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Demand estimation and optimal policies in lost sales inventory systems Ding, Xiaomei 2002

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D E M A N D ESTIMATION A N D O P T I M A L POLICIES IN LOST SALES I N V E N T O R Y S Y S T E M S by X I A O M E I DING B.Sc , Tianjin University, 1990 M.Sc , Tianjin University, 1992 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E OF D O C T O R OF P H I L O S O P H Y in T H E F A C U L T Y OF G R A D U A T E STUDIES Faculty of Commerce and Business Administration We accept this thesis as conforming to t^rygquired standard T H E U N I V E R S I T Y OF BRITISH C O L U M B I A January 2002 ©Xiaomei Ding, 2002 UBC Special Collections - Thesis Authorisation Form http://www.libraiy.ubc.ca/spcoll/thesauth.html In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y 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 t h e s i s for s c h o l a r l y 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 p u b l i c a t i o n of t h i s t h e s i s 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 The U n i v e r s i t y of B r i t i s h Columbia Vancouver, Canada l o f l 05/04/2002 2:27 PM A b s t r a c t In this thesis, we study the statistical issues in lost sales inventory systems, focusing on the complexity arising from the stochastic demand. We model the demand by the Zero Inflated Poisson (ZIP) distribution. The maximum likelihood estimator of the ZIP parameters taking censoring into account are derived separately for the newsvendor and the (s, S) inventory systems. We also investigate the effect of the estimation errors on the optimal policies and their costs. We observe from a simulation study that the M L E taking censoring into account performed the best in terms of cost as well as policy among various estimates. We then proceed to develop a Bayesian dynamic updating scheme of the ZIP parameters. It is applied to the newsvendor system. We perform a simulation study to investigate the advantage of the Bayesian updating approach over the traditional M L E approach. We conclude that the Bayesian approach offers a better learning technique when one lacks of good understanding of the demand pattern in the first few periods. • * Since inventory policy affects the information acquisition and-the demand distribution updating process, how to determine the optimal inventory policy when the demand distribution is yet to be learned is the focus of the latter part of the thesis. We investigate the effect of demand censoring on the optimal policy in newsvendor inventory models with general parametric demand distribution and unknown parameter values. We provide theoretical proof of the conjecture that it is better off to adopt a higher than the myopic optimal policy in the initial periods when demand is learned in a censoring system. We show that the newsvendor problem with observable lost sales reduces to a sequence of single-period problems while the newsvendor problem with unobservable lost sales requires a dynamic analysis. We explore the economic rationality for this observation and illustrate it with numerical examples. Contents Abstract • • • i a i List of Tables v i ? : List of Figures ij£ Acknowledgements x i i > : i -1 Introduction 1 1.1 Problem of the Thesis 1 1.2 Literature Review 4 1.3 Thesis Content 10 2 The Zero Inflated Poisson (ZIP) Distribution 14 2.1 Zero Inflated Poisson Density 15 2.2 Discretized Compound Poisson Process 15 2.3 ZIP Approximation to the Increments of the Discretized Compound Poisson Process 17 2.3.1 The Distribution of Z, 17 2.3.2 Discrete Time Approximation of the Compound Poisson Process 18 2.4 A Score Test for Zero Inflation in Sales Data - ZIP vs. Poisson Fi t to Data 30 2.4.1 The Data 31 2.4.2 A Score Test : 33 2.4.3 Testing Results 33 2.5 Goodness-of-fit Test for the Count Data of Daily Sales 34 2.5.1 The Goodness-of-fit Test 34 2.5.2 Test Results 35 2.5.3 Summary 36 3 M a x i m u m L i k e l i h o o d E s t i m a t i o n of Censored Zero Inflated Poisson D e m a n d 37 3.1 M a x i m u m L i k e l i h o o d E s t i m a t i o n for Uncensored Poisson and Z I P d i s t r ibu t ion 38 3.1.1 Poisson D i s t r i b u t i o n '. 39 3.1.2 Z I P D i s t r i b u t i o n 39 3.2 M L E For Z I P D e m a n d F r o m Sales D a t a 43 3.2.1 Z I P D e m a n d Censored A t a F i x e d Leve l 43 3.2.2 Z I P D e m a n d Censored A t Var iab le Levels 47 3.3 Effect of E s t i m a t i o n E r r o r on Cost and P o l i c y 52 3.3.1 Newsvendor P o l i c y and Cost 53 3.3.2 S imula t ion Design 54 3.3.3 Poisson D e m a n d 60 3.3.4 Z I P D e m a n d 74 3.3.5 S u m m a r y and Discuss ion 87 4 Bayes ian Analys i s For Z I P D e m a n d 89 4.1 Bayes ian Ana lys i s Overv iew 90 4.1.1 Bayes ian Framework 90 4.1.2 C o m p u t a t i o n a l Me thods 92 4 .2 ' Bayes ian A n a l y s i s For Poisson D i s t r i b u t i o n 95 4.2.1 Poster ior D i s t r i b u t i o n O f A W i t h Jeffreys' P r i o r 95 4.2.2 Poster ior D i s t r i b u t i o n O f A W i t h Conjugate P r i o r 95 4.3 Bayes ian Ana lys i s For the Z I P D i s t r i b u t i o n 102 4.3.1 Re-parameter iza t ion For Z I P 104 4.4 D y n a m i c Bayes ian Ana lys i s For Z I P from Sales 109 4.4.1 Posteriors of Z I P from Newsvendor Sales 109 4.4.2 Bayes ian Upda t e of the Z I P D i s t r i b u t i o n i n a Newsvendor Sys tem 110 4.4.3 S imula t ion S tudy of the D y n a m i c Newsvendor Sys tem 112 5 T h e Bayes ian Newsvendor wi th Unobservable Lost Sales 125 5.1 T h e Newsvendor Prob lems 126 5.1.1 T rad i t i ona l Newsvendor P r o b l e m 126 5.1.2 T h e Bayes ian Newsvendor P r o b l e m w i t h Observable Los t Sales 127 5.1.3 T h e Bayes ian Newsvendor P r o b l e m w i t h Unobservable Los t Sales 129 iV. 5.2 Optimal Policies for the Three Newsvendor Problems 131 5.2.1 Traditional Newsvendor Problem 131 5.2.2 The Bayesian Newsvendor Problem with Observable Lost Sales 131 5.2.3 The Bayesian Newsvendor Problem with Unobservable Lost Sales 132 5.3 The Key Result 133 5.4 Discussion of the Effect of Lost Sales on the Optimal Policy 141 5.4.1 Optimal Policy for B M D P Model 142 5.4.2 Marginal Analysis 142 5.4.3 Interpretation of the New Penalty 143 5.5 Illustrative Examples 144 5.5.1 Demand Updating and Cost Computation 145 5.5.2 Three Examples 147 5.6 Some Initial Work on N-Period Newsvendor Problem 149 5.6.1 Three Period Newsvendor with Poisson Demand 151 5.6.2 Numerical Examples 155 5.6.3 Some Discussion on Policy Cost - Observations from the Numerical Examples 177 5.6.4 The Traditional Newsvendor and the Cost 177 5.6.5 Cost Convergence - 2-Period vs 3-Period Problems 178 5.7 Conclusion 182 6 Fu tu r e Resea rch 184 6.1 Other Distribution Models to Account for Overdispersion 184 6.2 Extending the Bayesian Newsvendor Study 184 6.3 Seasonality and Trend in Demand 185 6.4 Other Factors That Affect Demand 185 6.4.1 Effect of Promotion Policies 186 6.4.2 Effect of Product Substitution 187 Bibll-ogaraphy; 188 Appendix A : Sample "Distribution" - A "Low" Demand Product 193 Appendix B : Sample "Distribution" - A "Medium" Demand Product 194 Appendix C: Sample "Distribution" - A "High" Demand Product 195 Appendix D: Sales "Distribution" for Product 200001 and Chi-sq Statistic 196 Appendix E : Sales "Distribution" for Product 200002 and Chi-sq Statistic 197 .v Appendix F: Sales "Distribution" for Product 203039 and Chi-sq Statistic 198 Appendix G: Posterior with Different Demand Observations 199 Appendix H: Marginal Analysis 200 Appendix I: Time Line of Events 201 v i List of Tables 2.1 T h e C P P increments densities versus the Z I P densities figures 20 2.2 T h e C P P increments densities versus the Z I P densities figures 20 3.1 Sample D a t a Generated F r o m ZIP(0 .7 , 5) 47 3.2 N o t a t i o n of Cos t Compar i son for Newsvendor P r o b l e m 56 3.3 Cos t St ructure and k 56 3.4 S imu la t i on Set t ing and Case Numbers for Poisson D e m a n d 60 3.5 P r o b a b i l i t y of Censor ing at y° 61 3.6 O p t i m a l P o l i c y y* 61 3.7 M e a n and S tandard Dev ia t i on of Expec t ed Costs w i t h A = 2 62 3.8 M e a n and S tandard Dev ia t i on of Expec t ed Costs w i t h A = 5 65 3.9 M e a n and S tanda rd D e v i a t i o n of E x p e c t e d Costs w i t h A = 15 66 3.10 E x p e c t e d Costs w i t h A = 5, n = 30 vs. n = 100 71 3.11 E x p e c t e d Costs w i t h A = 15, y° = 15 vs. y° = 5: 73 3.12 T h e In i t i a l Order Quant i t ies (y°) 74 3.13 M e a n and S tandard Dev ia t i on of Expec t ed Costs w i t h ZIP(0 .7 , 2) 75 3.14 M e a n and S tandard Dev ia t i on of Expec t ed Costs w i t h ZIP(0 .7 , 5) 75 3.15 M e a n and S tandard Dev ia t i on of Expec t ed Costs w i t h ZIP(0 .7 , 15) 76 3.16 E x p e c t e d Costs w i t h ZIP(0 .7 , 15), y° = 15 vs. y° = 5 ' 81 3.17 M e a n and S tandard Dev ia t i on of Expec t ed Costs w i t h Z IP (0 .3 , 5). . 84 3.18 M e a n and S tandard Dev ia t i on of Expec t ed Costs w i t h Z IP (0 .3 , 15) 84 4.1 M C M C s imula t ion results for Poisson rate w i t h G a m m a pr ior 97 4.2 M C M C s imula t ion results for Z IP (p , A) 103 4.3 M C M C s imula t ion results for Z IP (p , A) 105 V'i i 4.4 M C M C s imula t ion results for Z I P ( p , A) 107 4.5 A R a n d o m Sample of Newsvendor D e m a n d and Sales 115 4.6 D e m a n d E s t i m a t i o n from Sales 117 4.7 Inventory Costs 117 4.8 T h e D e m a n d and Sales Sample i n P e r i o d 4 119 4.9 Inventory Costs 120 4.10 D e m a n d E s t i m a t i o n from Sales; Compar i son of Bayes ian & C u m u l a t i v e M L E 121 4.11 D e m a n d E s t i m a t i o n from Sales; Compar i son of Bayes ian & C u m u l a t i v e M L E w i t h L o w F i r s t P e r i o d Order Quan t i t y 122 4.12 Inventory Costs 123 5.1 O p t i o n of Purchas ing A d d i t i o n a l Informat ion 143 5.2 Pol ic ies and their costs i n E x a m p l e 1. Observe that the B N po l icy is o p t i m a l i n per iod 1. . . 148 5.3 Pol ic ies and their costs i n E x a m p l e 2. T h i s table shows that the o p t i m a l order quant i ty is 5 whi le the B N pol icy is 3 at decision epoch 1 150 5.4 Pol ic ies and their costs i n E x a m p l e 3 showing that the o p t i m a l order quant i ty is 12 while the B N po l i cy is 11 at decision epoch 1 151 5.5 Some Probabi l i t i e s of N B ( 0 . 4 , 0.9) 157 5.6 P e r i o d 2 order quantit ies and their costs (part I w i t h x\ = 0 ) . Observe that the B N pol icy is o p t i m a l 158 5.7 P e r i o d 2 order quantit ies and their costs (part II w i t h x\ = 1 ) . Observe that the B N pol icy is o p t i m a l 159 5.8 P e r i o d 2 order quanti t ies and their costs (part II of y\ = 2 w i t h x± = 1 ) . Observe that the B N po l i cy is o p t i m a l 160 5.9 P e r i o d 2 order quantit ies and their costs (part III of y\ = 2 w i t h x\ = 2 ) . Observe that the B N pol icy is o p t i m a l 161 5.10 P e r i o d 2 order quanti t ies and their costs (part III of xj\ = 3 w i t h x\ = 2 ) . Observe that the B N po l i cy is o p t i m a l 162 5.11 P e r i o d 2 order quanti t ies and their costs (part I V of y\ = 3 w i t h X\ = 3). Observe that the B N pol icy is op t ima l 163 5.12 Pol ic ies and their costs i n E x a m p l e 1 showing that the o p t i m a l order quant i ty is 2 while the B N pol icy is 1 at decision epoch 1 164 v d d i 5.13 P e r i o d 2 order quantit ies and their costs (part I w i t h x\ = 0). Observe that the B N pol icy is o p t i m a l 166 5.14 P e r i o d 2 order quanti t ies and their costs (part II w i t h x\ = 1). Observe that the B N pol icy is op t ima l 167 5.15 P e r i o d 2 order quanti t ies and their costs (part III w i t h xx = 2). Observe that the B N pol icy is o p t i m a l 168 5.16 P e r i o d 2 order quantit ies and their costs (part I V w i t h x\ = 3 ) . Observe that the B N pol icy is o p t i m a l 169 5.17 P e r i o d 2 order quantit ies and their costs (part I V of y\ = 4 w i t h x± = 3). Observe that the B N pol icy is o p t i m a l 170 5.18 P e r i o d 2 order quanti t ies and their costs (part V of i / i = 4 w i t h x\ = 4). Observe that the B N po l i cy is o p t i m a l 171 5.19 P e r i o d 2 order quantit ies and their costs (part V of y\ = 5 w i t h x\ = 4 ) . Observe that the B N pol icy is op t ima l . . . 172 5.20 P e r i o d 2 order quantit ies and their costs (part V I of y\ = 5 w i t h x\ = 5 ) . Observe that the B N pol icy is op t ima l . T h e op t ima l order quant i ty y^BN^ and its cost «3 are omi t t ed 173 5.21 P e r i o d 2 order quantit ies and their costs (part V I of yi = 6 w i t h xi = 5 ) . Observe that the B N po l i cy is o p t i m a l 174 5.22 P e r i o d 2 order quantit ies and their costs (part V I I of y\ = 6 w i t h x\ = 6). Observe that the B N po l i cy is o p t i m a l 175 5.23 Pol ic ies and their costs i n E x a m p l e 2 showing that the o p t i m a l order quant i ty is 5 while the B N pol icy is 3 at decision epoch 1 176 5.24 Per iod ic Costs - E x a m p l e 1, 2-Per iod 179 5.25 Pe r iod ic Costs - E x a m p l e 1, 3-Per iod 180 5.26 T o t a l E x p e c t e d Costs for Future - E x a m p l e 1, 2 -Per iod 180 5.27 T o t a l E x p e c t e d Costs for Future - E x a m p l e 1, 3 -Per iod 180 5.28 Pe r iod ic Costs - E x a m p l e 2, 2-Per iod 181 5.29 Per iod ic Costs - E x a m p l e 2, 3-Per iod 181 5.30 T o t a l E x p e c t e d Costs for Future - E x a m p l e 2, 2-Per iod 182 5.31 T o t a l E x p e c t e d Costs for Fu ture - E x a m p l e 2, 3-Per iod 182 List of Figures 2.1 T h e C P P increments density and the Z I P ( A , /x) density w i t h fi = 2 and A = 0.2 21 2.2 T h e C P P increments density (line) and the Z I P ( A e _ A , /x) density (circles) w i t h /J = 2 and A = 0.2 21 2.3 T h e C P P increments density (line) and the Z I P ( A , /x) density (circles) w i t h \i = 2 and A = 0.3. 22 2.4 T h e C P P increments density (line) and the Z l P ( A e _ A , /x) density (circles) w i t h fi = 2 and A = 0.3 22 2.5 T h e C P P increments density (line) and the Z I P ( A , /x) density (circles) w i t h fj, = 2 and A = 0.4. 23 2.6 T h e C P P increments density (line) and the Z I P ( A e ~ A , fx) density (circles) w i t h /x = 2 and A = 0.4 . 23 2.7 T h e C P P increments density (line) and the Z I P ( A , /x) density (circles) w i t h \x = 2 and A = 0.5. 24 2.8 T h e C P P increments density (line) and the Z I P ( A e _ A , /x) density (circles) w i t h \x = 2 and A = 0.5 24 2.9 T h e C P P increments density (line) and the Z I P ( A , /x) density (circles) w i t h \i = 1 and A = 0.2. 25 2.10 T h e C P P increments density (line) and the Z I P ( A e _ A , /x) density (circles) w i t h / i = 1 and A = 0.2 25 2.11 T h e C P P increments density (line) and the Z I P ( A , ^) density (circles) w i t h / i = 3 and A = 0.2. 26 2.12 T h e C P P increments density (line) and the Z I P ( A e ~ A , /x) density (circles) w i t h [i = 3 and A = 0.2 ; . 26 2.13 T h e C P P increments density (line) and the Z I P ( A , fi) density (circles) w i t h / i = 4 and A = 0.2. 27 2.14 T h e C P P increments density (line) and the Z I P ( A e _ A , /x) density (circles) w i t h n = 4 and A = 0.2 27 2.15 T h e C P P increments density (line) and the Z I P ( A , /x) density (circles) w i t h /x = 5 and A = 0.2. 28 2.16 T h e C P P increments density (line) and the Z I P ( A e ~ A , /x) density (circles) w i t h fi = 5 and A = 0.2 28 2.17 His tograms of Sales of P r o d u c t 203039 (168 bulb) at 21 stores 32 3.1 Censor ing of Sales in a Newsvendor System 44 3.2 Censor ing of Sales in a (s, S) Sys tem 50 3.3 T h e cdf of Poisson(5) w i t h k « 0.67 61 3.4 T h e E s t i m a t e d O p t i m a l P o l i c y for Poisson(2) w i t h k = 0.4. True O p t i m a l P o l i c y y* — 1. . . 63 3.5 T h e E s t i m a t e d O p t i m a l P o l i c y for Poisson(2) w i t h k = 0.67. True O p t i m a l P o l i c y y* = 2. . . 64 3.6 T h e E s t i m a t e d O p t i m a l P o l i c y for Poisson(2) w i t h k = 0.87. True O p t i m a l P o l i c y y* - 4. . . 64 3.7 T h e Es t imates of A for Poisson(2) 67 3.8 T h e Es t imates of A for Poisson(5) 67 3.9 T h e Es t imates of A for Poisson(15) 68 3.10 T h e E s t i m a t e d O p t i m a l P o l i c y for Poisson(5) w i t h k = 0.4. True O p t i m a l P o l i c y y* = 4. . . . 69 3.11 T h e E s t i m a t e d O p t i m a l P o l i c y for Poisson(15) w i t h k = 0.4. True O p t i m a l P o l i c y y* = 14. . 69 3.12 T h e Es t imates of A for Poisson(5) from large samples 70 3.13 T h e E s t i m a t e d O p t i m a l P o l i c y for Poisson(5) from large samples (n = 100) w i t h k = 0.4. True O p t i m a l P o l i c y y* = 4 70 3.14 T h e Es t imates of A for Poisson(15) w i t h y° = 5 72 3.15 T h e E s t i m a t e d O p t i m a l P o l i c y for Poisson(15) w i t h y° = 5 (k = 0.4) 72 3.16 T h e Es t imates of A for ZIP(0.7,2) w i t h y° = 2 77 3.17 T h e Es t imates of p for ZIP(0.7,2) w i t h y° = 2 77 3.18 T h e Es t imates of A for ZIP(0.7,5) w i t h y° = 4 78 3.19 T h e Es t imates of p for ZIP(0.7,5) w i t h y° = 4 78 3.20 T h e Es t imates of A for ZIP(0.7,15) w i t h y° = \ \ 79 3.21 T h e Es t imates of p for ZIP(0.7,15) w i t h y° = 11 79 3.22 T h e Es t imates of A for ZIP(0.7,15) w i t h y° = 5 82 3.23 T h e Es t imates of p for ZIP(0.7,15) w i t h y° = 5 82 3.24 T h e E s t i m a t e d O p t i m a l P o l i c y for ZIP(0.7,15) w i t h y° = 5 (fc=0.87). . 83 3.25 T h e Es t imates of A for ZIP(0.3,5) w i t h y° = 2 : ' 85 3.26 T h e Es t imates of p for ZIP(0.3,5) w i t h y° = 2 85 3.27 T h e Es t imates of A for ZIP(0.3,15) w i t h y° = 5. . . 86 3.28 T h e Es t imates of p for ZIP(0.3,15) w i t h y° = 5 86 4.1 Poster ior Sample of A w i t h 7 = 1 98 X.i 4.2 Poster ior Sample of c w i t h 7 = 1 98 4.3 Poster ior Sample of A w i t h 7 = 5 99 4.4 Poster ior Sample of c w i t h 7 = 5 99 4.5 Poster ior Sample of A w i t h c = 5 100 4.6 Poster ior Sample of 7 w i t h c = 5 100 4.7 Pos ter ior Sample of A w i t h c = 10 101 4.8 Poster ior Sample of 7 w i t h c = 10 101 4.9 Poster ior Sample of A 105 4.10 Poster ior Sample of p 106 4.11 Poster ior Sample of 7 106 4.12 Poster ior Sample of A 107 4.13 Pos ter ior Sample of p 108 4.14 Pos ter ior Sample of 7 108 4.15 A Met ropo l i s Sample of the Z I P parameter p 116 4.16 A Met ropo l i s Sample of the Z I P parameter A 116 4.17 Inventory Costs C o m p u t e d Us ing Var ious Policies 118 4.18 Inventory Costs C o m p u t e d Us ing Var ious Policies 119 4.19 Inventory Costs C o m p u t e d Us ing Var ious Policies 122 x ; i i A c k n o w l e d g e m e n t s I would l ike to express my sincere grat i tude to my research supervisor, D r . M a r t i n L . Pu te rman , for his constant guidance and di rect ion throughout the selection of the research topic , the course of the research and the comple t ion of this thesis. l a m also grateful to my committee for their advice and helpful comments. F ina l l y , I would l ike to thank my husband and my parents for their continuous encouragement and devoted support du r ing the entire program. x i i i Chapter 1 Introduction Inventory management plays an impor tan t role i n the compet i t ive re ta i l industry . Efficient inventory control is a key to success. In recent decades, theories of inventory management have been well developed, and wide ly put into practice. However, some prac t ica l issues have not received due considerat ion. One of these is demand es t imat ion and its effect on op t ima l policies i n inventory systems. T h i s is especially true in lost sales inventory systems where demand can not be completely observed. A n o t h e r chal lenging issue is how to determine the inventory po l i cy when the demand d i s t r ibu t ion is unknown and learning is needed. 1.1 Problem of the Thesis In this thesis, we w i l l s tudy some stat is t ical issues i n retai l inventory management, focusing on the complexi ty ar is ing from the stochastic demand. In stochastic demand inventory systems, o p t i m a l policies depend on the demand d i s t r ibu t ion which is commonly assumed to be known . However , i n pract ice, the demand d is t r ibu t ion is rarely known, and has to be est imated. T h e most common approach is to observe some data, then estimate the unknown parameters of the demand d i s t r ibu t ion by, for example , m a x i m u m l ike l ihood es t imat ion ( M L E ) . T h i s prac t ica l recipe may not be op t ima l though. Nevertheless, to implement a policy, one must do demand es t imat ion either i m p l i c i t l y or expl ic i t ly . • T h i s research is mot iva ted by some serious problems facing re ta i l inventory managers. M a n y inventory control models are established based on assumptions that do not app ly to the retai l industry. For instance, it is often assumed that unmet demand is backordered and hence the demand is complete ly observable. Yet in re ta i l ing, unmet demand is often lost and unobserved even when backorder ing is al lowed. A l s o , the normal d i s t r ibu t ion is often assumed to describe the demand pat tern, wh ich is not t rue for many products , especially 1 those products w i t h sporadic demand. So the first problem that the retai l inventory managers face is how to estimate the u n k n o w n demand d is t r ibu t ion when the demand can not be complete ly observed, for instance, when there are stock-outs. Stock-outs refer to the case when the beginning inventory is sold out by the end of the per iod and hence the ending inventory is zero. Los t sales due to stock-outs is a b ig cost for retailers, and has been the subject of considerable research. In the case of stochastic demand, dynamic p rogramming is most often used to develop the op t ima l policies of when to order and how much to order after the demand d i s t r ibu t ion is obta ined. M o s t often, sales da ta is used i n es t imat ing the demand a l though this is not ideal W h e n the inventory is main ta ined at a sufficiently h igh level that it always provides enough for sale, sales are equivalent to demand. B u t this is usual ly not the case. Sales do not represent demand when stock-outs occur. Stock-outs usual ly indicate unobservable lost sales since some customers might go elsewhere to purchase the product or subst i tute it w i t h other products . T h i s might happen even at those re ta i l stores which backorder or issue r a in checks. In this case, the demand is at least as great as sales. Ano the r s i tua t ion i n which sales do not accurately reflect demand is when there is b r and or product subs t i tu t ion . W h e n i t em A is out of stock, subs t i tu t ion w i t h i t em B which is in stock increases the sales of i t em B . Hence sales of i t em B does not reflect its true demand. In these two cases when the sales and the demand are not equivalent, naively t reat ing sales as demand is no longer val id and results i n biased es t imat ion. We w i l l focus on the lost sales case i n this thesis. D e m a n d es t imat ion also depends on the inventory pol icy used since under different inventory policies there may be different stock-outs levels. W h e n the inventory pol icy i n use mainta ins low inventory levels stock-outs w i l l be prevalent. Obv ious ly stock-outs reduce the informat ion about the demand. Therefore, how to estimate the demand d i s t r ibu t ion using sales data , is the first question this thesis addresses. W e develop some parametr ic es t imat ion methods assuming the demand follows a specified p robab i l i ty d i s t r ibu t ion . We also assume that the parametr ic form of the density remains the same over t ime and the sales i n each per iod are independent. Sales da ta usual ly appears as count da ta which should be fit by a discrete d i s t r ibu t ion . T h e Poisson is one discrete d i s t r ibu t ion that is often used to fit count data . However, for products w i t h sporadic demand which have many periods of zero sales, overdispersion is evident i n sales da ta and has to be modeled. In da ta on da i ly sales obta ined from Canad i an T i r e Pacif ic Associates ( C T P A ) , there are many more zero observations than expected from a Poisson d is t r ibu t ion . T h i s mot iva ted us to use the zero inflated Poisson (ZIP) d i s t r ibu t ion . W e also found that the zero inflated Poisson d i s t r ibu t ion approximates the d is t r ibut ion of incrementa l j umps in uni t t ime intervals from a compound Poisson process when the a r r iva l rate is low. A s s u m i n g the demand follows a zero inflated Poisson d i s t r ibu t ion w i t h u n k n o w n parameters, we apply 2 m a x i m u m l ike l ihood es t imat ion and a Bayes ian approach for es t imat ion of the parameters, t ak ing lost sales into account. We consider inventory systems where the demand is t runcated at variable levels as well as at fixed levels. We are also interested i n the performance of these estimators w i t h respect to inventory cost since this addresses the question of whether the retailer should record the lost sales. W e w i l l see that considering lost sales i n es t imat ion greatly complicates the estimators and hence requires add i t iona l computa t iona l effort. T h i s na tura l ly raises another question. H o w would the inventory pol icy and cost respond i f we do not consider lost sales i n demand est imation? Therefore, fol lowing the demand es t imat ion, we need to investigate the impact of es t imat ion errors on the inventory policies and their costs. W h e n the demand d i s t r ibu t ion is unknown, one has to learn about i t th rough informat ion acquisi t ion. A s more da t a is obta ined over t ime, the demand d is t r ibu t ion es t imat ion should be updated , t ak ing advantage of the add i t iona l informat ion . T h e demand learning process is c r i t i ca l i n the inventory management process since the inventory pol icy should be adjusted accordingly based on the accumula ted knowledge. Fo l lowing this t r a in of thought, the first question we asked ourselves here is what is an efficient process of demand pa t te rn learning. T h a t is, how can one learn about the demand qu ick ly and adjust the estimates in the r ight d i rec t ion when new da t a becomes available. T h i s is a chal lenging issue since the demand informat ion acquis i t ion process interacts w i t h the inventory pol icy upda t ing process. Stock-outs reduce the informat ion about the demand, and ignor ing the da ta when stock-outs occurred or using i t wi thou t no t ing it was censored w i l l result i n bias i n demand es t imat ion. T h e downward biased estimate of the demand d i s t r ibu t ion means w i l l lead to ho ld ing less inventory and more lost sales. Hence, not consider ing lost sales results i n an inefficient demand learning process. Therefore, just as the demand learning process affects the inventory upda t i ng process, the inventory upda t ing process affects the demand learning process i n a lost sales inventory system since the inventory level determines at what level lost sales may occur. Unobservable lost sales can be considered as loss of in format ion from the demand es t imat ion perspective. T h i s loss of informat ion essentially penalizes the inventory management process i n the long run if not addressed correct ly and t imely. W h e n this loss of informat ion is inevi table , we need to unders tand how to min imize the effect on the inventory cost. Th i s requires considerat ion of the s tat is t ical issues of demand es t imat ion i n the context of op t imiza t ion . We show i n Chap te r 5 that informat ion brings i n value. Ineffective and inefficient u t i l i za t ion of informat ion results i n an inventory cost increase. 3 1.2 Literature Review In the enormous inventory control l i terature, few papers have addressed the effect of unobservable lost sales i n inventory systems w i t h unknown demand d is t r ibu t ion . There are numerous papers on op t ima l policies i n lost sales inventory systems assuming the demand d i s t r ibu t ion is known. W h e n the demand d is t r ibut ion is unknown, most of the research on lost sales inventory systems has concerned demand es t imat ion but not its effect on o p t i m a l policies. Some of the early works on s ta t is t ical es t imat ion of the unknown problem parameters i n inventory control models are [4], [12], [19], [28], [29], [51], and [62]. C o n r a d [16] dis t inguished sales and demand. He investigated the effect of lost sales on Poisson demand es t imat ion, and proposed an unbiased M L E of Poisson parameter compensat ing for lost sales. M o r e recently, N a h m i a s [46] s tudied no rma l demand es t imat ion i n lost sales inventory systems and proposed a procedure of sequential upda t ing of estimates of no rma l parameters. T h e performance of different est imators is evaluated th rough extensive s imulat ions. A g r a w a l and S m i t h [1] suggested that negative b i n o m i a l d i s t r ibu t ion provides a better fit than no rma l or Poisson dis t r ibut ions to discrete re ta i l sales da t a and developed a parameter es t imat ion method for base stock inventory systems where sales are t runca ted at a constant level. However, censoring of observations at a constant level is quite restr ict ive. In this dissertat ion, we study demand es t imat ion (Chapters 3 and 4) for two inventory systems, namely, a newsvendor system where censoring occurs at the constant order quant i ty level and an (s, S) system where censoring occurs at variable inventory levels. We assume that the newsvendor order quant i ty remains the same over t ime when the m a x i m u m l ike l ihood es t imat ion method for the newsvendor system was developed (Chapter 3). However, the method developed for the (s, S) system is applicable when the censoring level is not constant but variable. We later on relaxed the constant order quant i ty assumption for the newsvendor system and considered the dynamic upda t ing of the order quant i ty based on the updated demand es t imat ion (Chapter 4). W h e n demand for a product seems to be sporadic, a conventional discrete d i s t r ibu t ion may not fit wel l . G a r d e n [24] analyzed a large collect ion of da i ly sales da ta provided by C a n a d i a n T i r e Pacif ic Associates ( C T P A ) . U s i n g a score test for overdispersion due to zero inf lat ion developed by van den Broek [59] she concluded that there were an excessive number of zero sales days i n this data . She then studied the zero inflated Poisson regression model and investigated the effects of day of the week and store loca t ion on the sales patterns. We discuss this da ta i n Chapte r 2 and extend the Z I P mode l to take into account censoring i n subsequent sections. A n o t h e r appl ica t ion of the zero inflated Poisson regression mode l is given by L a m b e r t [38]. Dean [17] discussed three score tests for hypotheses test ing of an ord inary Poisson regression mode l against three al ternat ive models that account for overdispersion. 4 O u r m a i n focus i n the first part of this thesis w i l l be es t imat ion under demand censoring. Braden a n d Fre imer [10] categorized and characterized d i s t r ibu t ion families for different censoring situations. The objective was to "provide guidance to practi t ioners in the selection of d i s t r ibu t ions" . "If a modeler feels that no member of the families we characterized is a reasonable approx imat ion , then he w i l l a lmost surely encounter serious ana ly t ica l and computa t iona l problems i f his da ta include censored observations." ([10] p.1390) T h e y reviewed the results for the exponent ia l fami ly for wh ich a fixed d imens ional sufficient s tat is t ic exists when a l l observations are exact. T h e n they characterized the d i s t r ibu t ion families for which fixed d imensional sufficient statistics of purely censored observations exist, that is the case where a l l observations are censored. W h a t is most interesting is that they denned a newsboy d i s t r ibu t ion fami ly for which there is a sufficient s tat is t ic as a function of bo th exact and censored observations. T h i s sufficient statist ic was referred to as a newsboy-sufficient statistic. T h e y also l isted a wide variety of appl icat ions where censoring occurs and the observations consist of bo th exact and censored observations. T h e y appl ied Bayes ' theorem i n der iv ing the posterior dis t r ibut ions for a rb i t ra ry priors and showed how parameters were updated for the newsboy dis t r ibut ions . T h e y recommended that one consider us ing a newsboy d i s t r ibu t ion to model a continuous r andom variable when the observations consist of bo th exact and censored observations. M o s t of their results were obta ined assuming that the r andom variable has a continuous d i s t r ibu t ion . T h e y found that much of the analysis was s t i l l appl icable when this assumpt ion of continuous dis t r ibut ions was relaxed. However , their analysis for the discrete d i s t r ibu t ion case assumed that the r a n d o m variable has a support of a finite (count) set. T h i s is restr ict ive for us since we considered the zero inflated Poisson d is t r ibut ion which has infinite support . Z I P does not have a f ixed-dimensional sufficient s tat is t ic to represent censored observations. In their paper, B r a d e n and Freimer also derived useful condit ions for stochastic dominance of the pre-dict ive cdf (cumulat ive density function) of the newsboy d i s t r ibu t ion . T h i s is pa r t i cu la r ly interesting in the context of decision m a k i n g or op t imiza t ion since the stochastic dominance proper ty can be used to obta in ana ly t i ca l results about u t i l i t y functions w i t h certain properties. Chap te r 5 of this disser tat ion is an example. W e showed the key theoret ical results based on some stochastic order ing propert ies. A demand parameter es t imat ion related work by A n u p i n d i , D a d a and G u p t a [2] came to our at tention recently. T h e y s tudied product subst i tu t ion i n a stock-out s i tua t ion for the purpose of p rov id ing category management decision support . Specifically, "the es t imat ion of stock-out based subs t i tu t ion patterns between i tems i n a category". " A n impor tan t imp l i ca t i on of the occurrence of stock-outs and consumer subs t i tu t ion is that observed 5 sales of an i t em no longer equal its core consumer demand, that is, its demand when all i tems are available. T w o kinds of distort ions occur. F i r s t , sales of items that stock-out reflect demand that is right-censored because i t can be satisfied only to the extent of available stock. Second, sales of i tems other than the one(s) that stocked out may provide inflated estimates of their core demands because of consumer subst i tut ion." ([2], p.407) T h e y developed es t imat ion methods for the unknown demand rate and subs t i tu t ion rate based on the consumer ' choice observations (that is, the consumer preferences are not observed but their choices are). T w o key assumptions of their model construct are: the r andom customer a r r iva l process is a Poisson process, the consumer ' choice process is described by a subst i tu t ion p robab i l i ty m a t r i x and these two processes are independent of each other. T h e customer purchase process is derived based on these two assumptions. T h e y considered two types of inventory t rack ing systems from which different da ta were obtained. T h e perpetual inventory system tracks the times of sales occurrence and the quanti t ies. M a x i m u m l ike l ihood estimates were derived for this system. T h e other inventory system which they focused on is used to manage vending machines. State-of-the-art systems i n retai l vending provides only per iodic da t a since i t does not track the occurrence of sales, and hence the times of stock-outs are missing. T rea t ing the stock-out times as miss ing data , they appl ied the E M a lgor i thm to obta in the M L E ' s of the demand parameters. T h e i r results showed that the demand rates est imated naively by using observed sales rates, wi thout considering product subs t i tu t ion when stock-outs occur, are biased. T h e y started w i t h a 2-product case and extended to the general /V-product set t ing. T h i s work, s imi lar to ours i n Chapte r 3, develops methods for demand es t imat ion under stock-outs. We studied a discrete demand d is t r ibu t ion which is more flexible than Poisson , namely, the zero inflated Poisson d i s t r ibu t ion . W e developed the M L E method and a Bayes ian approach for es t imat ing the demand parameters and we invest igated the stock-outs effect on the inventory policies and the po l i cy costs. W h e n the demand d i s t r ibu t ion is not known w i t h certainty, it is desirable to update the demand dis t r ibu-t ion when new da ta becomes available wi thout keeping a l l the his tor ica l da ta . T h i s is na tu ra l ly implemented i n a Bayes ian fashion. T h e fol lowing is a sample of papers that s tudied Bayes ian demand d i s t r ibu t ion up-dat ing , [14], [19], [31], [34], [51] and [63]. K a r l i n [36], Scarf [52] and Iglehart [31] s tudied dynamic inventory po l i cy upda t ing w i t h the demand density being a member of either the exponent ia l family or the range fami ly and hav ing unknown parameters. T h e y showed that an adapt ive c r i t i ca l value (or order-up-to) pol icy is o p t i m a l where the c r i t i ca l value depends on the his tory pa th th rough a sufficient s tat is t ic . Scarf [52] and Iglehart [31] discussed the asymptot ic behavior of the adaptive po l i cy as the number of observations becomes large. A z o u r y [6] extended the work to more general demand dis t r ibut ions . A z o u r y [6] also investigated the 6 effect of dynamic Bayes ian demand upda t ing on op t ima l order quanti t ies. She concluded that Bayesian demand upda t ing , compared to non-Bayesian method, yields a more flexible o p t i m a l po l icy by a l lowing the oppor tun i ty to update the order quantit ies i n future periods. V a n Hee [60] appl ied the Bayes ian approach to general M a r k o v control system when there is some' uncertainty, and prov ided details o f formula t ing the p rob lem as a Bayes ian M a r k o v Decis ion Process ( B M D P ) model . A l t h o u g h the Bayes ian approach is con-sidered more appropria te as i t provides such a framework for dynamic demand upda t ing , it is generally more difficult to solve. Lovejoy [43] showed that a simple inventory po l i cy based on a c r i t i ca l fractile can be o p t i m a l or near -opt imal i n some inventory models. He also gave two numer ica l examples to i l lustrate the performance of the simple myopic policies. However, none of the above work addressed the issue of censoring (i.e. the demand observation is t runcated at a certain level) from lost sales inventory systems and its effect on o p t i m a l policies. Because of the appeal of the Bayes ian method, we investigate the effect of unobservable lost sales on the o p t i m a l po l icy by app ly ing a Bayes ian M D P model wh ich allows demand upda t ing and po l i cy upda t ing as more informat ion is gathered. However, a serious disadvantage of Bayes ian d y n a m i c p rog ramming model is its p robabi l i ty d i s t r ibu t ion valued state variable and computa t iona l in t rac tabi l i ty . Therefore, to keep the prob lem tractable, we s tudy the newsvendor inventory system. It is a s imple system because the ident ical state at each decision epoch eases the mathemat ica l task. Nevertheless, i t is not uncommon in practice. T h e benefits of a s imple operat ional decision rule make i t a t t ract ive i n implementa t ion . It is especially appl icable to the fashion goods and h igh technology industries where the products are characterized by short life cycles and long p roduc t ion lead times. E p p e n and Iyer [21] described a related inventory problem facing merchandis ing managers i n fashion industry. T h e problem is compl ica ted by a l lowing the opt ion of d u m p i n g a por t ion of stocks before the end of the season. T h e y used a newsvendor heuris t ic to avoid the computa t iona l issues from upda t ing the unknown demand d i s t r ibu t ion i n lost sales inventory systems. T h e concern that a s ta t is t ical ly good job of es t imat ion may not guarantee an o p t i m a l inventory pol icy has received at tent ion. There are some research results wh ich show that biased es t imat ion may yie ld lower inventory costs. K a t i r c i o g l u [37] i n his P h . D . thesis s tudied a class of inventory problems w i t h demand dis t r ibut ions that belong to the location-scale family. He addressed the issue of ca lcu la t ing op t ima l inventory policies i n stochastic inventory problems, when unknown demand parameters are es t imated from a sample of demand observations. H e developed a general framework for combin ing es t imat ion and op t imiza t ion for this class of inventory problems. T h e results show that bias ing the scale parameter estimates may give better inventory policies i n terms of bo th cost m in imiza t i on and service achievement objectives. However, there has been l i t t le work on the key issue of the es t imat ion errors effect on the objective 7 function's behavior. T h e most relevant works are [28], [65], [53], [54], and [49]. A m o n g these, Si lver and R a h m a n a [54] investigated the non-symmetr ic pat tern of s ta t is t ical estimates' effect on the inventory costs. T h e y pointed out that when the demand d is t r ibu t ion was unknown and the unknown parameters were esti-mated, the s ta t is t ical estimates were often assumed to be exact when comput ing the reorder point . Obviously , the es t imat ion effect on inventory costs needs to be unders tood. T h i s answers the question of whether a good s tat is t ical estimate is most desirable for op t imiza t ion purpose. It was shown that underes t imat ing the reorder point might produce a higher cost penal ty than an equivalent overest imation of i t . Hence they explored the possibi l i ty of deliberately bias ing the selection of the reorder point upward to reduce the ex-pected cost penal ty related to s ta t is t ical ly es t imat ing the reorder point . In Chapters 3 and 4, we developed es t imat ion methods for unknown demand d i s t r ibu t ion parameters. W e further investigated the estimates' effect on bo th inventory policies and costs. O u r objective was to unders tand the impact of estimates which do not take censoring into account. Since the M L E ' s we obtained do not have a closed form, we employed s imula t ion i n this invest igat ion. Lar iv ie re and Porteus [39] among other results address a special case of our model in Chapter 5. T h e y consider a dynamic newsvendor model w i t h unobservable lost sales i n the context of the newsvendor fami ly of d is t r ibut ions derived by B r a d e n and Freimer [10]. In par t icu lar they s tudy the properties of a model w i t h an exponential demand d is t r ibu t ion w i t h unknown parameter and a g a m m a pr ior d is t r ibut ion . U s i n g the specific upda t ing properties for these dis t r ibut ions they show i n Theo rem 3 that i t is op t ima l to "s ta lk informat ion" by sett ing the reorder point higher then that specified by the myopic policy. T h e results i n Chapte r 5 extend the work of Lar iv ie re and Por teus [39] to a more general demand d i s t r ibu t ion setting. In par t icular we drop the assumption of exponent ia l demand and gamma pr ior and under m i l d assumptions on the order ing of dis t r ibut ions under the u n k n o w n parameter show that i t is o p t i m a l to "s ta lk informat ion" i n a two-per iod model . After the comple t ion of Chap te r 5, but before submission of this thesis, the work was extended to N-per iods i n D i n g , P u t e r m a n and B i s i [18]. Gav i rnen i , K a p u s c i n s k i and T a y u r study the value of informat ion i n a two-echelon capacitated system [25]. T h e y model two levels of informat ion flow between a supplier and a customer and develop recursive procedures to compute the op t ima l solutions. T h e y investigate the value of informat ion by compar ing these two models to a t rad i t iona l mode l i n which no informat ion is p rovided from the customer. Interesting insights w i t h regard to when informat ion is most beneficial is obta ined th rough a computa t iona l study. A closely related paper is that of Harpaz , Lee and W i n k l e r [27]. T h i s paper uses a Bayesian dynamic p rogramming framework to s tudy the output decisions of a compet i t ive f i rm i n a market w i t h r andom demand i n wh ich the demand d is t r ibu t ion parameters are unknown. T h e y show that " learn ing by experiment ing" 8 which is analogous to " s t a lk ing informat ion" i n the inventory set t ing is op t ima l . M o r e specifically they show that the op t im iz ing f i rm should produce more output than a firm which ignored the future value of in format ion . T h i s result is s imi lar to that in Chapte r 5 and D i n g , P u t e r m a n and B i s i [18] however the context, mode l assumptions and method of proof are quite different. A n o t h e r related work [11] appears i n the product p r ic ing l i terature. B r a d e n and O r e n studied the firm's dynamic nonlinear p r i c ing prob lem when facing a market w i t h u n k n o w n consumer demand curves. The key interplay is between pr ic ing decisions and ob ta in ing informat ion . T h e paper studies the example of a company that learned about some d is t r ibu t ion (on which the p r i c ing schedule depends) through experience i n the market . " E a c h sale gives (the company) addi t iona l informat ion about the d i s t r ibu t ion .... A missed sale is infor-mat ive as wel l , but the qual i ty of the informat ion is lower, since (the) inferences ... aren't as sharp as when i t makes a sale. B y lowering its price schedule (the company) decreases the chances of exc luding a prospective customer from m a k i n g a purchase, thereby increasing its chance of generating high-qual i ty informat ion about the demand structure i t faces." ([11], p.311) T h i s paper invest igated two types of firms: myopic and exper iment ing. T h e y bo th face a market wi th u n k n o w n stochastic demand and make inferences by observing consumpt ion decisions of their customers. T h e myopic type derives their price schedule considering only the current per iod . T h e y do not take the benefit of learning into account when determining the price schedules. However, the exper iment ing type is forward-look ing and derives their price schedule to "produce informat ion" i n the sense that the demand uncertainty can be reduced i n future. T h e y used dynamic p rogramming model to formulate the exper iment ing firm's p r i c ing problem. Bayes ' rule was used to model the firm's learning process. T h e key finding was that "the exper iment ing f i rm produces informat ion about the market by lowering the price schedule". We found that this work was close to ours despite the fact that their appl ica t ion was different. T h e s imi la r i ty lies in p rob lem formula t ion of considering informat ion acquis i t ion in the decision m a k i n g process. In Chapte r 5, we considered informat ion acquis i t ion in inventory pol icy upda t ing process. W e not on ly gave the economical in terpre ta t ion of our findings but also quantified the marg ina l value of in format ion . A most recent paper by P e t r u z z i and D a d a [47] studied an extended newsvendor p rob lem and its infor-ma t iona l dynamics . T h e extended newsvendor p rob lem includes p r i c ing decision which affects demand, it also allows for mul t ip le suppliers and pooled procurement recourses, etc. T h i s paper analyzed the problem of inventory and p r i c ing decisions i n a two-stage re ta i l setting. T h e two-stages can either refer to two con-secutive t ime per iods .or two markets w i t h non-overlapping sell ing seasons. A t the beginning of the second stage, one faces an oppor tun i ty to refine informat ion about the uncer ta in demand based on the informat ion 9 obtained i n the fist pe r iod . T h e y showed that the compl ica ted two-stage decision problem (wi th a to ta l of 7 decision variables for the most complicated case) can be reduced to determining one decision variable which was called the stocking factor. The s tocking factor is defined the same as the censoring level (or the newsvendor order quant i ty) i n our work. It is basical ly the lower bound of the demand when the demand is not completely observed. T h a t is the case when the sales is observed to be same as the beginning inventory. T h e problem was formulated as a two-stage stochastic p rogram w i t h recourse. T h e y solved the first-stage problem by m a x i m i z i n g the to ta l revenue of the l inked markets and examined the benefit of incorpora t ing the informat ion dynamics between the two markets instead of t rea t ing them as independent markets . T h e key assumption of demand is that for each market , the demand is characterized as a decreasing function / o f price w i t h an addi t ive t e rm of the unknown variable. T h i s assumption implies that price influences the expected demand but not the variance. A l t h o u g h the decision prob lem they considered is more compl ica ted than ours w i t h more decision variables, our demand assumpt ion is more general. In addi t ion this paper gives a nice review of four relevant streams of research l i terature. 