"Business, Sauder School of"@en . "DSpace"@en . "UBCV"@en . "Slofstra, Anyu"@en . "2013-05-02T09:13:36Z"@en . "2013"@en . "Master of Science in Business - MScB"@en . "University of British Columbia"@en . "Red blood cells (RBCs) are the most common type of blood cells and the primary means of delivering oxygen throughout the body. They are perishable with a permitted storage time of forty-two days in Canada and in the United States. RBCs undergo a series of pathological changes while in storage. These pathological changes are known as storage lesions, and they have a negative impact on the amount of oxygen delivered to the tissue during transfusion. As a result, many studies have been conducted on the age of blood used in transfusion to patient outcomes over the past two decades. Although conflicting results have been found, most studies find that the age of blood used in transfusion plays a role in disease recurrence and mortality. Therefore, we are interested in studying hospital blood issuing policies, and in finding ones that can minimize hospital blood shortages and wastages while reducing the age of blood used in transfusion. In this thesis, we first formulate our problem as a Markov Decision Process (MDP) model, and find optimal policies that minimize blood shortages, wastages, and age of blood used in transfusion, individually. We then use simulation to compare eleven policies, including a Myopic policy derived from the MDP model. We find policies that minimize the average expected total cost of blood shortages, wastages, and age of blood used in transfusion for various shortage and wastage costs. We also perform sensitivity analyses of total costs with respect to varying threshold and cost parameters."@en . "https://circle.library.ubc.ca/rest/handle/2429/44417?expand=metadata"@en . "Optimal Issuing Policies for Hospital Blood Inventory by Anyu Slofstra B.Math, University of Waterloo, 2007 a thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in the faculty of graduate studies (Business Administration) The University Of British Columbia (Vancouver) April 2013 \u00C2\u00A9 Anyu Slofstra, 2013 Abstract Red blood cells (RBCs) are the most common type of blood cells and the primary means of delivering oxygen throughout the body. They are perishable with a permitted storage time of forty-two days in Canada and in the United States. RBCs undergo a series of pathological changes while in storage. These pathological changes are known as storage lesions, and they have a negative impact on the amount of oxygen delivered to the tissue during transfusion. As a result, many studies have been conducted on the age of blood used in transfusion to patient outcomes over the past two decades. Although conflicting results have been found, most studies find that the age of blood used in transfusion plays a role in disease recurrence and mortality. Therefore, we are interested in studying hospital blood issuing policies, and in finding ones that can minimize hospital blood shortages and wastages while reducing the age of blood used in transfusion. In this thesis, we first formulate our problem as a Markov Decision Process (MDP) model, and find optimal policies that minimize blood shortages, wastages, and age of blood used in transfusion, individually. We then use simulation to compare eleven policies, in- cluding a Myopic policy derived from the MDP model. We find policies that minimize the average expected total cost of blood shortages, wastages, and age of blood used in transfu- sion for various shortage and wastage costs. We also perform sensitivity analyses of total costs with respect to varying threshold and cost parameters. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Background Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Hospital A Blood Supply . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.2 Hospital A Blood Demand . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.5 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Models with Single Objectives . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1 Finite-Horizon MDP Formulation . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Shortages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Wastages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4 Age Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.5 Summary of Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3 Model with Combined Objective . . . . . . . . . . . . . . . . . . . . . . . . 42 3.1 Problem Formulation with Deterministic Supply and Demand . . . . . . . . 43 3.2 Curse of Dimensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 iii 3.3 A Myopic Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.4 Policies in Consideration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.5 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.6 Simulation Inputs and Outputs . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.7 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.7.1 Discussion of Each Individual Policy . . . . . . . . . . . . . . . . . . 52 3.7.2 Discussion of Optimal Policies . . . . . . . . . . . . . . . . . . . . . 66 3.7.3 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.8 Summary of Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4 Summary and Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . 88 4.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.2 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Appendix A Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 iv List of Tables Table 2.1 FIFO does not minimize wastages for every period . . . . . . . . . . . . . 30 Table 2.2 LIFO does not minimize the age factor for every period . . . . . . . . . . 35 Table 3.1 Optimal Policies for All Cost Parameters when Excess Demand is Lost . 67 Table 3.2 Optimal Policies for All Cost Parameters when Excess Demand is Backlogged 69 Table 3.3 Change in the Average Expected Total Wastage Cost with respect to One Unit Change in w when Excess Demand is Lost . . . . . . . . . . . . . . 75 Table 3.4 Change in the Average Expected Total Wastage Cost with respect to One Unit Change in w when Excess Demand is Backlogged . . . . . . . . . . 76 Table 3.5 Change in the Average Expected Total Cost with respect to One Unit Change in w when Excess Demand is Lost . . . . . . . . . . . . . . . . . 78 Table 3.6 Change in the Average Expected Total Cost with respect to One Unit Change in w when Excess Demand is Backlogged . . . . . . . . . . . . . 79 Table 3.7 Change in the Average Expected Total Shortage Cost with respect to One Unit Change in p when Excess Demand is Backlogged . . . . . . . . . . 82 Table 3.8 Excess Demand Lost vs. Excess Demand Backlogged in terms of Average Expected Total Wastages, Shortages and Age Factor when w= p= h= 1 83 Table 3.9 Excess Demand Lost vs. Excess Demand Backlogged in terms of Sensitiv- ities of Average Expected Total Wastage and Shortage Costs to w and to p, respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Table 3.10 Change in the Average Expected Total Cost with respect to One Unit Change in p when Excess Demand is Backlogged . . . . . . . . . . . . . 85 Table 3.11 Excess Demand Lost vs. Excess Demand Backlogged in terms of Sensitiv- ities of Average Expected Total Costs to w and to p . . . . . . . . . . . . 86 Table A.1 Average expected Shortage and Wastage Percentages, the average Ex- pected Total Age Factor Cost, and the average expected total costs for All Policies across all Simulation Runs for Both Excess Demand Cases . 94 v Table A.2 Average Expected Total Shortage, Wastage, and Age Factor Costs for All Policies for Each Cost Parameter when Excess Demand is Lost . . . . . . 99 Table A.3 Average Expected Total Shortage, Wastage, and Age Factor Costs for All Policies for Each Cost Parameter when Excess Demand is Backlogged . . 100 Table A.4 Percentage Change in Costs with respect to Change in Threshold r in the No. 4 Threshold Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Table A.5 Average Expected Total Wastage Costs for Each w for Selected Policies when Excess Demand is Lost . . . . . . . . . . . . . . . . . . . . . . . . . 102 Table A.6 Average Expected Total Wastage Costs for Each w for Selected Policies when Excess Demand is Backlogged . . . . . . . . . . . . . . . . . . . . . 103 Table A.7 Average Expected Total Costs for Each w for Selected Policies when Excess Demand is Lost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Table A.8 Average Expected Total Costs for Each w for Selected Policies when Excess Demand is Backlogged . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Table A.9 Average Expected Total Shortage Cost for Each p for Selected Policies when Excess Demand is Lost . . . . . . . . . . . . . . . . . . . . . . . . . 106 Table A.10 Average Expected Total Shortage Costs for Each p for Selected Policies when Excess Demand is Backlogged . . . . . . . . . . . . . . . . . . . . . 107 Table A.11 Average Expected Total Costs for Each p for Selected Policies when Excess Demand is Lost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 Table A.12 Average Expected Total Costs for Each p for Selected Policies when Excess Demand is Backlogged . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Table A.13 Change in the Average Expected Total Shortage Cost with respect to One Unit Change in p when Excess Demand is Lost . . . . . . . . . . . . . . 110 Table A.14 Change in the Average Expected Total Cost with respect to One Unit Change in p when Excess Demand is Lost . . . . . . . . . . . . . . . . . 111 vi List of Figures Figure 1.1 Hospital A Blood Supply Chart . . . . . . . . . . . . . . . . . . . . . . . 4 Figure 3.1 Average Percentages of Shortages and Wastages, and Average Expected Total Age Factor Costs for Non-Threshold Policies Excess Demand Back- logged vs. Excess Demand Lost . . . . . . . . . . . . . . . . . . . . . . . 55 Figure 3.2 Average Percentage of Shortages for the No. 1 Threshold Policy for both excess demand cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Figure 3.3 Average Percentage of Wastages for the No. 1 Threshold Policy for both excess demand cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Figure 3.4 Average Expected Total Age Factor Cost for the No. 1 Threshold Policy for both excess demand cases . . . . . . . . . . . . . . . . . . . . . . . . 56 Figure 3.5 Average Percentage of Shortages for the No. 2 Threshold Policy for both excess demand cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 Figure 3.6 Average Percentage of Wastages for the No. 2 Threshold Policy for both excess demand cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Figure 3.7 Average Expected Total Age Factor Cost for the No. 2 Threshold Policy for both excess demand cases . . . . . . . . . . . . . . . . . . . . . . . . 58 Figure 3.8 Average Percentage of Shortages for the No. 3 Threshold Policy for both excess demand cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Figure 3.9 Average Percentage of Wastages for the No. 3 Threshold Policy for both excess demand cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Figure 3.10 Average Expected Total Age Factor Cost for the No. 3 Threshold Policy for both excess demand cases . . . . . . . . . . . . . . . . . . . . . . . . 60 Figure 3.11 Average Percentage of Shortages for the No. 4 Threshold Policy for both excess demand cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Figure 3.12 Average Percentage of Wastages for the No. 4 Threshold Policy for both excess demand cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Figure 3.13 Average Expected Total Age Factor Cost for the No. 4 Threshold Policy for both excess demand cases . . . . . . . . . . . . . . . . . . . . . . . . 61 vii Figure 3.14 Average Percentage of Shortages for the No. 5 Threshold Policy for both excess demand cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Figure 3.15 Average Percentage of Wastages for the No. 5 Threshold Policy for both excess demand cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Figure 3.16 Average Expected Total Age Factor Cost for the No. 5 Threshold Policy for both excess demand cases . . . . . . . . . . . . . . . . . . . . . . . . 63 Figure 3.17 Average Percentage of Shortages for All Threshold Policies when Excess Demand Lost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Figure 3.18 Average Percentage of Wastages for All Threshold Policies when Excess Demand Lost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Figure 3.19 Average Expected Total Age Factor Cost for All Threshold Polices when Excess Demand Lost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Figure 3.20 Average Expected Total Cost for All Threshold Polices when Excess De- mand Lost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Figure 3.21 % Change in the Wastage Percentage with change in Threshold r of the No. 4 Threshold Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Figure 3.22 % Change in the Shortage Percentage with change in Threshold r of the No. 4 Threshold Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Figure 3.23 % Change in the Average Expected Total Age Factor Cost with change in Threshold r of the No. 4 Threshold Policy . . . . . . . . . . . . . . . 72 Figure 3.24 % Change in the Average Expected Total Cost with change in Threshold r of the No. 4 Threshold Policy . . . . . . . . . . . . . . . . . . . . . . . 