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Production Inventory Networks: Perspectives and Recent
Advances
Saif BenjaafarIndustrial & Systems Engineering Division
Department of Mechanical EngineeringUniversity of Minnesota
Presented at Chinese Academy of Sciences, June 16, 2007
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Benjaafar, S. and M. ElHafsi, “Production and Inventory Control of an ATO System
with Multiple Customer Classes,” Management Science, 51, 2006
Benjaafar, S., Y. Li and D. Xu, “Demand Allocation in Systems with Multiple
Inventory Locations and Multiple Demand Sources,” M&SOM, forthcoming, 2007
Gayon, J. P., S. Benjaafar and F. de Véricourt, “Using Imperfect Demand Information
in Production-Inventory Systems with Multiple Demand Classes,” M&SOM,
forthcoming, 2007
Benjaafar, S., William L. Cooper and J. S. Kim, “On the Benefits of Pooling in
Production-Inventory Systems,” Management Science, 51, 548-565, 2005
Benjaafar, S., M. Elhafsi and F. de Véricourt, “Demand Allocation in Multi-Product,
Multi-Facility MTS Systems,” Management Science, 50, 1431–1448, 2004
Some Recent Papers
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Benjaafar, S., E. Elahi and K. Donohue, “Outsourcing via Service Quality
Competition,” Management Science, 53, 241-259, 2007.
Benjaafar, S., J. S. Kim and N. Vishwanadham, “On the Effect of Product-Variety in
Production-Inventory Systems,” Annals of Operations Research, 126, 71-101, 2004
Gupta, D. and S. Benjaafar, “Make-to-order, Make-to-stock, or Delay Product
Differentiation? - A Common Framework for Modeling and Analysis,” IIE
Transactions, 36, 529-546, 2004
Benjaafar, S., M. ElHafsi, C. Y. Lee and W. Zhou, “Optimal Control of Assembly
Systems with Multiple Stages and Multiple Demand Classes,” Working Paper, 2006
Benjaafar, S., W. L. Cooper and S. Mardan, “Production-Inventory Systems with
Imperfect Advance Demand Information and Due-Date Updates,” Working Paper,
2006
Some Recent Papers (Continued…)
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Examples of production-inventory systems
Issues in design, analysis and control
Multi-stage assembly systems
Systems with imperfect advance demand information
Ongoing and future work
Production & Inventory Systems
Service Systems
Outline
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A Production-Inventory System
Production facility
Finished-goods
inventory
Customer orders
Raw Materials
Customer shipments
Production orders
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Tight coupling of production & inventory
Limited production capacity, items produced one unit at a time
Variability in both demand and production times
Supply lead times are affected by congestion at the production system
Characteristics
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Production and inventory treated separately
Capacity constraints are frequently ignored
Supply leadtimes are not affected by congestion
Inventory Literature
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How much inventory should we keep (should we produce to order or to stock)?
When should we place a production order and for how much?
When should we initiate production and for how long?
What is the impact of system parameters?
Issues
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Raw Materials
Production orders
Customer orders
Production orders
Production orders
Customer shipments
Supply orders
Supply orders
A Series System
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Where should we keep inventory and how much?
How should we coordinate production across stages?
Where should we invest in capacity and in variability reduction?
Issues
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Raw Materials
Customer orders from location 1
Customer orders from location 2
A Distribution System
Customer orders from location N
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How much inventory to stock of each product (which product to make to order and which to make to stock)?
How should priorities be assigned to different products?
What is the impact of inventory consolidation?
Issues
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Product 1
Product N
Product 2
Production facilities Components
An Assemble-to-Order System
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How much of each component to stock?
How should we coordinate the production of different components?
How should shared components be allocated?
What is the impact of various parameters?
