Abstract - ETDA

The Pennsylvania State University

The Graduate School

College of Business Administration

AVAILABILITY MANAGEMENT FOR CONFIGURE-TO-ORDER

SUPPLY CHAIN SYSTEMS

A Thesis in

Business Administration and Operations Research

by

Ching-Hua Chen-Ritzo

c© 2006 Ching-Hua Chen-Ritzo

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy

May 2006

The thesis of Ching-Hua Chen-Ritzo was reviewed and approved∗ by the following:

Terry P. Harrison

Professor of Supply Chain and Information Systems

Thesis Advisor, Chair of Committee

Tom M. Cavalier

Professor of Industrial Engineering

V. Daniel R. Guide, Jr.

Assistant Professor of Operations and Supply Chain Management

Susan H. Xu

Professor of Management Science and Supply Chain Management

John Tyworth

Professor of Supply Chain Management

Chair of Supply Chain & Information Systems Department

Tom Ervolina

Research Staff Member, IBM Research

Special Signatory

∗Signatures are on file in the Graduate School.

Abstract

In response to consumer demands, the range of products offered by manufacturers is

becoming increasingly complex. One of the ways that companies are dealing with this

challenge is to offer products that can be assembled from a collection of independent

components. One of the central difficulties facing the makers of such ‘configured-to-

order’ products is the following: The precise quantity and collection of resources required

to satisfy an order is not known until a customer has actually placed an order. The

supply chain tools currently available to practitioners are not prepared to deal with

such uncertainty and existing academic models do not adequately address the problem

in a practical manner. This thesis responds to this challenge by addressing a series

of optimization problems and framing them within the context of a popular business

process called sales and operations planning.

Within this context, three related optimization models are formulated and solved

using stochastic programming techniques. These models are referred to as the explosion,

implosion and component rationing problems, respectively. The explosion and implo-

sion problems both seek to maximize profit, but where the explosion model determines

component requirements given product demand, the implosion model determines the

appropriate product sales targets given certain restrictions on component supply. The

component rationing problem seeks to maximize revenue for a given component supply

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and product demand by appropriately setting the component threshold levels that will

reserve certain components for certain product orders. In all three problems, uncertainty

is present because the quantity of components required to satisfy an order is unknown

at the time that the decisions are made.

Computational studies performed using problem sets derived from data provided by

IBM show that, with respect to the explosion and implosion problems, there is significant

benefit to accounting for the uncertainty associated with how products are configured.

For the explosion problem, accounting for this uncertainty results in improved profit,

revenue and serviceability. For the implosion problem, accounting for this uncertainty

results in improved serviceability. As for the component rationing problem, it is shown

that rationing is often able to increase expected revenue over a first-come-first-served

allocation policy.

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Table of Contents

List of Figures x

List of Tables xii

Acknowledgments xvii

Chapter 1 Introduction 1

1.1 Availability Management . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Sales & Operations Planning . . . . . . . . . . . . . . . . . . . . . 3

1.2 Configure-to-Order Supply Chains . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Availability Management for CTO Systems . . . . . . . . . . . . . . . . . 9

1.3.1 The Explosion Problem . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.2 The Implosion Problem . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.3 The Component Rationing Problem . . . . . . . . . . . . . . . . . 14

1.4 Industry Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.5 Summary of Research Problem and Objectives . . . . . . . . . . . . . . . 16

Chapter 2 Literature Review 19

2.1 Inventory Ordering Policies in Assemble-To-Order Systems . . . . . . . . 19

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2.2 Available-To-Promise Scheduling . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Flexible Supply Contracting . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4 Inventory and Capacity Rationing . . . . . . . . . . . . . . . . . . . . . . 25

Chapter 3 The Explosion Problem 30

3.1 Model Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3 Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3.1 Deterministic Equivalent . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3.2 Scenario Generation . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3.3 Sample Average Approximation . . . . . . . . . . . . . . . . . . . . 40

3.3.3.1 Upper Bound Estimation . . . . . . . . . . . . . . . . . . 41

3.3.3.2 Lower Bound Estimation . . . . . . . . . . . . . . . . . . 42

3.3.3.3 Optimality Gap Estimation . . . . . . . . . . . . . . . . . 43

3.3.3.4 SAA Algorithm for T -Period Explosion Problem . . . . . 44

3.3.4 Myopic Approximation to the Explosion Problem . . . . . . . . . . 45

3.4 Computational Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.4.1 Problem Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.4.2.1 Assessing the Quality of the Myopic Approximation . . . 51

3.4.2.2 Value of the Stochastic Explosion Solution . . . . . . . . 52

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Chapter 4 The Implosion Problem 59


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4.1.1 Penalty Cost Structure for Deviations of the Commitment-to-Salesfrom Initial Sales Targets . . . . . . . . . . . . . . . . . . . . . . . 63


4.2.1 Implosion Problem with Fixed Component Supply . . . . . . . . . 64

4.2.2 Extension of Implosion Problem to Incorporate Flexible Supply . . 68

4.3 Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70




4.3.3.1 Bound and Optimality Gap Estimation . . . . . . . . . . 73

4.3.3.2 SAA Algorithm for T -Period Implosion Problem . . . . . 74

4.3.4 Myopic Approximation to the Implosion Problem . . . . . . . . . . 76


4.4.1 Problem Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.4.2.1 Assessing the Quality of the Myopic Approximation . . . 80

4.4.2.2 Value of the Stochastic Implosion Solution . . . . . . . . 82

4.4.2.3 Value of Supplier Flexibility . . . . . . . . . . . . . . . . 86

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Chapter 5 The Component Rationing Problem 111

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111



5.4 Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

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5.4.2 Search Algorithm for the Component Rationing Problem . . . . . 119



5.4.4.1 Bound and Optimality Gap Estimation . . . . . . . . . . 126

5.4.4.2 SAA Algorithm for the Component Rationing Problem . 127


5.5.1 Problem Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.5.2.1 Estimation of Optimality Gaps . . . . . . . . . . . . . . . 131

5.5.2.2 Assessing the Value of Component Rationing . . . . . . . 132

5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Chapter 6 The Availability Management Problem 138

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

6.2 Analysis of the Relative Contributions of Explosion, Implosion and Ra-tioning in Availability Management . . . . . . . . . . . . . . . . . . . . . . 139

6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Chapter 7 Validation and Verification of Models 161

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

7.2 Building Valid and Credible Models . . . . . . . . . . . . . . . . . . . . . 162

7.2.1 Regular Interaction with Subject-Matter Experts . . . . . . . . . . 162

7.2.2 Determining the Level of Model Detail . . . . . . . . . . . . . . . . 163

7.2.3 Empirical Analysis of Stationarity of Configuration Uncertainty . . 164

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7.3 Verification of Computer Models . . . . . . . . . . . . . . . . . . . . . . . 165

7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Chapter 8 Conclusions 169

Bibliography 176

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List of Figures

4.1 Percentage Improvement in Expected Profit for Various Levels of SupplierFlexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.2 Percentage Improvement in Expected Revenue for Various Levels of Sup-plier Flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.3 Percentage Change in Expected Ordering Cost for Various Levels of Sup-plier Flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.4 Percentage Change in Expected Holding Cost for Various Levels of Sup-plier Flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.5 Percentage Change in Expected Average Fill-Rate for Various Levels ofSupplier Flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.6 Percentage Change in Expected Cumulative Fill-Rate for Various Levelsof Supplier Flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.7 Percentage Improvement in Expected Profit for Various Levels of SupplierFlexibility with Initial Supply Given by Expected Value Explosion Model 105

4.8 Percentage Improvement in Expected Revenue for Various Levels of Sup-plier Flexibility with Initial Supply Given by Expected Value ExplosionModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.9 Percentage Change in Expected Ordering Cost for Various Levels of Sup-plier Flexibility with Initial Supply Given by Expected Value ExplosionModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.10 Percentage Change in Expected Holding Cost for Various Levels of Sup-plier Flexibility with Initial Supply Given by Expected Value ExplosionModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

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4.11 Percentage Change in Expected Average Fill-Rate for Various Levels ofSupplier Flexibility with Initial Supply Given by Expected Value Explo-sion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.12 Percentage Change in Expected Cumulative Fill-Rate for Various Lev-els of Supplier Flexibility with Initial Supply Given by Expected ValueExplosion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.1 Normal Probability Plot of Residuals for Component Rationing RegressionModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.2 Residuals vs. Fitted Values for Component Rationing Regression Model . 135

6.1 Illustration of Experimental Design . . . . . . . . . . . . . . . . . . . . . . 146

6.2 Cumulative Commitment-To-Sales and Expected Cumulative Number ofOrders Fulfilled for Treatment Mar-3d and Base Treatment Mar-3h . . . . 148

6.3 Commitment-To-Sales and Expected Fill Rate for Treatment Mar-3f andBase Treament Mar-3h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

6.4 Difference Between the Expected Volume of Each Product that is FulfilledUnder Treatment Mar-3g and Mar-3h . . . . . . . . . . . . . . . . . . . . 150

6.5 Difference Between the Expected Revenue Earned from Each ProductUnder Treatment Mar-3g and Mar-3h . . . . . . . . . . . . . . . . . . . . 151

7.1 Historical Order Quantities for Product-Component Pairs . . . . . . . . . 168

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List of Tables

3.1 Summary of Characteristics for Data Sets Used in Analysis . . . . . . . . 56

3.2 Comparison of 90% Confidence Interval for Optimality Gap Using SAAAlgorithm for T -period Explosion Problem vs. Myopic ApproximationProblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.3 Comparison of Computation Time (in Seconds) Using SAA Algorithm forT -period Explosion Problem vs. Myopic Approximation Problem . . . . . 57

3.4 Comparison of Relative Improvements in Profit, Revenue and Costs As-sociated with Stochastic Solution vs. Expected Value Solution for theExplosion Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.5 Comparison of Expected Fill-Rates Associated with Expected Value So-lution (EVS) vs. Stochastic Solution (SS) for the Explosion Problem . . . 58

4.1 Comparison of 90% Confidence Interval for Optimality Gap Using SAAAlgorithm for T -period Implosion Problem vs. Myopic ApproximationProblem, with Fixed Supply . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.2 Comparison of 90% Confidence Interval for Optimality Gap Using SAAAlgorithm for T -period Implosion Problem vs. Myopic ApproximationProblem, with 2% Supplier Flexibility . . . . . . . . . . . . . . . . . . . . 88



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4.5 Comparison of Computation Time (in Seconds) Using SAA Algorithm forT -period Implosion Problem vs. Myopic Approximation Problem, withFixed Supply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.6 Comparison of Computation Time (in Seconds) Using SAA Algorithm forT -period Implosion Problem vs. Myopic Approximation Problem, with2% Supplier Flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.7 Comparison of Computation Time (in Seconds) Using SAA Algorithm forT -period Implosion Problem vs. Myopic Approximation Problem, with5% Supplier Flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.8 Comparison of Computation Time (in Seconds) Using SAA Algorithm forT -period Implosion Problem vs. Myopic Approximation Problem, with10% Supplier Flexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.9 Comparison of Relative Improvements in Profit, Revenue and Costs As-sociated with Stochastic Solution vs. Expected Value Solution for theImplosion Problem with Fixed Supply . . . . . . . . . . . . . . . . . . . . 91

4.10 Comparison of Expected Fill-Rates Associated with Expected Value So-lution vs. Stochastic Solution for the Implosion Problem with Fixed Supply 91

4.11 Comparison of Relative Improvements in Profit, Revenue and Costs As-sociated with Stochastic Solution vs. Expected Value Solution for theImplosion Problem with Fixed Supply when Supply is Given by the Solu-tion to the Expected Value Explosion Model . . . . . . . . . . . . . . . . . 92

4.12 Comparison of Expected Fill-Rates Associated with Expected Value Solu-tion vs. Stochastic Solution for the Implosion Problem with Fixed Supplywhen Supply is Given by the Solution to Expected Value Explosion Model 92

4.13 Comparison of Relative Improvements in Profit, Revenue and Costs As-sociated with Stochastic Solution vs. Expected Value Solution for theImplosion Problem with 2% Supplier Flexibility . . . . . . . . . . . . . . . 93

4.14 Comparison of Expected Fill-Rates Associated with Expected Value Solu-tion vs. Stochastic Solution for the Implosion Problem with 2% SupplierFlexibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.15 Comparison of Relative Improvements in Profit, Revenue and Costs As-sociated with Stochastic Solution vs. Expected Value Solution for theImplosion Problem with 2% Supply Flexibility when Committed Supplyis Given by the Solution to the Expected Value Explosion Model . . . . . 94

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4.16 Comparison of Expected Fill-Rates Associated with Expected Value So-lution vs. Stochastic Solution for the Implosion Problem with 2% SupplyFlexibility when Committed Supply is Given by the Solution to ExpectedValue Explosion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.17 Comparison of Relative Improvements in Profit, Revenue and Costs As-sociated with Stochastic Solution vs. Expected Value Solution for theImplosion Problem with 5% Supplier Flexibility . . . . . . . . . . . . . . . 95



4.20 Comparison of Expected Fill-Rates Associated with Expected Value So-lution vs. Stochastic Solution for the Implosion Problem with 5% SupplyFlexibility when Committed Supply is Given by the Solution to ExpectedValue Explosion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.21 Comparison of Relative Improvements in Profit, Revenue and Costs As-sociated with Stochastic Solution vs. Expected Value Solution for theImplosion Problem with 10% Supplier Flexibility . . . . . . . . . . . . . . 97



4.24 Comparison of Expected Fill-Rates Associated with Expected Value Solu-tion vs. Stochastic Solution for the Implosion Problem with 10% SupplyFlexibility when Committed Supply is Given by the Solution to ExpectedValue Explosion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.1 Characteristics of Problem Instances Analyzed in the Component Ra-tioning Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

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5.2 Global Parameter Settings for the SAA Algorithm for the ComponentRationing Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5.3 Average Optimality Gaps for the Search Algorithm for the ComponentRationing Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5.4 Parameter Settings and Results for the Application of the SAA Methodto the Component Rationing Problem . . . . . . . . . . . . . . . . . . . . 137

5.5 Percentage Improvements Observed With Component Rationing as Com-pared to Without Rationing . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6.1 Comparison of Improvements in Expected Performance of Stochastic Ex-plosion, Stochastic Implosion and Threshold Rationing for the Mar-3Problem Set Relative to the Mar-3h Case . . . . . . . . . . . . . . . . . . 147



6.4 Comparison of Improvements in Expected Performance of Stochastic Ex-plosion, Stochastic Implosion and Threshold Rationing for the May-1Problem Set Relative to the May-1h Case . . . . . . . . . . . . . . . . . . 154

6.5 Comparison of Improvements in Expected Performance of Stochastic Ex-plosion, Stochastic Implosion and Threshold Rationing for the May-2Problem Set Relative to the May-2h Case . . . . . . . . . . . . . . . . . . 155

6.6 Comparison of Improvements in Expected Performance of Stochastic Ex-plosion, Stochastic Implosion and Threshold Rationing for the Jul-1 Prob-lem Set Relative to the Jul-1h Case . . . . . . . . . . . . . . . . . . . . . . 156



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Acknowledgments

I am deeply grateful to my thesis advisor, Terry Harrison, for gently, yet firmly guidingme through the dissertation process and for providing me with the opportunities that Ineeded to achieve my personal and professional goals. Through a perfect combinationof patience and prodding, he has helped to make this journey a truly enjoyable andeducational one for me.

My mentor, Tom Ervolina, has been an absolute inspiration to me. His technical,moral and professional support have been instrumental in seeing me through many chal-lenges. I could not have succeeded without him.

A hearty thanks to my committee members for taking the time to read my work, forbeing supportive, and for lending me their ears and their advice.

My special thanks to Barun Gupta for his interest in this research, and for takingthe time to share his insights and provide valuable feedback. I am very grateful as wellto Jeff Linderoth for volunteering to help me explore solution methods that I would haveotherwise been incapable of exploring, and for his patience in answering so many of myquestions.

For providing the computing resources essential to performing my research, I amindebted to the Penn State High Performance Computing Group (Vijay Agarwala, PhilSorber, Adam Focht, Jason Holmes and the rest of HPC group), and to the RIIT Group(Fred Houlihan) at the Smeal College of Business Administration.

I am grateful to the professors in the SC&IS (formerly MS&IS), Marketing, IndustrialEngineering, and Statistics departments at Penn State, whose influences on me throughtheir lectures, research collaborations and other interactions have been so importantto my intellectual well-being. In particular, Susan Xu has been a wonderful mentorand role-model to me, and I am constantly learning from her. I also want to expressmy sincere appreciation to Hemant Bhargava, who helped to plant the seeds of my

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interest in configure-to-order systems, and who first introduced me to the larger researchcommunity. I am appreciative, as well, to Russell Barton for being a staunch supporterof the Smeal Ph.D. Association, my involvement with which has added so much richnessto my graduate experience.

My sincere thanks to Beth Bower and Alice Young, for their confidence in me andfor making life at the office pleasant and convenient. I am grateful for the friendship ofthe graduate students that I have shared classes and offices with, and the comraderie ofthe students involved with the Smeal Ph.D. Association. I am particularly grateful toHari Natarajan for teaching me how to use the CPLEX callable library.

I want to acknowledge the IBM Integrated Supply Chain, which provided fundingfor a large part of this research, the Supply Chain & Information Systems departmentfor funding my graduate studies, and the Laboratory for Economics Management andAuctions at the Smeal College of Business Administration, not only for its financialsupport, but for providing me with a valuable and unique research experience in myearly years.

As it is with almost everything, I owe so much to my family who, with their uncon-ditional love and support, serve as constant reminders to me of who I am and what’simportant. Many expressions of gratitude and appreciation are owed to my husband,who is the meaning in my life, and who fed me without getting fed up with me as Ilumbered toward the finish line.

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Chapter 1

Introduction

“You can have it in any color, as long as it’s black,” is a phrase attributed to Henry

Ford back when Ford Motor Company first introduced the Ford Model T in 1908. It

was The Ford Motor Company that popularized the practice now referred to as ‘mass

production’, or the practice of manufacturing identical products in large quantities and

at low cost. Nearly a century later, an improved paradigm is emerging, in which products

are customized rather than standardized, in large quantities and at low cost. This new

paradigm is referred to as ‘mass customization’.

The practice of mass customization is motivated by an increasingly competitive mar-

ketplace and is enabled by flexible manufacturing technologies, modern business pro-

cesses, and new information technologies such as the World Wide Web. Papers by Lam-

pel and Mintzberg (1996), Gilmore and Pine (1997), Silveira et al. (2001), and Zipkin

(2001) provide greater insight into mass customization.

Mass customization, along with the economic and technological environments that

support and enable it, has brought change to supply chain management practice. The

traditional ‘make-to-stock’ supply chain system in which products are produced in antic-

2

ipation of actual customer orders is, in many cases, being replaced by the ‘make-to-order’

supply chain system. In the make-to-order supply chain system, products are produced

in direct response to actual customer orders. The latter system, while not new, is par-

ticularly suited to the production of customized products since such products cannot be

completed prior to receiving a customer order.

Effectively matching supply and demand (a process referred to as ‘availability mange-

ment’) in an uncertain environment is challenging regardless of whether one is managing

a make-to-stock or make-to-order supply chain. Due to the increasing popularity of mass

customization, and hence relevance of make-to-order supply chains, this dissertation ad-

dresses the problem of availability management for a specific type of a make-to-order

system, called a ‘configure-to-order’ (CTO) supply chain. Both the CTO system and

the process of availability management are described in greater detail in the sections

that follow.

The overarching objective of this research is to define and solve optimization problems

in availability management for CTO systems that fit into a realistic decision-making

framework. The specific decision-making framework, or business process, chosen in this

research is that of the Sales and Operations Planning Process, which is described in

§1.1.1.

1.1 Availability Management

Generally speaking, the term ‘availability management’ is a decisionmaking process per-

formed by a goods or services provider for the purpose of ensuring that the supply of

product is in alignment with their customers’ demand for product. It constitutes an

integral part of the total effort that is required to provide effective and efficient customer

service. In some situations, demand may be affected by completely exogenous factors,

3

and so availability management is restricted to the management of resources (e.g., raw

materials, equipment, labor). In other situations, demand may be affected by endoge-

nous factors (e.g., advertizing, promotions and discounts) as well. In such situations,

availability management can be used to not only manage supply, but demand as well.

This creates the opportunity for more effective alignment between supply and demand.

In this dissertation, the latter situation is assumed to be the case.

Effective availability management is critical to the successful operation of any goods

or services provider because the ability to match supply and demand has a direct effect

on order fulfillment and the earned revenue. For a publicly owned company, consistent

revenue growth is an important signal of the company’s financial health. Therefore,

investors pay close attention to the revenue forecasts announced by a company every

financial quarter. While aggressive forecasts encourage investors, causing a stock’s price

to rise, ultimately, the failure to meet forecasts at the end of a quarter can have dele-

terious effects on investor confidence, causing a stock’s price to fall. In light of this, it

is important for a company to be aggressive yet realistic about its ability to achieve its

revenue targets. As mentioned earlier, clever management of supply and demand can be

crucial in this regard. Profitability and customer satisfaction are also important metrics

for gauging the health of a company. These factors are also accounted for in availability

management.

1.1.1 Sales & Operations Planning

Sales & Operations Planning (S&OP) is an availability management concept that is

becoming more popular in industry, as evidenced by a recent industry roundtable (Center

for Digital Strategies 2004) and some recent white papers on the subject (Manugistics

2001, Aberdeen Group 2005). The APICS1 Dictionary (Cox and Blackstone 2002) defines

1The Association for Operations Management

4

S&OP as “a process to develop tactical plans that provide management the ability to

strategically direct its businesses to achieve competitive advantage on a continuous basis

by integrating customer focused marketing plans for new and existing products with

the management of the supply chain. The process brings together all the plans for

the business (sales, marketing, development, manufacturing, sourcing and financial) into

one integrated set of plans.” The main premise of S&OP is that demand and supply

matching can be done most effectively if all parts of the business collaborate.

S&OP may require radical change in organizations that are used to allowing these

different business functions to be managed independently. In such organizations, it is not

uncommon that the incentives of the individual business functions are in direct opposi-

tion to each other. For example, it is often the case that manufacturing is interested in

keeping inventory levels lower so as to reduce costs. At the same time, sales is interested

in keeping inventory levels higher so that more sales can potentially be met. As a result,

sales representatives tend to provide manufacturers with a deliberate overstatement of

demand. The manufacturer, expecting this kind of behavior, tells the sales representa-

tives that they can support their requests, but deliberately produce less than what the

sale representatives ask for. In the midst of all of this, a clear, unbiased view of the true

supply and demand situation is lost. The S&OP process seeks to promote collaborative

decision making so that an unbiased picture of supply and demand will emerge, and the

incentives of all business functions can be aligned. Managerial perspectives on S&OP

are provided in the literature (Palmatier and Crum 2003, Ling and Goddard 1988).

There is no standard for the sequence of decisions made in the S&OP process, but

a general representation of such a sequence may be given as: demand planning, supply

planning, review of demand and supply plans, execution of plans. This sequence of

decisions is typically repeated every few weeks to a month, and each time, the demand

and supply plans span several months. The overlapping planning horizons that results

from such practice are collectively referred to as a ‘rolling planning horizon’.

5

Demand planning refers to the process of forecasting demand over the planning hori-

zon. Historical sales data and knowledge about market trends, planned promotions and

new product introductions are used to determine an initial, unbiased forecast of demand.

This unbiased forecast represents the decision-makers’ best guess as to what demand will

be. That is, it is neither deliberately overstated nor understated, as might occur in a

decentralized decisionmaking setting. In practice, it is not uncommon for the unbiased

forecast to be adjusted, sometimes arbitrarily so, to achieve a desired level of forecasted

revenue. The difference between the adjusted forecast and the unbiased forecast can

be partly bridged through demand ‘shaping’ activities performed by the marketing de-

partment. These adjusted numbers constitute the ‘demand plan’, which provides point

estimates for the quantity of each product that the company targets to sell in each period,

over the planning horizon. In this sense, the demand plan reflects both the operational

and the financial objectives of the organization. Thus, it is perhaps more appropriate to

think of these quantities as sales targets rather than just forecasts. Although realistic,

the notion of treating demand as a targeted sales level is largely unaddressed in the

research literature.

In the demand planning stage, only a general notion of supply constraints may be

taken into account. However, once the demand plan has been established, detailed supply

plans can be made in the supply planning stage. In the supply planning stage, the sales

targets in the demand plan are translated into the corresponding component require-

ments. As will be described in the following sections, this translation is not necessarily

straightforward in a CTO system. Nevertheless, after the component requirements have

been determined, they are requested from the suppliers. The supply plan is a statement

of the supply quantities that the suppliers have agreed to deliver in each period of the

planning horizon. Typically, the supply plan will differ from the supply that was initially

requested, depending on the capacities and lead time capabilities of the suppliers.

At this point, the supply and demand plans are reviewed to detemine if they are in

6

alignment. To this end, either the demand plan or supply plan, or both, may need to be

adjusted. Depending on the relationships between the manufacturer and its suppliers,

modifications to the supply plan may be somewhat restricted. The final, adjusted set of

sales targets is referred to as the ‘commitment-to-sales’ because it represents the quantity

of products that the supply chain makes a commitment to support in sales over the

planning horizon. The reliability of the commitment-to-sales has important implications

for the company to meet its financial objectives. Determining the optimal commitment-

to-sales is not trivial in a CTO system with order configuration uncertainty, and is one

of the main challenges addressed in this research. The generation of the commitment-

to-sales completes the planning cycle. Until the next cycle, the company will execute its

marketing and manufacturing operations in accordance with the commitment-to-sales. In

this way, S&OP serves to create a link between the operational, financial and marketing

objectives of the organization.

So far, the discussion of S&OP has deliberately been kept general. However, the

primary objective of this research is to study availability management in a CTO supply

chain. Therefore, the following sections describe CTO supply chains and how availability

management, by way of S&OP, can be applied to such systems.

1.2 Configure-to-Order Supply Chains

A ‘configure-to-order’ (CTO) system is a supply chain in which the product being sold

may be configured by the customer. A configurable product is one which is composed of,

or assembled from, several modular components. While the configuration (i.e., a subset

of modular components) may be specified by the customer (with possibly compatibility

restrictions), the components themselves are standardized. For example, a customer

ordering a mainframe computer may specify the speed and number of central processing

units (CPUs), size and number of disk drives and size and number of random access

7

memory units, that will be installed in the final system. As an example of a configurable

product in the services industry, consider a travel package that includes a flight, lodging

and car rental, where the customer is allowed to pick the airline and flight schedule, the

number of nights of lodging and type of room, and the size of car to rent. Riess (1996)

provides a non-technical overview of supply chain planning in a CTO environment. Refer

to §2.1 for a comparison of the CTO system with the more general assemble-to-order

(ATO) system.

In a CTO system that supports multiple products, such as the one considered in this

research, it is important to distinguish between a configurable product and a product

configuration. A product configuration refers to a unique combination of components.

A configurable product is a classification for several similar configurations, or configu-

rations whose potential component requirements satisfy similar criteria. For example,

a company selling modular homes may classify their products as ‘Cape Cod’, ‘Colonial’

and ‘Split-Level’, since each of these home styles may have distinctive features and ma-

terial requirements. However, each of the styles can be configured into several different

unique configurations satisfying that particular style of home.

In practice, a CTO supply chain is really a hybrid of the pure make-to-stock and

make-to-order systems, because the supply-facing end of the supply chain operates in

a make-to-stock fashion, while the customer-facing end of the supply chain operates in

a make-to-order fashion. This hybrid system defines a ‘push-pull’ boundary2 that lies

between the make-to-stock and make-to-order based portions of the end-to-end supply

chain system. In a CTO system, it is perhaps natural to position the push-pull boundary

at the level of the configurable components, although this does not necessarily need to

2On the supply facing side of a push-pull boundary, sub-assemblies are assembled according to a planprior to receiving actual customer orders whereas assemblies on the customer facing side of a push-pullboundary are assembled in response to actual customer orders. Refer to Simchi-Levi et al. (2002) formore on on the topic of ‘push-pull’ boundaries in supply chains.

8

be the case.3 As with its close relative, the assemble-to-order system, a CTO system

stocks inventory of the modular components and holds no finished goods.

In this dissertation, we focus on products whose configurations can be completely

specified in terms of a vector of non-negative integer quantities, where the length of the

vector is equal to the number of unique standardized components that can be selected for

a given product, and each element in the vector represents the quantity of a particular

standardized component to be used in a given product. Additionally, it is assumed

that vector elements are independently distributed. While most customizable products

can be adequately defined in this way, some products are more suited to this format of

expression than others, and some customizable products may not fit into this format at

all (e.g., dentures!). A particularly challenging aspect of managing CTO supply chains is

dealing with the uncertainty associated with how customers will want to configure their

products. That is, prior to receiving a customer order, the quantity of each component

that will be requested to fulfill the product order is uncertain. This uncertainty is referred

to as ‘order configuration uncertainty’.

In practice, stochastic models are rarely used to account for uncertainty of any kind,4

and are even less likely to be used to account for configuration uncertainty. Even the

academic treatment of this particular source of uncertainty is limited. As will be shown

in Chapter 2, the academic literature has tended to focus on product demand uncertainty

in assemble-to-order systems, assuming that usage quantities of product components is

fixed. One of the main contributions of this thesis is to use stochastic optimization to

addresses configuration uncertainty in the context of availability management for a CTO

supply chain.

3Swaminathan and Tayur (1998a) perform an interesting study on identifying this boundary in aCTO setting in the computer industry.

4Stochastic models are not commonly used in practice due to the additional complexity required tounderstand such models.

9

1.3 Availability Management for CTO Systems

In a CTO supply chain system, several stages in the S&OP process become challenging.