73 Figure 3.25 Average Change in Average Expected Total Cost with One Unit Change in w, Excecss Demand Lost vs. Excess Demand Backlogged . . . . . . . 80 Figure 3.26 Average Change in Average Expected Total Cost with One Unit Change in p, Excecss Demand Lost vs. Excess Demand Backlogged . . . . . . . 86 viii Acknowledgments First of all, I would like to thank my supervisors Professors Tim Huh and Steven Shechter for their advice, encouragement, and support in my study, research, and personal decisions for the past two years. I also appreciate their invaluable comments and suggestions for improving my thesis. This thesis could not be completed without them. I would also like to thank Professor Hao Zhang for being my examining committee member, for his thorough reading of my thesis, and for his invaluable suggestions. I also appreciate Professor Maurice Queyranne for all of his comments and suggestions for im- proving this thesis. I benefited greatly from the help of several office mates. I thank Antoine Saur\u00C3\u00A9, for lending me his books for so long and for spending time answering all my questions; Greg Werker, for answering my questions; Xin Geng, for lending me his course notes and helping me with math problems; Sha Liao, for lending me her textbook, and for helping me figure out an inventory model on a Saturday morning in her PJs; Reza Skandari, for being my only companion into the program and for many course and non-course related discussions. I would like to thank everyone mentioned above, as well as Yi Duo, Charles Martineau, Alireza Sabouri and Xiaonan Sun for making my time at UBC enjoyable. I am indebted to Elaine Cho for all her help and assistance in many administrative issues. She has made my life at UBC so much easier. I would like to use this opportunity to express my sincere gratitude to my undergraduate supervisor and mentor, Professor Steve Drekic from the University of Waterloo, for his nurture, support, and invaluable advice in both professional and personal aspects of my life ix over the years. His scholarship, generosity, and humility have inspired me not only to do math research, but also to be a better person. Last but not least, I thank my friends and family, in particular my husband William for his constant love, support, and encouragement. I am very fortunate to have you in my life. x To William xi Chapter 1 Introduction 1.1 Background Information Red blood cells (RBCs) are the most common type of blood cells and the primary means of delivering oxygen throughout the body. As the name suggests, they also give blood its red colour [3]. In Canada [7] and in the United States [1], RBCs have a permitted maximum storage time (or shelf life) of forty-two (42) days if refrigerated according to guideline. However, according to Zubair [36], while blood is in storage its ATP, pH, and 2,3- Diphosphoglycerate levels decrease. These pathological changes are collectively known as \u00E2\u0080\u009Cstorage lesions\u00E2\u0080\u009D. Storages lesions may have negative impact on the amount of oxygen delivered to tissue during transfusion. As a result, researchers have conducted various studies to find the impact of age of blood used in transfusion on key performance factors such as recurrence of diseases and patient survival rates. Results are inconclusive as conflicting results have been found in different studies. For example, Spinella et al. [31] studied 202 traumatic injury patients, and divided them into two groups: one group received RBCs less than or equal to 27 days old, and the other group received RBCs greater than or equal to 28 days old. They found that patients in the second group have higher mortality incidences from multi-organ failure compared to patients in the first group. Weinberg et al. [35] studied 1,624 patients with mild and moderate injuries over a period of 7.5 years, and found that patients who received blood older than 2 weeks are \u00E2\u0080\u009Cindependently associated 1 with mortality, renal failure, and pneumonia\u00E2\u0080\u009D. In contrast, Edna and Bjerkeset [11] studied 336 patients who had undergone colorectal cancer surgery with a median follow-up time of 5.8 years, and found that age of blood had no effect on local recurrences and distant metastases. Mynster and Nielsen [18] studied 740 patients who had undergone elective resection for primary colorectal cancer with a median follow-up time of 6.8 years. They found that patients who exclusively received older (\u0015 21 days) blood had higher survival years (3.7 vs. 2.5) than patients who exclusively received younger (<21 days) blood during the operation. They also observed disease recurrence rates for 532 patients who received curative resection. After comparing the two transfused groups with patients who did not receive blood transfusion, they found that patients who received older blood had the same rate of disease recurrence as patients who did not receive blood transfusion, whereas patients who received younger blood had 1.5 times the rate of disease recurrence as patients who did not receive blood transfusion. Wang et al. [33], reviewed 1,348 articles that compared patients who had received younger and older blood. After elimination, they performed a meta-analysis on 21 studies, involving 409,966 patients, that compared survival rates with various ages of blood used in transfusion. They concluded that there was a significantly increased (16% with 95% confidence interval) risk of mortality if older blood was used for transfusion. However, their results were questioned by Warkentin and Elkelboom [34]. In short, although many studies have been conducted, the impact of age of blood used in transfusion on patients\u00E2\u0080\u0099 mortality rates and disease recurrence rates is still inconclusive. However, most studies do suggest that the age of blood used in transfusion plays a role in disease recurrence and mortality, and that it has different impact on different types of patients. Therefore, it is important to study different types of blood issuing policies, and gain insights on how they affect wastages, shortages, and other aspects of hospital blood inventory. 1.2 Problem Description The problem background and data in this thesis are provided by a physician and his staff members at a large hospital in North America. For anonymity, we will refer to this hos- 2 pital as hospital A. The current blood issuing policy at hospital A is to issue the oldest blood first (known as the FIFO policy in literature) unless specified by physicians for their elective surgeries. However, due to research results that have shown the potential harm of transfusing older blood, the doctor is concerned about the current blood issuing practice at hospital A. Hospital A maintains a low percentage of blood wastage due to outdating because of the current blood issuing policy and high RBC demand. However, the doctor and his colleagues are interested in finding another issuing policy which could balance blood wastage, blood shortage, and the age of blood used in transfusion. The main purpose of this thesis is to evaluate a number of blood issuing policies, and find the policies that can balance all three factors. To better understand the problem, we will describe the blood supply and demand processes at hospital A below. 1.2.1 Hospital A Blood Supply There is only one blood supplier in the region where hospital A is located. Each hospital in the region has a target blood inventory level that the blood supplier agrees to fulfill. The blood supplier routinely delivers blood to hospitals during weekdays, and on weekends for emergency circumstances. However, when the blood supplier has excess blood supply, instead of keeping the blood in its storage, it delivers the excess blood to hospital A and one other nearby large hospital, regardless of whether they need it. Hospital A shares its blood supply with two smaller hospitals in the region, meaning that the blood supplier delivers blood to hospital A, and hospital A delivers blood to the two smaller hospitals. When the two smaller hospitals have excess blood supply, they ship the excess blood back to hospital A. Below is a flow chart of the blood supply process at hospital A and the two smaller hospitals. Because of the stochastic nature of excess supply from the blood supplier and the two smaller hospitals, the supply process of hospital A is also stochastic. 3 Figure 1.1: Hospital A Blood Supply Chart 1.2.2 Hospital A Blood Demand Besides pre-scheduled elective surgeries and other procedures, most of the daily blood de- mand at hospital A is not known in advance because of unforeseen emergency events. As a result, the demand process at hospital A is also stochastic. 1.3 Literature Review Our goal in this thesis is to find optimal blood issuing policies. Therefore, research on inventory depletion policies is related to our problem. Since blood is a perishable product, research on inventory management of perishable products is also relevant to our problem. The study of inventory depletion policies began with the paper by Greenwood [14] in 1955. His paper led to much theoretical development on this topic from the late 1950s to the 1970s ([10], [17], [4], [12], [13], [26], [25], [5], [6], [19], [2]). These papers focus on the study of one basic problem and some variations of the basic problem. In the basic problem there are n items in a stockpile. The ith item in the stockpile is of age Si. The field life of an item is the age of the item upon release into the field from the stockpile, and is represented by a function L(S). The goal is to find the issuing order of items in the stockpile that maximizes the total field life. In these papers, items in the stockpile are released one at a time, and they are only issued if the current item in the field is not useful anymore. New items are not added into the stockpile once the process has started. The main objective of these papers is to find sufficient conditions of the field life function L(S) under which the FIFO (issuing the oldest item first) and/or the LIFO (issuing the youngest item first) policies are optimal. Problems considered in these papers are very simplified versions of the 4 problem in this thesis, and so their results cannot be generalized and directly apply to our problem. During the above mentioned period (late 1950s to the 1970s), Pierskalla and Roach [29] study a blood issuing problem that is very similar to our problem in this thesis. We both are interested in finding optimal issuing policies that minimize total blood shortages and total blood wastages for more than one decision period. We both also assume that the supply of blood can be of any age, and can be added into the stock at every decision period. We both also allow issuing more than one unit of blood at any decision period, and let the issuing quantity in a period depend on the demand and inventory level of the period. However, our work differs in the following ways: (1) demand and supply are stochastic in this thesis, whereas in their paper demand and supply are deterministic. (2) Besides investigating the optimal issuing policies for minimizing shortages and wastages, we are also interested in finding the optimal policy that minimizes the total age of blood used in transfusion, as well as in finding optimal policies that minimize the weighted sum of shortages, wastages, and age of blood used in transfusion. The latter two objectives are not considered in [29]. Instead, Pierskalla and Roach are interested in finding the optimal policy that maximizes the total utility of the system. Differences between this objective and our objective of finding the optimal policy that minimizes the total age of blood used in transfusion are discussed in detail in Section 2.4 of this thesis. (3) Pierskalla and Roach divide demand into various age categories so that ordering demand from a specific age category is possible, and demand can only be satisfied from blood of the requested age category. However, we do not allow that in this thesis. Demand can be satisfied with blood of any age. (4) Pierskalla and Roach also include an age category that contains all blood units that have passed the maximum shelf life, and include them in their inventory counts in their proofs. However, we discard all blood units that have passed their maximum shelf life from our inventory counts. Tetteh [32] also considers a blood issuing problem that is very similar to our problem in his PhD thesis. In his thesis, Tetteh uses simulation to compare the FIFO, LIFO, and a mixed FIFO and LIFO (different from our threshold policy) policies in terms of wastages, 5 shortages, and average age of blood used in distribution. Tetteh does not develop any analytical results for wastages, shortages, and average age of blood used, whereas we develop analytical results for each of the three in Chapter 2 of this thesis. He compares the FIFO, LIFO, and a mixed FIFO and LIFO policies using a heuristic method, and implements it using Microsoft Excel for blood of ages 3, 4, and 5. He also formulates his problem as a linear programming (LP) model with rolling horizon N, and simulates his model using LINGO for N = 1;3;5;7;9, and for blood of ages 3, 4, and 5. He then compares simulation results from the heuristic method to results from the LP method for each N. We formulate our problem by a Markov Decision Process (MDP) model and perform simulation in Matlab to compare eleven policies for 200 