).
44
See Dillenberger et al. (1993, p. 112). More recently, this model has been investigated by
Gopalakrishnan et al. (2001, p. 851) and Suerie and Stadtler (2003, p. 1039).
45
Note that this classification – although sufficient for our subsequent argumentation – is too rough
to capture the difference between the DLSP and the CSLP, which is the all-or-nothing condition
required by the DLSP and relaxed in the CSLP.
46
Note that we state here whether the basic formulations of these models (which have been
reported in Drexl and Kimms (1997, p. 221) and (Suerie, 2005a, p. 16)) include sequencedependent setups. This does not always apply to the model type in general; e.g., Fleischmann
(1994, p. 397) proposes a variant of the DLSP which allows sequence-dependent setups.
47
See Suerie (2005c, p. 164) for computational results for campaign planning models based on the
PLSP and the CLSPL. Regarding the best solutions found, however, the CLSPL can outperform
16
2 Supply Chain Planning and Coordination
are usually not included at the Master Planning level;48 hence, there is no need for
a computationally more demanding model like the GLSP that additionally covers
this issue.
Concerning the restrictions on feasible campaigns, we limit here to single-item
campaigns with fixed batch sizes.49 For sake of simplicity, we neither include
minimum campaign lengths50 nor batch availability.51
Further discussion deserves the modeling of lead times in the multi-level version
of the CLSPL considered here. Lead times of zero, which have implicitly been assumed for the MLCLSP, may cause infeasibility of solutions obtained by multi-level
lot-sizing models with the preservation of the setup states. Such infeasibility may
arise if a successor item is not produced at the end of a period (i.e., its setup state is
not carried over into the next period) and one of its predecessor items is produced
at the end of the same period (i.e., its setup state is carried over into the next period). With insufficient inventories of the predecessor item at the beginning of the
period, the secondary demand of the successor item might not be fulfilled in time.52
In order to exclude the generation of infeasible solutions, we choose in analogy to
Kimms (1996)53 lead times equal to or greater than one period length for intermediate items.54
Below we present the adaptation of the MLCLSP to the preservation of setup
states and campaign restrictions. This model extends the single-level formulation
proposed by Suerie (2005b)55 to a multi-level BOM structure.
min CMLCLSP
(MLCLSPL-C) s.t. (2.4)–(2.11), (2.15)
LSjt C BLjt C Ijt1 C Xjtb C Xjte D djt C BLjt1 C Ijt
8j 2 J E ; t 2 T
(2.16)
the PLSP (without campaign restrictions) in spite of the larger optimality gaps, see Suerie (2005c,
p. 157).
48
This holds, e.g., for Timpe and Kallrath (2001) and Grunow et al. (2003), the papers cited in
Sect. 2.1.2 as examples for lot-sizing decisions in Master Planning.
49
The modeling of these campaigns in lot-sizing has been introduced by Kallrath (1999, p. 330)
and further been improved by Suerie (2005a, p. 49).
50
For their modeling, see, e.g., Suerie (2005c, p. 95).
51
For the modeling of batch availability, see, e.g., Brüggemann and Jahnke (1994, p. 755) for the
DLSP and Suerie (2005c, p. 98) for the CLSPL.
52
Note that this problem is only relevant for models with the preservation of setup states and,
hence, does not apply to the MLCLSP.
53
See the formulation of the Multi-Level Proportional Lot-sizing and Scheduling Problem
(MLPLSP) in Kimms (1996, p. 87).
54
This modeling is only exact if exogenous lead times of this duration occur. For single-machine
problems, Stadtler (2008) has proposed an extension of the MLPLSP, which allows the modeling
of zero lead times. For multi-machine problems (like the problem considered here), however, we
are not aware of any corresponding practicable formulation.
55
Suerie (2005b, p. 102).
2.2 Model Formulations for Master Planning
17
X
prev
b
e
C Ijt
Ijt 1 C ijt
D
rjk Xkt
C Xkt
k2Sj
8j 2 J nJ ; t D 1; : : : ; j
E
X
b
e
Ijt 1 C Xjt
j C Xjt j D
b
e
C Ijt
rjk Xkt
C Xkt
(2.17)
k2Sj
8j 2 J nJ ; t D j C 1; : : : ; jT j
X
X
b
e
C
amj Xjt
C Xjt
stj Yjt kmt C Omt
E
j 2J
(2.18)
8m 2 M; t 2 T (2.19)
j 2J
amj >0
e
Xjt
bjt Yjt
8j 2 J; t 2 T
(2.20)
b
bjt Wjt 1
Xjt
8j 2 J; t 2 T
Wjt Yjt C Wjt 1 8j 2 J; t 2 T n fjT jg
X
Wjt 1 8m 2 M; t 2 T n fjT jg
(2.21)
(2.22)
(2.23)
j 2J
amj >0
Wjt 1 C Wjt Yjt C Ykt 2 8m 2 M; j; k 2 J; k ¤ j;
amj ; amk > 0; t 2 T n fjT jg
(2.24)
b
CAM jt CAM jt 1 C Xjt
C bjt Yjt
8j 2 J; t 2 T
(2.25)
b
CAM jt CAM jt 1 C Xjt
bjt Yjt
8j 2 J; t 2 T
(2.26)
CAM j 0 cam0j 8j 2 J
e
CAM jt Xjt
C bjt 1 Yjt
e
8j 2 J; t 2 T
CAM jt Xjt
(2.27)
8j 2 J; t 2 T
(2.28)
(2.29)
b
CAM jt 1 C Xjt
bsj Y Imt 8m 2 M; j 2 J; amj > 0; t 2 T
(2.30)
b
D bsj Rjt C Sjt
CAM jt 1 C Xjt
8j 2 J; t 2 T n f1g
Sjt bsj .1 Y Imt / 8m 2 M; j 2 J; amj > 0; t 2 T n f1g
(2.31)
(2.32)
Y Imt Yjt 8m 2 M; j 2 J; amj > 0; t 2 T
X
Y Imt Yjt 8m 2 M; t 2 T
(2.33)
(2.34)
j 2J
amj >0
Wj 0 D w0j
8j 2 J
(2.35)
CAM jt 0 8j 2 J; t 2 T [ f0g
Rjt 2 N0 8j 2 J; t 2 T
(2.36)
(2.37)
Wjt 2 f0; 1g 8j 2 J; t 2 T n fT g
Sjt 0 8j 2 J; t 2 T n f1g
(2.38)
(2.39)
YI mt 0 8m 2 M; t 2 T:
(2.40)
18
2 Supply Chain Planning and Coordination
Data
bsj Batch size for item j
cam0j Initial campaign quantity for item j
prev
ijt
Inflow of item j in period t originating from production in earlier periods
stj Setup time for item j
Lead time for item j
j
w0j Initial setup state of item j
Variables
CAM jt Campaign variable for item j in period t (quantity of the current
campaign up to period t)
Integer number of full batches produced in the current campaign of
Rjt
item j up to period t
Quantity of the last batch of item j in period t which is not
Sjt
finished in t
Wjt Setup state indicator variable (D 1 if item j is set up at the end of
period t, D 0 otherwise)
Xjtb Production quantity of item j at the beginning of period t (i.e., of the
first campaign in t)
e
Xjt Production quantity of item j that is not produced at the beginning
of period t (i.e., not part of the first campaign in t)
YI mt Setup operation indicator for resource m in period t (D 1 if a setup
occurs on resource m in period t, D 0 otherwise)
The objective function and some constraints are taken from the MLCLSP. The inventory balance constraints (2.16)–(2.18) now account for nonzero production lead
prev
times.56 In the left-hand side of constraints (2.16), inflows ijt
from earlier production periods have been considered, which are relevant if the MLCLSPL-C is applied
with rolling schedules. As a prerequisite for modeling production campaigns, we
b
have replaced Xjt , the standard variables denoting the production quantities, by Xjt
e
and Xjt in (2.16)–(2.18). The same change has been applied to constraints (2.19),
where setup times have additionally been included.57
The aim of constraints (2.20)–(2.24) is to ensure that in case of production of item
j in period t the corresponding resource has been set up for j . Constraints (2.20)
force variables Yjt to 1 if item j is produced in period t, but not at the beginning
of t.58 The alternative to a setup operation is a setup carry-over from the preceding
period. With a setup carry-over for item j , i.e., Wjt 1 D 1, the production of j can
56
Note that we have omitted inventory holding of items during their production lead times. The
only relevant effect of these inventories is on the relative profitability of items. This, however, can
be easily considered by altering the unit penalty costs for lost sales accordingly.
57
Note that for the validness of these constraints, we implicitly assume that setup times for an item
j on a machine m only incur with nonzero variable capacity load for the production of j on m.
58
Note that in order to avoid infeasibilities, the maximum lot size bj has to exceed the batch size
bsjt .
2.2 Model Formulations for Master Planning
19
take place at the beginning of period t [see constraints (2.21)]. Constraints (2.22)–
(2.24) ensure correct values for Wjt . According to constraints (2.22), Wjt can only
take 1 if a setup for item j has been performed in period t or if the corresponding
resource has already been set up for j at the end of t 1. Moreover, each resource
can be set up at most for one item at period boundaries [see constraints (2.23)].59
Finally, we have to exclude a positive setup state for item j both at the beginning
and the end of a period if another item k is produced in this period and no setup has
been performed for j [see (2.24)].
The properties of feasible campaigns are defined by constraints (2.25)–(2.32).
For that purpose, we have introduced variables CAMjt indicating the quantity of
the campaign of item j that is finished in the period where, relative to t, the last
setup has been performed for j (i.e., in one of the periods t, t 1, . . . ).60 Constraints (2.25) and (2.26) fix the campaign quantities of item j in period t to those
of t 1 augmented by the production amount of j at the beginning of t, given that
there has been no setup for j in t. Otherwise, these constraints are inactive. The
campaign quantities are initialized by the batch size in (2.27).61 Further bounds on
the campaign quantities are provided by (2.28)–(2.30). Constraints (2.28) ensure
that the campaign quantity of item j at the end of t must not exceed the quantity
of j produced after a setup of j in t. A lower bound for the campaign quantities
is set by (2.29). Constraints (2.30) ensure that at least the batch size is produced in
each campaign. Moreover, constraints (2.31) and (2.32) restrict the campaign length
to an integer number of batches. Constraints (2.31) identify the quantity of item j
in period t within an unfinished batch (i.e., the production quantity within an unfinished campaign that exceeds an integer multiple of bsj ) and assign this value to
variable Sjt . Constraints (2.32), in turn, imply the end of the current campaign (i.e.,
Sjt D 0) if there has been a setup on the resource on which j is produced.
The next sets of constraints, (2.33) and (2.34), fix the values for Y Imt . Due to (2.33),
variable Y Imt has to take a value greater than or equal to 1 if resource m has been set
up in period t.62 Furthermore, Y Imt is set to zero if no setup occurs on m in t (2.34).
59
Motivated by the real-world test data of Sect. 6.5, we have formulated this constraint differently
to Suerie (2005c, p. 102), where a setup state of 1 is strictly required for all periods T n fjT jg. A
setup state of 0 in periods T n fjT jg can be superior to a setup state of 1 if setups in these periods
lead to suboptimal (e.g., due to zero demand in early periods) or even infeasible solutions (e.g.,
if the available capacity in period 1 is smaller than the corresponding setup time, e.g., due to prescheduled maintenance).
60
Note that variables CAM jt therefore do not necessarily reflect the actual campaign quantity of
item j in the current period t ; i.e., CAM jt may take positive values although j is not produced in t .
This information, however, is sufficient for modeling purposes here.
61
Again we have chosen a formulation that differs somewhat from that of Suerie (2005c, p. 95).
Suerie (2005c) has initialized CAM j 0 by 0 if Yjt D 1, which, however, leads to wrong results if
the minimum batch size exceeds one period length. Then, constraints (2.30) would require that the
complete batch quantity is produced within a single period if there has not been any setup carryover into this period [which holds, e.g., in period 1 due to constraints (2.35)]. This may result in
infeasible plans or excessive overtime costs.
62
Note that due to constraints (2.32), variables YI mt cannot exceed 1.
20
2 Supply Chain Planning and Coordination
Variables Wjt are initialized by (2.35). The remaining constraints (2.36)–(2.40)
define decision variables as nonnegative, binary, and integer, respectively.
2.3 Decentralized Planning and Coordination
In this section, we shift our focus from centralized to decentralized supply chain
planning. We analyze the inherent lack of coordination in decentralized planning,
point out the resulting drawbacks, and motivate the need for coordination, specifically collaborative planning. We start with some basic definitions which are related
to coordination and used throughout this work.
2.3.1 Basic Definitions
Supply Chain Coordination
According to Horváth (2001), the term coordination is among the most “dazzling”
ones of business economics.63 First, we consider the meaning of coordination within
organizational theory, a classical field of economic research, where coordination is
of crucial importance.64 There, the need for coordination is a direct consequence of
the division of labor, which creates single activities with interdependencies among
them.65 Thus, coordination can be regarded as complementary to the division of
labor: The re-adjustment of these single activities in order to reach superordinate
aims.66 The essence of this definition is commonly accepted in the literature,67 although some authors prefer application-oriented definitions, such as: “Coordination
is the meshing and balancing of all factors of production or service and of all the
departments and business functions so that the company can meet its objectives.”68
With respect to supply chains, the definitions of coordination in the literature
are somewhat more concrete than those mentioned above, but differ substantially
from each other. A first stream of literature defines a supply chain as coordinated
if and only if actions leading to the supply chain optimum are implemented.69
A weaker requirement for the actions implemented is an improvement for the whole
supply chain compared to the default solution, i.e., the solution that would have
63
Horváth (2001, p. 113).
See, e.g., Horváth (2001, p. 113).
65
See, e.g., Laux and Liermann (1993, p. 7).
66
See, e.g., Frese (1975, column 2263).
67
See, e.g., Hansmann (2001, p. 255), Horváth (2001, p. 114), and Kieser and Walgenbach (2003,
p. 101).
68
Horngren et al. (2006, p. 180) and similar Bhatnagar et al. (1993, p. 142).
69
See, e.g., Cachon (2003, p. 230), who limits the set of contracts that coordinate supply chains
to those where “the set of supply chain optimal actions is a Nash equilibrium, i.e., no firm has a
profitable unilateral deviation from the set of supply chain optimal actions.”
64
2.3 Decentralized Planning and Coordination
21
been implemented without coordination. This definition is implicitly supported by
Corbett and de Groote (2000), who devise a menu of contracts which results in
systemwide costs that are equal to or lower than those of the default solution,
but not necessarily equal to those of the systemwide optimum, and call the output
“coordinated.”70 Note that the usual default solution in supply chains is characterized by double marginalization, where the more powerful parties decide on actions
with impact for the whole supply chain and thereby consider their own margins
only.71 A third alternative is to regard any systemwide feasible solution as coordinated, which seems to be favored by Schneeweiß.72
We adopt a variant of the alternative secondly mentioned and define that:
actions coordinate a supply chain if the resulting systemwide profit is greater than that in
the default solution.
Our reasoning for this is as follows: The identification of plans which assuredly
lead to the exact systemwide optimum is rather an exception in the context of supply chain planning, especially if information asymmetries are present. Information
asymmetry means that (at least) one party has information that is relevant for decisions concerning all parties but not known by at least one other party.73 Hence, it
would seem misleading to classify the vast majority of approaches which only lead
to improvements as non-coordinating. Therefore, we regard the implementation of
the optimal actions as a special case of coordination (“optimal coordination”). Note
that also in organizational theory,74 usually no emphasis is placed on the question
whether the readjustment of the single actions does lead to the global optimum.
The third alternative, i.e., only to require feasibility for coordination, seems least
appropriate since it completely ignores aspects of solution quality.
Coordination Scheme, Coordination Mechanism
Next, we introduce two further terms that are closely related to the establishment of
coordination and of central importance in this work.75 We begin with the definition
of a coordination scheme:
A coordination scheme is a set of rules specifying actions whose implementation by
decentralized parties potentially coordinates a system.
70
See Corbett and de Groote (2000, p. 449); for a detailed description of this approach, see
Sect. 3.2.2.
71
This phenomenon has been recognized first by Spengler (1950, p. 347), much earlier than
corresponding advances in the field of supply chain management.
72
“Worst-case coordination,” see Schneeweiss (2003, p. 278).
73
Explicit definitions of information asymmetry are rare in the literature. See Schneeweiss (2003,
p. 219) for a notion of this term similar to ours.
74
See the literature cited above.
75
Here, we limit to the definitions of these terms. For examples refer to Chap. 3. Note that in
the following, we will abbreviate “coordination mechanism” by “mechanism” and “coordination
scheme” by “scheme” if the meaning of these terms is clear from the context.
22
2 Supply Chain Planning and Coordination
Note that we do not require here that the application of a coordination scheme
always leads to the coordination of a system, which would reduce the scope of
this term considerably. Furthermore, a coordination scheme does not necessarily incentives for decentralized parties to follow its rules, i.e., to implement the actions
proposed. The definition of a coordination mechanism goes a step further in this respect. It is built on the definition of a mechanism in the field of mechanism design,76
where a mechanism constitutes a framework that specifies the outcomes (e.g., surplus sharing) of decentralized parties depending on the actions undertaken by them
(e.g., their information disclosed):77
A coordination mechanism is a mechanism for which the implementation of the optimal
strategies by decentralized, self-interested parties may lead to a coordinated outcome and
neither violates the individual rationality of the participating parties nor the budget balance
of the system.
Hence, we regard a coordination mechanism as a specific mechanism requiring a
potential coordinated outcome and two basic properties in mechanism design, individual rationality and budget balance.78 (Ex post) individual rationality requires
that no party is worse off by participating in the mechanism, i.e., that the profits of
all participating parties are at least equal to their profits achieved in the default solution. Budget balance means that the payments of parties in the mechanism sum up to
zero, i.e., that the mechanism does not require an outside subsidy.79 Moreover, note
that a coordination mechanism extends the tasks of a coordination scheme; hence,
as in the work, a scheme is often embedded in a mechanism.
Collaborative Supply Chain Planning
The term “collaborative planning” is generally known as a part (and sometimes even
regarded as a synonym80) of the business practice “Collaborative Planning, Forecasting and Replenishment.” As a formalized process, CPFR has been worked out
by the standardization committee VICS (Voluntary Interindustry Commerce Standards) and implemented within over 300 companies.81 The CPFR process model
consists of eight planning tasks, which can be subsumed under four main activities:
Strategy and planning, demand and supply management, execution, and analysis.82
76
Mechanism design is an area of game theory concerned with the aggregation of unobservable
individual preferences into a collective decision, see, e.g., Mas-Colell et al. (1995, p. 857).
77
Somewhat more technically, in Jackson (2003, p. 2), a mechanism is defined as a “specification
of a message space for each individual and an outcome function that maps vectors of messages
into social decisions and transfers.”
78
See, e.g., Chu and Shen (2006, p. 1215), who argue that these properties are necessary for mechanisms to be practicable.
79
See, e.g., Chu and Shen (2006, p. 1215).
80
See, e.g., Li (2007, p. 159).
81
See VICS (2008).
82
See VICS (2008).
2.3 Decentralized Planning and Coordination
23
“Planning” in this context does not refer to the alignment of operational plans, but
to the identification and communication of events which may affect demand, such
as promotional activities or product introductions.
Whereas collaboration is restricted to mere information exchange in the original
process model, some authors broaden the scope of this term to decision-making.
Raghunathan (1999) explicitly includes production scheduling within CPFR.83
Danese (2005) mentions a concept called “limited CPFR collaboration,” where
plans are jointly synchronized by the partners (e.g., the replenishment between a
central company and a distribution center).84 Akkermans et al. (2004) describe a
business process called “collaborative planning,” where companies “jointly take
decisions regarding production and shipments for a large part of their collective
supply chains.”85
This notion of collaboration is also included in the definition proposed here:
Collaborative supply chain planning is a joint decision making process for the alignment of
plans of independent, legally separated supply chain parties.
This definition is closely related to that of Stadtler (2009), except for two minor
differences. First, we have provided a definition for the term “collaborative supply
chain planning” instead of “collaborative planning” in order to avoid potential contradictions to other fields of science dealing with planning for objects different to
supply chains. However, since we are only concerned with supply chains in this
work, we will use the terms “collaborative supply chain planning” and “collaborative planning” as synonyms in the following. Second, we do not require information
asymmetry. In fact, information asymmetry is one of the main reasons for the application of collaborative planning. However, it seems reasonable not to exclude
important developments in the field of supply chain coordination that cover uncertain demand or issues of coalition forming, but no asymmetric information.86
Finally, it is important to note that a coordination of supply chains made up by
legally separated parties cannot be achieved without collaboration. Such supply
chains do not comprise a central entity which is entitled to determine the actions
of the decentralized parties directly or by means of incentive schemes.87 As a
83
See Raghunathan (1999, p. 1054); note that the terminology CFAR instead of CPFR is used
there.
84
See Danese (2005, p. 458).
85
Akkermans et al. (2004, p. 445).
86
For a survey of these approaches, see Sect. 3.1.
87
Mechanisms relying on such entities have been proposed by, e.g., Lee and Whang (1999, p. 633),
Pfeiffer (1999, p. 319), Bukchin and Hanany (2007, p. 273), and Kutanoglu and Wu (2006, p. 421).
However, these approaches do not ensure the individual rationality of parties. This is feasible if
a central entity can entirely determine the costs allocated to the decentralized parties, but seems
problematic for general independent parties, who want to improve their gains compared to a default
solution. Note in this context that we do not categorically exclude the participation of a central
entity in collaborative planning. We only exclude central entities that influence the default solution
and thus the individual rationality of parties, but not those that act as mediators and only determine
the allocation of the systemwide surplus from coordination (see Sect. 3.3 for examples).
24
2 Supply Chain Planning and Coordination
consequence, we will also use the terms “collaborative (supply chain) planning”
and “coordination mechanism for supply chain planning” as synonyms in this work.
2.3.2 Decentralized Supply Chain Planning
In this section, we analyze the impact of decentralization on supply chain planning.
We argue that decentralized planning may result in suboptimality for a supply chain
as a whole and identify the major drivers for that. For ease of exposition, we focus on
supply chains consisting of one buyer and one supplier that run the Master Planning
models presented in Sect. 2.2. Thereby, we adopt the convention to consider the
buyer as female (“she”) and the supplier as male (“he”).88 Note that most of our
analysis is equally relevant for more general settings, i.e., supply chains with more
than two parties and other planning levels.
2.3.2.1 Models for Decentralized Planning
Due to the increasing concentration on core competencies and subsequent outsourcing activities, many supply chains are not run by a single enterprise, but by several,
legally separated business units. Under these conditions, an integrated planning for
a supply chain as a whole might fail. A major reason for this is information asymmetry: Usually, none of the decentralized parties has the knowledge about all data
required for integrated planning. Sharing local production data among decentralized
parties, however, is problematic.89 Private data may be sensitive (especially capacity data90 ) and constitute a strategic advantage for bargaining, which is lost after
revelation.
Although this problem is fundamental and matters in most real-world supply
chains, current APS do not offer any convincing solution for it. In fact, APS often include tools for collaboration in their planning suites;91 however, these tools
only facilitate the exchange of information, but do not account for the reluctance of
decentralized parties to interchange detailed (production) data.
Hence, instead of solving the centralized model, a decentralized party can only
determine plans that are valid for its own planning domain, i.e., the “part of the
supply chain and the related planning processes that is under the control and in the
responsibility of one planning organization.”92 For developing mathematical models
88
This convention is also used within the supply chain contracting literature, e.g., Cachon (2003,
p. 230).
89
This has also been pointed out by , e.g., Arikapuram and Veeramani (2004, p. 111).
90
As an indicator for the reluctance of practitioners for exchanging production capacities, see the
empirical study of Kersten (2003, p. 332).
91
E.g., SAP (2008) and Oracle (2008).
92
Kilger et al. (2007, p. 263).
2.3 Decentralized Planning and Coordination
25
for decentralized planning, consider the interdependencies between the decentralized models that are made up by the inventory balance constraints for the items
supplied:93
Ijt 1 C Xjt D
X
rjk Xkt C Ijt
8j 2 J D ; t 2 T:
(2.41)
8j 2 J D ; t 2 T;
(2.42)
ISjt1 C Xjt D XSjt C ISjt
8j 2 J D ; t 2 T;
(2.43)
Ijt D ISjt C IBjt
D
8j 2 J ; t 2 T ;
(2.44)
XBjt D XSjt
D
8j 2 J ; t 2 T ;
(2.45)
IBjt 0
D
8j 2 J ; t 2 T;
(2.46)
ISjt 0
D
8j 2 J ; t 2 T;
(2.47)
XBjt 0
8j 2 J ; t 2 T;
(2.48)
XSjt 0 8j 2 J ; t 2 T:
(2.49)
k2Sj \J B
Sets
J B Set of items produced by the buyer
J D Set of items supplied
These constraints can be reformulated as
X
IBjt1 C XBjt D
rjk Xkt C IBjt
k2Sj
\J B
D
D
Variables
IBjt Inventory of the (supplied) item j at the buyer’s site in period t; IBj 0 D 0
ISjt Inventory of item j at the supplier’s site in period t; ISj 0 D 0
XBjt Amount of item j delivered to the buyer in period t
XSjt Amount of item j delivered by the supplier in period t
(2.42) and (2.43) are inventory balance constraints for the decentralized models.
Constraints (2.44) are necessary for the correct computation of the systemwide costs
for inventory holding. Constraints (2.45) exclusively comprise the interdependent
decisions and link the decentralized models. Constraints (2.46)–(2.49) ensure the
nonnegativity of the new variables IBjt ; ISjt ; XBjt , and XSjt .
The formulations for the decentralized models of buyer and supplier can be derived from a given centralized model by limiting the items and resources modeled
to those owned by buyer and supplier and augmenting these models by (2.42) and
(2.43), respectively. We denote the decentralized versions of the models of buyer
and supplier by adding “S” and “B,” respectively, to the model names. Throughout
this dissertation, we abbreviate “buyer” by “B” and “supplier” by “S.”
93
With “items supplied,” we mean the subset of items that may be ordered by the buyer (and
subsequently supplied by the supplier)
26
2 Supply Chain Planning and Coordination
We take GM as the base model in the following. Since (2.45) comprise decisions
of both planning domains, a direct inclusion of these constraints is neither possible
in GM-S nor in GM-B. Instead, for modeling decentralized planning, targets for the
supply quantities have to be incorporated into the decentralized models. I.e., GM-B
has to be augmented by
xtjt D XBjt
8j 2 J D ; t 2 T
(2.50)
xtjt D XSjt
8j 2 J D ; t 2 T:
(2.51)
and GM-S by
Data
xtjt Target for the supply quantity of item j in period t
Without the knowledge of the solution to the centralized model, the target quantities leading to the supply chain optimum can only be determined by chance. In
practice, myopic procedures for (unilaterally) determining these targets are employed. Before dealing with these procedures in Sect. 2.3.3, we will point out the
major reasons why such myopic procedures often fall short.
2.3.2.2 Drivers for Suboptimality of Decentralized Planning
We have identified a couple of major drivers for systemwide suboptimality due to
inappropriate targets for the supply quantities, thereby focusing on the Master Planning models presented in Sect. 2.2. These drivers correspond to the characteristics
of these targets that cause elevated costs for one decentralized party and, thus, potentially for the whole supply chain. The knowledge of these drivers is useful for
both characterizing settings with a need for coordination and classifying collaborative planning approaches; in fact, all approaches mentioned in the literature review
of Chap. 3 can be classified according to the drivers identified here.
In Table 2.3, we provide an overview of the drivers and state their effects on the
parties’ costs. Three main drivers can be distinguished. The category “time” refers
to a temporal over- or undersupply, “lot cycles” to a misalignment of the lot cycles in
the supply target, and “quantity” to inappropriate choices of the absolute quantities
ordered within the planning interval. In the following, we will explain in more detail
how these drivers can lead to increases in the supplier’s costs.
Regarding the driver T, two main effects can be distinguished. First, the target
supply quantities in early periods of the planning horizon may be too large to be
Table 2.3 Drivers for increases in the supplier’s costs
Driver
Impact on parties’ costs
Time (T)
Lot cycles (L)
Quantity (Q)
Increase of overtime, setup, holding, and backorder costs
Increase of holding, setup costs
Increase of costs for overtime, lost sales
2.3 Decentralized Planning and Coordination
27
covered by the supplier’s stock and production with normal capacity. In order to
fulfill this target, the supplier has to employ costly overtime (see Example 2.1). If
the supplier’s costs for early supply exceed the costs of the buyer for a potential
delay, suboptimality for the whole supply chain results.
Example 2.1 Consider the supply target for an item in the left of Fig. 2.4. Assume
zero initial inventories. Since the possible build-up of stock of this item in period
1 is smaller than the excess supply in period 2, the use of (expensive) overtime is
necessary for the supplier.
Second, the contrary may happen, i.e., the supply may take place too late from the
supplier’s point of view. Then, elevated holding costs for the supplier may incur if
the supplier has to prepone part of his production due to, e.g., restricted production
capacity in a later period or scale effects of lot-sizing. The systemwide costs will
increase, too, provided that these additional holding costs exceed the cost increase
for the buyer due for an earlier order (see Example 2.2).
Example 2.2 In the right of Fig. 2.4, the supply target for item 3 is depicted. Since
the supplier’s capacity is partially reserved in period 2, a part of the production
of item 3 has to take place in period 1. Let there be end items 1 and 2 with equal
holding costs and demand. Let each of these end items require one period length
and item 1 one unit (U) of the predecessor item 3 for production. Then the buyer
is indifferent which item to produce first, and his choice, i.e., producing item 1 in
period 2 (instead of period 1), increases systemwide costs.
The driver L has two manifestations, too. On the one hand, considering single items,
their TBO (D time between orders) may be inappropriate. If an item is ordered too
often, the additional costs for supplier’s setups or inventory holding may cause systemwide suboptimality (see the following example).
Example 2.3 Consider a serial supply chain with one end item and one item supplied. For the production of 1[U] of the end item, 1[U] of the item supplied is
required. Assume six periods with level end item demand of 1[U] per unit time
(UT=1 period), holding costs of 1 monetary unit (MU) per unit and unit time, and
setup costs of 2[MU] for both parties. Let the TBO for the supply target be 2[UT]
(see Fig. 2.5), as preferred in the local solution of the buyer. This target leads to
Fig. 2.4 Example for early and late supply
28
2 Supply Chain Planning and Coordination
Fig. 2.5 Example for inappropriate TBO
Fig. 2.6 Example for misalignment of lot sizes
systemwide costs of 15[MU] (composed by setup costs of 6[MU] for buyer and supplier as well as holding costs of 3[MU] for the buyer), which exceed 14[MU], the
systemwide optimal costs for a TBO of 3[UT].
On the other hand, a misalignment of lot cycles among successor items may further
increase the supplier’s costs (see Example 2.4).
Example 2.4 Consider Fig. 2.6. Assume equal demand for all end items (1,2,3) at
the end of period 2 and zero initial inventories. Let the production capacity in period
2 be half the capacity in period 1 and setup and inventory holding costs be equal
for all end items. Assume that the supply target is one of the buyer’s locally optimal
plans where items 1 and 3 are produced in period 1 and item 2 in period 2. This plan
involves a positive secondary demand for items 4 and 5 in period 1 and for item 4
in period 2. Due to restricted production capacities, the supplier has to set up item
4 twice during the planning horizon to fulfill this demand. The systemwide optimal
order, in contrast, goes along with the production of items 1 and 2 in period 1 and
the production of item 3 in period 2. This plan involves the same costs for the buyer,
but lower costs for the supplier. Since item 4 is exclusively demanded in period 1,
the supplier can limit to one setup of item 4 and thus reduce his setup costs.
The last driver for elevated costs of the supplier is the quantity, i.e., inappropriate
relations among the absolute quantities ordered. First, consider one single item supplied. Here, too large target quantities are suboptimal if the resulting overtime of the
2.3 Decentralized Planning and Coordination
29
Fig. 2.7 Example for production program as driver
supplier is more expensive than the buyer’s lost sales that would occur for smaller
supply quantities. Second, for several interdependent items, the relation of their target quantities may be inappropriate as illustrated by the subsequent example.
Example 2.5 Consider one period and two items supplied. Assume that the buyer
produces two items (1,2) with contribution margins of 4[MU] and 3[MU], respectively. Let the maximum demand for these items be 5[U]. The capacity of the buyer’s
resource is 5 capacity units (CU). Both items 1 and 2 require 1[CU/U] of this
resource for their production as well as 1[U] of their predecessor items 3,4 (see
Fig. 2.7). Items 3 and 4 are produced on a resource with a capacity of 10[CU]. The
production of item 3 requires 4[CU/U] on this resource, and the production of item
4 1[CU/U]. The capacity of this resource can be extended by overtime at costs of
1[MU/U]. Consider a supply target of 5[U] of item 3 and 0[U] of item 4. Although
maximizing the buyer’s profits, this target is not optimal for the whole supply chain
(systemwide profits: 4 5 10 D 10 [MU]). Due to the elevated overtime costs
for the supplier, the production of 1.25[U] of item 3 and 5[U] of item 4 is more
profitable instead (4 1:25 C 3 3 D 14[MU]).
For the buyer, analogous drivers can be identified, with the difference that early and
late supply have contrary consequences here and that backorders and lost sales may
become necessary apart from overtime.
Note that often combinations of these drivers occur, e.g., for the MLCLSP, which
comprises capacity restrictions and multi-level lot-sizing. In the presence of specific
model characteristics, the impact of the drivers can be considerably aggravated (see
Example 2.6).
Example 2.6 Assume a BOM with two end items which require each 1[U] of an
intermediate item for production (see Fig. 2.8a). Let the target supply quantities be
equal to the secondary demand for the items supplied (Fig. 2.8b). An implementation of this target avoids overtime at a buyer planning based on the MLCLSP94
(Fig. 2.8c), an insight that proved useful for the design of coordination schemes.95
With the presence of large setup times, however, overtime at the buyer may be required if the capacity consumption of a setup for item 2 is greater than the available
94
95
See Sect. 2.2.2 for this model.
See the scheme of Dudek and Stadtler (2005, p. 673) and that proposed in Chap. 4.
30
2 Supply Chain Planning and Coordination
A
B
C
D
E
Fig. 2.8 Aggravation of the suboptimality of decentralized planning with setup times and
campaign restrictions
slack capacity (Fig. 2.8d). Even worse may be the consequences of minimum campaign quantities (Fig. 2.8e). Here, the campaign for item 1 cannot be started since
the supply of item 3 is smaller than the minimum campaign quantity.
2.3.3 Upstream Vs. Collaborative Planning
A common procedure for determining targets for supply quantities is upstream
planning, i.e., sequential planning starting with the locally optimal plan for the
downstream party (here: the buyer). Then this party derives an order for the raw materials required to fulfill her plan and communicates this order to the upstream party
(here: the supplier).96 Depending on the supplier’s leeway for order fulfillment, two
cases can be distinguished, full and voluntary compliance by the supplier.
Forced compliance means that the supplier fulfills the buyer’s order without
any changes. In the literature, this assumption underlies simulation experiments for
evaluating the systemwide suboptimality due to decentralization97 as well as coordination schemes.98 As a motivation for the forced compliance case, refer to the use of
96
In supply chains with more than two tiers, the upstream party would act as an upstream party
again and communicate his order to the party at the next tier upstream, and so on.
97
See Simpson and Erenguc (2001, p. 119) and Simpson (2007, p. 127). Note that in these papers, other terminologies (“pull style planning,” “local planning”) have been employed instead of
“upstream planning.”
98
See, e.g., Dudek and Stadtler (2005, p. 668). There, the term “upstream planning” has been
introduced.
2.3 Decentralized Planning and Coordination
31
Fig. 2.9 Upstream planning with forced compliance by the supplier
quantity flexibility contracts in practice. There, the supplier has to fulfill the buyer’s
order completely if the order quantities are kept within a prespecified corridor.99
We illustrate upstream planning with forced compliance by an example (see also
Fig. 2.9).
Example 2.7 In order to fulfill her end item demand (1), the buyer determines the
production plan that is optimal for her planning domain (2). She prefers to produce
the demand of period 3 already in period 2 since the associated setup cost reduction
exceeds the additional inventory holding costs. Next, the buyer derives via BOMexplosion the amounts of the intermediate items which are necessary to fulfill her
production plan and places orders for these intermediate items (3). Finally, the supplier determines a production plan for his planning domain with the restriction of
fulfilling the buyer’s orders (4). Note that the production of the supplier may differ from the order schedule. Here, the supplier has to prepone a part of production
because of tight capacities.
The mathematical modeling of this planning process is as follows: First, the buyer
solves GM-B, thereby assuming an unrestricted delivery of the items supplied. Then
she communicates to the supplier the supply quantities xtjt , which are determined by
X
xtjt D
rjk Xkt :
k2Sj \J B
Next, the supplier solves GM-S augmented by (2.51).
An obvious drawback of this procedure is that the supply target is unilaterally determined by the buyer. This renders the resulting solution vulnerable for
the drivers for suboptimality affecting the supplier’s costs.100 For uncapacitated
lot-sizing problems, the resulting suboptimality has been evaluated by simulation
studies. For different parameter settings, average costs increases of 11.5% and 4%
compared to upstream planning have been reported.101 If capacity restrictions come
into play, greater suboptimalities can be expected since the overtime costs caused by
99
See Tsay (1999, p. 1340) for examples for quantity flexibility contracts in industry.
See Table 2.3 on p. 26.
101
See Simpson and Erenguc (2001, p. 123) and Simpson (2007, p. 133).
100
32
2 Supply Chain Planning and Coordination
Fig. 2.10 Upstream planning with voluntary compliance by the supplier
inappropriate orders are usually of a higher magnitude than the costs for inventory
holding and setups.102 Usually, this suboptimality increases in case small safety
stocks are held by the supplier.
Although forced compliance allows a straightforward determination of the target
supply quantities, this assumption cannot always be sustained. Apart from different
contractual agreements between buyer and supplier that concede the supplier some
(or even infinite) leeway for shortages,103 the supplier simply might not be able to
fulfill the buyer’s order – even if he is willing to do so. This occurs, for example, if
the supplier has scarce production capacities and is not able to extend them.104
Therefore, we further consider voluntary compliance, which means that the supplier is not obliged to exactly fulfill the buyer’s order, but can freely choose the
extent of his order fulfillment. Below, we provide an example for this (Fig. 2.10).
Example 2.8 Analogously to Example 2.7, the buyer combines the demands of
periods 1 and 2 (1) in her production plan (2) in order to save setup costs and
places a corresponding order (3). Here the supplier cannot fulfill this order completely. His optimal production plan (4) results in a reduced supply in period 1 (5).
The order fulfillment by the supplier determines the initial, uncoordinated solution.
Here, this solution is feasible for the whole supply chain, provided that the resulting
lost sales are feasible for the buyer.
Again, we provide the mathematical modeling for this procedure. First, the buyer
solves GM-B and communicates the supply quantities xtjt to the supplier. Next,
the supplier determines the extent he is willing to fulfill the buyer’s order, thereby
taking into account penalties for shortages and lost sales. For this purpose, he solves
GM-S with the modified objective function
102
Some indications on the resulting suboptimality in capacitated settings provide our computational tests in Chap. 6.
103
As an example for such agreements, consider the contractual practice in the video rental
industry, see, e.g., Cachon and Lariviere (2001b, p. 20).
104
This may be due, e.g., to technical constraints on the production process such as batch
production (see Sect. 2.2.2.2). There, the batch sizes (e.g., volumes of tanks) might not be
extendable in medium term.
2.3 Decentralized Planning and Coordination
X X
j 2J S t 2T
hj Ijt C
X X
ocm Omt C
m2M S t 2T
33
X X
X X
blcj BLjt C
j 2J D t 2T
lscj LSjt ;
j 2J D t 2T
(2.52)
S
with M as the set of resources of the supplier and modified inventory balances for
the items supplied:
Ijt 1 C Xjt C BLjt C LSjt D BLjt 1 C Ijt C xtjt
8j 2 J D ; t 2 T: (2.53)
Finally, the buyer solves GM-B augmented by
0
D
xtjt
X
k2Sj
rjk Xkt
8j 2 J D ; t 2 T;
\J B
0
derived from the outcome of the supplier’s
with modified supply quantities xtjt
model:
0
xtjt
D xtjt BLjt LSjt C BLjt 1 :
Like in the forced compliance case, systemwide suboptimality may result from inappropriate orders of the buyer. Moreover, inappropriate shortages by the supplier
may trigger elevated costs for the buyer. A further potential drawback of this procedure is the missing guarantee for feasibility in the presence of particular restrictions
such as shelf-life or storage capacity restrictions (see Example 2.9).
Example 2.9 Assume that the supplier delivers two intermediate items that are assembled by the buyer to an end item. Given zero initial inventories, the buyer will
order these items in an equal relation, of course. Assume that the supplier can only
fulfill the order of one of these items due to restrictions on his production capacities.
Since the buyer cannot perform the assembly then, she would have to keep the supplier’s delivery in stock. If her storage capacities are too scarce, then the supplier’s
delivery might be infeasible for her. Simply choosing a lower supply than offered
by the supplier, however, does not necessarily establish systemwide feasibility if the
supplier’s storage capacity is restricted, too.
Summarizing, upstream planning may lead to suboptimal plans and even bears the
risk of generating infeasible solutions. In this work, we present a remedy for this:
Collaborative planning. We propose that decentralized parties do not settle for the
solution from upstream planning, but collaborate in order to identify a feasible supply target with costs coming near to the systemwide optimum.
While upstream planning is based on unilateral targets, collaborative planning
takes the preferences of all decentralized parties into account. It aims to combine
the advantages of centralized and decentralized planning, i.e., to reach an appropriate solution quality without exchanging detailed information about the decentralized
parties’ production processes. The collaborative planning mechanisms proposed in
this work operate as add-ons to improve a default solution, which, e.g., may result from upstream planning. They try to identify an improvement by an iterative
34
2 Supply Chain Planning and Coordination
Fig. 2.11 Collaborative planning
exchange of several supply proposals, with the chance that one of these proposals
is superior to the target determined by upstream planning. We illustrate this process
by an example (see also Fig. 2.11).
Example 2.10 Consider upstream planning with forced compliance. The buyer’s
demand (1) and purchase order (2) are taken from Example 2.7. Now, apart from
upstream planning (2), parties iteratively interchange supply proposals in order to
identify an improved solution for the whole supply chain. Here the supplier proposes
a more equal temporal distribution of the supply quantities to increase his capacity
utilization (3), and the buyer adapts this proposal by aggregating the supply of periods 3 and 4 (4). Steps (5),(6), : : : stand for further proposals generated, which are
not depicted in Fig. 2.11 for ease of exposition.
The main challenges for carrying out such a collaborative planning process are the
identification of potentially coordinating proposals and the assurance that the coordination process is supported by the incentives of self-interested decentralized
parties. Both issues are tackled by the new mechanisms proposed in Chaps. 4 and 5.
Chapter 3
Coordination Mechanisms for Supply
Chain Planning
In recent years, a large number of papers have been produced that propose and
analyze mechanisms for supply chain coordination. The number of existing surveys
about the literature of this area is considerable, too.1 These surveys, however, are
not exhaustive; surprisingly few emphasis has been placed on a central topic for the
design of coordination mechanisms, the determination of appropriate incentives for
decentralized parties in light of information asymmetry.2
In this chapter, we provide a literature review, which comprises new classifications as well as explanations of central ideas behind different types of mechanisms. We focus on mechanisms that are directly applicable or transferable to
the coordination of Master Planning. In addition, we include approaches that address operational planning (e.g., scheduling) and provide interesting, novel ideas
for the design of coordination mechanisms, but exclude those dealing with strategic
planning tasks.3
3.1 Symmetric Information
In this section, we review coordination mechanisms which require that at least one
party has access to the information that is necessary for solving the centralized
planning models. In principle, coordination in such settings could be achieved easily: The party disposing of all relevant data could determine the optimal plan for
the whole system and offer the resulting supply target to the other parties together
with some share of the surplus from coordination. This approach, however, is not
regarded satisfactory in the literature; the largest body on supply chain coordination
mechanisms has been developed for settings with complete information.
1
See, e.g., Whang (1995, p. 413), Fugate et al. (2006, p. 129), Cachon and Netessine (2004, p. 1),
and the references provided below.
2
One exception is the paper by Cachon and Netessine (2004, p. 13), which, however, completely
omits the literature on bilateral information asymmetry provided in Sect. 3.3.
3
See, e.g., Van Mieghem (1999, p. 954), Böckem and Schiller (2004, p. 219), Plambeck and Taylor
(2007, p. 1872), and Ha and Tong (2008, p. 701).
M. Albrecht, Supply Chain Coordination Mechanisms: New Approaches
for Collaborative Planning, Lecture Notes in Economics and Mathematical Systems 628,
c Springer-Verlag Berlin Heidelberg 2010
DOI 10.1007/978-3-642-02833-5 3, 35
36
3 Coordination Mechanisms for Supply Chain Planning
First, this approach may fall short if multiple parties are to be coordinated. Then,
parties may not be willing to follow the advice of a central planner if some of them
can form subcoalitions which yield greater benefits than those to be allocated by the
central planner. For the design of coordination mechanisms in this setting, the application of cooperative game theory is required; we review corresponding approaches
in Sect. 3.1.2.
Second, some research accounts for the inexpediency of central targets in decentralized organizations and advocate the establishment of coordinating contracts
instead of these targets. One advantage of contracts is that parties can reduce the
administrative burden by handling over some effort for decision-making and data
collection (e.g., the generation of demand forecasts) to the responsibility of the
other parties. This may lead to substantial simplifications in the planning, e.g., a
buyer can usually collect demand information easier than a supplier acting as a central coordinator. Moreover, the establishment of contracts has the huge benefit that
parties’ incentives are kept untouched, e.g., a buyer has potentially a better motivation for optimally determining sales quantities and prices on her own since she is
directly affected by the results of these decisions.4 The effects of these contracts are
investigated using concepts from non-cooperative game theory in the literature.5 We
address them in the following subsection.
3.1.1 Non-cooperative Game Theory
Mechanisms based on non-cooperative game theory usually propose the establishment of coordinating contracts. We review these approaches separately according
to the drivers for suboptimality identified in Sect. 2.3. In Sect. 3.1.1.1, we survey
approaches related with the drivers Q (quantity) and T (time), and in Sect. 3.1.1.2
those related with L (lot sizes).6
4
As a further obstacle in this context, note that if one party determines the optimal solution for
the whole supply chain and asks the other parties for an implementation of this solution, the other
parties might not consent due to their missing involvement into the decision-making process.
5
Non-cooperative and cooperative game theory are branches of game theory that basically differ
by their focus: Whereas non-cooperative game theory yields a prediction about the outcome of the
game (in monetary terms) without exactly specifying the actions taken, cooperative game theory
focuses on the specific actions taken by players assuming that parties can negotiate effectively, see,
e.g., Cachon and Netessine (2004, p. 26).
6
Note that in the section at hand, we will also include simple contracting schemes, where no sophisticated game theoretical analysis is needed since neither uncertainty nor coalition building is
considered. This is a convention taken here since these approaches could be subsumed under both
cooperative and non-cooperative game theory in principle. However, we do not include approaches
that, instead of relying on contracts, advocate a unilateral determination of the systemwide optimum by one party and disregard the design of incentives for its implementation (e.g., Shirodkar
and Kempf, 2006, p. 420; and Barbarosoĝlu, 2000, p. 732).
3.1 Symmetric Information
37
3.1.1.1 Coordination on Quantities and Time
Except for those focusing on the coordination of order cycles (see the next subsection), most coordination mechanisms requiring symmetric information are built on
the newsvendor model and extensions. To illustrate the basic ideas of these mechanisms, we exemplarily discuss the application of revenue-sharing contracts in this
setting.
Consider a buyer–supplier supply chain with a single item supplied. This item is
sold by the buyer at a price p in an external market (see Fig. 3.1). Thereby, the buyer
faces a newsvendor problem:7 The market demand is stochastic and the quantities
ordered by the buyer have to be specified before the start of the selling season.
Denote the market demand as a random variable D with the probability density
function f .D/ and the cumulated density function F .D/, the quantity ordered by
the buyer by Q, and the quantity sold to the market by M .Q/. Neither the order
of the buyer nor the market demand have to be fulfilled completely, i.e., voluntary
compliance is assumed for both buyer and supplier. Furthermore, the buyer has the
possibility to sell leftover inventory I.Q/ D .Q M.Q//C at a salvage value v.
Production costs at the supplier’s site are denoted by c0 .
In the default planning process without coordination, the supplier specifies a
wholesale price w for the traded good. Unfortunately, such a wholesale price (contract) does not coordinate the supply chain under standard assumptions (amongst
them, risk-neutrality and self-interest of parties8 ). This can be recognized by the
following analysis.9 Consider gB .Q/, the expected profit of the buyer dependent
on Q.
Z Q
gB .Q/ D .p w/ Q .p v/
f .D/ dD:
(3.1)
0
gB .Q/ is made up by the profit for selling all units ordered in the external market
less the difference of selling price and salvage value for the salvaged units. Since
Fig. 3.1 Coordination by revenue sharing
7
For an introduction to the newsvendor model, see, e.g., Silver et al. (1998, p. 385).
See, e.g., Cui et al. (2007, p. 1303), for a paper that does not rely on these standard assumptions.
There, it is shown that wholesale price contracts may lead to supply chain coordination, given fair
(and not self-interested) behavior of parties.
9
Similar analyses can be found, e.g., in Cachon (2003, p. 233), Lariviere (1999, p. 233), and
Thonemann (2005, p. 215).
