LIFE-SAVING DECISIONS: A Model for Optimal Blood Inventory Management. Lindsey Cant. Advised By: Professor Warren Powell

Transcription

1 LIFE-SAVING DECISIONS: A Model for Optimal Blood Inventory Management Lindsey Cant Advised By: Professor Warren Powell Submitted in partial fulfillment Of the requirements for the degree of Bachelor of Science in Engineering Department of Operations Research and Financial Engineering Princeton University April 17 th, 2006

2 I hereby declare that I am the sole author of this thesis. I authorize Princeton University to lend this thesis to other institutions for the purpose of scholarly research. Lindsey Cant I further authorize Princeton University to reproduce this thesis by photocopying or by other means, in total or in part, at the request of other institutions or individuals for the purpose of scholarly research. Lindsey Cant

3 Dedication To my family

4 Acknowledgments First I would like to thank my advisor, Professor Warren Powell, for his incredible accessibility this year. I truly appreciate all of the guidance and support he has given me for this thesis. I would like to thank Dr. Eric Senaldi at the Blood Center of New Jersey for taking time to meet with me as well as for the data he provided. I would also like to thank Phillip Schiff at the American Association of Blood Banks, Jamie Blietz at the National Blood Exchange, Leo DeBandi at the American Red Cross, and LTC Ron Fryar of the Armed Services Blood Program Office of the United States Army for their contributions to this thesis. To all of my friends, thank you so much for your support and friendship these last four years. To the ORFE s, thanks for the comic relief and entertainment during those late nights in Friend. To everyone else, thanks for tolerating all of the ORFE talk. Special thanks to Lawrence Azzaretti for inspiring my thesis topic. Finally to my parents, my brother, and my sister, thank you for your endless encouragement, patience, and love. You have given me such incredible opportunities to succeed and I am so grateful to have such a devoted and loving family.

5 Table of Contents CHAPTER 1 THE NEED FOR BLOOD IMAGINE THIS BLOOD ORGANIZATIONS The American Red Cross America s Blood Centers Military BLOOD BASICS White Blood Cells Platelets Plasma Blood Uses Red Blood Cells CURRENT BLOOD SITUATION Growth in Demand Supply Characteristics Situation Today ISSUES TO ADDRESS Blood Substitution Blood Import Policies REVIEW CHAPTER 2 BACKGROUND INFORMATION RED BLOOD CELL BASICS Blood Types Storage Requirements Frozen RBC Specific Types of RBC Products BLOOD DONATION Donor Requirements Types of Donors Types of Donations Loss of Donor Blood BLOOD TRANSFUSION Crossmatching... 23

6 2.3.2 Blood-Type Substitution in Practice LITERATURE REVIEW Simulation and Regression Analysis Markov Chain Analysis Dynamic Programming Recent Research REVIEW CHAPTER 3 SINGLE PERIOD BLOOD SUBSTITUTION ANALYSIS MODEL FRAMEWORK AND ASSUMPTIONS VARIABLE DEFINITIONS Resource Variable Decision Variables Exogenous Information THE NETWORK CONSTRAINTS CONTRIBUTION FUNCTION SINGLE PERIOD MODEL Data Contribution Parameters Results for Demand of 770 Units SINGLE PERIOD MODEL WITH VARIABLE DEMAND IMPLICATIONS REVIEW CHAPTER 4 MULTI-PERIOD MODEL FRAMEWORK VARIABLE DEFINITIONS Resource Variable Decision Variable Exogenous Information TRANSITION FUNCTION THE NETWORK CONSTRAINTS VALUE FUNCTION Exponential Smoothing of Value Functions CAVE Algorithm Stepsizes CONTRIBUTION FUNCTION AND OPTIMALITY EQUATION... 55

7 4.7 ASSESSING CONVERGENCE DATA REVIEW CHAPTER 5 MULTI-PERIOD MODEL RESULTS CONVERGENCE OPTIMAL POLICY ANALYSIS Total Shortages and Outdates Shipping and Holding Behavior Age Distribution Behavior Substitution Behavior Summary of Optimal Behaviors and Policies SPECIFIC EXAMPLE OF MODEL BEHAVIOR General Scenario Particular Optimal Behaviors Analysis of A-positive, O-positive, and O-negative Blood Behaviors REVIEW CHAPTER 6 IMPORT POLICY SIMULATION AND ROBUSTNESS IMPORT POLICY SIMULATION Import Policy Results Other Import Policies HOLIDAYS AND SUMMER POLICIES General Behavior Patterns Total Shortages and Outdates Summary DISASTER SITUATION POLICIES General Behavior Patterns Total Shortages and Outdates Summary REVIEW CONCLUDING REMARKS APPENDIX A BIBLIOGRAPHY

8 Every 2 seconds, someone in America needs a blood transfusion. [4] Chapter 1 The Need for Blood 1.1 Imagine This Imagine yourself getting into your car, coffee in hand, to drive to work just like you do every other morning. It is 7:25am as you pull out of your driveway and begin to head for the highway. You look down to turn on the radio for the traffic report, but as you reach for the dial, a car swerves and hits you head on at 50mph. The next thing you know, you are in the hospital. The car accident left you with extreme blood loss, but the blood transfusions given to you in the ER have saved your life. You are going to be okay. Imagine you are just diagnosed with leukemia. You will need blood. Imagine your best friend has sickle cell disease. He or she needs blood on a regular basis. Imagine your mother needs a liver transplant. She will need blood. Imagine your father needs open heart surgery. He will need blood. Imagine you and your spouse just had a premature baby. Your child will most likely need blood. While nobody likes to imagine these scenarios, they are everyday realities for many people in the United States. Every two seconds, someone in the US needs blood. Aggregated, thirty Americans each minute, 1,800 each hour, will require blood transfusions (American Red Cross, 2005). There is a 97% chance that in your lifetime, someone you know will need blood [5]. These staggering statistics illuminate the ever-pressing need for blood in the US. Blood is a unique resource in that it is supplied through volunteer donors. It is not something that can be manufactured or produced by machines. It must be given by healthy, charitable adults who are willing to donate their own blood for the use of complete strangers. In the United States, approximately eight million people donate blood each year, forming the blood supply. In terms of units, approximately 16 million units are collected

9 Chapter 1: The Need for Blood every year from volunteer donors [32]. In 2001, it is estimated that hospitals transfused almost 14 million units of blood to 4.9 million patients nationwide. On average, 38,000 units of red blood cells are demanded each day for various uses such as for accident victims, surgeries, transplants, cancer treatments, and for people with diseases such as sickle cell anemia [1]. In other terms, 4,000 gallons of red cells are used every day, with the average patient receiving approximately 4 pints, or units, of blood [13]. This large volume of donations accompanied by the urgent and frequent demands for blood requires an efficient network of blood collection facilities and hospitals. 1.2 Blood Organizations Blood centers throughout the US collect 93% of donated blood while the remaining 7% is collected at hospitals [1]. All blood collection centers are licensed by the Food and Drug Administration [32]. The massive inflow and simultaneous demand of blood in the US creates a complex inventory management problem. Various organizations exist to transfer this life-saving resource of blood from donors to recipients The American Red Cross The American Red Cross emerged during the World War II era as a supplier of blood. In 1940, a national blood collection program was created, with participation from the American Red Cross helping to collect about 13 million units during the war [1]. In 1947, after the end of World War II, the Red Cross president, Basil O Connor, declared that the American Red Cross would develop a blood collection program to control all United States blood donation facilities. Independent blood collection centers objected to this statement and organized the American Association of Blood Banks, an organization initially created to resist the Red Cross [8]. The independent centers were able to maintain some control in the blood banking market, but the Red Cross is still an integral part of blood collection today. The Red Cross operates 36 regional blood centers and five testing facilities across the United States. Their highly centralized approach to blood collection allows them to manage their inventory through a hub in St Louis, Missouri [8]. Some regions are net importing areas while others are net exporters and routine movements are scheduled annually, broken down by month. Using historical usage data, a one day of supply for each blood type is

10 Chapter 1: The Need for Blood calculated for each location and then inventory targets are set accordingly. A forecasting model also forecasts two weeks into the future by day to foresee inventory problems three to four days in advance [12]. Their ability to monitor all inventory levels at all locations allows blood to be moved around the country very efficiently. Currently, the American Red Cross collects about half of the blood in the United States [32] America s Blood Centers Amidst the conflict between the Red Cross and the independent blood centers, the AABB was created to support non-red Cross collection facilities. In 1962, America s Blood Centers was founded and is now an organization of 77 community-based blood centers with more than 600 collection facilities. Currently, ABC provides Canada with its entire blood supply and the US with about half of its blood supply [2]. Members of Americas Blood Centers participate in resource sharing programs to help balance inventories across locations. The National Blood Exchange is the primary program, helping to move over 400,000 units of blood each year [1]. Similar to the Red Cross, members of Americas Blood Centers can schedule routine shipments of blood from other regions. Often, large metropolitan areas and large university hospitals consume more than the local community collects, thus they always face a shortfall. Therefore, a routine movement of blood is essential to support hospital functions [6]. According to The Blood Center of New Jersey, their center imports approximately 23% of their blood supply. In general, blood moves from the center of the country towards the coasts, as the ratio of supply to demand is higher in the central United States [34]. Thus, it is apparent that this resource sharing and routine movement of blood is essential to successful hospital function Military An interesting aspect to blood banking is that the United States Department of Defense (DoD) manages its own, entirely separate blood network. Unlike the civilian population that is highly regionalized, the DoD has longer distribution lines as they are concerned with overseas facilities. With 22 collection centers in the United States and overseas and two distribution hubs in the US, one on each coast, the military is able to collect

11 Chapter 1: The Need for Blood blood for all of their operations. Both hubs retype the red blood cells and package them for shipment overseas with routine flights lasting from 36 to 48 hours. The Blood Transshipment Center receives the blood at the air field and then distributes blood to a Blood Supply Unit which is the final distributor, allocating the blood to hospitals, coalition partnership hospitals, navy ships etc. On the battlefield, different echelons of care exist, with smaller teams up front. The rear of the battlefield holds larger teams with more resources. Blood is pushed as far forward as possible, but lab capabilities are not always available to test the recipient s blood, so all front lines receive only the universal O-negative blood type. The military thus has different parameters guiding their management of blood including a larger demand for O-negative blood, longer lag times between donation and transfusion, as well as more frequent emergency situations. To protect soldiers during large attacks, the military must have reliable readiness and they thus maintain larger inventories than civilian blood centers. Additionally, when future operations are planned, the risk to soldiers can be calculated and factored into inventory targets [16]. The operations of the military present a unique niche in the blood banking industry as they remain completely separate from civilian blood collection. 1.3 Blood Basics Understanding the basic physics of blood is critical to accurately modeling any part of the blood supply chain. On average, each adult has approximately 14 to 18 pints of blood in circulation, comprising about 7% of total body weight. The circulation of blood throughout the body is a critical function as it provides nutrients and oxygen throughout the body, carries carbon dioxide and various body waste products to the kidneys, liver, and lungs for disposal, helps to fight infection, and facilitates the healing of wounds. To serve all of the above purposes, blood is comprised of many components: water, various types of cells, hormones, enzymes, nutrients, and body waste. The primary types of cells in the bloodstream are white blood cells, red blood cells, and platelets. The remainder of the blood is the plasma that consists of water, proteins, and salts [7].