Issues
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A Multi-Stage Assembly System
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Benjaafar et al. (MS, 2005; AOR, 2004): Impact of inventory
pooling in a distribution system
Benjaafar et al. (MS, 2006): Optimal control of ATO systems
Benjaafar et al. (MS, 2004; MSOM 2007): Demand allocation
in a network with multiple production facilities/inventory
locations
Benjaafar et al. (M&SOM, 2007): Optimal control of systems
with ADI
Benjaafar et al. (IIE Transactions, 2004): MTS versus MTO
systems
Benjaafar et al. (MS, 2007): Competition among MTS
suppliers
Related Papers
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Raw Materials
Customer orders from location 1
Customer orders from location 2
Benjaafar et al. (AOR 2004, MS 2005)
Customer orders from location N
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Raw Material
s
Customer orders
Customer
shipments
Make-to-stock segment Make-to-order segment
Benjaafar and Gupta (IIE Transactions, 2004)
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Benjaafar et al. (MS, 2004; MSOM 2007)
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Production facilities Components
Benjaafar and ElHafsi (MS 2006)
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Benjaafar et al. (MS, 2007)
Competing suppliers
A buyerthat allocates
demand
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Benjaafar et al. (MSOM, 2007)
Orders are announced
Demand leadtime process
Orders are due
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Benjaafar et al. (2007): Multi-stage assembly systems
with multiple stages with multiple classes
Gayon et al. (2006): Systems with imperfect advance
demand information
Benjaafar et al. (2006): Systems with imperfect advance
demand information and due-date update
Papers Related to this Talk
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Optimal Control of Multi-Stage Assembly Systems with
Multiple Demand Classes
(Joint work with Mohsen Elhafsi, Larry Zhou and Chung-Yee Lee)
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A Multi-Stage Assembly System
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Motivation
Assembly permeates most manufacturing and
multi-stage assembly is a feature of most
manufactured products
Modeling, analysis, and control of assembly
systems is notoriously difficult
Few exact analytical results and the structure of
the optimal policy is largely unknown
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“Little is known about the forms of optimal policies for multi-period models. The research to date mostly assumes particular policy types. It would be valuable to learn more about truly optimal policies. Even partial characterizations would be interesting. Also, better heuristic policy forms would be useful.”
--Song and Zipkin (2003)
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Challenges
Demands for different items (components and
sub-assemblies) are correlated
Production and order fulfillment depends on the
availability of multiple items
Production leadtimes of different items can be
different
Costs of different items can be different
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Research Questions
What can we say about the structure of optimal
policies?
What is the benefit of using optimal policies
instead of common heuristics such as static
base-stock policies?
Are there simple but effective heuristics that can
serve as substitutes to optimal policies?
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The Setting Multiple items (components & intermediates)
progressively assembled into a single product
Each item can have multiple predecessors and a
single successor
Each item is produced (assembled) on an
independent production facility
Demand for the end item arises from n customer
classes
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1
2
3
4
5
6
8
9
production facility inventory location
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1
2
3
4
5
Single-Stage Assembly Systems
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1235
Series Systems
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The Demand Classes
Demand for the end-item emanating from class l
occurs continuously one unit at a time and follows
a Poisson process with rate l
If demand cannot be satisfied, it is lost and incurs
a lost sales cost cl
WLOG, c1 c2 … cn
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Item Production Items are produced on independent facilities in a make-to-stock
fashion