decision days for blood of ages 1 to 42. Overall, our simulation and analysis are much more involved. Another relevant work is Parlar et al. [24]. In their paper, Parlar et al. compare FIFO and LIFO issuing policies in terms of maximizing long-term net average profit. In this thesis, we also compare FIFO and LIFO issuing policies. However, we also compare nine other policies in terms of minimizing costs. Parlar et al. also assume that the arrival times of demand and supply of perishable inventories follow a Poisson distribution, and hence the time between successive demands (and supplies) is exponentially distributed. We assume that daily demand and supply of blood follow some general stochastic distributions in our analysis, and use distributions derived from the actual hospital A demand and supply data in our simulation. We also explore a much greater number of issuing policies in this thesis, whereas Parlar et al. only consider two policies, FIFO and LIFO. Parlar et al. conclude that FIFO dominates LIFO, except for when the holding cost is high, or when the purchase cost is low, in which cases LIFO dominates FIFO. These results are due to the fact that items usually stay longer on the shelf under the FIFO policy. In this thesis, we show that the expected total/average age of blood used in transfusion is maximized under the FIFO policy. This is a similar result to their results mentioned above. Much of the research on inventory management of perishable products focuses on finding the optimal ordering policies ([27], [28], [20], [21], [22], [30]). In 1982, Nahmias [23] did a comprehensive review on ordering policies for perishable inventories. In 2007, Deniz [9] did 6 an up-to-date comprehensive review on the perishable supply chain management literature in his PhD thesis. Most of the papers mentioned in his thesis focus on finding optimal ordering policies. While studies on finding optimal ordering policies do not directly apply to our problem of finding optimal issuing policies, most studies on finding optimal ordering policies assume an underlying issuing policy. Therefore, we are interested in knowing which underlying issuing policy/policies these studies use, and whether there are any relevant comparison results when they assume more than one issuing policy. We find that most ordering policy papers assume a FIFO issuing policy. A small number of papers also assume the LIFO policy, or the LIFO policy and a threshold policy, and they compare results for different issuing policies. These papers are listed below. Haijema et al. [15] use a Markov Dynamic Programming model with simulation approach to find the optimal/near-optimal ordering policies to minimize a combination of costs such as production costs, outdating costs, and shortage costs for a number of issuing policies (FIFO, LIFO, and a combination of FIFO and LIFO). Although they show some results for a few different issuing policies, they do not spend much effort on comparing and analyzing these results. They conclude that the FIFO policy should be used for their \u00E2\u0080\u009Cany\u00E2\u0080\u009D demand category to reduce outdating. We also show in this thesis that the FIFO policy minimizes wastages/outdates. Their work is similar to ours in the sense that in Chapter 3 of this thesis, we also use a Markov Dynamic Programming model with simulation approach to find optimal policies that minimize a combination of costs. Among perishable supply chain management papers, their work is the closet to our work in Chapter 3 in terms of model and methodology used. Our work differs in the following ways: (1) Haijema et al. assume that fresh blood supply is of age 1, whereas we use the actual age of blood supply. (2) They allow two demand categories, \u00E2\u0080\u009Cyoung\u00E2\u0080\u009D and \u00E2\u0080\u009Cany age\u00E2\u0080\u009D. We only allow the \u00E2\u0080\u009Cany age\u00E2\u0080\u009D demand category. (3) They also incorporate day of the week into their model, and the supply and demand distributions are day-of-the-week dependent, making their dynamic programming model periodic. However, we use historical data to obtain daily demand and supply distributions that are not periodic. (4) While finding the optimal ordering policy is the main focus of their paper, this thesis is focused on finding the optimal issuing policy. 7 Cohen and Pekelman [8] are interested in finding the optimal ordering policy that max- imizes the after tax profit under FIFO and LIFO tax valuation schemes. Lee et al. [16] compared the FIFO, the LIFO, and a heuristic issuing policies in terms of the optimal ordering quantity and a revenue function from their modified Economic Manufacturing Quantity (EMQ) model in their unpublished technical report. Models and objectives of these two papers are very different from ours, and so their results cannot be applied to our problem. 1.4 Contributions Our contributions are: \u00E2\u0080\u00A2 We find optimal issuing policies for minimizing/maxmizing the total shortage, wastage and age factor individually with the assumption that both demand and supply are stochastic, and the product is perishable with a fixed shelf-life. \u00E2\u0080\u00A2 We find the optimal issuing policy for minimizing/maximizing the average age of blood used in transfusion when excess demand is lost. \u00E2\u0080\u00A2 We explore three other age penalty functions, and find optimal policies that mini- mize/maximize the total age factor for each age penalty function. \u00E2\u0080\u00A2 We develop a Markov Decision Process (MDP) model to find optimal policies that minimize the combined average total expected shortage, wastage and age factor costs. \u00E2\u0080\u00A2 We compare eleven issuing policies for our MDP model using simulation with real demand and supply data from hospital A, and find the best policy for each set of cost parameters. Our analysis of simulation results should provide some insights for hospital decision makers into blood issuing. 1.5 Structure The structure of this thesis is as follows: in Chapter 2, we first introduce our problem formulation (Section 2.1). We then consider objectives of miminizing the expected total 8 blood shortages (Section 2.2), wastages (Section 2.3) and the age of blood used in transfusion (Section 2.4) individually. We determine the best and the worst policies for each objective analytically. In Chapter 3, we consider minimizing the weighted sum of the three factors. We first formulate our problem using an integer linear programming model for deterministic demand and supply (Section 3.1). We then discuss the curses of dimensionality of our finite- horizon MDP model that prevent us from calculating the optimal policy (Section 3.2). We introduce a myopic policy in Section 3.3 for our MDP formulation. In Section 3.4 and Section 3.5, we describe policies and data used in simulation. In Section 3.6, we describe input and output variables for our simulation. In Section 3.7, we display and analyze simulation results. Lastly, in Chapter 4, we summarize our findings, discuss limitations, and suggest future research directions. 9 Chapter 2 Models with Single Objectives The main objective of this thesis is to find policies that minimize the expected total costs of shortages, wastages, and total age of blood used in transfusion. In this chapter, we begin by introducing details of our problem, including assumptions, sequence of events, and notations we use. We then formulate our problem as a finite-horizon Markov Decision Process model (MDP) in Section 2.1. Afterwards, we consider policies that minimize and maximize the expected total shortages (Section 2.2), wastages (Section 2.3), and age of blood used in transfusion (Section 2.4), individually, for excess (or unsatisfied) demand lost and backlogged cases. To minimize the expected total age of blood used in transfusion, we penalize each unit of blood issued by its age, by using the age as a multiplier. The penalty for each decision period is the sum of the product of the amount of age i blood issued and age i for all blood ages. Therefore the older the blood issued, the greater the penalty. We define age factor as the total penalty for each decision period, and we would like to find the policy that minimizes the expected total age factor for all decision periods. In Section 2.4, we also find the optimal policy that minimizes the expected average age of blood used in transfusion for all decision periods, as well as optimal policies that minimize/maximize the expected total age factor for three other age penalty functions. We begin by making the following assumptions: 10 \u00E2\u0080\u00A2 Daily supply and demand of blood are random. \u00E2\u0080\u00A2 Since blood demand and supply are expressed in terms of non-negative integer units, they can only take on non-negative integer values. \u00E2\u0080\u00A2 The maximum shelf-life of blood is M days. Blood of age greater than M days are thrown out and are not counted towards inventory. \u00E2\u0080\u00A2 Reservation of blood is not allowed. If there is any outstanding demand, and that the current inventory is not yet depleted, current inventory must be used to satisfy the outstanding demand. \u00E2\u0080\u00A2 Demand is non-age-specific. That is, any unit of blood in inventory can be used to satisfy a unit demand of blood. \u00E2\u0080\u00A2 If there is inventory left at the current period after satisfying demand, inventory of age less than M will be carried on into the next period, and their age will be one period older than their age in the current period. Remaining inventory of age M in the current period will be thrown out and recorded as wastage. Based on assumptions above, we define a feasible policy as follows: Definition 2.0.1. A blood issuing policy is feasible if it does not reserve any blood for the next period. It releases blood whenever there is outstanding demand in the current period until inventory has been depleted. We also assume the following sequence of events for a decision maker in each decision period. At the start of each period, a decision maker counts the amount of inventory for each blood age. She then receives the supply of blood for the period. Afterwards, demand for the period occurs, and she determines the issuing amount for each age of blood to satisfy the demand. She also counts the amount of stockouts if any. At the end of the period, she updates the inventory level for each blood age. She records the ending inventory level of blood of age M as wastage and throw them out if there is any. 11 Stockout occurs when there is more demand than the total updated inventory level in a decision period, and the difference between demand and the total updated inventory level is called excess demand. We deal with excess demand in two ways in this thesis: excess demand lost and excess demand backlogged. In the excess demand lost case, we discard all excess demand in the current period, and do not fulfill them in any future periods. In the excess demand backlogged case, we carry excess demand into the next period, and count them towards demand in the next period. We use the following notations to express our problem mathematically. \u00E2\u0080\u00A2 i: Index blood age, a subscript. \u00E2\u0080\u00A2 t: Index time period, a subscript. \u00E2\u0080\u00A2 T: The end of the decision horizon. \u00E2\u0080\u00A2 P: Index policy P, a superscript. \u00E2\u0080\u00A2 P: The set of all feasible blood issuing policies based on Definition 2.0.1. \u00E2\u0080\u00A2 M: The maximum age of blood that can be used in transfusion. \u00E2\u0080\u00A2 xPi;t : The beginning inventory of blood of age i in decision epoch t for policy P. We assume that all policies have the same beginning inventory in period 1, and that xPi;1 \u0015 0 for all i= 1;2; : : : ;M. \u00E2\u0080\u00A2 Qi;t : The supply of age i blood in period t, Qi;t \u0015 0 for all i= 1;2; : : : ;M and all t \u0015 1. The supply of blood is the same for all policies. \u00E2\u0080\u00A2 Dt : The demand of blood for period t, Dt \u0015 0 for all t \u0015 1. It is also the same for all policies. \u00E2\u0080\u00A2 DPt : Blood demand of period t under policy P. When unsatisfied demand is lost, DPt = Dt for all P 2 P. When unsatisfied demand is backlogged, DPt = Dt+SPt\u00001, where SPt is defined below. 