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LOGS:
AN OPTTMIZATION SYSTEM
FOR LOGISTICS PLANNING*
by
William D. Northup
and
Jeremy F. Shapiro
OR 089-79
October 1979
*This is a revision of an earlier version dated June 1979.
ABSTRACT
This paper reports on the design, implementation and use in a large
process manufacturing firm of a general purpose logistics planning system,
called LOGS.
A novel and essential component of LOGS is a set of basic
decision elements for describing logistics models; they provide an easy
to use interface between the data in the firm's information systems and
mathematical programming modeling and optimization capabilities.
Included
in the paper are discussions of experience with the system applied to production allocation, supply contracting, analysis of a acquisition.,. and
other logistics planning problems.
LOGS:
AN OPTIMIZATION SYSTEM FOR LOGISTICS PLANNING
Introduction
Computer hardware and software have developed to the point where all
large manufacturing companies have extensive information systems containing
data about present and projected operations at all levels from purchasing
to sales.
In spite of the availability of data, logistics planning decisions
remain difficult to evaluate because of the complex interactions between
manufacturing levels and the geographical regions in which the company operates.
Typical decisions are those about the size and location of new
facilities, product mix, terms for contracts with suppliers and customers,
as well as many operational decisions of production and distribution.
In recent years, a variety of mathematical programming models have
been proposed to aid in logistics planning.
The models are attractive
because they describe a manufacturing system in its entirety, thereby facilitating a global optimization that attempts to reconcile important tradeoffs between the profit contributions of different products, production and
distribution costs at different manufacturing facilities, contractual
arrangements with different suppliers, and so on.
However, the models have
not yet been widely integrated into logistics planning at the corporate level,
in part because of the technical difficulties in constructing them from the
relevant data and in using them in a flexible and expeditious manner.
This paper gives details about a logistics planning system, called LOGS,
that has been successfully implemented in a large process manufacturing firm.
LOGS analyzes a given logistics planning problem by automatically generating
an appropriate mixed integer programming (MIP) model from the relevant data,
and then optimizing it. This model is generated by the Logistics Modeling
2
Program that provides an easy to use interface between the data in the
company's information systems and mathematical programming modeling and
optimization capabilities.
An analyst using LOGS is not required to know
anything about mathematical programming modeling techniques or algorithms
in order to specify and analyze a logistics planning problem.
LOGS uses
cost, activity and resource data about the company's distribution network,
purchasing, manufacturing and sales activities plus additional management
policy constraints that cut across individual facilities.
LOGS also writes reports based on the MIP solutions.
Many of these
reports are straight translations of the solutions to other formats although
some further accounting calculations are also performed.
An example of
the latter are fair transfer prices reflecting accrued costs on products
as they flow through the stages of the manufacturing process.
Moreover, the existence of LOGS permits and encourages the establishment, maintenance and use of common data bases in analyzing related logistics
planning problems.
Models for strategic, tactical and operational planning
may share a description of system-wide interactions.
For example, the pro-
duction and transportation data used to determine monthly production plans
can also be used to determine and revise quarterly budgets and production
standards.
The form of the models used in these two cases is identical.
The differences are in the lengths of the planning horizons and the degree
of certainty in sales projections.
LOGS was developed for a specific company but its applicability is much
more general.
It is capable of generating and optimizing mixed integer
programming models for a wide range of decision problems faced by many
manufacturing firms.
LOGS was implemented in PL/I for interactive or batch
use on IBM 370 computers.
Figure 1 gives a schematic representation of the
system; the ellipses correspond to data and the rectangles to programs.
3
The plan of this paper is the following.
The next section contains a
detailed discussion of the Logistics Modeling Program.
A sample problem
illustrating many of the modeling constructs is given in the section after
that.
We then discuss features of the MIP optimization performed by LOGS.
The following section is devoted to our experience with LOGS applied to
actual logistics planning problems.
The final section contains a few
concluding remarks directed mainly at future extensions.