1.3 Thesis Content T h i s dissertation makes three p r imary contr ibut ions. F i r s t , it develops es t imat ion methods (the M L E and the Bayes ian approach) for the zero inflated Poisson demand d i s t r ibu t ion parameters under stock-outs. Unobservable lost sales change the form of the estimates th rough their contr ibut ions to the l ike l ihood . W e observed that naively ignor ing censoring i n the sales da ta results i n downward biased estimates. D y n a m i c parameter upda t ing procedure for the Z I P parameters is developed using the Bayes ian approach. Second, the effect of censoring on the inventory policies and the pol icy costs are investigated. Resul ts from a s imula t ion s tudy supports the use of M L E ' s that take censoring into account since they y ie ld lower costs compared to those estimates that do not consider censoring. D y n a m i c order quant i ty upda t ing for the newsvendor system allows the order quant i ty to change based on the parameter es t imat ion upda t ing . A s imula t ion s tudy reveals that as the parameter upda t ing progresses the order quant i ty and the newsvendor cost bo th move toward the baseline case where the true demand d i s t r ibu t ion is known. T h i r d , l ike many theoret ical analyses, the newsvendor p rob lem study that incorporates informat ional dynamics (given i n Chapte r 5) provides more insights t han precise policies. E v e n though the numer ica l examples do not show at t ract ive cost savings, the ana ly t i ca l results serve a valuable purpose of enhancing managers ' i n tu i t ion . For instance, managers concerned w i t h test ing market a im at learning about demand, and hence they should be relieved from concerning themselves too much w i t h inventory cost m i n i m i z a t i o n i n the in i t i a l test ing periods. T h e ma in body of the dissertat ion is s t ructured as follows. In the fol lowing chapters we explore demand 10 models for inventory systems w i t h par t icular a t tent ion pa id to explor ing es t imat ion of mode l parameters and its effect on policies and costs. We start by mot iva t ing the mode l l ing of demand by the zero inflated Poisson (ZIP) d i s t r ibu t ion . In Chapte r 2 we investigate the compound Poisson process, a continuous t ime stochastic process, observed at discrete t ime points. We found that under cer ta in condi t ions, the d i s t r ibu t ion of the increments of a discret ized compound Poisson process can be approximated by the zero inflated Poisson d i s t r ibu t ion . D a t a analysis of da i ly sales provided from C a n a d i a n T i r e Pacif ic Associates serves as another mot iva t ing factor. Some pre l iminary analysis showed overdispersion i n this da ta . A s ta t is t ical test proved evidence against a Poisson fit. Hence, this dissertat ion focuses on the Z I P parametr ic mode l for the demand es t imat ion in inventory systems. Chap te r 3 studies m a x i m u m l ike l ihood es t imat ion for the Z I P parameters i n lost sales inventory systems. A n M L E is developed t ak ing lost sales into account. T h i s is an extension of Conrad ' s [16] work for the Poisson demand. T w o types of inventory systems are considered, the newsvendor and the (s, S) systems. T h e nature of censoring differs for these two systems. In the newsvendor system, demands are censored by the constant order quant i ty i n a l l stock-out periods i f the order quant i ty remains unchanged. In this system, the inventory is set at a constant level y° at the start of each p lann ing per iod . Regardless of whether there is excess inventory or a stock-out has occurred at the end of the previous per iod , a l l the demands higher than the fixed order quant i ty y° w i l l be censored as long as y° is in place. T h i s simplifies the prob lem since the da t a censoring is at the constant level. O n the contrary, the (s, S) system has variable censoring levels w i t h i n the inventory cycle. T h i s system is implemented as follows. T h e inventory level is reviewed per iodica l ly and it is compared to s. If it is lower than s, then an order to b r ing the inventory to S is placed. Otherwise, no order is placed and the inventory continues to deplete. Because of the different censoring, the estimators w i l l also differ. T h e M L E ' s of the Z I P parameters t ak ing censoring into account are derived separately for the newsvendor and the (s, S) inventory systems i n Sect ion 3.2. A numer ica l example is given for each system to i l lus t ra te the es t imat ion. In Sect ion 3.3, we investigate the effect of es t imat ion error on the o p t i m a l policies and their costs. Th i s was mot iva ted by the question of whether i t is worthwhile to consider censoring i n es t imat ion, since not considering censoring simplifies es t imat ion. W e focus on the newsvendor system because of its closed form o p t i m a l pol icy. W e perform a s imula t ion study to compare the o p t i m a l policies and the expected costs where various est imators of the unknown parameters are used. Poisson demand is also considered for comparison purposes. T h e computa t ion results are presented and summar ized for Po isson and Z I P demand separately. Brief ly , the M L E tak ing censoring into account performs the best i n terms of cost as well as po l icy among various estimates. It closely approximates the inventory cost computed based on demand da ta though it 11 also has the most compl ica ted form. Ignoring the presence of censoring i n da t a results i n downward biased parameter estimates. U s i n g these estimates generally gives lower order quanti t ies . T h i s devia t ion results in an inventory cost increase. T h e impacts of the in i t i a l order quant i ty and the sample size are investigated as wel l . W e observe that the M L E t ak ing lost sales into account sustains its performance when the in i t i a l order quant i ty is low whi le the other two estimators not considering censoring deteriorate in their cost performances. In Chap te r 4, we proceed to the Z I P parameter upda t ing by the Bayes ian method . W e further investigate the demand d i s t r ibu t ion es t imat ion prob lem from the Bayes ian perspective since the B a y s i a n approach provides a framework for dynamic upda t ing of the unknown parameters when more da ta is collected. T h e B a y s i a n method can also incorporate expert judgement into a pr ior belief about the unknown parameters. W h e n new da ta becomes available, the new da ta is used to update the pr ior to the posterior. One can use this scheme to learn about the demand pat tern over t ime and the inventory control process would be enhanced when the inventory pol icy is updated accordingly. In this chapter, Bayes ian analysis is appl ied to bo th Poisson and Z I P dis t r ibut ions . We start w i t h the case where there is no censoring and then extend results to the case where censoring occurs. A Bayes ian d y n a m i c upda t ing scheme for the Z I P parameters is developed and appl ied to the newsvendor system due to its s t ruc tura l appeal . We again conducted a s imula t ion s tudy to investigate the advantage of the Bayes ian upda t ing approach over the t r ad i t iona l M L E approach. We conclude that the Bayes ian approach offers a better learning technique when one lacks good unders tanding of the demand pat tern i n the first few periods, especially when the i n i t i a l guess is on the low side, and before there is enough da ta to correct the guess. It performs well even w i t h very l imi t ed informat ion. Fur thermore , we made a very impor tan t observation. It seems that the i n i t i a l order quant i ty plays a c ruc ia l role for the entire s tudy per iod . N o t only does i t determine the cost as a direct impact , it also sets off a demand informat ion gather ing process as an indirect impact . T h e direct effect on the cost is short t e rm while the indirect effect is long te rm. A n d the subtle long te rm effect may outweigh the short t e rm effect. Hence, it is preferable to i n i t i a l l y overestimate the demand than underest imate i t . Overes t imat ion results in a higher than op t ima l order quanti ty. A l t h o u g h a higher order quant i ty may incur higher ordering cost, it allows more exact demand observations i n general which help to learn the demand d i s t r ibu t ion better and consequently y ie ld a lower cost i n the long run . Therefore, the next chapter focuses on the order quant i ty 's effect on the demand es t imat ion. We provide a proof of the conjecture that it is better to adopt a higher than o p t i m a l order quant i ty i n the in i t i a l periods when demand is learned i n a lost sales system. A s we have discussed earlier, the inventory pol icy affects informat ion collect ion and the demand d i s t r ibu t ion upda t ing process. T h e inter-woven nature of these 12 two processes requires considering the s ta t is t ical problem of demand es t imat ion i n the rea lm of inventory op t imiza t ion . Hence, this chapter investigates the effect of demand censoring on the o p t i m a l pol icy i n newsvendor inventory models w i t h general parametr ic demand d i s t r ibu t ion and u n k n o w n parameter values. We show that the newsvendor p rob lem w i t h observable lost sales reduces to a sequence of single-period problems whi le the newsvendor problem w i t h unobservable lost sales requires a dynamic analysis. Us ing a Bayes ian M a r k o v Decis ion Process approach we show that the op t ima l inventory level i n the presence of pa r t i a l ly observable demand is higher t han when demand is completely observed. We explore the economic ra t iona l i ty for this observation and i l lustrate i t w i t h numer ica l examples. T h e last chapter summarizes areas for future research. 13 Chapter 2 The Zero Inflated Poisson (ZIP) Distribution Very often a continuous t ime stochastic process is observed discretely. For instance, i n an inventory system, the demand is observed dai ly or per iodica l ly which may be considered as at discrete t ime points . Therefore, the parameter es t imat ion of the demand d is t r ibu t ion is often based on the accumula ted da i ly sales. T h i s raises the fol lowing question, how does the observable discrete process agree w i t h the under l in ing continuous process. For instance, for one par t icular i t em at a grocery store, i f customers arr ive fol lowing a Poisson process and the quant i ty of purchase at the ar r iva l follows a general non-negative d i s t r ibu t ion , then the accumulated sales process of this i t em is a C P P . A l t h o u g h the sales may be generated by a C P P , i t may not be prac t ica l for the store to continuously moni tor sales and hence capture the sales by bo th the t ime when i t occurs and the quanti ty. A s a matter of fact, da i ly sales is the most common da t a that is used for merchandise management today. So the question we asked ourselves here is how wel l does the dai ly sales data, or more generally the da ta that is obtained at discrete t ime points reflect the under ly ing C P P of sales. Section 2.3 shows that under certain condit ions, zero inflated Poisson models provide a good approx imat ion to the increments of the discretized C P P , that is the C P P increments observed at discrete t ime. We consider two Z I P densities w i t h various parameter settings to i l lus t ra te the qual i ty of the approx ima t ion . In practice, s ta t is t ical tests should be performed, in the first place, to verify that the overdispersion i n da ta indeed arises from the ex t ra zeros. Section 2.4 gives a score test of the Poisson fit versus the alternative Z I P hypothesis. A n d the score test is appl ied to the C T P A data . Fo l lowing that , a goodness-of-fit test of the Z I P dis t r ibut ions is given i n Section 2.5 where again the C T P A da ta is used. We start w i t h the definitions of the Z I P d i s t r ibu t ion and the compound Poisson process i n the next two 14 We start w i t h the definitions of the Z I P d i s t r ibu t ion and the compound Poisson process i n the next two sections. 2.1 Zero Inflated Poisson Density A r a n d o m variable X follows the zero inflated Poisson (ZIP(p , A)) d i s t r ibu t ion i f P r ( X = 0) = ( l - p ) + p e - * ; P r ( X = x) =pe-xXx/x\ (for x = 1, 2 , 3 , . . . ) . where 0 < p < 1 and A > 0. T h i s implies that there is an ex t ra p robab i l i ty mass (1 - p ) ( l - e~ A ) at X = 0. T h e expecta t ion and variance of X are as follows. oo E(X) = ^2x-Pr(X = x)=p\ (2.2) x=0 and V a r ( X ) = E ( X 2 ) - [ E ( X ) ] 2 = p A ( l + A - p A ) . (2.3) Observe that the variance exceeds the mean i f 0 < p < 1 T h e Z I P d i s t r ibu t ion can be interpreted as a B e r n o u l l i mix tu re of Poissons where one of the Poissons is degenerate at zero. T h a t is, the r andom variable X follows Poisson(A) w i t h p robab i l i ty p and takes a single value 0 w i t h p robab i l i ty 1 — p. We refer to p as the B e r n o u l l i parameter and A as the Poisson parameter. A n example s i tua t ion where the Z I P d i s t r ibu t ion is appropria te follows. Suppose an ' Ivory ' soap customer at Safeway, a grocery chain i n B r i t i s h C o l u m b i a , purchases soap w i t h p robab i l i ty p at her vis i t to the store, and does not buy w i t h p robab i l i ty 1 - p . T h e number of packages that she buys follows a Poisson(A) d i s t r ibu t ion . T h e n the uni t sales from this customer follows the Z I P d i s t r ibu t ion . A Poisson d i s t r ibu t ion does not fit the demand well i n this case. If the store predicts that the average number of customers v i s i t ing the store on a day is N, then the expected uni t sales of the ' Ivory ' soap on that day is N * p * A. A more general case than the Z I P d i s t r ibu t ion is the m u l t i n o m i a l mix ture of Poissons. W e w i l l explore this later on. 2.2 Discretized Compound Poisson Process Let Ti denote the r andom interar r ival t ime of a stochastic a r r iva l process and let Xi denote the j u m p size at the ith a r r iva l . Xi is independent of Tj. If T; exp(X), and Xi * '~ d ' (j>(fi,a), where </>(/u,cr) is a general non-negative d i s t r ibu t ion w i t h mean p and s tandard devia t ion a, then the stochastic process Z = {Zt\ t > 0}, 15 where ZT = Xi, NT is the number of arrivals up to a given t ime point (denoted by t) and ZQ = 0, is a C o m p o u n d Poisson process. T h e mapp ing £ —> Zt(ui) for any w i n is r ight continuous and real valued, w i t h ZQ{U) = 0 and ZT increases by jumps only. Obv ious ly the a r r iva l process N = {Nt;t>0} (2.4) is a Po isson process w i t h rate A£ since the in terar r ival t ime X i follows the exponent ia l d i s t r ibu t ion w i t h parameter A. Hence the p robabi l i ty density functions (pdf) of T i and Nt are fTi(t) = Xe-xt ; e-xt(Xt)n fm (n) = T h e expecta t ion of the C P P is E(Zt) = E[E(Zt\Nt)} = E\fiNt] = fiXt . (2.5) W e now look at the C P P at times t,t + 1, Since the a r r iva l process Nt is Po i s son(Ai ) , by definit ion, the increments of Nt are independent of each other and depend only on the length of the t ime interval but not the s ta r t ing t ime point of the interval . Therefore, P r ( i V t + 1 -Nt=n)= PriNx = n) (2.6) for £ = 1 , 2 , . . . . T h a t is, Nt+i — Nt is equivalent to i V i i n d i s t r ibu t ion . For the increments of the C P P , we have P r ( Z , t+i = P r ^ Xi = rnj = P r ( Z i = m). (2.7) W h e n observing the C P P at successive discrete t ime points , that is £ = 1 , 2 , 3 , . . . , a sequence of r andom variables {Yi — Zi - for i = 1 , 2 , . . . can be obtained and Y0 = 0. T h e y are the accumulated amount of j u m p s i n successive uni t t ime intervals [i - 1, i) for i = 1,2, It can be shown that Yi's are independent and ident ica l ly d is t r ibuted , and Yi is equivalent to Zx i n d i s t r ibu t ion for i = 1,2, In the next section, we w i l l derive the p robab i l i ty d i s t r ibu t ion of Z\. 16 2.3 ZIP Approximation to the Increments of the Discretized Com-pound Poisson Process A s discussed at the beginning of this chapter, a good discrete approx ima t ion of C o m p o u n d Poisson process may be p rac t i ca l as well as convenient in s ta t is t ical analysis. We derive the d i s t r ibu t ion of Z\, the accu-mula ted jumps i n the first uni t t ime per iod i n Section 2.3.1 before we investigate the Z I P approx imat ion to the increments of the discret ized C P P i n Sect ion 2.3.2. W e also give condi t ions for good approximat ions in Sect ion 2.3.2. 2.3.1 The Distribution of Zx For s impl ic i ty , we consider the special case where the j u m p size Xi is Poisson(/ i ) d is t r ibuted for i — 1,2, — L e t z > 0, we have where Xi is i . i . d . Po(p) for i = 1 , 2 , . . . , n . Note , we use Po to denote a Poisson d i s t r ibu t ion for short throughout this thesis. Therefore, for z > 0, (2.8) T h i s is based on n (2.9) OO O O Pr(Zi = z) = P r ( ^ i =z,Wi = k) ^ P r ( Z i = z | JVi = k) P r ( i V i = k) k=0 k=l k=l (2.10) For z = 0. k=l k=0 = e-\a+e-n (2.11) 17 f(X,p,y) = e « - A f > - * " (^p) =e»-^-»V{Ky) ( 2 - 1 3 ) T h e p robab i l i ty density of Z\ for the general case where X ~ 4>(p, a) can be derived i n the same manner. For the process { Y i ; i > 0 } defined above when Xi is Poisson(/j) , we have for y > 0 f rom ( 2 . 1 0 ) that P r ( K i = y)= P r ( Z x = y) = e - " ^ / ( A ) M ) l / ) ( 2 . 1 2 ) where, '\kkv w i t h i f ~ P o ( A e - ' i ) . A n d i t can be shown that y - i / _ i \ E(KV) = ( A e - " ) ( V • ) E ^ " - 1 - ' ) < oo ( 2 . 1 4 ) »=o ^ * ' for y = 1 , 2 , . . . w i t h E ( # ) = A e _ " and E(K2) = \2e~2^ + A e _ " . T h e expecta t ion and the variance of Y i (i = 1 , 2 , . . . ) are E(Yi) = E(Z1)=E[E(Z1\N1)]=»\; ( 2 . 1 5 ) V a r ^ ) = V a r ( Z i ) = E{Zl) - E2(Z1) = ^ 2 A + / J A ( 2 . 1 6 ) where E(Z2) is obta ined as follows. E(Z2) = E[E(Z2\N1)] = E(N2v2+Nl(j,) = / i 2 ( A 2 + A ) + / x A . ( 2 . 1 7 ) 2.3.2 Discrete Time Approximation of the Compound Poisson Process W e again consider the special C P P where the r andom j u m p X at arr ivals follow Poisson(/x). We w i l l extend the approx ima t ion at the end of this section. A s ment ioned earlier, we assume the arrivals are independent of the jumps . W h e n we observe the C P P at discrete t ime points , a sequence of i . i . d . Y i , i = 1 , 2 , . . . is obtained. F r o m another perspective, Y i can also be considered as approximate ly a " m u l t i n o m i a l " 1 mix ture of Poisson dis t r ibut ions : w i t h probabi l i ty po, there is no a r r iva l i n the uni t t ime interval [i — l , i ) and hence, Y i = 0 w i t h certainty; w i t h probabi l i ty p\, 1li Nt is a truncated Poisson process, i.e., at a given t ime t, Nt is discrete and bounded from above by some positive finite number w i th certainty, then Yi can be described precisely by a mul t inomia l .mix ture of Poisson. However, when Nt is a conventional Poisson process, at a given time t, Nt is infinite wi th a positive probabili ty. T h e term "mul t inomia l" mixture of Poisson is not accurate then. 1 8 . there is one a r r iva l i n the unit t ime interval [i - l,i) and hence, Yi ~ Po(p); s imi lar ly , w i t h probabi l i ty Pk (k > 1), there are k arrivals in the uni t t ime interval [i — and hence, Yi ~ Po(kp) given k arrivals. No te that , Pr(7Vi > 2) = 1 - P r ( / V i < 2) = 1 - e~x - Xe~x = 0{\2) . (2.18) T h a t is, when A is smal l , E[T] = ^  is large and hence, i t is unl ike ly that there is more than one arr ival i n any uni t t ime interval [i — 1, i ) , for i = 1 , 2 , . . . ) . Neglect ing the smal l probabi l i t ies of mul t ip le arrivals i n a uni t t ime interval , the "mul t inomia l " mix tu re of Poisson can be approx imated by the B e r n o u l l i mix tu re of Poissons. F r o m the pd f of the compound Poisson d is t r ibu t ion , expressions (2.10) and (2.11), we have for y > 0, fe=i = e xX + e ^^-j-e x— + ... y\ y\ 2! = Ae-Ae~M4 + ° ( A ) y-= P r ( y ; = y) + o(A) (2.19) where Yi ~ ZIP(Xe~x,p). T h a t is, Xe~x is the B e r n o u l l i parameter and p the Poisson parameter. For y = o, P r ( y i = 0) = e~x + e-»e-xX + e-2)1e-x^ + ... = e~x + e~'1e-xX + o(A) (2.20) Rear rang ing (2.18), we have e~x = 1 - Xe~x + o(A). T h u s the above becomes PvCYi = 0) = 1 - A e ~ A + Xe-xe-» + o(A) = P r ^ ' = 0) + o(A) (2.21) where again, Y- ~ ZIP{Xe~x,p). C o m b i n i n g (2.19) and (2.21), we have P r ( y , = y)= P r ( y ; =y)+ o(A) (2.22) for any i / £ { 0 , 1 , 2 , . . . } . Therefore, the Z I P pd f approximates well the pd f of the increments of the discretized C P P when A is smal l . T h e expectat ion and the variance of Yi are as follows. E(Y-) = A e ~ V ; V a r ( y i ' ) = A / i e - A ( l + ^ - A / i e - A ) . (2.23) 19 H = 2 A = 0.2 A = 0.3 A = 0.4 A = 0.5 Z I P ( A , / i ) F igure 2.1 F igure 2.3 F igu re 2.5 F igu re 2.7 Z I P ( A e - A , / x ) F igure 2.2 F igure 2.4 F igu re 2.6 F igu re 2.8 Table 2.1: T h e C P P increments densities versus the Z I P densities figures A = 0.2 / x = l /IX = 3 / i = 4 H = 5 ZIP(A, /x ) F igure 2.9 F igure 2.11 F igure 2.13 F igure 2.15 Z I P ( A e - \ / i ) F igure 2.10 Figure 2.12 F igure 2.14 F igure 2.16 Table 2.2: T h e C P P increments densities versus the Z I P densities figures W h e n A is smal l , that is when A ->• 0+, e _ A - » 1_. Hence, A e _ A aa A. Therefore, Y ; can be further approx imated by Y-' where Y-' ~ Z I P ( A , / i ) . Obvious ly , Y ^ ' approximates Y i i n a s impler form. T h e expecta t ion and the variance of Yi are as follows. E(Yi') = Xn; Var(Y-; ' ) = A / x ( l + / i - A / i ) . (2.24) Note these come closer than (2.23) to the expectat ion and the variance of Y i . Please refer to expressions (2.15) and (2.16). Intui t ively, i f the demand arr iva l process is Poisson w i t h a smal l rate A, i t is unl ike ly to have mul t ip le arr ivals i n any uni t t ime per iod . M o s t often either there are no arr ivals or jus t one a r r iva l . T h e realized to ta l demand i n each per iod can be well described by the Z I P d i s t r ibu t ion i n this case. Since the Z I P d is t r ibu t ion does not account for the overdispersion from mul t ip le arr ivals , the app rox ima t ion becomes worse when A is large. We compare the densities of the increments from some C P P ' s versus Z I P densities i n the following. Table 2.1 and 2.2 give the case settings and the corresponding figure numbers. Sixteen figures are included. T h e first set has different A values but fixed /J. T h e second, different /x values but fixed A. We chose these ranges for A and /x since we ant ic ipated improv ing approximat ions as fi increases and A decreases. We w i l l see that when /x takes a relat ively low value, /x = 2, the approx ima t ion is good w i t h a low A, i.e. A = 0,2. A n d w i t h A = 0.2, the approx imat ion is good when \i is as large as 5. T h e o d d number figures show Z I P ( A , \x) while the even number figures show Z I P ( A e _ A ,\x). In a l l figures, the open circles are from the Z I P density whi le the line is from the C P P increments density. It can be seen that the Z I P densities lie farther away from the C P P increments densities as A gets larger w i t h a fixed \i. However, the approx imat ion improves as /x increases w i t h A fixed at a low value. In addi t ion , compar ing the two sets of figures suggests that Z I P ( A e ~ A , fi) approximates the C P P increments densities 20 Figure 2.1: T h e C P P increments density and the Z I P ( A , /i) density w i t h JJ, = 2 and A = 0.2. •8 S o Figure 2.2: T h e C P P increments density (line) and the Z I P ( A e x , n) density (circles) w i t h p = 2 and A = 0.2. 21 Figure 2.3: T h e C P P increments density (line) and the Z I P ( A , fx) density (circles) w i t h fi = 2 and A — 0.3. F igu re 2.4: T h e C P P increments density (line) and the Z I P ( A e A , LL) density (circles) w i t h LI — 2 and A = 0.3. 22 0 2 4 6 8 10 x Figure 2.5: T h e C P P increments density (line) and the Z I P ( A , LI) density (circles) w i t h LL = 2 and A = 0.4. o i 1 1 1 1 [-0 2 4 6 8 10 x Figure 2.6: T h e C P P increments density (line) and the Z I P ( A e ~ A , LL) density (circles) w i t h LL = 2 and A = 0.4. 23 •S o Figure 2.7: T h e C P P increments density (line) and the Z I P ( A , LL) density (circles) w i t h LI — 2 and A = 0.5. F igu re 2.8: T h e C P P increments density (line) and the Z I P ( A e A , fx) density (circles) w i t h /j, = 2 and A = 0.5. 24 Figure 2.9: T h e C P P increments density (line) and the Z I P ( A , LL) density (circles) w i t h LL = 1 and A = 0.2. F igure 2.10: T h e C P P increments density (line) and the Z I P ( A e  X, LL) density (circles) w i t h LL — 1 and A = 0.2. 25 Figure 2.11: T h e C P P increments density (line) and the Z I P ( A , LI) density (circles) w i t h LI = 3 and A = 0.2. F igu re 2.12: T h e C P P increments density (line) and the Z I P ( A e A , LI) density (circles) w i t h LI = 3 and A = 0.2. 26 Figure 2 . 1 3 : T h e C P P increments density (line) and the Z I P ( A , /J ) density (circles) w i t h LI = 4 and A = 0 . 2 . Figure 2 . 1 4 : T h e C P P increments density (line) and the Z I P ( A e A , fi) density (circles) w i t h LI = 4 and A = 0 . 2 . 2 7 Figure 2.15: T h e C P P increments density (line) and the Z I P ( A , LI) density (circles) w i t h LI = 5 and A = 0.2. F igu re 2.16: T h e C P P increments density (line) and the Z I P ( A e A , LI) density (circles) w i t h LI = 5 and A = 0.2. 28 better than Z I P ( A , LL), i n general, regardless of bo th A and LL. W h e n A is sma l l , Z I P ( A , LL) also approximates the C P P increments density well (see F igure 2.1, 2.2 and F igure 2.9 th rough 2.16). T h i s point was discussed earlier w i t h A e ~ A « A as A -> 0 + . It is also notable that Z I P ( A , LL) gives the p robab i l i ty mass at zero closer to that f rom the C P P . F r o m these two sets of figures, we observe that Z I P ( A e ~ A , LL) and Z I P ( A , LL) approximate C P P wel l under some condit ions. T h e condit ions are A is sma l l enough and LL is moderate or large. G o o d approximat ions are found w i t h A = 0.2 and LL — 2 and the approx ima t ion improves as LL goes up to 3, 4 and 5. T h e condi t ion of smal l A is especially true for Z I P ( A , LI) since A has to be smal l to show a good approx ima t ion . A s discussed earlier, when a stochastic inventory system is observed discretely, the parameter estima-t ion of the demand d i s t r ibu t ion is often based on the accumula ted da i ly sales. If the demand ar r iva l is a Poisson process w i t h rate Xt while the amount of each ar r ived demand has pd f 4>(fj,,o), we saw in the pr ior section that the accumula ted da i ly demand follows the C o m p o u n d Poisson d i s t r ibu t ion . T h e sequence Y i , Y%, •.. , Yt describes the C P P Zt- We have just shown for the case where the size of each arr ived demand also follows Poisson(/x), the accumulated da i ly demand Yi can be approx imated by ZIP(A, /x ) provided the a r r iva l rate of demand A is low. In the inventory system, Y i is often the r a n d o m variable that is observed. A good approx ima t ion of Yi to Y i enables us to estimate the unknown parameters A and LL. W e extend the app rox ima t ion i n the fol lowing. For a general case where the r andom interarr ival t ime of a stochastic process follows a general non-negative d i s t r ibu t ion w i t h mean A and s tandard devia t ion er, i.e., T; ip(X,a) and the j u m p size Xi follows Poisson . L e t Nt denote the number of arrivals up to t ime t, assuming Xi is independent of Nt, we have the fol lowing (by (2.10) and (2.11)). For y > 0, O O O O P r C X =!/) = £ P r ( Y i = y, Nt = k) = £ P r ( Y ; = y\Ni = k) P r ( i V i = k) k=0 k=l = P r ( Y i = y\Ni = 1) Pr(JV 4 = 1) + P r ( y j = y\Ni = 2) P r ( W i = 2) + P r ( Y i = i / | i V i = 3 ) P r ( J V i = 3 ) - r . . . (2.25) For y = 0, 0 0 / k \ P r ( Y ; = 0) = P r ( i V i = 0) + ^ P r ^ X{ = 0\Nt = k P r ( ^ = k) k=l \i=l / = Pi(Ni = 0) + P r ( Y i = 0|JVi = 1) Pr (Af i = 1) + Pr(Yi=0\Ni = 2)Pv(Ni = 2) +... (2.26) If P r ( Y i = y\Nt = 2) P r ( i V i = 2) 4- P r ( Y i = y\N{ = 3) P r ( A ^ = 3) + . . . can be reduced to o(A) for y > 0, that 29 is, when it is unl ike ly to have more than two arrivals i n any t ime interval i, then the above two equations can be wr i t t en as For y > 0, P r ( K i = y) = e-^Pv(Ni = l) + o{\) y-« e - ^ p r ( A T i = i ) = P r ( y ; =y )Pr (JV i = l ) (2.27) For y = 0, P r ( Y ; = 0) = P r ( ^ = 0) + P r ( y s = 0 | ^ = 1) P r ( i V ; = 1) + o(A) a 1 - Pr(JVi = 1) + e-» Pr (JV 4 = 1) = 1 - Pr(ty = 1) + P r ( y / = 0) P r ( i V ; = 1) (2.28) where Y- ~ PO(LI). Fur thermore , i f Pr(JVi = 1) « A, then (2.27) and (2.28) become P r ( y i = y ) « A e - " ^ (2.29) for y > 0 and P r ( Y i = 0) sa 1 - A + e-"A (2.30) for y = 0. T h a t is again Y , can be approximated by ZIP(A,/z). Therefore, under cer ta in condit ions, the Z I P d i s t r ibu t ion can be used to approximate this more general case where the in terar r iva l t ime T» follows a general non-negative d i s t r ibu t ion . No te the process of accu-mula ted jumps , Zt, is no longer a C P P . T h i s is quite obvious, for instance, when P r ( i V i = 2) = 1. The condi t ion for Z I P being a good approx imat ion is that the p robabi l i ty of more t han one arrivals i n [i — l,i) (i = 1 , 2 , . . . ) , P r ( i V i > 1), is smal l . T h a t is when 1/A is relat ively large and a is smal l . However, if there is more than one a r r iva l i n some unit t ime intervals, the Z I P approx imat ion cannot be used. 2.4 A Score Test for Zero Inflation in Sales Data - ZIP vs. Poisson Fit to Data W e gave a defini t ion of Z I P density and discussed the mot iva t ion for us ing i t i n the previous sections. In the fol lowing two sections, we provide s ta t is t ical tests to verify i f the Z I P provides good fit to a da ta set. Ga rden [24] s tudied the zero inflated Poisson regression model and gave a thorough da t a analysis on Canad i an T i r e 30 Pacif ic Associates sales data . We start w i t h a score test i n this section to test the count da ta set for zero inf la t ion. Sect ion 2.4.1 describes the data . Section 2.4.2 introduces a score test by van den Broek [59]. Sect ion 2.4.3 presents the test results. 2.4.1 The Data A n enormous da ta set of da i ly sales was provided by C a n a d i a n T i r e Paci f ic Associates ( C T P A ) . C T P A owns and operates 21 re ta i l stores i n the lower ma in l and of B r i t i s h C o l u m b i a . T h e stores, carry approximate ly 30,000 products wh ich are p r imar i l y automotive parts and accessories, home hardware, housewares and spor t ing goods. T h e products are delivered per iodica l ly to loca l re ta i l stores from a central warehouse in B u r n a b y which is also owned by C T P A . T h e central warehouse receives products from Toronto and keeps inventory. D a i l y sales da ta was collected for 458 regular ly stocked p roduc t s 2 over a pe r iod of 307 days from Oct . 10, 1992 to A u g . 18, 1993 at a l l 21 retai l stores. T h e sales observations took non-negative integer values. Since C a n a d i a n T i r e Pacif ic Associates adopted inventory policies that kept very h igh stock on-hand levels du r ing that pe r iod , the incidence of stock-outs was very low, i f not zero. Therefore, the sales da ta collected was believed to well represent the true demand d is t r ibu t ion . T h e or ig ina l da ta was i n the form of t ime series. W i t h some da ta process ing 3 the empi r ica l demand "dis-t r ibu t ions" were available. A p p e n d i x A , B and C present some typ ica l "d is t r ibut ions" of " low" , " m e d i u m " , and "high" demand products respectively. It can be seen that the "dis t r ibut ions" present the t ime series da i ly sales as count data , which is convenient for the s ta t is t ical tests. T h e categorizat ion was based on the shapes of the empi r ica l histograms. T h e "high" demand dis t r ibut ions seem to be well approximated by Poisson dis t r ibut ions . T h e "low" demand products have zero demand on most of the days, and low posi t ive demands that seldom exceed 3 i tems on the remain ing days. We are ma in ly interested i n the products w i t h "medium" demands since Poisson d i s t r ibu t ion does not fit well in this case. His tograms of the da i ly sales of one product , a #168 bulb, from a l l 21 stores are plot ted i n F igure 2.17. We see a large p ropor t ion of zeros i n each of the histograms. Z I P may be used to mode l the ex t ra zeros i f it provides a good overal l fit to the da i ly sales. B u t first of a l l , we need to test that there are more zeros than what we expect from a Po isson d i s t r ibu t ion . 2 These products were chosen because demand was believed to be non-seasonal, prices are relatively constant and this product line contained both fast and slow moving items. 3 Refe r to Kapa lka , Kar t i r c iog lu and Pute rman [35] for details of the original data set, data processing, and some prel iminary data analysis. 31 demand demand demand demand demand Figure 2.17: His tograms of Sales of P r o d u c t 203039 (168 bulb) at 21 stores. 32 2.4.2 A Score Test van den B r o e k [59] s tudied generalized linear models where there is some overdispersion i n the d is t r ibut ion coming f rom ex t ra zeros. T h e under ly ing mechanism of the Z I P d i s t r ibu t ion was discussed and some data examples were given to i l lustrate the use of Z I P d is t r ibu t ion . He proposed a score test for zero inflation and invest igated the properties of the score statist ic. W h e n there are no covariates, the score stat ist ic s was simplif ied as s = ( m - ^ (2.31) n where n is the sample size of data , m is the number of 0 observations i n the d a t a set and rno is the expected number of 0 observations from a Poisson fit. To calculate mo, let A = n~l Yl7=i X i w n e r e X j are the observed values. If the number of 0 observations is denoted by M o , under the Poisson model , E ( M 0 ) = Pr(Xi = 0) • n = e~x • n . (2.32) T h e n mo = e~x • n. T h e score stat is t ic s approximate ly follows a x\ d i s t r ibu t ion . La rge values of this s tat is t ic indicate the presence of zero inf la t ion. 2.4.3 Testing Results We apply the above score test to the C T P A dai ly sales data . T h e hypotheses are as follows. HQ : no zero inf la t ion i n the demand d i s t r ibu t ion Hi : zero inf la t ion presents in the demand d i s t r ibu t ion G a r d e n [24] discussed the score test for generalized l inear models. Some i n i t i a l da ta analysis indicated that overdispersion is present. She investigated the overdispersion ar is ing from the possibil i t ies of the zero inf la t ion as wel l as the effect of two factors, day of the week and the store loca t ion . A pre l iminary Poisson regression mode l w i t h these effects was found to be inadequate for some of the "med ium" demand products. L a c k of fit was suggested by the relat ively large residual deviance compared to the degrees of freedom (the residual deviance is 9032 on 6567 degrees of freedom), wh ich also indica ted that overdispersion might be a p rob lem even after those two factors were taken into account. G a r d e n [24] calcula ted the test statist ic based on the vector of the parameter estimates from a Poisson regression mode l . T h e asympto t ica l ly xl dis t r ibu ted test s tat is t ic had the value of 346.54. T h i s showed strong evidence against ord inary Poisson fit 33 to each "medium" demand product at a l l 21 stores. She appl ied the zero inflated Poisson regression model and showed that i t provided a better fit to the data . 2.5 Goodness-of-fit Test for the Count Data of Daily Sales N o w that the score test shows strong evidence against the o rd inary Poisson fit, the next question we ask ourselves is does the Z I P d i s t r ibu t ion provide a good overal l fit for the da ta set? Since Z I P is only effective i n model ing the overdispersion that is accounted for by ex t ra zeros, i t is not appropria te for a general case of overdispersion. So we apply a goodness-of-fit test i n this section to see i f the Z I P d i s t r ibu t ion provides an adequate fit to the da ta set. T h e test ing results follow the descr ipt ion of the test. 2.5.1 The Goodness-of-fit Test Since the empi r ica l "d i s t r ibu t ion" of da i ly sales are i n the format of count da ta , we use a Pearson's goodness-of-fit ( G O F ) test to test the overal l goodness-of-fit of a Z I P d i s t r ibu t ion . Pearson 's s tat is t ic is given as follows. where is the number of demands observed to be i, and i = 0 ,1 , . . . , dj for a par t icu lar product at store j; dj is last category of d is t inct sales at store j w i t h sales above this amount combined; n° is the expected number of demands of i units from a Z I P fit. Therefore, the Pearson's G O F test is based on compar ing the observed number of demands, to the expected demand, from the Z I P fit. It can be shown that i n wh ich dfj denotes the degrees of freedom and a is the c r i t i ca l value. Obvious ly , we w i l l have to fit the sales of a product at each store w i t h a Z I P d i s t r ibu t ion and estimate the Z I P parameters before we can apply the x 2 test since the computa t ion of the expected number of a cer ta in demand observat ion requires the Z I P probabi l i ty density. Once the Z I P parameters are available, n ° is easily obta ined by of da i ly sales observations. We fit the sales w i t h a Z I P density and the m a x i m u m l ike l ihood est imat ion (2.33) X) ~ x 2 ( ^ - , a ) n ° = P r ( X = i) • n where P r ( X = i) is the p robabi l i ty density function (pdf) of Z IP (p , A) at X — i and n is the to ta l number 34 ( M L E ) is used to obta in the parameter estimates. We derive and discuss M L E ' s of the parameters of the Z I P d i s t r ibu t ion i n Sect ion 3.1.2. Note censoring is not a concern yet. R e c a l l , we consider the sales da ta as representing true demand since there is almost no incidence of stock-outs du r ing the da ta collection per iod. 2.5.2 Test Results W e apply the x2 test to the empir ica l "dis t r ibut ions" of sales for three "medium" demand products . We arb i t r a r i ly chose product 200001, 200002 and 203039. T h e count da ta of sales and x2 statistics for product 200001, 200002 and 203039 at a l l 21 stores are presented, respectively, i n A p p e n d i x D , E and F . C o m p a r i n g X 2 statistics to the cr i t i ca l quanti le of x2(dj + 1, a ) w i t h a = 0.05 for j - 1,2,... , 21, we conclude about the Z I P fit to the sales by product i n the fol lowing. P r o d u c t 200001 A p p e n d i x E shows that Z I P density provides a good fit to the sales at store 2, 3, 4, 7, 8, 9, 10, 12, 13, 15, 16, 17, 18, 19, 20, and 21 where Xj < x 2(4, 0.05). T h e estimate of the B e r n o u l l i parameter p ranges from 0.235 at store 1 (wi th A = 0.749) to 0.863 at store 7 (wi th A = 0.426). T h e estimate of the Poisson parameter A ranges from 0.250 at store 16 (wi th p = 0.676) to 0.963 at store 3 (wi th p = 0.643). P r o d u c t 200002 A p p e n d i x F shows that Z I P density provides good fit to the sales at a l l stores except at store 7 since X2 = 21.27 > x 2 (3, 0.05) = 7.81. T h e estimate of the B e r n o u l l i parameter p ranges from 0.216 at store 5 (wi th A = 0.317) to 0.508 at store 11 (wi th A = 0.115). T h e estimate of the Poisson parameter A ranges from 0.115 at store 11 (wi th p = 0.508) to 0.628 at store 7 (wi th p = 0.342). P r o d u c t 203039 Since at most stores (except at store 4, 5, 12 and 13) there are no sales exceeding 11 units , we determine the number of sales categories to be 12 from 0, 1, to 10 and at or above 11. T h a t is df = 12. A p p e n d i x E shows that Z I P density is a good fit only at store 6, 15 and 20 where X? < x 2(9, 0.05) = 16.92. T h e estimate of the B e r n o u l l i parameter p ranges from 0.248 at store 1 (wi th A = 2.112) to 0.651 at store 7 (wi th A = 2.047). T h e estimate of the Poisson parameter A ranges from 1.503 at store 6 (wi th p = 0.251) to 2.149 at store 13 (wi th p = 0.605). C o m b i n i n g the results from the above three products , one notes that Z I P fits well i n general for "medium" demand products 200001 and 200002, yet it on ly occasional ly fits for product 203039. For this product , even 35 though the score test shows evidence against conventional Poisson, the Z I P d i s t r ibu t ion does not account well for the overdispersion. Fur ther invest igat ion is needed here. Zero inflated negative b inomia l model may be one of the candidates to consider. A negative b inomia l regression mode l is a common approach to incorporate r andom effects i n an ord inary Poisson model and hence accommodates the overdispersion (Jansen [33]). Lawless [41] gave a thorough discussion on the negative b i n o m i a l regression model and its performance. We leave this for future research. 