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 © 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, 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. 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 Summary of Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.8 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 . . . . . . . . . . . . . . Table 3.4 Change in the Average Expected Total Wastage Cost with respect to One Unit Change in w when Excess Demand is Backlogged . . . . . . . . . . Table 3.5 . . . . . . . . . . 82 Excess Demand Lost vs. Excess Demand Backlogged in terms of Average Expected Total Wastages, Shortages and Age Factor when w = p = h = 1 Table 3.9 79 Change in the Average Expected Total Shortage Cost with respect to One Unit Change in p when Excess Demand is Backlogged Table 3.8 78 Change in the Average Expected Total Cost with respect to One Unit Change in w when Excess Demand is Backlogged . . . . . . . . . . . . . Table 3.7 76 Change in the Average Expected Total Cost with respect to One Unit Change in w when Excess Demand is Lost . . . . . . . . . . . . . . . . . Table 3.6 75 83 Excess Demand Lost vs. Excess Demand Backlogged in terms of Sensitivities 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 Sensitivities of Average Expected Total Costs to w and to p . . . . . . . . . . . . 86 Table A.1 Average expected Shortage and Wastage Percentages, the average Expected Total Age Factor Cost, and the average expected total costs for All Policies across all Simulation Runs for Both Excess Demand Cases . v 94 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 . . . . . . . . . . . . . . . . . vi 111 List of Figures Figure 1.1 Hospital A Blood Supply Chart . . . . . . . . . . . . . . . . . . . . . . . Figure 3.1 Average Percentages of Shortages and Wastages, and Average Expected 4 Total Age Factor Costs for Non-Threshold Policies Excess Demand Backlogged vs. Excess Demand Lost . . . . . . . . . . . . . . . . . . . . . . . Figure 3.2 Average Percentage of Shortages for the No. 1 Threshold Policy for both excess demand cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.3 58 Average Percentage of Shortages for the No. 3 Threshold Policy for both excess demand cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.9 58 Average Expected Total Age Factor Cost for the No. 2 Threshold Policy for both excess demand cases . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.8 57 Average Percentage of Wastages for the No. 2 Threshold Policy for both excess demand cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.7 56 Average Percentage of Shortages for the No. 2 Threshold Policy for both excess demand cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.6 56 Average Expected Total Age Factor Cost for the No. 1 Threshold Policy for both excess demand cases . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.5 56 Average Percentage of Wastages for the No. 1 Threshold Policy for both excess demand cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 3.4 55 59 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 . . . . . . . . . . . . . . . . . . . . . . . . vii 61 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 Demand 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 . . . . . . . viii 86 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 improving this thesis. I beneﬁted greatly from the help of several oﬃce mates. I thank Antoine Sauré, 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 2,3Diphosphoglycerate levels decrease. These pathological changes are collectively known as “storage lesions”. 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 “independently associated 1 with mortality, renal failure, and pneumonia”. 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 (≥ 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’ 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 hos2 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 demand 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 “any” 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, “young” and “any age”. We only allow the “any age” 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 maximizes 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: • 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. • We ﬁnd the optimal issuing policy for minimizing/maximizing the average age of blood used in transfusion when excess demand is lost. • We explore three other age penalty functions, and ﬁnd optimal policies that minimize/maximize the total age factor for each age penalty function. • 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. • 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 ﬁnitehorizon 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 • Daily supply and demand of blood are random. • Since blood demand and supply are expressed in terms of non-negative integer units, they can only take on non-negative integer values. • 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. • 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. • Demand is non-age-speciﬁc. That is, any unit of blood in inventory can be used to satisfy a unit demand of blood. • 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. • i: Index blood age, a subscript. • t: Index time period, a subscript. • T: The end of the decision horizon. • P: Index policy P, a superscript. • P: The set of all feasible blood issuing policies based on Deﬁnition 2.0.1. • M: The maximum age of blood that can be used in transfusion. P : The beginning inventory of blood of age i in decision epoch t for policy P. We • xi,t assume that all policies have the same beginning inventory in period 1, and that P ≥ 0 for all i = 1, 2, . . . , M. xi,1 • Qi,t : The supply of age i blood in period t, Qi,t ≥ 0 for all i = 1, 2, . . . , M and all t ≥ 1. The supply of blood is the same for all policies. • Dt : The demand of blood for period t, Dt ≥ 0 for all t ≥ 1. It is also the same for all policies. • DtP : Blood demand of period t under policy P. When unsatisﬁed demand is lost, P , where DtP = Dt for all P ∈ P. When unsatisﬁed demand is backlogged, DtP = Dt + St−1 StP is deﬁned below. 12 • yPi,t : The total inventory level of blood of age i in period t for policy P after blood P +Q . supply has been realized. This satisﬁes yPi,t = xi,t i,t • 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 P P demanded, i.e., ∑M i=1 zi,t ≤ Dt . We also cannot release more blood than the amount of blood available, i.e., 0 ≤ zi,t ≤ yi,t , for i = 1, 2, . . . , M. P : The remaining inventory of age i blood in period t for policy P, i.e., I = y − z ≥ • Ii,t i,t i,t i,t 0. Note that P P xi+1,t+1 = Ii,t , for all i = 1, 2, . . . , M − 1 , and t ≥ 1 , P x1,t = 0 for all t ≥ 2 . (2.1) (2.2) P is the wastage amount in period t under policy P. Also note that IM,t P • StP : Shortage amount for period t under policy P: StP = max{0, DtP − ∑M i=1 yi,t }. This is the amount of lost demand in the excess demand lost case, and the amount of backlogged demand in the backlogging case. S0P = 0 for all P ∈ P. 2.1 Finite-Horizon MDP Formulation The MDP formulation for our problem is as follows: • Decision epoch: {1, 2, . . . , T }, T < ∞. • 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 ) ∈ X ⊂ (N ∪ {0})M+1 . • 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 ) ∈ AXt ⊂ A ⊂ (N ∪ {0})M . 13 • Transition probabilities: Let fs and gd be probability distributions for supply and demand, respectively. Then ∑ pt ( j|Xt , at ) = fs (Qt+1 )gd (Dt+1 ) . (Qt+1 ,Dt+1 ): j=(Q1,t+1 ,y1,t −z1,t +Q2,t+1 ,...,yM−1,t −zM−1,t +QM,t+1 ,Dt+1 ) • 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 M Ct (Xt , at ) := h · ∑ zi,t · i + w · IM,t + p · St (2.3) i=1 where IM,t = yM,t − zM,t (2.4) 0 ≤ zi,t ≤ yi,t , i = 1, . . . , M max{0, Dt − ∑M yi,t }, i=1 St = max{0, Dt + St−1 − ∑M i=1 yi,t }, Dt , M when excess ∑ zi,t ≤ Dt + St−1 , when excess i=1 (2.5) when excess demand is lost (2.6) when excess demand is backlogged demand is lost (2.7) demand is backlogged. S0 = 0 . • Optimality Equation: { Vt (Xt ) = min at ∈AXt } C(Xt ) + ∑ pt ( j|Xt , at )Vt+1 ( j) (2.8) j∈X where VT +1 = 0. Our objective is to ﬁnd policy P ∈ P such that the expected total cost over the decision making horizon T is minimized. Let VTP (X1 ) represent the expected total cost under some 14 policy P ∈ P with starting state X1 in decision epoch 1. It is deﬁned by { VTP (X1 ) := EPX1 T } ∑ Ct (XtP , atP ) . (2.9) t=1 We would like to ﬁnd P∗ ∈ P such that { } ∗ VTP (X1 ) = minP∈P VTP (X1 ) . 2.2 (2.10) 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. FIFO ≥ n xP , for Lemma 2.2.1. For each period t ≥ 1, and for blood age n = 1, . . . , M, ∑ni=1 xi,t ∑i=1 i,t all P ∈ 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 ≥ 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 n i=1 i=1 FIFO P . ≥ ∑ xi,1 ∑ xi,1 Second, t = 2: By Equation 2.2 and Equation 2.1, we have that the beginning inventory P , is 0, and the begining inventory level of blood of level of blood of age 1 for period 2, x1,2 P for i = 2, . . . , M, is the same as I P each other age, xi,2 i−1,1 , 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, P P Ii,1 = xi,1 + Qi,1 − zPi,1 = yi,1 − 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 ≥ ∑M i=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 n i=1 i=1 FIFO P , for n = 1, . . . , M, and all P ∈ P . = 0 ≥ 0 = ∑ xi,2 ∑ xi,2 Case 2: The blood demand in period 1 is less than the total updated inventory level in period 1, i.e., D1 < ∑M i=1 yi,1 . In this case, not all inventory needs to be used for demand { } P D1 . Note that by Equation 2.11, max ∑ni=1 Ii,1 = ∑ni=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 FIFO = y blood for age M, and release 0 units of blood for all other ages. Then IM,1 M,1 − D1 , and FIFO = y Ii,1 i,1 for i = 1, . . . , M − 1. Then for any feasible policy P other than the FIFO policy, and ages n = 2, . . . , M, n n n−1 i=1 i=2 j=1 FIFO FIFO = 0 + ∑ xi,2 = 0 + ∑ I FIFO ∑ xi,2 j,1 n−1 = { } n−1 ∑ y j,1 = max ∑ j=1 j=1 I Pj,1 ≥ n−1 n n j=1 i=2 i=1 P P = ∑ xi,2 . ∑ I Pj,1 = 0 + ∑ xi,2 Since the beginning inventory of blood of age 1 is 0 for all periods by Equation 2.2, we have 17 FIFO ≥ n xP for all ages n = 1, . . . , M. that ∑ni=1 xi,2 ∑i=1 i,2 M Subcase 2.2: Suppose that for some age m, 1 < m ≤ M, ∑M i=m yi,1 ≤ D1 < ∑i=m−1 yi,1 . Based on the FIFO policy, we would release yi,1 units for blood of ages i = m, . . . , M, release D1 − ∑M i=m yi,1 units for blood of age m−1, and not release any unit for blood of ages i = 1, . . . , m−2. Then the ending inventory is 0 for blood of each age m to M, the ending inventory is ∑M i=m−1 yi,1 − D1 for age m − 1 blood, and is yi,1 for blood of each age i = 1, . . . , m − 2. Now consider any feasible policy P other than the FIFO policy. Since the demand, D1 , the beginning inventory, xi,1 , and the supply of blood, Qi,1 , for all i = 1, . . . , M, in period 1 are the same for all feasible policies, and no reservation of blood is allowed by our assumption, P the total amount of blood issued, ∑M i=1 zi,1 , is equal to D1 for all feasible policies, and the M P total amount of blood remaining, ∑M i=1 Ii,1 , is ∑i=1 yi,1 − D1 for all feasible policies for period 1. Then for any age n, 1 ≤ n ≤ M, we have that M P n P ∑M n i=1 yi,1 − D1 = ∑i=1 Ii,1 ≥ ∑i=1 Ii,1 FIFO ∑ Ii,1 = { } i=1 n FIFO = n y = max P P ∑ni=1 Ii,1 I ∑i=1 i,1 ∑i=1 i,1 ≥ ∑ni=1 Ii,1 for m − 1 ≤ n ≤ M for 1 ≤ n < m − 1 . FIFO ≥ n I P for all n = 1, . . . , M. As a result, for ages n = 2, . . . , M, Therefore, ∑ni=1 Ii,1 ∑i=1 i,1 n n−1 n−1 n i=1 j=1 j=1 i=1 P FIFO . = 0 + ∑ I FIFO ≥ 0 + ∑ I Pj,1 = ∑ xi,2 ∑ xi,2 j,1 Since the beginning inventory of age 1 blood is 0 for all periods by Equation 2.2, we have FIFO ≥ n xP . that for period 2, and any age n = 1, . . . , M, ∑ni=1 xi,2 ∑i=1 i,2 FIFO ≥ n xP , for Induction hypothesis: suppose that for some period t, t ≥ 2, ∑ni=1 xi,t ∑i=1 i,t FIFO ≥ n xP ages n = 1, . . . , M, and all feasible polices P ∈ P. We will show that ∑ni=1 xi,t+1 ∑i=1 i,t+1 , for ages n = 1, . . . , M, and all feasible policies P ∈ P. FIFO ≥ n xP , we get that n (xFIFO + Note: given that for all feasible policies P ∈ P, ∑ni=1 xi,t ∑i=1 i,t ∑i=1 i,t } { P = P +Q ), or n yFIFO ≥ n yP for ages n = 1, . . . , M. Moreover, max Qi,t ) ≥ ∑ni=1 (xi,t ∑ni=1 Ii,t ∑i=1 i,t ∑i=1 i,t i,t ∑ni=1 yPi,t for n = 1, . . . , M. Now for any feasible policy P other than the FIFO policy, we consider the following three cases. 18 Case 1: The blood demand in period t is greater than or equal to the total updated FIFO . Then the blood demand inventory in period t under the FIFO policy, i.e., Dt ≥ ∑M i=1 yi,t 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 n i=1 i=1 FIFO P = 0 ≥ 0 = ∑ xi,t+1 , n = 1, . . . , M . ∑ xi,t+1 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 M P FIFO . Then similar to the inventory in period t under policy P, i.e., ∑M i=1 yi,t ≤ Dt < ∑i=1 yi,t 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 FIFO > 0. Hence, i ∈ {1, . . . , M}, we have that Ii,t n n i=1 i=1 ∑ Ii,tFIFO ≥ 0 = ∑ Ii,tP , 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 FIFO ≥ 0 = n xP counted in period t + 1, we have that ∑ni=1 xi,t+1 ∑i=1 i,t+1 , for any age n = 1, . . . , M. Case 3: The blood demand in period t is less than the total updated blood inventory M P FIFO as well. Similar to the in period t under policy P, i.e., Dt < ∑M i=1 yi,t . Then Dt < ∑i=1 yi,t 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 < yFIFO M,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 − 1. As a FIFO units of blood result, there would be yFIFO M,t − Dt units of blood left for age M, and yi,t 19 left for each age i = 1, . . . , M − 1 at the end of period t. We then have that for age n = M, FIFO = M yFIFO − D ≥ M yP − D = M I P , and for age n = 1, . . . , M − 1, we have ∑i=1 i,t ∑i=1 i,t ∑i=1 i,t ∑M t t i=1 Ii,t FIFO = n yFIFO ≥ n yP ≥ n I P . Carrying on into to the next period, we that ∑ni=1 Ii,t ∑i=1 i,t ∑i=1 i,t ∑i=1 i,t then obtain that for age n = 2, . . . , M, n n−1 n−1 n i=1 j=1 j=1 i=1 P FIFO ≥ 0 + ∑ I Pj,t = ∑ xi,t+1 = 0 + ∑ I FIFO ∑ xi,t+1 j,t Now again, since the inventory level for age 1 blood is 0 for all periods by Equation 2.2, FIFO ≥ n xP we have that ∑ni=1 xi,t+1 ∑i=1 i,t+1 for all n = 1, . . . , M. FIFO ≤ D < M Subcase 3.2: Suppose that for some age m, 1 < m ≤ M, ∑M . ∑i=m−1 yFIFO t i=m yi,t i,t Then under the FIFO policy, we would release yFIFO units of blood for each age i = m, . . . , M, i,t FIFO units of blood for age m − 1, and release no blood for ages 1 to m − 2. release Dt − ∑M i=m yi,t As a result, there would be no inventory remaining for blood of ages m to M. There would FIFO − D units remaining for age m − 1 blood, and yFIFO units remaining for be ∑M t i=m−1 yi,t i,t each age i = 1, . . . , m − 2. Since ∑ni=1 yFIFO ≥ ∑ni=1 yPi,t for n = 1, . . . , M, we have that the total i,t remaining inventory for blood of ages 1 to n at the end of period t under FIFO n { } P ≥ n IP , ∑ni=1 yFIFO ≥ ∑ni=1 yPi,t = max ∑ni=1 Ii,t ∑i=1 i,t i,t n = 1, . . . , m − 2 FIFO − D ≥ M yP − D = M I P ≥ n I P , ∑M ∑i=1 i,t ∑i=1 i,t ∑i=1 i,t t t i=1 yi,t n = m − 1, . . . , M . ∑ Ii,tFIFO = i=1 FIFO ≥ n I P . Then using similar arguments Therefore, for all n = 1, . . . , M, we have ∑ni=1 Ii,t ∑i=1 i,t FIFO ≥ n xP as before, we get that ∑ni=1 xi,t+1 ∑i=1 i,t+1 for n = 1, . . . , M. FIFO ≥ n xP So to conclude, when excess demand is lost, ∑ni=1 xi,t+1 ∑i=1 i,t+1 for all n = 1, . . . , M. By induction, the inequality is true for all t ≥ 1. FIFO ≥ n xP , Lemma 2.2.2. For each period t ≥ 1, and for blood age n = 1, . . . , M, ∑ni=1 xi,t ∑i=1 i,t for all P ∈ 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 ≥ 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., StFIFO ≤ StP for all P ∈ 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 P . Also, as introduced in the notation section, we assume period t −1, i.e., DtP = Dt +St−1 that S0P = 0 for all feasible policies P ∈ P. 2. The shortage amount of period t under any feasible policy P is 0 if there is enough in{ } M P P P P ventory to satisfy demand, and is DtP − ∑M i=1 yi,t if otherwise, so St = max 0, Dt − ∑i=1 yi,t . 3. If the amount of backlogged demand is the same for two diﬀerent feasible policies in period t − 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 following: 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 n i=1 i=1 FIFO P ≥ ∑ xi,1 . ∑ xi,1 Also, since the blood demand and supply of period 1 are the same for all policies, we have that { S1FIFO = max 0, D1 + S0FIFO − { M = max 0, D1 − ∑ yi,1 i=1 = S1P 21 } } M ∑ i=1 yFIFO i,1 Second, period t = 2. Since S1FIFO = S1P , the demand of period 2 for each policy P, DP2 , is the same for all P ∈ P. Therefore, as proved in the excess demand lost case, we have FIFO ≥ M xP . Hence, FIFO ≥ M yP since blood supply is the same for that ∑M ∑i=1 i,2 ∑M ∑i=1 i,2 i=1 xi,2 i=1 yi,2 all policies. Using this relation, we have that { } M S2FIFO = max 0, D2 + S1FIFO − ∑ yFIFO i,2 i=1 { = max 0, D2 + S1P − ≤ max ∑ yFIFO i,2 i=1 { 0, D2 + S1P − } M } M ∑ yPi,2 i=1 = S2P FIFO ≤ SP , and n xFIFO ≥ n xP , for Now, suppose that for some period t, t ≥ 2, St−1 ∑i=1 i,t ∑i=1 i,t t−1 FIFO ≥ n xP n = 1, . . . , M. We will show that StFIFO ≤ StP , and ∑ni=1 xi,t+1 ∑i=1 i,t+1 for n = 1, . . . , M. By the induction hypothesis, we immediately obtain ∑ni=1 yFIFO ≥ ∑ni=1 yPi,t , and i,t { StFIFO = max FIFO 0, Dt + St−1 − ∑ yFIFO i,t i=1 { ≤ max } n P 0, Dt + St−1 − n ∑ } yPi,t i=1 = StP FIFO ≥ n xP Now we will show that ∑ni=1 xi,t+1 ∑i=1 i,t+1 . Case 1: The backlogged demand under the FIFO policy is the same as the backlogged FIFO = SP . Then DFIFO = DP . Therefore, demand under policy P for period t − 1, i.e., St−1 t t t−1 FIFO ≥ n xP as shown in the excess demand lost case, we have that ∑ni=1 xi,t+1 ∑i=1 i,t+1 , for all ages n = 1, . . . , M. Case 2: The backlogged demand under the FIFO policy is less than the backlogged FIFO < SP . Then DFIFO < DP . We will use a demand under policy P for period t − 1, i.e., St−1 t t t−1 similar approch as approach used in the excess demand lost case, and consider the following 22 subcases. Subcase 2.1: The demand in period t under FIFO is greater than or equal to the total FIFO ≥ M yFIFO . We updated inventory level under FIFO in period t, i.e., DtFIFO = Dt + St−1 ∑i=1 i,t FIFO ≥ M yP . Therefore, under both policy then have the relation DtP > DtFIFO ≥ ∑M ∑i=1 i,t i=1 yi,t 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 FIFO = 0 ≥ 0 = of period t + 1 for all blood ages is 0 under both policies. We have that ∑ni=1 xi,t+1 P for all n = 1, . . . , M. ∑ni=1 xi,t+1 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 FIFO < M yFIFO . Since DP > DFIFO , we get P under policy P in period t, i.e., ∑M ∑i=1 i,t t t i=1 yi,t ≤ Dt P that DtP > ∑M i=1 yi,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 FIFO , not all inventory needs to be used to satisfy policy P. However, since DtFIFO < ∑M i=1 yi,t demand. Therefore, there is inventory remaining for blood of some age i in the age set {1, . . . , M} at the end of period t under FIFO. Hence, for any age n = 1, . . . , M, we have FIFO ≥ 0 = n I P for all n = 1, . . . , M. Then using similar arguments as before, that ∑ni=1 Ii,t ∑i=1 i,t FIFO ≥ n xP we have that ∑ni=1 xi,t+1 ∑i=1 i,t+1 for all n = 1, . . . , M. Subcase 2.3: The demand of period t under FIFO is less than the total updated inventory P FIFO < M yFIFO as level of period t under policy P, i.e., DtFIFO < ∑M ∑i=1 i,t i=1 yi,t . Then we have Dt 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., DtFIFO < yFIFO M,t . Then according to the FIFO policy, we release FIFO units of age M blood remaining. DtFIFO units of age M blood, and so there is yFIFO M,t − Dt For all other blood ages i = 1, . . . , M − 1, we do not release any blood, and so there are units of blood remaining at the end of period t for each age i blood. Then for all yFIFO i,t 23 n = 1, . . . , M − 1, we have that n n n n i=1 i=1 i=1 i=1 P ≥ ∑ yPi,t ≥ ∑ Ii,t . ∑ Ii,tFIFO = ∑ yFIFO i,t FIFO ≥ n xP Therefore, using similar arguments as before, we get that ∑ni=1 xi,t+1 ∑i=1 i,t+1 for all n = 1, . . . , M. P We would like to note that when n = M, if DtP < ∑M i=1 yi,t , we have that M M M M M i=1 i=1 i=1 i=1 i=1 P − DtFIFO ≥ ∑ yPi,t − DtFIFO ≥ ∑ yPi,t − DtP = ∑ Ii,t . ∑ Ii,tFIFO = ∑ yFIFO i,t P If DtP ≥ ∑M i=1 yi,t , there is no inventory remaining at the end of period t under policy P, so FIFO > 0 = M I P . As a result, the total remaining inventory of period t under FIFO, ∑M ∑i=1 i,t i=1 Ii,t FIFO ≥ n I P . for all n = 1, . . . , M, we that that ∑ni=1 Ii,t ∑i=1 i,t FIFO ≤ DFIFO < M Second, suppose for some blood age m, 1 < m ≤ M, ∑M . ∑i=m−1 yFIFO i=m yi,t t i,t Then similar to the excess demand lost case, we do not release any blood of ages 1 to m − 2, FIFO units of age m − 1 blood, and release FIFO units of we release DtFIFO − ∑M ∑M i=m yi,t i=m−1 yi,t blood for ages i = m, . . . , M. Hence, there is yFIFO units of blood remaining for each age i,t FIFO − DFIFO units of blood remaining for age m − 1, and no i = 1, . . . , m − 2, and ∑M t i=m−1 yi,t blood remaining for ages m to M. Then for n = 1, . . . , m − 2, ∑ FIFO Ii,t =∑ yFIFO i,t i=1 i=1 { n n n ≥∑ yPi,t = max i=1 } n ∑ P Ii,t n P ≥ ∑ Ii,t i=1 i=1 P Now for n ≥ m − 1, if DtP ≥ ∑M i=1 yi,t , then there would be no inventory remaining at the end of period t for policy P. As a result, n M n i=1 i=1 i=1 P − DtFIFO > 0 = ∑ Ii,t . ∑ Ii,tFIFO = ∑ yFIFO i,t P If DtP < ∑M i=1 yi,t , we have that n M M M n i=1 i=1 i=1 i=1 i=1 P P − DtFIFO ≥ ∑ yPi,t − DtP = ∑ Ii,t ≥ ∑ Ii,t . ∑ Ii,tFIFO = ∑ yFIFO i,t 24 FIFO ≥ n I P for all n = 1, . . . , M. Using similar arguments as before, we have Then ∑ni=1 Ii,t ∑i=1 i,t FIFO ≥ n xP that ∑ni=1 xi,t+1 ∑i=1 i,t+1 . FIFO ≥ n xP In conclusion, when excess demand is backlogged, ∑ni=1 xi,t+1 ∑i=1 i,t+1 and StFIFO ≤ StP for all n = 1, . . . M. By induction, the two inequalities hold for all t ≥ 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 ≥ 1, and for all policies P ∈ P, StFIFO ≤ StP 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 DtP = DtFIFO = Dt for all t and for all feasible policies P ∈ P. Therefore, for all periods t ≥ 1, { } M StFIFO = max 0, Dt − ∑ yFIFO i,t { i=1 } M ≤ max 0, Dt − ∑ yPi,t by Theorem 2.2.1 i=1 = StP 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 LIFO ≤ n xP , for all Lemma 2.2.4. For each period t ≥ 1, and age n = 1, . . . , M, ∑ni=1 xi,t ∑i=1 i,t policies P ∈ 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 ≥ 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. LIFO ≤ n I P for all n = 1, . . . , M, and all t ≥ 1. We To proof this lemma, we show that ∑ni=1 Ii,t ∑i=1 i,t 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 } { P = n yP − D level for blood of ages 1 to n minus the demand in period t, i.e., min ∑ni=1 Ii,t ∑i=1 i,t t 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. LIFO ≤ n xP , for all Lemma 2.2.5. For each period t ≥ 1, and age n = 1, . . . , M, ∑ni=1 xi,t ∑i=1 i,t policies P ∈ 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 ≥ 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, StLIFO , are greater than or equal to shortages under policy P for each period t, StP , for all policies P ∈ 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 ≥ 1, and all policies P ∈ P, StLIFO ≥ StP 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 DtP = DtLIFO = Dt for all periods t and for all policies P ∈ P. Therefore, { } M StLIFO = max 0, Dt − ∑ yLIFO i,t { i=1 M ≥ max 0, Dt − ∑ yPi,t } by Lemma 2.2.4 i=1 = StP 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. P FIFO ≥ t Lemma 2.3.1. For each period t, t ≥ 1, for all policies P ∈ P, ∑tn=1 ∑M ∑n=1 ∑M i=1 zi,n i=1 zi,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 ∈ P, the total amount of blood released 27 P P P plus shortages in period t equals to the total demand in period t, i.e., ∑M i=1 zi,t +St = Dt +St−1 , for both excess demand cases. P , For both excess demand cases, if the total demand in a period t under policy P, Dt +St−1 P is greater than or equal to the total updated inventory ∑M i=1 yi,t , then all inventory needs to M P P be released to satisfy demand, so ∑M i=1 zi,t = ∑i=1 yi,t , and shortages in period t under policy P − M yP . Therefore, P P P P, StP , equals to Dt + St−1 ∑i=1 i,t ∑M i=1 zi,t + St = Dt + St−1 . Note that when P = 0 for all t. excess demand is lost, St−1 P , is less than the total updated If the total demand in period t under policy P, Dt + St−1 P inventory level, ∑M i=1 yi,t , then the amount of blood we need to release in period t equals to P P P the total demand, and no shortage occurs. Therefore, ∑M i=1 zi,t + St = Dt + St−1 . Since the equality is true for each period, it is also true for cumulative periods of 1 to t for t ≥ 1 such that t M t t t n=1 n=1 n=1 P . ∑ ∑ zPi,n + ∑ SnP = ∑ Dn + ∑ Sn−1 n=1 i=1 Therefore, t t t t M P − ∑ SnP ∑ ∑ zPi,n = ∑ Dn + ∑ Sn−1 n=1 i=1 = n=1 n=1 n=1 ∑tn=1 Dn − ∑tn=1 SnP when excess demand is lost ∑tn=1 Dn − StP when excess demand is backlogged. t P But when excess demand is backlogged, StP = ∑tn=1 SnP . Therefore, ∑tn=1 ∑M i=1 zi,n = ∑n=1 Dn − ∑tn=1 SnP for both excess demand cases. Then by Theorem 2.2.3, we have that for all policies P ∈ P, t M t t t t n=1 n=1 n=1 n=1 t M = ∑ Dn − ∑ SnFIFO ≥ ∑ Dn − ∑ SnP = ∑ ∑ zPi,n . ∑ ∑ zFIFO i,n n=1 i=1 n=1 i=1 FIFO when excess P ≥ t Theorem 2.3.2. For each period t ≥ 1, all policies P ∈ P, ∑tn=1 IM,n ∑n=1 IM,n 