In particular, determining the optimal supply requirements and the optimal commitment-

to-sales, in the presence of configuration uncertainty, are two problems that this thesis

addresses. These problems are referred to as the ‘explosion’ problem and ‘implosion’

problems, and are introduced in §1.3.1 and §1.3.2, respectively. A third problem, called

the component rationing problem, deals with optimizing a policy for allocating the com-

ponents that have been promised by suppliers to the products committed to sales. This

problem is introduced in §1.3.3.

1.3.1 The Explosion Problem

In a CTO system with configuration uncertainty, the explosion problem involves the

determination the profit-maximizing quantity of components to request from suppliers

over the planning horizon, given the initial demand plan. The term ‘explosion’ is used

to describe this problem because it entails expanding the configurable product into its

detailed component requirements.

The fact that the demand plan is treated deterministically in the explosion problem

needs to be explained. The primary reason for this assumption is that in many prac-

tical situations, the data required to accurately estimate product demand distributions

is either difficult to identify, not available,5 or both. This is because product demand

distributions are affected by a complex combination of exogenous factors (e.g., compe-

tition, consumer tastes, seasonality) and endogenous factors (e.g., pricing, promotions,

new product introductions), some of which may be difficult to quantify. Additionally,

the importance of the role of the demand plan in the organization cannot be overstated.

5For example, companies typically keep track of actual orders or shipments, but not those sales thatmay have been lost due to insufficient stock.

10

The financial objectives of the company are closely tied to the demand plan. In view of

this, there needs to be clarity in its statement and decision makers are prone to insert

their better judgement into the final demand plan. For all these reasons, demand plans

are often treated, in practice, as deterministic point estimates with risk factors quali-

tatively factored into the point estimate. While it may not be ideal to ignore product

demand distributions, it is often not practical to accurately model them. In such circum-

stances, models that assume the availability of distributional demand data are likely to

lose their effectiveness and/or applicability. On the other hand, product configurations

are relatively insulated from the various endogenous and exogenous factors that affect

product demand. Additionally, empirical distributions can be easily obtained using the

order history for a product.

Even assuming a deterministic demand plan, the explosion problem is not straight-

forward. This is because for a given demand plan, it is uncertain as to what the detailed

component requirements are prior to receiving actual orders for the product. This is due

to configuration uncertainty. The optimal set of supply requirements to request from

suppliers will need to appropriately balance the risk of carrying too much inventory of

components against the risk of not carrying enough. This calculation becomes more

complicated when several products have certain components in common, a circumstance

referred to as ‘component commonality’, because of the combined effects of the config-

uration uncertainty associated the products involved. Within the S&OP process, the

explosion problem would be addressed after the demand plan is in place, and prior to

the determination of the supply plan. The explosion problem is modeled and analyzed

in §3.

11

1.3.2 The Implosion Problem

In the review stage of the S&OP process, decision-makers are given the current demand

and supply plans. While the demand plan is stated at the product level (i.e., in terms

of forecasted quantities of various configurable products), in a CTO system, the supply

plan is stated at the bottom of the bill-of-materials (BOM) structure. This is because

the supply plan reflects the quantities of items that need to be procured from suppliers.

As stated earlier, for the sake of simplifying the problem setting, in this research it

is assumed that the components used to configure the products lie at the bottom of

the BOM structure and are directly procured from suppliers. The implosion problem

involves the reconciliation of the demand and supply plans in such a way as to maximize

the expected profit over the planning horizon. The reconciled demand plan provides the

final set of sales targets that is the commitment-to-sales.

If the demand and supply plans are well-aligned, then there may be no need to modify

either plan, and the commitment-to-sales would simply be identical to the demand plan.

However, if the demand and supply plans are misaligned, either or both plans may

need to be adjusted. Since it is unlikely that suppliers will commit more supply than

was requested, it will typically be the case that misalignments are caused by supply

shortages. In this case, the optimal commitment-to-sales will need to balance the risks

of over-committing and under-committing to sales. While over-commitments result in

reduced service-levels,6 under-commitments fail to capture the full market potential.

The implosion problem is similar in concept to the abtract inverse newsvendor problem

studied by Carr and Lovejoy (2000).

To convey the complexity involved in coming up with an optimal commitment to

sales, first consider the case of a single non-configurable product (i.e., all component

requirements are fixed). In this case, given the current supply plan, the commitment-to-

6The service-level is a measure of the percentage of orders that are satisfied on time

12

sales would simply be the maximum quantity of product that is satisfiable from the given

supply. If there are multiple non-configurable products that do not share components,

then the commitment-to-sales is just as easily determined. However, in the presence of

component commonality, tradeoffs may need to be made regarding the preferred alloca-

tion of available components among products. Now consider again the single product

case but relax the requirement that it has a fixed configuration. In the presence of con-

figuration uncertainty, it is no longer clear what quantity of the product can be sold,

even given the exact number of components that are available. When there are multiple

products to consider, component shortages that force a reduction in the commitment-

to-sales (relative to the demand plan) for one product may result in an excess volume

of the other components that that product also uses.7 In such a situation, it may be

advantageous to increase the commitment-to-sales for another product that also uses

the components that are now in excess. Therefore, the optimal commitment-to-sales

quantities may be either greater than, or less than, the quantities stated in the original

demand plan. In the multi-product CTO environment considered in this research, both

configuration uncertainty and component commonality make determining the optimal

commitment-to-sales from a given supply and demand plan a difficult problem.

The solution to the implosion problem is influenced by the supply plan insofar as

the commitment-to-sales quantities are limited by the availability of components. The

supply plan, in turn, is influenced by the result of the explosion process insofar as sup-

pliers will agree to deliver components based on the quantities that have been requested.

Meanwhile, the quantity of components requested from suppliers is obtained by solving

the explosion problem. In this way, the explosion problems and implosion problems are

related. The fact that detailed supply constraints may not be available until a request for

supply has been made implies that modeling approaches that presume the availability of

7This excess is created when products cannot be partially configured, so that shortages in one com-ponent result in excesses of complementary components.

13

detailed supply constraints when determining the optimal ordering quantities may not

be practical. In following with this scenario, the explosion and implosion are solved in se-

quence, rather than solving a single problem that simultaneously finalizes the component

order quantities and the commitment-to-sales. This being said, reasonable adjustments

to the component order quantities, based on the original supply plan determined via the

explosion process, may be allowed in the implosion problem.

In this research, both the case where the supply plan is fixed and the case where

it is adjustable, or flexible, are considered. The latter case is considered because the

quantities that suppliers agree to deliver over the planning horizon may be subject to

change, particularly for the periods futher out in the future. Downward supply flexibility

is beneficial in a CTO system since supply constraints on a component may reduce the

need for a complementary component. Upward supply flexibility is beneficial when the

planned supply is not sufficient to achieve the sales target. Suppliers may accomplish

supply adjustments by simply redistributing the quantity of components supplied over

the planning horizon, or by changing the overall production quantities. The ability and

willingness of a supplier to perform such adjustments would depend on several factors,

such as the pre-negotiated contract terms, production capacity and the perceived value

of the business relationship. Sometimes, supply appears to be flexible because a supplier

has previously misrepresented or miscalculated his actual supply capacity and, given the

rolling planning horizon, corrects for these errors in a future planning period. Whatever

the case may be, if there is a tendency for supply to exhibit flexibility, then it is reasonable

to take this into account when optimizing the commitment-to-sales in the implosion

problem. In this case, the implosion problem also needs to determine the optimal supply

adjustments to be made, subject to limited flexibility around the original supply plan.

The implosion problem is modeled and analyzed in §4.

14

1.3.3 The Component Rationing Problem

After the commitment-to-sales has been determined, the S&OP cycle is complete and

the company must now execute its operations in accordance with the plans set forth

by the S&OP process. An important part of this execution is that of order promising.

In practice, order promising is performed with using an Available-to-Promise (ATP)

schedule. As defined in the APICS dictionary (Cox and Blackstone 2002), an ATP

schedule in a manufacturing setting refers to “the uncommitted portion of a company’s

inventory and planned production maintained in the master schedule to support customer

order promising.” Therefore, when an order arrives, it is promised in the period that

the customer requests it, or if there is insufficient inventory to do so, the earliest period

thereafter.

Traditionally, the ATP schedule is stated at the product level, with the available

quantity of a product being stated period by period over a given time horizon. These

quantities are decremented accordingly whenever an order for the product is placed.

In this way, the ATP schedule ensures that demand will not exceed supply. Stating

availability at the product level makes sense in a make-to-stock system where orders are

directly satisfied from inventory. In a CTO setting, however, it is difficult to apply the

ATP in this manner since no finished goods inventory is kept and order configurations

are uncertain. Instead, it makes sense to make the ATP schedule a statement of available

components. Still, there remains the challenge of ensuring that the available components

are allocated to orders in such a manner as to maintain alignment with the commitment-

to-sales, which is stated at the product level.

If used in conjuction with an effective component allocation scheme, the ATP schedule

of available components can be used to complement the commitment-to-sales. A com-

ponent rationing policy specifies when certain product orders should not be promised

in a given period, even if there is sufficient component supply to fulfill it. The purpose

15

of doing this would be to reserve the remaining component supply in anticipation of

the arrival of more valuable orders8 that may require the same components. In this

research, the component rationing problem involves determining the optimal rationing

levels to use, given a particular policy structure. The particular policy structure that is

optimized as part of this research is called a ‘threshold policy’.9 This policy structure

has been shown to be optimal for similar supply chain settings in the research literature,

as reviewed in §2.4. The details of the threshold policy are described in Chapter 5.

To summarize availability management in the context of sales and operations plan-

ning for a CTO system:

1. A demand plan is created by decision-makers based on the financial goals of the

business.

2. The Explosion problem is solved to determine the corresponding supply needs.

3. A supply commitment is received from suppliers.

4. The Implosion problem is solved, resulting in a modified demand plan called the

commitment-to-sales that is used to guide sales efforts.

5. Orders arrive and are satisfied according to some allocation policy.

1.4 Industry Context

The specific research problems addressed by this thesis are motivated by real supply

chain issues being faced by the Systems and Technology Group at International Business

8One order may be more valuable than another by virtue of generating higher revenue, higher profit,or being requested by a more important customer.

9Under this policy structure, an order is no longer fulfilled if fulfilling the order will cause theinventory level of any of its necessary components to fall below a specified threshold level.

16

Machines (IBM). IBM is a global information technology company. Its Systems and

Technology Group is primarily responsible for the manufacture and sale of IBM’s server

computers, storage systems and microelectronic devices. In early 2005, IBM held the

largest share (approximately 30%) in the computer server market (Gonsalves 2005).

Roughly one fifth of IBM’s total revenue in 2004 was generated by the activities of the

Systems and Technology Group (IBM 2004).

The manufacture of server computers at IBM occurs within in a CTO supply chain

environment, where IBM’s customers are allowed to to configure their own servers. These

products are considered to be highly configurable because there is a considerable amount

of flexibility in the way components can be combined in a final product. Therefore,

there is significant configuration uncertainty present in this environment. Availability

management plays a central role in the planning of supply and demand in the Systems

and Technology Group and S&OP is a major driver of the company’s marketing and

manufacturing efforts. The success or failure of the S&OP process has the potential to

have a significant impact on IBM’s ability to reach its annual revenue targets.

This research was conducted in close collaboration with IBM’s Systems and Technol-

ogy Group, as well as with IBM’s world renowned research division. It has been funded

by a research grant from IBM and the data used in this research is based on data relevant

to IBM’s operations.

1.5 Summary of Research Problem and Objectives

The various problems studied in this thesis relate to availability management in a mul-

tiple period, multiple product CTO supply chain with order configuration uncertainty.

Within this context, this thesis identifies and solves three main problems: the explosion

problem, the implosion problem and the component rationing problem. Each of these

17

problems is defined to fit within the framework of sales and operations planning, which

is a practiced business process. Together, the three problems address three sequential

stages within S&OP, allowing for the analysis of their combined impact on expected

revenue, profit and serviceability (i.e., order fill rate).

While the general concepts behind the explosion problem and component rationing

problem are not new to the field, order configuration uncertainty is often ignored in the

literature in these areas. Also, compared to existing work, this dissertation addresses

both of these problems on a larger, more realistic scale and places them within a prac-

tical decision-making framework. The concepts that underlie the implosion problem are

largely unaddressed by the existing literature. Additionally, the topic of S&OP, de-

spite its increasing popularity in practice, has been largely ignored, or unnoticed, by the

academic community.

The research objectives of this thesis may be summarized as follows:

1. Model and solve the explosion problem for a multiple product CTO supply chain

with configuration uncertainty, and demonstrate the value of accounting for this

uncertainty when it does in fact exist.

2. Model and solve the implosion problem for a multiple product CTO supply chain

with configuration uncertainty, for both the fixed and flexible supply plan cases.

Demonstrate the value of flexible supply and of accounting for configuration un-

certainty when it does in fact exist.

3. Model and solve the inventory rationing problem, and compare the performance of

the resulting rationing policy with a first-come-first-serve policy.

18

4. Demonstrate the value of using the explosion, implosion and inventory rationing

models developed in this thesis in concert and in isolation, for sales & operations

planning.

The rest of this thesis is organized as follows: Chapter 2 provides a comprehensive

review of the related literature. Chapter 3 describes the explosion problem. Chapter 4

describes the implosion problem. Chapter 5 describes the component rationing problem.

Chapter 6 analyzes the broader impact of solving all three problems in sequence. Chapter

7 provides a decription of the methods used to develop credible, valid and correct models.

Finally, Chapter 8 concludes the thesis with a brief statement of the potential value

provided by this research.

Chapter 2

Literature Review

The work in this dissertation is related to existing research in the areas of stochas-

tic inventory control for assemble-to-order systems, available-to-promise scheduling and

flexible supply contracting. In the following sections, these research areas are reviewed

from the point of view of the research at hand.

2.1 Inventory Ordering Policies in Assemble-To-Order Sys-

tems

An assemble-to-order (ATO) system is one in which the final assembly of a product

from several components is postponed until an order for the product is received. A

configure-to-order (CTO) system is a distinct type of ATO system in that the customer

determines the components that will be included in the product ordered. This is not

necessarily true for an ATO system in general. It is typically argued that the assembly

of final products is delayed in an ATO system because assembly times are negligible and

this delay provides a risk pooling advantage when several ATO products have common

components. However, in a CTO system, due to the need for customer involvement in

20

the configuration process, the assembly of the final product is delayed even if assembly

times are significant and doing so poses no risk advantage. This is because it is impossible

to assemble the final product until the customer configures it. Despite these practical

differences between ATO and CTO systems, from a purely analytic perspective, ATO

and CTO systems are similar.1 Therefore, the research done on ATO systems is relevant

to the CTO system studied in this research.

In the stochastic inventory control literature for ATO systems, significant focus is

placed on identifying and/or evaluating the performance of various inventory ordering

policies in an ATO system. This focus is most relevant to the explosion problem ad-

dressed in this research, and the literature having this focus is reviewed in this section.

Another area of focus in the literature is on the allocation of components among prod-

ucts. This second focus area is most relevant to the implosion and component rationing

problems addressed in this research. A review of the literature having the latter focus is

provided in §2.4.

Song and Zipkin (2003) provide a comprehensive review of the research that has been

done in the area of stochastic inventory control in ATO systems. It is clear from the

literature that ATO systems are complex to analyze, and optimal inventory ordering

policies have been determined only for special cases (e.g., zero component replenishment

leadtimes, or positive leadtimes for a single product only). Furthermore, it is necessary to

assume stationary demand distributions or an otherwise steady state condition in order

for these policies to apply in multiple period or continuous review settings. In many

practical settings, however, the demand environment is highly non-stationary. This may

be especially true for fashion goods or technology goods, which have relatively short life

1For example, if a CTO system is modeled such that all possible product configurations are definedas separate, uniquely configured products, then there is little evidence from the model (aside from thepotentially exponentially large number of products), that the products are configured-to-order. Likewise,if an ATO system that does not allow customers to configure their orders is modeled such that similarproducts are aggregated into set of product families so that each product family consists of a variety ofproduct configurations, the model may appear to be representing a CTO system when in pratice it isnot.

21

cycles. Aside from unexpected events, seasonal and fiscal calendar effects can also create

non-stationary demand conditions.2 Therefore, the optimal policies developed in the

research assume conditions that are unlikely to hold in practice. Typically, in situations

where an optimal policy is not available, a particular stationary inventory ordering policy

is assumed. Heuristics are then proposed to optimize and evaluate the performance of

the assumed policy. Again, these policies are evaluated assuming stationarity of the

problem data.

Only a small portion of the ATO literature explicitly models uncertainty in product

configurations, a key focus in this research. In particular, Cheng et al. (2002) and Lu

et al. (2003) do consider order configuration uncertainty, in addition to product demand

uncertainty. Cheng et al. (2002) study the inventory-service tradeoff in a CTO setting

with multiple demand classes, where each demand/product class requires a different

level of service. For given base-stock levels, Lu et al. (2003) study the impact of product

structure, demand and leadtime variability, and advance demand information on system

performance. Both of these papers, like most of the literature, assume stationary base

stock inventory ordering policies in a stationary demand environment. Additionally, they

are concerned with cost minimization as opposed to profit maximization, which is the

objective of the explosion problem.

The preoccupation of the literature with finding and evaluating stationary inventory

ordering policies, optimal or otherwise, is perhaps excessive since stationary policies

are frequently not optimal and their simplicity does not always override the need to

account for non-stationarites. As an exception to the norm, Swaminathan and Tayur

(1998b) use a series of nested stochastic programs to solve a multi-period inventory

ordering and production problem for an ATO system with non-stationary demand. Their

solution approach is a myopic one since each of the nested subproblems is a single

2For example, IBM observes significant demand surges for its server products towards the end ofeach fiscal quarter and towards the end of every year.

22

period, two-stage stochastic inventory ordering problem that does not consider future

periods. In an earlier study, Srinivasan et al. (1992) analyze a similar problem, but

include probabilistic service level constraints. They show that solution methods that

ignore component commonality in component ordering decisions is ‘vastly inferior’ to

methods that do not. In both cases, as is the case in the explosion problem explored

in this research, structured inventory ordering policies are neither sought nor evaluated.

However, unlike the explosion problem, the problems addressed by Srinivasan et al.

(1992) and Swaminathan and Tayur (1998b) treat product demand as uncertain and

order configurations as fixed.3 They are also concerned with cost minimization rather

than profit maximization.

Finally, the literature reviewed in this section solve relatively small problems, typ-

ically consisting of no more than 10 products and 20 components. As an exception,

Swaminathan and Tayur (1998b) and Srinivasan et al. (1992) consider problems as large

as 50 products and 50 components. In this dissertation, problems with 50 products and

50 components are solved, as well as problems as large as 21 products and 569 compo-

nents are solved, corresponding to data sets obtained from the Systems & Technology

Group at IBM.

2.2 Available-To-Promise Scheduling

The ATP scheduling literature relates to both the implosion problem and the component

rationing problems studied in this dissertation.

Ball et al. (2004) provide an excellent overview and review of the research in the area

of ATP scheduling, and much of this section draws upon the insights provided therein.

They point out that the conventional ATP systems that were associated with traditional

3The potential advantages of treating order configuration as uncertain rather than product demandare explained in §1.1.1 and §1.2.

23

make-to-stock supply chains are being updated to accomodate the make-to-order supply

chains prevalent today. They classify ATP models as being either ‘push based’ or ‘pull

based’. Push based ATP models allocate resources to products or demand classes prior

to receiving orders, while pull based ATP models perform the allocation in response to

incoming orders.4 The primary advantages of push based scheduling over pull based

scheduling is that order promising decisions can incorporate long term objectives and

can be provided to customers immediately. However, pull based ATP scheduling has

the advantage over push based scheduling in that it can be responsive to disparaties

between actual demand and forecasted demand. Since pull based ATP scheduling makes

resource allocations after demand is realized, there is the need to repeatedly determine

allocations. This process could be time consuming, depending on the complexity of the

allocation problem. Additionally, the more frequently the allocation problem is solved,

the more myopic it becomes.

The implosion problem (i.e., the problem of determining the optimal commitment-

to-sales quantities that can be supported by the available component supply) and com-

ponent rationing problem (i.e., the problem of determining how to reserve limited com-

ponents for higher priority orders) studied in this research are related to the research in

push based ATP scheduling since the commitment-to-sales and the threshold rationing

policy is set prior to order arrivals. Ervolina and Dietrich (2001) studied deterministic

push based ATP scheduling models for CTO systems with multiple products, compo-

nents and time periods. Their work is based on the resource allocation software engines

developed at IBM Research, and provides an important part of the foundation for the

research developed in this dissertation. They sketch two different heuristic approaches

for determining the ATP schedule, but do not provide any computational results. While

4It is important to note that this classification can be independent of the underlying supply chainsystem. That is, push based ATP scheduling can be implemented in pull based (or make-to-order) supplychains. Similarly, pull based ATP scheduling can be implemented in push based (or make-to-stock) supplychains.

24

Ervolina and Dietrich (2001) acknowledge that product configurations are uncertain,

they use an ‘average box demand’ to represent them. The implosion problem studied

in this dissertation extends their problem to include stochastic product configurations.

Therefore, the implosion problem can be classified as a stochastic push based ATP prob-

lem.

In the deterministic pull-based ATP scheduling realm, Chen et al. (2002) consider a

rolling horizon, ATP problem for a CTO product. In their model, orders are batched

over some pre-specified period of time, after which order commitment dates and the

production schedule are obtained by solving a multi-period mixed-integer program. In

addition to specifying a due date, customers are allowed to specify a range of acceptable

delivery quantities and a set of substitutable suppliers for a given component. They

find that while profits initially increase with the length of the batching period due to

increased information about demand, profits eventually drop as more orders with shorter

due dates are lost. In an earlier work, Chen et al. (2001) consider a similar problem where

customer due dates are flexible and charge a penalty for allowing component inventory

levels to drop below a pre-specified reserve level at the end of each batching interval. In

any given run, they use the same reserve level for all resources. An experimental study

shows the use of such a reserve level can increase profits by anticipating the arrival of

more profitable orders in future batching intervals.

The stochastic push based ATP problem is related to the inventory and capacity

rationing literature, which is reviewed in §2.4.5 Stochastic push based ATP scheduling

is also related to yield management (McGill and van Ryzin 1999, Belobaba 1987). Yield

management problems, however, tend to emphasize the use of pricing as a method of

controlling the allocation of fixed, perishable resources. In ATP scheduling, resources

are typically not perishable and pricing is typically not treated as a decision variable.

5The ‘line’ between rationing problems and pull based ATP scheduling is a thin one, and where it iscurrently drawn has more to do with the problem context and the modeling approach as opposed to thefundamental nature of the problem itself.

25

2.3 Flexible Supply Contracting

In this dissertation, the implosion problem is extended to consider the case where supply

commitment quantities from component suppliers are flexible over the planning horizon.

Therefore, it is related to the literature on flexible supply contracting. Tsay and Lovejoy

(1999) study quantity flexible supply contracts for a multi-echelon supply chain in a

rolling horizon setting. In their model, supply flexibility is a function of the most recent

supply request and the time difference between the current period and period in which

supply is being requested. Bassok and Anupindi (1997) and Anupindi and Bassok (1998)

analyze supply contracts which have a minimum procurement commitment. That is,

while periodic supply requests may be unconstrained or flexible, the total cumulative

supply request over a finite horizon must satisfy some minimum level. In a similar

vein, Li and Kouvelis (1999) study ‘time-flexible’ supply contracts in which a minimum

commitment quantity, or liability, is required over a period of time. Their problem is

motivated by price stochasticity rather than demand stochasticity. Plambeck and Taylor

(2004) study the implications of designing quantity flexible supply contracts with future

regotiations in mind. A more thorough review of the supply contracting literature is

provided by Tsay et al. (1998). However, none of the literature that deals with flexible

supply contracting addresses these issues in either an ATO or CTO setting, as is done

in this dissertation.

2.4 Inventory and Capacity Rationing

In the stochastic inventory control literature, several have studied the inventory rationing

problem associated with serving several demand classes from a single stockpile of inven-

tory (Veinott 1965, Topkis 1968, Nahmias and Demmy 1981, Cohen et al. 1988, Ha

1997, Moon and Kang 1998, Ha 2000, Cattani and Souza 2002, Deshpande et al. 2003).

26

Both periodic and continuous review models have been considered and excepting Veinott

(1965) and Topkis (1968), demand is assumed to be stationary and independent from

period to period. Other than Veinott (1965),6 these papers all recommend the use of a

stationary (and not necessarily optimal) hedging point, or threshold, type of rationing

policy. Under a threshold rationing policy, where orders from lower priority demand

classes are either rejected (lost sales case) or backlogged when the inventory level falls

below a certain threshold level. The optimal threshold level is typically decreasing in

demand class priority, so that orders from lower priority customers are rejected before

those of higher priority demand classes as inventory drops below each threshold level.

de Vericourt et al. (2000) explore a similar problem from a machine capacity schedul-

ing perspective. In their problem, incoming orders from competing demand classes must

be processed on a single machine. Like in the inventory control research, an optimal

hedging point policy for scheduling machine capacity is characterized. Also in the ca-

pacity rationing literature, Balakrishnan et al. (1999) study what is essentially a dynamic

threshold capacity rationing policy for a two product make-to-order manufacturing firm

and find this policy to be quite robust with respect to demand forecast error.

Unlike the previously cited literature, in which different demand classes are dis-

tinguished by priority or backorder cost, Frank et al. (2003) categorize demand for a

single item according to whether the demand is deterministic (e.g., committed orders)

or stochastic (e.g., unexpected or after-market orders). In their model, deterministic

demand must be satisfied immediately while stochastic demand need not be, even if

inventory is available. Their rationing policy simply specifies the level of deterministic

demand that is to be satisfied.

The rationing problem has not been well studied for ATO systems. Gerchak et al.

6Veinott (1965) suggests the use of non-stationary base-stock policies where all demands of higherpriority must be satisfied before lower priority demands, and earlier demands are satisfied before laterdemands.

27

(1988) were the first to consider inventory rationing in the context of an ATO system

consisting of more than one product and component commonality. They consider a single

period, two product, three component setting where one of the components is common

between the products. It is only after observing demand for the two products that they

determine the optimal portion of each demand to satisfy.

More recently, Agrawal and Cohen (2001) determine the optimal base stock ordering

levels for components in an ATO system with multiple products and stationary demand.

They assume a fair share component allocation scheme in which equal fractions of avail-

able components are allocated to different orders when there are shortages in component

supply. Additionally, components may be released for delivery for an order even if some

of the components necessary to complete the order are unavailable at the time. While

these allocations rules are not optimal, they are simple and practical. Akcay and Xu

(2004) jointly optimize inventory ordering and component allocation for an ATO system

with multiple products and stationary demand. Additionally, orders may be charac-

terized by due date windows. They show that significant benefit is gained by jointly

optimizing a base-stock component ordering policy and the allocation of components to

products. They develop an efficient algorithm for optimizing the component allocation

subproblem, which is essentially a general multidimensional knapsack problem. As a

point of interest, Mieghem and Rudi (2002) define a class of models called ’newsven-

dor networks’. These models are quite general, and can be used to model component

allocation problems in assembly systems, among other things.

Grotzinger et al. (1993) solve a similar problem to the implosion problem studied

in this dissertation. However, they consider only a single common component shared

between N products. They jointly determine the appropriate order up to level for the

common component and the allocation of the components among products. In their

problem, the objective is to minimize finished goods inventory holding costs subject to

a minimum service level. A bisection procedure is used to select candidate base-stock

28

levels, and a combination of Monte Carlo simulation and nonlinear optimization is used

to estimate the aggregate service level achieved for the current base-stock level under

a pre-determined allocation policy. In their allocation policy, when there is insufficient

inventory to satisfy all projected demand, the available inventory is allocated to reduce

the backorders for each product by an equal percentage. Unlike the implosion problem,

they do not consider configuration uncertainty and do not rank products by priority

or revenue. Additionally, the implosion problem includes backorder costs rather than

explicitly including minimum service level requirements. Finally, the implosion problem

considers multiple common components, which makes it considerably more complex.

Benjaafar and ElHafsi (2004) characterize the optimal component ordering and ra-

tioning policy for a single product, multiple component, multiple demand class ATO

system with continuous review and stochastic leadtimes. They show that a state de-

pendent base-stock policy is optimal for ordering components, and a state-dependent

multi-level threshold policy is optimal for component rationing.

While Gerchak et al. (1988), Agrawal and Cohen (2001), Mieghem and Rudi (2002)

and Akcay and Xu (2004) determine component allocations after a batch of demand

for products is realized, the threshold rationing policy characterized by Benjaafar and

ElHafsi (2004) is determined prior to demand arrivals. The latter approach is adopted for

the component rationing problem addressed in this dissertation. The difference between

the former and latter of these two approaches is similar to the differences between pull

and push based ATP scheduling, respectively. The advantages of this approach are,

again, similar to the advantages of push based ATP scheduling, as described in §2.2.

Given the predominance of threshold type rationing policies in the inventory ra-

tioning literature in a variety of problem settings, the same policy structure is assumed

in the component rationing problem studied in this dissertation. The component ra-

tioning problem studied here is different from that of Benjaafar and ElHafsi (2004) in

29

that it considers multiple products and product configuration uncertainty. Also, the

component rationing problem studied here treats component supply as fixed, its focus

being to determine the optimal rationing thresholds for all product-component pairs.

While optimizing the the rationing threshold levels independently from the component

ordering levels may be sub-optimal, the increased tractability of the problem allows for

the solving of more realistic problem sizes.

Chapter 3

The Explosion Problem

This chapter provides the technical details of how the explosion problem is formulated

and solved. Recall that in the explosion problem, the decision maker seeks to determine,

for every period in the planning horizon, the profit maximizing component ordering levels

given a multi-product demand plan. Refer to §1.3.1 for a more detailed, non-technical

introduction to the explosion problem and how it fits into the sales and operations

planning framework.