2.5.3 Summary W e in t roduced he zero inflated Poisson d i s t r ibu t ion i n this chapter. It was mot iva ted by the presence of overdispersion i n a re ta i l sales data. A score test is discussed and appl ied to show evidence of zero inflat ion. G i v e n the evidence of zero inf la t ion, the overall fit of the Z I P model is tested by a goodness-of-fit test. T h e re ta i l sales da ta was used i n this testing. We saw that the Z I P d i s t r ibu t ion provides a good fit for two products at various locat ions. However, i t is not the case for P r o d u c t 203039. Fur ther invest igat ion is needed. T h i s chapter gives just i f icat ion of the Z I P model . W e w i l l focus on the Z I P demand model i n the rest of the thesis. It w i l l provide prac t ica l convenience i f a continuous t ime stochastic process can be well approximated by a discrete t ime process since very often the continuous process is moni to red and control led at discrete t ime. A business example is, even though the sales process at a re ta i l store is usual ly a continuous process in nature, inventory pos i t ion is checked per iodica l ly and based on that , an appropr ia te inventory control pol icy takes place. V e r y often, an inventory system is adopted due to ease of implementa t ion . A retai l store may not keep records of the continuous sales process but just the da i ly sales. T h i s mot iva ted us to investigate the Z I P app rox ima t ion to a general compound Poisson process. W e saw that Z I P provides good approx imat ion when the Poisson a r r iva l rate is smal l and the mean j u m p at each ar r iva l is re la t ively large. W e have seen that Z I P provides f lexibi l i ty i n dealing w i t h count da t a by accommoda t ing the ex t ra zeros. A l t h o u g h zero inf lat ion may not be uncommon i n re ta i l inventory management environments, we are not aware of any l i terature on Z I P demand i n par t icular i n a retai l inventory system. In the next two chapters, we w i l l develop es t imat ion methods for Z I P demand and investigate the es t imat ion error 's effect on inventory po l i cy and cost. 36 Chapter 3 Maximum Likelihood Estimation of Censored Zero Inflated Poisson Demand A s we have discussed i n Chap te r 1, es t imat ion of the demand d i s t r ibu t ion using sales da ta is a challenging s ta t i s t ica l p rob lem i n lost sales inventory systems. If the sales data , wh ich is often the relevant da ta available, represents the demand completely, that is, the demand is fully observable, the p rob lem can be solved using the s tandard m a x i m u m l ike l ihood es t imat ion ( M L E ) method . However, i n re ta i l inventory systems, unmet demand is usual ly lost and unobservable. W h e n sales i n a par t icu lar t ime per iod equal the inventory on hand at the beginning of that per iod, it is l ikely that the demand i n the per iod is higher than the inventory level wh ich is the amount available for sales. Hence, the demand informat ion is considered as "incomplete" when only the satisfied demand is observed as the sales. T h i s is referred to as the demand being censored by available inventory. In this case, sales do not represent demand. T h e question of how to use the sales da ta which consists of bo th exact observations and censored observations to estimate the unknown parameters of the demand d i s t r ibu t ion is explored i n the next two chapters. In essence we derive estimators which take into account censoring by lost sales. Th i s is mot iva ted by the questions of whether a retailer should record lost sales da ta and what can be done i f this da ta can not be easily obtained. C o n r a d [16] dis t inguished sales from demand. He studied Poisson demand es t imat ion in a newsvendor system. A n M L E of the parameter was proposed by inc lud ing in the l ike l ihood function the probabi l i ty of observing a demand censored at the order quant i ty which is the amount available for sales. We extend his 37 work to Z I P demand. A s a Poisson mix tu re , the Z I P d i s t r ibu t ion requires some ex t ra effort i n the m a x i m u m l ike l ihood estima-t ion der ivat ion. T h i s is especially true when it is coupled w i t h the issue of censoring i n the data. Poisson demand is considered for compar ison purposes. We derive the m a x i m u m l ike l ihood estimators for bo th the Poisson and Z I P parameters using sales data . T w o types of inventory systems are considered, namely, newsvendor and (s, S) systems. T h e newsvendor system is a re la t ively s imple inventory system i n terms of implementa t ion and control . T h i s is also true when es t imat ion is concerned. In this chapter we assume that the same order quant i ty is used i n a newsvendor system. T h a t means demands are censored by the constant order quant i ty i n sell-out periods. We w i l l discuss the order quant i ty upda t ing and demand upda t ing in the next chapter. In the (s,S) system, demands are censored by the variable inventory levels even w i t h the same inventory pol icy i n place. Refer to the censoring por t ion of Sect ion 3.2.1 and 3.2.2 for details. Since the nature of censoring differs for these two systems, the estimators w i l l also differ. A n i terat ive numerical me thod is needed since the l ike l ihood equations are nonlinear i n the parameters when censoring exists. In general, i t requires more computa t iona l effort to consider lost sales than not the case where demand is complete ly observable. Hence, an interesting question we ask of ourselves is, is i t wor thwhi le to make the ex t ra effort of considering censoring i n est imation? T h a t is, w i l l i t result i n a significant increase i n inventory cost i f we ignore censoring? H o w sensitive are the op t ima l po l icy and its cost to the es t imat ion errors? These questions mot ivate us to investigate the effect of es t imat ion error on the o p t i m a l policies and their costs. To address these issues, we focus on the newsvendor system because of its closed form o p t i m a l policy. T h i s chapter is organized as follows. We give the M L E ' s for the Poisson and the Z I P dis t r ibut ions in Sect ion 3.1. We extend Conrad ' s work to the Z I P demand i n Section 3.2. T h e M L E ' s of the Z I P parameters t ak ing censoring into account are derived separately for the newsvendor and the (s, S) inventory systems. F ina l l y , we investigate i n Sect ion 3.3 the effect of the es t imat ion errors from not considering lost sales by cost and pol icy comparisons i n the newsvendor system where various est imators of the unknown parameters are used. 3.1 Maximum Likelihood Estimation for Uncensored Poisson and ZIP distribution W e assume Poisson and Z I P d is t r ibuted demand w i t h unknown parameter values. We start w i t h the case where the demand is fully observable. T h e M L E ' s of the Poisson and Z I P parameters are computed next. 38 3.1.1 Poisson Distribution T h e Poisson d i s t r ibu t ion can be considered as a special case of the Z I P d i s t r ibu t ion . Since the Z I P d is t r ibu t ion is a B e r n o u l l i m ix tu re of Poisson dis t r ibut ions , the Z I P d is t r ibu t ion becomes Poisson when the Bernou l l i parameter p = 1 and i t degenerates to a mass at 0 when p = 0. Hence the Poisson d i s t r ibu t ion can be considered as one extreme of the Z I P d is t r ibu t ion . Suppose the dai ly demand Xi is Poisson dis t r ibuted w i t h unknown parameter A. Le t Xi be the observed demand on day i, % = 1, 2 , . . . , n. It is well known that En i=l Xi n is the m a x i m u m l ike l ihood est imator of A. 3.1.2 ZIP Distribution R e c a l l , i f a r a n d o m variable X has Z I P d i s t r ibu t ion w i t h parameters p and A, then oo E(X) = % • P*(X = x) = p\ (3.1) x = 0 and Var (A: ) = E{X2) - [E(X)}2 = PX(1 + A - pX). (3.2) Le t xi, x2, • • • , xn be an independent r andom sample of da i ly demand from the Z I P ( p , A) d is t r ibut ion . T h e l ike l ihood funct ion of Xi, i = 1 , 2 , . . . , n, L(p, X\xi,... ,xn) is given by L(p,X\xu... ,xn) = n?=1 Pr(Xi = Xi) = [(l-p)+Pe- xy\pe-'r-i (3.3) where j is the number of Xi's wh ich equal zero, and z = Y17=i Xi' * n e t o ta l demand. T h e log- l ike l ihood denoted by I is given by I = In L(p,\\xi,... ,xn) = j In [(1 -p)+ pe~x] + (n - j) [In (p) - A] n + z l n ( A ) - ] T l n ( a : i ! ) . (3.4) i=l 39 T h e l ike l ihood equations are obta ined by sett ing the pa r t i a l derivatives of the log- l ike l ihood w i t h respect to the parameters equal to zero. T h a t is, m = - i ( l - e ~ A ) | n-j = Q dp 1 — p + pe~x p H- = i " J T \ - ( " - i ) + % i i = o . a A 1 - p + pe A A After rearranging terms, the above simplifies to P = 1—- ; (3-5) n e A - 1 A e A = SLlfi . (3.6) e A - 1 n-j Obvious ly when j = n, p = 0 and A is arbi t rary. In this case, no posi t ive sales have been observed, hence no informat ion is obta ined regarding the demand d i s t r ibu t ion . T h i s is the case when the Z I P d is t r ibut ion degenerates to a single value 0 w i t h p robabi l i ty 1. T h e other extreme case is when j = 0. W i t h our assumpt ion of 0 < p < 1, the Z I P l ike l ihood function given i n 3.3 is m a x i m i z e d at p = 1 and A = E"=i xi/n-Note , A is the same as the M L E of the Poisson parameter, hence this is the case that the Z I P d is t r ibut ion reduces to a regular Poisson d is t r ibu t ion . Since the l ike l ihood equations are nonlinear i n the parameters and the closed-form solut ion of the l ike l i -hood equations can' t be found, an i terat ive numerica l me thod is needed to ob ta in p and A. We use ' n l m i n ' , a rout ine i n S - P L U S 1 . It finds a loca l m i n i m u m of a non-linear function using a general quas i -Newton opt i -mizer . W e use ' n l m i n ' to min imize —I. T w o i l lus t ra t ive numer ica l examples are given i n Sect ion 3.2. In bo th examples, we use the s tar t ing point (0.1,0.1) from which convergence is obta ined. A s w i t h many non-linear p rog ramming a lgor i thms, va ry ing the s tar t ing point generally result i n different solutions. Fur thermore , it may not y ie ld convergence at a l l . Since 0 < p < 1 is a parameter for probabi l i ty , we chose to start from a close to zero value for bo th parameters. S imi l a r l y for A which should have a posi t ive value. T o be complete, we use second order condit ions to verify that we are indeed m a x i m i z i n g the l ike l ihood funct ion, rather than finding some other inflection point . T h e second derivatives are as follows. 02l _ -j(l-e~x)2 n-j dp2 (1 — p + p e ~ A ) 2 p 2 ' d2l je~x dpd\ ( l - p + p e ~ A ) 2 d2l _ j p e - A ( l - p ) E"=i a ; i <9A2 ( l - p + p e - A ) 2 A 2 1 S - P L U S , a statistical system is used for data generation and computat ion in this thesis research. 40 T h e condit ions for I being m a x i m i z e d at the M L E (p, A) are: H = A B B C > 0 A < 0 wnere A - 9 p 2 | ( ^ a p a A i ( p , A) _ a A a P l ( p , A) a n Q ° _ 9A*l(p, A)-T h e fol lowing equations can be derived from the l ike l ihood equations (3.5) and (3.6). 1 — p + pe 1 - e 3_. n re - j np U t i l i z i n g the above equations, A, B and C are obtained as follows. (n - j)n A = JP ,2 ' B = - - ( r e — ) ; J P c _ P(l-P) ,2 _ (n-j)n _ rip Hence, H = A - C — B2 can be simplif ied to: « 2 AC - B = L e t pp2 (np - n + j)[-(n - j)(l - p) - (np - n + j)] + re r i - -\ n ~ 3^ = ^ [ - ( " P - " + J ) + - ^ - ] G = -(np - n + j) + n-j n2(n — i p A Since by (3.8), we have, np = n - J l - e ~ x G = ( „ - j ) [ i - _ J - T + l ] 1 - e~ A A 1 - e~'x - Xe~'x = (re - j) (l-e-x)X It can be seen that the denominator is greater than zero for any A > 0. L e t g(X) = 1 - e~ A - Xe~x. 41 Since g'(X) = Xe~x > 0 for a l l A > 0, g(X) is increasing i n A when A > 0. Since g(0) — 0, we can conclude that g(X) > 0 for A > 0. Hence, G > 0, and furthermore, H > 0 w i t h any posit ive A. W e have shown H > 0. Obv ious ly A < 0 is true. Therefore, the log- l ike l ihood function I is local ly m a x i m i z e d at the M L E (p, A) . If the l ike l ihood equat ion (3.6) can be shown to y ie ld a unique A, then the M L E (p, X) uniquely maximizes the l ike l ihood funct ion. L e t M A ) = - A e -1 It can be seen that h(X) is increasing from h(Q+) — 1 to h(+oo) = +co . Note , h(0+) = 1 can be obtained by L ' H o p i t a l ' s rule. Hence, a unique solut ion to the l ike l ihood equat ion (3.6) exists. We can conclude that (p, A) uniquely maximizes / . In the fol lowing, we give the informat ion m a t r i x of the Z I P parameters. I = I\p I\\ where I - El ^ 1— p + pe~x p d2l \ ne A 1 — p + pe - A = np dX2J 1 e-x(l-p) ^X 1 — p + pe~x t In general, the m a x i m u m l ike l ihood est imator vector (p, A) has approx imate ly a no rma l d i s t r ibu t ion in a large finite sample. T h a t is, (1 )~*<(5),[/-» I-1 can be approx imated by X) which is obta ined by inser t ing the est imated parameter values p and A into the ana ly t ic expression for the inverse of the informat ion ma t r ix . 42 It is not ha rd to see that i f there is posi t ive remain ing inventory each per iod , the observed sales yields complete informat ion about demand. If this is the case, the demand is represented by the sales and the two are equivalent. We say then, the demand data is available i n which case the above m a x i m u m l ike l ihood es t imat ion can be appl ied. We refer to this case as estimation using the demand data. However, i n lost sales inventory systems the equivalence is no longer va l id . In the following sections, we w i l l look at the case where the demand da ta is not available. 3.2 M L E For ZIP Demand From Sales Data In this section, we extend Conrad ' s work to the Z I P demand. We consider a newsvendor system and an (s, 5 ) system. These systems present two different cases of da ta censoring, one w i t h stock-outs occurr ing at a constant level, the other at variable levels. For each system, we w i l l first describe the da ta censoring. 3.2.1 ZIP Demand Censored At a Fixed Level We start w i t h the newsvendor system where demands are censored at a constant level . R e c a l l that i n the newsvendor system, remaining inventory is not carr ied to the next per iod . T h e order quant i ty at the beginning of each per iod is the amount available for sales for that per iod. Hence, i f we denote the order quant i ty by y°, demand is censored at y° when there are stock-outs. Thus , the censoring level remains constant at y° as long as this ordering pol icy is i n place. F igure 3.1 i l lustrates the censoring i n a newsvendor system. T h e dash l ine broken by "o" is the order quanti ty. It is set to 6 for each day. T h e sol id line is the sales. T h e sales on day 4 and 7 hits the order quant i ty which means that they are censored at 6. T h e sales on a l l other days can be considered as fully representing demand since they fal l below the order quant i ty level. A m o n g them, sales on day 3 and 12 are zero. Censor ing W h e n a l l of the stock is sold in a per iod, the demand is known to be greater than or equal to sales. In such cases, we say that the demand is censored at sales. Le t the demand i n t ime per iod i be denoted Xt. Suppose {Xi}i>i are independent and ident ical ly d is t r ibuted (i.i .d.) and follow the Z I P d i s t r ibu t ion w i t h parameters p, A. Define the observed demand i n i, i n other words, the sales i n i, as X i =mm{Xi,y°) (3.9) 43 o o o o o o o o o ro O order quantity o 2 4 6 8 10 12 14 time Figure 3.1: Censor ing of Sales i n a Newsvendor Sys tem. where y° is predetermined and fixed. Hence, the true demand is greater than or equal to the sales, i.e. Xi > Xi. We now introduce a non-censoring indica tor variable. Le t Hence, the sales da t a set consists of pairs of the sales observation and the non-censoring indicator , (xi,Si), for i — 1 , 2 , . . . , n . W e rearrange the da ta so that the censored observations follow the uncensored, and among the uncensored observations the posi t ive demands follow the zero demands. Note that none of the zero demands in a newsvendor system are censored i f the pre-set i n i t i a l order quant i ty y° that determines the censoring level is posi t ive. However, i n the extreme case where it is not economical to ho ld stock, a l l demands are censored w i t h y° set to zero. T h i s is a degenerate case i n which no informat ion is available. L e t j denote the number of zero demands and r denote the number of uncensored observations. No te the r uncensored observations include the j zero demands. T h e number of censored demand observations is hence n — r. T h e L i k e l i h o o d If the ful l demand is observed, indica ted by di = 1, then Xi = a^. In this instance the cont r ibu t ion of the sales observat ion Xi to the l ike l ihood function is f(xi), where /(•) is the p robab i l i ty mass function (pmf) of the Z I P d i s t r ibu t ion . O n the other hand, when the demand is t runcated by the order quanti ty, which is 1 i f Xi < y° 0 if Xi > y° (not censored) (censored) (3.10) 44 indica ted by Si = 0, then we know X ; > xt = y° and the sales observat ion Xi contributes S(y° - 1) to the l ike l ihood function where S(y° - 1) is the surv iva l function of the Z I P demand at y° — 1 given by y° S(y°) F{y°) = 1 - F(y°) = 1 - £ P r ( X = x) r = 0 (3.11) x=0 where F ( - ) is the cumula t ive density function (cdf) of the Z I P d i s t r ibu t ion , F\(-) is the cdf of the Poisson d i s t r ibu t ion , and p and A are the parameters of the Z I P d i s t r ibu t ion . T h e fol lowing gives the l ike l ihood funct ion for (p, A) given (xi,Si),i = 1 , 2 , . . . , n , L = UUfixitSiy"-!)1-^ oc [(1 - p ) +pe-x]jpn-je-x^Xz[l - Fx(y° - l)]n-r where z = 521= i xi- Accord ing ly , the log- l ikel ihood is given by l = \nL = j\n[{l-p)+pe-x] + (n-j)\n(p)-\(r-j) + z \n(X) + (n - r) \n[l - Fx(y°-1)}. T h r o u g h differentiation and rearrangement, the l ike l ihood equations are obta ined as follows. A P = n ex - I (3.12) (3.13) (3.14) '-^ n ex — 1 We solve this system using the n l m i n routine i n S - P L U S . T h e informat ion m a t r i x is given as follows. Ipp Ip\ (3.15) 45 where I p p ~ E \ dp2 d2r ) n ( l - e~ A ) p(l - p + pe~x) ipx = hp=E{'MP:) ,-x 1 — p + pe x dx2J pFx(y° - : A fx(y°-l) e-x{l-p) l-Fx(y°-l) l - p + pe~xi' w i t h F\(-) and f\(-) the cdf and pd f of Poisson(A) . A N u m e r i c a l E x a m p l e Suppose we generate 30 i . i . d . da i ly demands d i , cfe, . . . , d 3o from a Z I P d i s t r ibu t ion w i t h p = 0.7, A = 5. Note , we use di to denote demand sample and s» to denote sales sample i n the fol lowing, to avoid confusions. R e c a l l , Z I P ( p , A) can be wr i t t en as a B e r n o u l l i mix tu re of Poisson(A) and zero w i t h p robabi l i ty p and 1 - p respectively. W e generate Z I P demand da ta i n this way. A n i n i t i a l order quant i ty y° is needed for generating sales data . W e set y° = 6 i n this example which yields an expected p ropor t ion of 0.27 of censored observations. T h i s is the o p t i m a l po l i cy for a par t icular cost s tructure. T h e sales da ta s i , S2, • • • , S30 are generated by Si = min(di,y°) In the generated da ta set from one par t icu lar repl icat ion of the s imula t ion , there are 18 exact sales observa-tions among which 5 are zero, and the remaining 12 observations are a l l censored at the order quant i ty 6. Table 3.1 gives the generated demand (di; i = 1 , 2 , . . . , 30) and sales (s,, i = 1 , 2 , . . . , 30). Note that i n this par t icu lar r andom sample, there are more censored observations than expected. T h e p ropor t ion of censored observations (12/30=0.4) is much higher than the expected p ropor t ion . T h e M L E ' s of the parameters are given as follows. Note , subscript d on estimates means that the demand da ta was used whi le the subscript s means that the sales da ta was used. Hence, so lv ing the l ike l ihood equations (3.5) and (3.6) us ing the demand da ta where there is no censoring, we obta in the M L E ' s of the parameters denoted by pd and A^ to be pd = 0.837, Ad = 5.54. Compara t ive ly , us ing the sales da ta where 46 i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 di 0 11 5 0 5 2 0 0 4 3 2 2 4 6 4 Si 0 6 5 0 5 2 0 0 4 3 2 2 4 6 4 i 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 di 4 6 10 6 3 8 10 5 7 0 7 4 6 6 9 Si 4 6 6 6 3 6 6 5 6 0 6 4 6 6 6 Table 3.1: Sample D a t a Genera ted F r o m ZIP(0 .7 , 5). there is censoring, so lv ing the l ike l ihood equations (3.14) and (3.15) gives ps = 0.837, A s = 5.45. Note that even though these estimates differ somewhat from the actual values, there is not much difference in the M L E ' s of the parameters from the demand da ta and the sales da ta despite the fact that the sales da ta represents the t runca ted demand i n 12 days out of a 30-day per iod . Fur thermore , the mean demand psXs = 4.56 is close to pdXd = 4.64 = n _ 1 Yli=i ^i, but differs considerately from the true mean demand p\ = 0.7 * 5 = 3.5. We w i l l i l lus t ra te how wel l the est imator compensates for the lost sales th rough a s imula t ion study i n Section 3.3. 3.2.2 ZIP Demand Censored At Variable Levels In this section, we derive the M L E ' s of the Z I P demand i n a more wide ly used re ta i l inventory system. W h e n there are fixed order ing costs in a system and inventory is carr ied over f rom one per iod to the next, an (s, S) po l icy is often used to control the inventory level. Such inventory systems are often referred to as (s, 5 ) systems. A n (s, S) po l icy is implemented as follows. Inventory levels are reviewed periodical ly, at discrete t ime points . T h e review t ime which is the interval between inventory review t ime points is usual ly pre-determined. If the inventory at the review point is less than or equal to s, an order that w i l l raise the inventory up to S is placed. Otherwise , no order is placed and the current inventory is mainta ined. In most p rac t ica l s i tuat ions, there is a lead t ime for the replenishment stock to arr ive, that is, when the order can not be fil led immediate ly . L e a d t ime is measured as length of the dura t ion between p lac ing a replenishment order and its a r r iva l . T h i s formula t ion of the inventory system is mot iva ted by the C a n a d i a n T i r e example. It was used for da t a analysis discussed i n Sect ion 2.4. 47 The Inventory Level W e assume a basic t ime uni t such as a day i n a week and assume bo th the review t ime and the lead t ime are fixed at some constant mul t ip le of this basic t ime uni t . For instance, i f we use one day as the basic t ime uni t , review t ime of 4 means the inventory pos i t ion is reviewed every 4 days, and lead t ime of 2 means that every replenishment order takes 2 days to arrive. We introduce some no ta t ion as follows using day as the basic t ime uni t . However, the model and the derivat ion i n the entire section remain va l id w i t h any general definit ion of the basic t ime uni t . Vi: the inventory level at the beginning of day i, w i t h Vi being the i n i t i a l inventory level, at the beginning of day 1; s: the reorder point , i t sets the level below which a replenishment order is placed; S: the level of the desired inventory when the current level is at or below s; Xf. the demand variable for day i (same as before); xf. the observed sales for day i (same as before); Sii the non-censoring indica tor for day i (same as before); rv: the review t ime which is the number of basic t ime units between reviews, e.g. rv = 4 days; Id: the lead t ime, e.g. Id = 2 days; Pi\ the reordering indica tor at the beginning of day i given by _ J 1 i f Vi < s, and i = 1 + k • rv, k = 0 , 1 , 2 , . . . (order) , ^ l \ 0 otherwise (do not order) , Oi: the amount to order at the beginning of day i. G i v e n the fixed review t ime and lead t ime, the relat ionship of consecutive inventory levels for any i is Vi+i = Vi-Xi + pi-idOi-id (3.16) where _ f S-Vi if Vi < s and i = 1 + k • rv for k = 0 , 1 , 2 , . . . , 1 ~ \ 0 otherwise. Thus , the inventory level at the beginning of day i + \ (Vj+i) is the pending order wh ich has arr ived at that same moment plus the remain ing inventory stock at the end of the pr ior day (i). T h e remain ing inventory is obvious ly Vi — X i . If i t happens that the inventory level was reviewed at the beginning of day % — Id and an order of S — Vi-u was placed to b r ing Vi-id which was below s to S, then that order wh ich has taken Id to arr ive should be added. T h e inventory level at the beginning of i — Id is reviewed only when i — Id equals one plus an integer mul t ip le of the review t ime, i.e. i — Id = 1 + A; • rv for k = 0 , 1 , 2 , — For instance, s ta r t ing from day one, the inventory level is reviewed every four days i f rv = 4. 48 C e n s o r i n g : A n E x a m p l e W e define the sales and the non-censoring indica tor variables as i n Sect ion 3.2.1. Since inventory is carried over f rom one per iod to the next, the inventory level is changing over t ime. So is the censoring level of observed sales. F igu re 3.2 i l lustrates the censoring i n a (s, 5 ) system w i t h a four day review per iod and a two day lead t ime. T h e dash l ine broken by "S" is the order-up-to quanti ty. It is set to 12. T h e dash line broken by "s" is the re-order level which is set to 5. T y p i c a l l y , i n a per iodic review (s, 5 ) system, inventory pos i t ion is checked only at the review t ime points . For instance, the inventory pos i t ion is reviewed every four days i f rv = 4. However, we assume here that the inventory is changed by either sales or ar r ived orders. Since the da i ly sales are observed, the inventory pos i t ion can be t racked on a da i ly basis as given by (3.16). T h e purpose of t racking inventory pos i t ion on a da i ly basis i n this example is so that the demand censoring level is known . Therefore, the (s, 5 ) pol icy is implemented as follows. S ta r t ing w i t h V\ = 10 at the beginning of day 1, the on-hand stock (the sol id line) is consumed by sales (the x- l ine or the sol id line w i t h x 's ) , and forms an inventory t rack ing pa th over t ime. T h e inventory level is reviewed s tar t ing from day 1 and every add i t iona l four days, that is, at the beginning of day 1, day 5, day 9 and so on . Since V\ = 10 is greater than s = 5, no order is placed on day 1. O n the contrary, at the beginning of day 5, V 5 = 3 falls below s, so an order of 5 — V 5 = 9 is placed to raise the inventory level to S. It arrives two days later, at the beginning of day 7. T h e inventory level is reviewed again at the beginning of day 9. Since Vg = 6 is greater than s, no order is placed. Hence, the on-hand stock w i l l be consumed for the next four days before the inventory level is checked against s. F ina l l y , V13 = 0 < s, so an order of 12 is placed. Whenever the sales meets the inventory level, we say that the sales are censored. In this figure, sales are censored whenever the so l id l ine overlays w i t h the x- l ine. We can see, in this example , that the sales on day 6, 12, 13, 14 and 17 are censored. B o t h sales on day 6 and 12 are censored at 1. A t the beginning of day 13, an order of 12 is placed as the inventory level has dropped to zero. Since this order w i l l not arr ive for two days and there is no stock available for sale, the sales on day 13 and 14 are censored at zero. T h e sales on day 17 is censored at 5. T h e L i k e l i h o o d Since inventory is carr ied over f rom one per iod to the next, the inventory level is changing over t ime. So is the censoring level of observed sales. It is obvious that the t runca t ion of the demand in per iod i, Xi, depends on the his tory pa th of sales Xi, . . . ,x\. Hence, Xi is no longer independent of Xj for j < i since Xi = mm(Xi,Vi). However, Xi\Vi is independent of Xj\Vj for i,j = 1 , 2 , . . . ,n and i ^ j . T h e joint 49 s — s — s — s — s — s — s — s — s — s — s — s — s — s - -s — s s s s -Figure 3.2: Censor ing of Sales i n a (s, S) Sys tem. p robab i l i ty of (xi, 5i) for i = 1 , 2 , . . . , n is therefore, Pr((xi,6i),i = 1 ,2 , . . . ,n\Vn,Vn-i,... ,V0) = Pi((xn, 5n)\Vn) Pr((a:n_i, <f„_i)|V„_i)... W i M V i ) = n^Jixrf'Sixi)1-5*. T h e second equal i ty i n the above is va l id since PrfY-r A \\V-\ - / f(Xd l f 5 i = l j t h a t i S X i = x i < Vi> ri((Xi,Oi)\Vi) - | s ^ . f §i = 0^ t h a t - s x . > x . = y._ for i = 1 , . . . , n. It can be seen that the censoring level Xi varies as the inventory level Vi varies. A s was done i n the newsvendor system, we reorder the sales da ta so that the uncensored (exact) sales observations come before the censored ones, and among either of the groups zero observations are first. Le t r denote the number of uncensored observations, j the number of uncensored zero sales, and k the number of censored zero sales. T h e l ike l ihood is obta ined for Z I P demand observed from an (s, 5 ) retai l inventory system w i t h fixed review t ime and lead t ime. Loc[(l-p)+pe-AyPn-:>e l Y l = n _ r _ k { l - F x { x i - l ) ) (3.17) where z = Ei=i xi a n ^ F\(xi — 1) = 0 when Xi = 0. 50 T h e l ike l ihood equations are given by fp = =r' ' x e A - l ' - i p e " A - (r - i) 4- ^i=l X i + T " Fx(XJ-1)-Fx(X{-2) _ n +k+l 1-Fx(xi-1) - u A g a i n we use the n l m i n routine to ob ta in the M L E ' s . T h e informat ion m a t r i x is given as follows. where I\p 1XX E[-dp^] n ( l - e~x) p(l -p + pe~x) dXdp' ,-x 1 — p + pe' 'XX = E[-= np | d\2i l - p + PFx{y° - 1 ) e - A ( l - p ) A 1 — p + pe~x + E ( . ^ J l - t t f e - l ) + ( l - F A ( a ; l - l ) ) ] ) F x ( X i - l ) w i t h F \ ( - ) and /A(-) the cdf and pd f of Poisson(A) . A N u m e r i c a l E x a m p l e A s imulated example is given below to i l lustrate parameter es t imat ion i n the (s, 5 ) system. T h i r t y days of i . i . d . demand d\, d2, • • • , d^o are generated from ZIP(0 .7 ,5 ) . T h e i n i t i a l (s, S) inventory po l i cy is a rb i t r a r i ly set as s = 5, S = 12 so that a moderate p ropor t ion of the demand is censored. Inventory is reviewed every four days and the lead t ime is two days (i.e. rv = 4, Id =2). Suppose the in i t i a l inventory Vo is 4. T w o sets of da ta are available, di the da i ly demand, and s* the observed sales for i = 1 ,2 , . . . , 30. In the generated da ta set from one par t icular run of the s imula t ion , there are 13 uncensored sales observations, and among these there are 4 zero demands. O n one day, there is no sale (i.e. x = 0) because of the zero stock. 51 A s for the newsvendor system i n the previous example, two estimates are obtained; one using the demand da ta and the other using the sales data . T h e estimate w i t h subscript d is the estimate from the demand da ta set while that w i t h subscript s is from the sales data . T h e es t imat ion results are: pd = 0.837, Ad = 5.54; ps = 0.872, \ s = 5.11. Note again psXs = 4.46 is close to pdXd = 4.64 = ' , the sample mean of demands. In this example, the point estimate of A from the sales da ta is closer to the true value compared to that from the demand data . B u t i t is not the case for p. Note the two sets of estimates are not as close as i n the previous example where the demand is censored at fixed inventory level. W h e n the demand is censored at a fixed posi t ive level, only the demand exceeding the fixed level is t runcated. In this example, where inventory level varies over t ime, low demand could also be censored because of low stock on shelf. Some demands are even censored at zero because no stock is left for sale at the start of that per iod . If this occurs, the sales observation which is censored at zero gives no informat ion for es t imat ion since it does not contr ibute to the l ike l ihood . Therefore, when the demand is censored at variable inventory levels, it is quite l ikely that there is more informat ion loss for es t imat ion purposes, which results i n less accurate estimates. A s a mat ter of fact, less accurate estimates are expected i n the fol lowing two cases: first, when there is a h igh chance of censoring; second, when there are many sales censored at zero because of no stock on hand . A n example for the first case is that the in i t i a l order quant i ty for the newsvendor system is set at a very low value. Obviously , the chance of censoring and the magni tude of censoring bo th affect parameter es t imat ion. Performance of the estimators w i l l be investigated th rough s imula t ion i n Section 3.3. 3.3 Effect of Estimation Error on Cost and Policy In Sect ion 3.2, we considered m a x i m u m l ike l ihood es t imat ion for the Z I P demand d i s t r ibu t ion from sales da ta t ak ing unobservable lost sales into account. Ignoring censoring i n sales da ta results i n an estimator w i t h negative bias, that is, on average the est imator underestimates the a r r iva l rate A i f the demand is Poisson d is t r ibuted . However, ignor ing censoring may not have as much impact on the to ta l expected cost. T h i s section explores the effect on the o p t i m a l policies and their expected costs of not considering lost sales i n parameter es t imat ion. T h e objective of this section is to investigate the sensi t ivi ty of the expected cost to es t imat ion error. W e perform a s imula t ion s tudy to compare the o p t i m a l policies and the expected costs using different parameter est imators. A newsvendor inventory system is considered for this purpose w i t h Poisson and Z I P demand, respectively. Sect ion 3.3.1 briefly reviews the newsvendor pol icy and its expected cost. Sect ion 3.3.2 presents the s imula t ion design. Section 3.3.3 reports the cost compar ison results for Poisson demand and Section 3.3.4 52 reports this compar ison for Z I P demand. The key observations from the s imula t ion s tudy are summarized in Sect ion 3.3.5. W e choose not to analyse an (s, S) system because of the absence of easily computed op t ima l values for s and S. T h a t is, an i terat ive computa t ion method is need for an ( s , 5 ) system. 3.3.1 Newsvendor Policy and Cost Reca l l , a newsvendor does not carry over inventory from one per iod to the next due to fast obsolescence of the product . Hence, when the demand d i s t r ibu t ion is known, op t imiza t ion of the newsvendor can be reduced to a single per iod model . T h e op t ima l order quant i ty can be obtained by m i n i m i z i n g the one per iod expected cost. Intui t ively, the o p t i m a l order quant i ty is the amount at which the expected cost of ordering too l i t t le balances w i t h the expected cost of ordering too much. Refer to Sections 5.2 and 5.3 for a detailed discussion of the newsvendor p rob lem and its model formulat ion. We give the o p t i m a l po l i cy and its expected cost in the fol lowing for convenience. T h i s op t ima l po l i cy is often referred to as the newsvendor po l i cy ( N P ) . L e t c denote a variable ordering cost (the cost per uni t of products ordered); let h denote the salvage value (the revenue brought i n by each uni t of products i f it is unsold); let p denote the penalty cost (the penal ty per uni t of products in shortage). It is reasonably and commonly assumed that h < c < p. T h e o p t i m a l policy, denoted by y(N\ is given by the fol lowing formula . yW=F- 1(fc; 6) for continuous demand and y (N) = min {y.yyF-1 (k; 0)} (3.18) y&z+ for discrete demand w i t h Z+ the set of non-negative integers. In bo th equations, k = i 0 < fc < 1, is referred to as the critical fractile; and F(x; 6) is the cdf of the demand d i s t r ibu t ion w i t h parameter 6. Let r(x,y) denote the cost of any order quant i ty y w i t h demand x. r(x,y) = h-h(;y-X} [iX^V> (3.19) [cy+p(x-y) ifx>y. T h e expected cost of the pol icy for a discrete demand is oo R(y,6) = ^ r f o i / M s ; 9) 2 = 0 y oo = cy-h £ ( y - x)f(x; 6) + p £ ( x - y)f(x; 6) (3.20) x=0 x=y where f(x; 6) is the pd f of the i . i . d . demand w i t h 0 denot ing the demand d i s t r ibu t ion parameters. T h e values of the costs c, h, p and the cr i t i ca l fractile k i n the s imula t ion s tudy are given in the next section. 53 3.3.2 Simulation Design A prac t i ca l me thod to compute the inventory pol icy and its cost when the demand d i s t r ibu t ion is unknown is to subst i tute the estimates of unknown parameters i n the o p t i m a l po l i cy and the expected cost formulas. For instance, (3.18) and (3.20) can be used for discrete demand newsvendor w i t h 6 replaced by its estimate. However , the po l i cy calculated this way may not be op t ima l . W e refer to this po l i cy as the estimated op t ima l pol icy. T h e s imula t ion s tudy starts w i t h da ta generation, proceeds to parameter es t imat ion followed by op t ima l pol icy computa t ion , and finishes w i t h cost comparison. In this section, we first define four est imated expected costs wh ich w i l l be compared to a baseline cost. We then specify the exper imenta l factors followed by the s imula t ion procedure. I. Expected Costs W h e n the unknown parameters are est imated, the op t ima l po l icy depends on the estimate of the demand d i s t r ibu t ion and so does the expected cost (refer to (3.18) and (3.20)). Thus , we compute the "estimated" o p t i m a l policies us ing various estimates of the unknown parameters and compare the expected costs of the policies. We describe four estimates of unknown parameters, define the o p t i m a l policies for each of the est imated dis t r ibut ions and the corresponding expected costs of the policies below. W e s tudy the fol lowing estimates: 1. M L E obta ined by using the unobserved but known demand data . 2. M L E using sales da t a considering censoring. 3. M L E using sales da ta after deleting a l l censored observations. 4. M L E using sales da ta w i t h censoring ignored. T h a t is, a l l sales are considered as if they are exact demand observations. We subst i tute the aforementioned estimates of the parameters i n (3.18) to compute the order policies. T h e corresponding expected cost w i t h y replaced by each of these policies is computed using (3.20) and it is referred to as the estimated expected cost. T h e mot iva t ion for this is as follows. W e assume that an in i t i a l newsvendor pol icy y0 is i n effect at the t ime da ta is collected. T h e n after observing this data , we estimate the demand d i s t r ibu t ion , recompute the newsvendor inventory pol icy y us ing (3.18), and then use it over the foreseeable future. To assess its performance, we compute its expected inventory cost using (3.20). In the fol lowing, we first explore four types of cost as the potent ia l candidates for compar ison purposes. T h e n we discuss why we choose to use the first two costs for the s imula t ion study. 54 1. Basel ine cost: T h e expected cost of the op t ima l po l icy w i t h respect to the true demand dis t r ibut ion , denoted by R(y*,90). It is given by (3.20). T h e op t ima l policy, y*, is computed using the true demand density, f(x; 90) where 90 denotes the true parameters. T h i s can only be obtained i n the ideal case where the demand d i s t r ibu t ion is completely known. 2. E s t i m a t e d expected cost: T h e expected cost of the estimated o p t i m a l po l i cy w i t h respect to the true demand d i s t r ibu t ion . It is denoted by R(y,90)- T h e estimated o p t i m a l po l i cy y is calculated by (3.18) where 9 is replaced by one of the four aforementioned M L E ' s . Refer to Tab le 3.2 for details. 3. E s t i m a t e d real cost: T h e mean real cost of the estimated op t ima l pol icy. It is computed as follows. R(y) = ^ = i r ^ y ) (3.21) n where r(xi,y) is given by (3.19). A n in i t i a l da ta set is used to calculate the estimated op t ima l pol icy y. T h e n y is set i n place i n an n per iod newsvendor inventory system from which a random sample of n demands is obtained. Note , this cost uses the real demand da ta whi le the first two costs use the demand d i s t r ibu t ion . T h e cost computed by. (3.21) can be considered as an estimate of the estimated expected cost. 4. E s t i m a t e d cost: T h e expected cost of the estimated o p t i m a l po l icy w i t h respect to the estimated demand d i s t r ibu t ion . It is denoted by R(y,9). For instance, the Z I P density w i t h the est imated parameters is used for comput ing bo th the cost and the order policy. Since the baseline cost is the true picture of the real wor ld , we use i t as a benchmark. In contrast, the est imated cost R(y,9) may be misleading since it does not respect the true demand d i s t r ibu t ion . Consider a Poisson demand. Ignoring censoring yields a negatively biased estimate of A. T h i s results in a lower order quanti ty. T h e est imated cost i n this case turns out to be less than the baseline cost which is better because of the incorrec t ly est imated demand d is t r ibu t ion . A l t h o u g h this may be what the newsvendor pract ica l ly expects, i t does not reflect the real wor ld . It is equivalent to m a k i n g the r ight decision based on wrong assumptions, which of course does not give a good result. Since i t is not even comparable to the baseline cost, we do not consider this cost in the s imula t ion study. It can be seen that the est imated real cost serves as a na tu ra l est imate of the baseline cost when the demand d i s t r ibu t ion parameters are unknown. However, since we are m a i n l y interested i n the effect of the parameter es t imat ion errors i n the op t ima l po l icy and that effect carr ied into the expected cost, we should el iminate other sources of error as much as possible. For this reason, we use the est imated expected cost R(y,9o) instead of the est imated real cost R(y). It is compared to the baseline cost R(y*,6o)- We emphasize 5 5 Parameter D e m a n d D i s t r i b u t i o n O p t i m a l Expec t ed Es t imates Poisson Z I P Pol ic ies Costs F r o m D e m a n d D a t a A d (Pd.Ad) lid R(vd) F r o m censoring considered A s (Ps, A„) Vs R(ys) Sales delete a l l censored da ta A s N Aa Vs R(vs) D a t a treat censored da ta as exact A s (Ps, A s ) Vs R(ys) t rue value of parameters A (P. A) y* R(y*) "This estimator is not considered for ZIP demand. Table 3.2: N o t a t i o n of Cost Compar i son for Newsvendor P r o b l e m . c 1 1 1 h 0.25 0.5 0.7 P 1.5 2 3 p—tl 0.4 0.67 0.87 Table 3.3: Cost Structure and k. here that the true demand density is used i n computa t ion of bo th expected costs. G i v e n that there is no significance ca r ry ing 80 i n the nota t ion of these costs, we omi t it for convenience. For the same reason, we say the expected cost when referring to the est imated expected cost where it does not cause confusion. Simi lar ly , we say the o p t i m a l po l icy when referring to the est imated o p t i m a l pol icy . W e summar ize the nota t ion in Table 3.2. II. F o c u s o f t h e S i m u l a t i o n S t u d y It is obvious that the expected cost depends on two things, the demand d i s t r ibu t ion parameters and the cost s tructure. Express ion (3.18) shows that the op t ima l po l i cy depends on the cost s tructure through the c r i t i ca l fractile k. Hence, we specify k at three levels, namely k = 0.4,0.67 and 0.87. Table 3.3 gives the specification of the cost structure and the corresponding c r i t i ca l fractile k. Since the est imated expected cost we chose to compare w i t h the baseline cost also uses the true demand d i s t r ibu t ion , i t depends on the d i s t r ibu t ion parameter estimate th rough the est imated op t ima l po l icy y. W e specify the Poisson parameter A at three levels; 2, 5, and 15 for low, med ium and high intensities. For a predetermined number of demand observations (i.e. a fixed sample size), we expect to have more censored observations w i t h a low order quant i ty than w i t h a h igh order quanti ty. T h i s can be easily seen 56 since M(x) is monotonica l ly non-increasing i n x where M(x) denotes a cumula t ive density function. T h a t is, 1 — M(xi) > 1 — M{x2) for x\ < x-2- Hence, the in i t i a l order po l i cy which determines the censoring level and the chance of censoring affects the es t imat ion and as a result, the expected costs. Table 3.11 summarizes the effect of the in i t i a l order pol icy on the expected costs. We w i l l discuss the effect in detai l later. V e r y often, sample size influences exper imental results as wel l . We include i t i n a pre l iminary study for Poisson demand. T w o levels are specified, 30 for smal l samples and 100 for large ones. However, we observed l i t t le change i n the results due to sample size (see Table 3.10). Therefore, we do not investigate this effect further. III. S imulat ion Procedure T h e procedure consists of four steps; da ta generation, parameter es t imat ion, o p t i m a l po l icy determinat ion and expected cost computa t ion . W e describe the details of each step for Po isson demand a l though we also carry out this analysis for Z I P data . Step 1: Data Generation D e m a n d da t a of sample size n, i.e. d \ , d 2 , . . . ,dn, I S generated from a Poisson density w i t h parameter value fixed at each level. Sales da ta is then obtained by compar ing each demand observat ion to the predetermined order quant i ty y° (or the in i t i a l po l i cy ) . T h a t is, Si = min(di,y°) for i = 1 , . . . , n . R e c a l l that sales s; is a censored demand, i.e. s, < di, when Si = y° whereas i t is a fully observed demand, i.e. Sj = di, when Si < y°. General ly , we set y° in such a way that it would give a moderate number of censored observations. For instance, we set y° = A for Po isson demands. It is above the newsvendor pol icy ( N P ) calcula ted w i t h the true value of A i n some cases whi le below the N P i n other cases depending on the cost constant k. Al te rna t ive ly , we could set y° at the N P based on the true d i s t r ibu t ion which varies w i t h k. For instance, y° = 4, 6, 8 at k = 0.4, 0.67, 0.87 for Poisson(5) . However, since we are not interested i n the sensi t ivi ty analysis of the in i t i a l order quanti ty, we use y° = A. A s a mat ter of fact, one wou ld not p rac t ica l ly know the demand d i s t r ibu t ion in the first place. Hence the in i t i a l order quant i ty w i l l be an estimate at best. W e generate 100 such samples of demand and sales for each level of the parameter. Step 2: Parameter Estimation We give i n detai l the four estimators described earlier. 57 i) Xa: the s tandard M L E of Poisson parameter. A s s u m i n g the r andom demand is fully observable, we have Elk*. (3.22) A . n T h i s is appl icable when demand can be observed completely. For instance, in some inventory systems every demand order is recorded and unmet demand is backlogged. In a l ighter vein, for a newsvendor system, the newsvendor may stay a round after the stock-out and see how many papers she would have sold. However, unmet demand is lost and the magni tude of lost sales is usual ly unobservable i n retai l inventory systems inc lud ing the newsvendor system. A fundamental question is "How much is gained by observing lost sales?". ii) A s : the M L E from sales da ta considering censoring. Its computa t ion follows the same line of logic discussed i n Sect ion 3.2. U s i n g the same nota t ion i n Sect ion 3.2, the l ike l ihood funct ion for A given (xi, Si),i = 1 , 2 , . . . ,n, is L = n ^ / f c ) * ^ 0 - ! ) 1 - ' ' oc e - r A A z [ l - F A ( y ° - l ) ] n - r (3.23) where z = YH=i xi a n d is the cdf of the Poisson d i s t r ibu t ion . Accord ing ly , the log- l ike l ihood is given by I = In L = -rX + z ln(A) + (n - r ) l n [ l - Fx(y° - 1)]. (3.24) T h e M L E A s can be obtained by solving the fol lowing equation. § — • ! - < - ' > i ^ V * where f\(-) is the pd f of the Poisson d is t r ibu t ion . A g a i n , we solve this system using the n l m i n rout ine i n S - P L U S . i i i) As: the est imator using exact sales observations only. In this case, we delete a l l censored observations. We then compute the M L E using the new data set. Equiva lent ly , \ _ E?