8
38
3 Coordination Mechanisms for Supply Chain Planning
gB .Q/ is strictly concave, QB
, the optimal order quantity of the buyer, can be
obtained by differentiating (3.1). Setting
@gB .Q/
D p w .p v/F .Q/
@Q
equal to zero yields
QB
D F 1
pw
:
pv
(3.2)
To find out whether the choice of a profit-maximizing wholesale price is a Nash
equilibrium,10 compare (3.2) to QSC
, the order quantity optimal for the whole sup
ply chain, which can be derived analogously to QB
:
QSC
D F 1
p c0
pv
:
It is easy to see that c0 is the only wholesale price which induces the buyer to place
a systemwide optimal order. Such a choice, however, would lead to a profit of zero
for the supplier. Hence, for a rational supplier, the Nash equilibrium is to choose a
wholesale price w > c0 . This results in double marginalization since the buyer will
order inadequate (and too low) quantities then.
In light of this problem, several contracts have been designed which actually provide incentives for the buyer to place a systemwide optimal order. Among those are
revenue-sharing contracts which are used within the video retail industry.11 There,
the supplier subsidizes larger order quantities by offering a lower wholesale price
and requires a share s of the revenue generated by the buyer for compensation (see
also Fig. 3.1). Like w, the share s is fixed before the start of the selling season.
It can be shown that properly designed revenue-sharing contracts lead to an optimal coordination of the supply chain, independently from all other parameters like
the market demand, and so on. With such a contract, the buyer’s gains become:
Z Q
gB .Q/ D .sp w/ Q .sp v/
F .D/ dD:
(3.3)
0
can be obtained by the partial derivative of (3.3) to Q:
Again, QB
D F 1
QB
sp w
:
sp v
(3.4)
10
A Nash equilibrium is a central solution concept in non-cooperative game theory, where “each
player’s strategy choice is a best response [. . . ] to the strategies actually played by his rivals,”
Mas-Colell et al. (1995, p. 246).
11
See, e.g., Cachon and Lariviere (2001a, p. 20).
3.1 Symmetric Information
39
Setting the RHS (right-hand side) of (3.4) equal to the RHS of (3.2) leads to a conditional equation for s and w with real-valued solutions for these parameters. As a
consequence, always a contract can be designed where the choice of the systemwide
optimal order quantity is a Nash equilibrium and both parties obtain greater profits
than in the default solution (i.e., the mere application of a wholesale price contract).
The distribution of the surplus depends on the concrete choices of s and w and has
to be determined by additional negotiations between the parties.
Table 3.2 provides a review of seminal and recent papers proposing and analyzing
coordination by contracts. Note, however, that this review is not exhaustive; we refer
to the extensive surveys by Tsay et al. (1998),12 Lariviere (1999),13 and Cachon
(2003),14 for additional research done in this area. Moreover, Table 3.1 provides
some nomenclature used for the classification.
Most papers are based on the newsvendor model and, hence, address the driver Q
(quantity) characterizing the suboptimality of the default setting. The buyer is usually assumed as the leader deciding about the order quantities. An exception is the
paper of Corbett and DeCroix (2001),15 where the supplier decides about his supply
of hazardous solvents causing additional disposal costs for the buyer. A further nonstandard setting is that considered by Ferguson et al. (2006).16 There, false failure
returns by the retailer cause systemwide suboptimality. To incorporate these returns
into our classification, they can be interpreted as inappropriate quantities required,
which cause elevated production costs for the supplier.
Papers that focus on the driver T (time) mostly consider multi-period inventory
models with base stock policies adopted by the decentralized parties. Both discrete17 and continuous time structures with finite production capacities18 have been
addressed.
Concerning the supply chain structures, extensions on assembly supply chains
and multiple retailers have been investigated apart from the basic buyer–supplier
setting. The increased number of players at one tier results in competition effects,
Table 3.1 Nomenclature for
the classification (structure)
12
See Tsay et al. (1998, p. 299).
See Lariviere (1999, p. 233).
14
See Cachon (2003, p. 227).
15
See Corbett and DeCroix (2001, p. 881).
16
See Ferguson et al. (2006, p. 376).
17
E.g., Cachon and Zipkin (1999, p. 936).
18
E.g., Caldentey and Wein (2003, p. 1).
13
Group
Parameter
Comment
Structure
1-1
n-1
1-n
Arb.
Buyer–supplier setting
Multiple suppliers, one buyer
One supplier, multiple buyers
Arbitrary structure
40
3 Coordination Mechanisms for Supply Chain Planning
Table 3.2 Coordination on quantities and time with symmetric information
Author(s)
Driver Structure Type of contract
Bernstein and Federgruen (2005) Q
1-1
(Nonlinear) price discount
Bernstein and Federgruen (2007) Q
1-n
(Nonlinear) price discount
Bernstein et al. (2006)
Q
1-n
Wholesale price
Burer et al. (2008)
Q
1-n
Transfer payment
Cachon (2004)
Q
1-1
Wholesale, advanced purchase
Cachon and Lariviere (2005)
Q
1-n
Revenue sharing
Corbett and DeCroix (2001)
Q
1-1
Shared-savings
Cui et al. (2007)
Q
1-1
Wholesale, quantity discount
Dana Jr. and Spier (2001)
Q
1-n
Revenue sharing
Dong and Zhu (2007)
Q
1-1
Wholesale, advanced purchase
Ferguson et al. (2006)
Q
1-1
Target rebate
Gerchak and Wang (2004)
Q
n-1
Revenue sharing
Lariviere and Porteus (2001)
Q
1-1
Wholesale
Pasternack (1985)
Q
1-1
Buyback
Sethi et al. (2004)
Q
1-1
Quantity flexibility
Taylor (2001)
Q
1-1
Return policies
Taylor (2002)
Q
1-1
Target rebate
Tsay (1999)
Q
1-1
Quantity flexibility
Bassok and Anupindi (2008)
T
1-1
Quantity flexibility
Barnes-Schuster et al. (2002)
T
1-1
Option
Bernstein and DeCroix (2006)
T
n-1
Transfer payment
Cachon and Zipkin (1999)
T
1-1
Transfer payment
Caldentey and Wein (2003)
T
1-1
Transfer payment
Gupta and Weerawat (2006)
T
1-1
Two-part revenue sharing
Zhang (2006)
T
n-1
Transfer payment
which – depending on the characteristics of the setting considered – may mitigate19
or exacerbate20 the suboptimality of (pure) wholesale price contracts.
Although some contract types have been shown to dominate others or to be equivalent in some specific settings,21 no general preference for particular contract types
can be recognized, a result, which underlines the activeness of this field of research.
3.1.1.2 Coordination of Lot Cycles
A further stream of coordination mechanisms that assume symmetric information
deals with the alignment of production and order lot cycles. The basic insight,
that an independent or unilateral determination of lot cycles may cause systemwide
19
See, e.g., the analysis of Cachon (2003, p. 271).
See, e.g., Gerchak and Wang (2004, p. 29).
21
See, e.g., Cachon and Lariviere (2005, p. 32), for the equivalence of revenue-sharing and
buyback contracts in a buyer–supplier newsvendor setting.
20
3.1 Symmetric Information
41
suboptimality, has already been recognized by Goyal (1976).22 For illustrating the
manifestation of double marginalization in this setting, consider a supply chain with
one buyer and one supplier planning based on the EOQ model.23 Denote the buyer’s
and supplier’s setup costs by scB ; scS > 0 and the buyer’s holdings costs by hB .
Moreover, assume that the supplier follows a lot-for-lot policy, i.e., he directly produces each order of the buyer without any inventory holding. Then the order lot size
chosen by the buyer under upstream planning (i.e., without coordination),
s
QB
D
2scB d
;
hB
is smaller than the systemwide optimal order lot size
s
D
QSC
2 .scB C scS / d
:
hB
Based on this insight, several authors have addressed the determination of
coordinated production policies for more complex settings24 and the design of
quantity discounts for their implementation.25 Compared to the mechanisms presented in the previous subsection, these approaches seem less appropriate for
decentralized coordination since they usually require the computation of the centralized optimum by the party offering the rebate and – in contrast to those analyzed
in the previous subsection – do not provide the advantage of handling over some
effort for data collection and decision-making to the other party. Due to this fact
and to the existence of several recent and comprehensive surveys on this topic,26 we
omit a detailed listing of related papers.
3.1.2 Cooperative Game Theory
Two basic objects in cooperative game theory are bargaining under complete
information and coalition building. While few applications for supply chain
coordination exist regarding the former,27 a considerable number of coordination
22
See Goyal (1976, p. 107).
For the EOQ model, see also Sect. 2.2.2.1.
24
E.g., the relaxation of the lot-for-lot assumption (Lee and Rosenblatt 1986, p. 1177) and the
coordination of production and shipment policies (e.g., Hill 1999, p. 2463).
25
E.g., Monahan (1984, p. 720) and Joglekar and Tharthare (1990, p. 492).
26
See, e.g., Sucky (2004b, p. 110), Sarmah et al. (2006, p. 1), Li and Wang (2007, p. 1), and
Ben-Daya et al. (2008, p. 726).
27
Among those are Kohli and Park (1989, p. 693), Gjerdrum et al. (2002, p. 586), and the analysis
in the paper by Nagarajan and Sos̆ic (2008, p. 725).
23
42
3 Coordination Mechanisms for Supply Chain Planning
mechanisms based on alliance formation have been proposed in the recent years.
There, the question is whether the surplus from coordination can be shared such
that all parties have incentives to participate in the grand coalition, i.e., the coalition
comprising all parties, which maximizes systemwide profit. This is not necessarily
the case, as shown in the following example.
Example 3.1 Consider three parties that can achieve a joint surplus of 300[MU]
from cooperating in the grand coalition. If two parties form a subcoalition, the
surpluses would be 250[MU] for this coalition and 0[MU] for the remaining party.
Although the grand coalition maximizes the systemwide surplus, a binding agreement for the formation of this coalition cannot be formed. For any rule for surplus
sharing in the grand coalition (e.g., 100,100,100), two parties would benefit from
building a subcoalition (e.g., with a surplus sharing of (125,125) among them).
A formal answer to this question, i.e., whether an allocation of the surplus exists
that provides incentives for forming the grand coalition, is given by the concept of
the core. The core is defined as the set of utility vectors u D .u1 ; : : : ; uI / of parties
i 2 P D f1; : : : ; I g that satisfy
X
i 2S
ui v .S / 8S P and
X
ui v .P / ;
i 2P
with v .S / ; v .P / as the surpluses (in terms of utility)28 from forming the subcoalition S and the grand coalition P , respectively, that can be allocated among parties
(e.g., by a central planner).
Apart from a unique utility vector, the core may comprise multiple elements or
be empty (as in Example 3.1). Some approaches applying this concept to supply
chain coordination indeed prove emptiness or non-emptiness of the core,29 whereas
others calculate core cost allocations explicitly.30
The potential non-emptiness is an important disadvantage of the core. Therefore,
other concepts such as the Shapley value and the nucleolus31 have been elaborated,
which – in contrast to the core – can always be determined, but are not necessarily
chosen by self-interested parties.
In the last years, many application of cooperative game theory to supply chain
coordination have been proposed. Note, however, that a transferability of these approaches to general settings with asymmetric information does not seem possible
since cooperative game theory requires the specification of the parties’ concrete actions, which is only possible under complete information. Moreover, several current
28
Coalition analysis in cooperative game theory usually relies on games with transferable utility
(TU games), where utility (most often money) is freely transferable among players, see, e.g.,
Myerson (1991, p. 422).
29
E.g., Dror and Hartman (2007, p. 78).
30
See, e.g., Drechsel and Kimms (2008) for capacitated lot-sizing and Houghtalen et al. (2007) for
freight alliances.
31
See, e.g., Myerson (1991, p. 452) for more on these concepts.
3.2 One-Sided Information Asymmetry
43
surveys of this area are available. Therefore, instead of providing a detailed listing
of corresponding papers, we refer to Meca and Timmer (2008)32 for a review of
cooperative lot-sizing models and to Nagarajan and Sos̆ic (2008)33 for a general
review about supply chain coordination based on cooperative game theory.
3.2 One-Sided Information Asymmetry
Since symmetric information often does not reflect business reality properly,
research on supply chain coordination mechanisms has been extended to information asymmetry. As a first step towards this extension, one-sided information
asymmetry has been addressed by a series of papers. This assumption is particularly
attractive since it allows for the transfer of well-known concepts that originally
have been developed in economic theory to overcome the phenomenon of adverse
selection. Adverse selection arises with private information held at least by one
party and distorts the optimal strategies of the uninformed parties from the systemwide optimum.34 This phenomenon has been recognized by Akerlof (1970)35
in the context of markets for used cars, where qualitatively good cars can only be
sold at low prices since buyers cannot distinguish the actual conditions of cars in
advance. Basic mechanisms to mitigate the resulting inefficiencies are signaling
and screening. In the following subsections, we present papers that apply these
mechanisms to supply chain coordination.
3.2.1 Signaling
Signaling has first been proposed by Spence (1973)36 in the context of labor
markets. There, employers cannot directly infer the types of applicants (i.e., their
skills), which leads to inefficiencies in these markets. A common remedy for that
is signaling: Skillful applicants signal by investing into their education (which
unskillful workers are not willing to do) in order to credibly communicate their
types to potential employers.
Cachon and Lariviere (2001a) have transferred this idea to supply chain
coordination.37 They consider a setting reversed to that outlined in Sect. 3.1.1.1.
The buyer determines the wholesale price for the item supplied and the supplier
32
See Meca and Timmer (2008, p. 1).
See Nagarajan and Sos̆ic (2008, p. 719).
34
For a similar definition, see Mas-Colell et al. (1995, p. 436).
35
See Akerlof (1970, p. 488).
36
See Spence (1973, p. 355).
37
See Cachon and Lariviere (2001a, p. 629).
33
44
3 Coordination Mechanisms for Supply Chain Planning
Fig. 3.2 Setting considered by Cachon and Lariviere (2001a)
bears the demand risk: His capacity limits the maximum supply quantity and, hence,
the fulfillment of market demand; a build-up of capacity is possible, but costly and
has to take place before the start of the selling season. The buyer, in turn, can place
her order after observing the market demand (see also Fig. 3.2).
In this setting, parties have conflicting goals with respect to the build-up of
capacity by the supplier. The buyer prefers a high build-up; she has no extra costs
and, if demand turns out to be high, she can fulfill a larger share of it. The supplier,
however, favors this build-up less since he might have spent the corresponding effort
unnecessarily in case of low demand. Moreover, assume that the forecast of market
demand is private information of the buyer and that the buyer communicates this
forecast to the supplier. Then, in light of the above conflict, the forecast information
provided by the buyer may be not credible for the supplier since the buyer benefits
from an inflated forecast without any repercussions on her profit.
Cachon and Lariviere (2001a) analyze the parties’ best strategies in this setting
with the simplification that only two possible types 2 fH; Lg (H D high, L D
low) for the buyer’s demand forecast exist: The forecast is D D X , with X as
a publicly known random variable with the distribution function F .X /. Denote the
level of the capacity built up by the supplier by k. Then the buyer’s profit dependent
on and k becomes
gB
.k/ D .p w/ M.k/;
with M.k/ as the expected sales volume defined by
M.k/ D E Œmin fk; D g :
The buyer’s profit crucially depends on the supplier’s capacity choice, which in turn
depends on the supplier’s believe about D and, hence, about . The buyer can
follow two main strategies here. First, she can design a contract (e.g., choose an appropriate level of the wholesale price) without regard to the question whether the
supplier believes the demand forecast. This strategy results in a pooling equilibrium, where the supplier has to rely on his prior knowledge about
in order to
determine his optimal capacity build-up. If the (buyer’s) forecast is of type H, this
equilibrium may be unfavorable since it may imply a low build-up of capacity and
result in systemwide forgone profits.
The second strategy for the buyer is signaling, i.e., to provide a credible forecast
to the supplier. To achieve this, the simplest possibility is to alter the wholesale
3.2 One-Sided Information Asymmetry
45
Fig. 3.3 Signaling
price (see Fig. 3.3). Assume that the forecast is of type H. In order to convince the
supplier that the type is H, the buyer has to increase w to such a value that would be
suboptimal for a buyer of the type L – even if the supplier believed the signal and
built up capacity accordingly. That means, the optimal wholesale price w has to be
chosen such that
H
@gB
.w ; H /
D 0;
@w
L
.w ; H /
@gB
< 0;
@w
H
L
.w ; H / ; gB
.w ; H / as the gains for a buyer of types H and L,
with gB
respectively, provided that the supplier believes that the buyer’s type is H .38
Apart from the increased wholesale price, Cachon and Lariviere (2001a) point
out further signaling strategies for the supplier like burning money, minimum quantity commitments, and option contracts, that can be used with forced compliance
of the supplier. Cachon and Lariviere (2001a) also show the preference of some
strategies over the others, but did not determine the optimal signaling strategy under
voluntary compliance. Further insights into this setting provides the paper of Özer
and Wei (2006),39 who take the perspective of the supplier and propose contracts
that induce signaling of the buyer.
3.2.2 Screening
Screening is a mechanism for the uninformed party to improve market efficiency.40
There, the uninformed party proposes a menu of choices and the party holding the
private information self-selects, i.e., it selects the choice with the highest benefits
and thereby reveals the private information. Often, an infinite number of potential (continuous or discrete) menus exist. Fortunately, the well-known revelation
principle41 allows reducing the number of these menus considerably: It states that
the optimal menu is incentive-compatible, i.e., the selecting party has incentives to
reveal its private information truthfully. In the above example for labor markets,
38
See Cachon (2003, p. 325).
See Özer and Wei (2006, p. 1238).
40
Among the first to study screening have been Rothschild and Stiglitz (1976, p. 629), who applied
this mechanism for coordination in insurance markets.
41
See Myerson (1979, p. 61).
39
46
3 Coordination Mechanisms for Supply Chain Planning
Fig. 3.4 Screening
a possibility for screening is to offer contracts specifically designed for different
types of applicants (e.g., regarding their stay in the organization).42
Relying on the paper of Corbett and de Groote (2000),43 we will exemplarily
describe how screening can be applied for supply chain coordination. We consider
the setting outlined in Sect. 3.1.1.2, with the difference that the buyer’s unit holding
cost hB is incomplete information for the supplier. Incomplete information means
that “some players do not know the payoffs of the others.”44 Here we assume that the
supplier has some estimates about this cost in the form of a cumulated probability
distribution F .hB / lower bounded by hB .
Further assume that the supplier has sufficient bargaining power to perform
screening, i.e., to propose a menu of quantity discount contracts as a take-it-orleave-it offer to the buyer. Each of these contracts specifies a pair .Q; P .Q// made
up by the order quantity Q and the corresponding price P .Q/. Choosing one of
these contracts, the buyer reveals her private information hB (see Fig. 3.4).
C
Further assume given reservation values of parties tcB
, tcSC , i.e., their costs
(which can also be negative with profitable outside opportunities, e.g., trading with
other parties) incurring if the buyer accepts neither of the contracts offered. Moreover, assume that the supplier opts to refuse trading with the buyer if hB exceeds
a limit hB . Then the optimization problem for determining the optimal menu of
contracts to be offered by the supplier can be stated as follows:
scS d
C P .hB / C EhB >hB tcSC
(3.5)
Q .hB /
hB
scB d
C
Q .hB / P .hB / 8hB 2 hB ; hB (3.6)
s.t.
tcB
Q .hB /
2
scB d
hB
@P .hB /
D
:
(3.7)
@hB
2
Q .hB /2
min EhB hB
The objective function (3.5) minimizes the sum of the expected costs for the supplier
in case of acceptance and refusal of trade. We write P .hB / in short for P .Q .hB //
here. Constraints (3.6) are individual rationality constraints. They ensure that the
buyer’s costs from accepting one of the contracts offered (the RHS of 3.6) do not
42
See, e.g., Salop and Salop (1976, p. 619).
See Corbett and de Groote (2000, p. 444).
44
See, e.g., Fudenberg and Tirole (1991, p. 209). The pioneering work for the modeling of
incomplete information in game theory is the three-part essay of Harsanyi (1967, p. 159), Harsanyi
(1968b, p. 486), and Harsanyi (1968a, p. 320), where it is shown how to model a game with
incomplete information by a game with imperfect information. For a recent review on this topic,
see Myerson (2004, p. 1818).
43
3.2 One-Sided Information Asymmetry
47
exceed the buyer’s reservation value, provided that hB 2 hB ; hB . The second
constraint (3.7) determines the incentive compatibility of the menu. This constraint
ensures that the buyer selects the contract designed for her private information (here:
hB ). Here, this is achieved by setting the partial derivative of the quantity discount
function P .hB / equal to the partial derivative of the buyer’s costs (the RHS of 3.7).
Based on this model, Corbett and de Groote (2000) derive the optimal set of
contracts that can be offered by the supplier. This set is continuous, i.e., for any
holding cost rate reported by the buyer, a different contract is offered.45
The structure of the above model with the individual rationality and the incentive
compatibility constraints is shared by all supply chain coordination mechanisms
based on screening. In Table 3.3, we provide a classification of these mechanisms.
Note that the criteria chosen for this classification differ from those for symmetric
information.46 Since all approaches deal with buyer–supplier settings and rely on
screening as the contract type, we omit these parameters and incorporate the type of
the unknown data instead.
As with symmetric information, most of these papers focus on the driver Q and
base their analyses on newsvendor models47 or bilateral monopoly settings.48 However, also the other drivers mentioned in Sect. 2.3.2.2 have been covered.49 The
variety of the unknown data considered has been substantial, too. Note that the
Table 3.3 Coordination mechanisms based on screening
Author(s)
Driver Private data
Arya and Mittendorf (2004)
Burnetas et al. (2007)
Corbett et al. (2005)
Corbett et al. (2004)
Ha (2001)
Özer and Wei (2006)
Li et al. (2007)
Schenk-Mathes (1995)
Taylor (2006)
Zhang et al. (2008)
Cachon and Zhang (2006)
Lutze and Özer (2008)
Corbett (2001)
Corbett and de Groote (2000)
Sucky (2004a)
45
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
T
T
T, L
L
L
Demand (consumer’s valuations)
Market demand (distribution)
No private, but non-contractible data
Variable costs of the buyer
Variable costs of the buyer
Market demand (distribution)
Market demand (distribution)
Supplier’s type (cost function)
Market demand (distribution)
Buyer’s inventory level
Supplier’s costs for lead time reduction
Shortage costs of the buyer
Buyer’s setup, backorder costs
Buyer’s holding costs
Buyer’s type (setup and holding costs)
For an approach that assumes discrete choices, see, e.g., Schenk-Mathes (1995, p. 176).
See Table 3.2 on p. 40.
47
E.g., Burnetas et al. (2007, p. 465).
48
E.g., Corbett et al. (2004, p. 550).
49
The driver T has been considered by Corbett (2001, p. 487), Cachon and Zhang (2006, p. 881),
and Lutze and Özer (2008, p. 898), and L by Corbett and de Groote (2000, p. 444), and Sucky
(2004a, p. 493). Note in this context that there are further publications of Sucky (e.g., Sucky 2006)
with almost identical scopes. We did not include them separately into our classification.
46
48
3 Coordination Mechanisms for Supply Chain Planning
approach by Corbett et al. (2005)50 plays a particular role in this context. There,
all information is commonly known, but some relevant parameters, the parties’ efforts, are non-contractible ex ante. Corbett et al. (2005) show that screening can
coordinate decisions in this setting, too. They propose that the supplier offers to
the buyer an optimal set of contracts with the buyer’s non-contractible effort as the
parameter to be selected by her.
3.3 Multilateral Information Asymmetry
In this section, we review mechanisms designed for multilateral information
asymmetry. We begin with an introduction to the concept of auctions, that plays
a central role for the determination of the decentralized parties’ incentives and
the allocation of the surplus, and provide an overview of related papers focusing
on supply chain coordination (Sect. 3.3.1). In Sect. 3.3.2, we address papers that
additionally deal with the generation of coordinating proposals, a crucial issue
that is not considered by conventional auctions, and describe the Dantzig–Wolfe
decomposition as a basic approach, which underlies many of these papers.
3.3.1 Auctions and Their Application to Supply
Chain Coordination
Standard Auctions
Consider a single good traded, an auctioneer, and a set of potential buyers with
private information about their valuations. To determine the allocation of the good
among the buyers, four basic auction procedures can be applied.51 In the ascendingbid auction, the auctioneer successively raises prices until a single bidder remains,
who has to buy the good at the final price. The descending-bid auction works the
other way round. There, the auctioneer decreases prices, and the first buyer that
accepts a price obtains the good. Further auction types are first-price sealed bid and
second-price sealed bid (Vickrey) auctions, where bidders simultaneously submit
sealed bids and the object is sold to the buyer with the highest bid at the highest
and second-highest price, respectively.
Two important topics in the design of (optimal) auctions are revenue maximization and incentive compatibility. A famous insight is the revenue equivalence
theorem52 here: If the buyers are risk-neutral and if their private valuations of the
good are independently drawn from some random distributions, then any auction
mechanism where the highest bid wins the good and bidders with valuations at the
50
See Corbett et al. (2005, p. 653).
E.g., Klemperer (1999, p. 229).
52
See Myerson (1981, p. 65).
51
3.3 Multilateral Information Asymmetry
49
lowest possible level obtain zero benefits yields the same expected revenue. Truthtelling is ensured if the good is sold at the second-highest price, which holds both
for the ascending-bid auction and the second-price sealed bid auction.53
Double Auctions
A double auction is a variant of auctions that lifts the requirement of a central auctioneer and treats buyers and sellers of the traded good(s) symmetrically. Its simplest
form is bilateral trade. There, a potential buyer and a potential seller of an indivisible
good simultaneously submit sealed bids consisting of purchasing and selling prices
(b and s), respectively. After breaking the seals, the conditions of trade are determined by the following rule: If the purchasing bid is lower than the selling bid, i.e.,
if b < s, nothing happens. Otherwise, the good is sold at the price kb C.1k/s. The
factor k determines the allocation of the surplus among parties. A natural choice for
k is 1=2, leading to a price equal to the average of seller’s and buyer’s bids.
In the literature, this mechanism has been investigated extensively.54 A basic insight is that buyer and seller have incentives to underbid and overbid, respectively.
The fact, that they will not exactly bid their reservation values, their valuations of
the traded good (which they receive if trade does not take place), can also be interpreted that parties do not truthfully reveal their private information.
Analyses of this mechanism assume that the probability distributions of parties’
reservation values are common knowledge. This common knowledge does not only
refer to the first-order believes, i.e., the knowledge of parties about the probability
distributions of the other parties’ reservation values, but also to higher-order believes, i.e., the additional knowledge that the own distribution is known by the other
party, that the latter knowledge is known by the other party, and so on.55 Then, parties’ best bidding strategies are interdependent since the bids of the seller depend
on his expectations about the buyer’s bids and vice versa. It has been shown that the
resulting game has multiple Nash equilibria.56 Fortunately, players prefer the most
efficient equilibrium in empirical tests,57 they adopt the linear equilibrium strategies
first characterized by Chatterjee and Samuelson (1983).58 For uniformly distributed
prior knowledge with equal probability density functions of parties, the efficiency
of this equilibrium is 27=32 D 84:375% on average.59
53
See, e.g., Klemperer (1999, p. 230). The proof of truth-telling in the second-price sealed bid
auction is due to Vickrey (1961, p. 8) and is among the most famous results in auction theory.
54
See Chatterjee and Samuelson (1983, p. 835) and Myerson and Satterthwaite (1983, p. 265) for
seminal papers on this topic.
55
For a formal description of this type of knowledge, see Aumann (1976, p. 1236).
56
See Satterthwaite and Williams (1989, p. 107) for characterizations of equilibria when parties
play differential strategies and Leininger et al. (1989, p. 63) when parties’ strategies are stepfunctions.
57
See Radner and Schotter (1989, p. 179) and Rapoport and Fuller (1995, p. 179).
58
See Chatterjee and Samuelson (1983, p. 838).
59
See Chatterjee and Samuelson (1983, p. 844).
50
3 Coordination Mechanisms for Supply Chain Planning
This mechanism is regarded to capture main features of bargaining in both small
and large markets.60 In fact, its generalization to multiple sellers and buyers can
be carried out straightforwardly. We refer to Williams (1993),61 Rustichini et al.
(1994),62 and Cripps and Swinkels (2006)63 for the analysis of efficiency and parties’ equilibrium strategies in these settings, and Chu and Shen (2006)64 and to
Chu and Shen (2008)65 for recently developed variants inducing truth-telling of
participants.
Multiunit and Combinatorial Auctions
A recent focus of auction research is on multiunit auctions. Even in the simplest
case, where each bidder wants to purchase one of several nonidentical items, additional problems compared to standard auctions have to be addressed, like the
question, which item is allocated to which bidder.66 Combinatorial auctions form
a more complex variant of multiunit auctions, which is of considerable relevance in
practice.67 They can be defined as “simultaneous multiple-item auctions that allow
submission of all or nothing bids for combinations of the items being sold.”68
Compared to other auction types, combinatorial auctions involve new challenges
like the non-existence of market prices and the computational difficulty of the winner determination problem. In principle, extensions of standard auction mechanisms
can be applied (e.g., ascending-bid or Vickrey auctions),69 but their analysis and
optimal design become considerably more difficult here.
Table 3.4 provides an overview about auctions applied for supply chain coordination. All approaches address inappropriate quantities as the driver for systemwide
suboptimality. Here, we again take the supply chain structure as a criterion for the
classification. Thereby, we write an “(e)” behind the identifier of the structure if a
central entity, i.e., a third-party auctioneer, is required. Moreover, we distinguish
between the types of the auctions and state their additional properties.
60
See, e.g., Satterthwaite and Williams (1989, p. 108).
See Williams (1993, p. 1101).
62
See Rustichini et al. (1994, p. 1041).
63
See Cripps and Swinkels (2006, p. 47).
64
See Chu and Shen (2006, p. 1215).
65
See Chu and Shen (2008, p. 102). This paper, in fact, synthesizes double auctions with multiunit
auctions (see the next paragraph) since the buyers bid on bundles of goods there.
66
See, e.g., Milgrom (2004, p. 251).
67
See, e.g., the application of combinatorial auctions for the US spectrums for telephone services
reported by Milgrom (2004, p. 297).
68
See Pekec and Rothkopf (2003, p. 1485). For recent surveys on combinatorial auctions, refer
further to Abrache et al. (2007, p. 131), de Vries and Vohra (2004, p. 247), de Vries and Vohra
(2003, p. 284), and Milgrom (2004, Chap. 8).
69
See, e.g., Milgrom (2007, p. 935).
61
3.3 Multilateral Information Asymmetry
Table 3.4 Auctions for supply chain coordination
Author(s)
Type
Beil and Wein (2003)
Ascending
Chen (2007)
All standard formats
Chen et al. (2005)
Second-price
Gallien and Wein (2005)
Ascending
Mishra and Veeramani (2006)
Ascending
Mishra and Veeramani (2007)
Descending
Parkes and Kalagnanam (2005) Iterative
51
Structure
1-n
1-n
1-n(e)
1-n(e)
1-n(e)
1-n(e)
1-n(e)
Additional properties
Multiattribute
Object: contracts
Multiunit
Multiunit
Multiunit
Multiunit
Multiattribute
All papers address procurement auctions, and most of them rely on standard
auctions formats. The aim of the auctions and, hence, the structures of the underlying supply chains, differ: Some papers aim at maximizing the buyer’s profit, and
others at maximizing the profit of the whole system and, hence, are designed for use
with a third-party auctioneer, e.g., an e-marketplace.70
Regarding the objects of the auctions, most often multiple units of items or items
with multiple attributes are traded. An exception is Chen (2007),71 who applies
auctions for determining supply contracts in a newsvendor-type setting. Interestingly, neither combinatorial auctions nor double auctions have been proposed in
this context yet.
Although they do not consider the auctioning of bundles, most approaches rely
on multiunit auctions. Since the cost functions assumed are rather simple, the
generation of potentially coordinating purchase quantities does not involve major
difficulties. In fact, the approach of Gallien and Wein (2005)72 is the only to include
one bottleneck in form of a capacity restriction at the supplier. The applicability
of this approach, however, is limited by the additional assumption that the auctioneer can estimate the suppliers’ capacity constraints, i.e., the capacity needed for the
production of items and the total available capacity at the suppliers’ bottlenecks.
In Master Planning, however, where a plenty of decisions and interdependencies
exist, a direct application of these approaches seems less appropriate.
3.3.2 Mechanisms with Focus on Proposal Generation
If optimization problems with multiple bottlenecks (e.g., multiple periods) are used
for modeling the decisions of the decentralized parties, a multidimensional set of
potentially coordinating supply quantities has to be considered. Here, apart from
designing incentives for the decentralized parties to reveal their costs and from determining the allocation of the surplus, the generation of potentially coordinating
supply proposals becomes a further task to resolve.
70
See Elmaghraby (2004, p. 213) for a survey about the application of auctions in e-marketplaces.
See Chen (2007, p. 1562).
72
See Gallien and Wein (2005, p. 76).
71
52
3 Coordination Mechanisms for Supply Chain Planning
We begin with a description of dual decomposition as a generic approach for
generating such proposals. Subsequently, we survey the corresponding literature on
coordination mechanisms. Since these mechanisms are most closely related to the
topic of this work, we describe in detail the characteristics of each of them. Finally,
we discuss to what extent generic requirements for the coordination of Master Planning are fulfilled by these approaches and the other mechanisms reviewed in this
chapter.
3.3.2.1 Outline of Dual Decomposition
Consider a generic centralized decision model.
(C) min ZC D
s.t.
X
X
ciT xi
i 2P
Ai xi b0
(3.8)
i 2P
Bi xi bi
8i 2 P
(3.9)
xi 0 8i 2 P :
(3.10)
Indices and Sets
i Decentralized parties, i 2 P D f1; : : : ; I g.
Data
Ai Use of the central resources by decisions xi , Ai 2 Qnoi
Bi Use of the decentralized resources by decisions xi , Bi 2 Qmi oi
b0 Total amount of the central resources, b0 2 Rn
bi Total amount of the decentralized resources, bi 2 Rmi
ci Costs associated with decisions xi , ci 2 Roi
Variables
xi Decision variables, xi 2 Roi
The objective function of C minimizes the costs resulting from the decisions xi .
The first constraint (3.8) links the interdependent decisions of the decentralized
parties i by establishing an upper bound on the use of the central resources.
Note here that the rationality of the constraint matrix Ai is necessary and sufficient for ensuring the consistency of (3.8).73 Constraints (3.9) and (3.10) capture the decentralized restrictions for variables xi . By skipping the interdependent constraint (3.8), C decomposes into the decentralized problems of parties i :
(DPi ) min ciT xi
s.t. Bi xi bi
xi 2 Xi :
73
See Meyer (1974, p. 223).
8
(3.11) <
8i 2 P
(3.12) :
3.3 Multilateral Information Asymmetry
53
Fig. 3.5 Dantzig–Wolfe decomposition
To illustrate the main ideas of dual decomposition, we will present Dantzig–
Wolfe decomposition as a basic procedure and outline subgradient optimization, a
variant which is frequently used here.
The idea of Dantzig–Wolfe decomposition is that the decentralized parties solve
DSi , i.e., DPi augmented by (dual) prices for central resource use and communicate
the costs (ciT xi ) and the central resource use (Ai xi ) of the resulting solution (xi )
to a central entity. Using a master model (MP), the central entity recombines the
proposals for central resource use obtained so far and derives new prices to be sent
to the decentralized parties. All parties carry out these steps iteratively until neither
DPi nor MP yield any new solutions and, hence, the optimal solution to C has been
identified (see also Fig. 3.5).74
In the following, we will provide a formal motivation for this procedure:75 Consider C with a Lagrangian relaxation of constraints (3.8).
.LRu / min
X
ciT xi C u
i 2P
X
!
Ai xi b0
i 2P
(3.9), (3.10):
Variables
u Prices for central resource use, u 2 Rn
For an arbitrary choice of u, the optimal solution to LRu is a lower bound on the
costs of C. For a given u, LRu can be decomposed into the subproblems solved by
the decentralized parties:76
(DSi ) min ZDSi D ciT xi C uAi xi
s.t. (3.11), (3.12):
74
8
<
:
8i 2 P
This procedure can be initialized, e.g., be solving DPi with prices of zero for central resource
use, see Ho and Loute (1981, p. 306).
75
With our exposition of this approach, we follow Holmberg (1995, p. 61) and Klose (2001, p. 74).
Alternative descriptions provide, e.g., Dantzig and Wolfe (1960, p. 101) and Wolsey (1998, p. 185).
76
Note that the constant ub0 is omitted in the objective functions of these problems for ease of
exposition.
54
3 Coordination Mechanisms for Supply Chain Planning
Next, consider the maximum of LRu for u 0:
(LD)
max min
u0
xi
X
ciT xi C u
i 2P
X
!
Ai xi b0
i 2P
s.t. (3.9), (3.10):
Due to Lagrangian duality,77 both the objective function values and the optimal
solutions to C and LD are equal. Obviously, for each party i , the optimal solution to LD will include one of the solutions to DSi that are optimal for a given u.
These solutions are vertices of the polyhedron defined by (3.11) and (3.12), and
are successively generated in the course of the decomposition scheme; hence, before the termination of the scheme, only a subset of the vertices has been identified.
Accounting for that, LD can be rewritten as
!
X
X
0
T v
v
(LD ) max vminvs
(3.13)
ci xi C u
Ai xi b0 :
u0 x 2X
i
i
i 2P
i 2P
Sets
Xivs Set of vertex solutions identified so far
Data
xiv Value taken by xi in the vertex v identified so far, xiv 2 Xivs
Taking the dual of LD0 , we obtain the master problem that is solved by the central
entity.
v
X X
min ZMP D
ciT
i v xiv
i 2P
(MP) s.t.
X
ATi
i 2P
v
X
v
X
vD1
i v xiv b0
(3.14)
vD1
i v xiv D 1
8i 2 P
(3.15)
vD1
i v 0 8i 2 P; v D 1; : : : ; v:
Index
v Vertex identified so far, v D 1; : : : ; v
Variables
i v Variables indicating the share of the vertex solution v of party i that is
included in the new allocation determined by MP
77
See, e.g., Minoux (1986, p. 212).
(3.16)
3.3 Multilateral Information Asymmetry
55
In MP, the central entity identifies a systemwide feasible convex combination
of the proposals previously submitted by the decentralized parties with minimum
systemwide costs.78 Constraint (3.14) ensures that the central resource use by this
convex combination does not exceed the totally available capacity. Constraints
(3.15) and (3.16) determine that a convex combination must be chosen.
During the course of the decomposition scheme, neither the optimal (dual) prices
u nor the set of all relevant vertex
Psolutions (and the proposals derived from them)
are known. The P
former implies i 2P ZDSi ZC , whereas the latter ZMP ZC . We obtain i 2P ZDSi D ZMP after the identification of all vertices that
are not dominated in terms of central resource use. This is also the reason for finite
convergence of Dantzig–Wolfe decomposition: Since the number of vertices is finite
and at least one new vertex solution is found in each iteration, a finite number of
iterations is sufficient for identifying the optimal solution to C.
Apart from solving MP, the prices for the use of the central resources can be
determined via subgradient optimization.79 There, these prices are updated by a step
into the direction of the subgradient of the optimal objective function value of LD
in each iteration. This procedure has asymptotic convergence towards the optimal
prices and the optimal objective function value of MP, but does not provide the
primal solution, which has to be determined by an additional heuristic then.80
Since both Dantzig–Wolfe decomposition and subgradient optimization are
based on the solution of LD0 , they are subsumed under the term dual decomposition.
Apart from dual decomposition, a couple of further mathematical decomposition
techniques have been developed.81 Among those is primal decomposition,82 where
parties follow a reversed methodology, i.e., the central entity determines the allocation of the central resources relying on dual prices supplied by the decentralized
parties. We will not deal in-depth with these approaches here since they – with
one exception83 – have not been used for the design of supply chain coordination
mechanisms yet and show drawbacks similar to Dantzig–Wolfe decomposition due
to the exchange of dual information.84
3.3.2.2 Literature
For classifying the literature of this subsection, we again distinguish between the
structures of the supply chains considered (“Arb.” stands for arbitrary) and the
Note that for this purpose, it is sufficient for the central entity to know proposals Ai xiv and the
cost changes ciT xiv , which are actually supplied by the decentralized parties (instead of xiv ).
79
See Held et al. (1974, p. 62).
80
See Holmberg (1995, p. 76).
81
For an overview of these techniques, see, e.g., Holmberg (1995, p. 61).
82
Within primal decomposition, Benders decomposition (see Benders 1962, p. 238) is the counterpart to Dantzig–Wolfe decomposition.
83
See the paper by Arikapuram and Veeramani (2004) discussed below.
84
See also our discussion of these drawbacks at the end of this section.
78
56
3 Coordination Mechanisms for Supply Chain Planning
Table 3.5 Nomenclature for the classification (self-interest)
Parameter Comment
SI Disc.
The applicability of the mechanism is discussed in light of self-interested behavior by
parties, e.g., by means of simulation studies
SI Proof
Analytically proven results for self-interest
Team
Assumption of team behavior
drivers for suboptimality. Thereby, we include “GLP” as a generic driver which
characterizes mechanisms that can be applied for general LP problems and comprises the drivers T and Q. Further note that we have included two approaches that
have been developed for scheduling (Sched.), but whose methodologies seem relevant for the coordination of Master Planning in principle.
In addition, we state (if applicable) the decomposition procedures the mechanisms are based on and distinguish between the underlying assumptions about
parties’ behavior. With the exception of Cui et al. (2007), all approaches analyzed in
the previous sections assume self-interest of parties. Here, in contrast, most authors
postulate team behavior, which allows significant simplifications for the design of
the mechanisms.85 In summary, three representations of self-interest can be distinguished (see Table 3.5).
The classification is provided in Table 3.6. Note that there are further surveys available in the literature like Stadtler (2009) for collaborative planning and
Heydenreich et al. (2007)86 for scheduling, which employ somewhat different criteria for their classifications and only cover a part of the literature listed here.
Two properties are shared by most papers: The assumption of team behavior and
the application of mathematical decomposition procedures based on the exchange
of dual information. We describe the approaches grouped according to the parties’
behavior, starting with team behavior.
The paper of Arikapuram and Veeramani (2004)87 proposes a coordination
scheme based on the L-shaped method. The L-shaped method is a decomposition procedure, which builds on ideas from primal decomposition and is usually
applied within stochastic programming.88 There, the central entity iteratively generates primal solutions, and the subproblems supply constraints, which are added to
the master problem. Arikapuram and Veeramani (2004) propose customized model
formulations for the application of the L-shaped method to supply chain planning
and present the results of corresponding computational tests.
Dudek and Stadtler (2005, 2007)89 present coordination schemes relying on an
iterative exchange of supply proposals and their associated cost effects between
85
Team behavior trivially satisfies the important requirement that parties should have incentives to
implement the actions specified by the mechanism.
86
See Heydenreich et al. (2007, p. 437).
87
See Arikapuram and Veeramani (2004, p. 111).
88
See, e.g., Birge and Louveaux (1997, p. 155).
89
See Dudek and Stadtler (2005, p. 668) and Dudek and Stadtler (2007, p. 465).
3.3 Multilateral Information Asymmetry
Table 3.6 Mechanisms focusing on proposal generation
Author(s)
Driver Behavior
Arikapuram and Veeramani (2004) GLP
Team
Dudek and Stadtler (2005)
T
Team
Dudek and Stadtler (2007)
T
Team
Ertogral and Wu (2000)
T
Team
Jeong and Leon (2002)
GLP
Team
Jung and Jeong (2005)
T
Team
Karabuk and Wu (2002)
T
Team
Schneeweiss and Zimmer (2004)
T
Team
Walther et al. (2008)
Q
Team
Chu and Leon (2008)
L
SI Disc.
Fink (2006)
T
SI Disc.
Kutanoglu and Wu (1999)
Sched. SI Disc.
Lee and Kumara (2007)
T
SI Disc.
Guo et al. (2007)
GLP
SI Disc.
Fan et al. (2003)
GLP
SI Proof
Nisan and Ronen (2001)
Sched. SI Proof
57
Structure
n-1(e)
1-1
1-n
1-1-1(e)
Arb.
1-1
1-1
1-1
n-1
1-n
2-1(e)
N.a.
Arb.
Arb.(e)
Arb.(e)
N.a.
Procedure
Primal decomposition
Other
Other
Dual decomposition
Dual decomposition
Other
Dual decomposition
Other
Dual decomposition
Other
Other
Dual decomposition
Other
Dual decomposition
Dual decomposition
Other
decentralized parties. The aim of this exchange is to improve upstream planning
in supply chains, where parties plan based on the MLCLSP. The proposals are
generated by optimization models that penalize deviations from the last proposal
communicated by the other party. Dudek and Stadtler (2005, 2007) evaluate the effectiveness of their schemes by computational tests. These papers differ from each
other by the assumptions on the supply chain structures and on the cost exchange
(bilaterally, unilaterally).
The papers of Ertogral and Wu (2000),90 Jeong and Leon (2002),91 and Walther
et al. (2008)92 are based on dual decomposition with subgradient optimization for
the solution of LD and mainly differ by their focus. Whereas Ertogral and Wu (2000)
address the coordination of decentralized MLCLSP with a fairness objective and an
artificially introduced central agent, Walther et al. (2008) coordinate a recycling
network with multiple bottlenecks, where the focal company can naturally serve
as the party updating the prices sent to the recycling companies. Jeong and Leon
(2002), in turn, address a general decentralized environment consisting of different
types of agents.
Jung and Jeong (2005) propose a simplistic scheme for the coordination of
plans of a manufacturer and a distributor.93 They address a supply chain where
orders determined by the distributor cause infeasibilities or high penalty costs at
90
See Ertogral and Wu (2000, p. 931).
See Jeong and Leon (2002, p. 789).
92
See Walther et al. (2008, p. 334).
93
See Jung and Jeong (2005, p. 167). Note that these authors have published further papers (e.g.,
Jung et al. 2008) relying on essentially the same setting and the same approach. Hence, we did not
include these papers separately in our classification.
91
58
3 Coordination Mechanisms for Supply Chain Planning
the manufacturer. The scheme consists of the communication of feasible production
plans by the manufacturer, which are incorporated in the form of additional
constraints into the planning of the distributor thereafter.
Karabuk and Wu (2002) present a further approach using dual decomposition.94
They address the capacity planning between marketing and manufacturing departments and model this problem with formulations similar to the Master Planning
models presented in Sect. 2.1.2. Their coordination scheme relies on the Augmented
Lagrangian procedure, a variant of dual decomposition, with a dual problem that differs from LD by an additional quadratic perturbation term in the objective function.
Karabuk and Wu (2002) propose to solve this dual problem by an updating procedure similar to subgradient optimization and show the effectiveness of their scheme
by a case study for a real-world semiconductor capacity allocation problem.
Finally, Schneeweiss and Zimmer (2004) present a coordination scheme based on
hierarchical anticipation.95 There, the buyer estimates the supplier’s model parameters in order to identify a supply plan with low systemwide costs. This paper relies
on an extreme requirement regarding team behavior: It not only assumes truthful
information exchange, but also the willingness of parties to accept solutions inferior
to the initial solution, provided that a systemwide improvement is obtained.
Chu and Leon (2008) address the coordination of uncapacitated, static single item
lot-sizing problems of one supplier and multiple buyers.96 They provide a heuristic
for the solution of the centralized problem and a coordination scheme that yields
the same solution, given truth-telling by parties. In addition, they show by means of
(deterministic) simulations that the outcome of their mechanism is not affected by
strategic (and potential untruthful) behavior of parties.
Fink (2006)97 proposes a scheme for the coordination of supply chains of two
suppliers and one buyer. This scheme relies on a mediator, that is informed about
the restrictions, but not about the objective functions of the decentralized problems.
This mediator generates systemwide feasible solutions, which are either accepted
or rejected by the decentralized parties. Fink (2006) shows in computational tests
that a good performance of this mechanism can be achieved if parties are willing
to accept a substantial share of the proposals generated. As a potential incentive
for this, he suggests minimum acceptance ratios, however without further exploring
their concrete effects.
Guo et al. (2007) propose a coordination scheme along the lines of Dantzig–
Wolfe decomposition.98 They assume a central auctioneer with initial inventories
of the goods traded. This auctioneer iteratively recombines proposals of the decentralized parties and determines market-clearing prices. Moreover, Guo et al. (2007)
investigate the effects of different bidding strategies where parties do not follow the
94
See Karabuk and Wu (2002, p. 743).
See Schneeweiss and Zimmer (2004, p. 687).
96
See Chu and Leon (2008, p. 484).
97
See Fink (2006, p. 351).
98
See Guo et al. (2007, p. 1345).
95
3.3 Multilateral Information Asymmetry
59
rules of the scheme, but try to usurp greater shares of the surplus by distorted cost
reporting. Since these strategies only showed a minor impact on the performance of
the scheme in their computational study, Guo et al. (2007) infer that their mechanism is robust regarding strategic behavior of parties.
Similarly, Kutanoglu and Wu (1999) also rely on dual decomposition, but propose subgradient optimization to determine the prices communicated to the decentralized parties.99 Apart from a general introduction into the concept of combining
auctions with dual decomposition, Kutanoglu and Wu (1999) provide customized
model formulations for scheduling. Finite convergence of their scheme, however,
cannot be guaranteed due to the presence of binary variables in the scheduling problems considered there.
Lee and Kumara (2007)100 propose a coordination mechanism based on a “double auction market.” They formulate a Master Planning model with inventory
decisions and call it a lot-sizing problem, which seems incorrect since this model
neither includes any lot-sizing constraints of the type of (2.14) nor any binary variables. Their scheme requires that, apart from orders, the suppliers communicate
their inventory cost functions and inventory capacities to the buyers. Lee and Kumara (2007) claim that parties adopt a truth-telling behavior, but they did not provide
any analytical proof for that.101
Finally, there are two papers that include analytical proofs for the incentive compatibility of the mechanisms presented there. Fan et al. (2003) propose a mechanism
based on ideas from dual decomposition.102 Similarly to Guo et al. (2007), decentralized parties iteratively generate and communicate proposals and their associated
costs to a central entity, which determines the market-clearing prices. Fan et al.