12 Chapter 1: The Need for Blood White Blood Cells White blood cells, also called leucocytes, protect the body against infection. There are many types of white blood cells that perform different functions, but in general, white blood cells kill bacteria and parasites, produce antibodies that destroy foreign bodies, and destroy old and damaged body cells [32]. White blood cells are larger than red blood cells and platelets, but have a shorter life cycle that ranges from days to a couple weeks. When the body is fighting an infection, the number of white blood cells in the body can increase [7] Platelets Much smaller than white blood cells and red blood cells, platelets comprise approximately five to seven percent of an adult s total blood volume. Their primary function is to form clots to stop and bleeding [7]. Also called thrombocytes, the platelets break down when exposed to the air and release chemicals that transform fibrinogen, or a specific type of protein in the blood, into fibrin which are long threads that combine with red blood cells to form the blood clot [32]. Thus, platelets are critical in major trauma patients who have significant blood loss. The platelets themselves are formed in bone marrow and circulate in the blood for about 10 days. When donated and stored, they can only be kept for up to five days before they must be discarded [1]. This short shelf-life makes them a valuable product of donations Plasma The plasma is the contents of blood excluding the various types of cells. It is 95% water with the remainder consisting of proteins, hormones, and various nutrients such as oxygen. Plasma comprises 55% of the total volume of blood in an adult. Carrying nutrients, vitamins, and enzymes, maintaining blood pressure, and supplying proteins (such as various clotting factors) are the chief functions of plasma [7]. Plasma itself is not transfused in whole but is instead often separated into various products for transfusion. After donation, plasma is separated from the cells and is frozen within a few hours of collection. It can be stored for one to seven years and is often used for transfusion in patients with bleeding disorders [1].

13 Chapter 1: The Need for Blood Source: Griffith, Mitch [20] Blood Uses Whole blood is an extremely complex resource with many uses. When blood is donated at a blood center, it is typically separated into its components: red blood cells, white blood cells, platelets, and plasma. Each component serves a different purpose. For example, a liver transplant requires red blood cells, platelets, and plasma, a bone marrow transplant requires red blood cells and platelets, and an accident victim typically needs only red blood cells. The quantity of each product needed also varies as well as the usable lifespan of each product once donated [18]. In general, plasma is useful for trauma victims, burn victims, organ transplant recipients, and patients with clotting disorders. Platelets are useful for patients with bleeding disorders, often resulting from leukemia, cancer therapy, and/or open heart surgery [13]. The following table illustrates the specific blood needs for various conditions: Table 1-1: Blood Requirements for Particular Surgeries and Diseases Condition Red Blood Cells Platelets Plasma Liver Transplant 6-10 units 10 units 20 units Open Heart Surgery 2-6 units 1-10 units 2-4 units Car Accident 4-40 units - - Leukemia 2-6 units 6-8 units (daily for 2-4 weeks) - Sickle Cell Disease units - - While each component is extremely important for various uses described above, red blood cells are the critical component from whole blood donations as they are the most widely demanded and used. The remainder of this thesis will focus on the use and management of red blood cells. Source: Donate-Blood Website [13]

14 Chapter 1: The Need for Blood Red Blood Cells The distinctive red color of blood is a result of the hemoglobin in the red blood cells. These cells carry the hemoglobin, or a complex protein that transports oxygen, throughout the body and thus supply all tissues with necessary oxygen. The hematocrit, or the percentage of total blood that is red blood cells, is slightly over 40%. To give a more specific depiction of the distribution of the various components in blood, in a few drops of blood there are about one billion red blood cells and for each 600 red blood cells, there are approximately one white blood cell and 40 platelets [1]. The large presence of red blood cells in whole blood makes them a crucial component to study when examining blood donation and collection. Red blood cells, also called erythrocytes, survive in the blood stream for up to 120 days. When donated and stored, they have a shelf life of 42 days. After this period, the blood must be discarded [32]. Therefore, even though overall yearly supply still may exceed demand, the outdating of blood, or the necessity to discard after 42 days, creates time periods and specific regions with shortages. These shortages often occur with seasonality as summer and holiday months usually see a smaller number of donations [7]. Red blood cells are useful in many scenarios such as for patients with chronic anemia, kidney failure, gastrointestinal bleeding, acute blood loss from trauma, and surgery [1]. Thus, the wide range of uses for red blood cells makes it an attractive blood product to analyze. Additionally, the presence of eight different types of red blood cells with various substitution possibilities adds complexities to the management decisions made when requesting and transfusing red blood cells. Chapter 2 will discuss some more of the specific details of red blood cell storage, types, and products. 1.4 Current Blood Situation The current trends in supply and demand of blood in the United States present a major problem that needs to be addressed. The growth rate of supply is significantly smaller than the growth in demand. Although total supply of blood (as an aggregate statistic) exceeds total demand in the US, the disparity between growth rates suggests that major shortage situations are imminent. Understanding the reasons for these growth trends should help identify how to address impending shortages.

15 Chapter 1: The Need for Blood Growth in Demand The United States is facing a large growth in demand for blood transfusions. Advances in healthcare and medicine are extending the average human lifespan through improvements in surgical procedures such as organ transplants as well as progress in oncology. However, these medical procedures often require blood transfusions. Thus, the favorable advances in medical procedures are causing a strain on the supply of blood in the United States [13]. Improved hospital care is supporting a growing population of elderly people in the United States. These people are often no longer able to donate blood themselves and instead require blood. People that are over the age of 69 account for only 10% of the population, yet it is estimated that they require 50% of all whole and red blood cell transfusions [1]. Thus, the elderly population consumes a significant portion of the nation s blood supply. Another estimate is that over 85% of Americans that reach the age of 75 will require a blood transfusion [13]. This large surge in a specific population requiring large amounts of blood presents a serious problem for blood banking in the future. It is estimated that the demand for blood is growing at a rate of 6% per year nationwide [1]. However, individual regions estimate different figures based on their local population growth and demand trends. The Red Cross Badger-Hawkeye Blood Services, providing blood in parts of Wisconsin, Iowa, Illinois, and Michigan, estimates their growth in demand to be 11% per year [5]. The Blood Center of New Jersey saw a rise in transfusions of 8.8% from 1996 to 2000 [34]. These increases in demand will only rise as medical technologies continue to improve and as life spans are prolonged Supply Characteristics If the rate of increase in blood donation matched the increase in demand, then future blood shortages would not be an issue. However, the demographics of the United States presents a mismatch with the demand. The average donor is a college educated white male between the ages of 30 and 50 who is married with an above average income [1]. The median age for all blood donors in the US is 38 years old [7]. Thus, it is immediately apparent that those donating blood are not the same people as those receiving blood. Additionally, the WWII veterans are the most frequent blood donating population and

16 Chapter 1: The Need for Blood today s youth donate the least amount of blood. The following diagram illustrates these age specific donation trends: Years (WWII) Years (WWII) Years (WWII) Years (Baby Boomers) Years (Gen X) Years (Baby Bust) Years (Gen Y/Echo) Years (Gen Y) Figure 1-1: Graph of Average Donations by Age Group from Dr. Eric Senaldi at The Blood Center of New Jersey [34] The height of each bar is the average donation visits per year for each age group. The decreasing frequency of donations towards the younger generations illustrates a trend in today s world that younger people are less likely to donate blood now and in the future as they have not adopted blood donation as part of their lifestyle. Thus, as the WWII veterans age and stop donating blood, the growth in supply will be stunted further since younger generations are not making up for the lost donations. Due to various restrictions, only 60% of the overall US population is eligible to give blood. However, only 5% actually donate [49]. This staggeringly small percentage of the population is providing the entire supply of blood for the country. The Blood Center of New Jersey experienced an increase in supply of 3.1% from 1996 to This is significantly smaller than the previously stated rise in demand of 8.8% [34]. The Red Cross Badger- Hawkeye Blood Services estimates that their supply is growing at a rate of 8% per year, also lower than their 11% per year increase in demand. Increasing restrictions for donation are also greatly reducing supply [5]. Growing concerns regarding SARS, Creutzfeldt-Jakob Disease, HIV, and many others only increase the number of deferred donors and thus the

17 Chapter 1: The Need for Blood eligible donor pool is shrinking. The Blood Center of New Jersey estimates their deferral rate to be approximately 13.5% [34]. Additionally, different ethnic groups have different distributions of blood types. Much like the age disparity, there is a discrepancy between ethnic groups donating blood and receiving blood. As stated previously, the average donor is Caucasian. However, the demand for blood should generally match the ethnic distribution of the country. Since Caucasians make up only 75% of the population in the United States, the other 25% of demand has a different blood-type distribution than what is on average being supplied [38]. Thus, a distribution mismatch exists between supply and demand Situation Today The mismatch of supply and demand as well as the shrinking gap between available supply and demand is a realistic cause for concern in the future. The National Blood Data Resource Center s Comprehensive Report on Blood Collection and Transfusion in the United States in 2001 states that in 1989, the margin between supply and demand was 1,870,000 units, or 13.8% of the supply. However, in 2001, this margin had shrunk to 7.9% of the supply, or a margin of only 1,161,000 units [11]. The decrease in the margin in both percentage and absolute terms reveals the real danger of blood shortages in the future for the United States. Furthermore, the aggregate surplus does not reflect shortages by blood type or region. The country as a whole has a small surplus of red blood cells, but individual regions may face frequent shortages of some or all blood types. As the aggregate margin shrinks, these individual shortages will most likely become larger and more frequent. Therefore, the management of blood from blood center to hospital must be closely monitored to ensure efficient resource allocation. Additionally, this shrinking margin between available and requested blood is causing real problems in hospitals today. Hospitals categorize surgeries based upon urgency. If the hospital has an inadequate supply of blood, then various surgeries must be cancelled or postponed. The elective surgeries are the first to be delayed since they are the least critical. In 2000, 7% of hospitals had to postpone surgeries because of insufficient blood inventories [13]. As transfusion demand increases and more procedures are performed routinely in

18 Chapter 1: The Need for Blood hospitals, proper blood inventory management will be critical to minimizing the postponement of surgeries. 1.5 Issues to Address With the current blood-banking environment, efficient inventory management is critical. Making blood-type substitutions, when possible, is an important behavior for optimal shipment and transfusion decisions. Additionally, understanding the different value of holding each blood type in inventory should help blood banks more efficiently fill demand and hold blood for future use Blood Substitution The current state of blood banking in the United States warrants a careful analysis as the need for blood will soon meet and possibly surpass the amount of available blood. One critical aspect to study is blood type substitution. Although demands are characterized by the blood type of the recipients, the requests may be filled with other blood types more readily available in the hospital s inventory. Although not possible in all scenarios (as will be discussed in the next chapter), blood type substitution offers incredible alternatives to filling demand when supply is constricted. Freeing up a unit of one type can lead to many other substitutions that may allow what previously would be characterized as unmet demand to be filled. Thus, blood substitution analysis is a new and important facet to study when considering options to mitigate the impending blood supply shortage. Additionally, the current research and development regarding synthetic blood substitutes warrants detailed investigation. Although these substitutes only serve certain functions such as hemoglobin replacement and can only be transfused in certain situations, they offer universal compatibility, long-term storage, lower probability of disease contamination, and large-scale supply [33]. Many of the products under development still face multiple problems regarding safety and efficiency, so further research is required before these products can be fully accepted into medical practice [40]. Nevertheless, successful development of a universal blood substitute, even if only applicable in limited situations, would relieve the stress on the current blood supply. Therefore, investigating the effects of

19 Chapter 1: The Need for Blood the introduction of a universal blood substitute into medical practice is an important and constructive study Blood Import Policies Since some areas of the country generally receive more donations than are locally demanded while other metropolitan areas generally need more blood than is available through community blood drives, a pattern of imports and exports has developed. This large-scale movement of blood represents significant portions of supply for some blood centers. Once blood centers optimize inventory behaviors with locally donated blood, the optimal import (or export) quantity can be determined. These import policies can then be simulated to assess the effects of import quantity with respect to total shortages. The combination of optimal inventory shipping and holding decisions along with an optimal import (or export) policy should allow blood centers to minimize the expected total shortages at their center through a better understanding of overall optimal inventory levels for their center. 1.6 Review The current situation in the United States regarding the blood supply warrants immediate attention and investigation. If existing growth trends in supply and demand continue, the country will face an extensive shortage. Thus, creating a model that helps blood centers understand the effects of optimal blood substitution should reduce some shortages. Developing better estimates for the value of holding blood in inventory should affect the behavior patterns of blood centers to more efficiently minimize shortages and outdates of blood. Hospitals require consistent and reliable shipments from blood centers. Thus, reducing variation in available inventory despite unpredictable donation levels is of critical concern for blood centers. Additionally, the sensitivity of the centers to shifts in supply and demand trends is important for long-term inventory planning and emergency response strategies. Knowing whether current inventory management policies can support a shift in donation or demand levels is important information for blood center managers. Finally, structuring adequate import and export policies should minimize shortages across regions.

