Supply Chain Optimization
KUBO Mikio
Agenda
Definition of Supply Chain (SC) and Logistics
Decision Levels of SC
Classification of Inventory
Basic Models in SC




Logistics Network Design
Inventory
Production Planning
Vehicle Routing
Def. of SCM
Council of SCM Professionals
Supply chain management encompasses the planning
and management of all activities involved in
sourcing and procurement, conversion, and all
logistics management activities. Importantly, it
also includes coordination and collaboration with
channel partners, which can be suppliers,
intermediaries, third party service providers, and
customers. In essence, supply chain management
integrates supply and demand management within
and across companies.
Def. of Logistics
Council of SCM Professionals
Logistics management is that part of
supply chain management that plans,
implements, and controls the efficient,
effective forward and reverses flow and
storage of goods, services and related
information between the point of origin
and the point of consumption in order
to meet customers' requirements.
What’s Supply Chain
IT(Information Technology＋Logistics
＝Supply Chain
Real System, Transactional IT,
Analytic IT
brain
解析的IT
処理的IT
nerve
Analytic IT
Model＋Algorithm=
Decision Support System
Transactional IT
POS, ERP, MRP, DRP…
Automatic Information Flow
Real System=Truck, Ship, Plant, Product, Machine, …
muscle
実システム
Levels of Decision Making
Strategic Level
A year to several years; long-term decision making
Analytic IT
Tactical Level
A week to several months; mid-term decision making
Operational Level
Transactional IT
Real time to several days;
short-term decision making
Models in Analytic IT
Supplier
Plant
Retailer
DC
Logistics Network Design
Strategic
Multi-period Logistics Network Design
Tactical
Operational
Inventory
Production
Safety stock allocation
Inventory policy
optimization
Lot-sizing
Scheduling
Transportation
Delivery
Vehicle Routing
Models in Analytic IT
Supplier
Strategic
Plant
Retailer
DC
Logistics Network Design
Multi-period Logistics Network Design
Tactical
Operational
Inventory
Production
Safety stock allocation
Inventory policy
optimization
Lot-sizing
Scheduling
Transportation
Delivery
Vehicle Routing
Models in Analytic IT
Supplier
Plant
Strategic
Retailer
DC
Logistics Network Design
Multi-period Logistics Network Design
Tactical
Operational
Inventory
Safety stock allocation
Inventory policy
optimization
Production
Lot-sizing
Scheduling
Transportation
Delivery
Vehicle Routing
Inventory=Blood of Supply Chain
Inventory acts as glue connecting optimization systems
Supplier
Raw material
Plant
Work-in-process
DC
Retailer
Finished goods
Time
Classification of Inventory
In-transit (pipeline) inventory
Inventories that are in-transit of products
Trade-off: transportation cost or production speed
->Logistics Network Design (LND)
Seasonal inventory
Inventories for time-varying (seasonal) demands
Trade-off: resource acquisition or overtime cost -> multi-period LND
Trade-off: setup cost -> Lot-sizing
Cycle inventory
Inventories caused by periodic activities
Trade-off : transportation (or production) fixed cost -> LND
Trade-off: ordering fixed cost-> Economic Ordering Quantity (EOQ)
Lot-size inventory
Cycle inventories when the speed of demand is not constant
Trade-off: fixed cost ->Lot-sizing, multi-period LND
Safety inventory
Inventories for the demand variability
Trade-off: customer service level >Safety Stock Allocation, LND
Trade-off: backorder (stock-out) cost ->Inventory Policy Optimization
In-transit (pipeline) Inventory
Inventory that are in-transit of products
Trade-off: transportation cost or
transportation/production speed
->optimized in Logistics Network Design (LND)
Seasonal Inventory
Inventory for time-varying (seasonal) demands
Trade-off: resource acquisition or overtime cost
-> optimized in multi-period LND
Trade-off: setup cost
-> optimized in Lot-sizing
Demand
Resource Upper Bound
Period
Cycle Inventory
Inventory caused by periodic activities
Trade-off : transportation fixed cost -> LND
Trade-off: ordering fixed cost
-> Economic Ordering Quantity (EOQ)
Inventory
Level
demand
Cycle Time
Lot-size Inventory
Cycle inventory when the speed of
demand is not constant
Trade-off: fixed cost
->Lot-sizing, multi-period LND
Time
Safety Inventory
Inventory for the demand variability
Trade-off: customer service level
->Safety Stock Allocation, LND
Trade-off: backorder (stock-out) cost
->Inventory Policy Optimization
Classification of Inventory
Seasonal Inventory
Cycle Inventory
Lot-size Inventory
Safety Inventory
In-transit (Pipeline) Inventory
Time
It’s hard to separate them but…
They should be determined separately to optimize the trade-offs
Logistics Network Design
Decision support in strategic level
Total optimization of overall supply chains
Example
 Where should we replenish pars?