(2003) show the incentive compatibility of this mechanism, but do not provide
any (analytical or empirical) insights about the convergence behavior, which does
not allow further conclusions about the potential practical applicability of this
mechanism.103
The paper of Nisan and Ronen (2001) provides a theoretical framework and analytical results regarding the design of incentive-compatible mechanisms in the area
of task scheduling. Moreover, Nisan and Ronen (2001) propose a new mechanism
based on randomization and derive a lower bound for its efficiency.
99
See Kutanoglu and Wu (1999, p. 813).
See Lee and Kumara (2007, p. 4715).
101
Lee and Kumara (2007, p. 4724), argue that truth-telling is ensured since the final allocation is
determined by a Vickrey-type auction. However, amongst others, they did not address the suppliers’
incentives to reveal their inventory cost functions and capacities truthfully.
102
See Fan et al. (2003, p. 1).
103
The latter drawback has also been pointed out by Guo et al. (2007, p. 1346), who in addition
provide a simple numerical example, for which the scheme developed by Fan et al. (2003) does
not converge.
100
60
3 Coordination Mechanisms for Supply Chain Planning
3.3.2.3 Discussion and Conclusions
Summarizing the findings from our literature review, a series of generic requirements for the decentralized coordination of Master Planning can be recognized.
In the following, we mention four of them, which seem mandatory for coordination
in many practical settings and have been advocated in part by the literature, but have
never been simultaneously covered within a single approach:
Bilateral information asymmetry
Self-interested behavior by parties
No involvement of a third party
Applicability for complex LP or MIP
Bilateral information asymmetry, which is the most natural information status of
independent parties in supply chains,104 has only been assumed by a small share of
coordination mechanisms. Most of these mechanisms overcome the resulting difficulties by postulating team behavior of parties. Such behavior has the obvious
advantage in case of information asymmetry that reliable data can be exchanged
additionally, which helps the decentralized parties to identify coordinated solutions.
Unfortunately, team behavior does not seem justifiable if a supply chain consists
of independent, profit-maximizing (and potentially competing) parties. The party
adopting team behavior by, e.g., freely disclosing the cost changes of proposals,
runs the risk that the other party shows unfair behavior and usurps the lion’s share
of the surplus generated.
Moreover, many approaches rely on the involvement of a third party. Important
advantages of such a third party are that its existence allows a direct application of
Dantzig–Wolfe decomposition and that the third party can participate in bidding and
surplus sharing.105 However, a third party is a strong assumption and not naturally
given in supply chains. An artificial establishment of a (trusted) third party and its
acceptance by all participating parties seems very problematic.106
A final and trivial requirement for coordinating Master Planning is the applicability of the coordination mechanism to complex LP or MIP. Complex LP can be
tackled by mathematical decomposition procedures, whereas their transfer to MIP
raises additional problems. For the application of primal decomposition, the identification of dual functions would become necessary. Their identifiability for MIP
has be shown;107 however, both the associated computational effort for their generation108 and the amount of data involved and, hence, exchanged among parties may
cause difficulties, like the risk that parties’ private data can be reconstructed by other
104
See, e.g., Arikapuram and Veeramani (2004, p. 111).
See, e.g., Wu (2003, p. 67), for a generic description of the possibilities of third parties to mediate bargaining processes.
106
See, e.g., Chu and Leon (2008, p. 484), for a paper supporting this argument.
107
E.g., Wolsey (1981, p. 173).
108
E.g., Guzelsoy and Ralphs (2008, p. 118).
105
3.3 Multilateral Information Asymmetry
61
parties.109 In dual decomposition, in turn, dual prices are not an effective guidance
towards a systemwide improvement if duality gaps are large. A final argument
against the application of approaches based on classical decomposition schemes is
that not all managers have an understanding about the meaning of dual information
and, hence, may be reluctant towards a disclosure of information of that kind.
The lack of existing mechanisms that cover all of these requirements for the
coordination of Master Planning constitutes the starting point for this thesis: In
the following, we present novel mechanisms incorporating these requirements
and evaluate the properties of these mechanisms by analytical reasoning and
computational tests.
109
E.g., using inverse optimization, see Troutt et al. (2006, p. 422).
Chapter 4
New Coordination Schemes
This chapter and Chap. 5 contain the core of this thesis, new mechanisms for
collaborative supply chain planning. Recapitulating the definitions of Sect. 2.3.1, a
coordination mechanism is a contractual framework for coordinating the outcomes
of (self-interested) actions of the decentralized parties. Each of the mechanisms
presented here comprises a scheme specifying the proposal generation and the
information exchange.
For all schemes developed in this thesis, two variants will be provided that cover
the different requirements for organizing the information exchange raised by the
contractual frameworks presented in Chap. 5: An iterative, unilateral exchange of
cost changes and a one-shot exchange by both parties. This allows us to describe
and analyze the different schemes and frameworks separately.
In this chapter, different schemes are presented and customized for the Master
Planning models described in Sect. 2.2. We begin with schemes for the coordination
of general decentralized LP problems (Sect. 4.1) and of one buyer and several suppliers planning based on uncapacitated dynamic lot-sizing models (Sect. 4.2), and
derive analytical results about their convergence behavior.
For a practical application of these schemes, however, two major limitations have
to be considered: First, the scopes of the schemes are restricted to specific problem
classes, i.e., LP and the MLULSP, which do not include some Master Planning problems such as the MLCLSP. Second, the solution quality of the generic scheme for
LP decreases considerably with an increase of the underlying optimization models.1
Therefore, apart from adapting the schemes to a generic Master Planning setting
in a two-party supply chain (Sect. 4.3), we additionally propose (heuristic) modifications, which yield a favorable performance even for complex MIP. Further
extensions to capacitated lot-sizing, voluntary compliance by the supplier, lost sales,
and Master Planning with several suppliers are provided in Sect. 4.4.
Figure 4.1 illustrates the relationships between the different schemes proposed.
Note that the single extensions presented in Sect. 4.4 can again be regarded as building blocks, e.g., the scheme for the MLCLSP is not restricted to full compliance, but
can be applied in a voluntary compliance setting, too, as done in our computational
study of Sect. 6.4.
1
See also the results of our computational tests in Sect. 6.1.
M. Albrecht, Supply Chain Coordination Mechanisms: New Approaches
for Collaborative Planning, Lecture Notes in Economics and Mathematical Systems 628,
c Springer-Verlag Berlin Heidelberg 2010
DOI 10.1007/978-3-642-02833-5 4, 63
64
4 New Coordination Schemes
Fig. 4.1 Relationships between Master Planning models
4.1 Generic Scheme for Linear Programming
and Analytical Results
In this section, we outline two versions of the generic scheme for LP that differ
by their requirements on the information exchange. In the first, cost changes are
reported by all but one party in an iterative fashion, whereas the second relies on a
one-shot disclosure of cost changes by all parties.
4.1.1 Version with Iterative, Unilateral Exchange
of Cost Information
Consider a decentralized organization with the centralized and decentralized problems outlined in Sect. 3.3.2.1. In this version of the scheme, decentralized parties
iteratively exchange proposals for the use of the central resources by parties i D
1; : : : ; I 1 compared to an initial solution.2 In addition to these proposals, the associated cost changes are reported unilaterally, i.e., parties i D 1; : : : ; I 1 report
to party I both the cost changes due to their own proposals and the cost changes
resulting from a (potential) feasible implementation of party I ’s proposals.
Exemplarily for a three-party setting, the resulting information exchange is depicted in Fig. 4.2. Here, party I D #3 determines the first proposal (action 1), and
so on.3 One iteration of the scheme comprises the submission of one proposal by
each party and of the associated cost changes by the cost-reporting parties.
For ease of exposition, we have assumed in Fig. 4.2 that the information is
processed in a synchronous fashion. If proposals are exchanged asynchronously
(e.g., the proposal by #1 and the associated cost changes are known to #3 earlier
than the proposal by #2), the scheme can be applied without further changes.
2
For the scheme, the initial solution might be chosen arbitrarily; when applying the mechanisms
proposed in Chap. 5, the initial solution will correspond to the default solution, the solution established without coordination.
3
This sequence is chosen for ease of exposition; other variants are possible, too (see Example 4.2).
4.1 Generic Scheme for Linear Programming and Analytical Results
65
Fig. 4.2 Information exchange within the generic scheme with unilateral exchange of cost
information
Remarkably, the information exchange is restricted to proposals and their
associated cost changes compared to the initial solution. Thereby, sensitive information like capacities and absolute (unit) costs are kept private. Both supply
proposals and cost changes can be subsumed under primal information. The limitation of the exchange to primal information is one of the major advantages of
the scheme proposed here since difficulties arising from the generation and the
exchange of dual information can be avoided that way.4
The crucial feature that determines the performance of this scheme as well as
the performance of any coordination scheme in general is the proposal generation. An obvious requirement for new proposals is that they potentially result in
systemwide improvements. Moreover, the number of proposals generated in the
course of the scheme should be modest since each proposal generated and each cost
change disclosed involve transaction costs for the participating parties and some
rough knowledge about the others’ modeling data.
In all the schemes presented in this work, the proposals are generated by appropriately designed optimization models. In the scheme at hand, we use two different
models for that purpose. The first of them, CS1i , is run by parties i D 1; : : : ; I 1.
Its aim is twofold: Proposals are generated that show a potential for systemwide
savings and address regions of the solution space that have not been investigated in
previous iterations of the scheme.5
(4.1)
.CS1i / max ZCS1i D ciT xist xi C kiT Ai xist xi
s.t. ATi .xist xie /ki ciT .xist xie /
for e D 1; : : : ; e i
ki M
ki 0
(3.11); (3.12):
Sets
XiE Set of proposals found so far
4
5
See our corresponding discussion at the end of Sect. 3.3.
See Sect. 3.3.2.1 for basic models and input data used for the exposition of this scheme.
(4.2)
(4.3)
(4.4)
66
4 New Coordination Schemes
Data
Mi Vector consisting of big numbers that exceed marginal cost savings
resulting
in the central
from increases
resource use, i.e.,
MiT Ai xi1 xi2 > ciT xi1 xi2 with Mi 2 Rn
ˇ
ˇ
e i Number of different solutions found so far for party i (e i D ˇXiE ˇ)
xie Solution previously found, xie 2 XiE ; xist denotes the starting solution
Variables
ki Prices for changes in central resource use, ki 2 Rn
The objective function of CS1i maximizes the potential savings of a new solution
xi compared to a fixed starting solution xist .6 Potential starting solutions for party
i comprise the solutions obtained by CS1i in previous iterations of the scheme and
those resulting from an evaluation of proposals generated by the other parties.7 The
second term of the objective function comprises penalties or bonuses for changes in
the use of the central resources. Variables ki are prices for these changes (i.e., unit
penalty costs and bonuses, respectively). They are determined by constraints (4.2)
such that ZCS1i 0 with the repetition of a previous solution xie 2 XiE .8 In order
to avoid unboundedness, constraint (4.3) establishes an upper bound for variables
ki . Finally, (4.4) is a nonnegativity constraint for ki .
After
solving
CS1i , the new supply proposal i D Ai xi together with i D
ciT xi xii ni t , the associated cost change compared to xii ni t , are communicated
to party I . xii ni t is the initial solution and is determined based on the initial resource
allocation ii ni t . This data is retained by parties in form of the ordered sets ˘iE , the
proposals previously identified for party i , and iE , the associated cost changes.
Note that with fixed values for ki , CS1i corresponds to the subproblem of dual
decomposition DSi .9 The fact that ki is determined endogenously in CS1i and
not supplied by a master problem (or a central updating procedure like subgradient optimization) is the crucial point that allows us to avoid an exchange of dual
information.
Like in the subproblem of dual decomposition, the optimal solutions to CS1i are
elements of Xiv , the set of vertices of DPi . To prove this, we start with a preliminary
result characterizing the solutions to CS1i .
f
f
Lemma 4.1 Consider CS1i with xi fixed to an arbitrary value xi . For ki ,
T
the optimal solution to this problem, there are scalars i e 0 with kif
6
Note here the different meaning of “starting solution,” which is the solution that is used as the
starting point for proposal generation, compared to that of “initial solution,” which is the first
solution used in the scheme.
7
See also p. 71 for the statement of the model used for this evaluation.
8
Note that for ease of exposition, the redundant constraint with e D st has been included here.
9
See p. 53 for this model.
4.1 Generic Scheme for Linear Programming and Analytical Results
67
T
T
P ei
Pei st
xi xie
Ai xist xif D M T i 0 eD1
ci xist xie i e and Ti0 C eD1
T
ATi i e xist xif
ATi .
Proof. Write CS1fi for CS1i with xi D xif . CS1fi can be reduced to
n
max kiT Ai xist xif j kiT Ai xist xie ciT xist xie
ki
8e D 1; : : : ; ei ; ki Mi ; ki 0g :
CS1fi -D, the problem dual to CS1fi , is
8
ei
<
X
T
T
ci xist xte i e j
min Mi i 0 i :
eD1
X
ei
Ti0 C
e T
xist xt
T
i e ATi xist xif
ATi ; i e 0
8e D 0; : : : ; ei
eD1
9
=
;
:
According to duality theory, the objective function values of the optimal solutions
to CS1fi and CS1fi -D are equal, which proves this lemma.
In Lemma 4.2, we show that the outcome of CS1i is a feasible vertex solution,
provided that the starting solution is not dominated by a convex combination of
already existing solutions and that the objective function takes a positive value.
Lemma 4.2 Let DPi only comprise continuous variables. Then:
P ei
1. CS1i has a feasible solution if ciT xist ciT eD1
i e xie for all i e 0 with
P
P
ei
ei
Ai xist Ai eD1
i e xie and eD1
i e D 1.
2. For xi , the optimal solution to CS1i , xi 2 Xiv holds.
Proof. 1. Since C and, hence, DPi are feasible, it suffices to show that constraints
(4.2)–(4.4) do not render CS1i infeasible. Since these constraints do not depend
on xi , CS1i is feasible, provided the feasibility of CS1fi for at least one xif . To
demonstrate the latter, we prove the feasibility and the boundedness of the dual
problem CS1fi -D, which has been stated in the proof of Lemma 4.1. CS1fi -D has
a feasible solution since i 0 can take any value greater than or equal to zero. The
optimal solution to CS1fi -D is bounded, e.g., if
ei
X
ei
T
X
st
f
ciT xist xie ie 0 8i 0; Ti0 C
ATi :
xi xie ATi ; ie xist xi
eD1
eD1
(4.5)
Due to the assumptions (1) of this lemma,
ei
X
eD1
ciT
i
i
X
X
st
xi xie i e 0 8i 0; Ai
i e xist xie 0;
ie D 1: (4.6)
e
e
eD1
eD1
68
4 New Coordination Schemes
P ei
P ei
The condition eD1
i e D 1 can be omitted in (4.6) since (4.6) implies eD1
P
ei
ciT xist xie i e ˛ 0 and Ai eD1
i e xist xie ˛ 0 for any scalar ˛ 0
P ei
with eD1 ˛i e D 1. Hence, together with (4.5) we get that CS1fi -D is bounded if
T
ATi 8Ti0 0. This inequality is fulfilled for, e.g., xif D xist .
Ti0 xist xif
Hence, CS1fi -D has a feasible, bounded solution and, according to duality theory,
also CS1fi , and thus CS1i .
2. Since we only consider the properties of xi , we can omit (4.2)–(4.4). Then, CS1i
reduces to
max ciT xist xi C kiT Ai xist xi
s.t. (3.11); (3.12):
Each solution xi to this problem is a vertex of the polytope defined by inequalities
(3.11), (3.12), and hence, a vertex solution to DPi .10
We further illustrate the effect of CS1i by an example.
Example 4.1 Assume that there are two dimensions of central resource use, i.e., two
dimensions of vector Ai xi contain non-zero elements.11 Denote the value of these
dimensions by the scalars d1 and d2 . Moreover, set d1st D 0 and d2st D 0, i.e., the
starting proposal is situated at the origin of the
coordinate system.
corresponding
Further assume two proposals C D d C D d1C ; d2C and D D d D obtained in
previous iterations of the scheme (see the left of Fig. 4.3). Note that this example
comprises a special case, where the coordinates of new proposals always exceed
those of the starting proposal. Hence, only penalties ki and no bonuses incur here
(i.e., ki 0).
We consider two new proposals, differing by the question whether they can be
expressed by a scalar multiple of a convex combination of the proposals identified
so far.12 First, we consider a proposal
by such a
PNew1, whiche can be
Peexpressed
New1
scalar multiple ˛ > 0, i.e., d New1 D eeD1 New1
d
with
D ˛. The
e
eD1 e
penalty costs for New1 are determined endogenously in CS1i to ˛ times the savings
resulting from a convex combination of these proposals compared to the starting
proposal. The determination whether New1 has a potential for being found as a
new proposal (which is the case if the associated penalty costs are smaller than
the savings compared to the starting proposal) is illustrated in the right of Fig. 4.3.
Compare the costs of New1 with the costs of the starting solution and the costs of Lk,
10
Note that DPi and CS1i may have several optimal solutions including non-vertex solutions. Here
we assume that CS1i is solved using an algorithm that limits its search to vertex solutions (e.g., a
primal simplex).
11
For ease of exposition, we skip the index i within this example.
12
A proposal Ai xin can be expressed by this multiple if and only if Ai xin lies within the cone
spanned by the starting proposal and the proposals found so far (here: C, D), see the left of Fig. 4.3.
4.1 Generic Scheme for Linear Programming and Analytical Results
69
->
Fig. 4.3 Proposal generation by CS1i
P
Pe
e
Lk
a scalar multiple of New1, for which d Lk D eeD1 Lk
e d with
eD1 e D 1. If
the cost change at New1 relative to its distance to the starting proposal is smaller
than the relative cost change at Lk (like in the right of Fig. 4.3), then New1 has a
potential for being found as a new proposal.
Next, consider the proposal New2, for which no New2
exists such that d New2 D
e
Pe
New2 e
d . This is because in one dimension (d2 ), the coordinates of New2
eD1 e
are not sufficiently large to be covered by a linear combination of proposals already
found, unless incurring excess resource use in the other dimension (d1 ). When determining the penalty costs, CS1i will internally attribute all cost changes between
the starting proposal and the other proposals identified to the other dimension (d1 ).
For New2, we have an extreme case. Since the d2 -coordinate is zero, the penalty
costs attributed to New2 will be zero, too.
What is finally done in model CS1i , is an endogenous evaluation of the (real)
costs and the penalty costs of all points of the solution space (among those New1
and New2). The solution with the smallest sum of the real costs and penalty costs
will be chosen as the new proposal.
The second model for proposal generation, CS2I , is only run by party I . Its mathematical formulation is as follows:
min ZCS2I D cIT xI C
ei
XX
e
i i e
i 2P; eD1
i ¤I
.CS2I / s.t. AI xI C
ei
XX
ie i e b0
(4.7)
i e D 1 8i 2 P; i ¤ I
(4.8)
i 2P; eD1
i ¤I
ei
X
eD1
i e 0 8i 2 P; i ¤ I; e D 1; : : : ; e i
(3.11); (3.12):
(4.9)
70
4 New Coordination Schemes
Variables
i e Decision variables defining a convex combination of previous proposals
about the central resource use of party i
The objective function of CS2I minimizes the costs of party I that are due to
xI and to a convex combination (defined by variables i e ) of the cost changes of
other parties’ previous proposals. Constraint (4.7) takes care that the total capacity
of the central resource exceeds the systemwide resource use, i.e., the sum of the
new proposal of party I and the convex combination of the other parties’ proposals determined by i e . Constraints (4.8) ensure that i e actually define a convex
combination of previous proposals, and constraints (4.9) enforce the nonnegativity
of i e .
Pei
Note that in (3.11) and (3.12), i is set to I here. CS2I yields i D eD1
i e ie ,
the new proposals for the use of the central resources by parties i D 1; : : : ; I 1.
These proposals are communicated to parties i D 1; : : : ; I 1 in each iteration of
the scheme.
CS2I is closely related to the master problem of Dantzig–Wolfe decomposition.
The basic difference is that, instead of using two separate optimization models, the
proposal of party I and the allocation of the central resources are determined simultaneously in CS2I (see also Fig. 4.4). The importance of this technical trick is
stressed by its general applicability to other (classical) decomposition techniques. In
principle, common primal and dual decomposition procedures can be transformed
analogously and subsequently used for decentralized coordination without a central
entity.
Note, however, that not all disadvantages of classical decomposition techniques
for supply chain coordination can be removed that way. When coordinating MIP
models, the difficulties due to the exchange of dual information13 persist. Moreover,
the mechanisms presented in Chap. 5 cannot be applied in combination with dual
decomposition procedures since these procedures require bilateral or multilateral
exchange of cost information (either directly or in form of dual prices), which is not
feasible in these mechanisms.
Fig. 4.4 Comparison of classical decomposition techniques with the scheme with unilateral cost
exchange
13
See p. 60 for a discussion of these difficulties.
4.1 Generic Scheme for Linear Programming and Analytical Results
71
Apart from CS1i and CS2I , a model for the evaluation of the other parties’ proposals is run by parties i D 1; : : : ; I 1.
(CS-EVALi ) min ciT xi
s.t. Ai xi il
(4.10)
(3.11); (3.12):
Data
il Last proposal communicated by party I to i
This model extends DPi by the additional constraint (4.10), which ensures that
the resource use by party i does not exceed il . Relevant outcomes of CS-EVALi are
l
T
i ni t
, the cost changes of a potential implementation of il , and the
i D ci xi xi
solution xi , that is used by party i as the starting solution for the next application of
CS1i .
Based on these models, a coordination scheme can be devised which identifies
the systemwide optimum in a finite number of steps. We first present a basic version of this scheme in Algorithm 1, which requires all parties to run LP models.
After that, we show how this scheme can be easily adapted such that convergence is
achieved even if one party (party I) runs an MIP model.
The function solve( ), that is used Algorithm 1, runs the corresponding
models and extracts their outputs as specified above. Note that for reasons of computational effectiveness, CS1i should only be run if ZCS1i > 0 for party i in the last
application of CS1i ; we did not include this in Algorithm 1 for ease of exposure.
In Table 4.1, we summarize the input and output data for the models run within
the scheme. The superscript new denotes the new proposals generated by CS1i . The
solutions XiE remain private knowledge of parties i during the scheme. The set ˘iE
consisting of proposals inew generated by party i and proposals il generated by
party I , as well as the set of cost changes iE become known bilaterally among
parties i and I .
Algorithm 1: GenericSchemeLPUnilateralCostExchange
for i
1 to I 1 do
solve(CS-EVAL i ) with i D ii nit
/* initialization */
repeat
/* iteration */
for i
1 to I 1 do
solve(CS1i ) with xist as the last outcome of CS-EVALi and communicate i
and i to I
solve(CS2I ) and communicate i separately to i D 1; : : : ; I 1
for i
1 to I 1 do
solve(CS EVALi ) with i as the last proposal of I and communicate i to I
until ZCS1i 0 8i and ZCS2I has not been improved compared to the last run of CS2I .
72
4 New Coordination Schemes
Table 4.1 Input and output
for the models solved within
the generic scheme
CS1i
CS-EVALi
CS2I
Input
xil
XiE
il
˘iE
iE
Output
inew
l
i
xil
il
new
i
xinew
Finite convergence of the scheme is shown by Theorem 4.1.
Theorem 4.1 Following the scheme specified above, the optimal solution to C can
be identified within a finite number of iterations if all decentralized decision problems can be formulated as LP models.
Proof. The proof of this theorem is structured into five parts. In (a), we prove that
CS1i always yields a feasible solution. This allows us to derive a lower bound on
the costs of the solutions xi 2 Xi that have not been identified after the termination
of the scheme (b). Based on that, we can characterize a subset of the solutions
to C for
systemwide costs are greater than or equal to the costs of
which the
x l D x1l ; : : : ; xIl , the systemwide solution resulting from the last run of CS2I (c).
Since this subset comprises an r-neighborhood of x l , the optimality of x l follows
(d). Last (e), we show that the convergence is finite.
P ei
(a) By Lemma 4.2, CS1i yields a feasible solution if ciT xist ciT eD1
i e xie
Pei
P ei
st
e
for all i e 0 with Ai xi Ai eD1 i e xi and
eD1 i e D 1. As a contraPei
diction, assume the existence of scalars i e 0 with ciT xist > ciT eD1
i e xie ,
P
P
P
0
e
e
e
i
i
i
e
Ai xist Ai eD1
i e xie and
eD1 i e D 1. Set xi D
eD1 i e xi . Since starting solutions xist are the optimal solutions for the last run of CS-EVALi ,
0
ciT xist ciT xi holds for all feasible xi with Ai xist Ai xi . Hence, ciT xist ciT xi D
P ei
ciT eD1
i e xie and a contradiction results.
(b) Hence, with the termination of the scheme, ZCS1i 0 holds. Then we get
f
ciT xist xi C kiT Ai xist xi 0 for any (fixed) solution xi . Setting xi D xi
we get by Lemma 4.1 that there are i e 0 with
ciT xi ciT xist C i 0 MiT ci
ei
X
st
T
xi xie i e ;
eD1
Ti0 C
ei
X
st
T
T
xi xie ATi i e xist xi ATi :
(4.11)
eD1
Next, we show by contradiction that there are i e 0 .i D 1; : : : ; e i / with
ciT xi ciT xist ci
ei
ei
X
X
st
T
st
T
T
xi xie ie ;
xi xie ATi ie xist xi ATi : (4.12)
eD1
eD1
4.1 Generic Scheme for Linear Programming and Analytical Results
73
T 0
Pei st
0
xi xie i e 8i e 0
Assume the contrary, i.e., that ciT xi < ciT xist ci eD1
T
T
Pei st
0
xi xie ATi i e xist xi ATi . Then we get with (4.12) that
with eD1
there are i e 0 with
i 0 MiT < ci
ei
X
xist xie
T 0
i e i e
eD1
0
X
T
xist xie ATi i e i e :
ei
0
8i e 0; Ti0 (4.13)
eD1
Due to the definition of Mi , MiT Ai xi1 xi2 > ciT xi1 xi2 holds 8xi1 ; xi2 2
T 0
T
Pei st
Pei st
xi xie i e and xi2 D eD1
xi xie i e , we
Xi . Setting xi1 D eD1
T
Pei 1
P ei
xi xi2 . i 0 MiT MiT Ai .xi1 xi2 / > ciT eD1
obtain i 0 Ai eD1
st
0
xi xie i e i e follows, which contradicts (4.13).
(c) Consider Xis , the subset of feasible solutions to CS1i characterized by (4.12)
P ei
and eD1
i e 1. Obviously, 8xi 2 Xis , there are
i e 0 with ciT xi P ei
P
0
0
T
ei
xist xie i e C xist xist i e 8i e 0, eD1
i e 1,
ciT xist ci eD1
Pei st
0
T
T
T
and eD1 xi xie ATi i e C xist xist ATi i e xist xi ATi . Set
P ei
0
0
0
i e for g D st, g D 0 otherwise, and define i e D i e C i e
g D 1 eD1
P
T
ei
xist xie i e ,
8e D 1; : : : ; ei . Since xist 2 XiE , ciT xi ciT xist ci eD1
P
Pei st
T
ei
e T
ATi i e xist xi ATi , and eD1
i e D 1 8xi 2 Xis .
eD1 xi xi
s
Hence, 8xi 2 Xi , there are i e > 0 with
ciT xi ci
ei
ei
ei
X
X
e T T T T X
e T T
xi i e ; xi Ai xi Ai i e ;
i e D 1:
eD1
eD1
(4.14)
eD1
With the termination
of the scheme, party I is not able to identify an improvement
l
l
compared to xI ; 11 ; : : : ; li 1;ei , the last solution determined by CS2I . Then,
cIT xIl C
ei
XX
i 2P eD1
i ¤I
li e ie cIT xI C
ei
XX
i e ie ; 8i e 0; xI 2 XI ; i ¤ I;
i 2P eD1
i ¤I
b0 AI xI C
ei
XX
i 2P eD1
i ¤I
i e ie :
(4.15)
74
4 New Coordination Schemes
P ei
Let xil be the optimal solution to CS-EVALi subject to i D eD1
l A x e . Re
Pi e i Ti l
e
T
e
i ni t
e
e
and i D Ai xi in (4.15), we get that i 2P ci xi placing i D ci xi xi
P
P ei
T
T
cI xI C i 2P ci
eD1 i e xi 8i e 0 for i ¤ I and xI 2 XI with b0 P i ¤I Pei
P
P
AI xI C i 2P Ai eD1 i e xie . With (4.14), we get that i 2P ciT xil i 2P ciT xi
i ¤I
P
8xi 2 Xis with b0 i 2P Ai xi .
P ei
i e D > 1,
(d) For any xi … XIs characterized by (4.12) with xist D xil and eD1
k
s
k
l
l
there is a xi 2 Xi with xi D xi C xi xi k and k D 1=. Consider Br xist ,
an r-neighborhood of xil , with an Euclidean metric and r > 0 chosen such small
2
that r < xik xil 8xik . This neighborhood comprises solely xi 2 Xis . Then,
P
P
T l
T B
8xiB 2 Xi . Hence, on each edge outgoing
i 2P ci xi i 2P ci xi followsP
P
l
from xi , there is a solution xi with i 2P ciT xil i 2P ciT xi . Since no further
improvement is possible when moving from xist in the direction of any of its outgoing edges for any i 2 P , x l is the optimal solution to C.
(e) By Lemma 4.2, only vertices and, hence, a finite number of different solutions
are identified by CS1i . Since each application of CS1i yields a new solution, provided that ZCS1i > 0, CS1i and, hence, CS2I are run a finite number of times. Theorem 4.1 provides the mathematical foundation of the scheme: For a large
class of optimization problems, the systemwide optimal solution can be identified
within a finite number of steps. This indicates that the scheme proposed is built on an
analytically solid basis and suggests a wide applicability of the scheme. Of course,
the finiteness does not guarantee that the optimum is always found within a practicable number of iterations. To speed up the convergence rates for huge problems,
some (heuristic) modifications are provided in Sects. 4.3 and 4.4.
To further illustrate the single steps undertaken by the scheme, we provide a
numerical example.
Example 4.2 Consider the following centralized optimization problem with a
block-angular structure.
max 5x1 C 6x2 C 8x3 C 5x4 C 5x5 C 6x6
s.t. x1 C 4x2 C 2x3 C 4x4 C 6x5 C 4x6 24
2x1 C 3x2 C 6x3 C 4x4 C 3x5 C 4x6 25
x1 C x2
x1 C 2x2
4x3 C 3x4
(4.16)
5
8
(4.17)
(4.18)
12
x5 C 2x6 6
(4.19)
(4.20)
x1 ; x2 ; x3 ; x4 ; x5 ; x6 0:
4.1 Generic Scheme for Linear Programming and Analytical Results
75
Fig. 4.5 Graphical representations of DPi
Assume that there are three decentralized parties #1, #2, and #3 that respond for
decisions .x1 ; x2 /, .x3 ; x4 /, and .x5 ; x6 /, respectively. Then the decentralized problems DP#1 , DP#2 , and DP#3 can be stated as follows:
max 5x1 C 6x2
s.t. (4.17); (4.18)
x1 ; x2 0 (4.21)
max 8x3 C 5x4
s.t. (4.19)
x3 ; x4 0
max 5x5 C 6x6
s.t. (4.20)
x5 ; x6 0:
In Fig. 4.5, we display the solution spaces and objective functions of DPi
graphically.
Further assume that the initial solution is determined by upstream planning and
that parties #1 and #2 report their cost changes to party #3 during the scheme. Let
#3 be the leader determining his central resource use first and #2 and #1 successively determine their production plans using the remaining central resources. The
optimal solution to DP#3 is to produce x5 D 6 and x6 D 0. Although this proposal
violates (4.16) and, hence, is infeasible for the whole system, it can be used as the
initial solution for running the scheme. As the responses of #2 and #1 to this proposal, we assume x1 D x2 D x3 D x4 D 0 [other responses would lead to a
greater violation of (4.16)]. The resulting proposals 11 D .0; 0/ and 21 D .0; 0/
are communicated to #3.
Next, models CS1#1 and CS1#2 are run. Since x1 D x2 D x3 D x4 D 0,
these models correspond to DP#1 and DP#2 in this step. Their optimal solution is
to produce x1 D 2, x2 D 3, x3 D 3, and x4 D 0, leading to objective function
values of 28 and 24, respectively. Resulting are proposals about central resource use
12 D .2 C 12 D 14; 4 C 9 D 13/ and 22 D .6; 18/ and cost changes 12 D 28,
2
1
2 D 24 to the initial solution that are communicated to #3 by #1 and #2 ( 1 and
1
2 are 0).
76
4 New Coordination Schemes
Next, #3 solves CS2#3 . With c5 D 5 and c6 D 6, this model can be formulated
as follows:
max 011 C 021 C 2812 C 2422 C 5x5 C 6x6
s.t.
011 C 1412 C 021 C 622 C 6x5 C 4x6 24
011 C 1312 C 021 C 1822 C 3x5 C 4x6 25
11 C 12 D 1
21 C 22 D 1
11 ; 21 ; 12 ; 22 0:
The optimal solution to this model is x5 D 0, x6 D 2:25, 11 D 0, 12 D 1,
21 D 0:833, 22 D 0:167, leading to a new lower bound of 45.5 on the systemwide
optimum. #3 communicates the resulting proposals 13 D .14; 13/ and 23 D .1; 3/
to #1 and #2, respectively.
Then, the turn is again of parties #1 and #2. They evaluate these proposals by
models CS-EVALi . Exemplarily, we provide the formulation of CS-EVAL#1 .
max 5x1 C 6x2
s.t. x1 C 4x2 14
2x1 C 3x2 13
(4.17); (4.18); (4.21):
Models CS-EVALi yield x1 D 2, x2 D 3, x3 D 0:5, x4 D 0, with associated cost
changes of 13 D 28 and 23 D 4, which are communicated to #3 in turn.
The
next step of the scheme is that #1 and #2 run models CS1i again. Noting that
14
AD
, we can exemplarily state CS1#1 below.
23
max 4 .x1 2/ C 6 .x2 3/ C k1 ..2 x1 / C 4 .3 x2 // C
k2 .2 .2 x1 / C 3 .3 x2 //
s.t. ..2 0/ k1 C 4 .3 0// C .2 .2 0/ k2 C 3 .3 0// 28 0 (4.22)
..2 2/ k1 C 4 .3 3// C .2 .2 2/ C 3 .3 3// k2 28 0 (4.23)
(4.24)
k1 4:00001
k2 2:00001
(4.25)
k1 ; k2 0
(4.17), (4.18), (4.21):
Since the starting solution is equal to one of the solutions previously found, (4.23)
is redundant. The upper bounds for k1 and k2 in (4.24) and (4.25) have been set to
4.00001 and 2.00001, respectively. We exemplarily provide the reasoning for determining the upper bound for k1 : An increase in the use of the first central resource
4.1 Generic Scheme for Linear Programming and Analytical Results
77
by 1 can be caused by increases of 1x1 or 1=4x2 , which correspond to increases
in the objective function of DP#1 of 4 and 3=2, respectively. Hence, the maximum
marginal cost savings are 4. Noting that the upper bound for k1 has to exceed this
value, we have set this bound to 4:00001.14
The optimal solution to this model is the point of the solution space where #1
expects the maximum increase in the systemwide savings compared to the initial
solution. Since only one proposal different from the starting proposal has been found
so far,15 the complete cost difference between the proposals will be (endogenously)
attributed to one single resource in (4.22). That way, the penalties imputed in the
objective function can be minimized and the bonuses can be maximized. Here it
is optimal to attribute this cost difference to the first resource, i.e., to set k1 to its
maximum (4) and k2 to 0. The optimal solution is x1 D 5, x2 D 0 with an objective
function value of 61 consisting of gains of 25 from the decentralized decisions and
a bonus of 36. Analogously, the optimal solution to CS1#2 can be determined to
x3 D 0, x4 D 4. As before, the resulting proposals 14 D .5; 10/, 24 D .16; 16/
and the cost changes 14 D 25, 24 D 20 are communicated to #3.
The subsequent solution of CS2I by #3 yields the systemwide optimum: x1 D 5,
x2 D 0, x3 0:23, x4 D 0, x5 1:64, x6 2:18, with an objective function
value of 48:09. This solution comprises a vertex of #1, and solution vectors of the
interiors of the solution spaces of DP#2 and DP#3 .
However, the scheme is not finished here. In the last iteration, the vertex solution
x1 D 0, x2 D 4 is identified by CS1#1 . After that, the scheme terminates since neither CS2i nor CS1i yield further new solutions. Table 4.2 summarizes the proposals
generated together with the associated cost changes and solutions in each step of
the scheme.
Next, we show how to extend this scheme such that it also yields finite convergence if party I runs an MIP. The mere change is that in case CS1i yields
no new solution, CS1i is run based on other starting solutions among XiE . For
proving convergence, no specific rule for determining the starting solutions is
Table 4.2 Single steps in the numerical example
Iteration Models
x1 x2 x3
x4 x5
x6
0
2
2
0
3
3
0
3
0.5
0
0
0
6
–
0
0
–
2.25
2
.0; 0/
.6; 18/
.1; 3/
2
Initialization
CS1i
CS2I , CS-EVALi
1
.0; 0/
.14; 13/
.14; 13/
1
0
1
1
0
28
28
0
24
4
2
2
CS1i
CS2I , CS-EVALi
5
5
0
0
0
0.2
4
0
–
1.6
–
2.2
.5; 10/
.5; 10/
.16; 16/
.0:4; 1:2/
25
25
20
1.6
3
3
CS1i
CS2I , CS-EVALi
0
5
4
0
0.2
0.2
0
0
–
1.6
–
2.2
.16; 12/
.5; 10/
.0:4; 1:2/
.0:4; 1:2/
24
25
1.6
1.6
14
15
Other possible choices are, e.g., 4.00000001, 5, or even 1,000.
This is because the proposals determined by CS-EVAL#1 and CS1#1 have been equal here.
78
4 New Coordination Schemes
Algorithm 2: GenericSchemeUnilateralCostExchangeOneMIP
for i
1 to I 1 do
xii nit ˚solve(CS-EVAL i ) with i D ii nit
XiE
xii nit
/* initialization */
repeat
/* iteration */
for i
1 to I 1 do
XiST
XiE
repeat
xist
random(XiST )
XiST
XiST nxist
xinew
solve(CS1i ,xist )
until ZCS1i > 0 or XiST D fg
XiE
XiE [ xinew
communicate inew and inew to I
solve(CS2I ) and communicate i separately to i D 1; : : : ; I 1
for i
1 to I 1 do
solve(CS EVALi ) with i as the last proposal of I and communicate i to I
until ZCS1i 0 8i D 1; : : : ; I 1 and ZCS2I has not been improved compared to the last
run of CS2I .
necessary; they, e.g., can be chosen randomly as done in our computational study of
Sect. 6.1.16 The scheme terminates if no new solution can be found for any choice
of xist 2 XiE . Algorithm 2 describes the extended scheme formally. There, the
function random(XiST ) randomly chooses the starting solution among those previously found, except for such solutions which have already yielded ZCS1i 0 in
earlier steps.17
Theorem 4.2 shows that the extended scheme yields finite convergence even in
case of one decentralized MIP. Define Xivd as a set of vertex solutions to DPi that
weakly dominates Xiv , i.e., for all xi 2 Xiv , there are scalars i e with ciT xi P ei
Pei
Pei
vd
T e
e
e
eD1 i e ci xi ; Ai xi eD1 i e Ai xi , xi 2 Xi , and
eD1 i e D 1. (Note
that several dominating sets Xivd may exist for each Xiv ; the choice among them is
irrelevant for our subsequent demonstration.)
Theorem 4.2 Following the scheme specified above, the optimal solution to C can
be identified within a finite number of iterations if the decision problems of the
cost-reporting parties can be formulated as LP models. The decision problem of the
remaining party may be of an LP or MIP type.
16
A further potentially favorable rule is to choose the initial solution as the starting solution, and
only if no new proposal can be found, to randomly select another solution.
17
If ZCS1i 0 has held in earlier steps, this obviously extends to the further steps of the scheme,
where the only difference in CS1i is that XiE is augmented by additional solutions.
4.1 Generic Scheme for Linear Programming and Analytical Results
79
Proof. The proof is structured into three steps. We begin with demonstrating that
all elements xi of a set Xivd are identified (a). Based on that, we show that CS2i
identifies the optimal solution to C (b). Finiteness is proven in (c).
(a) We prove by contradiction that all xi 2 Xivd for all i D 1; : : : ; I 1 have been
identified if ZCS1i 0 8i D 1; : : : ; I 1. As the contradiction, assume that there
P ei
is a xif , for which no scalars i e 0 exist with ciT xif ciT eD1
i e xie ; Ai xif P
P ei
ei
Ai eD1 i e xi e , xie 2 Xivd , and eD1
i e D 1. Recall the following insight from
Theorem 4.1 [see (4.11)]: For each solution xi , and hence, for each xif , there are
scalars i e 0 with
ciT xif ciT xist ci
ei
ei
T
X
X
st
T
st
T
xi xie ie ;
xi xie ATi i e xist xif
ATi :
eD1
eD1
(4.26)
As the next step of the proof, we show that for each xif , there is at least one xist
P ei
with i e and eD1
i e 1 complying with (4.26). For that purpose, we consider a
set XiP XiE for whose elements xip scalars ip > 0 exist with
ciT
X
xip ip ciT
p
ei
X
xie 0i e ; Ai
eD1
p2Xi
80i e ; Ai
ei
X
X
xip ip Ai xif ;
p
ip D 1
p
p2Xi
p2Xi
0i e xie Ai xif ;
eD1
X
ei
X
0i e D 1:
(4.27)
eD1
Define scalars i e with i e D i e for e ¤ st and xie 2 XiP , i e D 0 otherwise.
Then (4.27) can be transferred to
ciT xist ci
ei
X
T
xist xie i e ciT xist ci
eD1
ei
X
T
xist xie 0i e ;
eD1
ei
X
i e < 1
eD1
ei
T
X
st
T
f
xi xie ATi i e xist xi
ATi ;
eD1
80i e ;
ei
ei
T
X
X
st
T
f
xi xie ATi 0i e xist xi
ATi ;
0i e D 1:
eD1
(4.28)
eD1
T
Pei st
xi xie i e in (4.28) corresponds to the obThe expression ciT xist ci eD1
jective function of CS1-fi 18 with i e D i e for e D 1; : : : ; ei and i 0 D 0. Due to
18
See Lemma 4.1 for this model.
80
4 New Coordination Schemes
Pei
the convexity of CS1-fi in i e , (4.28) is also valid for all 0i e with eD1
0i e > 1.
Pei
st
P
0
Hence, for any xi 2 Xi , if (4.26) holds for a scalar i e 0 with eD1 0i e > 1,
P ei
then there are scalars i e 0 with eD1
i e 1 for which (4.26) holds. As a conP ei
sequence, there is at least one xist 2 XiE , for which there are i e with eD1
i e 1
that comply with (4.26).
T
Pei
Pei st
Since ciT xist ci eD1
xi xie i e remains constant if
eD1 i e 1
and only i e with e D st is varied, there is at least one xist with i e 0 such
P ei
Pei
Pei
f
that eD1
i e D 1, ciT xi ciT xist ci eD1
.xist xie /T i e , and eD1
.xist P ei
f T T
e T T
st
e
E
T f
T
xi / Ai i e .xi xi / Ai . Hence, there are xi 2 Xi with ci xi ci
eD1
P ei
P ei
f
i e xie , Ai xi Ai eD1
i e xie , and eD1
i e D 1, which contradicts the definif
tion of xi .
(b) Next, we prove by contradiction that solving CS2i yields the optimal solution to
C, given that all elements of Xivd have been identified. As the contradiction, assume
that the decisions xi of (at least) one decentralized party i , though being optimal
in C, cannot be expressed by a linear combination of elements among Xivd . For
each xi … Xivd , there is a convex combination of xie 2 Xivd leading to the same or
P ei
P ei
lower costs as xi , i.e., ciT eD1
i e xie ciT xi with eD1
i e Ai xie Ai xi and
P ei
vd
eD1 i e D 1 hold due to the definition of Xi . Substituting xi in the optimal
Pei
solution to CS2i by eD1 i e xie yields an objective function value smaller than or
equal to that for xi and does not violate the constraints of CS2i , (4.7), (4.8), (4.9),
(3.11), and (3.12). Hence, a contradiction results.
ˇ vˇ
ˇ
(c) Since CS1i yields vertex solutions only, CS1i is run at most ˇX
ˇ i ˇtimes for each
proposal generated. Since the number of vertices and, hence, ˇXiv ˇ is finite, the
validness of this theorem follows directly.
Note that in contrast to the base version of the scheme, a considerably larger share
of the solution space has to be investigated in order to prove optimality. All nondominated vertices adjacent to all solutions previously found have to be generated
here, whereas the base version only requires the identification of the non-dominated
vertices within the neighborhood of the best solutions found so far.
4.1.2 Version with One-Shot Exchange of Cost Information
The scheme outlined above can only be applied in combination with coordination mechanisms that allow for an iterative disclosure of cost information. In the
following, we provide a modification that is applicable for a different form of information exchange, a one-shot disclosure of cost changes by all parties. This version
shows somewhat increased computational complexity, but finite convergence, provided that all decentralized problems can be modeled as LP problems.
4.1 Generic Scheme for Linear Programming and Analytical Results
81
Here we also assume an (iterative) exchange of supply proposals among parties,
but without the disclosure of any cost effects throughout the proposal generation.
Model CS1i is used for proposal generation, too. However, this model is applied by
all parties and the proposal generation terminates if models CS1i yield no further
improvement for any party i D 1; : : : ; I and any starting solution, which is chosen
among the solutions previously generated – analogously to the scheme for one decentralized MIP (see Sect. 4.1.1). After proposal generation, parties simultaneously
communicate the cost effects of their proposals to each other (see Fig. 4.6).
The proposal finally implemented can be determined by the master model MP.19
This model can be solved by any decentralized party since all parties hold the information needed. We summarize the single steps of this scheme in Algorithm 3.
Theorem 4.3 shows that for this version finite convergence holds, too, provided
that all decentralized problems can be modeled as LP.
Fig. 4.6 Generic scheme for LP with one-shot exchange of cost information
Algorithm 3: GenericSchemeLPWithOneShotExchangeOfCostChanges
for i
1 to I do
xii nit ˚solve(CS EVALi ) with i D ii nit
XiE
xii nit
repeat
for i
1 to I do
XiST
XiE
repeat
xist
random(XiST )
XiST
XiST nxist
xinew
solve(CS1i ,xist )
until ZCS1i > 0 or XiST D fg
communicate inew to the other parties i
XjE
XiE C xinew
until XiST D fg 8i
communicate all cost changes among parties
solve(MP)
19
See p. 54.
/* initialization */
/* iteration */
82
4 New Coordination Schemes
Theorem 4.3 Following the scheme outlined above, the optimal solution to C can
be identified within a finite number of iterations if the decision problems of all parties can be formulated as LP models.
Proof. This proof is closely related to that of Theorem 4.2. From the proof of
Theorem 4.2 we know that all xi 2 Xivd are identified by iteratively running CS1i
and that this only requires a finite number of steps. What remains to show here is
that MP identifies the optimal solution to C after the generation of all xi 2 Xivd
for all parties i . We proof this by contradiction. As the contradiction, assume the
existence of a xi … Xivd for which MP yields a lower objective function value.
For each xi … Xivd , there is a convex combination of xie 2 Xivd leading to the
P ei
P ei
same or lower costs, i.e., ciT eD1
i e xie ciT xi with eD1
i e Ai xie Ai xi and
P ei
vd
eD1 i e D 1 hold due to the definition of Xi . Substituting xi in the optimal
P ei
solution to MP by eD1
i e xie yields an objective function value smaller than
or equal to that for xi and does not violate the central resource constraints (4.7).
Hence, a contradiction results.
Compared to the scheme for LP with an iterative exchange of cost changes,
the different form of information exchange has been gained at the expense of the
same disadvantage valid for the extended scheme for one decentralized MIP: All
non-dominated vertices have to be generated. Anyway, like in Dantzig–Wolfe decomposition, the identification of all (relevant) vertices is too demanding in general
since their number increases exponentially with the problem size; computational results for interrupting the schemes after a given number of iterations are provided
in Sect. 6.1.2. Moreover, for tackling larger problem instances and multiple MIP
models, heuristic versions are proposed in Sects. 4.3 and 4.4.
4.2 Scheme for Uncapacitated Dynamic Lot-Sizing
and Analytical Results
In contrast to the generic scheme for LP, the scheme for uncapacitated lot-sizing
only requires an exchange of proposals, which are directly evaluated in terms of a
systemwide improvement. Moreover, the proposals are only generated by the buyer.
Thereby, no knowledge about the others’ costs changes is needed, which makes this
scheme directly applicable for all forms of information exchange specified by the
mechanisms presented in Chap. 5. The scheme can be applied by one buyer performing single sourcing with one or multiple suppliers. We depict the information
flow for a setting of one buyer and two suppliers in Fig. 4.7. The optimization model
for the buyer’s proposal generation is:
min .2.120 /
(MLULSP-CS) s.t. .2.20 /; .2.60 /; .2.80 /; .2.110 /; .2.130 / .2.150 /
Y Sjt Ykt
8j 2 J D ; k 2 Sj \ J B ; t 2 T (4.29)
4.2 Scheme for Uncapacitated Dynamic Lot-Sizing and Analytical Results
83
Fig. 4.7 Scheme for uncapacitated lot-sizing
X X
j 2J D t 2T
YSjt 0
Y Sjt X
ysjup w
(4.30)
j 2J D
8j 2 J D ; t 2 T:
(4.31)
Data
w
Target for the reduction of the number of orders for the items supplied
ysjup Number of orders for item j in the proposal from upstream planning
Variables
Y Sjt Indicator variable (=1 if item j is ordered in period t, =0 otherwise)
The objective function of MLULSP-CS and constraints (2.20 ), (2.60), (2.80),
(2.110), (2.130)–(2.150) have been taken from the MLULSP20 and modified such that
they only comprise items of the buyer’s domain, i.e., j 2 J B . Constraints (4.29) imply that variable YSjt takes a value of 1 if one of the successor items of j has been
set up in period t.21 Constraint (4.30) establishes an upper bound for the number of
orders for the supplied items. (4.31) are nonnegativity constraints.