20 Chapter 1: The Need for Blood The following chapters address all of these issues. Chapter 2 discuses some background information regarding donation and transfusion practices, red blood cell details, and previous literature and research on blood inventory management practices. Chapter 3 sets up the basic network of blood flow through a blood center and investigates the effects of O-negative or synthetic blood substitution with a single period model. Chapter 4 extends the analysis to a multi-period approximate dynamic programming model, introducing all relevant functions and notation. Chapter 5 presents the results of this multi-period model, providing an analysis of optimal behavior patterns from the model in comparison to resulting behaviors and total shortages when value function approximations are not used. Finally, Chapter 6 investigates the robustness of the model solution and studies the effects of various import policies on total shortages.

21 Few people realize that blood is perishable and cannot be stored indefinitely. Blood centers function more as pipelines than banks, and there is a steady need for donors. [3] Chapter 2 Background Information Before creating a model, it is important to understand the physics of blood because many intricacies exist in terms of the types of blood, the types of RBC products, and the storage requirements. Blood is a perishable resource with a specific useful lifespan and thus must be modeled appropriately. The various types of red blood cells as well as the different red blood cell products require a detailed understanding before a realistic model can be formulated. Thus, the first section of this chapter explains the most important aspects of blood physics that pertain to the formulation of the model. In addition, the donation and transfusion processes are quite complicated and they thus also require explanation so that a theoretical model can accurately reflect reality. Many different types of donations occur in today s blood banks and the current focus on blood purity and safety has added extra requirements to the blood donation process. The transfusion process involves a matching step in which a patient s blood is tested with donor blood to ensure compatibility. This step as well as physician preferences complicate bloodtype substitution in practice. Therefore, Sections 2.2 and 2.3 outline these important details necessary for the creation of a theoretical model. Many studies have been completed regarding blood inventory management and optimal inventory levels. Although current research is extremely limited, research done in the 1960 s and 1970 s provides powerful insight and conclusions useful for blood substitution analysis. Section 2.4 summarizes these findings and emphasizes applicable conclusions as well as limitations and weaknesses of the previous studies.

22 Chapter 2: Background Information Red Blood Cell Basics Although only one of many components of human blood, red blood cells themselves are quite complex. The eight different blood types provide interesting substitution possibilities critical to hospital and blood bank functionality. Additionally, new technology has advanced storage capabilities not possible when much previous research was completed. Therefore to accurately represent activity at a blood center or hospital, these processes must be described so that appropriate assumptions and simplifications can be made Blood Types A major complication to the blood collection and transfusion process is the existence of different blood types among humans. In the 1600 s, animal blood was transfused into humans but these attempts were disastrous and it was not until the 1900 s with the work of Karl Landsteiner that the four main blood types were identified and the ABO typing system was thus established. The existence of two different antigens, A and B, found on red blood cells characterize these different blood groups. If a person has the A antigen, then their blood type is classified as A. The same is true for antigen B. If a person has both antigens, then their blood type is AB and if neither antigens are present, then the blood type is classified as O [7]. Accompanying each antigen are proteins in the plasma called antibodies. The A antigen is accompanied by the B antibody and vice versa. Type A antibodies will destroy any type A red blood cells and type B antibodies will destroy any type B blood cells. Therefore, if a recipient with type B blood is given type A blood in a transfusion, the recipient s A antibodies will stick to the A antigens of the donor blood. The resulting reaction can be fatal [17]. Therefore, donor and recipient blood much be matched appropriately prior to transfusion to avoid adverse reactions such as death. The following table created by the University of Utah s Genetic Science Learning Center illustrates the ABO typing system:

23 Chapter 2: Background Information Figure 2-1: Diagram of the ABO Blood System In addition to the A and B antigens, there are over 600 other antigens that can be present on the red blood cell, the most important of which is the Rh antigen. If it is present, the blood type is specified with a + and if it is not present, the type is specified with a - [7]. Therefore, eight types of blood exist in humans: A+, A-, B+, B-, AB+, AB-, O+, and O-. Types A+ and O+ are the most common while type AB- is the rarest [32]. These types have a complex substitutability pattern in that there is a universal donor, O-, and a universal recipient, AB+, but the other types can only be given to people with certain other types according to the antigens and antibodies present in the donor and recipient blood. The following diagram illustrates the substitution rules: Figure 2-2: Possible Blood Type Substitutions (Source: Northern Ohio Blood Services Region website [28]) The distribution of these blood types differs among different demographic groups and populations around the world. The following table illustrates the variability of blood-type distribution among populations. Each ethnic group is broken down according to the percent of population with each blood type:

24 Chapter 2: Background Information Table 2-1: Blood Type Distribution by Ethnic Group Ethnic Group Type O Type Type B Type AB US (Caucasian) US (African American) Hawaiian Chinese-Canton Brazilians Germans French Peru (Indians) Russians Source: Blood Book Website [2] The diversity in the United States produces variability in blood-type distributions among different population groups. This presents some problems when matching donor blood to recipient blood. An approximation to the general blood-type distribution in the Unites States is as follows: Table 2-2: Blood-Type Distribution of Population in the United States Blood Type O + A + B + O - A - AB + B - AB - % of US Population Source: Blood Book Website [2] Storage Requirements Red blood cells can live in the blood stream for up to 120 days, but when donated and stored, the current shelf life is only 42 days. There exist many reasons for this expiration. The first of which is that red blood cells, from now on referred to as RBC s, need constant metabolic energy to survive. Glycolosis is the process by which this energy is supplied and can be measured through ATP content in the blood. As time progresses, this process slows as glucose is consumed, thus reducing the ATP concentration. This reduction in ATP results in membrane loss in RBC s that decreases their viability in the blood stream after transfusion [22]. To help maintain glyocolosis, a specific ph range must be maintained in the RBC storage solution. Additive solutions are used to keep the ph between 6.4 to 7.2. When the ph rises above or below this range, less ATP is produced [21]. Therefore, the additive solutions used today maintain this narrow ph range during the shelf-life of the RBC s.

25 Chapter 2: Background Information During storage, many damaging changes can take place in the RBC s. Some of these changes are bacterial contamination, red cell lysis/destruction, and a decrease in the potency of red blood cell antibodies [9]. These morphological and biochemical changes reduce the survival time of RBCs after transfusion into a patient. Depending upon the storage substance, the lifetime of RBCs can be either 35 or 42 days, with most RBCs stored with a 42 day shelf-life. The actual ability of the RBC to deliver oxygen throughout the body has not been conclusively shown to decrease over storage times [31]. Therefore, the lifetime of RBCs in storage is limited to 42 days by the ATP concentrations and the various changes that can take place during storage described above. One final important requirement for storage is a proper temperature range. Red blood cells must be stored between 2 C and 6 C. Therefore, adequate refrigerator space must be maintained at blood banks to properly store all inventory or RBC s. If the temperature rises above or below this range, the red cell metabolism changes and alters the composition of the unit of blood. An additional note is that once removed from the refrigerator, transfusion must begin within 30 minutes [41]. These ph and temperature parameters produce quite specific storage requirements to maintain safety and viability of the RBC sample. The actual shelf-life of 42 days is determined by the requirement that 24 hours after transfusion, at least 75% of the transfused red cells must still survive. Thus, red blood cells are stored in anti-coagulant solutions that contain sodium citrate, citric acid, and glucose [37]. The additive solution and the RBC s are placed into polyvinyl chloride and polyolefin blood bags. This appropriate plasticizer allows the RBC s to achieve their full shelf life [23]. These storage solutions in use today are efficient and safe and are 95% effective [22]. These major advances in the understanding of blood storage requirements have increased the shelf life of red blood cells to their current 42- day life span Frozen RBC Source: Red Gold [32] Red blood cells can also be frozen, thereby extending their shelf life to up to 10 years [1]. This method would solve most inventory problems. However, frozen blood has many

26 Chapter 2: Background Information limitations that greatly reduce its attractiveness to hospitals. The first is that the blood must be thawed which takes anywhere from 60 to 90 minutes. In a trauma situation, this time frame is unacceptable and fresh (not frozen) RBC units must be used. Additionally, once it has been thawed, it must be used within 24 hours [7]. This further requirement complicates the use of frozen blood in a hospital setting. Blood must be frozen in a glycerol solution to prevent ice crystals that can later tear open the red cells. The thawing process then requires a washing step to re-suspend the cells into a saline solution which contributes to the thawing time [16]. Thus, the freezing process produces many disadvantages described above in addition to added costs. The processes of preparing, freezing, storing, and thawing the blood costs at least twice as much as the ordinary storage procedures for fresh blood. Furthermore, there is an increased risk for bacterial contamination during the 24 post-thawing period [24]. Fresh blood is therefore a much more attractive product to hospitals. Generally, frozen blood is only used for autologous blood donations that constitute a very small portion of total donations. Autologous blood donation refers to the donation of blood in which the donor and the recipient will be the same [1] Specific Types of RBC Products When RBCs are prepared into blood bags for storage and transfusion, different products are created. Some of the primary products are red cells in additive solution, washed red blood cells, and leukocyte-depleted red blood cells. The red cells in the additive solution are the RBCs that have been discussed previously. Washed RBCs are produced by placing a whole blood sample in a centrifuge to remove the plasma (the same process that occurs for RBCs in additive solution) but are then washed in an isotonic solution. Washed red cells have more limited use in patients. Leukocyte-depleted red cells are the same as the red cells in additive solution except that the leucocytes, or white blood cells, have been removed [37]. The reason for leukodepletion is that the white cells break apart in storage and can cause adverse reactions in patients. Thus, filtering the blood at a blood center and removing the leucocytes right after transfusion avoids these adverse reactions. The Blood Center of New Jersey estimates that two-thirds to three-quarters of all distributed RBCs are leukodepleted [34].

