UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Optimal issuing policies for hospital blood inventory Slofstra, Anyu 2013

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
24-ubc_2013_fall_slofstra_anyu.pdf [ 2MB ]
Metadata
JSON: 24-1.0073844.json
JSON-LD: 24-1.0073844-ld.json
RDF/XML (Pretty): 24-1.0073844-rdf.xml
RDF/JSON: 24-1.0073844-rdf.json
Turtle: 24-1.0073844-turtle.txt
N-Triples: 24-1.0073844-rdf-ntriples.txt
Original Record: 24-1.0073844-source.json
Full Text
24-1.0073844-fulltext.txt
Citation
24-1.0073844.ris

Full Text

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&sectionid=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. Management Science, 19(11): 1292–1294, Jul 1973. [6] M. Brown and H. Solomon. Optimal issuing policies under stochastic field lives. Journal of Applied Probability, 10(4):761–768, Dec 1973. [7] Canadian Blood Services. Blood shelf life. http://www.blood.ca/CentreApps/Internet/ UW_V502_MainEngine.nsf/page/Blood+Shelf+Life?OpenDocument. Last visited: 02/14/2013. [8] M.A. Cohen and D. Pekelman. Optimal inventory ordering policy with tax payments under fifo and lifo accounting. Management Science, 25(8):729–743, Aug 1979. [9] B. Deniz. Essays on Perishable Inventory Management. Ph.D. dissertation, Carnegie Mellon University, Pittsburgh, PA, 2007. [10] C. Derman and M. Klein. Depletion management. Management Science, 4(4): 450–456, Jul 1958. [11] T.H. Edna and T. Bjerkeset. Perioperative blood transfusions reduce long-term survival following surgery for colorectal cancer. Diseases of the Colon and Rectum, 41 (4):451–459, April 1998. [12] S. Eilon. Fifo and lifo policies in inventory management. Management Science, 7(3): 304–315, Apr 1961.  91  [13] S. Eilon. Obsolescence of commodities which are subject to deterioration in store. Management Science, 9(4):623–642, Jul 1963. [14] J.A. Greenwood. Issue priority: Lifo vs fifo as a method of issuing items from supply storage. Nava Research Logistics Quarterly, 2(4):251–268, 1955. [15] R. Haijema, J. van der Wal, and N.M. van Dijk. Blood platelet production: Optimization by dynamic programming and simulation. Computers and Operations Research, 34(3):760–779, Mar 2007. [16] Y-M. Lee, S. Mu, Z. Shen, and M. Dessouky. Optimal ordering and issuing for perishable inventory management problem with a minimum inventory volumne constraint. 2009. [17] G.J. Lieberman. Lifo vs fifo in inventory depletion management. Management Science, 5(1):102–105, Oct 1958. [18] T. Mynster and H.J. Nielsen. Storage time of transfused blood and disease recurrence after colorectal cancer surgery. Diseases of the Colon and Rectum, 44(7):955–964, July 2001. [19] S. Nahmias. Inventory depletion management when the field life is random. Management Science, 20(9):1276–1283, May 1974. [20] S. Nahmias. Optimal ordering policies for perishable inventory-ii. Operations Research, 23(4):735–749, Jul 1975. [21] S. Nahmias. Myopic approximations for the perishable inventory problem. Management Science, 22(9):1002–1008, May 1976. [22] S. Nahmias. Higher-order approximations for the perishable-inventory problem. Operations Research, 25(4):630–640, Jul 1977. [23] S. Nahmias. Perishable inventory theory: A review. Operations Research, 30(4): 680–708, Jul 1982. [24] M. Parlar, D. Perry, and W. Stadje. Fifo versus lifo issuing policies for stochastic perishable inventory systems. Methodology and Computing in Applied Probability, 13 (2):405–417, Jun 2011. [25] W.P. Pierskalla. Inventory depletion management with stochastic field life functions. Management Science, 13(11):877–886, Jul 1967. [26] W.P. Pierskalla. Optimal issuing policies in inventory management. Management Science, 13(5):395–412, Jan 1967. [27] W.P. Pierskalla and S. Nahmias. Optimal ordering policies for a product that perishes in two periods subject to stochastic demands. Naval Research Logistics Quarterly, 20(2):207–229, Jun 1973.  92  [28] W.P. Pierskalla and S. Nahmias. Optimal ordering policies for a perishable inventory - i. In Proceedings from the XXth International Meeting for the Institute of Management Sciences, volume 2, pages 485–493, Israel, Jerusalem Academic Press, 1975. [29] W.P. Pierskalla and C.D. Roach. Optimal issuing policies for perishable inventory. Management Science, 18(11):603–614, Jul 1972. [30] G.P. Prastacos. Lifo distribution systems. The Journal of the Operational Research Society, 30(6):539–546, Jun 1979. [31] P.C. Spinella, C.L. Carroll, I. Staff, R. Gross, J. McQuay, L. Keibel, C.E. Wade, and J.B. Holcomb. Duration of red blood cell storage is associated with increased incidence of deep vein thrombosis and in hospital mortality in patients with traumatic injuries. Critical Care, 13(5):R151, 2009. [32] G.A. Tetteh. Optimal Allocation of Blood Products. Ph.D. dissertation, New Jersey Institute of Technology, Newark, NJ, 2008. [33] D. Wang, J. Sun, S.B. Solomon, H.G. Klein, and C. Natanson. Transfusion of older stored blood and risk of death: a meta-analysis. Transfusion Practice, 52:1184–1195, June 2012. [34] T.E. Warkentin and J.W. Elkelboom. Old blood bad? either the biggest issue in transfusion medicine or a nonevent. Transfusion Practice, 52:1165–1167, June 2012. [35] J.A. Weinberg, G. Jr. McGwin, M.B. Marques, S.A. 3rd. Cherry, D.A. Reiff, J.D. Kerby, and L.W. 3rd. Rue. Transfusions in less severely injured: does age of transfused blood affect outcomes? Journal of Trauma, 65(4):794–8., October 2008. [36] A.C. Zubair. Clinical impact of blood storage lesions. 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  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.24.1-0073844/manifest

Comment

Related Items