=i si*I(si<y°) s~ ZUKsi<y) (3-26) where 7(s, < y°) = 1 i f s, < y°, otherwise I(si < y°) = 0 for i = 1 , . . . ,n. I(-) is referred to as an indicator funct ion. Since the new d a t a only consists o f values smaller than y°, i t is obvious tha t A s is biased downward. Note , a l l demands w i l l be censored at zero when the order quant i ty is set at zero. In this case, this method cannot be used for es t imat ion. 58 iv) A s : the est imator obta ined from ignor ing censoring. Sales da t a is used i n es t imat ion as i f i t completely represented demand. T h a t is, censored da ta is treated as exact. Hence, En A s = i = i S i , (3.27) n Since s; < di, Xs is a negatively biased est imator of A as wel l . It is clear tha t A s has a larger negative bias than A s . Step 3: Estimated Optimal Myopic Policies W e compute the op t ima l policies yd, ys, Vs and ys (summarized i n Table 3.2) using (3.18) w i t h F(-) replaced by the est imated cdf 's. For instance, yd = m i n {j/ : y > F^ik)} where Fd(y) = ZtoeXP{~)t){'Xd)i-Note , these policies are op t ima l w i t h respect to the corresponding estimated demand dis t r ibut ions . W h e n more da ta is collected over t ime, the M L E ' s w i l l be re-computed and accordingly the est imated op t ima l po l i -cies are updated . W e discuss i n Chap te r 5 how demand es t imat ion upda t ing and po l icy upda t ing interact under a Bayes ian framework when demand is not completely observable. T h i s characterizes retai l inven-tory systems. In this framework, the es t imat ion upda t ing and po l icy upda t ing are considered as stochastic processes as t ime evolves and more informat ion is gathered. We show that for a two per iod newsvendor system the o p t i m a l order quant i ty w i t h respect to the Bayes ian estimate of demand density is not op t imal if the effect of es t imat ion effort on future cost is considered. D i n g , P u t e r m a n & B i s i [18] extended these results to n-periods. However, here we are concerned w i t h the effect of es t imat ion error on the current-per iod cost using M L E w i t h no demand and pol icy updat ing . Step 4: Expected Costs T h e corresponding expected costs can be obtained, as discussed earlier, by subs t i tu t ing the four order policies i n (3.20). W e give R(yd) below as an example. oo R(Vd) = ^2r(x,yd)f(x) x=0 yd oo = cy-hJ2(yd-x)f(x)+pYi^-yd)Hx). (3.28) x—0 x—yd R e c a l l , the true demand density f(x) is used here as opposed to the est imated density i n op t ima l policies de terminat ion . Hence, R(ys), R(ys), R(ys) a r e of the same form as (3.28). 59 k A 2 5 15 0.4 case 1 case 4 case 7 0.67 case 2 case 5 case 8 0.87 case 3 case 6 case 9 Table 3.4: S imula t ion Set t ing and Case Numbers for Po isson D e m a n d . T h e procedure for Z I P demand is s imi lar . F i r s t , demand da ta is generated by Z IP (p , A) w i t h the pair of parameters set at different levels. Sales da ta is generated as described i n step 1. T h e parameter pair is es t imated i n four ways as for the Poisson case given i n step 2. Fo r an explana t ion of how lost sales is considered for i n the Z I P case, refer to Sect ion 3.2.1. T h e next two steps, o p t i m a l po l icy determinat ion and expected cost computa t ion remain exact ly the same as the Poisson case. 3.3.3 Poisson Demand T h i s section reports the po l i cy and the cost compar ison results for each combina t ion of the factors. Table 3.4 enumerates the nine settings of a two-way layout . We summar ize the s imula t ion results by the level of A in Table 3.7, 3.8, 3.9. Table 3.7 compares the costs w i t h A = 2 across c r i t i ca l fractiles k. S imi la r ly , Tables 3.8 and 3.9 report results for A = 5 and A = 15, respectively. One hundred samples of size 30 are r andomly generated for each case. T h e i n i t i a l order quanti ty (y°) was set to A i n each table to y ie ld a good propor t ion of censored sales observations. Table 3.5 shows the p robab i l i ty of censoring at the in i t i a l order quant i ty for each demand density. Note , k affects bo th the costs and the o p t i m a l order quantit ies but not the parameter es t imat ion. T h e i n i t i a l order quant i ty y° is fixed for each A across k so that the r andom samples remain the same i n each table, which enables us to focus on the effect of k on costs. Usua l l y y° is est imated from the h is tor ica l demand pat tern given k. However, since y° only plays a role i n generating sales data , we fix it regardless of k. O n the contrary, y* depends on k. F igure 3.3 gives the cdf of Poisson(5) from which we can easily determine y* = 4, 6, 8 for k = 0.4, 0.67, 0.87. Table 3.6 gives the true o p t i m a l order quant i ty for each combina t ion of A and k. B r i e f S u m m a r y o f R e s u l t s A brief summary of the key observations is given here, followed by the detai led discussion on the effects of the d i s t r ibu t ion parameter, the c r i t i ca l fracile, the sample size and the i n i t i a l order pol icy. C o m b i n i n g the observations about cost and pol icy w i t h respect to the various estimates of the Poisson 60 A 2 5 15 y° 2 5 15 Vx(X > y°) 0.59 0.56 0.53 Table 3.5: P r o b a b i l i t y of Censor ing at y°. 1 1 1 1 r-0 5 10 15 20 x Figure 3.3: The cdf of Poisson(5) w i t h k « 0.67. A k 2 5 15 0.4 1 4 14 0.67 2 6 16 0.87 4 8 19 Table 3.6: O p t i m a l P o l i c y y*. 61 R o w M e a n k N u m b e r Es t ima te 0.4 0.67 0.87 1 Xd = 1.998 2.672 2.818 2.785 M L E - d e m a n d (0.251 a ) (0.004) (0.007) (0.014) 2 Xs = 2.013 2.672 2.821 2.793 M L E - s a l e s (0.317) (0.004) (0.030) (0.036) 3 A s = 0.669 2.996 3.219 3.597 Delete censored (0.120) (0.033) (0.112) (0.504) 4 I s = 1.453 2.669 2.812 2.828 Ignore censoring (0.122) (0) (0) (0.106) 5 baseline (A = 2) 2.669 2.812 2.773 "The numbers in the parentheses are standard deviations. Table 3.7: M e a n and S tandard Dev ia t i on of Expec t ed Costs w i t h A = 2. parameter for a l l three demand intensities, we conclude that despite the fact that we see a s trong effect of the est imators on the o p t i m a l policy, the effect on the expected cost is less d ramat ic . T h i s is because the cost funct ion is re la t ively flat i n the neighborhood of y* for a l l three levels of the cost constant k we specified i n the s imula t ion . W e see from Table 3.7, 3.8 and 3.9, the s imple est imator A s , ignor ing censoring, is well-suited for expected cost computa t ion . Therefore, one may favor A s for its s impl ic i ty wi thout sacrificing much i n the expected cost. However, deleting censored observations is problemat ic (row 3), especially when k is large. Fur thermore , compar ing the expected costs from sales da ta while consider ing censoring (row 2) and those from sales da ta ignor ing censoring (row 4), we can conclude that censoring does not affect cost as much as i t does the o p t i m a l policies. T h i s is a key observation. In other words, the expected cost is not as sensitive as the o p t i m a l po l icy to the es t imat ion errors from not considering censoring. T h i s is t rue in a l l the newsvendor systems w i t h Poisson demands considered i n this section. In fact, these two costs are fair ly close i n a l l nine cases. We w i l l see that this holds true for Z I P demand i n the next section. Effect of the Parameter and the Critical Fractile Table 3.7 gives the mean and the s tandard devia t ion of the expected costs for case 1, 2 and 3 where A = y° = 2 (see Table 3.4 for reference on the case numbers) . We observe the fol lowing from this table, i) W e first note that among the four estimates of the Poisson parameter A, the first two have a mean estimate close to the true value (A = 2) and the last two bias downward , especially the t h i r d one which is most biased. Between the first two estimates, the second one has a greater s tandard devia t ion . 62 1 2 from demand 1 2 from sales 1 2 3 ignore censoring Figure 3.4: T h e E s t i m a t e d O p t i m a l P o l i c y for Poisson(2) w i t h A; = 0.4. True O p t i m a l P o l i c y y* = 1. ii) T h e mean costs corresponding to the four estimates are a l l greater than the baseline cost of the op t ima l po l icy w i t h respect to the true demand d i s t r ibu t ion (row 5). T h e mean costs obta ined from using sales da ta (row 2) are very close to those from using demand da ta (row 1) for a l l three levels of k, and they are both close to the baseline costs. T h e costs from sales da ta have larger s tandard deviat ions i n general than those from demand. F igu re 3.4 shows very s imi lar histograms of the es t imated o p t i m a l policies obtained from using demand da ta and sales da ta for the case where k = 0.4. T h i s s imi la r i ty remains for k — 0.67 and k — 0.87; see Figures 3.5 and 3.6 for these two cases respectively. Note , the true o p t i m a l po l icy y* for each (A, k) combina t ion is given i n Table 3.6. i i i) T h e mean costs from ignor ing censoring, i.e. t reat ing censored da ta as exact (row 4) are close to those from demand da ta (row 1), sales da ta (row 2) and baseline (row 5) despite the fact that the mean estimate A s = 1.453 is quite different from the true value, 2. T h e s tandard devia t ion of the cost at k = 0.4 when we ignore censoring is zero because the op t ima l order quantit ies are the same for a l l samples. It can be seen from F igure 3.4 that the est imated order po l icy equals 1 for a l l the generated samples (ys = 1). T h i s is due to the discreteness of demand even though the estimate of A varies from sample to sample. S imi l a r l y when k = 0.67. Therefore, this est imator seems to be fair ly robust except when k is large. iv) In contrast , the mean costs from deleting censored da ta (row 3) are different from a l l others, especially for large k. T h e s tandard deviations are also not iceably larger. In addi t ion , the s tandard devia t ion (row 3) increases d ramat ica l ly as k increases. T h e est imated o p t i m a l po l i cy equals zero (ys = 0) except for few 1 2 delete censored 63 1 2 from demand 1 2 from sales delete censored ignore censoring Figure 3 .5 : T h e E s t i m a t e d O p t i m a l P o l i c y for Poisson(2) w i t h k = 0.67. True O p t i m a l P o l i c y y* = 2. 2 3 4 from demand 2 3 4 from sales delete censored ignore censoring Figure 3.6: T h e E s t i m a t e d O p t i m a l P o l i c y for Poisson(2) w i t h k = 0.87. True O p t i m a l P o l i c y y* = 4. 64 M e a n k Es t ima te 0.4 0.67 0.87 \ d = 5.023 6.063 6.267 6.193 (0.391 a ) (0.031) (0.045) (0.042) A s = 5.054 6.071 6.276 6.197 (0.479) (0.046) (0.052) (0.050) \ s = 3.027 6.432 6.836 7.076 (0.279) (0.167) (0.277) (0.345) I s = 4.134 6.144 6.340 6.331 (0.210) (0.084) (0.087) (0.123) baseline (A = 5) 6.046 6.240 6.181 "The numbers in the parentheses are standard deviations. Table 3.8: M e a n and S tandard Dev ia t i on of E x p e c t e d Costs w i t h A = 5. samples (See F igu re 3.4). v) L o o k i n g at the table column-wise, we can see that the gap between each of the mean costs (row 1-4) and the baseline costs (row 5) increases as k increases. A l s o , the s tandard devia t ion of each cost (row 1-4) consistently increases i n k. T h i s arises because the o p t i m a l order quant i ty becomes more sensitive to es t imat ion as k increases. vi) A s expected, the discrepancies between the mean costs (row 1-4) and the baseline costs (row 5) seem to be consistent w i t h the difference between the mean estimates and the true value of the unknown parameter A for large k. C o m b i n i n g this point and point ( iv) , we can conclude that the difference between the expected cost and the baseline cost becomes larger as the es t imat ion error increases. It gets worse when k is large. S imi la r ly , Table 3.8 gives the mean costs and the corresponding s tandard deviat ions for cases 4, 5, 6 where A = y° = 5 and Table 3.9 for cases 7, 8, 9 where A — y° = 15. W e make s imi la r observations as those from Table 3.7 for bo th demand densities except that point (iv) no longer applies. W e reiterate a common observation. Genera l ly judg ing from the mean costs i n row 4, ignor ing censoring results i n higher expected cost t han that of the M L E considering censoring. It is especially so w i t h h igh cost constant (k = 0.87). E v e n though the mean costs i n row 4 may s t i l l be close to the baseline cost, the s tandard deviations are much greater than those i n row 1 and 2. To unders tand how the es t imat ion error of the unknown parameter A affects the expected cost, we further investigate the effect on the est imated op t ima l policy. T h e est imated po l i cy refers to the op t ima l 65 M e a n k Es t ima te 0.4 0.67 0.87 \ d = 15.009 16.871 17.175 17.001 (0 .659°) (0.041) (0.046) (0.044) A s = 15.086 16.874 17.186 17.013 (0.755) (0.052) (0.063) (0.057) A s = 11.672 17.431 18.110 18.132 (0.487) (0.188) (0.338) (0.425) I s = 13.473 16.984 17.337 17.192 (0.356) (0.082) (0.116) (0.136) baseline (A = 15) 16.839 17.153 16.975 "The numbers in the parentheses are standard deviations. Table 3.9: M e a n and S tandard Dev ia t i on of Expec t ed Costs w i t h A = 15. po l i cy der ived from using an estimate of demand density. W e first compare the four estimators. Figures 3.7, 3.8 and 3.9 give the boxplots of the estimators w i t h A = 2, 5, 15, respectively. It can be seen from a l l three figures that A s takes lost sales into account very well since A s is close to Xd- However, As has a larger s tandard devia t ion . In each case, A^  and As are w i th in 2 s tandard deviat ions of the true value of A. However, bo th A s and A s are clear ly biased to the low side, and A s is more biased than A s . In addi t ion , A s has a larger s tandard devia t ion for m e d i u m and h igh levels of A. Figures 3.10 and 3.11 give histograms of the est imated o p t i m a l policies w i t h k = 0.4 and A = 5 and 15, respectively. T h e est imated o p t i m a l policies corresponding to A^ ("from demand") and A s ("from sales") are very s imi lar for the three demand intensities (Figures 3.4, 3.10 and 3.11) w i t h the except ion of ys which has a larger s tandard devia t ion for A = 5. T h e other two "estimated" policies ("delete censored" and "ignore censoring") bo th have a negative bias. Fur thermore , s imi lar to Figures 3.7, 3.8 and 3.9, ys has a greater bias. For instance of A = 2, the true op t ima l po l icy at k = 0.4 is one. F igu re 3.4 shows that while both yd and ys are evenly d is t r ibu ted at 1 and 2, ys concentrates on 1. O n the other hand, ys takes value 0 most often whi le i t is 1 on ly occasionally. Note , whi le the bias of the policies results f rom the underest imat ion of the parameter , the fact that the demand density is discrete has an effect as wel l . Since the order quant i ty must be an integer, when k is between F(0) and F ( l ) , the order quant i ty takes value 1. T h i s explains why ys is constant. Since ys equals the true op t ima l policy, the cost corresponding to As has a mean value equal to the baseline cost w i t h zero s tandard devia t ion . There are s imi lar observations about the "estimated" 66 from demand from sales delete censored ignore censoring Figure 3.7: T h e Es t imates of A for Poisson(2) . from demand from sales delete censored ignore censoring Figure 3.8: T h e Es t imates of A for Poisson(5) . 67 from demand from sales delete censored ignore censoring Figure 3.9: T h e Es t imates of A for Poisson(15) . o p t i m a l policies when A = 5 and 15. To conclude, we observe a s imi la r pa t te rn i n the "estimated" policies corresponding to various estimates of the parameter. Hence, we may conjecture that the es t imat ion error has a consistent effect on the "estimated" op t ima l policies. Effect of the Sample Size To see the effect of sample size on costs, we generate 100 samples of size 100 for A = 5 (the m e d i u m demand) and compute the costs. T h e costs are given i n Table 3.10 i n para l le l to those f rom Table 3.8. C o m p a r i n g the costs column-wise for each level of k, we see there is ha rd ly any effect of sample size. Large sample (n = 100) s l ight ly reduces the s tandard deviations of the four costs at k = 0.4 and 0.6, while the reduct ion becomes large at k — 0.87. T h e mean costs corresponding to the two M L E ' s , A^ and A s , are s l ight ly closer to the baseline cost across A;. A s far as es t imat ion is concerned, we see the same pa t te rn i n F igure 3.12 as i n F igu re 3.8 but the s tandard devia t ion is apparently reduced by the large samples. Hence, just as expected, we have a s imi lar observation when compar ing F igure 3.13 to F igu re 3.10. T h a t is, large sample size has no effect other than reducing the s tandard deviations of the cost of the "est imated" policies, as well as, the s tandard deviat ions of the estimators of the parameter. Therefore, i t seems that large sample size does not p lay a significant role as far as cost is concerned. However, the s tandard dev ia t ion of a l l four estimates w i t h n = 100 reduces to about half of the s tandard deviations of those w i t h n = 30. 68 3 4 from demand 3 4 5 from sales delete censored ignore censoring Figure 3.10: T h e E s t i m a t e d O p t i m a l P o l i c y for Poisson(5) w i t h k = 0.4. True O p t i m a l P o l i c y y 10 12 14 from demand 10 12 14 from sa les 10 12 14 delete censored 10 12 14 ignore censoring Figure 3.11: T h e E s t i m a t e d O p t i m a l P o l i c y for Poisson(15) w i t h k = 0.4. True O p t i m a l P o l i c y y 69 from demand from sales delete censored ignore censoring Figure 3.12: T h e Es t imates of A for Poisson(5) from large samples. 3 4 from demand 3 4 from sa les 3 4 delete censored 2 3 4 5 ignore censoring Figure 3.13: T h e E s t i m a t e d O p t i m a l P o l i c y for Poisson(5) from large samples (n = 100) w i t h k = 0.4. True O p t i m a l P o l i c y y* = 4. 70 M e a n k M e a n Es t ima te 0.4 0.67 0.87 Es t ima te (n = 30) n = 30 n = 100 n = 30 n = 100 n = 30 n = 100 (n = 100) Xd = 5.023 6.063 6.057 6.267 6.245 6.193 6.184 A d = 5.038 (0 .391°) (0.031) (0.021) (0.045) (0.020) (0.042) (0.003) (0.220) Xs = 5.054 6.071 6.057 6.276 6.251 6.197 6.184 A 5 = 5.035 (0.479) (0.046) (0.021) (0.052) (0.031) (0.050) (0.003) (0.257) I s = 3.027 6.432 6.466 6.836 6.806 7.076 7.018 I s = 3.018 (0.279) (0.167) (0.154) (0.277) (0.262) (0.345) (0) (0.161) I s = 4.134 6.144 6.156 6.340 6.316 6.331 6.358 I s = 4.134 (0.210) (0.084) (0.081) (0.087) (0) (0.123) (0.115) (0.116) baseline 6.046 6.240 6.181 baseline "The numbers in the parentheses are standard deviations. Table 3.10: Expec t ed Costs w i t h A = 5, n = 30 vs. n = 100. Effect of the Initial Order Policy A s we ment ioned earlier, the i n i t i a l order quant i ty plays a role i n parameter es t imat ion as it determines the level of censoring. To investigate this effect, we set a very low order quant i ty for the h igh demand setting, namely y° = 5 for A = 15. T h i s censoring level is so low that a l l da t a generated from one run of s imula t ion was censored (Note P r ( X > 5) = 0.999 for Poisson(15)). Since a l l sales i n each sample were censored at the order quant i ty 5, a l l samples are ident ical . Hence, y° = 5 is an extreme case since 100 r andom samples can be considered as one sample consist ing of a single censored observat ion as far as es t imat ion is concerned. T h i s explains why A s has the same value (17.042) for a l l samples. S i m i l a r l y for A s (see F igure 3.14). N o w that there were no exact observations i n the sales samples, we d id not consider est imator Xs and the costs. E v e n i f there are a few exact observations i n generated sales samples, A s can be expected to be a poor est imator . Thus , we no longer consider i t . F igure 3.15 shows the est imated o p t i m a l policies corresponding to the four estimates of A at k = 0.4. We see that ys overestimates the true o p t i m a l po l icy (14) while ys underestimates i t w i t h a large bias. A g a i n , bo th ys and ys have no var ia t ion . Table 3.11 compares the costs from the case y° — 5 to those i n Table 3.9 when y° = 15. T h e costs corresponding to the estimate from demand da ta (the first row) are kept for compar ison between rows. Costs when there is a low order quant i ty are the same as those when there is a h igh order quant i ty because the generated demand samples are ident ical i n the two cases. C o m p a r i n g the costs corresponding to A s (in 71 from demand from sales delete censored • ignore censoring Figure 3.14: T h e Es t imates of A for Poisson(15) w i t h y° = 5. 5 10 from demand 5 10 from sales 5 10 delete censored 5 10 ignore censoring Figure 3.15: T h e E s t i m a t e d O p t i m a l P o l i c y for Poisson(15) w i t h y° = 5 (A; = 0. 72 M e a n k M e a n Es t ima te 0.4 0.67 0.87 Es t ima te (y° = 15) y ° = 15 y ° = 5 y° = 15 y ° = 5 i , 0 = 15 y° = 5 (y° = 5) Xd = 15.009 16.871 16.871 17.175 17.175 17.001 17.001 \ d = 15.009 (0 .659°) (0.041) (0.041) (0.046) (0.046) (0.044) (0.044) (0.659) I s = 15.086 16.874 17.131 17.186 17.506 17.013 17.275 Xs = 17.042 (0.755) (0.052) (0) (0.063) (0) (0.057) (0) (0) I s = 13.473 16.984 20.500 17.337 24.006 17.192 29.068 I s = 5 (0.356) (0.082) (0) (0.116) (0) (0.136) (0) (0) baseline 16.839 17.153 16.975 baseline "The numbers in the parentheses are standard deviations. Table 3.11: Expec t ed Costs w i t h A = 15, y° = 15 vs. y° = 5. the second row) from two order quantit ies, we see that the mean costs f rom y° = 5 are s l ight ly higher (by less than 2%) across k despite the fact that the mean estimate A s i n this case is almost 13% higher than its counterpart . T h i s is consistent w i t h our former observation that the relat ive difference between the costs are smaller t han that between the estimates. However, the mean costs i n the t h i r d row show a large discrepancy at each level of k between the two cases of order quanti ty. T h i s discrepancy increases w i t h k. In addi t ion , the difference between the costs i n row 3 and the costs i n row 2 when y° = 5 becomes large at med ium and h igh k. T h e mean cost i n row 3 is 37%, 68% higher than that i n row 2 when k = 0.67 and 0.87, respectively. Hence, a l though the s imple est imator A s wh ich does not take censoring into account yields generally good enough expected costs across k, its performance was poor i n some extreme cases. T h i s is evident in the present example where y° is set very low. A n in i t i a l po l icy as low as 5 yields bad sales samples i n terms of s ta t i s t ica l es t imat ion since a l l sales are censored at the same low order quanti ty. Obviously , A s seriously underestimates A, wh ich results i n underest imated policies (see F igure 3.15). T h e effect is drastic when the c r i t i ca l fractile k is large. Hence, i t shows a s trong effect on the expected costs when censoring is not considered. Nevertheless, Xs performs wel l , at a computa t iona l cost, even i n the extreme case. S tandard deviat ions of 0 i n row 2 and 3 at y° = 5 are due to no var ia t ion i n data . 73 A 2 5 15 y° (P = 0.7) 2 4 11 P r ( X > y°) 0.42 0.51 0.62 y° (P = 0.3) 2 2 5 P r ( X > y°) 0.18 0.29 0.30 Table 3.12: T h e In i t i a l Order Quant i t ies (y°). 3.3.4 ZIP Demand R e c a l l , the density function of Z IP (p , A) is as follows. Pr(AT = 0) = (1 -p) +pe~x, Pr(X = x)= pe~xXx/x\ for x = 1, 2 , . . . . T h u s E(X) = p • A. In the s imula t ion s tudy for Z I P demand, we set p at two levels; p = 0.3 and 0.7. T h e settings of parameter A and cost s tructure remain the same as those for Poisson demand (see Table 3.4). T h e i n i t i a l order quant i ty is set, i n general, at a level close to the expected demand, i.e. the smallest integer greater than p- A (except for ZIP(0.3,2)) so that a number of censored sales w i l l be generated. Table 3.12 gives the order quantit ies and the probabil i t ies of demand to be censored at these quantit ies. Since the s imula t ion s tudy for the Poisson demand shows that A s , the estimate that deletes censored observations, is not good for cost computa t ion , we no longer consider this est imator here. O n the other hand , a new density estimate is in t roduced which is obta ined by using a Poisson fit to the Z I P d is t r ibu t ion . T h e new Poisson est imator is denoted A p , and is obta ined from (3.22) us ing demand data. Hence, the Z I P ( p , A) density is est imated to be P o i s s o n ( A p ) . In this section, we investigate the effect on cost of density es t imat ion ignor ing the ex t ra p robabi l i ty mass at zero of Z I P , as wel l as that of ignor ing censoring i n data . It can prac t ica l ly reduce the es t imat ion effort i f a s impler d i s t r ibu t ion fit or ignor ing censoring does not affect the cost much . T h i s is impor tan t and i t is the key mot iva t ion of the s imula t ion study. T h e o p t i m a l policies and their expected costs are computed i n a s imi lar manner as Poisson demand. A g a i n , 100 r andom samples of size 30 are generated i n each case. I. p = 0.7 T h e results for p = 0.7 w i t h A = 2, 5 and 15 are given i n Table 3.13, 3.14, and 3.15, respectively. We make the fol lowing observations. 74 R o w M e a n k N u m b e r Es t imates 0.4 0.67 0.87 1 A d = 1.952 (0.351 a ) 2.099 2.302 2.258 pd = 0.722 (0.113) (0.023) (0.075) (0.057) 2 A 5 = 2.011 (0.583) 2.103 2.322 2.288 p~s = 0.729 (0.128) (0.039) (0.103) (0.100) 3 I s = 1.131 (0.161) 2.097 2.375 2.427 ps = 0.914 (0.155) (0.003) (0.063) (0.069) 4 A p = 1.398 (0.286) 2.098 2.295 2.303 Poisson fit (0.032) (0.051) (0.127) 5 baseline (A = 2, p = 0.7) 2.093 2.268 2.231 "The numbers in parentheses are standard deviations. Table 3.13: M e a n and S tandard Dev ia t i on of E x p e c t e d Costs w i t h ZIP(0 .7 , 2). R o w M e a n k N u m b e r Es t imates • 0.4 0.67 0.87 1 A d = 5.036 (0.535°) 5.096 5.221 5.005 pd = 0.703 (0.097) (0.086) (0.064) (0.060) 2 A s = 5.142 (0.738) 5.102 5.234 5.026 ps = 0.703 (0.097) (0.090) (0.100) (0.085) 3 I s = 3.487 (0.203) 5.096 5.404 5.342 ps = 0.721 (0.099) (0.078) (0.147) (0.151) 4 I p = 3.538 (0.596) 5.050 5.272 5.212 Poisson fit (0.041) (0.130) (0.255) 5 baseline (A = 5, p = 0.7) 5.025 5.171 4.961 "The numbers in the parentheses are standard deviations. Table 3.14: M e a n and S tandard Dev ia t i on of E x p e c t e d Costs w i t h ZIP(0 .7 , 5). 75 R o w M e a n k N u m b e r Est imates 0.4 0.67 0.87 1 Xd = 14.918 (0.792 a ) 14.833 14.426 14.417 pd = 0.714 (0.089) (0.417) (0.088) (0.121) 2 A s = 15.720 (2.783) 14.969 14.672 14.665 ps = 0.714 (0.089) (0.636) (0.785) (0.788) 3 I s = 10.737 (0.197) 14.960 15.322 15.318 ps = 0.714 (0.089) (0.349) (0.255) (0.259) 4 A p = 10.645 (1.429) 14.673 14.986 14.982 Poisson fit (0.097) (0.547) (0.555) 5 baseline (A = 15, p = 0.7) 14.598 14.363 13.583 "The numbers in the parentheses are standard deviations. Table 3.15: M e a n and S tandard Dev ia t i on of E x p e c t e d Costs w i t h ZIP(0 .7 , 15). i) F igures 3.16, 3.18 and 3.20 give boxplots of the estimates of A. F igure 3.17, 3.19, and 3.21 give boxplots of the estimates of p. A s imilar pat tern is presented from this set of figures compared to the Poisson case discussed i n Sect ion 3.3.3. T h a t is, the mean estimate of the parameter pai r (A, p) f rom sales da ta while consider ing lost sales, ( A s , ps), is close to that from demand data , (Xd, pd), but w i t h a larger s tandard devia t ion . B o t h are close to the true value of the parameters. For A = 15, the s tandard devia t ion of A s is not iceably large (see Table 3.15). E s t i m a t i o n ignor ing censoring gives a negat ively biased estimate of A w i t h a smaller s tandard devia t ion for a l l three cases. For A = 5 and 15, the estimate of p, ps, has a s imilar d i s t r ibu t ion as the other two estimates, pd and ps, w i t h the mean estimate close to the true value 0.7. B u t i t is not true for A = 2. It has a larger s tandard devia t ion . W h a t is p roblemat ic is that ps is occasionally calcula ted to be greater than 1. It is outside of the va l id range for a p robab i l i ty parameter est imator. Th i s is because of the negatively biased estimate A s . Reca l l , ps is given by expression (3.14) which does not guarantee a value between 0 and 1. Trea t ing the censored sales as exact yields an estimate of A biased downward. A smal l est imate for A consequently gives a large estimate of p wh ich may go beyond 1. Cons t ra ined M L E may be used i n this case. In general, the estimates of the B e r n o u l l i parameter p seem to vary l i t t le across various es t imat ion methods as A increases. In contrast, the estimates of A change d ramat i ca l ly w i t h the est imat ion method . A n explanat ion for this is that when A is large, ps is approximate ly the p ropor t ion of non-zero observations (see (3.14)). T h a t is, when A is large, e~x is smal l , so the p robab i l i ty of a zero observation is approx imate ly 1 — p. 76 from demand from sales ignore censoring Po isson fit Figure 3.16: T h e Es t imates of A for ZIP(0.7,2) w i t h y° = 2. from demand from sales ignore censoring Figure 3.17: T h e Es t imates of p for ZIP(0.7,2) w i t h y° = 2. 77 from demand from sales ignore censoring Poisson fit Figure 3.18: T h e Es t imates of A for ZIP(0.7,5) w i t h y° = 4. from demand from sales ignore censoring Figure 3.19: T h e Es t imates of p for ZIP(0.7,5) w i t h y° = 4. 78 from demand from sales ignore censoring Poisson fit Figure 3.20: T h e Es t imates of A for ZIP(0.7,15) w i t h y° = 11. from demand gnore censoring Figure 3.21: T h e Es t imates of p for ZIP(0.7,15) w i t h y° = 11. 79 ii) W e observe a pat tern i n the est imated op t ima l po l i cy d is t r ibut ions s imi la r to that i n the boxplots of A estimates. T h a t is, considering censoring yields a mean estimate of the order quant i ty very close to that obta ined from using completely observed demands. T h e d i s t r ibu t ion of the "estimated" order quanti ty corresponding to the estimate of not t ak ing censoring into account shifts to the left. Same observation was obta ined for the Poisson fit. i i i ) T h e mean costs corresponding to both M L E ' s us ing demand da ta and sales da t a are a bi t higher than the baseline costs across k for a l l levels of A. T h e mean costs corresponding to ( A s , ps) f rom sales da ta are s l ight ly higher than those corresponding to (Xd, Pd) from demand data , w i t h larger s tandard deviations. iv) T h e Poisson fit gives a mean estimate of A close to the expecta t ion. T h e Poisson est imator yields a mean cost (row 4) comparable to that corresponding to the M L E ' s of Z I P , ( A S , ps), across k for a l l levels of A. W h e n k is low, the mean cost w i t h the Poisson fit is even closer to the baseline cost than that w i t h the Z I P fit (row 1), and the s tandard devia t ion is smaller for A = 5 and 15. W h e n k is m e d i u m or high, the Poisson fit yields good cost estimates as wel l . T h e mean levels are just s l ight ly higher than those in row 1 w i t h re la t ively greater s tandard deviat ions. T h e largest difference of the mean cost from the baseline cost is 10% for A = 15 and k = 0.87. T h e largest difference of the mean cost f rom the baseline cost w i t h Z I P fit is 4% for A = 5 and k = 0.87 v) For Poisson demand, the estimate ( A s ) obtained wi thout considering censoring has good performance as far as cost is concerned. For Z I P demand, a s imi lar impact of es t imat ion on cost was observed. Ignoring censoring i n es t imat ion does not necessarily result i n large increase i n cost. A s a mat ter of fact, despite the fact that the mean estimates (ps, Xs) are very different than the actual values when A = 2, the mean cost (in row 3) at k = 0.4 is even smaller than that corresponding to the more compl ica ted M L E ' s ( A s , ps) (in row 2) w i t h a smaller s tandard devia t ion at each level of A (see Table 3.13). W h e n k = 0.67 and 0.87, the mean cost i n row 3 increases a l i t t le . S imi l a r l y for the mean cost i n row 2. T h e largest discrepancy between row 2 and row 3 is 6% at A = 5 and k = 0.87 and the discrepancy between row 2 and the baseline cost increases to almost 13% at A = 15 and k = 0.87. C o m b i n i n g (iv) and (v), we can conjecture that the expected cost becomes more sensitive to the estimate of the demand density when k is high and A is large if 1 — p is smal l . T h e effect of censoring level on cost for Z I P demand is investigated. S imi l a r to the Poisson demand case, we consider on ly the h igh demand, i.e. A = 15. T h e in i t i a l order quant i ty y° is again set to 5. A s i n the Po isson case, this order quant i ty yields a great number of censored sales observations. The costs are reported i n Table 3.16. Since y° determines the censoring level in sales, a low y° results i n a different sales sample whi le the demand sample remains the same. Hence, only the costs obta ined from using sales da ta 80 M e a n k M e a n Es t ima te 0.4 0.67 0.87 Es t ima te (y° = 15) y" = 15 y° = 5 y ° = 15 y° = 5 y° = l5 y° = 5 (y° = 5) A s = 15.720 14.969 14.845 14.672 14.406 14.665 13.593 A s = 15.491 (2.783 a ) (0) ps = 0.714 (0.636) (0.416) (0.785) (0.037) (0.788) (0.015) ps = 0.714 (0,089) (0.089) I s = 10.737 14.960 15.441 15.322 ' 18.284 15.318 22.413 X s = 4.965 (0.197) (0) Ps = 0.714 (0.349) ( 0.152) (0.255) (0.152) (0.259) (0.284) ps = 0.719 (0.089) (0.089) baseline 14.598 14.363 13.583 baseline . "The numbers in the parentheses are standard deviations. Table 3.16: Expec t ed Costs w i t h ZIP(0 .7 , 15), y° = 15 vs. y° = 5. i n parameter es t imat ion are of interest here. T h e cost from using demand data , either w i t h Z I P fit or w i t h Po isson fit, is not affected by y°. Therefore, the costs corresponding to ZIP(p<i, Ad) and Po i s son(A p ) are omi t ted . Because of the low censoring level, y° = 5, the generated sales samples consist of most ly censored observation values of 5 and exact observation values of 0. Therefore, the s tandard deviations of bo th A s and A s are zero as a l l posi t ive observations are censored at 5, whi le bo th ps and ps have a smal l s tandard devia t ion (0.089) since the number of 0 observations varies from sample to sample. Trea t ing the censored observations as exact results i n a seriously negative-biased estimate of A (4.965). Since the estimate of p is m a i n l y determined by the p ropor t ion of zeros i n the sample, when A is es t imated to be moderate or high (above 5), bo th ps and ps have a mean value close to the true value 0.7 w i t h smal l s tandard deviations. See F igure 3.22 and F igure 3.23 for the boxplots of the parameter estimates i n this case. Interestingly, the costs w i t h y° = 5 corresponding to the M L E ' s ( A s , ps) have a lower mean level wi th a smaller s tandard devia t ion than those w i t h y° = 15 across k. T h i s is due to the smaller variance of the "est imated" o p t i m a l order quantit ies when y° = 5. O n the other hand, the pair of est imator, ( A s , ps) is not acceptable at m e d i u m or h igh levels of k. T h e mean cost is 65% higher t han bo th the baseline cost and the mean cost w i t h respect to Z I P ( A S , ps) at k = 0.87. F igure 3.24 shows that ignor ing censoring results i n an "est imated" order quant i ty greatly biased downward. In fact, almost the whole d i s t r ibu t ion lies to the 81 f r o m d e m a n d f r o m s a l e s i g n o r e c e n s o r i n g P o i s s o n fit Figure 3.22: T h e Es t imates of A for ZIP(0.7,15) w i t h y° = 5. f r o m d e m a n d f r o m s a l e s i g n o r e c e n s o r i n g • F igure 3.23: T h e Es t imates of p for Z I P ( 0 . 7 , 1 5 ) w i t h y° = 5. 82 f r o m d e m a n d I g n o r e c e n s o r e d Figure 3.24: T h e E s t i m a t e d O p t i m a l P o l i c y for ZIP(0.7,15) w i t h y° = 5 (fc=0.87). left of those of y& and ys. Since the low in i t i a l order quant i ty determines the chance and the magni tude of censoring i n the sales data , the Poisson fit using the demand da ta is not affected at a l l . I I . p = 0.3 T h e s imula t ion results for p = 0.3 are given i n Table 3.17 and Table 3.18. No te P r ( X = 0) > 0.7 when p = 0.3, so there are a lot of 0 observations i n a sample. T h e case where A = 2 is not interesting since generated demand observations either have value zero or a smal l posi t ive value from which the demand is es t imated to be low by various estimators. Hence the est imated o p t i m a l policies are zero, same as the o p t i m a l po l i cy w i t h respect to the true density for k — 0.4 and 0.67. Therefore, we only consider A = 5 and 15 i n this case w i t h a low p. W e see from bo th tables that , when k — 0.4, the costs obta ined using various Z I P estimates have the same mean level as the baseline cost w i t h zero s tandard deviat ions. T h i s is i n spite of the fact that the estimates of A are quite different. T h i s occurs because P r ( X = 0) > k regardless of the estimates of A, which re turn 0 for the est imated op t ima l policies, as well as, for the o p t i m a l po l i cy w i t h respect to the true density. In other words, w i t h a h igh chance of zero demand, i t is not economical at a l l to be i n business when the ho ld ing cost is so h igh (relative to the penalty cost) that the c r i t i ca l fractile k is as low as 0.4. For the case where A = 5, it can be seen that Xs is two times of true value, and the s tandard devia t ion is large too. We suspect this is because of the smal l samples (sample' size is 30) and large p ropor t ion of censored observations 83 R o w M e a n k N u m b e r Es t imates 0.4 0.67 0.87 1 I r f = 4.909 (0.808 s ) 2.25 3.067 3.235 pd = 0.310 (0.076) (0) (0.117) (0.146) 2 Xs = 10.709 (4.398) 2.25 3.653 4.651 Ps = 0.310 (0.074) (0) (1.162) (0.857) 3 I s = 1.546 (0.096) 2.25 3.016 3.810 p, = 0.390 (0.094) (0) (0.024) (0.133) 4 Xp = 1.524 (0.462) 2.671 3.124 3.488 Poisson fit (0.189) (0.054) (0.193) 5 baseline (A = 5, p = 0.3) 2.25 3 3.155 "The numbers in the parentheses are standard deviations. Table 3.17: M e a n and S tandard Dev ia t i on of E x p e c t e d Costs w i t h Z IP (0 .3 , 5). R o w M e a n k N u m b e r Est imates 0.4 0.67 0.87 1 A d = 14.819 (1 .316°) 6.75 9.194 8.864 pd = 0.306 (0.079) (0) (0.357) (0.306) 2 A s = 17.366 (0) 6.75 9.262 8.920 ps = 0.306 (0.079) (0) (0.061) (0.175) 3 I , = 4.965 (0) 6.75 9.035 11.575 ps = 0.308 (0.080) (0) (0.061) (0.374) 4 Xp = 4.552 (1.322) 8.191 9.269 10.806 Poisson fit (0.474) (0.082) (0.605) 5 baseline (A = 15, p = 0.3) 6.75 9 8.710 °The numbers in the parentheses are standard deviations. Table 3.18: M e a n and S tandard Dev ia t i on of E x p e c t e d Costs w i t h ZIP(0 .3 , 15). 84 f r o m d e m a n d f r o m s a l e s i g n o r e c e n s o r i n g P o i s s o n fit Figure 3.25: T h e Est imates of A for ZIP(0.3,5) w i t h y° = 2. f r o m d e m a n d f r o m s a l e s i g n o r e c e n s o r i n g Figure 3.26: T h e Es t imates of p for ZIP(0.3 ,5) w i t h y° = 2. 85 f r o m d e m a n d f r o m s a l e s i g n o r e c e n s o r i n g P o i s s o n fit Figure 3.27: T h e Es t imates of A for ZIP(0.3,15) w i t h y° = 5. f r o m d e m a n d f r o m s a l e s i g n o r e c e n s o r i n g Figure 3.28: T h e Est imates of p for ZIP(0.3,15) w i t h y° = 5. i n each sample. Hence, the costs from this estimate are higher for bo th k = 0.67 and k = 0.87, compared to the costs from the other three estimates. W e have the s imi lar observation for the case where A = 15. B u t i t is on ly true for k = 0.67. W h e n there is a h igh chance of censoring, the effect of sample size on the M L E needs to be investigated. Figures 3.25, 3.26 and 3.27, 3.28 give the d is t r ibut ions of the parameter estimates. W i t h the Po isson estimate of Z I P density, the cost is h igh when A is h igh (A=15) . T h i s may be due to the large variance i n the estimate of A, as well as the overal l poor fit of Poisson d i s t r ibu t ion to Z I P d is t r ibut ion when there is a large probabi l i ty mass at 0. O n the other hand, recal l that the Poisson fit performs better than the Z I P fit but ignor ing censoring when p = 0.7. 3.3.5 Summary and Discussion W e focused on the Z I P demand i n a newsvendor system from the M L E perspective i n this chapter. T w o aspects were considered, the demand es t imat ion i n a lost sales inventory system and its impact on inventory po l i cy as well as cost. We developed an M L E of the Z I P parameters wh ich takes censoring into account, extending Conrad ' s work on Poisson demand [16]. In addi t ion to the newsvendor, we also studied an (s, S) inventory system where the demand is censored at variable levels. It is quite obvious that not considering censoring simplifies es t imat ion. T h e simple form of estimate is favorable f rom the computa t ion perspective. Ye t we must determine i f this increases cost. Hence, we moved forward to answer the fol lowing questions. H o w would es t imat ion wi thout consider ing censoring affect the inventory po l i cy and more impor tant ly , its cost. T h e answer to this question provides guidance on what cost advantage is gained at the computa t iona l expense due to the compl ica ted M L E . For this purpose, a newsvendor system is considered because of the existence of the closed form o p t i m a l po l i cy and consequently, ease of computa t ion of the inventory pol icy and pol icy cost. S imula t ion studies were performed to investigate the impac t on po l i cy and cost using four estimates. T h e o p t i m a l po l i cy and its corresponding expected cost were computed based on those four parameter estimates. We compared various costs and summar ized the results for Z I P i n Sect ion 3.3.4. For compar ison purposes, Poisson demand was also studied; these results were reported i n Sect ion 3.3.3. T h e impacts of the low in i t i a l order quant i ty and the sample size were also investigated. Deta i ls of the s imula t ion design and the procedures can be found i n Sect ion 3.3.2. We summar ize i n the fol lowing the key findings and observations from the s imula t ion s tudy for bo th Z I P and Poisson demands. T h e M L E using sales da ta while t ak ing censoring into account performs the best in terms of cost as well as po l i cy among various estimates. T h i s M L E closely approximates those based on demand da ta in most cases. However, i t has a complicated form and consequently requires add i t iona l computa t iona l effort 87 compared to the other two which do not compensate for censoring. T h i s answers the question raised at the beginning of this chapter, what can be done i n case lost sales da ta can not be easily recorded by the retailer. Cor rec t ly es t imat ing the unknown parameters helps reduce inventory cost. B o t h estimates wi thout considering censoring underestimate the u n k n o w n parameters and hence, gen-eral ly y ie ld lower than op t ima l order quantities. T h e histograms of the "est imated" op t ima l policies (e.g. F igu re 3.11) show that these two estimates shift the d i s t r ibu t ion of the "est imated" pol icy to the left. T h i s devia t ion from the o p t i m a l po l icy results i n an increase i n the expected cost (e.g. Table 3.9). Trea t ing cen-sored observations as exact seems to be a better op t ion than completely delet ing the censored observations from the sales sample since the shift of the "estimated" op t ima l po l icy d i s t r ibu t ion and subsequently the cost increase appear to be smaller i n magnitude. Genera l ly speaking, the est imator ignor ing censoring performs well i n po l i cy and cost computa t ion for bo th Poisson and Z I P demand. However, ex t ra caut ion is required when dealing w i t h extreme cases. For instance, when the in i t i a l order quant i ty is set to be very low which provides sales samples w i t h a larger p ropor t ion of censored observations, the est imator based on ignor ing censoring may y ie ld a seriously underest imated o p t i m a l po l icy which i n t u rn results i n a large increase i n the expected cost. T h i s is especially true when the cost s tructure is such that k is h igh . T h i s applies for bo th Poisson and Z I P demand. T h e effect on the expected cost of not t ak ing censoring into account i n parameter es t imat ion is reduced compared to that on the o p t i m a l order quanti ty. T h i s is due to the flatness of the cost function near its m i n i m u m . Obvious ly , the cost structure plays a key role here. F i n a l l y , i n the case where demand is completely observable, a s imple Po isson fit for Z I P demand performs wel l i n po l i cy and cost computa t ion w i t h a h igh level p. O n the contrary, when p is low, which means there is a large p ropor t ion of ex t ra zeros, the Poisson fit does not do wel l given A is h igh w i t h either a low or a h igh c r i t i ca l fractile k. It seems that sample size does not have much effect on mean cost. However , M L E generally requires a large sample size. Its effect on the M L E of Z I P w i t h a low p needs further invest igat ion. T h e above is established based on one hundred generated samples of size t h i r t y each i n a to ta l of nine cases for bo th Poisson and Z I P demand. In these s imula t ion studies, the M L E was computed for each sample separately. We are also interested i n how M L E performs i n a cumula t ive manner when the parameter estimates are upda ted as more da ta is collected. In other words, how does M L E perform in the learning process of the demand pat tern . T h i s w i l l be investigated i n the next chapter. W e w i l l use the M L E and the newsvendor cost computa t ion developed in this chapter and compare them w i t h the Bayes ian alternative. 