27 Chapter 2: Background Information Blood Donation Blood in the United States today is collected from a variety of people that qualify based on given specifications regarding, age weight, travel etc. When donating blood, people also have a variety of options in terms of the method they choose for donation. The blood supply therefore becomes segmented, further confusing what is considered available supply Donor Requirements In the United States, there are guidelines and regulations in place that describe who is able to donate blood. While many parameters exist, the general rules are that a donor must be at least 17 years old, must weigh more than 110 pounds, must be in good health, and must pass a health and physical examination [1]. This examination ensures that the quality of the blood donor is safe. An extensive list of conditions is in place to describe reasons for donor ineligibility. For instance, a person cannot donate blood if he/she has been in a correctional facility in the last 12 months, has had their ears pierced outside of a doctor s office in the last year, has had acupuncture in the last year, or has one of a long list of diseases and disorders [7]. These guidelines are in place to help keep the US blood supply safe, but they do decrease the pool of eligible donors Types of Donors In the United States today, blood donors are all volunteer donors. They do not receive any sort of payment for their donation and donate purely out of altruistic reasons. In 1972, the head of the U.S Department of Health, Elliot Richardson, placed blood banks under the control of the FDA. This then created a motion for a safer blood supply and in 1978, the FDA passed regulation that all paid donations must be marked as such on the blood bag. Hospitals almost instantly stopped using paid donor blood because they did not want to be responsible for the increased risk of hepatitis [8]. Current FDA regulation updated on December 12, 2005 states that all blood products intended for transfusion must be labeled paid if the donor received any sort of monetary compensation for donation. Reimbursement costs for donation are not considered payment and certain donor gifts and incentives are allowed [10]. Allowable gifts can include tee-shirts, cups, and raffle tickets

28 Chapter 2: Background Information [15]. The usual lower quality of paid donor s blood has eliminated the payment for donation of products intended for transfusion in the US. Additionally, different ethnic and racial groups donate blood with different frequencies. In the United States, the Caucasian population donates the most amount of blood. Blood centers have a more difficult time recruiting other groups such as African- Americans [34]. A study of first-time donors from 1991 to 1996 illustrates that 73.8% of donors were White, 8.6% were Black, 8.5% were Hispanic, 4.4% were Asian, and the rest were another race/ethnicity [49]. However, the 2000 US census found that the actual population distribution is as follows: 75.1% White, 12.3% Black or African-American, 5.3% Hispanic or Latino, 3.6% Asian, and 3.7% other races [38]. Although the Caucasian and Asian percentages are somewhat similar, the disparity between donors and actual population percentages for the Black and Hispanic populations is significant. Additionally, even small differences in these population representations can cause divergence at a large collection facility Types of Donations The first method for blood donation is called autologous donation, meaning derived from the patient s own body. With this process, blood donors can donate their own blood with the intention of having it transfused back into them at a later time. This is often done before surgery to have blood on hand as a replacement for blood loss during the procedure. Autologous donations and transfusions are advantageous as they reduce the risk of acquiring a disease from foreign blood [1]. However, autologous blood donation represents a somewhat small portion of all blood donations. A 1999 study by the National Blood Data Resource Study on 2040 blood centers in the US found that 4.7% of all collected blood, or 651 units was collected through autologous donations [36]. Autologous blood cannot be used by the general public and is collected for the sole use of the donor. These donations therefore will not be considered in modeling since they do not contribute to the available supply for the general population. New technology created by the Haemonetics Corporation allows for another donation process called apheresis donation. This is a process by which specific components of blood are donated through an extraction process, such as platelets or red blood cells. The

29 Chapter 2: Background Information donated product does not need to be processed like a whole blood donation and is immediately ready for transfusion [13]. The same study by the National Blood Data Resource Center found that only 0.8% of all donated RBCs are from this apheresis procedure [36]. The apheresis procedure takes one to two hours to complete whereas the ordinary whole blood donation takes 10 to 20 minutes [1]. Therefore, the majority of donors give red blood cells through regular donation methods. When blood donors donate whole blood for general use by anyone, their donation is referred to as an allogenic donation. These types of donations are the primary source of blood inventory in the United States. In 1999, 93.2% of all donated blood was collected through allogenic collections [36]. This large percentage signifies the substantial role that general, whole blood donations play in the blood supply system Loss of Donor Blood The first source of lost blood is in pre-donation deferrals. Often when a blood drive is conducted, roughly 75 to 80 percent of the scheduled appointments attend, thus resulting in an initial loss of expected blood from the drive. Of the people who do attend, approximately 15 to 20 percent are often deferred because of failed health and physical exams. The top three reasons for deferrals are a lack of hemoglobin in the blood stream, recent traveling, and colds. The first of these reasons is the primary grounds for deferral and occurs because some people do not have enough RBCs in their bloodstream to donate useful whole blood samples. These three factors account for approximately half of all deferrals [34]. Thus, a large percentage of scheduled donors do not actually donate any units of blood. Another important aspect to blood donation is the process of testing for infectious diseases. Once blood is donated, it is sent to testing labs that usually perform 11 to 12 tests, 9 of which are required by the World Health Organization. If any diseases are identified in the blood, the sample is immediately discarded, thus reducing the supply of blood to a level lower than total donations [7]. The American Association of Blood Banks estimates that approximately 2% of blood donated is discarded due to failed tests [1]. Testing procedures are not generally time consuming as about 95% of donations are tested within 24 hours of arriving at the testing facility [7]. Thus, the major effect of testing is in the slight reduction of supply, not in the time delay.

30 Chapter 2: Background Information Blood Transfusion Much like donation, the process of blood transfusion involves many intricacies. New technology and medical research has advanced the understanding of the effects of blood type substitution. A quote from the Mayo Clinic found on the Blood Book website exemplifies the new complications in blood substitution: In the past, a person with blood type O-negative blood was considered to be a universal donor. It meant his or her blood could be given to anyone, regardless of blood type, without causing a transfusion reaction. This is no longer a relevant concept because of a better understanding of the complex issues of immune reactions related to incompatible donor blood cells" [7]. Therefore, the following sections discuss the blood-type matching process as well as the practical aspects of blood-type substitution Crossmatching One of the most unique aspects to blood collection and transfusion is the process of crossmatching, a process which matches donor and recipient blood samples. Although the ABO typing system dictates which blood types can be allocated to which patients, the patients serum and the donor s red blood cells are tested directly for compatibility of other antibodies in the blood. The crossmatching process takes about an hour to complete. When a physician requests blood for a patient, the recipient s blood is tested and matched with particular blood units from the available inventory. This test is especially important when the patient has a clinically significant red blood cell antibody other than the A,B, and Rh already discussed [41]. However, physicians always request more blood than is necessary in case of emergency. Thus, a new variable must enter the supply chain model: the transfusioncrossmatch ratio that relates the amount of blood actually transfused to the amount crossmatched, or assigned. The blood originally assigned but not transfused is returned to available inventory after a period of time, referred to as the crossmatch-release period. [25]. This extremely unique aspect to blood banking complicates any modeling as assigned inventory can later be moved back into available inventory, but only after a period of time that must be factored into the age of those specific units of blood.

31 Chapter 2: Background Information Blood-Type Substitution in Practice The blood type substitution diagram in Section is only accurate in certain situations. Some patients and procedures are especially sensitive to proper matching of donor and patient blood type. For instance, pregnant women and newborns are more sensitive and often must only receive blood of their own type. Additionally, certain procedures such as open heart surgery require an exact blood type match. Therefore, some situations do not allow for any blood substitution while other blood demands are more flexible. For example, in an emergency, a person can receive a positive blood type even if they are Rh negative without any significant adverse reactions. However, a second time mismatch may cause adverse reactions [34]. Therefore, blood substitution while possibly very powerful in alleviating shortages does have some limitations. This presents a significant modeling challenge as substitution practices are highly variable from physician to physician. Simulating demands in which substitution is allowed as well as demands that require an exact blood type match can solve this issue. Some physicians will only substitute when absolutely necessary while others will do so much more freely. This wide range of behaviors makes it impossible to define normal substitution behavior. Thus, assumptions and generalizations must be made about average substitution frequency. 2.4 Literature Review Over the last fifty years, a somewhat large body of research has been completed regarding blood inventory management as well as supply chain analysis at a regional level. The studies primarily implement simulation, regression, and markov chain analyses to examine various policies at a blood bank. The bulk of this research was conducted in the 1960 s and 1970 s and current studies are lacking. Additionally, no one has implemented dynamic programming methods that include both type and age to analyze blood inventory management. Therefore, the limited current research paired with the recent advances in blood storage lifetime as well as the lack of thorough dynamic programming models provides an opportunity for new exploration.

32 Chapter 2: Background Information Simulation and Regression Analysis Using inventory models to manage the distribution and collection of blood began with the work of Millard in He was the first person to apply inventory models to the supply chain of whole blood. The next contributors to this area of research are Elston and Pickerel who used statistical methods to analyze the effects on a single hospital of altering the age composition and the size of blood supply [26]. They also analyzed the effect of issuing policies, specifically LIFO and FIFO policies, on expected shortages and outdates, concluding that the first-in first-out policy reduces both shortages and outdates. They use the negative binomial distribution to model the demand. In 1975, they expanded their model to create tables of operating characteristics of the system at various order-up-to levels that help a hospital pick the optimal order-up-to quantity [39]. In 1973, Jennings was the first to expand this analysis to a regional level instead of a single hospital. Using simulation analysis, he examines a system with a number of identical individual hospital blood banks making decisions based on the same policies. He studies two cases, no blood sharing between hospitals and blood transferring between hospitals, and concludes that when blood is shared, shortages and outdates are reduced. Thus, managing blood on a regional scale with available transfers increases overall efficiency [39]. This simulation also takes into account the distinction between assigned blood and available, unassigned blood [26]. Jennings was the first researcher to understand the true importance of this distinction. The next contribution came from Brodheim, Hirsch, and Prastacos in 1976 when they studied operational data to develop relationships between demand and inventory levels at specific shortage rates [26]. Using regression from 49 sets of daily demand data, they examined the fits of linear, piecewise linear, quadratic, and piecewise quadratic equations and were successful in accurately predicting inventory levels when compared to actual data from 21 hospitals across the country. Their average errors ranged from 4.2% to 19% depending on the size of the hospital and they thus concluded that all hospitals could use the model to accurately set inventory for a specific, desired shortage rate [39]. A limitation of the analysis is that they analyzed the behavior at a single hospital instead of a regional blood network, which would exhibit different behavior based on the results from Jennings.

33 Chapter 2: Background Information Cohen and Pierskalla in 1975 attempted to use simple equations to set optimal inventory levels. Their analysis uses many variables, such as rates of demand, the return of unused assigned blood back into available inventory, and thus the effects of crossmatching [26]. Using simulation, they determine an optimal inventory level based on all significant variables and they also analyze various ordering policies, issuing policies, and crossmatching policies while viewing the supply of blood from a regional perspective. They additionally examined the sensitivity of different variables to the total costs and the optimal inventory using outdate costs of $25 and shortage costs of $55 [39]. This work united many important variables into a single simulation analysis and explored the importance and effect of changing specific values such as the shortage rate. The next big contribution to the analysis of supply management of blood using simulation came from Abbot in This model was the first to incorporate the various types of blood while all previous work had used aggregate blood or dealt with each type individually. He thus included the substitutability of blood types into his simulation therefore reducing shortages and representing a more realistic model of actual operating procedures at hospitals. He also concluded that older blood has a higher probability of being used if moved to a larger hospital [39]. After Abbot s work, no large contributions have been made in using simulation to analyze the inventory management of blood Markov Chain Analysis Markov chain analysis has also been applied to this perishable inventory problem. Pegels and Jelmert developed a Markov chain model to analyze the effects of issuing policies, but their studies were criticized by many [26]. They used an absorbing state Markov chain to see how inventory levels and average age of transfused blood is affected by various issuing policies. Their model requires the transition probabilities that a unit of blood will be transfused given its age. They also take into account the effects of crossmatching in another model, but neither model is truly dependable as their fixed transition probabilities in reality do change as the issuing and ordering policies change. Additionally, their model was criticized by Kolesar who pointed out that the transition probabilities are also a function of additional variables other than age alone, such as demand rate and current inventory level [39]. Thus, their model is flawed and not reliable.