3.1 Model Environment

A manufacturer produces a set of configurable products P = p | p = 1, 2, . . . , P. These

products are configured from a set of common components C = c | c = 1, 2, . . . , C.

Let ξcp be the random quantity of component c that is required to configure an or-

der for one unit of product p. A realization of a configurable product p is given by

ξξξp(ω) = [ξ1p(ω), ξ2p(ω), . . . , ξCp(ω)]⊤ with ξcp(ω) ∈ Z+ and ω ∈ Ω, where Ω is the set

of all random events. The ξcp are assumed to be independent random variables. The

uncertainty associated with these random variables is referred to as ‘configuration uncer-

31

tainty’. Since not all components are used to configure all products, let Cp ⊆ C be the set

of components that can be used to configure product p. Then, ξcp(ω) = 0 ∀ (c, p) /∈ Cp×P

and ω ∈ Ω.

In the explosion problem, the manufacturer’s objective is to determine the quantity

of each component to request from each supplier in each period so as to meet sales targets

and maximize expected profit over the planning horizon. The manufacturer’s decision is

made in the face of order configuration uncertainty. The fundamental trade-off made in

this problem is between requesting too many components (i.e., over-stocking) versus too

few components (i.e., under-stocking) from suppliers. The planning horizon is T periods

long and periods in the horizon are indexed by t.

The explosion problem is formulated as a two-stage stochastic program with com-

plete recourse. A two-stage stochastic program with complete recourse has the following

basic charateristics. In the first stage problem, certain decisions are made prior to the

realization of any random events. After the first stage decisions have been made, a par-

ticular random outcome is realized and another set of decisions are made in light of this

new information. However, the decisions made in the second stage are restricted by the

decisions made in the first stage. The second stage decision problem is referred to as the

‘recourse problem’, and complete recourse refers to the fact that the ‘recourse problem’

is feasible for any set of feasible first stage decisions. Birge and Louveaux (1997) provide

the definitive introduction to stochastic programming.

In the first stage of the explosion problem, the manufacturer is given dt = [d1t, d2t, . . . , dPt]⊤,

where dpt ∈ Z+ is the targeted sales quantity for product p ∈ P in period t. Recall from

§1.1.1 that dt constitutes the initial demand plan determined in the demand planning

stage of the S&OP process. In the presence of order configuration uncertainty, the

manufacturer makes a decision, ut = [u1t, u2t, . . . , uCt]⊤, where uct is the quantity of

component c ∈ C that is requested to be delivered at the beginning of period t. Note

32

that supply requests for all T periods are determined simultaneously. Without loss of

generality, components are assumed to be sole-sourced (i.e., each component is supplied

by only one supplier) and suppliers are assumed to be uncapacitated.1

At the end of the first stage, orders are realized for all products and all periods. It is

assumed that every order is for a unit quantity.2 Thus, an order for product p consists of

a realized configuration ξξξp(ω). It is also assumed that the total number of orders for each

product in each period is equal to the targeted sales quantity for that product in that

period.3 Define θcpt to be the average quantity of component c that is used to configure

all orders for product p in period t. θcpt is used to ‘summarize’ the order configuration

uncertainty associated with ξcp in period t and is distributed as the sampling distribution

of the mean for ξcp, with sample size, Spt = dpt, where dpt is the targeted sales quantity

for product p in period t. The θcpt are independent random variables, which is a direct

result of the assumption of the independence between the ξcp variables. Realizations of

θcpt are given by θcpt(ω) ∈ R+, where ω ∈ Ω. Since not all components are used to

configure all products, let Cp ⊆ C be the set of components potentially used to configure

product p. Then, θcpt(ω) = 0 ∀ (c, p) ∈ Cp × P, for all t = 1, 2, . . . , T and for all ω ∈ Ω.

In the second stage the manufacturer decides how to allocate the requested compo-

nent supply, ut, among the configured orders so as to maximize profit over the planning

horizon. This allocation decision is captured by vtt′ = [vt

1t′ , vt2t′ , . . . , v

tP t′ ]

⊤, where vtpt′ is

the number of orders for product p that are received in period t′ and fulfilled in period

t, where t′ ≤ t ≤ T . All required components must be available in order for an order to

be fulfilled. Assembly capacity is assumed to be infinite (i.e., assembly lead times are

1This assumption is used because supply capacities are unknown at this stage in the S&OP process.In fact, they become known only after suppliers respond to the supply request. Hence, supply constraintsare ignored here, but are considered in the subsequent optimization problem, the implosion problem.

2The assumption that orders are configured one unit at a time is appropriate for products that arerelatively expensive and highly customized (e.g., high-end server computers).

3Justification of the assumption of fixed product demand is provided in §1.1.1.

33

negligible).

Over the planning horizon, revenue is earned for every fulfilled order. Let r =

[r1, r2 . . . , rP ]⊤, where rp ∈ R+ is the revenue earned for every order for product p that

is fulfilled. At the end of each period, unfulfilled orders are backordered and a backorder

cost is incurred for each backordered unit per period. Let btt′ = [bt

1t′ , bt2t′ , . . . , b

tP t′ ]

⊤,

where bt′

pt ∈ R+ is the number of units of product p that are ordered in period t′ and

backordered at the end of period t, where t′ ≤ t ≤ T . Let q = [q1, q2 . . . , qP ]⊤, where

qp ∈ R+ is the cost incurred for each unit of product p that is backordered in each

period. Note that backorder costs are incurred for product shortages as opposed to

component shortages.4 Let It = [I1t, I2t, . . . , ICt]⊤ ∀ t, where Ict ∈ R+ is the quantity

of component c that would remain in inventory at the end of period t. The initial

inventory levels for components are given by I0 = [I10, I20, . . . , IC0]⊤, where Ic0 ∈ Z+ is

represents the initial inventory level for component c. Per unit ordering costs are given

by o = [o1, o2 . . . , oC ]⊤, where oc is the ordering cost for component c. Holding costs are

incurred for the quantity of each component that would remain in inventory at the end

of each period. Let h = [h1, h2, . . . , hC ]⊤, where hc ∈ R+ is the holding cost per unit

of component c that remains in inventory at the end of each period. Finally, a discount

factor, γ, is applied over the planning horizon to account for the time value of money.

The second stage problem is modeled as a T period deterministic linear program,

with the first stage decisions (i.e., the component supply request, ut, for t = 1, 2, · · · , T )

as its parameters. The two-stage stochastic formulation of the explosion problem is

presented in the following section.

4Song (2002) studies the impact of order-based backorders in ATO systems. Lu and Song (2005)compare order-based and item-based (i.e. component-based) backorder costs in ATO systems.

34

3.2 Model Formulation

This section begins with a summary of the key notation introduced in §3.1.

Sets:

C = Set of components = c | c = 1, 2, . . . , C

P = Set of products, indexed by p = p | p = 1, 2, . . . , P

Ω = Set of all random events

Deterministic Parameters:

dt = Targeted sales quantities for products, in period t

o = Per unit ordering costs for components

h = Per unit holding costs for components held in inventory at the end of

each period

q = Per unit backorder costs for products, per period

r = Per unit revenues for products fulfilled

I0 = Initial inventories of components

γ = Discount factor

Random Parameters:

θcpt = Average quantity of component c used to configure product p in period t

First Stage Decision Variables:

ut = Quantities of components requested to be delivered by suppliers at the

35

beginning of period t

Second Stage Decision Variables:

vtt′ = Numbers of product orders that are received in period t′ and fulfilled

in period t

btt′ = Numbers of product orders that are received in period t′ and backordered

at the end of period t

It = Quantity of component c that remains in inventory the end of period t

The explosion problem (EXP) is given by

[EXP]

Z∗EXP = max

ut≥0 ∀tQ(u1,u2, . . . ,uT ) (3.1)

In (3.1), the recourse function Q(u1, . . . ,uT ) is a piecewise linear concave function that

represents the expected profit over the planning horizon. This function also defines the

second stage optimization problem. That is,

Q(u1, . . . ,uT ) = E[Q(u1, . . . ,uT ,ΘΘΘ1, . . . ,ΘΘΘT )] (3.2)

where ΘΘΘt is the C × P random matrix of average order configurations in period t, as

given by

ΘΘΘt =

θ11t θ12t · · · θ1Pt

θ21t · · · · · · θ2Pt

.... . .

...

θC1t θC2t · · · θCPt

(3.3)

36

Let ΘΘΘt(ω) be a realization of ΘΘΘt, where θcpt(ω) ∈ R+ and ω ∈ Ω. Then,

Q(u1, . . . ,uT ,ΘΘΘ1(ω), . . . ,ΘΘΘT (ω))

= maxbt,It,vt∀t

T∑

t=1

γt−1

[

−o⊤ut − h⊤It +T

∑

t′=t

(

r⊤vtt′ − q⊤bt

t′

)

]

(3.4)

s.t. It = It−1 + ut −t

∑

t′=1

Θt′(ω)vtt′ ∀ t (3.5)

btt′ = bt−1

t′ − vtt′ ∀ t, t′ ≤ t (3.6)

bt−1t = dt ∀ t (3.7)

It ≥ 0 ∀ t (3.8)

btt′ ,v

tt′ ≥ 0 ∀ t, t′ ≤ t (3.9)

The objective function (3.4) captures the total discounted profit for a particular set

of realizations ΘΘΘ1(ω), . . . ,ΘΘΘT (ω). The flow of component inventory from one period

to the next is constrained by (3.5). Backorder constraints are given by (3.6), with

constraints (3.7) initializing the number of orders that are demanded at the start of

period t. For all periods and all products, the number of orders demanded at the start

of the period is initialized to the targeted sales quantity, dpt. Constraints (3.8) and (3.9)

are non-negativity constraints, where 0 is an appropriately sized vector of zeroes.

3.3 Solution Method

To solve the explosion problem, the deterministic equivalent formulation of the explosion

problem is solved myopically using the sample average approximation (SAA) method.

37

3.3.1 Deterministic Equivalent

A two-stage stochastic program for a problem with a finite number of possible random

outcomes, or scenarios, may be expressed as its ‘deterministic equivalent’ (Birge and

Louveaux 1997).5 The deterministic equivalent formulation for the explosion problem

is a linear program. Before this form of the explosion problem is presented, additional

notation is required. Let K = k | k = 1, 2, · · · ,K be the set of all scenarios and

let ΘΘΘt(ωk) be the realization of average product configurations in period t in the kth

scenario. The probablity that scenario k occurs is given by πk. Therefore,∑

k∈K πk = 1.

Additionally, allow an additional subscript, k, to be appended to each of the second stage

decision variables to denote the scenario for which these decisions are made. Then, the

deterministic equivalent formulation of the explosion problem (DEEP) is given below.

[DEEP]

max.T

∑

t=1

γt−1

−o⊤ut +K

∑

k=1

πk

[

−h⊤Itk +T

∑

t′=t

(

r⊤vtt′k − q⊤bt

t′k

)

]

(3.10)

s.t. Itk = It−1,k + ut −t

∑

t′=1

Θt′(ωk)vtt′k ∀ t, k (3.11)

btt′k = bt−1

t′k − vtt′k ∀ t, k, t′ ≤ t (3.12)

bt−1tk = dt ∀ t, k (3.13)

Itk ≥ 0 ∀ t, k (3.14)

btt′k,v

tt′k ≥ 0 ∀ t, k, t′ ≤ t (3.15)

5If the number of possible outcomes is not finite (e.g., when the random variables are continuous),then the deterministic equivalent formulation is an approximation of the stochastic program, obtainedby appropriately discretizing the sample space.

38

3.3.2 Scenario Generation

In the explosion problem, every scenario k corresponds to a particular set of realizations

of product configurations, ΘΘΘ1(ωk), . . . ,ΘΘΘT (ωk). Recall that the average quantity of

component c that is used to configure all orders for product p in period t is given by θcpt.

θcpt is distributed as the sampling distribution of the mean for the random variable ξcp

with sample size Spt = dpt, the targeted sales quantity for product p in period t.

The alias method (Law and Kelton 2000) is used to generate the scenarios used

in the deterministic equivalent for the explosion problem. This method is generally

used to generate random variables from arbitrary discrete distributions with a finite

support. The discrete distributions that are sampled from for the explosion problem are

the empirical distributions for the random variables ξcp. Let Ppc(w) be the empirical

probability that m units of component c are ordered in product p. This probability is

defined for all c ∈ C, p ∈ P and w = 0, 1, 2, . . . ,Wpc, where Wpc is the maximum quantity

of component c that is ordered in product p. Thus,∑Wpc

w=1 Ppc(w) = 1 ∀ p, c.

To initialize the alias method, quantities referred to as ‘aliases’ and ‘cutoffs’ must be

computed. Let Apc(w) and Cpc(w) be the alias and cutoff quantities, respectively, for

product p, component c, where w = 0, 1, 2, . . . ,Wpc. Also, the notation X ∼ DU(a, b)

implies that the random variable X has a discrete uniform distribution with a lower and

upper support bound of a and b, respectively. The notation U(a, b) is used to refer to

the continuous uniform distribution with similar support. The following algorithm illus-

trates how the alias method is used to generate a realization, θcpt(ωk), for component c

and product p for period t in the kth scenario. For clarity of exposition, the details of

how the alias and cutoff quantities are to be computed in the initialization step of the

algorithm are postponed.

39

Algorithm 1. Generation of θcpt(ωk) for the Explosion Problem

Begin

Step 0 Initialize i← 0 and θcpt(ωk)← 0.

Step 1 If i = Spt, go to End.

Otherwise, compute Apc(w) and Cpc(w) for w = 1, 2, · · · , Wpc.

Step 2 Set i← i + 1 and generate X1 ∼ DU(0, Wpc) and X2 ∼ U(0, 1),

where X1 and X2 are independently generated.

Step 3 If X2 ≤ Cpc(X1), set θcpt(ωk)← θcpt(ωk) + X1.

Otherwise, set θcpt(ωk)← θcpt(ωk) + Apc(X1).

If i = Spt, go to Step 4. Otherwise, go to Step 2.

Step 4 Set θcpt(ωk)← θcpt(ωk)/Spt.

End

In order to generate the random parameters necessary to populate DEEP, the above

procedure must be repeated for all p ∈ P, c ∈ C, k ∈ K and t = 1, 2, · · · , T . Originally

written by Walker (1977), the following algorithm illustrates how the alias and cutoff

quantities are computed for product p and component c in Step 1 of Algorithm 1.

Algorithm 2. Setting Alias and Cutoff Values for Product p and Component c

Begin

Step 0 Set Apc(w)← w, Cpc(w)← 0 and B(w)← Ppc(w) − 1Wpc+1

for w = 0, 1, 2, . . . , Wpc. Initialize i← 0.

Step 1 If∑Wpc

w=0 |B(w)| < ǫ, then go to End.

Otherwise, set B← minwB(w), ℓ← argmin

w

B(w)

and set B← maxwB(w), ℓ← argmax

w

B(w)

and go to Step 2.

40

Step 2 Set Apc(ℓ)← ℓ, Cpc(ℓ)← 1 + B(Wpc + 1)

and set B(ℓ)← 0, B(ℓ)← B + B and go to Step 1.

End

A tolerance of ǫ = 10−9 was used to define the stopping criteria in Step 1 of Algo-

rithm 2. Also in this step, ties may be broken arbitrarily.

3.3.3 Sample Average Approximation

The Sample Average Approximation (SAA) method is a statistical approach for dealing

with stochastic programs with extremely large sets of scenarios (Kleywegt et al. 2001,

Linderoth et al. 2002, Verweij et al. 2003, Akcay and Xu 2004). It is a simulation based

approach, as opposed to a decomposition based approach such as the well-known L-

shaped method of Slyke and Wets (1969) and other methods (Mulvey and Ruszczynski

1995, Rockafellar and Wets 1991). Yet other methods attempt to combine decomposition

and simulation (Infanger 1992, Higle et al. 1994, Higle and Sen 1996).6 The SAA method

is the method of choice here because of its generality, simplicity and effectiveness for the

problem at hand.7

For a two-stage stochastic program with recourse, the SAA method samples scenarios

from the set of all possible scenarios, and seeks to optimize the average of the second-

stage objective across the sampled scenarios. The sampling done in the SAA method is

referred to as an ‘exterior’ approach because all samples are generated prior to solving

6The stochastic decomposition method of Higle and Sen (1996) generally requires a fixed recoursematrix whereas the explosion probem possesses a random recourse matrix. Modifications to the stochasticdecomposition algorithm to allow for random recourse matrices have been suggested (Higle and Sen 1999)but they are non-trivial to implement if several columns of the recourse matrix are random, essentiallyrequiring the application of a separate decomposition algorithm to a subproblem in order to generatevalid cuts in each iteration.

7The L-shaped method was implemented for the explosion problem but was found not be as effectiveas the SAA method.

41

a set of deterministic optimization problems. This combination of sampling followed

by deterministic optimization is also used by other simulation based optimization tech-

niques such as sample-path optimization (Gurkan et al. 1994, Plambeck et al. 1996) and

retrospective optimization (Healy and Schruben 1991).

The SAA problem for the explosion problem (SAA-EXP) is given by

[SAA-EXP]

νM = maxut≥0 ∀ t

1

M

M∑

m=1

Q(u1, . . . ,uT ,ΘΘΘ1(ωm), . . . ,ΘΘΘT (ωm))

(3.16)

where M is the number of scenarios sampled. Typically, M << K so that the SAA-EXP

is tractable. SAA-EXP is a linear program. Assuming, without loss of generality, that

each of the scenarios in the explosion problem occurs with equal probability, then if

M = K, (3.16) is equivalent to DEEP. Since the SAA method produces an approximate

solution to the true explosion problem, EXP, bounds with respect to Z∗EXP should be

provided. Since M < K, the bounds obtained using the SAA method are statistical

estimates. The method of estimation of these bounds for the explosion problem is based

on the work of Mak et al. (1999).

3.3.3.1 Upper Bound Estimation

In the SAA method, one first solves N , independent M -scenario instances of SAA-EXP to

obtain N independent candidate solutions to the explosion problem. Given N candidate

solutions, un1 , un

2 , . . . , unT

Nn=1, with corresponding objective values ν1

M , ν2M , . . . , νN

M , an

unbiased estimate of an upper bound for the true explosion problem is given by the

average of these objective values. That is,

ZuEXP =

1

N

N∑

n=1

νnM (3.17)

42

is an unbiased estimate of E[νM ], and

E[νM ] = E

maxut≥0 ∀ t

1

M

M∑

m=1


≥ maxut≥0 ∀ t

E

1

M

M∑

m=1


= maxut≥0 ∀ t

Q(u1,u2, . . . ,uT )

= Z∗EXP. (3.18)

The estimator ZuEXP satisfies the Central Limit Theorem (CLT) as follows

√N

[

ZuEXP − E[νM ]

]

⇒ N(0, σ2u) as N →∞ (3.19)

where σ2u = Var[Q(u∗

1, u∗2, . . . , u

∗T )]. The sample variance of Zu

EXP is given by

σ2u =

1

(N − 1)

N∑

n=1

(νnM −Zu

EXP)2 . (3.20)

3.3.3.2 Lower Bound Estimation

Each of the candidate solutions un1 , un

2 , . . . , unT Nn=1 can be evaluated for a single L-

scenario problem, where L >> M and the L scenarios are independently generated from

those used to obtain the candidate solutions.8 Let u∗1, u

∗2, . . . , u

∗T be the candidate

solution that yields the highest objective value for the L-scenario problem out of all N

candidates. That is,

u∗1, u

∗2, . . . , u

∗T = argmax

un1 ,...,un

TN

n=1

1

L

L∑

l=1

Q(un1 , . . . , un

T ,ΘΘΘ1(ωl), . . . ,ΘΘΘT (ωl))

(3.21)

8In general, any solution that is feasible to the true explosion problem qualifies as a candidatesolution.

43

Since u∗1, u

∗2, . . . , u

∗T is feasible to the true explosion problem, an unbiased estimate of

a lower bound for Z∗EXP is given by

ZℓEXP =

1

L

L∑

l=1

Q(u∗1, . . . , u

∗T ,ΘΘΘ1(ωl), . . . ,ΘΘΘT (ωl)) (3.22)

In order for this lower bound estimate to be unbiased, the sample used to evaluate

(3.22) must be independent of the sample used to identify u∗1, u

∗2, . . . , u

∗T . Therefore,

in this lower bound estimation procedure, two samples of size L must be generated. The

estimator ZℓEXP satisfies the CLT as follows

√L

[

ZℓEXP −Q(u∗

1, u∗2, . . . , u

∗T )

]

⇒ N(0, σ2ℓ ) as L→∞ (3.23)

where σ2ℓ = Var[Q(u∗

1, u∗2, . . . , u

∗T )]. The sample variance of Zℓ

EXP is given by

σ2ℓ =

1

(L− 1)

L∑

l=1

[

Q(u∗1, . . . , u

∗T ,ΘΘΘ1(ωl), . . . ,ΘΘΘT (ωl))−Zℓ

EXP

]2. (3.24)

3.3.3.3 Optimality Gap Estimation

The optimality gap for the true, T -period explosion problem is estimated by

ZgEXP = Zu

EXP −ZℓEXP (3.25)

As noted by Linderoth et al. (2002), ZgEXP is a biased estimator of the true optimality

gap because ZuEXP and Zℓ

EXP are unbiased estimators of E[νM ] and Q(u∗1, u

∗2, . . . , u

∗T ),

respectively, and

E[νM ]−Q(u∗1, u

∗2, . . . , u

∗T ) ≥ Z∗

EXP −Q(u∗1, u

∗2, . . . , u

∗T ) (3.26)

44

Therefore, ZgEXP overestimates the true optimality gap by E[νM ] − Z∗

EXP. The size of

this gap can be reduced by increasing the size of M in the SAA-EXP problem. Due to

sampling error, a negative optimality gap may be observed. Using results obtained in

§3.3.3.1 and §3.3.3.2, an approximate (1−α)-level confidence interval for the optimality

gap at u∗1, u

∗2, . . . , u

∗T is given by

[

0, [ZgEXP]+ +

tN−1, α2σu√

N+

tL−1, α2σℓ√

L

]

(3.27)

where in general, [a]+ = max0, a and tn−1, α2

is the t-value for a two-tailed Student’s

t-distribution with n− 1 degrees of freedom and tail probability of α.

3.3.3.4 SAA Algorithm for T -Period Explosion Problem

The SAA algorithm for the T -period explosion problem, as described in the previous

sections, is summarized below.

Algorithm 3. SAA Algorithm for T -period Explosion Problem

Begin

Step 0 Select values for N, M and L. Initialize n← 0

Step 1 Set n← n + 1. For m = 1, 2, . . . , M , independently generate

a random sample θcpt(ωm) ∀ c ∈ C, p ∈ P and t = 1, 2, . . . , T .

Step 2 Construct the SAA-EXP problem using the random sample last

generated in step 1 and solve for νnM and un

1 , un2 , . . . , un

T .

If n = N , set ZuEXP = 1

N

∑N

n=1 νnM , and go to Step 3.

Otherwise, go to Step 1.

Step 3 For l = 1, 2, . . . , L, independently generate a random

sample θcpt(ωl)∀ c ∈ C, p ∈ P and t = 1, 2, . . . , T .

45

Step 4 Construct the SAA-EXP problem using the random sample generated

in Step 3 and for n = 1, 2, . . . , N , evaluate the SAA-EXP problem using

the solution un1 , un

2 , . . . , unT to obtain νn

L.

Step 5 Identify n = argmaxn

νnL and set u∗

1, u∗2, . . . , u

∗T = un

1 , un2 , . . . , un

T .

Step 6 For l = 1, 2, . . . , L, independently generate

a random sample θcpt(ωl) ∀ c ∈ C, p ∈ P and t = 1, 2, . . . , T .

Step 7 Construct the SAA-EXP problem using this random sample and

evaluate it at u∗1, u

∗2, . . . , u

∗T to obtain Zℓ

EXP.

End

In Algorithm 3, a total of N , M -scenario SAA problems are solved and N + 1,

L-scenario SAA problems are evaluated for a given solution.

3.3.4 Myopic Approximation to the Explosion Problem

In order to obtain reasonably sized confidence intervals (3.27) for the optimality gap,

M may need to be so large that the linear program representing the SAA problem

(3.16) is inefficient or difficult to solve. Therefore, a myopic approach is proposed. In

this approach, a myopic solution to the true T -period explosion problem is obtained

by solving a sequence of single period versions of the explosion problem. This myopic

approach is similar to the one used by Swaminathan and Tayur (1998b). For a given

value of t, the myopic version of the explosion problem is referred to as EXP-[t] and is

given by

[ EXP-[t] ]

maxut≥0Qt(ut) (3.28)

where Qt(ut) is similar to the recourse function in (3.2) but defined only for a single

46

value of t. That is,

Qt(ut) = E[Qt(ut,ΘΘΘt)] (3.29)

where Qt(ut,ΘΘΘt(ω)) defines a second-stage problem analogous to (3.4)-(3.9) but for a

fixed period, t. For all t > 1, the inventory and backorders that remain after solving EXP-

[t−1] are used to initialize EXP-[t]. The collection of problems EXP-[t] for t = 1, 2, . . . , T

is referred to as the myopic approximation to the explosion problem.

Since the myopic problem still consists of a large number of scenarios, the SAA

method is used to derive a solution to the myopic problem. The myopic SAA problem

for period t is given by

[ SAA-EXP-[t] ]

νM = maxut≥0

1

M

M∑

m=1

Qt(ut,ΘΘΘt(ωm))

. (3.30)

Let N be the number of candidate solutions generated when applying the SAA method to

the myopic approximation to the explosion problem. Additionally, let M be the number

of scenarios used to obtain the candidate solutions, and let L be the number of scenarios

used to evaluate the candidate solutions. Suppose that the ‘best’ myopic solution (col-

lected over all time periods) obtained using the SAA method is given by u∗1, u

∗2, . . . , u

∗T .

This solution is feasible to the true T -period explosion problem. Therefore, an unbiased

estimate of a lower bound for Z∗EXP is given by

ZℓEXP =

1

L

L∑

l=1

Q(u∗1, . . . , u

∗T ,ΘΘΘ1(ωl), . . . ,ΘΘΘT (ωl)). (3.31)

If common random numbers are used for both the myopic and the T -period SAA prob-

lems, and if N = N ,M = M and L = L, then ZℓEXP ≤ Zℓ

EXP. Otherwise, the relationship

47

between ZℓEXP and Zℓ

EXP is not determined. The sample variance of ZℓEXP is given by

σ2ℓ =

1

(L− 1)

L∑

l=1

[

Q(u∗1, . . . , u

∗T ,ΘΘΘ1(ωl), . . . ,ΘΘΘT (ωl))− Zℓ

EXP

]2. (3.32)

The SAA algorithm as applied to the myopic approximation problem is similar to

Algorithm 3 but with a few minor adjustments. For the sake of clarity, the SAA algo-

rithm for the myopic problem is provided below.

Algorithm 4. SAA Algorithm for Myopic Approximation to the Explosion Problem

Begin

Step 0 Select values for N , M and L. Initialize n← 0.


a random sample θcpt(ωm) ∀ c ∈ C, p ∈ P and t = 1, 2, . . . , T .

Step 2 For t = 1, 2, . . . , T , construct the SAA-EXP-[t] problem using the

random sample last generated in Step 1 and solve for unt and νn

tildeM (t).

If n = N , initialize t← 0 and go to Step 3.


Step 3 Set t← t + 1. For l = 1, 2, . . . , L, independently generate a

random sample θcpt(ωl) ∀ c ∈ C and p ∈ P .

Step 4 Construct the SAA-EXP-[t] problem using the random sample

generated in Step 3 and for n = 1, 2, . . . N , evaluate it

at unt to obtain νn

L(t). If t = T go to Step 5. Otherwise,

go to Step 3.


∑T

t=1 νn

L(t) and set u∗

t = unt ∀ t = 1, 2, . . . , T .

Initialize t← 0.

48


random sample θcpt(ωl) ∀ c ∈ C and p ∈ P .

Step 7 Construct the SAA-EXP problem using this random sample and

evaluate it at u∗1, u

∗2, . . . , u

∗T to obtain Zℓ

EXP.

End

In Step 4 of Algorithm 4, the candidate solutions are evaluated using the formulation

for the myopic approximation problem rather than the T -period problem for the sake

of computational efficiency (computation times are reported in Table 3.3). Lustig et al.

(1991) suggest a method of reformulating the deterministic equivalent problem (e.g.,

SAA-EXP-[t]) so that it is more conducive to being solved using interior point methods.

This technique was implemented and tested but found not to be more effective than the

simplex method for solving the explosion problem.

3.4 Computational Studies

3.4.1 Problem Sets

The problem sets used in this study are based on actual planning data obtained from

IBM Systems and Technology Group in March, May and July of 2005. Table 3.1 displays

some basic characteristics of the data sets analyzed. In particular, it displays the BOM

ratio, inverse BOM ratio, product commonality (PCOMM) and average usage quantity

coefficient of variation. The BOM ratio refers to the number of components potentially

used to configure a component. The inverse BOM ratio refers to the number of prod-

ucts that a component is potentially configured in. Product commonality refers to the

number of other products with which a product has components in common. The BOM

ratio, inverse BOM ratio, PCOMM and average usage quantity coefficient of variation

49

(CoV) are reported in columns 4-6 of Table 3.1. In each of these columns, three values

are displayed per row. The first value provides the minimum statistic for the current

characteristic, and the second and third value provide the average and maximum statis-

tic for the current characteristic. For example, in row 1 and column 4 of Table 3.1,

the entry 15/80.25/128 indicates that for the Mar-1 problem set, the minimum BOM

ratio observed is 15 components per product, the average BOM ratio observed is 80.25

components per product and the maximum BOM ratio observed is 128 components per

product. Finally, the average usage quantity coefficient of variation (CoV) is reported in

the last column of 3.1. For a given product-component pair, the usage quantity coeffi-

cient of variation is calculated by dividing the standard deviation of the usage quantity

for that component for that product by the average quantity of that component ordered

for that product. The quantity displayed in the final column of Table 3.1 is the average of

the quantities, as taken over all product-component pairs in the problem set of interest.