12 \u00E2\u0080\u00A2 yPi;t : The total inventory level of blood of age i in period t for policy P after blood supply has been realized. This satisfies yPi;t = xPi;t +Qi;t . \u00E2\u0080\u00A2 zPi;t : The amount of age i blood released in period t to satisfy demand according to policy P. Note that we are not allowed to release more blood than the amount demanded, i.e., \u00C3\u00A5Mi=1 zPi;t \u0014 DPt . We also cannot release more blood than the amount of blood available, i.e., 0\u0014 zi;t \u0014 yi;t , for i= 1;2; : : : ;M. \u00E2\u0080\u00A2 IPi;t : The remaining inventory of age i blood in period t for policy P, i.e., Ii;t = yi;t\u0000zi;t \u0015 0. Note that xPi+1;t+1 = I P i;t ; for all i= 1;2; : : : ;M\u00001 ;and t \u0015 1 ; (2.1) xP1;t = 0 for all t \u0015 2 : (2.2) Also note that IPM;t is the wastage amount in period t under policy P. \u00E2\u0080\u00A2 SPt : Shortage amount for period t under policy P: SPt = maxf0;DPt \u0000\u00C3\u00A5Mi=1 yPi;tg. This is the amount of lost demand in the excess demand lost case, and the amount of backlogged demand in the backlogging case. SP0 = 0 for all P 2 P. 2.1 Finite-Horizon MDP Formulation The MDP formulation for our problem is as follows: \u00E2\u0080\u00A2 Decision epoch: f1;2; : : : ;Tg, T < \u00C2\u00A5. \u00E2\u0080\u00A2 States: the state of decision epoch t is represented by the updated inventory level of period t for each blood age, as well as demand of period t. Let Xt represent state of period t, and X be the state space, we have that Xt = (y1;t ;y2;t ; : : : ;yM;t ;Dt) 2 X \u001A (N[f0g)M+1. \u00E2\u0080\u00A2 Actions: at each decision epoch t, the decision maker needs to determine the amount of blood to be released for each blood age to satisfy demand after supply and demand have been realized. Let at represent the action of period t, and A be the action set, we have that at = (z1;t ;z2;t ; : : : ;zM;t) 2 AXt \u001A A\u001A (N[f0g)M. 13 \u00E2\u0080\u00A2 Transition probabilities: Let fs and gd be probability distributions for supply and demand, respectively. Then pt( jjXt ;at) = \u00C3\u00A5 (Qt+1;Dt+1): j=(Q1;t+1;y1;t\u0000z1;t+Q2;t+1;:::;yM\u00001;t\u0000zM\u00001;t+QM;t+1;Dt+1) fs(Qt+1)gd(Dt+1) : \u00E2\u0080\u00A2 Costs: Let h, w, p be cost (weight) parameters for the age factor, wastages, and shortages, respectively. We define cost function for decision epoch t as Ct(Xt ;at) := h \u0001 M \u00C3\u00A5 i=1 zi;t \u0001 i+w \u0001 IM;t + p \u0001St (2.3) where IM;t = yM;t \u0000 zM;t (2.4) 0\u0014 zi;t \u0014 yi;t ; i= 1; : : : ;M (2.5) St = 8><>: maxf0;Dt \u0000\u00C3\u00A5 M i=1 yi;tg; when excess demand is lost maxf0;Dt +St\u00001\u0000\u00C3\u00A5Mi=1 yi;tg; when excess demand is backlogged (2.6) M \u00C3\u00A5 i=1 zi;t \u0014 8><>: Dt ; when excess demand is lostDt +St\u00001; when excess demand is backlogged. (2.7) S0 = 0 : \u00E2\u0080\u00A2 Optimality Equation: Vt(Xt) = min at2AXt ( C(Xt)+\u00C3\u00A5 j2X pt( jjXt ;at)Vt+1( j) ) (2.8) where VT+1 = 0. Our objective is to find policy P 2 P such that the expected total cost over the decision making horizon T is minimized. Let VPT (X1) represent the expected total cost under some 14 policy P 2 P with starting state X1 in decision epoch 1. It is defined by VPT (X1) := EPX1 ( T \u00C3\u00A5 t=1 Ct(XPt ;a P t ) ) : (2.9) We would like to find P\u0003 2 P such that VP \u0003 T (X1) =minP2P \u0008 VPT (X1) : (2.10) 2.2 Shortages In this section, we show that for both excess demand cases, the FIFO policy minimizes the total shortages for all decision periods, and that the LIFO policy maximizes the total shortages for all decision periods, for each sample path. As a result, the FIFO policy minimizes the expected total shortages, and that the LIFO policy maximizes the expected total shortages for all decision periods. To show that the FIFO policy minimizes the total shortages for all decision periods for each sample path for both excess demand cases, we first show that it maximizes the total beginning inventory level of all blood ages for each decision period for each sample path for both demand cases, and then use the result to show that it minimizes shortages for each decision period for each sample path for both excess demand cases. Note that same shortage results are proven by Pierskalla and Roach [29] for both excess demand cases. However, Pierskalla and Roach assume that demand and supply are deterministic in proving their result, whereas we assume demand and supply are stochastic. Lemma 2.2.1. For each period t \u0015 1, and for blood age n= 1; : : : ;M, \u00C3\u00A5ni=1 xFIFOi;t \u0015\u00C3\u00A5ni=1 xPi;t , for all P2 P when excess demand is lost. In other words, for each sample path, the FIFO policy maximizes the total beginning inventory level of blood ages between 1 and n, n = 1; : : : ;M, for each period t \u0015 1, when excess demand is lost. Sketch of proof: To prove this lemma and the lemma for excess demand backlogged case below, we show that the total ending inventory of blood of ages 1 to n, n= 1; : : : ;M, is the 15 highest under the FIFO policy in all periods in both excess demand scenarios. We show this by demonstrating that the FIFO policy has the highest total ending inventory level for blood of ages 1 to n, n= 1; : : : ;M, in the first period, and subsquently has the highest total ending inventory level for for blood of ages 1 to n for all future periods. We use the fact that the maximum ending inventory any age of blood can have in a period is the amount of its updated inventory, and it is obtained when no blood is released from the age. Since we always release the oldest blood first under the FIFO policy, and all policies have the same demand and supply in all periods when excess demand is lost, we obtain the same or more total ending inventory for blood of ages 1 to n, n = 1; : : : ;M, in a period under the FIFO policy, given that we have the same or more total beginning inventory for blood of ages 1 to n, n= 1; : : : ;M under the FIFO policy than all other feasible policies. When excess demand is backlogged, we show that we get the least amount of backlogged demand under the FIFO policy for each period. We then use this result, and other results used in proving the excess demand lost case to show that the FIFO policy yields the highest total ending inventory for blood of age 1 to n, n= 1; : : : ;M, among all feasible policies in each decision period. Proof. We prove this lemma by induction. First, t = 1: We have that the beginning inventory level for each age of blood is the same for all feasible policies. Therefore, for all feasible policies P other than the FIFO policy, we get n \u00C3\u00A5 i=1 xFIFOi;1 \u0015 n \u00C3\u00A5 i=1 xPi;1 : Second, t = 2: By Equation 2.2 and Equation 2.1, we have that the beginning inventory level of blood of age 1 for period 2, xP1;2, is 0, and the begining inventory level of blood of each other age, xPi;2 for i = 2; : : : ;M, is the same as IPi\u00001;1, the ending inventory level of its corresponding age in period 1, for all feasible policies. Moreover, all policies have the same beginning inventory level and the same amount of supply in period 1. Then, for all feasible 16 policies P, and blood ages i= 1; : : : ;M, IPi;1 = x P i;1+Qi;1\u0000 zPi;1 = yi;1\u0000 zPi;1 : (2.11) Case 1: The blood demand in period 1 is greater than or equal to the total updated inventory level in period 1, i.e., , D1 \u0015\u00C3\u00A5Mi=1 yi;1. In this case, all inventory in period 1 needs to be used to satisfy demand D1. Hence, zPi;1 = yi;1, for each age i = 1; : : : ;M, and for all feasible policies P. So the ending inventory for each age of blood in period 1 is 0 for all feasible policies. As a result, the beginning inventory for each age of blood in period 2 is also 0 for all feasible policies, and we get that n \u00C3\u00A5 i=1 xFIFOi;2 = 0\u0015 0= n \u00C3\u00A5 i=1 xPi;2; for n= 1; : : : ;M; and all P 2 P : Case 2: The blood demand in period 1 is less than the total updated inventory level in period 1, i.e., D1 < \u00C3\u00A5Mi=1 yi;1. In this case, not all inventory needs to be used for demand D1. Note that by Equation 2.11, max n \u00C3\u00A5ni=1 IPi;1 o = \u00C3\u00A5ni=1 yi;1 when zPi;1 = 0 for i= 1; : : : ;n, and n= 1; : : : ;M. We consider the following subcases. Subcase 2.1: The blood demand in period 1 is less than the updated inventory level of blood of age M in period 1, i.e., D1 < yM;1. Therefore, the blood supply of age M in period 1 is sufficient to satisfy D1. Then according to policy FIFO, we would release D1 units of blood for age M, and release 0 units of blood for all other ages. Then IFIFOM;1 = yM;1\u0000D1, and IFIFOi;1 = yi;1 for i= 1; : : : ;M\u00001. Then for any feasible policy P other than the FIFO policy, and ages n= 2; : : : ;M, n \u00C3\u00A5 i=1 xFIFOi;2 = 0+ n \u00C3\u00A5 i=2 xFIFOi;2 = 0+ n\u00001 \u00C3\u00A5 j=1 IFIFOj;1 = n\u00001 \u00C3\u00A5 j=1 y j;1 =max ( n\u00001 \u00C3\u00A5 j=1 IPj;1 ) \u0015 n\u00001 \u00C3\u00A5 j=1 IPj;1 = 0+ n \u00C3\u00A5 i=2 xPi;2 = n \u00C3\u00A5 i=1 xPi;2 : Since the beginning inventory of blood of age 1 is 0 for all periods by Equation 2.2, we have 17 that \u00C3\u00A5ni=1 xFIFOi;2 \u0015 \u00C3\u00A5ni=1 xPi;2 for all ages n= 1; : : : ;M. Subcase 2.2: Suppose that for some age m, 1><>>: \u00C3\u00A5Mi=1 yi;1\u0000D1 = \u00C3\u00A5Mi=1 IPi;1 \u0015 \u00C3\u00A5ni=1 IPi;1 for m\u00001\u0014 n\u0014M \u00C3\u00A5ni=1 IFIFOi;1 = \u00C3\u00A5 n i=1 yi;1 =max n \u00C3\u00A5ni=1 IPi;1 o \u0015 \u00C3\u00A5ni=1 IPi;1 for 1\u0014 n< m\u00001 : Therefore, \u00C3\u00A5ni=1 IFIFOi;1 \u0015 \u00C3\u00A5ni=1 IPi;1 for all n= 1; : : : ;M. As a result, for ages n= 2; : : : ;M, n \u00C3\u00A5 i=1 xFIFOi;2 = 0+ n\u00001 \u00C3\u00A5 j=1 IFIFOj;1 \u0015 0+ n\u00001 \u00C3\u00A5 j=1 IPj;1 = n \u00C3\u00A5 i=1 xPi;2 : Since the beginning inventory of age 1 blood is 0 for all periods by Equation 2.2, we have that for period 2, and any age n= 1; : : : ;M, \u00C3\u00A5ni=1 xFIFOi;2 \u0015 \u00C3\u00A5ni=1 xPi;2. Induction hypothesis: suppose that for some period t, t \u0015 2, \u00C3\u00A5ni=1 xFIFOi;t \u0015 \u00C3\u00A5ni=1 xPi;t , for ages n = 1; : : : ;M, and all feasible polices P 2 P. We will show that \u00C3\u00A5ni=1 xFIFOi;t+1 \u0015 \u00C3\u00A5ni=1 xPi;t+1, for ages n= 1; : : : ;M, and all feasible policies P 2 P. Note: given that for all feasible policies P2P, \u00C3\u00A5ni=1 xFIFOi;t \u0015\u00C3\u00A5ni=1 xPi;t , we get that \u00C3\u00A5ni=1(xFIFOi;t + Qi;t)\u0015\u00C3\u00A5ni=1(xPi;t+Qi;t), or \u00C3\u00A5ni=1 yFIFOi;t \u0015\u00C3\u00A5ni=1 yPi;t for ages n= 1; : : : ;M. Moreover, max \u0008 \u00C3\u00A5ni=1 IPi;t = \u00C3\u00A5ni=1 yPi;t for n= 1; : : : ;M. Now for any feasible policy P other than the FIFO policy, we con- sider the following three cases. 18 Case 1: The blood demand in period t is greater than or equal to the total updated inventory in period t under the FIFO policy, i.e., Dt \u0015 \u00C3\u00A5Mi=1 yFIFOi;t . Then the blood demand in period t is also greater than or equal to the total updated inventory in period t under policy P. As a result, all updated inventory in period t is released to satisfy demand Dt for all feasible policies. Hence, there is 0 inventory remaining at the end of period t for all feasible policies. Therefore, using similar arguments as before, we get that n \u00C3\u00A5 i=1 xFIFOi;t+1 = 0\u0015 0= n \u00C3\u00A5 i=1 xPi;t+1; n= 1; : : : ;M : Case 2: The blood demand in period t is less than the total updated blood inventory in period t under the FIFO policy, but greater than or equal to the total updated blood inventory in period t under policy P, i.e., \u00C3\u00A5Mi=1 yPi;t \u0014 Dt < \u00C3\u00A5Mi=1 yFIFOi;t . Then similar to the case above, there is 0 inventory remaining at the end of period t under policy P. However, since demand is less than the total updated inventory under the FIFO policy in period t, not all inventory needs to be released to satisfy Dt under FIFO. Therefore, for some age i 2 f1; : : : ;Mg, we have that IFIFOi;t > 0. Hence, n \u00C3\u00A5 i=1 IFIFOi;t \u0015 0= n \u00C3\u00A5 i=1 IPi;t ; n= 1; : : : ;M : Now, since the beginning inventory of age 1 blood in period t+1 is 0 by Equation 2.2, and the ending inventory of age M blood in period t expires in period t + 1, and thus is not counted in period t+1, we have that \u00C3\u00A5ni=1 xFIFOi;t+1 \u0015 0= \u00C3\u00A5ni=1 xPi;t+1, for any age n= 1; : : : ;M. Case 3: The blood demand in period t is less than the total updated blood inventory in period t under policy P, i.e., Dt < \u00C3\u00A5Mi=1 yPi;t . Then Dt < \u00C3\u00A5Mi=1 yFIFOi;t as well. Similar to the period 2 case, we consider the following subcases. Subcase 3.1: The demand in period t is less than the updated inventory level of age M blood, i.e., Dt < yFIFOM;t . Then according to FIFO, Dt units of blood would be released for age M blood, and no blood would be released for blood of ages 1 to M\u0000 1. As a result, there would be yFIFOM;t \u0000Dt units of blood left for age M, and yFIFOi;t units of blood 19 left for each age i = 1; : : : ;M\u00001 at the end of period t. We then have that for age n =M, \u00C3\u00A5Mi=1 IFIFOi;t = \u00C3\u00A5 M i=1 y FIFO i;t \u0000Dt \u0015 \u00C3\u00A5Mi=1 yPi;t \u0000Dt = \u00C3\u00A5Mi=1 IPi;t , and for age n = 1; : : : ;M\u0000 1, we have that \u00C3\u00A5ni=1 IFIFOi;t = \u00C3\u00A5ni=1 yFIFOi;t \u0015 \u00C3\u00A5ni=1 yPi;t \u0015 \u00C3\u00A5ni=1 IPi;t . Carrying on into to the next period, we then obtain that for age n= 2; : : : ;M, n \u00C3\u00A5 i=1 xFIFOi;t+1 = 0+ n\u00001 \u00C3\u00A5 j=1 IFIFOj;t \u0015 0+ n\u00001 \u00C3\u00A5 j=1 IPj;t = n \u00C3\u00A5 i=1 xPi;t+1 Now again, since the inventory level for age 1 blood is 0 for all periods by Equation 2.2, we have that \u00C3\u00A5ni=1 xFIFOi;t+1 \u0015 \u00C3\u00A5ni=1 xPi;t+1 for all n= 1; : : : ;M. Subcase 3.2: Suppose that for some age m, 1< m\u0014M, \u00C3\u00A5Mi=m yFIFOi;t \u0014 Dt < \u00C3\u00A5Mi=m\u00001 yFIFOi;t . Then under the FIFO policy, we would release yFIFOi;t units of blood for each age i=m; : : : ;M, release Dt\u0000\u00C3\u00A5Mi=m yFIFOi;t units of blood for age m\u00001, and release no blood for ages 1 to m\u00002. As a result, there would be no inventory remaining for blood of ages m to M. There would be \u00C3\u00A5Mi=m\u00001 yFIFOi;t \u0000Dt units remaining for age m\u0000 1 blood, and yFIFOi;t units remaining for each age i= 1; : : : ;m\u00002. Since \u00C3\u00A5ni=1 yFIFOi;t \u0015 \u00C3\u00A5ni=1 yPi;t for n= 1; : : : ;M, we have that the total remaining inventory for blood of ages 1 to n at the end of period t under FIFO n \u00C3\u00A5 