34 Chapter 2: Background Information Brodheim, Derman, and Prastacos also applied Markov chains to analyze blood supply and demand in Their model also requires transition probabilities, but was reliable enough to be used to create the Programmed Blood Decision System, a computer model that helps to schedule deliveries to hospitals. In 1976, Cohen modeled blood inventory management as a stochastic process using FIFO issuing policy, taking into account backlogged demand, but assumed that all units assigned are used, which is not in fact the case due to crossmatching. Jagannathan and Sen later, in 1986, specifically studied the crossmatch demand process [39]. Their analysis assumes constant, deterministic demand though, and is thus not very realistic Dynamic Programming Finally, Fries and Nahmias have each applied dynamic programming to the perishable inventory management problem, both publishing work in Their studies do not pertain to blood specifically and thus the particular shelf-life and substitution patterns are not incorporated into their research. Both models are very similar, the main difference being that Nahmias applies outdating costs for a specific decision as part of that one-period cost whereas Fries includes the costs after a number of transitions equal to the lifetime of the blood. Their analysis has many valuable conclusions regarding perishable inventory in general: increasing current inventory by one unit decreases optimal order quantity by less than a single unit, and the optimal order quantity is not as sensitive to changes in the older inventory as it is to newer inventory. This age sensitivity requires that for an accurate model, all ages present in the inventory must be taken into account, and therefore a vector with a dimension equal to the shelf life of the product is required. Nahmias thus concludes that working with inventory that has an expected lifetime greater than 3 periods is extremely difficult [39] Recent Research In recent years, few operations research or management science studies on managing the blood supply chain have been carried out. Since the 1980 s, research on this topic dropped significantly, partly due to a cutback in federal funding to support such research. Instead, the focus has shifted towards increasing the safety of the blood supply. Since the

35 Chapter 2: Background Information management of blood involves so many complications such as substitutability and crossmatching, accurate models are quite difficult to create and new research in the area of blood banking has focused on developing better information systems to facilitate faster and easier screening, testing, and ordering. In 1991, Sirelson and Brodheim created a platelet inventory model using simulation to predict shortage and outdate rates based on demand and inventory levels. The scheduling of donors has also been studied recently with Brennan s work in studying the psychology of donors and collection facility management. Using simulation, donor registration and screening was improved [29]. Pierskalla s paper (2004) on the supply chain management of blood banks involves simulation analysis and regression analysis in modeling optimal inventory levels at hospital blood banks and community blood centers and also examines issuing and assignment policies. However, the bulk of the research on blood bank management was done in the 1960 s, 1970 s and 1980 s. The dramatic changes in blood supply and demand characteristics since the 1980 s coupled with the dearth of new models and studies warrants new research into supply chain operations of blood banks and hospitals. 2.5 Review Dynamic programming analysis regarding blood inventory management, especially taking into account the substitution possibilities with blood types and the age of the blood, has not been completed. Therefore, this thesis will examine the effects of optimal blood substitution, initially focusing on the value of O-negative substitution as it is the universal donor. This has implications for supporting the development of synthetic blood substitutes. Then, non-linear value function approximations for holding blood for the future will be applied to the model to better understand the effects of value distinction by type when making shipment and transfusion decisions. Additionally, this model will address Nahmias concern over the difficulty in accurately representing blood as a resource. The model will divide blood into six age buckets, by week, thus providing an accurate dynamic programming model. Although all 42 days will not be individually monitored, many blood banks make shipments on a weekly basis, thus a weekly age is often sufficient to accurately make shipment decisions. Therefore, the multi-period model in this thesis investigates the utility of applying dynamic programming methods to blood inventory management decisions.

36 The arena of artificial blood offers a powerful forum where the fields of science and engineering combine to illuminate the ingenuity of mankind. [35] Chapter 3 Single Period Blood Substitution Analysis The American Red Cross and Americas Blood Centers manage the allocation and distribution of blood quite differently, yet both methodologies for filling demand involve distribution by a community or regional blood center within specific regions to hospital blood banks. Therefore, the most simplistic model for blood distribution is at the regional level, with a given set of hospital blood banks and a community blood center. Although different regions may have different numbers of hospital blood banks or different supply or demand characteristics, all regions have the basic structure of blood donation at the community blood center from which regular shipments are sent to the hospital blood banks where transfusions occur. The following chapter develops a single period model to examine blood substitution effects at a single location. Using a linear programming solver to optimize flow through the blood substitution pattern, the effects of increasing or decreasing the supply of the universal donor, O-negative, can be evaluated. This will help illustrate how blood type substitutions, particularly with the universal donor, can optimally conserve inventory of all types for future use. The framework for this model also sets the foundation for the multi-period dynamic programming model established in Chapter Model Framework and Assumptions Within a single region, each location must be precisely defined in terms of what decisions can be made and what activities can occur at that particular location type. Additionally, the timing of these decisions and activities must also be clearly defined to properly model the information known by agents at each location when making decisions.

37 Chapter 3: Single Period Blood Substitution Analysis Agents are defined as people at each location that have access to a specified set of information helping them to make ordering, shipping, and transfusion decisions. The primary function of a hospital blood bank is to transfuse blood. Although some hospital blood banks (HBB s) collect a small portion of blood, this model assumes that no blood is donated at the HBB. The exogenous information pertaining to an HBB location is the amount of blood demanded during each time period. The role of a community blood center (CBC) is to collect donated blood and then distribute it to hospitals for transfusion. Thus, the exogenous information pertaining to the blood center is the amount of donations during each time period. Since this analysis focuses on inventory at a single location, activities at a community blood center will be discussed. The crossmatching activities occur in hospitals so they are thus not a part of this model. The blood in inventory at a blood center ranges in age from one to forty-two days, but this model assumes ages of one to six weeks. Another important assumption is that all donated blood is in fact allogenic. Autologous blood donations are available only to the donor or requested/intended recipient so these donations are not part of the general, available inventory pool at a hospital or blood center. Therefore, this model assumes that all blood donated can be used for any recipient. Also, this model aggregates all RBC products into one category such that no distinctions are made between washed RBCs, leukodepleted RBCs etc. All RBCs are assumed to be fresh blood, not frozen, since frozen blood is not frequently used, is not desirable, and poses complications to modeling because it takes time to thaw and then it must be used within a 24- hour period. Since the model investigates weekly supply and demand, frozen RBCs are ignored. Additionally, the exogenous supply at the CBC represents available red blood cells after donor deferral and testing rejections. This is an appropriate assumption because even though expected supply may include all bookings for a blood drive, actual supply seen by a blood center does not include deferred donors. Since such a small percentage of blood is rejected due to testing, this parameter is also ignored. More assumptions will be made when the model is extended to a multi-period time frame, but for the initial blood substitution model, these are all of the necessary assumptions. The choice of using a community blood center is made because more accurate data on supply and demand was available. However,

38 Chapter 3: Single Period Blood Substitution Analysis this model can easily be translated for a hospital location with simple changes to decision definitions. 3.2 Variable Definitions Before substitution patterns can be studied, proper notation must be developed to describe the resource state and the decisions at a blood center. The inclusion of attribute vectors to describe all characteristics of resources and decisions provides an elegant notational framework that extends to the multi-period model. The attribute vectors can also be applied to the exogenous supply and demand information to provide consistent notational framework. The resource and decision variables as well as the exogenous information provide the basic foundation for this model Resource Variable Each location has its own inventory of blood of different types and ages. Since this chapter investigates a single time period model, a time index is not necessary for the notation. The resource state variable includes as attributes the location, blood type, and age of blood. The age attribute is included because blood is still identified by age even though decisions are not made based on age. The attribute vector, a, provides an elegant method of representing all this information with one variable: R a = the number of units of blood with attribute a " $ a = $ $ # a type a location a age % ' ' ( A ' & Therefore ( R a ) a A R = is the resource state. The age attribute refers to the age of the blood by week, the location attribute refers to the location of the blood in terms of which blood center or hospital, and the type attribute refers to the blood type. Since the state of the system is purely defined by the number of units of blood with given attributes, and no other information is pertinent for decision-making, the state of the system is simply the resource state in this model.

39 Chapter 3: Single Period Blood Substitution Analysis Decision Variables The decisions to order and transfuse blood are made at the HBB location while the decisions to assign and ship blood are made at the CBC. Thus the decisions of interest in this analysis are the shipping and holding of blood denoted by the following notation: D = D D "D M = the set of all decision classes where D D = { satisfy demand} D M = { hold} A decision attribute is also defined to capture all relevant information pertaining to decisions: ' b % % b b = % % b % & b type location surgery type substitution $ " " " B " " # With this framework, the following notation is introduced for decisions: Thus, x ad = the # of units of blood with attribute a acted on by a decision of type d with decision attribute b x = ( x ad ) a "A,d "D d " D M # decision to hold blood d " D D # decision to satisfy demand type b " B The consistency of subscripts for each type of decision allows for summations over all decision types, which is useful when implementing a transition function. The attribute b type refers to the blood type of a demand, more specifically the blood type of the patient requesting blood. The attribute b location is the location requesting the blood. In this model, demand at a community blood center is the aggregated orders from all hospitals. Hence, the attribute b location surgery type refers to all hospital locations collectively. The attribute b classifies demands as either urgent or elective. Finally, the attribute b substitution classifies the demand according to whether or not blood-type substitution is permissible. This notational framework produces a model similar to blood center practice as CBC s must fill standing orders, taking into consideration requests by type. They also face

40 Chapter 3: Single Period Blood Substitution Analysis regular (elective) demands as well as emergency (urgent) demands for blood. Therefore, the overarching structure of the model represents relatively well the substitution patterns and levels of urgency present in real shipment decisions Exogenous Information With the attribute notation established for resource and decision variables, the exogenous information notation can be introduced. The supply is the number of donated units of each blood type during the week at the CBC and the demand is the aggregated weekly orders by type from all hospitals in that given blood center s region. This single period model ignores noise in the exogenous supply and demand. Again, the attribute notation is implemented to succinctly write out variables: R ˆ a = the number of units of blood donated with attribute a " a type % $ ' a = $ a location ' $ # a age ' & B ˆ b = the number of units of blood demanded with attribute b " b type % $ ' $ b location ' b = $ surgery type b ' $ # b substitution ' & The hat above each variable denotes this information as exogenous and clearly distinguishes it from endogenous information. When random sample realizations are used, often times the notation w is used to denote a single sample realization, R ˆ ("). However, this can become clumsy and thus is left off. 3.3 The Network Figure 3-1 shows the basis for the single-period blood substitution model.

41 Chapter 3: Single Period Blood Substitution Analysis Figure 3-1: Network for Single-Period Model Since eight blood types exist, the network is comprised of eight supply nodes, eight demand nodes in which no blood type substitution is allowed, eight demand nodes in which substitution is allowed, and eight nodes representing holding blood for use in future time periods. All nodes in the middle column flow into a single super-sink node to ensure conservation of flow through the network. When urgent demands are filled, a higher reward is earned since urgent blood requests usually translate to situations where lives are at risk. Non-urgent blood requests, either for elective surgeries or for general inventory replacement, are less important and thus the reward for filling elective demand is much lower. Although hospitals vary in terms of elective surgeries as a percent of total procedures, the model assumes that 15% of procedures are elective, a general figure provided by the Department of Health of Australia [14]. (No such figure could be found for the United States, but hospital behavior in the two developed countries can be assumed to be similar). Hospitals in metropolitan areas probably have a smaller elective percentage as the crime rate is often higher and more emergency surgeries are performed whereas hospitals in suburban areas may have a slightly higher percentage as emergency trauma is a smaller part of hospital activity. Thus, this percentage varies across hospitals, but 15% is assumed in this model to be a reasonable representation of average hospital activity. The total weekly demand is classified by blood type. Then demand for each blood type is divided between demands in which substitution is allowed and demands requiring an exact blood type match. This percentage is highly variable, but the blood substitution model assumes this percentage to be 50%, keeping in mind that demands allowing substitution can still be filled with the matching blood type. The demands are then split further to differentiate between elective procedures and urgent surgeries, using the 15% estimate.