An important
aspect of the use of LOGS that we will not discuss in this paper are the
front end programs that feed data to LOGS in the formats accepted by the
Logistics Modeling Program.
The management of these information systems,
as well as the implications and consequences to the company of centralized
logistics planning, will be discussed in a separate paper.
Automated Logistics Modeling
The Logistics Modeling Program accepts basic decision elements describing a logistics planning problem and creates an appropriate MIP model
for optimizing it.
These basic decision elements are a novel and essential
feature of LOGS that permits a problem to be described in natural terms by
an analyst.
The exact specification of the basic decision elements resulted
from extensive discussions with analysts experienced in the use of logistics
planning models in the company and elsewhere.
As a result, LOGS has the
capability to create MIP models describing a wide variety of logistics
planning problems.
The generic logistics planning problem that can be analyzed by LOGS
is comprised of three components.
First, there is a distribution network
that describes product flows at all levels from the suppliers of raw materials through intermediate stages of manufacturing and warehousing to the
4
Figure
5
markets.
Superimposed on the distribution network are production activities
that describe how the products are bought, transformed, stored and sold.
Finally, there are logical constraints cutting across the network that link
and restrict production activities.
An example is a constraint stating
that a particular product can be produced in at most three plants.
Network terminology is used extensively by the Logistics Modeling Program in the description of logistics planning problems.
However, the
general class of models that can be created by the program includes many
that are not pure network optimization models because of the presence of
non-convex cost functions and constraints that are not easily represented
as flow balance constraints.
The generic model is defined over a distribu-
tion network consisting of nodes and arcs (i,j;p) describing directed
connections between nodes i and j by product type p. The nodes in the
distribution network represent facilities considered in a broad sense to
include plants, machines, suppliers, warehouses and markets.
Products
include raw materials, work in process and finished goods to be sold on
the market.
Products are converted at nodes by processes.
Let Q denote
the set of arcs and let x. denote the variable flow of product p in the
z3
arc (i,j;p).
q at node i.
Let yq
denote the variable production activity for process
6
As shown in Figure 2, each node i in the distribution network can be
expanded to a set of nodes 0Q corresponding to the processes at i.
Arcs for
PI
I
l_
(i)
P2
. I
L
(9
Figure 2
flows within a node and their associated flow conservation constraints
are generated automatically as needed.
For simplicity, these arcs will
be considered elements of a in the following discussion.
Processes include conversion of input product to output products at
intermediate manufacturing nodes, conversion of finishing product to
income at the market nodes, and conversion of costs to raw materials at
the supply nodes.
Five basic desision elements can be used to define and
constrain these process activities and to describe their costs:
7
(NA)
*; node activity
*
'.=
-:
material balance
(MB)
recipe
(RE)
fixed ratio
(FR)
product conservation
(PC)
For expositional reasons, we will discuss these relationships using standard
mathematical programming terminology but we wish to emphasize again that
the analyst does not describe a logistics. planning problem in this way.
The
following section gives a sample logistics planning problem and its description. in terms of its basic decision elements.
The node activity is the product flow or combination of product flows
and process activities that are used to compute costs at the node.
If
there
is no explicit specification of node activity, material balance, recipe or
product conservation, then by default the node activity is the variable
0
yi given by
0
(i;p)
L
EC,
ixp.
(i, i:;P) e 0
Process 0 (and other processes specified for a node) can be used.to specify
a general cost function of the type shown in Figure 3.
The equality between
total flow into node i and out of i indicates that the default option also
includes an implicit material balance equation for the node.
8
COST FUNCTION
_
_
Uosr
i
i
F
-1%
r-Lm
III.. !
II
L
,
F O - close dow
F11
-
!
II.
I-_
Node Activity
ll
ML
LL = lower limit
cost
UL - upper limit
fixed cost
Figure 3
The explicit (non-default) specification of node activity (NA) can be
used to define a new activity for node i
r
Yi =
( ,k;pvPxp
PCi
where the weights
( ,k;p) £
t
P, z q , vP are specified by the analyst.
