I N V E N T O R Y C O N T R O L IN T H E R E T A I L S E C T O R : A C A S E STUDY O F C A N A D I A N TIRE PACIFIC ASSOCIATES

by

BRIAN A N T H O N Y K A P A L K A

B.Sc.(C.E.), The University of Manitoba, 1992 B.Sc., The University of Manitoba, 1992

A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF

T H E REQUIREMENTS FOR T H E D E G R E E OF

MASTER OF SCIENCE (BUSINESS ADMINISTRATION)

in

T H E F A C U L T Y OF G R A D U A T E STUDIES

(Department of Commerce and Business Administration)

We accept this thesis as conforming to the required standard

T H E UNIVERSITY OF BRITISH COLUMBIA

April 1995

Brian Anthony Kapalka, 1995

In presenting t h i s thesis i n p a r t i a l f u l f i l l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by h i s or her representatives. I t i s understood that copying or pub l i c a t i o n of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission.

Department of Cotv^errg- ftni)> ^osmecs AAnmn\a-WoA\or' The University of B r i t i s h Columbia Vancouver, Canada

Date ZS A y r A \ 9 9 5

Abstract

Canadian Tire Pacific Associates owns and operates 21 retail stores in the lower

mainland of British Columbia and a central warehouse in Burnaby. In this thesis, we formulate

a single-product, single-location model of its inventory system as a first step in developing an

integrated, interactive inventory control system. Specifically, we formulate a Markov chain

model for a periodic review system with a deterministic lead time and lost sales. The model

utilizes empirical demand data to calculate the long-run average cost of inventory for a given

(s,S) policy. We then develop a heuristic that locates a "near" optimal policy quickly. The

heuristic incorporates a constraint on the customer service level, makes use of an updating

technique for the transition probability matrix, and is based on assumptions regarding the

properties of the solution space. Next, we create a prototype of the interface that enables

managers to use the model interactively. Finally, we compare the existing inventory policy to

the optimal policy for each of 420 products sold at one of the stores. This thesis finds that

Canadian Tire Pacific Associates is currently holding excessively large in-store inventory and

that it could reduce its cost of inventory by approximately 40% to 50%. We estimate that

implementing optimal inventory control in the stores would result in annual savings of between

$5.5 and $7 million.

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Table of Contents

Abstract ii

Table of Contents iii

List of Tables v

List of Figures vi

Acknowledgement vii

I. INTRODUCTION 1

II. T H E INVENTORY SYSTEM A T CANADIAN TIRE PACIFIC ASSOCIATES . . . . 8 A. Background 8 B. The Problem .11 C. The Project 15 D. The Demand Data 17 E . The Cost Data 18

III. M O D E L FORMULATION 20 A. Terminology 20 B. Assumptions 21 C. A Markov Chain Model 22 D. The Transition Probability Matrix 25 E . The Steady State Probabilities 27 F. The Cost of Inventory . 29 G. The Customer Service Level 31 H . A Methodology for Evaluating an (s,S) Policy 34

IV. A N ALGORITHM FOR OBTAINING T H E OPTIMAL (s,S) POLICY 35 A. Introduction 35 B. A Grid Search 36 C. A Technique for Updating the Transition Probability Matrix 41

A new policy (s+m,S) 41 A new policy (s,S+m) 42 The modified algorithm 43

D. A Lower Bound on S . . . 45 E . A Heuristic Search 48

V . T H E INTERFACE 52

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VI. RESULTS A N D SENSITIVITY ANALYSIS 55 A. Comparison of Current Policies to Optimal Policies 55 B. Sensitivity Analysis on the Ordering Cost and Holding Rate 62 C. Sensitivity Analysis on the Demand Probability Mass Function 65

VII. CONCLUSION 70

Afterword 73

Bibliography 74

Appendix A: A Portion of the "Sales" File for Product 200001, a 30-amp

Inline Fuse 76

Appendix B: Sample "Distribution" Files 77

Appendix C: A Proof that the Limiting Distribution is Independent of the

Initial State 79

Appendix D: A Measure for the Steady State Customer Service Level 83

Appendix E: The Calculation of the Conditional Expected Demand Not Satisfied

During a Period of T Consecutive Days 86

Appendix F: Justification of the Updating Technique for the New Policy (s+m,S) . . . . 87