From Table 3.1, it can be observed that the data sets analyzed represent problems

with varied characteristics. For example, problem sizes range from 4 to 39 products and

from 55 to 569 components. Average BOM ratios range from 4.73 to 147.17 components

per product and average product commonality range from 1 to 24.51. By analyzing

varied problem sets, it is possible to make an observation regarding the consistency

or generality of the results. The results of the computational study for the explosion

problem are reported in §3.4.2.

Configuration data was provided by IBM in the form of a twelve month historical

log of orders shipped. Each entry in this log provides the order number, the name of the

product ordered (all orders in unit quantities), the name of a component configured in the

product, and the quantity of the component used to configured the product. Therefore,

the number of line entries in the log of shipped orders corresponding to a give order

number is equal to the number of different components used to configure the product

ordered. Empirical distributions for all product-component pairs were derived from this

50

historical log. These empirical distributions are used to generate the various problem

scenarios via the alias method, which was described in §3.3.2.

Unit ordering costs for all components were obtained from IBM. The total cost for

a product was estimated by multiplying the cost of each component by the average

historical usage quantity of the component for that product. The revenue for a product

was derived by applying a profit margin factor to the total cost of the product. The profit

margin factor was randomly sampled from a set of actual profit margins for a family of

servers manufactured by the IBM Systems and Technology Group. The unit inventory

holding cost for each component was determined by multiplying the unit component

ordering cost by a factor of 0.01, or 1 percent. The backorder cost for each product

per unit and per unit time, was determined by multiplying the revenue of the product

by a factor of 0.05, or 5 percent. These factors were deemed to be appropriate by an

IBM subject-matter expert. The number of periods, T is equal to 20 weeks.9 The

weekly forecast numbers (i.e., the demand plan) for all products were provided by IBM.

A discount factor of 0.997 was used.

3.4.2 Results

In this section two key findings are reported. First, it is shown that the quality of the

myopic approximation to the explosion problem is very good. In fact, applying the SAA

algorithm to the myopic approximation problem provides a better lower bound to the ex-

plosion problem as compared to applying it to the T -period problem. This is because the

improved tractability of the myopic approximation problem allows for a larger number

of scenarios to be considered when generating candidate solutions, resulting in candidate

solutions of a higher quality. Second, it is shown that there is significant value in using

9In the IBM Systems and Technology Group, a 32 week rolling planning horizon is used. The numberof periods was reduced to 20 by aggregating some of the later periods. Specifically, weeks 1 through 12are not aggregated, weeks 13 through 24 are aggregated into 6 periods (i.e., 2 weeks per period) andweeks 25 - 32 are aggregated into 2 periods (i.e., 4 weeks per period).

51

the stochastic explosion solution that takes configuration uncertainty into account, as

opposed to the expected value solution, which does not.

3.4.2.1 Assessing the Quality of the Myopic Approximation

To assess the quality of the myopic approximation solution, a comparison of the size of

the 90% confidence interval for the optimality gap for the explosion problem using the

lower bound generated by Algorithm 3 and the lower bound generated by Algorithm

4. That is, a comparison of the confidence interval given by (3.27) and the following

confidence interval

[

0, [ZuEXP − Zℓ

EXP]+ +tN−1, α

2σu√

N+

tL−1, α2σℓ√

L

]

(3.33)

is made, where α = 0.1. This comparison reveals the quality of the lower bound generated

by the myopic approximation, and is done for problem sets generated using data taken

from March 2005. For the remaining problem sets, it is assumed that the quality of

the approximation will be similar. For the comparison, common random numbers are

used across the different solution algorithms. Additionally, N = N = 20, M = 100

and L = L = 2000. Two values of M are considered, M = 100 and M = 500. The

purpose of varying the value of M is to show how the estimate of a lower bound to

the explosion problem can be improved by increasing the number of scenarios used to

generate candidate solutions.

Table 3.2 shows that when M = 500, the SAA algorithm for the myopic explosion

problem produces confidence intervals for the optimality gap which are less than half

that of those produced by the SAA algorithm for the T -period explosion problem. The

upper limits of the confidence intervals for the algorithm with M = 500 range from 0.67%

to 1.89%, as compared to 1.42% to 4.93% when the SAA algorithm is applied to the T -

period problem with M = 100, and 1.42% to 5.13% when the SAA algorithm is applied

52

to the myopic problem with M = 100. These results confirm the earlier statement that

ZℓEXP ≤ Zℓ

EXP when M = M and common random numbers are used. At the same time,

it is observed that when M is increased to 500, the quality of the lower bound on Z∗EXP

is significantly better when the SAA algorithm is applied to the myopic approximation

problem rather than the T -period problem (with M = 100). The faster solution times

of the SAA algorithm when applied to the myopic problem makes it more efficient to set

M >> M . Solution times for the March 2005 problems sets are provided in Table 3.3.

All computations were run on a Sun v40z server equipped with four, 2.6 GHz, 64-bit

Opteron processors and 32 GB of random access memory. ILOG’s CPLEX mathematical

optimization software was used to solve the linear programs in the SAA method for the

explosion problem. In Table 3.3, computational times are reported in units of seconds.

Each table entry consists of two values, the value to the left of the backslash is the

time to obtain the N candidate solutions and the value to the right of the backslash is

the time to identify the best candidate solution. It is observed that the solution times

when applying the SAA algorithm to the myopic approximation problem are well under

a fifth of those when applying the SAA algorithm to the T -period problem. Using the

myopic approximation with M = 500, the smallest problem set, March-1, takes just

under 2 hours to solve and the two larger problems, March-2 and March-3, take around

3.5 hours each to solve. In contrast, solving the T -period problem takes about 6 hours

for problem set March-1, 36 hours for problem set March-2 and 23 hours for problem

set March-3. As a reminder, within the S&OP process, the explosion problem would

typically need to be re-solved every few weeks.

3.4.2.2 Value of the Stochastic Explosion Solution

Based on the findings reported in the previous section, the stochastic explosion solution

for the different problems sets are generated by applying the SAA algorithm for the

myopic approximation problem (i.e., by using Algorithm 4), with N = 20, M = 500 and

53

L = 2000. The value of the stochastic solution is measured by the relative improvement

in expected profit that it yields over the expected value solution, for an independently

generated set of 2000 scenarios. The expected value solution is obtained by solving a

single scenario version of DEEP, where in this scenario, the usage quantity for a compo-

nent in a product is set equal to the average usage quantity for that component in that

product. Comparisons between the stochastic solution and the expected value solution

are also made with respect to expected revenue, component ordering costs, inventory

holding costs and fill rate. Only relative improvements of the stochastic solution over

the expected solution for expected profit, revenue and costs are reported here for reasons

of confidentiality.

Table 3.4 shows the relative improvement in expected profit achieved by the stochastic

solution over the expected value solution. Note that profit values reported in this table

do not include backorder costs or penalty costs, since these costs are not actual costs

in the supply chain. Across the ten problem sets studied, an average improvement of

41.12% in expected profit is achieved by the stochastic solution over the expected value

solution. Noticeably smaller improvements are observed for problem sets May-2 and Jul-

4. Between these two problems sets, an average improvement of the stochastic solution is

just 4% over the expected value solution. This is in contrast to an average improvement

of approximately 50% for the remaining problem sets. Referring to Table 3.1, it appears

that a distinguishing characteristic of problem sets May-2 and Jul-4 is that they have

significantly fewer components and lower BOM Ratios when compared to those of the

other problem sets. Therefore, there are fewer random variables in these two problem

sets, which is likely to contribute to the observation that the stochastic solution provides

less of an improvement over the expected value solution. Still, a 4% improvement in

expected profit for a business that is valued in the hundreds of millions to billions of

dollars is not insignificant.

Table 3.4 also breaks down the relative improvement of the stochastic solution over

54

the expected value solution into relative improvements in revenue, component ordering

and inventory holding costs. The average relative improvement in revenue is roughly

24%. This is the result of higher fill-rates when using the stochastic solution as compared

to the expected value solution. A comparison of the fill-rate performance is reported in

Table 3.5. In this table, the average fill-rate refers to the average fraction of orders that

are fulfilled in the same period in which they are demanded. The cumulative fill-rate

refers to the fraction of all orders that are fulfilled by the end of the planning horizon.

It is observed from Table 3.5 that the stochastic solution (SS) yields an improvement in

expected average fill-rate of 42%, on average, and an improvement in cumulative fill-rate

of 20%, on average, over the expected value solution (EVS).

Comparing the results for the component ordering and inventory costs in Table 3.4,

it is observed that the stochastic solution incurs higher component ordering costs than

the expected value solution. However, while ordering costs are roughly 8% higher on

average, the savings in inventory holding costs is roughly 51% on average. Additionally,

as noted earlier, fill rates are significantly better under the stochastic solution. Therefore,

the stochastic solution orders a greater volume of components, but also orders them in

such a way as to enable the supply chain to fulfill more orders while at the same time

reducing inventory levels.

3.5 Summary

The explosion problem addresses the problem of determining the quantity of components

to order over the planning horizon, in the face of order configuration uncertainty. The

purpose of studying this problem is to show the value of dealing with the uncertainty

associated with the order configurations rather than assuming average order configura-

tions. Towards this end, the explosion problem was formulated as a two-stage stochastic

program and was solved by applying the sample average approximation method to the

55

myopic approximation of the explosion problem. Ten problem sets were analyzed using

data obtained from the IBM Systems and Technology Group. The results of the compu-

tational studies show significant advantage to using the stochastic solution as opposed to

the expected value solution. This advantage is observed with respect to expected profit

and improved fill-rates.

56

Table 3.1. Summary of Characteristics for Data Sets Used in Analysis

Name of Number of Number of Inverse Avg. UsageProblem Set Products Components BOM Ratio BOM Ratio PCOMM Qty. CoV

Mar-1 4 181 15/80.25/128 1/1.77/4 3/3.00/3 3.55Mar-2 6 421 16/147.17/201 1/2.10/6 5/5.00/5 14.49Mar-3 15 319 1/56.27/160 1/2.66/10 1/8.80/12 4.04May-1 10 265 1/59.60/160 1/2.25/7 1/5.60/8 4.27May-2 51 55 1/4.73/23 1/4.38/27 1/20.12/39 7.61Jul-1 19 256 1/36.95/138 1/2.74/9 1/9.16/17 3.74Jul-2 8 434 24/116.38/179 1/2.15/8 7/7.00/7 13.03Jul-3 2 129 90/94.50/99 1/1.47/2 1/1.00/1 2.54Jul-4 39 74 1/8.18/36 1/4.31/23 12/24.51/34 5.72Jul-5 21 569 1/41.00/174 1/1.51/9 1/7.43/17 6.76

57

Table 3.2. Comparison of 90% Confidence Interval for Optimality Gap Using SAA Algorithmfor T -period Explosion Problem vs. Myopic Approximation Problem

T -period SAA Myopic SAA Myopic SAA

Problem Set M = 100 M = 100 M = 500

Mar-1 [ 0, 2.44% ] [ 0, 2.64% ] [ 0, 1.09% ]Mar-2 [ 0, 4.93% ] [ 0, 5.13% ] [ 0, 1.91% ]Mar-3 [ 0, 1.42% ] [ 0, 1.42% ] [ 0, 0.67% ]

Table 3.3. Comparison of Computation Time (in Seconds) Using SAA Algorithm for T -periodExplosion Problem vs. Myopic Approximation Problem


Problem Set M = 100 M = 100 M = 500

Mar-1 13108/7474 267/4689 1737/4653Mar-2 112428/18389 679/7834 6619/7313Mar-3 58139/25772 628/7653 4761/7729

Table 3.4. Comparison of Relative Improvements in Profit, Revenue and Costs Associated withStochastic Solution vs. Expected Value Solution for the Explosion Problem

Problem Set Profit Revenue Ordering Cost Inventory Cost

Mar-1 47.86% 33.85% -10.47% 57.06%Mar-2 105.58% 40.51% -6.13% 80.54%Mar-3 26.89% 22.33% -9.83% 61.22%May-1 25.03% 21.31% -10.90% 53.67%May-2 6.18% 5.75% -4.91% 8.31%Jul-1 49.48% 30.77% -9.96% 58.29%Jul-2 98.71% 50.15% -7.74% 75.39%Jul-3 26.08% 12.21% -3.80% 32.26%Jul-4 1.80% 1.66% -1.57% 41.96%Jul-5 23.62% 21.79% -17.31% 14.24%

58

Table 3.5. Comparison of Expected Fill-Rates Associated with Expected Value Solution (EVS)vs. Stochastic Solution (SS) for the Explosion Problem

Average Fill-Rate Cumulative Fill-RateProblem Set EVS SS Improv. EVS SS Improv.

Mar-1 0.54 0.99 82.65% 0.79 0.99 26.32%Mar-2 0.52 0.94 81.22% 0.69 0.99 43.00%Mar-3 0.72 0.95 30.40% 0.84 0.99 17.90%May-1 0.68 0.86 27.65% 0.80 0.90 12.40%May-2 0.78 0.87 12.22% 0.88 0.92 4.92%Jul-1 0.86 0.93 7.81% 0.92 0.99 7.75%Jul-2 0.50 0.94 86.16% 0.65 0.99 51.87%Jul-3 0.61 0.91 49.50% 0.89 1.00 12.09%Jul-4 0.82 0.90 9.54% 0.91 0.95 3.84%Jul-5 0.71 0.94 32.52% 0.79 0.98 23.99%

Chapter 4

The Implosion Problem

This chapter describes how the implosion problem is formulated and solved. The implo-

sion problem is the converse problem to the explosion problem. While in the explosion

problem the manufacturer determines component requirements given sales targets, in

the implosion problem the manufacturer determines adjusted sales targets (i.e., commit-

ment to sales (CTS) quantities) given component supply. As in the explosion problem,

the manufacturer makes these decisions in the face of order configuration uncertainty

with the objective of maximizing the expected profit over the planning horizon. The

fundamental trade-off made in the implosion problem is between committing too many

versus too few products to sales. Like the explosion problem, the implosion problem is

formulated as a two-stage stochastic program with complete recourse. Refer to §1.3.2

for a non-technical introduction to the implosion problem and how it fits into the sales

and operations planning framework.

60


In the implosion problem, the product structure is modeled similarly to that in the

explosion problem. That is, a manufacturer produces a set of configurable products

P = p | p = 1, 2, . . . , P. These products are configured from a set of components

C = c | c = 1, 2, . . . , C. Let ξcp be the random quantity of component c that is required

to configure an order for one unit of product p. A realization of a configurable product p is

given by ξξξp(ω) = [ξ1p(ω), ξ2p(ω), . . . , ξCp(ω)]⊤ with ξcp(ω) ∈ Z+ and ω ∈ Ω, and where Ω

is the set of all random events. The ξcp are assumed to be independent random variables.

Since not all components are used to configure all products, let Cp ⊆ C be the set of

components that can be used to configure product p. Then, ξcp(ω) = 0 ∀ (c, p) /∈ Cp ×P

and ω ∈ Ω.

The planning horizon is T periods long and periods in the horizon are indexed by t.

At the beginning of the first stage, the manufacturer is provided with an initial set of

sales targets for all products and periods, dt = [d1t, d2t, . . . , dPt]⊤, where dpt ∈ Z+ is the

targeted sales quantity for product p ∈ P in period t. He is also provided with the com-

ponent supply commitments for all components and periods, st = [s1t, s2t, . . . , sCt]⊤ ∀ t,

where sct ∈ Z+ is the quantity of component c that the supplier has committed to supply

at the beginning of period t.1 It is assumed that each component is single-sourced (i.e.,

each component is supplied by exactly one supplier). At this point in time, the configu-

rations of the orders that will be demanded over the planning horizon are uncertain.

In the first stage, the manufacturer determines the CTS quantities for every product

in every period. Let xt = [x1t, x2t, . . . , xPt]⊤ ∀ t, where xpt is the quantity of prod-

uct p committed to sales in period t. In each period, a penalty cost is incurred when

the cumulative CTS quantity for a product deviates from the cumulative targeted sales

1In the sales and operations planning process, the component supply commitments may be obtainedas responses to the supply requests that are determined by solving the explosion problem.

61

quantity for that product by more than some pre-determined limit. The purpose of

applying this penalty is to ‘encourage’ the CTS quantities to be close to the targeted

sales quantities, particularly when alternative solutions of similar quality exist. By al-

lowing the cumulative CTS quantity to deviate, within limits, from the cumulative sales

targets without penalty, optimal trade-offs between high and low revenue products, and

between products with high and low order configuration uncertainty can be freely made.

Additionally, allowing for such limited, penalty-free deviations is an acknowledgement

of the uncertainty associated with the initial sales targets, which are partly forecasts of

product demand. Details of how the penalty costs are applied are provided in §4.1.1.

At the end of the first stage, it is assumed that the orders for all products are realized

for all periods. It is also assumed that the total number of orders for a product p in

period t is equal to the corresponding CTS quantity, xpt.2 Let θcp be the average quantity

of component c that is used to configure one unit of product p, taken over all T periods.

θcp is a random variable having the distribution of the sampling distribution of the mean

for ξcp. θcp is used to ‘summarize’ the order configuration uncertainty associated with

the quantity of component c that is required to fulfill a single order for configurable

product p (i.e., ξcp) over the planning horizon. As a direct result of the assumption of

the independence between the ξcp variables, the θcp are independent random variables.

At the end of the first stage, θcp(ω) ∈ R+, where ω ∈ Ω, is realized for all p ∈ P and

c ∈ C. Since not all components are used to configure all products, let Cp ⊆ C be the set

of components potentially used to configure product p. Then, θcp(ω) = 0 ∀ (c, p) ∈ Cp×P

and for all ω ∈ Ω.

To avoid the technical difficulties3 associated with decision-dependent random vari-

ables, the sample size, Sp, associated with the sampling distribution of the mean for

2This is akin to assuming that the sales team is 100% effective in selling the exact quantity of productthat was committed to them by the manufacturer.

3Existing techniques for solving stochastic programs with decision dependent random variables isextremely limited (Jonsbraten et al. 1998).

62

ξcp is approximated by the total initial targeted sales quantity instead of the total CTS

quantity. That is, it is assumed that Sp =∑T

t=1 dpt. For any given product, the strength

of this approximation depends on the magnitude of the difference between the total CTS

quantity and the total targeted sales quantity for that product. In a system with tighter

supply, this approximation is likely to be weaker since it is unlikely that the optimal CTS

quantities will be able to achieve values near to the targets sales quantities. However, in

a system with mild or no supply constraints, it is reasonable to expect this approxima-

tion to be strong, particularly in the presence of the aforementioned penalty costs. For

any given product, the effect of any difference between the total CTS quantity and the

total targeted sales quantity on the quality of this approximation diminishes as both of

these quantities become large.

In the second stage, the committed supply, st, is allocated among the configured

orders so as to maximize profit over the planning horizon. This allocation decision is

captured by vt = [v1t, v2t, . . . , vPt]⊤, where vpt is the number of orders for product p that

are fulfilled in period t. All required components must be available in order for an order

to be fulfilled. Assembly capacity is assumed to be infinite (i.e., assembly lead times

are negligible). Unfulfilled orders are backordered. Let bt = [b1t, b2t, . . . , bPt]⊤, where

bpt ∈ R+ is the number of units of product p that are backordered at the end of period t.

Incurring backorder costs for a product may be indicative of having committed a larger

quantity to sales than can be supported by the supply commitment, for a given realization

of orders. Note that in contrast to the explosion problem, the variables representing the

allocation and backorder decisions are not indexed by the period of origination for the

order. This is because for a given scenario, θcp(ω), the realization of the average quantity

of component c that is used to configure one unit of product p, is the same for all periods

in the planning horizon.

The initial inventory levels for components are given by I0 = [I10, I20, . . . , IC0]⊤,

where Ic0 is the initial inventory level for component c. Per unit ordering costs are given

63

by o = [o1, o2 . . . , oC ]⊤, where oc is the ordering cost for component c. Holding costs are

incurred per unit of each component remaining in inventory at the end of each period.

Let h = [h1, h2, . . . , hC ]⊤, where hc ∈ R+ is the holding cost per unit of component c that

remains would remain in inventory at the end of each period. Incurring holding costs

for certain components may be indicative of having committed too few of the products

that use those components to sales, resulting in missed sales opportunities, for a given

realization of orders. Finally, a discount factor, γ, is applied over the planning horizon

to account for the time value of money.

4.1.1 Penalty Cost Structure for Deviations of the Commitment-to-

Sales from Initial Sales Targets

The details of the penalty costs associated with deviations between the CTS quantities

and the initial targeted sales quantities are provided in this section. First, define the

following general notation for parameters and variables representing quantities that are

accumulated over time:

a = ai | i = 0, 1, . . . , T, a0 = 0, and at =

t∑

i=1

ai

In every period t, a linear penalty cost ρ+pt ∈ R+ (ρ−pt ∈ R+) is assessed on a portion of

the positive (negative) deviation that may exist between the cumulative CTS quantity,

xpt, and the cumulative targeted sales quantity, dpt, for product p in period t. Let α−pt

and α+pt be the fractions of the cumulative initial sales target quantities for product p

that are not charged with penalty costs in period t when the deviation is positive and

negative, respectively. Meanwhile, β−pt and β+

pt are the fractions of the cumulative initial

sales target quantities for product p and period t that are charged with penalty costs

when the deviation is positive and negative, respectively. To simplify the notation, define

64

a penalty function for period t, Ht(·), as follows:

Ht(βββ−t ,βββ+

t ) =

P∑

p=1

(

ρ−ptdptβ−pt + ρ+

ptdptβ+pt

)

(4.1)

To limit the deviation quantities that are not penalized, a ‘penalty-free region’ is defined,

the upper and lower boundaries of which are delineated using the constants µ−pt ∈ R+

and µ+pt ∈ R+, respectively. That is, it is required that α−

pt ≤ µ−pt and α+

pt ≤ µ+pt.

Details of how this is modeled is provided in the section §4.2. Let µµµ−t = [µ−

1t, . . . , µ−Pt]

⊤,

µµµ+t = [µ+

1t, . . . , µ+Pt]

⊤, ααα−t = [α−

1t, . . . , α−Pt]

⊤, ααα+t = [α+

1t, . . . , α+Pt]

⊤, βββ−t = [β−

1t, . . . , β−Pt]

⊤

and βββ+t = [β+

1t, . . . , β+Pt]

⊤.

The limits of the penalty-free region may be chosen to reflect the range within which

actual product demand is expected to lie. In other words, they may reflect the uncer-

tainty associated with the initial sales targets. Alternatively, if using a simple newsvendor

approach to determining the sales target for a product, the limits of the penalty-free re-

gion may represent the region within which the expected profit varies insignificantly from

the optimal expected profit.


4.2.1 Implosion Problem with Fixed Component Supply

This section begins with a summary of the notation introduced in §4.1.

Sets:


P = Set of products, indexed by p = p | p = 1, 2, . . . , P


65

Deterministic Parameters:

dt = Targeted sales quantity for products, in period t

ρρρ+t = Penalty costs associated with positive deviations of CTS quantity from initial

sales target for products, in period t

ρρρ−t = Penalty costs associated with negative deviations of CTS quantity from initial

sales target for products, in period t

µµµ+t = Upper limits on the penalty-free region for products, in period t as a percentage

of the cumulative initial sales target

µµµ−t = Lower limits on the penalty-free region for products, in period t as a percentage

of the cumulative initial sales target

st = Quantities of components committed to be supplied at the beginning of

period t

o = Per unit ordering costs for components

h = Per unit holding costs for components held in inventory at the end of

each period

q = Per unit backorder costs for products, per period

r = Per unit revenues for products fulfilled

I0 = Initial inventories of component c

γ = Discount factor

Random Parameters:

θcp = Average quantity of component c used to configure product p

over the planning horizon for a sample of size Sp =∑T

t=1 dpt

66


xt = Quantities of products committed to sales at the beginning of period t

ααα+t = Penalty-free positive percentage deviations of CTS quantities from

cumulative sales targets for products, in period t

ααα−t = Penalty-free negative percentage deviations of CTS quantities from


βββ+t = Penalized positive percentage deviations of CTS quantities from


βββ−t = Penalized negative percentage deviations of CTS quantities from



vt = Numbers of product orders that are fulfilled in period t

bt = Numbers of product orders that are backordered at the end of period t

It = Quantities of components that remains in inventory the end of period t

The implosion problem (IMP) is given by

[IMP]

Z∗IMP = max

xt∀t

T∑

t=1

γt−1Ht(βββ−t ,βββ+

t ) +Q(x1, . . . ,xT ) (4.2)

s.t. xt ≥ (1−ααα−t − βββ−

t )⊤ dt ∀ t (4.3)

xt ≤ (1 + ααα+t + βββ+

t )⊤ dt ∀ t (4.4)

xt − xt−1 ≥ 0 ∀ t (4.5)

ααα−t ≤ µµµ−

t ∀ t (4.6)

67

ααα+t ≤ µµµ+

t ∀ t (4.7)

ααα−t ,ααα+

t ,βββ−t ,βββ+

t ≥ 0 ∀ t (4.8)

In the objective function, Ht is as defined in equation (4.1), and Q(x1, . . . ,xT ) is the

recourse function representing the expected profit (excluding the sunk ordering costs)

over the planning horizon. Constraints (4.3) and (4.4) determine the portion of the CTS

quantity for each product that will be penalized in each period. Constraints (4.6) and

(4.7) limit the portion of the CTS quantity for each product that is not penalized in each

period. In these constraints, 1 is a column vector of P ones. Constraint (4.5) ensure that

the cumulative CTS quantities are nondecreasing. The objective function (4.2) contains

expressions for the penalty costs and ordering costs.4

As mentioned earlier, the function Q(x1, . . . ,xT ) in (4.2) represents the expected

profit (excluding ordering costs) over the planning horizon. It is a piecewise concave

function and defines the second stage problem as follows:

Q(x1, . . . ,xT ) = E [Q(x1, . . . ,xT ,ΘΘΘ)] (4.9)

where ΘΘΘ is the C × P random matrix of average order configurations over the planning

horizon, as given by

ΘΘΘ =

θ11 θ12 · · · θ1P

θ21 · · · · · · θ2P

.... . .

...

θC1 θC2 · · · θCP

(4.10)

and

Q(x1, . . . ,xT ,ΘΘΘ(ω)) = maxIt,bt,vt∀t

T∑

t=1

γt−1(

r⊤vt − q⊤bt − h⊤It − o⊤st

)

(4.11)

4The ordering costs here are sunk costs, but are included for the purposes of comparison with theextended implosion model presented in §4.2.2.

68

s.t. It = It−1 + st −Θ(ω)vt ∀ t (4.12)

bt = xt −t

∑

t′=1

vt′ ∀ t (4.13)

It ≥ 0 ∀ t (4.14)

bt ≥ 0 ∀ t (4.15)

The second stage objective function (4.11) captures the total profit earned for a

particular set of realized orders, ΘΘΘ(ω), excluding the ordering costs already accounted

for in the first stage. The flow of the committed supply of components from one period

in the ATP schedule to the next is constrained given by (4.12). Backorder constraints

are given by (4.13).

4.2.2 Extension of Implosion Problem to Incorporate Flexible Supply

In the IMP model, the quantity of components that may be allocated to configured

orders in each period of the second stage is fixed to be equal to the supply commitment

quantity in that period. In many practical situations, these supply commitments are not

‘hard’ constraints. Rather, these commitments may be flexible in the sense that their

suppliers may be amenable (due to contracted agreements or otherwise) to adjusting the

committed quantity of components over the planning horizon. This is particularly true

for supply commitments made for periods lying farther out in the planning horizon.5

To model supplier flexibility, define yt = [y1t, y2t, . . . , yCt]⊤, where yct is the quantity

of component c that is available to promise at the beginning of period t. The decision yt

is made in the first stage of the implosion problem, in the presence of order configuration

uncertainty. In the second stage, orders are fulfilled based on the available to promise

quantities rather than the committed supply quantities. In every period, the cumulative

5Refer to Chapter 1 for a detailed description of supplier flexibility.

69

available to promise quantities, yt, are bounded both from above and from below. That

is,

yt ≤ (1 + δδδ+t )⊤st ∀ t (4.16)

yt ≥ (1− δδδ−t )⊤st ∀ t (4.17)

In addition, it is required that

yt − yt−1 ≥ 0 ∀ t (4.18)

so that the cumulative commitment-to-sales quantities are non-decreasing with time. In

(4.16), δδδ+t = [δ+

1t, δ+2t, . . . , δ

+Ct]

⊤, where δ+ct ∈ R+ is the maximum percentage above the

cumulative supply commitment that the available to promise quantity for component c

can achieve in period t. Similarly, in (4.17), δδδ−t = [δ−1t, δ−2t, . . . , δ

−Ct]

⊤, where δ−ct ∈ (0, 1)

is the maximum percentage below the cumulative supply capacity that the available

to promise quantity for component c can achieve in period t. The extended implosion

problem (IMP′) is given by

[IMP′]

Z∗IMP′ = max

xt,yt∀t

T∑

t=1


t ) +Q′(x1, . . . ,xT ,y1, . . . ,yT ) (4.19)

s.t. (4.3) − (4.8) and (4.16) − (4.18)

with Q′(x1, . . . ,xT ,y1, . . . ,yT ) = E [Q′(x, . . . ,xT ,y1, . . . ,yT ,ΘΘΘ)] and

Q′(x1, . . . ,xT ,y1, . . . ,yT ,ΘΘΘ(ω))

= maxIt,bt,vt∀t

T∑

t=1

γt−1(

r⊤vt − q⊤bt − h⊤It − o⊤ (yt − yt−1))

(4.20)

s.t. It = It−1 + yt − yt−1 −Θ(ω)vt ∀ t (4.21)

70

bt = xt −t

∑

t′=1

vt′ ∀ t (4.22)

It ≥ 0 ∀ t (4.23)

bt ≥ 0 ∀ t (4.24)

4.3 Solution Method

To solve the implosion problem, the deterministic equivalent formulation of the implo-

sion problem is solved myopically using the framework provided by the sample average

approximation (SAA) method.