The proposals generated by MLULSP-CS show two characteristics favorable for
the coordination of lot-sizing problems: First, they comprise a reduction of the number of orders YSjt and, hence, of the number of the setups necessary for the supplier
to produce the items supplied without any inventory holding. Second, this reduction
affects the buyer’s costs as little as possible since the buyer can choose freely the
items for which she performs the required reduction of her order frequency. The
rationale behind the reduction of Y Sjt is that the supplier’s setup and holding costs
may be reduced that way considerably (see also Example 4.3).
In each iteration of the scheme, MLULSP-CS is successively run for a given target w for the reduction of the number of the buyer’s orders until a new proposal is
found.22 Thereby, we initialize w by 1 (or by 0 if the default solution is different
20
See Sect. 2.2.2.1.
Note that values greater than 1 are precluded by the internal logic of the optimization model.
22
Note here that MLULSP-CS does not always yield a proposal different to all previously found.
If there is more than one predecessor item, a reduction of the number of setups for a buyer’s item
might go along with a reduction of the number of setups for several items of the supplier and,
hence, cause an overfulfillment of the setup target [i.e., some slack in (4.30)]. A proposal that has
been generated under such an overfulfillment will be repeated at least once since further increases
of the setup target by 1 will not enforce an additional reduction of the number of orders.
21
84
4 New Coordination Schemes
Algorithm 4: ProposalGenerationUncapacitatedLotsizing
ys up
solve(MLULSP-B)
w
1
˘E
fg
repeat
new solve(MLULSP-CS, x st ,w)
w
wC1
˘E
˘ E [ new
ˇ ˇ
P
up
ˇ Dˇ
until new … ˘ E or
j 2J D ysj w < J
from upstream planning) and increase this parameter by 1 with each application of
MLULSP-CS. A description of the single steps for proposal generation is provided
by Algorithm 4. Analogously to GM-B, MLULSP-B denotes the decentralized
model of the buyer.
After the termination of the proposal generation, the proposals are split according to the origin of the items supplied (i.e., which supplier delivers the items) and
separately communicated to the suppliers, who evaluate the costs of these proposals thereafter. According to the requirements of the coordination mechanisms the
scheme is embedded in, the cost effects can either be exchanged unilaterally by the
suppliers or in form of a one-shot disclosure by all parties.
Below, we illustrate the scheme by a numerical example for a buyer–supplier
setting.
Example 4.3 Consider a serial BOM structure with one item supplied. Assume
a planning horizon of jT j=6[UT], setup costs of 8[MU], and holding costs of 1
[U/ (UT MU)] for both parties. Table 4.3 provides the demand data, the solution
resulting from upstream planning, and the proposals generated by the scheme.
In upstream planning, it is optimal for the buyer to place an order in each period.
The same holds for the supplier when answering the buyer’s proposal, which in sum
results in systemwide costs of 96[MU]. In her first proposal, the buyer is enforced
to reduce her order frequency, and she aggregates the orders of the periods with
the lowest demand, where the additional costs for the aggregation are lowest. This
decreases the systemwide costs to 89[MU]. In the second proposal, the buyer is
indifferent whether to aggregate the orders of periods 1 and 2 or 3 and 4; here she
chooses periods 1 and 2. The third proposal constitutes the systemwide optimum.
There, the buyer performs a further aggregation resulting in systemwide costs of
77[MU]. Interestingly, the decrease in the systemwide costs of 19[MU] compared
to upstream planning went along with a modest cost increase of only 5[MU] for the
buyer. The fourth and fifth proposals, in turn, show again greater systemwide costs
due to the large increases in the holding costs incurred by the buyer. The scheme
terminates after the generation of the fifth proposal.
4.2 Scheme for Uncapacitated Dynamic Lot-Sizing and Analytical Results
85
Table 4.3 Example for the scheme for the coordination of dynamic
lot-sizing
t D1
2
3
4
5
6
Proposal
Party d D 10 10 10 10 9
9 costs[MU]
0 (default) B
10
10 10 10 9
9
48
0
S
10
10 10 10 9
9
48
1
1
B
S
10
10
10
10
10
10
10
10
18
18
0
0
49
40
2
2
B
S
20
20
0
0
10
10
10
10
18
18
0
0
51
32
3
3
B
S
20
20
0
0
20
20
0
0
18
18
0
0
53
24
4
4
B
S
30
30
0
0
0
0
28
28
0
0
0
0
73
16
5
5
B
S
58
58
0
0
0
0
0
0
0
0
0
0
149
8
For the special case of a two-stage serial or assembly supply chain with level end
item demand,23 we are able to derive analytical results regarding the convergence
behavior of the scheme. Note in this context that the condition of level end item
demand is of special relevance for coordination. In simulation experiments for uncapacitated lot-sizing, the greatest peaks of the suboptimality with upstream planning
occurred for test instances with level demand.24 First of all, we introduce a common
assumption:
Assumption 4.1 Marginal holding costs are greater than or equal to zero.
Marginal holding costs are defined as the differences between the unit holding costs
of items and the sum of the unit holding costs of their predecessor items. If holding
costs are only made up by the costs for the capital bound,25 then the statement of
Assumption 4.1 holds directly.
Further necessary is the introduction of some basic notation
˚ (Fig. 4.8). We denote
a single supply proposal for an item j 2 J D by j D xtj1 ; xtj 2 ; : : : ; xtj jT j .
Furthermore, let be the proposal (matrix) comprising the single proposals j for
all j 2 J D . Hence, an (optimal) implementation of a proposal is equivalent to the
implementation of a given supply target.26 Note that an optimal implementation of a
23
Note that we consider end item demand only, i.e., we assume zero demand for intermediate
items. A further (implicit) assumption is the time independence of holding costs hj .
24
See, e.g., Simpson (2007, p. 136) and our computational results for the MLULSP (p. 172).
25
In fact, other costs like those for shelf space use are often minor compared to the costs of capital
bound in inventory, e.g., Schneeweiß (1981, p. 69).
26
See p. 26 for the modeling of the supplier’s implementation of a supply target.
86
4 New Coordination Schemes
SJD ∩ J B
NB
NS
JD
items
set of items
J
B
buyer
J S supplier
„refers to this set“
Fig. 4.8 Illustration of some basic notation
proposal always leads to a nested solution here, where the supplier performs setups
in the order periods of the buyer only. This characteristic is shared by all optimal
solutions to the MLULSP, provided a serial or an assembly BOM.27
Denote the systemwide costs resulting from an implementation of by csys ./.
Obviously, there is at least one proposal for which csys . / D csys
holds, with
csys as the costs of the optimal solution to the MLULSP. Moreover, define NjS as
the number of the supplier’s setups in the planning interval for items j 2 J D and
NkB as the number of setups for their successor items k 2 SJ D \ J B .
Moreover, define PB as the set of supply proposals optimal for the buyer subject to any N B . I.e., PB is composed by proposals that are obtained by solving
MLULSP-B augmented by
X
Ykt NkB
8k 2 SJ D \ J B :
(4.32)
t 2T
PB contains all proposals resulting from any combination
of NkB for all k 2
ˇ
ˇ
Sj \ J B . Therefore, jPBj increases exponentially with ˇJ D ˇ, which makes a direct
enumeration of all elements of PB computationally expensive. Moreover, define
PB
csys
D min2PB csys ./ as the systemwide costs for the implementation of the
best proposal out of PB, CS as the set of solutions identified by the scheme, and
CS
csys
as the costs for the best solution out of CS .
In the following, we derive our analytical results about the convergence behaviour of this scheme. We begin with considering the quality of solutions resulting
from the implementation of the best proposal among PB. In Lemma 4.3, we focus
on the special case that NjB =NjS 2 N and jT j =NjS 2 N for each j 2 J D in the
systemwide optimal solution.
Lemma 4.3 For a two-stage serial or assembly supply chain with zero initial inventories and level end item demand, the following implication holds: If NkB =NjS 2 N
27
See Love (1972, p. 329) for the proof of this property for a serial BOM and Crowston and Wagner
(1973, p. 16) for an assembly BOM.
4.2 Scheme for Uncapacitated Dynamic Lot-Sizing and Analytical Results
87
and jT j =NjS 2 N for each j 2 J D and each k 2 Sj \ J B in the systemwide
optimal solution, 2 PB holds.
Proof. We begin with considering a serial supply chain with one item supplied. Write
N S and N B for the number of the supplier’s and buyer’s setups in the systemwide
optimal solution. Denote the starts of the periods in which setups of the supplier ocS
cur by tiS , with i D 1; : : : ; N S . Set tN
S C1 D jT j and define the number of periods
between subsequent setups of the supplier by tiS D tiSC1 tiS . Further define oi as
the number of the buyer’s orders within tiS ; tiSC1 . Analogously, denote the buyer’s
order periods of the systemwide optimal solution by tijB with i D 1; : : : ; N S and
B
B
j D 1; : : : ; oi . Define tijB D tijBC1 tijB and set ti;e
i C1 D ti C1;1 (see also Fig. 4.9).
Then the buyer’s holding costs can be written as:
t B 1
S
dhB
oi
N X
ij
X
X
t;
t D1
i D1 j D1
with hB as the unit holding cost of the item produced by the buyer. Obviously, the
buyer prefers equal tijB for all i D 1; : : : ; N S and j D 1; : : : ; oi , which also holds
for each 2 PB then. For the supplier, this is not necessarily true. In the following,
we will show that the potential influence of the supplier’s costs is not sufficient to
affect the systemwide optimality of equal tijB .
Subsequently, we show that equal lengths for tijB ; : : : ; tiBoi are systemwide
optimal for all i D 1; : : : ; N S and j D 1; : : : ; oi . The systemwide costs for an
interval tiS ; tiSC1 are
i
csys
B
tijB ; : : : ; tio
i
t B 1
D oi scB C dhB
ij
oi
X
X
t C scS C dhS
j D1 tD1
oX
i 1
tijB
tiS j D1
j
X
!
tikB
:
kD1
(4.33)
hS denotes the holding cost of the item produced by the supplier and scS , scB
the corresponding unit setup costs. The last term of (4.33) comprises the supplier’s
holding costs. It is based on the rationale that the supplier’s inventory in periods
Δt1S
N S=2
Δt2S
x
N B=4
e.g., e1=2
x: setups supplier, e.g., t2S = 6
x
Δt11B
Δt12B
B
Δt21
...
x
x
x
x
B
t (T=8)
1
Fig. 4.9 Illustration of some notation
5
B
x: setups buyer, e.g., t12 = 3, t21 =6
88
4 New Coordination Schemes
between two adjacent orders of the buyer depends both on the duration of this order
interval and on the inventory required for the fulfillment of subsequent orders within
S S PtijB 1
ti ; ti C1 . Dissolving some of the sums, e.g., by t D1
t D tijB tijB 1 =2,
Poi 1 B
and setting tiBoi D tiS kD1
ti k , (4.33) can be written as
i
B
csys
tijB ; : : : ; tio
i 1
tiS oP
i 1
kD1
B
tik
1
tijB tijB 1
@
AC
D oi scB C scS C dhB
2
j D1
oX
i 1
tiS oP
i 1
kD1
B
tik
1
C dhS
2
0
oX
i 1
tijB tiS j D1
(4.34)
!
j
X
B
tik
:
kD1
Calculate the partial derivative of (4.34) with respect to a tijB with j ¤ oi :
i
tijB ; : : : ; tiBoi 1
@csys
@tijB
dhB
D
2
!
oi 1
2tijB 1 2tiS C 2
X
kD1
oi 1
CdhS
tiS tijB X
tiBk C 1
!
tiBk
:
kD1
We obtain:
i
@csys
tijB ; : : : ; tiBoi 1
@tijB
!
oi 1
D d .hB hS /
tiS C tijB C
X
tiBk
:
kD1
(4.35)
The associated Hessian matrix contains constants of 2d .hB hS / at its diagonal
and of d .hB hS / at all other entries. Using Gaussian elimination, a matrix with
entries of 0 except at its diagonal can be obtained. The entries at the diagonal are
constant and take the sign of hB hS . Since hB hS due to Assumption 4.1, the
Hessian matrix is positive definite and a minimum is obtained by equating (4.35) to
zero. It follows tijB D tiBoi for all j ¤ oi . Since tiS and tijB are integer due
to the assumptions of this lemma, equal values for all tijB are feasible and, hence,
systemwide optimal.
Next, we show that also the setups of the supplier are equidistant in the systemwide optimal solution. We assume an optimal choice for tijB , i.e., tijB D tiBk
for all i D 1; : : : ; N S and j; k D 1; : : : ; oi , and consider the systemwide costs for
the whole planning horizon:
2
tijB 1
tiS
B 1
tij
3
7
S
X6
X
7
6 ti hB X
csys tiS D N S scS C N B scB C d
t C hS
tijB tiS jtijB 7 :
6
B
5
4 tij
iD1
tD1
j D1
NS
4.2 Scheme for Uncapacitated Dynamic Lot-Sizing and Analytical Results
89
At first, we limit to the extreme case hB D hS . There, we obtain after some transformations (amongst others, dissolving some of the sums as above):
NS
dhS X S 2
S
S
S
B
csys t1 ; : : : ; tN S D N scS C N scB C
ti
tiS :
2
i D1
(4.36)
PN S 1 S
S
ti , we obtain for the partial derivatives:
With tN
S D jT j i D1
S
@csys t1S ; : : : ; tN
S
@tiS
S
D dhS tiS tN
S
(4.37)
for all i ¤ N S . Setting (4.37) to zero yields the minimum of (4.36) since the associS
ated Hessian matrix is positive definite. We obtain tiS D tN
S as the condition for
S
B
this minimum. Since ti and tij are integer due to the assumptions of this theorem, the optimality of equal order intervals, the buyer’s preferred solution, follows.
The same holds for the case hB > hS a fortiori. Then the buyer’s preferences have
an even greater impact such that the optimality of the buyer’s preferred solution is
not affected.
This result directly extends to general two-level BOM structures. In structures
with more than one predecessor item (i.e., assembly), the sum of unit holding costs
for the supplier’s predecessor items is equal to or smaller than the unit holding
cost for the successor item due to Assumption 4.1. Since the supplier’s costs are
not sufficient for any N B and N S to cause a deviation from the buyer’s preferred
solution, this extends also for the above case, where the supplier’s holding costs are
partitioned among different items. With more than one successor item, this result
obviously extends due to the even greater impact of the buyer’s preferences.
PB
Based on this result, we derive a general upper bound for csys
in Lemma 4.4.
Lemma 4.4 In serial and assembly supply chains where buyer and supplier comprise one production stage each,
inventories are zero, and
demand is time initial
p p p PB
invariant, csys =csys 18=17 11 4 6 25 6= 24 C 11 6 1:53.
Proof. We have structured the proof of this lemma into five parts. First of all, we
describe the basic methodology and provide an lower bound for csys
(a). In (b)–(d),
we derive an upper bound for a serial structure, thereby distinguishing between
three subcases depending on the relationships between N B , N S , and jT j. In (e),
finally, we show that this bound extends to assembly structures.
PB
PB
(a) Obviously, cB
cB
with cB
as the buyer’s costs for PB , the proposal
B
S
out of PB with the same N and N as in the systemwide optimal solution, and
cB
as the buyer’s costs of the systemwide optimal solution. cSPB , the costs for an
implementation of PB by the supplier, however, may exceed cS , the supplier’s
90
4 New Coordination Schemes
costs in the systemwide optimum. Hence, the extreme case for the cost excess of a
proposal out of PB is reached for hB D hS , a setting, which we assume for the rest
of this proof.
In the following, we investigate an upper bound for the excess over the systemwide optimum, i.e., we determine
ub PB
csys
csys
:
(4.38)
A lower bound for csys
can be determined via (4.36), replacing tiS by jT j =N S
B
B
and tij by jT j =N there:
scB N B C scS N S C
csys
d
jT j2 hB
2
1
1
:
S
N
jT j
(4.39)
Next, we prove the result of this lemma for a serial supply chain with one item
PB
supplied. To determine an upper bound for csys
, a variety of cases has to be distin B
˘
˘
guished. Introduce the parameters r D N =N S and z D jT j =N B . In the rest
of the proof for the serial structure, we distinguish the following cases and subcases:
(b) r D 1 and z D 1, (c) r > 1 or z > 1 and T =N B 2 N, and (d) r > 1 or z > 1
and T =N B … N.
PB
Our basic methodology for determining the upper bound for csys
is to consider
PB
the worst-case order pattern that may result from running MLULSP-CS for a
given N B .28 For the supplier, a possible, not necessarily implementation of PB is
considered.
(b) Assume r D 1 and z D 1. Since the buyer prefers setup intervals with lengths
differing at most by 1, each PB only comprises order intervals of the lengths 1
and 2. W.l.o.g., assume that in his implementation of PB , the supplier will only
choose setup intervals that include 1 or 2 orders of the buyer. If N B =N S is integer
(i.e., 1), the supplier will place setups in all order periods. The resulting costs directly correspond to those of the systemwide optimum then.
Hence, we focus on the case that N B =N S is not integer. Consider the implementation of PB , where setup intervals contain either one or two orders of the buyer.
For the intervals that comprise only one order, the supplier’s holding costs are zero.
Hence, cSPB can only deviate from cS for setup intervals comprising two orders.
cSPB exceeds cS if larger order intervals are subsumed into these setup intervals in
the implementation of PB . This is due to the disproportional increase of the supplier’s costs with increased length of the setup intervals, an effect, which can be
recognized by considering cSPB :
28
Especially with level demand, there may be several equally optimal outcomes of MLULSP-CS;
which of them are determined by the solver running this model, depends on external factors like
the model structure and the solver characteristics and cannot be determined a priori.
4.2 Scheme for Uncapacitated Dynamic Lot-Sizing and Analytical Results
S
cSPB N S scS C hS d
NB
NB
1
S
N NX
X
91
tijB
i D1 j D1
NS
X
tiBk :
(4.40)
kDj C1
The value of the RHS of (4.40) depends on the temporal distribution of larger order
intervals B
ij within the supplier’s setup pattern (note that some notation introduced
in the proof of Lemma 4.3 is used here). Consider
NB
NB
S 1
NX
NS
X
S
S
B
ch ti D
tij
tiBk ;
j D1
kDj C1
the supplier’s holding costs between two subsequent setups dependent on tiS , the
length of the setup interval. These costs increase disproportionally
with an increase
S S of tiS , which implies an increase of any B
ij within ti ; ti C1 , i.e.,
chS tiS 1 C 1=tiS chS tiS 1 C 1=tiS :
(4.41)
As a consequence, the choice of setup intervals with equal lengths minimizes the
supplier’s costs. However, there are two potential obstacles for such a choice by the
supplier.
First, order intervals of the buyer with same lengths may be adjacent, while a
short order follows a large one and vice versa in the systemwide optimum. With
equal-sized adjacent orders, the supplier may be enforced to replicate such a sequence in his order fulfillment and, hence, incur larger costs due to (4.41). The
extreme case for that is reached for jT j ! 1 since this diminishes the effects of
the fact that at least one setup interval contains only one order in both the implementation of PB and the systemwide optimum. The latter adds constant factors to
nominator and denominator of (4.38) and, hence, increases this ratio.
ub for jT j ! 1 is upper bounded by ub for the constellation r D 2 and z D 1
with integer N B =N S , with proposals comprising two adjacent orders of the length
1 and two further adjacent order intervals of the length 2 (see the left of Fig. 4.10)
(Here each setup interval of the supplier contains an equal number of orders; the
absolute value of jT j does not matter since this equally affects nominator and denominator of the RHS of (4.38)). In the systemwide optimum, instead, the buyer’s
and supplier’s setups of period 3 would be postponed to period 4 there. Hence, the
supplier’s holding costs are augmented by 5hB d 4hB d D hB d . Here as well as
in further parts of this proof, the buyer’s and supplier’s setup costs can be omitted
since they constitute constant factors in nominator and denominator of the RHS of
(4.38) and, hence, do not increase this ratio. With the buyer’s holding costs of 2hB d ,
we obtain as an upper bound
5hB d C 2hB d
7
D :
4hB d C 2hB d
6
92
4 New Coordination Schemes
N B/ N S integer
N B / N S not integer
z =1, r = 2, |T | = 6
z =1, r =1, |T | =4
Setups B
Setups B
Setups S
Setups S
t
|T |
t
|T|
PB
Fig. 4.10 Cases with maximum csys
=csys
in 1(a), (b)
Second, the buyer’s pattern may comprise two short order intervals and a large
one in between them. Then this large interval and one of the shorter have to be
combined by the supplier within one single setup. The case with an extreme cost
increase over the systemwide optimum is reached for jT j D 4, N B D 3, and N S D
2 (see the right of Fig. 4.10). There, the supplier’s costs exceed by 2hB d hB d D
hB d the systemwide optimum, where the buyer’s and supplier’s setups would take
place in period 2 instead of 3. Again omitting the buyer’s and supplier’s setup costs,
we obtain as an upper bound
ub1 D
3
hB d C 2hB d
D :
hB d C hB d
2
Since 3=2 > 7=6, we get for ubb , the upper bound for the case (b), ubb D 3=2.
(c) Assume that z > 1 or r > 1 and that jT j =N B is integer, but not necessarily
N B =N S . For the proof of this case, we express the costs of the base setting (with
N B and N S ) by a weighted sum of the costs of settings 10 and 20 , where N B =N S
is integer. Consider a feasible, but not necessarily optimal implementation of the
buyer’s proposal that comprises o D N B rN S setup intervals of the supplier with
r C 1 orders of the buyer and N S o setup intervals with r orders of the buyer.
Define 10 and 20 as variations of the base setting differing by the number of buyer’s
0
0
orders, N B1 D N B .r C 1/ and N B2 D rN B as well as by the number of time
0
0
horizons, T 1 D jT j .r C 1/ and T 2 D jT j r, but with equal numbers of supplier’s
0
0
setups, N S1 D N S2 D N B (as an illustration, see Fig. 4.11).
In these settings, each setup interval comprises an equal number of buyer’s orders
(r C 1 and r, respectively). Since jT j =N B is integer, the production schedule of the
base setting and its associated costs can be directly expressed by the combination
10
20
of appropriate fractions of the schedules of 10 , 20 (and their costs csys
, csys
), which
comprises the same number of orders as the base setting. Hence, we obtain
PB
csys
o 10
N S o 20
c
C
c :
sys
NB
N B sys
4.2 Scheme for Uncapacitated Dynamic Lot-Sizing and Analytical Results
93
Base setting (| T | =6, N B = 3)
N B =3
N S =2
Setups B
Setups S
t
|T|
Setting 2‘ (T 2‘ = 6, NB2‘= 3, NS2‘= 3)
Setting 1‘ (T 1‘=12, N B1‘ = 6, N S1‘ = 3)
Setups B
Setups S
t
t
T 2‘
T 1‘
Fig. 4.11 Illustration of settings 10 and 20
This expression can be written as
PB
csys
y
.1 y/ 20
0
c1 C
csys ;
r C 1 sys
r
(4.42)
rel;10
rel;20
and csys
as the lower bounds
with y D .r C 1/ 1 rN S =N B . Define csys
for the costs with the relaxation of the integrality of tiS and tijB for settings 10
and 20 , respectively. Combining (4.42) with (4.38) and (4.39), we obtain as an upper
bound:
1
r
1
2scB N B C 2scS N S C T 2 dhB y rC1
C
.1
y/
B
B
jT j
jT j
N
N
: (4.43)
2
1
1
B
S
2scB N C 2scS N C jT j dhB N S jT j
Omitting the setup costs as above and introducing a new parameter ˛ with 0 ˛ 1 and N B D .r C ˛/N S , we obtain the upper bound:
z r 2 C ˛ C 2r˛ r ˛
:
.r C ˛/ .r C ˛z 1/
(4.44)
The partial derivative of (4.44) with respect to r is
.˛ 1/ ˛z .1 C 2z .r C ˛//
.r C ˛/2 .z .r C ˛/ 1/
:
(4.45)
Since 0 < ˛ < 1, z 1, r 1, and either z > 1 or r > 1, the value of (4.45) is
always smaller than zero. We further calculate the partial derivative of (4.44) with
respect to z:
˛ .˛ 1/
0:
.r C ˛/ z .r C ˛/2 1
Hence, ubc takes its maximum with either z D 1 and r D 2 or z D 2 and r D 1. In
the former case, we obtain the bound
94
4 New Coordination Schemes
2 C 4˛
:
2 C 3˛ C ˛ 2
(4.46)
p
Equation (4.46) takes its maximum with ˛ D 1=2
3 1 , leading to an upper
p p bound of 4 3= 3 C 2 3 1:07. For z D 2 and r D 1, we obtain the bound
1 C 5˛
:
1 C 3˛ C 2˛ 2
(4.47)
p
p
Equation (4.47) takes its maximum with ˛ D 1=5
6 1 . Since 25 6=
p p p p p 24 C 11 6 > 4 3= 3 C 2 3 , ubc D 25 6= 24 C 11 6 1:2.
(d) Assume that z > 1 or r > 1 and that jT j =N B is not integer. The strategy for
PB
this part of the proof is to express an upper bound on csys
by the costs of settings
B
with time horizons of integer multiples of N . Upper bounds for the costs in these
settings are ubc ubsys;U;L
, with ubsys;U
and ubsys;L
as upper bounds on the systemwide costs
in
U
and
L,
which
can
be
calculated
analytically
˙
˘ by (4.39). Define
T U D N B jT j =N B and T L D T U N B D N B jT j =N B . Then the number
of larger order intervals, i.e., those with tijB D T U =N B , is equal to jT j T L .
Hence, the buyer’s costs for PB with the number of buyer’s setups N B are
0
1
TU
TL
B 1
B 1
NX
NX
B
C
PB
cB
D N B scB C hB d @ jT j T L
t C T U jT j
tA :
t D1
t D1
Define U and L as variations of the base setting with the same parameterizations
except for jT j; the time horizons of U and L are T U and T L , respectively (see
Fig. 4.12 for an example). Denote the buyer’s costs for PB subject to N B in
these settings by cB;U
and cB;L
, respectively. Introduce a new parameter ˇ D
U
L
L
with 0 < ˇ < 1 indicating the weights of the characjT j T = T T
teristics of the setting U within the base setting solution. Then we obtain
0
TU
1
NB
TL
1
NB
t D1
t D1
1
X C
B B X
PB
cB
D N B scB ChB d B
t C .1 ˇ/ N B
tC
@ˇN
A D ˇcB;U C.1 ˇ/ cB;L :
Next, consider an implementation of PB by the supplier, where the number of
buyer’s orders contained in the setup intervals of the supplier differ by one at most.
Due to (4.41), the supplier will try to accumulate the larger order intervals of the
buyer into setup intervals comprising a smaller number of orders.
4.2 Scheme for Uncapacitated Dynamic Lot-Sizing and Analytical Results
95
Base setting (|T|=5)
Setups B
Setups S
t
N B=4
N S=3
|T |
Setting L (TL =4)
U
Setting U (T =8)
Setups B
Setups S
t
T
TU
L
t
Fig. 4.12 Example for settings U and L
Preliminarily, we will assume that the supplier can accomplish that there is no
setup interval that comprises both more orders and a greater share of the larger
setup intervals than other intervals (this assumption is relaxed below). Recall that
for cSPB holds
S
cSPB N S scS C hS d
NB
NB
1
S
N NX
X
i D1 j D1
tijB
NS
X
tiBk :
(4.48)
kDj C1
Due to (4.41), for the supplier’s holding costs of a setup interval tiS with n1 orders
of the buyer with lengths of tijB and n2 orders with lengths of tijB C 1 holds:
chS tiS n1
n2
chS n1 tijB C
chS n2 tijB C 1 :
n1 C n2
n1 C n2
(4.49)
Due to the above assumption that there is no setup interval that comprises both
more orders and a greater share of the larger setup intervals than
the others,
we
can set tijB C 1 D T U =N B , tijB D T L =N B , n1 D N S jT j T L , and
n2 D N S T U jT j . (If T U =N B is not integer, this setting does not violate the
validness of (4.48) since the supplier could then include shorter order intervals of
the buyer into the setup intervals containing more orders of the buyer, which reduces
his costs for the implementation of PB .) Then, combining (4.48) with (4.49), we
obtain:
0
B
N BS 1 U N
NS
NS
S
L NX
X
X
T
N
jT
j
T
TU
B
cSPB N S scS C hS d
C
@
NB
NB
NB
i D1
j D1
1
NB
N BS 1
NS
N S T U jT j NX T L X
TL C
A:
NB
NB
NB
j D1
kDj C1
kDj C1
96
4 New Coordination Schemes
This expression can be written as
cSPB ˇcS;U
C .1 ˇ/ cS;L
;
(4.50)
; cS;L
as the supplier’s costs in settings U and L, respectively.
with cS;U
Next, we consider the (more general) case where a setup interval of the supplier exists that contains a greater share of larger orders. This is only enforced if
a large setup interval is located between two shorter ones within the buyer’s order pattern (see the right of Fig. 4.10). Again the extreme case is reached here for
N S D 2 and N B D 3. (For other values of N S , N B , additional constant cost factors would be included for both buyer and supplier, which would lower the resulting
relative suboptimality of PB .) In the systemwide optimal solution, the supplier
will gather the two smaller order intervals into a single setup interval, resulting
in costs of hB d z .z C 1/. The supplier’s costs of setting U, in turn, will amount
up to hB d.z C 1/2 and of setting L, up to hB d z2 . Since ˇ D 1=3 holds here,
the weighted
sum of the costs
of settings U and L (see 4.50) will amount up to
hB d 1=3.z C 1/2 C 2=3z2 . Hence, the maximum excess of the supplier’s costs in
the systemwide optimum over this weighted sum becomes
ex D 2
z .z C 1/
z2 C 13 .z C 1/2
3
:
(4.51)
p
Equation (4.51) takes its maximum with z D 1 C 2. Since z 2 N and z > 1, the
maximum cost excess is reached for z D 2. Then,
D 18=17 follows.
ex
PB
PB
PB
Hence, we obtain an upper bound of 18=17 xcU
csys
C .1 x/ cL
, with
PB
PB
PB
cU and cL as the systemwide costs for the implementation of in settings U
PB
PB
PB
PB
and L, respectively. As in (c), for cU
and cL
, cU
ubc cU
and cL
ubc cL
holds, where cU and cL can be calculated by analytically via (4.36). Hence, we
obtain as a general upper bound:
scB N B C scS N S
18ubc
C
17
scB N B C scS N S C jT j2 hB d2 N1S jT1 j
L 2 1
2
d
1
1
1
C
.1
ˇ/
T
h ˇ TU
S
U
S
L
2 B
N
T
N
T
:
2
1
1
d
B
S
scB N C scS N C jT j hB 2 N S jT j
(4.52)
Since the supplier’s and the buyer’s setup costs are constants in numerator and denominator, their omittance does not affect the validity of (4.52). With z D T L =N B
and r D N B =N S , we obtain the upper bound
rz2 z ˇ C rˇ C 2rˇz
18
ubc
:
17
.ˇ C z/ .r .ˇ C z/ 1/
(4.53)
4.2 Scheme for Uncapacitated Dynamic Lot-Sizing and Analytical Results
97
The partial derivative of (4.53) with respect to z is
18
r .ˇ 1/ ˇ .2r .ˇ C z/ 1/
ubc
:
17
.ˇ C z/2 .r .ˇ C z/ 1/2
(4.54)
Since 0 < ˇ < 1, z 1, and r 1, (4.54) takes only values smaller than zero. The
partial derivative of (4.53) with respect to r is
18
.ˇ 1/ ˇz .2rz C z 1/
ubc
0:
17
.ˇ 1 C r .z C rz 1//2
As above, we first consider the case z D 1 and r D 2. Then we obtain the bound
18
.3 ˇ/ .1 C ˇ/
ubc
1:
17
3Cˇ
p
This expressiontakes its maximum
with ˇ D 2 3 3, leading to an upper bound
p
of 18=17 ubc 7 4 3 .
With r D 1 and z D 2, we obtain the upper bound
18
2 C 3ˇ 2ˇ 2
ubc
:
17
2Cˇ
p
This expression
takesits maximum with ˇ D 6 2. Since the resulting bound
p
p
18ubc =17 11 4 6 > 18=17ubc 7 4 3 ,
p ubd D 18ubc =17 11 4 6 . Since ubd > ubb and ubd > ubc ,
p ubd D 18ub2a =17 11 4 6 1:53 is a general upper bound for a serial BOM.
(e) For assembly structures, assume the following transformation of the BOM into a
set of serial substructures (see Fig. 4.13): Each of the buyer’s items j is replaced by
a set of items jk , with k 2 Rj and Rj as the set of the predecessor items of j , such
that each item supplied can be combined with one specific successor
P item. Moreover, determine the unit holding costs of items jk to hkj D hj hk = i 2Rj hi , with
P
hk as the holding costs of the respective predecessor item k. Since hj i 2Rj hi
due to Assumption 4.1, hkj hk holds. We assume any repartition of the buyer’s
setup costs among the buyer’s items in the transformed BOM. If the same N B
is chosen, then the costs for PB (for the worst possible setup pattern) of the
transformed BOM are greater than or equal to those of the original BOM. For the
transformed BOM, the costs of the systemwide optimal solution are smaller than or
equal to those for the original BOM. Since the buyer’s setup costs are not needed
for the proof for the serial structure presented above, ubd extends for each of these
serial substructures and, hence, for the original BOM.
98
4 New Coordination Schemes
j= 1
Rj = {2,3,4}
2
1
Transformation
12
13
14
Buyer
3
4
2
3
4
Supplier
items
Fig. 4.13 Example for the transformation of the BOM
The main result of this section is presented in Theorem 4.4.29
Theorem 4.4 For a serial or assembly two-stage supply chain with level end item
demand, zero initial inventories, and the production portfolio of the buyers consists
of one item, the scheme identifies a solution
costs exceed those ofthe sys whose
p p p
temwide optimum at maximum by 18=17 11 4 6 35 6 11 C 24 6 1 1:53. The number of iterations required increases linearly with the number of periods and the number of items supplied.
Proof. At first, we show that (in the worst case) proposals of CS dominate the corresponding proposals out of PB, provided that the buyer produces one item only.
Define N B;up as the number of setups in the upstream planning solution. Successively decreasing the number of orders for items j 2 J D leads to the coverage
of all setup frequencies N B N B;up which are not dominated by other setup
frequencies.30 To prove that the upper bound derived in Lemma 4.4 extends to
N B > N B;up , consider
1
1
1
B ;
cSrel N B D scS N S C d jT j2 hS
S
2
N
N
the supplier’s costs subject to N B and relaxed integrality for tiS and tijB .31 Since
cSrel N B increases with N B and the buyer’s costs are greater for N B > N B;up
than for N B;up , the denominator of (4.38) is greater for N B than for N B;up . Hence,
the upper bound derived in Lemma 4.4 for a proposal out of PB with N B;up is also
valid with an order frequency of N B > N B;up in the systemwide optimal solution.
29
Note that a corollary to this theorem is stated in Albrecht (2008) (called Theorem 2 there),
where convergence is shown under the additional requirements on N B =N S and jT j =N B specified
in Lemma 4.3.
30
Since the buyer’s costs for a dominating frequency are lower, proposals with dominated frequencies – which are not identified by our procedure – are obviously inferior.
31
See also equation (4.39).
4.3 Application to Master Planning
99
What remains
ˇ ˇ to show is that the number of proposals generated increases linearly with ˇJ D ˇ and jT j. In our procedure for proposal generation, w is initialized
ˇ
ˇ by
P
1 (or 0) and increased in each iteration by 1 at most until w j 2J D ysjup D ˇJ D ˇ.
ˇ
ˇ
ˇ
ˇ
P
Since j 2J D ysjup ˇJ D ˇ jT j, the maximum number of steps is ˇJ D ˇ jT j.
4.3 Application to Master Planning
In this and the next section, we present the necessary steps for customizing the
generic scheme for LP to the Master Planning models presented in Sect. 2.2. We
begin with a description of the linearization of the objective function of CS1i , a
basic requirement for tackling this model with a standard solver. Next, we show
how to adapt the generic scheme for LP to GM as the underlying model and provide
modifications that improve the applicability of this scheme for MIP models and
accelerate the convergence rate.
4.3.1 Linearization
Model CS1i is nonlinear since two vectors of variables are multiplied in its objective
function. Therefore, the optimal solutions to CS1i cannot be identified by standard
MIP software like CPLEX or Xpress-MP, that can efficiently tackle large-scale practical optimization problems,32 but falls short in case of general nonlinearities like
those in CS1i . Thus, the identification of an appropriate solution procedure for CS1i
is an additional issue, that needs to be covered here.
For nonlinear problems comprising continuous variables only (NLP), a broad
range of solution procedures has been elaborated in the literature. A procedure frequently used is successive linear programming (SLP). SLP has been successfully
applied for the solution of optimization models with few nonlinear terms, which are
employed, e.g., in the petrochemical industry, where quality and volume of products
have to be determined simultaneously.33 In SLP, a series of linear approximations
of the nonlinear terms (by their first-order Taylor series expansions) is solved that
often converges to a local optimum of the original problem.
Many practical supply chain planning problems, however, comprise integer or
binary variables. For the resulting nonlinear mixed-integer programming problems
(MINLP), computational difficulties are considerably greater than for NLP.34 Most
solution procedures for MINLP are based on Branch & Bound, Generalized Benders
32
E.g., ILOG (2008) and Dash (2008).
See Baker and Lasdon (1985, p. 264).
34
See, e.g., Kallrath and Wilson (1997, p. 376).
33
100
4 New Coordination Schemes
Decomposition, and Outer Approximation.35 Like in the standard case (pure MIP),
Branch & Bound uses the solutions of the subproblems with relaxed integrality constraints as lower bounds for the search tree.36 The computational effort of Branch
& Bound for MINLP, however, is considerably greater than for MIP because of
the limited possibilities for exploiting the information generated at nodes previously investigated. The two latter procedures, Generalized Benders Decomposition
and Outer Approximation, iterate between solving a master problem and nonlinear subproblems with fixed integer variables. As the name indicates, Generalized
Benders Decomposition37 directly extends Benders Decomposition to the solution
of MINLP. In Outer Approximation,38 the master problem uses primal information
from the nonlinear subproblems and constructs a linearization of the nonlinear terms
in the objective function and the constraints around the primal solution. Global convergence of these procedures can only be shown for some additional conditions
about the problem structure. One of them is concavity of the objective function for
fixed values of the integer variables.39 Unfortunately, this condition is not fulfilled
in CS1i , as shown by the following example.
Example 4.4 Assume that the integer variables in CS1i have been fixed to arbitrary
values and that the vector k comprises
one dimension only. With ci D 0 and scalars
k 0 D kiT and d 0 D Ai xist xi , the objective function of CS1i becomes k 0 d 0 .
A sufficient and necessary condition for the concavity of k 0 d 0 is the positive definiteness of the associated Hessian matrix
0
0
H.k d / D
0 1
:
1 0
The eigenvalues of this Hessian matrix are the roots of the following equation:
ˇ
ˇ
ˇ 1 ˇ
ˇ
ˇ
ˇ 1 ˇ D 0:
The roots are 1 D 1 and 2 D 1. Since both positive and negative eigenvalues
exist, H .k 0 d 0 / is indefinite, from which non-concavity of the objective function of
CS1i follows.
The principal aim of the computational tests of this work40 is to evaluate whether
the schemes are able to identify practicable solutions. For this purpose, a piecewise
linear approximation of the nonlinear term of the objective function of CS1i turned
35
See, e.g., Floudas (1995, p. 112) and Kallrath and Wilson (1997, p. 379).
See, e.g., Gupta and Ravindran (1985, p. 1534).
37
See Geoffrion (1972, p. 237).
38
See Duran and Grossmann (1986, p. 307).
39
See, e.g., Floudas (1995, p. 114 and p. 144).
40
See Chap. 6.
36
4.3 Application to Master Planning
101
out to be appropriate.41 In contrast to the procedures for MINLP mentioned above,
this approximation can be carried out using standard mathematical programming
software and showed sufficient accuracy for obtaining meaningful results in our
computational tests.
For ease of exposition, we apply the following transformation for the nonlinear
term in the objective function of CS1i : Write kj0 for the value of the j -th dimension
of vector kiT and dj0 for the value of the j -th dimension of vector Ai xist xi .
P
0 0
Then, kiT Ai xist xi D N
j D1 kj dj holds. Following a well-known programming trick,42 kj0 dj0 can be transformed into two quadratic expressions:
2
2
1 0
1 0
kj C dj0
kj dj0
kj0 dj0 D
:
(4.55)
2
2
0
0
0
0
ubjk , ubjd , lbjk , and lbjd , the upper and lower bounds for kj0 and dj0 , are determined
0
0
as follows: ubjd are lbjd are the maximum (one-sided) deviations of proposals about
0
central resource use from the starting proposal in CS1i . ubjk is set equal to M , a
0
large number, and lbjk equal to zero.43
2
Next, we address the linear approximations of f f kj0 ; dj0 D 1=2 kj0 C dj0
2
and f s kj0 ; dj0 D 1=2 kj0 dj0
, the quadratic terms of (4.55). Instead of
focusing specifically on f f kj0 ; dj0 and f s kj0 ; dj0 ,44 we explain our methodology for a general quadratic function f .x/ D x 2 . We divide the curve of f .x/ into
several intervals and approximate each of them by a linear function. In the example
of Fig. 4.14, we have chosen three intervals for the approximation.
The line-segments OA, AB, and BC approximate f .x/ for x 2 ŒxO ; xC . The
nodes O and C are the lower and upper bounds of f .x/, and the nodes A and B are
b .x/,
breakpoints. xO , xA , xB , and xC are the values of these nodes on the x-axis. f
the function defined by these line-segments, can be written as a weighted sum of the
function values of the nodes:
b .x/ D O x 2 C A x 2 C B x 2 C C x 2 :
f
O
A
B
C
O ,A ,B , and C are additional variables and can be interpreted as the weights of
the nodes. If, e.g., x is the average of xB and xC , the weights for xB and xC will
41
A thorough investigation of the effectiveness of different MINLP solution procedures for CS1i ,
in turn, does not seem necessary for that purpose.
42
See, e.g., Williams (1993, p. 152).
0
0
43
For the specific determination of ubjd , lbjd , and M for GM used here, see p. 110 and p. 114.
44
In Sect. 4.3.2, we present a model where these functions are explicitly linearized for GM.
102
4 New Coordination Schemes
Fig. 4.14 Piecewise linear
approximation of f .x/
C
f(x)
f (x)
fˆ(x)
B
A
O
xO
x
xA
xB
xc
be 1=2 each. The correct determination of these weights is assured by the following
constraints:
x D O xO C A xA C B xB C C xC ;
(4.56)
O C A C B C C D 1;
O ; A ; B ; C 0;
(4.57)
(4.58)
O ; A ; B ; C
(4.59)
SOS 2:
Constraint (4.56) sets x equal to the weighted sum of the arguments of the functions
used for the approximation, whereas (4.57) normalizes the sum of the weights to 1.
Constraints (4.58) ensure the nonnegativity of the weights, and constraints (4.59)
restrict variables O ,A ,B , and C to a Special Ordered Set of type 2 (SOS2),
which implies that at most two adjacent variables within the SOS2 (e.g., B and
C ) can be nonzero.45 Alternatively to the use of SOS2, constraints (4.59) could be
reformulated using additional (standard) binary variables. Regardless of which formulation is chosen, the computational difficulty of the underlying model increases
considerably due to the additional discrete decisions. Thereby, the SOS2 characteristic can be exploited by the optimizer solving the problem, which usually reduces
solution time.
Further
note that constraints (4.59) can be omitted for the linearization of
f s kj0 ; dj0 . Due to the negative sign of this term in (4.55), both the objective
function value of CS1i and the slopes of the corresponding line-segments are decreasing with greater values for x. Hence, greater weights for breakpoints with
smaller x are preferred in the optimal solution to CS1i , which ensures
that the
correct line-segment is chosen. The opposite holds for f f kj0 ; dj0 due to its concavity. Without (4.59) and small x, the optimizer solving the problem might choose,
e.g., a value greater than zero for O and C , and one of zero for A and B .
45
See Beale and Tomlin (1970, p. 447).
4.3 Application to Master Planning
103
The lower and upper bounds for f f kj0 ; dj0 and f s kj0 ; dj0 are ubd =4 and
1=4 .ubk C ubd /. Further consideration deserves the choice of the breakpoints and
the expected approximation error. Both issues are dealt with in Sharpe (1971).46
There it is shown that, given equal probabilities of each x 2 ŒxO ; xC , the (absolute)
expected error for the approximation of a quadratic function can be minimized by
choosing equal interval lengths.
In this work, we follow this reasoning and also rely on equal interval lengths.
Define the average approximation error by
1
ub lb
Z ub
b .x/ f .x/ dx;
f
lb
with ub and lb as the upper and lower bounds of the range that is subject to the
approximation.47 With equal interval lengths, the approximation error becomes
.ub lb/2
;
6m2
with m as the number of approximation intervals. With, e.g., m D 5, we get an
average approximation error of .ub lb/2 =150. Comparing this with the average of f .x/ within these bounds and assuming that each value for x is equally
probable within the range considered, the average approximation error becomes
1=75 1:33% of the average of the approximated function.
4.3.2 Adaptation to Master Planning
Next, we describe how to adapt the generic scheme for LP to a two-party supply
chain planning based on GM.48 Let each party dispose of a subset of resources and
produce a subset of items. The buyer uses items produced by the supplier as input
for her production process. The interdependence between the parties’ submodels are
(2.45), the inventory balance constraints for the items supplied. To make the central
resources explicit, we reformulate these constraints as
XBjt XSjt
8j 2 J D ; t 2 T;
(4.60)
XBjt XSjt
8j 2 J ; t 2 T:
(4.61)
D
Based on this decomposition, the optimization models used within the scheme can
be derived directly.
46
See Sharpe (1971, p. 1269).
In the preceding example, lb D xO and ub D xC hold.
48
See Sect. 2.2.1 for the mathematical formulation of this model.
47
104
4 New Coordination Schemes
First, we address the version of the scheme with unilateral iterative exchange of
cost information. We assume for ease of exposure that the supplier communicates
his cost changes to the buyer during the execution of the scheme. Then, models
GM-CS2B , GM-CS1B , and GM-CS1S are run within the scheme.49
(GM-CS2B ) min CBd C
eN
X
e cs e
eD1
s.t. .DC-B/
eN
X
XBjt D
e
e xbjt
8j 2 J D ; t 2 T
eD1
eN
X
e D 1
(4.62)
eD1
e 0
for e D 1; : : : ; e:
N
X X
C
st
(GM-CS1B ) min CBd C
Kjt
XBjt xtjt
C
Kjt
(4.63)
j 2J D t 2T
X X X Tjt C X TjtC
j 2J D t 2T
s.t. .DC-B/
X X
j 2J D t 2T
X X
e
C
st
xtjt xtjt
cb st cb e 8e D 1; : : : ; eN
Kjt
(4.64)
st
e
xtjt xtjt
cb st cb e 8e D 1; : : : ; eN
Kjt
(4.65)
j 2J D t 2T
st
X TjtC X Tjt XBjt xtjt
8j 2 J D ; t 2 T
CBd C csys
C
Kjt
mjt
Kjt mjt
C
Kjt
; Kjt
0
(4.67)
D
8j 2 J ; t 2 T
(4.68)
D
(4.69)
8j 2 J ; t 2 T
8j 2 J D ; t 2 T
X TjtC ; X Tjt 0
49
(4.66)
D
8j 2 J ; t 2 T :
(4.70)
(4.71)
For a reversed information flow, GM-CS2S is applied instead of GM-CS2B . Further note that
for adapting the models stated below to the MLCLSP and MLCLSP-C presented in Sect. 2.1.2,
these models simply have to be augmented by the additional features for lot-sizing and campaign
planning.
4.3 Application to Master Planning
105
X X
C
st
Kjt
XSjt xtjt
C
(GM-CS1S ) min CSd C
Kjt
j 2J D t 2T
X X X Tjt C X TjtC
j 2J D t 2T
s.t. .DC-S/; (4.68)–(4.71)
X X
e
C
st
xtjt xtjt
cs st cs e 8e D 1; : : : ; eN
Kjt
j 2J D t 2T
X X
st
e
xtjt xtjt
cs st cs e 8e D 1; : : : ; eN
Kjt
(4.72)
(4.73)
j 2J D t 2T
st
8j 2 J D ; t 2 T:
X TjtC X Tjt XSjt xtjt
(4.74)
Data
cb e Buyer’s costs change of the previous proposal e compared to the initial
solution (e D 1; : : : ; e; st denotes the starting proposal)
cs e Supplier’s costs change of the previous proposal e
Unit penalty costs for arbitrary deviations, small number, e.g., 0.000001
hbj Buyer’s unit costs for inventory holding of the supplied item j
mjt Big number, denoting the maximum cost change per unit deviation
in the supply quantities
e
xtjt Amount of item j supplied in period t in the previous proposal e
Variables
CBd Costs for the decisions of the buyer’s planning domain
CSd Costs for the decisions of the supplier’s planning domain
e
Variables defining a convex combination of previous proposals e
C
Endogenously determined unit prices for positive deviations from the
Kjt
st
starting proposal xjt
of item j in period t
st
Kjt Unit prices for negative deviations of item j in period t from xjt
st
X TjtC Increase in the supply of item j in period t compared to xjt
X Tjt Decrease in the supply of item j in period t
Model GM-CS2B can be directly derived from CS2I and GM. For ease of exposition, (DC-B) and (DC-S) abbreviate the decentralized restrictions of GM-B and
GM-S, respectively, augmented by the restrictions for an explicit determination of
the supply quantities. (DC-B) comprises constraints (2.1)–(2.11) with the items and
resources limited to those of the buyer as well as (2.42), (2.46), and (2.48). (DC-S)
comprises constraints (2.1)–(2.3), (2.6), (2.8), (2.10), and (2.11) with the items and
resources limited to those of the supplier, (2.47), and50
50
The modeling of these constraints instead of (2.43) and (2.44) – which have been introduced for
clearness of exposition in Sect. 2.3.2 – is more direct and allows to reduce the number of variables
used.