 In which plant or on which production line
should we produce products?
 Where and by which transportation-mode
should we transport products?
 Where should we construct (or close) plants
or new distribution centers?
Trade-off in Facility Location Model:
Ｎumber of Warehouses v.s.
Number
of warehouses
輸送中在庫費用
•
•
•
•
•
Service lead time ↓
Inventory cost ↑
Overhead cost ↑
Outbound
輸送費用 transportation cost ↓
Inbound transportation cost ↑
Trade-off:
In-transit inventory cost v.s. Transportation cost
輸送中在庫費用
In-transit
inventory cost
輸送費用
Transportation
cost
Multi-period logistics network design model
Decision support in tactical level
An extension of MPS (Master Production System) for
production to the Supply Chain
Treat the seasonal demand explicitly
Demand
Period (Month)
Trade-off:
Overtime v.s. Seasonal Inventory Cost
資源超過ペナルティ
作り置き在庫費用
Overtime
penalty Seasonal
inventory
（残業費）
Demand
Resource Upper Bound
Period
Constant
Production
Inventories
Overtime
Variable
Production
Model: MIP+Concave Cost Minimization
BOM or Recipie
×
３
Safety Stock Cost
Warehouses Customer Gropus
Plant s
Suppliers
Product ion Lines
Safety Stock Allocation
Decision support in tactical level
Determine the allocation of safety
stocks in the SC for given service levels
安全在庫費用
Safety Stock
サービスレベル
Service Level
＋統計的規模の経済
+Statistical Economy of Scale
（リスク共同管理）
or Risk Pooling
Basic Principle of Inventory
Economy of scale in statistics: gathering
inventories together reduces the total
inventory volume.
-> Modern supply chain strategies

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risk pooling
delayed differentiation
design for logistics
Where should we allocate safety stocks to minimize the
total safety stock costs so that the customer service level
is satisfied.
Lead-time and Safety Stock
Normal distribution with average demand μ，
standard deviation σ
Service level （the probability with no lost
sales） 95%->safety stock ratio 1.65
Lead-time (the time between ordering and
arrival of item） L
Max Inv. Volume ＝
  L＋Safety Stock Ratio    L
The relation between lead-time and
(average,safety,maximum) inventory
3000
2500
2000
Average
Max.
Safety
1500
1000
500
0
0
5
10
Lead-time
15
20
Safety Stock and Guaranteed
Lead-time
Guaranteed Lead-time (LT)：Each
stocking point guarantees to deliver the
item to his customer within the
guaranteed lead-time
Safety stock
＝２ days
２
Guaranteed LT
of upstream point
=1 day
= Entering LT
1
Guaranteed LT to
downstream point
＝２ days
２
Production time＝３
Stocking point
An example: Serial multi-stage model
Average demand=100 units/day
Standard deviation of demand=100
Normal distribution (truncated), Safety stock ratio=1
Guaranteed lead-times of all stocking points =0
Production time
3 days
Inventory cost
10$
Safety stock cost
1732 $
2 days
１day
１day
20$
30 $
40 $
2828 $
3000 $
4000 $ Total 11560 $
Optimal Solution
Guaranteed LT=3
Entering LT=2
Safety stock=3-(2+1)=0 day
Production time
3 days
Guaranteed LT
0 day
Safety stock cost
1732 $
2 days
１ day
１ day
2 days
3 days
0 day
0$
0$
8000$
Total 9732$ （16% down）
Algorithms for Safety Stock Allocation
Concave cost minimization using piecewise linear approximation
Dynamic programming (DP) for tree
networks
Metaheuristics
（Local Search, Iterated LS, Tabu Search）
A Real Example: Ref.