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 ∈ P, and for each period t ≥ 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 t M t M n=1 i=1 M P P . + ∑ Ii,t ∑ ∑ Qi,n = ∑ ∑ zPi,n + ∑ IM,n n=1 i=1 n=1 i=1 We know that for both excess demand cases, for all feasible policies P ∈ P, and for each period t ≥ 1, the total beginning inventory plus the total supply minus the total amount of M P P P blood released is the total ending inventory for the period, i.e., ∑M i=1 (xi,t +Qi,t −zi,t ) = ∑i=1 Ii,t . Therefore, M M i=1 i=1 ∑ Qi,t = ∑ (zPi,t + Ii,tP − xi,tP ) . Then by Equations 2.1 and 2.2, we get that t ∑ M ∑ Qi,t = n=1 i=1 = t ∑ t M ∑ zPi,n + ∑ ∑ Ii,nP − ∑ M P ∑ xi,n n=1 i=1 t M n=1 i=1 t M n=1 i=1 t M M ∑ ∑ zPi,n + ∑ ∑ Ii,nP − ∑ ∑ Ii−1,n−1 − ∑ xi,1 n=2 i=2 n=1 i=1 t M M P P zPi,n + IM,n + Ii,t − xi,1 n=1 i=1 n=1 i=1 i=1 n=1 i=1 t M = t M ∑∑ ∑ ∑ ∑ i=1 , Note that in the proof of Lemma 2.2.1 and Lemma 2.2.2, we showed that for each t ≥ 1, M M i=1 i=1 ∑ Ii,tFIFO ≥ ∑ Ii,tP (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 M t M M n=1 i=1 i=1 t M t M M n=1 i=1 i=1 P P FIFO FIFO + ∑ Ii,t − ∑ xi,1 = ∑ ∑ zFIFO + ∑ IM,n + ∑ Ii,t − ∑ xi,1 . ∑ ∑ zPi,n + ∑ IM,n i,n n=1 i=1 n=1 i=1 29 Therefore, t t n=1 n=1 P FIFO − ∑ IM,n ∑ IM,n ( = t M ∑∑ n=1 i=1 zFIFO − i,n t ) M ∑∑ zPi,n ( M ∑ + n=1 i=1 FIFO Ii,t − i=1 ) M ∑ P Ii,t ( + i=1 M M i=1 i=1 ) ∑ xi,1 − ∑ xi,1 ≥0 by Lemma 2.3.1 and Equation 2.12. P , is greater than or equal to the Hence, the cumulative wastages of policy P, ∑tn=1 IM,n FIFO , for all periods t ≥ 1, and for all feasible cumulative wastages of the FIFO policy, ∑tn=1 IM,n policies P ∈ 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 ≥ 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 zi,1 0 1 1 0 Ii,1 20 9 19 10 xi,2 0 20 0 19 Qi,2 0 0 0 0 zi,2 0 10 0 10 Ii,2 0 10 0 9 xi,3 0 0 0 0 D1 = 1 D2 = 10 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. P ≤ t LIFO when Theorem 2.3.3. For each period t ≥ 1, and all policies P ∈ P, ∑tn=1 IM,n ∑n=1 IM,n excess demand is lost and when excess demand is backlogged. In other words, LIFO maximizes 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 ∈ P, t ∑ M = ∑ zLIFO i,n n=1 i=1 t t n=1 n=1 ∑ Dn − ∑ SnLIFO ≤ t t n=1 n=1 ∑ Dn − ∑ SnP = t M ∑ ∑ zPi,n . n=1 i=1 Then using the same steps as steps used in the proof of Theorem 2.3.2, we get that t t P LIFO − ∑ IM,n ∑ IM,n n=1 n=1 ( = t M t M ∑ ∑ zPi,n − ∑ ∑ zLIFO i,n n=1 i=1 n=1 i=1 ) ( + M M i=1 i=1 ∑ Ii,tP − ∑ Ii,tLIFO ) ( + M M i=1 i=1 ) ∑ xi,1 − ∑ xi,1 ≥0 . LIFO , is greater than or equal Hence, the cumulative wastages under policy LIFO, ∑tn=1 IM,n P , for all periods t ≥ 1, and for all policies to the cumulative wastages under policy P, ∑tn=1 IM,n P ∈ 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. LIFO · i ≤ t P Theorem 2.4.1. For each period t ≥ 1, ∑tn=1 ∑M ∑n=1 ∑M i=1 zi,n i=1 zi,n · i for all policies P ∈ 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. P Proof. Let RtP := ∑tn=1 ∑M i=1 zi,n · i, for all periods t ≥ 1, and feasible policies P ∈ P. We will show that for any policy P other than the LIFO policy through period n, we can construct another policy P′ which is one unit closer to LIFO through period n, and such that the ′ cumulative age factor amount, RtP , under policy P′ , is not greater than the cumulative age factor amount, RtP , under policy P. For simplicity, we use Rt to represent RtP , and Rt′ to ′ represent RtP . Consider a policy P ∈ P under which unit q of age m, 1 < m ≤ M, is used to ﬁll a unit of demand in period n, for 1 ≤ n ≤ t, and there is a unit q′ of age m′ , m′ < m, that can be used to ﬁll the same unit demand. We will show that there is a feasible policy P′ which ﬁlls the unit demand with q′ , and its cumulative age factor R′n is less than or equal to the cumulative age factor of policy P, Rn , for each n = 1, . . . ,t. We say that R′ is less than or equal to R if and only if R′n ≤ Rn for all n = 1, . . . ,t. Similar to Pierskalla and Roach’s proof, we deﬁne • j := period in which policy P issues item q′ . Note that j ≥ n + 1. • l := period in which policy P′ issues item q, or item q expires. Note that l ≥ n + 1. 33 Note that item q′ is younger than item q, so item q expires before item q′ . Case 1: j < l, i.e., P issues q′ and there is no intervening stockouts under P for which P′ issues q. For all periods u < n, we have that R′u = Ru since they had been issuing under the same policy P. For all periods u, n ≤ u < j, we have that Ru − R′u = m − m′ ≥ 0. For period j, let P′ issues q to satisfy the unit demand for which P issues q′ . Then R′j = R j . As a result, each policy has satisﬁed the same demands through j. Moreover, they have the same inventory remaining. Therefore, R′u = Ru for all u > j. In conclusion, R′ ≤ R when j < l. Case 2: j ≥ l. Again, for all periods u, u < n, we have that R′u = Ru , and for all periods u, M ′ n ≤ u < l, Ru − R′u = m − m′ ≥ 0. Note that ∑M i=1 yi,l = ∑i=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 P′ issues it. Then the remaing inventory under policy P′ is one less than the remaining inventory under policy P. But all other inventories are the same for P and P′ . From construction above and that excess demand is lost, we have that R′u ≤ Ru for all u > l. Suppose P′ issues q in period l to meet demand when q is of age m + (l − n). ′ ′ ′ • If Dl ≥ ∑M i=1 yi,l , then P also issues q in period l. Thus, Rl − Rl = (m + (l − n) − m − (l − n)) + (m − m′ ) = 0. Moreover, the ending inventory for period l is 0 for both P and P′ , and therefore, R′u = Ru , for all u > l. ′ ′ • If Dl < ∑M i=1 yi,l , and if P issues q in period l, then let P issues q for the unit of demand for which P issues q′ . Then Rl − R′l = (m′ + (l − n) − m − (l − n)) + (m − m′ ) = 0. Then as the case above, we get R′u = Ru , for all u > l. If P does not issue q′ in period l, then hold oﬀ issing q in period l for policy P′ , and issue according to P. So Rl − R′l is still m − m′ . Since Dl < ∑M i=1 yi,l , unit q remains in the inventory at the end of period l. Continue this practice until P issues q′ , or unit q expires, or stockout occurs. If P issues q′ ﬁrst, then we are back to the case in which P′ issues q for the unit of demand for which P issues q′ . If unit q expires ﬁrst, then again the remaing inventory under policy P′ is one less than the remaining inventory under policy P. But all other inventories are the 34 same for P and P′ . From construction above and that excess demand is lost, R′u ≤ Ru for all u > l. If stockout occurs ﬁrst, say in period k, k > l, we are back to the case of ′ D k ≥ ∑M i=1 yi,k , with both P and P having 0 inventory at the end of period k. Hence, R′u − Ru = m − m′ for all l ≤ u < k, and R′u = Ru , for all u ≥ k. Therefore, R′ ≤ R if j ≥ l. Consequently, we have constructed a policy P′ 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 ≥ 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 Age xi,1 Qi,1 D1 = 6 zi,1 Ii,1 xi,2 Qi,2 D2 = 4 zi,2 Ii,2 1 0 5 LIFO 2 0 5 3 0 5 1 0 5 (2, 1, 3) 2 0 5 3 0 5 5 0 0 0 1 4 0 0 0 5 4 0 1 4 0 0 5 0 4 0 0 5 0 0 0 0 0 0 4 0 0 0 4 0 0 0 We see in the example that the age factor under LIFO for period 2 is 12 = 4 · 3, greater than the age factor under policy (2, 1, 3), which is 8 = 4 · 2. However, both have cumulative age factor of 19. P FIFO · i ≥ t Theorem 2.4.2. For each period t ≥ 1, ∑tn=1 ∑M ∑n=1 ∑M i=1 zi,n · i for all policies i=1 zi,n P ∈ P 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 P′ which is one unit closer to FIFO, and that RtP ≥ RtP . 35 Consider a policy P ∈ P under which unit q of age m, 1 < m ≤ M, is used to ﬁll a unit demand in period n, for 1 ≤ n ≤ t, and there is a unit q′ of age m′ , m′ > m, that can be used to ﬁll the same unit demand. We will show that there is a feasible policy P′ which ﬁlls the unit demand with q′ , and R′n ≥ Rn , for each n = 1, . . . ,t. Note that age of q′ is older than age of q, so q′ expires before q does. We deﬁne • j := period in which policy P issues item q′ , or item q′ expires. Note that j ≥ n + 1. • l := period in which policy P′ issues item q. Note that l ≥ n + 1. Case 1: j < l. We have that P issues item q′ , or item q′ expires before P′ issues item q. For periods u < n, we have that R′u = Ru . For all periods u, n ≤ u < j, we have that R′u − Ru = m′ − m ≥ 0. For period j, suppose P issues q′ , then let P′ issues q to satisfy the unit demand for which P issues q′ . Then R′j = R j . As a result, each policy has satisﬁed the same demands through j. Moreover, they have the same inventory remaining. Therefore, R′u = Ru for all u > j. Now, suppose q′ expires in period j, then the remaining inventory under policy P′ is one unit more than the remaining inventory under policy P. But all other inventories are the same for P and P′ . From construction above and that excess demand is lost, we get R′u ≥ Ru for all u > l. In conclusion, R′ ≥ R when j < l. Case 2: j ≥ l. Again, for periods u < n, we have that R′u = Ru . For all periods u, M ′ n ≤ u < l, we have that R′u − Ru = m′ − m ≥ 0. Note that ∑M i=1 yi,l = ∑i=1 yi,l since both policies have issued the same amount of blood up until period l. ′ • If Dl ≥ ∑M i=1 yi,l , then P also issues q in period l. Then similar to the arguments used in the proof of Theorem 2.4.1, we get that Rl − R′l = 0, and that R′u = Ru , for all u > l. ′ ′ • If Dl < ∑M i=1 yi,l , and if P issues q in period l, then let P issues q for the unit of demand for which P issues q′ . Then Rl − R′l = (m′ + (l − n) − m − (l − n)) + (m − m′ ) = 0. Then as the case above, we get R′u = Ru , for all u > l. If P does not issue q′ in period l, then hold oﬀ issing q in period l for policy P′ , and issue according to P. So R′l − Rl is still m′ − m. Since Dl < ∑M i=1 yi,l , unit q remains in the inventory at the end of period l. We 36 hold oﬀ issuing q until P issues q′ , or unit q′ expires, or stockout occurs. If P issues q′ ﬁrst, then we are back to the case in which P′ issues q for the unit of demand for which P issues q′ . If unit q′ expires ﬁrst, then again the remaing inventory under policy P′ is one unit more than the remaining inventory under policy P. But all other inventories are the same for P and P′ . From construction above and that excess demand is lost, we get R′u ≥ Ru for all u > l. If stockout occurs ﬁrst, say in period k, k > l, we are ′ back to the case of Dk ≥ ∑M i=1 yi,k , with both P and P having 0 inventory at the end of period k. Hence, R′u − Ru = m − m′ for all l ≤ u < k, and R′u = Ru , for all u ≥ k. Therefore, R′ ≥ R when j ≥ l. Consequently, we have constructed a policy P′ 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)C1 B ≤ E2 +(M+1)C2 , B and E1 ≤ E2 , B > C1 ≥ C2 , B − C1 ≤ E1 ≤ (M + 1)(B − C1 ), B − C2 ≤ E2 ≤ (M + 1)(B − C2 ), where B > 0,C1 ,C2 , E1 , E2 are all non-negative, and M ≥ 1. Then Proof. First, E1 B−C1 ≤ E1 +(M+1)C1 B E2 B−C2 . ≤ E2 +(M+1)C2 B implies that E1 + (M + 1)C1 ≤ E2 + (M + 1)C2 , and so (M + 1)(C1 − C2 ) ≤ E2 − E1 . Since C1 ≥ C2 and M + 1 > 1, we get that C1 − C2 ≤ E2 − E1 . Therefore, E1 +C1 ≤ E2 +C2 . Now since B > 0, we also have that both sides of the numberator by C2 , we get that we get that E1 +(C1 −C2 ) B−C2 ≤ E2 B−C2 . (M + 1)(C1 −C2 ) + E1 . Therefore, Second, we show that E1 B−C1 ≤ E1 +(C1 −C2 ) B ≤ E1 +C1 B E2 B. ≤ E2 +C2 B . Subtracting Moreover, since B > C2 , From (M + 1)(C1 − C2 ) ≤ E2 − E1 , we also get that E2 ≥ E1 +(C1 −C2 ) B−C2 ≤ E1 +(M+1)(C1 −C2 ) B−C2 E1 +(M+1)(C1 −C2 ) . B−C2 ≤ E2 B−C2 . It is true if and only if E1 · B − E1 ·C2 ≤ B · E1 −C1 ·E1 +(M +1)(C1 −C2 )(B −C1 ), if and only if C1 ·E1 −E1 ·C2 ≤ (M +1)(C1 −C2 )(B −C1 ), 37 if and only if E1 (C1 −C2 ) ≤ (M +1)(C1 −C2 )(B−C1 ), if and only if C1 = C2 , or E1 ≤ (M +1)(B− C1 ). Since we are given E1 ≤ (M + 1)(B −C1 ), we get that E1 B−C1 E1 +(M+1)(C1 −C2 ) B−C2 ≤ ≤ E2 B−C2 . LIFO + (M + 1) · t Lemma 2.4.4. For each period t ≥ 1, ∑tn=1 ∑M ∑n=1 SnLIFO ≤ ∑tn=1 ∑M i=1 i · zi,n i=1 i · zPi,n +(M +1)· ∑tn=1 SnP 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. t t P P Proof. As shown in Lemma 2.3.1 that ∑tn=1 ∑M i=1 zi,n = ∑n=1 Dn − ∑n=1 Sn for all feasible poliP cies P and t ≥ 1. Therefore, ∑tn=1 SnP = ∑tn=1 Dn − ∑tn=1 ∑M i=1 zi,n . Then for t ≥ 1, we have that t ∑ M t ∑ i · zPi,n + (M + 1) · ∑ SnP = n=1 i=1 n=1 = t ∑ M t t ∑ i · zPi,n + (M + 1) · ( ∑ Dn − ∑ n=1 i=1 t M n=1 M ∑ zPi,n ) n=1 i=1 t ∑ ∑ (i − (M + 1))zPi,n + (M + 1) ∑ Dn . n=1 i=1 n=1 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 P that the LIFO policy minimizes ∑tn=1 ∑M i=1 (i − (M + 1))zi,n . Moreover, since Dn is the same P for all policies for all period n, we get that the LIFO policy minimizes ∑tn=1 ∑M i=1 i · zi,n + (M + 1) · ∑tn=1 SnP . Theorem 2.4.5. For each period t ≥ 1, and when excess demand is lost ∑tn=1 i · zLIFO ∑tn=1 i · zPi,n i,n ≤ . ∑tn=1 zPi,n ∑tn=1 zLIFO i,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 ≥ 1, t LIFO + (M + 1) · t P P ∑n=1 SnLIFO ∑tn=1 ∑M ∑tn=1 ∑M i=1 i · zi,n i=1 i · zi,n + (M + 1) · ∑n=1 Sn ≤ . ∑tn=1 Dn ∑tn=1 Dn 38 LIFO Now, let B = ∑tn=1 Dn , C1 = ∑tn=1 SnLIFO , C2 = ∑tn=1 SnP , E1 = ∑tn=1 ∑M i=1 i · zi,n , and E2 = P ∑tn=1 ∑M i=1 i · zi,n . Then B, C1 , C2 , E1 , and E2 satisfy conditions speciﬁed in Lemma 2.4.3. As a result, we get that ∑tn=1 i · zLIFO ∑tn=1 i · zPi,n i,n . ≤ ∑tn=1 zPi,n ∑tn=1 zLIFO i,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 · i + c, a > 0, be a function of blood age i. Then for LIFO · f (i) ≤ t P each period t ≥ 1, ∑tn=1 ∑M ∑n=1 ∑M i=1 zi,n i=1 zi,n · f (i) for all policies P ∈ 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: • In Case 1 where j < l, we have that Ru − R′u = a · m + c − a · m′ − c > 0 for all periods u, n ≤ u < j since a > 0 and m > m′ . • In cases where we have P′ issues q in the same period u as P issues q′ , for u ≥ n + 1. We have that Ru − R′u = f (m) − f (m′ ) + f (m′ + (u − n)) − f (m + (u − n)) = 0. Theorem 2.4.7. Let function f (i) = a · i + c, a > 0, be a function of blood age i. Then for P FIFO · f (i) ≥ t each period t ≥ 1, ∑tn=1 ∑M ∑n=1 ∑M i=1 zi,n · f (i) for all policies P ∈ P when excess i=1 zi,n 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: • In Case 1 where j < l, we have that R′u − Ru = a · m′ + c − a · m − c > 0 for all periods u, n ≤ u < j. • In cases where we have P′ issues q in the same period u as P issues q′ , for u ≥ n + 1. We have that R′u − Ru = f (m′ ) − f (m) + f (m + (u − n)) − f (m′ + (u − n)) = 0. Theorem 2.4.8. Let function f (i) be an increasing concave function of blood age i. Then for LIFO · f (i) ≤ t P each period t ≥ 1, ∑tn=1 ∑M ∑n=1 ∑M i=1 zi,n i=1 zi,n · f (i) for all policies P ∈ 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 (m′ ) for m > m′ , and that f (m) − f (m′ ) ≥ f (m + (u − n)) − f (m′ + (u − n)) for u − n > 0. Therefore, in cases where we have P′ issues q in the same period u as P issues q′ , for u ≥ n + 1. We have that Ru − R′u = f (m) − f (m′ ) − ( f (m + (u − n)) − f (m′ + (u − n))) ≥ 0. As a result, R′k ≤ Rk for all k ≥ n. Theorem 2.4.9. Let function f (i) be an increasing concave function of blood age i. Then for FIFO · f (i) ≥ t P each period t ≥ 1, ∑tn=1 ∑M ∑n=1 ∑M i=1 zi,n i=1 zi,n · f (i) for all policies P ∈ 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 P LIFO · f (i) ≥ t each period t ≥ 1, ∑tn=1 ∑M ∑n=1 ∑M i=1 zi,n i=1 zi,n · f (i) for all policies P ∈ 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 (m′ ) > f (m) for m′ < m, and that f (m′ ) − f (m) ≥ f (m′ + (u − n)) − f (m + (u − n)). Therefore, in cases where we have P′ issues q in the same period u as P issues q′ , for u ≥ n + 1. We have that Ru − R′u = f (m) − f (m′ ) − ( f (m + (u − n)) − f (m′ + (u − n))) ≤ 0. As a result, R′k ≥ Rk for all k ≥ n. Theorem 2.4.11. Let function f (i) be a decreasing convex function of blood age i. Then for FIFO · f (i) ≤ t P each period t ≥ 1, ∑tn=1 ∑M ∑n=1 ∑M i=1 zi,n i=1 zi,n · f (i) for all policies P ∈ 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 m′ > 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 minimizes 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. T Min M t=1 i=1 s.t. T T t=1 t=1 h · ∑ ∑ zi,t · i + w · ∑ IM,t + p · ∑ St 0 ≤ zi,t ≤ 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 − zM,t ,t = 1, 2, . . . , T Dt , M when excess z ≤ i,t ∑ Dt + St−1 , when excess i=1 max{0, Dt − ∑M yi,t }, i=1 St = max{0, Dt + St−1 − ∑M i=1 yi,t }, demand is lost demand is backlogged. when excess demand is lost 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 · 1181 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 · N 41 · 118 states, and our action space has L42 states. The 175 in 175 · N 41 · 118 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 ≥ 2 is 0. Therefore, we only need to count variations in the supply of age 1 blood. The 118 in 175 · N 41 · 118 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 , M Ct (Xt , at ) = h · ∑ zi,t · i + w · IM,t + p · St i=1 M = h · ∑ zi,t · i + w · (yM,t − zM,t ) + p · St = h· i=1 M−1 ∑ zi,t · i + (h · M − w)z M,t i=1 44 + w · yM,t + p · St . (3.1) 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 · yM,t and p · St become constants, and Equation 3.1 depends on zt . We observe Equation 3.1 that the coeﬃcient of zi,t is i · h, for i = 1, . . . , M − 1, and the coeﬃcient of zM,t is M · h − w. Each coeﬃcient represents the weight of of releasing one unit of zi,t . Let l be the smallest positive integer that follows M · h − w, 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 − 1, M, l, . . . , M − 1. Note that if l = 1, we issue blood according to age order M, 1, 2, . . . , M − 1. Proof. Let 1′ , 2′ , . . . , M ′ represent the increasing weight order for blood of ages 1 to M, and let wi′ be equal to the weight of the age of blood i′ represents, for i′ = 1′ , 2′ , . . . , M ′ . Then w1′ ≤ w2′ ≤ · · · ≤ wM′ . If demand Dt of period t is greater than or equal to the total updated inventory level ∑M i=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 · ∑M i=1 yi,t · i + p · St , regardless of the order we issue blood, and therefore, CtMyopic (Xt , at ) ≤ Ct (Xt , at ) for all other blood issuing orders. ′ If demand Dt < ∑ki′ =1′ yi′ ,t for k′ ≤ M ′ , where yi′ ,t = yr,t for i′ represents blood of age r. Then under the Myopic policy, the cost of period t, CtMyopic (Ct , at ) k′ k′ −1 = ∑ wi · yi ,t + wk · ( ∑ yi ,t − Dt ) . ′ ′ i′ =1′ ′ ′ i′ =1′ Now suppose for 1′ ≤ j′ ≤ k′ , and k′ ≤ r′ ≤ M ′ , we release one unit of blood of age represented 45 by r′ , and release one unit less of blood of age represented by j′ , then Ct′ (Xt , at ) j′ −1 = k′ −1 ∑ wi · yi ,t + w j · (y j ,t − 1) + wr · 1 + ∑ ′ ′ ′ ′ ′ i′ =1′ i′ = j′ +1 k′ −1 = k′ wi′ · yi′ ,t + wk′ · ( ∑ yi′ ,t − Dt ) i′ =1′ k′ ∑ wi · yi ,t + wk · ( ∑ yi ,t − Dt ) + (wr − w j ) ′ i′ =1′ ′ ′ ′ ′ ′ i′ =1′ ≥ CtMyopic (Xt , at ) since wr′ ≥ w j′ . 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 − 1, M, are 1, 2, . . ., M − 1, M − w, respectively, and that l is the smallest positive integer that follows M − w. Hence, l − 1 < M − w ≤ l. Therefore, when l = 1, we have M − w ≤ 1 < 2 < · · · < M − 1, and thus should issue blood according to age order M, 1, 2, . . . , M − 1 to minimize cost of period t. When l > 1, we have that 1 ≤ · · · ≤ l − 1 < M − w ≤ l ≤ · · · ≤ M − 1, and thus should issue blood according to age order 1, 2, . . . , l − 1, M, l, . . . , M − 1 to minimize cost of period t. 3.4 Policies in Consideration In this section, we will introduce 11 policies that we evaluate in simulation. Each policy we consider needs to be simple, and has a consistent pattern for a decision-maker to implement on a day-to-day basis. The FIFO and LIFO policies we consider below have been considered in Pierskalla and Roach [29], Haijema et al. [15], and Parlar et al. [24] as mentioned earlier. The threshold policy considered in Haijema et al. [15] is the same as the No. 3 threshold policy described below. However, Pierskalla and Roach [29] only considers analytical results of two of the single-objective models we consider in Chapter 2. Haijema et al. [15] implements the FIFO, LIFO, and a threshold issuing policies in order to ﬁnd the optimal ordering policy under each issuing policy. Therefore, their results do not directly apply to our problem. Parlar et al. [24] assume that demand and supply are independent Poisson processes, and use a queueing model with simulation to compare the FIFO and LIFO policies. We use historical data to develop demand and supply distributions, and use 46 an MDP model with simulation to compare the 11 policies below. We consider the FIFO, LIFO, and the No. 3 threshold policies since they have been considered by other researchers, and we would like to see how they perform under our model. Besides the three policies, We are also interested in knowing the outcome of issuing blood age randomly, the outcome of issuing blood based on the inventory level of each blood age, the outcome of the Myopic policy developed earlier, and the outcome of using other types of threshold policies. Note that each threshold policy described below has M − 2 thresholds. Therefore, each threshold policy consists of M − 2 policies. 1. FIFO policy: issue blood according to age order M, M − 1, . . . , 1 at each decision epoch t. 2. LIFO policy: issue blood according to age order 1, 2, . . . , M at each decision epoch t. 3. Random policy: randomly decide on the issuing order of blood age at each decision epoch t. 4. Max inventory policy: order blood based on the inventory level of each blood age at decision epoch t after daily supply has been realized, then issue age of blood that has the highest inventory level ﬁrst. 5. Min inventory policy: order blood in the same way as above, but issue age of blood that has the lowest inventory level ﬁrst. 6. Myopic policy: as described in the previous section. 7. No. 1 threshold policy (threshold age r): issue blood according to age order r, r + 1, . . . , M, r − 1, r − 2, . . . , 1 for r = 2, ..., . . . , M − 1 at each decision epoch t. 8. No. 2 threshold policy: issue blood according to age order M, M − 1, . . . , r, 1, 2, . . . , r − 1 for r = 2, ..., . . . , M − 1 at each decision epoch t. 9. No. 3 threshold policy: issue blood according to age order r −1, r −2, . . . , 1, r, r +1, . . . , M for r = 2, ..., . . . , M − 1 at each decision epoch t. 47 10. No. 4 threshold policy: issue blood according to age order 1, 2, . . . , r − 1, M, M − 1, . . . , r for r = 2, ..., . . . , M − 1 at each decision epoch t. 11. No. 5 threshold policy: issue blood according to age order r, r + 1, . . . , M, 1, 2, . . . , r − 1 for r = 2, ..., . . . , M − 1 at each decision epoch t. 3.5 Data The supply and demand distributions we used for simulation are based on daily blood delivery and tranfusion data from hospital A for the period of April 1, 2010 to March 31, 2011. The delivery data from hospital A contain the date, time, age of blood, blood source (blood supplier or re-distribution from the two smaller hospitals) for each RBC unit delivered. The transfusion data contain the date and time for each RBC unit transfused during that period of time. From the supply data, we are able to generate distributions for daily supply units, age of blood supply, and the number of diﬀerent ages in a daily supply. We then use the three distributions to generate the number of supply units for each blood age for each decision epoch in our simulation. We use the transfused data to obtain the demand distribution for daily RBC units since we were told that the transfused data is a good representation of demand at hospital A. 3.6 Simulation Inputs and Outputs In this section, we ﬁrst describe simulation inputs and outputs. We then display simulation results and compare the above listed policies for both excess demand lost and backlogged cases. Simulation Inputs • T : time horizon, set to 730 days. • Number of runs for simulation: set to 200. • M: maximum blood shelf life, set to 42 days. • p: cost (weight) parameter for shortage. 48 • w: cost (weight) parameter for wastage. • h: cost (weight) parameter for age factor. • Demand distribution per day: provides the probability distribution for the number of RBC units in demand per day. • Supply distribution per day: provides the probability distribution for the number of RBC units delivered per day. • Supply distribution by age: provides the probability distribution for the age of RBC units. • Supply distribution for number of diﬀerent ages: provides the probability distribution for the number of diﬀerent ages in a daily supply. • Excess demand type: backlogging or lost sale. Simulation Outputs • Expected total wastage cost, calculated by adding up the total wastage cost of T periods for each simulation run, and then dividing the sum by the number of runs. Note that by Theorem 2.3.2 in Chapter 2, the FIFO policy yields the minimum expected total wastage cost. • Expected total shortage cost, calculated by adding up the total shortage cost of T periods for each simulation run, and then dividing the sum by the number of runs. Note that by Theorem 2.2.3 in Chapter 2, the FIFO policy yields the minimum expected total shortage cost. • Expected total age factor cost, calculated by adding up the total age factor cost of T periods for each simulation run, and then dividing the sum by the number of runs. Note that by Theorem 2.4.1 in Chapter 2, the LIFO policy yields the minimum expected total age factor cost when excess demand is lost. • Expected total cost, calculated by summing the above costs. 