The variable y.
can be used to compute costs as in Figure 3 and upper and lower bounds on it
9
can be specified.
Note that this broad definition of node activities per-
mits the acc=ulation of costs incurred at several levels of a physical
faclity; e.g., indirect plant costs, indirect machine costs and direct
costs for specific processes..
Alternatively, the Logistics Modeling Program accepts the specification
of a material balance (MB) equation
(i k;p)R
ea
(Ji;p a 1
When the material balance option is used for process ,
the weighted sum of
inputs (equal to the weighted sum of outputs) is automatically used as the
node activiy
y
The recipe option (RP) is used to describe process q whereby input
products are converted to output products at node i. Let Sl,... ,sa denote
a set of input products and t,
.. ,~ denote a set of output products.
product is a reference product; we assme this to be s
by specifying the quantities a,
Q-
2,...,c
.
and bim, m
A recipe is given
1,...,,
implying the equation
aH
b
Z
a
5
£
im ,Qi;s 1 )EQ i
Moreover, we define the variable y
for Q
sm
(i,k;tm) Q
I
ri;s)ea
t
i
m..,. for
xS.
j
One
2...,a
thereby
10
The fixed ratio (FR) option constrains flows of different products
into or out of a node, or processes at a node.
This option is needed
to model, for instance, constraints imposed by outside suppliers.
The
constraints have the form
r
St
t
(i,
(i,k;s)ea
t)e
.
The final option, product conservation, permits the specification of
individual constraints by product of the form
r
-
±
(i,k),a
Upper and lower bounds can be imposed on these sms.
Product conservation
is used to modeL, for example, the flows of individual products through a
warehouse.
Additional information required to create a logistics planning problem
is-a specification of unit costs, upper and lower bounds for each arc
(i,J;p)c O.
The current implementation of LOGS does not permit nonlinear
costs on arc flows.
These can be modeled by the creation of dummy nodes, one
for each arc for which the costs are nonlinear.
The final type of input is a specification of side constraints that
cut across nodes such as multiple choice, logical implication and logical
11
equivalence constraints.
These constraints are optional and need not be
given for a particular logistics planning problem.
The Logistics Modeling Program reads the nodes, arcs and logical
constraints files discussed above and creates an MIP model file to represent it
in standard MPS format.
The translation is done one node at a
time for which the required variables and constraints are defined as indicated by the basic decision elements.
specified arcs are created.
Then variables corresponding to
The rules used to generate the
are straightforward although sometimes complicated.
A significant effort
was made to find effective and compact model representations;
a
mnimal. ntbe
IP models
for example,
of zero-one variables are used to represent a non-convex
cost curve such as the one shown in Figure 3.
MIP problem implied b
the input fies
More importantly, if the
is too large for efficient optimiza-
tion, then a second, smaller HIP problem is created as a surrogate to be
used in optimizing the true one.
More details on how this is done are
give= in the section on MIP optimtzation.
Sample Problem
Figura 4 shows a logistics planning problem that we use for' illustrative purposes.
8B,. CC.
The SP nodes are the suppliers of. the input products AA,
These are converted at the mill node MIL to the finished products
DD and EE which are transshipped through the warehouse WAR to be sold at
the markets
R1 and MR2.
12
SALE LOGISTICS PLANNING PROBLEM
BB
DD
DD
Figure 4
13
Table 1 is a listing of the input files (nodes and arcs) for a logistics plannng problem defined over the network of Figure 4.
The first line
of the nodes file gives the costs of supply of AA and BB from SP1.
In
addition, the FPR specification states that AA and BB must be bought in
equal. quantities.
Figure 5 depicts the implied cost function for SP1.
$
Fixed
cost.
AA + BB
5
12
Fixed Product Ratio
AA tBB
Cost Function - Supplier 1
Figure 5
The cost function for SP2 uses flow of BB as the node activity.