Managing the Supply Chain –The
Definitive Guide for the Business Professional –by Simchi-Levi,
Kaminski,Simchi-Levi
15 x2
37
5
28
Part 4
Malaysia ($180)
37
3
Part 5
Charleston ($12)
58
4
Part7
Denver ($2.5)
29
58
37
8
Part 6
Raleigh ($3)
Part 2
Dallas ($0.5)
39
37
15
17
Part 3
Montgomery ($220)
Part 1
Dallas ($260)
30
15 15
30
Final Demand
N(100,10)
Guaranteed LT
=30 days
43,508$ (40%Down)
What if analysis:
Guaranteed LT=15 days ->51,136$
Inventory Policy Optimization
Decision support in operational level
Determine various parameters for
inventory control policies
品切れ費用
Safety
Stock
安全在庫費用
Lost
Sales
Classical Newsboy Model
発注（生産）固定費用
Cycle Inventory
サイクル在庫費用
Fixed
Ordering
Classical Economic Ordering
Quantity Model
Economic Ordering Quantity
(EOQ) Model
Given
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d (items/day): a constant demand rate.
Q (items): order quantities.
K (yen): a fixed set-up cost of an order.
h (yen/day・item): an inventory holding cost per
item per day.
Find the optimal ordering policy minimizing
total ordering and inventory carrying cost
over infinite planning horizon.
Inventory
d
Q
Cycle Time (T days)
Cost over T days =
f(T)= Cost per day =
Time
Find the optimal ordering
quantify
Minimize f(T)
positive
So f(T) is convex. By solving f’=0, we get:
EOQ (Harris’) formula
Newsboy Problem
inventory cost
backorder (lost sales) cost
demand of newspaper (random variable)
Distribution function of the demand
Density function
Expected Value of Total Cost
Expected cost when the ordering amount is
s:
Optimal Solution
First-order differentiation:
Second-order differentiation :
is convex!
Base-stock Policy
Base stock level＝Target of the
inventory position
Inventory position=
In-hand inventory+In-transit inventoryBackorder
Base stock policy: Monitoring the inventory
position in real time; if it is below the base
stock level, order the amount so that it
recovers the base stock level
Base Stock Policy (Multi Stage
Model)
n serial inventory stocking points
demand point is 1
final supplier is n+1 that has enough inventory
Notations (1)
time index
local stock at the i-th point
backorder at the i-th point
net inventory at the i-th point
Notations (2)
inventory on order
inventory in transit (transit inventory)
Notations (3)
inventory ordering position
inventory transit position
Notations (4)
：lead time
：demand between time interval (s,t]
：base stock level
：backorder cost
：inventory cost
Inventory Flow Conservation Equation
base stock level
s’i
By using
ITP’i(t)
=>random demand
L’i
IN’i(t+L’i)
Recursive Equation
：equilibrium value of stationary demand
during lead time
Using
i+1
i
can compute B’ from n+1 to 1.
=>cannot compute the opt.
base stock levels
Echelon Inventory Model
：echelon
inventory at the i-th point
：system backorder
：net echelon inventory
Echelon Inventory Model
Notations (Cont’d)
：echelon inventory ordering position
：echelon inventory transit position
：echelon base stock level at the i-th point
Echelon base stock policy:
Order the amount so that the inventory ordering position
recovers the echelon base stock level.