106
4 New Coordination Schemes
Ijt 1 C Xjt D XS jt C Ijt
8j 2 J D ; t 2 T:
Moreover, we have introduced additional variables representing the decentralized
costs of parties:
CBd D
X X
hj Ijt C
j 2J B t 2T
C
X X
j 2J B t 2T
lscj LSjt C
j 2J E t 2T
CSd D
X X
j 2J S t 2T
X X
hj Ijt C
X X
scj Yjt C
X X
m2M B t 2T
X X
blcj BLjt C
j 2J E t 2T
hbj IBjt ;
j 2J D t 2T
X X
ocm Omt C
ocm Omt C
m2M S t 2T
X X
scj Yjt :
j 2J S t 2T
Sets
J S Set of items produced by the supplier
M B Set of resources of the buyer
M S Set of resources of the supplier
For GM-CS1B and GM-CS1S , some further explanations are necessary. For each
set of central resources (4.60) and (4.61), different price variables have been introC
duced. Their values Kjt
, Kjt
are determined by constraints (4.64), (4.65), (4.66),
(4.67), (4.77), and (4.73) in analogy to the generic scheme.
Thereby, the upper bounds for mjt have to be set to large numbers exceeding
the potential cost changes with changes in the central resource use, i.e., shortages
or excess supply. In the following, we show how mjt can be determined for GM
without lost sales and zero backorders at the end of the planning interval, the basic
version which we have tested computationally in Sect. 6.1.
For the buyer, changes in the central resource use can affect several cost types.
First, shortages in the supply may lead to backorders for end items. For determining
the resulting costs, we introduce a new parameter blcj denoting the maximum costs
for backorders caused by a shortage in the supply or the production of item j . We set
b
b
blcj D blcj C "
for end items, with " as an arbitrarily small number, and determine this parameter
recursively by
maxk2Sj blck
blcj D
C"
rjk
b
b
for intermediate items. Moreover, overtime may become necessary if production has
to be increased in later periods due to shortages of supplied items. This may affect
all successors of the items with shortages in their supply. We introduce a further new
cum
parameter, rjk
, the amounts of the predecessor item j that are needed to produce
cum
D rjk if k is a
item k (where k is not necessarily a direct successor of j ). rjk
direct successor of j . If k is an indirect successor of j ,
4.3 Application to Master Planning
107
Y
cum
rjk
D
cum
rjl
;
l2Gj k
with Gjk as the set of items that are located at one of the connecting lines between
items j and k in the BOM. Assuming that overtime and backorder costs are of a
greater order of magnitude than holding costs, we can set
8
<
b
mjt D max blcj ;
:
X X
cum
rjk
amk ocm
m2MS k2J B
9
=
;
C ":
For the supplier, overtime and holding costs can be affected by changes in the
supply pattern. Assuming that overtime costs are of a greater order of magnitude,
we can set mjt equal to the maximum overtime costs caused by increases in the
supply quantities. Hence, we can define for all j 2 J D
mjt D
X X
cum
rkj
amk ocm C ":
m2MS k2J S
In GM-CS1S and GM-CS1B , three further features have been incorporated in
order to cut off inferior solutions. First, additional penalty cost terms have been included into the objective functions. These terms consist of fixed unit penalty costs
" multiplied with the absolute deviation from the starting supply pattern (expressed
by variables X TjtC and X Tjt ). Such penalizations are favorable if several optimal
solutions to GM-CS1S or GM-CS1B exist that differ by the supply quantities of particular items in some periods. Most probably, the other party prefers the proposal
where these supply quantities come nearest to the starting proposal; hence, penalizing arbitrary deviations by very small costs helps to avoid unnecessary changes
in the supply pattern that could cause increases in the other party’s costs. If " is
sufficiently small, the convergence of the scheme is not affected. Note that a further
favorable consequence of these penalty costs is that penalties for negative deviations
for the buyer and positive deviations for the supplier and, hence, constraints (4.65)
and (4.72) become redundant, given zero inventories at the beginning of the planning interval.51
Second, constraint (4.67) cuts off unfavorable solutions by keeping the decentralized costs below the costs of the systemwide best solution found so far. Note
that this constraint can only be applied by the party that is informed about the other
party’s cost changes.
Third, in order to avoid excess inventory holding of supplied items at the buyer’s
site, additional holding costs for the supplied items have been included into the
objective functions of the buyer’s problems.52
51
This insight has also been used in our computational tests of Chap. 6.
In this context, note that inventory holding of supplied items at the buyer’s site will become
more expensive than at the supplier’s if the holding costs comprise capital costs only with identical
52
108
4 New Coordination Schemes
Recall that models GM-CS1B and GM-CS1S cannot be directly tackled by a standard solver due to their nonlinear objective functions. The linearizations of these
objective functions can be carried out along the lines of Sect. 4.3.1, but have not
been included in GM-CS1B and GM-CS1S for ease of exposition. Next, we exemplarily provide a formulation for GM-CS1B with an explicit linearization of the
objective function.
(GM-CS1LB ) min CdB C
X X p;C
p;
Cjt C Cjt
C X Tjt C X TjtC
j 2J D t 2T
s.t. .DC-B/; (4.64)–(4.71)
nod NX
2
2
p;C
f;C
s;C
s;C
f;C
8j 2 J D ; t 2 T
Cjt x
x
njt
njt
njt
njt
nD1
(4.75)
p;
Cjt
nod NX
2
2
f;
f;
s;
njt xnjt
8j 2 J D ; t 2 T
s;
njt xnjt
nD1
(4.76)
X f;C f;C
1 C
st
Kjt C XBjt xtjt
D
njt xnjt 8j 2 J D ; t 2 T (4.77)
2
nD1
N nod
NX
1 C
s;C
st
D
Kjt XBjt C xtjt
D
s;C
njt xnjt 8j 2 J ; t 2 T
2
nD1
nod
nod
NX
f;C
njt D 1
(4.78)
8j 2 J D ; t 2 T
(4.79)
D
s;C
njt D 1 8j 2 J ; t 2 T
(4.80)
nD1
nod
NX
nD1
s;C
njt
for n D 1; : : : ; N nod
SOS 2 8j 2 J D ; t 2 T
(4.81)
N nod
X f; f;
1 st
Kjt XBjt C xtjt
D
njt xnjt 8j 2 J D ; t 2 T
2
nD1
(4.82)
nod
NX s; s;
1 st
Kjt C XBjt xtjt
D
njt xnjt 8j 2 J D ; t 2 T
2
nD1
(4.83)
interest rates for parties and if the purchase prices for these items exceed the supplier’s production
costs.
4.3 Application to Master Planning
nod
NX
109
D
f;
njt D 1 8j 2 J ; t 2 T
(4.84)
s;
njt D 1
(4.85)
nD1
nod
NX
8j 2 J D ; t 2 T
nD1
for n D 1; : : : ; N nod SOS 2
s;
njt
P;C
P;
; Cjt
0 8j 2 J D ; t 2 T
Cjt
f;
s;C
s;
f;C
njt ; njt ; njt ; njt 0
8j 2 J D ; t 2 T
for n D 1; : : : ; N nod ; 8j 2 J D ; t 2 T :
(4.86)
(4.87)
(4.88)
Index
n Index for the nodes used for the approximation (including the borders of the
interval that is to be approximated); n D 1; : : : ; N nod
Data
f;C
C
st
xnjt
Node n (x-coordinate) for the linearization of f f Kjt
; XBjt xbjt
f;
st
Node n for the linearization of f f Kjt
; XBjt xbjt
xnjt
s;C
C
st
xnjt
Node n for the linearization of f s Kjt
; XBjt xbjt
s;
st
xnjt
Node n for the linearization of f s Kjt
; XBjt xbjt
Variables
p;C
Cjt
Penalty or bonus for greater supply of item j in period t
p;
Cjt
Penalty or bonus for less supply of item j in period
t
f;C
C
st
; XBjt xbjt
njt Weight for node n for the linearization of f f Kjt
f
st
f;
K
Weight
for
node
n
for
the
linearization
of
f
;
XB
xb
jt
njt
jt
jt s;C
C
s
st
njt Weight for node n for the linearization of f Kjt ; XBjt xbjt
s
st
K
Weight
for
node
n
for
the
linearization
of
f
;
XB
xb
s;
jt
njt
jt
jt
d k;C
d k;
, Cjt
The objective function of GM-CS1LB comprises new variables Cjt
denoting penalties or bonuses due to changes in the supply pattern. Their values
are determined by (4.75) and (4.76), respectively.53 Constraints (4.77)–(4.81) correspond to (4.56)–(4.59) and assure
a correct determination of the weights for the
C
st
piecewise linearization of Kjt XBjt xbjt
. Analogously, (4.82)–(4.86) are used
st
for the linearization of Kjt
XBjt xbjt
. (4.87) and (4.88) are nonnegativity
53
The introduction of these additional variables helps to avoid potential errors in the approximation
C
leading to negative values for actually positive Kjt
, Kjt
. Such effects are natural for the approximation applied and would cause some degradation in the solution quality in case of an alternative
modeling without these variables.
110
4 New Coordination Schemes
constraints. The upper and lower bounds for the linearization are determined as
described in Sect. 4.3.1. As upper bounds for the deviations XBjt xbjt and
xbjt XBjt , the cumulated demand for item j in the planning interval can be
taken.
In case backorders of the buyer are never chosen (e.g., if backorders of the buyer
are much more costly than a corresponding amount of overtime at the supplier), the
performance of the coordination process can be somewhat enhanced by imposing
a lower limit on the cumulated supply quantities in GM-CS1S .54 If this limit is set
equal to the secondary demand for item j in period t, it can be guaranteed that the
supplier’s proposals complying with this limit will not cause any backorders at the
buyer.55 For this purpose, we augment GM-CS1S by
t
X
cum
XSj ejt
8j 2 J D ; t 2 T;
D1
cum
with ejt
as the cumulated secondary demand for item j in period t, which can be
determined in the same way as bjt .56 Since the end item demand is usually private
cum
information of the buyer, she has to communicate ejt
additionally to the supplier
(e.g., at the beginning of the scheme). However, we do not regard this exchange as
critical since, in contrast to absolute costs or capacities, demand data is usually not
perceived as confidential by decision makers.57
For sake of completeness, we state the models needed for the evaluation of the
last proposal of the other party.
(GM-CS-EVALB ) min CBd
l
s.t. xtjt
D XBjt
.DC-B/:
8j 2 J D ; t 2 T
(GM-CS-EVALS ) min CSd
l
s.t. xtjt
D XSjt
8j 2 J D ; t 2 T
.DC-S/:
If the version with the one-shot exchange of cost information is applied, all parties run models GM-CS1i without (4.67) for proposal generation. Moreover, instead
of GM-CS2I , GM-MP is run once at the end of the scheme. This model can be
directly derived from MP; we state it below without further explanations.
54
Note that this extension has also been used in the scheme of Dudek and Stadtler (2005, p. 677).
In our computational tests, we apply this extension in Sect. 6.3, when we compare the performance
of the scheme proposed here with that of Dudek and Stadtler (2005).
55
For a further illustration of this idea, see Example 2.6 on p. 29.
56
For the determination of bjt , see p. 13 of this work.
57
See, e.g., Kersten (2003, p. 332).
4.3 Application to Master Planning
(GM-MP) min
111
eN X
se cs e C be cb e
eD1
s.t.
eN
X
e
se xsjt
D
eD1
eN
X
eN
X
e
be xbjt
8j 2 J D ; t 2 T
eD1
se D 1
eD1
eN
X
be D 1
eD1
se 0; be 0
for e D 1; : : : ; e:
N
4.3.3 Generic Modifications
As it will be shown in our computational study of Sect. 6.1, a straightforward application of the schemes presented in the previous subsection involves two major
difficulties: Fast convergence and modest computing times for models GM-CS1B
and GM-CS1S can only be achieved for rather small test instances. To mitigate these
difficulties, we propose a couple of modifications to the schemes. In this section, we
introduce a set of generic modifications, that can be directly transferred to other decision problems and defer to the next subsection the modifications specific to Master
Planning. Since the option for lost sales imposes additional complexity in the exposition of the models, we initially assume that costs for lost sales are such large that
the buyer never chooses this option and defer adaptations to lost sales to Sect. 4.4.3.
As the base for the modifications proposed, we present models GM-CS10 B and
GM-CS10 S .58
X X
C
C
Kjt
C Kjt
C XBjt
C XBjt
(GM-CS10 B ) min CBd C
j 2J D t 2T
s.t. .DC-B/; (4.66); (4.67); (4.68); (4.71)
X X
e
C
st C
xtjt xtjt
Kjt
cs e cs st 8e D 1; : : : ; eN (4.89)
j 2J D t 2T
X X
st
e C
xtjt xtjt
Kjt
cs e cs st 8e D 1; : : : ; e:
N (4.90)
j 2J D t 2T
58
For ease of exposition, we will limit to the formulations with nonlinear objective functions. Their
linearizations can be formulated analogously to GM-CS1LB .
112
4 New Coordination Schemes
X X
C
C
Kjt
(GM-CS10 S ) min CSd C
C Kjt
C XSjt
C XSjt
j 2J D t 2T
s.t. .DC-S/; (4.70); (4.71); (4.74)
X X
e
C
st C
xtjt xtjt
Kjt
cs st cs e 8e D 1; : : : ; eN (4.91)
j 2J D t 2T
X X
st
e C
xtjt xtjt
Kjt
cs st cs e 8e D 1; : : : ; e:
N (4.92)
j 2J D t 2T
The basic difference to GM-CS1B and GM-CS1S is that constraints (4.89), (4.90),
C
(4.91), and (4.92) imply non-negative values for Kjt
, Kjt
and, hence, only penalties
and no bonuses for changes in the central resource use incur. Hence, only proposals
with cost savings compared to the starting solution are generated, which increases
the probability for the identification of a systemwide improvement. The restriction
to solutions with one-sided (positive) deviations from the starting solution generates new limiting hyperplanes in the solution spaces of GM-CS10 B and GM-CS10 S .
Since the positions of these hyperplanes depend on the starting solutions chosen, the
same holds for the vertices of the solution spaces of GM-CS10 B and GM-CS10 S that,
amongst others, are determined by these hyperplanes. These “new” vertices, however, potentially become new proposals in further iterations of the scheme and might
be employed as starting solutions themselves. Hence, the number of proposals generated may grow to infinity. Therefore, the above modification renders the scheme a
heuristic also for LP problems, and finite convergence cannot be guaranteed.
With the buyer as the cost-reporting party, these models (and analogously further
models presented in this section) are modified as follows: First, (4.67) is skipped
from the buyer’s model and a constraint equivalent to (4.67) is included into the
supplier’s model. Second, constraints (4.89) and (4.90) are modified to
X X
e
C
st C
xtjt xtjt
Kjt
cb st cb e
j 2J D t 2T
X X
st
e C
xtjt xtjt
Kjt
cb st cb e
8e D 1; : : : ; e;
N
8e D 1; : : : ; e;
N
j 2J D t 2T
and constraints (4.91), and (4.92) modified to
X X
j 2J D t 2T
e
C
st C
xtjt xtjt
Kjt
cb e cb st
X X
st
e C
xtjt xtjt
Kjt
cb e cb st
8e D 1; : : : ; e;
N
8e D 1; : : : ; e:
N
j 2J D t 2T
Apart from improving the convergence rate, this modification considerably reduces
the time for running the scheme. The main reason is the renunciation on bonuses for
unfavorable deviations in the supply quantities. Then the optimizer solving these
4.3 Application to Master Planning
113
models has a significantly greater leeway to endogenously determine feasible valC
C
ues for Kjt
and Kjt
. Difficulties can be avoided that arise if Kjt
and Kjt
have to
account for both the correct determination of bonuses for solutions with unfavorable deviations in the supply quantities and penalties for solutions with favorable
deviations. Although, in theory, there is a feasible way to correctly determine the
bonuses and penalties if decentralized models only contain linear variables,59 the
computational effort increases considerably with the number of iterations.60
Second, model GM-CS2B is modified to allow additional deviations from the
convex combination of previous proposals of the other party at the expense of
penalty costs.
(GM-CS2B 0 ) min CBd C
eN
X
e cs e C
X X
. C cp/ X TjtC C X Tjt
j 2J D t 2T
eD1
s.t. .DC-B/; (4.62)–(4.63), (4.71)
XBjt C X Tjt D
eN
X
e
e xtjt
C X TjtC
8j 2 J D ; t 2 T:
eD1
By allowing such deviations, the limitation on exact recombinations inherent in
GM-CS2B is removed, such that the probability is increased that this model yields a
new proposal. Unit penalty costs cp are defined by
cp D
cs i ni t cs f
C :
P
P
f
i ni t
xt
xt
D
j 2J
t 2T
j
jt
(4.93)
Index
f Superscript denoting the first proposal generated
The rationale behind (4.93) is to perform a rough anticipation of the other party’s
cost changes with a deviation from the convex combination. For each unit of deviation in the supply, an equal share of the cost difference between the initial solution
and the first solution identified in the scheme is attributed. Note that this idea is similar to the goal programming approach, which has been used by Dudek and Stadtler
(2005).61
59
See Lemma 4.2 on p. 67.
See Chap. 6. A further difficulty experienced in computational tests is that the optimizer running
models GM-CS1B and GM-CS1S might declare barely feasible problems as infeasible. This can
be caused by a large downscaling of the matrix elements in case of huge differences between the
upper and lower bounds used for the linearization.
61
See Dudek and Stadtler (2005, p. 677). The difference to our approach is that their penalty costs
are not fixed, but updated by exponential smoothing in each iteration. We did not follow their
approach because it did not prove significantly better in computational tests, while augmenting the
complexity of the scheme.
60
114
4 New Coordination Schemes
Although in infrequent cases the deviations of new proposals from the starting
solution may be greater, the denominator of (4.93) can serve as an appropriate proxy
for the maximum of these deviations. Hence, we can set
X X
ubd D
f
i ni t
xtjt
xtj
C
j 2J D t 2T
when performing the linearizations of the objective functions of CS10i and
extensions.62
4.3.4 Modifications for Master Planning
In this subsection, we introduce further modifications of the scheme that are specific
for Master Planning and both aim at speeding up the convergence rate and decreasing the computational complexity of the underlying optimization models.
First, we consider deviations in terms of the cumulated supply quantities instead
of deviations in absolute supply quantities.63 This can be modeled by replacing constraints (4.60) and (4.61) by
t
X
XBj D1
t
X
D1
XBj t
X
XSj
D1
t
X
XSj
8j 2 J D ; t 2 T;
(4.94)
8j 2 J D ; t 2 T:
(4.95)
D1
The benefit of this modification is that penalizations of complimentary changes in
the supply pattern can be avoided. We illustrate the effect of constraints (4.94) by
an example.
Example 4.5 Assume that the starting proposal specifies a supply of 3[U] of an
item in periods 3 and 5 (see Fig. 4.15). Further assume that the supplier has tight
capacity in period 3 and that he prefers a different supply schedule with only 2[U] in
period 3 and 4[U] in period 5. Hence, in period 5, the cumulated supply quantities of
the schedule preferred by the supplier correspond to those of the starting proposal,
which does not hold for the absolute supply quantities. Since the buyer has to make
use of backlogging if she cannot fulfill the demand, the late delivery in period 5 is
superior to no delivery or an even later delivery in a subsequent period. Hence, the
change to 4[U] in period 5 is in line with the needs of the buyer given the shortage
in period 3, and the concentration on cumulated quantities grasps this issue better.
62
63
See also p. 101.
A similar idea has been used by the scheme of Dudek and Stadtler (2005, p. 677).
4.3 Application to Master Planning
115
New proposal
Starting proposal
6
Absolute supply
quantities
4
4
2
2
3
4
Cumulated
supply
quantities
t
5
3
4
5
t
Fig. 4.15 Absolute vs. cumulated supply quantities
Constraints (4.95) account for savings from earlier supply. Such savings may
result, e.g., if the supplier has a lack of capacity in a period and slack capacity in
an earlier one. Then he will prefer earlier supply in order to reduce his costs for
inventory holding.
With a direct application of (4.94) and (4.95), the savings for a solution previously found with both earlier and delayed supply compared to the starting solution
could be explained in models GM-CS10 B and GM-CS10 S (endogenously) by reductions of inventory holding and of shortages (i.e., lower amounts of backorders and
C
overtime needed). However, for a meaningful determination of variables Kjt
and
Kjt , contributing savings to earlier supply is less probable because the unit costs
for capacity shortages usually exceed the unit costs for inventory holding. To account for this, we apply constraints (4.95) only to proposals without any delay in
supply.
cum
We denote the resulting model formulations by GM-CS1cum
and
B and GM-CS1S
state them below.
X X d
(GM-CS1cum
KjtC C XT C
jt C Kjt C XT jt
B ) min CB C
j 2J D t 2T
(4.96)
s.t. .DC-B/; (4.67); (4.70); (4.71)
t
X X
X
e
C
xtj xtstj
KjtC
cs e csst 8e 2 DS
j 2J D t 2T
D1
X X
t
X
(4.97)
j 2J D t 2T
Kjt
e
xtst
j xtj
C
cse csst 8e 2 NDS (4.98)
D1
XT C
jt XT jt D
t
X
D1
XBj xtstj 8j 2 J D ; t 2 T:
(4.99)
116
4 New Coordination Schemes
X X C
C
d
K
(GM-CS1cum
)
min
C
C
C
XT
C
K
C
XT jt
S
jt
jt
jt
S
j 2J D t 2T
s.t. .DC-S/; (4.70); (4.71)
t
X
X X
e
C
KjtC
xtj xtstj csst cse 8e 2 NES
j 2J D t 2T
D1
X X
t
X
(4.100)
Kjt
j 2J D t 2T
st
C
xtj xtej csst cse 8e 2 ES (4.101)
D1
XT C
jt XT jt D
t
X
XSj xtstj 8j 2 J D ; t 2 T:
(4.102)
D1
Sets
NDS Set of proposals without delayed supply compared to the starting proposal
DS Set of proposals with delayed supply
NES Set of proposals without early supply
ES Set of proposals with early supply
Since NDS \DS D ¿ and NES \ES D ¿, we can state the linking constraints
C
that determine Kjt
and Kjt
as sets of separate constraints (4.97), (4.98) and (4.100),
(4.101), respectively, and thus reduce the number of nonlinear variables.
Moreover, also model GM-CS20 B is adapted to GM-CS2cum
B :
d
(GM-CS2cum
B ) min CB C
eN
X
X X
e cs e C
j 2J D t 2T
eD1
s.t.
t
X
XBj C XT jt D
D1
. C cp/ XT C
C
XT
jt
jt
eN
t X
X
e
e xtj
C XT C
jt
8j 2 J D ; t 2 T
D1 eD1
.DC-B/; (4.62)–(4.63), (4.71);
with the unit penalty costs cp defined by
cp D
cs i ni t cs f
C :
P
P
Pt i ni t
f
xt
xt
D
D1
j 2J
t 2T
j
j
Again, the denominator of this term is used for determining ubd .64
Second, in order to generate a broader range of proposals, we introduce two
further modifications, which are successively skipped if no improvement can be
64
See p. 101.
4.3 Application to Master Planning
117
identified for any starting proposal. The first of them is to replace constraints (4.94)
and (4.95) by
t
XX
XBj t 2T D1
t
XX
XBj t
XX
XSj
8j 2 J D ;
XSj
8j 2 J D :
t 2T D1
t
XX
t 2T D1
t 2T D1
The effect of this modification is an aggregation of the deviations in the supply
pattern with respect to the time periods. Then, no attention is paid to the specific
periods in which these deviations occur.
Example 4.6 Consider a two-party supply chain with one item supplied. Assume
that a single solution has been identified previously, with cost savings of 6[MU] for
the supplier and a reduction in the cumulated supply of 3[U] in one period and a
reduction of 4[U] in another. Then, a unit penalty cost of Kj D 6=7[MU/U] for
any deviation in the supply of this item will be charged in the supplier’s model for
proposal generation.
Of course, this only provides a rough picture of the potential cost changes caused by
deviations in the supply quantities. However, especially if the cost sensitivities regarding such deviations differ for particular items substantially from the sensitivities
for other items, such an aggregation turns out to be effective.65 Then this modification allows concentrating on the principal question, which product should be
delayed, and not in which period the delay should take place, which is of minor
importance there.
To integrate this new formulation of the central resource constraints into the modcum
els run within the scheme, we modify GM-CS1cum
as follows:
B and GM-CS1S
agg
(GM-CS1B ) min CBd C
X X agg;C
agg;
C
Kj
C Kj
C XT jt C XT jt
j 2J D t 2T
s.t. .DC-B/; (4.67); (4.71); (4.99)
t
X X agg;C X
e
C
xtj xtstj cse csst 8e 2 DS (4.103)
Kj
j 2J D t 2T
X X
j 2J D t 2T
D1
agg;
Kj
t
X
C
cse csst 8e 2 NDS (4.104)
D1
agg;C
agg;
Kj
; Kj
0
65
xtstj xtej
See the sensitivity analyses in Sect. 6.1.3.
8j 2 J D :
(4.105)
118
4 New Coordination Schemes
X X
agg
(GM-CS1S ) min CSd C
agg;C
Kj
agg;
C Kj
C
X Tjt C X TjtC
j 2J D t 2T
s.t. .DC-S/; (4.71); (4.102); (4.105)
t
X X agg;C X
e
st C
Kj
cs st cs e 8e 2 NES
xtj xtj
j 2J D t 2T
D1
X X
t
X
(4.106)
agg;
Kj
j 2J D t 2T
st
e
xtj
xtj
C
cs st cs e 8e 2 ES:
D1
(4.107)
Variables
Kjagg;C Endogenously determined unit penalty costs for shifts of the supply of
item j to later periods compared to the starting supply pattern
Kjagg; Unit penalty costs for shifts of the supply of item j to earlier periods
The penalty costs are now determined by constraints (4.103), (4.104), (4.106),
and (4.107). This modification reduces the number of the linking variables in large
part and, hence, the number of SOS2 needed for the piecewise linear approximation.
Since the modified models are often able to find new proposals, the time for running
the scheme can be reduced significantly.66
As a second modification, additional penalty costs for deviations from the startagg-P
agg-P
and GM-CS1S
for the
ing supply pattern are included. We write GM-CS1B
agg
agg
resulting models that differ from GM-CS1B and GM-CS1S merely by the new
objective functions
min CBd C
X
j 2J D
min CSd C
X
j 2J D
C Kjagg;C
X
t 2T
X TjtC C
!
X
agg;
C Kj
C cp
X Tjt ;
t 2T
!
X
X
agg;C
agg;
C
C Kj
X Tjt C C Kj
C cp
X Tjt :
t 2T
t 2T
The modifications introduced above form building blocks that can be freely comagg-P
bined with each other. As a promising sequence, first GM-CS1i
is tried for
proposal generation, and if no new solution can be found for any starting proposagg
als, GM-CS1i and GM-CS1cum
are run instead.67 In the subsequent procedure, the
i
proposal generation is summarized exemplarily for the buyer.
66
See also the results of our computational study in Sects. 6.1 and 6.3.
Note that the use of all levels of modifications proposed is not always advisable. E.g., for GM
agg-P
investigated in our computational tests (Sect. 6.1), the omittance of GM-CS1i
turns out to be
superior.
67
4.3 Application to Master Planning
119
Procedure proposalGenerationBuyer(iteration, ˘ E )
input : iteration, ˘ E
output: new
new
tryModelClass(GM-CS2cum
,˘ E )
i
new
E
if 2 ˘ and iteration > 1 then
agg-P
new
tryModelClass(GM-CS1i
,˘ E )
/* add. penalty costs */
if new 2 ˘ E then
agg
new
tryModelClass(GM-CS1i ,˘ E )
/* aggregation */
if new 2 ˘ E then
new
tryModelClass(GM-CS1cum
,˘ E )
i
/* only cumulation */
The output of this algorithm is a new proposal that is not element of ˘ E , the
agg-P
is not applied in the first
set of proposals already found.68 Note that GM-CS1S
iteration since the penalty costs cannot be determined yet in this stage of the coordination process. The successive solution of the single models for different starting
solutions x st is subsumed by the procedure tryModelClass stated below.
Procedure tryModelClass(modelClass, ˘ E )
input : modelClass, ˘ E
output: new
˘ ST
˘E
repeat
x st
random()
new solve(modelClass, x st )
˘ ST
˘ ST nx st
until new … ˘ E or ˘ ST D fg
The procedure solve( ) renders the proposals generated by means of a model
st
of the type modelClass (e.g., GM-CS1cum
or x l as a further input (if
i ) with x
new
applicable). This proposal is assigned to the vector . Finally, we state the algorithm for the whole scheme (Algorithm 7).
We assume here that the initial solution is determined by upstream planning; for
other choices of the initial solution, the scheme can be modified straightforwardly.
The procedure ProposalGenerationSupplier( ) can be formulated analogously to ProposalGenerationBuyer( ), keeping in mind that the supplier
does not run CS2cum
since he is not informed about the buyer’s cost changes. The
i
parameter nIterations, which can be exogenously specified by decision makers
using the scheme, indicates the number of iterations for which the scheme is run.
68
Since in a two-party setting only one cost-reporting party exists, we can omit the index i here
(as for ˘ E ).
120
4 New Coordination Schemes
Algorithm 7: CoordinationSchemeModified
input : nIterations
solve(GM-B)
new
˘E
new
for it D 1 to nIterations do
/* for each iteration */
solve(GM-CS-EVALS , new )
new
proposalGenerationSupplier(it,˘ E )
˘E
new [ ˘ E
solve(GM-CS-EVALB , new )
new
proposalGenerationBuyer(it,˘ E )
E
˘
new [ ˘ E
When implementing the scheme, a further enhancement regarding computing
time is applied, which for ease of exposure has not been included in the algorithms
stated above: Combinations of modifications and starting solutions that have not
yielded new proposals are discarded in future iterations. Since the unit penalty costs
imputed by the models do not decrease in the course of the scheme, this refinement
does not cut off potentially optimal solutions.
Note that by straightforward adaptations, this scheme and further modifications of it (see the next subsection) can be employed for a one-shot disclosure
cost information at the end of the scheme, which is required by the mechanism
presented in Sect. 5.3. Then, parties skip constraints (4.67) and rely on their own
cost changes for determining the penalty costs (K; cp) in models GM-CS1i and
extensions. Of course, models GM-CS2i are not run by any party then. Instead, like
in the generic scheme, GM-MP is run once at the end of the coordination process.
4.4 Customizations
4.4.1 Master Planning with Lot-Sizing
In principle, the scheme outlined in Sect. 4.3 can also be applied for Master Planning
models that include lot-sizing decisions. However, the performance of the scheme
can additionally be improved by a further optimization model CS-LOT that is able
to capture specific characteristics of lot-sizing models. CS-LOT synthesizes ideas
of the scheme for uncapacitated lot-sizing with those of the modified version of the
generic scheme outline in Sect. 4.3. The mathematical formulation of CS-LOT is69
69
Note that we use the MLCLSP as the base model here since CS-LOT only applies for lot-sizing
models and, thus, not for GM.
4.4 Customizations
121
X X
X Tjt C X TjtC
min CBd C .cp C /
(4.108)
j 2J D t 2T
(CS-LOT) s.t. .DC-B/; (2.14), (2.15), (4.29)–(4.31), (4.67), (4.71), (4.99):
In CS-LOT, the objective function and constraints have been adapted from
CS1cum
and MLULSP-CS. This model identifies proposals which involve a smaller
B
number of orders for the supplied items and thereby anticipates potential increases
in the supplier’s costs for earlier supply. The latter issue is modeled by the penalty
costs cp in (4.108) and helps to make ideas of the scheme for uncapacitated lotsizing applicable for capacitated problems. Without these penalty costs, the resulting
solution quality would be affected since the new proposal generated by the buyer
would completely ignore the effects of early orders on the supplier’s costs.
CS-LOT can be used as an add-on to the scheme outlined in the previous section,
i.e., CS-LOT is tried for the identification of a new proposal before applying GMagg-P
CS1B , and so on. CS-LOT should only be run if a real benefit can be expected
thereof. To decide about this, we apply a criterion similar to that for the inclusion of
cum
constraints (4.98) and (4.100) into GM-CS1cum
B and GM-CS1S , respectively. These
models aim to identify proposals with reduced backlog and overtime costs first,
and only after this has been achieved, to consider potential reductions in inventory
holding costs. Analogously to that, we run model CS-LOT if and only if a proposal
with the envisaged reduction of backlog and overtime costs has been identified. As
an indicator for this, we take the existence of two proposals (e1 and e2) with the
following property: One of these proposals (e1) shows earlier supply compared to
the other (e2) and yields a cost increase for the buyer, which is smaller than or equal
to the holding costs for the supplied items due to the early supply. I.e., we require
X
e1
xtjt
t D1
X
e2
xtjt
t D1
for any j 2 J D , 2 f1; : : : ; jT jg and
0 < cb e1 cb e2 X
j 2J D
hj
X
e1
e2
xtjt
xtjt
C
:
t 2T
Then it can be inferred that this cost increase is due to early supply and that there
is a potential benefit from reducing the buyer’s order frequency. This conclusion is
supported by our computational tests, where this rule proved to be effective.
In the procedure proposalGenerationBuyerLotsizing( ), we summarize the proposal generation for this extension. Along the reasoning discussed
above, the function potentialLotsizing() determines whether to employ
CS-LOT.
122
4 New Coordination Schemes
P
up
Procedure proposalGenerationBuyerLotsizing( j 2J ysj , ˘ E )
output: P
new
up
input : j 2J ysj , ˘ E
if potentialLotsizing() then
˘ ST
˘ E repeat
w
1
x st
random()
˘ ST
˘ ST nx st
repeat
new solve(CS-LOT,w,x st )
w
wC1
ˇ ˇ
P
up
ˇ Dˇ
until new … ˘ E or
j 2J ysj w < J
ST
new
E
until ˘ D fg or …˘
/* lot-sizing */
if new 2 ˘ E then
/* additional penalty costs */
agg-P
tryModelClass(MLCLSP-CS1B ,˘ E )
if new 2 ˘ E then
agg
tryModelClass(MLCLSP-CS1B ,˘ E )
/* aggregation */
if new 2 ˘ E then
E
tryModelClass(MLCLSP-CS1cum
B ,˘ )
/* only cumulation */
4.4.2 Voluntary Compliance
If the compliance of the supplier to the buyer’s proposals is voluntary, a direct implementation of the buyer’s proposals is not assured. Then, three major changes
have to be applied to the scheme. First, the buyer’s proposals are not directly evaluated in terms of a systemwide improvement; analogously to upstream planning with
voluntary compliance, the supplier determines the extent of his fulfillment of the
buyer’s proposal, thereby taking into account potential penalty costs for backorders
and lost sales, and communicates the resulting supply quantities to the buyer. The
supplier’s model for evaluating the buyer’s proposals corresponds to that used for
upstream planning with voluntary compliance.70 I.e., the supplier solves GM-S with
the modified objective function (2.52) and the additional set of constraints (2.53).
up
We abbreviate this model by GMS in the following. Define ˘ up as the set of proposals identified by this model and ˘ S , ˘ B as the proposals generated by supplier
and buyer, respectively. The resulting information flow is depicted in Fig. 4.16 exemplarily for the supplier as the cost-reporting party.
The second change is a consequence of the first: In each iteration, two different
proposals are determined: One that is communicated by the buyer to the supplier,
and a further that indicates the extent to which the supplier is willing to fulfill the
buyer’s proposal.
70
See p. 30.
4.4 Customizations
123
Fig. 4.16 Information exchange with voluntary compliance of the supplier
Table 4.4 Origin of proposals used as input data for CS1S ; CS1B and
CS2S ; CS2B (and modifications) with voluntary compliance
Cost-reporting party
S
B
Starting proposals for CS1B
Starting proposals for CS1S
Proposals for Kjt -penalty costs in CS1S ; CS1B
Proposals for recombination in CS2S ; CS2B
˘ S [ ˘ up
˘ S [ ˘ up
˘ S [ ˘ up
˘ S [ ˘ up
˘ S [ ˘ up
˘S [ ˘B
all
˘S [ ˘B
Third, not all proposals previously generated are used as input for the models
run within the scheme. The determination of which of them are used depends on
the choice of the cost-reporting party (see Table 4.4). A careful distinction has to be
made about the starting proposals for models CS1i and their modifications stated in
the previous sections. With the supplier as the reporting party, cost changes are only
known for the proposals generated by the supplier since the buyer’s proposals are not
directly implemented. Then, only proposals among ˘ S [ ˘ up can be considered.
If the buyer reports the cost changes, infeasibility of proposals does not matter since
parties rely on the buyer’s cost changes, which are determined for all proposals.
Here, all previous proposals could be chosen in principle. However, it proves favorable to concentrate on the proposals among ˘ S [˘ B as starting proposals for CS1S
and the extensions of this model. In contrast to ˘ up , these proposals incorporate
to some degree the requirements of the buyer (either due to their generation by the
buyer or due to the penalty costs in CS1S ). This increases the probability of the identification of a systemwide improvement. Similar arguments hold for the choice of
the starting proposals for CS1B and for the proposals for the recombination in CS2S .
We summarize the single steps of the scheme by Algorithm 9.
4.4.3 Lost Sales
In this section, we show how to adapt the scheme to the simultaneous presence of
backorders and lost sales. Then, apart from the temporal distribution of the supply
quantities, their total amounts may vary, too. This extension enlarges the scope of
the scheme considerably. Since differences in the cumulated supply are allowed
124
4 New Coordination Schemes
Algorithm 9: CoordinationSchemeVoluntaryCompliance
input : nIterations
˘ B D fg,˘ S D fg, ˘ UP D fg
/* initialization */
new
solve(GM-B)
˘B
new C ˘ B
for it D 1 to nIterations do
/* iteration */
up
up
solve(GMS , new )
˘ UP
up C ˘ UP
new
ProposalGenerationSupplier(it,˘ S [ ˘ B [ ˘ up )
˘S
new C ˘ S
new
ProposalGenerationBuyer(it,˘ S [ ˘ B [ ˘ up )
B
˘
new C ˘ B
then, the scheme can also be applied for the coordination of parties maximizing
their contribution margins, which is required by most of the real-world instances
investigated in Chap. 6.5.
The extension of the models presented above to lost sales can be modeled by an
additional central resource, the deviations in the supply cumulated within the planning interval. We exemplarily show how to adapt GM-CS1cum
to GM-CS1cum,ls
, their
i
i
agg
versions including lost sales. An adaptation of the other models like GM-CS1i can
be carried out analogously.
(GM-CS1cum,ls
) min CBd C
B
X X (4.109)
C
ls ls
Kjt
C X TjtC C Kjt
C X Tjt C Kjt
Djt
j 2J D t 2T
s.t. .DC-B/; (4.67); (4.70); (4.71); (4.98)
!
t
X X
X
C
C
e
st
ls
Kjt
xtj
xtj
d lb ej C Kjt
d lb ejt j 2J D t 2T
D1
8e 2 DS
cs cs
t t
X
X
C
st
ls
X Tj
D
XBj xtj
C Djt
X Tj
e
D1
st
(4.110)
D1
8j 2 J D ; t 2 T
C
ls
ls
Kjt
; Kjt
Kjt
8j 2 J D ; t 2 T
Kjt
(4.111)
(4.112)
ls
0
Kjt
(4.113)
8j 2 J D ; t 2 T :
(GM-CS1cum,ls
) min CSd C
(4.114)
S
X X C
ls ls
Kjt
C X TjtC C Kjt
C X Tjt C Kjt
Djt
j 2J D t 2T
s.t .DC-S/; (4.70); (4.71); (4.100); (4.112); (4.113)
4.4 Customizations
125
X X
j 2J D t 2T
Kjt
t
X
st
e
e C
ls
e
xtj
xtj
d lsj
C Kjt
d lsjt
!
D1
cs st cs e 8e 2 ES
t t
X
X
C
st
ls
X Tj
D
XSj xtj
Djt
X Tj
D1
(4.115)
D1
8j 2 J D ; t 2 T:
(4.116)
Data
e
d lbjt
Deviation of proposal e that is due to lost sales and relevant for the buyer
e
d lsjt Deviation of proposal e due to lost sales for the supplier
Variables
ls
Difference in the supply quantity of item j in period t due to lost sales
Djt
ls
Kjt Penalty costs for lost sales of item j in period t
By the introduction of constraints (4.111) and (4.116), models GM-CS1cum,ls
and
B
allow
for
cumulated
deviations
in
the
supply
quantities.
In
order
to
GM-CS1cum,ls
S
determinate the penalty costs for these deviations, constraints (4.97) and (4.101)
have been modified to (4.110) and (4.115). Thereby, the decision whether to impute
the cost difference between xist and a xie to lost sales is determined endogenously.
Of course, the models will preferably declare shortages occurring in early periods
as lost sales since the backorder costs can be minimized that way. Otherwise, if
shortages occurring in later periods were declared as lost sales, more backorders in
earlier periods would incur. Assuming that the penalty costs for lost sales exceed
the penalties for backorders, we have introduced the redundant constraints (4.112)
e
in order to sharpen the linearization applied.71 Moreover, new parameters d lbjt
,
e
d lsjt
have been used, which denote the deviation in lost sales between xist and
presume that lost sales occur as early as
xie . As noted above, models GM-CS1cum,ls
i
e
possible. In Algorithm 10, we exemplarily illustrate the determination of d lbjt
for
e
a given item j and a given proposal e. The determination of d lsjt is analogous,
e
st
st
e
xjt
is replaced by xjt
xjt
.
with the difference that xjt
71
Further modifications that help to somewhat sharpen the linearization are the introduction of adp;C
p;
ditional variables for the costs for lost sales (analogous to Cjt ,Cjt ) and of constraints assuring
that, in case of shortages for a single item, the penalty costs for lost sales are directly imputed to
this item. These modifications have been considered in our computational study for the real-world
planning problems in Sect. 6.5, but not in the models stated above for ease of exposition.
126
4 New Coordination Schemes
e
Algorithm 10: DeterminationOfd lbjt
e
output: d lbjt
P
e
st
diff
t2T xjt xjt
t
1
while diff > 0 do
n
e
st
earlySupply
max 0; xjt
xjt
e
max fdiff; earlySupplyg
d lbjt
diff
diff earlySupply
t
t C1
o
4.4.4 Multiple Suppliers
The scheme proposed can be straightforwardly adapted to multiple suppliers. For
the suppliers’ optimization models, no changes are necessary. Regarding the buyer’s
models, we exemplarily provide the modification of GM-CS1cum
B .
(GM-CS1-Mcum
B ) min (4.96)
s.t. .DC-B/; (4.67); (4.70); (4.71); (4.99)
t C
X X
X
e;i
st;i
C
xtj
Kjt
xtj
csie csist
j 2J D t 2T
D1
8e 2 DS; i 2 PS
(4.117)
t
X X
X
C
st;i
e;i
xtj
Kjt
xtj
csie csist
j 2J D t 2T
D1
8e 2 NDS; i 2 PS :
(4.118)
Index
i Suppliers, i 2 PS , with PS as the set of suppliers
Data
csie Costs of proposal e for supplier i
e;i
xtjt
Supply quantity of item j in period t delivered by supplier i and
specified by proposal e
The necessary modifications are limited to the constraints (4.117) and (4.118),
where the knowledge about the cost changes of the single suppliers is used to deterC
mine the penalty costs Kjt
, Kjt
more precisely.
4.4 Customizations
127
The adaptation of GM-CS2cum
B to multiple suppliers is:
d
(GM-CS2-Mcum
B ) min CB C
eN
XX
i e csie C
(4.119)
i 2P eD1
X X
i
C cpB
X TjtC C X Tjt
j 2J D t 2T
s.t. .DC-B/; (4.71)
t
X
XBj C X Tjt D
eN
t XX
X
e;i
i e xtj
C X TjtC
D1 i 2P eD1
D1
D
8j 2 J ; t 2 T
eN
X
i e D 1 8i 2 PS
eD1
i e 0
8i 2 PS ; e D 1; : : : ; e:
N
Variables
i e Variable indicating for supplier i the share of previous proposals e used for
the recombination
Data
i
Penalty costs for supplier i
cpB
Here, the only change is that proposals recombined originate from several suppliers. This implies that several suppliers evaluate and generate proposals and that
i
the penalty costs cpB
can be determined more precisely for each single supplier.
Apart from that, no further adaptations of the scheme are needed.
Chapter 5
New Coordination Mechanisms
Apart from the identification of improved solutions within an acceptable number
of iterations, a further requirement for practicable coordination schemes is their
applicability by rational, self-interested parties, which implies that the schemes can
be embedded into suitable coordination mechanisms.
In the following sections, we outline three contractual frameworks that form
building blocks for the resulting mechanisms in combination with the schemes
proposed in Chap. 4. All frameworks rely on compensation payments among parties
as incentives for the implementation of coordinated solutions. First, these payments are necessary for ensuring individual rationality in the mechanisms. Often,
the implementation of coordinated solutions involves cost increases for at least one
party. Such increases necessarily occur if a party acts as the leader and unilaterally
determines the allocation of the central resources in the default solution.1 Unless
several optimal solutions exist for the leader’s problem, the implementation of a
coordinated proposal will force the leader to deviate from his individually optimal
solution, and, hence, to implement a solution with increased costs. Second, such
payments are a straightforward way to align parties’ incentives with the actions
required by the schemes.2
Apart from establishing individual rationality, these compensation payments
specify the sharing of the surplus from coordination, i.e., the difference between the
systemwide costs of the default solution and the systemwide costs for implementing
a coordinated proposal. At the same time, the rule for surplus sharing is the one of
1
An example for this is upstream planning with forced compliance (see p. 30), where the buyer
acts as the leader.
2
A renunciation of such payments – as sporadically advocated in the literature, e.g., by Gjerdrum
et al. (2002, p. 592) – might only be an option if no party incurs any losses from implementing
a coordinated proposal. This may occur to a limited extent in voluntary compliance settings, as
indicated by our computational tests of Sect. 6.4. Even then, however, it seems difficult to ensure
that parties will actually follow the rules of the underlying scheme. The main problem is that –
depending on the rule for determining the proposal implemented – parties will have incentives not
to accept proposals with small own savings in order to increase the probability of the acceptance of
proposals that are more lucrative for these parties (though less advantageous for the other parties).
If all parties pursued this strategy, the overall coordination performance would suffer strongly since
a great number of favorable proposals would be declined then.
M. Albrecht, Supply Chain Coordination Mechanisms: New Approaches
for Collaborative Planning, Lecture Notes in Economics and Mathematical Systems 628,
c Springer-Verlag Berlin Heidelberg 2010
DOI 10.1007/978-3-642-02833-5 5, 129
130
5 New Coordination Mechanisms
the major differences between the mechanisms proposed. In the first mechanism,
the surplus sharing is solely determined by the party which receives information
about the others’ cost changes during the execution of the scheme. In the second,
the surplus is shared by a previously fixed lump-sum payment, and in the third, by
the outcome of a sealed bid double auction. Our descriptions and analyses of these
mechanisms are structured as follows: At first, we consider a two-party setting and
a one-shot application of the mechanisms. Built on that, we discuss extensions to
more than two parties and to a repeated application, which seems most relevant for
practice since fixed costs frequently incur for setting up coordination. All mechanisms have specific advantages, which are summarized and discussed in Sect. 5.4.
In Sect. 5.5, we outline how to adapt the mechanisms for rolling schedules.
5.1 Surplus Sharing Determined by the Informed Party
In this and the next section, we assume – according to the generic scheme outlined in
Chap. 4 – a party that recombines the proposals of the others [party I , abbreviated by
IP (informed party) in the following] and a set of cost-reporting parties [abbreviated
by RPi ]. Analogously to Sect. 2.3, we assume different sexes for the IP and the RPi ;
we consider the IP as female and the RPi as male.
As mentioned above, we initially focus on a two-party case with one RP3 and one
IP. In the section at hand, we discuss the most straightforward possibility for surplus
sharing, which is to let the IP determine the allocation of the surplus.4 If we additionally postulate that the RP reports his cost changes truthfully, the generic scheme
can be applied directly. Of course, the assumption of truthful information exchange
is questionable, particularly from the point of view of (neoclassical) economic theory. We will discuss this issue in light of experimental investigations below.
The structure of this mechanism is simple (see also Fig. 5.1): Parties exchange
proposals as specified by the scheme. There, the cost reporting by the RP is the only
action that influences parties’ shares of the surplus. After the termination of the
scheme, the best solution found is determined by the IP and implemented. Moreover,
Fig. 5.1 Mechanism with
surplus sharing determined
by the IP
3
4
We omit the index i when we consider a single cost-reporting party.
Note that this possibility has been mentioned by Dudek and Stadtler (2007, p. 478).
5.1 Surplus Sharing Determined by the Informed Party
131
the IP specifies compensation payments, which, apart from ensuring individual
rationality of parties, allocate shares of the surplus to the parties, e.g., using a rule
like 80:20 or 50:50. We illustrate this procedure by an example.
Example 5.1 Using the scheme, two parties have determined an improved supply
schedule with a cost increase of 7[MU] for the IP and a decrease of 11[MU] for the
RP. Resulting is a surplus of 4[MU]. Its allocation is determined by the IP using the
50:50 rule. With a compensation payment of 11 2 D 9[MU] by the RP to the IP,
both parties obtain gains from coordination of 2[MU] (see Table 5.1, row “Basic”).