88 Chapter 4 Bayesian Analysis For ZIP Demand In the previous chapter, m a x i m u m l ike l ihood es t imat ion for Z IP (p , A) demand from the frequentist point of view is discussed. It is more desirable that the his tor ica l informat ion regarding the demand d is t r ibu t ion is captured and the estimates are updated when new da ta becomes available. A s opposed to the M L E , Bayesian methodology provides a framework for dynamic upda t ing of the demand d i s t r ibu t ion es t imat ion. Th i s enables us to improve on the inventory pol icy determinat ion process based on the updated demand d is t r ibu t ion . We can learn about the demand pat tern down the road as more informat ion is gathered. Fur thermore , expert judgement about the d i s t r ibu t ion parameters or some pre l iminary knowledge about the parameters based on previous experience can be incorporated in the es t imat ion upda t ing process. Usua l ly , the expert judgement or the his tor ica l informat ion is summar ized as a pr ior on the unknown parameters. D a t a is used to update the pr ior to the posterior wh ich can then be used to produce estimates of the parameters. Hence, the demand d i s t r ibu t ion estimate is revised as the pr ior is updated to the posterior. Bayes ' theorem which is applied to ob ta in the posterior serves as the foundat ion of the Bayes ian framework. T h e framework and its workflow w i l l be discussed i n detai l i n this chapter. W e w i l l start w i t h an overview section on Bayes ian analysis. Bayes ' T h e o r e m and its appl ica t ion wi l l be discussed. Some computa t iona l methods w i l l be explored. In the fol lowing sections, the Bayes ian approach is appl ied to b o t h the Poisson and the Z I P dis t r ibut ions . A g a i n , the Poisson d i s t r ibu t ion is s tudied as a special case for compar i son purposes. We study the case where there is no censoring (or demand da ta is available) before extending to the case where censoring occurs and only sales da ta is available. A Bayes ian dynamic upda t ing scheme of the Z I P parameters is described i n the last section. It is appl ied to the newsvendor system because of the existence of a closed form op t ima l policy. A s imula t ion s tudy is presented to i l lustrate a compar ison between the Bayes ian newsvendor upda t ing process and the t r ad i t iona l M L E process. In an 89 (s, S) inventory system, the pol icy is obtained by some heurist ic computa t ion . T h i s add i t iona l computa t ion requirement makes the s imula t ion s tudy significantly more difficult, especially when only censored sales da ta is available for demand d i s t r ibu t ion es t imat ion. A s we have discussed earlier, i n this case, the inventory pol icy affects informat ion collect ion and the demand d i s t r ibu t ion upda t ing process. T h e inter-woven nature of these two processes requires intensive computa t ion i n a mul t i -pe r iod (s, 5 ) system and thus, we use the newsvendor system to investigate the advantage of Bayes ian analysis over the M L E . 4.1 Bayesian Analysis Overview Class ica l statistics is directed towards the use of sample informat ion i n m a k i n g inferences about an unknown quant i ty wh ich is commonly called the state of nature [9]. A n example of the state of nature here is the unknown demand d i s t r ibu t ion parameters. However, i n addi t ion to the sample informat ion , there are other types of in format ion which can be used i n the decision m a k i n g or the s ta t i s t ica l inference processes. One source of such nonsample informat ion that is relevant is cal led prior information. T h i s informat ion arises from sources other than the s ta t is t ical invest igat ion. Very often, pr ior informat ion comes from past experience about s imi lar s i tuat ions invo lv ing the unknown quantity. T h e approach to statist ics wh ich formal ly seeks to ut i l ize pr ior in format ion is called Bayesian analysis [9]. A s discussed, the capabi l i ty of m a k i n g use of the impor tan t pr ior in format ion when it is available is a major advantage of the Bayes ian methodology. In the context of inventory management when the demand d i s t r ibu t ion is unknown , one has to propose a form before an inventory po l i cy can be derived. T h e pol icy affects the demand informat ion gathering i n a lost sales inventory system. T h i s in teract ion between pol -icy de terminat ion and demand d i s t r ibu t ion es t imat ion w i l l be thoroughly discussed i n Chap te r 5. In this chapter, we w i l l focus on how the pr ior informat ion can be used i n po l i cy der ivat ion and how the demand d i s t r ibu t ion es t imat ion is upda ted using sample informat ion. A disadvantage of the Bayes ian method is often the computa t iona l challenge since the posterior d i s t r ibu t ion could have a compl ica ted form. In this section, we w i l l start w i t h the general Bayes ian framework and its major components. T h e n we w i l l explore some M o n t e C a r l o M a r k o v C h a i n ( M C M C ) methods that are often used to ob ta in the posterior samples. 4.1 .1 Bayesian Framework Bayes ian analysis is performed by combining the prior informat ion and the sample informat ion to ob ta in a posterior d i s t r ibu t ion for the unknown parameter. T h e Bayes ' theorem, der ived by Bayes (1763), is used for this purpose [9]. It is typ ica l ly phrased i n terms of disjoint events Ai, A2,... , An, whose union has 90 probab i l i ty one. P r i o r probabi l i t ies P r ( A j ) as well as the condi t iona l p robab i l i ty Pi(B\Ai) for each Ai are assumed known . A n event B occurs. T h e n Bayes ' Theo rem states that r i l l 117) P r ( f l W P r W ( 4 1 ) There are three major elements in the Bayes ian framework. T h e y are the pr ior , the sample informat ion which is often represented i n the form of l ike l ihood, and the posterior. Bayes ' theorem is appl ied to obta in the posterior from the pr ior using the sample informat ion. P r i o r Information P r i o r informat ion is an impor tan t element. It represents an expression of belief about the unknown quanti ty before da t a is available. There are various ways of construct ing a pr ior d i s t r ibu t ion of the unknown parameter, noninformat ive priors , conjugate priors , subjective priors , m a x i m u m entropy pr iors , priors determined using marg ina l d is t r ibut ions and hierarchical priors [9] are some examples. In this chapter, we w i l l only discuss the first two types of priors since it is not our major interest to investigate the impac t of the priors on the es t imat ion. T h e Bayes ian approach can be used even when no or m i n i m a l pr ior in format ion is available. W h a t is needed i n such si tuations is a noninformative prior. T h i s means a pr ior wh ich contains no informat ion about the u n k n o w n quanti ty, say 6. For instance, i n testing between two s imple hypotheses, the pr ior which gives p robab i l i ty 1/2 to each of the s imple hypotheses is clearly noninformat ive. Sometimes it is reasonable to give equal weight to a l l possible values of 9 w i t h parameter space 0 = ( - c o , oo). T h u s a t yp i ca l noninformative pr ior , denoted by ir(6), is chosen that satisfies TT(9) = 1. T h i s is often referred to as a flat pr ior . It is also cal led Jeffreys' p r ior because of Jeffreys' cont r ibu t ion to this pr ior cons t ruc t ion approach (referred to as Jeffreys' rule) [58]. A noninformative pr ior would suggest that any value is reasonable. However, it frequently happens that the na tu ra l noninformative pr ior is an improper pr ior since i t has infinite mass. We w i l l discuss Bayes ian analysis for the Poisson d i s t r ibu t ion w i t h a flat pr ior i n the next section. T h r o u g h out this chapter, we use the c o m m o n Bayes ian abuse of nota t ion, for instance, TT, / , etc. These generic symbols are used to denote more than one function. A c o m m o n approach to specify a pr ior is to choose a density function ir(9) such that the posterior density, denoted by n(9\x), where x represents the sample informat ion has the same form. Such priors are called conjugate priors. Note , 7r here is an example of nota t ion abuse, i t is used for two different functions. Th i s approach simplifies ca lcu la t ion of the posterior TT(8\X). T o give an example, a no rma l pr ior for the mean p is conjugate for the no rma l density (wi th known a). Obvious ly , besides p rov id ing for easy calcula t ion, conjugate priors a l low one to begin w i t h a certain funct ional form for the pr ior and end up w i t h a posterior 91 of the same funct ional form, but w i t h parameters upda ted by the sample informat ion . T h i s upda t ing of parameters also provides an easy way of seeing the effect of the pr ior and the sample informat ion. Hence, this makes the conjugate priors appeal ing. However, only a l im i t ed number of sampl ing models belong to the exponent ia l fami ly for wh ich conjugate priors exist [58]. We w i l l discuss Bayes ian analysis for the Poisson w i t h a G a m m a pr ior wh ich is conjugate. It can be seen that when there is censoring the G a m m a prior is no longer conjugate due to the change in the l ike l ihood function. Posterior Bayes ' T h e o r e m is used to ob ta in the posterior as follows. posterior = l ike l ihood * pr ior / marg ina l oc l ike l ihood * pr ior T h a t is, the posterior is p ropor t iona l to the product of the l ike l ihood and the pr ior . L e t n(8) denote the pr ior density, f(x\8) the l ike l ihood of observing x and g(x) the marg ina l density funct ion of x. G i v e n a r a n d o m sample Xi,i = 1 , . . . , n the posterior density f(6\xi,i = 1 , 2 , . . . , n) is: nr = 1 <?(zi |0M0) f{9\xui = 1,2,... ,n) n ? = 1 3 ' ( ^ ) cx UU9{xi\e)ir{6) • (4.2) T h e posterior can be considered as an est imated density of 9 combin ing the in format ion from the pr ior and the r a n d o m sample, Xi, i — 1,2,... ,n. Hence, the shape of the posterior d i s t r ibu t ion is determined by the product of the l ike l ihood and the pr ior . 4.1.2 Computational Methods A s ment ioned earlier, computa t ion becomes a challenge for Bayes ian analysis since the posterior density may be compl ica ted . W e w i l l see i n Sect ion 4.4.1 that the posterior of the Z I P parameter A becomes complicated when there is censoring and the G a m m a pr ior is no long conjugate. In this case, one may use some sampl ing a lgor i thms to tackle the problem. T h e G i b b s sampler and the Me t ropo l i s a lgo r i thm are two examples of M o n t e C a r l o M a r k o v C h a i n ( M C M C ) s imula t ion methods which are often appl ied . B o t h the G ibbs sampler and the Met ropo l i s a lgor i thm y ie ld a M a r k o v chain whose equ i l ib r ium d i s t r ibu t ion (under certain regulari ty condit ions) is the normal ized l ike l ihood function or the posterior density of interest. Hence, bo th algori thms produce correlated samples from a function of interest [58]. T h e y can be used to ob ta in a sample of parameter 92 values from the l ike l ihood function or from the posterior density. T h e s imula ted samples can then be used to ob ta in an estimate of a lower d imensional marg ina l of the posterior, to estimate a moment of this marg ina l or, more generally, to estimate some function of the posterior. Since we are not focusing on the techniques for assessing convergence and hand l ing such si tuations i n inference, we w i l l not explore various ways to obta in the parameter estimate from the posterior samples. W e use sample means f rom the generated random samples i n this chapter. It is wor thwhi le to point out that while M L E usual ly requires a moderate sample size, the Bayes ian approach is of great value to the analysis of a smal l sample p rob lem. Tanner [58] has one chapter on the M C M C methods where details of G i b b s sampl ing and Met ropo l i s a lgo r i thm can be found. We w i l l briefly discuss these two algor i thms and i l lustrate how they are appl ied i n the fol lowing. T h e Gibbs S a m p l e r W h e n the parameter of interest is a vector of d imension d (d > 1) and the posterior of the parameter vector n{8i,92, • • • &d\xi, i = 1 , 2 , . . . , n) is not i n a s tandard mul t ivar ia te density form, the G i b b s sampler may be appl ied . T h e a lgor i thm is as follows. 1. G i v e a s ta r t ing point (9^ ,9{20),.. .9(d]). 2. A t the ith i terate, sample 9[l+1^ from n(6i\62\ • • . 8 d \ x ) where 7r(• [-) represents the condi t ional proba-b i l i t y funct ion. 3. Sample 8{2i+1) f rom ir(82\8{i+1) ,8(3i}, • • .ef ,X) 4. Repeat the above un t i l 8d+1^ is sampled from ^[8d\9^+l\ .. .8d^,X). Iterate steps 2 to 4 i n the above procedure t t imes. T h e n the vectors (8^),(8^),... ,(8^) where 0W = (9^),9^,...edi))T f o r i = 1 , 2 , . . . ,t are a real iza t ion of a M a r k o v chain w i t h t rans i t ion probabi l i ty from (0') to (6>), T(9',9) = T T ^ I ^ , . . . ,ffd,X)ir{e2\6uV3... ,9'd,X)... (4.3) Tt{od\oue2... ,ed-ltx). Under some regular i ty condit ions, the jo int d i s t r ibu t ion of (9[l\ 92 \ • • • 9d^) converges to TT(9I,92, ... 8d\X) geometr ical ly as i —> co. In addi t ion , - E / ( 0 ( f ) ) ^ / f{9)*{9\X)d9 (4.4) as n goes to oo for any integrable function / where #W is the sampled value at i te ra t ion i. 93 M e t r o p o l i s M e t h o d T h e Me t ropo l i s me thod is another s imula t ion a lgor i thm which can be used to construct a M a r k o v chain wi th equ i l i b r ium d i s t r ibu t ion 7r(-). General ly , a M a r k o v chain w i t h t rans i t ion m a t r i x T w i l l have an equi l ib r ium d i s t r ibu t ion 7r i f and only i f TT = 7r T. Once T is constructed, we can sample from the equ i l ib r ium dis t r ibut ion 7r by runn ing a M a r k o v chain w i t h the t rans i t ion ma t r i x T un t i l the chain appears to have settled down to equ i l ib r ium. One of the contr ibut ions of Me t ropo l i s ' is i n proposing a general way of cons t ruc t ing T to realize a M a r k o v chain w i t h equ i l ib r ium d is t r ibu t ion n [58]. We give the Met ropo l i s a lgor i thm for the continuous case i n the fol lowing. L e t ir(9) be the density of the parameter of interest and T(9,6') be a symmet r ic t rans i t ion probabi l i ty funct ion, that is T{9,9') =T(9',9) (4.5) T h e n , 1. If the chain is current ly at 9n = 9, generate a candidate value 9' for next loca t ion 9n+\ f rom T(9,9'). 2. W i t h p robab i l i ty a ( M ' ) = m i n { ^ l , l } (4.6) accept the candidate value and move the chain to 8n+i = 9'. Otherwise , reject and let 9n+i = 9. T h e Met ropo l i s a lgor i thm also yields a sequence of dependent realizations forming a M a r k o v chain w i t h ir as the equ i l i b r ium d i s t r ibu t ion [58]. Some possible candidates for the t rans i t ion p robab i l i ty function T are mul t ivar ia te no rma l or mul t ivar ia te t. W e w i l l apply bo th G i b b s sampl ing and the Met ropo l i s a lgor i thm i n the next two sections i n Poisson and Z I P demand parameter es t imat ion. T h e G i b b s sampler generates samples f rom a condi t iona l p robabi l i ty funct ion. Hence, we use G i b b s sampler when the condi t iona l posterior follows a s tandard d is t r ibu t ion . Otherwise , we use the Met ropo l i s me thod to construct a t rans i t ion m a t r i x first. Sometimes, they are applied s imultaneously to generate posterior samples. We w i l l see that some of the condi t iona l posteriors follow a s tandard d i s t r ibu t ion while the others do not. We use G i b b s sampler for the ones which have a s tandard d i s t r ibu t ion and the Met ropo l i s me thod for the others. We use the sample mean as the parameter estimates. In general, caut ion is needed when using the simulated samples for inference purposes since each sample is a correlated series of realizat ions. 94 4.2 Bayesian Analysis For Poisson Distribution Simi l a r to Chap te r 3, we assume that the r andom demand variable X follows a Poisson d is t r ibu t ion w i t h unknown parameter A . T h a t is, the condi t ional d i s t r ibu t ion is X | A ~ P o ( A ) . (4.7) T h e u n k n o w n parameter A is assumed to have a pr ior d i s t r ibu t ion 7r (A) . T h e posterior can be considered as an es t imated density of A from the r andom sample, •^ii 1 — 1 , 2 , . . . ,72. 4.2.1 Posterior Distribution Of A With Jeffreys' Prior W e first specify the mode l as follows. 1. condi t iona l d i s t r ibu t ion : X | A ~ P o ( A ) , 2. pr ior : TT (A) = ^ , A > 0. W e ob ta in the posterior i n the fol lowing by subst i tu t ing into (4.2). f (\\xui = 1,2,...,n) cx \&^Xi-& e~nX (4.8) T h a t is, the posterior density of A is G a m m a ( J ^ ^ 1 — \ , Hence, the posterior mean and s tandard devia t ion of A , denoted by A and <f\ respectively are: n » = i 1 n We use a s imula t ion example to i l lustrate the analysis. A r andom sample of size 20 is generated from Poisson(5) . T h e posterior mean is A = 5.125 w i t h <f\ = 0.506 for this par t icu lar sample. These are obtained from the above two expressions respectively. 4.2.2 Posterior Distribution Of A With Conjugate Prior Same as the above case, we first give the model as follows. 1. condi t iona l d i s t r ibu t ion : X\X ~ P o ( A ) , 2. pr ior : A ~ G a m m a ( c , 7) where c and 7 are fixed. 1 If a random variable X follows Gamma(c,7J distribution, then its pdf is as follows, f(x) = x r ( ^ y c — • 95 It yields a G a m m a posterior density. T h a t is, n x \\c,j,Xi,i = 1 , 2 , . . . ,n ~ G a m m a ( ^ i i + c, (n H — ( 4 - 1 1 ) i = l ^ T h e hyperparameters of the G a m m a prior , c and 7, can be model led i n the same way assuming a "flat" pr ior for t hem and they are independent of each other. For instance, TT(C) = 1; 7r(7) = 1. T h e corresponding posteriors of c and 7 are obta ined as follows, respectively. c\X,j,Xi,i = 1 , 2 , . . . , n ~ — r ( c)7 c ' ^4'12^ E - A / 7 7 | A , c , a ; i , i = 1 , 2 , . . . , n ~ — — . (4.13) It can be seen that hyperparameter c has a non-standard d i s t r ibu t ion whi le 7 follows the Inverse G a m m a d i s t r ibu t ion . T h i s is referred to as hierarchical Bayes. F i x i n g one hyperparameter 2 , G i b b s sampl ing (for A and 7) or Me t ropo l i s sampl ing (for c) can be used to generate samples f rom the posterior densities. T h e sample mean of those posterior draws after the burn- in per iod can be taken as the estimate of the posterior mean wi thout losing much accuracy 3 . Ve ry often it takes a whi le for the real ized M a r k o v chain to stabil ize. T h e burn- in pe r iod refers to the part before the chain seems to settle down. T h e es t imat ion of A is of major interest. T h e estimate of a hyperparameter can be obta ined at the same t ime. W e perform a s imula t ion s tudy again. A r andom sample of size 20 is generated from X\X ~ Po(X) where A ~ G a m m a ( 5 , 1 ) (that is, c = 5, 7=1). T h e Poisson parameter A generated f rom this pr ior is A = 5.687. A n estimate of A close to this value is desired. T w o cases are considered when using M C M C simulat ions to generate the posterior samples. In each case, one hyperparameter , either c or 7, is fixed. We specify the fixed hyperparameter at the true value, then at a far off value for each Case. So we have four scenarios i n to ta l . T h e posterior samples of A and the non-fixed hyperparameter (c or 7) are generated using (4.11) and (4.12) or (4.13). W e set the posterior sample size to be 2000 for bo th parameters i n a l l scenarios. T h a t is, 2000 sample points are generated for each parameter us ing its derived posterior. A s mentioned above, we 2 We are not interested in various prior specifications, especially for the hyperparameters. Hence, we set one hyperparameter at a fixed value while using a simple prior definition on the other hyperparameter, just to illustrate that should prior of hyperparameters becomes interesting, the same method can be followed. 3Since there might be autocorrelation among the sample points generated from a M C M C method, a more accurate estimator may be considered. 96 Parameters Sampl ing Sample M e a n E s t i m a t e d M e t h o d F r o m Poster ior 7 = 1 A G i b b s A = 5.767 (fixed at the true value) c Met ropo l i s c = 6.732 7 = 5 A G i b b s A = 5.767 (fixed at a wrong value) c Met ropo l i s 5 = 2.117 c=5 A G i b b s A = 5.758 (fixed at the true value) 7 G i b b s 7 = 1.93 c=10 A G i b b s A = 5.755 (fixed at a wrong value) 7 G i b b s 7 = 0.722 Table 4.1: M C M C s imula t ion results for Poisson rate w i t h G a m m a pr ior estimate the parameters by the posterior sample mean. We exclude the first 100 sample points in calcula t ing the sample means to ensure that the burn- in per iod is not inc luded. It can be seen from a l l the posterior sample figures that 100 sample points al low enough burn- in . Note , the same r a n d o m sample of X (of size 20) generated from Po(5.687) is used i n a l l four scenarios. Table 4.1 summarizes the es t imat ion results from each of the four scenarios described above. It can be seen that the posterior means of A from a l l four scenarios are very close to the generated value 5.687. T h e first two s imula ted posterior samples give the same rounded estimate of A, despite the fact that in the second scenario, 7 is specified at a value far from the true value. However, the posterior means of the hyperparameter , c, are quite different. We have s imi lar observations for the last two scenarios. We observe that the posterior mean of A is insensitive to the s tar t ing point of the hyperparameters . It seem to give a robust est imate of A. Since the hyperparameter c does not follow a s tandard d i s t r ibu t ion , the Met ropo l i s a lgor i thm is used to generate the samples from its posterior while the G i b b s sampl ing method is used for the hyperparameter 7 wh ich follows an Inverse G a m m a d is t r ibu t ion . Figures 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.7 and 4.8 give the s imula ted posterior samples of the Poisson parameter A and the hyperparameter (either c or 7) for each scenario, i n the same order as shown i n Table 4.1. In each figure, 2000 (see z-axis) sample points are p lo t ted . 97 Figure 4.2: Poster ior Sample of c w i t h 7=1. 98 99 Figure 4.5: Poster ior Sample of A w i t h c = 5. 1000 1 :nn Figure 4.6: Poster ior Sample of 7 w i t h c = 5. 100 "1 1 1 1 1 0 500 1000 1500 2000 Figure 4.7: Poster ior Sample of A w i t h c = 10. 0 500 1000 1500 2000 1 :nn Figure 4.8: Poster ior Sample of 7 w i t h c = 10. 101 4.3 Bayesian Analysis For the ZIP Distribution W e now assume that the condi t ional d i s t r ibu t ion of r andom variable X g iven the parameters p and A is Z I P ( p , A) . W e specify the pr ior densities of p and A. Interest lies i n the posterior estimates of the parameters p and A. Throughou t this section, we specify the pr ior of p to be a B e t a d i s t r ibu t ion (i.e., p ~ Beta(a, /3)) since B e t a d i s t r ibu t ion is a conjugate pr ior . A s i n the previous section for the Poisson parameter, we assume A follows G a m m a ( c , 7). In contrast to the Poisson case, the G a m m a pr ior of A is no longer conjugate for the Z I P d i s t r ibu t ion . We assume that p and A are independent. T h e hyperparameters a, (3, and c, 7 can be considered as nuisance parameters. In the fol lowing analysis, a , (3, c are fixed whi le 7 is est imated (wi th a "flat" pr ior ) . S imi la r ly , another hyperparameter can be est imated whi le the rest of t hem are fixed. T h e mode l described above is summar ized as follows. 1. condi t iona l d i s t r ibu t ion : X\p, A ~ ZIP (p , A) . 2. pr iors: ir(p) oc p Q _ 1 ( l - p ) 0 " 1 , (4.14) * « = w <4-15> W i t h a r a n d o m sample Xi (i = 1 , 2 , . . . ,n), re-order the sample da ta such that a l l the zero observations are followed by the non-zero observations. T h a t is, 2^(1) ? ^ ( 2 ) ? • • • •> • * • > * (^n) w i t h = Z ( 2 ) = . . . = X(fi = 0 where j is the number of zero observations. T h e l ike l ihood of Xi ( i = 1 , 2 , . . . , n) is derived in the fol lowing. fp, X2, . • • , Xn) OC fpi A ( Z ( 1 ) , XI2), • • • , £ ( „ ) ) (4.16) np~ x Ax(0 = [(l-p)+pe-*]mi=j+1P , . •*•(»)• B y Bayes ' Theorem, the joint posterior is obtained as follows, n(p,\,l\xi,i = 1 , 2 , . . . , n ) (4.17) assuming the hyperparameters a, /3, c are fixed and 7 has a "flat" pr ior , 7r(7) = 1. F r o m this jo int posterior, we give the condi t iona l posterior densities i n the fol lowing. 7r(p|A,7; Xi,i = 1 , . . . ,n) oc [ { l - p ) + p e - x ] i p a + n - i - 1 { l - p ) 1 3 - 1 , (4.18) 102 parameters est imated sampl ing method posterior estimate true value A Metropo l i s 4.991 4.742 P Metropo l i s 0.687 0.696 7 G i b b s 0.613 1 Table 4.2: M C M C s imula t ion results for Z I P ( p , A ) T T ( A | P,7; Xi,i = l,... ,n) oc [ ( l - p ) - t - p e - A ] J ' A ( S ^ + ' x ( i > + c - 1 ) e - ( 7 l - J ' + 1 / 7 ) A , (4.19) 7 r ( 7 | p , A ; Xi,i = 1 , . . . , n ) oce~x^/jc . (4.20) It can be seen, from (4.18) and (4.19), that the posterior densities of p and A are non-s tandard. Express ion (4.20) shows that the posterior of 7 follows the Inverse G a m m a d i s t r ibu t ion . A p p l y i n g M C M C methods, namely G i b b s sampl ing and Met ropo l i s sampl ing , s imula ted samples from the posterior d is t r ibut ions can be generated. T h e n any function of each of the parameters can be estimated from these s imula ted samples. We apply M C M C simulat ions using the above posterior densities to obta in the moments of the posteriors. We estimate the parameters p and A and the hyperparameter 7 by the means of the s imula ted samples. A l l other hyperparameters are fixed. T h e posterior estimates of the parameters from a M C M C s imula t ion s tudy are given in Table 4.2. A r a n d o m Z I P sample of 100 is generated. The Z I P parameters p and A are generated as follows assuming they follow B e t a and G a m m a dis t r ibut ions respectively. A ~ G a m m a ( 5 , 1 ) ; and p ~ Beta(0.7 ,0 .3) (4.21) T h e hyperparameters a, f3, c and 7 are fixed. T h a t is, a = (3 = 0.5, c = 10 and 7 = 1. T h e posterior sample size is set to 500. . W e are m a i n l y interested i n the posterior estimates of the Z I P parameters p and A . We estimate 7 in the s imula t ion only to i l lus t ra te that the hyperparameters can also be est imated by the Bayes ian method should i t be of interest. It can be seen from Table 4.2 that the posterior estimates A and p are fairly close to the true values. W e can s impl i fy the posterior densities by re-parameterizat ion. T h i s is explored i n the fol lowing and the s imula t ion is performed for the re-parameterized Z I P d is t r ibu t ion . 103 4.3.1 Re-parameterization For ZIP To s impl i fy the condi t iona l posterior densities, let q denote the probab i l i ty of X = 0, that is, q = p r ( X = 0) = (1 - p) + pe~x. (4.22) T h u s we re-parameterize the Z I P d is t r ibu t ion of the r andom variable X as Z I P ( g , A) . W e assume the pr ior of q follows the B e t a d i s t r ibu t ion , ir(q) = qa~l(l — ? ) / 3 _ 1 • T h e pr ior of A remains G a m m a dis t r ibuted as given by (4.15). T h e jo in t posterior density of the parameters q, A and 7 is derived as follows. -K(q,X,j\xi;i = 1 , . . . ,n ) A s s u m i n g a "flat" pr ior of the hyperparameter 7,7r(7) = 1, whi le f ix ing the rest of the hyperparameters, the corresponding condi t iona l dis t r ibut ions are: n{q\Xi,i = l,...,n) oc qa+^ {I - q ) ^ ' ^ 1 , (4.23) Tr(X\q,r,Xi,i = 1 , . . . , n ) oc ( e A - l ) - ( " - ^ A S ^ + i x ( i ) + c - 1 e - A / 7 , (4.24) e - V 7 7r(j\\;xi,i = 1 , . . . ,n) oc — — . (4.25) It can be seen that by re-parameter izat ion, the posterior of q is s implif ied to a B e t a d i s t r ibu t ion . The hyperparameter 7 again follows the Inverse G a m m a d i s t r ibu t ion as before. T h e condi t iona l posterior of A is s t i l l i n a compl ica ted form. However, when A is "large" (10 can be considered "large" enough since e 1 0 = 22026.466), approx imat ion of e A — 1 by ex i n (4.24) can s implify the condi t iona l posterior density of A to the fol lowing. Pr(X\q,T,Xi,i = 1 , . . . ,n) oc e ~ x { n - j + ^ X ^ = ^ x ^ + C ~ 1 (4.26) which is G a m m a ( ^ " = i + 1 x ( i ) + c,(n-j + ^ )_1). B y the defini t ion of q, Therefore, a sample of p can be obtained once the samples from the posterior densities of q and A are generated. T h e n the sample mean of p can be considered as the re-parameterized "posterior" estimate. A M C M C s imula t ion s tudy is performed for the re-parameterized Z I P ( p , A) d i s t r ibu t ion . T h e est imat ion results are summar ized i n Tables 4.3 and 4.4. Table 4.3 gives the s imula t ion results wi thout approx imat ion 104 parameters est imated sampl ing method posterior est imate true value A Met ropo l i s 5.083 4.742 P G i b b s for q 0.686 0.696 7 G i b b s 0.643 1 Table 4.3: M C M C s imula t ion results for Z I P ( p , A) -1 1 1 I 1 r— 0 100 200 300 400 500 Figure 4.9: Poster ior Sample of A. and Table 4.4 gives the results using the approx imat ion . We use the s imi lar settings as given earlier. A r a n d o m Z I P sample of size 100 is generated. T h e Z I P parameters q and A are generated by the fol lowing. A ~ G a m m a ( 5 , 1 ) ; and q ~ Beta(0 .7 ,0 .3) (4.28) T h e hyperparameters a, (3, c and 7 are fixed, i.e., a = (3 = 0.5, c = 10 and 7= 1 . T h e posterior sample size is set to 500. Note this analysis is based on reparametr iza t ion. Figures 4.9, 4.10 and 4.11 give the s imulated posterior samples of the est imated parameters A, p and 7 respectively wi thout the approx imat ion of A. W e also considered the case w i t h a large A. Hence we kept the same settings except that A ~ G a m m a ( 1 0 , 1 ) . (4.29) A l s o the hyperparameters a, (3, c and 7 are fixed at the same values except tha t c = 5. 105 106 i 1 1 1 1 r~ 0 100 200 300 400 500 Figure 4.12: Poster ior Sample of A. parameters est imated sampl ing method posterior est imate true value A G i b b s 7.110 6.784 P G i b b s for q 0.674 0.644 7 G i b b s 0.923 1 Table 4.4: M C M C s imula t ion results for Z I P ( p , A) Figures 4.12, 4.13 and 4.14 give the s imulated posterior samples of the es t imated parameters A, p and 7 respectively w i t h the approx imat ion of A. It can be seen from Table 4.3 and 4.4 that the posterior means of the parameter p and A are close to the "true" values used to generate the da ta sets, though the estimate of the hyperparameter 7 are quite far from the corresponding "true" value. Since our ma in interest is the estimates of the parameters p and A, the s imula t ion results are considered satisfying. In addi t ion , i t can be seen from these two tables that the misspecif icat ion of the hyperparameters a, (3, c (i.e., f ixing them at values far f rom the "true" values) does not affect much the posterior estimates of the Z I P parameters. T h i s is desired since we are not l ikely to know the "true" value of hyperparameters i n reality. 107 Figure 4.13: Poster ior Sample oi p. Figure 4.14: Poster ior Sample of 7. 108 4.4 Dynamic Bayesian Analysis For ZIP from Sales In this section, we apply Bayes ian analysis to the newsvendor system. Hence, es t imat ion of Z I P demand for a lost sales inventory system is s tudied from the Bayes ian point of v iew. La te r , we introduce the dynamic update of the demand d i s t r ibu t ion based on the Bayes ian analysis. F r o m the previous section, we know that re-parameterizat ion simplifies the condi t iona l posterior densities. T h e B e t a pr ior is conjugate for the parameter q. A n d when A is "large", the posterior of A can be well approx imated by the G a m m a density, that is, the G a m m a pr ior is approx imate ly conjugate for "large" A . However, when censoring occurs so that the da ta set consists of censored observations as well as exact observations, the G a m m a pr ior is no longer conjugate for A even when A is "large". T h e newsvendor system is considered here since, as mentioned at the beginning of this chapter, i t has a closed form op t ima l policy. 4.4.1 Poster iors of Z I P f rom Newsvendor Sales Once the sales da ta is available, reorder the da ta such that a l l the uncensored observations are before the censored observations, and the zero demand come before the posi t ive demands among the uncensored (note that the zero demand is always uncensored i n the newsvendor system since only those exceeding y° w i l l be censored). A g a i n , let r denote the number of uncensored observations; and j , the number of zero demands. T h e n , the l ike l ihood of Xi, i = 1, 2 , . . . , n for the re-parameterized Z I P d i s t r ibu t ion is: x(xi,i = 1 , 2 , . . . , n ) (4.30) oc qj(l - q)n-j{ex - l)-(n-i)e^n-^X^'-i  Xi (1 - Fp(y° - l))n~r A s s u m i n g B e t a and G a m m a priors for q and A respectively, the marg ina l posteriors are: T T ( 9 | A ; X i , i = 1 , 2 , . . . , n) a qi+a~l (1 - g ) " - ^ " 1 (4.32) ir(\\q;xi,i - 1 , 2 , . . . ,n) (4.33) oc ( e A - i ) i - » e - M » - n + i ) A E r = i ^ + C - 1 [ « 7 ( A ) ] " - R where g(X) = 1 - Fp(y° - 1) and Fp(y° - 1) denotes the cdf of the Poisson at y° - 1. Note that the posterior of A can no longer be approximated by the G a m m a d i s t r ibu t ion . Nevertheless, we can s t i l l use the M C M C methods to sample from the above posterior densities. S imi la r ly , we use the sample 109 means as the posterior estimates of the parameters. We do not give a s imula t ion i l lus t ra t ion of this analysis since we w i l l conduct an intensive s imula t ion s tudy based on the me thod proposed here. In the fol lowing sections, we w i l l in t roduce the concept of dynamic update of the density parameters before we simulate the interact ive processes of the demand d i s t r ibu t ion es t imat ion and the inventory po l i cy determinat ion. O u r m a i n interest is to see i f this dynamic upda t ing process w i l l outperform the M L E and its corresponding pol icy. W e compare the to ta l inventory cost for these two processes. 4.4.2 Bayesian Update of the ZIP Distribution in a Newsvendor System A s discussed at the beginning of this chapter, an advantage of the Bayes ian me thod is that it can capture h is tor ica l informat ion and ut i l ize new informat ion i n the analysis. In the fol lowing we first introduce the dynamic Bayes ian upda t ing framework for the demand d i s t r ibu t ion . T h e n , we derive the posterior densities before we give the procedures. F i n a l l y we compare the dynamic Bayes ian analysis and the M L E analysis. D y n a m i c Bayes ian Framework We briefly i l lus t ra te in the fol lowing the dynamic Bayes ian upda t ing of the demand d i s t r ibu t ion i n a mul t i -pe r iod inventory system. Deta i led discussions are given i n Chap te r 5. In this section, the order quanti ty is obta ined i n a myopic manner. T h a t is, i t is determined by m i n i m i z i n g the current pe r iod expected cost, wi thou t consider ing the imp l i ca t ion of this decision on the fol lowing periods. In the next chapter, we w i l l model the demand upda t ing w i t h the objective of m i n i m i z i n g the to ta l expected cost of a l l t ime periods. Sect ion 5 of La r iv i e re and Por teus ' paper [39] is pa r t i cu la r ly interesting. T h e y considered mul t ip le markets where "a single decision maker s imultaneously stocks mul t ip le products i n independent markets" . T h e markets are independent and are of different sizes. T h e y concluded that the u n k n o w n parameter upda t ing process developed by Braden and Freimer [10] s t i l l ho ld , and the o p t i m a l inventory level vector is obtained by scal ing the o p t i m a l vector of the or ig inal normal ized problem. T h e y gave an example of a two-period, two-market p rob lem w i t h exponent ia l demand. Information dynamics of the two markets were i l lus t ra ted. "In short , there is a second oppor tun i ty to gain informat ion and a second oppor tun i ty to use informat ion, and these forces p u l l the stock levels in opposite direct ions" when the s tocking decision is jo in t ly made for the two markets . Hence, the example suggested that the real gain is from efficiently using the informat ion that is gathered as opposed to op t imiz ing the s tocking levels loca l ly or myopica l ly . V e r y interestingly, they also invest igated the cost of informat ion gathering from different markets. T h e y concluded that i t was cheaper to acquire informat ion from s tocking more i n the smaller market . W e have s imi lar conclusions regarding the trade-off of the informat ion gathering and the op t imiza t ion . T h i s is discussed i n Chap te r 5. We give 110 economic interpretat ions of our result. A n d specifically, we quantify the marg ina l informat ion cost. In this section, we do not consider the informat ion dynamics , and separate the demand upda t ing and the order quant i ty upda t ing processes. T h e his tor ica l informat ion up to the current pe r iod is summarized i n the pr ior of the parameters. A myopic op t ima l po l i cy is calculated based on the pr ior and is appl ied when the sales of the current pe r iod are observed. Use the da ta of the current per iod to update the pr ior to the posterior, then take the posterior as the pr ior for the next per iod . T h e myopic o p t i m a l po l i cy is updated based on the new pr ior . W h e n the da ta of the next per iod is available, we follow the same procedure of upda t ing the pr ior to the posterior for the next per iod . T h i s i terat ive process is then repeated for the subsequent periods. T h e posterior estimates of the parameters are updated per iodical ly . F r o m the computa t iona l perspective, we use M C M C methods to generate samples from the posterior densities i n each per iod and then estimate the u n k n o w n parameters by these posterior sample means. D y n a m i c Bayes ian U p d a t i n g of the Posteriors W e first give the probabi l i ty model . Af ter reparameter iz ing the Z I P demand, we have the following for per iod t. 1. condi t iona l d i s t r ibu t ion : X\qt, A t ~ ZlP(qt, A 4 ) . 2. pr iors: r r fe ) oc q f - ^ l - q t ) * - 1 , (4.34) *{XT) = TS?- (4-35) where at,Pt,ct and -ft are hyperparameters w i t h fixed in i t i a l values an , Po, cn and 70. T h e posterior densities can be derived in the same way given i n section 4.4.1 by app ly ing the calculations for q and A i n expression (4.32) and (4.34). N o w we propose the upda t ing procedure of the posteriors for Z I P demand i n the mul t i -pe r iod newsvendor system. T h e mul t i -pe r iod newsvendor system here refers to that the newsvendor order quant i ty is updated each per iod after the demand es t imat ion is updated. Remember , censoring occurs at the order quant i ty level, y%, i n the newsvendor system. Define for per iod t + 1: cet+i = at + jt and an = ct, a s tar t ing point; Pt+i = Pt+nt- jt and L% = P\ ct+i = ct + X)i=i xi and c 0 = c - 1; 7t+i = l t + n - n t and 7 o = h 111 st+i =st+nt- jt and s0 = 0; Vt+i =r)t+nt-rt and r)0 = 0. T h e no ta t ion of nt, jt and rt is the same as given earlier i n this chapter except that i t is for per iod t now. For instance, nt is the to ta l number of observations i n per iod t. T a k i n g the posterior of current per iod t as the pr ior for the next pe r iod , the posterior of the next per iod can be upda ted to the fol lowing when the da ta from per iod t + 1 is available: Tr(qt+1\\t+1;xl+1,i = 1 , 2 , . , . , n t + 1 ) (4.36) oc qOti+i+ji+i-i^ _ q^Pt+i+n.t+i-jt+i-1 n(Xt+1\qt+i;xii+\i = 1 , 2 , . . . ,nt+1) (4.37) r t + l t + 1 oc A ^ 1 " * " ^ ^ ^ 1 ^ + 1 e ~ A t + 1 ( 7 l + 1 + r ' t + 1 _ " ' + 1 ) ( e ^ + 1 — i )~( s '+ 1 +™'+ 1 ~-?«+ 1 ) [g(\t+i)]nt+1+nt+1-rt+l where g(\t+i) — 1 - Fp(y%+1 — 1) and Fp(y°+1 - 1) denotes the cdf of P o i s s o n ( A t + i ) at y%+1 - 1. Set t to t + 1 and repeat the above. T h i s procedure updates the posterior densities one per iod at a t ime. In each per iod , the posterior estimates are obta ined by M C M C simulat ions. 4.4.3 Simulation Study of the Dynamic Newsvendor System After exp lor ing the Bayes ian analysis and the dynamic upda t ing of the unknown demand d is t r ibu t ion , our interest lies i n how this me thod is appl ied i n an inventory system, and more impor tan t ly , i f the per iodica l ly upda ted inventory pol icy based on the Bayesian learning process of demand yields a lower cost over t ime compared to its M L E counterpart . We w i l l perform a s imula t ion s tudy for this purpose. In Chapte r 5, we w i l l show that this process may be subopt imal . Intui t ively, once the unknown parameters are est imated, the inventory po l i cy can be derived using the est imated demand d i s t r ibu t ion . W e follow this t h ink ing i n the s imula t ion study. W e apply the Bayesian me thod to calculate the posterior estimates of the parameters. T h e n we calculate the newsvendor pol icy based on these estimates. W e refer to this po l icy as the Bayesian newsvendor policy. R e c a l l the order quant i ty determines the sales censoring level for the next per iod. We also compute the M L E of the parameters and calculate the corresponding inventory cost. We repeat this for a number of periods and compare the accumula ted cost. We have two options when ca lcula t ing the M L E of the parameters. F i r s t , we use the single sample from each per iod to calculate the M L E . Second, we use the accumula ted samples up to per iod 112 t to calculate the M L E for that per iod . W e refer to this as the cumula t ive M L E . W e w i l l use bo th methods and report the results. It w i l l not be surpr is ing if the cumulat ive M L E performs better and yields lower to ta l cost. We assume B e t a and G a m m a priors for the reparameterized Z I P parameters q and A w i t h fixed in i t i a l hyperparameter values (a, /?, c and 7). For the first per iod , the Bayes ian newsvendor pol icy is obtained based on the priors before any demand is observed. We set the M L E pol icy i n the first per iod to be the same as the Bayes ian po l i cy so that bo th pol icy upda t ing processes have the same in i t i a l order quant i ty to start w i t h . F r o m the in format ion acquis i t ion perspective, this also ensures that the demand upda t ing processes start w i t h the same censoring level. In the real wor ld , this reflects the inventory management requirement for a new product where m i n i m a l demand informat ion is available at the beginning. T h e merchandise manager has to decide on the order quant i ty based on either a best guess of the demand pat tern or the his tor ical demand da t a for a comparable product . E i the r way, the priors can be set such that the i n i t i a l guesstimate of the demand is well represented. We w i l l give the pr ior assumpt ion details later. Simulat ion Procedure T h e fol lowing details the procedure of the s imula t ion study. 1. For per iod t, calculate the Bayes ian newsvendor order quanti ty, yfN, based on the prior of q and A. 2. Ca lcu la te the M L E order quanti ty, y ^ L E , based on the M L E that takes lost sales into account. Refer to Chap te r 3 for the ca lcula t ion of the newsvendor pol icy using the es t imated demand d is t r ibut ion . 3. Generate a demand sample from ZIP (p , A) w i t h A = 5 and p = 0.705 which yields about 30% zero observations (q = 0.3 from expression (4.22)). 4. O b t a i n the sales samples by compar ing the demand observations to the order quanti t ies. Note we w i l l have two sales samples at different censoring levels, yfN and yf^LE • W e refer to these as the Bayesian sales sample and M L E sales sample respectively. 5. Use the Bayes ian sales da ta to ob ta in the posterior estimates of the Z I P parameters as out l ined i n the previous section, and update the hyperparameters. Set the posterior as the pr ior for per iod t + 1. 6. Use the M L E sales da ta to ob ta in the new M L E ' s of p and A considering censoring. Refer to Chapter 3 for details. W e start w i t h the single sample M L E . Single sample here refers to one sample of 14 generated da ta points for per iod t w i t h the order quant i ty y ^ L E . Hence the M L E for per iod t in this case is calcula ted by using only the generated sales sample of that pe r iod (of sample size 14). T h e n we calculate the cumulat ive M L E by using the accumulated sample data . T h a t is a l l the sample da ta 113 generated from per iod 1 to per iod t, a to ta l of 14 * t da ta points . In a special single sample M L E case, i f the entire generated sales sample (i.e., a l l 14 sample sales) are 0, then a l l the demands are exactly observed to be 0 (wi th y £ i L B > 0). T h e M L E ca lcula t ion can s t i l l be used and bo th the M L E ' s of p and A are 0. 7. Ca lcu la te the inventory costs for this per iod . Three costs are calculated, the baseline cost w i t h the o p t i m a l newsvendor pol icy based on the real demand d i s t r ibu t ion ZIP(0 .705,5) , the Bayes ian cost w i t h the Bayes ian newsvendor pol icy and the M L E cost w i t h the M L E pol icy. In contrast to Section 3.3, the generated demand da ta is used here i n ca lcula t ing these costs. Hence the costs are what would be incurred i f a such pol icy were i n place. Refer to Chapte r 3 for newsvendor cost computat ions . 8. A d d the current pe r iod cost to the accumulated costs from per iod 1 to per iod t — 1. 9. Set t to t + 1 and go to step 1. Repeat the procedure t i l l the last per iod is complete. We w i l l ob ta in a series of Bayes ian newsvendor policies and a series of M L E policies. We w i l l also have three series of inventory costs. Note , we use the real demand i n cost computa t ion , despite the fact that real demand may not be available i n a real newsvendor system (when the demand exceeds the order quant i ty) . A reasonable al ternat ive is the expected cost w i t h respect to the es t imated demand dis t r ibut ions . Obviously , this al ternative involves the es t imated d i s t r ibu t ion direct ly i n add i t ion to the pol icy. In order to isolate the pol icy ' s effect on the cost, we use the real demand. Th i s enables us to focus on the policies while compar ing various costs. T h i s ind i rec t ly involves the demand es t imat ion since the policies are derived based on i t . S i m u l a t i o n S e t t i n g s We list the settings and assumptions i n the fol lowing. 1. T h e pr ior for the re-parameterized Z I P ( g , A ) is: Beta(a , /9) (4.38) A G a m m a ( c , 7). (4.39) T h e i n i t i a l values of the hyperparameters are set as follows. a = 0 .5; 0 = 0.5 (4.40) T h i s yields E(</)=0.5 which indicates a rough guess on q wi thout favoring either extreme, 0 or 1. c = l ; 7 = 10 (4.41) 114 1 2 3 4 5 6 7 8 9 10 11 12 13 14 demand 4 7 5 0 2 4 7 4 0 3 7 5 0 0 sales 4 4 4 0 2 4 4 4 0 3 4 4 0 0 Table 4.5: A R a n d o m Sample of Newsvendor D e m a n d and Sales T h i s yields E ( A ) = 1 0 and uar (A)=100. W e set them like this i n order to l i m i t the influence of the pr ior . T h i s re la t ively 'flat ' set t ing representing a w i l d guess would be a safer choice for the case where the demand knowledge is very l imi t ed . 2. T h e cost s tructure of the newsvendor systems is: cc = 15 the uni t order ing cost, h = 2 the uni t ho ld ing cost, pn — 30 the uni t penal ty cost. (4.42) (4.43) (4.44) T h i s gives k — (pn — cc)/(pn — h) = 0.536. Note , this rat io plays an impor tan t role i n determining the newsvendor order quanti t ies. A detailed discussion about this ra t io can be found i n Chapte r 5. 