Echelon Inventory Flow Conservation
Equation
：echelon inventory cost at the i-th point
Flow conservation equation for echelon inventory:
Recursive Equation
：equilibrium value of stationary demand
during lead time
=>can compute net inventory from n to 1
Objective Function
Local inventory model
Echelon inventory model
Derivation of Optimal Solution
(1)
：expected cost for 1 to i points when
INi+1 is x
：expected cost for 1 to i points when
INi is x
：expected cost for 1 to i points when
ITPi is y
=>Convex Function
Derivation of Optimal Solution (2)
expected cost for 1 to i points when INi is x
The minimum cost to the i-1st
point when the echelon net
inventory at the i-th point is x
i
i-1
=>Linear＋Convex＝Convex
Derivation of Optimal Solution (3)
expected cost for 1 to i points when ITPi is y
y=ITPi
The minimum cost to the
i-th point when the
echelon net inventory is
y- Di
=>random demand Di
L’i
IN
=> Expectation of convex functions => convex
Derivation of Optimal Solution
(4)
expected cost for 1 to i points when INi+1 is x
i+1
i
Echelon net inventory
Minimum cost when
x =INi+1
=y
Derivation of Optimal Solution (5)
Echelon base stock level:
C is convex
Since echelon base stock level is non-decreasing,
The optimum local base stock level:
where
is
Basic Formula of SCM
Is convex
Basic formula of
SCM
(Q,R) and (s,S) Policies
If the fixed ordering cost is large, the ordering
frequency must be considered explicitly.
(Q,R) policy：If the inventory position is below
a re-ordering point R, order a fixed quantity Q
(s,S) policy：If the inventory position is below
a re-ordering point s, order the amount so
that it becomes an order-up-to level S
Periodic Ordering Policy
Check the inventory position periodically;
if it is below the base-stock level, order
the amount so that it recovers the basestock level
Order
Mon.
Tue.
Wed.
Thu.
Demand
Arrival of the order of Mon.（Lead-time＝１day）
Algorithms for Inv. Policy Opt.
Base-stock，(Q,R)， and (s,S) policies
->DP
Periodic ordering policy
-> Infinitesimal Perturbation Analysis
During simulation runs, derivatives of
the cost function are estimated and are
used in non-linear optimization
Lot-size Optimization
Decision support in tactical level
Optimize the trade-off between the set-up cost and the
lot-size inventory
段取り費用
Lot-size
Inv.
Setup
Cost
在庫費用
Basic Single Item Model (1)
Parameters
T : Planning horizon (number of periods)
dt : Demand during period t
ft : Fixed order (or production set-up) cost
ct : Per-unit order (or production) cost
ht : Holding cost per unit per period
Mt: Upper bound of production (capacity) in
period t
Basic Single Item Model (2)
Variables
It : Amount of inventory at the end of
period t (initial inventory is zero.)
xt : Amount ordered (produced) in
period t
yt : =1 if xt >0, =0 otherwise (0-1
variable), i.e. , =1 production is
positive, =0 otherwise (it is called “setup variable.”)
Basic Single Item Model (3)
Formulation
Lot-sizing (Basic Flow) Model
Production
x(t)
Inventory
I(t-1)
I(t)
t
Demand d(t)
Week formulation
x(t)≦ “Large M” ×y(t) [set-up variable]
I（t-1)+x(t) = d(t)+I（t)
0-1 variable
Valid Inequality
Then the inequality (called the (S,l) inequality)
is valid.