It
consists
of a fixed cost of 3.75 with a unit cost of 1.625 up to 10 units followed by
a unit cost of 1.875 up to the upper limit of 20 units.
There are two processes specified at MIL denoted by $P1 and $P2.
These
I 14
*
.
NODES
FLE: ODUBOS
,· .-
..
.
T~~~~~~~~
1
. <88
20.
50
50
L.Z78
SPL <
FR:<AA L.0
1.875
SPZ <
2.35
SP3 <
2.48
SP4 <
0.33
"NA
-- $Pt
.i,.
-L.
1.18 SPL
0.8
15 ·
-RP:>8
L.4-7
M -L
-
>AA:
0.05
RPWA>8
WAR,
NA :
IC: DO
';'-----:'
50
-t
SOD
>0D
Sit
>E
F '-;.tLE~
SP
wA"2
' r
-
0.33
<EE
0.41
1. 5
-
-
- EE.
4:
*-
-- 39 .8
- -204.
-:T
.
-- :
';,.-
--
-.-
:':*
8.
38..
...
$00>OI.3'-
-'-"
40
-
-35. 4
BO&r
..
.
Z.
: A,
-
-173.1
20
-'
-.
..
.
'
-
. '--.
.,....
'
-.-'--
'"-7-
-.
t.2
·
-
.1t
.--
fltL- 2-.-t:
B M L 2. - 617
S
.:- .
r IAQ 3
.'!t,
W.
EMIL
JA . !X' K t 3.42
X 14'
0.37
'.(00'
<EE
-21S.8
; -,
CC
0. 4.
<00
-=
S
-1t92.2
SPL AA ,4tL
S-
-"-"--
-
-
2
29-
-223.8a-
>E
.
.. ._....
>CC- 1 L
3..Z5
.4>CC-
<EE
-.
SE
..
. - P2 - 1.2
. 2'.21
.5
-
'
..... 6Z5
3.75
6.0
.... .
·-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
<DO
.
.'
1.455-
5
-
-5.7
--
1.05
SMY >00-t93.
.q RL
.
.
;
4.0
.
.
:
.
.
.
.
MR I 3.57.
E M2 2 . 6
D~o M 2 2.80
Em-
.
Listing of. Input Files - Nodes and -Arcs -
'or Logistics Plamning Problem
Table
a
- ..-
15
are depicted in Figures 6 and 7.
equal to
The node activity at the MIL level is
xnode activity of $P1 + 1.2xnode activity of $P2
Thus, indirect costs at the MIL level are given by 0.33 (AA
AA + 1.2BB.
+ 1.2BB).
The income (negative cost) functions at the market nodes are given in
the final. lines of the nodes file.
pieces and for
The functions consist of a number of
R1, there are shut-down costs of 38.7 and 47.3 for not
supplying DD and EE, respectively, at levels equal to the minimal specificaonsna of 4 and 2 units.
The arcs file gives the unit transportation costs
for flows in the indicated arcs; e.g., 1.1 for each unit of AA sent from
SPL to MIL.
The. Logistics Modeltng Program created a mixed integer programming
problem from the input files that consisted of 32 rows, 9 zero-one variables
and 35 continuous variables.
PSX/370 was used to optimize the problem and
also- to locate alternative optima within 50% of the maximal profit attainable.
16
PROCESS
:
$
jI
II
I
_
r
I
RECIPE:
Inputs
1.0 AA, 0.8
Outputs
0.4 DD, 0.33 EE
Figr
BB,
AA
1.1C
6
P3ROCESS 2:
I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Fixed
Cost
RECIPE:
Inputs
Outputs
1.05 AA, 1.0
.37 DD,
Figure 7
BB,
.41 EE
1.17 CC
17
Mixed Integer Programming Optimization
LOGS uses the IM system MPSX/370 to perform the mixed integer programing (MIP) optimization on the problems created by the Logistics
Modeling Program.