The main obstacle for this approach is that parties might have incentives for distorted
statements of the cost changes. For the RP, stating higher costs than actually incurred
might be advantageous. Then the RP might receive a greater compensation payment
or pay less for the implementation of a coordinated solution (see the example in
Table 5.1, “Case A,” where the RP exaggerates his costs by 3[MU]). A potential
drawback of such behavior, however, is the risk that the IP will prefer the default
solution if the costs of the IP are greater than anticipated by the RP and, hence, the
cost exaggeration exceeds the potential surplus (e.g., Table 5.1, “Case B”). Then,
none of the parties will receive any benefit from coordination.
Moreover, also the IP might exaggerate her costs and assign a smaller share of
the surplus to the RP (e.g., Table 5.1, “Case C”) or simply use another allocation
rule than the announced. This is even more appealing than an exaggeration of cost
changes by the RP since the IP does not incur any risk of breakdown of coordination.
The IP can use her knowledge to exaggerate her costs to such an amount that she
can usurp the whole surplus except for an arbitrarily small share allocated to the RP.
At first glance, especially the second objection, the potential opportunistic behavior by the IP, seems to affect the applicability of this mechanism seriously. If the
share allocated to the RP becomes arbitrarily small, the RP might prefer not to
participate in future coordination processes or to substantially exaggerate his cost
changes in order to obtain a significant share of the surplus.
However, there are actually some reasons for the IP to assign a substantial share
of the surplus to the RP. Apart from inciting the RP for participating in future coordination processes, the IP might concede some share of the surplus for reasons of
fairness. It has been shown in numerous behavioral investigations that real subjects
prefer solutions that are fair to a certain degree. Examples are dictatorial bargaining
Table 5.1 Effects of different cost reporting strategies (all data in [MU])
Cost changes
Payment
Gains
Gains from
stated
from
exaggeration
RP
IP
RP to IP
RP
IP
RP
IP
Basic
11
7
9
2
2
–
–
Case A
8
7
7.5
3.5
0.5
1.5
–
Case B
8
10
–
–
–
2
–
Case C
11
10.9
10.95
0.05
3.95
–
1.95
132
5 New Coordination Mechanisms
Fig. 5.2 Dictatorial bargaining game and trust game
games and trust games. In the dictatorial bargaining game,5 one party, the dictator,
decides about the allocation of a monetary amount between itself and another, less
powerful party (see Fig. 5.2). The experimental outcome of this game sharply contradicts the self-interest hypothesis of neoclassical economic theory. For different
experimental conditions (e.g., framing, demographic parameters, culture),6 the share
allocated to the less powerful party turned out to be substantial, and about 20% of
the total monetary amount on average.7 This and related insights gave even rise to
the development of new economic theories such as inequality aversion.8
Even more similar to the mechanism proposed is the trust game.9 There, the dictatorial bargaining game is extended by a potential investment of the less powerful
party. A multiple of this investment (e.g., a triple) is allocated by the dictator to both
parties, and there is no guarantee that the investment is paid back to the investor.
In experiments, and again in contrast to the self-interested behavior postulated by
theory, a large part of investors choose nonzero investments, which were paid back
on average. Corresponding experiments have also been conducted for repeated trust
games.10
For transferring the trust game to our mechanism, the investment can be interpreted as the commitment of the RP (here: the investor) not to exaggerate his cost
changes. Compared to non-truthful reporting, this commitment increases the total
5
See Kahneman et al. (1986, p. 728).
See, e.g., Kagel and Roth (1995, p. 256) and Camerer (2003, p. 59).
7
See Camerer (2003, p. 113).
8
This theory has been developed by Fehr and Schmidt (1999, p. 817) and been applied to supply
chain coordination by Cui et al. (2007, p. 1303). For more information about behavioral theories
explaining these phenomena, see Camerer (2003, p. 101).
9
This game has initially been proposed Berg et al. (1995, p. 122) and extensively investigated in the
literature afterwards. For variants of this game recently examined, see, e.g., Bracht and Feltovich
(2008, p. 39) and Falk (2007, p. 1501).
10
Repeated trust games have first been investigated by Camerer and Weigelt (1988, p. 1) and recently by Engle-Warnick and Slonim (2004, p. 553) and Engle-Warnick and Slonim (2006, p. 603).
6
5.2 Surplus Sharing Determined by Lump-Sum Payments
133
surplus, which is to be allocated by the IP.11 However, the behavior of real subjects
in this setting is difficult to foresee since thorough empirical analyses for trust games
in a supply chain environment have not been undertaken yet.
Finally, note that the mechanism presented can be straightforwardly transferred
to settings with three or more parties. Then several RPi report their cost changes,
and the IP allocates to each of them shares of the surplus as a bonuses for their
participation.
5.2 Surplus Sharing Determined by Lump-Sum Payments
Although there are some experimental insights in favor, the mechanism presented
in the preceding section shows basic deficits when analyzed from the perspective
of game theory. Again consider a two-party setting first. Using that mechanism, a
purely rational, self-interested IP will try to keep the complete surplus. Anticipating
this, a rational, self-interested RP will considerably exaggerate his cost changes
(e.g., by a constant markup), with the consequence that the rules of the scheme will
not be kept. This has two negative implications: First, the best solution identified
by the scheme might not be recognized as an improvement of the default solution.
Second, the quality of the coordination process might suffer due to the internal logic
of the scheme. We illustrate the latter issue by an example.
Example 5.2 Assume that decentralized problems can be modeled as LP problems.
Further assume that the proposals generated have two dimensions, d1 and d2 (e.g.,
two items supplied and one time period). Consider the starting proposal S, proposals
A and B, which have been identified in previous iterations of the scheme, and C as a
potential new proposal (see Fig. 5.3). Proposals A, B, and C constitute systemwide
improvements compared to S. Let A, B, and C go along with cost reductions for
the IP. The real changes in the systemwide costs due to the implementation of these
Fig. 5.3 Suboptimal
outcome of the scheme due to
cost exaggeration
11
As an illustration for the effects of distorted reporting of cost changes, see Example 5.2 in the
next section.
134
Table 5.2 Cost data for
proposals A, B, and C
5 New Coordination Mechanisms
Real overall cost changes
Markup by the RP
Overall cost changes perceived by the IP
A
B
C
10
12
2
8
12
4
13
12
–
Fig. 5.4 Mechanism with lump-sum payment
proposals are given in the first row of Table 5.2. Let the cost changes reported by
the RP comprise additional markups (see Table 5.2). Since these markups exceed
the real cost changes, the IP erroneously considers S superior to A and B. When the
IP runs CS1i or a modification of this model, the endogenously determined penalty
costs for C would exceed her savings for this proposal. These penalty costs increase
linearly with a movement from S into the direction of C, whereas the costs of the IP
decrease linearly or less due to the convexity of decision problems. Hence, C could
only be found with a different starting proposal (e.g., A). It is remarkable that C will
not be found starting from S even though the systemwide surplus of C exceeds the
markup chosen by the RP.
In the following, we present a mechanism for which truth-telling is a weakly
dominant strategy for the RP, such that the difficulty illustrated by Example 5.2 can
be avoided. In this mechanism, the RP determines the amount of a lump-sum payment in advance (see also Fig. 5.4). This lump sum is paid by the IP to the RP in
addition to the (positive or negative) payment for reestablishing parties’ costs of
the default solution, provided that a solution resulting from coordination is implemented. The IP can determine whether to accept a coordinated solution and to pay
the lump sum to the RP or to keep the default solution. With an exaggeration of his
costs, the RP runs the risk of losing the lump sum, which he obtains if coordination
has been successful; if the surplus is lower than the lump sum, the IP will prefer the
default solution, of course.
In Fig. 5.5, the actions specified by the mechanism are displayed on a time axis.
In the first step of the mechanism (t0), the amount of the lump-sum payment is
determined by the RP. The scheme is applied in the second step (t1), and last (t2), the
IP decides whether to implement a proposal generated by the scheme. We illustrate
the mechanism by an example.
5.2 Surplus Sharing Determined by Lump-Sum Payments
135
Fig. 5.5 Time bar of the
mechanism
Example 5.3 Assume that the best proposal found by the scheme yields a cost
change of 6[MU] for the RP and a cost change of 3[MU] for the IP compared to
the default solution. Let the previously fixed lump sum be 2[MU]. If the RP reports
his cost changes correctly, the IP will opt for the implementation of the coordinated
solution. Then 6 2 D 4[MU] is paid by the RP to the IP. Hence, shares of 2[MU]
and 1[MU] are obtained by the RP and the IP, respectively. If the RP exaggerated
his costs too much, e.g., by stating a cost reduction of only 4[MU], the IP would
reject the coordinated solution since the lump-sum payment exceeds the systemwide
cost savings perceived by her.
Hence, the risk of losing the lump-sum payment limits the cost exaggeration
by the RP. In the following, we will analyze the properties of this mechanism for
different sets of assumptions. Below, we introduce our basic set.
Assumption 5.1 The RP has prior, incomplete knowledge about the surplus from
coordination.
Such knowledge can be derived by the general experience of decision makers or
can be acquired by learning if coordination is undertaken repeatedly (e.g., once a
month).12 Denote this knowledge by the density distribution f .S / over the interval
Œa; b for S , the random variable denoting the surplus from coordination.
Assumption 5.2 In the second step of the mechanism, parties can maximize their
expected surpluses by implementing the actions as specified by the scheme.
This assumption requires that the scheme used in the second step of the mechanism is the most efficient among all known schemes based on the exchange of
primal information. This holds, up to our knowledge, for the schemes specified in
Chap. 4 when Master Planning is to be coordinated. Note that we do not recommend
schemes with an exchange of dual information for use with this mechanism. Dual
information communicated to the RP may entitle him to estimate the cost changes
of the IP and to increase his gains by a distorted reporting of his cost changes, such
that strategy proofness would get violated.
Assumption 5.3 The information exchange required by the scheme does not violate
individual rationality of parties.
Within supply chain management, an (iterative) exchange of order information (or
forecasts) is common practice, e.g., in CPFR.13 We argue that an exchange of a
12
A discussion of potential learning effects with a repeated application of the mechanism is
provided at the end of this section.
13
See e.g., Aviv (2001, p. 1327).
136
5 New Coordination Mechanisms
modest number of such orders and their associated, aggregated cost effects does
not prejudice the party disclosing this information, as it might do an exchange of
production capacities.
We model the mechanism as a dynamic game with the commitment of parties
to the decisions taken in previous steps of the game. Thereby, the last two steps
correspond to a Stackelberg game with the RP as the leader.14 First, consider the
parties’ decision problems in the second step (t1). Obviously, the IP can maximize
her gains by following the rules of the scheme since the probability for improvements is highest then. The determination of the best strategy of the RP requires
a somewhat deeper analysis. The gains of the RP comprise the lump sum and his
markup for the proposal implemented. Due to the information status of the RP,15 we
do not consider different markups in single iterations. We denote the lump sum by
L and the markup by l. Hence, the decision problem of the RP at t1 is to maximize
gRP .l/, the gains of the RP from coordination subject to l:
Z b
max g RP .l/ D max
l
l
.L C l/ f .S / dS;
LClCr.l/
with r .l/ as the function that maps the expected reduction of S with l.
By Assumption 5.2, r .l/ 0 holds. By inspection, we get that the RP weakly
prefers a lump-sum payment L D LO C lO together with l D 0 to any payment LO
with a markup lO > 0. Hence, truth-telling is a weakly dominant strategy for the RP
since the RP is allowed to determine L at t0.
Assuming l D 0, the optimization problem of the RP at t0 becomes
Z b
max g RP .L/ D max
L
L
Lf .S / dS:
(5.1)
L
To determine L , the optimal solution to (5.1), we calculate the first-order derivative
of (5.1):
@g RP .L/
D
@L
Z b
f .S / dS Lf .L/ :
(5.2)
L
To facilitate our analysis, we assume that F .S /, the cumulated density function
of f .S /, has an increasing generalized failure rate (IGFR).16 Then, the RHS of
(5.2) is decreasing in L for all ranges where the RHS of (5.2) takes a value greater
14
A Stackelberg game is a dynamic game in which one party is the (Stackelberg) leader and moves
before the follower, see, e.g., Myerson (1991, p. 187).
15
See Assumption 5.1. We discuss an alternative assumption about the information status of the
RP below.
16
This assumption is natural since most standard probability distributions, e.g., normal, uniform,
gamma, and Weibull distributions, show this property, see Lariviere and Porteus (2001, p. 296).
5.2 Surplus Sharing Determined by Lump-Sum Payments
137
than zero.17 If L0 , the root of (5.2), is smaller than a, g RP .L/ takes its minimum
at a since the RHS of (5.2) decreases for ranges where it takes positive values.18
Otherwise, if f .L0 / is defined, i.e., if L0 2 Œa; b ,19 L is equal to the root of (5.2).
Then we obtain
Z b
(5.3)
f .S / dS D L f L
L
as the condition for the optimality of L . For general probability distributions f .S /,
a closed-form solution to (5.3) is not available. In spite of that, some quantitative
insights can be obtained assuming that the prior knowledge of the RP is uniformly distributed.20 With f .S / uniformly distributed between Œa; b , f .L / D
1= .b a/ holds. Given risk neutrality of the RP,21 we obtain
L
b L
D
ba
ba
as the condition for the optimality, given that L0 2 Œa; b . Hence, we get
b
:
L D max a;
2
Hence, the expected surplus that can be realized by the mechanism is
Z b
g
mech
D
L
Sf .S / dS D
o
n
2
b 2 max a2 ; b4
2 .b a/
bCa
3b 2
D min
;
:
8 .b a/
2
Next, we compare gmech with g max , the expected surplus of the best solution
identified by the scheme:
Z b
g
max
D
Sf .S / dS D
a
bCa
b 2 a2
D
:
2 .b a/
2
R1
R1
An IGFR for f .S/ means that Lf .L/ = L f .S/ dS is increasing in L. Since L f .S/ dS
R1
R1
is decreasing in L, L f .S/ dS Lf .L/ also decreases then, provided that L f .S/ dS Lf .L/ > 0.
18
A potential increase of the RHS of (5.2) for ranges where the RHS takes negative values does
not affect the optimality of L . The best value for L within such a range is the lower bound of this
range, which either corresponds to a or to the upper bound of the adjacent positive range.
19
Values of L > b can be excluded since the gains of RP become zero there.
20
This assumption is often used within mechanism design, see, e.g., Chatterjee and Samuelson
(1983, p. 842) and Baldenius (2000, p. 32), if – as it is the case here – otherwise no meaningful
analytical results can be derived.
21
For a risk-averse RP, L is lower. Risk averseness can be modeled straightforwardly here, e.g.,
analogously to the modeling by Chatterjee and Samuelson (1983, p. 848) for their sealed bid double
auction.
17
138
5 New Coordination Mechanisms
Since g mech =g max 3b 2 = .8 .b a// = ..b C a/ =2/ 3b 2 =8b = .b=2/ D 3=4,
at least 3=4 of the maximum surplus can be realized by the mechanism on average.
Assuming a D 0, which will hold if also instances with arbitrarily small surpluses
exist, the surplus allocated to the RP is
b b b2
b
g RP D
D :
L f .S / dS D L f .S / b L D
2b
4
L
Z b
Then a share of the surplus of gRP =g mech D .b=4/ = 3b 2 =8b D 2=3 is allocated
to the RP and 1=3 to the IP on average. We summarize our main results in the
following theorem.
Theorem 5.1 The coordination schemes proposed in Chap. 4 with unilateral cost
reporting can be embedded into an individually rational, strategy-proof, and budgetbalanced mechanism that does not require the involvement of a third party. Given a
two-party setting with uniformly distributed prior knowledge about the surplus from
coordination, at least 3/4 of the systemwide surplus will be realized on average.
Note that the choice which party takes the role of the IP and which party the role
of the RP has to be determined before applying the mechanism. For the two-party
supply chain considered, both roles have their advantages. The share of 2/3, which
holds for a risk-neutral RP with prior knowledge that is uniformly distributed within
the interval Œ0; b , seems to favor the RP. However, there are reasons for the RP to
claim smaller shares than anticipated by our model. First, the RP will do so if he
is risk-averse. Second, experimental studies22 indicate that parties (and especially
the less informed ones, which holds for the RP here) tend to claim considerably
smaller shares of the overall profit in sealed bid double auctions, a setting related to
that analyzed here. Third, settling for smaller shares strengthens long-term collaboration. This may induce the IP to put more effort into corresponding coordination
activities and result in future increases of marginal surpluses. Moreover, note that
the efficiency of the mechanism enhances strongly if the RP claims a smaller share
(see Example 5.4).
Example 5.4 Assume that the RP settles for half of the
I.e., the
surplus.
RP
determines L such that gRP =g mech D Lf .S / .b L/ = f .S / b 2 L2 =2 D 1=2.
With the
resulting lump sum L D b=3, the efficiency of the mechanism rises from
3=4 to b 2 b 2 =9 =b 2 D 8=9. The real decrease in the RP’s savings, however, is
only .2=3 3=4 1=2 8=9/ = .2=3 3=4/ D 1=9 of his savings for a lump sum
of 1=3.
Next, we consider the extension of the mechanism to settings with more than
two parties, i.e., one IP and several RPi , as required by the schemes with unilateral
exchange of cost information. The structure of the mechanism remains unchanged
here. The IP applies the mechanism bilaterally with each RPi . If the IP opts for
22
See e.g., Rapoport et al. (1998, p. 221) and Seale et al. (2001, p. 187).
5.2 Surplus Sharing Determined by Lump-Sum Payments
139
the implementation of a solution with different resource use by an RPi compared
to the default solution, this RPi will receive a lump-sum payment Li apart from
reestablishing his costs of the default solution. Consequently, the IP will agree on
implementing a different solution for an RPi if and only if the marginal surplus
from coordination S i , i.e., the increase of the systemwide surplus that results from
the participation of this RPi in the coordination process, is equal to or greater than
Li . To model these decisions, we propose to introduce additional binary variables
i;def i;i ni t and to augment the objective functions of CS2I and extensions by
Li i;def . That way, potential incentives of an RPi for claiming higher lump sums
and understating his cost changes can be avoided. Such incentives might particularly
arise if the sum of the marginal surpluses from coordination goes below the surplus
for the whole system.23 Then an RPi might obtain a benefit by understating his costs
since an increased profitability of solutions preferred by this RPi might induce the
IP to focus on the solution space preferred by this and not by other RPi . This would
result in an increased probability that solutions favorable to this RPi , but not to the
others, are found by the scheme. Since further possibilities to influence the savings
of other RPi do not exist due to the privateness of the decentralized data of the RPi ,
we can assume:
Assumption 5.4 In the second step of the mechanism, implementing the actions as
specified by the scheme maximizes the expected marginal surpluses of all parties.
Since the mechanism is applied bilaterally within subsets consisting of the IP and
one RPi , we can analyze the mechanism separately for each of these subsets.
In
case parties’ prior knowledge is uniformly distributed within an interval ai ; b i ,
the characterization
of the optimal lump sum derived above directly extends, i.e.,
˚
Li D max ai ; b i =2 . This is also true for the minimum foregone profits for each
RPi (1=4). Hence, the average systemwide efficiency is
P
1
i
i 2P E S
4S sys
;
with S sys as the expected surplus for the whole system. The marginal surplus of
a party’s participation in coordination can be calculated by the difference between
the (maximum) systemwide surplus with and without the participation of this party.
The sum of marginal surpluses is not necessarily equal to S sys ; it rather depends on
whether coordinated solutions of the RPi are complementary and the prior knowledge of the RPi about that. Consider the following examples.
Example 5.5 Consider two different RPi A and B, that are competing about the use
of one central resource, which is expandable by the IP at maximum by 3[CU] at
costs of 1[MU/CU]. Moreover, let A have offered within the scheme two proposals
A1 and A2 with overuses of 1[CU] and 2[CU] of this resource by A compared to
the default solution and resulting cost savings of 2[MU] and 5[MU], respectively.
B1 and B2, the proposals generated by B, show overuses of 1[CU] and 3[CU] of
23
We provide an example for such a setting below (Example 5.5).
140
5 New Coordination Mechanisms
this resource by B and savings of 2[MU] and 5[MU], respectively. The best solution
for the whole system is the simultaneous implementation of A2 and B1 resulting
in a systemwide surplus of 5 2 C 2 1 D 4[MU]. Marginal surpluses are 4 .5 3/ D 2[MU] for A (only B2 is implemented if A does not participate) and
4 .5 2/ D 1[MU] for B. Here, the sum of the marginal surpluses goes below the
surplus realized by the scheme.
Example 5.6 Consider two different RPi A and B and two central resources that are
both expandable by the IP at costs of 1[MU/CU]. Let A have generated a proposal
that additionally requires 1[CU] of the first resource by A compared to the default
solution, 1[CU] less of the second, and yields a benefit of 2[MU]. Further assume
a proposal of B with a benefit of 3[MU] and deviations in the use of the central
resources by B complementary to the proposal of A. Then, the systemwide surplus
is 2 C 3 D 5[MU] and the marginal surpluses are 5 .3 1/ D 3[MU] for A
and 5 .2 1/ D 4[MU] for B. The sum of the marginal surpluses is 7[MU] and
exceeds the surplus for the whole system.
Finally, we explicitly discuss the consequences of a repeated application of
this mechanism. For a two-party supply chain, we illustrate the resulting decision
situation of the RP by an example.
Example 5.7 Consider two parties that have applied the mechanism six times in
previous coordination activities. The history of the lump sums required and the decisions about the acceptance of coordinated proposals by the IP is displayed in
Table 5.3. The question for the RP which lump sum to choose in the current application of the mechanism (7th), cannot be answered unambiguously. Based on his
previous experiences, the RP may construct a cumulative probability distribution
function for the minimum surplus (see Fig. 5.6). The RP knows, e.g., that a lump
sum smaller than or equal to 3[MU] has led to acceptance in more than half of all
previous applications of the mechanism. Perhaps, this might induce him to choose
such a lump sum for the current application of the mechanism. A completely different and likewise rational strategy, however, is to claim a much greater lump sum in
order to explore the solution space further; the acceptance of a lump sum of 3[MU]
Table 5.3 Example of historical data used for learning in the mechanism
Application of mechanism
1st
2nd
3rd
4th
5th
6th
Lump sum (MU)
Accepted
Fig. 5.6 Probability
distribution of the minimum
surplus
3
Yes
6
No
4
No
3
Yes
5
Yes
4
No
5.3 Surplus Sharing by a Double Auction
141
may go along with, e.g., a surplus of exactly 3[MU] or even 100[MU]! The choice
of real subjects, however, is far from clear and crucially depends on the particular
learning strategies adopted by them.
Potential learning strategies include reinforcement learning, belief learning, and
experience-weighted attraction learning, which is a synthesis of the two former
ones.24 Of course, learning behavior is subject-specific and difficult to assess a
priori. Experimental studies for a related setting, the sealed bid double auction,25
provide some indications. There, the strategies of subjects observed in laboratory
experiments could appropriately be modeled by reinforcement learning.26
Besides learning, signaling strategies become relevant with a repeated application of the mechanism since the information learnt about the surplus from coordination depends on the lump sums previously requested by the RP as well as on the IP’s
previous decisions about the acceptance of coordinated solutions. E.g., it might be
beneficial for the IP to reject coordinated solutions providing her very small shares
of the surplus (after subtracting the lump sum) in order to build up reputation and
to induce the RP to require smaller lump sums in future applications of the mechanism. Unfortunately, meaningful analytical results about parties’ best strategies are
very difficult to obtain. Even if parties’ learning behavior were known, the problem
of multiple equilibria would persist. There is some empirical evidence about signaling strategies in the presence of multiple equilibria;27 like in the previous section,
however, we are not aware of corresponding (analytical or empirical) research for
settings similar to that considered by us.
To summarize, a repeated application may influence the efficiency of the mechanism proposed; the exact effect, however, cannot be assessed without significant
additional research effort. It is important to note, however, that a repeated application of the mechanism only affects the parties’ prior knowledge and, hence, the
amounts of the lump sums that will be required by them. The preference for truthful
reporting of cost changes by the RPi , instead, remains unchanged.
5.3 Surplus Sharing by a Double Auction
In the following, we show how to apply the sealed bid double auction mechanism28
to a two-party supply chain; extensions to more than two parties are discussed
afterwards. We rely on the variant of the scheme with a one-shot disclosure of
24
See, e.g., Camerer et al. (2002, p. 137). For a survey of learning theories and associated
experimental results, see Camerer (2003), Chap. 6.
25
See also Sect. 3.3.1.
26
See, e.g., Rapoport et al. (1998, p. 226) and Seale et al. (2001, p. 192).
27
See Brandts and Holt (1992, p. 1350) as a reference for a seminal paper on this topic and Sect. 8
of Camerer (2003).
28
See Sect. 3.3.1.
142
5 New Coordination Mechanisms
information29 in combination with this mechanism. To adapt this mechanism to
this scheme, the purchasing price can be interpreted as the cost savings (optionally
decreased by a markdown chosen by rational parties) of one party with a proposal
generated by the scheme, and the selling price as the cost increase (optionally augmented by a markup) of the other party. One assumption of bilateral trade, however,
does not fit with the scheme: Multiple proposals are generated within the scheme
for each party, while only one offer is processed in the sealed bid double auction.
At first glance, an appealing way to tackle this issue is to carry out sealed bid
double auctions for each proposal sequentially. After a new proposal has been
generated, the outcome of this auction determines whether the proposal is regarded
as a systemwide improvement by parties. This approach, however, has a severe
drawback: A strategy potentially adopted by parties would be then to overstate the
own costs when evaluating a proposal of the other party; the knowledge about the
other party’s last bid could be used to design a proposal similar to the last and to
submit a bid which very little exceeds the negative value of the other party’s previous bid – with the aim to usurp the lion’s share of the surplus. Anticipating this, the
other party will also considerable exaggerate its bids on new proposals, which will
result in an inferior and little predictable performance of the mechanism.
Instead, we propose to use the sealed bid double auction only once after the termination of the scheme, but simultaneously for all proposals generated. The resulting
mechanism can be described as follows: As a first step, parties (due to the symmetry
of the mechanism, the roles of parties are identical; we abbreviate them by #1 and
#2 in the following) generate a set of promising proposals ˘ along the lines of the
scheme with one-shot disclosure of cost information.30 After that, parties simultaneously submit a set of sealed bids containing one bid for each proposal k 2 ˘ . Each
bid bk1 , bk2 for #1 and #2, respectively, is the sum of the potential savings.31 sk1 ,sk2
of parties with k and markdowns m1k , m2k .32 After breaking the seals, the parties
can determine the systemwide benefits from implementing
the proposals. If the best
˚
proposal, i.e., the proposal j with j 2 arg maxk bk1 C bk2 , yields systemwide
gains compared to the default solution, i.e., bj1 C bj2 > 0, then this proposal is im
plemented. By a (positive or negative) compensation payment of bj1 bj2 =2 by
#1 to #2, an equal sharing of the surplus from coordination can be achieved (see also
Fig. 5.7 for the sequence of actions in the mechanism).
We illustrate the mechanism by an example.
29
See Sect. 4.1.2.
Note that, like in the other mechanisms presented in this work, we assume the existence of a
default solution, which is implemented without coordination.
31
Note that these “savings” can also take negative values if the implementation of a proposal
involves a cost increase for a party. For ease of exposition, however, we keep this terminology also
in this case.
32
Such markdowns are always chosen by rational parties under mild conditions. This result has
been proven by Myerson and Satterthwaite (1983, p. 265) for the bilateral trade mechanism and is
directly valid for the setting considered here if j˘ j D 1.
30
5.3 Surplus Sharing by a Double Auction
143
Fig. 5.7 Double auction mechanism
Table 5.4 Data for
Example 5.8
#1
#2
Bids [MU]
1
2
3
4
5
6
2
3
2
4
6
3
4
5
3
1
1
1
Resulting payment
by #1 to #2 [MU]
3
Example 5.8 Assume that parties have generated six proposals using the scheme
and passed sealed bids for these proposals to each other (see Table 5.4). Proposal 4
yields the greatest surplus. The mechanism implies the implementation of this proposal and a compensation payment of 3[MU] by #2 to #1, such that both parties
obtain a share of the surplus of 1[MU].
As a prerequisite for analyzing the properties of this mechanism, we assume analogously to Assumption 5.3 that parties’ individual rationality is not violated by
the information exchange required. If the mechanism is applied repeatedly, parties
will obtain knowledge about their leeways for the single proposals, i.e., about the
maximum possible markdowns which allow that the corresponding proposals are
recognized as systemwide improvements. Formally, define the leeway for a proposal k as lk1 D sk1 C bk1 . For our analysis of the mechanism, we assume that parties
already have this kind of knowledge when applying the mechanism:
Assumption 5.5 For each proposal generated, parties know the probability distributions of their leeways.33
This prior knowledge differs from the incomplete knowledge about the other parties’
savings, which is commonly assumed for analyzing bilateral trade.34 We argue that
Assumption 5.5 is more realistic in our context, since the knowledge about leeways
can be induced by observing the others’ bids in previous application of the mechanism (which does not hold for the knowledge about the other parties’ savings).
33
In contrast to the previous section, a more detailed prior knowledge (i.e., for each proposal) is
natural here since parties submit bids about the savings for all proposals within the auction, and
these bids will become globally known afterwards.
34
E.g., Myerson and Satterthwaite (1983, p. 265).
144
5 New Coordination Mechanisms
We denote the knowledge about the leeways by random variables L1j and their
probability density functions lj1 L1j , which are upper bounded by lj1 . Define
f1 D L1 ; : : : ; L1 with the associated probability density functhe vector L
1
k
1
Qk
1 f
1
1
e
tion l L D kD1 l Lk and the lower and upper support vectors le1 D
e l11 ; : : : ; lk1 and li1 D li1 ; : : : ; l 1 , respectively. Moreover, denote parties’ exk
pected gains from the mechanism by g 1 and g 2 .
For analyzing the properties of the mechanism, we further introduce
random
variables Bk2 and Bk1 with the probability density functions fk1 Bk2 and fk2 Bk1 ,
respectively. Due to the symmetry of the mechanism, we limit our analysis to the
best bidding strategy of #1. Of course,
all resultsderived hold equally for #2. Set
f1 D B 1 ; : : : ; B 1 with the associated probability
k D j˘ j. Define the vector B
1
k
Q
1
k
f
1
1
1
e B D
density function f
f
B
and
the lower and upper support veckD1 k
k
e
tors be1 D b11 ; : : : ; b 1 and b 1 D b11 ; : : : ; b 1 , respectively. The expected gains
k
k
of #1 from the mechanism can be expressed by
Z be1 Z bj2
1
1
1
g b1 ; : : : ; bk D
1
be
bj1
sj1 bj1 Bj2
2
!
f1 d B
f1
e1 B
fj1 Bj2 dBj2 f
if bj1 bj2
D 0 otherwise;
(5.4)
˚
with j 2 arg maxk bk1 C Bk2 . We apply the following transformation. First, note
that for the markdown chosen by #1 for proposal j , m1j D sj1 bj1 holds. Then,
(5.4) can be written as
g
1
m11 ; : : : ; m1k
Z be1 Z b 2
D
j
e1
b
sj1 Cm1
j
sj1 C m1j C Bj2
2
if sj1 C m1j bj2
f1 d B
f1
e1 B
fj1 Bj2 dBj2 f
(5.5)
D 0 otherwise;
˚
with j 2 arg maxk sk1 C Bk2 m1k . Obviously, in the optimal solution to (5.5),
m1j 0 holds for all j . Otherwise, if m1j < 0, the benefits for proposals which also
would be accepted in case of m1j 0 will decrease, and there will be a positive
probability that a proposal is implemented which yields a loss for #1 compared to
the initial solution.
Since the proposal j is implemented only if m1j L1j , the gains of #1 can be
expressed by
5.3 Surplus Sharing by a Double Auction
145
Z e
l 1 Z lj1 m1 C L1
j
j 1 1
f1 d L
f1
lj Lj dL1je
g 1 m11 ; : : : ; m1k D
l1 L
1
1
e
2
l
mj
D 0 otherwise;
if m1j lj1 ;
(5.6)
˚
with j 2 arg maxk L1k m1k . Unfortunately, the derivation of an analytical
solution to this multidimensional optimization problem seems not possible. For a
special form of prior knowledge – namely a uniform distribution – we are able to
determine a lower bound on the efficiency of the mechanism proposed here.
A possible, not necessarily optimal choice by #1 are equal markdowns, i.e., m1i D
1
m 8i . Then, (5.6) simplifies to
g m1 D
Z l1 1
m1
m1 C L1
2
h1 L1 dL1 ;
(5.7)
˚
with L1 as the largest order statistic of L1j , i.e., L1 D max L11 ; : : : ; L1n . Then,
the optimization problems to determine parties’ best response strategies, i.e., markdowns m1 , m2 , become
max g 1 m1 D max
Z l1
m1 C L1 1 1 1
l L dL ;
2
m1
m1
m1
Z l2 2
m C L2 2 2 2
l L dL :
max g 2 m2 D max
2
m2
m2
m2
(5.8)
(5.9)
This insight relates the mechanism proposed here to the bilateral trade mechanism:35
Parties’ problems in the latter mechanism
to (5.8) and (5.9),
can betransformed
too.36 In bilateral trade, however, l 1 L1 and l 2 L2 are not given
directly, but
only the knowledge about parties’ reservation values, which l 1 L1 and l 2 L2
depend on.
Moreover, this insight allows us to derive a lower bound on the efficiency of the
mechanism proposed, given uniformly distributed prior knowledge:
Theorem 5.2 Consider a setting where l 1 L1 and l 2 L2 are uniformly distributed and one proposal generated corresponds to the systemwide optimum. If
parties choose constant markdowns, the efficiency of the mechanism is at least
3/4. Under general bidding strategies (including unequal markups), the efficiency
is at least 1/3.
Proof. This proof is structured as follows: First (a), we derive a general lower bound
on gmech , the sum of parties’ gains from the mechanism. Since the prior knowledge
about the leeways and, hence, about the other parties’ bids is (exogenously) given,
35
See Sect. 3.3.1 for more details on this mechanism.
These problems correspond to (5.4) with only one proposal generated and the reservation value
of the buyer (seller) as the (negative) savings of these parties. See Chatterjee and Samuelson (1983,
p. 838).
36
146
5 New Coordination Mechanisms
the impact of the strategic behavior of #2 on the gains of #1 is already included there.
Hence, to determine a lower bound, we can rely on any strategies of parties. Here we
w.l.o.g assume that parties choose constant markdowns m1 and m2 for all proposals.
Second (b), we determine a general upper bound on gmax , the systemwide surplus
from implementing the best proposal generated. Comparing this to the lower bound
on gmech , we obtain a lower bound on the efficiency of the mechanism. Third (c),
we tighten this bound for the assumption that equal markdowns are actually parties’
best strategies.
(a) Define the random variables Sk as the systemwide surpluses from implementing
˚
k and S as the corresponding largest order statistic, i.e., S D max S1 ; : : : ; Sk .
Let s .S / be the probability density function of S with the support Œs; s . We, w.l.o.g.,
assume s D 0. Negative values for s do not affect our results regarding the efficiency
of the mechanism; proposals with negative surpluses, i.e., losses, will neither lead to
the systemwide optimum nor be chosen within the mechanism. For all realizations
of S smaller than m1k or m2k , parties obtain zero gains from the mechanism despite
a positive systemwide surplus. Hence, the maximum share of the surplus is cut off
by the markdowns (and, hence, the efficiency of the mechanism is lowest) if s D 0.
Note that the leeway of #1 for a proposal k corresponds to the systemwide
surplus for k less the markup which is chosen by and uniquely allocated to #2,
i.e., L1k D Sk m2k . This also is valid for the largest order statistic of L1k , i.e.,
L1 D S m2 .
Since mi 0 8i , parties keep with their initial solutions in case of a negativevalued realization of L1 , which results in zero gains for both of them. Hence, (5.7)
can be written as
g m1 D P L1 0
Z l1 m1 C L1 01 1 1
l L dL ;
(5.10)
2
m1
˚
with l 01 L1 D 1= l 1 l 01 , l 01 D max 0; l 1 , and P L1 0 as the
1
probability that L1 0.
Analogously to the calculation of the optimal lump sum in Sect. 5.2, the optimal
solution to (5.10) can be
by the first-order condition if the cumulated
determined
density function of l 01 L1 has an IGFR, which holds for uniform distributions.
Then the partial derivative,
!
Z l 1 01 1 1
1
@g 1 m1
l L
1
1 01
dL m l m
;
DP L 0
@m1
2
m1
(5.11)
0
1
is decreasing for ranges
i this derivative takes a positive value. If for m , the
h where
0
root to (5.11), m1 2 l 01 ; l 1 holds, we obtain for the optimal markdown m1 :
Z l1
m1
l 01 L1 dL1 D 2m1 l 01 m1 :
(5.12)
5.3 Surplus Sharing by a Double Auction
0
147
i
h
If m1 … l 01 ; l 1 , m1 D l 01 in analogy to the preceding section. Replacing
l 01 L1 D 1= l 1 l 01 and applying a simple transformation to (5.12), we get
for the optimal markdown
)
(
1
1
01 l
m D max l ;
:
3
If m1 D l 01 , the markdown does not cut off any potential improvements and the
efficiency of the mechanism is maximized. Hence, for calculating the lower bound,
m1 D l 1 =3 is relevant. Consequently,
Z l1 l1
C L1 01 1 1
l L dL D
l1
2
3
2
2
2
2
1
1
1
1 l1
2
C
l
l1 l 9
3
3
2
1
1
l1
:
DP L 0
DP L 0 2 l 1 l 01
3 l 1 l 01
g1 P L1 0
3
(b) Next, we calculate an upper bound on g max . Since gmax D
Rs
s s .S / dS , S D
max
2
2
2
2
l ,g
is maximized for m D l . With m D l 2 , l 1 D l 1
m C L , and m
follows. Hence, we obtain:
2
1
2
Z l1
g max l 2 C
l 01 L1 D l 2 :
l 1
Analogously, with S D m C L , we get
1
2
g max l 1 :
1
Also the probability P L 0 decreases with m2 . The minimum is P L1 0 D
1=2 for m2 D l 2 . Since si D 0, l 01 D 0 holds. Hence, we obtain g mech l 1 =6 C l 2 =6. Since g max l 1 and g max l 2 , g max 0:5 l 1 C l 2 . Hence, the
minimum efficiency of the mechanism becomes 1=6= .1=2/ D 1=3.
(c) Consider the case that only one proposal has been generated. From L1 D S m2
and L2 D S m1 , we obtain L1 C m2 D L2 C m1 . Since m1 increases with the
upper bound of L1 , this equality is only fulfilled for L1 D L2 and m1 D m2 . Hence,
g1 D g 2 holds. Since m1 D m2 , L1 D S m1 D S l 1 =3. Since s D 0, l 01 D 0,
and hence, l 1 D l 1 =3, P L1 0 D l 1 = l 1 =3 C l 1 D 3=4. With g1 D g 2 , the
lower bound on the gains from the mechanism becomes l 1 =2. We compare this to
Z s
sCs
s2 s2
:
s .S / D
D
g max D
s
s
2
s
148
5 New Coordination Mechanisms
With s D 0 and s D l 1 Cl 1 =3, gmax D 2l 1 =3 follows. Hence, at least .1=2/=.2=3/ D
3=4 of the surplus will be realized by the mechanism on average.
This result shows that minimum profits are guaranteed for both parties, which
underlines the applicability of the mechanism. These profits are substantial, especially if parties choose equal markdowns, which they will do if the probability
distributions of the leeways are equal for the each proposal. Equal probability distributions seem frequently a suitable approximation; due to the privateness of the
decentralized data, pronounced and unambiguous differences in the advantageousness of the single proposals are not to be expected. For unequal markups, a lower
performance bound could be proven. Note, however, that the decrease in the performance is mainly due to the complexity of the underlying mechanism, which does
not permit the determination of a tighter bound. We expect a considerably better
performance of the mechanism when applied for real-world coordination.
Next, we present the extension of the mechanism to several parties. Here the
participating parties submit sealed bids on all proposals. The determination of the
proposals implemented, however, is somewhat more involved and requires the solution of an additional optimization model. Like in the two-party case, this model can
be solved by each party since the necessary data is globally known after revealing
the bids submitted. If the decentralized problems can be modeled as LP problems,
the proposals implemented can be determined by MP.37 For MIP problems, combinations of proposals are not allowed. Then the “winning” proposals have to be
determined along the lines of a combinatorial auction.38 For this purpose, we present
a model based on a set packing problem.39
max
kN
XX
sji Zij
(5.13)
ij Zij b0
(5.14)
i 2P j D1
N
(CS-COM) s.t.
k
XX
i 2P j D0
Zij 2 f0; 1g
N
8i 2 P; j D 0; : : : ; k:
(5.15)
Data
ij Resource use for the j th proposal generated by party i ; i 0 denotes the
resource use of the default solution
sji Savings of party i with proposal ij
Variables
Zij Binary variable (D 1 if proposal i of party j is implemented, D 0 otherwise)
37
See Sect. 4.1.2.
See also Sect. 3.3.1.
39
For the definition of the set packing problem and an analysis of its properties, see, e.g., Balas
and Padberg (1976, p. 710).
38
5.4 Comparison of Mechanisms and Discussion
149
The objective function (5.13) minimizes the sum of the savings for the proposals
implemented. Constraint (5.14) assures that the corresponding central resource use
does not exceed b0 . Constraints (5.15) define variables as binary. Fortunately, the
solution of CS-COM does not impose major computational difficulties. Available
procedures allow the solution of problems with up to 30; 000 variables,40 which
considerably exceeds usual problems sizes for the mechanism proposed (e.g., 20
proposals generated and two parties require only 42 variables).
After the implementation of the winning proposals, a share of the surplus of
1= jP c j is allocated to each party among P c , with P c as the set of parties for which
proposals different to the initial solution have been chosen. If we assume prior
knowledge of parties about their potential leeway for bids, parties’ best strategies
can be determined like in the two-party case. Analogously to the marginal surpluses,
the leeway Li for bids of party i can be defined as the difference of the systemwide
surplus with and without the implementation of a proposal different to the default
for party i . Since the leeway for P
the whole system
is equal to the systemwide sur
plus, we obtain that at least 1 2 i 2P E Li = .3S sys / of the systemwide surplus
can be obtained on average, provided that Li is uniformly distributed.41
A further relevant issue for this mechanism is the interplay of reputation and
learning effects42 if coordination is undertaken repeatedly.43 Analogously to the
preceding sections, however, neither theoretical nor empirical analyses of sealed
bid double auctions have been carried out yet for this dynamic environment.
5.4 Comparison of Mechanisms and Discussion
In the preceding sections, we have presented three different mechanisms for
implementing the coordination schemes proposed in this work. Without further
assumptions about the context of their application, none of these mechanisms can
be strictly preferred over the others. The most important differences between them
are summarized in Table 5.5.
For the mechanism with surplus sharing determined by the IP, the total surplus
from coordination can be realized, provided truthful cost reporting by the RP.
40
See, e.g., Andersson et al. (2000, p. 6).
Analogously
to Theorem 5.2, with constant markups chosen by parties, the lower bound is
P
1 i2P E Li = .4S sys / of the systemwide surplus. Further note that the sum of parties’
leeways may exceed S sys or go below Lsys , depending on the complementarities of the central
resource use in coordinated solutions; for an analogous discussion, see Sect. 5.2.
42
For literature on learning and signaling strategies, see the corresponding discussion of the
preceding section.
43
For experimental studies of learning in repeated (anonymous, i.e., without signaling effects)
sealed bid double auctions under the standard assumption of common knowledge about the distributions of parties’ reservation values, see Rapoport et al. (1998, p. 221), Seale et al. (2001, p. 177),
and Daniel et al. (1998, p. 133).
41
150
5 New Coordination Mechanisms
Table 5.5 Comparison of coordination mechanisms
Surplus sharing by
IP
Lump sum
Double auction
Total savings realized?
No
No
Yes
Rather equal
No
Equal
Strategy proofness
Rights in savings sharing
Yes (given
truth-telling)
No
Unequal
A natural drawback of this mechanism, however, is the missing strategy proofness.
Moreover, the inability of the RP to influence surplus sharing might constitute a
further obstacle for the acceptance of this mechanism in practice.
An obvious advantage of the second mechanism (lump sum) is that truth-telling
is a weakly dominant strategy for parties. Moreover, parties’ rights in surplus sharing are allocated more equally despite the asymmetric procedure applied. Surplus
sharing does not depend on the IP only, but on the messages of all parties sent within
the mechanism. For two risk-neutral parties with uniformly distributed prior knowledge about the surplus from coordination, a share of 1/3 of the surplus is allocated to
the IP and 2/3 to the RP. Moreover, we could show that with slightly less aggressive
bidding by the RP, surplus sharing becomes more equal and the efficiency of the
mechanism improves substantially. However, since coordinated solutions are only
accepted if the surplus from coordination exceeds the lump sum, not all potentially
profitable proposals can be implemented.
This drawback also applies to the sealed bid double auction, where rational, selfinterested parties include markups or markdowns into their bids in order to obtain
larger parts of the surplus. Since a disclosure of cost effects during the proposal generation is not possible, the performance of the underlying schemes may be somewhat
affected.44 Moreover, a double auction has the advantage of equal rights in surplus
sharing, i.e., the shares received by parties only depend on the actual surplus and
on the parties’ skills in choosing optimal bids. A further advantage of this mechanism is that real-world decision makers are familiar with auctions since auctions are
frequently used within B2B (business-to-business) relationships.45
To further illustrate the advantages and the different scopes of the mechanisms,
we outline three hypothetical settings for their practical application. First, consider
a supply chain consisting of an OEM (original equipment manufacturer) and one of
its component suppliers. In this setting, parties’ roles are fairly natural if one of the
mechanisms presented in Sects. 5.1 and 5.2 is applied. The OEM will serve as the IP
determining the proposal implemented and the supplier as the cost-reporting party.
Assigning these roles differently seems inferior; then a supplier would determine
the proposal finally implemented (for himself and other suppliers), which might not
be accepted by other suppliers.
44
See also our computational tests of Sect. 6.1, p. 168.
See e.g., Elmaghraby (2004, p. 214) for examples of auctions in B2B marketplaces and Hohner
et al. (2003, p. 23) for a case study about the use of procurement auctions at Mars.
45
5.5 Application with Rolling Schedules
151
Whether the lump sum payment will be preferred, depends, amongst others, on
the negotiation power of the OEM. Assume a very powerful OEM that has insight
into the cost calculations of the supplier, e.g., due to open-book accounting agreements.46 Then an incorrect reporting by the supplier becomes less probable, which
mitigates the principal disadvantage of the mechanism presented in Sect. 5.1. Moreover, a very powerful position of the OEM is in line with the procedure for savings
sharing applied in this mechanism.
With less negotiation power of the OEM, the mechanism based on the lumpsum payment might be preferred. If coordination is not enforced by the OEM, but
introduced as a voluntary improvement project, the suppliers might not consent to
leave the allocation of the surplus completely to the OEM. In order to cover their
transaction costs, they might require a minimum reward if a coordinated solution
is implemented. Advantages of this mechanism compared to the double auction in
this setting are the guarantee of truth-telling that supports long-term collaboration,
the option for the OEM to keep his cost data completely private, and the higher
performance of the underlying scheme.47
As a third setting, consider parties in a rather loose relationship with equal rights
for each of them. In such a setting, equitable possibilities for obtaining benefits by
coordination may be crucial such that a mechanism is perceived as fair by parties.
Here we propose the use of a sealed bid double auction. It captures the independence
of supply chain parties and provides each party with equal rights in information
disclosure and surplus sharing.
5.5 Application with Rolling Schedules
To mitigate unfavorable effects of uncertainties inherent in demand forecasts,
practical supply chain planning is often based on rolling schedules.48 Hence,
the applicability for rolling schedules is a major requirement for practicable
coordination mechanisms. The mechanisms presented in this work do not need
significant adjustments; in this section, we outline how they can be embedded in the
corresponding planning processes.
A natural application of these mechanisms is the identification of a systemwide
near optimal plan if an operational replanning has to be undertaken. Such replanning
becomes often necessary when targets from a long-term contract have to be implemented in operational planning. Due to uncertainties in the end item demand, the
quantities fixed in a long-term contract may become inefficient for the whole supply
chain at a later point of time. Even if they have been optimal subject to the information status at the time of their negotiation, this may change with an additional
revelation of information in later periods. Consider the following example.
46
See, Agndal and Nilsson (2008, p. 154) for a compilation of approaches for open-book
accounting in supply chains.
47
See our computational results of Sect. 6.1.
48
See also Sect. 2.1.2.
152
5 New Coordination Mechanisms
Example 5.9 Assume a two-party supply chain with one item supplied, for which
a long-term supply contract has been negotiated between buyer and supplier. This
contract specifies the quantities that have to be delivered by the supplier in each
period as well as the purchase prices for these items. Assume that we are at the
end of period 2 and that the operational planning for periods 3–14 has to be undertaken. In Fig. 5.8, the supply quantities fixed in the long-term contract and those
needed to fulfill the current demand forecast are displayed. In period 4, the supply
quantities required exceed those agreed in the long-term contract. Hence, in this
period, greater quantities might be preferred for the whole supply chain, given that
the supplier holds enough inventories and that his overtime costs for temporally
greater production are lower than the buyer’s backorder costs. On the contrary, if
demand is lower than expected (periods 3 and 5), lower supply may be preferable if
holding costs of the buyer or overtime costs of the supplier can be saved. In later periods, the flexibility of production is usually greater and forecasts are less concise,
such that no changes in the supply quantities may be necessary. In this example,
the supply quantities of the long-term contract are (near) optimal for the buyer in
periods 6–14.