Valid Inequality，Cut，Facet
Inequality of week formulation
Facet
(valid inequality)
Relaxed solution x*
Solution x
Integer Polyhedron
Cut
Extended (Strong) Formulation
Notations
Xst : ratio of the amount produced in period s to satisfy
demand in period t (
)
The cost produced in period s
to satisfy demand in period t
Formulation
Facility Location Formulation
=> Strong formulation; it gives an integer hull of solutions
Lot-sizing Model
Facility Location Model
Ratio of the amount produced
in period s to satisfy demand in period t
Xst
s
t
Xst ≦y(t)

s t
Xst = 1
d(t)
Extended Formulation and
Projection
is a formulation of X
= Q is an extended formulation of X
Facility Location Formulation and
Projected Polyhedron
Extended Formulation
(Facility Location Formulation)
Projection
Integer Polyhedron
of Original Formulation
Comparison of Size and
Strength
Standard Formulation
# of var.s
# of
const.s
O(T )
Facility Location Formulation
O(T )
Week
formulation
# of var.s O(T 2 )
# of const.s
2
O(T )
added
const.s
(S, l) ineq.s
O(2T )
cut
Strong formulation
Strong formulation
linear prog. relax.
=integer polyhedron
T: # of periods
Dynamic Programming
for the Uncapacitated Problem
Upper bound of production (capacity) Mt is large enough.
F(j) : Minimum cost over the first j periods (F(0)=0)
O(T2) or O(T log T) time algorithm
Silver-Meal Heuristics
Define:
Let t=1. Determine the first period j (>=t) that satisfies:
(If such j does not exist, let j=T.) The lot size produced in
period t is the total demand from t to j. Let t=j+1 and repeat the
process until j=T.
Least Unit Cost Heuristics
Let t=1. Determine the first period j (>=t) that
satisfies:
(If such j does not exist, let j=T.) The lot size
produced in period t is the total demand from t to j.
Let t=j+1 and repeat the process until j=T.
Example: Single Item Model
Period （day，week，month，hour）：１，２，３，４，５ （５ days）
setup
production
Setup cost： ３ $
demand
： 5,7,3,6,4 （tons）
Inventory cost ： １ $ per day
Production cost ： 1,1,3,3,3 $ per ton
Comparison with ad hoc methods
Product at once：
setup (3)+production(25)+inventory(20+13+10+4)=75
Ｊust-in-time production：setup(15)+prod.(51)+inv.(0)=66
Optimal production：setup(9)+prod.(33)+inv.(15)=57
Comparison with heuristics
Silver-Meal heuristics
Determine the lot-size so that the cost per period is minimized.
setup(9)+prod.(45)+inventory(7)=61
Least unit cost heuristics
Determine the lot-size so that the cost per unit-demand
is minimized. setup(9)+prod(51)+inventory(14)=74
Algorithms for Lot-sizing
Metaheuristics using MIP solver



Relax and Fix
Capacity scaling
ＭＩＰ neighbor local search
Scheduling Optimization
Decision support in operational level
Optimization of the allocation of activities (jobs,
tasks) over time under finite resources
What is the scheduling?
Allocation of activities (jobs, tasks) over
time


Resource constraints. For example, machines,
workers, raw material, etc. may be scare
resources.
Precedence relation. For example., some
activities cannot start unless other activities
finish.
Solution methods for scheduling
Myopic heuristics


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Active schedule generation scheme
Non-delay schedule generation scheme
Dispatching rules
Constraint programming
Metaheuristics
Vehicle Routing Optimization
Customers
earliest time
latest time
Customer
Depot
waiting
time
service time
Routes
service time
Algorithms for Vehicle Routing
Saving (Clarke-Wright) method
Insertion method
Guided Local Search
Iterated Local Search
History of Algorithms for Vehicle Routing
Problem
Approximate Algorithm
Genetic Algorithm
AMP
Tabu
Search
Local Search
Simulated Annealing
Sweep
Method
Generalized
Assignment
Construction Method
(Saving, Insertion)
(Adaptive Memory
Programming)
Location Based
Heuristics
Route Selection
Heuristics
GRASP
(Greedy Randomized
Adaptive Search Procedure)
Exact Algorithm
Set Partitioning Approach
State Space Relax.
Cutting Plane
K-Tree Relax.
1970
1980
1990
2000
Hierarchical
Building Block
Method
Conclusion
Decision Levels of SC
Classification of Inventory
Basic Models in SC




Logistics Network Design
Inventory
Production Planning
Vehicle Routing