Alternatively, any mathematical programming system for
370 computers that accepts problems in MPS format could be used.
It is
important to emphasize that our experience thus far indicates that many MIP
problems can be predictably and dependably solved if the number of integer
variables is not too large (less than 150) and if the number of constraints
is not too large (less than 2000).
following section.
Detailed results are discussed in the
This is somewhat contrary to the widely held feeling
that MIP computation is almost always extremely difficult.
Nevertheless,
IP problems with many zero-one variables, or those with
several thousand rows, can be implicitly proposed by users of LOGS with
large logistics planning problems to be analyzed.
For this reason, LOGS
will automatically aggregate an MIP problem if it is judged to be too large.
The analyst can also tell LOGS to aggregate the problem if desired.
The
criterion for automatic aggregation is the anticipated size of the linear
programming subproblems to be analyzed during the branch and bound stage
of MPSX/370.
Aggregation is desirable only if
the resulting linear pro-
gramming subproblems are sufficiently reduced in size to more than compensate
for the additional overhead.
The aggregated MIP problem is used as a surro-
gate to optimize the original MIP problem.
LOGS aggregates large MIP models by reducing the number of distinct
market-product locations.
A typical warehouse distribution problem could
have 150 markets to which 40 products are distributed.
bute 6000 rows to the model, and it
This would contri-
is usually possible to reduce this
18
number to 1000 by aggregation without introducing great approximation
errors.
Specifically, nodes are distributed into aggregate nodes on the
basis of similar transportation costs from plants and warehouses to the
markets, and products are clustered into aggregate products on the basis
of similar market sales functions.
Unit costs for arcs in the aggregate
network are always selected to understate the true cost relative to the
original logistics planning problem.
flect bounds on market demand.
Bounds on aggregate arc flows re-
Thus, the aggregation is always constructed
so that it provides lower bounds on the minimal cost in the original MIP
problem.
The aggregate MIP problem is optimized using MPSX/370 and each
feasible solution is disaggregated to generate a solution to the original
MIP problem.
The reader is referred to Geoffrion [1] and Zipkin [4] for
similar approaches to aggregating logistics planning problems.
Experience with the System
LOGS is fully implemented and being used in a batch mode by a large
process manufacturing firm to study a variety of logistics planning problems.
Table 2 gives a few representative results.
The purchasing problems
1 and 2 address company strategy in making contracts with suppliers of
primary inputs to production.
A minimal cost strategy is difficult to
find because of complicated supply cost functions involving fixed costs and
price breaks for high volume purchases.
In addition, some suppliers impose
fixed ratio (FR) restrictions on supplies due to the nature of their own
processes for making these supplies.
It is well worth noting that the
optimal solution to problem 1 was $4 million lower than the cost of the
19
Problem
Number
Description
Rows
Zero-One
Variables
Continuous
Variables
Seconds to
Generate
Seconds to
Optimize
(MPSX/370)
1
Purchasing - Item one
290
40
340
4
7
2
Purchasing - Item two
330
38
600
5
8
3
Monthly Production Plamning
2000
0
8000
53
288
Monthly Production Plaaning
With Fixed Costs:
4a
Unaggregated
2150
70
10400
56
540
4b
Agregated
1410
70
7000
110
925
Table 2
Time
are on an IM 370/168
20
proposed strategy obtained prior to the use of LOGS.
This large savings
represented real dollars to be saved by the company in its purchases for
the coming year.
Moreover, the Purchasing Department was able to use the
results of the model to negotiate better terms with some of the suppliers.
The monthly production planning models are aggregate models used to
assign forecasted demand for all products to specific plants and machines
across the U.S.
Problem 3 is a linear programming (LP) version of the
problem that had previously been generated and solved by a commercially
available matrix generation language and LP optimizer with a generalized
upper bound (GUB) option.
This run was made to validate LOGS and to com-
pare its performance with the other system.