i=1 IFIFOi;t = 8>><>>: \u00C3\u00A5ni=1 yFIFOi;t \u0015 \u00C3\u00A5ni=1 yPi;t =max \u0008 \u00C3\u00A5ni=1 IPi;t \t\u0015 \u00C3\u00A5ni=1 IPi;t ; n= 1; : : : ;m\u00002 \u00C3\u00A5Mi=1 yFIFOi;t \u0000Dt \u0015 \u00C3\u00A5Mi=1 yPi;t \u0000Dt = \u00C3\u00A5Mi=1 IPi;t \u0015 \u00C3\u00A5ni=1 IPi;t ; n= m\u00001; : : : ;M : Therefore, for all n= 1; : : : ;M, we have \u00C3\u00A5ni=1 IFIFOi;t \u0015 \u00C3\u00A5ni=1 IPi;t . Then using similar arguments as before, we get that \u00C3\u00A5ni=1 xFIFOi;t+1 \u0015 \u00C3\u00A5ni=1 xPi;t+1 for n= 1; : : : ;M. So to conclude, when excess demand is lost, \u00C3\u00A5ni=1 xFIFOi;t+1 \u0015 \u00C3\u00A5ni=1 xPi;t+1 for all n = 1; : : : ;M. By induction, the inequality is true for all t \u0015 1. Lemma 2.2.2. For each period t \u0015 1, and for blood age n= 1; : : : ;M, \u00C3\u00A5ni=1 xFIFOi;t \u0015 \u00C3\u00A5ni=1 xPi;t , for all P 2 P when excess demand is backlogged. In other words, for each sample path, the FIFO policy maximizes the total beginning inventory level of blood ages between 1 and n, n= 1; : : : ;M for each period t \u0015 1, when excess demand is backlogged. Proof. The proof for this lemma is similar to the proof for Lemma 2.2.1 ablove. However, 20 in this proof we also show that the shortage amount under the FIFO policy is less than or equal to the shortage amount under any other feasible policy P for each period t, i.e., SFIFOt \u0014 SPt for all P 2 P in each t. This shortage result for the excess demand lost case is shown in Theorem 2.2.3 below. Note that: 1. The demand for period t under any feasible policy P is the sum of the new demand of period t and the backlogged demand (or the shortage amount), from the previous period t\u00001, i.e., DPt =Dt+SPt\u00001. Also, as introduced in the notation section, we assume that SP0 = 0 for all feasible policies P 2 P. 2. The shortage amount of period t under any feasible policy P is 0 if there is enough in- ventory to satisfy demand, and is DPt \u0000\u00C3\u00A5Mi=1 yPi;t if otherwise, so SPt =max \u0008 0;DPt \u0000\u00C3\u00A5Mi=1 yPi;t . 3. If the amount of backlogged demand is the same for two different feasible policies in period t\u0000 1, then the total demand is the same in period t for the two policies. In this situation, proof of the excess demand backlogged case is the same as proof of the excess demand lost case. Let policy P be any feasible policy other than the FIFO policy. We consider the follow- ing: First, period t = 1. In this period, the beginning inventory level for each blood age is the same for all policies. Then as proved in the excess demand lost case, we have that n \u00C3\u00A5 i=1 xFIFOi;1 \u0015 n \u00C3\u00A5 i=1 xPi;1 . Also, since the blood demand and supply of period 1 are the same for all policies, we have that SFIFO1 =max ( 0;D1+SFIFO0 \u0000 M \u00C3\u00A5 i=1 yFIFOi;1 ) =max ( 0;D1\u0000 M \u00C3\u00A5 i=1 yi;1 ) = SP1 21 Second, period t = 2. Since SFIFO1 = SP1 , the demand of period 2 for each policy P, DP2 , is the same for all P 2 P. Therefore, as proved in the excess demand lost case, we have that \u00C3\u00A5Mi=1 xFIFOi;2 \u0015 \u00C3\u00A5Mi=1 xPi;2. Hence, \u00C3\u00A5Mi=1 yFIFOi;2 \u0015 \u00C3\u00A5Mi=1 yPi;2 since blood supply is the same for all policies. Using this relation, we have that SFIFO2 =max ( 0;D2+SFIFO1 \u0000 M \u00C3\u00A5 i=1 yFIFOi;2 ) =max ( 0;D2+SP1 \u0000 M \u00C3\u00A5 i=1 yFIFOi;2 ) \u0014max ( 0;D2+SP1 \u0000 M \u00C3\u00A5 i=1 yPi;2 ) = SP2 Now, suppose that for some period t, t \u0015 2, SFIFOt\u00001 \u0014 SPt\u00001, and \u00C3\u00A5ni=1 xFIFOi;t \u0015 \u00C3\u00A5ni=1 xPi;t , for n= 1; : : : ;M. We will show that SFIFOt \u0014 SPt , and \u00C3\u00A5ni=1 xFIFOi;t+1 \u0015 \u00C3\u00A5ni=1 xPi;t+1 for n= 1; : : : ;M. By the induction hypothesis, we immediately obtain \u00C3\u00A5ni=1 yFIFOi;t \u0015 \u00C3\u00A5ni=1 yPi;t , and SFIFOt =max ( 0;Dt +SFIFOt\u00001 \u0000 n \u00C3\u00A5 i=1 yFIFOi;t ) \u0014max ( 0;Dt +SPt\u00001\u0000 n \u00C3\u00A5 i=1 yPi;t ) = SPt Now we will show that \u00C3\u00A5ni=1 xFIFOi;t+1 \u0015 \u00C3\u00A5ni=1 xPi;t+1. Case 1: The backlogged demand under the FIFO policy is the same as the backlogged demand under policy P for period t\u0000 1, i.e., SFIFOt\u00001 = SPt\u00001. Then DFIFOt = DPt . Therefore, as shown in the excess demand lost case, we have that \u00C3\u00A5ni=1 xFIFOi;t+1 \u0015 \u00C3\u00A5ni=1 xPi;t+1, for all ages n= 1; : : : ;M. Case 2: The backlogged demand under the FIFO policy is less than the backlogged demand under policy P for period t\u00001, i.e., SFIFOt\u00001 < SPt\u00001. Then DFIFOt DFIFOt \u0015 \u00C3\u00A5Mi=1 yFIFOi;t \u0015 \u00C3\u00A5Mi=1 yPi;t . Therefore, under both policy FIFO and policy P, all inventory needs to be released to satisfy demand in period t, and so there is no remaining inventory at the end of period t. Hence the beginning inventory level of period t+1 for all blood ages is 0 under both policies. We have that \u00C3\u00A5ni=1 xFIFOi;t+1 = 0\u0015 0= \u00C3\u00A5ni=1 xPi;t+1 for all n= 1; : : : ;M. Subcase 2.2: The demand of period t under FIFO is less than the total updated inventory level under FIFO in period t, but greater than or equal to the total updated inventory level under policy P in period t, i.e., \u00C3\u00A5Mi=1 yPi;t \u0014 DFIFOt < \u00C3\u00A5Mi=1 yFIFOi;t . Since DPt > DFIFOt , we get that DPt >\u00C3\u00A5Mi=1 yPi;t . Therefore, all inventory needs to be released to satisfy demand in period t under policy P, and so there is no inventory remaining at the end of period t under policy P. However, since DFIFOt < \u00C3\u00A5Mi=1 yFIFOi;t , not all inventory needs to be used to satisfy demand. Therefore, there is inventory remaining for blood of some age i in the age set f1; : : : ;Mg at the end of period t under FIFO. Hence, for any age n = 1; : : : ;M, we have that \u00C3\u00A5ni=1 IFIFOi;t \u0015 0 = \u00C3\u00A5ni=1 IPi;t for all n = 1; : : : ;M. Then using similar arguments as before, we have that \u00C3\u00A5ni=1 xFIFOi;t+1 \u0015 \u00C3\u00A5ni=1 xPi;t+1 for all n= 1; : : : ;M. Subcase 2.3: The demand of period t under FIFO is less than the total updated inventory level of period t under policy P, i.e., DFIFOt < \u00C3\u00A5Mi=1 yPi;t . Then we have DFIFOt < \u00C3\u00A5Mi=1 yFIFOi;t as well. Similar to the excess demand lost case, we consider the following scenarios: First, suppose the demand of period t under FIFO is less than the updated inventory level of age M blood, i.e., DFIFOt < yFIFOM;t . Then according to the FIFO policy, we release DFIFOt units of age M blood, and so there is yFIFOM;t \u0000DFIFOt units of age M blood remaining. For all other blood ages i = 1; : : : ;M\u0000 1, we do not release any blood, and so there are yFIFOi;t units of blood remaining at the end of period t for each age i blood. Then for all 23 n= 1; : : : ;M\u00001, we have that n \u00C3\u00A5 i=1 IFIFOi;t = n \u00C3\u00A5 i=1 yFIFOi;t \u0015 n \u00C3\u00A5 i=1 yPi;t \u0015 n \u00C3\u00A5 i=1 IPi;t : Therefore, using similar arguments as before, we get that \u00C3\u00A5ni=1 xFIFOi;t+1 \u0015 \u00C3\u00A5ni=1 xPi;t+1 for all n= 1; : : : ;M. We would like to note that when n=M, if DPt < \u00C3\u00A5Mi=1 yPi;t , we have that M \u00C3\u00A5 i=1 IFIFOi;t = M \u00C3\u00A5 i=1 yFIFOi;t \u0000DFIFOt \u0015 M \u00C3\u00A5 i=1 yPi;t \u0000DFIFOt \u0015 M \u00C3\u00A5 i=1 yPi;t \u0000DPt = M \u00C3\u00A5 i=1 IPi;t : If DPt \u0015 \u00C3\u00A5Mi=1 yPi;t , there is no inventory remaining at the end of period t under policy P, so the total remaining inventory of period t under FIFO, \u00C3\u00A5Mi=1 IFIFOi;t > 0=\u00C3\u00A5Mi=1 IPi;t . As a result, for all n= 1; : : : ;M, we that that \u00C3\u00A5ni=1 IFIFOi;t \u0015 \u00C3\u00A5ni=1 IPi;t . Second, suppose for some blood age m, 1 < m \u0014M, \u00C3\u00A5Mi=m yFIFOi;t \u0014 DFIFOt < \u00C3\u00A5Mi=m\u00001 yFIFOi;t . Then similar to the excess demand lost case, we do not release any blood of ages 1 to m\u00002, we release DFIFOt \u0000\u00C3\u00A5Mi=m yFIFOi;t units of age m\u0000 1 blood, and release \u00C3\u00A5Mi=m\u00001 yFIFOi;t units of blood for ages i = m; : : : ;M. Hence, there is yFIFOi;t units of blood remaining for each age i = 1; : : : ;m\u0000 2, and \u00C3\u00A5Mi=m\u00001 yFIFOi;t \u0000DFIFOt units of blood remaining for age m\u0000 1, and no blood remaining for ages m to M. Then for n= 1; : : : ;m\u00002, n \u00C3\u00A5 i=1 IFIFOi;t = n \u00C3\u00A5 i=1 yFIFOi;t \u0015 n \u00C3\u00A5 i=1 yPi;t =max ( n \u00C3\u00A5 i=1 IPi;t ) \u0015 n \u00C3\u00A5 i=1 IPi;t Now for n\u0015 m\u00001, if DPt \u0015 \u00C3\u00A5Mi=1 yPi;t , then there would be no inventory remaining at the end of period t for policy P. As a result, n \u00C3\u00A5 i=1 IFIFOi;t = M \u00C3\u00A5 i=1 yFIFOi;t \u0000DFIFOt > 0= n \u00C3\u00A5 i=1 IPi;t : If DPt < \u00C3\u00A5Mi=1 yPi;t , we have that n \u00C3\u00A5 i=1 IFIFOi;t = M \u00C3\u00A5 i=1 yFIFOi;t \u0000DFIFOt \u0015 M \u00C3\u00A5 i=1 yPi;t \u0000DPt = M \u00C3\u00A5 i=1 IPi;t \u0015 n \u00C3\u00A5 i=1 IPi;t : 24 Then \u00C3\u00A5ni=1 IFIFOi;t \u0015 \u00C3\u00A5ni=1 IPi;t for all n = 1; : : : ;M. Using similar arguments as before, we have that \u00C3\u00A5ni=1 xFIFOi;t+1 \u0015 \u00C3\u00A5ni=1 xPi;t+1. In conclusion, when excess demand is backlogged, \u00C3\u00A5ni=1 xFIFOi;t+1 \u0015\u00C3\u00A5ni=1 xPi;t+1 and SFIFOt \u0014 SPt for all n= 1; : : :M. By induction, the two inequalities hold for all t \u0015 1. Note that by setting n=M in the theorem above, we get that the FIFO policy maximizes the total beginning inventory level for all blood ages for each decision period. We will use this result to show that the FIFO policy minimizes shortages for each decision period below. Theorem 2.2.3. For all periods t \u0015 1, and for all policies P 2 P, SFIFOt \u0014 SPt when excess demand is lost and when excess demand is backlogged. In other words, for each sample path, FIFO minimizes the shortage amount among all feasible policies for each decision period. Proof. We have shown in the proof of Lemma 2.2.2 that the statement is true when excess demand is backlogged. We will show that it is also true when excess demand is lost. When excess demand is lost, the demand of period t under any feasible policy P is the same as the demand occuring in period t, so DPt = DFIFOt = Dt for all t and for all feasible policies P 2 P. Therefore, for all periods t \u0015 1, SFIFOt =max ( 0;Dt \u0000 M \u00C3\u00A5 i=1 yFIFOi;t ) \u0014max ( 0;Dt \u0000 M \u00C3\u00A5 i=1 yPi;t ) by Theorem 2.2.1 = SPt Now, since for each sample path the FIFO policy minimizes shortages for every decision period, it minimizes the total shortages for all decision periods. Using similar steps as above, we show that for each sample path, the LIFO policy maximizes total shortages below. 25 Lemma 2.2.4. For each period t \u0015 1, and age n = 1; : : : ;M, \u00C3\u00A5ni=1 xLIFOi;t \u0014 \u00C3\u00A5ni=1 xPi;t , for all policies P 2 P when excess demand is lost. In other words, for each sample path, the LIFO policy minimizes the total beginning inventory level for blood ages between 1 and n, n= 1; : : : ;M for each period t \u0015 1 when excess demand is lost. Proof. The proof of this lemma follows the same steps as steps in the proof of Lemma 2.2.1. To proof this lemma, we show that \u00C3\u00A5ni=1 ILIFOi;t \u0014\u00C3\u00A5ni=1 IPi;t for all n= 1; : : : ;M, and all t \u0015 1. We also use the fact that when there is enough inventory to satisfy demand in a period t, the smallest possible total ending inventory for blood of ages 1 to n is the total updated inventory level for blood of ages 1 to n minus the demand in period t, i.e., min \u0008 \u00C3\u00A5ni=1 IPi;t =\u00C3\u00A5ni=1 yPi;t\u0000Dt in the proof. The rest of the details in proving this lemma are similar to those in the proof of Lemma 2.2.1, and so are not provided here. Lemma 2.2.5. For each period t \u0015 1, and age n = 1; : : : ;M, \u00C3\u00A5ni=1 xLIFOi;t \u0014 \u00C3\u00A5ni=1 xPi;t , for all policies P 2 P when excess demand is backlogged. In other words, for each sample path, the LIFO policy minimizes the total beginning inventory level for blood ages between 1 and n, n= 1; : : : ;M for each period t \u0015 1 when excess demand is backlogged. Proof. The proof of this lemma follows the same steps as steps in the proof of Lemma 2.2.2. Similar to the proof of Lemma 2.2.2, we also show that shortages under the LIFO policy for each period t, SLIFOt , are greater than or equal to shortages under policy P for each period t, SPt , for all policies P2 P. Moreover, shortage result for the excess demand lost case are shown in Theorem 2.2.6 below. The main differences between the proof of this lemma and the proof of Lemma 2.2.2 are the same as the main differences between the proof of Lemma 2.2.4 above and the proof of Lemma 2.2.1. These differences are listed in the proof of Lemma 2.2.4 above. The rest of the details in proving this lemma are similar to those in the proof of Lemma 2.2.2, and so are also not provided here. Theorem 2.2.6. For all periods t \u0015 1, and all policies P 2 P, SLIFOt \u0015 SPt when excess demand is lost and when excess demand is backlogged. In other words, LIFO has the most shortages among all feasible policies for each decision period under each sample path. 26 Proof. The proof of this theorem for the excess demand backlogged case is included in the proof of Lemma 2.2.5 above. When excess demand is lost, we have that DPt = DLIFOt = Dt for all periods t and