In a manner similar to that in §3.3.1, the deterministic equivalent problem for the ex-

tended implosion problem is presented in this section. First, let K = k | k = 1, 2, · · · ,K

be the set of all scenarios and let ΘΘΘ(ωk) be the realization of average product configura-

tions in the kth scenario. Additionally, allow an additional subscript, k, to be appended

to each of the second stage decision variables to denote the scenario for which these

decisions are made. Then, the deterministic equivalent linear programming formulation

of the extended implosion problem (DEIP) is given below.

[DEIP]

max.

T∑

t=1

γt−1

[

Ht(βββ−t ,βββ+

t ) + o⊤ (yt − yt−1) +

K∑

k=1

πk

(

r⊤vtk − q⊤btk − h⊤Itk

)

]

(4.25)


t )⊤ dt ∀ t


t )⊤ dt ∀ t

71

yt ≤ (1 + δδδ+t )⊤st ∀ t

yt ≥ (1− δδδ−t )⊤st ∀ t

xt − xt−1 ≥ 0 ∀ t

yt − yt−1 ≥ 0 ∀ t


t ∀ t


t ∀ t

ααα−t ,ααα+


t ≥ 0 ∀ t

Itk = It−1,k + yt − yt−1 −Θ(ωk)vtk ∀ t, k

btk = xt −t

∑

t′=1

vt′k ∀ t, k

Itk ≥ 0 ∀ t, k

btk ≥ 0 ∀ t, k


Scenario generation for the implosion problem is done in a similar manner to that for

the explosion problem. Unless otherwise defined, the notation in this section is similar

to that in §3.3.2. In the implosion problem, every scenario k corresponds to a particular

set of realizations of product configurations, as given by the matrix ΘΘΘ(ωk). Recall that

the average quantity of component c that is used to configure all orders for product p

over the planning horizon is given by θcp and that θcp is distributed as the sampling

distribution of the mean for ξcp with sample size Sp =∑T

t=1 dpt. Realizations of θcp are

obtained by repeatedly sampling from the empirical distributions for ξcp. As was done

for the explosion problem, the alias method is used to perform such sampling. Algo-

rithm 5 illustrates how the alias method is used to generate a realization, θcp(ωk), for

component c and product p in the kth scenario, using the empirical distribution of ξcp.

In Algorithm 5, Wpc is the maximum quantity of component c that is ordered in product

72

p. Apc(w) and Cpc(w) are alias and cutoff values, respectively, which can be computed

using Algorithm 2 in §3.3.2.

Algorithm 5. Generation of θcp(ωk) for the Implosion Problem

Begin

Step 0 Initialize i← 0 and θcp(ωk)← 0.

Step 1 If i = Sp, go to End.

Otherwise, compute Apc(w) and Cpc(w) for w = 1, 2, · · · , Wpc.

Step 2 Set i← i + 1 and generate X1 ∼ DU(0, mpc) and X2 ∼ U(0, 1),


Step 3 If X2 ≤ Cpc(X1), set θcp(ωk)← θcp(ωk) + X1.

Otherwise, set θcp(ωk)← θcp(ωk) + Apc(X1).

If i = Sp, go to Step 4. Otherwise, go to Step 2.

Step 4 Set θcp(ωk)← θcp(ωk)/Sp.

End

In order to generate the random parameters necessary to populate DEIP, the above

procedure must be repeated for all p ∈ P, c ∈ C and k ∈ K. Algorithm 5 differs from

Algorithm 1 with respect to how the sample size and random variables are defined.


As with the explosion problem, the sample average approximation (SAA) method is

used to address the implosion problem. Some of the details of the SAA method that are

provided in §3.3.3 are not repeated here. The SAA problem for the extended implosion

problem (SAA-IMP′) is given by the following deterministic linear program:

73

[SAA-IMP′]

νM = maxxt,yt∀t

T∑

t=1


t ) +

1

M

M∑

m=1

Q′(x1, . . . ,xT ,y1, . . . ,yT ,ΘΘΘ(ωm))

(4.26)

s.t. (4.3) − (4.8) and (4.16) − (4.18)

where M << K is the number of scenarios sampled. Note that SAA-IMP′ is a variant of

DEIP, with M scenarios instead of K, and with each scenario occurring with equal prob-

ability. Since the SAA method produces an approximate solution to the true extended

implosion problem, IMP′, bounds with respect to Z∗IMP′ should be provided. Since the

derivation of the bounds and optimality gap for the implosion problem is similar to that

for the explosion problem (refer to §3.3.3), the presentation of these expressions is more

concisely provided in the following sections.

4.3.3.1 Bound and Optimality Gap Estimation

An unbiased estimate of an upper bound on Z∗IMP′ is given by

ZuIMP′ =

1

N

N∑

n=1

νnM (4.27)

where ν1M , ν2

M , . . . , νNM are the objective values obtained by solving N , independent

M -scenario instances of SAA-IMP′. Let σ2u be the sample variance of Zu

IMP′ . Let

xn1 , . . . , xn

T , yn1 , . . . , yn

T Nn=1 be the candidate solutions generated from these N , M -

scenario problem instances. Additionally, let Hn1 , . . . , Hn

T be the values of the penalties

corresponding to the nth candidate solution. An independently generated set of L >> M

scenarios is used to select the ‘best’ candidate solution as follows:

x∗1, . . . , x

∗T , y∗

1, . . . , y∗T

74

= argmaxxn

1 ,...,xnT

,yn1 ,...,yn

TN

n=1

[

T∑

t=1

γt−1Hnt +

1

L

L∑

l=1

Q′(xn1 , . . . , xn

T , yn1 , . . . , yn

T ,ΘΘΘ(ωl))

]

(4.28)

Let H∗t be the value of the penalty function for period t, corresponding to the best

candidate solution. Then, an unbiased estimate of a lower bound on Z∗IMP′ is given by

ZℓIMP′ =

T∑

t=1

γt−1H∗t +

1

L

L∑

l=1

Q′(x∗1, . . . , x

∗T , y∗

1, . . . , y∗T ,ΘΘΘ(ωl))

(4.29)

with sample variance equal to σ2ℓ .

The optimality gap for the T -period extended implosion problem is estimated by

Zg

IMP′ = ZuIMP′ −Zℓ

IMP′ (4.30)

with a (1− α) confidence interval at x∗1, . . . , x

∗T , y∗

1, . . . , y∗T given by

[

0, [Zg

IMP′ ]+ +

tN−1, α2σu√

N+

tL−1, α2σℓ√

L

]

, (4.31)

where in general, [a]+ = max0, a and tn−1, α2

is the t-value for a two-tailed Student’s

t-distribution with n − 1 degrees of freedom and tail probability of α. By way of a

similar explanation to that provided in §3.3.3.3, Zg

IMP′ is positively biased by an amount

E[νM ]−Z∗IMP′ .

4.3.3.2 SAA Algorithm for T -Period Implosion Problem

The SAA algorithm for the T -period implosion problem is summarized below.

75

Algorithm 6. SAA Algorithm for T -period Implosion Problem

Begin

Step 0 Select values for N, M and L. Initialize n← 0


a random sample θcp(ωm) ∀ c ∈ C and p ∈ P .

Step 2 Construct the SAA-IMP′ problem using the random sample last

generated in step 1 and solve for νnM and xn

1 , . . . , xnT , yn

1 , . . . , ynT .

If n = N , set ZuIMP′ = 1

N

∑N



Step 3 For l = 1, 2, . . . , L, independently generate a random

sample θcp(ωl) ∀ c ∈ C and p ∈ P .

Step 4 Construct the SAA-IMP′ problem using the random sample generated

in Step 3 and for n = 1, 2, . . . , N , evaluate the SAA-IMP′ problem using

the solution xn1 , . . . , xn

T , yn1 , . . . , yn

T to obtain νnL.


νnL and

set x∗1, . . . , x

∗T , y∗

1 , . . . , y∗T = xn

1 , . . . , xnT , yn

1 , . . . , ynT .

Step 6 For l = 1, 2, . . . , L, independently generate

a random sample θcp(ωl) ∀ c ∈ C and p ∈ P .

Step 7 Construct the SAA-IMP′ problem using this random sample and

evaluate it at x∗1, . . . , x

∗T , y∗

1, . . . , y∗T to obtain Zℓ

IMP′ .

End

In Algorithm 6, a total of N , M -scenario SAA problems are solved and N + 1,

L-scenario SAA problems are evaluated at a given solution.

76

4.3.4 Myopic Approximation to the Implosion Problem

Since the T -period implosion problem DEIP can still be very large and therefore difficult

to solve, a myopic approximation to the implosion problem is presented in this section.

Define the following single period implosion problem

[ IMP′-[t] ]

Z∗IMP′-[t] = max

xt,yt


t ) +Q′t(xt,yt) (4.32)


t )⊤ dt (4.33)


t )⊤ dt (4.34)

yt ≤ (1 + δδδ+t )⊤st (4.35)

yt ≥ (1− δδδ−t )⊤st (4.36)

xt − xt−1 ≥ 0 (4.37)

yt − yt−1 ≥ 0 (4.38)


t (4.39)


t (4.40)

ααα−t ,ααα+


t ≥ 0 (4.41)

where

Q′t(xt,yt) = E[Q′

t(xt,yt,ΘΘΘ)] (4.42)

and Q′t(xt,yt,ΘΘΘ(ω)) defines a second-stage problem analogous to that defined by (4.20)

- (4.24) but for a fixed period, t. For all t > 1, the inventory and backorders that remain

after solving IMP′-[t − 1] are used to initialize IMP′-[t]. The collection of problems

IMP′-[t] for t = 1, 2, . . . , T is referred to as the myopic approximation to the implosion

problem.

Since the myopic problem still consists of a large number of scenarios, the SAA

77

method is used to derive a solution to the myopic problem. The myopic SAA problem

for period t is given by

[ SAA-IMP′-[t] ]

νM = maxxt,yt≥0


t ) +1

M

M∑

m=1

Q′t(xt,yt,ΘΘΘ(ωm)) (4.43)

s.t. (4.33) − (4.41)

Let N be the number of candidate solutions generated when applying the SAA method

to the myopic approximation to the explosion problem. Additionally, let M be the

number of scenarios used to obtain the candidate solutions, and let L be the number

of scenarios used to evaluate the candidate solutions. Suppose that the ‘best’ myopic

solution (collected over all time periods) obtained using the SAA method is given by

x∗1, . . . , x

∗T , y∗

1, . . . , y∗T . Additionally, let H∗

1 , . . . , H∗T be the corresponding collec-

tion of penalty function values. This solution is feasible to the true T -period explosion

problem. Therefore, an unbiased estimate of a lower bound for Z∗IMP′ is given by

ZℓIMP′ =

T∑

t=1

γt−1H∗t +

1

L

L∑

l=1

Q′(x∗1, . . . , x

∗T , y∗

1, . . . , y∗T ,ΘΘΘ(ωl)). (4.44)

If common random numbers are used for both the myopic and the T -period SAA prob-

lems, and if N = N ,M = M and L = L, then ZℓIMP′ ≤ Zℓ

IMP′ . Otherwise, the rela-

tionship between ZℓIMP′ and Zℓ

IMP′ is not determined. Let σ2ℓ be the sample variance of

ZℓIMP′ . The SAA algorithm for the myopic approximation to the implosion problem is

provided below.

Algorithm 7. SAA Algorithm for Myopic Approximation to the Implosion Problem

Begin

78

Step 0 Select values for N, M and L. Initialize n← 0.


a random sample θcp(ωm) ∀ c ∈ C and p ∈ P .

Step 2 For t = 1, 2, . . . , T , construct the SAA-IMP′-[t] problem using the

random sample last generated in Step 1 and solve for xnt , yn

t and νn

M(t).

If n = N , initialize t← 0 and go to Step 3.



random sample θcp(ωl) ∀ c ∈ C and p ∈ P .

Step 4 Construct the SAA-IMP′-[t] problem using the random sample

generated in Step 3 and for n = 1, 2, . . . N , evaluate it at xnt , yn

t

to obtain νn

L(t). If t = T , go to Step 5. Otherwise, go to Step 3.


∑T

t=1 νn

L(t) and set x∗

t = xnt and y∗

t = ynt

∀ t = 1, 2, . . . , T . Initialize t← 0.


random sample θcp(ωl) ∀ c ∈ C and p ∈ P .

Step 7 Construct the SAA-IMP′ problem using this random sample and

evaluate it at x∗1, . . . , x

∗T , y∗

1, . . . , y∗T to obtain Zℓ

IMP′ .

End

In Step 4 of Algorithm 7, the candidate solutions are evaluated using the formulation

for the myopic approximation problem rather than the T -period problem for the sake of

computational efficiency.

79


4.4.1 Problem Sets

The data sets used to generated the problem sets studied for the implosion problem are

identical to the ones used to study the explosion problem. That is, ten problems sets

were generated from data taken from the IBM Systems and Technology Group in March,

May and July of 2005. The data characteristics are displayed in Table 3.1, and defined

in §3.4.1. Problem parameters are the same as those used in the explosion problem, with

the exception that in the implosion problem, additional parameters are needed to specify

the quantities of each component that are committed to be delivered in each period, the

supplier flexibility factors, and the penalties for deviations of the CTS quantities from

the demand plan.

The implosion problem takes as input the quantities that suppliers have committed

to deliver in each period. Unless otherwise stated, in the test problems studied, these

quantities were set to be equal to the optimal solution to the corresponding explosion

problem (i.e., the explosion problem with the same product structure and demand plan).

To maintain independence between the explosion and implosion problems, the product

configuration scenarios were independently generated. To analyze the impact of supplier

flexibility, three levels of supply flexibility are considered: 2%, 5% and 10%. That is, the

cases δ+ct = δ−ct = 0.02 ∀ c, t, and δ+

ct = δ−ct = 0.05 ∀ c, t, and δ+ct = δ−ct = 0.1 ∀ c, t. The

choice of these supplier flexibility factors is based on conversations with an IBM subject-

matter expert. The same set of scenarios is used under the various supplier flexibility

factors.

Again, based on conversations with an IBM subject-matter expert, penalties on cu-

mulative deviations of the CTS quantities from the cumulative initial sales targets are

applied as follows: Deviations of the cumulative CTS quantity for a each product are not

80

penalized if they are at least 50% of, and no more than 5% greater than, the cumulative

initial sales target. Additionally, an infinite penalty cost is applied to CTS quantities

exceeding the upper bound of this penalty-free region. From a modeling perspective,

this implies setting β+pt = 0 ∀ p, t. The penalty cost for negative deviations is 3.75%

of product revenue, per product and period. Note that the penalty cost is chosen to

be less than the backorder cost (i.e., 5% of product revenue) to avoid solutions where

products are committed-to-sales even though their orders can/will never be fulfilled.

All computations were run on a Sun v40z server equipped with four, 2.6 GHz, 64-bit

Opteron processors and 32 GB of random access memory. ILOG’s CPLEX mathematical

optimization software was used to solve the linear programs in the SAA method for the

implosion problem.

4.4.2 Results

In this section, three key findings are reported. First, it is shown that the quality of the

myopic approximation to the explosion problem is very good when obtain using the SAA

method with M = 500. Second, it is shown that it is generally beneficial to account for

configuration uncertainty when solving the implosion problem. Third, supplier flexibility

is beneficial with diminishing returns.

4.4.2.1 Assessing the Quality of the Myopic Approximation

To assess the quality of the myopic approximation solution, a comparison of the size of

the 90% confidence interval for the optimality gap for the implosion problem using the

lower bound generated by Algorithm 6 and the lower bound generated by Algorithm 7 is

made. That is, a comparison of the confidence interval given by (4.31) and the following

81

confidence interval

[

0, [ZuIMP′ − Zℓ

IMP′ ]+ +tN−1, α

2σu√

N+

tL−1, α2σℓ√

L

]

(4.45)

is made, where α = 0.1. This comparison reveals the quality of the lower bound generated

by the myopic approximation, and is done for problem sets generated using data taken

from March 2005. For the remaining problem sets, it is assumed that the quality of

the approximation will be similar. For the comparison, common random numbers are

used across the different solution algorithms. Additionally, N = N = 50, M = 100

and L = L = 2000. Two values of M are considered, M = 100 and M = 500. The

purpose of varying the value of M is to show how the estimate of a lower bound to

the explosion problem can be improved by increasing the number of scenarios used to

generate candidate solutions.

Table 4.1 through Table 4.4 provide the 90% confidence intervals obtained by applying

the SAA method to the T -period problem and the myopic approximation problem with

M = 100 and M = 500, for no supplier flexibility, 2%, 5% and 10% supplier flexibility,

respectively. The results in these tables show that the SAA method applied to the myopic

approximation problem with M = 500 produces improved optimality gaps over both the

SAA method applied to the T -period problem and the myopic approximation problem

with tilde = 100. These improvements increase as the degree of supplier flexibility

incerases. There is a roughly 25% reduction, on average, in the optimality gap when

the SAA method is applied to the myopic approximation problem with M = 500 as

opposed to the T -period problem when there is no supplier flexibility. This reduction in

optimality gap increases to approximately 70%, on average.

In Table 4.5 through Table 4.8, computational times are reported in units of seconds.

For each of these tables, each entry consists of two values, the value to the left of the

backslash is the time to obtain the N candidate solutions and the value to the right

82

of the backslash is the time to identify the best candidate solution. These tables show

that the time to solve the myopic approximation problem with M = 500 is roughly 25%

less than that required to solve the T -period problem, even though, as shown earlier, it

produces better optimality gaps. This is because the improved tractability of the myopic

approximation problem allows for a larger number of scenarios to be considered when

generating candidate solutions, resulting in candidate solutions of a higher quality. Using

the myopic approximation with M = 500, the solution times for the test problems with

no supplier flexibility, 2%, 5% and 10% supplier flexbility are 5 hours, 13 hours, 15 hours

and 18 hours, on average, respectively. Considering that the implosion problem may

need to be solved every two weeks, these solve times, while not short, are still practical.

4.4.2.2 Value of the Stochastic Implosion Solution

Based on the findings reported in the previous section, the stochastic implosion solution

for the different problems sets are generated by applying the SAA algorithm for the

myopic approximation problem (i.e., by using Algorithm 7), with N = 50, M = 500 and

L = 2000. The value of the stochastic solution is measured by the relative improvement

in expected profit that it yields over the expected value solution. The expected value

solution is obtained by solving a single scenario version of the T -period DEIP problem,

where in this scenario, the usage quantity for a component in a product is set equal to

the average usage quantity for that component in that product. This single scenario

implosion model is referred to as the expected value implosion model. Comparisons

between the stochastic solution and the expected value solution are also made with

respect to expected revenue, component ordering costs, inventory holding costs and fill

rate. Only relative improvements in expected profit, revenue and costs are reported here

for reasons of confidentiality.

Table 4.9 shows the relative improvement in expected profit achieved by the stochas-

83

tic solution over the expected value solution when the component supply is fixed. It

also breaks down the relative improvement of the stochastic solution over the expected

value solution into relative improvements in revenue, component ordering and inventory

holding costs. Note that profit values reported in this table do not include backorder

costs or penalty costs, since these costs are not actual costs in the supply chain. Across

the ten test problems, the average improvement in expected profit is roughly 0.2% when

using the stochastic solution as opposed to the expected value solution. A significant

contributor to the improvement in expected profit is the result of savings in inventory

costs, which average around 4%. In the fixed supply case, the ordering cost is identical

for the stochastic solution and the expected value solution. The improvement in revenue

averages 0.13%.

Table 4.10 displays the average fill-rates and cumulative fill-rates for the expected

value solution (EVS) and stochastic solution (SS). In this table, the average fill-rate refers

to the average fraction of orders that are fulfilled in the same period in which they are

demanded. The cumulative fill-rate refers to the fraction of all orders that are fulfilled by

the end of the planning horizon. It is observed from this table that the average fill-rate

worsens by 3.2%, on average, and the cumulative fill-rate worsens by 0.2%, on average,

when using the SS versus the EVS. Despite the shortcomings of the stochastic solution

with respect to average fill-rate observed here, it should be noted that these do not come

at the expense of the expected profit or expected revenue.

The differences in performance between the expected value solution and the stochas-

tic solution are small when supply is fixed. This is perhaps expected since the stochastic

explosion solution tends to order more components than would be deemed adequate

by the expected value implosion model. Given the rather tight upper bound on the

commitment-to-sales quantity (i.e., 5% above the demand plan), the commitment-to-

sales quantity provided by the expected value implosion solution would be close to the

demand plan. The same would be true for the stochastic implosion solution since the

84

component supply was generated from the stochastic explosion model. Therefore, addi-

tional comparisons of the stochastic and expected value implosion models were performed

using a component supply generated by the expected value explosion model. In the pres-

ence of order configuration uncertainty, this component supply will tend to underestimate

the actual component need. The results of this new comparison are reported in Table

4.11 and Table 4.12.

In Table 4.11 and Table 4.12, significant differences in performance are observed

between the stochastic implosion solution and the expected value implosion solution.

Namely, Table 4.11 reports a 2% decrease in expected profit and a 0.8% decrease in

expected revenue, on average across problem sets, for the stochastic implosion solution

over the expected value implosion solution. Holding costs are only slightly lower (0.2%

on average) for the stochastic implosion solution. To explain these results, refer to

Table 4.12, which reports the associated fill rates. From this table, it is clear that the

stochastic solution trades off profit and revenue for serviceability. The average expected

improvement in average fill rate over that of the expected value solution is 57.46%.

From these results, it is clear that, when the component supply is insufficient to match

the demand plan, the primary value provided by the stochastic implosion model is in

improving serviceability. This is achieved by setting more realistic sales targets. This is

important because, as explained in Chapter 1, the failure to meet revenue targets could

have a negative impact on share price.

The value of the stochastic solution improves when suppliers are flexible, both when

the the initial supply commitment is given by the expected value explosion solution

and when it is given by the stochastic explosion solution. For the 2%, 5% and 10%

levels of supplier flexibility, the expected profit, revenue, ordering cost and holding costs

are displayed in Table 4.13, Table 4.17 and Table 4.21, respectively, when using the

stochastic explosion solution as the initial supply commitment. Comparing the value of

the stochastic solution over the expected value solution with respect to expected profit,

85

these tables show that the average improvement is 1.1%, 3.9% and 6.9% when supplier

flexibility is 2%, 5% and 10%, respectively. The corresponding improvements in revenue

are 1.1%, 2.8% and 5.4%. Ordering costs are higher when using the stochastic solution

as compared to the expected value solution, by 1.7%, 3.7% and 5.6% for the respective

flexibility levels. Inventory holding costs are increased by 1.1% and 0.2% for the 2%

and 5% flexibility cases respectively, and reduced by 8.9% in the case of 10% supplier

flexibility. A similar examination of Table 4.15, Table 4.19 and Table 4.23 shows that

the value of the stochastic solution when supply is flexible is even more significant when

the initial supply commitment is given by the expected value explosion solution.

Additionally, significant improvements in fill-rates obtained by the stochastic implo-

sion solution as opposed to the expected value implosion solutions are also observed

as supply flexibility increases. The average and cumulative fill-rates for each level of

supplier flexibility are displayed in Table 4.14, Table 4.18 and Table 4.22, respectively.

These tables show a 7.2%, 23.2% and 33.7% improvement in average fill-rate when using

the stochastic solution as opposed to the expected value solution for supplier flexibility

levels of 2%, 5% and 10%, respectively. Correspondingly, the improvements in cumula-

tive fill-rate are 2.0%, 4.2% and 6.9%. A similar examination of Table 4.16, Table 4.20

and Table 4.24 shows that the value of the stochastic solution with respect to fill rate

when supply is flexible is even more significant when the initial supply commitment is

given by the expected value explosion solution. The reason that the value of the stochas-

tic solution generally increases with supplier flexibility is two-fold: First, the stochastic

implosion model is able to better align supply and demand since both the component

supply and the commitment-to-sales may be adjusted. Second, by allowing component

supply to be modified, the expected value implosion model, which seeks to optimize the

solution for only a single, expected value scenario, has the natural tendency to shift the

component supply in such a manner that it becomes less robust to order configuration

uncertainty.

86

4.4.2.3 Value of Supplier Flexibility

To examine the value of supplier flexibility, comparisons of the expected profit, revenue,

ordering costs, inventory holding costs, average fill-rate and cumulative fill-rate for the

various supplier flexibility levels (i.e., 2%, 5% and 10%) are made with respect to the case

where there is no supplier flexibility. Figure 4.1 through Figure 4.6 illustrate these com-

parisons. In regard to expected profit, Figure 4.1 shows diminishing returns on expected

profit as supplier flexibility is increased, although the returns are always positive, rang-

ing between 0 and 1 percent. The rate at which this return diminishes varies depending

on the test problem. A similar observation is made for expected revenue in Figure 4.2,

although the range of returns is smaller. Notice how, in this figure, the improvement in

expected revenue for problem set ‘Jul-5’ worsens as supplier flexibility increases from 5%

to 10%. While this may seem to be counter-intuitive, it can be explained by means of the

SAA method. Recall that in this method, the candidate solutions are determined using

independent samples of 500 scenarios. These scenarios are independent from the sample

of 2000 scenarios that are eventually used to produce the results presented in Figure 4.2.

While this observation is not expected, it is certainly possible in theory. In regards to

the remaining measures (i.e., ordering costs, inventory holding costs and fill-rate), there

appears to be no general trend as supplier flexiblity is increased, as evidenced by Figure

4.3 through Figure 4.6. While the observed changes in revenue with supplier flexibility

may seem small, it should be noted that IBM’s server business is valued in the tens of

billions of dollars in revenue.

The results in Figure 4.1 through Figure 4.6 assume that the initial component

supply levels (i.e., the supply levels before supplier flexiblity is applied) are given by

the solution to the stochastic explosion model. For purposes of comparison, the same

types of comparisons performed in Figure 4.1 through Figure 4.6 are repeated for the case

where the initial component supply is given by the expected value explosion model. These

87

results are presented in Figure 4.7 through Figure 4.12. The most prominent observation

to be made from these additional charts is that the value of supplier flexibility is much

more significant, particularly in regards to expected profit and revenue. Moreover, in

contrast to what was previously observed, the improvements in expected profit and

revenue show no sign of slowing down as supplier flexibility approaches 10%. This

reveals the extent to which the supply levels provided by the expected value explosion

solution are inappropriate in the presence of order configuration uncertainty.

In general, the results presented in this section may be used to guide decisions re-

garding supply contracting, by suggesting the potential value of negotiating for greater

supplier flexibility. As is apparent from this computational study, the value of supplier

flexibility varies depending on the problem characteristics and the appropriateness of the

initial component supply levels.

4.5 Summary

The implosion problem addresses the problem of determining the quantity of products to

commit-to-sales over the planning horizon, in the face of order configuration uncertainty.

Additionally, in the presence of supplier flexibility the solution to the implosion problem

also provides updated component supply requests. One of the purposes of studying this

problem was to show the value of dealing with the stochasticity of the order configurations

rather than assuming average order configurations. Towards this end, the implosion

problem was formulated as a two-stage stochastic program and was solved by applying

the sample average approximation method to the myopic approximation of the explosion

problem. Ten problem sets were analyzed using data obtained from the IBM Systems and

Technology Group. The results of the computational studies show that it is advantageous

from both a profit as well as revenue perspective to use the stochastic solution as opposed

to the expected value solution. Another purpose for studying the implosion problem was

88

to understand the value of supplier flexibility. Towards this end, results show that

supplier flexibility improves profitability and revenue. Additionally, it was shown that

the value of the stochastic implosion solution increases as supplier flexibility increases. In

general, the benefits provided by the stochastic solution were dependent on the problem

characterisitics and the quality of the initial component supply levels.