If the costs for the supply plan of the current period and that specified by the
long-term contract differ significantly, the buyer will communicate a new order to
the supplier. The supplier’s decision whether to agree to this change depends on the
leeway for the supplier’s order fulfillment fixed in contractual agreements49 and the
supplier’s cost changes due to this new order. If he does not agree, the resulting
conflict can be resolved by one of the coordination mechanisms proposed.50 Taking the supply quantities specified by the long-term contract as the initial proposal,
Supply quantity [MU]
80
70
60
50
40
30
20
10
0
2
4
6
8
10
12
14
16
Periods
Current demand forecast
Long-term contract
Fig. 5.8 Fixed long-term contract and current deviations
49
Such leeway is often specified by a flexibility range, e.g., Tsay (1999, p. 1341).
Of course, such conflicts only arise if the supplier holds too little inventory to fulfill the buyer’s
orders without a significant increase in his costs. Hence, a further interpretation of the mechanisms
proposed is that of instruments to reduce inventory at decentralized parties.
50
5.5 Application with Rolling Schedules
153
the mechanisms generate proposals for a systemwide improvement and establish
compensation payments if one of these proposals is accepted.
The same idea can be applied with rolling schedules. Here, a replanning may
become beneficial for both parties; for the buyer, e.g., if demand fluctuates more
than anticipated, and for the supplier, e.g., if a machine failure occurs or if urgent
orders of other customers have to be preponed. If replanning results in a conflict
of parties’ interests, this conflict can be resolved by the coordination mechanisms
proposed. Then the initial proposal corresponds either to the quantities fixed in the
long-term contract or to the outcome of an earlier operational replanning process
(see the subsequent example).
Example 5.10 Assume a time frame of 18 periods covered by a long-term contract
and a horizon of operational planning of six periods (see Fig. 5.9).51 If the buyer
initiates coordination for the first time at period 2, the initial proposal corresponds
to the supply quantities of periods 2–7 as fixed in the long-term contract. Here, the
outcome of coordination results in smaller supply in periods 3 and 5 and greater
supply in periods 4 and 6. Assume that the next plan change takes place in period 5.
Then, the default solution comprises the quantities of periods 5–7 as agreed in the
previous coordination process and those of periods 8–10 as fixed in the long-term
contract, and so on.
The approach to use the supply quantities fixed in the last coordination process
as the default solution is not affected by concerns raised by Dudek (2004)52 about
the application of his coordination scheme for rolling schedules. Dudek was worried by the fact that solutions determined by coordination can be discarded in future
coordination processes. First, he was concerned about the consequences on compensation payments. Since the scheme proposed by him does not allow for backorders
of the buyer, he had to design an extension, which includes additional penalty costs
for potential infeasibilities for the buyer arising with an evaluation of the proposals
Fig. 5.9 Application of the mechanisms with rolling schedules
51
52
For ease of exposition, the existence of a frozen horizon is omitted here.
See Dudek (2004, p. 116).
154
5 New Coordination Mechanisms
generated by the supplier.53 Since these penalty costs have no meaningful (monetary) explication, they obviously cause problems when determining compensation
payments. This issue, however, is not relevant for the schemes proposed here since
they allow for modeling backlogs and lost sales, the real consequences of uncertain
demand and shortages.54
Dudek’s second concern was that discarding coordinated proposals in future periods represents a waste of coordination effort. We argue, however, that such changes
are natural and simply reflect the actual information statuses of parties facing a planning problem subject to uncertainty. If, e.g., tendencies for an increase of demand
are recognized in an earlier period and a plan change including compensation payments is arranged, coordination effort and payments are not wasted; they fix the new
status quo which – if not directly implemented – forms the base for future replanning
and coordination. For parties spending positive payments for the implementation of
a coordinated proposal, this new status quo solution comes closer to the expected
optimum than the previous solution. Hence, it can be expected that these parties will
have to incur lower payments in future coordination processes. For the whole supply
chain, in turn, less deviations of the current plan from the systemwide optimum and,
hence, less coordination effort can be expected.
53
54
See Dudek (2004, p. 120).
See Sect. 4.4.3.
Chapter 6
Computational Tests of Coordination Schemes
In this chapter, we analyze the performance of the schemes proposed for several
classes of randomly generated test instances as well as for real-world planning
problems. In Sect. 6.1, we begin with basic insights about the coordination performance for the generic Master Planning model with forced compliance by the
supplier. Section 6.2 deals with the coordination of uncapacitated multi-level lotsizing. In the further sections, the model complexity is successively increased:
Sect. 6.3 addresses capacitated lot-sizing and Sect. 6.4 campaign planning under
voluntary compliance. For the real-world data of Sect. 6.5, we additionally include
lost sales and a series of further restrictions such as inventory capacity constraints
and minimum transportation lot sizes.
6.1 General Master Planning Model
This section and Sects. 6.2–6.4, which deal with randomly generated test instances,
are structured as follows: First, we describe the generation of the test instances
unless the instances have already been proposed in the literature. Second, we
present basic results regarding the coordination performance and compare them
with upstream planning as the default procedure without coordination. Third, we
conduct sensitivity analyses for different parameter settings and specifications of
the schemes.
6.1.1 Generation of Test Instances and Performance Indicators
The generation of hard test instances is crucial for obtaining reliable insights into
the performance of algorithms or model formulations. Therefore, this question has
received broad attention in the related literature.1 Surprisingly, the literature on
1
See, e.g., Kolisch and Sprecher (1995, p. 1693) for the generation of hard instances for the
resource-constrained project scheduling problem.
M. Albrecht, Supply Chain Coordination Mechanisms: New Approaches
for Collaborative Planning, Lecture Notes in Economics and Mathematical Systems 628,
c Springer-Verlag Berlin Heidelberg 2010
DOI 10.1007/978-3-642-02833-5 6, 155
156
6 Computational Tests of Coordination Schemes
coordination mechanisms does not emphasize this issue, but is limited to specific
settings derived from real data,2 seemingly arbitrary random data,3 and instances
developed for monolithic optimization problems.4
However, when using such approaches for test data generation, two problems
may arise which lead to test instances that are rather easy to coordinate. Both problems are due to large cost asymmetries. First, proposals by the party with the larger
costs may come very near to the systemwide optimum without further need for coordination. Second, large cost asymmetries among single items may allow the scheme
to concentrate on the high-cost items, which has a similar effect like reducing the
problem size. We illustrate each of these problems by an example.
Example 6.1 Consider a supply chain with one item supplied, which is directly sold
by the buyer to an external market. Let the production of the item require the use of
several capacity-constrained resources at the supplier’s site. Moreover, assume that
the backorder costs for the buyer are very low compared to the overtime costs of
the supplier (for the production of an equivalent amount of the item). In this setting,
deviations from the orders preferred by the buyer may lead to large savings in the
supplier’s overtime costs and only require small increases in the buyer’s backorder
costs. Hence, upstream planning with forced compliance by the supplier may result
in large suboptimalities, which can easily be removed by a proposal that does not
imply overtime for the supplier.
Example 6.2 Consider a supply chain with two items supplied (A, B), which are
directly sold by the buyer to the market. These items are produced by the supplier
on a single resource with restricted capacity. Thereby, the production coefficient of
A exceeds significantly that of B. Moreover, let the backorder costs for A exceed
those for B by a small amount. Further assume that the sum of the order quantities
of A and B is restricted for the buyer. In this setting, upstream planning with forced
compliance may lead to a suboptimal outcome since the buyer may inappropriately
prefer delays of B instead of A. In the systemwide optimal solution, in turn, only
delays of A, but no delays of B will be included. In case of further items with characteristics similar to that of B, the structure of the optimal solution (i.e., to produce
less of A), and, hence, the coordination effort remains unchanged. That way, cost
asymmetries among items act in a similar way like a reduction of the problem size.
To avoid these problems, we propose the generation of test instances where the
sensitivities of parties’ major cost factors with respect to central resource use
are equal on average. Below, we show how this can be achieved for GM as the
underlying optimization model.
2
E.g., Karabuk and Wu (2002, p. 753) and Walther et al. (2008, p. 343).
E.g., Arikapuram and Veeramani (2004, p. 119).
4
E.g., Ertogral and Wu (2000, p. 937), Dudek and Stadtler (2005, p. 681), and Dudek and Stadtler
(2007, p. 465), all rely on instances generated for the MLCLSP by Tempelmeier and Derstroff
(1996, p. 743).
3
6.1 General Master Planning Model
157
Master Planning problems that do not comprise lot-sizing or scheduling activities
can be tackled by standard solvers even for large-scale instances. This explains that
few test instances specifically addressing Master Planning without discrete variables have been developed in the literature.5 This holds a fortiori for test instances
for the coordination of Master Planning; the only test set we are aware of is that of
Arikapuram and Veeramani (2004),6 which, however, addresses a strongly simplified setting with one item and one constrained resource per supplier and does not
include considerations about coordination hardness.
Hence, we had to design a completely new set of test instances for this study.
Assuming Master Planning with forced compliance and omitting the option for lost
sales,7 the major cost components with impact on central resource use are backorder
costs for the buyer and overtime costs for the supplier. In order to avoid easy coordination, we set the cost sensitivities of parties for changes in the supply to be equal on
average. I.e., we set increases in the supplier’s overtime costs, which are potentially
necessary to avoid shortages in the supply, equal to the average backorder costs of
the successor items at the buyer’s site. Additionally, we conduct sensitivity analyses
for different relations between average overtime and backorder costs (1:5, 5:1).
For this purpose, we define the cumulated capacity requirements of an item j as
aj D
X
M
X
rkj ak C
amj :
(6.1)
mD1
k2Rj
Backorder costs for one unit of item j in a period are set to
blcj D ajK ;
(6.2)
with K as a constant of an arbitrary value (e.g., 1) used for the correct transformation
of the unit of aj . To focus on the temporal distribution of the supply quantities, we
assume zero backorders at the end of the planning interval. Overtime costs for one
unit of capacity increase on resource m are set to
P
ocm D
j 2Jm
blcj
amj
jJm j
;
with Jm as the set of items produced on resource m. In order to determine unit costs
for inventory holding, we set the value of an end item equal to its backorder costs8
5
One of the few exceptions is Scholl (2001, p. 314), who has designed instances for testing robust
optimization procedures.
6
See Arikapuram and Veeramani (2004, p. 119).
7
In order to decrease complexity, the modeling of lost sales is deferred to Sect. 6.5.
8
Of course, this relation between backorder and overtime costs of items does not always hold in
practice. Here we rely on this assumption to account for the fact that items with larger backorder
costs are often more valuable and, hence, require higher capital costs for their holding in inventory.
158
6 Computational Tests of Coordination Schemes
and the value of a preliminary item equal to its cumulated capacity requirements in
monetary units. Assuming monthly planning periods and only capital costs (10%
p. a.) for inventory holding, we get
hj D
aj K
:
jT j 10
(6.3)
Like in Sects. 6.2 and 6.3, zero initial inventories are assumed. The average capacity
utilization is set to 90%. By default, we assume that overtime is not restricted and
continuously available. In order to assess the sensitivity of the scheme regarding
discrete decisions, we additionally consider a setting where overtime can only be
taken in an integer number of shifts of one-eighth of the period capacity. For this
purpose, GM is extended by declaring variables Omt as integer and multiplying
these variables both in the objective function of GM and in the capacity constraint
(2.3) by 1=8kmt .9
We investigate several test sets differing by their structures (see Fig. 6.1), the
number of time periods, the inclusion of integer variables (for modeling overtime),
|JD |=2
10
|JD|=4,
J=16,
4
1
M=7
7
11
8
5
2
12
9
6
3
9
5
10
13
24
14
25
15
7
1
26
16
8
2
9
3
27
17
28
18
10
4
11
5
12
6
1
S1
14
23
S1
S2
|JD |=4
13
|JD |=10
6
2
15
11
7
3
16
12
8
4
S3
S2
29
19
30
20
31
21
32
22
S4
Explanations:
j
items j
set of items that are produced
on a specific resource
2S: separate production of items by
S1; S2 (|JD|=4),
S1,S2; S3-S5 (|JD|=10)
S5
5S S1; S2; S3; S4; S5 (|JD|=10)
Fig. 6.1 Structures of test instances for GM
9
See Albrecht and Stadtler (2008) for an explicit modeling of this extension.
6.1 General Master Planning Model
Table 6.1 Notation for test sets
Characteristic
Parameter
ˇ ˇ
ˇ Dˇ
Structure
ˇJ ˇ D 2
ˇJ D ˇ D 4
ˇ ˇ
ˇJ D ˇ D 10
Time
jT j D np
Model class
LP
MIP
Ratio backorder
1:1
/ overtime costs
5:1
1:5
Cost exchange
Modifications
Number of suppliers
Initial solution
S!B
B!S
1-shot
All
Generic
agg-P
Without CS1i
agg-P
agg
Without CS1i
, CS1i
1S
2S
5S
Upstream
RAND
159
Comment
Structure with 2 items supplied
Structure with 4 items supplied
Structure with 10 items supplied
np periods
Overtime modeled by linear variables
Overtime modeled by integer variables
Equally distributed backorder and
overtime costs
Costs for backorders are five times the costs
for overtime on average
Costs for backorders are one fifth of
overtime costs
Unilateral reporting of cost changes by S
Unilateral reporting of cost changes by B
One-shot disclosure of cost changes
All levels of modifications
Generic scheme for LP
agg-P
All models except for CS1i
are run
agg-P
agg
All models except for CS1i
and CS1i
1 supplier producing all items supplied
Items produced by 2 suppliers
Items produced by 5 suppliers
Upstream planning with forced compliance
Random initial solution
and the exact specifications of the scheme (this includes the direction of the
agg-P
information exchange, the use of CS1i , etc.). The single options and their
abbreviations are summarized in Table 6.1. We ˇuse combinations
of them in order
ˇ
to describe the single settings investigated (e.g., ˇJ D ˇ D 4, jT j D 2).
For each test set, demands, production coefficients, and capacities have been
generated based on random numbers. All random numbers have been drawn from
normal distributions with base values of 1 and different combinations of coefficients of variation.10 Thereby, negative values have been replaced by 0 and all
random numbers have been divided by their averages to exclude any side-effects
by this replacement. Six variations of demand data have been considered, each
made up of a combination of a coefficient of variation (abbreviated by CV in the
following) C Vd 2 f0:1; 0:3; 0:5g and a seasonality component. This component
corresponds to a cosine (cos.) oscillation with an amplitude of 0 (no seasonality)
or 0.2. For production coefficients and period capacities, coefficients of variation
10
In order to avoid side effects due to fixed profiles for demand, capacities, etc., that have been
generated once and potentially comprise some extreme characteristics, we have generated new (and
hence, different) random data for each test instance.
160
6 Computational Tests of Coordination Schemes
Table 6.2 Input parameters for the test data generation for GM
Parameter
Base value
Coefficient of variation
Demand
1, 1, 1, 1 (cos.), 1 (cos.), 1 (cos.) 0.1, 0.3, 0.5, 0.1, 0.3, 0.5
Capacity requirements 1
0.001, 0.1, 0.2, 0.3, 0.4, 0.5
Production coefficients 1
0.2, 0.5, 0.7, 1, 1.5, 2
of CV coe 2 f0:2; 0:5; 0:7; 1; 1:5; 2g and CV cap 2 f0:001; 0:1; 0:2; 0:3; 0:4; 0:5g,
respectively, have been chosen. This yields 216 instances per test set. We summarize
their characteristics in Table 6.2.
To assess the coordination performance of the schemes proposed, we mainly rely
on three performance indicators. The first is the suboptimality of upstream planning
for the test instances generated. Formally, the average gap of the uncoordinated
solution11 (AGU) as be expressed by
PN
nD1
cunc;n cce n;n
cce n;n
N
:
Index
n Index of test instance
Data
cce n;n Costs of the solution to the centralized model for test instance n
cunc;n Costs of the uncoordinated solution (here: upstream planning) for n
ccor;i;n Costs after i iterations of the scheme for instance n
Second, we consider the average remaining gap (AGS) after the application of
the scheme:
P
ccor;i;n cce n;n
N
nD1
cce n;n
:
N
The third indicator is the average gap closure (AGC) after i iterations of the scheme:
PN
cunc;n ccor;i;n
nD1 cunc;n cce n;n
N
:
When displaying the performance of the scheme according to the number of
iterations, we rely on the remaining gaps that are equal to 1-AGC. The abbreviations for the performance measures as well as further indicators characterizing the
solutions obtained can be found in Table 6.3.
As a first insight into the properties of the test instances generated, Table 6.4
provides
ˇ D ˇ an overview about selected results for the AGU in the default setting
(ˇJ ˇ D 4, jT j D 12, LP, 1:1, Upstream). The column headers differing from
11
Note that in general, when we define a gap in this work, we consider the percentage deviation of
an upper bound (UB) to a lower bound (LB), i.e., (UB-LB)/LB.
6.1 General Master Planning Model
161
Table 6.3 Abbreviations for performance measures
Abbreviation Explanation
AGU
Average gap of the uncoordinated solution
AGS
Average gap after the application of the scheme
AGC
Average gap closure after the application of the scheme
N
Number of instances analyzed per test set
NOPT
Number of solutions found per test set, with costs equal to
or lower than those of the solution to the centralized model
TC
Average solution time for the centralized model (in seconds)
TS
Average time for running the scheme (in seconds)
Table 6.4 Suboptimality of upstream planning (AGU)
jT j D 3 Default MIP
jJ D j D 10 jT j D 24 5:1
1:5
10.9%
66.4%
15.1%
21.8%
28.1%
22.5%
8.9%
default indicate how the test sets have been altered (e.g., jT j D 3 means that
instances with three periods instead of 12 have been generated). It can be recognized
that increases in the model complexity due to a larger number of items, periods or
mixed-integer variables increase the average suboptimality of upstream planning
and, hence, the need for coordination. Moreover, varying the ratio among average
backorder and overtime costs considerably affects the AGU. With larger values of
this ratio, deviations in the costs from the systemwide optimum will lead to larger
cost increases for the buyer, but smaller decreases for the supplier. This will lead to
an increased probability that proposals unilaterally determined by the buyer come
near to the systemwide optimum, which results in smaller AGU.
Figure 6.2 depicts the cumulated probability distribution of the initial gaps for
the test set jJ D j D 10, jT j D 12, LP, 1:1. The shape of this curve is representative
for most instances generated in this study.12 The solutions from upstream planning
are characterized by a large number of solutions with small gaps and few solutions with very large gaps. This observation is interesting for two aspects: First, it
underlines that the need, and hence, the benefits of coordination may differ substantially according to the concrete choice of input parameters. Second, it reveals
that the assumption of a uniform distribution of the surplus, for which analytical
results could be obtained in Chap. 5, will only roughly reflect parties’ expectations. Anyhow, as argued there, parties’ behavior will additionally deviate due to
learning and signaling effects. Figure 6.2 in this context provides an indication
12
Note that for some other test sets, in particular for the MLULSP (see Sect. 6.2), the inclinations
of the corresponding curves differ far less for smaller and larger gaps (i.e., the curves resemble
more a straight line). The overall characteristic, however, that few instances with large gaps exist
is also valid there.
6 Computational Tests of Coordination Schemes
0.8
0.6
0.4
0.2
0.0
Cumulated distribution function
1.0
162
0
100
200
300
400
500
Gaps in %
Fig. 6.2 Distribution of the initial gaps for GM
about the general characteristics of parties’ prior expectations.13 Therefore, Fig. 6.2
constitutes a starting point for further analytical or empirical investigations about
the behavior of parties using the mechanism proposed.
6.1.2 Analysis of Solutions for the Generic Scheme
The generic scheme has the advantage of finding the systemwide optimal solution
within a finite number of iterations, provided that all but one decentralized problems
can be modeled as LP (in the version with unilateral exchange, even one problem may be of a MIP type). Since the maximum number of iterations increases
exponentially with the problems size, however, there is no guarantee that larger
problems can be tackled within a practicable number of iterations. To assess the
solution performance and the limitations of the generic scheme, we have carried out
a series of test runs for small and medium-sized problem instances.
As throughout this study, the main routine for running the scheme has been
implemented in the programming language Java (version jdk 1.6.0) using Eclipse
(version 3.3.2). Optimization problems have been solved using Xpress-MP, release
13
Interestingly, this equally extends when considering the absolute cost increases compared to the
upstream solutions; there, very similar curves can be obtained (with many instances with small
cost increases and few instances with large costs increases).
6.1 General Master Planning Model
Table 6.5 Solution performance of the generic scheme
AGU (%) AGC (%) AGC (%)
ˇ ˇ
ˇJ D ˇ D 3,jT j D 2
12.6
0.4
98.7
Default
10.9
2.1
62.7
MIP
21.1
16.1
22.7
jT
21.9
14.6
3.0
ˇ 6
ˇ jD
ˇJ D ˇ D 3,jT j D 2,1-shot 12.6
1.7
65.7
163
N
65
139
173
194
65
NOPT
59
20
2
0
9
TC
0.02
0.02
0.03
0.03
0.02
TS
467.9
1,550.8
474.3
1,689.3
1,176.6
2008A. The data exchange between Xpress-MP and Java has been partly managed
using the Java interface of Xpress-MP. In this section, tests have been run on a single
thread of an Intel SMP (single machine processor) with 1.99 GB RAM and a clock
speed of 2.61 GHz.
ˇ ˇ
As the default setting we assume here: ˇJ D ˇ D 4, jT j D 3, LP, 1:1, S!B,
generic, 1S, Upstream.14 With time limits of 30 s for all problems, all centralized
models have been solved to optimality. Since we aim at exploring the maximum coordination performance, we have chosen a very large number (500) of vertices for
the linearization described in Sect. 4.3.1. Throughout this section, we only investigate instances with a minimum suboptimality of 0.005%. Table 6.5 provides the
results obtained after 50 iterations of the scheme. In the smallest test set with two
periods and three items supplied for the version of the scheme with unilateral exchange of cost changes, the systemwide optimum could be identified for almost all
instances after 30 iterations. Also for the LP instances with four items supplied and
three periods (default), a large part of the suboptimality due to upstream planning
has been mitigated. This, however, changes drastically if decentralized parties run
MIP problems or if the model size is further increased. Then, less than one third of
AGU can be mitigated. This result is even worse when considering the second indicator AGC. Since the scheme performs better for test instances with greater AGU,
the ratio AGS/AGU exceeds the AGC here.
The version with one-shot exchange of cost information has been tested for the
smallest set. Although near optimal solutions have been obtained, too, some decrease compared to an unilateral exchange can be observed. We suggest that this
decrease is due to the different choice of the starting solutions, which – due to the
limitation of the information exchange – does not allow to concentrate on the most
favorable regions of the solution space. However, also in this case, the significant
gap closure has been possible with the identification of a small share of the vertex
solutions of the decentralized problems.
For both versions, the average time for running the scheme has been much greater
than that needed for the centralized solution. This is intuitive since LP problems can
14
We consider here the base version of the generic scheme with finite convergence for LP and not
its extension for finite convergence for one decentralized MIP due to the increased computational
complexity of this extension.
164
6 Computational Tests of Coordination Schemes
100
Remaining gap (1-AGC) (%)
80
60
40
20
0
0
10
20
30
40
50
Iterations
D
D
MIP,|J |=4,|T|=3
D
LP,|J |=4,|T|=6
LP,|J |=3,|T|=2
LP,|J |=4,|T|=3
D
Fig. 6.3 Convergence paths for generic scheme
be tackled very efficiently by standard solvers, whereas the linearization requires
SOS2, which significantly increases the computational effort.15
Further interesting insights about the coordination performance provide the
convergence paths, which comprise the remaining gaps after 1–20 iterations (i.e.,
1-AGC). For the scheme with unilateral exchange of cost information, the convergence paths are displayed (see Fig. 6.3). Interestingly and in contrast to the heuristic
modifications considered later on, the curves grow almost linearly with the number
of iterations for the LP models. The inclinations of the curves, however, decrease
considerably with increased problem sizes. The flattening of the curve for MIP after
36 iterations can be explained by the infeasibilities in CS1i that do not allow further
progresses in the convergence.
6.1.3 Analysis of Solutions for the Modified Scheme
As mentioned above, the modified version of the scheme shows advantages in
respect of computing time and convergence speed, especially when applied to the
15
Note in this context that the observed decrease in the solution time for MIP can be explained
by infeasibilities and, hence, a breakdown of the scheme for some instances. In this case, the best
solution found so far has been taken.
6.1 General Master Planning Model
Table 6.6 Solution performance of the modified scheme for GM
AGU (%) AGS (%) AGC (%) N
NOPT
ˇ ˇ
ˇJ D ˇ D 3, jT j D 2 12.6
0.8
91.2
65
24
jT j D 3
10.9
1.2
82.5
139 37
jT j D 3, MIP
21.2
3.5
63.5
173 17
Default
15.1
4.0
54.5
200 3
MIP
21.8
6.8
47.0
208 4
ˇ ˇ
ˇJ D ˇ D 10
28.1
8.8
47.6
212 0
ˇ ˇ
ˇJ D ˇ D 10, MIP
34.2
12.2
40.9
212 0
165
TC
0.00
0.05
0.05
0.01
135.3
0.06
320.2
TS
27.8
56.2
61.7
108.8
234.6
411.6
629.9
ˇ ˇ
coordination of MIP problems. As the default setting we assume here: ˇJ D ˇ D 4,
agg-P
jT j D 12, LP, 1:1, S!B, without CS1i , 1S, Upstream.
We applied 10 nodes for the linearization of the objective functions of CS1i .
Time limits have been set to 600 s for the centralized models and to 10 s for the
decentralized models. With these time limits, all instances of C have been solved to
optimality except of 34 instances for the MIP with jJ D j D 4 and 103 instances for
the MIP with jJ D j D 10.16 There, the average suboptimalities have been 0.53% and
0.56%, respectively. Table 6.6 provides basic results obtained after 20 iterations.
These results indicate that the modifications applied to the generic scheme substantially improve the coordination performance. The size of test instances turned
out to be the most important driver for the performance. For the smaller instances
with three periods or less, near optimal solutions could be obtained. This suggests
that managers applying the scheme should focus on a small number of key products and periods in order to obtain best coordination performance. The presence of
integer variables affects the performance, too, although the degradations in the solution quality have been far more modest than for the generic scheme. In general,
decreases in the solution time compared to the generic scheme can be observed.17
Like in the generic scheme, the reduction of the AGS compared to the AGU (i.e.,
the ratio .AGU AGS/=AGU) exceeds the AGC.
Further insights into the coordination process provide the convergence paths
displayed in Fig. 6.4. An overall characteristic that can be recognized is that the
suboptimalities can be mitigated in large part after only five iterations. This is in
strong contrast to the generic scheme, where the coordination performance increases
roughly linearly within the first 50 iterations (see Fig. 6.3). The modest number of
iterations favors considerably the applicability of the (modified) scheme in practice
since this allows decision makers to manually control the cost effects of the plans
generated.
16
Note that we assume here that the optimal solution has been found if the gap between the best
solution and the best bound goes below the standard solution tolerance of the optimizer of 0.01%.
17
Note that the different number of iterations, time limits and number of nodes applied for
the approximation may also explain this difference in part; in additional computational tests for
the modified version of the scheme with jT j D 3 and the same number of iterations, nodes, and
time limits as applied for the generic scheme, the average solution time was less than half the time
for running the generic scheme.
166
6 Computational Tests of Coordination Schemes
100
Remaining gap (%)
80
60
40
20
0
0
5
10
15
20
Iteration
|JD|=3,|T|=2,LP
D
|J |=4,|T|=3,LP
|JD|=4,|T|=3,LP
|JD|=4,|T|=12,MIP
D
|J |=10,|T|=12,LP
D
|J |=4,|T|=12,LP
Fig. 6.4 Convergence paths of the modified scheme
Table 6.7 Sensitivities regarding the generation of demand data
CVd , seasonality 0.1, 0 0.3, 0 0.5, 0 0.1, 0.2 0.3, 0.2 0.5, 0.2
AGU (%)
AGS (%)
AGC (%)
16.7
2.0
55.7
8.9
3.6
55.4
9.2
2.1
52.1
24.6
6.1
53.0
17.3
4.8
55.7
13.2
5.2
55.2
Next, we analyze the sensitivities in the performance of the scheme regarding
changes in the input parameters and the specification of the
Here we again
ˇ scheme.
ˇ
rely on the parameter settings of the default setting (i.e., ˇJ D ˇ D 4, jT j D 12, LP,
agg-P
1:1, S!B, CS1i , 1S, Upstream).
We start with investigating the effects of different input parameters. In Table 6.7,
we address the sensitivities regarding the generation of the demand data. Here a
somewhat counterintuitive trend can be recognized: The AGU decrease with greater
variations in the demand, but increase with the seasonality coefficient. Regarding
the AGS, the results are somewhat worse if the seasonality component is included,
which may be due to the larger AGU in that case. In general, however, the performance of the scheme seems stable with variations in the demand generation, which
is underlined by the small deviations in the AGC observed.
Table 6.8 provides the sensitivities according to the random generation of the
production coefficients. Here no unequivocal trend can be observed. Both the AGU
and AGS behave irregularly, but rather stably with changes in the coefficients of
variation.
6.1 General Master Planning Model
167
Table 6.8 Sensitivities
regarding the generation of
production coefficients
CVcoe
0.2
0.4
0.7
1
1.5
2
AGU (%)
AGS (%)
AGC (%)
15.3
3.2
48.1
8.1
3.2
58.2
15.4
5.6
52.1
13.9
3.4
53.8
24.1
5.2
56.9
15.3
3.7
58.6
Table 6.9 Sensitivities
regarding the generation of
capacity coefficients
CVcap
AGU (%)
AGS (%)
AGC (%)
0.001
11.6
3.4
57.4
0.1
32.8
5.4
60.2
0.2
11.7
5.0
56.7
0.3
10.1
4.1
52.9
0.4
9.1
2.4
47.9
0.5
16.2
3.9
52.6
Table 6.10 Sensitivities
regarding specifications of
the test instances
Default
jT j D 24
1:5
5:1
RAND
AGU (%)
15.1
22.5
66.4
8.9
37,007.7
AGS (%)
4.0
4.8
11.0
1.8
4.1
AGC (%)
54.5
54.1
55.6
59.1
99.4
Next, we consider the sensitivities to changes in the generation of the capacity
coefficients (Table 6.9). Also here, the deviations in the results seem rather stable,
which suggests that the performance of the scheme is not affected by variations of
this parameter.
In the following, we analyze the robustness of the default setting (see Table 6.10).
We start with considering increases in the number of time periods. Recall that in our
basic results (Table 6.6), we have observed a pronounced decrease in the performance when increasing the number of periods from 3 to 12. For further increases
to jT j D 24, however, the decreases in the AGC turned out to be modest (the ratio
AGU/AGS has even improved compared to the default setting), which indicates an
appropriate scalability of the scheme proposed.
Next, we consider variations in the ratio between the average backorder and overtime costs of parties. For both increases and decreases in this ratio, both the AGC
and the ratio AGU/AGS have improved compared to the default setting. We suggest
that this improvement is due to the fact that – as mentioned at the beginning of this
section – instances with less equilibrated costs (and, hence, a ratio between overtime
and backorder costs different to 1) are easier to coordinate.
Subsequently, the choice of a starting solution different to upstream planning
(RAND) did not significantly affect the AGS. Of course, the AGU and AGC increase sharply since the uncoordinated supply target does not consider the potential
cost increases for the decentralized parties (due to backorders or overtime). Apart
from the independence of the performance of the scheme from the starting solution
chosen, these results suggest that upstream planning is a rather effective heuristic
for decentralized coordination, which explains the frequent use of this procedure in
practice.
168
6 Computational Tests of Coordination Schemes
Next, we consider sensitivities regarding different designs of the scheme. We
begin with different specifications on the information exchange (see Table 6.11).
With the buyer as the cost-reporting party, the coordination performance even increases, which underlines the robustness of the results obtained above. The other
variant of the scheme with a one-shot exchange of information leads to a slight
decrease in the coordination performance, which, however, does not affect the applicability of the scheme proposed. The reason for this decrease is the missing
possibility for anticipating the other parties’ cost changes, which is due to the restricted information exchange during proposal generation.
A further important question is the optimal specification of the scheme. As
agg-P
the default setting, we applied the version with all modifications but CS1i . In
18
Table 6.12, we display results for other modifications. The default setting turned
out to be significantly superior to the other modifications. As indicated in Sect. 4.3.4,
the computing time could be reduced substantially compared to the omittance of
agg
CS1i .
Finally, we study the case that the production of items is undertaken in part by
several suppliers (see Table 6.13). At first glance counterintuitive, increasing the
number of suppliers improves the quality of the coordination process significantly.
This can be explained by the more detailed knowledge of the buyer about the supplier’s costs changes. Such knowledge allows the buyer to estimate the cost effects
of deviations in the supply target more concisely and provides greater leeway for
Table 6.11 Sensitivities
regarding different forms of
information disclosure
Default
B!S
1-shot
AGU (%)
AGS (%)
AGC (%)
15.1
15.1
15.1
4.0
2.9
4.4
54.5
58.1
50.6
Table 6.12 Sensitivity regarding the modifications of the scheme
AGU (%) AGS (%) AGC (%) TS
Default
All
agg-P
agg
Without CS1i , CS1i
15.1
15.1
15.1
4.0
5.1
5.7
54.5
49.4
42.8
108.8
88.4
325.3
Table 6.13 Solution performance for multiple suppliers
AGU (%) AGS (%) AGC (%) TS
Default
2S
ˇ ˇ
ˇ Dˇ
ˇJ ˇ D 10
ˇJ D ˇ D 10, 2S
ˇ ˇ
ˇJ D ˇ D 10, 5S
18
15.1
15.1
28.1
28.1
28.1
4.0
2.9
8.8
7.3
5.0
54.5
64.3
47.6
52.9
62.8
108.8
70.1
411.6
359.0
218.6
See further Albrecht and Stadtler (2008) for a larger number of test results of the scheme without
agg-P
agg
CS1i
and CS1i .
6.2 Uncapacitated Lot-Sizing Problem
169
recombining the proposals of the single suppliers in GM-CS2-MB . The improved
performance went along with a significant decrease in the computing time that is due
to the smaller models of the suppliers, notwithstanding the fact that the models of
two or five suppliers have to be solved here. These results underline the generic character of the scheme and let seem probable a further extendibility to different organizational structures such as supply chains with multiple buyers or more than two tiers.
Summarizing, the generic scheme only achieves convergence for rather small test
instances within a reasonable number of iterations. The modified version can successfully tackle much larger problems, even if they comprise integer variables. The
sensitivity analysis revealed that the performance of the scheme is robust to changes
in the input parameters or the application settings, such as multiple suppliers or
different specifications regarding the exchange of cost information between parties.
6.2 Uncapacitated Lot-Sizing Problem
6.2.1 Generation of Test Instances
For uncapacitated multi-level lot-sizing, the question of coordination hardness is
far less important than in Sect. 6.1. Here, reasonable test instances developed for
centralized procedures comprise balanced setup and holding costs, too; otherwise,
setups would either occur in all periods or only once at the beginning of the planning
interval.
Most literature on centralized models for uncapacitated lot-sizing relies on
arbitrarily fixed input data.19 In order to generate some additional structure useful
for analyzing sensitivities with changes in the input parameters, we fix the ratio between marginal holding and setup costs here. The rationale behind this is that, along
the lines of the EOQ formula, the expected TBO directly depend on this ratio.20
Here, we take the marginal holding costs and the TBO as given and determine the
setup costs scj to
1
(6.4)
scj D mhj .TBOj /2 e j :
2
e j is the average secondary demand defined by
ej D
ejcum
jT j
;
(6.5)
jT j
with ejcum
jT j as the cumulated primary and secondary demand of items as defined in
Sect. 2.2.2.1.21
19
E.g., Graves (1981, p. 95), Blackburn and Millen (1982, p. 51), Afentakis et al. (1984, p. 233),
Afentakis and Gavish (1986, p. 243), and Heinrich (1987, p. 174).
20
See, e.g., Derstroff (1995, p. 93) and Trigeiro et al. (1989, p. 353).
21
See p. 13.
170
6 Computational Tests of Coordination Schemes
A
B
1
2
1
2
3
4
5
6
7
8
9
10
11
12
C
1
3
2
13
14
15
16
17
18
19
20
21
22
23
24
20
26
27
28
29
30
6
5
E
1
D
1
8
9
2
3
4
5
6
7
8
9
10
2
1
4
7
12
7
13
14
8
15
16
5
9
17
5
4
3
6
6
10
F
2
3
4
18
24 25
10
19
26 27
20 21 22 23
28
8
7
11
29
30
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
Frontier Buyer-Supplier
Fig. 6.5 Structure of test instances
Other deterministic data include the planning horizon (jT j D 12 periods) and six
different BOM structures (see Fig. 6.5). Moreover, we have generated a couple of
random parameters drawn from a normal distribution with an expected value of 1.
Negative random values have been replaced by 0, except for the TBO, where we
have chosen a value of 0.00001. Demand series have been generated with different coefficients of variation (0, 0.2, 0.5, 0.2); the fourth series has been multiplied
by an additional seasonality component (generation based on a cosine oscillation
with an amplitude of 0.3). Random numbers for marginal holding costs are based
on coefficients of variation CV mh 2 f0:2; 0:5g. For the buyer’s items, the random
numbers directly correspond to mhj , whereas for the supplier’s items, these numbers are multiplied by an additional factor (1, 1.3, 2). This factor has been identified
as one of the major drivers for the suboptimality of upstream planning in two-stage
supply chains.22 The TBO profiles differ by their averages (1, 2, 4) and their coefficients of variation (CV TBO 2 f0:2; 0:5g). Resulting are 6 4 6 6 D 864 test
instances.
22
See Simpson (2007, p. 136).
6.2 Uncapacitated Lot-Sizing Problem
171
6.2.2 Analysis of Solutions
In this section as well as in Sects. 6.3 and 6.4, computational tests have been
performed on eight threads of an Intel SMP with a clock speed of 2.33 GHz and
8 GB RAM, using Xpress-MP 2008A parallel 64-bit. Details on the time limits and
the resulting suboptimalities of the centralized models can be found in Table 6.14.
We analyze the performance of the scheme for test instances with AGU >
0:01%.23 Among the 864 instances generated, 646 of them show this characteristic. Table 6.15 depicts the results after 20 iterations of the scheme, i.e., after 20
proposals generated.
On average, more than 88% of the gap of upstream planning could be closed by
the scheme. In more than half of the instances, even the optimal solution could be
identified. Further remarkably is the modest time required for proposal generation.
Less than 46.3% of the time necessary for solving the centralized model has been
needed on average.
Concerning the impact of the different BOM structures on the coordination
performance, several conclusions can be drawn. First of all, the size of the test sets
seems to be an important driver for the performance. The highest values for the AGC
have been obtained for the smaller test sets A, C, and E.
The structure of the BOM can be recognized as a further driver. The average gap
closure is highest (93.8% and 91.3%) for assembly and serial structures and considerably exceed the AGC for general structures (81.7%). A reason for the favorable
results in assembly structures is the limited leeway for changes in the supply pattern.
Table 6.14 Computing time and optimality gaps for the MLULSP
A
B
C
D
E
F
Centralized model:
– Time limits (s)
– Number of suboptimal solutions
– Average optimality gaps (%)
Decentralized model: time limits (s)
1,200
0
–
10
1,200
23
2.44
10
Table 6.15 Solution
performance of the scheme
for uncapacitated lot-sizing
23
1,200
0
–
10
A
B
C
D
E
F
1,200
9
0.99
10
1,200
0
–
10
1,200
14
3.34
10
AGU (%)
AGS (%)
AGC (%)
N
NOPT
4.96
3.89
3.88
2.30
7.44
4.07
0.01
0.35
0.18
0.12
0.37
1.47
98.98
85.48
95.92
91.73
92.45
64.72
104
138
86
90
140
88
102
41
77
64
88
25
Without this limitation, less meaningful results for the AGC for structure D would be obtained.
There, the AGC would rise up to 234% since one instance exists where the initial gap is
outperformed by 13,202.2% due to the suboptimality of the centralized solution.
172
6 Computational Tests of Coordination Schemes
Altering the setup frequencies for a small number of items of the buyer influences
a larger number of dependent items of the supplier. As a consequence, a larger part
of PB 24 can be covered by our procedure for proposal generation. The weaker performance of the scheme for general structures has already been indicated by the
scope of our analytical results of Sect. 4.2, which are only valid for special cases of
assembly and serial BOM.
Further insights provide the convergence paths. Figure 6.6 shows that a large
part of the savings can be reached after the generation of only 10 proposals. Except
for the large general test set F, the average remaining gaps have been smaller than
75% at that stage. Noteworthy is the convergence behavior in early iterations. For
the small set A and the assembly structures C and D, the gap to optimality could
be reduced to less than 20% after only five proposals generated, whereas for the
other instances, this gap ranged between 40% and 43%. This result underlines our
hypothesis that coordination can be achieved easier for smaller problems sizes and
assembly systems.
Next, we present sensitivity analyses for assessing how parameter settings for test
data generation influence the performance of the scheme. In Table 6.16, we consider
100
Remaining gap (%)
80
60
40
20
0
0
5
10
15
20
Iteration
A
B
C
D
E
F
Fig. 6.6 Convergence paths for the MLULSP
Table 6.16 Sensitivity
analysis for demand
24
See p. 82 for the definition of this set.
CVd , seasonality
AGU (%)
AGS (%)
AGC (%)
0, 0
5.4
0.50
86.6
0.2, 0
4.8
0.35
91.4
0.5, 0
4.1
0.40
88.8
0.2, 0.3
4.4
0.35
87.4
6.2 Uncapacitated Lot-Sizing Problem
173
the impact of different demand structures. Interestingly, level demand yields the
largest absolute gaps from upstream planning,25 but also many instances with low
coordination potential, which explains the modest differences in the values for the
AGU. According to the performance of the scheme, no unequivocal effects could be
recognized.
Table 6.17 presents the sensitivities regarding the assumptions on marginal
holding costs. The AGU are somewhat greater with increased coefficients of variations. The sensitivity of these gaps on the ratio between parties’ marginal holding
costs, however, turned out to be low. This result contrasts the study of Simpson
(2007), where this parameter had a significant impact on the observed suboptimality of upstream planning.26 We suggest that this difference is due to the ratio
between setup and holding costs, which is fixed in this study and randomly chosen
by Simpson (2007).27
In Table 6.18, we finally address the sensitivities regarding the TBO. Little intuitively, an increased CV for TBO leads to smaller AGU, a reversed effect compared
to the increase in the CV for the marginal holding costs. Moreover, there is a
trend for a better performance with large TBO. A potential explication of this phenomenon is that large TBO imply a smaller number of buyer’s setups in upstream
planning. This decreases the solution space where potential improvements might be
located and, hence, increases the probability that a (near) optimal solution is found.
Summing up, the scheme for uncapacitated lot-sizing problems allows the identification of near optimal solutions even for BOM structures comprising up to 30
items. Apart from somewhat weaker results for large general structures, the performance is robust regarding different parameterizations of the test instances.
Table 6.17 Sensitivities regarding the ratio between marginal holding
costs
CVmh , ratio mhj 0.2, 1 0.2, 1.3 0.2, 2 0.5, 1 0.5, 1.3 0.5, 2
AGU (%)
4.5
4.2
5.4
5.4
4.4
4.8
AGS (%)
0.37
0.37
0.37
0.48
0.42
0.35
AGC (%)
88.7
88.4
84.7
90.5
89.2
89.9
Table 6.18 Sensitivity analysis regarding the TBO
CVTBO , base value 0.2, 1 0.2, 2 0.2, 4 0.5, 1
AGU (%)
4.5
5.4
6.2
3.5
AGS (%)
0.23
0.72
0.04
0.34
AGC (%)
96.1
87.7
92.8
82.5
25
0.5, 2
4.5
0.67
82.7
0.5, 4
3.9
0.14
93.1
Note that this information is not drawn from Table 6.16, but from a separate analysis undertaken
by us. The same effect has also been recognized by Simpson (2007, p. 127).
26
See Simpson (2007, p. 134).
27
More exactly, Simpson (2007) does not explicitly consider this ratio; it depends on other parameters that are randomly generated, see Simpson (2007, p. 127).
174
6 Computational Tests of Coordination Schemes
6.3 Multi-level Capacitated Lot-Sizing Problem
In this section, we compare the performance of the scheme presented here with the
scheme by Dudek and Stadtler (2005).28 We have generated test data along the lines
of Dudek (2004).29 There, several problems structures (S1, S2, S3, M1, M2, L)
differing by the BOM and the number of the items supplied (2–7) have been investigated. For each of these structures, 126 instances have been randomly generated
based on different demand curves, expected TBO, and capacity utilization profiles.
We omit a detailed description of the generation of these instances and refer to
Dudek (2004) instead.30
agg-P
We have run the version of the scheme: S!B, CS1i , 1S, Upstream, with
10 nodes for the linearization of the objective functions of CS1i and extensions.
Table 6.19 displays information about the computing time and the optimality gaps
for the centralized models after aborting the solution process.31 As in Dudek
(2004),32 we only rely on capacity feasible solutions for our analysis.
In Table 6.20, we compare the performance of the scheme proposed by us
with the version of the scheme of Dudek (2004)33 that requires the same level of
28
Table 6.19 Computing time and optimality gaps for different structures
S1
S2
S3
M1
M2
L
Centralized model:
– Time limits (s)
– TC
– Number of capacity feasible solutions
– Number of suboptimal solutions
– Average optimality gaps (%)
Decentralized model: time limits (s)
600
601.5
101
101
5.8
15
200
11.7
113
1
1.3
5
200
23.2
116
5
1.3
5
200
8.7
117
0
0.0
5
400
168.7
121
25
3.6
10
400
166.4
110
22
3.9
10
See Dudek and Stadtler (2005, p. 681). Instead of the more aggregated presentation of the computational tests in this paper, we rely on the more detailed description of Dudek (2004, p. 165) in
the following.
29
See Dudek (2004, p. 168).
30
Note that we have adapted the test instance generation by Dudek (2004) in a minor degree. In
order to exclude potential distortions due to fixed demand profiles, the demand data has been generated randomly for each combination of problem parameters; however, both for this specification
and for another with fixed demand profiles used in preliminary test runs, the gaps after upstream
planning turned out as significantly larger than those reported by Dudek (2004, p. 180) (see also
Table 6.20).
31
Surprisingly, for the centralized model L, the average solution time observed exceeded the time
limit by a minor degree. A potential reason for that may lie in time lags due to the communication
of this data to the Java routine used.
32
See Dudek (2004, p. 178).
33
See Dudek (2004, p. 192).
6.3 Multi-level Capacitated Lot-Sizing Problem
Table 6.20 Comparison with Dudek (2004)
AGU AGS AGS/AGU It
AGU
(%)
(%)
(%)
(%)
Dudek (2004)
S1
S2
S3
M1
M2
L
25.6
33.9
18.3
18.8
15.3
26.6
2.7
1.6
1.2
1.5*
1.6*
2.0*
175
AGS4
(%)
AGS10
(%)
AGC10
(%)
AGS4 /AGU
(%)
2.0
1.9
1.3
1.8
1.7
1.9
67.1
46.4
61.0
71.1
64.6
79.4
6
7
4
9
7
8
Albrecht (2008)
11
5
7
8*
10*
8*
3.8
3.2
3.7
4.7*
4.3*
5.1*
39.1
35.4
33.3
25.1
36.0
34.0
2.4
2.6
1.5
2.2
2.5
2.7
information exchange.34 Unfortunately, Dudek (2004) has only tested the sets S1,
S2, and S3 with this version. Although not directly comparable, we additionally display for instances M1, M2, and L the results of a version of the scheme of Dudek
(2004) that includes a more demanding requirement on information exchange –
bilateral exchange of cost changes (we denote this by an “*” behind the corresponding results).35
Similarly to the scheme of Dudek (2004), the scheme presented here was able
to reduce most of the gaps of the uncoordinated setting. Unfortunately, the performance of both schemes is not directly comparable since the gaps of upstream
planning turned out to be substantially greater in the present study. As a resort, we
rely on the relative average reductions of AGU, the ratios AGS/AGU. In the scheme
proposed, AGS4 /AGU, the relative gap closure after four iterations – which is somewhat below the average number of iterations the scheme of Dudek (2004) is run36 –
is similar to that of Dudek (2004); it has been larger in half of the test sets.37 After
10 iterations, the scheme proposed here clearly outperforms the results reported by
Dudek (2004) with four iterations; unfortunately, results for running the scheme by
Dudek (2004) for 10 iterations are not available.38
Like in Sect. 6.2, we only consider test instances with AGU 0:01% when determining the
AGC in order to obtain meaningful results for this indicator.
35
Since the results of Dudek (2004) for M1, M2, and L with bilateral reporting of cost changes
are most probably superior to a limitation of the information exchange (as it has been the case for
S1, S2, and S3), the conclusion drawn below – that our results are at least comparable to those of
Dudek (2004) – is not affected. Moreover, note that we limit to the AGS here since the AGC are
not reported by Dudek (2004) for unilateral information exchange.
36
The number of iterations is determined in Dudek (2004) by a random stopping criterion along
the lines of simulated annealing, see Dudek (2004, p. 97).
37
Note that the different time limits chosen by Dudek (2004) (due to the different hard- and software used) do not affect this conclusion. The limits chosen here are rather unfavorable compared
to those of Dudek (2004); the ratio between the time limits for the decentralized and centralized
models has been chosen as 1:40 here instead of 1:10, and the average gaps to the optimal solution
to the centralized MLCLSP have been substantially lower (see Table 6.19).
38
Our preliminary computational experience with the scheme by Dudek (2004) has shown that
the convergence rate of this scheme does not improve significantly with a larger number of iterations due to the limited capabilities for generating proposals different to those previously found; a
comprehensive computational study, however, has not been carried out on this subject.