49 • Average percentage of demand that is unfulﬁlled. It is calculated by taking, for each simulation run, the ratio of the total units of unsatisﬁed demand of T periods divided by the total units of demand of T periods, and then adding up the ratios for each simulation run, and dividing by the number of runs. Let N be the total number of simulation runs. We can express the above description mathematically as follows: ∑Nn=1 ( T P St,n ∑t=1 T P ∑t=1 Dt,n ) N . • Average percentage of supply that is wasted. It is calculated by taking, for each simulation run, the ratio of the total waste cost of T periods divided by the total supply of T periods, and then adding up the ratios for each simulation run, and dividing by the number of runs. Again, we can express the above description mathematically as ( follows: ∑Nn=1 T P IM,t,n ∑t=1 T M ∑t=1 ∑i=1 QPi,t,n N 3.7 ) . Simulation Results We need to decide on values of cost parameters, w, p, and h, for wastage, shortage, and age factor, respectively. From Chapter 2, we know that the FIFO policy minimizes expected total wastages and shortages, and the LIFO policy minimizes expected total age factor, we would like to set the cost parameters such that none of the expected total costs of wastages, shortages, or the age factor are dominant, as otherwise results would clearly favour either the FIFO policy or the LIFO policy. We would also like to test a wide enough range of values for each parameter so that we can observe patterns for changes in each parameter. After a process of trial and error, we ﬁnd that when excess demand is lost, using wastage cost w = 100, 150, 200, . . . , 500, shortage cost p = 100, 900, . . . , 1700, and age factor cost h = 1 yield results that satisfy the above conditions. When excess demand is backlogged, using wastage cost w = 100, 150, 200, . . . , 500, shortage cost p = 10, 20, . . . , 100, and age factor cost h = 1 yield results that satisfy the above conditions. Note that we only use age factor cost 50 h = 1 because the age factor term itself is dominant as a result of the age penalty for releasing each age of blood. To illustrate, suppose for some decision period, we release one unit of blood from each age. Then the age factor for the period with the maximum blood shelf life being 42 days is 903 = (1 · 1 + 42 · 1) · 42/2, much higher than amounts of wastages and shortages normally occuring in one period without any penalties. So to test other values of the age factor cost, we would have to adjust the wastage and shortage costs accordingly to give us a balanced result. Since we already know that the only way to minimize the total age factor is to use as much young blood as possible, changing the cost of age factor would not give us more useful information. We would like to note that by deﬁnition of simulation, results generally diﬀer from one simulation to another. However, while costs of shortages and wastages vary with changes of their respective cost parameters p and w, the percentages of shortages and wastages do not. Moreover, since we set the age factor cost parameter h to 1 in all simulation runs, it does not take part in changes of the expected total age factor costs in simulation results. Therefore, we can take the average of percentages of shortages and wastages, and of the expected total age factor cost for all simulation runs for each policy, and compare policies based on these results. After running simulation with the above mentioned cost parameters for both excess demand lost and excess demand backlogged cases, we observe some consistent patterns for each policy. Hence, we organize this section as follows: we begin by describing and interpreting the consistent patterns found in each policy individually. We then display and discuss optimal policies for all sets of cost parameters. Lastly, we perform sensitivity analyses to expected costs with respect to threshold change in the optimal policy, with respect to change in cost parameter w, and with respect to change in cost parameter p. 51 3.7.1 Discussion of Each Individual Policy Discussion of Non-Threshold Policies 1. The FIFO policy: As expected from results in Chapter 2, the FIFO policy yields the lowest expected total wastage and shortage costs and percentages, and the highest expected total age factor costs for all simulation runs. From Table A.1 in A, we see that if we issue blood according to the FIFO policy, issuing the oldest blood ﬁrst, in hospital A over 730 days, we would have about 0.30% of unsatisﬁed demand and 7.42% supply wasted when excess demand is lost. When excess demand is backlogged, we would have about 5.17% of unsatisﬁed demand and 7.14% of supply wasted. 2. The LIFO policy: As expected from results in Chapter 2, the LIFO policy yields the highest expected total wastage and shortage costs and percentages, and the lowest expected total age factor costs for all simulation runs. As summarized in Table A.1, if we issue blood according to the LIFO policy, issuing the youngest blood ﬁrst, we would have about 6.93% of unsatisﬁed demand and 15.27% of supply wasted over 730 days in hospital A when excess demand is lost. When excess demand is backlogged, we would have about 9.15% of supply wastaed. However, close to half (49.16%) of the demand would be unsatisﬁed. Therefore, the LIFO policy is not a good policy to employ for hospital A. 3. The Random policy: If we issue blood age randomly, we would have about 2.14% of unsatisﬁed demand and 10.66% of supply wasted when excess demand is lost, and 26.54% of unsatisﬁed demand and 8.80% of supply wasted when excess demand is backlogged. These percentages, as well as the average expected total age factor costs, and the average expected total costs for both demand cases, are pretty close to their respective averages for all policies as shown in the last row of Table A.1. This says that the Random policy performs better than half of the policies we consider in this thesis. However, it is still not a good policy to employ for hospital A. 4. The Max inventory policy: If we issue blood according to this policy, ordering blood 52 age based on inventory, and issue blood age that has the highest inventory level ﬁrst, we would get about 1.53% and 10.00% of shortages and wastages, respectively, when excess demand is lost; and about 20.75% and 8.68% of shortages and wastages, respectively, when excess demand is backlogged. These percentages are smaller than those of the Random policy. But the average expected total age factor costs of the Max inventory policy for both excess demand cases are bigger than those of the Random policy. Overall, the Max inventory policy yields lower average expected total costs than the Random policy for both excess demand cases. Due to its high wastage and shortage percentages, this policy is still not a good policy for hospital A. 5. The Min inventory policy: If we issue blood in reverse order of the policy above, and issue blood age that has the lowest inventory level ﬁrst, we would get about 4.56% and 13.04% of shortages and wastages, respectively, when excess demand is lost; and about 41.32% and 9.05% of shortages and wastages, respectively, when excess demand is backlogged. These percentages are much higher than those of the Max inventory policy. Moreover, the average expected total costs of this policy are also much higher than those of the Max inventory policy for both excess demand cases. Therefore, issuing blood based on the inventory level of each blood age is not a good way to issue blood for hospital A. However, issuing blood age that has the highest inventory level ﬁrst is better than issuing blood age that has the lowest inventory level ﬁrst. 6. The Myopic policy: Based on cost parameters used, we would issue blood age according to order 42, 1, 2, . . . , 41 under the Myopic policy in all simulation runs. This order allows us to get rid of the blood that will expire in the next period, age 42 blood, ﬁrst, then issue the rest of the blood according to the LIFO policy in each period. By issuing age 42 blood ﬁrst, we are able to achieve 0.40% and 8.24% of shortages and wastages, respectively, when excess demand is lost; and 6.77% and 7.88% of shortages and wastages, respectively, when excess demand is backlogged. These percentages are much less than those of the LIFO policy and are closest to the FIFO policy among all non-threshold policies. However, this also results in a relatively high average expected 53 total age factor cost, comparing with the average for all policies for each excess demand case. But overall, average expected total costs of the Myopic policy are much lower than those averages of all policies for both excess demand cases. The Myopic policy can be a good policy to implement in hospital A. From discussions above and Figure 3.1 below, we observe that the average percentages of shortages of these non-threshold policies when excess demand is backlogged are higher than those when excess demand is lost as shown in Figure 3.1a. But the average percentages of wastages of these policies when excess demand is backlogged are lower than those when excess demand is lost as shown in Figure 3.1b. When excess demand is backlogged, the average expected total age factor costs of the FIFO and the Myopic policies are smaller than those of the two policies when excess demand is lost. But when excess demand is backlogged, the average expected total age factor costs of the other non-threshold policies are bigger than those when excess demand is lost. It seems that for policies that yield relatively low shortage and wastage percentages, their average expected total age factor costs are lower when excess demand is backlogged than when excess demand is lost. We will see later that observations above also apply to the Threshold policies. They make sense since when excess demand is backlogged, we would have more demand in general, causing more shortages, and more usage of supply, thus less wastages. Also, policies that use older blood ﬁrst usually yield higher total age factor because of higher penalties for the use of older blood. When excess demand is backlogged, we need to release more blood to satisfy demand than when excess demand is lost, so there would be less blood left for the next period, and especially less older blood for policies that use up older blood ﬁrst. As a result, the average expected total age factor costs for policies that release more older blood ﬁrst are lower when excess demand is backlogged than when excess demand is lost. 54 Figure 3.1: Average Percentages of Shortages and Wastages, and Average Expected Total Age Factor Costs for Non-Threshold Policies Excess Demand Backlogged vs. Excess Demand Lost (a) Average Shortage Percentage (b) Average Wastage Percentage (c) Average Expected Total Age Factor Cost Discussion of Threshold Policies 1. No. 1 Threshold Policy: we issue blood according to age order r, r + 1, . . . , 42, r − 1, r − 2, . . . , 1 in this policy. Therefore, the smaller the r, the more younger blood we issue ﬁrst. As r increases, it becomes closer and closer to the FIFO policy, except for that the issuing order of blood ages r, . . . , 42 is reversed from the FIFO policy. Hence, the average shortage and wastage percentages decrease as threshold r increases, and the average expected total age factor cost increases as threshold r increases for both excess demand cases as shown in ﬁgures 3.2, 3.3, and 3.4 below. 55 Figure 3.2: Average Percentage of Shortages for the No. 1 Threshold Policy for both excess demand cases Figure 3.3: Average Percentage of Wastages for the No. 1 Threshold Policy for both excess demand cases Figure 3.4: Average Expected Total Age Factor Cost for the No. 1 Threshold Policy for both excess demand cases 2. No. 2 threshold Policy: we issue blood according to age order 42, 41, . . . r, 1, 2, . . . , r − 1 under this policy. This policy always releases the oldest blood ﬁrst, then releases 56 the rest of the blood according to the LIFO policy. Therefore, similar to the Myopic policy, the average percentages of shortages and wastages cost of this policy are equal to or slightly higher than those of the FIFO policy, and the average expected total age factor costs are equal or slightly lower than those of the FIFO policy for both excess demand cases. The average percentages of shortages and wastages increase as threshold r increases, and the average expected total age factor cost descreases as threshold r increases as shown in ﬁgures 3.5, 3.6, and 3.7 below. Note that when threshold r = 2, this policy is the same as the FIFO policy. Also note that when threshold r = 41, we release blood according to age order 42, 41, 1, 2, . . . , 40. This is very similar to the Myopic policy discussed earlier, but besides issuing age 42 blood ﬁrst, it also issues age 41 blood before issuing the rest of the blood according to the LIFO policy. However, by releasing both age 42 and 41 blood, we can reduce the average percentages of shortages and wastages by 22.49% and 8.03%, respectively, when excess demand is lost, and by 25.45% and 8.62%, respectively, when excess demand is backlogged. But we also increase the average expected total age factor cost by about 12% for both excess demand cases. 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 Figure 3.7: Average Expected Total Age Factor Cost for the No. 2 Threshold Policy for both excess demand cases 3. No. 3 threshold Policy: we issue blood according to age order r − 1, r − 2, . . . , 1, r, r + 1, . . . , 42 under this policy. We always issue age 42 blood last. Therefore, the policy always yields higher percentages of shortages and wastages than the FIFO policy for all thresholds r. The average percentages of shortages and wastages decrease, and the average expected total age factor cost increases as r increases as shown in 3.8, 3.9, and 3.10 below. Note that when threshold r = 2, this policy is the same as the LIFO policy, having the highest percentages of shortages and wastages, and the lowest total age factor cost. Note that the smallest percentages of shortages occurs when threshold r = 41 in this policy for both excess demand cases, and they are still higher than those percentages of the Myopic policy. This means that even if we release age 1 to 41 blood ﬁrst according to the FIFO policy, we would still have fewer shortages 58 than if we just release age 42 blood ﬁrst, then release age 1 to 41 blood according to the LIFO policy. When threshold r = 41 in this policy, it yields relatively high average expected total age factor costs and average expect total costs in both excess demand cases. Overall, this policy is not a good policy for hospital A. Figure 3.8: Average Percentage of Shortages for the No. 3 Threshold Policy for both excess demand cases 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 4. No. 4 threshold Policy: we issue blood according to age order 1, 2, . . . , r − 1, 42, 41, . . . , 1 under this policy. We always release the youngest blood ﬁrst. When threshold r is high, this policy is similar to the LIFO policy, and when r is low, the policy is similar to the FIFO policy. When r = 22, the policy issues blood of ages less than or equal to 21 days with the youngest blood ﬁrst before issuing blood of ages older than 21 days with the oldest blood ﬁrst. It turns out that this policy with r around the value of 22 yields the minimum or close to the minimum average expected total cost in most simulation runs. We will give a more detailed discussion of these results in a later section. We can see in ﬁgures 3.11, 3.12, and 3.13 below that the average percentages of shortages and wastages increase, and the average expected total age factor cost decreases as r increases. 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 Figure 3.13: Average Expected Total Age Factor Cost for the No. 4 Threshold Policy for both excess demand cases 5. No. 5 threshold Policy: we issue blood according to age order r, r + 1, . . . , 42, 1, 2, . . . , r − 1 under this policy. We see in ﬁgures 3.14, 3.15, and 3.16 below that the average percentages of shortages and wastages decrease, and the average expected total age factor cost increases as threshold r increases. This policy is similar to the No. 2 threshold policy except for that the issuing order is reversed for blood of ages r, r +1, . . . , 42 in the No. 2 threshold policy. This diﬀerence makes the two policies behave very diﬀerently when r is small, but the policies get closer as r increases, as shown in ﬁgures 3.17, 3.18, and 3.19 below. When r = 41, this policy releases blood according to age order 41, 42, 1, 2, . . . , 40. It is interesting to compare this policy with the No. 2 threshold policy with r = 41 which releases blood according to age order 42, 41, 1, 2, . . . , 40 as mentioned earlier. By releasing age 41 blood before age 42 blood, we increase the av61 erage percentages of shortages and wastages by 14.97% and 6.27%, respectively, when excess demand is lost, and by 12.72% and 6.07%, respectively, when excess demand is backlogged. But the average expected total age factor cost is reduced by 7.68% when excess demand is lost, and by 7.55% when excess demand is backlogged from the No. 2 threshold policy with r = 41. From ﬁgures 3.17, 3.18, and 3.19 below, we can also see that this policy behaves very similar to the No. 1 threshold policy. This is because both policies release blood of ages r, r + 1, . . . , 42 ﬁrst. But this policy has equal or higher average percentage of shortages and wastages, and equal or lower average expected total age factor cost than the No. 1 threshold policy for the same r since this policy releases blood of ages 1, 2, . . . , r − 1 in a LIFO fashion, whereas the opposite is true for the No. 1 threshold policy. 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 Figure 3.16: Average Expected Total Age Factor Cost for the No. 5 Threshold Policy for both excess demand cases Similar to what we observed for the non-threshold policies, we observe from ﬁgures of all threshold policies above that the average percentages of shortages are higher when excess demand is backlogged than when excess demand is lost, and the average percentages of wastages are lower when excess demand is backlogged than when excess demand is lost. Moreover, policies that yield relatively lower shortage and wastage percentages have lower average expected total age factor costs when excess demand is backlogged than when excess demand is lost. We compare the ﬁve threshold policies in ﬁgures 3.17, 3.18, 3.19, and 3.20 below. These ﬁgures are for the excess demand lost case. However, when excess demand is backlogged, the ﬁgures for comparing the ﬁve policies look the same, only with diﬀerent scales. We observe that the No. 1 and No. 5 threshold policies behave very similar to each other as mentioned 63 earlier. They both have relatively high wastage and shortage percentages, and low average expected age factor costs when threshold r is less than 27, but relatively low wastage and shortage percentages, and high average expected total age factor costs when threshold r is greater than or equal to 27. Their average expected total costs are also relatively high when threshold r is less than 27, and relatively low when threshold r is greater than or equal to 27. These observations suggest that the No. 1 and No. 5 threshold policies may be good policies to implement in hospital A when threshold r is greater than or equal to 27. We see that the No. 2 threshold policy yields relatively low shortage and wastage percentages, and high average expected total age factor cost for all thresholds r. Its average expected total costs are also relatively low for all thresholds r, and are close to that of the FIFO policy. This policy can be a good policy to implement for hospital A. However, it would be simpler for a decision maker to implement the FIFO policy than to implement this policy. The No. 3 threshold policy has relatively high percentages of shortages, wastages, and average expected total costs, except for when its threshold r is greater than or equal to 39. However, as we can see in Table A.1 the Myopic policy is a better policy than the No. 3 threshold policy with r greater than or equal to 39 since it yields lower percentages of shortages, similar percentages of wastages, and lower average expected total costs. The No. 4 threshold policy has relatively low percentages of shortages and wastages, and high average expected total age factor costs when threshold r is less than 27. When excess demand is lost, its average expected total costs are less than that of the FIFO policy when threshold r is less than or equal to 25, and when excess demand is backlogged, its average expected total costs are less than that of the FIFO policy when threshold r is less than 25. Although having equal or slightly higher percentages of shortages and wastages than the FIFO policy when threshold r ≤ 25, it has equal or lower average expected total age factor cost than that of the FIFO policy when threshold r ≤ 25. Moreover, this policy allows a decision maker to release much more young blood ﬁrst before releasing the oldest blood. Therefore, the No. 4 threshold policy can be a good policy to implement in hospital A. 64 Figure 3.17: Average Percentage of Shortages for All Threshold Policies when Excess Demand Lost Figure 3.18: Average Percentage of Wastages for All Threshold Policies when Excess Demand Lost 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 Demand Lost 3.7.2 Discussion of Optimal Policies For each excess demand case, we obtain the optimal policy that minimizes the average expected total cost for each set of cost parameters w and p. Results are summarized in Table 3.1 and Table 3.2 below. Note that when excess demand is backlogged, we only run simulation for shortage cost parameter p = 10, 20, . . . , 100. This is because, as we can see in Table A.2 and Table A.3, the average expected total shortage costs across all policies for p = 10, 20, . . . , 100 when excess demand is backlogged are comparable to those for p = 100, 200, . . . , 1700 when excess demand is lost. This happens because we obtain much more shortages when excess demand is backlogged than when excess demand is lost. From Table 3.1, we observe that the optimal policies are the No. 4 threshold policy with various threshold r, except for when w = 500 and p = 1200, in which case the optimal policy is the No. 2 threshold policy with r = 41 (so blood is issued according to age order 42, 41, 1, 2, . . . , 40 as mentioned earlier). But from Table A.1, we see that values of the No. 2 threshold policy with r = 41 are comparable to those of the No. 4 threshold policy with r = 5 and with r = 6. Therefore, instead of implementing the No. 2 threshold policy with r = 41 when w = 500 and p = 1200, a decision maker can implement either the No. 4 threshold policy with r = 5 or No. 4 threshold policy with r = 6 to obtain close to optimal results. We also observe that for each p, the threshold r in the No. 4 threshold policy generally decreases as w increases; and for each w, the threshold r in the No. 4 threshold policy also generally decreases as p increases. Recall that the No. 4 threshold policy issues blood according to 66 age order 1, 2, . . . , r − 1, 42, 41, . . . , r. Our observations suggest that as p and w increase, we issue less and less young blood before issuing the oldest blood in inventory. This makes sense since as shortages and wastages become more and more expensive, we need to reduce costs of wastages and shortages by issuing as much older blood as possible before issuing younger blood. We also notice that when w = 250, and p = 1000 and 1100 in Table A.2 that the average expected total wastages, shortages and the age factor costs are not equal but very close to each other, and in Table 3.1, the corresponding optimal policies for these cost parameters are the No. 4 threshold policy with r = 20 and with r = 21. This tells us that when wastages, shortages, and the age factor cost are about the same, we can minimize the total cost of the three by issuing blood of ages 1 to 19 or 1 to 20 (blood that are less than 21 days old) according to the LIFO policy before issuing the rest of the blood according to the FIFO policy. This is interesting since 21 day is exactly half of the maximum shelf life of red blood cells. Table 3.1: Optimal Policies for All Cost Parameters when Excess Demand is Lost p=100 p=200 p=300 p=400 p=500 w=100 w=150 w=200 w=250 w=300 w=350 w=400 w=450 w=500 No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh r=28 r=27 r=26 r=25 r=24 r=23 r=22 r=21 r=20 No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh r=27 r=25 r=25 r=24 r=23 r=22 r=22 r=22 r=19 No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh r=25 r=25 r=24 r=23 r=23 r=22 r=21 r=20 r=18 No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh r=25 r=24 r=23 r=22 r=22 r=21 r=21 r=20 r=19 No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh r=24 r=23 r=23 r=22 r=22 r=21 r=20 r=20 r=14 and r=17 p=600 p=700 p=800 p=900 p=1000 p=1100 No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh r=23 r=23 r=22 r=22 r=22 r=22 r=20 r=18 r=17 No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh r=23 r=23 r=22 r=22 r=21 r=21 r=18 r=16 r=17 No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh r=22 r=22 r=22 r=21 r=20 r=20 r=20 r=17 r=18 No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh r=22 r=22 r=22 r=20 r=21 r=20 r=19 r=17 r=16 No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh r=22 r=22 r=21 r=20 r=19 r=19 r=18 r=18 r=13 No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh r=22 r=21 r=21 r=21 r=18 r=19 r=19 r=18 r=14 and r=15 Continued on next page 67 Table 3.1 – continued from previous page p=1200 p=1300 p=1400 p=1500 p=1600 w=100 w=150 w=200 w=250 w=300 w=350 w=400 w=450 w=500 No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 2 thresh r=22 r=21 r=21 r=20 r=19 r=19 r=18 r=17 r=41 No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh r=21 r=21 r=21 r=20 r=18 r=18 r=18 r=17 r=16 No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh r=21 r=21 r=20 r=21 r=19 r=17 r=17 r=12 r=15 No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh r=21 r=21 r=20 r=19 r=19 r=16 r=17 r=15 r=15 No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh r=21 r=21 r=19 r=20 r=18 r=18 r=18 r=17 r=12 and r=20 p=1700 No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh r=20 r=21 r=20 r=19 r=19 r=19 r=17 r=13 r=10 From Table 3.2 below, we see that the majority of the optimal policies for w and p are still the No. 4 threshold policy with various threshold r as in the excess demand lost case. Also as p and w increase, the threshold r of the No. 4 threshold policy generally decreases. We see that when w = 300 and p = 70, when w = 350 and p = 80, when w = 400 and p = 70, and when w = 450 and p = 80, the optimal policies are the No. 1 threshold policy with r = 35, 36, 36, 39, respectively, which release blood according to age order r, r + 1, . . . , 42, r − 1, r − 2, . . . , 1. From Table A.1, we see that results of the No. 1 threshold policy with r = 35 are comparable to those of the No. 4 threshold policy with r = 18 and with r = 19. Results of the No. 1 threshold policy with r = 36 are comparable to those of the No. 4 threshold policy with r = 17 and with r = 18. Results of the No. 1 threshold policy with r = 39 are comparable to those of the No. 4 threshold policy with r = 10 and with r = 11. We also see that when w = 400 and p = 60, and when w = 500 and p = 100, the optimal policy is the No. 5 threshold policy with r = 41 which releases blood according to order 41, 42, 1, 2, . . . , 40 as discussed before. When w = 500 and p = 60, the optimal policy is the No. 2 threshold policy with r = 41 which releases blood according to order 42, 41, 1, 2, . . . , 40. From Table A.1, we see that results of the No. 2 threshold policy with r = 41 are comparable to those of the No. 4 threshold policy with r = 5 and with r = 6, and results of the No. 5 threshold policy with r = 41 are comparable to those of the No. 4 threshold policy with r = 15 and with r = 16. Therefore, for cost parameters that yield optimal policies other than the No. 4 threshold policy, a decision-maker can ﬁnd some No. 4 threshold policies 68 that yield close to optimal results. We ﬁnd in Table A.3 that when w = 300 and 350, and when p = 60, the average expected total wastage, shortage, and age factor costs are very close to each other. Their corresponding optimal policies are the No. 4 threshold policy with r = 18 and with r = 19, respectively, as shown Table 3.2. This suggests that we can obtain close to equal wastages, shortages and age factor costs when excess demand is backlogged by ﬁrst releasing blood of ages 1, 2, . . . , 17 or 1, 2, . . . , 18 in the stated order, then releasing the rest of the blood according to the FIFO policy. Recall that when excess demand is lost, we should ﬁrst release blood that are less than 21 days, or half of the maximum RBC shelf life, in an increasing order to obtain close to equal wastage, shortage and age factor costs. To achieve the same objective, we need to release fewer younger blood ﬁrst when excess demand is backlogged. Table 3.2: Optimal Policies for All Cost Parameters when Excess Demand is Backlogged p=10 p=20 p=30 p=40 p=50 w=100 w=150 w=200 w=250 w=300 w=350 w=400 w=450 w=500 No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh r=27 r=27 r=26 r=26 r=26 r=24 r=25 r=23 r=21 No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh r=25 r=25 r=24 r=24 r=23 r=22 r=22 r=21 r=21 No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh r=24 r=23 r=24 r=23 r=22 r=21 r=22 r=19 r=19 No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh r=22 r=22 r=22 r=22 r=21 r=20 r=20 r=18 r=15 No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh r=22 r=21 r=23 r=20 r=20 r=19 r=19 r=16 r=17 and r=18 p=60 p=70 p=80 No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 5 thresh No 4 thresh No 2 thresh r=21 r=21 r=21 r=20 r=18 r=19 r=41 r=18 r=41 No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 1 thresh No 4 thresh No 1 thresh No 4 thresh No 4 thresh r=22 r=19 r=21 r=19 r=35 r=18 r=36 r=17 r=17 No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 1 thresh No 4 thresh No 1 thresh No 4 thresh r=21 r=20 r=18 r=19 r=19 r=36 r=16 r=39 r=16 and r=19 p=90 p=100 No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh r=18 r=20 r=20 r=21 r=16 r=18 r=14 r=16 r=10 No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 4 thresh No 2 thresh r=20 r=19 r=12 r=17 r=17 r=18 r=16 r=16 r=41 3.7.3 Sensitivity Analysis In this section, we perform sensitivity analysis to average expected total wastage costs, average expected total shortage costs, average expected total age factor costs, and average 69 expected total costs with respect to one unit change in threshold r of the No. 4 threshold policy, one unit change in wastage cost w, and one unit change in shortage cost p. We will focus our analysis on the No. 4 threshold policy since as shown in the above section it is the best policy, or close to the best policy for all of our simulation runs. Note that when r = 2 and 3, the No. 4 threshold policy gets the same results as the FIFO policy for hospital A because hospital A never receives blood of ages 1 or 2 days old. In other words, the probability that hospital A receives blood of ages 1 or 2 is 0. Change in Costs to Change in Threshold r of the No. 4 Threshold Policy We calculate percentage changes in average percentages of wastages and shortages, in average expected total age factor costs, and in average expected total costs with respect to one unit increase in threshold r of the No. 4 threshold policy for both excess demand cases, and summarize results in Table A.4. We also plot values of Table A.4 in ﬁgures 3.21, 3.22, 3.23, and 3.24 below. We see in Figure 3.21 that for each one unit increase in r, the percentage of wastages also increases. This makes sense since a unit increase in r means that we release the inventory of one more younger blood before releasing the oldest blood. As discussed earlier, this would increase wastages. We observe that percentage increases in the percentages of wastages ﬂuctuate with one unit increase in r, but in general, percentage increases rise then fall as r increases for both excess demand cases. We also see that the largest increase in percentages of wastages is 4.61%, which occurs when r changes from 30 to 31 when excess demand is lost. When excess demand is backlogged, the largest increase in percentages of wastages is only 1.34%, which occurs when r changes from 22 to 23. We see from Table A.4 that changes in percentages of wastages when excess demand is lost are higher than changes in percentages of wastages when excess demand is backlogged for r greater than 10. For r less than or equal to 10, changes in percentages of wastages when excess demand is lost are lower than changes in percentages of wastages when excess demand is backlogged. On average, with respect to one unit increase in r, or with respect to releasing one more younger blood before releasing the oldest blood, the percentage of wastages is increased by 1.86% when 70 excess demand is lost, and is increased by 0.64% when excess demand is backlogged. Figure 3.21: % Change in the Wastage Percentage with change in Threshold r of the No. 4 Threshold Policy We see in Figure 3.22 that for each unit increase in r, the percentage of shortages also increases, which is expected based on our discussions earlier. We observe that the increase in percentages of shortages generally rises then falls as r increases, and plots for the two excess demand cases have the same shape. The largest increase in percentages of shortages occurs when r increases from 29 to 30 for both excess demand cases: it is 26.87% when excess demand is lost, and 19.33% when excess demand is backlogged. For every one-unit increase in r, the percentage change in the average percentage of shortages when excess demand is lost is higher than those when excess demand is backlogged. On average, with respect to one unit increase in r, the average percentage of shortages is increased by 8.70% when excess demand is lost, and is increased by 6.11% when excess demand is backlogged. Both are higher than their respective percentages for wastages. This tells us that with one unit increase in r, on average, the percentage of shortages increases more than the percentage of wastages for both excess demand cases, and we see in Table A.4 that this observation is especially true for r greater than 10. 71 Figure 3.22: % Change in the Shortage Percentage with change in Threshold r of the No. 4 Threshold Policy We see in Figure 3.23 that the average expected total age factor decreases for each unit increase in r. This is also expected based on our discussions earlier. We observe that the decrease in the average expected total age factor generally rises then falls for both excess demand cases. The largest decrease in the average expected total age factor with one unit increase in r occurs when r increases from 22 to 23 for both excess demand cases: it is 2.46% when excess demand is lost, and 2.40% when excess demand is backlogged. Moreover, the average expected total age factor decreases more when excess demand is lost than when excess demand is backlogged for every one unit increase in r. On average, with respect to one unit increase in r, the average expected total age factor is decreased by 1.31% when excess demand is lost, and is decreased by 1.06% when excess demand is backlogged as in shown Table A.4. Figure 3.23: % Change in the Average Expected Total Age Factor Cost with change in Threshold r of the No. 4 Threshold Policy 72 We see in Table A.4 and Figure 3.24 below that the average expected total cost falls and then rises with one unit increase in r. Moreover, for both excess demand cases, the average expected total cost starts to rise when r increases from 20 to 21. This is interesting, since when r = 21 we release blood of ages exactly less than half of the maximum RBC’s shelf life according to the LIFO policy, before releasing the rest of the blood according to the FIFO policy. Due to diﬀerences in cost parameters used for both excess demand cases, it is not meaningful to compare the two excess demand cases in terms of the average expected total cost. Figure 3.24: % Change in the Average Expected Total Cost with change in Threshold r of the No. 4 Threshold Policy In summary, with one unit increase in threshold r of the No. 4 threshold policy, percentages of wastages and shortages increase, while the average expected total age factor decreases. However, the percentages of shortages increase more than the percentages of wastages for both excess demand cases, suggesting that shortages are more sensitive to one unit change in r than wastages. For both excess demand cases, the average expected total cost decreases for r less than or equal to 20, and increases when r is greater than 20. This means that across all cost parameters used in simulation, we obtain the minimum average expected total cost when we ﬁrst release age 1 to 19 blood in a LIFO fashion, then release the rest of the blood in a FIFO fashion, and that the overall optimal policy is the No. 4 threshold policy with r = 20. 73 Change in Costs to Change in Cost Parameter w As mentioned before, the wastage cost parameter w has no impact on the expected total shortage cost and the expected total age factor cost. It only has impact on the expected total wastage cost and the expected total cost. In this section, we will analyze how much the expected total wastage cost and the expected total cost change with one unit change in w for the No. 4 threshold policy, as well as the FIFO, the LIFO, and the Myopic policies. To do this, we obtain the average expected total wastage costs, and the average expected total costs across the shortage cost parameter p for both excess demand cases for the above mentioned policies. Results are summarized in Table A.5, Table A.6, Table A.7, and Table A.8. From tables A.5, and A.6, we see that the average expected total wastage costs increase as w increases for each policy as expected. We then calculate the change in the average expected total wastage cost with one unit increase in w by taking the diﬀerence of every adjacent average expected total wastage costs and dividing it by 50 for each policy for both excess demand cases. Results are summarized in Table 3.3 and Table 3.4 below. From these tables, we see that the average expected total wastage cost rises for each unit increase in w. As threshold r of the No. 4 threshold policy rises, the increase in the average expected total wastage cost with respect to one unit increase in w also rises for both excess demand cases. Also, increases in the average expected total wastage cost with respect to one unit increase in w when excess demand is lost are higher than when excess demand is backlogged. We know that the percentage of wastages increases as the threshold r of the No. 4 threshold policy increases, and that we get more wastages when excess demand is lost than when excess demand is backlogged. Therefore, these observations suggest that the higher the average expected total wastages a policy produces, the higher the average expected total wastage cost increases, with a one unit increase in cost parameter w. 74 Table 3.3: Change in the Average Expected Total Wastage Cost with respect to One Unit Change in w when Excess Demand is Lost w=100 w=150 w=200 w=250 w=300 w=350 w=400 w=450 Average to 150 to 200 to 250 to 300 to 350 to 400 to 450 to 500 Across w FIFO 3944 4108 4018 3848 4014 3756 3840 3985 3939 LIFO 8116 8237 8134 8049 8172 7956 8117 8138 8115 Myopic 4371 4545 4452 4273 4459 4187 4291 4396 4372 No 4 thresh r=2 3944 4108 4018 3848 4014 3756 3840 3985 3939 No 4 thresh r=3 3944 4108 4018 3848 4014 3756 3840 3985 3939 No 4 thresh r=4 3955 4119 4028 3860 4024 3766 3853 3995 3950 No 4 thresh r=5 3988 4151 4060 3893 4057 3795 3886 4028 3982 No 4 thresh r=6 4020 4181 4091 3924 4092 3822 3919 4059 4013 No 4 thresh r=7 4043 4205 4115 3947 4115 3846 3940 4086 4037 No 4 thresh r=8 4070 4232 4142 3973 4146 3871 3967 4112 4064 No 4 thresh r=9 4097 4260 4168 4000 4172 3899 3995 4135 4091 No 4 thresh r=10 4117 4280 4187 4019 4193 3918 4016 4153 4110 No 4 thresh r=11 4134 4297 4205 4037 4212 3935 4033 4171 4128 No 4 thresh r=12 4151 4314 4221 4053 4229 3952 4050 4188 4145 No 4 thresh r=13 4167 4330 4237 4070 4245 3970 4066 4203 4161 No 4 thresh r=14 4190 4353 4262 4091 4270 3993 4091 4225 4184 No 4 thresh r=15 4218 4380 4288 4120 4296 4022 4123 4251 4212 No 4 thresh r=16 4256 4422 4326 4162 4339 4061 4167 4291 4253 No 4 thresh r=17 4299 4463 4370 4206 4386 4100 4213 4336 4296 No 4 thresh r=18 4342 4510 4415 4246 4434 4143 4257 4377 4341 No 4 thresh r=19 4388 4558 4465 4292 4484 4188 4310 4424 4389 No 4 thresh r=20 4432 4605 4509 4334 4534 4231 4356 4478 4435 No 4 thresh r=21 4485 4662 4565 4388 4585 4286 4415 4538 4491 No 4 thresh r=22 4550 4729 4632 4457 4649 4356 4489 4609 4559 No 4 thresh r=23 4638 4817 4714 4552 4741 4440 4588 4696 4648 No 4 thresh r=24 4724 4900 4797 4646 4830 4514 4689 4780 4735 No 4 thresh r=25 4815 4997 4882 4748 4929 4590 4805 4873 4830 No 4 thresh r=26 4904 5090 4970 4845 5022 4666 4921 4947 4921 No 4 thresh r=27 5014 5194 5074 4958 5125 4764 5044 5041 5027 No 4 thresh r=28 5143 5316 5201 5084 5248 4885 5193 5156 5153 No 4 thresh r=29 5315 5481 5364 5253 5416 5047 5379 5310 5321 No 4 thresh r=30 5542 5699 5583 5474 5641 5267 5612 5532 5544 No 4 thresh r=31 5798 5956 5831 5735 5893 5533 5862 5805 5802 No 4 thresh r=32 6058 6213 6089 6009 6132 5808 6121 6073 6063 No 4 thresh r=33 6325 6484 6350 6287 6391 6094 6391 6339 6333 No 4 thresh r=34 6615 6763 6644 6586 6662 6390 6689 6621 6621 No 4 thresh r=35 6904 7050 6919 6889 6941 6703 6977 6887 6909 No 4 thresh r=36 7182 7323 7195 7162 7212 7002 7254 7168 7187 No 4 thresh r=37 7444 7576 7458 7405 7477 7279 7478 7437 7444 No 4 thresh r=38 7665 7792 7674 7609 7704 7503 7688 7663 7662 No 4 thresh r=39 7830 7967 7842 7773 7879 7676 7843 7842 7832 No 4 thresh r=40 7961 8090 7976 7897 8013 7801 7974 7967 7960 No 4 thresh r=41 8057 8181 8070 7995 8109 7896 8068 8072 8056 Average of All Policies 5793 5942 5838 5712 5866 5602 5762 5816 5791 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 w=100 w=150 w=200 w=250 w=300 w=350 w=400 w=450 Average to 150 to 200 to 250 to 300 to 350 to 400 to 450 to 500 Across w FIFO 3707 3877 3917 3804 3582 4073 3512 4176 3831 LIFO 4775 4954 4981 4874 4654 5140 4585 5236 4900 Myopic 4104 4278 4281 4200 3983 4453 3916 4566 4222 No 4 thresh r=2 3707 3877 3917 3804 3582 4073 3512 4176 3831 No 4 thresh r=3 3707 3877 3917 3804 3582 4073 3512 4176 3831 No 4 thresh r=4 3717 3887 3928 3814 3594 4083 3524 4185 3842 No 4 thresh r=5 3750 3919 3959 3848 3625 4115 3555 4212 3873 No 4 thresh r=6 3781 3949 3990 3879 3653 4150 3581 4248 3904 No 4 thresh r=7 3803 3973 4013 3905 3675 4175 3603 4271 3927 No 4 thresh r=8 3829 3998 4042 3931 3703 4200 3633 4295 3954 No 4 thresh r=9 3856 4024 4068 3960 3731 4226 3660 4317 3980 No 4 thresh r=10 3875 4043 4087 3978 3749 4244 3679 4336 3999 No 4 thresh r=11 3892 4059 4105 3995 3766 4263 3695 4357 4016 No 4 thresh r=12 3909 4075 4122 4010 3780 4280 3708 4376 4032 No 4 thresh r=13 3924 4091 4138 4026 3797 4295 3726 4390 4048 No 4 thresh r=14 3945 4115 4160 4046 3820 4315 3750 4415 4071 No 4 thresh r=15 3972 4140 4187 4071 3846 4342 3775 4442 4097 No 4 thresh r=16 4012 4178 4225 4108 3887 4377 3817 4476 4135 No 4 thresh r=17 4051 4218 4265 4151 3927 4418 3857 4512 4175 No 4 thresh r=18 4092 4259 4302 4191 3967 4457 3898 4544 4214 No 4 thresh r=19 4135 4301 4344 4233 4009 4498 3939 4578 4255 No 4 thresh r=20 4173 4341 4379 4271 4047 4536 3977 4615 4292 No 4 thresh r=21 4217 4385 4424 4314 4088 4582 4019 4661 4336 No 4 thresh r=22 4265 4438 4470 4363 4140 4630 4069 4711 4386 No 4 thresh r=23 4324 4496 4530 4420 4198 4688 4128 4766 4444 No 4 thresh r=24 4374 4543 4578 4466 4248 4734 4179 4820 4493 No 4 thresh r=25 4417 4590 4624 4510 4297 4776 4228 4869 4539 No 4 thresh r=26 4456 4629 4660 4549 4332 4818 4262 4905 4576 No 4 thresh r=27 4492 4667 4700 4586 4367 4858 4297 4943 4614 No 4 thresh r=28 4528 4706 4736 4624 4405 4894 4334 4978 4651 No 4 thresh r=29 4565 4747 4776 4665 4441 4937 4370 5020 4690 No 4 thresh r=30 4606 4787 4815 4708 4481 4979 4409 5064 4731 No 4 thresh r=31 4641 4823 4851 4741 4521 5009 4451 5099 4767 No 4 thresh r=32 4670 4850 4876 4771 4549 5037 4479 5130 4795 No 4 thresh r=33 4694 4873 4899 4796 4570 5063 4500 5156 4819 No 4 thresh r=34 4713 4894 4918 4817 4590 5083 4520 5176 4839 No 4 thresh r=35 4729 4911 4936 4829 4612 5094 4544 5186 4855 No 4 thresh r=36 4743 4924 4953 4838 4626 5105 4557 5200 4868 No 4 thresh r=37 4753 4935 4963 4850 4635 5118 4566 5211 4879 No 4 thresh r=38 4761 4943 4969 4858 4642 5126 4573 5220 4887 No 4 thresh r=39 4766 4947 4975 4865 4647 5132 4578 5227 4892 No 4 thresh r=40 4771 4950 4978 4869 4651 5136 4581 5232 4896 No 4 thresh r=41 4773 4953 4980 4871 4654 5137 4585 5235 4898 Average of All Policies 4345 4521 4549 4443 4223 4708 4154 4807 4469 From Table A.7 and Table A.8, we also see that the average expected total cost increases 76 as the cost parameter w increases. From the two tables, we calculate the change in the average expected total cost with respect to one unit increase in w, and summarize results in Table 3.5 and Table 3.6 below. Similar to observations earlier, the average expected total cost also increases with one unit increase in w, and rises of the average expected total cost with respect to one unit increase in w when excess demand is lost are also higher than those when excess demand is backlogged. This also suggests that the more wastages a policy produces, the higher the increase of the average expected total cost with respect to one unit increase in w. We also see that changes in the average expected total cost are close to changes in the average expected total wastage cost with respect to one unit increase in w. This means that if we increase the cost of wastage by 1, we would increase the average expected total cost by a similar amount to the increase of the average expected total wastage cost. We calculate the average increase of the average expected total cost with respect to one unit increase in w for both excess demand cases for each policy, and display results in the last columns of Table 3.5 and Table 3.6. We also plot the average increase of the average expected total cost for the No. 4 threshold policy for both excess demand cases in Figure 3.25, from which we see that the diﬀerence between the average changes of the two excess demand cases increases as r increases. 77 Table 3.5: Change in the Average Expected Total Cost with respect to One Unit Change in w when Excess Demand is Lost w=100 w=150 w=200 w=250 w=300 w=350 w=400 w=450 Average to 150 to 200 to 250 to 300 to 350 to 400 to 450 to 500 Across w FIFO 3935 4121 4018 3739 3962 3846 3937 3979 3942 LIFO 8026 8141 8134 8171 7967 8011 8763 7978 8149 Myopic 4234 4588 4487 4200 4419 4139 4527 4328 4365 No 4 thresh r=2 3935 4121 4018 3739 3962 3846 3937 3979 3942 No 4 thresh r=3 3935 4121 4018 3739 3962 3846 3937 3979 3942 No 4 thresh r=4 3946 4132 4028 3752 3972 3855 3950 3989 3953 No 4 thresh r=5 3980 4161 4062 3784 4006 3880 3988 4018 3985 No 4 thresh r=6 4015 4190 4094 3815 4046 3899 4029 4044 4017 No 4 thresh r=7 4040 4212 4120 3839 4072 3920 4055 4069 4041 No 4 thresh r=8 4067 4234 4148 3865 4105 3940 4086 4090 4067 No 4 thresh r=9 4092 4261 4176 3891 4135 3965 4120 4105 4093 No 4 thresh r=10 4111 4281 4195 3909 4158 3982 4146 4119 4113 No 4 thresh r=11 4127 4299 4216 3926 4178 3996 4167 4134 4130 No 4 thresh r=12 4142 4315 4233 3944 4195 4010 4189 4146 4147 No 4 thresh r=13 4156 4330 4251 3960 4210 4027 4211 4160 4163 No 4 thresh r=14 4176 4353 4280 3979 4236 4047 4242 4177 4186 No 4 thresh r=15 4199 4380 4308 4009 4261 4074 4282 4200 4214 No 4 thresh r=16 4229 4419 4349 4055 4301 4104 4338 4240 4254 No 4 thresh r=17 4261 4459 4393 4104 4348 4128 4401 4277 4296 No 4 thresh r=18 4294 4506 4438 4150 4395 4162 4461 4310 4340 No 4 thresh r=19 4322 4559 4486 4201 4443 4203 4522 4355 4386 No 4 thresh r=20 4336 4619 4532 4250 4481 4246 4574 4420 4432 No 4 thresh r=21 4364 4683 4600 4305 4520 4298 4643 4485 4487 No 4 thresh r=22 4410 4743 4666 4387 4572 4362 4745 4547 4554 No 4 thresh r=23 4470 4833 4744 4496 4671 4422 4874 4636 4643 No 4 thresh r=24 4549 4898 4840 4602 4760 4471 5009 4718 4731 No 4 thresh r=25 4613 5006 4919 4715 4873 4504 5172 4815 4827 No 4 thresh r=26 4680 5107 4998 4836 4963 4542 5338 4872 4917 No 4 thresh r=27 4802 5206 5107 4951 5061 4614 5504 4955 5025 No 4 thresh r=28 4937 5309 5256 5067 5179 4715 5711 5051 5153 No 4 thresh r=29 5145 5468 5402 5254 5341 4858 5951 5185 5326 No 4 thresh r=30 5393 5685 5603 5498 5559 5061 6233 5392 5553 No 4 thresh r=31 5638 5986 5810 5781 5799 5353 6496 5676 5817 No 4 thresh r=32 5894 6236 6064 6098 5998 5649 6779 5942 6082 No 4 thresh r=33 6154 6538 6290 6429 6214 5975 7071 6194 6358 No 4 thresh r=34 6471 6778 6627 6757 6424 6296 7420 6448 6653 No 4 thresh r=35 6763 7057 6873 7110 6675 6679 7698 6661 6940 No 4 thresh r=36 7041 7301 7160 7374 6912 7040 7990 6924 7218 No 4 thresh r=37 7337 7537 7436 7579 7215 7325 8190 7200 7477 No 4 thresh r=38 7580 7739 7640 7762 7467 7549 8404 7427 7696 No 4 thresh r=39 7728 7931 7797 7922 7654 7731 8520 7640 7865 No 4 thresh r=40 7870 8026 7957 8022 7807 7850 8643 7774 7994 No 4 thresh r=41 7971 8098 8056 8136 7892 7946 8741 7894 8092 Average of All Policies 5715 5921 5837 5724 5749 5615 6184 5714 5807 78 Table 3.6: Change in the Average Expected Total Cost with respect to One Unit Change in w when Excess Demand is Backlogged w=100 w=150 w=200 w=250 w=300 w=350 w=400 w=450 Average to 150 to 200 to 250 to 300 to 350 to 400 to 450 to 500 Across w FIFO 3715 3978 3985 3707 4065 3590 3996 3611 3831 LIFO 7587 3859 5001 5029 5694 4101 5624 2303 4900 Myopic 4056 4527 4333 4176 4307 4128 4240 4012 4222 No 4 thresh r=2 3715 3978 3985 3707 4065 3590 3996 3611 3831 No 4 thresh r=3 3715 3978 3985 3707 4065 3590 3996 3611 3831 No 4 thresh r=4 3725 3993 3989 3719 4077 3600 4006 3623 3842 No 4 thresh r=5 3755 4038 4005 3758 4102 3638 4032 3655 3873 No 4 thresh r=6 3784 4083 4023 3792 4122 3680 4051 3695 3904 No 4 thresh r=7 3804 4120 4032 3819 4147 3703 4075 3719 3927 No 4 thresh r=8 3829 4163 4043 3846 4172 3731 4101 3745 3954 No 4 thresh r=9 3857 4202 4059 3876 4192 3765 4121 3770 3980 No 4 thresh r=10 3877 4230 4071 3893 4206 3787 4136 3791 3999 No 4 thresh r=11 3896 4258 4077 3914 4221 3808 4149 3811 4017 No 4 thresh r=12 3914 4284 4083 3934 4230 3830 4159 3826 4032 No 4 thresh r=13 3931 4312 4084 3954 4243 3849 4172 3841 4048 No 4 thresh r=14 3959 4348 4092 3981 4263 3873 4191 3859 4071 No 4 thresh r=15 3993 4391 4096 4021 4266 3922 4195 3892 4097 No 4 