3. T h e sample size for each per iod is 14, that is two weeks of da i ly sales are observed. 4. T h e to t a l number of periods is 12. We w i l l see later that this is more t han enough to have the inventory costs s tabi l ize by the Bayes ian approach. 5. Me t ropo l i s samples are generated for q and A. A sample of p can then be obta ined by using (4.27). T h e sample size is set to 500. T h e sample mean after the burn- in pe r iod is used as the posterior estimate. F igu re 4.15 and F igure 4.16 give an example of sample paths of the Z I P parameters p and A respectively. Note the Met ropo l i s samples of A (Figure 4.16) are very variable, much more so than in the uncensored case given i n Section 4.3 (Figures 4.9 and 4.12). Table 4.5 gives an example of the demand and sales sample for one per iod where the order quant i ty is set to 4. Hence any demand exceeding 4 is censored. Simulation Results I - with Single Sample M L E Table 4.6 summarizes the demand es t imat ion results. T h e M L E is computed using the single sample of 14 observations i n each per iod . Table 4.7 gives the newsvendor policies for each per iod and the accumulated costs at the end of each per iod . 115 Figure 4.15: A Met ropo l i s Sample of the Z I P parameter p. Figure 4.16: A Met ropo l i s Sample of the Z I P parameter A. 116 T i m e P e r i o d Poster ior Es t imates M L E A P A * p A P A * p 1 4.5815 0.8497 3.8930 4.5631 0.86618 3.9525 2 4.1359 0.8589 3.5525 4.0996 0.87159 3.5732 3 4.3709 0.8138 3.5569 4.8781 0.71976 3.5111 4 4.6923 0.7529 3.5331 16.0632 0.57143 9.1790 5 4.9018 0.7310 3.5834 14.1816 0.64286 9.1168 6 4.6729 0.7277 3.4006 7.7452 0.71460 5.5347 7 4.7468 0.7297 3.4636 8.2541 0.71447 5.8973 8 4.7979 0.7345 3.5241 6.1218 0.78744 4.8206 9 4.6960 0.7458 3.5025 4.9445 0.79135 3.9129 10 4.7700 0.7495 3.5752 6.9666 0.78646 5.4789 11 4.9195 0.7321 3.6018 6.6951 0.57214 3.8305 12 4.7477 0.7299 3.4656 5.5046 0.71720 3.9479 Table 4.6: D e m a n d E s t i m a t i o n from Sales T i m e P e r i o d Bayes ian M e t h o d M L E M e t h o d Basel ine yBN cost yMLE cost y* cost 1 6 1256 6 1256 4 1172 2 4 2604 4 2604 4 2520 3 4 4004 4 4004 4 3920 4 4 5336 4 5336 4 5252 5 4 6822 12 7634 4 6738 6 4 8064 12 9912 4 7980 7 4 9258 7 11344 4 9174 8 4 10448 7 12744 4 10364 9 4 11550 5 13888 4 11466 10 4 12904 4 15242 4 12820 11 4 14200 6 16622 4 14116 12 4 15332 4 17754 4 15248 Table 4.7: Inventory Costs 117 1 1 i 1 1 r -2 4 6 8 10 12 period Figure 4.17: Inventory Costs C o m p u t e d U s i n g Var ious Pol ic ies . F igure 4.17 gives the plot of three accumulated costs. T h e sol id l ine is the baseline cost, the line w i t h empty circles is the Bayes ian newsvendor cost and the line w i t h sol id squares is the M L E cost. T h e figure shows that the Bayes ian cost is very close to the baseline cost while the accumula ted M L E cost is significantly higher. T h e M L E cost grows apart from the other two ma in ly i n per iod 5 and 6. T h i s can be seen from Table 4.7 and obviously, i t is due to the much higher order quant i ty i n these two periods. These much higher order quanti t ies are the result of overest imation of parameter A. F r o m Table 4.6, we can see that the M L E of A tr iples the true value. Table 4.8 gives the demand and sales samples i n per iod 4 4 . O u t of 14 sales, only 6 are exact. In this case, the demands are censored at 4. A m o n g these 6, there are three zero sales. It is not surpr is ing that such a sample yields a high M L E for A. In short , this sample shifts the m a x i m u m l ike l ihood est imated cdf to the right by a great deal wh ich i n t u r n results i n a larger order quant i ty for the next per iod . T h i s is supported by the s imula t ion results here. Table 4.7 shows that the Bayes ian order quant i ty becomes constant from the second per iod onward, and i t is the same as the o p t i m a l order quant i ty w i t h the true parameter values. T h a t is, the Bayes ian method and the baseline agree after only one per iod. T h e A * p row can be considered as an estimate of the expected demand. T h e expected demand i n this case is 5*0.7047=3.5237. 4 T h e random sales sample generated in period 4 is used to obtain the estimates first and then, the estimates are used to calculate the order quantity for period 5. 118 1 2 3 4 5 6 7 8 9 10 11 12 13 14 demand 9 8 11 3 0 6 5 0 8 0 7 3 2 9 sales 4 4 4 3 0 4 4 0 4 0 4 3 2 4 Table 4.8: T h e D e m a n d and Sales Sample i n P e r i o d 4. 2 4 6 8 10 12 period Figure 4.18: Inventory Costs C o m p u t e d U s i n g Var ious Pol ic ies . S i m u l a t i o n R e s u l t s II - w i t h C u m u l a t i v e M L E N o w we assume a l l the h is tor ica l da ta is available and the accumulated samples are used to compute the M L E . T h a t is, a l l da ta from per iod 1 to per iod t is used to ob ta in the M L E for pe r iod t. Hence the smal l sample p rob lem is avoided after the first few periods. A l l other details remain the same. Note , the history is summar ized i n the updated parameters i n the Bayes ian analysis and hence m i n i m a l da t a storage is needed. F igure 4.18 gives the plot of three accumulated costs for this case. T h e legend remains unchanged. T h e M L E cost becomes very close to the baseline cost this t ime. It seems that the es t imat ion problem due to the sma l l sample size is resolved. Table 4.9 gives the order quantit ies and the inventory costs. We inc luded the baseline and the Bayesian counterparts for ease of comparison. S imi la r ly , Table 4.10 gives the es t imat ion results for this case. It can be seen that this t ime, the M L E of A after per iod 4 is a lot closer to the true value. Hence the order quant i ty for pe r iod 5 is lowered to 2 which largely brings down the accumulated cost at the end of per iod 5. It is the 119 T i m e Bayes ian M e t h o d Cumula t i ve M L E Basel ine P e r i o d yBN cost yMLE cost y* cost 1 6 1256 6 1256 4 1172 2 4 2604 4 2604 4 2520 3 4 4004 4 4004 4 3920 4 4 5336 3 5378 4 5252 5 4 6822 2 6948 4 6738 6 4 8064 3 8148 4 7980 7 4 9258 3 9384 4 9174 8 4 10448 4 10574 4 10364 9 4 11550 4 11676 4 11466 10 4 12904 4 13030 4 12820 11 4 14200 4 14326 4 14116 12 4 15332 3 15500 4 15248 Table 4.9: Inventory Costs same for pe r iod 6. S i m u l a t i o n R e s u l t s I I I - w i t h L o w I n i t i a l O r d e r Q u a n t i t y W i t h the set t ing of the hyperparameters c and 7 which gives E ( A ) = 1 0 , 'we have an i n i t i a l order quanti ty higher than the op t ima l . W e are interested i n the other side of the story too; that is, when the first guess on demand is biased to the lower side. A s a result, the i n i t i a l order quant i ty is lower than the op t ima l . Since we believe that the cumulat ive M L E performs much better than the single sample M L E , which is also suppor ted by the previous two studies, we use the cumulat ive M L E here. W e r u n the s imula t ion again w i t h the fol lowing change. 7 = 4 . (4.45) T h i s gives E ( A ) = 4 and var(\) — 16. T h e i n i t i a l Bayes ian order quant i ty is reduced to 1. Obv ious ly this censors more demands in general. We plot three accumulated costs i n F igu re 4.19. It can be seen that the Bayes ian cost is higher than the baseline cost. T h e Bayes ian cost remains roughly paral le l to the baseline cost and i t is the first three periods which ma in ly contr ibute to the discrepancy. However, the M L E cost qu ick ly grows apart from the baseline cost s tar t ing from the second per iod and settles at a steady increasing rate after pe r iod 6 when it becomes paral le l to the baseline cost. To have a thorough unders tanding of this 120 T i m e P e r i o d Poster ior Es t imates C u m u l a t i v e M L E A P A * p A P A *p 1 4.5815 0.8497 3.8930 4.5631 0.8662 3.9525 2 4.1359 0.8589 3.5525 4.1153 0.8714 3.5859 3 4.3709 0.8138 3.5569 4.2996 0.7241 3.1134 4 4.6923 0.7529 3.5331 4.0910 0.5811 2.3775 5 4.9018 0.7310 3.5834 3.6680 0.6597 2.4198 6 4.9018 0.7277 3.4006 4.2971 0.7241 3.1117 7 4.7468 0.7297 3.4636 4.4923 0.7224 3.2451 8 4.7979 0.7345 3.5241 5.0836 0.7906 4.0192 9 4.6960 0.7458 3.5025 4.9771 0.7912 3.9377 10 4.7700 0.7495 3.5752 5.1137 0.7905 4.0422 11 4.9195 0.7321 3.6018 5.1137 0.5749 2.9398 12 4.7477 0.7299 3.4656 4.3872 0.7233 3.1732 Table 4.10: D e m a n d E s t i m a t i o n from Sales; C o m p a r i s o n of Bayes ian & C u m u l a t i v e M L E consequence, we give the es t imat ion results and their corresponding policies i n Table 4.11 and Table 4.12 respectively. It can be seen from Table 4.12 that the M L E order quantit ies are quite errat ic whi le the Bayesian newsvendor order quanti t ies settle at the op t ima l value half way th rough the twelve periods. T h e deviat ion of the order quanti t ies results i n a devia t ion of the costs. Therefore, the M L E cost is much higher than the baseline. F r o m Table 4.11, we observe that the changes of the Bayes ian estimates from per iod to per iod is less d ramat ic than the M L E . C o m p a r i n g the rows of A * p wh ich estimates the expected demand, the M L E is much higher than the Bayes ian estimates in periods 2, 3, 4, and 5. T h e Bayes ian order quant i ty y B N j umps from 2 to 7 i n per iod 3 as a result of overest imating the demand, consequently after observing a good number of censored sales for two periods due to the low order quanti t ies, 1 i n per iod 1 and 2 in per iod 2. T h i s over-correction helps to gradual ly reduce the est imat ion and the derived order quant i ty in the next four periods, s ta r t ing from per iod 3. Table 4.12 shows that y B N tapers down to the o p t i m a l in per iod 7 and remains there t i l l the last per iod . O n the contrary, the M L E order quant i ty y M L E more than triples the o p t i m a l i n per iod 4, after 2 periods of low y M L E and i t does not settle at the o p t i m a l t i l l there are only two periods left. T h e possible reasons for the observations summar ized above are as follows. F i r s t , this is based on only one repl ica t ion of the s imula t ion . Second, the sample size of 14 is too smal l for M L E . T h i r d , the extreme censoring i n the first few periods has greater influence on M L E ' s . T h e behavior of the Bayesian 121 period Figure 4.19: Inventory Costs C o m p u t e d U s i n g Var ious Pol ic ies . T i m e P e r i o d Poster ior Est imates C u m u l a t i v e M L E A P A * p A P A * p 1 2.50 0.83 2.07 3.00 0.60 1.80 2 8.36 0.64 5.38 15.18 0.71 10.84 3 8.11 0.64 5.23 12.56 0.64 8.08 4 6.72 0.70 4.68 8.34 0.86 7.15 5 5.78 0.73 4.20 6.68 0.86 5.73 6 5.16 0.75 3.88 5.50 0.86 4.74 7 4.86 0.76 3.68 5.23 0.79 4.13 8 4.70 0.74 3.47 5.09 0.57 2.93 9 4.75 0.73 3.47 4.57 0.65 2.93 10 4.64 0.73 3.36 4.55 0.72 3.29 11 4.82 0.72 3.49 4.92 0.72 3.54 12 4.85 0.74 3.58 4.95 0.86 4.27 Table 4.11: D e m a n d E s t i m a t i o n from Sales; Compar i son of Bayes ian & C u m u l a t i v e M L E w i t h L o w F i r s t P e r i o d Order Q u a n t i t y 122 T i m e P e r i o d Bayes ian M e t h o d Cumula t i ve M L E Basel ine yBN cost yMLE cost y* cost 1 1 1338 1 1338 4 1296 2 2 2582 2 2582 4 2428 3 7 4036 14 5226 4 3672 4 6 5544 10 7182 4 5236 5 5 6700 8 8744 4 6378 6 5 7724 6 9922 4 7304 7 4 9074 5 11286 4 8654 8 4 10182 5 12492 4 9762 9 4 11124 3 13448 4 10704 10 4 12132 3 14470 4 11712 11 4 13502 4 15840 4 13082 12 4 14680 4 17018 4 14260 Table 4.12: Inventory Costs approach versus the M L E approach should be the subject for further invest igat ion. Note that the derived order quantit ies may be the same when the estimates are close. T h i s is because the Z I P d i s t r ibu t ion is discrete which gives a step function cdf. We make the fol lowing conclusions based on these three s imula t ion studies. 1. It seems that the i n i t i a l order quant i ty plays a very impor tan t role for the entire s tudy per iod. No t only does it determine the cost as a direct impact , i t also sets off a demand informat ion gathering process as an indirect impact . T h e direct effect on the cost is short t e rm whi le the indirect effect is long te rm. A n d the subtle long term effect may very l ike ly outweigh the short t e rm effect. 2. It is preferable to i n i t i a l l y overestimate the demand than underest imate i t . Overes t imat ion results i n a higher t han o p t i m a l order quanti ty. A l t h o u g h it may incur higher order ing cost, it allows more exact demand observations i n general which helps to learn the demand d i s t r ibu t ion better and consequently yields a lower cost i n the long run . T h e order quant i ty 's effect on the demand es t imat ion w i l l be thoroughly discussed i n the next chapter. We also provide a theoret ical proof of the conjecture there that it is better off to adopt a higher than the o p t i m a l po l icy i n the i n i t i a l periods when demand is learned i n a censoring system. 3. T h e Bayes ian method offers a better learning technique when one lacks a good unders tanding of the 123 demand pat tern, especially when the i n i t i a l guess is on the lower side, and before there is enough data to correct the guess. 4. A s opposed to the M L E , the Bayes ian method performs well when l im i t ed informat ion is available, w i t h v i r t u a l l y no requirement on the sample size. T h i s is supported by the first study. 5. In a l l three studies, the Bayes ian cost stays close to the baseline whi le M L E cost deviates to the upper side i n the first and the t h i r d studies. 124 Chapter 5 The Bayesian Newsvendor with Unobservable Lost Sales We observed i n Chap te r 4 that adaptive Bayes ian upda t ing performs well i n one case. B u t the question arises whether our approach is op t ima l . T h i s chapter explores this question from the theoret ical perspective. Af ter comple t ion of this research the results were recently been extended i n D i n g , P u t e r m a n and B i s i [18]. In this chapter, we consider the s tat is t ical issues i n the context of op t imiza t i on . T h e s impl ic i ty of the newsvendor mode l enables us to focus on the value of informat ion i n stochastic inventory management. We show that unobservable lost sales result i n a higher op t ima l order quant i ty t han i n the fully observable case. T h i s is because lost sales result in a loss of informat ion i n demand upda t ing . Consequently, one orders more i n early periods to ob ta in addi t iona l informat ion. T h a t is, when lost sales cannot be observed, the trade off between m i n i m i z i n g cost and prov id ing satisfactory service w i t h observable demand extends to the trade off between m i n i m i z i n g cost and bo th p rov id ing satisfactory service and gather ing informat ion for future demand upda t ing . W e assume that possible lost sales are the only result of stock-outs. Other stock-out related issues, such as product subs t i tu t ion and the effect of stock-outs on goodwi l l of the company are not considered. Hence, we assume that the demand is independently and ident ical ly d is t r ibuted (iid) over t ime. T h e key result described above is proved i n Sect ion 5.3. We give, i n Sections 5.1 and 5.2, respectively, background and op t ima l policies for three versions of the newsvendor problem in progressively more realist ic, yet more compl ica ted settings. In Sect ion 5.4, we discuss some consequences of the argument. Some numer ica l examples are given i n Section 5.5. A n d Sect ion 5.6 gives some in i t i a l work on extending the key result developed for 2-period problem to A^-period problem. W e give some conclusions i n Sect ion 5.7. 125 5.1 The Newsvendor Problems In this section, we dis t inguish newsvendor problems under different assumptions about available demand informat ion . Hence, we have the fol lowing three problems. i) T h e t r ad i t iona l newsvendor p rob lem when the demand d i s t r ibu t ion is known; ii) A newsvendor p rob lem w i t h unknown demand d i s t r ibu t ion but i n which demand is completely observed; i i i) A newsvendor p rob lem w i t h unknown demand d i s t r ibu t ion i n which sales (as opposed to demand) is observed, and the quant i ty of lost sales is unobservable when stock-outs occur . In the u n k n o w n demand case, we assume that the density is known to belong to a par t icu lar family wi th parameters unknown. Trad i t iona l ly , the newsvendor p rob lem is treated as one-period prob lem assuming the demand d i s t r ibu t ion is known. However, when parameters are unknown, i t must be considered as a dynamic problem. T h e demand is updated when new data becomes available. T h e po l i cy is upda ted accordingly. We w i l l see that the t h i r d newsvendor p rob lem is no longer equivalent to a sequence of single-period problems which characterizes the t rad i t iona l newsvendor p rob lem w i t h known demand. Refer to Gal lego and M o o n [23] for a thorough discussion of the newsvendor p rob lem and its variants. W e assume non-negative continuous demand so that our a t tent ion w i l l not be diver ted by technicalities from discrete demand. However, most of the numerica l examples and mot iva t ion use discrete demand dis t r ibut ions . 5.1.1 Traditional Newsvendor Problem In a newsvendor inventory system, one cannot store inventory from one per iod to the next since the product lasts on ly one per iod . A s s u m i n g the demand is i i d over t ime, and the d i s t r ibu t ion is known, the problem reduces to a sequence of ident ical one-period op t imiza t ion problems. T h i s is because the t rans i t ion probabi l i ty d i s t r ibu t ion does not depend on the policy. We first give the cost s tructure. Le t c denote a variable order ing cost (the cost per uni t of products ordered); let h denote the salvage value (the revenue brought i n by each uni t of products i f it is unsold); let p denote the penalty cost (the penal ty per uni t of products i n shortage). It is reasonably and commonly assumed that h < c < p. T h e demand i n per iod n , an i i d r andom variable Xn, has p robab i l i ty density function (pdf) m(x) w i t h support R+ = [0, oo). L e t y be the order quant i ty and x the demand i n a per iod . T h e n the cost of the pol icy i n that pe r iod is: cy- h(y -x) i f x < y , cy + p(x — y) i f x > y . 126 T h e expected cost of the pol icy is R(y) = / r(x,y)m(x)dx JR+ fy f°° = cy - h (y - x)m(x)dx + p / (x - y)m(x)dx. (5.1) JO Jy It can be shown that R(y) is continuous and convex i n y. T h e o p t i m a l po l i cy y* can be obtained by m i n i m i z i n g the expected cost R(y) and i t remains the same for each per iod . 5.1.2 The Bayesian Newsvendor Problem with Observable Lost Sales A s discussed i n the in t roduc t ion , the demand d is t r ibu t ion is never t ru ly k n o w n i n practice. One has to estimate demand using sales data . In lost sales inventory systems, sales do not always represent demand. W h e n lost sales occur, we only know that the demand is at least as large as the s tock-on-hand at the start of that per iod . W e say i n that case, the demand is censored by the s tock-on-hand. Since inventory cannot be carr ied over i n the newsvendor system, when the demand is censored i n a pe r iod i t is censored by the order quant i ty for that per iod . Thus , the inventory pol icy determines the censoring level i n the newsvendor system. A s a s ta r t ing point , we assume that to ta l demand can somehow be observed. T h i s assumption might be realist ic when orders arrive through a ca l l center and accurate records of orders are retained. In a lighter vein, the newsvendor may stay a round after stock-out and see how many papers she would have sold. T h e demand is, hence, always observable as i t equals sales (i.e., satisfied demand) plus lost sales (i.e., unsatisfied demand) . Le t Xn £ R+ be the demand i n per iod n. T h e n the r andom variables {Xn, n— 1 , . . . , JV} are assumed i i d and observable. In each per iod, Xn is sampled from a d i s t r ibu t ion w i t h density f{-\Q), in which 0 £ G is the u n k n o w n parameter (6 can also be a vector of unknown parameters) , but otherwise f(-\9) is known. G i v e n an a rb i t ra ry pr ior d i s t r ibu t ion irn(8), the possible posterior d is t r ibut ions are irn+i(d\xn) where TTn+i{0\xn) = -r-r; r^r . . (5.2) We regard the sequence of posterior dis t r ibut ions as a stochastic process. T h e posterior process is the basis for dynamic Bayes ian demand upda t ing . W h e n the demand in per iod n is observed, i.e., Xn = xn, the pr ior wn is updated to the posterior 7r n +i , and we use the posterior at n as the pr ior for n + 1. T h e cycle continues i n the same manner. 127 Correspondingly , the marg ina l process is defined as {mn(x),n = 1,... , N} w i t h , mn{x) = [ f(x\6)nn{6\xn-.1)de. (5.3) Je T h e marg ina l is the Bayes ian estimate of the demand d i s t r ibu t ion i n per iod n. Hence, this marg ina l process can be considered as the demand upda t ing process. It evolves as add i t iona l in format ion is gathered over t ime. T h e cost s tructure remains the same as in the t r ad i t iona l newsvendor problem. We formulate this newsvendor p rob lem as a Bayes ian M D P ( B M D P ) model as follows. Le t TV be a finite number of decision epochs. Decis ion epochs refer to the times at which the order quant i ty is chosen. T h e informat ion available at that t ime is reflected i n the states. A t decision epoch n (n = 1,2,... ,N), S t a t e s : L e t Sn denote the state space at n , Sn C {/ : / is a probabi l i ty d i s t r ibu t ion on €)}. Theore t ica l ly , Si C {al l appropriate pr iors}. So the state space can be considered as the pr ior space at n = 1 and the posterior space at n = 2,... ,N. Since we are not pa r t i cu la r ly interested i n the effect of pr ior specification here, we assume a fixed p r i o r ' i n our model , i.e., si = ir\{6). T h e n Sn = n„(#) which is a set of a l l possible posteriors corresponding to the given pr ior 7Ti (#). A c t i o n s : As is a set of admissible actions (order quantities) i n state s. Specifically, As = [0,oo) for each state. C o s t s : A real-valued measurable function on Sn x As bounded from above. It is the Bayes ian expected cost w i t h respect to 7T n , ) = E W n [ r ( X n , j / n ) ] (5.4) = cyn-h (yn - x)mn(x)dx + p (x - yn)mn(x)dx Jo Jyn where mn(x) is defined i n (5.3), and yn is the quant i ty to order (action) i n state sn = 7r n. We assume zero t e rmina l reward , that is RB(^N+i,yN+i) — 0. A g a i n , RBi^niVn) is continuous and convex i n yn. T r a n s i t i o n p r o b a b i l i t y d e n s i t i e s : A t n, since xn occurs w i t h p robab i l i ty mn(xn), the t rans i t ion proba-b i l i t y density for the B M D P may be specified for any yn £ [0, oo) as p{irn+1\irn,yn) = mn(xn) when irn+1(0\xn) = T———— . T h e t rans i t ion p robab i l i ty density equals 0 for posteriors that cannot be a t ta ined from irn(6). Note that since the demand is completely observable, the t rans i t ion probabi l i ty is independent of the act ion yn. 128 Note , one needs proper measure spaces and subtle existence results to develop a dynamic model for d i s t r ibu t ion upda t ing . Several foundat ion papers i n the Bayes ian M a r k o v Dec is ion Process framework have developed the results. Hence, we adopt the common approach. T h a t is to regard the current pr ior as the "state" and then give a t rans i t ion rule which relates the subsequent pr ior (the posterior) to the current state given the ac t ion chosen. In reality, the pr ior is always parametr ized and the B M D P dynamics are in terms of the parameters of the model for which there is no difficulty. Unde r the to ta l expected reward cr i ter ion, the op t ima l i ty equations are given by WnUn) = m i n { i ? s (7T n ,y n ) + / u n + i ( 7 r n + i ( - | z ) ) m „ ( : r ) d a ; } (5.5) j R + for n = 1 , . . . ,N w i t h the boundary condi t ion UN+I{KN+I) = 0. T h e o p t i m a l po l i cy can be found by solving the op t ima l i ty equations given by (5.5). Since un depends on h is tory only th rough the current state 7 r „ , and 7rn is updated by the observed demands independent of actions {J/J; i •=!,... ,n - 1}, the B M D P can be reduced to a sequence of single per iod problems, w i t h the demand d i s t r ibu t ion being updated i n each per iod . Note , when the pr ior is degenerate at 9° corresponding to knowing the value of 9 w i t h certainty, this B M D P is further reduced to the t r ad i t iona l newsvendor model i n Sect ion 5.1.1. It is also possible that i t is not economical to stock at a l l w i t h some par t icular priors. Lovejoy [44] gave an example where the costs and the priors are such that this is the case. For a thorough discussion of the B M D P model w i t h unknown parameters, refer to van Hee [60]. 5.1.3 The Bayesian Newsvendor Problem with Unobservable Lost Sales In lost sales inventory systems, ac tual demands are rarely observable. T h i s is especially true when goods are sold th rough re ta i l systems. We now focus on this version of the newsvendor problem. In this p rob lem, the observed da ta is the to ta l sales i n each per iod . Before we extend the above B M D P to accommodate unobservable lost sales, we first define the sales by xn = min(Xn,yn), w i t h yn being the order quant i ty at decision epoch n. Obviously , demand is exact ly observed when sales are less than the order quanti ty, i.e., Xn = xn < yn; and the demand is censored at the order quant i ty when the observed sales equals yn, i.e., Xn > xn = yn. Therefore, by compar ing the sales and the order quanti t ies, we can conclude if the observed demand is greater than or equal to sales. T h e states, actions, costs and t rans i t ion probabi l i t ies are given i n the fol lowing, w i t h a l l other parameters and assumptions remain ing the same as the fully observable case. 129 S t a t e s : T h e state space Sn is the same as in Section 5.1.2, but the posterior p robab i l i ty n'n+1(6\xn) depends on b o t h the demand observation and the order quant i ty for n. It is given as follows TF' (0\Xn) = J7 r"+1(6'la ;«) i fZn<<M, ^ n + 1 " \ < + i ( % n ) i i x n = y n , where f~f(x\6)dxn'n(6) A p p e n d i x G i l lustrates the ca lcula t ion of the posterior given different demand observations from a discrete d i s t r ibu t ion . A s we can see, when the demand in n is fully observed indica ted by the sales being less than the order quant i ty (xn < yn), the state at n + 1 becomes n'n+1(6\xn) = irn+i(0\xn) wh ich is defined in (5.2). O n the other hand, i f the demand is censored at the order quant i ty (xn = yn), the state becomes < + i ( 0 | z n ) = <+i (0 |z /n) • T h e superscript c is for censoring i n this chapter. No te i n case of stock-outs, lost sales are taken into account by in tegrat ing f{x\6) over the range {yn, oo) to give the p robab i l i ty of hav ing a censored demand. A c t i o n s : Same as i n Sect ion 5.1.2. C o s t s : T h e Bayes ian expected cost w i t h respect to n'n is R B « , y n ) = EK[r(X,yn)} rVn r°° - c y n - h (yn - x)m'n(x)dx +p (x - yn)m'n{x)dx JO Jyn where m'n(x) = Je f (x\6)Tr'n(9\xn-i)dd. Since n'n is calculated differently w i t h an exact demand observation rather t han w i t h a censored observation as given i n (5.6), the marg ina l m'n is separated into the corresponding two cases as wel l , i.e. m ' (£) = \ mn(x) i f n'n(8\xn-i) = ^n{9\xn-i) i.e., when a ; n _ i < yn-i, ^ ^ \mcn(x) i f n'n(6\xn-i) = 7 r £ ( % n _ i ) i.e., when xn-i = yn-i, where m^(-) is the marg ina l pd f w i t h respect to 7r£(- |-) . T r a n s i t i o n p r o b a b i l i t y d e n s i t i e s : p(ir' | T T ' y ) - ) m ' n ^ X n ^ i f < + i ( % n ) = nn+i(6\xn) i-e., when xn < yn , "+1 " \l - Mn(yn) ii n'n+1(6\xn) = Tr^+1(e\yn) i.e., when xn = yn , where M'n(yn) = Vx(Xn < yn). T h e t rans i t ion probabi l i ty density equals 0 otherwise. It can be seen in contrast to the observed demand case (in Sect ion 5.1.2) that the t rans i t ion probabi l i t ies depend on actions. T h e op t ima l i t y equations i n the unobservable case become l-Vn « n K ) = m i n {RB(-K'n,yn) + / un+l{irn+i{-\x))m'n{x)dx yn€R+ Jo + «n+l«+1(-|j/n))[l - M'n{yn))} 130 for n = 1 , . . . , N, w i t h the same boundary condi t ion as i n Sect ion 5.1.2. 5.2 Optimal Policies for the Three Newsvendor Problems 5.2.1 Traditional Newsvendor Problem T h e o p t i m a l order quant i ty (y*) for this problem is the well known newsvendor pol icy : 2 / W = M - 1 ( f c ) where k = > 0 < fc < 1, is referred to as the c r i t i ca l fractile; and M(x) = Pi(X < x). T h i s result can be obta ined either by m i n i m i z i n g the expected cost R(y) i n (5.1) or by marg ina l analysis which is conceptually different (Silver, P y k e and Peterson [55]). Since the marg ina l analysis argument summarizes the economic jus t i f ica t ion for the o p t i m a l policy, we w i l l use marg ina l analysis to interpret some results later on . A p p e n d i x H summarizes the marg ina l analysis argument. Suppose the newsvendor has decided to order y units . N o w he is consider ing i f he should order the (y + l ) s t un i t . It w i l l not be sold i f the demand x is less than y + 1. A s a result, a cost c — h is incurred associated w i t h acqui r ing i t . O n the other hand , i t is sold i f x > y + 1, and we have avoided a cost p — c, or i n other words, we have a saving of p — c by having i t available. W i t h a par t icu lar (y + l ) s t unit , if the expected cost associated w i t h acquir ing i t is less than the expected saving from having i t on hand, we would want to order i t . Fur thermore , we might also want to order ' the (y + 2)nd, the (y + 3)rd, etc. Fol lowing this logic for continuous demand, we would want to order quant i ty y(N) where the expected cost equals the expected saving. T h i s leads to the fol lowing equat ion (c - ft)M(yW) = (p - c ) [ l - M(yW)}. (5.9) So lv ing this equat ion for y(N) gives the op t ima l newsvendor pol icy. 5.2.2 The Bayesian Newsvendor Problem with Observable Lost Sales For this version of the newsvendor p rob lem w i t h unknown demand, the o p t i m a l order quant i ty varies between decision epochs because the demand d i s t r ibu t ion is updated after observing demand. Since the t rans i t ion probabi l i t ies i n the B M D P model i n Sect ion 5.1.2 are independent of the actions, the p rob lem can be s implif ied. T h a t is, solving the op t ima l i ty equations i n (5.5) for (n = 1 , . . . , TV) is equivalent to so lv ing the fol lowing op t ima l i ty equations un{ttn) = m i n RB(irn,yn) for n = 1 , . . . , TV. yn€R+ 131 Note that these problems are s t i l l connected through the Bayes ian informat ion structure. Therefore, when the demand is fully observable the B M D P can be reduced to a series of single-period problems since the order ing pol icy affects only cost but not t rans i t ion probabi l i t ies . W e refer to bo th the t rad i t iona l newsvendor p rob lem and this version of the newsvendor p rob lem as the Single-Per iod-Equivalent Newsvendor ( S P E N ) problem. Intui t ively, the demand upda t ing process is independent of pol icy upda t ing . Thus , at each per iod, we need to first update demand and second, min imize the expected cost w i t h respect to the updated estimate of the demand d i s t r ibu t ion . T h a t is, at the end of each per iod n = 1 , . . . , N - 1, we update demand from the pr ior into posterior based on the observed to ta l demand i n that per iod . W e use the posterior as the pr ior for the next per iod and calculate Mn+i by integrat ing the density w i t h respect to the posterior except that in the first per iod , M i is calculated before observing xi by integrat ing over the pr ior d i s t r ibu t ion 7Ti. Fo l lowing the same line of the argument of marg ina l analysis for s ingle-period prob lem, we can obta in the o p t i m a l decision rule, for per iod n , from (5.9), w i t h M replaced by Mn. A p p e n d i x I gives the t ime line of events jus t described. Therefore, the o p t i m a l po l icy of this problem is obtained by the fol lowing procedure: i) at n , update pr ior 7r n _i (8 \x n -2) to posterior 7r n (#|a; n _i) us ing x n _ i by (5.2); ii) calculate Mn(x); i i i ) compute ynBN^ = M~1(k); iv) observe demand xn; v) increment n by 1 and re turn to i ) . W e refer to the o p t i m a l po l icy ynBN^ for n = 1 , . . . , N as the Bayes ian Newsvendor ( B N ) policy. 5.2.3 The Bayesian Newsvendor Problem with Unobservable Lost Sales Unobservable lost sales complicate the problem greatly. Because the t rans i t ion probabi l i t ies depend on the actions, the demand upda t ing now depends on the entire h is tory pa th Hn = { 5 „ _ i x An-i x Xn-i} for n = 2 , . . . ,N (wi th Hi = Si). So lv ing the op t imal i ty equations ana ly t i ca l ly becomes very difficult. Since the B M D P model i n Section 5.1.3 is an extension of the model i n Sect ion 5.1.2, we are interested i n the fol lowing questions: i) H o w do the o p t i m a l policies from the two models relate? ii) M o r e specifically, is the B N pol icy from Section 5.2.2 s t i l l o p t i m a l for this problem? i i i ) W h a t is the effect of unobservable lost sales on the op t ima l policies? 132 These are the questions we at tempt to answer i n the remainder of this chapter. C o m p a r i n g the two Bayes ian newsvendor problems described i n Sections 5.1.2 and 5.1.3, we note that wi thou t observing lost sales, demand upda t ing becomes dependent on the policies and the pol icy upda t ing is dependent on demand updat ing . A s a result, this newsvendor p rob lem can no longer be reduced to single-per iod problems. T h e economic just i f icat ion for S P E N problems breaks down here. A l t h o u g h the B N pol icy yields a m i n i m u m expected cost in the current per iod , it may result i n unobservable lost sales which provides inadequate informat ion for demand upda t ing in the next per iod . T h i s might have been avoided i f we had stocked a bi t more. Intui t ively, censoring implies loss of informat ion i n demand upda t ing which w i l l result i n a higher expected cost i n future. So there is a trade-off between reducing cost and gathering informat ion (this is referred to i n the cont ro l theory l i terature as dua l control or probing) . Consequently, the B N pol icy is no longer op t ima l . It is noteworthy, however, that the B N pol icy might be useful for compu t ing the op t imal pol icy. T h e procedure of comput ing the op t ima l po l icy for this problem is s imi la r to that i n Sect ion 5.2.2, except that i n step i) the pr ior is updated using bo th the demand observation and the order quanti ty, and in step i i i) we need to consider lost sales i n searching for the equ i l ib r ium at which the expected cost of hav ing ordered too much equals the expected cost of hav ing ordered too l i t t le . W e w i l l show i n Sect ion 5.3 that for a two per iod continuous demand problem, y\ > y[BN^; and y2 = y2BN^ • R e c a l l that y\BN^ is the o p t i m a l pol icy for unknown yet fully observable demand. W e w i l l see i n Sect ion 5.4 that w i t h some modif ica t ion , we can use the logic of marg ina l analysis to interpret this result. T h e proof i n the case n = 2 is very intr icate and it is not apparent how to extend this result to n > 2 at this t ime. We conjecture that for a general f ini te-horizon prob lem (TV > 2), y* > y \ B N ) for t = 1 ,2 , . . . , TV - 1; and y*N = y { B N ) . T h e argument concerning the op t ima l po l icy for a two-per iod p rob lem proceeds as follows: we might want to sacrifice cost m i n i m i z a t i o n at decision epoch 1 to obta in adequate in format ion for demand upda t ing at epoch 2, and we want to continue this up to a point where we w i l l not be better off from further sacrifice i n cost. Therefore, the op t ima l inventory pol icy is no less than the B N po l icy for n = 1. A n d at n = 2 when we do not have to worry about future demand updat ing , the B N po l icy becomes op t ima l . T h a t is, we w i l l stock more than we would in the fully observable case at the first per iod due to the possibi l i ty of incomplete ly observed demand. 5.3 The Key Result Consider a two per iod problem. Since the estimate of the demand d i s t r ibu t ion is updated after the demand i n pe r iod 1 is observed, the demand process can be considered as {Xi,X2} w i t h Xi hav ing pd f m\(x), X2 133 having pd f m'2{x) where m'2(x) is the marg ina l density corresponding to the posterior 7r 2(0|a;i) given by (5.6). To avoid technicali t ies, we consider non-negative continuous demand only. Some nota t ion follows: m2{x\x\) : the marg ina l of X2 corresponding to state ir2(9\xi) wh ich is the posterior when the demand i n per iod 1 is observed to be x\ (i.e. Xi = xi). m2{x\Vi) '• the marg ina l of X2 corresponding to state it2{9\y-C) which is the posterior when the demand i n per iod 1 is censored at the order quant i ty yi (i.e. Xx > j / i ) . M 2 ( a ; | i i ) = P r ( X 2 < x\Xi = n ) , M%{x\yi) = P r ( X 2 < x\Xx > Vl). y2 : o p t i m a l po l i cy for per iod 2 w i t h state ir2(9\xi). y2 : o p t i m a l po l icy for per iod 2 w i t h state 7r 2(f?|yi). Note , as ment ioned earlier, m 2 (a; |a;i) and m2{x\yi) depend on x\, y\ th rough TT2(8\XI) and Tt2(9\yi) respec-t ive ly (please refer to (5.8)). We w i l l need the fol lowing technical l emma. L e m m a 1. Let g(x,y) be a real valued function on R x R which satisfies: i) 9{a, b) = 0; ii) the partial derivatives gx and gy exist and are continuous near (a,b); iii) the partial derivative gy is not zero at (a,b). Lety = 4>(x) be a one-to-one function implicitly defined by g{x,y) = 0. Thus 4>{x) has a continuous derivative 4>'(x) given by 4>(x)= — 9y where the partial derivatives are evaluated at point (x,4>(x)). P r o o f . See B a r t l e [8] (p. 261-262). W e give definitions of stochastic order ing and l ike l ihood ra t io order ing (Ross [50]) before we introduce two proposi t ions . [ S t o c h a s t i c O r d e r i n g ] We say that the random variable X is stochastic larger than the random variable Y, written as X >st Y or fx >st / y if Pi(X > a) > P r ( y > a) for all a. [ L i k e l i h o o d R a t i o O r d e r i n g ] Let X and Y denote continuous non-negative random variables having respectively densities f and g. We say that X is larger than Y in the likelihood ratio sense, and write 134 X >LR Y or f > L R g if < 4T for all s < t. g(s) ~ g(t) That is X >LR Y if the ratio of their respective densities, f(s)/g(s), is nondecreasing in s. T h e next propos i t ion gives the stochastic ordering proper ty of demand dis t r ibut ions that are updated using a censored observation and an exact observation respectively. T h a t is, the censored demand observa-t ion results i n an estimate of demand d i s t r ibu t ion which is s tochast ical ly greater than that from an exact observat ion at the same level. P r o p o s i t i o n 1. Suppose X\ and X2 are non-negative iid random variables with density f(x\9), and suppose f{x\9) is likelihood ratio increasing in 6, that is, f(-\92) > L R f{-\9\) when 92 > 9\. Let iti(9) be given and suppose ff'2(9\xi) is given by (5.6). Then for any x\, X2\[X\ > x\\ is stochastically greater than X2\[X\ = xi], that is, M2(x\xi) < M2(x\xx). P r o o f . W e complete the proof i n three steps. F i r s t , we show that the posterior •K2{9\xi) is stochast ical ly increasing i n x\. Second, we show -K2(9\X\) >st n2(9\xi), that is, ir2(9\xi) is s tochast ical ly greater than 7T2(f9|a;i). T h i r d , we show that this implies the stochastic order ing of the marginals . [Step 1] A t any x\ and x[ such that, xi < x\, we have n2(9\xi) = — T — , n2{9\xi) = f-~T\—• mi(n) m i l x ' J W e w i l l show that Jt°° ^^x1^ > / ~ -K2{9\xx)d9 for any t. Since d9 mi(xi) and the c o m m o n denominator of the right hand side of the above equat ion is obvious ly posit ive, we only need to show that the numerator is greater than or equal to 0. Hence, consider /oo rOO f{x'1\9)w1(9)d9 • m i (n) - J f{xx\6)irx{0)dB • m ^ ) = rOO r OO /'CO rOQ / f(x'1\9)7r1(9)d9 f(x1\6')*1{9')d0'- f(Xl\9)^{9)d9 f{x,1\9'W{ff)dB'. Jt Jo Jt Jo T h e r ight hand side of the above equation can be reduced to, /•OO rt rOO ft / f(x[\9)7r1(9)d9 f i x ^ W W - f(xl\9)w1(9)d9 / (a ; ; |0>i(0 ' ) Jt Jo Jt Jo rOO rt = / / L/Vi\0)f(xi\#) - f(*il#)/K 19')}^(0)m(9')d9'd9. Je=t Je'=o 135 Since f(-\8) is a l ike l ihood ra t io increasing i n 8 , Hx'M^e') > f(xi\e)f(x[\e') for any x\ < x[ and 8' < 8. Therefore, TT2(8\XI) increases s tochast ical ly i n x\. [Step 2] Cxf{x\8)dx^{8) T h a t is, KW\XI) = mi(x)ir2(8\x)dx S™rni(x)dx (5.10) Note the right hand side of (5.10) can be considered as a weighted average of TT2(8\X) for x > x\. We showed i n step 1, it2{8\x) is s tochast ical ly increasing i n x. Therefore, (^^ Ixi) >st T^2{8\X\) for a rb i t ra ry x\. [Step 3] Consider the marginals where 8*' has density n2(8\xi). T h e condi t ion that f(x\8) is increasing i n l ike l ihood ra t io as 8 increases impl ies that f{x\8) is also s tochast ical ly increasing i n 8 since the l ike l ihood ra t io order ing is stronger t han stochastic ordering [50]. Therefore, 1 — F(x\8) is non-decreasing in 9. B y a s tandard theorem of stochastic order ing (see Propos i t ion 8.1.2 i n Ross [50], p. 252), we have (5.11) B y changing the order of integrat ion, (5.11) becomes, where 8* has density 7r2(#|a;i). Simi la r ly , Ee.[l - F(x\8)} < E ^ [ 1 - F ( x | 0 ' ) ] (5.12) 136 i f 6*' is s tochast ical ly greater than 6*. It follows from (5.12) that M2{x\xi) > M2{x\xx). P r o p o s i t i o n 2 further establishes a result about the o p t i m a l po l icy based on the stochastic ordering proper ty i n P r o p o s i t i o n 1. It says that censoring i n the first pe r iod results i n a greater op t ima l order quant i ty for the second per iod . P r o p o s i t i o n 2. Let y2 be the optimal order quantity for period 2 if in period 1, demand is observed to be x\, i.e. X\ = x\. Similarly, let y2 be the optimal order quantity if demand in period 1 is censored at x\, i.e. Xi >x\. Under the hypothesis in Proposition 1, y2 <y2. Proof . Since ^3(71-3) = R s i ^ , 2/3) = 0 by the assumption of the boundary condi t ion , u2W2) = mm{RB{-K2,y2) + 0 } . (5.13) 2/2 A s we have pointed out earlier, i?s(7r 2 , j /2) is convex i n y2. T h u s U2(7r2) is m i n i m i z e d at v\ = y{2BN] = W M x i ) ] - 1 where M2(x\xi) corresponding to 7r2(-|a;i) = 7T2(-|a;i), when X\ = x\ M2(x\xi) = 1 M2(x\x\) corresponding to 7r 2(-|a:i) = 7r 2(-|a;i), i.e. when Xi > Xi T h a t is, , = h i = [M2{k\x1)\-1 \ i X , = X l , \vc2 = MWx!))]-1 H X 1 > x 1 . Since bo th M2(x\xi) and M2(x\xi) are monotonica l ly increasing i n x, us ing the result of P ropos i t i on 1, i t is obvious that yc2 = [M^k^)]-1 > y\ = [M2{k\x{)]-\ W e now state the key result. T h e o r e m 1. Suppose the hypotheses in Proposition 1 hold, U2(7r2(-|a;i)) is differentiable in X\, and it is differentiable under an integral sign, that is, u2(iT2(-\x))mi(x)dx is differ entiable in y. For the Bayesian newsvendor problem with unobservable lost sales, >y[BN); and y*2=y[BN). Proo f . We w i l l s tart from per iod 2 and move backward to per iod 1 i n compu t ing the to ta l expected cost. 137 We have (5.13) from the boundary condi t ion . R e c a l l 7r 2 ( - | z i ) is calcula ted differently depending on whether Xi is fully observed or censored (refer to (5.6)). Hence, / / / , xx _ f w 2 ( 7 r 2 ( - | a ; i ) ) i f 7 r 2 ( - | z i ) = 7 r 2 ( 4 a ; i ) , i .e . when X i = z i < 3/1 U 2 2 1 ~ \w2(7r2(-|2/i)) i f 7 r 2 ( - | a ; i ) = 7 T ^ ( - | y i ) , i .e. when X i > a;i = j / i . Since 7r 2(-|a;i) can be considered as a function of -K\ and xi (refer to (5.2)), and 7 r 2 ( - | j / i ) as a function of 7Ti and r/i (refer to (5.7)), we redefine u 2 ( 7 r 2 ( - | a ; i ) ) as f i r , x x _ ( " 2 ( 1 - 1 , z i ) i f xi < 2/1 , [ W 2 ( T T I , 2 / I ) " xi = 2/1 • O m i t t i n g TTI as i t is fixed, we further simplify the above to , i,\ xx _ JMxi) i f xi < J/1 , W 2 ( T T 2 ( - M ) = < [uc2{yi) if i i = 2/1 • W e omi t 7 T i throughout the remainder of the proof. Obvious ly , the op t ima l i t y equat ion at n = 1 is U I ( T T I ) = mm{RB(yi) + / u2{xi)mi(xi)dxi + ^(2 / i ) [ l - M i(2 / i ) ] } i n wh ich the second te rm on the right hand side is the expected cost at n = 2 i f X i is fully observed (i.e. X\ < 2/1) and the t h i r d t e rm is the expected cost i f Xi is censored at 2/1 (i.e. Xi >y\). Let •/(yi) = RB(VI)+ u2(x)m1(x)dx+uc2(y1)[l-Afi(j/i)] ./o = i ? B ( 2 / i ) + / ( y i ) (5-15) then, the o p t i m a l po l i cy is obta ined from y* e arg m i n J (? / i ) . 2/1 To show 2/1* > 2 / i B W ) = M f ^ f c ) , i t suffices to show that m^\yi=yiBN) < 0 where dJ(yx) dRB{yx) , d/(t/i) 1 K x dj/i d y i dj/i = (c-p) + (p- h)Mi (2/1) + - Mid /O] - K(2 / i) - «2(2/i))mi(2/i). aj/i Since d R d y ^ \ y i = y ( B N ) = 0, we need only show that at 2/1 = y[BN\ dljy^ = di^yi) _ _ _ U 2 { y i ) ) m i i y i ) < o. ( 5 . 1 7 ) aj/i "1/1 138 A s a mat ter of fact, we w i l l show that this is true at any value of y\. We first rewri te U2(yi) and u2(yi) below. "2(2/1) = #5(712,2/2) = E ^ [ r ( X , y | ) ] = p xm2(x\y\)dx + h I xm2(x\y\)dx Jy* JO /•oo ryl = p[ xm2(x\yi)dx + / a;m2(a;|2/i)da;] •A/I Jvl + h[ xm2(x\yi)dx — / a;m2(a;|2/i)<ia;] (5.18) Jo Jy% Simi la r ly , u c M = RB{-Kc2,yc2)=E^[r(X,yl)] = p i xm2{x\y\)dx + h I xm2(x\y\)dx. (5.19) Jy% Jo rv2 >V2 Jo Subt rac t ing (5.18) from (5.19) gives roo ryl "2(2/1) - u2(2/1) = p x[m2(x\yi) - m2(x\y1)]dx + h / x[mc2{x\yx) - m2{x\yl))dx Jyl Jo — (p — h) xm2(x\yi)dx •A/! ' ' rvl = Dl-{p-h) I xm2{x\y1)dx, (5.20) Jy% where /•oo ryl Di=p x[ml(x\yi) - m2(x\yi)]dx + h x[m2(x\yi) - m2(x\yi)]dx Jvl Jo w i t h 7e mi (2/1) v ' > m c ( x l } = m ^ - ^ f i x ^ F j y m ^ d B 2 i J I • (5-22) We now calculate F r o m (5.19), we have by Le ibn iz ' s rule: du2(yi) f°° dm2(x\yi) , r a n \dy2 j 1 = P x 2i~jmdx-pyc2rnc2(yc2y1)-p-dyi Jyc dyi dy! = D2-(p- h)yc2mc2(yc2\yi)-/± , (5.23) dyi 139 where _ f°° dmc2(x\y1)J [V dm^\m) D2 = p x £T—•—-ax + h x -, ax Jv% dy± J0 i y$ JO dyi w i t h dmc2(x\yi) = mi(yQK(i) - / e f(x\8)F(yi \6)m{6)d6} dyi [ l - M i ( y i ) ] 2 Je/(s|fl)/(yi|flMfl)<M N o w , replace (5.20) and (5.23) i n (5.17), and rewrite l - M i ( y i ) = {[l-M1(y1)]D2-m1(y1)D1} + {m1(y1)(p-h) xm2(x\yi)d dyi Jyi F r o m (5.24) we have, Note , [ 1 - A f i f o i ) ] I>2 - m i ( j / i ) I > i = p | ~ ([1 - M i ( y i ) ] ^ | ^ - m i ( y ! ) K ( x | y i ) - m2(x\yi)]j xdx + h fJ' ( [1-MM]—"2™*' -mi(yi)[mZ{x\yi)-m2{x\yi)] ) xdx Since the integrands are zero (shown by (5.28)), we have [ l - M i d / O p a - m i ( i / i JUx = 0. (5.24) [1 - M1(y1)](p - h)yc2mc2(yc2\yi)^}. (5.25) ayi [ l - A Z i ( y i ) ^ = — ——r- - — — — — m i (y i ) dyi 1 - M i (2/1) 1 - M i (2/1) - / /(*|0)/(!