Table 4.1. Comparison of 90% Confidence Interval for Optimality Gap Using SAA Algorithmfor T -period Implosion Problem vs. Myopic Approximation Problem, with Fixed Supply


Problem Set M = 100 M = 100 M = 500

Mar-1 [ 0, 0.47% ] [ 0, 0.49% ] [ 0, 0.37% ]Mar-2 [ 0, 0.55% ] [ 0, 1.51% ] [ 0, 0.42% ]Mar-3 [ 0, 0.24% ] [ 0, 0.22% ] [ 0, 0.16% ]

Table 4.2. Comparison of 90% Confidence Interval for Optimality Gap Using SAA Algorithm forT -period Implosion Problem vs. Myopic Approximation Problem, with 2% Supplier Flexibility


Problem Set M = 100 M = 100 M = 500

Mar-1 [ 0, 0.64% ] [ 0, 1.07% ] [ 0, 0.35% ]Mar-2 [ 0, 2.17% ] [ 0, 4.08% ] [ 0, 1.69% ]Mar-3 [ 0, 0.56% ] [ 0, 0.47% ] [ 0, 0.06% ]

89



Problem Set M = 100 M = 100 M = 500

Mar-1 [ 0, 1.72% ] [ 0, 1.88% ] [ 0, 0.15% ]Mar-2 [ 0, 3.51% ] [ 0, 6.62% ] [ 0, 0.38% ]Mar-3 [ 0, 0.99% ] [ 0, 0.97% ] [ 0, 0.07% ]



Problem Set M = 100 M = 100 M = 500

Mar-1 [ 0, 2.84% ] [ 0, 2.95% ] [ 0, 0.15% ]Mar-2 [ 0, 6.28% ] [ 0, 9.35% ] [ 0, 1.76% ]Mar-3 [ 0, 1.28% ] [ 0, 1.26% ] [ 0, 0.07% ]

Table 4.5. Comparison of Computation Time (in Seconds) Using SAA Algorithm for T -periodImplosion Problem vs. Myopic Approximation Problem, with Fixed Supply


Problem Set M = 100 M = 100 M = 500

Mar-1 39482/9676 795/10437 4456/8563Mar-2 42754/15702 978/19778 4932/20676Mar-3 107588/34985 919/16134 1096/15520

90

Table 4.6. Comparison of Computation Time (in Seconds) Using SAA Algorithm for T -periodImplosion Problem vs. Myopic Approximation Problem, with 2% Supplier Flexibility


Problem Set M = 100 M = 100 M = 500

Mar-1 80821/8121 750/8838 16659/8563Mar-2 128479/16975 1601/18135 41622/20745Mar-3 283228/33014 1524/16144 40330/16871



Problem Set M = 100 M = 100 M = 500

Mar-1 98272/8655 792/9033 20102/8650Mar-2 174888/19323 1996/19868 59107/19086Mar-3 608431/38795 1647/15848 35739/16619



Problem Set M = 100 M = 100 M = 500

Mar-1 116717/9107 919/9205 23233/8536Mar-2 267880/20151 2521/18250 80924/19056Mar-3 1046880/41111 1748/15338 42155/15455

91

Table 4.9. Comparison of Relative Improvements in Profit, Revenue and Costs Associated withStochastic Solution vs. Expected Value Solution for the Implosion Problem with Fixed Supply


Mar-1 * * * *Mar-2 * * * *Mar-3 * * * 0.03%May-1 * * * -0.01%May-2 0.04% 0.02% * 2.65%Jul-1 0.15% * * 0.07%Jul-2 0.02% * * 1.35%Jul-3 * * * 0.14%Jul-4 0.04% 0.02% * 17.82%Jul-5 1.80% 1.23% * 16.68%

* These entries denote improvements whose absolute values are less than 0.01%

Table 4.10. Comparison of Expected Fill-Rates Associated with Expected Value Solution vs.Stochastic Solution for the Implosion Problem with Fixed Supply


Mar-1 0.96 0.96 * 0.98 0.98 *Mar-2 0.92 0.92 * 0.98 0.98 *Mar-3 0.93 0.93 * 0.98 0.98 *May-1 0.91 0.89 -2.81% 0.93 0.93 *May-2 0.78 0.69 -11.95% 0.91 0.91 *Jul-1 0.91 0.99 8.98% 0.99 0.99 *Jul-2 0.94 0.84 -10.35% 0.98 0.98 *Jul-3 0.97 0.96 -0.27% 0.99 0.99 *Jul-4 0.82 0.72 -12.79% 0.94 0.94 *Jul-5 0.88 0.85 -2.63% 0.99 0.97 -1.73%


92

Table 4.11. Comparison of Relative Improvements in Profit, Revenue and Costs Associatedwith Stochastic Solution vs. Expected Value Solution for the Implosion Problem with FixedSupply when Supply is Given by the Solution to the Expected Value Explosion Model


Mar-1 -2.66% -1.28% * -0.67%Mar-2 -5.83% -0.96% * -0.46%Mar-3 -1.32% -0.88% * -0.65%May-1 -1.99% -1.30% * -0.84%May-2 -0.12% -0.09% * 0.88%Jul-1 -4.01% -1.36% * -0.58%Jul-2 -4.47% -1.25% * -0.72%Jul-3 -0.69% -0.25% * -1.05%Jul-4 0.06% 0.03% * 5.88%Jul-5 -0.35% -0.25% * -0.21%


Table 4.12. Comparison of Expected Fill-Rates Associated with Expected Value Solution vs.Stochastic Solution for the Implosion Problem with Fixed Supply when Supply is Given by theSolution to Expected Value Explosion Model


Mar-1 0.07 0.27 263.16% 0.62 0.85 35.90%Mar-2 0.17 0.27 58.51% 0.51 0.75 46.62%Mar-3 0.49 0.55 12.20% 0.74 0.82 11.36%May-1 0.48 0.52 7.37% 0.73 0.85 17.41%May-2 0.74 0.67 -10.51% 0.87 0.87 0.37%Jul-1 0.69 0.91 32.37% 0.85 0.91 6.38%Jul-2 0.11 0.26 129.65% 0.46 0.70 54.45%Jul-3 0.32 0.62 90.72% 0.84 0.94 11.90%Jul-4 0.74 0.63 -15.87% 0.56 0.89 -0.26%Jul-5 0.49 0.53 7.04% 0.70 0.79 12.95%

93

Table 4.13. Comparison of Relative Improvements in Profit, Revenue and Costs Associated withStochastic Solution vs. Expected Value Solution for the Implosion Problem with 2% SupplierFlexibility


Mar-1 0.53% 1.00% -2.20% -2.69%Mar-2 1.09% 1.32% -1.13% -1.01%Mar-3 0.72% 0.92% -1.74% -2.72%May-1 0.59% 0.78% -1.56% -2.89%May-2 0.32% 0.72% -2.22% -13.90%Jul-1 2.79% 1.20% -1.56% -6.69%Jul-2 1.03% 1.21% -1.44% -1.85%Jul-3 1.42% 1.42% -0.78% 3.00%Jul-4 0.10% 0.26% -1.83% 4.06%Jul-5 2.47% 2.19% -1.89% 13.42%

Table 4.14. Comparison of Expected Fill-Rates Associated with Expected Value Solution vs.Stochastic Solution for the Implosion Problem with 2% Supplier Flexibility


Mar-1 0.83 0.93 12.73% 0.97 0.99 1.22%Mar-2 0.78 0.89 14.09% 0.97 0.98 1.63%Mar-3 0.86 0.91 5.90% 0.97 0.98 1.09%May-1 0.81 0.87 7.78% 0.87 0.93 6.14%May-2 0.74 0.75 1.70% 0.87 0.91 4.88%Jul-1 0.85 0.99 15.75% 0.98 0.99 0.55%Jul-2 0.80 0.84 5.31% 0.96 0.98 1.80%Jul-3 0.91 0.96 5.63% 0.98 0.99 1.46%Jul-4 0.82 0.80 -2.22% 0.92 0.93 1.39%Jul-5 0.82 0.86 5.45% 0.97 0.97 -0.55%

94

Table 4.15. Comparison of Relative Improvements in Profit, Revenue and Costs Associatedwith Stochastic Solution vs. Expected Value Solution for the Implosion Problem with 2% SupplyFlexibility when Committed Supply is Given by the Solution to the Expected Value ExplosionModel


Mar-1 3.08% 0.33% 2.04% 6.92%Mar-2 18.25% 1.25% 1.79% 5.60%Mar-3 1.79% 0.46% 2.00% 6.49%May-1 1.13% -0.02% 2.03% 5.96%May-2 1.90% 1.19% 0.18% 15.46%Jul-1 3.90% 0.07% 1.71% 5.59%Jul-2 10.58% 1.28% 2.04% 6.41%Jul-3 7.96% 1.56% 1.90% 21.52%Jul-4 0.80% 0.38% 0.74% 24.01%Jul-5 2.78% 1.49% 1.44% 8.73%

Table 4.16. Comparison of Expected Fill-Rates Associated with Expected Value Solution vs.Stochastic Solution for the Implosion Problem with 2% Supply Flexibility when CommittedSupply is Given by the Solution to Expected Value Explosion Model


Mar-1 0.08 0.26 245.10% 0.63 0.85 36.37%Mar-2 0.17 0.27 62.72% 0.51 0.75 47.81%Mar-3 0.49 0.49 12.73% 0.74 0.83 11.76%May-1 0.39 0.39 33.49% 0.64 0.86 34.27%May-2 0.73 0.73 -6.63% 0.84 0.88 4.03%Jul-1 0.69 0.69 32.75% 0.86 0.91 6.59%Jul-2 0.11 0.11 130.66% 0.45 0.71 55.57%Jul-3 0.32 0.32 81.30% 0.84 0.94 12.19%Jul-4 0.73 0.73 -8.82% 0.87 0.89 3.18%Jul-5 0.49 0.49 8.51% 0.70 0.79 13.55%

95



Mar-1 1.75% 2.73% -5.26% -8.85%Mar-2 3.35% 3.14% -3.06% 8.81%Mar-3 2.13% 2.52% -4.21% -3.24%May-1 1.97% 2.46% -4.47% -7.31%May-2 1.36% 1.96% -3.24% -20.08%Jul-1 8.72% 3.30% -4.69% -9.88%Jul-2 3.43% 3.33% -3.23% 2.51%Jul-3 5.18% 3.80% -2.84% 24.55%Jul-4 0.26% 0.44% -1.02% 1.24%Jul-5 4.62% 4.49% -4.60% 9.83%




96

Table 4.19. Comparison of Relative Improvements in Profit, Revenue and Costs Associatedwith Stochastic Solution vs. Expected Value Solution for the Implosion Problem with 5% SupplyFlexibility when Committed Supply is Given by the Solution to the Expected Value ExplosionModel


Mar-1 12.72% 3.26% 5.07% 17.76%Mar-2 55.22% 4.19% 4.49% 14.28%Mar-3 7.02% 2.86% 4.96% 16.80%May-1 6.63% 2.48% 5.01% 15.99%May-2 4.16% 3.00% -0.74% 24.71%Jul-1 15.50% 2.30% 4.18% 14.47%Jul-2 34.36% 5.07% 5.09% 16.30%Jul-3 19.28% 3.92% 4.40% 49.81%Jul-4 1.67% 0.80% 1.53% 42.29%Jul-5 7.27% 4.07% 3.29% 20.54%



Mar-1 0.08 0.31 277.88% 0.62 0.86 36.93%Mar-2 0.17 0.27 64.30% 0.50 0.75 49.79%Mar-3 0.48 0.56 16.37% 0.74 0.83 12.24%May-1 0.39 0.53 35.23% 0.64 0.86 34.94%May-2 0.74 0.74 0.02% 0.84 0.90 6.74%Jul-1 0.69 0.92 33.29% 0.86 0.92 6.95%Jul-2 0.12 0.27 126.40% 0.45 0.71 56.77%Jul-3 0.32 0.62 94.21% 0.84 0.95 13.13%Jul-4 0.74 0.73 -1.09% 0.87 0.90 4.23%Jul-5 0.49 0.54 10.15% 0.70 0.80 14.18%

97



Mar-1 3.81% 4.97% -8.10% -5.53%Mar-2 7.38% 5.87% -4.52% 22.90%Mar-3 4.58% 4.95% -6.65% 2.41%May-1 4.37% 4.95% -7.45% -4.26%May-2 3.79% 4.00% -4.48% -4.83%Jul-1 17.12% 6.55% -7.47% -6.02%Jul-2 7.05% 6.21% -4.89% 10.58%Jul-3 11.65% 6.82% -3.47% 41.82%Jul-4 0.73% 0.80% -1.14% 23.57%Jul-5 8.81% 8.55% -8.28% 8.05%




98

Table 4.23. Comparison of Relative Improvements in Profit, Revenue and Costs Associated withStochastic Solution vs. Expected Value Solution for the Implosion Problem with 10% SupplyFlexibility when Committed Supply is Given by the Solution to the Expected Value ExplosionModel


Mar-1 27.29% 7.85% 9.50% 30.95%Mar-2 118.88% 8.68% 8.87% 25.54%Mar-3 15.56% 6.94% 9.53% 29.78%May-1 14.67% 6.12% 9.65% 28.10%May-2 7.16% 5.76% -3.00% 23.14%Jul-1 35.92% 7.11% 7.48% 25.50%Jul-2 75.43% 11.42% 9.98% 29.17%Jul-3 28.98% 6.81% 5.23% 68.27%Jul-4 2.46% 1.72% 0.09% 51.02%Jul-5 13.34% 8.40% 3.03% 28.24%



Mar-1 0.10 0.30 277.88% 0.62 0.87 38.79%Mar-2 0.16 0.27 64.30% 0.50 0.77 54.34%Mar-3 0.49 0.56 16.37% 0.74 0.84 13.01%May-1 0.39 0.53 35.23% 0.64 0.87 35.04%May-2 0.74 0.74 0.02% 0.84 0.91 7.70%Jul-1 0.69 0.92 33.29% 0.86 0.92 7.84%Jul-2 0.12 0.27 126.40% 0.45 0.71 58.29%Jul-3 0.32 0.62 94.21% 0.84 0.97 15.90%Jul-4 0.74 0.73 -1.09% 0.87 0.92 6.27%Jul-5 0.49 0.54 10.15% 0.70 0.80 14.32%

99

Figure 4.1. Percentage Improvement in Expected Profit for Various Levels of Supplier Flexibility

0.00%

0.10%

0.20%

0.30%

0.40%

0.50%

0.60%

0.70%

0.80%

No Flex 2% Flex 5% Flex 10% Flex

Pe

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fit

Mar-1

Mar-2

Mar-3

May-1

May-2

Jul-1

Jul-2

Jul-3

Jul-4

Jul-5

100

Figure 4.2. Percentage Improvement in Expected Revenue for Various Levels of Supplier Flex-ibility

0.00%

0.05%

0.10%

0.15%

0.20%

0.25%

0.30%

0.35%

0.40%

0.45%

0.50%


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Mar-2

Mar-3

May-1

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Jul-1

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Figure 4.3. Percentage Change in Expected Ordering Cost for Various Levels of SupplierFlexibility

-0.60%

-0.40%

-0.20%

0.00%

0.20%

0.40%

0.60%

0.80%

1.00%

1.20%


Pe

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Ord

eri

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Co

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Mar-1

Mar-2

Mar-3

May-1

May-2

Jul-1

Jul-2

Jul-3

Jul-4

Jul-5

102

Figure 4.4. Percentage Change in Expected Holding Cost for Various Levels of Supplier Flexi-bility

-15.00%

-10.00%

-5.00%

0.00%

5.00%

10.00%

15.00%

20.00%


Perc

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Mar-1

Mar-2

Mar-3

May-1

May-2

Jul-1

Jul-2

Jul-3

Jul-4

Jul-5

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Figure 4.5. Percentage Change in Expected Average Fill-Rate for Various Levels of SupplierFlexibility

-10.00%

-5.00%

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%


Perc

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vera

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ill-

Rate

Mar-1

Mar-2

Mar-3

May-1

May-2

Jul-1

Jul-2

Jul-3

Jul-4

Jul-5

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Figure 4.6. Percentage Change in Expected Cumulative Fill-Rate for Various Levels of SupplierFlexibility

-1.00%

-0.50%

0.00%

0.50%

1.00%

1.50%


Pe

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Cu

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lati

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Fil

l-R

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Mar-1

Mar-2

Mar-3

May-1

May-2

Jul-1

Jul-2

Jul-3

Jul-4

Jul-5

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Figure 4.7. Percentage Improvement in Expected Profit for Various Levels of Supplier Flexibilitywith Initial Supply Given by Expected Value Explosion Model

0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

120.00%

NO Flex 2% 5.00% 10%

Pe

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Pro

fit Mar-1

Mar-2

Mar-3

May-1

May-2

Jul-1

Jul-2

Jul-3

Jul-4

Jul-5

106

Figure 4.8. Percentage Improvement in Expected Revenue for Various Levels of Supplier Flex-ibility with Initial Supply Given by Expected Value Explosion Model

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

NO flex 2.00% 5% 10%

Perc

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even

ue Mar-1

Mar-2

Mar-3

May-1

May-2

Jul-1

Jul-2

Jul-3

Jul-4

Jul-5

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Figure 4.9. Percentage Change in Expected Ordering Cost for Various Levels of SupplierFlexibility with Initial Supply Given by Expected Value Explosion Model

-12.00%

-10.00%

-8.00%

-6.00%

-4.00%

-2.00%

0.00%

2.00%

4.00%

NO Flex 2% 5.00% 10%

Pe

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Ord

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Mar-1

Mar-2

Mar-3

May-1

May-2

Jul-1

Jul-2

Jul-3

Jul-4

Jul-5

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Figure 4.10. Percentage Change in Expected Holding Cost for Various Levels of SupplierFlexibility with Initial Supply Given by Expected Value Explosion Model

-80.00%

-70.00%

-60.00%

-50.00%

-40.00%

-30.00%

-20.00%

-10.00%

0.00%

NO flex 2.00% 5% 10%

Perc

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ing

Co

st

Mar-1

Mar-2

Mar-3

May-1

May-2

Jul-1

Jul-2

Jul-3

Jul-4

Jul-5

109

Figure 4.11. Percentage Change in Expected Average Fill-Rate for Various Levels of SupplierFlexibility with Initial Supply Given by Expected Value Explosion Model

-10.00%

-5.00%

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

30.00%

NO Flex 2% 5.00% 10%

Pe

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Av

era

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Fil

l-R

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Mar-1

Mar-2

Mar-3

May-1

May-2

Jul-1

Jul-2

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Jul-4

Jul-5

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Figure 4.12. Percentage Change in Expected Cumulative Fill-Rate for Various Levels of Sup-plier Flexibility with Initial Supply Given by Expected Value Explosion Model

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

3.50%

4.00%

4.50%

5.00%

NO flex 2.00% 5% 10%

Pe

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Mar-1

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Jul-5

Chapter 5

The Component Rationing

Problem

5.1 Introduction

In the second stage of both the explosion and implosion problems, the allocation of

components to product orders is done for all T periods simultaneously. The implica-

tion of this is that orders due in earlier periods may be backordered, even though the

resources required to satisfy them are available, in order to reserve these resources for

more profitable orders in a later period. In practice, it is impossible to anticipate all

future orders with such precision. However, rationing policies can be implemented to

help improve the manufacturer’s ability to anticipate future orders. One such rationing

scheme is addressed in this chapter, and its formulation is referred to as the component

rationing problem.

In the literature (see Chapter 2), a popular rationing policy has a threshold structure.

Under this policy, in a single product make-to-stock environment, low priority orders are

112

backordered once the inventory level for the product drops below the threshold level set

for that priority class, and where the number of different priority classes may correspond

to the number of different customer types. In the multi-product, configure-to-order

environment considered here, this policy structure is extended so that low priority or-

ders are backordered if fulfilling the order would cause the inventory level of any of the

product’s required components to fall below a set threshold level. In the component

rationing model formulated in this chapter, threshold levels are designated for all rele-

vant component-product pairs, and priorities are associated with products, where higher

revenue products have higher priority. The solution to the component rationing problem

determines the set of revenue maximizing threshold levels.


A manufacturer produces a set of configurable products P = p | p = 1, 2, . . . , P.

These products are configured from a set of components C = c | c = 1, 2, . . . , C. Let

ξcp be the random quantity of component c that is required to configure an order for

one unit of product p. A realization of a configurable product p is given by ξξξp(ω) =

[ξ1p(ω), ξ2p(ω), . . . , ξCp(ω)]⊤ with ξcp(ω) ∈ Z+ and ω ∈ Ω, and where Ω is the set of all

random events. The ξcp are assumed to be independent random variables. The objective

of this problem is to determine the rationing threshold levels for all relevant product-

component pairs that will maximize revenue for a given period, t. The consideration of

relevant costs (e.g., component ordering and inventory holding costs) and service levels

(e.g., fill rates) are assumed to have been accounted for by the planning models (e.g.,

explosion and implosion models) which will determine the supply and demand inputs to

the rationing problem. The problem is constrained by the supply of components and the

rules of the threshold rationing policy. Since not all components are used to configure

all products, let Cp ⊆ C be the set of components that can be used to configure product

113

p. Then, ξcp(ω) = 0 ∀ (c, p) /∈ Cp × P and ω ∈ Ω.

The component rationing problem can be formulated as a single period, two-stage

stochastic mixed-integer program. In the first stage, for a fixed period t, the manufacturer

has estimates for the total demand for each product, dpt ∈ Z+∀ p ∈ P, and knows the

total initial supply for each component, Ic0 ∈ Z+∀ c ∈ C. In the larger picture of

availability management, the total demand could be set equal to the commitment-to-

sales quantities determined by solving the implosion problem. Additionally, the initial

component supply could be set equal to the supply request obtained by solving either

the explosion or implosion problem. In the first stage, the manufacturer faces order

configuration uncertainty. Additionally, he does not know the arrival sequence of the

orders, where orders are assumed to arrive in unit quantities. In the presence of these

uncertainties, the manufacturer sets the threshold level for each relevant component-

product pair (i.e., for each pair (c, p) ∈ Cp×P). The threshold levels are given by, τcpIc0,

where τcp ∈ [0, 1] ∀ (c, p) ∈ Cp × P. Let τcp be referred to as the ‘threshold factor’ for

component c and product p.

At the end of the first stage, order configurations and the order arrival sequence are

realized. It is assumed that the total number of orders for product p is equal to dpt, the

manufacturer’s estimated demand quantity for period t.1 Let J = j| j = 1, 2, . . . , J

be the set of all orders, where J =∑

p∈P dpt. The sequence of product orders is given

by φj ∈ P for j = 1, 2, . . . , J . Let φφφ be the random vector [φ1, φ2, . . . , φJ ]⊤. The

configurations for all orders are represented by the C × J random matrix given by

ΞΞΞ =

ξ1φ1 ξ1φ2 · · · ξ1φJ

ξ2φ1 · · · · · · ξ2φJ

.... . .

...

ξCφ1 ξCφ2 · · · ξCφJ

(5.1)

1The decision to treat demand deterministically is discussed in §1.3.1.

114

Let φφφ(ω) and ΞΞΞ(ω) be realizations of φφφ and ΞΞΞ respectively, where ω ∈ Ω. For a given

realization of orders, let Cj(ω) ⊆ C be the subset of components required to fulfill the

jth order (i.e., the subset of components for which ξcφj(ω)(ω) > 0, where ω ∈ Ω).

In the second stage, the manufacturer fulfills orders according to the threshold ra-

tioning policy, using the threshold levels set in the first stage. Let ηj be the binary

decision variable that is equal to 1 if the jth order is fulfilled, and equal to 0, otherwise.

The revenue associated with product p is given by rp. Since φj ∈ P, revenue rφjis earned

for fulfilling the jth order. The quantity of component c available after a decision regard-

ing the jth order is made is given by Icj for j = 1, . . . , J and for all c ∈ C. The initial

inventory level for component c is given by Ic0. Let R(τττ , I0) be the revenue associated

with a given threshold policy and a given set of initial inventory levels. The threshold

policy for a given realization of orders can be implemented using the following algorithm.

Algorithm 8. Implementation of a Given Threshold Rationing Policy for a Given Realiza-

tion of Orders

Begin

Step 0 Initialize j = 0.

Step 1 Set j ← j + 1, and I ′c,j ← Ic,j−1 − ξcφj(ω)(ω) ∀ c ∈ Cj(ω).

Step 2 If∏

c∈Cj(ω)(I′cj − τcφj(ω)Ic0) < 0, then set ηj ← 0 and Icj ← Ic,j−1∀ c ∈ C.

Otherwise, set ηj ← 1, Icj ← I ′cj ∀ c ∈ Cj(ω) and Icj ← Ic,j−1 ∀ c ∈ C − Cj(ω).

If j = J , go to Step 3. Otherwise, go to Step 1.

Step 3 Set R(τττ , I0)←∑J

j=1 ηjrφj(ω).

End

In Step 0 of Algorithm 8, the order count is initialized. In Step 1, the order count

is incremented and the inventory that would remain if the jth order is fulfilled, I ′cj is

115

calculated. In Step 2, if, for any component demanded in the jth order, the inventory

that would remain if the jth order is fulfilled is less than the rationing level set for that

component and the ordered product, φj(ω), then the order is not fulfilled. Otherwise, the

order is fulfilled and the inventory level of all components is appropriately updated. If

all orders have been considered, then the algorithm ends. Otherwise, Step 1 is repeated.


The component rationing problem can be formulated as a two-stage stochastic mixed-

integer program. The first stage decision variables are continuous and the second stage

consists of both binary and continuous variables. A summary of the variables and pa-

rameters used in the formulation is provided below.

Sets:


P = Set of products = p | p = 1, 2, . . . , P

J = Set of orders = j | j = 1, 2, . . . , J

Cj ⊂ C = Set of components required by order j ∈ J

Cp ⊂ C = Set of components potentially required by product p ∈ P


Parameters:

dpt = Total number of orders for product p in period t

Ic0 = Initial inventory of component c

Mcj = ‘Big’ M parameter for component c and the jth order

116

rp = Revenue associated with an order for product p

Random Parameters:

φj = Product type of the jth order

ξcφj= Quantity of component c required to fulfill jth order

rφj= Revenue associated with the jth order


τcp = Threshold factor for component-product pair (c, p) ∈ Cp × P


ηj =

1 if jth order is filled,

0 otherwise

Icj = Quantity of component c remaining after decision about whether or not to fill

jth order has been made

The component rationing optimization problem (RAT) is given by

[RAT]

Z∗RAT = max

0≤τcp≤1 ∀ (c,p)∈Cp×PQ(τττ) (5.2)

The objective function, Q(τττ ), represents the expected revenue earned during the period.

It is defined as

Q(τττ) = E[Q(τττ ,φφφ,ΞΞΞ)]

117

where Q(τττ ,φφφ(ω),ΞΞΞ(ω)) defines the second stage problem as follows:

max

J∑

j=1

rφj(ω)ηj (5.3)

s.t. Icj = Ic,j−1 − ξcφj(ω)(ω)ηj ∀ c, j (5.4)

0 ≤ Ic,j−1 − (ξcφj(ω)(ω) + τcφj(ω)Ic0)ηj ∀ (c, j) ∈ Cj(ω)× J (5.5)

Mcjηj − 1 ≥ Ic,j−1 − ξcφj(ω)(ω)− τcφj(ω)Ic0 −Mcj(1− ucj) ∀ (c, j) ∈ Cj(ω)× J

(5.6)

∑

c∈Cj(ω)

ucj = 1 ∀ j (5.7)

ηj ∈ 0, 1 ∀ j (5.8)

ucj ∈ 0, 1 ∀ (c, j) ∈ Cj(ω)× J (5.9)

In the objective function (5.3) expresses the revenue earned for a given realization of

order configurations and order sequence. The inventory flow constraints are given by

(5.4). Constraint set (5.5) ensures that the fulfillment of an order, j, does not cause

the inventory level of any one of its required components to drop below its threshold

level for the ordered product. Note that this constraint is redundant for unfilled orders

(i.e., whenever ηj = 0). In order to implement the threshold policy correctly, it is

necessary to fill an order as long as doing so will not violate constraint (5.5). This is the

purpose of constraints (5.6) and (5.7). In (5.6), the parameter Mcj can be set equal to

Ic0. It is an upper bound on the value of the right hand side of inequality (5.6) for a

given (c, j) ∈ Cj × J . Note that the second stage problem is a mixed-integer program

representation of Algorithm 8.

118

5.4 Solution Method


Let K = k | k = 1, 2, · · · ,K be the set of all scenarios. In each scenario, k ∈ K, let

φφφ(ωk) = φ1(ωk), . . . , φJ(ωk) be the realized order sequence and let ξcφj(ωk) be the re-

alized configuration of the jth order, for j = 1, 2, . . . , J . In formulating the deterministic

equivalent to RAT, allow an additional subscript, k, to be appended to each of the second

stage decision variables, denoting the scenario for which these decisions are made. Then,

the deterministic equivalent linear programming formulation of the rationing problem

(DERAT) is given below.

[DERAT]

max

K∑

k=1

πk

J∑

j=1

rφj(ωk)ηjk (5.10)

s.t. Icjk = Ic,j−1,k − ξcφj(ωk)(ωk)ηj ∀ c, j, k (5.11)

0 ≤ Ic,j−1,k − (ξcφj(ωk)(ωk) + τcφj(ωk)Ic0)ηjk ∀ (c, j) ∈ Cj(ωk)× J , k (5.12)

Mcjηjk − 1 ≥ Ic,j−1,k − ξcφj(ωk)(ωk)− τcφj(ωk)Ic0 −Mcj(1− ucjk)

∀ (c, j) ∈ Cj(ωk)×J , k (5.13)

∑

c∈Cj(ωk)

ucjk = 1 ∀ j, k (5.14)

ηjk ∈ 0, 1 ∀ j, k (5.15)

ucjk ∈ 0, 1 ∀ j, k, c ∈ Cj(ωk) (5.16)

DERAT is a large, two-stage, mixed-integer stochastic program with continuous first

stage variables and both continuous and binary second stage variables. Such problems

are generally difficult to solve, partly because integer restrictions permeate all second

stage scenarios, thereby magnifying the challenges normally associated with combinato-

119

rial optimization problems in non-stochastic settings, and partly because the recourse

function is not necessarily convex or continuous. Klein Haneveld and van der Vlerk

(1999) and Sen (2005) provide comprehensive overviews of the developments in stochastic

mixed-integer programming. While recent progress in two-stage stochastic mixed-integer

programing has been quite rapid, many approaches require the first stage decision vari-

ables to be binary (e.g., the integer-L shaped method of Laporte and Louveaux (1993)

and the methods explored by Sen and Higle (2005) and Sen and Sherali (2005)). Ahmed

et al. (2004) do allow for a mixed-integer first stage problem, but require a pure integer

second-stage problem. A branch-and-price method was used by Lulli and Sen (2004)

to solve a batch-sizing problem that was formulated as a multi-stage stochastic integer

program. However, the effectiveness of this approach is questionable when the number

of scenarios is very large.2

It turns out that a simple derivative free search algorithm is quite effective in produc-

ing good solutions to DERAT. After all, the implementation of the threshold rationing

policy for a given set of threshold levels is very straightforward algorithmically (see

Algorithm 8).