Appendix G: Justification of the Updating Technique for the New Policy (s,S+m) . . . . 92

Appendix H: A Hypothetical Consultation 97

Appendix I: The Source Code for the "Interface" Module 104

Appendix J: The Source Code for the "Main" Module 119

Appendix K: Current and Optimal Policies for Product Category 20 at Store 6 131

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List o f Tables

Table 1. Location and particulars of the 21 stores 10

Table 2. Execution times of the algorithms for product 200001, a 30-amp inline fuse 39

Table 3. Execution times of the algorithms for product 206917, a 6%

solder connector 40

Table 4. Products with insufficient existing policies 56

Table 5. Products with the largest potential absolute savings 60

Table 6. A comparison of "optimal" policies to "true optimal" policies 63

Table 7. The cost of existing policies and the relative savings of the "optimal" policies under various scenarios .64

Table 8. "True optimal" and "optimal" policies for each demand scenario 66

v

List of Figures

Figure 1. A typical sample path of the process with a 4-day review period

and a 2-day lead time 24

Figure 2. The flow chart of the grid search algorithm 38

Figure 3. The flow chart of the grid search algorithm utilizing the updating

technique for the transition probability matrix 44

Figure 4. An evaluation of the lower bound, Sm i n 47

Figure 5. The flow chart of the heuristic algorithm 50

Figure 6. The distribution of savings of optimal policies in (a) dollars and

(b) percentage of current cost 59

Figure A-1. The title screen 97

Figure A-2. The main menu 98

Figure A-3. Calculating the optimal policy and evaluating the current policy 99

Figure A-4. Displaying the results 100

Figure A-5. Entering an alternate policy 101

Figure A-6. The particulars of the alternate policy 102

Figure A-7. The main menu revisited 103

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Acknowledgement

The completion of this thesis was made possible by the encouragement and assistance

of a number of people.

I would like to express my sincere appreciation to my thesis supervisor, Professor

Martin Puterman, for all of his many efforts on my behalf. His help, advice, patience, charity,

and tolerance were very much appreciated.

I would like to acknowledge Professor Hong Chen and Professor Garland Chow for their

time and input as members of my thesis committee. In addition, I must acknowledge the

assistance of Ph.D. student Kaan Katircioglu for his insight and help on this project.

I offer many thanks to Professor Tom Ross for his kindness and friendship during the

past few years, especially during the writing of this work. I must also thank his family for

"adopting" me on many holidays. I still owe thanks to Professor Slobodan Simonovic at the

University of Manitoba for influencing me to attend graduate school in the first place and for

helping me to obtain funding.

I cannot thank my parents enough for their never-ending support, both emotional and

financial. Also, to my friends, especially Cathy, Dave, Lisa, Steve, and the "Philbuds": thanks

for giving me a life off campus and for picking up many a tab - the next one is on me.

I would like to give special thanks to my good friend and fellow M.Sc. student, Paul

Crookshanks, for allowing me to bounce ideas off of him and for being such a procrastinator

that, despite my finishing a year late, I only "lost" by two days.

Financial assistance received from the Natural Science and Engineering Research

Council (NSERC) and from Canadian Tire Pacific Associates was greatly appreciated.

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I. INTRODUCTION

The importance of inventory management has grown significantly over the years,

especially since the turn of this century. In colonial times, large inventories were viewed as

signs of wealth, and, therefore, merchants and policy makers were not overly concerned with

controlling inventory. However, during the economic collapse of the 1930s, managers began

to perceive the risks associated with holding large inventories. As a result, managers

emphasized rapid rates of inventory turnover (Silver and Peterson, 1985). Following the

Second World War, Arrow, Harris, and Marschak (1951) and Dvoretzky, Kiefer, and

Wolfowitz (1952a,b) laid the basis for future developments in mathematical inventory theory.

Shortly thereafter, new inventory control methodologies were widely applied in the private

manufacturing sector. More recently, when inflation and interest rates soared during the 1970s,

many organizations were forced to rethink their inventory strategies yet again. Today, the

control of inventory is a problem common to all org