/ i |0)7ri(f?)d0. (5.26) A n d from (5.22), l - M i ( i / i ) l - M i ( y i ) U s i n g (5.26) and (5.27), we get [ l - M i ( y i ) ] 2 ^ | y i ; - m i ( y i ) K ( a ; | y i ) - m 2 ( x | y i ) ] = - Jf(x\e)f(y1\e)ir1(6)d0 + m1(yi)Tn2(x\yi)=O. (5.28) 140 Therefore (5.25) reduces to dljyi) dyi J y -rV2 = m i ( y i ) ( p - h ) I xm2{x\yi)dx - [1 - Mi(yi))(p - h)yc2mc2(yc2\yi)^- (5-29) Note M2(x2\xi) is a function of bo th x2 and x\. It is a function of x\ t h rough 7r2. Since M | ( y 2 | y i ) = ^ i m p l i c i t l y defines the function y 2 ( y i ) , we have the fol lowing by L e m m a 1 dy2 _ dyi dyi rnc2{yc2\yi)' T h i s gives [1 - Mi{yiW2ml{yl\yi)d£- = - [ 1 - M 1 ( y 1 M d M ^ l y i ) . (5.30) Since Mi(yc2\yi) = mc2{x\yi)dx = l - M i ( y i ) ' (5.30) becomes -[1 - M ^ y O J y f ^ y ^ 0 = ySm^yO^d/Hy!) - M$(y°2\yi)]. (5.31) c y i B y L e m m a 2, we have y 2 < J/2 s o that mi(yi) / a;m2(a;|2/i)da; < m i ( y i ) y 2 / m 2 (x|yi)da; = m1(y1)yc2[M2(yc2\yi) - M2(ye2\yi)}- (5.32) F r o m (5.14), we know M^{y°2\yi) = M 2 ( y | | y i ) = k. Rep lac ing (5.31) and (5.32) i n (5.29), we have the fol lowing inequal i ty < (P- % 2 K ( y 1 ) ( M 2 ( y 2 c | y 1 ) - k) - m 1 ( y 1 ) ( M 2 ( y 2 c | y 1 ) - k)] = 0 which gives dJ}y^ \yt-iBN'>-M~l(k) — ®' T h a t is, ui(7Ti) is non-increasing at y[BN^. 5.4 Discussion of the Effect of Lost Sales on the Optimal Policy W e have shown that i n a 2-period newsvendor problem, when lost sales are unobservable, the op t ima l order quant i ty at n = 1 is greater than the B N order quanti ty, while i n the second pe r iod the B N pol icy is op t imal . In this section, we provide economic just i f icat ion for this result. W e w i l l first solve for the op t ima l pol icy by cost m i n i m i z a t i o n . T h e n , we w i l l apply marg ina l analysis to interpret the effect on the op t ima l po l icy of compensat ing for lost sales. 141 5.4.1 Optimal Policy for BMDP Model Because of the t e rmina l condi t ion , the B N pol icy is op t ima l for the last per iod . Since the last per iod is not of interest here, we only consider the first per iod of a two per iod newsvendor p rob lem i n this section. Theorem 2 suggests one way to compute the op t ima l order quant i ty for per iod 1 (yl). T h e o r e m 2. Under the hypotheses in Theorem 1, solving the following equation gives the optimal order quantity y* for a two period newsvendor problem: (c-h)M1{y1) = (5.33) P r o o f . Because of the convexity of the cost function J(yi) (defined i n (5.15)), we can obta in the op t ima l order quant i ty by solving for y\ i n dJJyv^ = 0 (refer to (5.16)). dJ}y^ — 0 gives equat ion (5.33) after rearrangement. 5.4.2 Marginal Analysis A l t h o u g h the B M D P model i n Section 5.3 cannot be reduced to a sequence of single per iod problems as we discussed earlier, we may s t i l l follow the concept of marg ina l analysis i n search of the o p t i m a l policy. T h a t is, we w i l l keep order ing un t i l we reach the point at wh ich the expected cost equals the expected savings. For the newsvendor p rob lem w i t h known demand, this argument is summar ized by (5.9) i n which (c — h)M(y) is the expected cost at y; (p—c)[l — M(y)] is the expected savings. No te i n (5.9), y* is non-decreasing in p. T h i s is in tu i t ive ly correct because we would order more when the penal ty increases. N o w we consider the newsvendor p rob lem w i t h unknown demand. For the newsvendor p rob lem w i t h observable lost sales, the marg ina l analysis argument is expressed by the fol lowing, (c - h)Mn{yn) = [1 - M „ ( i / „ ) ] ( p - c) for n = 1 , . . . ,N. Mn is the Bayes ian est imator of the demand density. T h i s also shows that the B N pol icy is o p t i m a l here. For the two per iod newsvendor w i t h unobservable lost sales, let // \ du^yx) m i ( j / i ) P M = P ~ ~dy~T~ + { U M ~ M y i ) ) l - M l i y i ) ' then (5.33) becomes (c - /I)MI(J/I) = [1 - M1(y1)]]p'(y1) - c]. (5.34) 142 decision sales demand prob . value function D o not buy x\ = 2/i * i >2/i 1 - M i ( y i ) B u y 2/i < xx < 2/1 + A y X i = x ^ X i > 2/i l-Mid/j) u 2 + A u ( i i ) B u y x\ = 2/i + A y Xi > 2/i + A j / I X i > 2/1 l -M 1(j/ 1)-/»i+ A "m 1( a:)d a: l - M i ( v i ) + Au° Table 5.1: O p t i o n of Purchas ing A d d i t i o n a l Informat ion Observe that we can view p'(yi) as a new policy-dependent variable penal ty cost. So this equat ion is i n accordance w i t h the marg ina l analysis argument. We showed, in Sect ion 5.3, du%(yi). dyi -[1 - MM] - (ucM - u2(2/i) W2/i) < 0. T h i s gives p'{y\) > p at any y\. Note that the solut ion of (5.34) increases i n p'(-). Therefore, the op t ima l po l i cy is greater than or equal to the Bayesian newsvendor po l icy y[BN^ since p'(-) is no less than p. 5.4.3 Interpretation of the New Penalty Let • / \ / c / \ /xx "ll(2/l) 2^(2/1) = K(2/i) - ! . M l ( g l ) -then, p'{yi) = p + 1(2/1)- Thus , the new penalty p' can be considered as consis t ing of the shortage penalty p and a posi t ive t e rm which varies w i t h po l icy 2/1- To unders tand how 1(2/1) relates to lost sales, we consider the fol lowing scenario. If xi = 2/1, i.e. the demand is censored, we would then apply the o p t i m a l po l i cy 2/2 for per iod 2 which gives the value function u2- N o w , suppose we have the opt ion of purchas ing some addi t iona l informat ion regarding whether the demand x\ lies below or above 2/1 + Ay where Ay is smal l and posi t ive. B u y i n g this piece of informat ion , we have an exact observation of the to ta l demand i n per iod 1 (i.e., X\ = xi) i f xi < 2/1 + A y , and a censored observation at a s l ight ly higher level (i.e., Xi > xi) i f x\ = y i + A y . Consequently, the value functions are u2 + Au(x) and u2 + Auc respectively. It is reasonable to define the value of the piece of informat ion as the difference between the expected cost wi thou t buy ing the informat ion and the expected cost from buy ing i t . Table 5.1 summarizes the scenario. 143 F r o m Table 5.1, we can calculate the expected cost i f we buy the add i t iona l in format ion . It is, V L + A \ , , M mi{x) , / c A C N 1 - Mi(yi) — AM where AM = J ^ + A y m\(x)dx. T h e expected value of the informat ion is, therefore, A y ) = ul - + A u ^ ^ ^ d x + K + A , * ) 1 " j ^ ^ " }• L e t t i n g A y go to 0, we obta in the marg ina l expected value of informat ion at j / i , h m —— = h m -r2 Ay-+o Ay A1/-+0 A y r A M Auc AM A ™ o | U 2 [ l - M i ( y i ) ] A y A y ( 1 - M 1 ( y 1 ) ) - ( u a + A u W , A M ... }. (5.35) ; [ l - M 1 ( y 1 ) ] A y ; Since, A u c A M Ay^o Ay 1- M i ( y i ) A u A M Ay->o A y 1 - M i ( y i ) Hence, (5.35) becomes ci{yx,Ay) m i (y i ) duc2(yi) } I M N A = ("2(2/1) - "2(2/1 h , . = i(yi). (5.36) Ay->o A y 1 - M i (y i ) d y i Therefore, we can say that the new penalty p1 consists of the shortage penal ty p and the marginal expected value of information i(yi). We can also th ink of i(yi) as the penalty from inadequate information at y i since the cost of in format ion is incurred due to the poss ibi l i ty of lost sales. Therefore, i n the newsvendor problem w i t h unobservable lost sales, jus t i f ica t ion for the higher-than-B N o p t i m a l po l i cy proceeds as follows. U n l i k e i n S P E N problems, the penal ty now varies w i t h the policy. Po ten t i a l unobservable lost sales results i n a higher penal ty which comes from the informat ion loss as well as the shortage. A s a consequence of the increased penalty, the expected cost equals the expected savings at a higher order quanti ty. 5.5 Illustrative Examples W e have been work ing w i t h continuous demand so far. For discrete demand, we have the op t ima l pol icy fol lowing the marg ina l analysis (5.34), j /J = m i n l y i : (c - / l ) M i ( y i ) > [1 - M i ( y i ) ] [ p ' ( y i ) - c]} yi€Z 144 where Z is the set of non-negative integers. A n d for the final per iod , y\ = yiBN) = m i n 0 , 2 : y2 > M'-\k)}. Since the cdf of discrete demand is a step-function, it is s t raightforward to conclude that the op t ima l pol icy before the last pe r iod is at least as large as the B N pol icy while they are equal for the last per iod . Hence, the key result proved for continuous demand in Section 5.3 remains true for discrete demand. We w i l l i l lustrate this w i t h some numer ica l examples. For the fol lowing examples, we assume Poisson demand. T h e demand upda t ing and cost computa t ion formulas are derived i n Sect ion 5.5.1. Three examples of the two per iod newsvendor problem are given in Sect ion 5.5.2. N u m e r i c a l examples of 3-period problems are given i n Sect ion 5.6 to i l lustrate the extension beyond 2-periods. 5.5.1 Demand Updating and Cost Computation W e assume i n a l l three examples that the demand has a Poisson d i s t r ibu t ion w i t h unknown parameter A. W e use the conjugate pr ior for s impl ic i ty i n demand upda t ing . R e c a l l tha t the G a m m a dis t r ibut ion is conjugate for the Poisson parameter A. Hence, the r andom demand Xi\X ~ Poisson(A) (i = 1,2.) w i t h A ~ G a m m a ( a , P). T h a t is, f(x\X) = ; (5.37) X a - 1 -X/0 MX) = - T £ 5 ] F - . (5.38) T h e hyper-parameters of the G a m m a prior , a and f3, are predetermined. T h i s assumpt ion yields the pdf of the marg ina l demand, JA e~xXx A " " 1 e - x l 0 A x\ T{a)p« mliX) = / — ; T"> / - \ /3^ v d X _ T(a + x) 1 P ~ T(a)x\ [p + l} {P+1} ' ( j It is equivalent to say X ~ N B ( a , ^ - ) since m\(x) is i n the form of a negative b i n o m i a l pmf. However, the size parameter of a negative b inomia l d i s t r ibu t ion is usual ly a posi t ive integer. T h e expectat ion and the variance of the marg ina l demand are E(X) = ap; Var(AT) = aP2 + ap. T h e demand d i s t r ibu t ion is only updated once i n the two per iod problem. L e t xi be the observed sales i n the first pe r iod . U p d a t i n g is separated into two cases: i) when x\ is an exact observation, and ii) when x\ 145 is censored at the order quant i ty for per iod 1 denoted y\. In the fol lowing, we derive the updated posterior and marg ina l d i s t r ibu t ion for the two cases. C A S E I - w i t h a n e x a c t o b s e r v a t i o n (Xi = x\ < yi) T h e posterior is / ( ^ | A ) Trx(A) Tr 2(A|a;i) ^/(nlA ' j T r ^ A O d A ' ^a+zi-l e-X(0+l)/0 T h a t is, \\xi ~ G a m m a ( a + x\, ^ r ) which also shows that 7r(A) is a conjugate pr ior . T h e upda ted marg ina l d i s t r ibu t ion of demand for per iod 2 is (5.40) m2{x\xi) = / /(a;|A)7r2(A|a;i)dA T(a + xx + x) p + 1 + P T(a + xi)x\ y2/3 + l> y2/3 + l' ' T h a t is, X\xi ~ NB(a + n, R e c a l l that the Bayes ian expected cost of any order quant i ty yn is given by (5.4) i n Sect ion 5.1.2. It is simplif ied by calculus. #s(>n,2/n) = (c-p)yn + (p-h)ynMn(yn) +pEn{X) - (p-h)^2xmn(x) (5.41) x=0 for n — 1,2. In the above equation, En(X) denotes the expectat ion of demand w i t h respect to mn(x), the updated pd f of the marg ina l demand d is t r ibu t ion at decision epoch n. Specif ical ly i n this case, Ei(X) = a(3 and E 2 ( X ) = (a + ziX^rr )• Mn{-) is the cdf of the marg ina l density at epoch n. Note that the cost function of the Bayes ian Newsvendor ( B N ) po l i cy y^N^ for continuous demand can be further reduced to vLBN) RB{*n,y{nBN)) = pEn(X) -(p-h)J2 » « W - ' (5-42) since some items cancel out w i t h Mn(yi^N^) = k = • However, i t is no longer va l id for discrete demand as y^W = mm {yn:yn>Mn-\k)} 146 C A S E I I - w i t h a c e n s o r e d o b s e r v a t i o n (X\ > X\ — y\) Censor ing complicates the demand d i s t r ibu t ion upda t ing formulas. T h e posterior is *2iMvi) - ———n (5.43) / A k i ( A ' ) - E ? i o l £ ^ r : i ^ ( A , ) ] d A ' To s implify the formula , we use g(X; a,/3) to denote the pd f of the G a m m a density given in (5.38). So the above equat ion becomes nc2(\\Vi) = (5-44) g(A; a , i9)-E?io 1 i^ff(gfr) < (gj t r) a a(X; a + i ,gf i ) i \pyi-i r(a+t) / 0 w i \„ , 1 Z^i=0 r(a ) i ! ^ Accord ing ly , the marg ina l density for per iod 2 is mc2(x\yi) = [ f(x\X)7Tc2(X\yi)dX (5.45) J A •: m(T- ry 1 N r(q-f-t)/ N j / 1 w ( , • \ _ " H x , "> /3+l/ Z^j=0 r(a)t! V/8+1/ ^/3+l' "H-*"i " ^ l> 2/3+1 > ~ 1 _ V 5 ' 1 - 1 r(a+i)/ /3 w 1 \ r a 1 Z^i=0 r(a)t! V /3+1 ^ V/J+l/ where m(:r; a, b) denotes the p m f of the negative b inomia l density (see (5.39) for example) . T h i s gives the corresponding expected demand, nft _ r ( ° + ' ) f 0 1 )a(n. + j\ E 2 ( X ) - y i - i r ( a + i ) r g y , t w • t 5 ' 4 6 ) 1 Zvi=0 r(a)i! v/3+i/ \f3+l> T h e Bayes ian expected cost i n either of the two periods has the same form as (5.41), but w i t h m n ( - ) and Mn{-) replaced by m°n(-) and M £ ( - ) . 5.5.2 Three Examples To solve the op t ima l i t y equat ion for y{, we need a search a lgor i thm. Nevertheless, discrete demand makes po l i cy enumerat ion prac t ica l . Therefore, we compute the Bayes ian Newsvendor po l i cy which is op t ima l for per iod 2 and enumerate the order quantities for per iod 1. We determine the o p t i m a l order quant i ty y\ by compar ing the to ta l expected cost. We vary the cost structure or the pr ior i n the three examples. E x a m p l e 1. T h e cost s tructure and the demand assumptions are as follows. 147 P e r i o d 1 P e r i o d 1 P r o b . O p t i m a l Order Cos t i n E x p e c t e d T o t a l Order Quan t i t y Sales Q u a n t i t y " i n P e r i o d 2 Cost i n Expec t ed and Cos t P e r i o d 2 under t/2 P e r i o d 2 Cost Xi P ( * i ) 2/2 E ( « 2 ) T E C Mr 1 (A:) = 1* xi=0 0.3832 2/1 = 0 0.5455 R B = 5.97906 X\ = 1 0.6168 2/2° = 3 8.8980 5.6973 11.6763* 2 i i = 0 0.3832 2/2e = 0 0.5455 R B = 6.1322 X\ = 1 0.1394 2/1 = 0 1.9091 zi = 2 0.4774 2/2c = 4 10.6267 5.5484 11.6806 3 i i = 0 0.3832 2/2e = 0 0.5455 R B = 6.3963 = 1 0.1394 2/1=0 1.9091 zi = 2 0.0887 1/1 = 1 3.0375 zi = 3 0.3887 2/2c = 5 12.0913 5.4445 11.8408 "The optimal order quantity in period 2 is conditional on the sales observation and the order quantity in period 1. bRg is the short form of RB(^I,HI)-Table 5.2: Pol ic ies and their costs i n E x a m p l e 1. Observe that the B N po l icy is op t ima l i n per iod 1. Costs: c = 1, h = 0.25, p = 1.5; hence, k = ^ = 0.4. D e m a n d : X\X ~ Poisson(A); TTI (A ) = Gamma(0.4,10). T h i s yields E ( X ) = 4 w i t h respect to 7Ti (A) . If h is tor ical da ta is available, the G a m m a parameters can be specified such that the expectat ion of demand is close to the mean of the demand data . W e set them i n these examples i n such a manner that the pr ior is flat. T h e pr ior here has a large variance, V a r ( A ) = 40, while E ( A ) = 4. F r o m Table 5.2, we see that the B N po l icy is o p t i m a l at n = 1. T h e m i n i m u m to ta l expected cost, 15.353, is achieved by y[BN^ = 1. E x a m p l e 2. In this example , we give a different cost structure while leaving the demand assumptions unchanged. Costs: c = 1, h = 0.5, p = 2; hence, k = = 0.67. D e m a n d : X | A ~ P o i s s o n ( A ) ; 7i"i(A) = Gamma(0.4,10). 148 T h e policies and their costs are given i n Table 5.3. T h e o p t i m a l order quant i ty y{ = 5 is greater than the B N policy, y[BN^ = 3. However, the relative cost savings of us ing the o p t i m a l order quant i ty instead of the B N po l icy is on ly 2%, (13.3709 - 13.2126)/13.3709. E x a m p l e 3 . A s s u m i n g the same cost structure as i n E x a m p l e 2 and change the demand assumptions. C o s t s : c = 1, h = 0.5, p = 2; and k = >~ = 0.67. D e m a n d : X\X ~ P o i s s o n ( A ) ; TTI (A ) = G a m m a ( 1 . 2 , 8 ) . Thus , E(X) = 9.6 w i t h respect to T T I ( A ) . In Table 5.4, we only give the enumerated policies and their corresponding costs for the two periods. We omit 2/2 to avoid the long list of a l l possible h is tor ica l paths at each 2/1 • A s i n E x a m p l e 2, the op t ima l order quant i ty is larger than the B N policy, i.e. 2/1 = 12 > y[BN^ = 11. T h e relat ive cost savings is almost none, only 0.2%, (27.3206 - 27.2659)/27.2659. These three numer ica l examples support our assertion about the o p t i m a l po l i cy for discrete demand at the beginning of this section. T h a t is, the op t ima l po l icy is higher than the Bayes ian newsvendor policy. 5.6 Some Initial Work on N-Period Newsvendor Problem T h e key result for the 2-period newsvendor p rob lem cannot be easily extended to the general A^-period prob lem w i t h unobservable lost sales since the proof of Theorem 1 does not d i rec t ly apply. To i l lustrate this loss of t rac tabi l i ty , we state the op t ima l i ty equations again. For the last pe r iod , UN(.IT'N) = m i n {RB^'N^VN)} , (5-47) where RB(^'N,UN) is the Bayes ian expected cost of the order quant i ty yjv w i t h respect to ir'N, the posterior at N. T h e boundary condi t ion is UN+I(TT'N+1) = 0. For n < N, u „ « ) = mmYNER+{RB{ir'n,yn) + J Q V N u„+ i (7r n + i ( - | a ; ) )m^(a ; )dx + u n + i « + i ( - | ! / n ) ) [ l - M'n{yn)]} , (5.48) where mn(-) is the pd f and Mn{-) is the cdf of the marg ina l demand density at decision epoch n. T h e o p t i m a l po l icy can be computed s tar t ing from the last per iod and cont inuing backward to the first per iod . A s i n the 2-period problem, the op t ima l order quant i ty for the last pe r iod is obta ined by m i n i m i z i n g 149 P e r i o d 1 Order Quan t i t y and Cos t P e r i o d 1 Sales P r o b . O p t i m a l Order Quan t i ty in P e r i o d 2 Cost i n P e r i o d 2 under y2 E x p e c t e d Cos t i n P e r i o d 2 T o t a l Expec t ed Cost 2 / i Xi P(Xi) V2 "2 E(u2) T E C 2 xx = 0 0.3832 3 , | = 0 0.7273 RB = 7.3587 x i = 1 0.1394 2/1 = 1 2.1521 = 2 0.4774 yc2 = 8 11.9609 6.2891 13.6478 M-\k) = 3 xi - 0 0.3832 2 / 1 = 0 0.7273 RB = 7.2755 X i = 1 0.1394 2/1 = 1 2.1521 i i = 2 0.0887 2 / 1 = 3 3.3372 x i = 3 0.3887 2/1 = 10 13.4297 6.0954 13.3709 4 xx = 0 0.3832 2/2e = 0 0.7273 = 7.2891 X l = 1 0.1394 2/2e = l 2.1521 x i = 2 0.0887 2/1 = 3 3.3372 x i = 3 0.0645 2 / 1 = 4 4.4627 x\ = 4 0.3242 2/1 = 11 14.7594 5.9483 13.2374 5* x i = 0 0.3832 2/1 = 0 0.7273 R}B = 7.3775 X i = 1 0.1394 2/1 = 1 2.1521 x i = 2 0.0887 2/1 = 3 3.3372 x i = 3 0.0645 2/1 = 4 4.4627 x i = 4 0.0498 2 / 1 = 5 5.5587 x i = 5 0.2744 2/1 = 12 16.0179 5.8351 13.2126* 6 x i = 0 0.3832 2 / 1 = 0 0.7273 RB = 7.5257 X i = 1 0.1394 2/1 = 1 2.1521 xx = 2 0.0887 2/1 = 3 3.3372 x i = 3 0.0645 2 / 1 = 4 4.4627 x i = 4 0.0498 2/1 = 5 5.5587 x i = 5 0.0399 2 / 1 = 6 6.6343 x i = 6 0.2345 2/1 = 13 17.2271 5.7446 13.2703 Table 5.3: Pol ic ies and their costs in E x a m p l e 2. Th i s table shows that the o p t i m a l order quant i ty is 5 while the B N po l icy is 3 at decision epoch 1. 150 P e r i o d 1 Expec ted Cost Expec t ed Cost T o t a l E x p e c t e d Order Quan t i ty i n P e r i o d 1 i n P e r i o d 2 Cos t for at y i 2 Per iods EO2) = £ ( u 2 p ) T E C 9 14.8431 12.7855 27.6286 10 14.7690 12.6704 27.4394 M f ^ j f c ) = 11 14.7526 12.5680 27.3206 12* 14.7885 12.4774 27.2659* 13 14.8717 12.3977 27.2694 Table 5.4: Pol ic ies and their costs i n E x a m p l e 3 showing that the o p t i m a l order quant i ty is 12 while the B N po l icy is 11 at decision epoch 1. its one-period expected cost. T h i s is shown by the op t ima l i ty equat ion (5.47). Since the Bayes ian Newsvendor pol icy UN*^ for the last per iod is op t ima l , the op t imal i ty equat ion for pe r iod N — 1 is equivalent to that for the first pe r iod i n the 2-period problem; the key result given in Theo rem 1 s t i l l holds. T h a t is 2/AT_I > V/v-i-For n < N — 1, our conjecture concerning the o p t i m a l order quant i ty is y*n > y^N^ . However, since we do not have the ana ly t i ca l solut ion to the op t ima l i ty equations for n = 1 , 2 , . . . , N — 1 (N > 2), we cannot prove that the key result remains true wi thout establishing some funct ional properties of un(ir'n). Th i s w i l l not be addressed here. However, i n this section we ini t ia te the effort to extend the key result to the general i V - p e r i o d prob lem by some numer ica l examples of the three per iod problem. A s i n the two per iod examples, we consider discrete demand which enables us to obta in the op t ima l order quanti t ies by po l i cy enumeration. In Sect ion 5.6.1, we define the prob lem and derive demand upda t ing and expected cost formulas. In Section 5.6.2, we give the results of two examples. 5.6.1 Three Period Newsvendor with Poisson Demand A s i n Sect ion 5.5.2, we consider Poisson demand w i t h unknown parameter A. Since it is conjugate, we again assume the G a m m a pr ior on A. Hence, Xn\X ~ Poisson(A) , A ~ G a m m a ( a , / 8 ) , 151 i n wh ich Xn denotes the i . i . d . r andom variable of per iodic demand for n = 1 ,2 ,3 . T h e hyper-parameters a and p are predetermined. R e c a l l this set t ing yields a negative b inomia l marg ina l demand density; namely, Xn ~ N B ( a , - , (5.49) w i t h its pd f m\(x) g iven by (5.39). In the 3-period problem, the demand is updated at decision epoch 2 and 3 after observing the sales in the first and the second per iod , x\ and x2 respectively. Refer to A p p e n d i x I for the t ime line of events. W h e n unmet demand is unobservable and the sales da ta is used, there are four cases of demand upda t ing at epoch 3. 1. when bo th x\ and x2 are exact observations, . 2. when x\ is exact but x2 is censored, 3. when X\ is censored but x2 is exact and 4. when b o t h x\ and x2 are censored. We now derive the demand upda t ing and the cost computa t ion formulas for each case. T h e costs r emain the same as described i n Sect ion 5.1.1 and a l l no ta t ion is the same as previously defined. C A S E I - E x a c t O b s e r v a t i o n s i n B o t h t h e F i r s t T w o P e r i o d s T h i s is the simplest case. W i t h two exact sales observations, the posteriors of A for the second and the t h i rd per iod are A | x i ~ G a m m a ( a + i i , + 1^' (5.50) A | ( x 2 , a ; i ) ~ G a m m a ( a + x\ + x2, ) (5.51) zp ~\~ 1 R e c a l l , the posteriors are the updated priors which form the basis of the demand d i s t r ibu t ion updat ing . Therefore, the marg ina l demand is updated and has densities at decision epoch 2 and 3, X2\Xl ~ N B t a + z j , A±L) , (5.52) X3\(x2>Xl) ~ NB(a + Xl+x2, . (5.53) T h e marg ina l d i s t r ibu t ion at decision epoch 1 is given by (5.39) before any sales is observed. 152 A g a i n , the Bayes ian expected cost of any order quant i ty yn, i nc lud ing the Bayes ian Newsvendor ( B N ) policy, is # B(7r „ , y n ) = (c-p)yn + (p-h)ynMn(yn) + pEn(X) Vn - [p-h)Y xmn{x) (5-54) x=0 for n = 1 ,2 ,3 . T h e expectations i n the above function can be easily computed i n this case since E ( X ) = ^ L f ^ (5.55) o for X ~ N B ( a , & ) . T h e cost function for the other three cases is in the same form as (5.54), but the marg ina l pdf and cdf, as well as, the corresponding expectat ion must be modified accordingly. C A S E II - E x a c t Observat ion in P e r i o d 1 and Censored Observat ion in P e r i o d 2 Since x\ is an exact observation, (5.50) yields the posterior ^ (A ^ x ) and (5.52) yields the marg ina l m 2(a;|a.' 1). W i t h a censored observation x2 = y2 where y2 is the order quant i ty for per iod 2, the posterior 7T3(-) is as follows. a i r > y ^ [ l - E ^ o V ^ W A M 7T3(A\X 2 > x2 = y2,X\ = x\) = — —= ' JA[l-E£o1/(i|A ')] 7r2(A'|a;1)rfA' g(A; a + xu jfc) - g ^ 1 ^ f e ^ ( 5 f e ) < ( f i & ) c ' + ^ , g ( * i a + ^ + ^ j f a ) 1 _ spy-1 r ( a + z i + i ) / 0 \ i ( 0+1 u ^ , x Z ^ i=0 T(a+xi)-i< ^2/3+1/ V 2/3+1/ i n wh ich g(X; a,/?) denotes the pd f of a G a m m a density. B y in tegrat ion the marg ina l density at n = 3 is f e~xXx mi{x\X2>x2 = y2,Xi=xi)= I — 7r3(A|X2 > y2,Xi = xx) d\ JA & _ m{xi a + xl, 2/3+1 ) EfcO r\a+x\yi\ (2/3+1 Y(2fi+l)a+Xl m{x> a + xl + h 30+1 ) 1 Z ^ i=0 r (a+a : i ) - t ! ^ 2/3+1' V 2/3+1 ' where 771(2;; a, 6) denotes the p m f of a negative b inomia l density. T h a t is, T h e cost funct ion remains the same as (5.54). T h e expected demand i n per iod 3 is more complicated, ^ ^ 1 r v - i r ( » + . + i ) ( /3 w / W 1 Z ^ i=0 r ( a + x i ) - i ! 2/3+1 / V 2/3+1/ E i ( X ) and E 2 ( X ) remain the same as Case I. •.. 153 C A S E I I I - C e n s o r e d O b s e r v a t i o n i n P e r i o d 1 a n d E x a c t O b s e r v a t i o n i n P e r i o d 2 T h i s is the case where X i is censored at y\ whi le X 2 is observed to be x2 • T h e demand d i s t r ibu t ion updat ing at epoch 3 can be shown to be equivalent to Case II . Since the demand Xi is i . i . d . , the posterior at per iod 3 is the same as (5.56) except that x\ and x2 are interchanged. S imi l a r l y w i t h the upda ted marg ina l density and its expecta t ion. T h e posterior , the marg ina l , and the expected demand at epoch 2 are given i n (5.44), (5.45), and (5.46), respectively. Refer to Sect ion 5.5.2 for the derivations. C A S E I V - C e n s o r e d O b s e r v a t i o n s i n B o t h t h e F i r s t T w o P e r i o d s W h e n bo th sales are censored, possibly at different levels, the updated demand density at epoch 3 has the most compl ica ted form. Since, , , i M y i ) = P-EK'/HWliW the posterior 7T3(-) is quite tedious. Y A [ i -Er io 1 / (» |A ) ] [ i -EJio 1 / ( i |A) ] 7 r 1 (A) 7r3iA Ao > y2, A i > y\) = , — , (5.56) ' XJ1 - E^o 1 - EJlo 1 / ( i | A ) ] " I ( A ' ) d A ' [1 ~ E£oX fm ~ EJip 1 / ( i l A ) + E^o 1 E^o 1 / (»|A)/(j |A)]7Ti(A) / A [ l - Efio 1 /(»1A') - Efio 1 /01A') + E^o 1 E^o* /(i|A')/0'|A)] MX')dX' Due to the conjugacy of the G a m m a pr ior 7i"i(A), f(i\X)TTI(A) oc G a m m a ( a + i, ^ ^) . Hence, the above formula of 7:3 (•) becomes 7 r 3 ( A | X 2 > y2,Xi > j / i ) = g(A; a,/3) - E t o 1 C( t )G (» ,A) - Z%r0rC(j)G(j,\) + E^o* EJip 1 C(i,j)G(i,j,X) 1 - Efio 1 c(i) - E^o 1 m + Ef io 1 EJio 1 c(i , j ) , where , T(a + i) 1 /? j ° W - r(a)i! ' Tja + i + j) 1 /? - r(a)i!j! l2/3 + l j l 2/3+l j ' G(i, X) = g(X; a + i, -^-) , G(i,j,X) = g(X; a + i + j, ^~^) • 154 S i m i l a r l y for the marg ina l , we have, m3(x\X2 >y2,Xi > yi) = m(x; a, jfc) - E ^ p 1 C(i)N(i,x) - V ^ 1 C(j)N(j,x) + Eto' E ^ O * C{i,j)N(i,j,x) i - E r i o 1 - E%7 CO') + Eto1 E J i o 1 C(i,j) where N(i,x) = m{x\ a + i, ^jf^j) '•> N(i,j,x) = m{x\ a + i + j, ^ ^ *) . T h e expecta t ion is therefore, E (X) = a P ~ E ^ C ( i ) m " E ^ C U ) m + ^ E ^ C { i ' m i ' j ) i - E f i o 1 c(») - E J i o 1 cu) + Eto1 EfJo1 C(i,j) where E( t ) = ( a + t K ^ y ) ; E ( i , j ) = ( a + i + j ) ( ; ^ ' 2 / 3 + 1 ' 5.6.2 Numerical Examples In this section we give two i l lus t ra t ive examples. A s Section 5.5.2, we enumerate the order quantities for the first and the second per iod whi le comput ing the Bayes ian Newsvendor po l i cy for the last per iod. For n = 1, 2, we compute the B N policy. We then enumerate some policies that are greater than the B N policy. In essence, we compare the to ta l expected costs and determine the o p t i m a l order quant i ty that yields the m i n i m u m cost. W e enumerate the order quantit ies s tar t ing from the first pe r iod . However, we compute the cost i n a backward manner. T h a t is, after we obta in the B N pol icy for per iod 3, we move backward to per iod 2 and then to per iod 1. T h e cost computa t ion procedure for the 3-period prob lem follows. i) S tar t w i t h the expected cost i n per iod 3 w i t h respect to the his tory up to y 3 , i.e. { x \ y 3 , x 2 , y 2 , x i , y i } . Since "3(713) = mmy3€R+{RB(ir'3, J/3)} and the boundary condi t ion is zero, we have "3(713) = RB(TT'3, y3BN^)-R e c a l l that y 3 B N ^ minimizes i?B(7r 3 , y3). ii) C o m p u t e the expected cost E(u3) w i t h respect to m'2(x\xi,yi), or i n other words, w i t h respect to sales i n pe r iod 2 given the his tory up to y2. Note , u3 = u 3(7r 3(-|a;2,2/2,7r2)) where ir'3 summarizes a l l demand informat ion up to the sales observation i n per iod 2. Hence, { x 2 , y 2 , x \ , j / i } , / / , M \n(-\x2,TT2) when x2 < y2, 7r3{-\x2,y2,TT2) = < [^3Vm,^2) w h e n x2 = y2. 155 Therefore, E ( w 3 ) = Ex 2 (u3 (7 r 3 ) ) = ^U3(7r 3(-|a;,2/2,7r 2))m 2(a:|xi,2/i) 3/2 = ^2u3{^3(-\x,n2))m'2(x\xi,yi) x=0 +uz{Ttl{-\y2y2))[l-M!1{y2\x1,y1)} (5.57) i n wh ich X2 has pd f m'2(x\xi, 3/1). i i i ) N e x t , move back to find the m i n i m u m to ta l expected cost w i t h respect to his tory pa th {x\y2,xi,y\}, U 2 { T T ' 2 ) = m i n {RB(n'2,y2)+E(u3)} . (5.58) T h i s yields y2 wh ich depends on yi and X\ th rough TT'2. iv) S imi la r ly , ir'2 summarizes a l l demand informat ion up to { x i , 2 / i , ? r i } - Thus , J 7T 2C-|xi,7Ti) .when n < 3/1, [^(•|2/i,7ri) when i i = yx. Since TTI is predetermined and fixed i t is omi t ted i n most expressions. Therefore, E ( u 2 ) = E x 1 ( w 2 ( 7 r 2 ) ) = ^u 2 (7r 2 ( - | a;,3 / i ) )mi (a ; ) Xi yi = ^ « 2 ( 7 r 2 ( - | i ) ) m i ( a ; ) + U 2 ( 7 r 2 c ( - | 2 / i ) ) [ l - M i ( 3 / i ) ] (5.59) x=0 v) F ina l l y , wi(7Ti) = mm.yi£R+{RB(ni, yi) + E ( u 2 ) } is the m i n i m u m to ta l expected cost and the op t imal order quant i ty for the first per iod . T h e first step above requires that every possible real izat ion pa th be considered for an enumerated y\. T h i s results i n an explosion when the number of periods increases. T h a t is, the number of the enumerated paths quick ly grows as the number of periods increases. We w i l l see from the fol lowing examples that a 3-period prob lem requires much more computa t ion than a 2-period problem. E x a m p l e 1. Suppose the cost s tructure and the demand d is t r ibu t ion are the same as i n E x a m p l e 1 of the 2-period problem i n Sect ion 5.5.2. Thus , C o s t s : c = 1, h = 0.25, p = 1.5; hence k = = 0.4. D e m a n d : A" |A ~ Poisson(A); A ~ Gamma(0 .4 ,10) (we denote the pr ior of A by 7Ti(A)). 156 771! (0) 7771 (1) TO! (2) m i (3) 1 - M i (3) 0.3832 0.1394 0.0887 0.0645 0.3243 Table 5.5: Some Probabi l i t i es of N B ( 0 . 4 , 0.9). T h i s yields X ~ N B ( 0 . 4 , 0.09) so that , E 1 ( X " ) = 4 ; V a r i ( A - ) = 4 4 . Table 5.5 lists some marg ina l probabil i t ies of X. Hence, y[BN^ = M 1 _ 1 ( 0 . 4 ) = 1, and the enumerated first pe r iod order quanti t ies are y\ = 1,2,3. Since the order quanti t ies i n bo th the first and the second per iod are enumerated, and, as mentioned earlier, the enumerat ion starts from the first per iod, we explore the sales his tory up to the final per iod for each y\. We w i l l compute the to ta l expected cost i n a backward manner. T h e to ta l expected costs after per iod 2 are put i n tables for each fixed pair of y\ and x\. We start w i t h the B N pol icy y"BN^ for per iod 1. There are two sales possibi l i t ies w i t h y[BN^ = 1, namely x\ = 0 and x\ = 1. W e summarize a l l possible sales realizations i n per iod 2 for each enumerated y2 i n Tables 5.6 and 5.7 for x\ — 0, 1, respectively. T h e expected cost , 2/2) is computed for each y2. W h e n 7/1 = 2, i f x\ = 0, part I is the same as Table 5.6 except that RB{^I, J / I ) = 6.1322. For part II, x\ = 1, an exact observation, is displayed i n Table 5.8. A n d part III w i t h a censored observation, x\ = 2, is shown i n Table 5.9. S imi l a r to the case yi = 2, part I w i t h x\ = 0 for 7/1 = 3 is given by Table 5.6, while par t II w i t h x\ = 1 is shown i n Table 5.8. Table 5.10 gives P a r t III w i t h x\ = 2 (exact) and Table 5.11 reports par t I V w i t h x\ — 3 (censored). C o m b i n i n g Tables 5.6 and 5.7, we obta in the to ta l expected cost ( T E C ) for y[BN^ = 1. S imi la r ly , combin ing Tables 5.6, 5.8 and 5.9 we obta in the T E C for 7/1 = 2. A n d combin ing Tables 5.6, 5.8, 5.10, and 5.11 we have the T E C for 7/1 = 3. These to ta l expected costs are presented i n Table 5.12. C o m p a r i n g the to ta l expected costs of different order quantit ies i n Table 5.12, we see that 7/* = 2 is the o p t i m a l order quant i ty since it gives the m i n i m u m T E C (17.0067). Table 5.12 implies that the T E C is s t i l l decreasing at the B N po l icy y[BN^ = 1 while at yi = 2, it starts to increase. T h i s t rend continues as the order quant i ty 7/1 increases. Hence, we have y* > y^N^ i n this example. T h i s supports the conjecture stated at the beginning of this section which is y*n > y l B J V ^ for n = 1 , 2 , . . . , TV — 1 and y*N = y^BN\ However, the relative to ta l cost saving from using the o p t i m a l order quant i ty instead of the B N po l icy is less than 0.5% ^17.0914-^.0067 —0,00496). B y the conjecture, when n = 2, we expect to see y2 > y2BN^ at some (7/1, x±). 157 y \ B N ) = 1, RB = 5 .9790° ; xx = 0 (exact) Order P e r i o d 2 O p t i m a l Order M i n . Cost E x p e c t e d Cos t i n P e r i o d 2 Q u a n t i t y i n Sales Q u a n t i t y 6 i n in P e r i o d 3 Cos t i n under y2 P e r i o d 2 P e r i o d 3 under j/g P e r i o d 3 U2 = R%+ E ( u 3 ) 1/2 ^2 2/3* u3 = R% E ( u 3 )c u2 = miny2 U2 {BN) n 2/2 = 0 0 0 0.5455 0.5455 u2 = 1.0910 i ? | = 0 .5455 d 1 0 0 0.2857 R% = 1.0106 1 0 1.4254 1.2989 2.3095 2 0 0 0.2857 i ? ! = 1.6595 1 0 1.0000 2 1 2.1032 1.7033 3.3628 3 0 0 0.2857 R% = 2.3697 1 0 1.0000 2 1 1.7052 3 1 2.7165 1.9480 4.3177 A R 1 B is the short form of RB (7TI,2 / I ) -fcThe optimal order quantity in period 3 is the BN policy given by y^BN^ = Mg _ 1 (0 .4) . Note it is conditional on the sales observation and the order quantity in period 2 through TT'3. c E ( u 3 ) is computed by (5.57). DR2B is the short form of RB{^2,V2) where %2 is fixed in this table with the given xi and y\. Table 5.6: P e r i o d 2 order quanti t ies and their costs (part I w i t h xi = 0 ) . Observe that the B N pol icy is op t ima l . 158 y[BN) = 1, R B = 5.9790; xx = 1 (censored) Order P e r i o d 2 O p t i m a l Order M i n . Cost E x p e c t e d Cost i n P e r i o d 2 Quan t i t y i n Sales Quan t i ty i n i n P e r i o d 3 Cos t i n under y2 P e r i o d 2 P e r i o d 3 under 'yjj P e r i o d 3 U2 = R% + E ( u 3 ) 2/2 X2 2/3* «3 = E ( u 3 ) u2 = m i n ^ U2 yiBN) = 3 0 0 1.4254 R% = 8.8980 1 1 2.3832 2 2 3.3815 3 5 12.3419 8.4407 u2 = 17.3387 4 0 0 1.4254 i % = 9.3816 1 1 2.3832 2 2 3.3815 3 2 4.3480 4 6 13.5565 8.3263 17.7079 5 0 0 1.4254 R% = 9.9714 1 1 2.3832 2 2 3.3815 3 2 4.3480 4 3 5.3404 5 7 14.7521 8.2288 18.2002 Table 5.7: P e r i o d 2 order quantit ies and their costs (part II w i t h x\ = 1). Observe that the B N pol icy op t ima l . 159 2/i = 2, RB = 6.1322; xx = 1 (exact) Order P e r i o d 2 O p t i m a l Order M i n . Cost E x p e c t e d Cos t i n P e r i o d 2 Quan t i t y i n Sales Quan t i ty i n i n P e r i o d 3 Cos t i n under j / 2 P e r i o d 2 P e r i o d 3 under 2/3 P e r i o d 3 U2 = R?B + E ( « 3 ) J/2 2:2 2/3* u3 = R% E ( u 3 ) u 2 — m i n ^ U2 2/2 = 0 0 0 1.9091 1.9091 u2 = 3.8182 i ? 2 , = 1.9091 1 0 0 1.0000 = 1.9146 1 1 2.3832 2.2297 4.1443 2 0 0 1.0000 i ? ! = 2.2572 1 1 1.7052 2 1 2.9440 2.5057 4.7629 3 0 0 1.0000 R% = 2.7923 1 1 1.7052 2 1 2.2611 3 2 3.4921 2.6628 5.4551 Table 5.8: P e r i o d 2 order quantit ies and their costs (part II of 2/1 = 2 w i t h xi = 1). Observe that the B N po l icy is op t ima l . 160 y i = 2 , RB = 6.1322; J i = 2 (censored) Order P e r i o d 2 O p t i m a l Order M i n . Cost E x p e c t e d Cos t i n P e r i o d 2 Quan t i t y i n Sales Quan t i t y i n in P e r i o d 3 Cos t i n under j/2 P e r i o d 2 P e r i o d 3 under 2/3 P e r i o d 3 U2=R%+E(u3) J/2 X2 Vs u3 = R% E(«3) u2 = miny2 U2 y<™> = 4 0 0 2.1032 i ? | = 10.6267 1 1 2.9440 2 2 3.7909 3 3 4.6971 4 7 13.8613 10.1602 u2 = 20.7869 5 0 0 2.1032 R% = 11.1455 1 1 2.9440 2 2 3.7909 3 3 4.6971 4 3 5.5958 5 8 14.9389 10.0530 21.1985 6 0 0 2.1032 i % = 11.7714 1 1 2.9440 2 2 3.7909 3 3 4.6971 4 3 5.5958 5 4 6.5283 6 9 16.0370 9.9571 21.7285 Table 5.9: P e r i o d 2 order quanti t ies and their costs (part III of j / i = 2 w i t h X\ — 2). Observe that the B N pol icy is op t ima l . 161 y i = 3 , RB = 6.3963; x 1 = 2 (exact) Order P e r i o d 2 O p t i m a l Order M i n . Cost E x p e c t e d Cost i n P e r i o d 2 Quan t i t y i n Sales Quan t i ty i n i n P e r i o d 3 Cos t i n under y2 P e r i o d 2 Pe r iod 3 under y 3 P e r i o d 3 U2 = R2B + E ( u 3 ) 2/2 Z2 Vt u3 = R% E ( u 3 ) u2 = m i n y 2 U2 yiBN) = i 0 1 1.7052 R% = 3.0375 1 2 3.3815 3.1955 u2 = 6.2330 2 0 1 1.7052 R% = 3.1050 1 1 2.2611 2 2 3.7909 3.2846 6.3896 3 0 1 1.7052 i % = 3.4174 1 1 2.2611 2 1 2.8681 3 2 4.2823 3.3670 6.7844 4 0 1 1.7052 i 2 | = 3.9009 1 1 2.2611 2 1 2.8681 3 2 3.4281 4 3 4.7815 3.4089 7.3098 Table 5.10: P e r i o d 2 order quanti t ies and their costs (part III of y i = 3 w i t h x\ = 2). Observe that the B N po l icy is op t ima l . 162 yi = 3 , RB = 6.3963; xx - 3 (censored) Order P e r i o d 2 O p t i m a l Order M i n . Cost E x p e c t e d Cos t i n P e r i o d 2 Quan t i t y i n Sales Quan t i t y in i n P e r i o d 3 Cos t i n under y2 P e r i o d 2 P e r i o d 3 under j / * P e r i o d 3 U2=R% + E ( u 3 ) 2/2 X2 2/3 u3 = R% E ( « 3 ) u2 = m i n ^ U2 2/2BJV) = 5 0 1 2.7165 R% = 12.0913 1 2 3.4921 2 2 4.2823 3 3 5.0858 4 4 5.9458 5 8 15.2753 11.6031 u2 = 23.6944 6 0 1 2.7165 RB = 12.6555 1 2 3.4921 2 2 4.2823 3 3 5.0858 4 4 5.9458 5 4 6.8032 6 9 16.2632 11.5018 24.1573 7 0 1 2.7165 R% = 13.3253 1 2 3.4921 'a 6 5 7.6969 7 10 17.2899 11.4097 24.7350 "These rows are the same as the corresponding ones of yi = 6 above. Table 5.11: P e r i o d 2 order quanti t ies and their costs (part I V of j / i = 3 w i t h x\ = 3). Observe that the B N po l icy is op t ima l . 163 Order Qty . P e r i o d P r o b a b i l i t y M i n . Cos t E x p . Cos t T o t a l & E x p . Cos t 1 Sales of the Sales i n P e r i o d 2 i n P e r i o d 2 E x p . Cos t 2/i &RB Xi P r ( X ) u2a E(u2)b T E CC {BN) , y\ = 1 0 P r ( X = x i ) = 0.3832 1.0910 RlB = 5.9790 1 P r ( X > x i ) = 0.6168 17.3387 11.1126 17.0914 l / i=2 0 P r ( X = n ) = 0.3832 1.0910 RB = 6.1322 1 P r ( X = n ) = 0.1394 3.8182 2 P r ( X > xi) = 0.4774 20.7869 10.8740 17.0067* 2/i=3 0 P r ( X = z i ) = 0.3832 1.0910 RB = 6.3963 1 P r ( X = i i ) = 0.1394 3.8182 2 P r ( X = n ) = 0.0887 6.2330 3 P r ( X > i x ) = 0.3887 23.6944 10.7132 17.1105 aU2 is computed by (5.58). BE(u2) is computed by (5.59). In this table, that is E(u2) = Z ) u 2 P r ( X ) . c Due to the round-up error, T E C does not exactly equal the sum of RB and E ( « 2 ) given in the table. Table 5.12: Pol ic ies and their costs i n E x a m p l e 1 showing that the o p t i m a l order quant i ty is 2 while the B N po l icy is 1 at decision epoch 1. 164 However, from the tables of per iod 2 order quantit ies and costs, 2/2=2/2 a t every ( j / i , xi). We suspect that this is because of the discrete demand. R e c a l l , i n the 2-period problem w i t h the same demand and cost settings, since the demand dis t r ibut ion is discrete, we have y{ = y[BN^ (see E x a m p l e 1 in Section 5.5.2). It seems that the larger impact that the decision has on future demand est imat ion, the more should be ordered, for informat ion gathering purposes. In the 2-period prob lem, the first per iod order quant i ty affects the demand upda t ing , and as a result, the cost i n the second per iod . In the 3-period problem, the effect of the first pe r iod decision lasts for the next two per iods. Hence, we can say that the demand informat ion i n the first pe r iod is more valuable i n the 3-period p rob lem than i n the 2-period problem. A p p l y i n g the economic jus t i f ica t ion discussed in Section 5.4.3, we have the same interpretat ion. T h a t is, a higher op t ima l order quant i ty is resulted from an increase i n the penal ty associated w i t h the lack of informat ion in the 3-period problem. E x a m p l e 2 . We now modify the cost structure whi le keeping the demand assumpt ion the same as i n E x a m p l e 1. C o s t s : c = 1, h = 0.5, p = 2; hence k — = 0.67. D e m a n d : X\X ~ Poisson(A); TTI = Gamma(0 .4 ,10 ) . T h i s set t ing is exact ly the same as E x a m p l e 2 in Section 5.5.2. T h e marg ina l density is the same as that given in example 1, but the B N pol icy now becomes y[BN^ = M f 1 (0.67) = 3. Hence, we w i l l compute the to ta l expected cost for y\ — 3 , 4 , 5 , 6 . S imi l a r to E x a m p l e 1, the costs 112 are reported by the pair ( y i , x\) . Sta r t ing w i t h the B N po l icy y[BN^ — 3 for per iod 1, we summarize a l l possible sales realizations for each enumerated j / 2 i n Tables 5.13, 5.14, 5.15 and 5.16 w i t h x\ = 0 , 1 , 2 , 3 , respectively. T h e expected cost 2^(71-2,2/2) is computed for each w 2 . For the enumerated y\ — 4, parts I, II, and III are the same as Tables 5.13, 5.14, and 5.15, respectively except that RB(TTI, 2/1) = 7.2891. P a r t I V which is an exact observation case, x\ = 3, is given by Table 5.17. P a r t V , w i t h the censored observation x\ = 4, is reported i n Table 5.18. S imi l a r to the pr ior case, parts I, II, and III are shown i n Tables 5.13, 5.14, and 5.15, respectively, while par t I V w i t h x\ = 4 is given by Table 5.17. P a r t V w i t h x\ = 4 (exact) is given by Table 5.19 and part V I w i t h xi = 5 (censored) is shown in Table 5.20. Since the censored sales x\ — 5 yields y^N^ = 12, there is a long list of a l l possible sales realizations in per iod 2. We omit l i s t ing the t h i r d per iod op t ima l order quanti ty y3BN^ for each { x 2 | j / 2 , a ; i = 5 , j / i = 5} to make Table 5.20 concise. 165 y [ B N ) = 3 , RB = 7.2755; xx = 0 (exact) Order P e r i o d 2 O p t i m a l Order M i n . Cost E x p e c t e d Cos t i n P e r i o d 2 Q u a n t i t y i n Sales Quan t i ty i n in P e r i o d 3 Cost i n under y2 P e r i o d 2 P e r i o d 3 under j/3 P e r i o d 3 U2=R% + E ( u 3 ) 2/2 Z2 2/3* uz = R% E ( « 3 ) u2 = m i n y 2 U2 2/2 = 0 0 0 0.7273 0.7273 u2 = 1.4545 R% = 0.7273 1 0 0 0.3810 i ? | = 0.8854 1 1 1.6336 1.5954 2.4808 2 0 0 0.3810 = 1.2641 1 0 1.2029 2 2 2.3344 2.1863 3.4504 3 0 0 0.3810 R% = 1.7164 1 1 1.2029 2 1 1.8748 3 2 2.9730 2.6245 4.3409 Table 5.13: P e r i o d 2 order quantit ies and their costs (part I w i t h x\ = 0). Observe that the B N pol icy op t ima l . 166 y [ B N > = 3 , RB = 7.2755; xx = 1 (exact) Order P e r i o d 2 O p t i m a l Order M i n . Cos t E x p e c t e d Cos t i n P e r i o d 2 Quan t i t y i n Sales Quan t i ty in i n P e r i o d 3 Cost i n under y2 P e r i o d 2 P e r i o d 3 under y3 P e r i o d 3 U2 = R% + E ( u 3 ) 2/2 2/3 E ( u 3 ) u2 = m i n ^ U2 yiBN) = i 0 1 1.2029 i ? | = 2.1521 1 2 2.6372 2.5935 u2 = 4.7456 2 0 1 1.2029 i ? | = 2.1632 1 1 1.8748 2 3 3.2154 3.0492 5.2124 3 0 1 1.2029 R2B = 2.4053 1 1 1.8748 2 2 2.4738 3 3 3.7481 3.3719 5.7772 4 0 1 1.2029 R% = 2.7722 1 1 1.8748 2 2 2.4738 3 3 3.1029 4 4 4.3166 3.6407 6.4219 Table 5.14: P e r i o d 2 order quantit ies and their costs (part II w i t h x\ = 1). Observe that the B N pol icy op t ima l . 167 y\UN) = 3, RB = 7.2755; n = 2 (exact) Order P e r i o d 2 O p t i m a l Order M i n . Cost E x p e c t e d Cos t i n P e r i o d 2 Quan t i t y i n Sales Quan t i ty i n i n P e r i o d 3 Cos t i n under y2 P e r i o d 2 P e r i o d 3 under j/g P e r i o d 3 U2 = R2B + E ( « 3 ) y-2 X2 2/3* «3 = R% E ( u 3 ) u2 — m i n y 2 U2 y i B N ) = 3 0 1 1.8748 . R% = 3.3372 1 2 2.4738 2 3 3.1029 3 4 4.5589 4.1402 u2 = 7.4775 4 0 1 1.8748 i ? | = 3.5175 1 2 2.4738 2 3 3.1029 3 3 3.6594 4 4 5.0765 4.3412 7.8587 5 0 1 1.8748 R% = 3.8297 1 2 2.4738 2 3 3.1029 3 3 3.6594 4 4 4.2471 5 5 5.5804 4.4728 8.3025 Table 5.15: P e r i o d 2 order quantit ies and their costs (part III w i t h x\ = 2 ) . Observe that the B N pol icy op t ima l . 168 y i = 3 , RB = 7.2755; i i = 3 (censored) Order P e r i o d 2 O p t i m a l Order M i n . Cost E x p e c t e d Cost i n P e r i o d 2 Quan t i t y i n Sales Quan t i ty i n i n P e r i o d 3 Cos t i n under 2/2 P e r i o d 2 P e r i o d 3 under 2/3 P e r i o d 3 U2 = R% + E ( u 3 ) 2/2 ^2 2/3* u3 = R% E ( u 3 ) u2 = minj , 2 U2 2/2 - 10 0 2 2.9730 i ? | = 13.4297 1 3 3.7481 2 4 4.5589 3 5 5.4027 4 5 6.2744 5 6 7.1533 6 7 8.0648 7 8 9.0027 8 9 9.9614 9 10 10.9356 10 18 21.7954 11.8635 u2 = 25.2932 11 0 2 2.9730 R% = 14.0776 lo 10 11 11.9206 11 19 22.8557 11.7691 25.8467 12 0 2 2.9730 i ? | = 14.7485 :& 11 12 12.9186 12 20 23.9123 11.6873 26.4358 "These rows are the same as the corresponding ones of j/J • — 10 above. 'These rows are the same as the corresponding ones of y\ =11 above. Table 5.16: P e r i o d 2 order quantit ies and their costs (part I V w i t h x\ = 3 ) . Observe that the B N pol icy op t ima l . 169 yi = 4 , RB = 7.2891; xx = 3 (exact) Order P e r i o d 2 O p t i m a l Order M i n . Cos t E x p e c t e d Cos t i n P e r i o d 2 Quan t i t y i n Sales Quan t i ty i n i n P e r i o d 3 Cost i n under y 2 P e r i o d 2 P e r i o d 3 under y* P e r i o d 3 U2 = R% + E ( u 3 ) 2/2 Z2 2/3 u3 = R3B E ( u 3 ) u2 = m i n y 2 f/2 2/2 = 4 0 2 2.4738 i ? | = 4.4627 1 3 3.1029 2 3 3.6594 3 4 4.2471 4 5 5.8440 5.0424 u2 = 9.5051 5 0 2 2.4738 R% = 4.6073 1 3 3.1029 2 3 3.6594 3 4 4.2471 4 4 4.8037 5 6 6.3344 5.1378 9.7451 6 0 2 2.4738 R% = 4.8819 1 3 3.1029 2 3 3.6594 3 4 4.2471 4 4 4.8037 5 5 5.3666 6 6 6.8120 5.1912 10.0731 Table 5.17: P e r i o d 2 order quantit ies and their costs (part I V of y\ = 4 w i t h x i = 3). Observe that the B N po l icy is o p t i m a l . 170 yi =4, RB = 7.2891; xi = 4 (censored) Order Period 2 Optimal Order Min. Cost Expected Cost in Period 2 Quantity in Sales Quantity in in Period 3 Cost in under y2 Period 2 Period 3 under y3 Period 3 U2 = R2B+E(u3) .2/2 x 2 2/3 u3 = R3B E(u 3) tt2 = min y 2 U2 yfN) = i i 0 3 3.5602 i % = 14.7594 1 4 4.3166 2 4 5.0765 3 5 5.8440 4 6 6.6497 5 7 7.4893 6 8 8.3594 7 8 9^ 2361 8 9 10.1421 9 , 1 0 11.0723 10 11 12.0218 11 19 22.9140 13.2168 u2 = 27.9762 12 *a i? ! = 15.4564 i i 12 12.9861 12 20 23.9509 13.1213 28.5777 13 R2B = 16.1738 12 13 13.9613 13 21 24.9890 13.0378 29.2116 "These rows are the same as the corresponding ones of y\ — 11 above. 'These rows are the same as the corresponding ones of j/i = 12 above. Table 5.18: Period 2 order quantities and their costs (part V of y\ = 4 with x\ = 4). Observe that the BN policy is optimal. 