thresh r=16 4040 4446 4114 4069 4284 3981 4214 3933 4135 No 4 thresh r=17 4086 4509 4133 4134 4290 4054 4220 3972 4175 No 4 thresh r=18 4135 4570 4146 4202 4291 4133 4222 4011 4214 No 4 thresh r=19 4185 4629 4166 4284 4295 4212 4226 4040 4255 No 4 thresh r=20 4249 4668 4204 4347 4298 4285 4229 4059 4292 No 4 thresh r=21 4342 4715 4225 4424 4319 4351 4248 4062 4336 No 4 thresh r=22 4414 4812 4219 4508 4358 4411 4288 4076 4386 No 4 thresh r=23 4618 4828 4286 4585 4408 4477 4339 4009 4444 No 4 thresh r=24 4826 4817 4321 4633 4496 4486 4427 3935 4493 No 4 thresh r=25 5002 4790 4439 4667 4596 4477 4527 3812 4539 No 4 thresh r=26 5194 4712 4526 4684 4716 4433 4647 3700 4576 No 4 thresh r=27 5422 4667 4595 4715 4808 4417 4737 3548 4614 No 4 thresh r=28 5671 4521 4690 4762 4922 4376 4853 3411 4651 No 4 thresh r=29 5973 4417 4833 4686 5059 4319 4988 3248 4690 No 4 thresh r=30 6307 4231 4995 4720 5099 4360 5028 3107 4731 No 4 thresh r=31 6611 4203 5001 4750 5206 4323 5137 2905 4767 No 4 thresh r=32 6857 4140 5016 4777 5280 4306 5210 2777 4795 No 4 thresh r=33 7082 4080 4996 4829 5347 4286 5276 2654 4819 No 4 thresh r=34 7269 4054 5025 4851 5375 4298 5304 2534 4839 No 4 thresh r=35 7359 4021 5085 4808 5457 4248 5388 2474 4855 No 4 thresh r=36 7410 4002 5087 4861 5486 4245 5418 2438 4868 No 4 thresh r=37 7492 3969 5036 4919 5545 4208 5476 2385 4879 No 4 thresh r=38 7504 3925 5010 4976 5584 4184 5515 2393 4887 No 4 thresh r=39 7478 3959 4968 5045 5624 4155 5555 2353 4892 No 4 thresh r=40 7530 3928 4936 5064 5598 4189 5528 2394 4896 No 4 thresh r=41 7563 3902 4998 5012 5662 4129 5593 2329 4898 Average of All Policies 5693 4104 4566 4501 4920 4012 4851 3103 4469 Policies 79 Figure 3.25: Average Change in Average Expected Total Cost with One Unit Change in w, Excecss Demand Lost vs. Excess Demand Backlogged Change in Costs to Change in Cost Parameter p As with the wastage cost parameter w, the shortage cost parameter p has no impact on the expected total wastage cost and the expected total age factor cost. It only has impact on the expected total shortage cost and the expected total cost. Similar to Section 3.7.3 above, we will analyze how much the expected total shortage cost and the expected total cost change with one unit increase in p for the No. 4 threshold policy, as well as the FIFO, the LIFO, and the Myopic policies. To do that, we otain the average expected total shortage costs, and the average expected total costs across w for each p for each policy. Results for both excess demand cases are summarized in Table A.9, Table A.10, Table A.11, and Table A.12. We then calculate changes in the average expected total shortage costs with one unit increase in p for both excess demand cases using the same method as we calculate changes in the average expected total wastage costs, and summarize results in Table A.13 and Table 3.7. From Table A.9 and Table A.13, we see that the average expected total shortage cost does not always increases as p increases when excess demand is lost. It actually decreases for the No. 4 threshold policy with r = 8 to 18 when p increases from 1100 to 1200. Moreover, although consistent in most cases, we can see in Table A.13 that the change in the average expected total shortage cost with one unit change in p can vary a lot. This is especially true for p increases from 1100 to 1200, from 1200 to 1300, from 1300 to 1400, and from 1400 to 1500. However, similar to change in the average expected total wastage costs, the 80 more shortages a policy produces, the more change results in the average expected total shortage costs with respect to one unit change in p for the policy. From Table A.10 and Table 3.7, we see that the average expected total shortage cost always increases when p increases when excess demand is backlogged. However, Table 3.7 shows that similar to the excess demand lost case, the change in the average expected total shortage cost with one unit change in p can also vary a lot. For example, when p increases from 70 to 80, changes in the average expected total shortage costs with respect to one unit increase in p are a lot lower than their average changes for all p in the last column in the table for policies other than the LIFO policy. When p increases from 80 to 90, and from 90 to 100, changes in the average expected total shortage costs with respect to one unit increase in p are a lot higher than their average changes for all p. However, we still observe that the more shortages a policy produces, the more the average expected total shortage costs changes with respect to one unit change in p for the policy. Comparing average changes of the average expected total wastage costs for one unit change in w with the average changes of the average expected total shortage costs for one unit change in p, we see that the average change of the average expected total wastage cost is higher than the average change of the average expected total shortage cost when excess demand is lost for each policy. However, when excess demand is backlogged, the average change of the average expected total shortage cost is much higher than the average change of the average expected total wastage cost. These observations are due to the fact that when excess demand is lost, the average expected total wastages for hospital A are higher than those when excess demand is backlogged. When excess demand is lost, the average expected total shortages for hospital A are much lower than those when excess demand is backlogged. To illustrate, we see in Table 3.8 that the average expected total wastages for all policies when excess demand is lost are 32.92% more than those when excess demand is backlogged, and the average expected total shortages for all policies when excess demand is lost are 94.86% less than those when excess demand is backlogged. In Table 3.9, we see that the average expected total wastage cost and the average expected total shortage cost for all policies when excess demand is lost are 29.60% more and 94.66% less, respectively, 81 than those when excess demand is backlogged. Table 3.7: Change in the Average Expected Total Shortage Cost with respect to One Unit Change in p when Excess Demand is Backlogged Policies p=10 p=20 p=30 p=40 p=50 p=60 p=70 p=80 p=90 Average to 20 to 30 to 40 to 50 to 60 to 70 to 80 to 90 to 100 Across p FIFO 3549 2266 4739 3542 4465 6741 95 8244 7544 4576 LIFO 53857 50298 67869 38309 60616 55802 47994 67599 65657 56444 Myopic 4496 2878 7538 3151 6482 7542 1365 10433 9866 5972 No 4 Thresh r=2 3549 2266 4739 3542 4465 6741 95 8244 7544 4576 No 4 Thresh r=3 3549 2266 4739 3542 4465 6741 95 8244 7544 4576 No 4 Thresh r=4 3552 2270 4751 3526 4477 6734 108 8243 7554 4579 No 4 Thresh r=5 3573 2264 4788 3490 4514 6719 165 8235 7579 4592 No 4 Thresh r=6 3606 2249 4828 3453 4558 6701 191 8262 7595 4605 No 4 Thresh r=7 3632 2237 4853 3438 4588 6686 289 8200 7619 4616 No 4 Thresh r=8 3655 2230 4895 3417 4635 6663 420 8129 7674 4635 No 4 Thresh r=9 3681 2219 4940 3408 4662 6650 527 8128 7689 4656 No 4 Thresh r=10 3707 2211 4983 3391 4702 6627 603 8092 7745 4673 No 4 Thresh r=11 3732 2203 5014 3396 4728 6609 684 8056 7793 4691 No 4 Thresh r=12 3750 2199 5048 3408 4754 6589 797 8018 7822 4709 No 4 Thresh r=13 3769 2195 5086 3419 4767 6590 865 8014 7849 4728 No 4 Thresh r=14 3797 2206 5127 3461 4785 6582 1001 7977 7899 4759 No 4 Thresh r=15 3838 2191 5245 3479 4797 6561 1127 8076 7989 4811 No 4 Thresh r=16 3903 2219 5407 3478 4865 6571 1292 8198 8060 4888 No 4 Thresh r=17 3974 2244 5633 3498 4932 6646 1518 8465 8004 4990 No 4 Thresh r=18 4082 2319 5886 3515 5138 6726 1596 8861 8101 5136 No 4 Thresh r=19 4223 2433 6189 3567 5474 6761 1599 9633 8028 5323 No 4 Thresh r=20 4378 2622 6477 3677 5778 6886 1797 10187 8057 5540 No 4 Thresh r=21 4633 2911 6956 3802 6205 7132 1983 10877 8431 5881 No 4 Thresh r=22 5016 3399 7534 4046 6778 7662 2383 12080 8696 6399 No 4 Thresh r=23 5698 4131 8651 4386 7988 8485 2704 14042 8795 7209 No 4 Thresh r=24 6572 4956 10066 4539 9375 9495 3370 15633 9880 8209 No 4 Thresh r=25 7675 6028 11824 4907 11005 10721 4264 17702 11209 9482 No 4 Thresh r=26 8970 7148 13707 5424 12731 12115 4908 19974 13085 10896 No 4 Thresh r=27 10715 8346 16296 5777 14860 13921 5856 22616 15038 12603 No 4 Thresh r=28 12809 10311 19369 6187 17697 15972 7502 25819 16913 14731 No 4 Thresh r=29 15686 12684 23442 7557 20978 18561 10294 28856 20691 17639 No 4 Thresh r=30 19583 15999 28487 10115 24967 22321 13764 32909 25373 21502 No 4 Thresh r=31 23878 19863 33774 13027 29337 26903 17419 37423 30509 25793 No 4 Thresh r=32 27951 24006 38963 15699 34336 30026 22059 41847 34980 29985 No 4 Thresh r=33 31963 28097 43643 19116 38431 33763 26401 45576 40010 34111 No 4 Thresh r=34 36092 32317 48257 22361 42479 38143 30096 49526 45259 38281 No 4 Thresh r=35 39949 35979 52742 25936 46001 42263 34140 52861 50241 42235 No 4 Thresh r=36 43336 39786 56498 28934 49344 46197 37603 55962 54708 45819 No 4 Thresh r=37 46393 43068 59941 31288 52837 48654 40851 59643 57857 48948 No 4 Thresh r=38 48851 45599 62591 33481 55341 51244 43123 62777 60481 51499 No 4 Thresh r=39 50789 47372 64571 35440 57258 52852 45243 63832 63624 53442 No 4 Thresh r=40 52210 48677 66171 36827 58577 54131 46533 65558 64852 54837 No 4 Thresh r=41 53241 49570 67483 37488 60009 55078 47317 67016 64909 55790 Average of All Policies 24587 21976 32298 16891 28637 27112 20030 34269 31987 26421 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 Excess Demand Lost (LS) Excess Demand Backlogged (BK) (LS-BK)/BK Wastages 5825.20 4382.38 32.92% Shortages 1420.48 27636.77 -94.86% Age Factor 1454794.66 1490543.20 -2.40% Table 3.9: Excess Demand Lost vs. Excess Demand Backlogged in terms of Sensitivities of Average Expected Total Wastage and Shortage Costs to w and to p, respectively Excess Demand Lost (LS) Excess Demand Backlogged (BK) (LS-BK)/BK wastage cost parameter w 5791.31 4468.68 29.60% shortage cost parameter p 1411.42 26420.83 -94.64 % We see in Table A.13 and Table A.14 that the average expected total cost also does not always increase with respect to one unit increase in p when excess demand is lost. Moreover, we see in Table A.14 and Table 3.10 that in both excess demand cases, changes in the average expected total costs vary a lot for one unit increase of diﬀerent values of p. But again, as shown in Figure 3.26, the more shortages a policy produces, the more the average expected total cost changes with respect to one unit change in p. With a one unit increase in p, the average expected total cost changes much more when excess demand is backlogged than when excess demand is lost for each policy, suggesting that the average expected total cost is much more sensitive to one unit increase in shortage cost when excess demand is backlogged than when excess demand is lost. Comparing average changes of the average expected total costs for one unit change in w with those for one unit change in p, we see in tables A.14 and 3.10 that while the average change of the average expected total costs for one unit change in w is more than that for one unit change in p for each policy when excess demand is lost, the reverse is true when excess demand is backlogged. Like in Table 3.9, we summarize the average change in average expected total costs for one unit increase in p for both excess demand cases in Table 3.11. We see that results in Table 3.11 are very close to results in Table 3.9 and Table 3.8. This suggests that the average change in the average expected total wastage cost, and the average change in the average expected total cost with one unit increase in wastage cost w are close to the amount of average expected total wastages in hospital A for each excess demand case, and that the average change in the average expected total shortage cost, and the average 83 change in the average expected total cost with one unit increase in shortage cost p are close to the amount of average expected total shortages in hospital A for each excess demand case. We plot the average change in the average expected total cost with respect to one unit increase in p for the No. 4 threshold policy for both excess demand cases in Figure 3.26. We see that the diﬀerence between the average changes in average expected total costs also increases as threshold r increases. The two ﬁgures suggest that if a policy produces more shortages than another policy when excess demand is lost, it would produce even more shortages than the other policy when excess is backlogged; and if a policy produces more wastages than another policy when excess demand is backlogged, it would produce even more wastages than the other policy when excess demand is lost. According to Table A.1, this observation is true for most, but not all policies. 84 Table 3.10: Change in the Average Expected Total Cost with respect to One Unit Change in p when Excess Demand is Backlogged Policies p=10 p=20 p=30 p=40 p=50 p=60 p=70 p=80 p=90 Average to 20 to 30 to 40 to 50 to 60 to 70 to 80 to 90 to 100 Across p FIFO 2692 2960 2651 6203 2347 8168 213 8243 4798 4253 LIFO 52864 50749 66456 40586 58750 57080 48072 67706 63009 56141 Myopic 3628 3497 5502 5908 4047 9169 1511 10563 6809 5626 No 4 Thresh r=2 2692 2960 2651 6203 2347 8168 213 8243 4798 4253 No 4 Thresh r=3 2692 2960 2651 6203 2347 8168 213 8243 4798 4253 No 4 Thresh r=4 2687 2958 2678 6178 2359 8162 228 8232 4829 4257 No 4 Thresh r=5 2701 2949 2740 6127 2429 8111 292 8226 4857 4270 No 4 Thresh r=6 2740 2923 2800 6086 2472 8089 326 8247 4869 4283 No 4 Thresh r=7 2744 2919 2859 6053 2506 8052 446 8176 4891 4294 No 4 Thresh r=8 2763 2908 2914 6034 2551 8003 588 8106 4952 4313 No 4 Thresh r=9 2798 2877 2970 6029 2579 7991 693 8096 4971 4334 No 4 Thresh r=10 2816 2863 3031 6017 2624 7946 786 8042 5047 4352 No 4 Thresh r=11 2839 2829 3069 6031 2639 7931 862 8023 5083 4367 No 4 Thresh r=12 2844 2830 3106 6047 2660 7919 973 7974 5111 4385 No 4 Thresh r=13 2848 2826 3147 6076 2661 7912 1044 7974 5126 4401 No 4 Thresh r=14 2897 2813 3203 6107 2668 7916 1177 7932 5167 4431 No 4 Thresh r=15 2964 2784 3339 6111 2684 7904 1308 8016 5243 4484 No 4 Thresh r=16 3031 2800 3551 6080 2756 7936 1434 8158 5313 4562 No 4 Thresh r=17 3102 2798 3810 6093 2836 7989 1686 8411 5253 4664 No 4 Thresh r=18 3209 2858 4082 6110 3028 8067 1781 8800 5334 4808 No 4 Thresh r=19 3368 2930 4409 6160 3368 8094 1791 9584 5231 4993 No 4 Thresh r=20 3526 3126 4684 6272 3680 8226 2006 10138 5232 5210 No 4 Thresh r=21 3774 3404 5166 6396 4130 8463 2179 10826 5634 5552 No 4 Thresh r=22 4133 3867 5797 6618 4738 8973 2576 12038 5901 6071 No 4 Thresh r=23 4823 4600 6910 6949 5970 9812 2862 14007 6009 6882 No 4 Thresh r=24 5666 5438 8316 7104 7390 10796 3509 15631 7130 7887 No 4 Thresh r=25 6736 6521 10100 7454 9027 12021 4401 17716 8452 9159 No 4 Thresh r=26 8022 7660 11963 7970 10763 13406 5060 20009 10322 10575 No 4 Thresh r=27 9759 8846 14581 8309 12893 15214 5997 22650 12311 12284 No 4 Thresh r=28 11837 10821 17659 8706 15749 17239 7660 25852 14209 14415 No 4 Thresh r=29 14688 13231 21721 10072 19039 19844 10424 28900 17994 17324 No 4 Thresh r=30 18583 16534 26786 12617 23046 23603 13882 32951 22702 21189 No 4 Thresh r=31 22890 20407 32109 15481 27413 28186 17524 37501 27853 25485 No 4 Thresh r=32 26973 24524 37338 18130 32451 31279 22159 41939 32317 29679 No 4 Thresh r=33 30992 28602 42062 21518 36540 35020 26494 45674 37349 33806 No 4 Thresh r=34 35138 32791 46726 24730 40579 39423 30176 49627 42634 37980 No 4 Thresh r=35 38996 36439 51242 28281 44098 43567 34212 52949 47629 41935 No 4 Thresh r=36 42371 40243 55017 31256 47473 47479 37682 56050 52118 45521 No 4 Thresh r=37 45417 43527 58478 33577 50988 49928 40930 59730 55270 48649 No 4 Thresh r=38 47879 46049 61153 35756 53489 52523 43204 62873 57873 51200 No 4 Thresh r=39 49793 47832 63144 37698 55399 54150 45319 63942 60990 53141 No 4 Thresh r=40 51217 49122 64766 39094 56713 55412 46611 65673 62214 54536 No 4 Thresh r=41 52251 50014 66077 39760 58142 56354 47396 67130 62267 55488 Average of All Policies 23646 22528 30567 19377 26629 28461 20135 34346 29252 26105 85 Table 3.11: Excess Demand Lost vs. Excess Demand Backlogged in terms of Sensitivities of Average Expected Total Costs to w and to p Excess Demand Lost (LS) Excess Demand Backlogged (BK) (LS-BK)/BK wastage cost parameter w 5807.37 4468.68 29.96% shortage cost parameter p 1404.49 26104.60 -94.62 % Figure 3.26: Average Change in Average Expected Total Cost with One Unit Change in p, Excecss Demand Lost vs. Excess Demand Backlogged 3.8 Summary of Chapter 3 From results and analyses in this chapter, we learn that we get much higher unsatisﬁed demand, slightly lower wasted supply, and slightly higher total age factor if excess demand is backlogged than if excess demand is lost in hospital A over a two-year period. Intuitively, we may think that we should always issue the oldest blood ﬁrst to reduce wastages and shortages. However, as shown in this chapter, we can achieve a comparable level of wastages and shortages by following the No. 4 threshold policy for both excess demand cases. The No. 4 threshold policy is also a better policy to balance wastages, shortages and age factor. When excess demand is lost, we can equally balance the three factors by releasing blood that are less than half of the maximum RBC shelf life in an increasing age order before releasing the rest of the blood in a decreasing age order. When excess demand is backlogged, we can equally balance the three factors by releasing blood that are 17 or 18 days old in an increasing age order before releasing the rest of the blood in a decreasing age order. If wastages and/or shortages should weight more than the age factor, we can reduce the amount of younger blood we release before releasing older blood with the following in mind: 86 when excess demand is lost, the expected total cost is very sensitive to a unit cost increase in wastages, but not very sensitive to a unit cost increase in shortages. When excess demand is backlogged, the expected total cost is less but still sensitive to a unit cost increase in wastages; however, it is much more sensitive to a unit cost increase in shortages. Overall, with one unit increase in wastage cost, we increase the average expected total cost of a policy by an amount similar to the amount of wastages the policy produces, and with one unit increase in shortage cost, we increase the average expected total cost of a policy by a amount similar to the amount of shortages the policy produces. 87 Chapter 4 Summary and Concluding Remarks 4.1 Summary We have gained the following insights from this project: • The FIFO policy minimizes expected total blood wastages and shortages when excess demand is lost and when excess demand is backlogged. However, it maximizes the expected total age of blood used in transfusion when excess demand is lost. When excess demand is backlogged, it yields the highest expected total age of blood used in transfusion among all policies evaluated in simulation. Therefore, it is not a good policy to use when age of blood used in transfusion is a concern. • The LIFO policy minimizes the expected total and average age of blood used in transfusion when excess demand is lost. When excess demand is backlogged, it yields the lowest expected total age of blood used in transfusion among all policies evaluated in simulation. However, it maximizes expected total blood wastages and shortages for both excess demand cases. Therefore, it is not a good policy to use given that blood shortages and wastages are two big concerns most hospitals face. • The LIFO policy also minimizes the expected total age factor under increasing linear and increasing concave age penalty functions, and maximizes the expected total age factor under a decreasing convex age penalty function when excess demand is lost. 88 The reverse is true for the FIFO policy. • The No. 4 threshold policy with r ≤ 27 is generally the best policy to balance wastages, shortages and age factor by releasing some amount of younger blood ﬁrst in a LIFO fashion before releasing the rest of the blood in a FIFO fahsion. This policy works well because it allows us to release the youngest blood ﬁrst and thus reduces the age factor cost. However, when costs of wastages and shortages are high, we should release the oldest blood sooner to reduce wastages and shortages by reducing the amount of younger blood released ﬁrst. When costs of wastages and shortages are low, we should release the oldest blood later by increasing the amount of younger blood released ﬁrst. With one unit increase in threshold r of the No. 4 threshold policy, the average percentage of shortages and the average percentage of wastages increase while the average expected total age factor decreases. With one unit increase in threshold r of the No. 4 threshold policy, the average expected total cost ﬁrst decreases then increases, and it ﬁrst increases when r increases from 20 to 21. • The average expected total wastage cost, and the average expected total cost increase as the cost of wastages increases, and they both increase more when excess demand is lost than when excess demand is backlogged. Moreover, with one unit cost increase in wastages, both the average expected total wastage cost, and the average expected total cost of a policy are increased by roughly the amount of wastages the policy produces. • The average expected total shortage cost, and the average expected total cost of a policy do not always increase as the cost of shortages increases. However, they do increase on average across all cost parameters. They both increase much more when excess demand is backlogged than when excess demand is lost. Moreover, with one unit cost increase in shortages, both the average expected total shortage cost, and the average expected total cost of a policy are also increased by roughly the amount of shortages the policy produces. 89 • The average expected age factor does not change with change in cost of wastages or cost of shortages. When excess demand is lost, it is sometimes higher and sometimes lower than when excess demand is backlogged. But in general, policies that yield relatively lower shortages and wastages tend to have lower average expected age factor when excess demand is backlogged than when excess demand is lost. Overall for all policies considered in this thesis, the average expected total age factor when excess demand is lost is slightly (2.40%) less than that when excess demand is backlogged. 4.2 Concluding Remarks We can extend this project in many directions. The current project only considers issuing policies of one hospital. As mentioned in the introduction, hospital A has two networked hospitals. It would be helpful to consider issuing policies of all three hospitals. Given that there is only one blood distributor in the region where hospital A is located, we could also consider the blood distribution system in the region as a whole. It would be helpful to know how often the blood distributor should deliver blood to each hospital in the region, and how much it should deliver for each blood age. We did not allow blood reservation in our model. However, it is commonly used in pactice as physicians often reserve blood in advance for elective surgeries. It would be useful to incorporate blood reservation into our model in the future. It would also be useful to incorporate blood type into our model since it is an important element in blood transfusion. 90 Bibliography [1] AABB. Whole blood and blood components. http://www.aabb.org/resources/bct/bloodfacts/Pages/fabloodwhole.aspx. Last visited: 02/14/2013. [2] S.C. Albright. Optimal stock depletion policies with stochastic lives. Management Science, 22(8):852–857, Apr 1976. [3] BC Provincial Blood Coordinating Oﬃce. Red blood cells. http://www.pbco.ca/index. php?option=com_content&task=category§ionid=5&id=15&Itemid=55, February 2013. Last visited: 02/14/2013. [4] E.E. Bomberger. Optimal inventory depletion policies. Management Science, 7(3): 294–303, Apr 1961. [5] M. Brown and S.M. Ross. Optimal issuing policies. 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American Journal of Hematology, 85(2):117–22., Feburary 2010. 