Production Inventory Networks

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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

5

A Production-Inventory System

Production facility

Finished-goods

inventory

Customer orders

Raw Materials

Customer shipments

Production orders

6

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

7

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

9

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


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

13

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



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

36

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

38

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 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

57

For systems with lost sales, the IBR heuristic

performs remarkably well (within 3% of the

optimal policy in the vast majority of cases

tested)

58

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

59

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


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



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


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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

68

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

71

0

5

10

15

20

25

30

35

0 5 10 15 20 25 30 35x 1

x2


Produce items1 & 2

Produce item 1

but not 2Do not produce either item 1or 2

72

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

Technology

Production Inventory Networks