171 j / i = 5, RlB = 7.3775; x x = 4 (exact) Order P e r i o d 2 O p t i m a l Order M i n . Cost E x p e c t e d Cost i n P e r i o d 2 Q u a n t i t y i n Sales Quan t i ty i n i n P e r i o d 3 Cos t i n under y2 P e r i o d 2 P e r i o d 3 under y* P e r i o d 3 U2 = R2B + E ( u 3 ) 2/2 Z2 2/3* u3 = R3B E ( u 3 ) u2 = m\nV2 U2 2 / f N ) = 5 0 3 3.1029 R% = 5.5587 1 3 3.6594 2 4 4.2471 3 4 4.8037 4 5 5.3666 5 6 7.0896 5.7988 u2 = 11.3575 6 0 3 3.1029 2 % = 5.6804 1 3 3.6594 2 4 4.2471 3 4 4.8037 4 5 5.3666 5 5 5.9226 6 7 7.5432 5.8185 11.4989 7 0 3 3.1029 i ? | = 5.9275 1 3 3.6594 '.a 5 5 5.9226 6 6 6.4688 7 7 8.0186 5.8322 11.7597 "These rows are the same as the corresponding ones of yi = 6 above. Table 5.19: P e r i o d 2 order quantit ies and their costs (part V of y± = 5 w i t h x\ = 4 ) . Observe that the B N pol icy is op t ima l . 172 y i = 5 , RB = 7.3775; xx = 5 (censored) Order Quan t i t y i n P e r i o d 2 P e r i o d 2 Sales Expec t ed Cost i n P e r i o d 3 Cos t i n P e r i o d 2 under 2/2 U2 = R% + E ( u 3 ) 2/2 Z2 E ( u 3 ) u2 = m i n y 2 U2 yiBN) = 12 0 , 1 , . . . ,11 (exact) = 16.0179 12 (censored) 14.4882 u2 = 30.5062 2/2 = 13 0 , 1 , . . . ,12 (exact) i ? | = 16.7638 13 (censored) 14.3926 31.1564 2/2 = 14 0 , 1 , . . . ,13 (exact) R2B = 17.5277 14 (censored) 14.3085 31.8362 2/2 = 15 0 , 1 , . . . ,14 (exact) R2B = 18.3011 15 (censored) 14.2345 32.5356 Table 5.20: P e r i o d 2 order quantit ies and their costs (part V I of 2/1 = 5 w i t h x\ = 5 ) . Observe that the BN pol icy is op t ima l . T h e o p t i m a l order quant i ty y%BN^ and its cost u 3 are omi t ted . 173 yi = 6, RB = 7.5257 xx = 5 (exact) Order Quan t i ty i n P e r i o d 2 P e r i o d 2 Sales Expec t ed Cost i n P e r i o d 3 Cos t i n P e r i o d 2 under y2 U2 = R% + E ( u 3 ) 2/2 2?2 E ( u 3 ) u2 = m i n y 2 U2 2/2 = 6 0 , 1 , . . . ,5 (exact) i ? | = 6.6343 6 (censored) 6.4526 u2 = 13.0869 2/2 = 7 0 , 1 , . . . , 6 (exact) i ? | = 6.7403 7 (censored) 6.4317 13.1720 2 /2=8 0 , 1 , . . . , 7 (exact) = 6.9667 8 (censored) 6.4238 13.3905 2 /2=9 0 , 1 , . . . ,8 (exact) i % = 7.2820 9 (censored) 6.4161 13.6981 Table 5.21: P e r i o d 2 order quantit ies and their costs (part V I of j / i = 6 w i t h x\ = 5). Observe that the B N pol icy is op t ima l . S i m i l a r l y for y i = 6, the first five parts w i t h x\ = 0 , 1 , 2 , 3 , 4 are given by Tables 5.13, 5.14, 5.15, 5.17, and 5.19. P a r t V I w i t h x\ = 5, which is an exact observation, is given i n Table 5.21, and part V I I , w i t h the censored sales x\ = 6, is given i n Table 5.22. A g a i n , we omit l i s t ing some of the sales realizations to make the tables concise. C o m b i n i n g Tables 5.13, 5.14, 5.15 and 5.16, we obta in the to ta l expected cost ( T E C ) for y[BN^ = 3. C o m b i n i n g Tables 5.13, 5.14, 5.15, 5.17 and 5.18, we obta in the T E C for yx = 4. C o m b i n i n g 5.13, 5.14, 5.15, 5.17, 5.19 and 5.20, we have the T E C for yx = 5. C o m b i n i n g 5.13, 5.14, 5.15, 5.17, 5.19, 5.21 and 5.22 we have the T E C for j / i = 6. These to ta l expected costs are presented i n Table 5.23. Table 5.23 shows that y{ = 5 w i t h m i n i m u m T E C , u\ = 18.8100. Hence, the o p t i m a l order quant i ty is greater than the B N pol icy y[BN^ = 3. R e c a l l that the op t ima l order quant i ty i n the 2-period example in Sect ion 5.5.2 is also 5. S imi l a r to E x a m p l e 1, the relative cost saving is very smal l . In this example, it is 1% (i.e. (18.9902 - 18.8100)/18.9902), even less than i n the 2-period example. Therefore, bo th 3-period examples i n this section, as well as, the three 2-period examples i n Section 5.5.2 show that the o p t i m a l order quant i ty y{ is greater than or equal to the Bayes ian Newsvendor pol icy when 174 yi = 6, RB = 7.5257 xx = 6 (censored) Order P e r i o d 2 E x p e c t e d Cos t i n P e r i o d 2 Quan t i ty i n Sales Cost i n under y2 P e r i o d 2 P e r i o d 3 U2 = R%+ E ( u 3 ) 2/2 x2 E ( u 3 ) u2 = m i n y 2 U2 y2BN = 13 0 , 1 , . . . ,12 (exact) R% = 17.2271 13 (censored) 15.7050 u2 = 32.9321 2/2 = 14 0 , 1 , . . . ,13 (exact) i ? ! = 18.0214 14 (censored) 15.6098 33.6312 2/2 = 15 0 , 1 , . . . ,14 (exact) i ? ! = 18.8314 15 (censored) 15.5255 34.3569 2/2 = 16 0 , 1 , . . . ,15 (exact) R% = 19.6484 16 (censored) 15.4507 35.0991 Table 5.22: P e r i o d 2 order quantit ies and their costs (part V I I of j / i = 6 w i t h x\ = 6 ) . Observe that the B N pol icy is op t ima l . 175 Order Qty . & E x p . Cos t P e r i o d 1 Sales P r o b a b i l i t y of the Sales M i n . Cos t i n P e r i o d 2 E x p . Cos t i n P e r i o d 2 T o t a l E x p . Cost T E C 2/i & R B xx Pr (A- ) E ( u 2 ) (RB+E(u2)) y[BN) = 3 0 P r ( X = X l ) = 0.3832 1.4545 RB = 7.2755 1 Pr (A" = xx) = 0.1394 4.7456 2 P r (A" = xx)= 0.0887 7.4775 3 P r ( X > x i ) = 0.3887 25.2932 11.7147 18.9902 2 / 1= 4 0 Pi(X =xx) = 0.3832 1.4545 R}B - 7.2891 1 P r ( X = xx) = 0.1394 4.7456 2 P r (A" = xi) = 0.0887 7.4775 3 P r ( X = x i ) = 0.0645 9.5051 4 Pr(X > xx) = 0.3243 27.9762 11.5664 18.8556 2 / i = 5 0 P r ( X = xx) = 0.3832 1.4545 flj, = 7.3775 1 Pr(A~ = xx) = 0.1394 4.7456 2 P r ( X = x i ) = 0.0887 7.4775 3 Pr (A" = i 1 ) = 0.0645 9.5051 4 P r ( X = n ) = 0.0498 11.3575 5 Pr (A" > x j ) = 0.2744 30.5062 11.4325 18.8100* 2 / i = 6 0 P r ( X = X i ) = 0.3832 1.4545 i ? k = 7.5257 1 P r ( X = X l ) = 0.1394 4.7456 2 P r ( A " = x i ) = 0.0887 7.4775 3 Pr (A" = x i ) = 0.0645 9.5051 4 Pr (A" = x i ) = 0.0498 11.3575 5 P r ( X = x i ) = 0.0399 13.0869 6 Pr (A" > x i ) = 0.2346 32.9321 11.3071 18.8328 Table 5.23: Pol ic ies and their costs in E x a m p l e 2 showing that the o p t i m a l order quant i ty is 5 while the B N pol icy is 3 at decision epoch 1. 176 there is unobservable lost sales. In the 3-period problem, it is interest ing to note that i n bo th examples V\ — viBN^ f ° r a n y ( l / i i ^ i ) - T h i s is contrary to our belief that y2 > y2BN^ for some (yi,xi). Nevertheless, i t confirms our conjecture for the general iV-pe r iod newsvendor problems. W e speculate that this is due to the discreteness of the demand. 5.6.3 Some Discussion on Policy Cost - Observations from the Numerical Ex-amples A n interest ing question concerning pol icy cost is its convergence to the expected cost w i t h respect to the true demand d i s t r ibu t ion , as more informat ion is gathered. Note that the expected cost w i t h respect to the true demand d i s t r ibu t ion is the op t ima l cost which is on ly achievable when one has perfect knowledge about the demand d i s t r ibu t ion . We leave this convergence issue for further research. However , the numerica l examples shown i n Sect ion 5.5.2 and 5.6.2 lend some in tu i t ion . A s the upda ted demand d i s t r ibu t ion converges to the true demand d i s t r ibu t ion , we conjecture that the expected cost of the o p t i m a l order quant i ty converges to the "baseline" cost, namely, the expected cost w i t h respect to the true demand d i s t r ibu t ion . We re-visit some of the examples to see i f there is any v io la t ion against our conjecture. 5.6.4 The Traditional Newsvendor and the Cost Trad i t i ona l newsvendor p rob lem assumes the true demand d i s t r ibu t ion is known . See Sect ion 5.1.1 for details about the s imple t r ad i t iona l newsvendor. R e c a l l , we considered Poisson demand in the numerica l studies. Moreover , the demand structure in E x a m p l e 2 remains the same as i n E x a m p l e 1 w i t h expectat ion E ( X ) = 4. Hence, we assume that the true demand follows Po(4) so that the true demand also has expectat ion 4. N e x t , we give the op t ima l po l i cy and the "baseline" cost for two newsvendor examples w i t h Po(4) demand but different cost settings. E x a m p l e 1 Suppose X is the demand, X ~ Po(4) and the pd f and cdf of X are denoted f(x) and F(x). The cost s tructure is c = 1, h = 0.25, p = 1.5. Hence k = = 0.4. T h e o p t i m a l po l icy is given by y* =F~1(0A) = 3 . (5.60) 177 T h e one-period expected cost of y* is, y" R(y*) = (c-p)y* + (p-h)y*F(y*)+pE(X)-(p-h)^2xf(x) x=0 = 4.8095. (5.61) Since the t r ad i t iona l newsvendor w i t h known demand d i s t r ibu t ion can be model led as a single per iod op t imiza t ion prob lem, the to ta l expected cost in the 2-period and the 3-period systems are, TEC2 = 2R(y*) = 9.6190 , TEC3 = 3R{y*) = 14.4285 . E x a m p l e 2 T h e cost s t ructure is c = 1, h = 0.5, p = 2 which gives k = j*5f = 0.67. T h e o p t i m a l po l i cy is y* = F _ 1 ( 0 . 6 7 ) = 5. T h e one-period expected is R(5) - 4.2270. Hence, the T E C ' s of the 2-period and the 3-period problem are, TEC2 = 2R(y*) = 8.4540 , TEC3 = 2>R{y*) = 12.6809 . W e w i l l re-consider E x a m p l e 1 and E x a m p l e 2 i n Section 5.5.2 (where N = 2) as well as i n Section 5.6.2 (where N = 3). Some relevant costs are extracted from the tables and compared to the "baseline" cost in the fol lowing section. 5.6.5 Cost Convergence - 2-Period vs 3-Period Problems To see the t rend of the expected Bayes ian cost from the B M D P model , we compare it to the "baseline" cost R(y*) f rom the t r ad i t iona l s ingle-period model . T w o types of Bayes ian costs are considered. T h e y are the per iodic expected Bayes ian cost and the to ta l expected Bayes ian cost. T h e per iodic expected Bayesian cost is defined, E(Rn) = j j R B ( 7 r i ' 2 > i ) w h e n n = l , B ~ \E{RB(ir'n,yn)) o the rwi se , where -K'n is given by (5.6). T h i s is the cost one expects for the current per iod . T h e to ta l expected Bayesian cost is denoted E(un) for n > 1 and T E C = ux for n = 1. It is the expected cost for the rest of the horizon at decision epoch n. Fur thermore , we present the cost compar ison for a 2-period example and the corresponding 3-period example i n para l le l . 178 R o w # Comments Vi P e r i o d 1 Cos t P e r i o d 2 Cost 1 from B M D P 2/* = 1 RB = 5 .9790a E(R%) = 5.6973 6 2 Basel ine y* = 3 R(y*) = 4.8095 R{y*) = 4.8095 3 C Diff. of 1 & 2 i n % 24.3% 18.5% A R L B is the short form of RB{^I-,VI) which is given by (5.54). B E ( R B ) = E ( u 2 ) in the 2-period examples. c This row is computed as (row 1 - row 2) / row 3 * 100%. Table 5.24: Per iod ic Costs - E x a m p l e 1, 2 -Per iod E x a m p l e 1 We begin w i t h the per iodic Bayes ian cost. W h e n N = 2, the first pe r iod cost i n the B M D P model is •RBCTTI,?/*)- T h e second per iod cost is u2 = RB(^2,V2)- Note u2 as a function of ir'2 depends on the sales observat ion x\. T h u s the expected Bayes ian cost for the second per iod , E(i?jg(7r 2,y2))> a n expectat ion wi th respect to the predetermined pr ior is E(u2). T h e costs from the B M D P mode l as well as the costs from the simple s ingle-period mode l are presented i n Table 5.24. We see from Table 5.24 that the Bayesian cost for the first per iod is over 24% higher than the "baseline" cost. Fur thermore the relative difference is down by almost 6 points to 18.5% for the second per iod. N o w consider the example wi th N = 3. Note i n the last per iod , the expected cost is E ( . R ^ ) = Ex! [Ex2 ("3)]- It is calcula ted for this example as E X l [ E X 2 ( u 3 ) ] = 0.5455*0.3832 + 1.9091 * 0 . 1 3 9 4 + 10.1602*0.4774 = 5.3256 . A t n = 2, since u2 = RB + E x 2 ( " 3 ) which yields EXl (u2) = EXl (R2B) + E ^ [EX2("3)], we have EXl (R%) = E X l ( w 2 ) -E{R%). Hence, E(R2B) = 10.8740 - 5.3256 = 5.5484 . T h e costs are presented i n Table 5.25. Not ice that the relative cost difference is decreasing in n as the demand d i s t r ibu t ion is updated over t ime. T h e Bayes ian cost is 27.5% higher than the "baseline" cost i n per iod 1. T h e gap is larger when N = 3. T h e percent difference i n per iod 2 is 15.4% which is below the corresponding difference w i t h N = 2. We expect a difference of only 10.7% i n the final per iod . In summary, a l though the Bayes ian cost i n the first per iod is higher when N = 3 than TV = 2, i t decreases more rap id ly i n the fol lowing two periods. 179 Comments J/i P e r i o d 1 Cost Pe r iod 2 Cost P e r i o d 3 Cost from B M D P 2/1* = 2 i?k = 6.1322 E ( i ? | ) = 5 .5484° E(R%) = 5.3256 Basel ine y* = 3 R{y*) = 4.8095 R(y*) = 4.8095 R(y*) = 4.8095 Diff. i n % 27.5% 15.4% 10.7% "The subscript, X\, of the expectation is omitted for convenience. Table 5.25: Per iod ic Costs - E x a m p l e 1, 3-Per iod R o w . # Comments 2/1 T E C E ( u 2 ) 1 from B M D P 2 / i = l 11.6763 5.6973 2 Basel ine y* = 3 9.6190 4.8095 3 Diff. of 1 & 2 i n % 21.4% 18.5% Table 5.26: T o t a l Expec t ed Costs for Future - E x a m p l e 1, 2 -Per iod A n o t h e r way to compare the costs is to consider the to ta l expected cost for a l l future periods inc luding the current pe r iod at each decision epoch. We present the costs i n Tables 5.26 and 5.27 for the 2-period and the 3-period problems, respectively. A c o m m o n observat ion from bo th tables is that the relative cost difference is decreasing as t ime increases. C o m p a r i n g Tables 5.26 and 5.27, we see that the relative difference of the T E C from the 3-period example is smaller than that from the 2-period example. T h a t is, the T E C for a longer hor izon is relat ively closer to the "baseline" cost. S imi l a r l y for the to ta l expected cost from epoch 2 on, E ( u 2 ) - Hence, this implies that increasing the length of the hor izon reduces the relative difference of the T E C from any decision epoch. We speculate that the Bayes ian cost converges to the "baseline" cost as the demand d i s t r ibu t ion is updated. However, this needs future research. C o m b i n i n g these comparisons, we have the fol lowing conjecture regarding the convergence of the Bayes ian cost. R o w # Comments 2/i* T E C E ( u 2 ) E ( u 3 ) 1 from B M D P 2/i* = 2 17.0067 10.8740 5.3256 2 Basel ine y* = 3 14.4285 9.6190 4.8095 3 Diff. i n % 17.9% 13.1% 10.7% Table 5.27: T o t a l Expec t ed Costs for Future - E x a m p l e 1, 3 -Per iod 180 R o w # Comments P e r i o d 1 Cos t P e r i o d 2 Cos t 1 from B M D P 2/i* = 5 i ? k = 7 .3775° E ( i % ) = 5.8351 6 2 Basel ine y* = 5 R(y*) = 4.2270 R(y*) = 4.2270 3 C Diff. of 1 & 2 i n % 74.5% 38.0% ".Rjg is the short form of R B ^ I I V I ) which is given by (5.54). bE(R^) = E ( i t 2 ) in the 2-period examples. c This row is computed as (row 1 - row 2) / row 3 * 100%. Table 5.28: Per iod ic Costs - E x a m p l e 2, 2 -Per iod R o w # Comments 2/i* P e r i o d 1 Cost P e r i o d 2 Cos t P e r i o d 3 Cost 1 from B M D P 2/i* = 5 RB = 7.3775 E(R%) = 6.2346 E{R%) = 5.1979 2 Basel ine y* = 5 R(y*) = 4.2270 R(y*) = 4.2270 R(y*) = 4.2270 3 Diff. i n % 74.5% 47.5% 23.0% Table 5.29: Pe r iod ic Costs - E x a m p l e 2, 3 -Per iod i) T h e Bayes ian cost from the B M D P model converges to the "baseline" cost from the t r ad i t iona l newsven-dor mode l when the demand d is t r ibu t ion is known. T h i s holds regardless of i f the expected cost for the current per iod or the to ta l expected cost the rest of the hor izon is considered. ii) T h e to ta l expected cost is relat ively smaller w i t h a longer hor izon . T h a t is, despite the fact that the costs i n the i n i t i a l periods may be relat ively higher, after these periods, i t decreases at a faster rate so that the overal l cost eventually decreases. T h e higher cost i n the i n i t i a l periods is due to the larger shift of the order quant i ty above the Bayes ian Newsvendor po l icy to gather informat ion . Reca l l j / * = 1 i n the 2-period p rob lem while y\ = 2 i n the 3-period prob lem w i t h the same sett ing i n E x a m p l e 1. Therefore, one should commi t to learn about the demand as long as one's budget allows or un t i l one has obta ined good knowledge about the demand d i s t r ibu t ion . A l t h o u g h the investment i n the demand informat ion i n the earlier periods may incur an increase i n the cost, i t w i l l eventual ly pay off. E x a m p l e 2 We follow the steps i n E x a m p l e 1. T h e per iodic Bayes ian costs are presented i n Table 5.28 and Table 5.29 for N = 2 and N = 3, respectively. S imi la r ly , the to ta l expected costs for a l l future periods are presented in Tables 5.30 and 5.31. We make s imi la r observations as we had from E x a m p l e 1 which supports the conjecture above. Note 181 R o w # Comments Vi T E C E ( u 2 ) 1 from B M D P Vi = 5 13.2126 5.8351 2 Basel ine y* = 5 8.4540 4.2270 3 Diff. of 1 & 2 i n % 56.3% 38.0% Table 5.30: T o t a l Expec t ed Costs for Future - E x a m p l e 2, 2 -Per iod R o w # Comments Vi T E C E ( u 2 ) E ( u 3 ) 1 from B M D P y*i=5 18.8100 11.4325 5.1979 2 Basel ine y* = 5 12.6809 8.4540 4.2270 3 Diff. i n % 48.3% 35.2% 23.0% Table 5.31: T o t a l Expec t ed Costs for Future - E x a m p l e 2, 3 -Per iod that , i n E x a m p l e 1, the Bayes ian cost for the first per iod w i t h either TV" = 2 or N = 3 is almost 75% higher than the "baseline" cost despite the fact that the B M D P model yields the same o p t i m a l order quant i ty as the t r ad i t iona l newsvendor model . However, the relative difference decreases by a ha l f for the 2-period problem and two th i rds for the 3-period problem i n the final per iod . T h e to ta l expected costs for the two problems of different horizons are bo th approximate ly 50% above the "baseline" costs. W h e n N = 3, bo th the T E C for the entire hor izon and E ( u 2 ) , the future cost from per iod 2 onward, are re la t ively close to the "baseline" level compared to those when N = 2. In conclusion, as we expect that the Bayes ian updated demand density converges to the true density, we expect the Bayes ian cost converges to the "baseline" cost. B o t h examples show that bo th types of the Bayes ian expected cost move toward the "baseline" cost as t ime goes on. 5.7 Conclusion In this chapter, we s tudy newsvendor inventory systems w i t h unknown demand. O u r assessment of the demand d i s t r ibu t ion is updated dynamica l ly by the Bayes ian method . T h e newsvendor system is a simple example of lost sales inventory systems. W h e n demand is completely observable, the demand upda t ing is independent of the pol icy upda t ing process. T h i s reduces the B M D P to a sequence of s ingle-period problems. Thus , the Bayes ian Newsvendor ( B N ) po l i cy obta ined using the Bayes ian est imator of demand density i n each per iod is op t ima l . O n the other hand , when lost sales are unobservable, only the satisfied demand is observed. T h e interac-182 t ive demand upda t ing process and pol icy upda t ing process greatly complicates the problem. T h e B M D P can no longer be s implif ied into a sequence of single-period problems. For a two-per iod prob lem w i t h continuous demand, the B N po l icy remains op t ima l i n the second per iod . In the first per iod , the possibi l i ty of lost sales raises the penal ty cost by an amount which represents the marg ina l expected value of informat ion. Consequently, at the B N policy, the Bayes ian expected cost is s t i l l less than the expected savings. T h e op t ima l order quant i ty increases due to the higher penalty. T h e finiteness of the o p t i m a l order quant i ty y* is not yet established. We conjecture that this result applies to finite-horizon (N > 2) problems. General ly , lost sales can contr ibute to loss of informat ion in demand upda t ing . A n associated cost is incur red by potent ia l loss of informat ion. Therefore, it is in tu i t ive ly reasonable to order more to obta in add i t iona l in format ion for future demand upda t ing while sacrificing the cost m i n i m i z a t i o n i n the current per iod . A l t h o u g h this s tudy of the impact of lost sales on the o p t i m a l po l i cy is l im i t ed to the newsvendor inventory system, we expect to generalize the work for other inventory systems such as base stock inventory system. F r o m the numer ica l examples of bo th two-per iod and three-period problems, we notice that the difference between the to ta l cost of the op t ima l po l icy and that of the B N po l icy is very smal l . T h i s implies that the newsvendor po l i cy performs very well i n the system where there is unobservable lost sales. 183 Chapter 6 Future Research T h i s thesis starts w i t h s ta t is t ical problems of demand es t imat ion i n lost sales inventory systems using incom-plete in format ion , investigates the effects of es t imat ion error on the inventory policies and inventory costs, and finally develops op t imiza t ion results regarding the inventory po l i cy de te rmina t ion based on demand upda t ing . There are areas where further research is needed. We summarize these areas i n the fol lowing. 6.1 Other Distribution Models to Account for Overdispersion A s shown i n Sect ion 2.5, there are cases where the overdispersion cannot be wel l accommodated by the zero inflated Poisson mode l . T h e negative b inomia l model may provide a better fit than the Poisson since it is a member of a more general class. A s seen i n Chapte r 4, when the demand follows Poisson(A) where A is G a m m a d is t r ibu ted , the marg ina l d i s t r ibu t ion of the demand determined by integrat ing over the support of the G a m m a d i s t r ibu t ion w i t h respect to A is negative b inomia l . Hence, the negative b inomia l model is more general than the Poisson model . G a r d e n [24] tested the negative b inomia l versus the Poisson model fitted to the C T P A da ta and showed that the negative b inomia l regression model resulted i n significant improvement . However, a more general negative b inomia l mode l may not be adequate i n case where there is clear evidence of zero inf lat ion. A Zero Inflated Negat ive B i n o m i a l ( Z I N B ) model can expla in more accurately the overdispersion. S imi l a r ly to the Z I P , Z I N B has an ex t ra p robab i l i ty mass at zero. It w i l l be interest ing to see how Z I N B fit the sales of some of the C T P A products . T h e M L E of the Z I N B parameters can be developed s imi la r ly to the Z I P case. 6.2 Extending the Bayesian Newsvendor Study Some problems remain from our Bayes ian newsvendor s tudy shown i n Chap te r 4. 184 i) E x t e n s i o n of the result for 2-period prob lem to the case when N > 2. We give two numerica l examples i n Sect ion 5.6 of 3-period discrete demand newsvendor problem. A l l the results based on pol icy enumerat ion conform w i t h our conjecture about the iV-pe r iod problem. T h i s work is recently completed i n D i n g , P u t e r m a n and B i s i [18]. ii) Efficient numer ica l calculat ions of the op t ima l order quant i ty for continuous demand. i i i ) A p p l i c a t i o n of the B M D P to other lost sales inventory systems w i t h u n k n o w n demand such as the base-stock system and the (s,S) system. iv) Effect of lost sales on pol icy costs. A s we have seen from the numer ica l examples the op t ima l order quant i ty provides m i n i m a l savings when compared to the cost of the Bayes ian Newsvendor policy. Hence is the Bayes ian Newsvendor pol icy robust against unobservable lost sales in terms of cost? v) Convergence of the Bayes ian cost. T h e numerica l examples i m p l y that i t converges to the "baseline" cost of the t r ad i t iona l newsvendor w i t h known demand. Theore t ica l proof is needed. 6.3 Seasonality and Trend in Demand A s ta t ionary demand d i s t r ibu t ion is another common assumption used i n dynamic p rog ramming for i n -ventory management. In this thesis, we assumed s ta t ionary demand. However, most re tai l products have demands w i t h seasonal fluctuation and t rend. New products and products of low da i ly sales such as c lothing usual ly have the most variable demands. T h i s raises the question of how to estimate and update the de-m a n d d i s t r ibu t ion account ing for bo th the seasonality and t rend. Fur ther research is need on the Bayesian dynamic mode l that models the t rend i n demand upda t ing . A z o u r y [6] gave, i n Sect ion 3 of her paper, an example of model ing seasonal effect in demands. We feel that this extension of i nc lud ing the seasonality is s t ra ightforward assuming the effect is known and mul t ip l ica t ive . 6.4 Other Factors That Affect Demand Aside from seasonality and t rend i n demand represented by fluctuating da i ly sales, nonsta t ionar i ty in the demand d i s t r ibu t ion may come from some other factors, such as, to name a few, economic condit ions, consumers ' behavior , and advert is ing. A m o n g these, two interest ing factors are the p romot ion policies offered by the re ta i l store and the pat tern of product subs t i tu t ion determined by consumers ' behavior. 185 6.4.1 Effect of Promotion Policies In today 's h ighly compet i t ive retai l industry, occasional offering of p romot ion policies is widely pract iced in a t t empt ing to increase sales by a t t rac t ing new customers as well as p rov id ing incentives to old (or loyal) customers. T h e demand for a product is expected to respond to the p romot ion policies. If the promot ion policies have significant effects on the demand, then one wishes to estimate the magni tude of these effects so that the demand can be forecasted given a par t icular policy, or i n the same manner, the decision on p romot ion policies can be made i n order to reach the target of sales. O n the other hand, p romot ion policies are offered at costs. T h e direct cost of a po l icy is the discount offered to customers. Indirect costs are associated w i t h the increased demand the policies may br ing i n . There are inventory cost incurred by the increased demand i n the promot iona l per iod , and the low sales i n the fol lowing per iod because of the a t t ract ive forward buy ing dur ing the promot iona l per iod . So the question is: is offering p romot ion policies efficient or not? To answer this question, we need to look at a bigger pic ture . A n art icle by M c K e n n e y and C l a r k shown i n the H a r v a r d Business Rev iew [45] gave an example i n re ta i l indus t ry which i l lust ra ted the inefficiency of offering p romot ion policies as well as the inefficiency i n the supply chain . The article argued that the decreased s tabi l i ty (or increased var iabi l i ty) i n demand resul t ing from frequent offer of sales reduces the s tabi l i ty of the entire chain which consists of manufacturers, d is t r ibutors and retailers and hence i t reduces the efficiency of the chain. Cons ide r ing the p r i c ing prob lem w i t h i n the context of inventory management, there are two decision variables, one is the p r ic ing pol icy and the other is the inventory pol icy. T h e y should be developed jo in t ly to max imize the profit from sales of a par t icular product t ak ing into account the inventory costs. K a r l i n [36] proposed the op t ima l inventory policies for per iodic demand (i.e., the demands repeat every cycle of K periods) i n the discounted case. He showed that a per iodic c r i t ica l -number po l i cy is op t imal ; that is, there may be a different c r i t i ca l number op t ima l po l icy for each of the K period-types, but these numbers repeat every cycle. Z i p k i n [67] and Song and Z i p k i n [56] extended the s tudy and proved that the c r i t i ca l number policies (e.g., (s,S) pol icy) depending on the state of the w o r l d i n the per iod (i.e., (si,Si) i n pe r iod i) are o p t i m a l assuming the per iodic pat tern remains the same over the cycles (wi th K periods i n each cycle) . T h i s method can be appl ied to tackle a simplif ied version of the jo in t problem of pr ic ing and inventory management, simplif ied by assuming that the demand varies w i t h the per iodic promot iona l policies. 186 6.4.2 Effect of Product Substitution A s interest ing as the pr ic ing 's effect on demand, the pat tern of consumers ' behavior of product subst i tut ion affects the demand for products in the same category and accordingly, influences the inventory policies that max imize the to ta l profit from a re ta i l assortment, subject to given resource constraints. T h a t is, when cer ta in i tems are not stocked, the result ing subst i tut ions increase the demand for other i tems. In Smi th and Agrawa l ' s recent paper [57], they gave a general model to capture the effects of subs t i tu t ion on the p robab i l i ty d is t r ibut ions of demand for items in an assortment. T h e y also developed a methodology for j o in t ly o p t i m i z i n g the inventory levels of a set of items w i t h potent ia l for subs t i tu t ion . In short, p romot iona l policies and product subst i tu t ion are two of many interest ing issues in retai l inven-tory management. 187 Bibliography [1] A g r a w a l , N . and S m i t h , S. A . , "Es t ima t i ng Negative B i n o m i a l D e m a n d for R e t a i l Inventory Management w i t h Unobservable Los t Sales", Naval Research Logistics, V o l . 43, 1996, 839-861. [2] A n u p i n d i , R . , D a d a , M . and G u p t a , S., " E s t i m a t i o n of Consumer D e m a n d w i t h S tock -Out Based Subs t i tu t ion : A n A p p l i c a t i o n to Vend ing Mach ine P roduc t s " , Marketing Science, V o l . 17, N o . 4, 1998, 406-423. [3] A r c h i b a l d , B . C . , "Cont inuous Rev iew (s, S) Pol ic ies w i t h Los t Sales", Management Science, V o l . 27, N o . 10, October 1981, 1171-1177. [4] A r r o w , K . J . , K a r l i n , S. a n d Scarf, H . , Studies in the Mathematical Theory of Inventory and Production, Stanford Univers i ty Press, Stanford, Ca l i fo rn ia , 1958. [5] A z o u r y , K . S., " A Compar i son of O p t i m a l Order ing Levels of Bayes ian and Non-Bayes ian Inventory M o d e l s " , Management Science, V o l . 30, N o . 8, 1984, 993-1003. [6] A z o u r y , K . S., "Bayes Solu t ion to D y n a m i c Inventory Mode l s Unde r U n k n o w n D e m a n d D i s t r i b u t i o n " , Management Science, V o l . 31, N o . 9, 1985, 1150-1160. [7] Biggs , R . J . and C a m p i o n , W . M . , "The effect and Cos t of Forecast E r r o r B i a s for Mul t ip le -S tage Produc t ion- Inventory Systems", Decision Sciences, V o l . 13, 1982, 570-584. [8] B a r t l e , R . G . , The Elements of Real Analysis, J o h n W i l e y & Sons, 1964. [9] Berger , J . O . , Statistical Decision Theory and Bayesian Analysis, Second Edition, Springer Series in Stat is t ics , 1980. [10] B r a d e n , D . J . and Freimer, M . , "Informational Dynamics of Censored Observat ions" , Management Science, V o l . 37, N o . 11, 1991, 1390-1404. 188 [11] B r a d e n , D . J . and Oren , S. S., "Nonl inear P r i c i n g to P roduce Informat ion" , Marketing Science, V o l . 13, N o . 3, 1994, 310-326. [12] B r o w n , R . G . , Statistical Forecasting for Inventory Control, M c G r a w H i l l , N e w Y o r k , 1959. [13] C i n l a r , E . , Introduction to Stochastic Processes, Prent ice H a l l , 1965. [14] Cohen , C . D . , "Bayes ian Adjus tment of Sales Forecasts i n M u l t i - I t e m Inventory C o n t r o l Systems", Journal of Industrial Engineering, V o l . 17, 1966. [15] Cohen , M . A . , Kle indorfer , P . R . , and Lee, H . L . , "Service Cons t ra ined (s, S) Inventory Systems wi th P r i o r i t y D e m a n d Classes and Los t Sales", Management Science, V o l . 34, N o . 4, A p r i l 1988, 482-499. [16] C o n r a d , S. A . , "Sales D a t a and the E s t i m a t i o n of D e m a n d " , Operational Research Quarterly, V o l . 27, N o . 1, 1976, 123-127. [17] Dean , C . B . , "Test ing for Overdispers ion i n Poisson and B i n o m i a l Regression M o d e l s " , Journal of the American Statistical Association, V o l . 87, 1992, 451-457. [18] D i n g , X . , P u t e r m a n , M . L . and B i s i , A . , "The Censored Newsvendor and the O p t i m a l Acqu i s i t i on of Informat ion" , Operations Research, to appear. [19] Dvore tzky , A . , Kiefer J . and Wolfowi tz , J . , "The Inventory P r o b l e m : I. Case of K n o w n Di s t r i bu t ion of D e m a n d " , Econometrica, V o l . 20, 1952, 187-222. [20] E h r h a r d t , R . "The Power A p p r o x i m a t i o n for C o m p u t i n g (S, s) Inventory Po l i c ies" , Management Sci-ence, V o l . 25, 1979, 777-786. [21] E p p e n , G . D . and Iyer, A . V . , "Improved Fashion B u y i n g w i t h Bayes ian Upda tes" , Operations Research, V o l . 45 , ,No . 6, 1997, 805-819. [22] Ferguson, T . S., " A Bayes ian Ana lys i s of Some Nonparamet r i c P rob l ems" , The Annals of Statistics, V o l . 1, N o . 20, 1973, 209-230. [23] Gal lego , G . and M o o n , I. "The D i s t r i b u t i o n Free Newsvendor P r o b l e m : Rev iew and Extens ions" , Jour-nal of Operational Research Society, V o l . 44, N o . 8, 1993, 825-834. [24] G a r d e n , C , " M o d e l i n g Zero Inflated Coun t D a t a " , Master of Science Thesis, T h e Univers i ty of B r i t i s h C o l u m b i a , 1996. 189 [25] G a v i r n e n i , S., K a p u s c i n s k i , R . and Tayur , S., "Value of Informat ion i n Capac i t a t ed Supply Cha ins" , Working Paper, Gradua te School of Indus t r ia l A d m i n i s t r a t i o n , Carnegie M e l l o n Univers i ty , 1996. [26] Graves , S. C , R i n n o o y K a n , A . H . G . and Z i p k i n , P . H . , Logistics of Production and Inventory (chapter 1) - Handbooks in Operations Research and Management Science, Vol. 4, N o r t h - H o l l a n d , 1993. [27] Ha rpaz , G . , Lee, W . Y . , and W i n k l e r , R . L . , "Learn ing , E x p e r i m e n t a t i o n and the O p t i m a l Ou tpu t Decisions of a Compe t i t i ve F i r m " , Management Science, V o l . 28, N o . 6, June 1982, 589-603. [28] Hayes, R . H . , "Sta t i s t ica l E s t i m a t i o n Prob lems i n Inventory C o n t r o l " , Management Science, V o l . 15, 1969, 686-701. [29] Hayes, R . H . , "Sta t i s t ica l E s t i m a t i o n Mode l s for O p t i m a l Inventory C o n t r o l " , Ph.D. Thesis, Stanford Univers i ty , 1966. [30] Hjor t , N . L . , Invited Paper for the 5th International Meeting on Bayesian Statistics, Section of Bayesian Density Estimation, M a r c h 1994. [31] Iglehart , D . L . , "The D y n a m i c Inventory prob lem w i t h U n k n o w n D e m a n d D i s t r i b u t i o n " , Management Science, V o l . 10, N o . 3, 1964, 429-440. [32] Jacobs A . R . and Wagner , H . M . , "Reduc ing Inventory Systems Costs by U s i n g Robus t Demand E s t i -mators" , Management Science, V o l . 35, 1989, 771-787. [33] Jansen, J . , "Ana lys i s of Counts Involv ing R a n d o m Effects w i t h App l i ca t i ons i n E x p e r i m e n t a l B i o l o g y " , Biometrical Journal, V o l . 6, 1993, 745-757. [34] Johnson, G . and T h o m s o n , H . , " O p t i m a l i t y of M y o p i c Inventory Pol ic ies for C e r t a i n Dependent Demand Processes", Management Science, V o l . 21, 1975, 1303-1319. [35] K a p a l k a , B . A . , K a t i r c i o g l u , K . and P u t e r m a n , M . L . , "Re t a i l Inventory C o n t r o l w i t h Los t Sales, Service Cons t ra in t s , and Frac t iona l L e a d T i m e s " , Production and Operations Management, V o l . 8, N o . 4, 1999, 393-408. [36] K a r l i n , S., " D y n a m i c Inventory P o l i c y w i t h V a r y i n g Stochast ic Demands" , Management Science, V o l . 6, N o . 3, 1960, 231-258. [37] K a t i r c i o g l u , K . , "Essays i n Inventory C o n t r o l " Ph.D. Thesis, T h e Unive r s i ty of B r i t i s h C o l u m b i a , 1996. [38] L a m b e r , D . , "Zero Inflated Poisson Regression, W i t h an A p p l i c a t i o n to Defects i n Manufac tu r ing" , Technometrics, V o l . 34, 1992, 1-13. 190 [39] Lar iv ie re , M . A . and Porteus, E . L . "S ta lk ing Information: Bayes ian Inventory Management wi th Unobservable Los t Sales", Management Science, V o l . 45, N o . 6, 1999, 346-363. [40] Lawless , J . F . , Statistical Models and Methods for Lifetime Data, J o h n W i l e y & Sons, 1982. [41] Lawless , J . F . , "Negative B i n o m i a l and M i x e d Poisson Regression", The Canadian Journal of Statistics, V o l . 15, 1987, 209-225. [42] L o r d a h l , A . E . and Bookb inde r , J . H . , "Order Stat ist ics Ca l cu l a t i on , Costs and Service in an (s,Q) Inventory Sys tem" , Naval Research Logistics, V o l . 41, Feb. 1994, 81-97. [43] Lovejoy, W . S., " M y o p i c Pol ic ies for Some Inventory Mode l s w i t h U n c e r t a i n D e m a n d D i s t r i b u t i o n " , Management Science, V o l . 36, N o . 6, 1990, 724-739. [44] Lovejoy, W . S., "Subop t ima l Pol ic ies , w i t h Bounds , for Parameter A d a p t i v e Dec is ion Processes", Op-erations Research, V o l . 41, N o . 3, 1993, 583-599. [45] M c K e n n e y , J . L . and C l a r k , T . H . "Procter & Gamble : Improv ing Consumer Va lue T h r o u g h Process Redes ign" , Harvard Business Review, M a r c h 31,1995. [46] Nahmias , S., "Demand E s t i m a t i o n i n Los t Sales Inventory Systems", Naval Research Logistics, V o l . 41, 1994, 739-757. [47] P e t r u z z i , N . and D a t a , M . "Information and Inventory Recourse for a T w o - M a r k e t , Pr ice-Se t t ing Re-ta i ler" , Manufacturing & Service Operations Management, V o l . 3, N o . 3, Summer 2001, 242-263. [48] P u t e r m a n , M . L . , Markov Decision Process, J o h n W i l e y & Sons, 1994. [49] R i t chken , H . P . and Sankar, R . , "The Effect of E s t i m a t i o n R i s k i n Es t ab l i sh ing Safety Stock Levels in an Inventory M o d e l " , JORS, V o l . 35, 1984, 1091-1099. [50] Ross , S., Stochastic Processes, J o h n W i l e y & Sons, 1983. [51] Scarf, H . , "Bayes Solut ions of the Sta t i s t ica l Inventory P r o b l e m " , Annals of Mathematical Statistics, V o l . 30, 1959, 490-508. [52] Scarf, H . , "Some Remarks on Bayes Solutions to the Inventory P r o b l e m , " Naval Research Logistics Quarterly, V o l . 7, 1960, 591-596. [53] Si lver , E . A . and R a h m a n a , M . R . , "The Cost Effects of S ta t i s t ica l Sampl ing i n Selecting the Reorder Po in t i n a C o m m o n Inventory M o d e l " , JORS, V o l . 37, 1986, 705-713. 191 [54] Si lver , E . A . and R a h m a n a M . R . , "Biased Selection of the Inventory Reorder Po in t W h e n D e m a n d Parameters are S ta t i s t ica l ly E s t i m a t e d " , Engineering Costs and Product Economics, V o l . 12, 1987, 283-292. [55] Si lver . E . A . , P y k e , D . F . and Peterson, R . Inventory Management and Production Planning and schedul-ing, Third Edition, ch. 10, J o h n W i l e y & Sons, 1998. [56] Song, J . , and Z i p k i n , P . , "Inventory C o n t r o l i n a F l u c t u a t i n g D e m a n d E n v i r o n m e n t " , Operations Re-search, V o l . 41 , N o . 2, M a r c h - A p r i l 1993, 351-370. [57] S m i t h , S. A . , A g r a w a l , N . , "Management of M u l t i - i t e m R e t a i l Inventory Systems w i t h D e m a n d Subst i -t u t i on" , Operations Research, V o l . 48, Iss. 1, Jan . - Feb. 2000, 50-64. [58] Tanner , M . A . , Tools for Statistical Inference, Methods for the Exploration of Posterior Distributions and Likelihood Functions, Third Edition, ch. 6, Springer Series in Stat is t ics , 1996. [59] van den Broek , J . , " A Score Test For Zero Inflation in a Poisson D i s t r i b u t i o n " , Biometrics, 51, June 1995, 738-743. [60] van Hee, K . M . , Bayesian Control of Markov Chains, M a t h e m a t i s c h C e n t r u m , 1978. [61] Venables, W . N . , and Rip ley , B . D . , Modern Applied Statistics with S-Plus, Spr inger Series in Statistics and C o m p u t i n g , 1994. [62] Wagner , H . M . Statistical Management of Inventory Systems, W i l e y , N e w Y o r k , 1962. [63] W a l d m a n , K . H . , " O n the N u m e r i c a l Treatment of A Bayes ian Inventory M o d e l " , Bonner Mathematics Schriften, V o l . 98, 1977. [64] Wecker , W . E . , "P red ic t ing D e m a n d from Sales D a t a i n the Presence of Stockouts" , Management Science, V o l . 24, N o . 10, 1978, 1043-1054. [65] Weerahandi , S., "Decis ion Theore t ic E s t i m a t i o n of O p t i m u m Inventory Levels w i t h N a t u r a l Invariant Loss Func t ions" , Statistics and Decisions, V o l . 5, 1987. [66] West , M . , and Har r i son , J . Bayesian Forecasting and Dynamic Models, Second Edition, Springer Series i n Stat is t ics , 1997. [67] Z i p k i n , P a u l , " C r i t i c a l N u m b e r Policies For Inventory Mode l s W i t h Pe r iod ic D a t a " , Management Sci-ence, V o l . 35, N o . l , J an . 1989, 71-81. 192 CM O CO CM O CO T— o eo o CM CM LO O O co CM o CD CM CO T - LO O co a u (0 o e "c O O 3 "«3 (0 3 .a < £ S x £ i I ? ! co E a) a o O CO I— o =) Q o DC 0 . CO X o CO ID co CM" X o X o CO < O o CM O CD O T ~ T " o o tn o -^o T -o LO o co co o co co co CM CD cn CM LO CD CM co o co CM o CO CD cn CM O CD •>-o co CD cn CM o o co co LO co CM CO cn cn co CD O LO CM o CO LO CM o CM oc UJ oo ^ 00 O m O Q QC DC CL CL O CM O CO O CO O LO CM T -LO cn CM cn cn CM o co CM o CO CO CM w Q o H CO o o O O O O O O T - T - ^ T T - c o ^ r ^ f r - - o j T -T - CM ^ CM CM O O O O O O O O O T - C O - s t C O C M M - C O C M O T— CO LO CO CM O O o o o o o o o - i - c o - s T L n ^ t o c o r ^ - c o T - CM co o> T -o o O O O O O T - I - T - T - C D L O C O C D ^ - T - O D y— i— CO CO CD i— o o a —1 o o o CO 0 0 0 0 - < - O i - C M T - C O L r > - < t O ) C M C D r - ~ r - 1 - LO O y-CM O O O O T - O O O O C M I ^ C O I - O O O C O CM CM ^ T-CM 3 0 0 CO * 10 .ii 3 cn n 2 i » TJ Q. m < 0) Q a -S i a> D § o o D_ CO X O O CO 3 o O CO o I— o o o X o o g X § o o O O O O O O O T - T - C O C \ J C O T - O O O L O 1 - T - 00 T -o o o o T - T - c o T - ^ - c o o - ^ - c o c o L n - * T— CO CO CD T— • r - O O O O O - r - - s r r ^ C 0 C 0 C 0 r ^ C M C 0 C 0 T - CM CO t y-O O O O T - O O ^ - C M C M C D C D ^ - L O C O C M n N N T-o o o o o o c o T - L O T - ^ - ^ r c D ^ t c o i - -sr y~- y— CO 00 T - CO O O O O O O T - C O C O ^ O O C O C O T - L O O T - co co r- T -O O O O i - O i - C \ l - ' - l O C O O O O r - ~ C O C O i— co co co O O O O T - O O O C \ | T - O T - C \ J O O C \ J C O T - T- CO O CM W o o < a o o o o o o o i - c M T - c o T - c o « a - c o o c o r ^ T - y- CM co O O O O O O O O i - C M L O C O C O O r - - C O 1— CO t^" CM CO CO o CO o CM DC LU CQ ^ LO o 8 o ° . DC DC w Q o o o o o o o o o o co in o o ^ - i - o o o o c o o L o ^ t c o r - O L n T— co CM o o T - o o L n m - i - o i v - c o r ^ c o c D c o ^ -T - y~ y CM LO CM O O O O O C M C \ I C O i - N . O C O C D C O L O C O i— co co co O O O O O O O T - N - I - ^ O C N J O C O ^ C M i— i— co LO oo o o o o T - o o c M - ^ T - L o o ^ r o o ^ T - T - co CM LU • ^ • c o c M T - o c o o o r ^ c o L O ' ^ - c o c M T - o g y- y- y- y- ^ CO O O O ^ ^ O C O T f T f C D - i - T f C O C O T f T -T - T - C O L O C O f x . T f C M o o o o o o i - T - c o T - T - o r ^ r ^ t D o T - CM f S O) i f ) CM O C M O O T - O C O C D C O C M C O O T - CM LO r^ -T- -r- <x> CO LO T -o o o o c M o c o T - T j - T - r ^ o j i - i - C O LO C O LO LO CO f x . T f T -0 0 - > - 0 0 - > - C M r ^ 0 5 0 0 1 C M w n N O CD f x co fx. T— TJ (0 0) E « .a TJ a> is 0) (A =c I 0 3 1 g» o Jo $ X 'Z u | ? 5 Q. to a . T f < O I O. n E • co § "D O •a c IC E a> • O CO I-O ° => a g o 0. 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Q. < O "«-* CO CO CO CT CO •a c « o o o o CM a 3 •a o c o S CO O O 1 - CO LO CM CD T - CO CM o co fi CO CM ,_: o o CM T f CO Is-co co C M K O O O T f c O L O L O £ *" " CM fi CO C O O T - O T - O O L O C M £ C M s <0 LO s s CO C M T f C O T f 1— L O C O C M T f LO CD LO CM cn CD T f 00 CO CO CM CM o o C M £ <0 0 ) 0 0 o f- co T - 1 - 1 -1 - LO T f CM £ CO 00 o 1 -T— £ s CO K O O 1— £ CO C O o o £ s CO O T - CO CM CO T f LO CM T— r— C M O ) N C M C O O C M co cn co 1— co co CM O T - T f T - T -T f CD CM cn o 0 L O T f T f C M L O o CD cn a (A 0) (0 CO T f £ o CO O O T - T f 1^ co cn CM r^ - cn O T - T f T - O C D L O 1 - co 00 CO £ s CO C \ | o o £ s CO C D T f f~ L O T - L O C O C M T ~ O O CO CO N O) CM CD CM cn co o co o CM CD cn T f cri Lo" q o T t ^ T f _ cq -g co Ji^  ^ L O o 10 o o o o £ s CO 00 co co T f L O C M 1 - O 1 - L O C O C O T f C M L O C M C M £ s CO C O O O O T - C D £ CO C\j O O CM £ s CO cn LO CM N CD Ol O T - o CM T f o C O co CM cn T f cn "* A CM LO o q o CT . .CO o 00 L O o c o C O i O T f C O C M i - O ca CO CM 3 V CO h ~ Cr .<o •s c r CO CO CO LO T f -3? ca CO C O C M T - O _ o CM CtJ 1 -co .co CO -c 'sz O O O O O T -8 co CO o T - cn CM "tf O O O O O -r-T— s a tO CM O • * CM 00 CM CO CM to o o i - -tf oo "tf CM Ix. CM LO LO o o .2 ro CO cr w i !E o •o c ra CM O o o o LU CM I 2 I " < o c o Oi £ to O O O CM cn CD CM ix. CM en o o o co co co £ CM CO K •>— a CO o o rx. oo T— CO LO CM 3 CO a a to LO O a CO o o o co o CM co CM O i— i— CO CD T - co CM cn o co rx . CM CM cn •tf o co o co o o o CM to 0 ) 0 0 0 1— 2 a to CO 1— £ to o i - LO i -i - cn CM co co LO CM r x . CM o o o oo 00 T -CM CM K O to co a to o o co co co i - oo CM O O O CM LO O T - cn CM LO O O cn co LO co d cn d fx. cn 10 5 w a CO •w- O i - T- CO LO "tf £ CM p CO £ a CO O O O "tf C\j O O O "tf £ p 00 LO CM rx CM CO o CM 00 CM T - O £ a CO CO LO O O CM " -CM CO CM CD to C D " t f i r i o "tf o ix! cn A "tf LO o q o co cr ^ 2> co ^ T o O Jl^  LO o d "tf CO CM " - O n_ CM I CO "-tp CD Cr ~5" to to i n o o -i— co o co T - : CM oo £ a to \ r O O O "tf O CO CM 00 £ to CO O " -1— £ a to "— LO CM CO CO CD CM C\| O £ p to LO CD to O T- CO - i - CM CM 00 CM • t f CO CM "tf ix. d CO i n o rx' oo A CM LO CO CD d co cr o •% co x: o o Jl^  LO o o ^ o CM i CD "~ to co 6- cr .co to -c 'sz o o Appendix F Sales "Distribution" for Product 203039 and Chi -sq Statistic sales st 1 St 2 st 3 st4 st5 st 6 St7 st 8 st9 st 10 st 11 13 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 1 1 0 0 0 0 0 0 11 0 0 0 0 1 0 0 0 0 0 0 10 1 0 0 0 0 0 0 1 1 0 0 9 0 0 2 5 0 0 1 0 0 0 0 8 0 0 2 5 0 0 2 0 1 1 3 7 2 1 3 1 0 0 1 0 2 3 1 6 4 7 1 10 3 1 3 2 1 3 5 5 1 7 7 17 0 2 11 1 5 4 1 4 5 10 10 16 5 5 19 10 13 8 11 3 10 12 13 27 4 6 24 11 10 13 14 2 14 30 36 43 16 16 43 32 38 39 39 1 30 56 68 56 37 30 70 48 67 61 44 0 240 184 165 126 240 247 133 202 169 175 189 q-stat 247.5 38.3 262.3 144.0 20410 5.9 50.7 1136.6 600.6 58.0 85.2 q(12-1-2, 0.05)= •chisq(9, 0.05)=16.92 sales sr 12 st 13 st 14 st 15 st 16 st 17 st 18 st 19 St20 St 21 16 1 0 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 0 0 14 0 1 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 . 0 0 0 12 0 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 0 0 0 0 10 1 0 1 0 1 1 0 0 0 0 9 0 0 1 0 0 0 1 0 0 0 8 0 1 3 0 0 1 1 0 0 1 7 1 4 1 1 0 2 1 1 1 1 6 2 7 4 1 0 1 1 3 3 4 5 2 6 9 6 2 6 9 4 4 1 4 9 13 10 12 7 5 15 5 8 3 3 9 23 14 18 8 14 18 14 12 14 2 34 47 36 41 21 19 39 20 34 24 1 75 62 63 48 28 52 61 63 53 47 0 173 143 165 180 240 206 161 197 192 212 q-stat 10596 365.0 217.6 4.6 949.7 493.3 58.7 37.1 16.3 63.3 chisq(12-1-2, 0.05)=chisq(9, 0.05)=16.92 198 

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