5.4.2 Search Algorithm for the Component Rationing Problem

The search algorithm that is used to find a solution to the component rationing problem,

RAT, is based on the derivative-free cyclic coordinate pattern search method (Bazaraa

et al. 1993). Since the threshold levels are constrained to be between 0 and 1, the search

is constrained. Each threshold variable (i.e., each τcp ∀ (c, p) ∈ Cp × P) constitutes a

legitimate direction of search. Similar to the cyclic coordinate method, the search algo-

rithm used here explores each of the search directions in turn, in every search iteration.

2Lulli and Sen (2004) consider up to 32 scenarios. An attempt was made to apply the brach-and-priceapproach used by Lulli and Sen (2004) to solve the component rationing problem, but it was not foundto be effective.

120

The sequence of search directions is dictated by the relative revenue of the products.

Let p[1], p[2], . . . , p[P ] be an ordered list of products such that rp[1]≤ rp[2]

≤ · · · ≤ rp[P ].

Since components for lower revenue products are more likely to be rationed, directions

corresponding to products with lower revenues are searched before those correspond-

ing to products with higher revenues. Only improving solutions are accepted dur-

ing the search process. An appropriate starting solution to use here is the solution

τcp = 0 ∀ (c, p) ∈ Cp × P. This initial solution represents the case where no rationing

of components occurs. The search step size, γ, is gradually reduced by a factor, ρ, as

the iterations progress, until one of several stopping criteria is satisfied. One stopping

criterion stops the search when the step size is less than some tolerance level, TOL1.

A second stopping criterion stops the search when no better solution is found after

V1 consecutive iterations. A third stopping criterion stops the search when insufficient

improvement (defined using TOL2) is observed in the solution over V2 consecutive it-

erations. Additionally, if progress within an iteration stalls for V3 consecutive search

steps, the current iteration is ended. Let ZRAT be the revenue obtained by solving the

component rationing problem using this search algorithm. The details of this search

method are given by Algorithm 9.

Algorithm 9. Search Algorithm for an S-Scenario Component Rationing Problem

Begin

Step 0 Initialize τcp = 0 ∀ (c, p) ∈ Cp × P . Using these threshold

factors and initial inventory levels Ic0 ∀ c ∈ C, compute Rs(τττ , I0)

via Algorithm 8 for s = 1, 2, . . . , S.

Sort products in order of ascending revenue.

Select values for TOL1, TOL2, γ0, ρ, V1, V2 and V3.

Initialize Zinc ← 1S

S∑

s=1Rs(τττ , I0), q ← 0, γ ← γ0,

NOBETTER← 0, INSUFF← 0, and SLOW← 0.

121

Step 1 If γ < TOL1 go to Step 10.

Otherwise, set q ← q + 1, i← 0 and Zq ← Zinc.

Step 2 Set i← i + 1 and c← 0.

Step 3 If SLOW > V3 go to Step 8.

Step 4 Set c← c + 1. If c /∈ Cp[i], then if c < C repeat Step 4,

otherwise go to Step 7.

Step 5 If τp[i]c = 1, go to Step 6.

Otherwise, set τ ← τp[i]c and τp[i]c ← minτp[i]c + γ, 1 and

compute Rs(τττ , I0) for s = 1, 2, . . . , S via Algorithm 8.

If 1S

S∑

s=1Rs(τττ , I0) ≤ Zinc, set τp[i]c ← τ and go to Step 6.

Otherwise, set Zinc ← 1S

S∑

s=1Rs(τττ , I0) and SLOW← 0.

If c = C go to Step 2, otherwise go to Step 4.

Step 6 If τp[i]c = 0, set SLOW← SLOW + 1 and go to Step 3.

Otherwise, set τ ← τp[i]c and τp[i]c ← maxτp[i]c − γ, 0 and

compute Rs(τττ , I0) for s = 1, 2, . . . , S via Algorithm 8.

If 1S

S∑

s=1Rs(τττ , I0) ≤ Zinc, set τp[i]c ← τ and SLOW← SLOW + 1.

Otherwise, set Zinc ← 1S

S∑

s=1Rs(τττ , I0) and SLOW← 0.

If c = C go to Step 2, otherwise go to Step 3.

Step 7 If i = P go to Step 8. Otherwise, go to Step 2.

Step 8 If Zinc > Zq, then NOBETTER← 0 and go to Step 9.


Step 9 If γ < γ0, set γ ← ργ.

IfZinc−Zq

Zq< TOL2, set INSUFF← INSUFF + 1, else INSUFF← 0.

If INSUFF ≥ V2, go to Step 11. Otherwise, go to Step 1.

Step 10 Set γ ← ργ, NOBETTER← NOBETTER + 1 and

INSUFF← INSUFF + 1.

If NOBETTER ≥ V1 or INSUFF ≥ V2, go to Step 11.

122


Step 11 Set ZRAT = Zinc.

End

In Step 0 of the search algorithm, all threshold factors are set to zero (i.e., the case

where there is no rationing) and the corresponding expected revenue associated with

these threshold factors is used to initialize the incumbent objective value. Products are

sorted in order of ascending revenue. This ordering will guide the sequence in which the

search directions are explored. The decision to sort the orders in this way is inspired

by the idea that lower revenue orders are more likely to be rationing when supply is

constrained. Other parameters, such as the step size are also initialized in this step.

In Step 1 of the search algorithm, the current step size is checked to see if it satisfies

the tolerance stopping criterion (TOL1). If it does not, then the next iteration is started,

where an iteration encompasses an entire loop through all search directions. Step 2

simply defines the outer loop through the various search directions. Step 3 checks to

see if at least one of the steps made in the last V3 search directions was an improving step.

If one was not, then the current iteration is ended (i.e., the remaining search directions

are not explored in the current iteration). Step 4 checks if component-product pair

currently under consideration is legitimate (i.e., if the component in question can be

used to configure the product in question).

In Step 5, if the value of the current solution in the current search direction is less

than 1, a step is taken in the positive search direction such that the resulting value of the

solution in this direction is no greater than 1. If this proves to be an improving step, the

incumbent objective is updated and the next search direction is explored. Otherwise,

this step is reversed and the algorithm proceeds to Step 6. In Step 6, if the value of the

current solution in the current search direction is greater than 0, a step is taken in the

123

negative search direction, such that the resulting value of the solution in this direction

is no less than 0. If this proves to be an improving step, the incumbent objective is

updated and the next search direction is explored. Otherwise, this step is reversed and

the next search direction is explored.

Step 7 of the search algorithm checks if all legitimate search directions have been

explored. If all legitimate search directions have been explored, then the current iteration

ends. Step 8 checks if the most recent iteration results in a better solution than the

best solution found so far, and directs the search to either Step 9 (if a better solution is

found) or Step 10 (if a better solution is not found). If a better solution is found, then in

Step 9 the step size is reduced by a factor γ, but only if the current step size is less than

the initial step size. The relative improvement in the objective function is deemed to be

insufficient if it is smaller than some tolerance, TOL2. Additionally, if this is the V2-th

consecutive iteration in which insufficient improvement is observed, the search ends. On

the other hand, if no better solution was found, then in Step 10 the step size is reduced

by a factor γ. If this is the V1-th consecutive iteration in which no improved solution is

found, or if this is the V2-th consecutive iteration in which insufficient improvement is

observed, the search ends and, in Step 11, the best solution found so far is treated as

the solution to the rationing problem.


In the component rationing problem, both the sequence of order arrivals and the con-

figurations of the orders are uncertain. For a given period t, Algorithm 10 is used to

generate a random sequence of order arrivals. This algorithm essentially samples with-

out replacement from a collection of product orders, where the number of orders for a

particular product p ∈ P is given by dpt.

124

Algorithm 10. Generation of Order Sequence for Period t and Scenario k

Begin

Step 0 Initialize O ← J, j ← 0, F (0)← 0, F (p)←p∑

i=1

dit

O∀ p ∈ P

and dpt = dpt ∀ p ∈ P .

Step 1 Set j ← j + 1 and generate X ∼ U(0, 1).

Identify p satisfying F (p− 1) ≤ X ≤ F (p).

Set dpt ← dpt − 1, O← O − 1 and φj(ωk) = p.

Step 2 If j < J , set F (p)←p∑

i=1

dit

O∀ p ∈ P and go to Step 1.

Otherwise, go to End.

End

After the order arrival sequence has been determined for a given scenario k, the or-

der configurations for that scenario are generated using Algorithm 11. Similar to the

configuration generation performed for the explosion and implosion problems, the alias

method is used to generate configurations from the empirical distributions of ξcp ∀ c, p.

In Algorithm 11, Wpc is the maximum quantity of component c that is ordered in product

p. Apc(w) and Cpc(w) are alias and cutoff values, respectively, which can be computed

using Algorithm 2 in §3.3.2.

Algorithm 11. Generation of ξcφj(ωk)(ωk) for the Component Rationing Problem

Begin

Step 0 Initialize j ← 0. Compute Apc(w) and Cpc(w) ∀ (p, c) ∈ P × Cp

and w = 1, 2, · · · , Wpc.

Step 1 Set j ← j + 1 and initialize c← 0.

Step 2 Set c← c + 1. If c /∈ Cφj, then if c < C repeat Step 2,

otherwise go to Step 4.

125

Step 3 Generate X1 ∼ DU(0, mφjc) and X2 ∼ U(0, 1),


If X2 ≤ Cφjc(X1), set ξcφj(ωk)(ωk) = X1.

Otherwise, set ξcφj(ωk)(ωk) = Aφjc(X1).

If c = C go to Step 4. Otherwise, go to Step 2.

Step 4 If j = J , go to End. Otherwise, go to Step 1.

End

To generate a single scenario, k, Algorithm 10 is implemented once, and Algorithm

11 is implemented for all j = 1, 2, . . . , J and c ∈ Cj(ωk).


The sample average approximation (SAA) method is used to address the component

rationing problem. To reiterate the general approach of the SAA method, since the full

K-scenario problem DERAT may be prohibitively large, a subset of the scenarios in K

is considered, where the cardinality of this subset is given by S, where S << K. The

S-scenario SAA problem for the component rationing problem, SAA-RAT(S), is given

below.

[SAA-RAT(S)]

νS = max0≤τpc≤1 ∀ (c,p)∈Cp×P

1

S

S∑

s=1

Q(τττ ,φφφ(ωs),ΞΞΞ(ωs)) (5.17)

where Q(τττ ,φφφ(ωs),ΞΞΞ(ωs)) defines the integer program given by (5.3) - (5.9). The pa-

rameter S represents the number of scenarios to be included in the optimization. An

estimated upper bound to Z∗RAT, represented by Zu

RAT, can be derived by solving N , in-

dependent instances of the problem SAA-RAT(M). Then, using a larger set of scenarios,

126

L > M , an estimated lower bound to Z∗RAT is obtained by evaluating a feasible solution

to RAT in an L-scenario setting. Let this lower bound be represented by ZℓRAT. The

methods for obtaining bounds on Z∗RAT are slightly different from those used to generate

bounds for the explosion and implosion problems. These methods are described in the

following sections.

5.4.4.1 Bound and Optimality Gap Estimation

Suppose that ν1M , ν2

M , . . . , νNM are the objective values associated with N , independent in-

stances of the problem SAA-RAT(M). Let ν1M , ν2

M , . . . , νNM be upper bounds to ν1

M , ν2M , . . . , νN

M ,

respectively (i.e., νnM ≥ νn

M for n = 1, 2, . . . ,N). Then, an estimate of an upper bound

to Z∗RAT is given by

ZuRAT =

1

N

N∑

n=1

νnM (5.18)

When possible, SAA-RAT(M) is solved to optimality using a commercial off-the-shelf

mixed integer programming solver called CPLEX. In this case, νnM = νn

M . However,

when an optimal solution to SAA-RAT(M) is difficult to obtain, even in the case where

M = 1, an upper bound to νnM is used in equation (5.18). In such cases, the upper bound,

νnM is typically the objective value corresponding to a relaxed formulation of SAA-RAT.

In the worst case, νnM is the objective value corresponding to the linear relaxation of

SAA-RAT. In other cases, CPLEX is able to provide an improved upper bound on νnM

by generating useful cuts. In such cases νnM is set equal to this improved bound.

Let τpc ∀ (c, p) ∈ Cp × P (represented by τττ) be a feasible solution to the problem

SAA-RAT(L), obtained using the search algorithm described by Algorithm 9. Then, a

lower bound to Z∗RAT is given by

ZℓRAT =

1

L′

L′

∑

l=1

Q(τττ ,φφφ(ωl),ΞΞΞ(ωl)) (5.19)

127

where L′ > L and the L′ scenarios are independently generated. The value Q(τττ ,φφφ(ωl),ΞΞΞ(ωl))

in equation (5.19) can be computed using Algorithm 8. The resulting optimality gap at

τpc ∀ (c, p) ∈ Cp × P is estimated by

ZgRAT = Zu

RAT −ZℓRAT (5.20)

with a (1− α) confidence interval given by

[

0, [ZgRAT]+ +

tN−1, α2σu√

N+

tL′−1, α2σℓ√

L′

]

(5.21)

where σu is the sample variance for ZuRAT and σℓ is the sample variance for Zℓ

RAT.

5.4.4.2 SAA Algorithm for the Component Rationing Problem

A summary of the how the SAA method is applied to the component rationing problem

is provided by Algorithm 12, below.

Algorithm 12. SAA Algorithm for the Component Rationing Problem

Begin

Step 0 Select values for N, M, L and L′. Initialize n← 0.


a sequence of orders φφφ(ωm) and order configurations

ξcφj(ω)(ωm) ∀ j = 1, 2, . . . , J and c ∈ Cj(ωm).

Step 2 Construct the SAA-EXP(M) problem using the order sequence

and order configurations last generated in step 1 and solve for νnM .

If n = N , set ZuRAT = 1

N

∑N



Step 3 For l = 1, 2, . . . , L, independently generate a sequence of

128

orders φφφ(ωl) and order configurations ξcφφφj(ωl)(ωl) ∀ j = 1, 2, . . . , J

and c ∈ Cj(ωl).

Step 4 Construct the SAA-RAT(L) problem using the order sequence

and configurations generated in Step 3. Solve the SAA-RAT(L)

problem using Algorithm 9 to obtain τpc ∀ (c, p) ∈ Cp × P .

Step 5 For l = 1, 2, . . . , L′, independently generate a sequence

of orders φφφ(ωl) and order configurations ξcφφφj(ωl)(ωl) ∀ j = 1, 2, . . . , J

and c ∈ Cj(ωl).

Step 6 Construct the SAA-RAT(L′) problem using this random sample and

evaluate it at τpc ∀ (c, p) ∈ Cp × P to obtain ZℓRAT.

End

In Algorithm 12, a total of N , SAA-RAT(M) problems are solved, one SAA-RAT(L)

problem is solved and one SAA-RAT(L′) problem is evaluated at a given solution. Note

that in contrast to the SAA algorithm used for the explosion and implosion problems, the

solution used to estimate the lower bound, ZℓRAT, is not selected from the set of solutions

used to estimate the upper bound, ZuRAT. Rather, the solutions used to estimate the

upper and lower bounds are independently obtained.3

In a multi-period setting, the single-period rationing problem can be solved for each

period in chronological sequence, where any backorders and inventory remaining at the

end of a previous period would be carried over into the next period. In a subsequent

period, the oldest backorders could be assumed to arrive earliest in the order sequence.

In fact, this is the approach used in Chapter 6, which provides a holistic perspective of

3The solutions used to estimate the upper bound are generated by CPLEX, which solves relaxationsof SAA-RAT. Meanwhile, the solution used to estimate the lower bound is generated using Algorithm 9.The reason for this is that the number of scenarios that can be handled by CPLEX to estimate the upperbound, for the problem sets studied in this research, is very small. Therefore, the solutions generated byCPLEX are unlikely to be as robust as a lower bound solution that is generated by Algorithm 9, whichcan handle a much larger number of scenarios.

129

availability management.


5.5.1 Problem Sets

The data sets used to generated the problem sets studied for the implosion problem are

identical to the ones used to study the explosion problem. That is, ten problems sets

were generated from data taken from the IBM Systems and Technology Group in March,

May and July of 2005. For each data set, a problem instance was created by randomly

selecting one of the time periods for which data was available. The characteristics of

these problems are summarized in Table 5.1. The number of products and components

associated with each problem instance may be less than what they are for the problem set

as a whole (i.e., over all periods) because not all products are demanded in every period.

Table 5.1 reports on a data characteristic referred to as the ‘Average COVREV’. In order

to define this term, let COVREVc be the coefficient of variation4 of the total revenue

that is ‘supported’ by component c in the event that the component is constrained (i.e.,

when total demand for that component is greater than its availability). The total revenue

supported by a component is the sum of the revenues of all the orders that require it.

The variability in COVREVc stems from the fact that order configurations are uncertain,

so that a component may not be constrained in every scenario and the number of orders

which depend on it may also differ from scenario to scenario. The average COVREV is

simply the mean of COVREVc, taken over all components which experience shortages

in at least one scenario. It turns out that the average COVREV plays a significant role

in explaining the rationing benefits observed in §5.5.2. Finally, Table 5.1 also displays

the normalized average revenue per order for each problem instance. The normalization

4The coefficient of variation of a statistic is equal to its standard deviation divided by its mean.

130

process divides the average revenue per order for a problem instance by the minimum

average revenue per order over all problem instances. This normalization is performed to

maintain the confidentiality of the data obtained from the IBM Systems and Technology

Group. These values are displayed to show the relative differences in the average revenue

per order across problem instances.

The component rationing problem takes as input the initial inventory levels of each

component and the total demand quantity for each product, for the time period ad-

dressed. In each of the problem instances, the initial inventory levels used are the com-

ponent order quantities generated by solving the explosion problem for the corresponding

problem set. The demand quantities used are the commitment-to-sales quantities gener-

ated by solving the implosion problem with fixed supply (also using the order quantities

generated by solving the corresponding explosion problem). The settings for the various

parameters used to guide the search algorithm (Algorithm 9) are provided in Table 5.2.

In this table, note that V3 is equal to one-fifth of the total number of legitimate search

directions. This combination of settings for these parameters were determined to be

effective through trial-and-error experiments. All computations were run on a Sun v40z

server equipped with four, 2.6 GHz, 64-bit Opteron processors and 32 GB of random

access memory.

5.5.2 Results

In this section, the estimated optimality gaps produced by both the search algorithm

and the SAA method for the component rationing problem are presented. Additionally,

the value of component rationing is assessed by comparing its benefits over the case

where no rationing is performed. Additionally, some insight is provided in regards to

explaining when component rationing is likely to be more beneficial.

131

5.5.2.1 Estimation of Optimality Gaps

Obtaining upper bound estimates for the component rationing problem is challenging.

The presence of integer variables in the second stage problem causes the number of binary

variables in DERAT to become very large as the number of scenarios increases. In fact,

for some of the problem instances analyzed, even the single scenario problem was too

difficult for CPLEX to solve. It turns out that CPLEX could solve only three problem

instances to optimality (within a 24 hour period) for a small number of scenarios. Two

types of optimality gaps are estimated in this chapter. The first type of optimality gap

measures the gap between the optimal objective value for the SAA-RAT(S) problem

(νS) and the objective value produced by the search algorithm (ZRAT), for the same set

of S scenarios. The second type of optimality gap measured is the gap produced by the

SAA-method, ZgRAT, which is defined in equation (5.20).

Table 5.3 shows the average optimality gap of the search algorithm for the SAA-

RAT(S) problem for these three problem instances. To differentiate the single-period

problem instances from the larger 20-period problem sets from which they are derived,

the problem instances are named for their corresponding problem set, followed by a suffix

corresponding to the period being solved. For example, the problem instance solved from

problem set ‘Mar-1’ is named ‘Mar-1-19’. In addition to the average optimality gaps,

Table 5.3 also displays the number of trials over which the average optimality gap is

computed and the number of scenarios used in each trial. Based on the results of this

table, the average optimality gap of the search algorithm for solving the SAA-RAT(S)

problem is approximately 3.83%.

From a larger perspective, the 90% confidence intervals for the optimality gaps pro-

duced by the application of the SAA method to the component rationing problem are

reported in Table 5.4. This table also shows the values used for N , M , L, and L′ in

the SAA algorithm for the component rationing problem. The time limits reported in

132

this table are the time limits given to CPLEX for solving the SAA-RAT(M) problems

to obtain upper bounds on Z∗RAT. The choices of these values are based on the ability

of CPLEX to obtain either optimal solutions or reasonably good upper bounds within

a reasonable amount of time. In almost all cases, it is observed that CPLEX will either

obtain an optimal solution to the SAA-RAT(M) relatively quickly, or else settle at a

bound within the time limit specified in Table 5.3, and stay at that bound for several

hours or days without improvement. The average optimality gap produced by the SAA

method for the component rationing problem is around 7.7%.

5.5.2.2 Assessing the Value of Component Rationing

For each of the problem instances analyzed, the value of component rationing is measured

by the percentage improvement in revenue earned under the threshold policy determined

using the search algorithm as compared to the case where components are not rationed.

When components are not rationed, orders are satisfied as long as there is sufficient

availability of all required components at the time that the order arrives. The revenue

earned under the threshold component rationing policy is given by ZℓRAT, as defined in

equation (5.19). The revenue earned when no rationing is performed is given by R(000, I0I0I0),

which may be computed using Algorithm 8 for the same set of scenarios that was used

to evaluate ZℓRAT.

The results of this analysis are reported in Table 5.5, along with their computation

times, in seconds. The results in this table show that the average benefit of using

the threshold rationing policy is around 2.83%. Interestingly, there is little to no benefit

observed in the problem instances ‘Jul-2-7’ and ‘Jul-3-17’, while a relatively large benefit

is observed in the problem instances ‘Mar-3-13’ and ‘May-1-7’. This raises the general

question of when component rationing may be more beneficial. Intuitively, greater benefit

from rationing is expected when components are in short supply and there is a significant

133

difference in revenue associated with satisfying one order instead of one or more other

orders. The average COVREV, introduced in §5.5.1, captures these two aspects of

a problem. A problem instance possessing higher average COVREV is expected to

experience a greater benefit from component rationing than a problem instance with a

lower average COVREV.

It turns out that, by way of a multiple regression analysis, the average COVREV and

the average revenue per order are quite effective in explaning the variation observed in the

benefits of component rationing. In the regression analysis, the response variable is the

percentage improvement in revenue when using the threshold component rationing policy

than when not. The candidate dependent factors included the average COVREV, the

average revenue per order and the product of these factors (to account for interactions).

Using the best subsets method,5 a regression model with an adjusted R2 value of 94.7%

was selected, which possesses significant (the p-value for the interaction term is less

than 0.001 and the p-value of the average COVREV term is 0.053) positive coefficients

for the average COVREV term and the interaction term. This indicates that there is

a significant positive linear correlation between the benefit of the threshold component

rationing policy and these two factors.6 A normal plot of the residuals and a residual plot

are provided in Figure 5.1 and Figure 5.2, respectively. These figures show reasonable

compliance with the typical assumptions that are made in multiple regression.

5.6 Summary

The component rationing problem addresses the issue of reserving scarce components for

higher priority (e.g., higher revenue) orders in the presence of both order configuration

5See Neter et al. (1996).

6An attempt to sort components in descending order of COVREVc in Step 0 of the search algorithmfor the component rationing problem revealed no significant or consistent benefit, either in terms ofsolution quality or computation time.

134

uncertainty and order arrival sequence uncertainty. This problem was formulated as a

stochastic combinatorial optimization problem. It was solved by way of a derivative-

free hill-climbing pattern search algorithm, within the framework of the sample average

approximation method. Computational studies showed that tight upper bounds are

difficult to obtain. Still, the search algorithm was able to produce solutions to the

component rationing problem that resulted in an almost 3% improvement in revenue, on

average, as compared to a first-come-first-served allocation policy, for the various problem

instances studied. A multiple regression analysis revealed that much of the variation

observed in the percentage improvements can be explained by both the variability in

revenue across orders when components are constrained, and the average revenue per

order. While the problem instances considered in this chapter use the best solution to

the explosion solution as the initial inventory of components, and the best solution to

the implosion problem as the demand for products, alternative cases are considered in

Chapter 6.

Table 5.1. Characteristics of Problem Instances Analyzed in the Component Rationing Problem

Problem Period No. of No. of No. of Average NormalizedSet Solved Orders Products Components COVREV Avg. Revenue

Mar-1 19 158 4 160 0.0775 217.22Mar-2 11 409 5 261 0.0829 14.82Mar-3 13 163 12 254 0.3441 191.71May-1 7 20 6 146 0.1425 690.71May-2 17 1980 30 44 0.7438 1.00Jul-1 4 123 19 180 0.1873 82.53Jul-2 7 159 3 109 0.0096 6.63Jul-3 17 505 2 129 0.1379 23.23Jul-4 5 638 26 59 0.2590 5.53Jul-5 9 41 13 160 0.0639 57.61

135

Figure 5.1. Normal Probability Plot of Residuals for Component Rationing Regression Model

Residual

Pe

rce

nt

0.020.010.00-0.01-0.02

99

95

90

80

70

60

50

40

30

20

10

5

1

Figure 5.2. Residuals vs. Fitted Values for Component Rationing Regression Model

Fitted Value

Re

sid

ua

l

0.120.100.080.060.040.020.00

0.015

0.010

0.005

0.000

-0.005

-0.010

136

Table 5.2. Global Parameter Settings for the SAA Algorithm for the Component RationingProblem

Parameter Value

TOL1 0.01TOL2 0.001

γ0 0.55ρ 0.75V1 2V2 3V3 0.2

∑

p∈P

∑

c∈Cp

1

Table 5.3. Average Optimality Gaps for the Search Algorithm for the Component RationingProblem

Problem Number of Number of AverageInstance Scenarios Trials Optimality Gap

May-1-7 5 1967 4.43%Jul-1-4 1 473 6.38%Jul-5-9 10 394 0.60%

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Table 5.4. Parameter Settings and Results for the Application of the SAA Method to theComponent Rationing Problem

Problem Optimality Gap Time LimitInstance (90% C.I.) N M L L′ per N (sec.)

Mar-1-19 [ 0, 8.11% ] 400 1 500 2000 100Mar-2-11 [ 0, 5.59% ] 200 1 200 1000 100Mar-3-13 [ 0, 13.68% ] 400 1 500 2000 100May-1-7 [ 0, 13.51% ] 2000 5 1000 3000 30May-2-16 [ 0, 9.66% ] 200 1 200 1000 60Jul-1-4 [ 0, 15.30% ] 500 1 500 2000 120Jul-2-7 [ 0, 3.66% ] 300 1 500 * *Jul-3-17 [ 0, 4.97% ] 200 1 500 2000 120Jul-4-5 [ 0, 3.51% ] 500 5 500 2000 120Jul-5-9 [ 0, 6.62% ] 500 10 1000 4000 60

* These values are not applicable because the rationing solution produced by the searchalgorithm for problem instance Jul-2-7 is identical to the solution that implements the FCFSpolicy (i.e., all threshold factors are set to zero). Therefore there was no need for a comparisonstody between the two policies for this problem instance.

Table 5.5. Percentage Improvements Observed With Component Rationing as Compared toWithout Rationing

Problem Instance Improvement in Revenue Time(over no rationing) (in seconds)

Mar-1-19 1.38% 163Mar-2-11 0.43% 1245Mar-3-13 8.48% 1646May-1-7 10.65% 118May-2-16 2.10% 187Jul-1-4 1.18% 797Jul-2-7 0.00% 180Jul-3-17 0.17% 249Jul-4-5 2.24% 341Jul-5-9 1.67% 185

Chapter 6

The Availability Management

Problem

6.1 Introduction

While the previous chapters consider the explosion, implosion and rationing problems

somewhat in isolation of each other, this chapter provides a holistic perspective on avail-

ability management. In particular, an experimental analysis of the relative benefits of

performing stochastic explosion, stochastic implosion and component rationing, within

the sales and operations planning framework, is performed. The availability management

problem integrates the problems of determining component requirements (explosion), the

commitment-to-sales (implosion) and the allocation of components to orders (component

rationing). It should be noted that the explosion and implosion problems are planning

problems which help to define the supply and demand parameters for the rationing

problem, which is focused on revenue maximization.

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6.2 Analysis of the Relative Contributions of Explosion,

Implosion and Rationing in Availability Management

As a reminder, availability management under the sales and operations planning (S&OP)

process consists of the following sequence of decisions and information flows.

1. A demand plan is created by decision-makers based on the financial goals of the

business.

2. The Explosion problem is solved to determine the corresponding supply needs.

3. A supply commitment is received from suppliers.

4. The Implosion problem is solved, resulting in a modified demand plan called the

commitment-to-sales that is used to guide sales efforts.

5. Orders arrive and are satisfied according to some allocation policy (e.g., Threshold

rationing or First-Come-First-Served)

With respect to this planning process, the explosion, implosion and rationing models

developed in this thesis could be implemented in a modular fashion. For example, the

stochastic explosion model could be implemented without implementing the stochastic

implosion model or the threshold rationing model.1 In practice, there may be different

implementation costs associated with implementing the different models within the ex-

isting information technology infrastructure. This prompts the question of how each of

the different models contributes to improving the S&OP process for a CTO system. The

following sections address this question.

1Perceivably, there could exist planning frameworks different from that of S&OP which may haveuse for some or all of these models. The fact that the explosion, implosion and rationing models areself-contained lends itself well to the concept of componentizing services so that they can be convenientlyreused in different settings.

140

To capture the individual benefits of applying the explosion, implosion and rationing

models developed in the previous chapters to availability management fot CTO systems,

a simple experiment is designed to systematically examine these three factors for data

taken from the Mar-3 problem set which was characterized in §3.4.1 and §4.4.1. The

experiment is performed in three stages and assumes a 20 period planning horizon. In

the first stage of the experiment, the explosion problem is solved twice: once using

the stochastic optimization approach developed in Chapter 3 and once by solving the

deterministic equivalent explosion problem with a single, expected-value scenario. In the

second stage of the experiment, the implosion problem is solved four times: twice for

each explosion solution generated in the first stage of the experiment. Specifically, for

each of the explosion solutions (which are used as inputs to the implosion problem), the

implosion problem is solved once using the stochastic optimization approach developed

in Chapter 4 and once by solving the deterministic equivalent implosion problem with

a single, expected-value scenario. In this experiment, the implosion problem is solved

for fixed supply (i.e., the component supply is equal to the supply request generated by

the explosion problem). At the end of second stage, we use the explosion and implosion

solutions obtained in the first and second stages, respectively, to determine the product

demand and component supply to be used in the final stage of the experiment.