for all policies P 2 P. Therefore, SLIFOt =max ( 0;Dt \u0000 M \u00C3\u00A5 i=1 yLIFOi;t ) \u0015max ( 0;Dt \u0000 M \u00C3\u00A5 i=1 yPi;t ) by Lemma 2.2.4 = SPt 2.3 Wastages In this section, we also use a sample-path approach to show that the FIFO policy minimizes, and the LIFO policy maximizes the expected total wastages for all decision periods for both excess demand cases. To show that the FIFO policy minimizes the total wastages for all decision periods under each sample path for both excess demand cases, we first show that it maximizes the cumulative amount of blood transfused for both demand cases under each sample path, and then use the result to show that it minimizes cumulative blood wastages for each sample path for both excess demand cases. As a result, it minimizes the total wastages for all decision periods under each sample path for both excess demand cases, and thus it minimizes the expected total wastages for all decision periods for both excess demand cases. Note that the same result is also proven by Pierskalla and Roach [29] for both excess demand cases for deterministic supply and demand. Lemma 2.3.1. For each period t, t \u0015 1, for all policies P2 P, \u00C3\u00A5tn=1\u00C3\u00A5Mi=1 zFIFOi;n \u0015\u00C3\u00A5tn=1\u00C3\u00A5Mi=1 zPi;n when excess demand is lost and when excess demand is backlogged. In other words, FIFO maximizes the cumulative amount of blood transfused amount among all feasible policies for each sample path for both excess demand cases. Proof. First, we show that for all feasible policies P 2 P, the total amount of blood released 27 plus shortages in period t equals to the total demand in period t, i.e., \u00C3\u00A5Mi=1 zPi;t+SPt =Dt+SPt\u00001, for both excess demand cases. For both excess demand cases, if the total demand in a period t under policy P, Dt+SPt\u00001, is greater than or equal to the total updated inventory \u00C3\u00A5Mi=1 yPi;t , then all inventory needs to be released to satisfy demand, so \u00C3\u00A5Mi=1 zPi;t = \u00C3\u00A5Mi=1 yPi;t , and shortages in period t under policy P, SPt , equals to Dt + SPt\u00001\u0000\u00C3\u00A5Mi=1 yPi;t . Therefore, \u00C3\u00A5Mi=1 zPi;t + SPt = Dt + SPt\u00001. Note that when excess demand is lost, SPt\u00001 = 0 for all t. If the total demand in period t under policy P, Dt+SPt\u00001, is less than the total updated inventory level, \u00C3\u00A5Mi=1 yPi;t , then the amount of blood we need to release in period t equals to the total demand, and no shortage occurs. Therefore, \u00C3\u00A5Mi=1 zPi;t +SPt = Dt +SPt\u00001. Since the equality is true for each period, it is also true for cumulative periods of 1 to t for t \u0015 1 such that t \u00C3\u00A5 n=1 M \u00C3\u00A5 i=1 zPi;n+ t \u00C3\u00A5 n=1 SPn = t \u00C3\u00A5 n=1 Dn+ t \u00C3\u00A5 n=1 SPn\u00001 : Therefore, t \u00C3\u00A5 n=1 M \u00C3\u00A5 i=1 zPi;n = t \u00C3\u00A5 n=1 Dn+ t \u00C3\u00A5 n=1 SPn\u00001\u0000 t \u00C3\u00A5 n=1 SPn = 8>><>>: \u00C3\u00A5tn=1Dn\u0000\u00C3\u00A5tn=1 SPn when excess demand is lost \u00C3\u00A5tn=1Dn\u0000SPt when excess demand is backlogged. But when excess demand is backlogged, SPt =\u00C3\u00A5tn=1 SPn . Therefore, \u00C3\u00A5tn=1\u00C3\u00A5Mi=1 zPi;n=\u00C3\u00A5tn=1Dn\u0000 \u00C3\u00A5tn=1 SPn for both excess demand cases. Then by Theorem 2.2.3, we have that for all policies P 2 P, t \u00C3\u00A5 n=1 M \u00C3\u00A5 i=1 zFIFOi;n = t \u00C3\u00A5 n=1 Dn\u0000 t \u00C3\u00A5 n=1 SFIFOn \u0015 t \u00C3\u00A5 n=1 Dn\u0000 t \u00C3\u00A5 n=1 SPn = t \u00C3\u00A5 n=1 M \u00C3\u00A5 i=1 zPi;n : Theorem 2.3.2. For each period t \u0015 1, all policies P2P, \u00C3\u00A5tn=1 IPM;n\u0015\u00C3\u00A5tn=1 IFIFOM;n when excess demand is lost and when excess demand is backlogged. In other words, for each sample path, 28 FIFO minimizes the cumulative wastage amount among all feasible policies for both excess demand cases. Proof. First, we show that for both excess demand cases, for all feasible policies P 2 P, and for each period t \u0015 1, the total cumulative blood supply equals to the total cumulative amount of blood released plus cumulative blood wastage, and plus the total ending inventory of period t, i.e., t \u00C3\u00A5 n=1 M \u00C3\u00A5 i=1 Qi;n = t \u00C3\u00A5 n=1 M \u00C3\u00A5 i=1 zPi;n+ t \u00C3\u00A5 n=1 IPM;n+ M \u00C3\u00A5 i=1 IPi;t : We know that for both excess demand cases, for all feasible policies P 2 P, and for each period t \u0015 1, the total beginning inventory plus the total supply minus the total amount of blood released is the total ending inventory for the period, i.e., \u00C3\u00A5Mi=1(xPi;t+Qi;t\u0000zPi;t) =\u00C3\u00A5Mi=1 IPi;t . Therefore, M \u00C3\u00A5 i=1 Qi;t = M \u00C3\u00A5 i=1 (zPi;t + I P i;t \u0000 xPi;t) : Then by Equations 2.1 and 2.2, we get that t \u00C3\u00A5 n=1 M \u00C3\u00A5 i=1 Qi;t = t \u00C3\u00A5 n=1 M \u00C3\u00A5 i=1 zPi;n+ t \u00C3\u00A5 n=1 M \u00C3\u00A5 i=1 IPi;n\u0000 t \u00C3\u00A5 n=1 M \u00C3\u00A5 i=1 xPi;n = t \u00C3\u00A5 n=1 M \u00C3\u00A5 i=1 zPi;n+ t \u00C3\u00A5 n=1 M \u00C3\u00A5 i=1 IPi;n\u0000 t \u00C3\u00A5 n=2 M \u00C3\u00A5 i=2 Ii\u00001;n\u00001\u0000 M \u00C3\u00A5 i=1 xi;1 = t \u00C3\u00A5 n=1 M \u00C3\u00A5 i=1 zPi;n+ t \u00C3\u00A5 n=1 IPM;n+ M \u00C3\u00A5 i=1 IPi;t \u0000 M \u00C3\u00A5 i=1 xi;1 ; Note that in the proof of Lemma 2.2.1 and Lemma 2.2.2, we showed that for each t \u0015 1, M \u00C3\u00A5 i=1 IFIFOi;t \u0015 M \u00C3\u00A5 i=1 IPi;t (2.12) for both excess demand lost and backlogged cases. Since the blood supply, Qi;t , and the beginning inventory level in period 1, xi;1, for each age i= 1; : : : ;M, are the same for all feasible policies, we have that t \u00C3\u00A5 n=1 M \u00C3\u00A5 i=1 zPi;n+ t \u00C3\u00A5 n=1 IPM;n+ M \u00C3\u00A5 i=1 IPi;t \u0000 M \u00C3\u00A5 i=1 xi;1 = t \u00C3\u00A5 n=1 M \u00C3\u00A5 i=1 zFIFOi;n + t \u00C3\u00A5 n=1 IFIFOM;n + M \u00C3\u00A5 i=1 IFIFOi;t \u0000 M \u00C3\u00A5 i=1 xi;1 : 29 Therefore, t \u00C3\u00A5 n=1 IPM;n\u0000 t \u00C3\u00A5 n=1 IFIFOM;n = t \u00C3\u00A5 n=1 M \u00C3\u00A5 i=1 zFIFOi;n \u0000 t \u00C3\u00A5 n=1 M \u00C3\u00A5 i=1 zPi;n ! + M \u00C3\u00A5 i=1 IFIFOi;t \u0000 M \u00C3\u00A5 i=1 IPi;t ! + M \u00C3\u00A5 i=1 xi;1\u0000 M \u00C3\u00A5 i=1 xi;1 ! \u00150 by Lemma 2.3.1 and Equation 2.12. Hence, the cumulative wastages of policy P, \u00C3\u00A5tn=1 IPM;n, is greater than or equal to the cumulative wastages of the FIFO policy, \u00C3\u00A5tn=1 IFIFOM;n , for all periods t \u0015 1, and for all feasible policies P 2 P for both excess demand cases. Since FIFO minimizes cumulative wastages for each sample path, it minimizes the total wastages for all decision periods for each sample path. Note that the FIFO policy minimizes the cumulative wastages of periods 1 to t, for t \u0015 1. But it does not minimize wastages of each period. An example of this is provided below: Table 2.1: FIFO does not minimize wastages for every period FIFO Other Age 41 42 41 42 xi;1 0 0 0 0 Qi;1 20 10 20 10 D1 = 1 zi;1 0 1 1 0 Ii;1 20 9 19 10 xi;2 0 20 0 19 Qi;2 0 0 0 0 D2 = 10 zi;2 0 10 0 10 Ii;2 0 10 0 9 xi;3 0 0 0 0 As we see in the above example, the FIFO policy wasted 10 units of blood in period 2, 30 whereas the other policy wasted 9. But both policies wasted 19 units of blood in the first two periods. Now, we want to show that the LIFO policy maximizes cumulative wastages for each sample path. Theorem 2.3.3. For each period t \u0015 1, and all policies P 2 P, \u00C3\u00A5tn=1 IPM;n \u0014 \u00C3\u00A5tn=1 ILIFOM;n when excess demand is lost and when excess demand is backlogged. In other words, LIFO maxi- mizes cumulative wastages among all feasible policies for each sample path for both excess demand cases. Proof. First, by the proof of Lemma 2.3.1 and Theorem 2.2.6, we get that for all policies P 2 P, t \u00C3\u00A5 n=1 M \u00C3\u00A5 i=1 zLIFOi;n = t \u00C3\u00A5 n=1 Dn\u0000 t \u00C3\u00A5 n=1 SLIFOn \u0014 t \u00C3\u00A5 n=1 Dn\u0000 t \u00C3\u00A5 n=1 SPn = t \u00C3\u00A5 n=1 M \u00C3\u00A5 i=1 zPi;n : Then using the same steps as steps used in the proof of Theorem 2.3.2, we get that t \u00C3\u00A5 n=1 ILIFOM;n \u0000 t \u00C3\u00A5 n=1 IPM;n = t \u00C3\u00A5 n=1 M \u00C3\u00A5 i=1 zPi;n\u0000 t \u00C3\u00A5 n=1 M \u00C3\u00A5 i=1 zLIFOi;n ! + M \u00C3\u00A5 i=1 IPi;t \u0000 M \u00C3\u00A5 i=1 ILIFOi;t ! + M \u00C3\u00A5 i=1 xi;1\u0000 M \u00C3\u00A5 i=1 xi;1 ! \u00150 : Hence, the cumulative wastages under policy LIFO, \u00C3\u00A5tn=1 ILIFOM;n , is greater than or equal to the cumulative wastages under policy P, \u00C3\u00A5tn=1 IPM;n, for all periods t \u0015 1, and for all policies P 2 P for both excess demand cases. 2.4 Age Factor Note that our objective in this section is not to minimize the expected total amount of blood transfused. It is to minimize the expected total age penalty for all units of blood released to satisfy demand. It indirectly minimizes the expected total age of blood used in transfusion since the more young blood we use, the lower the total penalty. Also note that shortage occurs when demand is more than the sum of the total beginning inventory and 31 total supply in a period, and it occurs before a decision maker determines the amount of blood to be released for each age of blood. Therefore, the amount of shortages does not contribute to the age factor in each period. Intuitively speaking, we would like to issue the youngest blood possible in order to minimize the age factor. Hence the LIFO policy should be the best policy to minimize the objective function. We will show in the following theorem that the LIFO policy indeed minimizes the cumulative age factor when excess demand is lost for each sample path. Our intuition also tells us that the FIFO policy would receive the highest penalty and thus maximizes the cumulative age factor. We will show in Theorem 2.4.2 that the FIFO policy indeed maximizes the cumulative age factor when excess demand is lost for each sample path. In this section, we also show that the LIFO policy minimizes the average age of blood used in transfusion in Theorem 2.4.5 for each sample path when excess demand is lost. We also consider three alternative age penalty functions: increasing linear, increasing concave, and decreasing convex functions. We find that with an increasing linear, or an increasing concave age penalty function, the LIFO policy minimizes (Theorem 2.4.6, Theorem 2.4.8) and the FIFO policy maximizes (Theorem 2.4.7, Theorem 2.4.9) the cumulative age factor for each sample path when excess demand is lost. But with a decreasing convex age penalty function, the LIFO policy maximizes (Theorem 2.4.10), and the FIFO policy minimizes (Theorem 2.4.11) the cumulative age factor for each sample path when excess demand is lost. Theorem 2.4.1. For each period t \u0015 1, \u00C3\u00A5tn=1\u00C3\u00A5Mi=1 zLIFOi;n \u0001 i \u0014 \u00C3\u00A5tn=1\u00C3\u00A5Mi=1 zPi;n \u0001 i for all policies P2 P when excess demand is lost. In other words, the LIFO policy miminizes the cumulative age factor when excess demand is lost under each sample path. The proof of this theorem follows the same approach as the proof of Theorem 1 in Pierskalla and Roach [29] because of the similarity of our objective functions. However, the objective of Theorem 1 in their paper is to maximize the total utility of the system, defined as the value of all past demand filled plus the value of the current inventory in stock, with 32 younger blood having higher value. However, they assume that a unit of younger blood can be used to satisfy a unit demand of older blood, and when that occurs the value for filling the unit demand of older blood with the unit of younger blood is the same as the value for filling the demand with a unit of older blood. Therefore, their theorem 1 does not reduce the age of blood used in transfusion. Our objective is to minimize the penalty of all demand filled, with older blood having higher penalty. Hence, we can reduce the penalty by using as much younger blood as possible. Moreover, inventory in Pierskalla and Roach [29] do not expire, they only deteriorate. As a result, they do not retire any expired blood units from inventory counts in their proof. In our proof, we retire the expired blood units. Pierskalla and Roach prove that the FIFO policy maximizes the total system utility when excess demand is backlogged for deterministic demand and supply. We will show that when excess demand is lost, the LIFO policy minimizes the total age factor, and the FIFO policy maximizes the total age factor under each sample path for stochastic demand and supply. Proof. Let RPt := \u00C3\u00A5tn=1\u00C3\u00A5Mi=1 zPi;n \u0001 i, for all periods t \u0015 1, and feasible policies P 2 P. We will show that for any policy P other than the LIFO policy through period n, we can construct another policy P0 which is one unit closer to LIFO through period n, and such that the cumulative age factor amount, RP0t , under policy P0, is not greater than the cumulative age factor amount, RPt , under policy P. For simplicity, we use Rt to represent RPt , and R0t to represent RP0t . Consider a policy P 2 P under which unit q of age m, 1 < m \u0014M, is used to fill a unit of demand in period n, for 1 \u0014 n \u0014 t, and there is a unit q0 of age m0, m0 < m, that can be used to fill the same unit demand. We will show that there is a feasible policy P0 which fills the unit demand with q0, and its cumulative age factor R0n is less than or equal to the cumulative age factor of policy P, Rn, for each n = 1; : : : ; t. We say that R0 is less than or equal to R if and only if R0n \u0014 Rn for all n= 1; : : : ; t. Similar to Pierskalla and Roach\u00E2\u0080\u0099s proof, we define \u00E2\u0080\u00A2 j := period in which policy P issues item q0. Note that j \u0015 n+1. \u00E2\u0080\u00A2 l := period in which policy P0 issues item q, or item q expires. Note that l \u0015 n+1. 