Item k incurs a holding cost hk(xk), increasing convex in
inventory level xk
Production times for item k are exponentially distributed with
mean 1/k
An item can be produced (assembled) only if at least one unit
of all its predecessors items are available
Inventory of all items is continuously reviewed
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System state is described by the vector X(t)=(X1(t),…, Xm(t))
where Xk(t) is the inventory level of item k
Two types of decisions are made in each state Produce/not produce item k
Satisfy/reject an incoming order of class l
Objective: choose in each state the decision that minimizes
(over an infinite horizon) the total expected discounted cost
The average cost per period
A Markov Decision Process (MDP) formulation
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The Optimality Equation
* * *01 1
( ) ( ) ( ) ( )n m
ll k kl k
v h T v T v
x x x x
( )
( )
( ) if 0( )
min{ ( ), ( )} otherwise,
it P k
k
k P k
v xT v
v v
xx
x e e x
1
1
( ) if 0 ( )
min{ ( ), ( ) } otherwise.ll
l
v c xT v
v v c
xx
x e x
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Properties of the Value Function
( ) ( ) ( )j jv v v x x e x
, ( ) ( ) ( ) ( ),i j i j j i jv v v v x x x e x
( ) ( )( ) ( ) ( )k P k k P kv v v x x e e x
Define:
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Properties of the Value Function (Continued…)
*( ),1 ( ) 0i P i v x
*( ), ( ) ( ) 0i P i j P j v x
*1 1( )v c x
*( ), ( ) 0i P i jv x
*( ), ( ) 0i P i j lv x
*( ), ( ) 0i P i jv x
*, ( ) 0i jv x
*( ), ( ) ( ) 0i P i i P i v x
*, ( ) 0i jv x
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The Optimal Production Policy
A base-stock production policy with state-dependent
base-stock levels is optimal:
produce item k if xk < sk(x-k), x-k=(x1,…, xk-1, xk+1,…, xm)
do not produce if xk sk(x-k)
The base-stock level sk(x-k) is non-increasing in xj if j
S(k) and is non-decreasing in xj if j S(k)
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2
3
4
5
6
7
8
9
1
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2
3
4
5
6
7
8
9
1
Items on the path from item 5 to item 1, S(5)
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2
3
4
5
6
7
8
9
1
Items not on the path from item 5 to item 1
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The Optimal Production Policy (Continued…)
The optimal base-stock level of item k does not decrease
with the production completion of any other item
the closer an item in S(k) is to item k, the bigger the influence
it has on the base-stock level of item k,
for j S(k) and if l S(j)
It is never optimal to interrupt the production of an item once
it has been initiated
* *( ) ( )k k j k k ls s x e x e
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The Optimal Allocation Policy
An allocation policy with multiple state-dependent
rationing levels is optimal:
satisfy demand from class l if x1 rl(x-1)
do not satisfy demand from class l if x1 < rl(x-1)
The rationing level for class l rl(x-1) is non-increasing in
the inventory level xj of any item j ≠ 1
The rationing levels are ordered rn(x-1) … r1(x-1)=1
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1
2
3
4
An Example with 4 Items and 3 Demand Classes
class 1 demandclass 2 demandclass 3 demand
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0
5
10
15
20
25
30
35
40
0 5 10 15 20 25 30 35
x 2
x3
Produce item 3 but 2Produce
items2 & 3
Do not produce either items 2 or 3 Produce
item 2 but not 3
Produce item 3 but not 2
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0
5
10
15
20
25
30
35
40
0 5 10 15 20 25 30 35 40x 2
x3 Satisfy classes 1 & 2
Satisfy classes 1, 2 & 3
Sati
sfy
clas
s 1
only
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0
5
10
15
20
25
30
35
0 5 10 15 20 25 30 35x 1
x2
Produce item 2 but not 1
Produce items1 & 2
Produce item 1
but not 2Do not produce either item 1or 2
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0
5
10
15
0 5 10 15x 1
x2
Satisfy classes
1, 2 & 3
Sati
sfy
clas
ses
1 &
2 o
nly
Sati
sfy
clas
ses
1 on
ly
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x3
s3(x1, 3)
0
3
6
9
3 6 9 12 15 18 21x 1 or x 2
s 3
s 3(x 2, 3)
s 3(3, x 2)
s3(3, x2)
x3
s3(x1, 3)
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Special Cases
Single item, single class
Single item, multiple classes
Serial system, single class
Two stage assembly, single class
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Systems with Backorders(the Single Class Case)
1 1 1 12 1( ) ( ) ( ) ( ) ( ) ( )
m m
k k k kk kTv h x h x b x v T v
x x e x
( )
( )
( ) if 0( )
min{ ( ), ( )} o.w.