93 Appendix A Tables Table A.1: Average expected Shortage and Wastage Percentages, the average Expected Total Age Factor Cost, and the average expected total costs for All Policies across all Simulation Runs for Both Excess Demand Cases Excess Demand Lost Excess Demand Backlogged Policies Average Expected Shortage Wastage Total Age Total Shortage Wastage Total Age % % Factor Cost Cost % % Factor Cost Total Cost FIFO 0.30% 7.42% 1892429.41 3208658.82 5.17% 7.14% 1876802.22 3240161.11 LIFO 6.93% 15.27% 1126613.73 6577004.58 49.16% 9.15% 1237833.33 5716457.78 Random 2.14% 10.66% 1323414.38 3959736.60 26.54% 8.80% 1337551.11 4020532.22 Max Inventory 1.53% 10.00% 1371748.37 3638109.80 20.75% 8.68% 1376260.00 3719894.44 Min Inventory 4.56% 13.04% 1202784.31 5270718.30 41.32% 9.05% 1264484.44 4998940.00 Myopic 0.40% 8.24% 1648246.41 3141221.57 6.77% 7.88% 1637773.33 3188063.33 No. 1 Thresh r=2 6.93% 15.27% 1126613.73 6577004.58 49.16% 9.15% 1237833.33 5716457.78 No. 1 Thresh r=3 6.93% 15.27% 1126613.73 6577004.58 49.16% 9.15% 1237833.33 5716457.78 No. 1 Thresh r=4 6.91% 15.26% 1126951.63 6570255.56 49.13% 9.15% 1237933.33 5712438.89 No. 1 Thresh r=5 6.86% 15.21% 1128352.29 6541960.13 48.97% 9.15% 1238323.33 5696560.00 No. 1 Thresh r=6 6.77% 15.12% 1130881.70 6490488.24 48.70% 9.14% 1239020.00 5668716.67 No. 1 Thresh r=7 6.65% 15.01% 1134309.80 6421071.24 48.35% 9.14% 1239952.22 5632134.44 No. 1 Thresh r=8 6.49% 14.86% 1138684.31 6333118.95 47.89% 9.14% 1241147.78 5585931.11 No. 1 Thresh r=9 6.30% 14.68% 1143984.31 6227637.25 47.32% 9.13% 1242646.67 5529991.11 No. 1 Thresh r=10 6.09% 14.49% 1149906.54 6111254.25 46.68% 9.12% 1244367.78 5468362.22 No. 1 Thresh r=11 5.86% 14.28% 1156312.42 5987512.42 45.97% 9.11% 1246302.22 5401130.00 No. 1 Thresh r=12 5.63% 14.06% 1163115.03 5858789.54 45.20% 9.11% 1248450.00 5330095.56 No. 1 Thresh r=13 5.39% 13.84% 1170250.33 5726952.29 44.37% 9.10% 1250818.89 5255904.44 No. 1 Thresh r=14 5.14% 13.61% 1177942.48 5588898.04 43.45% 9.08% 1253536.67 5175511.11 No. 1 Thresh r=15 4.88% 13.36% 1186366.01 5442616.99 42.41% 9.07% 1256702.22 5087282.22 No. 1 Thresh r=16 4.59% 13.08% 1196026.80 5281303.27 41.19% 9.06% 1260563.33 4987043.33 No. 1 Thresh r=17 4.27% 12.78% 1207060.13 5105522.88 39.76% 9.04% 1265303.33 4873625.56 No. 1 Thresh r=18 3.93% 12.46% 1219479.08 4918256.21 38.10% 9.02% 1271066.67 4748685.56 No. 1 Thresh r=19 3.57% 12.12% 1233307.84 4723392.16 36.21% 8.99% 1277975.56 4611842.22 No. 1 Thresh r=20 3.21% 11.77% 1248324.84 4528007.84 34.12% 8.97% 1286122.22 4469582.22 Continued on next page 94 Table A.1 – continued from previous page Excess Demand Lost Excess Demand Backlogged Policies Average Expected Shortage Wastage Total Age Total Shortage Wastage Total Age % % Factor Cost Cost % % Factor Cost Total Cost No. 1 Thresh r=21 2.86% 11.42% 1264732.03 4333377.12 31.81% 8.93% 1295787.78 4320014.44 No. 1 Thresh r=22 2.50% 11.07% 1282792.81 4141003.27 29.27% 8.90% 1307312.22 4164081.11 No. 1 Thresh r=23 2.15% 10.72% 1302888.89 3952658.17 26.48% 8.85% 1321188.89 4005026.67 No. 1 Thresh r=24 1.83% 10.38% 1324505.23 3780006.54 23.63% 8.80% 1337264.44 3851196.67 No. 1 Thresh r=25 1.54% 10.08% 1347390.20 3628036.60 20.81% 8.74% 1355464.44 3708770.00 No. 1 Thresh r=26 1.29% 9.80% 1371152.29 3500526.80 18.18% 8.68% 1375450.00 3583494.44 No. 1 Thresh r=27 1.08% 9.56% 1395949.02 3395332.03 15.79% 8.62% 1397240.00 3475732.22 No. 1 Thresh r=28 0.91% 9.34% 1421954.90 3310098.04 13.64% 8.55% 1420902.22 3386223.33 No. 1 Thresh r=29 0.76% 9.14% 1449623.53 3241963.40 11.76% 8.47% 1446786.67 3312763.33 No. 1 Thresh r=30 0.64% 8.95% 1479251.63 3190033.99 10.14% 8.39% 1475045.56 3254015.56 No. 1 Thresh r=31 0.55% 8.79% 1510537.25 3153625.49 8.82% 8.30% 1505265.56 3211421.11 No. 1 Thresh r=32 0.48% 8.63% 1542994.77 3130990.20 7.82% 8.21% 1536834.44 3184080.00 No. 1 Thresh r=33 0.42% 8.49% 1576554.25 3119124.18 7.05% 8.11% 1569613.33 3167873.33 No. 1 Thresh r=34 0.38% 8.36% 1611287.58 3115469.93 6.47% 8.01% 1603582.22 3160655.56 No. 1 Thresh r=35 0.35% 8.23% 1646845.75 3118252.94 6.05% 7.90% 1638338.89 3160661.11 No. 1 Thresh r=36 0.33% 8.11% 1683011.11 3125841.18 5.75% 7.80% 1673647.78 3165755.56 No. 1 Thresh r=37 0.32% 7.99% 1719553.59 3136809.15 5.54% 7.68% 1709254.44 3174856.67 No. 1 Thresh r=38 0.31% 7.87% 1756056.21 3150111.76 5.40% 7.57% 1744752.22 3186608.89 No. 1 Thresh r=39 0.30% 7.75% 1792056.21 3164641.18 5.31% 7.46% 1779683.33 3199616.67 No. 1 Thresh r=40 0.30% 7.64% 1827243.14 3179707.84 5.24% 7.35% 1813760.00 3213385.56 No. 1 Thresh r=41 0.30% 7.53% 1861105.88 3194654.90 5.20% 7.24% 1846544.44 3227214.44 No. 2 Thresh r=2 0.30% 7.42% 1892429.41 3208658.82 5.17% 7.14% 1876802.22 3240161.11 No. 2 Thresh r=3 0.30% 7.42% 1892429.41 3208658.82 5.17% 7.14% 1876802.22 3240161.11 No. 2 Thresh r=4 0.30% 7.42% 1892429.41 3208658.82 5.17% 7.14% 1876802.22 3240161.11 No. 2 Thresh r=5 0.30% 7.42% 1892429.41 3208658.82 5.17% 7.14% 1876802.22 3240161.11 No. 2 Thresh r=6 0.30% 7.42% 1892429.41 3208658.82 5.17% 7.14% 1876802.22 3240161.11 No. 2 Thresh r=7 0.30% 7.42% 1892429.41 3208658.82 5.17% 7.14% 1876802.22 3240161.11 No. 2 Thresh r=8 0.30% 7.42% 1892429.41 3208658.82 5.17% 7.14% 1876802.22 3240161.11 No. 2 Thresh r=9 0.30% 7.42% 1892429.41 3208658.82 5.17% 7.14% 1876802.22 3240161.11 No. 2 Thresh r=10 0.30% 7.42% 1892429.41 3208658.82 5.17% 7.14% 1876802.22 3240161.11 No. 2 Thresh r=11 0.30% 7.42% 1892429.41 3208658.82 5.17% 7.14% 1876802.22 3240161.11 No. 2 Thresh r=12 0.30% 7.42% 1892429.41 3208658.82 5.17% 7.14% 1876802.22 3240161.11 No. 2 Thresh r=13 0.30% 7.42% 1892429.41 3208658.82 5.17% 7.14% 1876802.22 3240161.11 No. 2 Thresh r=14 0.30% 7.42% 1892429.41 3208658.82 5.17% 7.14% 1876802.22 3240161.11 No. 2 Thresh r=15 0.30% 7.42% 1892429.41 3208658.82 5.17% 7.14% 1876802.22 3240161.11 No. 2 Thresh r=16 0.30% 7.42% 1892429.41 3208658.82 5.17% 7.14% 1876802.22 3240161.11 No. 2 Thresh r=17 0.30% 7.42% 1892429.41 3208658.82 5.17% 7.14% 1876802.22 3240161.11 No. 2 Thresh r=18 0.30% 7.42% 1892429.41 3208658.82 5.17% 7.14% 1876802.22 3240161.11 No. 2 Thresh r=19 0.30% 7.42% 1892429.41 3208658.82 5.17% 7.14% 1876802.22 3240161.11 No. 2 Thresh r=20 0.30% 7.42% 1892429.41 3208658.82 5.17% 7.14% 1876802.22 3240161.11 No. 2 Thresh r=21 0.30% 7.42% 1892428.76 3208658.82 5.17% 7.14% 1876802.22 3240161.11 No. 2 Thresh r=22 0.30% 7.42% 1892428.76 3208658.82 5.17% 7.14% 1876802.22 3240161.11 No. 2 Thresh r=23 0.30% 7.42% 1892428.76 3208658.82 5.17% 7.14% 1876802.22 3240161.11 No. 2 Thresh r=24 0.30% 7.42% 1892428.76 3208658.82 5.17% 7.14% 1876801.11 3240161.11 No. 2 Thresh r=25 0.30% 7.42% 1892428.10 3208658.82 5.17% 7.14% 1876801.11 3240160.00 No. 2 Thresh r=26 0.30% 7.42% 1892427.45 3208658.82 5.17% 7.14% 1876801.11 3240157.78 No. 2 Thresh r=27 0.30% 7.42% 1892425.49 3208658.17 5.17% 7.14% 1876801.11 3240157.78 Continued on next page 95 Table A.1 – continued from previous page Excess Demand Lost Excess Demand Backlogged Policies Average Expected Shortage Wastage Total Age Total Shortage Wastage Total Age % % Factor Cost Cost % % Factor Cost Total Cost No. 2 Thresh r=28 0.30% 7.42% 1892423.53 3208657.52 5.17% 7.14% 1876796.67 3240155.56 No. 2 Thresh r=29 0.30% 7.42% 1892422.22 3208654.90 5.17% 7.14% 1876791.11 3240154.44 No. 2 Thresh r=30 0.30% 7.42% 1892421.57 3208650.98 5.17% 7.14% 1876786.67 3240148.89 No. 2 Thresh r=31 0.30% 7.42% 1892416.99 3208646.41 5.17% 7.14% 1876786.67 3240144.44 No. 2 Thresh r=32 0.30% 7.42% 1892407.84 3208638.56 5.17% 7.14% 1876781.11 3240134.44 No. 2 Thresh r=33 0.30% 7.42% 1892390.20 3208624.84 5.17% 7.14% 1876767.78 3240127.78 No. 2 Thresh r=34 0.30% 7.42% 1892364.71 3208601.96 5.17% 7.14% 1876737.78 3240094.44 No. 2 Thresh r=35 0.30% 7.42% 1892300.00 3208560.78 5.17% 7.14% 1876691.11 3240054.44 No. 2 Thresh r=36 0.30% 7.42% 1892175.82 3208456.21 5.17% 7.14% 1876551.11 3239952.22 No. 2 Thresh r=37 0.30% 7.42% 1891862.75 3208233.99 5.17% 7.14% 1876262.22 3239786.67 No. 2 Thresh r=38 0.30% 7.42% 1891074.51 3207735.29 5.17% 7.14% 1875461.11 3239331.11 No. 2 Thresh r=39 0.30% 7.43% 1888924.18 3206506.54 5.17% 7.14% 1873358.89 3238174.44 No. 2 Thresh r=40 0.30% 7.45% 1882205.88 3202989.54 5.18% 7.16% 1866724.44 3234968.89 No. 2 Thresh r=41 0.30% 7.53% 1854598.04 3190174.51 5.25% 7.24% 1839558.89 3223464.44 No. 3 Thresh r=2 6.93% 15.27% 1126613.73 6577004.58 49.16% 9.15% 1237833.33 5716457.78 No. 3 Thresh r=3 6.93% 15.27% 1126613.73 6577004.58 49.16% 9.15% 1237833.33 5716457.78 No. 3 Thresh r=4 6.93% 15.27% 1126613.73 6577004.58 49.16% 9.15% 1237833.33 5716457.78 No. 3 Thresh r=5 6.93% 15.27% 1126613.73 6577004.58 49.16% 9.15% 1237833.33 5716457.78 No. 3 Thresh r=6 6.93% 15.27% 1126613.73 6577004.58 49.16% 9.15% 1237833.33 5716457.78 No. 3 Thresh r=7 6.93% 15.27% 1126613.73 6577004.58 49.16% 9.15% 1237833.33 5716457.78 No. 3 Thresh r=8 6.93% 15.27% 1126613.73 6577004.58 49.16% 9.15% 1237833.33 5716457.78 No. 3 Thresh r=9 6.93% 15.27% 1126613.73 6577004.58 49.16% 9.15% 1237833.33 5716457.78 No. 3 Thresh r=10 6.93% 15.27% 1126613.73 6577004.58 49.16% 9.15% 1237833.33 5716457.78 No. 3 Thresh r=11 6.93% 15.27% 1126613.73 6577004.58 49.16% 9.15% 1237833.33 5716457.78 No. 3 Thresh r=12 6.93% 15.27% 1126613.73 6577004.58 49.16% 9.15% 1237833.33 5716457.78 No. 3 Thresh r=13 6.93% 15.27% 1126613.73 6577004.58 49.16% 9.15% 1237833.33 5716457.78 No. 3 Thresh r=14 6.93% 15.27% 1126613.73 6577004.58 49.16% 9.15% 1237833.33 5716457.78 No. 3 Thresh r=15 6.93% 15.27% 1126613.73 6577004.58 49.16% 9.15% 1237833.33 5716457.78 No. 3 Thresh r=16 6.93% 15.27% 1126613.73 6577005.23 49.16% 9.15% 1237833.33 5716457.78 No. 3 Thresh r=17 6.93% 15.27% 1126614.38 6577005.23 49.16% 9.15% 1237833.33 5716457.78 No. 3 Thresh r=18 6.93% 15.27% 1126614.38 6577005.88 49.16% 9.15% 1237833.33 5716457.78 No. 3 Thresh r=19 6.93% 15.27% 1126615.03 6577005.88 49.16% 9.15% 1237833.33 5716457.78 No. 3 Thresh r=20 6.93% 15.27% 1126615.69 6577005.88 49.16% 9.15% 1237833.33 5716458.89 No. 3 Thresh r=21 6.93% 15.27% 1126616.99 6577006.54 49.16% 9.15% 1237833.33 5716458.89 No. 3 Thresh r=22 6.93% 15.27% 1126618.30 6577006.54 49.16% 9.15% 1237835.56 5716460.00 No. 3 Thresh r=23 6.93% 15.27% 1126619.61 6577007.19 49.16% 9.15% 1237835.56 5716464.44 No. 3 Thresh r=24 6.93% 15.27% 1126622.88 6577009.15 49.16% 9.15% 1237835.56 5716466.67 No. 3 Thresh r=25 6.93% 15.27% 1126628.10 6577015.03 49.16% 9.15% 1237835.56 5716468.89 No. 3 Thresh r=26 6.93% 15.27% 1126635.95 6577012.42 49.16% 9.15% 1237842.22 5716467.78 No. 3 Thresh r=27 6.93% 15.27% 1126650.98 6576951.63 49.16% 9.15% 1237847.78 5716461.11 No. 3 Thresh r=28 6.93% 15.27% 1126683.66 6576742.48 49.16% 9.15% 1237856.67 5716373.33 No. 3 Thresh r=29 6.92% 15.26% 1126777.78 6575648.37 49.16% 9.15% 1237894.44 5715856.67 No. 3 Thresh r=30 6.92% 15.26% 1127116.99 6570954.90 49.14% 9.15% 1238018.89 5713573.33 No. 3 Thresh r=31 6.89% 15.23% 1128093.46 6556534.64 49.07% 9.15% 1238333.33 5707046.67 No. 3 Thresh r=32 6.83% 15.18% 1130192.81 6524625.49 48.93% 9.15% 1239015.56 5692333.33 No. 3 Thresh r=33 6.70% 15.05% 1134925.49 6452738.56 48.60% 9.14% 1240584.44 5659045.56 No. 3 Thresh r=34 6.39% 14.76% 1146634.64 6281510.46 47.75% 9.13% 1244705.56 5577021.11 Continued on next page 96 Table A.1 – continued from previous page Excess Demand Lost Excess Demand Backlogged Policies Average Expected Shortage Wastage Total Age Total Shortage Wastage Total Age % % Factor Cost Cost % % Factor Cost Total Cost No. 3 Thresh r=35 5.79% 14.18% 1170937.91 5955252.94 45.97% 9.09% 1254223.33 5414435.56 No. 3 Thresh r=36 4.72% 13.13% 1222622.88 5374122.88 42.04% 9.01% 1278757.78 5088611.11 No. 3 Thresh r=37 3.13% 11.49% 1326298.04 4525694.77 33.84% 8.79% 1342981.11 4522894.44 No. 3 Thresh r=38 1.76% 9.91% 1461773.86 3811395.42 23.17% 8.42% 1450317.78 3925567.78 No. 3 Thresh r=39 1.04% 8.95% 1575464.71 3460001.31 15.54% 8.07% 1555584.44 3571234.44 No. 3 Thresh r=40 0.67% 8.35% 1671701.31 3297189.54 10.77% 7.76% 1651213.33 3379935.56 No. 3 Thresh r=41 0.49% 7.97% 1748837.25 3236198.04 8.23% 7.53% 1729691.11 3297441.11 No. 4 Thresh r=2 0.30% 7.42% 1892429.41 3208658.82 5.17% 7.14% 1876802.22 3240161.11 No. 4 Thresh r=3 0.30% 7.42% 1892429.41 3208658.82 5.17% 7.14% 1876802.22 3240161.11 No. 4 Thresh r=4 0.30% 7.44% 1886150.98 3205741.83 5.18% 7.16% 1870706.67 3237462.22 No. 4 Thresh r=5 0.30% 7.50% 1867410.46 3197060.13 5.19% 7.22% 1852576.67 3229425.56 No. 4 Thresh r=6 0.30% 7.56% 1849006.54 3188591.50 5.21% 7.28% 1834761.11 3221593.33 No. 4 Thresh r=7 0.30% 7.61% 1835005.88 3182217.65 5.23% 7.32% 1821196.67 3215732.22 No. 4 Thresh r=8 0.30% 7.66% 1819015.03 3174982.35 5.25% 7.37% 1805723.33 3209213.33 No. 4 Thresh r=9 0.30% 7.71% 1802972.55 3167842.48 5.28% 7.42% 1790183.33 3202702.22 No. 4 Thresh r=10 0.30% 7.74% 1791501.96 3162818.95 5.30% 7.45% 1779073.33 3198123.33 No. 4 Thresh r=11 0.31% 7.78% 1780949.67 3158288.24 5.32% 7.49% 1768817.78 3193930.00 No. 4 Thresh r=12 0.31% 7.81% 1771100.65 3154111.11 5.35% 7.52% 1759272.22 3190147.78 No. 4 Thresh r=13 0.31% 7.84% 1761361.44 3150055.56 5.37% 7.55% 1749816.67 3186465.56 No. 4 Thresh r=14 0.31% 7.88% 1747580.39 3144491.50 5.41% 7.59% 1736432.22 3181446.67 No. 4 Thresh r=15 0.31% 7.94% 1731302.61 3138156.86 5.46% 7.64% 1720623.33 3175887.78 No. 4 Thresh r=16 0.32% 8.01% 1707432.03 3129449.02 5.56% 7.71% 1697408.89 3168192.22 No. 4 Thresh r=17 0.33% 8.09% 1682529.41 3121375.16 5.69% 7.79% 1673167.78 3161531.11 No. 4 Thresh r=18 0.34% 8.18% 1657430.72 3114609.80 5.86% 7.86% 1648673.33 3156338.89 No. 4 Thresh r=19 0.36% 8.27% 1630998.04 3109476.47 6.09% 7.94% 1622855.56 3152906.67 No. 4 Thresh r=20 0.38% 8.35% 1606667.32 3107004.58 6.36% 8.01% 1599136.67 3152152.22 No. 4 Thresh r=21 0.40% 8.46% 1578594.12 3107573.20 6.77% 8.09% 1571665.56 3155374.44 No. 4 Thresh r=22 0.45% 8.59% 1546256.86 3114167.97 7.38% 8.18% 1540123.33 3165735.56 No. 4 Thresh r=23 0.52% 8.75% 1508203.27 3133726.14 8.40% 8.29% 1503103.33 3191036.67 No. 4 Thresh r=24 0.60% 8.92% 1475742.48 3163881.70 9.59% 8.39% 1471738.89 3226878.89 No. 4 Thresh r=25 0.71% 9.09% 1445183.66 3208596.73 11.08% 8.47% 1442614.44 3278542.22 No. 4 Thresh r=26 0.83% 9.27% 1419400.65 3262298.69 12.66% 8.54% 1418446.67 3338724.44 No. 4 Thresh r=27 0.98% 9.47% 1393616.34 3335124.18 14.60% 8.61% 1394842.22 3416607.78 No. 4 Thresh r=28 1.18% 9.71% 1367656.86 3432540.52 16.96% 8.68% 1371746.67 3519082.22 No. 4 Thresh r=29 1.46% 10.02% 1338981.05 3575177.78 20.04% 8.76% 1347482.22 3661793.33 No. 4 Thresh r=30 1.85% 10.44% 1307981.70 3781388.24 23.91% 8.83% 1323100.00 3857127.78 No. 4 Thresh r=31 2.32% 10.92% 1278819.61 4034388.24 27.96% 8.90% 1302366.67 4078972.22 No. 4 Thresh r=32 2.81% 11.41% 1253857.52 4302339.22 31.61% 8.95% 1286586.67 4299375.56 No. 4 Thresh r=33 3.34% 11.91% 1231484.31 4586756.21 34.94% 9.00% 1274112.22 4517406.67 No. 4 Thresh r=34 3.90% 12.46% 1210288.24 4898534.64 38.06% 9.04% 1263865.56 4738463.33 No. 4 Thresh r=35 4.48% 13.00% 1191205.23 5215896.73 40.79% 9.07% 1255896.67 4948972.22 No. 4 Thresh r=36 5.04% 13.52% 1174373.20 5525696.08 43.11% 9.09% 1249874.44 5141072.22 No. 4 Thresh r=37 5.56% 14.01% 1159936.60 5814652.29 45.02% 9.11% 1245415.56 5310126.67 No. 4 Thresh r=38 6.00% 14.42% 1148436.60 6060676.47 46.47% 9.13% 1242358.89 5446445.56 No. 4 Thresh r=39 6.35% 14.73% 1139890.85 6253296.73 47.53% 9.14% 1240342.22 5549410.00 No. 4 Thresh r=40 6.61% 14.98% 1133670.59 6400190.85 48.29% 9.14% 1239061.11 5625474.44 No. 4 Thresh r=41 6.81% 15.16% 1129256.86 6509441.83 48.83% 9.15% 1238256.67 5681258.89 Continued on next page 97 Table A.1 – continued from previous page Excess Demand Lost Excess Demand Backlogged Policies Average Expected Shortage Wastage Total Age Total Shortage Wastage Total Age % % Factor Cost Cost % % Factor Cost Cost No. 5 Thresh r=2 6.93% 15.27% 1126613.73 6577004.58 49.16% 9.15% 1237833.33 5716457.78 No. 5 Thresh r=3 6.93% 15.27% 1126613.73 6577004.58 49.16% 9.15% 1237833.33 5716457.78 No. 5 Thresh r=4 6.91% 15.26% 1126951.63 6570255.56 49.13% 9.15% 1237933.33 5712438.89 No. 5 Thresh r=5 6.86% 15.21% 1128352.29 6541967.32 48.97% 9.15% 1238323.33 5696570.00 No. 5 Thresh r=6 6.77% 15.12% 1130875.82 6490592.81 48.71% 9.14% 1239020.00 5668776.67 No. 5 Thresh r=7 6.65% 15.01% 1134291.50 6421423.53 48.35% 9.14% 1239941.11 5632387.78 No. 5 Thresh r=8 6.49% 14.86% 1138632.68 6334094.77 47.89% 9.14% 1241135.56 5586541.11 No. 5 Thresh r=9 6.30% 14.69% 1143870.59 6229831.37 47.34% 9.13% 1242604.44 5531431.11 No. 5 Thresh r=10 6.09% 14.49% 1149694.12 6115385.62 46.71% 9.12% 1244287.78 5470894.44 No. 5 Thresh r=11 5.88% 14.29% 1155946.41 5994540.52 46.02% 9.11% 1246170.00 5405453.33 No. 5 Thresh r=12 5.65% 14.08% 1162520.26 5869955.56 45.28% 9.11% 1248244.44 5336830.00 No. 5 Thresh r=13 5.42% 13.87% 1169349.02 5743496.73 44.49% 9.10% 1250494.44 5266076.67 No. 5 Thresh r=14 5.19% 13.65% 1176605.23 5612761.44 43.63% 9.09% 1253024.44 5190492.22 No. 5 Thresh r=15 4.94% 13.42% 1184441.83 5476154.90 42.67% 9.08% 1255933.33 5109112.22 No. 5 Thresh r=16 4.67% 13.16% 1193281.70 5327668.63 41.58% 9.06% 1259400.00 5018330.00 No. 5 Thresh r=17 4.38% 12.89% 1203188.24 5168246.41 40.31% 9.05% 1263567.78 4916937.78 No. 5 Thresh r=18 4.07% 12.60% 1214194.77 5000032.68 38.87% 9.03% 1268500.00 4806575.56 No. 5 Thresh r=19 3.76% 12.30% 1226283.01 4825913.07 37.26% 9.01% 1274335.56 4686971.11 No. 5 Thresh r=20 3.44% 11.99% 1239273.86 4651430.07 35.50% 8.98% 1281073.33 4563324.44 No. 5 Thresh r=21 3.12% 11.68% 1253328.76 4476883.66 33.58% 8.96% 1288947.78 4433590.00 No. 5 Thresh r=22 2.80% 11.36% 1268688.89 4302655.56 31.46% 8.93% 1298195.56 4298283.33 No. 5 Thresh r=23 2.48% 11.04% 1285709.80 4128762.09 29.14% 8.89% 1309212.22 4157255.56 No. 5 Thresh r=24 2.17% 10.73% 1304060.78 3963749.02 26.71% 8.85% 1321944.44 4018114.44 No. 5 Thresh r=25 1.89% 10.44% 1323615.69 3812433.33 24.21% 8.80% 1336534.44 3882794.44 No. 5 Thresh r=26 1.63% 10.16% 1344055.56 3678591.50 21.77% 8.75% 1352710.00 3758604.44 No. 5 Thresh r=27 1.40% 9.91% 1365573.86 3561163.40 19.44% 8.70% 1370567.78 3645112.22 No. 5 Thresh r=28 1.20% 9.68% 1388292.16 3459724.18 17.24% 8.64% 1390218.89 3544552.22 No. 5 Thresh r=29 1.03% 9.46% 1412683.66 3372539.22 15.16% 8.57% 1412068.89 3454628.89 No. 5 Thresh r=30 0.87% 9.26% 1439116.99 3298872.55 13.24% 8.50% 1436476.67 3376442.22 No. 5 Thresh r=31 0.74% 9.07% 1467436.60 3240087.58 11.52% 8.42% 1463183.33 3311365.56 No. 5 Thresh r=32 0.64% 8.89% 1497119.61 3196012.42 10.09% 8.33% 1491627.78 3261441.11 No. 5 Thresh r=33 0.55% 8.73% 1528015.03 3165024.84 8.92% 8.25% 1521481.11 3224750.00 No. 5 Thresh r=34 0.49% 8.59% 1559932.68 3144619.61 7.98% 8.15% 1552511.11 3199380.00 No. 5 Thresh r=35 0.44% 8.45% 1592278.43 3133403.27 7.26% 8.06% 1584021.11 3183603.33 No. 5 Thresh r=36 0.40% 8.32% 1624275.82 3129252.29 6.72% 7.96% 1615307.78 3176160.00 No. 5 Thresh r=37 0.37% 8.21% 1654968.63 3129857.52 6.31% 7.87% 1645355.56 3173806.67 No. 5 Thresh r=38 0.35% 8.11% 1682744.44 3133722.88 6.05% 7.78% 1672433.33 3175815.56 No. 5 Thresh r=39 0.34% 8.03% 1705337.25 3138656.21 5.88% 7.71% 1694342.22 3179812.22 No. 5 Thresh r=40 0.34% 7.98% 1718590.85 3142559.48 5.83% 7.67% 1707180.00 3183248.89 No. 5 Thresh r=41 0.34% 8.00% 1712102.61 3142284.97 5.91% 7.68% 1700728.89 3183131.11 Average 2.90% 10.90% 1457939.26 4463137.30 24.88% 8.34% 1492397.29 4221217.69 98 Total Table A.2: Average Expected Total Shortage, Wastage, and Age Factor Costs for All Policies for Each Cost Parameter when Excess Demand is Lost w=100 w=150 w=200 w=250 w=300 w=350 w=400 w=450 w=500 Wastage 579937.38 874220.87 1159472.72 1483107.28 1760819.90 2039932.04 2331344.17 2619639.81 2952934.47 p=100 Shortage 140733.30 139221.15 140726.61 135439.47 141078.28 140362.89 140631.17 141241.94 138156.85 1456150.00 1460874.76 1458062.14 1461400.97 1455758.74 1455242.23 1458471.84 1456795.63 1460887.38 Wastage 586835.00 867182.09 1165166.36 1446801.41 1773151.46 2033409.22 2306257.77 2606212.14 2904563.11 p=200 Shortage 278694.96 281498.43 278200.44 270802.57 272427.16 280385.17 275843.45 281288.50 288786.66 1455945.63 1457965.05 1460183.01 1459165.53 1459603.40 1457295.15 1460097.57 1459963.59 1454427.67 Wastage 575678.01 868393.74 1147886.36 1460700.92 1764136.41 2058791.26 2305710.19 2635890.78 2852406.31 p=300 Shortage 424955.30 422828.86 431623.24 417475.96 419144.74 419559.71 418447.58 414603.73 428300.19 1456269.42 1457354.85 1457180.10 1458802.91 1458257.28 1458026.70 1458328.64 1456697.57 1456661.65 Wastage 580741.60 870941.60 1169191.02 1437747.43 1734786.41 2042481.07 2322040.29 2623141.26 2933731.55 p=400 Shortage 562207.21 561643.02 566612.26 586952.92 565225.84 565914.61 553368.40 561092.07 552308.83 1457317.48 1459479.61 1457066.99 1456380.58 1456739.81 1455846.60 1458033.50 1457229.13 1457688.83 Wastage 586193.01 877691.46 1189973.35 1455730.29 1762262.14 2015787.38 2291614.56 2630830.58 2916279.13 p=500 Shortage 688304.70 695590.00 686940.47 705151.82 689260.50 716668.82 711947.69 703863.98 707401.08 1459362.62 1459938.35 1459743.69 1455905.83 1459509.22 1456257.77 1458518.45 1457855.83 1458363.59 Wastage 578480.73 878204.90 1161478.11 1477701.94 1750502.91 2029946.12 2313809.22 2600736.89 2889666.50 p=600 Shortage 850931.41 848003.36 848713.00 822657.30 814640.80 839714.06 845631.83 855498.14 831529.91 1458211.17 1456660.19 1458489.81 1459864.56 1461028.64 1457949.03 1458032.04 1458282.52 1459388.35 Wastage 580621.41 873367.23 1181919.85 1442280.05 1749703.40 2037800.49 2272718.45 2615325.24 2874200.97 p=700 Shortage 967585.23 1007357.33 974534.03 991633.16 969493.41 977709.66 1008416.26 1006771.55 971859.22 1457883.98 1458298.54 1455991.75 1456813.59 1457423.79 1459919.90 1457572.33 1454937.86 1458676.70 580741.60 866448.50 1165802.04 1458546.75 1716668.93 2012876.70 2341191.75 2540451.46 2897003.88 1124415.34 1121210.63 1136905.29 1144405.05 1158334.42 1115919.69 1098200.58 1164404.56 1108098.54 1457317.48 1457906.31 1456732.52 1458376.21 1452132.52 1457625.24 1460460.19 1455489.81 1459197.09 586193.01 868254.81 1162250.92 1433202.72 1784804.37 2049265.53 2273827.18 2597784.95 2899448.54 1238941.75 1261265.15 1254182.23 1293157.96 1234002.52 1232898.35 1287496.46 1291269.90 1271706.84 1459362.62 1455222.33 1460587.86 1456358.74 1460147.57 1459007.28 1457559.22 1454641.26 1457468.45 578480.73 882510.44 1181038.83 1491478.64 1777875.24 2029562.62 2320583.98 2598158.25 2850450.97 1418218.93 1397176.94 1379329.61 1387557.04 1389723.83 1400567.86 1405094.71 1406228.40 1448003.64 1458211.17 1457672.33 1458581.55 1456231.07 1460271.36 1458552.91 1459206.31 1458987.86 1457610.19 580621.41 862011.46 1170439.90 1449225.19 1688898.06 2032291.26 2335305.34 2664365.53 2886205.34 1520491.36 1545914.27 1526231.50 1524236.07 1582990.39 1564584.66 1506123.35 1509920.78 1569645.29 1457883.98 1458082.04 1460430.58 1460528.64 1455306.31 1454594.17 1459103.40 1459633.50 1458135.92 572127.82 872897.23 1170964.47 1447159.71 1736150.97 2013666.50 2356559.71 2645567.96 2885993.20 1720111.65 1689543.54 1654220.15 1681141.80 1679489.13 1670094.42 1645301.31 1658732.43 1680320.87 1457522.82 1458000.49 1462597.09 1458833.01 1458495.63 1457599.51 1462497.57 1457197.57 1458898.54 573641.99 854672.86 1188082.28 1479187.38 1717314.08 2079297.57 2323738.83 2604150.97 2910152.91 1858074.32 1819161.31 1779754.85 1798879.81 1883304.51 1804253.93 1797983.64 1812585.05 1854332.23 1455071.84 1458794.66 1458088.83 1460550.49 1456411.17 1458492.72 1460292.72 1459110.68 1456280.10 581589.17 863046.84 1166540.63 1456372.04 1729756.31 2043929.61 2280358.25 2575133.98 2946678.64 1971031.46 1961507.18 1943736.46 1931856.89 1929985.49 1953449.17 1962230.68 2021481.94 1948429.81 1457271.84 1459289.81 1460025.24 1460721.36 1459464.08 1457287.38 1457677.18 1456491.26 1457673.30 578150.92 880309.61 1163187.43 1444434.90 1735690.29 2030013.11 2357241.75 2562760.68 2923369.42 2138868.93 2097943.11 2108013.98 2135649.85 2070624.03 2130730.19 2030414.03 2153328.45 2115903.16 1457090.29 1457337.86 1456295.63 1457976.70 1457704.37 1456527.18 1459737.38 1454139.81 1457800.00 587092.18 863751.94 1143082.96 1471540.29 1741955.34 2090674.27 2301960.19 2579921.84 2893736.41 2200722.43 2237748.45 2281843.50 2203544.51 2236517.67 2131227.86 2274669.42 2295199.85 2290336.46 1457911.65 1457009.22 1457661.65 1457723.30 1456630.10 1459565.53 1455754.37 1456239.81 1456716.50 580591.89 867593.25 1155567.23 1469458.74 1735770.39 2006616.02 2373609.71 2605299.51 2831829.13 Age Age Age Age Age Age Age Wastage p=800 Shortage Age Wastage p=900 Shortage Age Wastage p=1000 Shortage Age Wastage p=1100 Shortage Age Wastage p=1200 Shortage Age Wastage p=1300 Shortage Age Wastage p=1400 Shortage Age Wastage p=1500 Shortage Age Wastage p=1600 Shortage Age Wastage Continued on next page 99 Table A.2 – continued from previous page p=1700 Shortage Age w=100 w=150 w=200 w=250 w=300 w=350 w=400 w=450 w=500 2410718.06 2350076.07 2422583.98 2382519.42 2399193.45 2396534.47 2366409.90 2441070.00 2412996.80 1458633.01 1458427.67 1456093.69 1458026.21 1456209.71 1456916.02 1459465.53 1455766.02 1457562.62 Table A.3: Average Expected Total Shortage, Wastage, and Age Factor Costs for All Policies for Each Cost Parameter when Excess Demand is Backlogged p=10 w=100 w=150 w=200 w=250 w=300 w=350 w=400 w=450 w=500 Wastage 437915.39 677953.69 904093.35 1105460.19 1364391.75 1573263.59 1819183.01 2022769.90 2189577.18 Shortage 273882.07 245702.43 248359.95 250302.47 232643.79 273454.40 232643.79 273454.40 273882.07 1492460.68 1493506.80 1490739.32 1491085.44 1495911.17 1491256.80 1495911.17 1491256.80 1492460.68 Wastage 447181.80 669889.08 872530.05 1145285.49 1329734.47 1548792.23 1772995.15 1991311.17 2235890.78 Shortage 516184.17 494515.49 520175.92 482314.15 493423.88 500462.35 493423.88 500462.35 516184.17 1491956.31 1493166.99 1490302.91 1491908.74 1493846.60 1491923.79 1493846.60 1491923.79 1491956.31 Wastage 450380.00 687614.08 870539.37 1136652.72 1369722.33 1514038.35 1826296.60 1946627.67 2251907.28 Shortage 701792.67 724084.51 693932.28 692223.37 701418.60 789183.52 701418.60 789183.52 701792.67 1494925.24 1495499.03 1491650.97 1494649.03 1490946.60 1493381.07 1490946.60 1493381.07 1494925.24 Wastage 435836.84 649954.42 882033.69 1074961.94 1312337.38 1586506.31 1749763.59 2039791.26 2179193.69 Shortage 1062601.21 1109220.58 1007060.83 1020861.65 1086421.80 983342.48 1086421.80 983342.48 1062601.21 Age 1491000.49 1489598.54 1490724.27 1492629.13 1491783.98 1494753.40 1491783.98 1494753.40 1491000.49 Wastage 452517.48 667336.31 904181.12 1107944.13 1319239.81 1599231.55 1758992.72 2056169.90 2262595.15 Shortage 1177378.01 1317118.74 1174729.90 1316434.17 1192172.23 1187345.58 1192172.23 1187345.58 1177378.01 Age 1493563.11 1493005.34 1491867.96 1491670.87 1490672.33 1494407.77 1490672.33 1494407.77 1493563.11 Wastage 440047.96 648971.70 901843.79 1106932.62 1305485.44 1578638.35 