42 Chapter 3: Single Period Blood Substitution Analysis Since the decision notation includes this information in the attribute b, the probabilities are built in. Thus, when writing out constraints, it is not necessary to explicitly include these probabilities in the equations. 3.4 Constraints The constraints for the linear program specify flow conservation for all nodes in the middle of Figure 3-1. Additionally, all flows along arcs coming out of demand nodes and flowing into the final flow node must be less than or equal to the specified demand, given by a sample realization of exogenous information. All flow of a given blood type out of the supply nodes must be equal to the total supply available. Finally, all flows must also be non-negative. Before expressing the constraints in notation, an additional variable must be defined: y bd = the # of units of blood acted on by a decision of type d with attribute b In the network depicted in Figure 3-1, this variable refers to the flows on the arcs from the demand nodes into the final supply node. The following expresses these constraints in appropriate notation: # x ad = R a (3.1) d "D # x ad $ y bd = 0 (3.2) a "A y bd " ˆ B bd when d # D D (3.3) x ad " 0 (3.4) No constraint is needed for the super sink since this last constraint would be redundant. 3.5 Contribution Function There is a reward associated with each of the arcs in the network. contributions are defined by the following notation: These c ad = the unit contribution of acting on a resource of attribute a with decision type d The contribution function is defined as: ( ) = x ad C x # # c ad (3.5) a "A d "D

43 Chapter 3: Single Period Blood Substitution Analysis The actual contributions are set so that the utility function of physicians is accurately captured. Therefore, these contribution parameters are set on an ordinal basis and are then calibrated based on the performance of the model with respect to actual behaviors in practice over a large number of iterations. In this chapter, dynamic programming is not implemented. Therefore, the objective function simply maximizes the contribution function over all possible decisions. The optimal decisions are found according to the following equation: x = argmax x "X # # x ad c ad (3.6) a "A d "D 3.6 Single Period Model The goal of this single period model is to investigate the effects of increasing the supply of O-negative blood while holding the demand as well as the supply of other blood types constant. O-negative blood is chosen for the analysis since it is the universal donor and thus when substitution is tolerable, it can be used to fill any type of demand. Therefore, conclusions of this analysis extend to a better understanding of the effects of introducing a man-made blood substitute. Although no product is currently available, many companies such as Northfield, Sangart, and Baxter are developing various blood substitute products that could revolutionize the blood banking industry. Most of these products are in the clinical trial stage and are thus not approved for general use in the United States [33]. However, understanding how these substitutes would affect the blood banking industry is of great importance since some of these products are nearing the final development phases. Thus, the objective of this chapter is to investigate how additional units of universal blood improve blood bank performance. Additionally, the optimal level of additional units of a universal substitute can also be determined. Most blood banks have limited storage space, thus this optimal level for a universal substitute is valuable information. The analysis in this section is based on a constant, deterministic demand. Section 3.7 will explore deterministic demands at different levels as the effects of O-negative substitution depend on relative supply and demand levels.

44 Chapter 3: Single Period Blood Substitution Analysis Data The data regarding blood donations and blood transfusions is extremely private and only high level, aggregated figures are normally released to the public. The National Blood Data Resource Center conducts surveys periodically and produces a comprehensive report available to members and available for purchase to non-members. Thus, the extreme confidentiality of the blood banking data requires some extrapolation when performing simulation analysis. The Blood Bank of New Jersey has provided some weekly donation data as well as the average weekly shipments to hospitals, 770 units. This data is used as the basis of exogenous information sampling in this thesis. The supply distribution is based on 52 weeks of donation data by type. The weekly total donation amount is the average weekly total donation over those 52 weeks, or 630 units. Imported blood, used to fill the gap between excess demand over donated supply, is not included in these figures as the purpose of this section is to establish an understanding of how adjustments to the supply of O-negative blood affect a single CBC with given donation and demand levels. The demand distribution by type is based on the general population distribution of blood types. The assumption here is that the people demanding blood represent a sample of the entire US population such that their demand distribution by type is in fact the distribution of the entire US (previously given in Table 2-2). While this may not be entirely true, it is the most logical assumption to make and still illuminates the mismatch of supply and demand distributions. The following distributions of supply and demand by type are used in this analysis: Table 3-1: Supply and Demand Distributions by Blood Type in the United States Blood Type O + A + B + O - A - AB + B - AB - Percentage of Supply 39.82% 27.94% 11.63% 9.26% 5.17% 3.4% 2.13%.65% Percentage of Demand 38% 34% 9% 7% 6% 3% 2% 1% In this model, total demand remains constant as the supply of O-negative is increased from 5 to 505 units in increments of 5. Demand by type is found by multiplying the percentages in Table 3-1 by 770 units. The supplies of all other blood types other than O-

45 Chapter 3: Single Period Blood Substitution Analysis negative remain constant and are determined by the supply distribution shown in Table 3-1 along with the average 630 donated units Contribution Parameters To accurately represent blood banking in practice, a variety of behaviors must be captured by the model. Demand should always be filled with the FIFO method since previous research has found this method to be an optimal assignment method. Additionally, it is preferable to fill demand with the proper blood type, even when substitution is permissible. Thus, the contribution when the supply and demand blood types match should be higher than when they do not match. Since this model examines the effects of a universal substitute, a higher contribution should be realized when demand is filled with O-negative blood. This gives preference to O-negative blood whenever these substitutions are possible. In practice if a synthetic blood substitute is created, it will be preferable to use the substitute when possible to conserve real blood for situations in which the substitute may not be tolerable. The contributions for filling urgent and elective demands should be relatively high to reflect the importance of filling demand. Along the same lines, urgent demand should be given complete priority over elective demands. The value of holding blood is set to zero since this initial analysis does not include dynamic programming to update the approximate value functions of holding blood for the future. Given this required behavior, appropriate contributions are calibrated to the following values: Table 3-2: Contribution Parameter Settings for Single Period Model Conditions Description Value d " D M Holding 0 d " D D No Substitution 10 d " D D Substitution 5 d " D D O " Substitution 15 d " D D Filling Urgent Demand 40 d " D D Filling Elective Demand 20 when if a type = b type when if a type " b type when if a type = O " when if b surgery type =urgent when if b surgery type =elective when

46 Chapter 3: Single Period Blood Substitution Analysis Results for Demand of 770 Units Using Matlab s linear program solver that implements the Simplex method, an optimal solution can be found for a given set of supply and demands. O-negative substitution is found to be of value even when demand is entirely filled. Figure 3-2 shows the objective function value and slopes as a function of the supply of O-negative, shown here as a percentage of total supply. The slopes are calculated as the difference between successive objective function values: Figure 3-2: Objective Function verses Amount O-negative as a Percent of Total Supply. The numerous steps that are seen for the slopes correspond to different changes in substitution and demand-filling behavior. The first drop occurs when all O-negative demand is filled and the second drop occurs when the demand of all blood types is filled. The last drop is a series of steps reflecting O-negative substitutions of other blood types. Thus, demand is completely filled when O-negative is 25.5% of supply, or is 196 units. However, the objective function still has a positive slope beyond this point and reaches zero when O- negative is 30.6% of supply, or is 252 units. Thus, this illustrates that O-negative substitution has a positive marginal benefit even after supply surpasses the total demand of 770 units because of the blood-type substitution effects. Therefore, a blood center can conclude that maintaining an inventory of O-negative or of a man-made, universal blood substitute comprising 25% to 30% of the total supply should allow them to satisfy all demand. This conclusion is somewhat weak, though, since this model only considers one demand level, namely 770 units. Therefore, the following section explores the behavior when multiple demand levels are considered.

47 Chapter 3: Single Period Blood Substitution Analysis Single Period Model with Variable Demand Extending the previous analysis to better understand O-negative substitution effects across different demand levels, the demand in this analysis is increased from 575 units up to 950 units in increments of 50 units. For each of these demand levels, the supply of O- negative is increased in increments of five units, ranging from 5 to 345 units. Therefore, the total supply ranges from approximately 575 units up to 915 units. These two ranges of supply and demand provide a more comprehensive view of the effects of O-negative substitution, both ranging around the mean supply and demand observed at The Blood Center of New Jersey. Figure 3-3: Objective Function Values verses Amount of O-negative Blood as a Percent of Total Supply for Multiple Demand Levels. Figure 3-4: Slope of Objective Function verses Amount of O-negative as a Percent of Total Supply for Multiple Demand Levels.

48 Chapter 3: Single Period Blood Substitution Analysis The objective functions and slopes for all demand levels are plotted in Figures 3-3 and 3-4 to see the relative differences in O-negative substitution behavior across the different demand levels. An obvious expectation is that a higher demand level will benefit more from extra O-negative since more units of demand can be satisfied. Although the slope of the objective function when demand is 950 units appears to be non-linear, Figure 3-4 illustrates that across all demand levels, the slope of the objective function at various steps is the same. This means that increases in O-negative supply produce the same marginal benefit regardless of the demand level at each step. However, the placement of the steps varies across demand levels since different total demands must be filled in each case. The key finding here is that all demand levels still show the same behavior in that even after total supply meets total demand, there is a benefit to adding additional units of O-negative blood. This is reflected in the last step of each of the lines in the above graph where the slopes are at about five. Thus, regardless of the demand level, there is still a marginal benefit to additional units of O-negative blood beyond the point at which supply meets demand. However, the effects of an increase in O-negative beyond the point at which demand is initially filled varies as demand increases. The largest demand level, 950 units, shows an extremely small marginal benefit beyond the point at which demand is filled. This is an intriguing phenomenon and can better be visualized using three-dimensional analysis.. Figure 3-5: Three-Dimensional Representation of Objective Function verses Total Supply and Demand.