34
176
6 Computational Tests of Coordination Schemes
A further commonness to the scheme by Dudek (2004) is the relative low value of
AGC(10/ compared to the ratio .AGUAGS10 /=AGU, which is far more accentuated
than for GM and the MLULSP.39 This and the rather small AGS can be explained
by that fact that Dudek (2004) has generated instances that are relatively easy to
coordinate.40 One share of instances shows large initial gaps and involves elevated
overtime costs for the supplier after upstream planning. These overtime costs, however, are of a greater order of magnitude compared to the setup and holding costs
and can be saved rather easily by delayed orders of the buyer, which only require
some minor increases in the setup and holding costs. For the other instances comprising only setup and holding costs, the gaps (which are far smaller there) have
been closed more slowly and to a smaller extent. These insights are sustained by our
sensitivity analysis in Table 6.21 and the convergence paths displayed in Fig. 6.7.
Table 6.21 shows that the presence of overtime has a huge influence on the AGU;
the AGS, instead, are not significantly affected by that. This is also the reason why a
large part of the AGU could be closed after only one iteration (the AGC ranged between 33% and 52% there, see Fig. 6.7). The convergence paths for further iterations
of the scheme turned out as highly dependent on the size of test instances. While
for the smaller instances S1 and S2, almost the complete savings have been reached
after 3 iterations, for the larger instances (especially L), considerable shares could
be realized in later iterations. This result is intuitive. Larger instances are likely to
show greater difficulties for coordination since manifold interdependencies between
items have to be considered. Moreover, the large gap closure for set L indicates a
favorable scalability of the scheme. Note that this result has been favored by suboptimalities in the solutions to the centralized model (see also Table 6.19) due to the
time limits applied.
In Table 6.22, we compare the performance for different specifications of the
scheme after 10 iterations.41 With the exception of S3, the AGS are lowest for
the default version of the scheme, which includes all levels of modifications. The
Table 6.21 Sensitivities
regarding the need for overtime at the supplier with
upstream planning
39
S1
S2
S3
M1
M2
L
Without overtime
AGU (%) AGS10 (%)
With overtime
AGU (%) AGS10 (%)
5.4
1.8
2.7
4.4
3.6
3.6
103.2
113.2
75.8
65.4
94.9
84.3
2.2
1.0
1.4
2.2
1.6
1.4
1.5
4.0
1.1
1.5
1.8
2.7
See Sects. 6.1 and 6.2.
See Sect. 6.1.1 for a discussion on coordination hardness.
41
For the analysis of sensitivities regarding the input data for this test set, we refer to Dudek (2004,
p. 186).
40
6.3 Multi-level Capacitated Lot-Sizing Problem
177
100
Remaining gap (%)
80
60
40
20
0
0
2
4
6
8
10
Iteration
S1
S2
S3
M1
M2
L
Fig. 6.7 Convergence paths for the MLCLSP
agg-P
exception occurred for the version without CS1i , which showed the best results
for GM in Sect. 6.1. In spite of this exception, the hypothesis that the default specifiagg-P
cation yields the same AGS as the scheme without CS1i
can be rejected in favor
of assuming smaller AGS for the default at a significance level of 99% based on
a Wilcoxon matched pairs signed rank test simultaneously performed over all test
instances. Surprisingly, however, the second indicator considered behaves contrariagg-P
wise to AGS. AGC is mostly even greater if CS1i
is not used. Anyway, the use
agg-P
of CS1i
has only minor impact on the convergence.
For the other modifications, the differences are far more pronounced. The hypothesis that the default specification yields the same AGS as the scheme without
CS-LOT can be rejected in favor of assuming smaller AGS for the default at a significance level of 95% for each single test set based on the Wilcoxon matched pairs
signed rank test. The only benefits of not running CS-LOT are modest reductions
in TS. This, however, seems of minor relevance compared to the differences in the
solution quality.
Finally, without any additional modifications that are successively skipped if no
agg
agg-P
new solution can be obtained (i.e., without CS1i , CS1i , and CS-LOT), the
degradation of the convergence rate is substantially greater. Again applying the
Wilcoxon matched pairs signed rank test, the hypothesis that this specification yields
the same AGS as the default can be rejected in favor of assuming smaller AGS for
the default at a significance level of 99% for each single test set.
Summarizing, the performance of the scheme presented here is at least comparable to that developed by Dudek and Stadtler (2005), while showing a series of
S1
S2
S3
M1
M2
L
AGS (%)
2.0
1.9
1.3
1.8
1.7
1.9
AGC (%)
67.1
46.4
61.0
71.1
64.6
79.4
TS
26
24
28
140
68
320
AGS (%)
2.1
2.2
1.2
2.2
2.0
2.1
AGC (%)
69.7
56.7
66.3
61.4
64.3
89.5
TS
24
20
22
101
54
385
AGS (%)
2.0
2.0
1.5
2.5
2.2
2.6
AGC (%)
62.8
42.4
56.7
56.6
54.9
54.2
Table 6.22 Sensitivities regarding different specifications of the scheme applied to the MLCLSP
agg-P
Default
Without CS1i
Without CS-LOT
TS
15
12
19
120
50
328
agg-P
CS1i , CS-LOT
AGS (%) AGC (%)
2.9
62.3
4.1
41.9
1.6
56.9
2.8
54.9
2.4
54.7
2.9
62.7
agg
Without CS1i ,
TS
57
45
44
158
148
466
178
6 Computational Tests of Coordination Schemes
6.4 Models for Campaign Planning
179
advantages: It can coordinate settings with voluntary compliance by the suppliers,
it can be directly applied for planning based on rolling schedules, and it permits the
decision makers to determine the duration of the coordination process, instead of
relying on a random stopping criterion.
6.4 Models for Campaign Planning
6.4.1 Generation of Test Instances
Again, no comprehensive test sets for multi-level campaign planning have been reported in the literature.42 This holds a fortiori for decentralized instances that are
hard to coordinate. Hence, we have developed new test sets, thereby meshing ideas
from Sects. 6.1 and 6.2.
We start with a description of the underlying methodology and provide the relations between the input parameters that have been fixed. We define aj by (6.1) and
blcj by (6.2). Overtime costs for one unit of capacity on resource m are set to
P
ocm D
blcj
j 2Jm amj
jJm j conoc
;
with Jm as the set of items produced on resource m and conoc as a constant stating
the average ratio between backorder and overtime costs. Assuming that backorders
are more expensive than overtime, we set conoc D 2. In order to reduce the number
of necessary backlogs, we have introduced nonnegative initial inventories inij :
inij D 2ej :
capmt , the capacity of resource m in period t, is determined to
capmt D
fmt
P
j 2J amj dj
ut
T 2
;
T
(6.6)
with fmt as a randomly generated factor (generation see below), dj as the average
demand for item j , and ut as the average capacity utilization (here: 0.9). The last
factor of (6.6) accounts for the fact that the demand is partly covered by the initial inventories and, hence, does not need to be produced completely. Holding cost
factors hj are determined according to (6.3) and setup costs scj according to (6.4).
42
Related test sets include those developed by Trigeiro et al. (1989, p. 358), which have been
extended by Suerie (2005a, p. 59) to the modeling of (single-level) campaigns, and those by Tempelmeier and Buschkühl (2008) for the MLCLSPL without campaign restrictions.
180
6 Computational Tests of Coordination Schemes
mhj , the marginal holding costs, are a dependent factor here. They are recursively
determined by
X
mhj D hj rkj hk :
k2Rj
prev
Finally, for all items j , the initial setup states w0j and inflows ijt have been set
equal to zero and lead times equal to 1.
We rely on the BOM structures S1 and M1 of Dudek and Stadtler (2005). We
varied the remaining input parameters within a broad range. Demand data has been
generated randomly with coefficients of variation CV 2 f0:1; 0:3; 0:5; 0:3g; for the
last series, the random data has additionally been multiplied by a seasonality component consisting of a cosine oscillation with an amplitude of 0.3. All random numbers
have been taken from a normal distribution with an expected value of 1. As in the
previous sections, random values smaller than zero have been replaced by 0 here.
Random numbers for capacity requirements amj have been generated based on coefficients of variation of 0.3 and 0.5. Four coefficients of variation have been used for
generating production coefficients rjk : CV coe 2 f0:2; 0:5; 1; 1:5g. Similarly to Dudek
(2004),43 three TBO profiles have been generated, which determine the relation between inventory holding and setup costs. The average TBO for the buyer’s items are
4, 3, and 5, respectively, and 4, 5, and 3 for the supplier’s items. All of these profiles have been multiplied by random numbers generated based on a coefficient of
variation of 0.5. Analogously to the literature,44 the setup times have been set to fractions of period capacities. We have chosen fractions of 10% multiplied with random
numbers based on a coefficient of variation of 0.5. Finally, the batch sizes have been
varied. Apart from test instances without any batch size restrictions (NOBATCH),45
three profiles for batch sizes have been specified (all of them are subsumed under
“BATCH”). The average batch sizes for the buyer’s items are the amounts resulting
from production during 1, 1, and 2 period lengths (assuming full capacity utilization), respectively, and for the supplier 1, 2, and 1. Again, these values have been
multiplied by random numbers based on a coefficient of variation of 0.5.
By combining the profiles for all parameters, we obtain 768 test instances.
Table 6.23 summarizes the input parameters chosen.
6.4.2 Analysis of Solutions
For the computational study of this section, the same hard- and software and the
same specification of the scheme as in Sect. 6.3 have been used. Due to the elevated
43
See Dudek (2004, p. 172).
See, e.g., Porkka et al. (2003, p. 1139) and Suerie (2006, p. 882).
45
For a better comparability of the results for instances with and without batch size restrictions,
all tests have been run based on the formulation for the campaign planning model provided in
Sect. 2.2.2.2, but with arbitrary small values for the minimum batch sizes if no batch restrictions
have been present.
44
6.4 Models for Campaign Planning
181
Table 6.23 Parametrization of test instances for the MLCLSPL-C
Parameter
Base value
Coefficient of variation
Demand
1, 1, 1, 1 (cos.)
0.1, 0.3, 0.5, 0.3
Capacity requirements 1
0.3, 0.5
Minimum batch sizes
0 (NOBATCH), 1-1, 1-2, 2-1 0.5
Production coefficients 1
0.2, 0.5, 1, 1.5
0.5
Setup times
10% kmt
Structure
S1, M1
–
TBO
3-3, 3-5, 5-3
0.5
Table 6.24 Time limits and outcomes of the centralized models
S1
M1
NOBATCH BATCH NOBATCH
Centralized
– Time limits
1,200
1,200
2,400
– Number of instances
96
288
96
– Number of suboptimal solutions 72
282
91
– Average optimality gaps (%)
16.7
46.2
26.7
Decentralized time limits (s)
10
10
20
BATCH
2,400
288
283
57.7
20
Table 6.25 Solution performance for the coordination of campaign planning
AGU (%) AGS (%) AGC (%) NOPT TS
Win-win (%)
S1, NOBATCH
S1, BATCH
M1, NOBATCH
M1, BATCH
23.3
38.2
523.7
629.3
12.7
2.4
10.5
0.4
34.6
172.5
79.7
109.9
6
127
19
157
225.7
608.0
842.5
1171.5
11.9
35.2
20.7
28.9
problem complexity, we only chose five nodes for the linearization of the objective functions. The time limits and details about the centralized solutions for these
instances are summarized in Table 6.24.46
An overview of the results is given in Table 6.25. Apart from the performance indicators used in the previous sections, we provide the percentage of test instances for
which the coordinated proposals lead to cost reductions for both parties (win-win).47
The results differ significantly with respect to the sizes of instances and the presence of batch restrictions. The AGU for S1 turned out to be significantly lower than
the AGU for M1, whereas the AGS and AGC of S1 and M1 come much closer to
each other. The explanation for this is that the small structure S1 only comprises two
Like in Sect. 6.2, we only consider test instances with AGU 0:01% when determining the
AGC in order to obtain meaningful results for this indicator. AGC > 1 means that more than
the (complete) suboptimality from upstream planning can be mitigated, which holds if a solution
identified by the scheme is superior to the centralized solution.
47
In contrast to the test settings of the preceding sections, win-win situations may arise here due
to assumption of voluntary compliance.
46
182
6 Computational Tests of Coordination Schemes
production stages, whereas M1 comprises three stages. With two stages, the initial
inventories were sufficient to avoid backorders in large part in upstream planning.
This, however, does not extend to M1, where the necessary amount of backorders
and, hence, the AGU, increase sharply. This is due to the positive production lead
times causing shortages of items produced at the lower levels, an effect, which is
substantially aggravated by the three-level structure in M1. Somewhat surprisingly,
the suboptimalities due to these changes only affect the AGU in major degree; the
rather equal values of the AGS for S1 and M1 suggest that, irrespectively of the
problem structure, these suboptimalities can be mitigated almost completely by
the scheme.
A further counterintuitive issue is the decrease in the performance for instances
without batch size restrictions. As shown by the optimality gaps displayed in
Table 6.24, the presence of campaign restrictions augments the computational difficulties for the centralized models considerably. This, however, at the same time
decreases the difficulties for the scheme to identify solutions that are equivalent or
superior to those resulting from the centralized model. This is because campaign restrictions increase the computational difficulties less for the scheme than for the
centralized model, which substantially improves the relative performance of the
scheme. For both S1 and M1 with campaign restrictions, the best solutions identified
by the scheme outperformed the solutions found by the standard solver on average.
Thereby, less than half the computing time for solving the centralized model was
needed, which suggests that the scheme can be applied as an effective heuristic
for large-scale multi-level campaign planning problems. Some insights into problem structures where this heuristic is most effective are provided by our sensitivity
analysis below.
Moreover, we have evaluated the extent in which coordinated solutions lead to
improvements for both parties. The share of such solutions ranged between 11.9%
and 35.2%. On the one hand, this result underlines that even with voluntary compliance, coordination payments are needed if parties want to extract the maximum
benefits from coordination. On the other hand, the fact that coordinated solutions
can be implemented without recurring on compensation payments in some cases –
if this would be an issue48 – widens the scope of the scheme proposed.
Further insights provide the convergence paths (see Fig. 6.8). For M1, most savings are obtained within the two first iterations, whereas for S1, the savings realized
in later iterations are substantial. This supports our hypothesis that there are two levels of coordination difficulties: The mitigation of the (major) suboptimalities due to
insufficient initial inventories and nonzero lead time requires less coordination effort
and pays most (see also the differences in the AGU between S1 and M1). The alignment of the remaining (minor) suboptimalities, which are prevailing in S1, requires
greater coordination effort, instead. This also explains why larger relative savings
could be obtained in later iterations for S1 – in contrast to M1 and Sect. 6.3, where
almost all savings have been identified after two iterations due to the comparatively
easy mitigation of the lion’s share of the suboptimalities.
48
See our discussion at the beginning of Chap. 5.
6.4 Models for Campaign Planning
183
100
80
Remaining gap (%)
60
40
20
0
−20
−40
−60
−80
0
2
4
6
8
10
Iteration
S1, NOBATCH
S1, BATCH
M1, NOBATCH
M1, BATCH
Fig. 6.8 Convergence paths for the coordination of campaign planning
Table 6.26 Sensitivities
regarding demand
CVd , seasonality
0, 0
0.2, 0
0.5, 0
0.2, 0.3
AGU (%)
AGS (%)
AGC (%)
321.7
2.0
119.5
307.2
1.4
117.5
308.3
1.3
118.2
399.4
0.6
203.1
Next, we consider the sensitivities regarding changes in the input parameters for
test instance generation. Since the focus in this section is on the instances with batch
restrictions (BATCH), we skip the other instances for this analysis.
We start with changes in the demand (see Table 6.26). Here, no unequivocal
results can be recognized. Somewhat counter-intuitively, both the AGS and AGC
are largest for the seasonal cycle; however, the absolute differences in the AGS have
been minor for all parameterizations considered.
Since only two variations for capacities have been included, we omit an evaluation of their effects and consider the variations in the production coefficients instead
(Table 6.27). Here, a pronounced trend can be recognized: Both the AGU and AGS
increase with variations in the production coefficients, whereas the AGC decreases.
A reason for this is that greater variations favor inappropriate relations among the
quantities of the single items supplied ordered by the buyer. This increases the AGU,
which in turn favors increases in the AGS. The lower values for the AGC indicate
that the difficulties for coordinating such structures are somewhat greater in general.
184
Table 6.27 Sensitivities
regarding production
coefficients
6 Computational Tests of Coordination Schemes
CVcoe , base value
0.2
0.5
1
1.5
AGU (%)
AGS (%)
AGC (%)
233.5
3.1
201.0
351.1
4.1
154.6
370.3
0.2
106.0
381.2
1.8
96.1
Table 6.28 Sensitivities
regarding TBO
TBOB , TBOS
AGU (%)
AGS (%)
AGC (%)
4, 4
382.8
2.2
161.5
3, 5
358.4
0.6
110.0
5, 3
260.4
1.1
145.1
Table 6.29 Sensitivities
regarding batch sizes
Batch size B, batch size S
1, 1
1, 2
2, 1
AGU (%)
AGS (%)
AGC (%)
277.4
1.7
109.0
412.5
4.7
128.7
312.1
1.7
180.1
Although near optimal solutions could be obtained for all parameter settings, the
results suggest that it is most beneficial to apply the scheme as a heuristic if the
variations in the production coefficients are low.
Next, we consider the sensitivities regarding the TBO (Table 6.28). The AGU are
largest for equal TBO. Moreover, a trend for a decreasing coordination performance
with greater differences in the parties’ TBO can be recognized. A potential explanation for this is that coordination may become more difficult if the optimal TBO and,
hence, the expected production patterns of buyer and supplier, are less aligned.
Finally, Table 6.29 shows the sensitivities regarding different batch sizes. The
observed increase of the AGU with greater differences between the parties’ batch
sizes is intuitive since it seems less probable that upstream planning yields wellaligned plans in that case. For greater batch sizes of the buyer, both AGU and AGS
are largest; the difference in the AGS between the other choices for the batch sizes
is somewhat counterintuitive, but minor, and might be due to stochastic influences.
Summarizing the results for the MLCLSPL-C, the scheme is able to significantly
mitigate the gap resulting from upstream planning on average. If minimum batch
sizes come into play, the results are even superior to the centralized solution generated by the solver, with little more than half of the computational time needed. This
suggests a further potential application of the scheme as a heuristic for multi-level
batch production problems.
6.5 Real-World Supply Chain Planning Problems
To verify the applicability of the scheme for real-world supply chain planning,
we have tested it for selected problems from clients of the SAP AG, Walldorf. In
Sect. 6.5.1, we present the characteristics of these problems and the associated
6.5 Real-World Supply Chain Planning Problems
185
(centralized) model formulation. Moreover, we discuss the consequences of
decentralization on the planning processes for these instances and their modeling. The results of our tests and a numerical example are presented in Sect. 6.5.2.
6.5.1 Planning Problems and Model Formulation
We evaluated three real-world test instances (abbreviated by #1, #2, #3 in the following) provided by SAP. These data have been used as input for modeling customer
problems in the module SNP (Supply Network Planning) of SAP APO (Advanced
Planner and Optimizer).49 Table 6.30 gives an overview of the dimensions of these
instances. A new term introduced here is PPM, the abbreviation for “production
process model.” A PPM is a modeling feature used in SAP SNP to describe the
characteristics of production processes (e.g., inflows and outflows of items, resource
consumption). Modeling production planning problems using PPM is an alternative to the model formulations presented in Sect. 2.2. We rely on the modeling
by PPM here because joint production, which is relevant for #3, can be modeled
straightforwardly that way and additional adaptations of the real-world data can be
avoided.
It is remarkable that even the smallest test instance, #1, is considerably larger than
the theoretical instances considered in the previous sections due to the large number
of locations and time periods. The structures of the supply chains that underlie these
test instances are sketched in Fig. 6.9.
In Table 6.31, we list the characteristics of the instances with impact on the model
formulation used.50
Based on the MLCLSPL-C,51 we have developed a model formulation that covers all the characteristics mentioned in Table 6.31. Although differing from the
general model implemented in SAP SNP, this formulation is able to capture the
decisions and restrictions relevant for the real-world problems considered.
49
Table 6.30 Overview of real-world instances
Number of. . . Locations Items PPM Production resources
Periods
#1
#2
#3
32
35
104
12
7
8
4
8
69
6
18
62
6
12
13
For a description of the functionalities of this module, see Meyr et al. (2007a, p. 362).
Note that we have “cleaned” the original data in order to remove infeasibilities and to reduce
the problem sizes for some instances, such that they can be tackled by a standard solver. Amongst
others, we have investigated two variants of #3 that differ by the inclusion of binary variables
(#3-MIP and #3-LP). See Püttmann et al. (2007, p. 23) for details.
51
See Sect. 2.2.2.2.
50
186
6 Computational Tests of Coordination Schemes
Fig. 6.9 Structures of real-world test instances
min
XX
cvp Xpt C
p2P t 2T
C
XXX
a2A j 2J t 2T
X X XX
XX
(RW-C) s.t.
X
b
e
mf jp Xpt
C
X
ptp C
p
X
X
LSljst C
X
a2A
l2ABa
p2P
csslj SSljt
p2P
P 2Ol ;t > p
s2S
XT ajtaj D
a2A
l2AE a ;t > a
XT ajt C
lscljs LSljst (6.7)
l2L j 2J t 2T
s2S
X
X X XX
XXX
BLljst C Iljt1 C
X
hlj Iljt
l2L j 2J E s2S t 2T
l
csel ICElt C
l2L t 2T
XXX
l2L j 2J t 2T
blcljs BLljst C
l2L j 2J E s2S t 2T
l
C
ctaj X Tajt C
X
BLljst1 C Iljt
X
dljst C
(6.8)
s2S
8l 2 L; j 2 J; t 2 T
s2S
X
amp Xptb C Xpte C
stp Ypt kmt 8m 2 M; t 2 T
p2P
amp >0
(6.9)
6.5 Real-World Supply Chain Planning Problems
187
Table 6.31 Characteristics of real-world instances
#1
Production
Variable production costs
x
Variable capacity consumption
x
Production lead times
x
Setup times
Production batch sizes
Minimum production lot sizes
Storage
Holding costs
x
Storage limits
x
Penalty costs for increase of storage capacity x
Safety stocks
Penalty costs for shortages of safety stocks
Transportation
Transportation costs
x
Transportation lead times
x
Minimum transportation lot sizes
x
Maximum transportation lot sizes
x
Demand fulfillment
Backlogs
Lost sales
x
Maximum lateness
X
Iljt iclj icmax
C ICElt
l
#2
#3
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
8l 2 L; t 2 T
(6.10)
j 2J
ICElt icemax
l
8l 2 L; t 2 T
SSljt ssljt Iljt
BLljst t
X
(6.11)
8l 2 L; j 2 J; t 2 T
dljst2 8l 2 L; j 2 JlE ; s 2 S; t 2 T
(6.12)
(6.13)
t 2Dt lmaxljs C1
XT ajt xtmax
8a 2 A; j 2 J; t 2 T
ajt
e
Xpt bpt Ypt 8p 2 P; t 2 T
(6.14)
Xptb bpt Wpt1
(6.16)
(6.15)
8p 2 P; t 2 T
Wpt Ypt C Wpt1 8p 2 P; t 2 T n fjT jg
X
Wpt 1 8m 2 M; t 2 T n fjT jg
(6.17)
(6.18)
p2P
amp >0
Wpt1 C Wpt Ypt C Yp0 t 2 8m 2 M; p 2 P; p 0 2 P;
p ¤ p 0 ; amp > 0; amp0 > 0; t 2 T n fjT jg
(6.19)
CAM pt CAM pt1 C Xptb C bpt Ypt
8p 2 P; t 2 T
(6.20)
CAM pt CAM pt1 C Xptb bpt Yjt
8p 2 P; t 2 T
(6.21)
188
6 Computational Tests of Coordination Schemes
CAM p0 minlotp
CAM pt Xpte C bpt
CAM pt Xptb
8p 2 P
1 Ypt
(6.22)
8p 2 P; t 2 T
(6.23)
8p 2 P; t 2 T
(6.24)
CAM pt1 C Xptb minlotp YI mt
8m 2 M; p 2 P; amp > 0; t 2 T (6.25)
CAM pt1 C Xptb D bsp Rpt C Spt 8p 2 P; t 2 T n f1g
Spt bsp .1 Y Imt / 8m 2 M; p 2 P; amp > 0; t 2 T n f1g
(6.26)
(6.27)
YI mt Ypt 8m 2 M; p 2 P; amp > 0; t 2 T
X
YI mt Ypt 8m 2 M; t 2 T
(6.28)
(6.29)
p2P
amp >0
Wp0 D 0 8p 2 P
Rpt 2 N0 8p 2 P; t 2 T
(6.30)
(6.31)
CAM pt 0 8p 2 P; t 2 T
Wpt 2 f0; 1g 8p 2 P; t 2 T n f1g
(6.32)
(6.33)
Spt 0; 8p 2 P; t 2 T n f1g
YI mt 0 8m 2 M; t 2 T
(6.34)
(6.35)
BLljs0 D 0; BLljsjT j D 0; LSljs0 D 0
Ilj0 D inilj 8l 2 L; j 2 J
BLljst 0
8l 2 L; j 2 JlE ; s 2 S
8l 2 L; j 2 JlE ; s 2 S; t D 0; : : : ; jT j
(6.38)
Iljst 0 8l 2 L; j 2 J; s 2 S; t D 0; : : : ; jT j
LSljst 0 8l 2 L; j 2 J; s 2 S; t 2 T
(6.39)
(6.40)
Xpt 0 8p 2 P; t 2 T
8a 2 A; j 2 J; t 2 T:
XT ajt 2 f0g [ xtmin
ajt ; 1
(6.41)
(6.42)
Indices and Index Sets
a Arc linking two locations, a 2 A; ABa and AEa are the locations at the
beginning and the end of arc a, respectively
JlE Subset of items sold at location l
l Location, l 2 L
p PPM, p 2 P
s Customer class, s 2 S .
Data
amp Capacity needed on resource m for one unit of PPM p
bpt Big number indicating the maximum quantity of PPM p in period t 52
bsp Batch size for PPM p
blcljs Backorder costs for one unit of item j of customer class s in a period
at location l
52
(6.36)
(6.37)
This quantity can be calculated analogously to bjt , see p. 13.
6.5 Real-World Supply Chain Planning Problems
csslj
Penalty costs for one unit of stock below the required safety stock
of item j at location l
Costs for one unit of storage capacity increase at location l
csel
Transportation costs for one unit of item j along arc a
ctaj
Variable production costs of PPM p
cvp
Primary, gross demand for item j of customer class s in period t at
dljst
location l
Consumption of storage capacity at location l by one unit of item j
iclj
Maximum storage capacity at location l
icmax
l
Maximum extension of storage capacity at location l
icemax
l
Inventory of item j at location l at the beginning of the planning
inilj
interval
Holding cost for one unit of item j at location l in a period
hlj
lmaxljs Maximum lateness for demand fulfillment of item j of customer
class s at location l
Costs for lost sales of item j of customer class s at location l
lscljs
Material flow of item j from PPM p (a positive value corresponds
mf jp
to production, a negative value to consumption)
minlotp Minimal lot size for item p
Required level for the safety stock of item j at location l in period t
ssljt
p Production lead time of PPM p
aj Transportation lead time for item j along arc a
xtajmi n Minimum transportation quantity of item j along arc a in a period
xtajmax Maximum transportation quantity of item j along arc a in a period
Variables
BLljst Amount of backorders of item j of customer class s at location l
in period t
CAM pt Campaign variable for PPM p in period t (quantity of the current
campaign up to period t)
Amount of inventory of item j at location l at the end of period t
Iljt
ICElt Increase of storage capacity at location l in period t
LSljst Amount of lost sales of item j of customer class s at location l in
period t
Integer number of full batches produced in the current campaign of
Rpt
PPM p up to period t
Quantity of the last batch of PPM p in period t which is not
Spt
finished in t
SSljt Undershot of safety stock of item j at location l in period t
Setup state indicator variable (D 1 if PPM p is set up at the end of
Wpt
period t, D 0 otherwise)
Production quantity of PPM p at the beginning of period t
Xptb
189
190
6 Computational Tests of Coordination Schemes
Xpte
Production quantity of PPM p that is not produced at the beginning of
period t
XT ajt Transportation quantity of item j along arc a in period t occurs in
n
( xt mi
aj if item j is transported on arc a in period t; D 0 otherwise).
Ypt Binary setup variable (D 1, if PPM p is produced in period t,
D 0 otherwise)
The objective function (6.7) minimizes the sum of the variable production costs,
transportation costs as well as the costs for inventory holding, backlogging, lost
sales, and the expansion of storage capacity. Constraints (6.8) are inventory balance
constraints specifying equal inflows and outflows for each item in each (storage)
location. Apart from the inventories of the preceding period, the inflows consist of
unfulfilled demand in form of backorders or lost sales, the production output of
the PPM, and the quantities transported to the location considered. The outflows
are made up by the demand, the new inventory levels, the backorders taken over
from the preceding period, and the quantities transported to other locations. Constraints (6.9) are standard restrictions for production capacities. Constraints (6.10)
force the utilization of the storage capacity below the sum of the available storage
capacity and the expansion of the storage capacity chosen. For the maximum expansion of storage capacity, a limit is imposed by (6.11). If the lower bound on the safety
stock is violated, constraints (6.12) determine the number of units that fall below
this lower bound. Constraints (6.13) restrict the maximum lateness for backorders.
Campaigns are modeled through restrictions (6.15)–(6.35). For a detailed description of these constraints, we refer to Sect. 2.2.2.2. Constraints (6.36)–(6.37) are used
for initialization purposes and constraints (6.38)–(6.42) determine the scope of the
decision variables. Thereby, constraints (6.42) define transportation quantities using
semicontinuous variables, i.e., if an item is transported along an arc, its transportation quantity has to exceed a prespecified lower bound.
A prerequisite for the application of the scheme is a decentralized structure of
the organization considered. In the original modeling of the real-world instances,
the existence of a unique centralized decision maker is assumed. In order to generate
a decentralized structure in the test data, we have divided the test instances into
data for two different entities, a supplier and a buyer, which comprise a subset of
the relevant locations each. The splits applied are stated in Table 6.32. Note that
for instance #3, only two different splits have been relevant due to the asymmetric
distribution of items and PPM among the locations.
Table 6.32 Splits of test instances
Split 1
Split 2
#1
#2
#3
Supplier (S)
A–D
A, B
A, B
Buyer (B)
E–L
C–G
D–H
S
C, D
A–C
B
Split 3
B
A, B, E–L
D–G
A, C–H
S
A, B
B
–
Split 4
B
C–L
A, C–G
–
S
A–F, H
A–E
–
B
G, I–L
F, G
–
6.5 Real-World Supply Chain Planning Problems
191
The models used in the scheme can be formulated analogously to those for voluntary compliance53 with the additional option for lost sales.54 Two minor issues
have additionally to be taken into account here. First, the option for transporting the
items supplied has to be modeled. For this purpose, dummy locations indicating the
destinations of the items supplied have been included into the supplier’s model formulation. Second, with the presence of tight storage capacities, infeasibilities may
be caused by the restriction that the buyer – when evaluating the supplier’s proposals – must not use less than the quantities specified by these proposals. To avoid
such infeasibilities, we allowed the buyer to supply smaller quantities than specified in the supplier’s proposals, but charged high penalty costs to exclude that such
proposals are considered as a systemwide improvement.
All input data for the decentralized models could be directly derived from
the real-world data, except for the costs for lost sales and backlogs for the supplier. Analogously to the well-known dilemma of determining optimal transfer
prices,55 the values for these cost factors that are optimal for the whole system cannot be identified without the knowledge of the dual prices for the optimal solution
to the centralized model. Since this solution is not known a priori, these costs have
to be determined by some rules of thumb (in practice). These rules, of course, do
not guarantee that the optimal values for these costs are determined.56
In the following, we apply a straightforward rule of thumb: We determine the
costs for backorders and lost sales for intermediary items based on their potential
impact on the corresponding costs for the end items. Hence, we recursively set
blcj D min
k2Sj
blck
pre
nk rjk
!
;
(6.43)
with npre
as the number of items preceding item k. The assumptions underlying
k
(6.43) are the following: For BOM structures with more than one predecessor item,
the backorders costs of the successor item should be partitioned evenly among the
predecessor items. For structures with more than one successor item, it is favorable
in case of shortages of a preliminary item to resort on shortages of the successor
items with the lowest backorder costs. Therefore, the minimum of the costs of the
successor items has to be taken in (6.43). The costs for lost sales are determined
analogously.
53
See Sect. 4.4.2.
See Sect. 4.4.3.
55
See, e.g., Schweitzer and Küpper (1998, p. 475).
56
In other words, the outcome of these rules is affected by the driver Q for the suboptimality of the
default setting.
54
192
6 Computational Tests of Coordination Schemes
6.5.2 Analysis of Solutions
For the tests of this section, we rely on the same specification of the scheme that
has been applied in Sects. 6.3 and 6.4 (augmented by the extensions to voluntary
compliance and lost sales presented in Sects. 4.4.2 and 4.4.3).57 We have run the
scheme for 10 iterations. We applied time limits of 3,600 s for the centralized models
and 60 s for the decentralized models. These time limits have been defined such
that they are only valid after a feasible solution has been identified. All centralized
models could be solved to optimality, except of those for the variant of #3 including
the campaign restrictions (#3-MIP). We display our main results in Table 6.33. In
contrast to the preceding sections, we state the gaps of the uncoordinated solution
(GU) and the gaps after coordination (GC) separately for all instances.
The results differ substantially among the instances and splits performed. For the
first three splits (abbreviated by “spl.”) for the smallest instance, #1, near optimal solutions could be obtained. For split 4, however, no improvements have been found.
In principle, this is not really surprising; due to the elevated model sizes, the probability that no systemwide improvement can be found is much more elevated than
for our smaller academic test problems. However, we additionally could identify a
particular reason for this result. Split 4 is characterized by both a large number of
transportation arcs for the items supplied and by the simultaneous presence of tight
storage capacity restrictions and minimum transportation lot sizes. This reduces the
number of systemwide feasible solutions considerably since supply quantities complying with the minimum transportation lot sizes may lead to infeasibilities due
Table 6.33 Solution performance of the scheme for the
real-world data
57
Instances
#1, spl. 1
#1, spl. 2
#1, spl. 3
#1, spl. 4
GU (%)
56.32
4.37
4.27
144.20
GS (%)
2.35
0.00
1.29
144.20
TC
1.0
1.0
1.0
1.0
TS
43.8
65.1
75.8
26.5
#2, spl. 1
#2, spl. 2
#2, spl. 3
#2, spl. 4
5.48
7.31
6.20
43.74
5.41
7.31
3.47
23.98
0.7
0.7
0.7
0.7
35.6
21.1
18.2
120.7
#3-LP, spl. 1
#3-LP, spl. 2
#3-MIP, spl. 1
#3-MIP, spl. 2
19.53
19.17
60.39
74.49
16.89
9.02
9.03
15.93
215.1
210.1
3,600
3,600
1,891.5
1,793.2
1,853.1
1,744.2
Note when carrying out these tests within the EU-project InCoCo-S, we have included further
minor modifications like a somewhat different determination of the penalty costs factor cp in
agg-P
CSB . Preliminary computational tests showed that the effect of these modifications is negligible.
Hence, we did not consider it as necessary to rerun this study using the exact specifications
presented in Chap. 4.
6.5 Real-World Supply Chain Planning Problems
193
to excess inventory holding. In fact, when testing a modified model without the
restrictions on the storage capacities for split 4, the resulting gap from upstream
planning (144.9%) could be mitigated almost completely by the scheme (gap after
coordination: 3.11%).
For #2, the improvements for the two first splits have been low or non-existing.
Also the initial gaps after upstream planning have been relatively small here, which
usually makes the identification of further improvements more difficult. For the
larger GU in split 4, however, a considerable fraction could be mitigated.
For #3, a large part of the suboptimality from upstream planning has been mitigated on average. This result is somewhat surprising in light of the huge size of this
instance. A potential explanation is that the costs factors (e.g., the unit costs for lost
sales) differ substantially among items in #3, whereas for #1 and #2, the variations
of these costs and of other factors like production coefficients have been rather low.
We suppose that greater variations may increase the leeway for the identification of
improvements. Further note that for #3, the difference between the time for running
the centralized model and the scheme is substantially smaller than for the other
instances. This is due to the disproportionally increasing computational effort for
the solution of larger optimization models, an effect which has already been taken
into account when setting the time limits for the centralized and the decentralized
models.
In Fig. 6.10, we display the convergence paths for the test instances with a substantial improvement by the scheme, i.e., GS 5%. Interestingly, the convergence
Remaining gap (GS/GU) (%)
100
80
60
40
20
0
0
2
4
6
8
Iteration
1,spl.1
#2, spl. 3
#3-LP, spl. 2
#1, spl. 2
#2, spl. 4
#3-MIP, spl. 1
#1, spl. 3
#3-LP, spl. 1
#3-MIP, spl. 2
Fig. 6.10 Convergence paths for the real-world instances
10
194
6 Computational Tests of Coordination Schemes
Fig. 6.11 Data of #1, spl. 1
speed differs substantially among instances and splits. For some of them, e.g., #1,
spl. 2, almost the entire benefits from coordination have been identified after only
one iteration. For others (e.g., #2, spl. 4), substantial improvements have been found
even in the tenth iteration. All in all, most of the potential coordination benefits have
been obtained after five iterations, which indicates a modest effort for the corresponding coordination activities.
To illustrate how the real-world problems are tackled by the scheme, we provide a
numerical example for the smallest instance, #1. Figure 6.11 displays the structure of
#1 including the locations of the PPM, their inflows and outflows, and the locations
with end item demand, which, for ease of exposition, have been aggregated into one
client location (H). We consider split 1, i.e., we assume that locations A, B, C, and
D pertain to the supplier, and the other locations to the buyer. Note that there are
several possibilities to produce single items in #1, e.g., item 1 can be produced by
the supplier at location A or by the buyer at location F.
The supply plans for the first 16 periods of the solution from upstream planning
are displayed in Fig. 6.12.58 The column “initialization buyer” displays the buyer’s
orders for the supplied items and the column “evaluation supplier” the quantities
the supplier is willing (and able) to deliver. Here, the complete orders of the buyer
cannot be fulfilled. Due to positive production and transportation lead times, the
orders for items 1 and 4 cannot be delivered on time. The order for item 2 is not
fulfilled at all. The minimum transportation lot size for the arc between the locations
B and F (5 units) exceeds the quantities required per period. Since backlogging is
not allowed in #1, a supply of this item has not been possible. As a consequence, the
buyer had to incur additional costs for lost sales resulting in a large suboptimality of
upstream planning (56.32%).
The proposal generation within the scheme is illustrated by Fig. 6.13. In his first
proposal, the supplier offers to deliver item 2, but different quantities than required
58
Note that their absolute values have been altered for reasons of privacy.
6.5 Real-World Supply Chain Planning Problems
195
evaluation supplier
item 1 [units]
item 1 [units]
initialization buyer
10
5
0
10
5
0
2 4 6 8 10 12 14 16
periods
item 2 [units]
item 2 [units]
2 4 6 8 10 12 14 16
periods
10
5
0
10
5
0
2 4 6 8 10 12 14 16
2 4 6 8 10 12 14 16
periods
item 3 [units]
item 3 [units]
periods
10
5
0
10
5
0
2 4 6 8 10 12 14 16
2 4 6 8 10 12 14 16
periods
item 4 [units]
item 4 [units]
periods
10
5
0
10
5
0
2 4 6 8 10 12 14 16
periods
2 4 6 8 10 12 14 16
periods
item 1 [units]
Fig. 6.12 Supply quantities with upstream planning (#1, spl. 1)
1st proposal supplier
10
5
0
1st proposal buyer 2nd proposal supplier 2nd proposal buyer
3rd proposal supplier
10
10
10
10
5
5
5
5
0
item 2 [units]
4 8 12 16
item 3 [units]
0
4 8 12 16
0
4 8 12 16
4 8 12 16
10
10
10
10
10
5
5
5
5
5
0
0
4 8 12 16
0
4 8 12 16
0
4 8 12 16
0
4 8 12 16
4 8 12 16
10
10
10
10
10
5
5
5
5
5
0
0
4 8 12 16
item 4 [units]
0
4 8 12 16
0
4 8 12 16
0
4 8 12 16
0
4 8 12 16
4 8 12 16
10
10
10
10
10
5
5
5
5
5
0
0
0
0
0
4 8 12 16
4 8 12 16
4 8 12 16
4 8 12 16
4 8 12 16
periods
periods
periods
periods
periods
Fig. 6.13 Proposal generation by the scheme (#1, spl. 1)
196
6 Computational Tests of Coordination Schemes
by the buyer due to the minimum transportation lot sizes. This proposal realizes a
considerable percentage (85.5%) of the total surplus achieved by the scheme. At
first glance, the underlying idea of this proposal, i.e., to propose greater delivery
quantities and thereby to assume that the buyer will do some additional inventory
holding, seems trivial. In fact, we admit that this specific proposal or a similar one
could also have been found by manual negotiations. However, note that this idea
does not always work since larger supplies than needed might not always be feasible
for the supplier due to the tight restrictions on storage capacity in some locations.
Moreover, as we will show in the following, the scheme is able to identify further
and rather unobvious improvements in later iterations.
In her first proposal, the buyer complies with the reduction of the supply for items
1 and 3 in earlier periods proposed by the supplier, but asks again for a more even
supply of item 2. The rationale behind this is that a supply of item 2 in early periods
would substantially reduce the buyer’s costs for lost sales. Although a previous proposal with this characteristic (i.e., the initial solution) turned out to be inferior, the
buyer running the scheme is not aware of the exact reason for this. The potential reasons of the inferiority of upstream planning compared to the supplier’s first proposal
can be attributed to the changes in the supply of any of the items 1, 2, and 4 (compare the right column of Fig. 6.12 with the left column of Fig. 6.13). In the scheme,
the buyer charges internally determined penalty costs for a subset of these changes
to prevent that the entire changes are repeated in the new proposal generated. Here,
the inferiority of upstream planning has been attributed to the changes for item 1,
but not for items 2 and 4. This choice, however, did not result in a favorable new
proposal.
In his second proposal, the supplier tries different supply quantities for item 2, but
without success. In his next proposal, the buyer internally attributes some penalty
costs for changes in the supply of item 2. The resulting proposal addresses another
region of the solution space, i.e., the delivery of items 1 and 2 is partially replaced
by the delivery of item 3. This proposal did not lead to a direct improvement. The
knowledge, however, that also the supply of item 3 could be useful for the buyer
served as the base for the third proposal of the supplier. There, additional percentage
cost savings of 9.9% are obtained by incorporating the supply of this item. The rest
of the savings are found in later iterations, which are not illustrated here.
Summarizing, the results obtained in this section underscore the practical applicability of the scheme proposed. Although all real-world test instances comprise
different supply chain structures, decisions and restrictions, a substantial part of the
suboptimalities due to upstream planning could be mitigated at least for most of the
splits considered. As expected, the gap closures have been somewhat lower than for
the smaller-sized academic instances studied in the previous sections. The favorable
results for #3, however, suggest that the scheme can also be successfully applied for
problems comprising a huge number of decision variables and a series of difficult
restrictions.
Chapter 7
Summary and Outlook
This thesis has proposed new supply chain coordination mechanisms, which
incorporate a number of characteristics favoring their applicability in practice.
The mechanisms can coordinate plans generated on the basis of complex mathematical programming models on behalf of self-interested parties holding private
information, and do not require the participation of a third party. In the literature, all
of these characteristics have been regarded as important, but they have never been
covered within a single approach.
An introduction to the research problem tackled by this thesis is given in Chap. 2.
There, we have presented mathematical models for (intra-organizational) mid-term
supply chain planning (Master Planning) as well as basic definitions that are used
throughout this work. Moreover, we have shown how to model inter-organizational
planning processes formally. Last, we have determined drivers for suboptimalities
without coordination, which, e.g., result from the application of myopic procedures
such as upstream planning.
Chapter 3 provides a review of the state-of-the-art in supply chain coordination.
In contrast to existing surveys, which focus on specific types of mechanisms, our
review covers the whole spectrum of the related literature. The mechanisms and the
underlying ideas are described separately according to the assumptions on parties’
information statuses (no, unilateral, or multilateral information asymmetry) and the
concepts from game theory the mechanisms rely on.
The coordination mechanisms presented in Chaps. 4 and 5 constitute the principal contribution of this work. Two main tasks have to be resolved by them: The
identification of coordinated solutions, i.e., systemwide improvements compared to
a given initial solution, and the determination of incentives for their implementation.
In Chap. 4, several coordination schemes have been devised that identify coordinated solutions for a broad range of Master Planning problems and cover different
requirements on the information exchange.
At first, we have developed two variants of a generic scheme, which can be
applied for coordinating linear programming (LP) models in arbitrarily structured
decentralized organizations. In this scheme, parties iteratively exchange proposals
about the use of the central resources (i.e., supply quantities in supply chains).
Depending on the mechanism the scheme is embedded in, the cost effects to
these proposals are either reported iteratively by all but one decentralized party or
M. Albrecht, Supply Chain Coordination Mechanisms: New Approaches
for Collaborative Planning, Lecture Notes in Economics and Mathematical Systems 628,
c Springer-Verlag Berlin Heidelberg 2010
DOI 10.1007/978-3-642-02833-5 7, 197
198
7 Summary and Outlook
simultaneously disclosed by all parties. We have shown by analytical proofs that
both variants are able to identify the systemwide optimum in a finite number of iterations, provided that the decentralized problems can be formulated as LP models.
The variant with an iterative exchange of cost changes is even able to identify the
systemwide optimum if one decentralized model is of a mixed-integer programming
(MIP) type.
The second scheme developed in this work aims to coordinate supply chains
consisting of one buyer and one or multiple suppliers planning based on multi-level
uncapacitated lot-sizing problems (MLULSP). This scheme also relies on the exchange of supply proposals, but – in contrast to the generic scheme – does not
require specific assumptions on the exchange of cost changes. We can show that
the maximum number of proposals generated increases linearly with the number of
periods and the number of the items supplied. For the special case of time-invariant
end item demand, zero initial inventories, and the production portfolio of the buyer
limited to one item, we have derived an upper bound on the maximum gap between
the systemwide optimum and the best solution identified by this scheme.
Moreover, we have shown how to adapt the schemes for an effective coordination
of Master Planning in two-tier supply chains. First, the generic scheme for LP has
been customized for coordinating Master Planning among two parties. Second, we
have presented modifications that improve the convergence and allow an application
of this scheme to MIP models. Third, we have devised extensions that effectively
coordinate decentralized models with capacitated lot-sizing, voluntary compliance
by the supplier, lost sales, and multiple suppliers involved in the coordination.
For each of the schemes, extensive numerical tests have been conducted based
on randomly generated test instances. The results of these tests have been reported
in Chap. 6. In all settings considered, the systemwide costs have been significantly
reduced compared to upstream planning after 10 or 20 iterations. For small problem sizes or specific model classes (e.g., the MLULSP), near optimal solutions can
be obtained. For the campaign planning models, the scheme can even identify improvements over the solutions to the centralized model that have been found after
a given time limit, which is two times the limit for running the scheme on average.
In addition, we have tested the performance of the scheme for real-world data originating from customers of the SAP AG. Although these data comprise additional
constraints like storage capacity and transportation lot size restrictions and involve
huge model sizes, the scheme was able to identify significant improvements.
Apart from the identification of improved solutions, the second task to be tackled
by a coordination mechanism is the establishment of incentives for the implementation of these solutions. For this purpose, we have presented in Chap. 5 three different
contractual frameworks, where the schemes can be embedded.
First, the sharing of the surplus can solely be determined by the party that does
not report its cost changes during the execution of the scheme. For an efficient coordination, fair and truthful behavior by all parties is required then. Fairness, however,
has only been observed to a certain degree in behavioral experiments. In spite of
that, we believe that this mechanism can be employed in organizational structures
where one dominant party carries out auditing measures like open-book accounting.
7 Summary and Outlook
199
Second, we have devised a contractual framework where truthful cost reporting
is a Nash equilibrium for parties. There, the surplus is shared such that the costreporting parties obtain lump-sum payments, while one party (which pays these
lump sums) receives the remaining share of the surplus. Analyzing parties’ best
strategies in this mechanism under the assumption of prior incomplete knowledge
about the surplus, a lower bound on the efficiency of this framework has been derived: For two parties and uniformly distributed prior knowledge, the efficiency
exceeds 75% on average.
Third, the surplus can be shared by a sealed bid double auction. Here, parties
simultaneously submit their costs changes in form of sealed bids for all proposals
generated and implement the proposals with the lowest systemwide costs. The surplus is shared equally among parties. Again, a lower bound on the efficiency of this
auction can be derived.
Finally, we have outlined how the mechanisms can be adapted if production planning is based on rolling schedules.
Summarizing, the schemes and mechanisms proposed form generic, innovative
concepts for collaborative supply chain planning and include a series of features
favoring practical applications. Given the existence of an effective scheme, the
mechanisms can be applied for coordinating any decisions in decentralized systems.
The generic scheme is able to identify the systemwide optimum for any decentralized LP models within a finite number of steps. The applicability of (a modified
version) of this scheme to Master Planning problems of one buyer and one or several suppliers including binary variables has been shown by the computational study
of this work.
An investigation of the transferability of the schemes to other economic decision
problems seems worthwhile. This way has been pursued by Püttmann (2007),1 who
adapts the generic scheme to the coordination of intermodal freight transportation.
Apart from transportation, other potential applications are in areas like production
scheduling and controlling, where the cost allocation for several agents sharing a
common resource has to be resolved, as well as different organizational structures
like three-tier supply chains. Furthermore of interest is the determination of the
problem classes for which the scheme proposed constitutes an effective heuristic
(like for multi-level campaign planning problem with batch size restrictions).
Challenges for future research are analyses of the actions of parties using
repeatedly the mechanisms proposed. As mentioned in Chap. 5, learning and
signaling strategies have to be taken into account then, which considerably augments
the complexity of the resulting modeling. Both laboratory research to evaluate
the empirical behavior of parties and analytical characterizations of parties’ best
strategies constitute promising ways to advance this area of research.
1
See Püttmann (2007, p. 63) for the outline of her application setting.
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