1740640.78 2029679.13 2200238.35 Shortage 1517269.42 1523501.55 1481839.08 1447387.72 1549486.17 1456565.58 1549486.17 1456565.58 1517269.42 Age 1491418.93 1496097.09 1493399.03 1492483.01 1489453.88 1492581.07 1489453.88 1492581.07 1491418.93 Wastage 459155.05 662480.39 905251.55 1122135.34 1350303.88 1521083.50 1800393.69 1955651.46 2295771.84 Shortage 1671920.00 1812868.74 1699403.54 1779201.60 1742542.43 1909546.17 1742542.43 1909546.17 1671920.00 Age 1492073.30 1494029.61 1492232.52 1491846.60 1493422.82 1490702.43 1493422.82 1490702.43 1492073.30 Wastage 448421.46 660068.11 891660.92 1136749.13 1374047.57 1527033.50 1832069.42 1963331.07 2242116.99 Shortage 1906636.99 1976646.89 1992327.72 1936965.49 1936583.20 2074884.13 1936583.20 2074884.13 1906636.99 Age 1491581.07 1492195.15 1492259.22 1494715.05 1494555.83 1492653.88 1494555.83 1492653.88 1491581.07 Wastage 450302.67 654687.09 910909.81 1141321.46 1355171.84 1540395.63 1806902.43 1980514.56 2251533.01 Shortage 2282656.99 2307258.74 2175242.62 2092039.08 2453306.21 2389961.12 2453306.21 2389961.12 2282656.99 Age 1490493.20 1493391.75 1495106.80 1493563.59 1493558.74 1488589.81 1493558.74 1488589.81 1490493.20 446919.47 662215.34 858672.86 1098679.81 1316971.84 1520000.00 1755968.45 1954299.03 2234593.20 2337581.46 2604258.64 2926522.91 2903064.71 2561776.99 2736318.30 2561776.99 2736318.30 2337581.46 1494328.64 1490139.81 1489387.38 1490455.83 1491061.17 1491960.19 1491061.17 1491960.19 1494328.64 Age p=20 Age p=30 Age p=40 p=50 p=60 p=70 p=80 p=90 Wastage p=100 Shortage Age 100 Table A.4: Percentage Change in Costs with respect to Change in Threshold r in the No. 4 Threshold Policy Change in Average Expected No 4 Thresh % of Wastages % of Shortages Tot Age Factor Cost Backlog Lost Sale Backlog r=2 to 3 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% r=3 to 4 0.27% 0.28% 0.11% 0.10% -0.33% -0.32% -0.09% -0.08% r=4 to 5 0.81% 0.83% 0.37% 0.33% -0.99% -0.97% -0.27% -0.25% r=5 to 6 0.78% 0.81% 0.44% 0.38% -0.99% -0.96% -0.26% -0.24% r=6 to 7 0.59% 0.60% 0.38% 0.33% -0.76% -0.74% -0.20% -0.18% r=7 to 8 0.67% 0.68% 0.51% 0.43% -0.87% -0.85% -0.23% -0.20% r=8 to 9 0.66% 0.68% 0.61% 0.50% -0.88% -0.86% -0.22% -0.20% r=9 to 10 0.47% 0.48% 0.49% 0.42% -0.64% -0.62% -0.16% -0.14% r=10 to 11 0.43% 0.44% 0.50% 0.40% -0.59% -0.58% -0.14% -0.13% r=11 to 12 0.40% 0.40% 0.51% 0.44% -0.55% -0.54% -0.13% -0.12% r=12 to 13 0.40% 0.40% 0.55% 0.46% -0.55% -0.54% -0.13% -0.12% r=13 to 14 0.56% 0.56% 0.87% 0.73% -0.78% -0.76% -0.18% -0.16% r=14 to 15 0.66% 0.65% 1.20% 1.04% -0.93% -0.91% -0.20% -0.17% r=15 to 16 0.97% 0.94% 2.12% 1.81% -1.38% -1.35% -0.28% -0.24% r=16 to 17 1.02% 0.97% 2.77% 2.35% -1.46% -1.43% -0.26% -0.21% r=17 to 18 1.03% 0.96% 3.51% 2.99% -1.49% -1.46% -0.22% -0.16% r=18 to 19 1.11% 0.99% 4.65% 3.93% -1.59% -1.57% -0.16% -0.11% r=19 to 20 1.04% 0.90% 5.26% 4.39% -1.49% -1.46% -0.08% -0.02% r=20 to 21 1.25% 1.02% 7.40% 6.43% -1.75% -1.72% 0.02% 0.10% r=21 to 22 1.52% 1.16% 10.52% 9.06% -2.05% -2.01% 0.21% 0.33% r=22 to 23 1.95% 1.34% 15.87% 13.73% -2.46% -2.40% 0.63% 0.80% r=23 to 24 1.85% 1.11% 16.33% 14.19% -2.15% -2.09% 0.96% 1.12% r=24 to 25 1.98% 1.03% 17.96% 15.50% -2.07% -1.98% 1.41% 1.60% r=25 to 26 1.90% 0.84% 16.77% 14.34% -1.78% -1.68% 1.67% 1.84% r=26 to 27 2.17% 0.82% 18.41% 15.28% -1.82% -1.66% 2.23% 2.33% r=27 to 28 2.52% 0.81% 19.95% 16.15% -1.86% -1.66% 2.92% 3.00% r=28 to 29 3.25% 0.85% 23.56% 18.18% -2.10% -1.77% 4.16% 4.06% r=29 to 30 4.18% 0.87% 26.87% 19.33% -2.32% -1.81% 5.77% 5.33% r=30 to 31 4.61% 0.76% 25.51% 16.91% -2.23% -1.57% 6.69% 5.75% r=31 to 32 4.48% 0.59% 21.26% 13.07% -1.95% -1.21% 6.64% 5.40% r=32 to 33 4.43% 0.49% 18.49% 10.53% -1.78% -0.97% 6.61% 5.07% r=33 to 34 4.55% 0.42% 17.01% 8.92% -1.72% -0.80% 6.80% 4.89% r=34 to 35 4.36% 0.34% 14.74% 7.17% -1.58% -0.63% 6.48% 4.44% r=35 to 36 4.03% 0.28% 12.50% 5.68% -1.41% -0.48% 5.94% 3.88% r=36 to 37 3.58% 0.22% 10.33% 4.43% -1.23% -0.36% 5.23% 3.29% r=37 to 38 2.92% 0.16% 7.96% 3.24% -0.99% -0.25% 4.23% 2.57% r=38 to 39 2.21% 0.11% 5.76% 2.28% -0.74% -0.16% 3.18% 1.89% r=39 to 40 1.64% 0.08% 4.15% 1.59% -0.55% -0.10% 2.35% 1.37% r=40 to 41 1.20% 0.05% 2.96% 1.12% -0.39% -0.06% 1.71% 0.99% Average 1.86% 0.64% 8.70% 6.11% -1.31% -1.06% 1.86% 1.47% 101 Lost Sale Backlog Tot Cost Lost Sale Lost Sale Backlog Table A.5: Average Expected Total Wastage Costs for Each w for Selected Policies when Excess Demand is Lost Policies w=100 w=150 w=200 w=250 w=300 w=350 w=400 w=450 w=500 FIFO 395927 593148 798573 999475 1191882 1392565 1580359 1772376 1971624 LIFO 811768 1217559 1629400 2036106 2438553 2847141 3244965 3650806 4057682 Myopic 439544 658102 885335 1107929 1321559 1544512 1753847 1968418 2188229 No 4 thresh r=2 395927 593148 798573 999475 1191882 1392565 1580359 1772376 1971624 No 4 thresh r=3 395927 593148 798573 999475 1191882 1392565 1580359 1772376 1971624 No 4 thresh r=4 396991 594763 800725 1002139 1195147 1396329 1584606 1777271 1977024 No 4 thresh r=5 400171 599561 807106 1010108 1204741 1407588 1597329 1791624 1993035 No 4 thresh r=6 403285 604276 813341 1017884 1214065 1418659 1609771 1805700 2008665 No 4 thresh r=7 405664 607838 818109 1023845 1221218 1426982 1619294 1816294 2020612 No 4 thresh r=8 408380 611897 823482 1030581 1229253 1436547 1630082 1828424 2034024 No 4 thresh r=9 411082 615948 828956 1037365 1237371 1445947 1640912 1840647 2047394 No 4 thresh r=10 413012 618843 832829 1042176 1243118 1452753 1648653 1849435 2057071 No 4 thresh r=11 414830 621524 836386 1046618 1248453 1459053 1655788 1857435 2065994 No 4 thresh r=12 416485 624025 839715 1050765 1253424 1464853 1662435 1864941 2074318 No 4 thresh r=13 418139 626504 842992 1054835 1258347 1470588 1669065 1872382 2082553 No 4 thresh r=14 420489 629995 847632 1060712 1265271 1478794 1678459 1883018 2094265 No 4 thresh r=15 423275 634166 853166 1067571 1273571 1488388 1689465 1895635 2108176 No 4 thresh r=16 427441 640242 861341 1077665 1285765 1502729 1705788 1914141 2128700 No 4 thresh r=17 431802 646742 869867 1088376 1298688 1517971 1722976 1933618 2150394 No 4 thresh r=18 436235 653330 878826 1099559 1311865 1533547 1740718 1953582 2172453 No 4 thresh r=19 441096 660499 888402 1111629 1326212 1550412 1759806 1975282 2196506 No 4 thresh r=20 445731 667310 897575 1123029 1339718 1566400 1777965 1995782 2219688 No 4 thresh r=21 451285 675528 908650 1136912 1356324 1585565 1799853 2020594 2247494 No 4 thresh r=22 458203 685716 922153 1153776 1376641 1609071 1826859 2051288 2281729 No 4 thresh r=23 467125 699002 939859 1175547 1403159 1640218 1862206 2091606 2326406 No 4 thresh r=24 475692 711903 956921 1196747 1429053 1670571 1896288 2130724 2369712 No 4 thresh r=25 485051 725799 975652 1219765 1457153 1703594 1933082 2173318 2416965 No 4 thresh r=26 494248 739425 993947 1242471 1484735 1735859 1969141 2215171 2462529 No 4 thresh r=27 504817 755512 1015213 1268935 1516859 1773100 2011300 2263482 2515524 No 4 thresh r=28 517431 774557 1040347 1300400 1554612 1817012 2061282 2320912 2578735 No 4 thresh r=29 533996 799726 1073776 1341965 1604629 1875435 2127800 2396759 2662247 No 4 thresh r=30 556050 833130 1118088 1397253 1670976 1953006 2216335 2496947 2773524 No 4 thresh r=31 581459 871357 1169171 1460706 1747453 2042094 2318759 2611876 2902135 No 4 thresh r=32 607305 910191 1220841 1525276 1825712 2132329 2422724 2728765 3032429 No 4 thresh r=33 633940 950180 1274400 1591882 1906229 2225782 2530465 2850012 3166953 No 4 thresh r=34 662591 993329 1331459 1663641 1992935 2326024 2645547 2980012 3311071 No 4 thresh r=35 691361 1036547 1389071 1735029 2079471 2426500 2761635 3110471 3454806 No 4 thresh r=36 719302 1078400 1444553 1804300 2162418 2523041 2873129 3235841 3594229 No 4 thresh r=37 744848 1117053 1495859 1868759 2239018 2612882 2976835 3350741 3722600 No 4 thresh r=38 766556 1149788 1539371 1923065 2303524 2688741 3063894 3448282 3831418 No 4 thresh r=39 783492 1174988 1573329 1965412 2354076 2748035 3131835 3524006 3916129 No 4 thresh r=40 796315 1194341 1598853 1997659 2392488 2793159 3183224 3581906 3980241 No 4 thresh r=41 805832 1208706 1617765 2021282 2421053 2826494 3221306 3624724 4028306 Average of All Policies 580454 870088 1167179 1459099 1744720 2038020 2318110 2606198 2896979 102 Table A.6: Average Expected Total Wastage Costs for Each w for Selected Policies when Excess Demand is Backlogged Policies w=100 w=150 w=200 w=250 w=300 w=350 w=400 w=450 w=500 FIFO 383092 568439 762270 958114 1148300 1327410 1531070 1706670 1815230 LIFO 489977 728714 976390 1225450 1469130 1701850 1958840 2188100 2323150 Myopic 422244 627433 841321 1055390 1265380 1464510 1687180 1882960 2001940 No 4 thresh r=2 383092 568439 762270 958114 1148300 1327410 1531070 1706670 1915470 No 4 thresh r=3 383092 568439 762270 958114 1148300 1327410 1531070 1706670 1915470 No 4 thresh r=4 384157 570015 764366 960790 1151510 1331200 1535360 1711540 1920780 No 4 thresh r=5 387285 574767 770702 968658 1161060 1342290 1548060 1725820 1936440 No 4 thresh r=6 390387 579420 776881 976405 1170370 1353020 1560500 1739560 1951940 No 4 thresh r=7 392734 582861 781525 982199 1177440 1361200 1569940 1750100 1963670 No 4 thresh r=8 395389 586849 786768 988867 1185430 1370600 1580580 1762210 1976960 No 4 thresh r=9 398019 590841 792036 995419 1193410 1379940 1591220 1774220 1990090 No 4 thresh r=10 399895 593668 795808 1000169 1199060 1386530 1598720 1782690 1999470 No 4 thresh r=11 401649 596265 799222 1004465 1204220 1392500 1605640 1790380 2008240 No 4 thresh r=12 403243 598677 802431 1008522 1209010 1397990 1612010 1797410 2016200 No 4 thresh r=13 404827 601013 805581 1012482 1213770 1403600 1618340 1804620 2024140 No 4 thresh r=14 407082 604325 810097 1018080 1220390 1411380 1627150 1814640 2035390 No 4 thresh r=15 409706 608315 815333 1024672 1228240 1420560 1637670 1826430 2048540 No 4 thresh r=16 413507 614095 822997 1034264 1239650 1434020 1652860 1843730 2067540 No 4 thresh r=17 417469 620028 830933 1044160 1251690 1448030 1668910 1861760 2087350 No 4 thresh r=18 421382 625991 838951 1054040 1263610 1461960 1684800 1879690 2106890 No 4 thresh r=19 425461 632204 847271 1064460 1276100 1476550 1701460 1898410 2127300 No 4 thresh r=20 429246 637896 854969 1073900 1287430 1489790 1716600 1915460 2146220 No 4 thresh r=21 433598 644434 863663 1084850 1300540 1504960 1734040 1934970 2168000 No 4 thresh r=22 438588 651820 873736 1097250 1315420 1522410 1753920 1957380 2192940 No 4 thresh r=23 444367 660544 885337 1111840 1332860 1542750 1777130 1983540 2221830 No 4 thresh r=24 449267 667959 895131 1124020 1347310 1559730 1796410 2005350 2246330 No 4 thresh r=25 453882 674754 904278 1135470 1360950 1575780 1814580 2025980 2269420 No 4 thresh r=26 457631 680433 911895 1144890 1372340 1588920 1829820 2042920 2288150 No 4 thresh r=27 461362 685978 919337 1154340 1383620 1601970 1844850 2059680 2306830 No 4 thresh r=28 465062 691485 926762 1163550 1394770 1615010 1859710 2076410 2325330 No 4 thresh r=29 469024 697288 934618 1173440 1406700 1628740 1875600 2094100 2345120 No 4 thresh r=30 473111 703411 942745 1183500 1418920 1642950 1891910 2112370 2365560 No 4 thresh r=31 476700 708757 949900 1192430 1429480 1655540 1905980 2128550 2383500 No 4 thresh r=32 479530 713036 955551 1199330 1437880 1665340 1917180 2141140 2397660 No 4 thresh r=33 481869 716563 960194 1205130 1444930 1673420 1926580 2151560 2409350 No 4 thresh r=34 483884 719512 964216 1210140 1450970 1680460 1934620 2160620 2419430 No 4 thresh r=35 485510 721961 967518 1214330 1455790 1686390 1941070 2168250 2427540 No 4 thresh r=36 486837 723969 970150 1217810 1459730 1691040 1946310 2174170 2434180 No 4 thresh r=37 487877 725516 972273 1220430 1462910 1694640 1950550 2178850 2439380 No 4 thresh r=38 488654 726687 973848 1222320 1465220 1697320 1953640 2182290 2443270 No 4 thresh r=39 489205 727523 974875 1223620 1466850 1699210 1955790 2184700 2446040 No 4 thresh r=40 489590 728122 975619 1224520 1467950 1700480 1957270 2186330 2447950 No 4 thresh r=41 489843 728490 976115 1225110 1468670 1701360 1958220 2187480 2449210 Average of All Policies 446868 664117 890172 1117612 1339741 1550898 1786321 1994015 2234342 103 Table A.7: Average Expected Total Costs for Each w for Selected Policies when Excess Demand is Lost Policies w=100 w=150 w=200 w=250 w=300 w=350 w=400 w=450 w=500 FIFO 2417718 2614465 2820535 3021424 3208394 3406471 3598765 3795606 3994553 LIFO 4953312 5354629 5761676 6168400 6576935 6975282 7375853 7814018 8212935 Myopic 2266829 2478506 2707929 2932282 3142306 3363271 3570241 3796612 4013018 No 4 thresh r=2 2417718 2614465 2820535 3021424 3208394 3406471 3598765 3795606 3994553 No 4 thresh r=3 2417718 2614465 2820535 3021424 3208394 3406471 3598765 3795606 3994553 No 4 thresh r=4 2412653 2609953 2816541 3017924 3205518 3404094 3596853 3794341 3993800 No 4 thresh r=5 2397512 2596529 2804576 3007653 3196865 3397176 3591165 3790582 3991482 No 4 thresh r=6 2382618 2583382 2792894 2997576 3188324 3390635 3585606 3787041 3989247 No 4 thresh r=7 2371476 2573453 2784041 2990018 3181953 3385535 3581529 3784259 3987694 No 4 thresh r=8 2358888 2562259 2773971 2981388 3174629 3379900 3576912 3781206 3985688 No 4 thresh r=9 2346388 2551000 2764071 2972894 3167447 3374194 3572435 3778441 3983712 No 4 thresh r=10 2337488 2543029 2757094 2966865 3162324 3370200 3569294 3776571 3982506 No 4 thresh r=11 2329394 2535724 2750671 2961482 3157782 3366688 3566494 3774835 3981524 No 4 thresh r=12 2321924 2529035 2744794 2956435 3153618 3363347 3563865 3773335 3980647 No 4 thresh r=13 2314700 2522500 2738994 2951529 3149529 3360035 3561382 3771924 3979906 No 4 thresh r=14 2304494 2513276 2730912 2944924 3143888 3355694 3558065 3770159 3979012 No 4 thresh r=15 2292765 2502718 2721712 2937118 3137547 3350582 3554265 3768341 3978365 No 4 thresh r=16 2276312 2487747 2708706 2926153 3128888 3343935 3549147 3766071 3978082 No 4 thresh r=17 2260053 2473094 2696053 2915724 3120941 3338335 3544747 3764782 3978647 No 4 thresh r=18 2244771 2459471 2684771 2906671 3114188 3333918 3542041 3765088 3980571 No 4 thresh r=19 2230641 2446765 2674700 2898982 3109035 3331194 3541347 3767435 3985188 No 4 thresh r=20 2219782 2436600 2667571 2894176 3106682 3330729 3543041 3771741 3992718 No 4 thresh r=21 2210012 2428212 2662341 2892335 3107588 3333588 3548512 3780659 4004912 No 4 thresh r=22 2204053 2424547 2661706 2894982 3114312 3342935 3561024 3798294 4025659 No 4 thresh r=23 2206712 2430200 2671859 2909035 3133812 3367353 3588465 3832147 4063953 No 4 thresh r=24 2219882 2447335 2692253 2934259 3164371 3402371 3625906 3876341 4112218 No 4 thresh r=25 2246400 2477065 2727341 2973276 3209024 3452688 3677871 3936488 4177218 No 4 thresh r=26 2282400 2516412 2771747 3021653 3263447 3511582 3738688 4005582 4249176 No 4 thresh r=27 2333435 2573547 2833841 3089194 3336735 3589782 3820471 4095688 4343424 No 4 thresh r=28 2405518 2652388 2917835 3180618 3433965 3692900 3928671 4214200 4466771 No 4 thresh r=29 2513276 2770535 3043924 3314012 3576688 3843753 4086671 4384235 4643506 No 4 thresh r=30 2674047 2943706 3227953 3508112 3782988 4060918 4313971 4625612 4895188 No 4 thresh r=31 2875006 3156906 3456200 3746682 4035747 4325676 4593312 4918094 5201871 No 4 thresh r=32 3090576 3385288 3697088 4000306 4305188 4605076 4887506 5226465 5523559 No 4 thresh r=33 3320094 3627806 3954700 4269194 4590629 4901312 5200071 5553641 5863359 No 4 thresh r=34 3572806 3896335 4235235 4566582 4904447 5225665 5540441 5911447 6233853 No 4 thresh r=35 3832076 4170229 4523100 4866771 5222253 5555988 5889941 6274824 6607888 No 4 thresh r=36 4087165 4439206 4804265 5162271 5530976 5876553 6228535 6628047 6974247 No 4 thresh r=37 4323641 4690476 5067335 5439147 5818100 6178835 6545088 6954612 7314635 No 4 thresh r=38 4526165 4905153 5292088 5674094 6062218 6435582 6813035 7233212 7604541 No 4 thresh r=39 4685382 5071771 5468341 5858176 6254288 6637012 7023571 7449571 7831559 No 4 thresh r=40 4806676 5200153 5601465 5999312 6400429 6790782 7183282 7615453 8004165 No 4 thresh r=41 4896965 5295500 5700388 6103176 6509982 6904576 7301853 7738912 8133624 Average of All Policies 3303538 3589267 3885294 4177140 4463346 4750804 5031575 5340787 5626486 104 Table A.8: Average Expected Total Costs for Each w for Selected Policies when Excess Demand is Backlogged Policies w=100 w=150 w=200 w=250 w=300 w=350 w=400 w=450 w=500 FIFO 2469910 2655680 2854570 3053800 3239160 3442430 3621910 3821710 4425970 LIFO 4652630 5031970 5224930 5474980 5726430 6011120 6216150 6497370 13406160 Myopic 2337420 2540200 2766530 2983190 3191970 3407340 3613760 3825770 4607830 No 4 thresh r=2 2469910 2655680 2854570 3053800 3239160 3442430 3621910 3821710 4002280 No 4 thresh r=3 2469910 2655680 2854570 3053800 3239160 3442430 3621910 3821710 4002280 No 4 thresh r=4 2465090 2651360 2850990 3050460 3236420 3440250 3620270 3820590 4001730 No 4 thresh r=5 2450700 2638470 2840390 3040640 3228520 3433610 3615530 3817110 3999860 No 4 thresh r=6 2436720 2625940 2830090 3031220 3220800 3426920 3610920 3813480 3998250 No 4 thresh r=7 2426150 2616340 2822350 3023960 3214910 3422240 3607390 3811160 3997090 No 4 thresh r=8 2414240 2605710 2813880 3016020 3208340 3416920 3603490 3808530 3995790 No 4 thresh r=9 2402290 2595130 2805220 3008180 3201980 3411560 3599790 3805830 3994340 No 4 thresh r=10 2393850 2587720 2799210 3002760 3197410 3407730 3597100 3803890 3993440 No 4 thresh r=11 2386070 2580850 2793750 2997600 3193280 3404310 3594700 3802140 3992670 No 4 thresh r=12 2378920 2574630 2788820 2992970 3189660 3401180 3592670 3800600 3991880 No 4 thresh r=13 2371960 2568520 2784120 2988340 3186020 3398160 3590590 3799210 3991270 No 4 thresh r=14 2362130 2560080 2777490 2982090 3181120 3394260 3587920 3797490 3990440 No 4 thresh r=15 2351010 2550670 2770240 2975040 3176070 3389370 3585490 3795240 3989860 No 4 thresh r=16 2335350 2537360 2759640 2965360 3168830 3383020 3582050 3792740 3989380 No 4 thresh r=17 2320130 2524430 2749880 2956540 3163220 3377710 3580430 3791430 3990010 No 4 thresh r=18 2306460 2513230 2741740 2949040 3159130 3373700 3580330 3791430 3991990 No 4 thresh r=19 2293970 2503210 2734650 2942970 3157190 3371960 3582550 3793840 3995820 No 4 thresh r=20 2284400 2496840 2730260 2940480 3157840 3372750 3586980 3798430 4001390 No 4 thresh r=21 2277000 2494100 2729870 2941140 3162350 3378320 3595890 3808300 4011400 No 4 thresh r=22 2275900 2496620 2737210 2948140 3173560 3391460 3612030 3826440 4030260 No 4 thresh r=23 2284790 2515680 2757060 2971350 3200590 3420990 3644850 3861780 4062240 No 4 thresh r=24 2306600 2547880 2788740 3004810 3236480 3461270 3685570 3906900 4103660 No 4 thresh r=25 2344120 2594240 2833720 3055660 3289010 3518810 3742660 3969030 4159630 No 4 thresh r=26 2392840 2652520 2888100 3114380 3348580 3584370 3806020 4038350 4223360 No 4 thresh r=27 2457200 2728310 2961670 3191410 3427140 3667520 3888360 4125220 4302640 No 4 thresh r=28 2547430 2830990 3057020 3291540 3529620 3775740 3994530 4237170 4407700 No 4 thresh r=29 2675440 2974070 3194920 3436580 3670860 3923800 4139770 4389160 4551540 No 4 thresh r=30 2855510 3170880 3382430 3632200 3868220 4123190 4341200 4592590 4747930 No 4 thresh r=31 3062090 3392660 3602820 3852850 4090330 4350640 4566810 4823650 4968900 No 4 thresh r=32 3271320 3614160 3821160 4071970 4310800 4574810 4790110 5050600 5189450 No 4 thresh r=33 3479670 3833780 4037800 4287610 4529050 4796380 5010700 5274490 5407180 No 4 thresh r=34 3691370 4054830 4257550 4508820 4751360 5020090 5235010 5500230 5626910 No 4 thresh r=35 3896770 4264740 4465800 4720030 4960440 5233300 5445720 5715120 5838830 No 4 thresh r=36 4085110 4455600 4655680 4910040 5153110 5427410 5639680 5910560 6032460 No 4 thresh r=37 4250820 4625440 4823890 5075710 5321660 5598900 5809310 6083090 6202320 No 4 thresh r=38 4386650 4761860 4958110 5208630 5457420 5736630 5945840 6221610 6341260 No 4 thresh r=39 4488350 4862250 5060190 5308590 5560830 5842030 6049780 6327520 6445150 No 4 thresh r=40 4563920 4940400 5136810 5383630 5636820 5916710 6126140 6402560 6522280 No 4 thresh r=41 4617840 4995980 5191080 5440990 5691600 5974680 6181140 6460790 6577230 Average of All Policies 3284040 3568697 3773897 4002191 4227235 4473228 4673814 4916346 5071511 105 Table A.9: Average Expected Total Shortage Cost for Each p for Selected Policies when Excess Demand is Lost 106 Policies FIFO LIFO Myopic No 4 Thresh r=2 No 4 Thresh r=3 No 4 Thresh r=4 No 4 Thresh r=5 No 4 Thresh r=6 No 4 Thresh r=7 No 4 Thresh r=8 No 4 Thresh r=9 No 4 Thresh r=10 No 4 Thresh r=11 No 4 Thresh r=12 No 4 Thresh r=13 No 4 Thresh r=14 No 4 Thresh r=15 No 4 Thresh r=16 No 4 Thresh r=17 No 4 Thresh r=18 No 4 Thresh r=19 No 4 Thresh r=20 No 4 Thresh r=21 No 4 Thresh r=22 No 4 Thresh r=23 No 4 Thresh r=24 No 4 Thresh r=25 No 4 Thresh r=26 No 4 Thresh r=27 No 4 Thresh r=28 No 4 Thresh r=29 No 4 Thresh r=30 No 4 Thresh r=31 No 4 Thresh r=32 No 4 Thresh r=33 No 4 Thresh r=34 No 4 Thresh r=35 No 4 Thresh r=36 No 4 Thresh r=37 No 4 Thresh r=38 No 4 Thresh r=39 No 4 Thresh r=40 No 4 Thresh r=41 Average of All Policies p=100 14940 332847 19864 14940 14940 14956 15007 15063 15114 15176 15251 15315 15376 15437 15507 15610 15778 16070 16469 17008 17740 18618 19975 21977 25341 29364 34439 40062 47357 56774 70083 88821 111380 135118 160151 187438 215006 241908 267101 288337 304940 317598 327048 p=200 28199 666183 37830 28199 28199 28222 28316 28438 28533 28666 28820 28950 29081 29209 29361 29605 29943 30547 31346 32405 33860 35584 38140 42019 48700 56896 67211 78606 93263 112178 138796 176151 221564 268930 319270 374486 430368 484466 534306 576698 609750 635298 654358 p=300 43705 1005370 58953 43705 43705 43744 43888 44052 44184 44364 44598 44797 45026 45236 45451 45812 46311 47210 48565 50341 52762 55603 59691 65952 76107 88490 104357 121964 144233 172439 212763 269497 337803 409296 484383 565981 649229 730794 806186 870713 921174 959431 987818 p=400 59096 1342178 80167 59096 59096 59182 59461 59802 60059 60396 60814 61184 61543 61888 62232 62784 63561 64811 66461 68550 71689 75368 81038 89623 104039 120936 142221 165468 194894 232167 285578 360709 451270 545922 646232 756440 868352 976899 1077144 1163267 1230133 1280889 1318689 p=500 69770 1674200 94821 69770 69770 69837 70090 70400 70683 71032 71443 71774 72111 72476 72890 73483 74385 76006 78027 80818 84588 89101 95783 106182 123296 143601 169767 198403 235364 282591 349576 445072 559053 678322 804910 941868 1082278 1218422 1343678 1449756 1533267 1597189 1644567 p=600 83552 2005756 116643 83552 83552 83648 83988 84375 84713 85180 85720 86146 86573 87041 87615 88431 89575 91793 94669 98326 103218 108898 117011 129078 149512 174023 205181 238956 283288 340484 421594 535288 671190 812878 964208 1128167 1294000 1455911 1608489 1736756 1836644 1913422 1969878 p=700 99973 2347756 139502 99973 99973 100095 100535 101062 101564 102267 103040 103665 104248 104888 105559 106633 108052 110732 114286 118663 124536 131453 141477 156741 180539 209171 245809 286529 338838 405572 499202 631743 791511 958480 1134833 1326689 1522311 1711956 1888511 2037600 2154722 2242200 2307300 p=800 117464 2689289 161056 117464 117464 117584 118068 118637 119142 119826 120644 121340 122073 122837 123670 124918 126570 129459 133257 138163 144769 152100 163171 179593 207736 240953 283499 330450 389128 464219 570741 721838 904439 1094889 1297044 1517778 1740389 1957256 2159522 2330611 2465711 2567100 2642533 p=900 127779 3012578 171822 127779 127779 127910 128333 128842 129312 129890 130621 131224 131827 132466 133098 134193 135753 138496 142066 146980 153884 162267 174244 192481 222560 259883 308080 360274 427213 513553 636719 808567 1015519 1230756 1456056 1701122 1949633 2192600 2419622 2610356 2758589 2873544 2959911 p=1000 145281 3339633 199230 145281 145281 145432 146003 146713 147303 148079 149018 149762 150516 151254 152127 153421 155150 158453 163162 168936 176919 186780 200982 222759 259156 301136 354251 412174 486628 581493 714374 904224 1131700 1368244 1615822 1886911 2161967 2430678 2680500 2893389 3059500 3187111 3281122 p=1100 156527 3674233 213558 156527 156527 156726 157408 158228 158927 159893 161038 161916 162779 163648 164500 165932 167933 171663 176558 182711 190877 200811 215601 238110 275452 318903 375634 437368 516028 619732 766154 973321 1221978 1485589 1762800 2064211 2370678 2668300 2945278 3181878 3364633 3504367 3608900 p=1200 157302 4015778 217698 157302 157302 157459 157939 158511 159037 159767 160721 161462 162261 163122 163986 165464 167476 170969 175746 182136 190933 201826 217623 242178 283432 332548 396613 468319 559144 674326 836034 1065367 1341689 1630322 1932522 2260933 2594044 2918956 3220933 3478756 3680078 3832167 3945267 p=1300 188107 4344989 258171 188107 188107 188350 189134 189989 190766 191694 192816 193721 194757 195790 196912 198818 201482 206062 212092 219913 230831 242597 260763 287778 332316 385271 452717 526282 622411 745344 919248 1163867 1458411 1767433 2092889 2448789 2808400 3159356 3484378 3763333 3981167 4146022 4269511 p=1400 189870 4684889 261337 189870 189870 190057 190819 191714 192432 193494 194707 195702 196774 197872 198989 200992 203633 208199 214160 222106 232900 245793 264097 291848 338746 394454 466443 546752 649688 783391 974674 1242200 1564633 1901611 2255867 2642489 3031467 3410267 3762778 4061689 4295089 4473644 4605256 p=1500 215528 5030189 293650 215528 215528 215769 216579 217484 218293 219393 220764 221804 222952 224042 225347 227340 230054 234854 241329 249824 261334 274741 294339 