In the final stage of the experiment, the rationing problem is solved for all 20 periods

for each of the four different outcomes of the implosion problem.2 Additionally, for

each of the four outcomes of the implosion problem, the solution corresponding to a

first-come-first-served (FCFS) component allocation policy is evaluated. In all, eight

different 20-period availability management problems are solved, corresponding to the

eight different experimental treatments. These treatments are named Mar-3a, Mar-3b,

2As noted in Chapter 5, in a multi-period setting, the single-period rationing problem is solved foreach period in chronological sequence, where any backorders and inventory remaining at the end of aprevious period are carried over into the next period. In a subsequent period, the oldest backorders areassumed to arrive earliest in the order sequence.

141

Mar-3c, Mar-3d, Mar-3e, Mar-3f, Mar-3g and Mar-3h.

An illustration of the design of this experiment is provided by the tree-like diagram

in Figure 6.1. In this diagram, the three stages of the experiment (explosion, implosion

and rationing) are shown as separate levels in the tree. The differently patterned ar-

rows that connect one stage to the next indicate the solution method employed within

a stage. Finally, the ‘leaves’ of the ‘tree’ contain the name of each experimental treat-

ment. According to this diagram, in the treatment Mar-3b, both explosion and implosion

are solved stochastically and no rationing of components is performed. As another ex-

ample, the treatment Mar-3c solves the availability management problem by solving a

single scenario expected-value explosion problem, a stochastic implosion problem and

the threshold rationing problem. Treatment Mar-3h is considered to the base treatment

in which the expected-value problem is solved for both explosion and implosion and no

rationing is performed.

The revenue and cost parameters used in the experiment are the same as those that

have been consistently used thoughout the previous chapters. The final performance of

each treatment is characterized by the expected profit, revenue, costs and fill rate, evalu-

ated over 2000 scenarios of randomly generated order sequences and order configurations.

The generation of these scenarios is performed using Algorithm 10 and Algorithm 11 in

Chapter 5. When evaluating each scenario, the component supply is given by the solution

to the explosion problem (with non-integer quantities rounded up to the nearest integer),

the product demand is given by the solution to the implosion problem (with non-integer

quantities rounded to the nearest integer). Unit orders are considered one-by-one in

their generated sequence, and are promised according to the threshold rationing policy

determined by the solution to the component rationing model, or else according to a

first-come-first-serve policy, depending on the particular experimental treatment.

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6.3 Results

Several observations can be drawn from the results of the experiment described in the

previous section. Table 6.1 presents the expected improvements in profit, revenue, hold-

ing and ordering costs, fill rate and cumulative fill rate for the various treatments relative

to the base treatment. The expected fill rate captures the proportion of orders that are

fulfilled in the same period in which they arrived. The expected cumulative fill rate cap-

tures the proportion of all orders that are fulfilled by the end of the planning horizon.

The largest improvement in expected profit and expected revenue over the base treat-

ment is observed for treatment Mar-3a, at 8.37% and 8.57%, respectively. In treatment

Mar-3a, the explosion and implosion problem are solved stochastically and a threshold

rationing policy is utilized. Treatment Mar-3b differs from Mar-3a in that no rationing

is performed (i.e., orders are fulfilled in a FCFS fashion). This results in better fill rates

but lower profit despite improved holding costs. This is because higher-revenue orders

are not given preference over lower-revenue orders in treatment Mar-3b. To more clearly

observe the separate effects of the experimental factors, the following comparisons with

the base treatment are made: Mar-3d and Mar-3h are compared to observe the effect of

utilizing the stochastic explosion model, Mar-3f and Mar-3h are compared to observe the

effect of utilizing the stochastic implosion model, and Mar-3g and Mar-3h are compared

to observe the effect of utilizing a threshold rationing policy.

Figure 6.2 shows the cumulative commitment-to-sales (aggregated over all products)

and expected cumulative number of orders filled under treatments Mar-3d and Mar-

3h, over the planning horizon. It is observed that while the commitment-to-sales is

similar in in both cases, a significantly greater amount of orders are filled in treatment

Mar-3d, resulting in improved revenue and a much improved expected fill rate. This is

because the stochastic explosion solution is able to account for the uncertainty in order

configurations. While this results in higher ordering and holding costs, profit is still

143

improved over the base treatment.

Figure 6.3 shows the commitment-to-sales (aggregated over all products) and ex-

pected fill rate under treatments Mar-3f and Mar-3h, over the planning horizon. It

is observed that, given the same component supply (i.e., the same explosion problem

solution), the stochastic implosion solution provides a more conservative commitment-

to-sales, resulting in a 51.80% improvement in expected fill rate. This is because it is

able to better account for order configuration uncertainty. Expected revenue improves

by 2.84% under treatment Mar-3f over the base treatment, but expected profit declines

due to a doubling in inventory holding costs.

Figure 6.4 and Figure 6.5 illustrate the benefit of rationing in treatment Mar-3g

relative to the base treatment, Mar-3h. In both figures, the horizontal axis categorizes

the 15 different products included in problem set Mar-3, sorted (from left to right) in

order of descending revenue. Figure 6.4 shows difference in the expected volume of

each product that is fulfilled under the two treatments, where a positive difference for

a given product indicates that a larger expected volume of that product is fulfilled in

treatment Mar-3g than in the base treatment. It is observed that in the treatment in

which threshold rationing is performed (Mar-3g), more high revenue product orders are

fufilled and less lower revenue product orders are fulfilled. It appears from Figure 6.4

that, in the treatment with threshold rationing, only a few high revenue product orders

are fulfilled at the expense of many more lower revenue product orders. However, Figure

6.5 provides another perspective of the allocation differences between the two treatments.

Figure 6.5 shows the difference in relative expected revenue earned from the different

products under the different treatments. It is observed from this figure that the revenue

associated with the products ranked first and second in terms of revenue is significantly

higher than the revenue associated with the products which are rationed for the sake

of the higher revenue product orders. This explains the 2.84% increase in revenue in

treatment Mar-3g over the base treatment. Since holding costs are similar in both cases,

144

treatment Mar-3g also results in a 3.53% expected increase in profit.

A similar analysis was run for the remaining problem sets. These results are pre-

sented in Table 6.2 through Table 6.10. The naming conventions for the different cases

studied for a given problem set are analogous to those used for the Mar-3 problem set,

as illustrated in Figure 6.1. Overall, the results generated for the other problem sets

display similar characteristics to those of the Mar-3 problem set.

If one could choose to implement either the stochastic explosion, stochastic implosion

or threshold rationing model, the results of this computational analysis seem to suggest

that the stochastic explosion model may provide the greatest benefit. However, the

implementation of the threshold rationing policy may be the most economical, both in

terms of computation time and implementation cost. At the same time, the concept of

rationing requires that the company be willing to potentially turn away lower priority

orders. These are the sorts of issues that need to be dealt with by decision-makers.

The objective of this thesis is to provide viable optimization models for availability

management in a configure-to-order supply chain and to provide insight to managers as

to the benefits of the stochastic explosion and implosion models over their deterministic

counterparts, and the benefits of rationing components to optimize revenue.

6.4 Summary

In this chapter, the explosion, implosion and component rationing problems are inte-

grated to address the availability management problem. Since the explosion, implosion

and rationing models need not all be implemented to solve availability management, a

simple experiment is run to examine the expected benefits of each of each of the models

with respect to availability management as a whole. Performance of the models was

evaluated by applying a given allocation policy to independently generated sequences of

145

individual orders, where the quantity of orders and the quantity of components in each

period were determined by solutions to the explosion and implosion problems, respec-

tively. It is shown that the utilization of the stochastic explosion model has the primary

effect of improving inventory levels to support a given demand plan with high service

levels (i.e., fill rates) so that higher profit and revenue are achieved. Meanwhile, for a

given supply of components, the stochastic implosion model has the primary effect of

better aligning the commitment-to-sales with the available supply so that good service

levels are achieved, perhaps at the expense of revenue. Finally, implementing a thresh-

old component rationing policy for a given component supply and commitment-to-sales

results in improved revenue earnings. These computational findings complement those

of the previous three chapters.

146

Figure 6.1. Illustration of Experimental Design

Mar-3a

Mar-3b

Mar-3c

Mar-3d

Mar-3e

Mar-3f

Mar-3g

Mar-3h

Stochastic Explosion

Expected Value Explosion

Stochastic Implosion

Expected Value Implosion

Threshold Rationing

No Rationing (FCFS)

LEGEND:

Explosion Implosion Rationing

147

Table 6.1. Comparison of Improvements in Expected Performance of Stochastic Explosion, Stochastic Implosion and Threshold Rationingfor the Mar-3 Problem Set Relative to the Mar-3h Case

Mar-3a Mar-3b Mar-3c Mar-3d Mar-3e Mar-3f Mar-3g Mar-3h

Profit 8.37% 8.27% 8.35% 8.24% -23.71% -23.71% 3.53% 0.00%Revenue 8.57% 8.49% 8.56% 8.47% -18.35% 2.84% 2.84% 0.00%Holding Cost -22.72% -22.08% -23.64% -22.87% -201.44% -201.43% 0.64% 0.00%Ordering Cost -9.16% -9.16% -9.16% -9.16% 0.00% 0.00% 0.00% 0.00%Cumulative Fill Rate 4.41% 4.46% 4.41% 4.46% 4.44% 4.44% -1.70% 0.00%Fill Rate 40.55% 46.34% 41.18% 46.11% 50.44% 51.80% -5.39% 0.00%

Notes E, I, R E, I E, R E I, R I R –

Explanation of Notes:

E - Stochastic Explosion UtilizedI - Stochastic Implosion UtilizedR - Threshold Rationing Utilized

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Figure 6.2. Cumulative Commitment-To-Sales and Expected Cumulative Number of OrdersFulfilled for Treatment Mar-3d and Base Treatment Mar-3h

0

200

400

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1200

1400

1600

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Period

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mb

er o

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rder

s

Mar-3d CTS Mar-3h CTS Mar-3d Orders Filled Mar-3h Orders Filled

149

Figure 6.3. Commitment-To-Sales and Expected Fill Rate for Treatment Mar-3f and BaseTreament Mar-3h

0

50

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

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Figure 6.4. Difference Between the Expected Volume of Each Product that is Fulfilled UnderTreatment Mar-3g and Mar-3h

4.649

3.282

0.376

-7.7125

-1.834

-0.0345 -0.3325

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Figure 6.5. Difference Between the Expected Revenue Earned from Each Product Under Treat-ment Mar-3g and Mar-3h

14849

4877

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Profit 8.20% 8.18% 8.26% 8.24% -46.53% -36.53% 1.84% 0.00%Revenue 8.59% 8.58% 8.64% 8.62% -32.65% -32.64% 1.33% 0.00%Holding Cost -8.34% -7.64% -8.17% -7.59% -162.53% -162.43% -1.74% 0.00%Ordering Cost -9.64% -9.64% -9.64% -9.64% 0.00% 0.00% 0.00% 0.00%Cumulative Fill Rate 4.11% 4.16% 4.15% 4.18% 3.92% 3.93% -0.26% 0.00%Fill Rate 67.23% 68.13% 67.76% 68.52% 60.68% 60.78% 2.63% 0.00%




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Profit 1.93% 1.88% 1.91% 1.85% -59.44% -59.44% 1.10% 0.00%Revenue 3.75% 3.72% 3.74% 3.71% -31.07% -31.07% 0.61% 0.00%Holding Cost -26.09% -25.97% -26.12% -26.04% -509.64% -509.63% 2.46% 0.00%Ordering Cost -5.90% -5.90% -5.90% -5.90% 0.00% 0.00% 0.00% 0.00%Cumulative Fill Rate 3.35% 3.35% 3.34% 3.34% 3.30% 3.30% 0.11% 0.00%Fill Rate 36.73% 35.90% 36.84% 36.00% 31.90% 32.00% -1.51% 0.00%




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Table 6.4. Comparison of Improvements in Expected Performance of Stochastic Explosion, Stochastic Implosion and Threshold Rationingfor the May-1 Problem Set Relative to the May-1h Case

May-1a May-1b May-1c May-1d May-1e May-1f May-1g May-1h

Profit 6.99% 6.21% 7.11% 6.53% -26.85% -26.86% 2.49% 0.00%Revenue 7.76% 7.15% 7.84% 7.38% -20.87% -20.88% 2.02% 0.00%Holding Cost -35.44% -40.03% -33.80% -34.18% -187.17% -187.01% -4.51% 0.00%Ordering Cost -10.41% -10.41% -10.41% -10.41% 0.00% 0.00% 0.00% 0.00%Cumulative Fill Rate -8.28% 2.26% -12.24% -2.46% 3.92% 3.94% -1.89% 0.00%Fill Rate 33.65% 42.69% 67.28% 71.93% 78.26% 83.83% -9.73% 0.00%




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Table 6.5. Comparison of Improvements in Expected Performance of Stochastic Explosion, Stochastic Implosion and Threshold Rationingfor the May-2 Problem Set Relative to the May-2h Case

May-2a May-2b May-2c May-2d May-2e May-2f May-2g May-2h

Profit 3.63% 3.16% 3.60% 3.20% -2.49% -3.09% 0.53% 0.00%Revenue 3.75% 3.43% 3.74% 3.46% -1.67% -2.09% 0.37% 0.00%Holding Cost -4.96% -6.17% -7.80% -8.64% -22.98% -24.98% 0.50% 0.00%Ordering Cost -4.02% -4.02% -4.02% -4.02% 0.00% 0.00% 0.00% 0.00%Cumulative Fill Rate -3.73% -3.30% -3.74% -3.30% -1.54% 0.05% -1.67% 0.00%Fill Rate -11.22% -10.84% 1.51% -0.29% -22.23% -10.09% -7.95% 0.00%




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Table 6.6. Comparison of Improvements in Expected Performance of Stochastic Explosion, Stochastic Implosion and Threshold Rationingfor the Jul-1 Problem Set Relative to the Jul-1h Case

Jul-1a Jul-1b Jul-1c Jul-1d Jul-1e Jul-1f Jul-1g Jul-1h

Profit 11.18% 10.92% 11.14% 10.88% -42.94% -42.97% 1.58% 0.00%Revenue 7.21% 7.05% 7.18% 7.02% -25.09% -25.11% 0.97% 0.00%Holding Cost 22.03% 21.50% 22.39% 21.88% -106.45% -106.43% -0.06% 0.00%Ordering Cost -1.62% -1.62% -1.62% -1.62% 0.00% 0.00% 0.00% 0.00%Cumulative Fill Rate 2.96% 3.36% 3.02% 3.33% 3.38% 3.38% -0.19% 0.00%Fill Rate 26.67% 30.86% 34.95% 38.46% 56.30% 56.54% 0.06% 0.00%




157



Profit 0.65% 0.65% 0.65% 0.64% -57.19% -57.19% 0.15% 0.00%Revenue 3.02% 3.02% 3.03% 3.03% -36.30% -36.30% 0.10% 0.00%Holding Cost -54.20% -53.76% -56.02% -55.99% -407.82% -407.82% 0.31% 0.00%Ordering Cost -7.14% -7.14% -7.14% -7.14% 0.00% 0.00% 0.00% 0.00%Cumulative Fill Rate 2.47% 2.47% 2.47% 2.47% 2.50% 2.50% 0.01% 0.00%Fill Rate 27.13% 27.08% 28.72% 28.61% 29.27% 29.28% 0.36% 0.00%




158



Profit 2.38% 2.83% 2.36% 2.36% -18.02% -18.02% 0.04% 0.00%Revenue 3.17% 3.17% 3.16% 3.16% -7.85% -7.85% 0.02% 0.00%Holding Cost -21.20% -21.19% -21.35% -21.34% -131.22% -131.14% -1.11% 0.00%Ordering Cost -3.75% -3.75% -3.75% -3.75% 0.00% 0.00% 0.00% 0.00%Cumulative Fill Rate 2.96% 2.96% 2.95% 2.95% 3.02% 3.03% -0.08% 0.00%Fill Rate 53.07% 53.03% 52.84% 52.74% 24.87% 24.55% 0.07% 0.00%




159



Profit 0.57% -0.79% 0.51% -0.44% 0.21% -0.49% 0.51% 0.00%Revenue 0.79% -0.23% 0.76% 0.03% 0.15% -0.38% 0.39% 0.00%Holding Cost 5.89% -11.27% -2.52% -2.44% 9.31% 1.88% 3.09% 0.00%Ordering Cost -1.56% -1.56% -1.56% -1.56% 0.00% 0.00% 0.00% 0.00%Cumulative Fill Rate -4.46% -2.30% -3.72% -2.08% -0.90% 0.30% -1.14% 0.00%Fill Rate -26.92% -22.36% -8.33% -1.11% -28.64% -19.85% -7.94% 0.00%




160



Profit 8.01% 7.93% 5.16% 5.12% -19.21% -19.24% 0.30% 0.00%Revenue 8.73% 8.67% 6.77% 6.73% -14.31% -14.33% 0.23% 0.00%Holding Cost -18.12% -18.26% -48.90% -49.07% -65.83% -65.63% -0.22% 0.00%Ordering Cost -10.91% -10.91% -10.91% -10.91% 0.00% 0.00% 0.00% 0.00%Cumulative Fill Rate 5.30% 5.31% 5.37% 5.38% 5.30% 5.30% -0.01% 0.00%Fill Rate 15.75% 20.31% 25.14% 27.28% 28.81% 30.86% -1.06% 0.00%




Chapter 7

Validation and Verification of

Models

7.1 Introduction

While the methods and models presented in this thesis are general to sales and opera-

tions planning for configure-to-order systems, their development has been guided by the

insights provided by supply chain planners at IBM, and by the data structures used by

IBM for availability management. Therefore, the validity and credibility of these models

are measured against the data and feedback provided by IBM. According to Law and

Kelton (2000), the validity of a model refers to how true the model is to the system that

it is supposed to represent. Meanwhile, the credibility of a model refers to the degree

to which the model is accepted its end-users (e.g., managers, decision makers, clients).

This chapter discusses the measures taken to improve the validity and credibility of the

models developed in this thesis. Methods for verifying the correctness of the computer

implementations of the models are also discussed.

162

7.2 Building Valid and Credible Models

The inspiration behind the models developed in this thesis, and the sales and operations

planning (S&OP) process in which they are positioned, came from weeks of discussions

with several key decision makers (including managers, and developers and users of the

current existing availability management tools) at IBM Research and IBM Integrated

Supply Chain. From these discussions, it was apparent that while the current ana-

lytic tools being used at IBM treat order configurations as deterministic, actual order

configurations may be quite variable. It was agreed that it would be interesting to

IBM to explore methods that could incorporate the uncertainty associated with order

configurations. Therefore, the models developed in this thesis have a pointed focus on

order configuration uncertainty. For the 2004-2005 academic year, this research was fully

funded by IBM Integrated Supply Chain.

7.2.1 Regular Interaction with Subject-Matter Experts

Beginning in late 2003, regular meetings with key personnel at IBM were held during the

conceptual stages of the explosion, implosion and rationing models. This was done to

ensure that the basic model objectives, assumptions and data requirements were realistic.

In early 2004, a detailed presentation of the key premises, assumptions and objectives

of the proposed models was made in Fishkill, NY, to several IBM Integrated Supply

Chain personnel whose primary job responsibilities relate to availability management for

the Systems and Technology Group. This presentation was well-received and provided

positive feedback on the validity and credibility of the explosion, implosion and rationing

models. Since then, weekly meetings were held with a key subject-matter expert at IBM

Research in order to validate the progress made during the developmental stages of

the proposed models. The final results reported in this thesis were presented to IBM

subject-matter experts for final validation.

163

7.2.2 Determining the Level of Model Detail

Since models cannot capture all the intricacies of the actual system being modeled,

careful decisions need to be made regarding which aspects of the system will be accounted

for by the model. In this case, given that the focus of this research was to examine

models for dealing with order configuration uncertainty, it was agreed that assuming a

bi-level bill-of-materials consisting only of products and the modular components used

to configure them was an appropriate simplification of the product structure. This is

because the uncertainty in order configurations exists between the first and second levels

of the actual bill-of-materials, with the bill-of-materials below the level of the configurable

components being essentially fixed. Additionally, for the new line of x-series servers, the

actual bill-of-materials consists of just two-levels, since the supply of the configurable

components are outsourced.

Another decision made to simplify the model was in regards to the source of compo-

nent supply. In reality, the supply of a given part may be provided by more than one

supplier. However, our models assume that each part is supplied by a single ’world-wide’

supplier. Leaving out this detail in the models studied here was deemed acceptable

by the IBM personnel for two main reasons. First, the model only considers a bi-level

bill-of-materials, while parts are typically procured at the bottom of a multi-level bill

of materials. Therefore, it makes sense to assume a single source of supply since there

would not exist any real data to support a model with multiple sources of supply at the

second level in the bill-of-materials. Second, individual supplier characteristics do not

factor into the objectives of this study except from the perspective of supplier flexibility.

Since actual data of suppliers’ flexibility is not available or well understood within IBM,

an aggregated treatment of supply when examining supply flexibility is acceptable.

164

7.2.3 Empirical Analysis of Stationarity of Configuration Uncertainty

One of the underlying assumptions made in all three models is that the distribution of the

quantity of each component in a product is stationary. From the perspective of an IBM

subject-matter expert, this assumption is reasonable based on general observations of

actual orders. Additionally, over the life of a product, the configuration of the product

is less likely to undergo the kind of life-cycle and seasonal trends experienced by the

product itself. While engineering changes may require changes in product components

over time, this is more the exception than the norm.

To test the validity of this assumption empirically, ten product-component pairs were

randomly selected from a twelve month history of orders received by the IBM Systems

and Technology Group. The age of the order history lies somewhere between March

2004 and July 2005, depending on the age of the data set being used. For each of

the ten product-component pairs, a series of box plots was created. These plots are

shown in Figure 7.1. In each of the ten sub-plots shown in this figure, a series of boxes

representing the different quantities of a given component that were ordered for a given

product in the months in which orders for the product was received. As an exception, the

right-most box in each plot represents this quantity aggregated over all of these months.

Therefore, the right-most box in each sub-plot is labeled ‘Overall’, and is darkly shaded.

The aggregated data represented in the right-most boxes are the data used to create the

empirical distributions used for scenario generation. Figure 7.1 provides a visual analysis

of the homogeneity and stationarity of the quantity of a component in a product, over

time. For some product-component pairs, less than twelve months of data are plotted.

This is because in some months no orders for a product were received.

A statistical test called Mood’s median test1 (Conover 1980) is used to perform a

non-parametric multiple comparison of the medians of the component order quantity

1Mood’s median test is a more robust version of the Kruskal-Wallis test.

165

associated with a product, for the ten product-component pairs previously mentioned.

For these tests, the null hypothesis is that the medians for all independent data sets for

a given product-component pair are equal. The alternative hypothesis is that at least

one of these medians is not equal to another. The p-values for these tests are reported

in the bottom left-hand corner of each sub-plot in Figure 7.1. The p-value represents

the probability that one would observe the given data if the null hypothesis were true.

Therefore, smaller p-values suggest that the distribution of the order quantity of a given

component in a given product are less likely to be the same for all twelve months. Six

out of the ten tests are significant at the 95% level. However, a test would be significant

even if the data from one month showed a significant difference from any one of the

other data sets. Additionally, some months experienced a very low volume of demand

for a product-component pair (i.e., a volume of less than 5). These small data sets may

reduce the accuracy of Mood’s median test. A visual analysis of the data in Figure 7.1

suggests that it is not unreasonable to merge the data over time and use the resulting

empirical distribution for scenario generation.

7.3 Verification of Computer Models

To verify the correctness of the computer implementations of the various models devel-

oped in this thesis, the models were run under simplifying assumptions for which the

solution is ‘known’. IBM’s Supply Capability Engine is the software tool currently being

used to solve the deterministic explosion problem for the Systems & Technology Group.

This tool has been used and trusted by IBM for several years. To verify the correctness

of the implementation of the explosion model, the solution generated by the model for

a single scenario, expected value problem was compared to the solution generated by

the Supply Capability Engine for the same set of input data and found to be identical.

To verify the multi-scenario capability of the stochastic explosion model, multiple iden-

166

tical scenarios (i.e., identical to the expected value scenario) were input to the model.

It was observed that the resulting solution was identical to that of the single-scenario

implementation.

Similarly, if the single-scenario, expected value implosion model with fixed supply

is provided with the single-scenario, expected value explosion solution as supply input,

then for the same demand plan, the optimal objective value for the implosion problem

should be identical to the optimal objective value for the explosion problem used to

generate the supply input. This is, in fact, what was observed. To verify the multi-

scenario capability of the stochastic implosion model, multiple identical scenarios (i.e.,

identical to the expected value scenario) were input to the model. It was observed that

the resulting solution was identical to that of the single-scenario implementation. To

support the correctness of the extended implosion model which accounts for supplier

flexibility, it was observed for all of the test problems that, under identical scenario sets,

the objective value produced by the sample average approximation method was non-

decreasing for increasing levels of flexibility. Additionally, the solution obtained when

flexibility is set to zero was observed to be identical to the solution obtained from the

implosion problem with fixed supply.

For the component rationing problem, the objective value evaluated for a given solu-

tion was found to be identical for both the stochastic integer programming formulation

of the problem (as evaluated by CPLEX) and Algorithm 82, for a given set of scenarios.

It was also observed that when the rationing threshold factors are set to be equal to one,

the resulting objective value to the component rationing problem is zero. Finally, toy

problems for which the optimal solution could be determined via inspection produced the

correct result when both the search algorithm and the stochastic integer programming

models were implemented.

2This algorithm implements the inventory rationing policy for a given threshold solution and isdocumented in Chapter 5.

167

7.4 Summary

This chapter provided a description of the various actions taken and analyses performed

to improve the validity, credibility and correctness of the explosion, implosion and ra-

tioning models developed in this thesis. The close collaboration with IBM subject-matter

experts throughout the research process was essential in this regard. As is generally

agreed in the scientific community, ‘all models are wrong, but some models are use-

ful.’ In accordance with this point of view, some simplifications of the actual IBM

system were made. However, close attention was paid as to avoid simplifications that

would signficantly impair the usefulness of these models. In the final evaluation by IBM

subject-matter experts, the research produced by this thesis was found to be both rel-

evant and promising. Further collaborations with IBM will focus on how these models

could be integrated into IBM’s sales and operations planning process.

168

Figure 7.1. Historical Order Quantities for Product-Component Pairs

Da

ta

Overall

12

11

10987654321

9

8

7

6

5

4

3

2

1

0

Product-Component Pair # 1

Da

ta

Overall

12

11

10987654321

20.0

17.5

15.0

12.5

10.0

7.5

5.0


Da

ta

Overall

12

11

10987654321

8

7

6

5

4

3

2

1

0


Da

ta

Overall

12

11

10987654321

6

5

4

3

2

1

0

Product-Configuration Pair # 5

Da

ta

Overall54321

12

10

8

6

4

2

0


Da

ta

Overall

12

11

1098764321

5

4

3

2

1

0


Da

ta

Overall

12

11

10987654321

20

15

10

5

0


Da

ta

Overall

10987654321

16

14

12

10

8

6

4

2

0


Da

ta

Overall

12

11

10987654321

4

3

2

1

0


Da

ta

Overall

12

11

10987654321

14

12

10

8

6

4

2

0


p = 0.36 p = 0.019

p = 0.064 p < 0.001

p < 0.001 p = 0.792

p < 0.001 p < 0.001

p = 0.786 p < 0.001

Chapter 8

Conclusions

The topic of this dissertation is very relevant to the kinds of challenges being faced by

managers of configure-to-order supply chains today. Though the customization of prod-

ucts is increasingly common, the use of stochastic models for dealing with the challenges

imposed on the supply chain due to order configuration uncertainty is not common, ei-

ther in practice or in the academic literature. This research provides practical models

and methods for dealing with order configuration uncertainty and illustrates their bene-

fits within an increasingly popular business process called sales and operations planning.

The fact that the models presented in this thesis were developed under the endorsement

of, and in close collaboration with, an industry partner enhances the likelihood that they

can be implemented in practice.

Although this research was performed with specific supply chain applications in mind,

models and concepts presented here are closely related to those in the growing sevices

industry. Many services, such as business process outsourcing, consulting and business

transformation services must deal with uncertainty not unlike the order configuration

uncertainty studied in this thesis. For example, the need for consultants and skilled

technicians is not known precisely prior to receiving a service request. In fact, they

170

may not be precisely known until the service is completely delivered. In this sense, the

problem of matching employees to projects (e.g., through hiring, firing, training, and

assignment) then becomes somewhat analogous to that of the availability management

problems studied in this thesis. Of course, there are many intricacies to human resource

management that would require the current models to be extended. For example, re-

sources, once used, would not leave the system, but rather, become available again for

future projects. It is apparent that human resource management could stand to bene-

fit from some of the insights gains from supply chain management. In a company like

IBM, where both goods and services are significant contributors to revenue, the research

developed here possesses much potential for dealing with a broad range of problems.

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Vita

Ching-Hua Chen-Ritzo

EDUCATION

The Pennsylvania State University, University Park, PA 2006

Ph.D. in Business Administration with a dual title in Operations Research

The Pennsylvania State University, University Park, PA 1999

M.S. in Architectural Engineering

University of North Carolina, Chapel Hill, NC 1997

B.S. in Physics (minor in Chemistry)

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