33 Note that item q0 is younger than item q, so item q expires before item q0. Case 1: j < l, i.e., P issues q0 and there is no intervening stockouts under P for which P0 issues q. For all periods u < n, we have that R0u = Ru since they had been issuing under the same policy P. For all periods u, n \u0014 u < j, we have that Ru\u0000R0u = m\u0000m0 \u0015 0. For period j, let P0 issues q to satisfy the unit demand for which P issues q0. Then R0j = R j. As a result, each policy has satisfied the same demands through j. Moreover, they have the same inventory remaining. Therefore, R0u = Ru for all u> j. In conclusion, R0 \u0014 R when j < l. Case 2: j\u0015 l. Again, for all periods u, u< n, we have that R0u = Ru, and for all periods u, n\u0014 u< l, Ru\u0000R0u = m\u0000m0 \u0015 0. Note that \u00C3\u00A5Mi=1 y0i;l = \u00C3\u00A5Mi=1 yi;l since both policies have issued the same amount of blood up until period l. Suppose item q expires in period l before P0 issues it. Then the remaing inventory under policy P0 is one less than the remaining inventory under policy P. But all other inventories are the same for P and P0. From construction above and that excess demand is lost, we have that R0u \u0014 Ru for all u> l. Suppose P0 issues q in period l to meet demand when q is of age m+(l\u0000n). \u00E2\u0080\u00A2 If Dl \u0015\u00C3\u00A5Mi=1 yi;l, then P also issues q0 in period l. Thus, Rl\u0000R0l = (m0+(l\u0000n)\u0000m\u0000 (l\u0000 n))+(m\u0000m0) = 0. Moreover, the ending inventory for period l is 0 for both P and P0, and therefore, R0u = Ru, for all u> l. \u00E2\u0080\u00A2 If Dl <\u00C3\u00A5Mi=1 yi;l, and if P issues q0 in period l, then let P0 issues q for the unit of demand for which P issues q0. Then Rl\u0000R0l = (m0+(l\u0000n)\u0000m\u0000 (l\u0000n))+(m\u0000m0) = 0. Then as the case above, we get R0u= Ru, for all u> l. If P does not issue q0 in period l, then hold off issing q in period l for policy P0, and issue according to P. So Rl\u0000R0l is still m\u0000m0. Since Dl < \u00C3\u00A5Mi=1 yi;l, unit q remains in the inventory at the end of period l. Continue this practice until P issues q0, or unit q expires, or stockout occurs. If P issues q0 first, then we are back to the case in which P0 issues q for the unit of demand for which P issues q0. If unit q expires first, then again the remaing inventory under policy P0 is one less than the remaining inventory under policy P. But all other inventories are the 34 same for P and P0. From construction above and that excess demand is lost, R0u \u0014 Ru for all u> l. If stockout occurs first, say in period k;k > l, we are back to the case of Dk \u0015 \u00C3\u00A5Mi=1 yi;k, with both P and P0 having 0 inventory at the end of period k. Hence, R0u\u0000Ru = m\u0000m0 for all l \u0014 u< k, and R0u = Ru, for all u\u0015 k. Therefore, R0 \u0014 R if j \u0015 l. Consequently, we have constructed a policy P0 which is one unit closer to the LIFO policy and is at least as good as policy P. Therefore, the LIFO policy minimizes the cumulative age factor amount for each sample path when excess demand is lost. Note that similar to wastages, the LIFO policy also does not minimize the age factor for each period, but the cumulative age factor for periods 1 to t, t \u0015 1. An example of this is provided in Table 2.2 below. For simplicity, we set M = 3. Table 2.2: LIFO does not minimize the age factor for every period LIFO (2;1;3) Age 1 2 3 1 2 3 xi;1 0 0 0 0 0 0 Qi;1 5 5 5 5 5 5 D1 = 6 zi;1 5 1 0 1 5 0 Ii;1 0 4 5 4 0 5 xi;2 0 0 4 0 4 0 Qi;2 0 0 0 0 0 0 D2 = 4 zi;2 0 0 4 0 4 0 Ii;2 0 0 0 0 0 0 We see in the example that the age factor under LIFO for period 2 is 12= 4 \u00013, greater than the age factor under policy (2;1;3), which is 8= 4 \u00012. However, both have cumulative age factor of 19. Theorem 2.4.2. For each period t \u0015 1, \u00C3\u00A5tn=1\u00C3\u00A5Mi=1 zFIFOi;n \u0001 i \u0015 \u00C3\u00A5tn=1\u00C3\u00A5Mi=1 zPi;n \u0001 i for all policies P2P when excess demand is lost. In other words, the FIFO policy maximizes the cumulative age factor under each sample path when excess demand is lost. Proof. The proof of this theorem is very similar to the proof of Theorem 2.4.1. In this proof, we show that for any policy P other than the FIFO policy, we can construct another policy P0 which is one unit closer to FIFO, and that RP0t \u0015 RPt . 35 Consider a policy P 2 P under which unit q of age m, 1 < m \u0014M, is used to fill a unit demand in period n, for 1\u0014 n\u0014 t, and there is a unit q0 of age m0, m0 >m, that can be used to fill the same unit demand. We will show that there is a feasible policy P0 which fills the unit demand with q0, and R0n \u0015 Rn, for each n= 1; : : : ; t. Note that age of q0 is older than age of q, so q0 expires before q does. We define \u00E2\u0080\u00A2 j := period in which policy P issues item q0, or item q0 expires. Note that j \u0015 n+1. \u00E2\u0080\u00A2 l := period in which policy P0 issues item q. Note that l \u0015 n+1. Case 1: j < l. We have that P issues item q0, or item q0 expires before P0 issues item q. For periods u < n, we have that R0u = Ru. For all periods u, n \u0014 u < j, we have that R0u\u0000Ru = m0\u0000m \u0015 0. For period j, suppose P issues q0, then let P0 issues q to satisfy the unit demand for which P issues q0. Then R0j = R j. As a result, each policy has satisfied the same demands through j. Moreover, they have the same inventory remaining. Therefore, R0u = Ru for all u > j. Now, suppose q0 expires in period j, then the remaining inventory under policy P0 is one unit more than the remaining inventory under policy P. But all other inventories are the same for P and P0. From construction above and that excess demand is lost, we get R0u \u0015 Ru for all u> l. In conclusion, R0 \u0015 R when j < l. Case 2: j \u0015 l. Again, for periods u < n, we have that R0u = Ru. For all periods u, n\u0014 u< l, we have that R0u\u0000Ru =m0\u0000m\u0015 0. Note that \u00C3\u00A5Mi=1 y0i;l =\u00C3\u00A5Mi=1 yi;l since both policies have issued the same amount of blood up until period l. \u00E2\u0080\u00A2 If Dl \u0015 \u00C3\u00A5Mi=1 yi;l, then P also issues q0 in period l. Then similar to the arguments used in the proof of Theorem 2.4.1, we get that Rl\u0000R0l = 0, and that R0u = Ru, for all u> l. \u00E2\u0080\u00A2 If Dl <\u00C3\u00A5Mi=1 yi;l, and if P issues q0 in period l, then let P0 issues q for the unit of demand for which P issues q0. Then Rl\u0000R0l = (m0+(l\u0000n)\u0000m\u0000 (l\u0000n))+ (m\u0000m0) = 0. Then as the case above, we get R0u = Ru, for all u> l. If P does not issue q0 in period l, then hold off issing q in period l for policy P0, and issue according to P. So R0l\u0000Rl is still m0\u0000m. Since Dl <\u00C3\u00A5Mi=1 yi;l, unit q remains in the inventory at the end of period l. We 36 hold off issuing q until P issues q0, or unit q0 expires, or stockout occurs. If P issues q0 first, then we are back to the case in which P0 issues q for the unit of demand for which P issues q0. If unit q0 expires first, then again the remaing inventory under policy P0 is one unit more than the remaining inventory under policy P. But all other inventories are the same for P and P0. From construction above and that excess demand is lost, we get R0u \u0015 Ru for all u > l. If stockout occurs first, say in period k;k > l, we are back to the case of Dk \u0015\u00C3\u00A5Mi=1 yi;k, with both P and P0 having 0 inventory at the end of period k. Hence, R0u\u0000Ru = m\u0000m0 for all l \u0014 u< k, and R0u = Ru, for all u\u0015 k. Therefore, R0 \u0015 R when j \u0015 l. Consequently, we have constructed a policy P0 which is one unit closer to the FIFO policy, and has cumulative age factor greater than or equal to the cumulative age factor under policy P. Therefore, the FIFO policy maximizes the cumulative age factor amount for each sample path when excess demand is lost. We will show below that the LIFO policy also minimizes, and that the FIFO policy also maximizes the average age of blood used in transfusion for each sample path when excess demand is lost. We begin by proving the following lemmas: Lemma 2.4.3. If E1+(M+1)C1B \u0014 E2+(M+1)C2B , and E1 \u0014 E2, B >C1 \u0015C2, B\u0000C1 \u0014 E1 \u0014 (M+ 1)(B\u0000C1), B\u0000C2 \u0014 E2 \u0014 (M+ 1)(B\u0000C2), where B > 0,C1,C2, E1, E2 are all non-negative, and M \u0015 1. Then E1B\u0000C1 \u0014 E2 B\u0000C2 . Proof. First, E1+(M+1)C1B \u0014 E2+(M+1)C2B implies that E1+(M+ 1)C1 \u0014 E2+(M+ 1)C2, and so (M+ 1)(C1\u0000C2) \u0014 E2\u0000E1. Since C1 \u0015 C2 and M+ 1 > 1, we get that C1\u0000C2 \u0014 E2\u0000E1. Therefore, E1+C1 \u0014 E2+C2. Now since B> 0, we also have that E1+C1B \u0014 E2+C2B . Subtracting both sides of the numberator by C2, we get that E1+(C1\u0000C2)B \u0014 E2B . Moreover, since B >C2, we get that E1+(C1\u0000C2)B\u0000C2 \u0014 E2 B\u0000C2 . From (M+ 1)(C1\u0000C2) \u0014 E2\u0000E1, we also get that E2 \u0015 (M+1)(C1\u0000C2)+E1. Therefore, E1+(C1\u0000C2)B\u0000C2 \u0014 E1+(M+1)(C1\u0000C2) B\u0000C2 \u0014 E2 B\u0000C2 . Second, we show that E1B\u0000C1 \u0014 E1+(M+1)(C1\u0000C2) B\u0000C2 . It is true if and only if E1 \u0001B\u0000E1 \u0001C2 \u0014 B \u0001 E1\u0000C1 \u0001E1+(M+1)(C1\u0000C2)(B\u0000C1), if and only if C1 \u0001E1\u0000E1 \u0001C2\u0014 (M+1)(C1\u0000C2)(B\u0000C1), 37 if and only if E1(C1\u0000C2)\u0014 (M+1)(C1\u0000C2)(B\u0000C1), if and only ifC1=C2, or E1\u0014 (M+1)(B\u0000 C1). Since we are given E1 \u0014 (M+1)(B\u0000C1), we get that E1B\u0000C1 \u0014 E1+(M+1)(C1\u0000C2) B\u0000C2 \u0014 E2 B\u0000C2 . Lemma 2.4.4. For each period t \u0015 1, \u00C3\u00A5tn=1\u00C3\u00A5Mi=1 i \u0001 zLIFOi;n +(M+1) \u0001\u00C3\u00A5tn=1 SLIFOn \u0014 \u00C3\u00A5tn=1\u00C3\u00A5Mi=1 i \u0001 zPi;n+(M+1) \u0001\u00C3\u00A5tn=1 SPn when excess demand is lost. In other words, the LIFO policy minimizes the sum of the total age factor and M+ 1 times the total shortages for each sample path when excess demand is lost among all feasible policies. Proof. As shown in Lemma 2.3.1 that \u00C3\u00A5tn=1\u00C3\u00A5Mi=1 zPi;n =\u00C3\u00A5tn=1Dn\u0000\u00C3\u00A5tn=1 SPn for all feasible poli- cies P and t \u0015 1. Therefore, \u00C3\u00A5tn=1 SPn = \u00C3\u00A5tn=1Dn\u0000\u00C3\u00A5tn=1\u00C3\u00A5Mi=1 zPi;n. Then for t \u0015 1, we have that t \u00C3\u00A5 n=1 M \u00C3\u00A5 i=1 i \u0001 zPi;n+(M+1) \u0001 t \u00C3\u00A5 n=1 SPn = t \u00C3\u00A5 n=1 M \u00C3\u00A5 i=1 i \u0001 zPi;n+(M+1) \u0001 ( t \u00C3\u00A5 n=1 Dn\u0000 t \u00C3\u00A5 n=1 M \u00C3\u00A5 i=1 zPi;n) = t \u00C3\u00A5 n=1 M \u00C3\u00A5 i=1 (i\u0000 (M+1))zPi;n+(M+1) t \u00C3\u00A5 n=1 Dn : Now, as we will show in Theorem 2.4.6 that the LIFO policy minimizes the cumulative age factor for increasing linear age penalty functions when excess demand is lost, we get that the LIFO policy minimizes \u00C3\u00A5tn=1\u00C3\u00A5Mi=1(i\u0000 (M+ 1))zPi;n. Moreover, since Dn is the same for all policies for all period n, we get that the LIFO policy minimizes \u00C3\u00A5tn=1\u00C3\u00A5Mi=1 i \u0001zPi;n+(M+ 1) \u0001\u00C3\u00A5tn=1 SPn . Theorem 2.4.5. For each period t \u0015 1, and when excess demand is lost \u00C3\u00A5tn=1 i \u0001 zLIFOi;n \u00C3\u00A5tn=1 zLIFOi;n \u0014 \u00C3\u00A5 t n=1 i \u0001 zPi;n \u00C3\u00A5tn=1 zPi;n : In other words, the LIFO policy minimizes the average age factor for each sample path when excess demand is lost. Proof. First, by Lemma 2.4.4, we get that when excess demand is lost, for all t \u0015 1, \u00C3\u00A5tn=1\u00C3\u00A5 M i=1 i \u0001 zLIFOi;n +(M+1) \u0001\u00C3\u00A5tn=1 SLIFOn \u00C3\u00A5tn=1Dn \u0014 \u00C3\u00A5 t n=1\u00C3\u00A5 M i=1 i \u0001 zPi;n+(M+1) \u0001\u00C3\u00A5tn=1 SPn \u00C3\u00A5tn=1Dn : 38 Now, let B = \u00C3\u00A5tn=1Dn, C1 = \u00C3\u00A5tn=1 SLIFOn , C2 = \u00C3\u00A5tn=1 SPn , E1 = \u00C3\u00A5tn=1\u00C3\u00A5Mi=1 i \u0001 zLIFOi;n , and E2 = \u00C3\u00A5tn=1\u00C3\u00A5 M i=1 i \u0001 zPi;n. Then B, C1, C2, E1, and E2 satisfy conditions specified in Lemma 2.4.3. As a result, we get that \u00C3\u00A5tn=1 i \u0001 zLIFOi;n \u00C3\u00A5tn=1 zLIFOi;n \u0014 \u00C3\u00A5 t n=1 i \u0001 zPi;n \u00C3\u00A5tn=1 zPi;n : Now, we would like to investigate policies that minimize and maximize the total age factor when we penalize the issuance of age i blood by an increasing linear, an increasing concave, and a decreasing convex function of blood age, respectively. Theorem 2.4.6. Let function f (i) = a \u0001 i+ c, a> 0, be a function of blood age i. Then for each period t \u0015 1, \u00C3\u00A5tn=1\u00C3\u00A5Mi=1 zLIFOi;n \u0001 f (i)\u0014\u00C3\u00A5tn=1\u00C3\u00A5Mi=1 zPi;n \u0001 f (i) for all policies P 2 P when excess demand is lost. In other words, if we penalize the issuance of age i blood by an increasing linear function f (i), then LIFO minimizes the cumulative age factor for each sample path when excess demand is lost. Proof. The proof of this theorem is very similar to the proof of Theorem 2.4.1. Note that Theorem 2.4.1 is a special case of this theorem with f (i) = i. To prove this theorem, we follow the same steps in proof of Theorem 2.4.1 with changes