ii P kk
k P k
v xT v
v v
xx
x e e x
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A base-stock policy with state-dependent base-
stock levels is optimal
The base-stock levels retain all the properties
observed in the lost sales case
The Optimal Policy
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Optimal Policy versus Heuristics
Base-stock policies with independent and fixed
base-stock and rationing levels (IBR)
Dynamic linear base-stock and rationing policies
(LBR)
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Comparisons with The IBR Policy (Lost Sales)
0
1
2
3
4
5
6
0 5 10 15 20 25 30
Data set
Perc
enta
ge d
iffer
ence
from
op
timal
cos
t
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For systems with lost sales, the IBR heuristic
performs remarkably well (within 3% of the
optimal policy in the vast majority of cases
tested)
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Comparisons with The IBR Policy (Backorders)
13
14
15
16
17
18
10 30 50 70 90backorder cost, b
Perc
enta
ge d
iffer
ence
from
op
timal
cos
t
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Comparisons with The IBR Policy (Backorders)
0
2
4
6
8
10
12
14
16
4.5 6.5 8.5 10.5 12.5
Service rate, 1
Per
cent
age
diffe
renc
e fr
om o
ptim
al
cost
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For systems with backorders, the IBR policy can
perform poorly (an average of 13% in the cases
tested and up to 40% in some cases)
The performance of the IBR policy is particularly
poor when Backorder costs are low
Utilization of the production facilities is high
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0
10
20
30
40
50
60
70
80
90
-50 -40 -30 -20 -10 0 10 20 30 40x 1
x3
Produce both items 1 & 3
Produce item 1 but not item 3
Produce item 3 but not item 1
Do
not p
rodu
ce e
ithe
rit
em 1
or i
tem
3
-
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0
10
20
30
40
50
60
70
80
90
-50 -40 -30 -20 -10 0 10 20 30 40x 1
x3
Produce both items 1 & 3
Produce item 1 but not item 3
Produce item 3 but not item 1
Do
not p
rodu
ce e
ithe
rit
em 1
or i
tem
3
-
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0
5
10
15
20
25
30
35
40
0 5 10 15 20 25 30 35
x 2
x3
Produce item 3 but 2Produce
items2 & 3
Do not produce either items 2 or 3 Produce
item 2 but not 3
Produce item 3 but not 2
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Linear Base-Stock Production Policies (LBP)
Produce component k if
do not produce otherwiseThe LBP policy has the same structural properties as the
optimal policy
The LBP policy can be evaluated using simulation and the
parameters sk, jk, jk obtained via a search
( ) ( )k jk j jk j kj S k j S kx x x s
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Linear Rationing Policies (LRP)
Satisfy demand from class l
do not satisfy demand otherwise
1 1 j j ljx x r
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Summary of Work so Far
A formulation of the multi-stage/multi-class assembly
problem
A characterization of the structure of the optimal
policy
Preliminary numerical results
A class of simple heuristic policies
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Ongoing and Future Work
A more comprehensive numerical study
Extensions to assembly systems withdemand for all items
multiple unit requirements
variable order size
Multiple products
General networks
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Production-Inventory Systems with Imperfect Advance Demand Information
and Due-Date Updates
Saif BenjaafarGraduate Program in Industrial Engineering
Department of Mechanical EngineeringUniversity of Minnesota
(Joint work with William Cooper, Jean-Philippe Gayon, Setareh Mardan, and Francis de Véricourt)
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Other Research
Production-Inventory systems with both backorders
and lost sales
Production planning and scheduling for process
industries
An item-customer approach to modeling, analysis
and control of stochastic inventory systems
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Comparisons with the IBR Policy (Continued…)
5
7
9
11
13
15
5 7 9 11 13
Holding cost, h 1
Perc
enta
ge d
iffe
renc
e fr
om o
ptim
al c
ost
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0
5
10
15
20
25
30
35
0 5 10 15 20 25 30 35x 1
x2
Produce item 2 but not 1
Produce items1 & 2
Produce item 1
but not 2Do not produce either item 1or 2
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0
5
10
15
0 5 10 15x 1
x2
Satisfy classes
1, 2 & 3
Sati
sfy
clas
ses
1 &
2 o
nly
Sati
sfy
clas
ses
1 on
ly