49 Chapter 3: Single Period Blood Substitution Analysis The demand levels in Figure 3-5 range from 505 units to 905 units in increments of 5 units. The supply levels have the same range as in the previous section. Figure 3-5 clearly shows that as demand increases, the effects of additional units of O-negative blood beyond the point at which demand is filled are smaller. The red line separating the green and the blue/teal sections on the right is the point at which increasing O-negative blood has no marginal benefit. The beginning of the blue section is the point at which demand is first filled. In examining the specific decisions being made, it is observed that when the demand level is high, the supply is primarily used to fill non-substitutable demand. As the supply of O-negative increases previously unsatisfied substitutable demand are immediately filled with O-negative blood. Therefore, once all demand is filled, there is little room for improvement with additional units of O-negative since most substitutable demand is already being filled with O-negative blood. However, when demand is low, the initial supply fills both non-substitutable and substitutable demands. Thus the point at which total demand is first filled corresponds to a situation in which blood types other than O-negative are filling substitutable demand. Therefore, many O-negative substitutions can be made even though total demand is satisfied. This behavior explains the shrinking width of the blue/teal section of Figure 3-5. This conclusion is valuable in that blood centers with varied average supplies and demands will respond to additional units of O-negative, or a universal substitute, differently. Blood centers in which average supply is close to average demand do not have a strong requirement for additional units of O-negative, but many substitution possibilities exist. However, blood centers with large demands have a great need for additional units and immediately implement substitution behaviors, but do not consequently free up units of blood types since they are filling what would otherwise be shortages. Another way to visualize the effects of adding O-negative blood is to examine the shortages with respect to the total supply across all demand levels. Since demand is distinguished by substitutable and non-substitutable demand, the total shortages here represent only shortages for demands in which substitution is allowed. As the quantity of O- negative blood increases, the linear program finds solutions that allocate other blood types optimally given the growing supply of O-negative. The highest purple peak is when O- negative urgent demand is not filled, the next green plateau is when O-negative elective

50 Chapter 3: Single Period Blood Substitution Analysis demand is not met, and the next small yellow plateau and drop off occurs as a result of O- negative demand filling other blood type demands. The point at which this graph turns red and flattens at zero corresponds to the boundary between the red and green areas of Figure 3-5. The shortage graph does not illuminate the effects of adding O-negative blood beyond the point at which total supply equals total demand because at that point, all shortages of substitutable demand go to zero. However, the dramatic initial decrease in total shortages as the supply of O-negative increases can be undoubtedly identified in Figure 3-6. Figure 3-6: Total Shortages as a Function of Supply and Demand 3.8 Implications Analyzing optimal blood substitution, even in a situation in which only half of demand can be met through substitution, is an important aspect to optimal management of blood at a community blood center. Although the analysis in this chapter only focuses on a single time period, it is apparent that there is a value to O-negative substitution beyond the point at which demand is filled. With a mean supply of 630 units and an average demand of 770 units, the optimal inventory level for O-negative blood is approximately units. However in other blood centers, the magnitude of the benefit of additional units of O- negative blood depends upon the relationship between average supply and demand of that particular blood center. This analysis of O-negative substitution extends to implications regarding the creation and use of synthetic blood substitutes. This examination clearly shows that such a substitute would greatly improve the inventory shortages. Although blood substitutes cannot be used in

51 Chapter 3: Single Period Blood Substitution Analysis all scenarios [34], they have a significant effect even when they can only be used 50% of the time, as in this chapter. The ability of the substitute to free up units of blood to fill urgent, non-substitutable demands greatly moderates shortages and also can build up stored inventory without a corresponding increase in donations. The availability and long shelf-life of a blood substitute would thereby greatly reduce shortages and would relieve the mismatch of blood types in the supply and demand distributions. 3.9 Review The analysis presented thus far illustrates the results of increasing the supply of O- negative blood. In creating a network that considers blood type, substitutability, and demand urgency, the linear program is able to find optimal solutions that reflect how a community blood center should fill demands. The results illustrate that the model will perform bloodtype substitution to put aside inventory for future use. Although the behavior differs when average supply and demand vary, there is always value to having more O-negative blood. This has important implications for the development of a universal blood substitute. The biggest limitation of this chapter is in the assumption that the reward for holding all blood types is zero. In reality, some blood types are more valuable than others in terms of how many different kinds of demand they can fill, how much demand there is for them, as well as how much is generally donated. Additionally, the optimal decisions are not made based on the age of blood- another considerable weakness. Therefore, Chapter 4 will develop a multi-period dynamic programming model that updates the value of holding blood in the future based on both age and blood type. This dynamic programming model should help extend the conclusions made thus far regarding optimal blood bank behaviors. Unlike many other substitutable resources, blood is a resource that has unpredictable supply and demand and when demand is not met, lives may be lost. Thus, maintaining adequate inventory to fill demand is of critical importance. The inclusion of value estimates for holding blood should provide insight for optimal inventory levels and shipping behaviors.

52 If you began donating blood at age 17 and donated every 56 days until you reached 76, you would have donated 48 gallons of blood, potentially helping save over 1,000 lives [18]. Chapter 4 Multi-Period Model Framework The analysis in Chapter 3 identifies optimal behaviors in a single period, but does not take into account a long-term horizon. Blood center managers must make weekly, or sometimes daily, shipping decisions but must simultaneously realize the importance of holding an adequate amount of supply for future use. Estimating the value of holding blood for the future is thus an important aspect to optimal inventory management. These estimations as well as shipment decisions must take into consideration both the age and blood type of inventory. This chapter will incorporate these considerations into a multi-period approximate dynamic programming model that develops non-linear value function approximations for holding each type and age of blood. This value function information should affect the way in which decisions are made to fill demand, to hold blood, as well as to send blood to another blood center. This larger model should provide a more realistic analysis of long-term decisions facing agents at actual blood centers. The framework from Chapter 3 provides a concise and elegant notation that can be implemented in this multi-period model, but additional notation must also be introduced. Constructing value function approximations requires adequate descriptions of their general structure as well as explanations of the value function updating technique. Furthermore, the optimal decisions must be redefined to include these value function approximations. Thus, this chapter adapts the previous notation and also establishes all additional variables and equations needed to accurately model multi-period blood banking activities. The results and analysis of this model are then discussed in Chapters 5 and 6.

53 Chapter 4: Multi-Period Model Framework Variable Definitions All resource and decision variables in this chapter are essentially identical to those in Chapter 3. The following section will describe the required modifications for the resource variables, decision variables, and the exogenous information Resource Variable The attribute notation from Chapter 3 is identical here since the age is already included in the resource and decision attributes. The transition from a single period to a multi-period model requires time indexing for all variables. Thus a time subscript must be added to all variables to denote the information content at a particular point in time. This model uses the pre-decision state variable, defined as the resources and information known just before a decision is made. The resource state is defined as follows: R t = the number of units of blood available when making The information arrives as a continuous process, but decisions are made at distinct points in time. So, x t is based upon information from time period t but not information from time period t +1. In the blood banking context, information refers to the exogenous supply of blood donated at the blood center. Figure 4-1 helps illustrate the timing of the information flow and the decisions. x t Figure 4-1: Timeline of Information and Decisions Processes Decision Variable With this framework in place, all resources and decisions over time can be modeled accurately. The following describes all possible decisions:

54 Chapter 4: Multi-Period Model Framework X t " ( ) = a function that gives the optimal decision vector x R t based on information known in state R t " ( X t ) " #$ = all possible functions The optimal decision vector is found through the use of a linear programming solver in Matlab (as in Chapter 3) by solving the optimality equation that will be discussed in Section 4-6. A difference from the previous framework is that the blood center now has the option to send blood to another CBC, allowing this model to better reflect actual blood center operation. This decision is incorporated into the class of decisions in which d " D M. Thus, a redefinition of D M is necessary, but no change to other notation is needed: D = D D "D M D D = satisfy demand = the set of all decision classes where { } D M = { hold, send to another CBC} Policies for filling demand must be discussed with the inclusion of different ages in the decision attribute vector. The research discussed in Section describes Elston and Pickerel s studies regarding FIFO verses LIFO policies. In terms of minimizing shortages and outdates, the optimal policy was found to be FIFO. This is the current practice at blood centers and hospital blood banks around the country today as well. Thus, the model will implement a FIFO policy so that the oldest blood is used first. Enforcing such a policy is done by manipulating the contribution parameters for shipping blood by age. The framework for resource and decision variables is now complete for this multi-period dynamic programming analysis Exogenous Information The supply of blood always enters the system with an age of one week. This assumption is reasonable as donated blood must be tested, typed, and packaged. This process usually takes 24 to 48 hours, but since the time periods here are weekly, the blood therefore enters the system with an age of one week. Although the demand random variable includes an age attribute, it is important to realize that in practice, demand is not specified by a given age. This is a valid assessment because blood with an age anywhere between one and six weeks is safe to transfuse. The policy to throw out blood after 42 days reflects the time

55 Chapter 4: Multi-Period Model Framework span that the ph can be maintained to sustain adequate ATP concentrations in the red blood cells. Therefore, there is no significant deterioration process that occurs while the blood is on the shelf. The conclusion in Section regarding no decrease in oxygen-carrying capacity of red blood cells over the storage lifetime supports the assumption that one week old blood is deemed the same as six week old blood from a demand perspective. Thus, understanding that demand does not require a certain age is an important insight to realize. However, the age in the attribute vector notation is included to provide consistency with other attribute vectors since all of them must be identical for proper constraint and transition function notation. 4.2 Transition Function To update the resource state variable in each time period, the resource state from the previous period, any decisions made, and any exogenous information must be known. This can be expressed in functional notation: R t +1 = f M ( R t, x t, R ˆ t +1 ) (4.1) The order of the parameters in parenthesis reflects the order of occurrence. The equation above represents the pre-decision state variable used in this model. The aging process of blood must be captured by the transition function as well so that discarding decisions are made appropriately. Therefore, a shift function is defined that increases the age of the blood by one week when applied: indicator function must be defined to identify which decisions are incorporated into the transition function: f shift function increases the age attribute of the resource variable by one week. ( ) = " a' t,a,d ( ). The application of this $ 1 If decision d acting on a resource with attribute a produces & a resource with attribute a' for d # { fill demand} or % & d # { ship to another CBC} & ' 0 Otherwise R ta Additionally, an Therefore, the indicator function equals 1 for all decisions except holding. This indicator function definition presents a clear definition of the post decision state variable defined here simply for clarity in describing the transition from one time period to the next:

56 Chapter 4: Multi-Period Model Framework x R ta' = $ $ " a' ( t,a,d)x tad (4.2) a #A d #D Equation 4.2 transitions the notation from the pre-decision state to the post-decision state, or from R ta to R x ta'. This transition does not include any exogenous information. Thus, an additional equation must be written to illustrate the transition of the resource state with the addition of new, exogenous information. This equation implements the shift function defined previously: R t +1,a' = f shift x ( R ta' ) + ˆ R t +1,a' (4.3) Therefore, the notation and equations developed in this section fully describe the transition of the resource state over time. The indicator function captures all blood sent to another CBC or shipped to hospitals. The post decision state variable then contains only the blood remaining after this process. The shift function automatically discards all six-week old blood. Thus any six-week old blood held is no longer included in the resource state at t+1. The donated blood entering the system has an age of one week, so the full age range of blood in R t +1,a' is one to six weeks. An important assumption here is that there is no backlogged demand. Therefore, if demand is not fully met in a period, the number of units of demand not filled is recorded as a shortage and is not carried over to the next period. This is somewhat reasonable because when hospitals request blood, they often need the blood for a particular surgery. If the blood center cannot fill the order, the hospital might ask a different blood center for that blood. Thus, that demand is lost to the first CBC. While this is not always the case, especially for less urban areas where there are fewer blood centers, it is often a reflection of reality. This assumption simplifies the problem since no transition function for demand is required. 4.3 The Network The same overall structure for the network as shown in Figure 3-1 is utilized in this multi-period model with respect to supply nodes, demand nodes, holding nodes, and the final flow node. The difference in this model is that forty-eight different nodes are created for each of the supply, demand, and holding blocks in Figure 3-1 to account for each blood type and age combination. This allows the linear program solver to make specific shipping and

57 Chapter 4: Multi-Period Model Framework holding decisions with respect to the age of each blood type. To accommodate the additional option to send blood to another CBC, all supply nodes also have arcs for this decision. Another modification to the network is the creation of ten parallel arcs linking each holding node to the final flow node. The purpose of these parallel arcs is to create a piecewise linear value function approximation for holding blood. Each arc corresponds to a single piece, or interval, of the value function and thus has a specified flow limit. When the flow on one arc reaches the limit, flow continues on the next arc, corresponding to the next piece of the value function. The last arc has unbounded flow (in the actual algorithm, this upper bound is set to 100,000 units, well above any possible flow in this model). The value functions will be discussed in more detail in Section 4.5, but this is the basic connection between the parallel arcs and the piecewise value function approximations. 4.4 Constraints The constraints for this model are similar for those stated in Chapter 3. However, the inclusion of the parallel arcs requires an additional variable definition. Each parallel link for a given holding node is represented by the following notation: z t,adk = the # of units of blood with attribute a acted on by decision d " { hold} flowing on the k th parallel arc at time t u adk = a paramter specifying the maximum # of units of blood with attribute a acted on by decision d " { hold} on the k th parallel arc If all intervals of the piecewise linear functions were the same length, for instance 10 units of blood, then the upper bound parameter would not need to depend upon the resource attribute. However, the large variation among the amounts of each blood type in a blood center s inventory as well as the large variation of ages existing in inventory necessitates this notation. The y tbd variable is still a part of this network but is only applicable when These two extra variables allow for explicit constraints to be written: # x tad = R ta (4.4) d "D y tbd " ˆ B tbd when d # D D (4.5) d " D D. " x tad # y tbd = 0 when d $ D D (4.6) a