325521 377366 438742 517111 603348 714907 858244 1060597 1345678 1688700 2048844 2425422 2837556 3255300 3660722 4039022 4360189 4609389 4800756 4942544 p=1600 228483 5344189 305803 228483 228483 228672 229332 230103 230828 231811 233027 233982 235004 236033 237240 239039 241598 245666 251787 259954 271454 285447 306206 339131 394000 459543 544269 639418 759239 910877 1127567 1428133 1790322 2171500 2569756 3004822 3446178 3877478 4280222 4622900 4893500 5098022 5250333 p=1700 246854 5715733 334187 246854 246854 247082 247913 248923 249811 250970 252421 253647 254834 256111 257546 259788 262807 268404 275197 283790 295421 309397 331772 366558 425719 495519 584110 681854 808271 970944 1200856 1525200 1918422 2327833 2761144 3230878 3706478 4165478 4591922 4954156 5239411 5456533 5616711 139732 278659 421882 563925 700570 839702 986151 1130210 1262769 1403545 1538904 1675439 1823148 1958190 2109053 2239090 2398011 Table A.10: Average Expected Total Shortage Costs for Each p for Selected Policies when Excess Demand is Backlogged Policies p=10 p=20 p=30 p=40 p=50 p=60 p=70 p=80 p=90 FIFO 66377 388552 343091 146263 181688 226336 293741 294691 377128 452564 LIFO 872949 5035122 5011444 2270344 2653433 3259589 3817611 4297556 4973544 5630111 Myopic 90768 527806 435308 204888 236401 301220 376642 390290 494617 593281 No 4 Thresh r=2 40722 76211 98869 146263 181688 226336 293741 294691 377128 452564 No 4 Thresh r=3 40722 76211 98869 146263 181688 226336 293741 294691 377128 452564 No 4 Thresh r=4 40772 76288 98990 146504 181768 226536 293877 294954 377381 452919 No 4 Thresh r=5 40921 76651 99287 147162 182058 227197 294383 296034 378389 454177 No 4 Thresh r=6 41100 77156 99642 147924 182458 228041 295053 296962 379582 455529 No 4 Thresh r=7 41268 77589 99960 148489 182867 228751 295610 298504 380502 456690 No 4 Thresh r=8 41460 78011 100309 149259 183430 229778 296412 300609 381900 458639 No 4 Thresh r=9 41711 78519 100712 150110 184192 230809 297306 302579 383857 460751 No 4 Thresh r=10 41903 78974 101087 150913 184827 231842 298114 304144 385063 462514 No 4 Thresh r=11 42097 79416 101447 151587 185551 232830 298923 305767 386329 464261 No 4 Thresh r=12 42318 79815 101803 152281 186363 233906 299793 307760 387941 466162 No 4 Thresh r=13 42563 80252 102201 153064 187257 234931 300833 309482 389620 468111 No 4 Thresh r=14 42938 80908 102966 154239 188848 236694 302512 312519 392284 471278 No 4 Thresh r=15 43467 81842 103753 156203 190992 238962 304568 315842 396599 476489 No 4 Thresh r=16 44394 83425 105616 159689 194467 243121 308831 321754 403731 484328 No 4 Thresh r=17 45691 85427 107867 164201 199180 248496 314952 330130 414781 494820 No 4 Thresh r=18 47164 87988 111181 170046 205196 256571 323833 339796 428408 509421 No 4 Thresh r=19 49220 91449 115780 177670 213336 268079 335686 351679 448004 528288 No 4 Thresh r=20 51490 95266 121481 186249 223014 280797 349656 367627 469494 550063 No 4 Thresh r=21 54963 101295 130409 199968 237991 300038 371357 391186 499958 584269 No 4 Thresh r=22 60213 110375 144368 219707 260167 327942 404567 428392 549197 636153 No 4 Thresh r=23 68937 125917 167227 253733 297591 377474 462328 489371 629791 717744 No 4 Thresh r=24 79034 144756 194313 294971 340358 434104 529050 562747 719081 817878 No 4 Thresh r=25 91868 168617 228901 347146 396218 506271 613482 656127 833148 945237 No 4 Thresh r=26 105439 195136 266618 403689 457932 585243 706394 755472 955216 1086062 No 4 Thresh r=27 122237 229388 312846 475801 533570 682172 821381 879944 1106108 1256489 No 4 Thresh r=28 142539 270632 373740 567430 629299 806269 965988 1041007 1299200 1468333 No 4 Thresh r=29 171091 327949 454786 689208 764774 974559 1160167 1263111 1551667 1758578 No 4 Thresh r=30 208143 403969 563960 848832 949987 1199656 1422867 1560511 1889600 2143333 No 4 Thresh r=31 250656 489437 688071 1025814 1156089 1449456 1718489 1892678 2266911 2572000 No 4 Thresh r=32 291538 571048 811110 1200744 1357733 1701089 2001344 2221933 2640400 2990200 No 4 Thresh r=33 332001 651634 932607 1369033 1560189 1944500 2282133 2546144 3001900 3402000 No 4 Thresh r=34 372602 733527 1056696 1539267 1762878 2187667 2569100 2870056 3365311 3817900 No 4 Thresh r=35 411396 810884 1170678 1698100 1957456 2417467 2840100 3181500 3710111 4212522 No 4 Thresh r=36 446813 880170 1278033 1843011 2132356 2625800 3087767 3463800 4023422 4570500 No 4 Thresh r=37 477639 941572 1372256 1971667 2284544 2812911 3299456 3707967 4304400 4882967 No 4 Thresh r=38 502051 990566 1446556 2072467 2407278 2960689 3473133 3904367 4532133 5136944 No 4 Thresh r=39 520546 1028440 1502156 2147867 2502267 3074844 3603367 4055800 4694122 5330367 No 4 Thresh r=40 533781 1055878 1542644 2204356 2572622 3158389 3699700 4165033 4820611 5469133 No 4 Thresh r=41 543781 1076189 1571889 2246722 2621600 3221689 3772467 4245633 4915789 5564878 Average of All Policies 256036 501905 721670 1044653 1213564 1499930 1771055 1971350 2314043 2633911 107 p=100 Table A.11: Average Expected Total Costs for Each p for Selected Policies when Excess Demand is Lost 108 Policies FIFO LIFO Myopic No 4 Thresh r=2 No 4 Thresh r=3 No 4 Thresh r=4 No 4 Thresh r=5 No 4 Thresh r=6 No 4 Thresh r=7 No 4 Thresh r=8 No 4 Thresh r=9 No 4 Thresh r=10 No 4 Thresh r=11 No 4 Thresh r=12 No 4 Thresh r=13 No 4 Thresh r=14 No 4 Thresh r=15 No 4 Thresh r=16 No 4 Thresh r=17 No 4 Thresh r=18 No 4 Thresh r=19 No 4 Thresh r=20 No 4 Thresh r=21 No 4 Thresh r=22 No 4 Thresh r=23 No 4 Thresh r=24 No 4 Thresh r=25 No 4 Thresh r=26 No 4 Thresh r=27 No 4 Thresh r=28 No 4 Thresh r=29 No 4 Thresh r=30 No 4 Thresh r=31 No 4 Thresh r=32 No 4 Thresh r=33 No 4 Thresh r=34 No 4 Thresh r=35 No 4 Thresh r=36 No 4 Thresh r=37 No 4 Thresh r=38 No 4 Thresh r=39 No 4 Thresh r=40 No 4 Thresh r=41 Average of All Policies p=100 3114011 3905400 3005578 3114011 3114011 3110944 3101711 3092744 3085856 3077856 3069867 3064133 3058922 3054089 3049356 3042522 3034789 3023333 3011856 3000789 2989311 2979567 2969300 2959211 2951133 2948178 2950056 2956978 2969844 2990844 3024967 3078522 3147722 3223789 3306556 3398533 3492678 3585511 3673133 3747422 3805889 3850911 3884678 p=200 3110956 4230367 3007500 3110956 3110956 3107944 3098967 3090211 3083456 3075744 3068011 3062589 3057444 3052822 3048211 3041722 3034089 3023000 3011833 3001122 2990422 2981433 2972300 2963933 2959267 2961233 2969078 2981900 3002567 3033367 3081178 3153600 3246722 3346433 3455211 3575989 3699311 3819822 3931611 4027278 4102056 4159856 4203244 p=300 3123167 4567744 3022944 3123167 3123167 3120178 3111167 3102222 3095511 3087789 3080144 3074700 3069689 3065056 3060489 3053900 3046278 3035533 3024900 3015033 3005500 2997922 2990656 2984922 2983522 2989233 3002722 3022144 3050689 3090533 3152444 3244667 3360489 3484733 3617378 3763322 3913878 4062511 4200378 4318933 4411967 4482667 4535122 p=400 3143144 4908433 3051156 3143144 3143144 3140189 3131278 3122667 3115956 3108467 3100911 3095633 3090811 3086278 3081800 3075678 3068489 3058011 3047844 3038133 3029300 3022200 3016589 3013689 3017322 3028033 3047200 3072267 3107889 3156756 3231289 3341256 3478467 3625400 3783344 3958767 4138367 4313489 4475911 4615944 4724956 4807956 4869956 p=500 3153933 5246389 3063578 3153933 3153933 3150900 3142011 3133333 3126767 3119167 3111622 3106289 3101467 3096889 3092567 3086456 3079489 3069633 3059667 3050622 3042389 3036511 3031744 3030422 3036178 3049733 3073478 3103956 3147300 3206322 3294556 3426122 3587856 3760411 3945856 4148211 4357322 4560533 4748356 4907800 5033600 5130011 5201744 p=600 3165922 5569522 3085856 3165922 3165922 3162978 3154167 3145422 3138967 3131533 3124067 3118778 3114022 3109644 3105400 3099467 3092589 3083367 3074300 3066211 3059200 3054222 3050844 3050544 3059367 3076933 3105256 3140256 3190667 3259600 3362444 3511522 3694278 3888289 4097956 4326900 4559800 4788122 5004211 5186456 5328533 5437867 5518267 p=700 3173556 5904922 3100756 3173556 3173556 3170622 3161856 3153344 3146956 3139822 3132611 3127567 3122967 3118822 3114744 3109100 3102656 3093878 3085700 3078500 3072544 3068944 3067711 3071200 3083400 3105489 3139667 3182300 3241022 3319711 3434756 3602722 3809856 4029611 4264022 4520733 4784100 5040589 5279944 5482478 5641911 5760922 5849778 p=800 3184789 6242600 3115167 3184789 3184789 3181856 3173333 3164878 3158567 3151344 3144300 3139378 3134944 3130911 3126944 3121522 3115144 3106733 3098733 3091978 3086711 3083278 3082722 3087022 3103667 3130511 3170878 3220067 3285044 3371800 3499422 3685867 3915778 4159111 4419700 4705911 4995944 5279367 5544722 5769511 5947467 6081322 6180900 p=900 3204211 6574433 3132656 3204211 3204211 3201322 3192544 3184067 3177733 3170422 3163189 3158144 3153633 3149433 3145200 3139611 3133278 3124433 3116100 3109256 3104411 3102422 3102756 3109033 3127433 3158056 3204000 3258067 3331300 3429789 3575256 3783333 4038722 4307056 4590222 4900289 5215778 5525144 5815067 6058789 6248656 6396033 6506822 p=1000 3231622 6904822 3171489 3231622 3231622 3228722 3219956 3211556 3205311 3198122 3191067 3186044 3181500 3177311 3173356 3167811 3161500 3153111 3145744 3139522 3135244 3134589 3137344 3147467 3172900 3208467 3259311 3318778 3399144 3505533 3659411 3884933 4159578 4448422 4753078 5088811 5430878 5766033 6078456 6344911 6553167 6713356 6831378 p=1100 3238878 7234789 3180000 3238878 3238878 3236000 3227589 3219500 3213333 3206456 3199767 3195033 3190700 3186778 3182856 3177456 3171344 3163411 3156222 3150478 3146478 3145722 3148944 3159500 3185222 3221600 3275411 3338133 3422189 3537444 3704844 3947344 4242767 4559389 4894822 5261189 5634867 5998867 6338167 6628489 6853311 7025422 7154300 p=1200 3241044 7584178 3186000 3241044 3241044 3238178 3229433 3220778 3214378 3207089 3200144 3195178 3190844 3186778 3182767 3177411 3171411 3163122 3155611 3150022 3146544 3146722 3150967 3163656 3193622 3236411 3298422 3372200 3469433 3596833 3779989 4045611 4370211 4712100 5072522 5465800 5866333 6257600 6622289 6934156 7177667 7361667 7498667 p=1300 3276744 7912722 3232178 3276744 3276744 3273911 3265544 3257256 3251200 3244278 3237444 3232667 3228411 3224522 3220756 3215900 3210689 3203611 3197656 3193578 3192267 3193222 3199956 3214833 3247767 3293878 3358689 3433556 3535578 3670178 3865278 4145011 4486656 4848400 5231600 5652667 6079489 6497022 6884400 7217367 7477500 7674700 7822367 p=1400 3264867 8246900 3222044 3264867 3264867 3261989 3253622 3245622 3239389 3232478 3225778 3221211 3217111 3213400 3209778 3205067 3199756 3192711 3186789 3182922 3182000 3184511 3191444 3207033 3242433 3291244 3360733 3442856 3552267 3698378 3911711 4215033 4585389 4976100 5389156 5841467 6297944 6743478 7158778 7511122 7786544 7997311 8152844 p=1500 3294111 8593356 3258344 3294111 3294111 3291311 3283067 3274900 3268867 3262244 3255711 3251056 3246978 3243111 3239644 3234889 3229589 3222900 3217544 3214567 3214189 3217033 3225033 3244611 3285022 3339756 3415533 3503378 3621167 3776678 4000544 4321389 4711733 5124978 5559478 6036744 6522044 6994022 7434656 7809478 8100678 8324644 8490833 p=1600 3307989 8906333 3272611 3307989 3307989 3305089 3296589 3288356 3282167 3275211 3268400 3263678 3259533 3255722 3252156 3247278 3241856 3234033 3228200 3224356 3223700 3226867 3235900 3256578 3299889 3358578 3440900 3538044 3664200 3827900 4066711 4403022 4812678 5246667 5702544 6202467 6710889 7208622 7674189 8070711 8383944 8620789 8797500 p=1700 3318256 9276167 3292911 3318256 3318256 3315478 3307189 3299200 3293289 3286678 3280289 3275844 3271922 3268322 3264922 3260567 3255722 3249811 3244678 3241278 3240889 3243911 3254533 3277200 3325200 3388656 3474811 3574200 3706811 3881522 4133222 4493644 4935700 5398878 5891411 6427289 6970622 7496100 7984822 8400656 8728200 8977811 9162111 3353628 3480244 3620461 3767325 3906330 4040640 4180096 4318538 4460036 4607487 4738134 4879070 5029074 5154778 5307917 5437853 5591723 Table A.12: Average Expected Total Costs for Each p for Selected Policies when Excess Demand is Backlogged Policies p=10 p=20 p=30 p=40 p=50 p=60 p=70 p=80 p=90 p=100 FIFO 3137367 3320811 3126667 3153178 3215211 3238678 3320356 3322489 3404922 3452900 LIFO 3637556 7624567 7634956 4961967 5367822 5955322 6526122 7006844 7683900 8313989 Myopic 3045133 3330633 3270100 3090367 3149444 3189911 3281600 3296711 3402344 3470433 No 4 Thresh r=2 3070144 3097067 3126667 3153178 3215211 3238678 3320356 3322489 3404922 3452900 No 4 Thresh r=3 3070144 3097067 3126667 3153178 3215211 3238678 3320356 3322489 3404922 3452900 No 4 Thresh r=4 3067389 3094256 3123833 3150611 3212389 3235978 3317600 3319878 3402200 3450489 No 4 Thresh r=5 3058811 3085822 3115311 3142711 3203978 3228267 3309378 3312300 3394556 3443122 No 4 Thresh r=6 3050378 3077778 3107011 3135011 3195867 3220589 3301478 3304733 3387200 3435889 No 4 Thresh r=7 3044033 3071478 3100667 3129256 3189789 3214844 3295367 3299822 3381578 3430489 No 4 Thresh r=8 3036778 3064411 3093489 3122633 3182978 3208489 3288522 3294400 3375456 3424978 No 4 Thresh r=9 3029444 3057422 3086189 3115889 3176178 3201967 3281878 3288811 3369767 3419478 No 4 Thresh r=10 3024167 3052322 3080956 3111267 3171433 3197678 3277133 3284989 3365411 3415878 No 4 Thresh r=11 3019444 3047833 3076122 3106811 3167122 3193511 3272822 3281444 3361678 3412511 No 4 Thresh r=12 3014933 3043378 3071678 3102733 3163200 3189800 3268989 3278722 3358467 3409578 No 4 Thresh r=13 3010589 3039067 3067322 3098789 3159544 3186156 3265278 3275722 3355467 3406722 No 4 Thresh r=14 3004244 3033211 3061344 3093378 3154444 3181122 3260278 3272044 3351367 3403033 No 4 Thresh r=15 2996656 3026300 3054144 3087533 3148644 3175489 3254533 3267611 3347767 3400200 No 4 Thresh r=16 2985722 3016033 3044033 3079544 3140344 3167900 3247256 3261600 3343178 3396311 No 4 Thresh r=17 2974733 3005756 3033733 3071833 3132767 3161122 3241011 3257867 3341978 3394511 No 4 Thresh r=18 2963678 2995767 3024344 3065167 3126267 3156544 3237211 3255022 3343022 3396367 No 4 Thresh r=19 2952344 2986022 3015322 3059411 3121011 3154689 3235633 3253544 3349389 3401700 No 4 Thresh r=20 2942167 2977422 3008678 3055522 3118244 3155044 3237300 3257356 3358733 3411056 No 4 Thresh r=21 2931311 2969056 3003100 3054756 3118711 3160011 3244644 3266433 3374689 3431033 No 4 Thresh r=22 2920033 2961367 3000033 3058000 3124178 3171556 3261289 3287044 3407422 3466433 No 4 Thresh r=23 2909056 2957289 3003289 3072389 3141878 3201578 3299700 3328322 3468389 3528478 No 4 Thresh r=24 2902500 2959156 3013533 3096689 3167733 3241633 3349589 3384678 3540989 3612289 No 4 Thresh r=25 2900022 2967378 3032589 3133589 3208133 3298400 3418611 3462622 3639778 3724300 No 4 Thresh r=26 2900722 2980944 3057544 3177178 3256878 3364511 3498567 3549167 3749256 3852478 No 4 Thresh r=27 2905167 3002756 3091211 3237022 3320111 3449044 3601189 3661156 3887656 4010767 No 4 Thresh r=28 2913544 3031911 3140122 3316711 3403767 3561256 3733644 3810244 4068767 4210856 No 4 Thresh r=29 2929633 3076511 3208822 3426033 3526756 3717144 3915589 4019833 4308833 4488778 No 4 Thresh r=30 2954511 3140344 3305689 3573544 3699711 3930167 4166200 4305022 4634533 4861556 No 4 Thresh r=31 2986822 3215722 3419789 3740878 3895689 4169822 4451678 4626922 5001933 5280467 No 4 Thresh r=32 3020233 3289967 3535211 3908589 4089889 4414400 4727189 4948778 5368167 5691333 No 4 Thresh r=33 3055156 3365078 3651100 4071722 4286900 4652300 5002500 5267444 5724189 6097678 No 4 Thresh r=34 3091433 3442811 3770722 4237978 4485278 4891067 5285300 5587056 6083322 6509667 No 4 Thresh r=35 3127200 3517156 3881544 4393967 4676778 5117756 5553422 5895544 6425033 6901322 No 4 Thresh r=36 3160589 3584300 3986733 4536900 4849456 5324189 5798978 6175800 6736300 7257478 No 4 Thresh r=37 3190189 3644356 4079622 4664400 5000167 5510044 6009322 6418622 7015922 7568622 No 4 Thresh r=38 3213767 3692556 4153044 4764578 5122133 5657022 6182256 6614300 7243033 7821767 No 4 Thresh r=39 3231989 3729922 4208244 4839689 5216667 5770656 6312156 6765344 7404767 8014667 No 4 Thresh r=40 3245011 3757178 4248400 4896056 5287000 5854133 6408256 6874367 7531100 8153244 No 4 Thresh r=41 3254944 3777456 4277600 4938367 5335967 5917389 6480933 6954889 7626189 8248856 Average of All Policies 3092608 3329066 3554346 3860020 4053794 4320084 4604694 4806044 5149501 5442021 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 FIFO LIFO Myopic No 4 Thresh r=2 No 4 Thresh r=3 No 4 Thresh r=4 No 4 Thresh r=5 No 4 Thresh r=6 No 4 Thresh r=7 No 4 Thresh r=8 No 4 Thresh r=9 No 4 Thresh r=10 No 4 Thresh r=11 No 4 Thresh r=12 No 4 Thresh r=13 No 4 Thresh r=14 No 4 Thresh r=15 No 4 Thresh r=16 No 4 Thresh r=17 No 4 Thresh r=18 No 4 Thresh r=19 No 4 Thresh r=20 No 4 Thresh r=21 No 4 Thresh r=22 No 4 Thresh r=23 No 4 Thresh r=24 No 4 Thresh r=25 No 4 Thresh r=26 No 4 Thresh r=27 No 4 Thresh r=28 No 4 Thresh r=29 No 4 Thresh r=30 No 4 Thresh r=31 No 4 Thresh r=32 No 4 Thresh r=33 No 4 Thresh r=34 No 4 Thresh r=35 No 4 Thresh r=36 No 4 Thresh r=37 No 4 Thresh r=38 No 4 Thresh r=39 No 4 Thresh r=40 No 4 Thresh r=41 Average of All Policies p=100 to 200 133 3333 180 133 133 133 133 134 134 135 136 136 137 138 139 140 142 145 149 154 161 170 182 200 234 275 328 385 459 554 687 873 1102 1338 1591 1870 2154 2426 2672 2884 3048 3177 3273 p=200 to 300 155 3392 211 155 155 155 156 156 157 157 158 158 159 160 161 162 164 167 172 179 189 200 216 239 274 316 371 434 510 603 740 933 1162 1404 1651 1915 2189 2463 2719 2940 3114 3241 3335 p=300 to 400 154 3368 212 154 154 154 156 157 159 160 162 164 165 167 168 170 173 176 179 182 189 198 213 237 279 324 379 435 507 597 728 912 1135 1366 1618 1905 2191 2461 2710 2926 3090 3215 3309 p=400 to 500 107 3320 147 107 107 107 106 106 106 106 106 106 106 106 107 107 108 112 116 123 129 137 147 166 193 227 275 329 405 504 640 844 1078 1324 1587 1854 2139 2415 2665 2865 3031 3163 3259 p=500 to 600 138 3316 218 138 138 138 139 140 140 141 143 144 145 146 147 149 152 158 166 175 186 198 212 229 262 304 354 406 479 579 720 902 1121 1346 1593 1863 2117 2375 2648 2870 3034 3162 3253 p=600 to 700 164 3420 229 164 164 164 165 167 169 171 173 175 177 178 179 182 185 189 196 203 213 226 245 277 310 351 406 476 556 651 776 965 1203 1456 1706 1985 2283 2560 2800 3008 3181 3288 3374 p=700 to 800 175 3415 216 175 175 175 175 176 176 176 176 177 178 179 181 183 185 187 190 195 202 206 217 229 272 318 377 439 503 586 715 901 1129 1364 1622 1911 2181 2453 2710 2930 3110 3249 3352 p=800 to 900 103 3233 108 103 103 103 103 102 102 101 100 99 98 96 94 93 92 90 88 88 91 102 111 129 148 189 246 298 381 493 660 867 1111 1359 1590 1833 2092 2353 2601 2797 2929 3064 3174 p=900 to 1000 175 3271 274 175 175 175 177 179 180 182 184 185 187 188 190 192 194 200 211 220 230 245 267 303 366 413 462 519 594 679 777 957 1162 1375 1598 1858 2123 2381 2609 2830 3009 3136 3212 1389 1432 1420 1366 1391 1464 1441 1326 1408 p=1000 to 1100 112 3346 143 112 112 113 114 115 116 118 120 122 123 124 124 125 128 132 134 138 140 140 146 154 163 178 214 252 294 382 518 691 903 1173 1470 1773 2087 2376 2648 2885 3051 3173 3278 1354 p=1100 to 1200 8 3415 41 8 8 7 5 3 1 -1 -3 -5 -5 -5 -5 -5 -5 -7 -8 -6 1 10 20 41 80 136 210 310 431 546 699 920 1197 1447 1697 1967 2234 2507 2757 2969 3154 3278 3364 1365 p=1200 to 1300 308 3292 405 308 308 309 312 315 317 319 321 323 325 327 329 334 340 351 363 378 399 408 431 456 489 527 561 580 633 710 832 985 1167 1371 1604 1879 2144 2404 2634 2846 3011 3139 3242 1477 p=1300 to 1400 18 3399 32 18 18 17 17 17 17 18 19 20 20 21 21 22 22 21 21 22 21 32 33 41 64 92 137 205 273 380 554 783 1062 1342 1630 1937 2231 2509 2784 2984 3139 3276 3357 1350 p=1400 to 1500 257 3453 323 257 257 257 258 258 259 259 261 261 262 262 264 263 264 267 272 277 284 289 302 337 386 443 507 566 652 749 859 1035 1241 1472 1696 1951 2238 2505 2762 2985 3143 3271 3373 1509 p=1500 to 1600 130 3140 122 130 130 129 128 126 125 124 123 122 121 120 119 117 115 108 105 101 101 107 119 136 166 208 272 361 443 526 670 825 1016 1227 1443 1673 1909 2168 2412 2627 2841 2973 3078 1300 p=1600 to 1700 184 3715 284 184 184 184 186 188 190 192 194 197 198 201 203 207 212 227 234 238 240 240 256 274 317 360 398 424 490 601 733 971 1281 1563 1914 2261 2603 2880 3117 3313 3459 3585 3664 1589 Average Across p 145 3364 196 145 145 145 146 146 147 147 148 149 150 150 151 153 154 158 162 167 174 182 195 215 250 291 344 401 476 571 707 898 1129 1370 1626 1902 2182 2452 2703 2916 3084 3212 3306 1411 Table A.14: Change in the Average Expected Total Cost with respect to One Unit Change in p when Excess Demand is Lost 111 FIFO LIFO Myopic No 4 Thresh r=2 No 4 Thresh r=3 No 4 Thresh r=4 No 4 Thresh r=5 No 4 Thresh r=6 No 4 Thresh r=7 No 4 Thresh r=8 No 4 Thresh r=9 No 4 Thresh r=10 No 4 Thresh r=11 No 4 Thresh r=12 No 4 Thresh r=13 No 4 Thresh r=14 No 4 Thresh r=15 No 4 Thresh r=16 No 4 Thresh r=17 No 4 Thresh r=18 No 4 Thresh r=19 No 4 Thresh r=20 No 4 Thresh r=21 No 4 Thresh r=22 No 4 Thresh r=23 No 4 Thresh r=24 No 4 Thresh r=25 No 4 Thresh r=26 No 4 Thresh r=27 No 4 Thresh r=28 No 4 Thresh r=29 No 4 Thresh r=30 No 4 Thresh r=31 No 4 Thresh r=32 No 4 Thresh r=33 No 4 Thresh r=34 No 4 Thresh r=35 No 4 Thresh r=36 No 4 Thresh r=37 No 4 Thresh r=38 No 4 Thresh r=39 No 4 Thresh r=40 No 4 Thresh r=41 Average of All Policies p=100 to 200 -31 3250 19 -31 -31 -30 -27 -25 -24 -21 -19 -15 -15 -13 -11 -8 -7 -3 0 3 11 19 30 47 81 131 190 249 327 425 562 751 990 1226 1487 1775 2066 2343 2585 2799 2962 3089 3186 p=200 to 300 122 3374 154 122 122 122 122 120 121 120 121 121 122 122 123 122 122 125 131 139 151 165 184 210 243 280 336 402 481 572 713 911 1138 1383 1622 1873 2146 2427 2688 2917 3099 3228 3319 p=300 to 400 200 3407 282 200 200 200 201 204 204 207 208 209 211 212 213 218 222 225 229 231 238 243 259 288 338 388 445 501 572 662 788 966 1180 1407 1660 1954 2245 2510 2755 2970 3130 3253 3348 p=400 to 500 108 3380 124 108 108 107 107 107 108 107 107 107 107 106 108 108 110 116 118 125 131 143 152 167 189 217 263 317 394 496 633 849 1094 1350 1625 1894 2190 2470 2724 2919 3086 3221 3318 p=500 to 600 120 3231 223 120 120 121 122 121 122 124 124 125 126 128 128 130 131 137 146 156 168 177 191 201 232 272 318 363 434 533 679 854 1064 1279 1521 1787 2025 2276 2559 2787 2949 3079 3165 p=600 to 700 76 3354 149 76 76 76 77 79 80 83 85 88 89 92 93 96 101 105 114 123 133 147 169 207 240 286 344 420 504 601 723 912 1156 1413 1661 1938 2243 2525 2757 2960 3134 3231 3315 p=700 to 800 112 3377 144 112 112 112 115 115 116 115 117 118 120 121 122 124 125 129 130 135 142 143 150 158 203 250 312 378 440 521 647 831 1059 1295 1557 1852 2118 2388 2648 2870 3056 3204 3311 p=800 to 900 194 3318 175 194 194 195 192 192 192 191 189 188 187 185 183 181 181 177 174 173 177 191 200 220 238 275 331 380 463 580 758 975 1229 1479 1705 1944 2198 2458 2703 2893 3012 3147 3259 p=900 to 1000 274 3304 388 274 274 274 274 275 276 277 279 279 279 279 282 282 282 287 296 303 308 322 346 384 455 504 553 607 678 757 842 1016 1209 1414 1629 1885 2151 2409 2634 2861 3045 3173 3246 1266 1402 1469 1390 1343 1395 1384 1415 1475 p=1000 to 1100 73 3300 85 73 73 73 76 79 80 83 87 90 92 95 95 96 98 103 105 110 112 111 116 120 123 131 161 194 230 319 454 624 832 1110 1417 1724 2040 2328 2597 2836 3001 3121 3229 1306 p=1100 to 1200 22 3494 60 22 22 22 18 13 10 6 4 1 1 0 -1 0 1 -3 -6 -5 1 10 20 42 84 148 230 341 472 594 751 983 1274 1527 1777 2046 2315 2587 2841 3057 3244 3362 3444 1409 p=1200 to 1300 357 3285 462 357 357 357 361 365 368 372 373 375 376 377 380 385 393 405 420 436 457 465 490 512 541 575 603 614 661 733 853 994 1164 1363 1591 1869 2132 2394 2621 2832 2998 3130 3237 1500 p=1300 to 1400 -119 3342 -101 -119 -119 -119 -119 -116 -118 -118 -117 -115 -113 -111 -110 -108 -109 -109 -109 -107 -103 -87 -85 -78 -53 -26 20 93 167 282 464 700 987 1277 1576 1888 2185 2465 2744 2938 3090 3226 3305 1257 p=1400 to 1500 292 3465 363 292 292 293 294 293 295 298 299 298 299 297 299 298 298 302 308 316 322 325 336 376 426 485 548 605 689 783 888 1064 1263 1489 1703 1953 2241 2505 2759 2984 3141 3273 3380 1531 p=1500 to 1600 139 3130 143 139 139 138 135 135 133 130 127 126 126 126 125 124 123 111 107 98 95 98 109 120 149 188 254 347 430 512 662 816 1009 1217 1431 1657 1888 2146 2395 2612 2833 2961 3067 1299 p=1600 to 1700 103 3698 203 103 103 104 106 108 111 115 119 122 124 126 128 133 139 158 165 169 172 170 186 206 253 301 339 362 426 536 665 906 1230 1522 1889 2248 2597 2875 3106 3299 3443 3570 3646 1539 Average Across p 137 3358 191 137 137 137 138 138 139 140 141 141 142 143 144 145 147 150 154 159 165 173 186 206 241 282 334 392 466 563 699 890 1123 1365 1621 1899 2178 2446 2697 2911 3079 3207 3300 1404
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Optimal issuing policies for hospital blood inventory Slofstra, Anyu 2013
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Title | Optimal issuing policies for hospital blood inventory |
Creator |
Slofstra, Anyu |
Publisher | University of British Columbia |
Date Issued | 2013 |
Description | 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. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2013-05-02 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0073844 |
URI | http://hdl.handle.net/2429/44417 |
Degree |
Master of Science - MSc |
Program |
Business Administration |
Affiliation |
Business, Sauder School of |
Degree Grantor | University of British Columbia |
Graduation Date | 2013-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
Aggregated Source Repository | DSpace |
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