in the following details: \u00E2\u0080\u00A2 In Case 1 where j < l, we have that Ru\u0000R0u = a \u0001m+c\u0000a \u0001m0\u0000c> 0 for all periods u, n\u0014 u< j since a> 0 and m> m0. \u00E2\u0080\u00A2 In cases where we have P0 issues q in the same period u as P issues q0, for u \u0015 n+1. We have that Ru\u0000R0u = f (m)\u0000 f (m0)+ f (m0+(u\u0000n))\u0000 f (m+(u\u0000n)) = 0. Theorem 2.4.7. Let function f (i) = a \u0001 i+ c, a> 0, be a function of blood age i. Then for each period t \u0015 1, \u00C3\u00A5tn=1\u00C3\u00A5Mi=1 zFIFOi;n \u0001 f (i)\u0015\u00C3\u00A5tn=1\u00C3\u00A5Mi=1 zPi;n \u0001 f (i) for all policies P 2 P when excess demand is lost. In other words, if we penalize the issuance of age i blood by an increasing linear function f (i), then FIFO maximizes the cumulative age factor for each sample path when excess demand is lost. 39 Proof. The proof of this theorem is very similar to the proof of Theorem 2.4.2. Note that Theorem 2.4.2 is also a special case this theorem with f (i) = i. To prove this theorem, we follow the same steps in proof of Theorem 2.4.2 with changes in the following details: \u00E2\u0080\u00A2 In Case 1 where j < l, we have that R0u\u0000Ru = a \u0001m0+c\u0000a \u0001m\u0000c> 0 for all periods u, n\u0014 u< j. \u00E2\u0080\u00A2 In cases where we have P0 issues q in the same period u as P issues q0, for u \u0015 n+1. We have that R0u\u0000Ru = f (m0)\u0000 f (m)+ f (m+(u\u0000n))\u0000 f (m0+(u\u0000n)) = 0. Theorem 2.4.8. Let function f (i) be an increasing concave function of blood age i. Then for each period t \u0015 1, \u00C3\u00A5tn=1\u00C3\u00A5Mi=1 zLIFOi;n \u0001 f (i)\u0014\u00C3\u00A5tn=1\u00C3\u00A5Mi=1 zPi;n \u0001 f (i) for all policies P 2 P when excess demand is lost. In other words, if we penalize the issuance of age i blood by an increasing concave function, then LIFO minimizes the cumulative age factor for each sample path when excess demand is lost. Proof. The proof of this theorem is very similar to the proof of Theorem 2.4.6. Note that since f (i) is an increasing concave function, we have that f (m) > f (m0) for m > m0, and that f (m)\u0000 f (m0) \u0015 f (m+(u\u0000 n))\u0000 f (m0+(u\u0000 n)) for u\u0000 n > 0. Therefore, in cases where we have P0 issues q in the same period u as P issues q0, for u \u0015 n+1. We have that Ru\u0000R0u = f (m)\u0000 f (m0)\u0000 ( f (m+(u\u0000 n))\u0000 f (m0+(u\u0000 n))) \u0015 0. As a result, R0k \u0014 Rk for all k \u0015 n. Theorem 2.4.9. Let function f (i) be an increasing concave function of blood age i. Then for each period t \u0015 1, \u00C3\u00A5tn=1\u00C3\u00A5Mi=1 zFIFOi;n \u0001 f (i)\u0015\u00C3\u00A5tn=1\u00C3\u00A5Mi=1 zPi;n \u0001 f (i) for all policies P 2 P when excess demand is lost. In other words, if we penalize the issuance of age i blood by an increasing concave function, then FIFO maximizes the cumulative age factor for each sample path when excess demand is lost. Proof. The proof of this theorem follows arguments in the proof of Theorem 2.4.7 and arguments in the proof of Theorem 2.4.8. 40 Theorem 2.4.10. Let function f (i) be a decreasing convex function of blood age i. Then for each period t \u0015 1, \u00C3\u00A5tn=1\u00C3\u00A5Mi=1 zLIFOi;n \u0001 f (i)\u0015\u00C3\u00A5tn=1\u00C3\u00A5Mi=1 zPi;n \u0001 f (i) for all policies P 2 P when excess demand is lost. In other words, if we penalize the issuance of age i blood by a decreasing convex function, then LIFO maximizes the cumulative age factor for each sample path when excess demand is lost. Proof. The proof of this theorem is similar to the proof of Theorem 2.4.8 except for that the signs are reversed. When f (i) is a decreasing convex function, we have that f (m0) > f (m) for m0 < m, and that f (m0)\u0000 f (m) \u0015 f (m0+(u\u0000 n))\u0000 f (m+(u\u0000 n)). Therefore, in cases where we have P0 issues q in the same period u as P issues q0, for u \u0015 n+1. We have that Ru\u0000R0u = f (m)\u0000 f (m0)\u0000 ( f (m+(u\u0000 n))\u0000 f (m0+(u\u0000 n))) \u0014 0. As a result, R0k \u0015 Rk for all k \u0015 n. Theorem 2.4.11. Let function f (i) be a decreasing convex function of blood age i. Then for each period t \u0015 1, \u00C3\u00A5tn=1\u00C3\u00A5Mi=1 zFIFOi;n \u0001 f (i)\u0014\u00C3\u00A5tn=1\u00C3\u00A5Mi=1 zPi;n \u0001 f (i) for all policies P 2 P when excess demand is lost. In other words, if we penalize the issuance of age i blood by a decreasing convex function, then FIFO minimizes the cumulative age factor for each sample path when excess demand is lost. Proof. Using similar arguments used in proof of Theorem 2.4.10 above with m0 > m. We can show that FIFO minimizes the cumulative age factor for each sample path when excess demand is lost. 2.5 Summary of Chapter 2 From results obtained in this chapter, we can see that it would be challenging to balance between the age factor and shortages and wastages. This is because although LIFO min- imizes the expected total age factor, it maximizes expected total shortages and expected total wastages, and the reverse is true for the FIFO policy. Moreover, FIFO and LIFO are in reverse issuing order of each other. In the next chapter (Chapter 3), we will explore other policies and use simulation to see how well they are able to balance the three objectives. 41 Chapter 3 Model with Combined Objective Our main objective in this chapter is to use simulation with demand and supply distributions derived from historical hospital A data to evaluate eleven (11) policies for minimizing the expected total costs of shortages, wastages, and the age factor for both excess demand lost and backlogged cases. We first present an integer linear programming formulation of our problem with known supply and demand in Section 3.1. We then discuss curses of dimenionality of our MDP model introduced in Section 2.1. We introduce a myopic policy based on the single-period cost function of our MDP model in Section 3.3. After that, we introduce the policies (Section 3.4) and data (Section 3.5) used for simulation, and introduce simulation input and output variables in Section 3.6. We display and discuss simulation results in Section 3.7. Lastly, we summarize our findings in Section 3.8. 42 3.1 Problem Formulation with Deterministic Supply and Demand We can formulate our problem as follows when demand and supply in each period are deterministic. Min h \u0001 T \u00C3\u00A5 t=1 M \u00C3\u00A5 i=1 zi;t \u0001 i+w \u0001 T \u00C3\u00A5 t=1 IM;t + p \u0001 T \u00C3\u00A5 t=1 St s:t: 0\u0014 zi;t \u0014 yi;t ; i= 1;2; : : : ;M; t = 1;2; : : : ;T yi;t = xi;t +Qi;t ; i= 1;2; : : : ;M; t = 1;2; : : : ;T IM;t = yM;t \u0000 zM;t ; t = 1;2; : : : ;T M \u00C3\u00A5 i=1 zi;t \u0014 8><>: Dt ; when excess demand is lostDt +St\u00001; when excess demand is backlogged. St = 8><>: maxf0;Dt \u0000\u00C3\u00A5 M i=1 yi;tg; when excess demand is lost maxf0;Dt +St\u00001\u0000\u00C3\u00A5Mi=1 yi;tg; when excess demand is backlogged S0 = 0 xi;t ;yi;t ;zi;t ;St ;Dt ;Qi;t non-negative integers for all i= 1;2; : : : ;M; t = 1;2; : : : ;T The problem above is an integer linear programming problem and can be solved with standard solution algorithms. 3.2 Curse of Dimensionality From our MDP formulation in Section 2.1 in Chapter 2, we can see that the state in each period t, Xt , has M+1 dimensions, and the action in each period t, at , has M dimensions. In each period t, we also have random supply and demand that are of dimension M+1 in total. In our problem, M is 42, demand in each period has 118 different integer values to take on, and supply for each age of blood has 175 different integer values to take on. We derive 118 and 175 from data provided by hospital A. They are calculated by taking the total demand and total supply for each day. We get that the total daily demand in hospital 43 A has 118 different values, and the total daily supply in hospital A has 175 different values. But daily supply in hospital A can be for any age of blood, so daily supply for each age of blood in hospital A can take on 175 different values. As a result, supply and demand in each period have 17542 \u00011181 outcomes. We would need to evaluate 43 nested summations to evaluate the expectation. Moreover, each yi;t , i = 2;3; : : : ;M, in Xt also depends on the ending inventory of the previous period, and can take on a large number of non-negative integer values. Each zi;t in at depends on yi;t and Dt , and can also take on a large number of non-negative integer values. Suppose each yi;t can take on N different values, and each zi;t can take on L different values, then our state space has 175 \u0001N41 \u0001118 states, and our action space has L42 states. The 175 in 175 \u0001N41 \u0001118 is the number of different integer values daily supply for each age of blood can take on. It is 175, not N for age 1 blood because the initial inventory level for age 1 blood for all periods t \u0015 2 is 0. Therefore, we only need to count variations in the supply of age 1 blood. The 118 in 175 \u0001N41 \u0001118 is the number of different integer values daily demand can take on for hospital A as mentioned above. Both N and L are large. As a result, it is intractable to evaluate all possible values of at . Hence, we cannot evaluate the optimality equation Equation 2.8 of our MDP model. Instead, we come up with eleven policies to evaluate in simulation. These policies are introduced in Section 3.4 below. 3.3 A Myopic Policy In this section, we will present a myopic policy that minimizes the cost function defined in Equation 2.3 for each decision epoch t after supply and demand have been realized. At decision epoch t with begining inventory xt , Ct(Xt ;at) = h \u0001 M \u00C3\u00A5 i=1 zi;t \u0001 i+w \u0001 IM;t + p \u0001St = h \u0001 M \u00C3\u00A5 i=1 zi;t \u0001 i+w \u0001 (yM;t \u0000 zM;t)+ p \u0001St = h \u0001 M\u00001 \u00C3\u00A5 i=1 zi;t \u0001 i+(h \u0001M\u0000w)zM;t +w \u0001 yM;t + p \u0001St : (3.1) 44 Note that yM;t and St do not depend on zt . The term yM;t depends on xM;t and QM;t , and St depends on Dt , xt , and Qt . Therefore, after demand and supply have been realized in period t, w \u0001 yM;t and p \u0001St become constants, and Equation 3.1 depends on zt . We observe Equation 3.1 that the coefficient of zi;t is i \u0001h, for i= 1; : : : ;M\u00001, and the coefficient of zM;t is M \u0001h\u0000w. Each coefficient represents the weight of of releasing one unit of zi;t . Let l be the smallest positive integer that follows M \u0001h\u0000w, we claim that Theorem 3.3.1. The cost of period t, Ct(Xt ;at), is minimized if we order ages of blood based on their weights, and release the age of blood that has the lightest weight first. When h= 1, we issue blood according to age order 1;2; : : : ; l\u00001;M; l; : : : ;M\u00001. Note that if l = 1, we issue blood according to age order M;1;2; : : : ;M\u00001. Proof. Let 10;20; : : : ;M0 represent the increasing weight order for blood of ages 1 to M, and let wi0 be equal to the weight of the age of blood i0 represents, for i0 = 10;20; : : : ;M0. Then w10 \u0014 w20 \u0014 \u0001\u0001 \u0001 \u0014 wM0 . If demand Dt of period t is greater than or equal to the total updated inventory level \u00C3\u00A5Mi=1 yi;t in period t, then based on our assumption, all inventory needs to be used to satisfy demand Dt . In this case, Ct(Xt ;at) = h \u0001\u00C3\u00A5Mi=1 yi;t \u0001 i+ p \u0001 St , regardless of the order we issue blood, and therefore, CMyopict (Xt ;at)\u0014Ct(Xt ;at) for all other blood issuing orders. If demand Dt < \u00C3\u00A5k 0 i0=10 yi0;t for k0 \u0014 M0, where yi0;t = yr;t for i0 represents blood of age r. Then under the Myopic policy, the cost of period t, CMyopict (Ct ;at) = k0\u00001 \u00C3\u00A5 i0=10 wi0 \u0001 yi0;t +wk0 \u0001 ( k0 \u00C3\u00A5 i0=10 yi0;t \u0000Dt) : Now suppose for 10 \u0014 j0 \u0014 k0, and k0 \u0014 r0 \u0014M0, we release one unit of blood of age represented 45 by r0, and release one unit less of blood of age represented by j0, then C0t(Xt ;at) = j0\u00001 \u00C3\u00A5 i0=10 wi0 \u0001 yi0;t +w j0 \u0001 (y j0;t \u00001)+wr0 \u00011+ k0\u00001 \u00C3\u00A5 i0= j0+1 wi0 \u0001 yi0;t +wk0 \u0001 ( k0 \u00C3\u00A5 i0=10 yi0;t \u0000Dt) = k0\u00001 \u00C3\u00A5 i0=10 wi0 \u0001 yi0;t +wk0 \u0001 ( k0 \u00C3\u00A5 i0=10 yi0;t \u0000Dt)+(wr0 \u0000w j0) \u0015CMyopict (Xt ;at) since wr0 \u0015 w j0 : Therefore, the cost of period t, Ct(Xt ;at), is minimized if we order ages of blood based on their weights, and release the age of blood that has the lightest weight first. Now when h= 1, weights of blood of ages 1, 2, : : :, M\u00001, M, are 1, 2, : : :, M\u00001, M\u0000w, respectively, and that l is the smallest positive integer that follows M\u0000w. Hence, l\u0000 1 < M\u0000w \u0014 l. Therefore, when l = 1, we have M\u0000w \u0014 1 < 2 < \u0001 \u0001 \u0001 < M\u0000 1, and thus should issue blood according to age order M;1;2; : : : ;M\u00001 to minimize cost of period t. When l > 1, we have that 1\u0014 \u0001\u0001 \u0001 \u0014 l\u00001 "Thesis/Dissertation"@en . "2013-11"@en . "10.14288/1.0073844"@en . "eng"@en . "Business Administration"@en . "Vancouver : University of British Columbia Library"@en . "University of British Columbia"@en . "Attribution-NonCommercial-NoDerivatives 4.0 International"@en . "http://creativecommons.org/licenses/by-nc-nd/4.0/"@en . "Graduate"@en . "Optimal issuing policies for hospital blood inventory"@en . "Text"@en . "http://hdl.handle.net/2429/44417"@en .