We found that MPSX/370 without
the GUB option was able to find an optimal solution to the model in approximately the same length of time as the LP optimizer with GUB despite the
fact that 80% of the rows were GUB rows.
This relative efficiency of
MPSX/370 without GUB is not sustained, however, for larger GUB problems.
We were able to overcome this deficiency of MPSX/370 by procedures for choosing better starting bases and also by putting into the objective function
those constraints that are merely accounting balances.
In general, LOGS
generates these LP and MIP models at least 10 times faster than the generator
written in the matrix generation language.
Problem 4 addresses the same company-wide production planning problem
as problem 3, but with fixed costs and minimal activity levels on certain
production activities.
production costs.
These fixed costs contribute significantly to total
More generally, it is hoped that the richer MIP models
generated and optimized by LOGS will result in a better reconciliation of
the company's medium term (monthly) production planning and short term (weekly)
21
production planning at the plant-and machine levels.
MIP modeling is also
needed for certain logical constraints such as one permitting production
of product B at a certain plant only if at least a minimal amount of
product A is made.
The aggregated version of problem 4 was run for experimental purposes.
The added time to generate the aggregated MIP model is a certainty because
LOGS generates it after generating the unaggregated version.
The longer
time to optimize indicates either that aggregation is not attractive for
MIP models of the stated size or that more work is required to efficiently
use the aggregation to solve the unaggregated model.
However, an optimal
solution to the aggregated MIP model easily provided an optimal solution
to the unaggregated model.
At the time of this writing, LOGS has been in operational use in the
firm for almost a year.
and optimize hundreds
and several others.
During that time, it has been used to generate
f LP and MIP models in the above applications areas
One application establishes budgeting standards and
variances by use of quarterly versions of the monthly production planning
models.
Another application was an analysis of the impact on the company
of a proposed acquisition.
Other applications include process design, cost/
service tradeoff analysis and plant layout.
There has naturally been some
evolution of LOGS as the result of this experience with it.
The evolution
is exclusively in the optimization program and in the corresponding model
representations.
There have been no changes needed thus far in LOGS' model-
ing capability.
Conclusions
We have discussed in some detail the design, implementation and use of
22
LOGS within a single company.
We feel its success thus far in over a
half-dozen applications indicates its potential value for other logistics
planning problems.
Some new applications currently being contemplated
would require extensions of the descriptive capability of the basic decision elements.
For example, the use of LOGS to analyze multi-period
distribution models would require the addition of a time dimension and
implied inventory balance equations.
The efficiency and ease of use of LOGS is derived in part from the
fact that it addresses a specific class of MIP models, rather than the
general class of models addressed by languages for matrix generators.
There are classes arising in other applications that would appear to be
good candidates for the same approach.
U. S. coal supply and demand markets.
One such class includes models of
In their full form, these are multi-
period, multi-regional LP and MIP models that describe the logistics of
coal extraction, transportation and use in generating electricity (see
Shapiro and White [3]).
Although the general purpose code MPSX/370 has generally performed well,
extensions to the optimization programs of LOGS may be worthwhile.
Included
are algorithms and methods for exploiting GUB and pure network structures
(see Glover et al [2]).
Special structures such as these are particularly
easy for LOGS to identify from the basic decision elements describing a
logistics planning model.
23
References
[1]
Geoffrion, A. M., "Customer aggregation in distribution modeling,"
Western Management Science Institute Working Paper No. 259, UCLA,
October, 1976.
[2]
Glover, F., J. Hultz and D. Klingman, "Improved computer-based planning
techniques: Part II,"
[3]
Interfaces, 9, pp. 12-20, 1979.
Shapiro, J. F. and D. E. White, "Decomposition and integration of
coal supply and demand models,"
[4]
(in preparation).
Zipkin, P., "A priori bounds for aggregated linear programs with
fixed-weight disaggregation," paper presented at TIMS/ORSA conference,
San Francisco, May, 1977.