58 Chapter 4: Multi-Period Model Framework K x tad "# z t,adk = 0 when d $ hold k= 0 { } (4.7) z t,adk " u adk when d # { hold} (4.8) x tad " 0 (4.9) Equation 4.7 specifies that the flow into a holding node equals the total flow over all parallel arcs ranging from k =1 to k = K. In this model, ten parallel arcs are used, therefore K=10. Equation 4.8 provides the upper bound on the parallel arcs corresponding to the limits of each interval of the piecewise linear value functions. Equations 4.4, 4.5, 4.6, and 4.9 are the same constraints from Chapter 3, but include the time indexing. 4.5 Value Function This model seeks to estimate the value of holding blood for future use, assuming a concave value function approximation with respect to the total number of units of blood being held. The functions are piecewise linear, thereby providing separable properties that are helpful when solving the optimality equation. The concavity assumption is reasonable since the total value should increase as more blood is being held, up until a point at which additional units cause storage problems for the blood center. In other words, additional units of blood before reaching a maximum capacity are accompanied with decreasing marginal value since the last unit of blood held is probably not as useful as the first unit. Beyond the maximum capacity, additional units of blood have a negative marginal value. Therefore, the concavity assumption is a valid representation of reality. Since the value of holding blood varies by age and type, individual value functions are created for all forty-eight holdings. Instead of estimating the actual value function itself, this model produces estimates of the slopes of each value function along every piecewise interval. This process provides the same general information as the actual function but has some advantages. The use of slopes allows for a range of sample realizations to update a single slope parameter [30]. All sample realizations that visit states within a particular range provide an update to the slope over that specific interval. Therefore, each sample realization provides information that updates the slope of one piece on each of the forty-eight value functions. The proper notation for the value of being in a particular state can now be written as:

59 Chapter 4: Multi-Period Model Framework V ak ( ) = the slope of the value function on the k th interval for blood with attribute a R a Since this model is a steady-state, infinite horizon model, the time periods discussed thus far correspond directly with iterations. The time indexing is useful when addressing the resource dynamics of the system, but when defining value function notation, iteration indices are shown instead. The variable definitions thus far containing time indices are accurate, but the time index will be dropped. All indexing of variables and functions from this point forward will therefore use iterations. This model uses a discrete representation of the value function in that there is no need to loop over all states. The dynamic program implements a forward pass and is thus termed an approximate dynamic program. When a specific state is visited, the slope of the value of being in that particular state is updated. Since all states are not visited in each iteration, the value functions must first be initialized properly to reflect concavity properties and then steps must be taken to ensure that concavity is maintained. All value function slopes are initialized such that they preserve the concavity requirements. The following is an illustration of an initialized value function according to the framework of this model: Figure 4-2: Form of Value Function Approximations However, the information provided by a sample realization causes a change in the slope of one of the intervals of the function that can possibly lead to a violation of concavity. Section will address this issue.

60 Chapter 4: Multi-Period Model Framework This model applies a pure exploitation strategy in that decisions are made such that the maximum possible reward is obtained with the given information. Therefore, no attempts are made to explore all possible decisions [30]. Although a drawback to a pure exploitation strategy is that the solution can converge to a local solution, this method should provide sufficient insight into optimal blood inventory management behavior. The updating and smoothing of the value function approximations through iterations is discussed in the next three sections Exponential Smoothing of Value Functions The linear programming tool in Matlab supplies dual variables for all nodes. These duals provide an estimate of the slope of the value function with respect to the resource state, another reason that estimating slopes instead of the actual value function is advantageous. These duals are stochastic gradients of dual variables: V ak ( ). The following notation is introduced for the R a v ˆ n a = the dual variable for a specific age/type given by attribute a based on the sample realization " n v ˆ n = ˆ n ( v a ) a #A represents all dual variables With this notation in place, the exponential smoothing for the value function approximations can be introduced: V n ak ( R a ) = V n"1 ak R a ( )( 1"# n ) + # n ˆ v a n (4.10) where k is determined by R a. The " n refers to a stepsize that depends on the iteration. Section will discuss the choice of stepsize in more detail CAVE Algorithm Since the exponential smoothing expressed by Equation 4.10 might violate the concavity requirement, a method must be executed to preserve the concavity of the value function. This model updates the slope, thus slopes must decrease as the total number of resources increases. A provably convergent algorithm known as the CAVE (Concave Adaptive Value Estimation) algorithm provides a simple and efficient method to preserve concavity. This algorithm looks to the left and right of the piece updated by the smoothing.

61 Chapter 4: Multi-Period Model Framework If concavity is violated, the dual variable is smoothed into the surrounding pieces of the function as well [19]. The algorithm can be described as follows: if V n n a,k < V a,k +1 then the following smoothing is performed : n n n V a,k +1 = ( 1"# n )V a,k +1 ( R a ) + # n v ˆ a n if V a,k"1 n V a,k"1 > V n a,k then the following smoothing is performed : n n = ( 1"# n )V a,k"1 ( R a ) + # n v ˆ a Equation 4.11 is addressed for k to K and Equation 4.12 is addressed for k to Stepsizes (4.11) (4.12) Stepsizes are of critical importance to accurately monitor convergence as well as for appropriate value function smoothing. For a stochastic gradient algorithm such as this model, stepsizes must adhere to the following properties for the algorithm to be provably convergent [30]: # $ " n = # (4.13) n=1 # $ (" n ) 2 < # (4.14) n=1 " n # 0 (4.15) This model will implement a deterministic stepsize with the following format: " n = a a + n #1 (4.16) This stepsize rule satisfies Equations 4.13, 4.14, and 4.15 required for convergence. It has similar behavior to a 1/n stepsize, but as a increases, the rate at which the stepsize declines with respect to iterations slows. The reason for including n-1 instead of n in the denominator is that the iterations begin at 1, thus in order to have the first stepsize equal one, this n-1 in the denominator is necessary. Constant stepsizes are not used because an extremely large number of iterations would have to be run to see convergence. The stepsize given by Equation 4.16 with a=8 is an ideal stepsize to monitor convergence and to implement for smoothing. This stepsize will be used not only in Equation 4.10 but also later in Equation 4.19 to assess convergence of the

62 Chapter 4: Multi-Period Model Framework contribution function. The choice of a is determined through analysis of the following graph: Figure 4-3: Stepsizes for a/(a+n-1) while varying a The yellow line in this figure illustrates the stepsizes when a=8, which is the choice for this model. Since this model implements a forward pass, it is desirable to prevent the stepsize from converging to zero too quickly because a learning phase is present in the beginning iterations. Therefore, the stepsize with a=8 is ideal since it approaches zero much more slowly than when a=1, but shrinks fast enough to achieve convergence within a few hundred iterations. 4.6 Contribution Function and Optimality Equation The contribution function requires no notational change from Chapter 3. The contribution parameters are set to similar values as in Chapter 3, but values in the multiperiod model must be specified by age when filling demand. The contribution parameters previously defined in Table 3-2 for shipping blood to hospitals are set as the rewards for filling demand with six-week old blood in this model. For each blood type, the contribution values for shipping blood decrease by 0.1 as the age decreases by one week (for example in the case of no substitution, filling a demand with three week old blood would correspond to a contribution parameter of 9.7). This initialization procedure guarantees that demands are filled with the oldest blood before using newest blood, thus imposing the FIFO policy.

63 Chapter 4: Multi-Period Model Framework The sending contribution parameters are of negligible importance in this model since multiple locations are not considered. However, the framework is established with this decision possibility to more accurately model actual behavior. Thus, these parameters are set such that oldest blood is sent before newest blood. The contribution parameters for filling urgent and elective demand remain the same in this model, 40 and 20 respectively. The penalty for discarding blood, modeled as holding blood that is six weeks old or sending sixweek old is set to -60. The multi-period model attempts to maximize the single period contribution plus the value of the future resource state. Since this is a steady state problem, the model recursively solves the following optimality equation: ( ) = E max V R { C (#,R, x) + $V( R M (#,R,x)) R} (4.17) x "X There is an expectation since the future value is uncertain. However, substituting the approximate value functions V ak ( ), discussed in Section 4.5, for R a x n = argmax C # n, R n, ( x d ) d "D + $V n%1 D f M # n, R n, x d x "X V ( R) eliminates the need for this expectation. Thus, the optimal decisions are found by solving the following equation: ( ( ( ) d " { hold )) (4.18) } The discount rate, ", is incorporated because the approximate value function estimates a future value. Thus, this model includes a discount factor of Assessing Convergence The optimality equation is solved in each iteration to determine the optimal solution given the set of contribution parameters and value function slopes. The decision vector that is returned from the LP solver is the optimal set of decisions based on the one period contribution plus the discounted future value of any blood held to the next period. The purpose of including the value functions (or more specifically the slopes) is to improve the quality of the solution. To measure convergence, however, the contribution function is examined. Therefore, plotting the contribution function over all iterations provides a reasonable reflection of convergence. With the large amount of noise in the random variable simulation, the contribution function does not converge quickly. Therefore, smoothing is performed after every iteration

64 Chapter 4: Multi-Period Model Framework to more easily assess convergence. An additional variable must be defined to indicate the smoothed contribution function: C n = ( 1"# n )C n"1 + # n C n ( $ n,r n,x n ) (4.19) The term C n refers to the smoothed contribution function after the one period contribution value from iteration n is smoothed into C n"1. Observing these smoothed values is a more efficient method of judging convergence since it evens out noise and thus requires fewer iterations to adequately judge the point at which convergence is reached. The stepsize implemented in this smoothing is the same stepsize as described in Section Since there is a learning phase as the value function slopes are being updated, it is important to maintain a large stepsize for the beginning iterations. Thus, the same stepsize choice is made for the contribution function smoothing as was made for the exponential smoothing on the value function approximations. In addition to keeping track of the smoothed contribution functions, the variance can also be calculated as follows: " n = ( 1#$ n )" n#1 + $ n C % n,r n, x n ( ( ) # C n ) 2 (4.20) The smoothed contribution function estimates are assumed to be normally distributed. Since this estimate is a sum of many function values, the law of large numbers allows for the normality assumption to be made. The stepsize given in Equation 4.16 with a=8 is used for the variance smoothing as well. This variance is initialized at zero and is updated throughout all iterations. The reason for updating the variance estimate is that it allows for a 95% confidence interval to be calculated by the following equations: * $ 95% CI :, C n "1.96 & +, % # n n ' $ ),C n & ( % # n n '- / ) (./ (4.21) This confidence interval gives an indication of the accuracy of the smoothed contribution functions as well as an indication of convergence as the interval shrinks.

65 Chapter 4: Multi-Period Model Framework Data This model incorporates noise into the supply and the demand exogenous information, unlike the model from Chapter 3. Donations at a CBC are highly variable and fluctuate throughout the year. The noise around the mean supply reflects this uncertainty of weekly donations. The demand is the aggregated standing orders from all hospitals in a blood center s region. The standing orders are set prior to the actual week in which blood is preferred to be received. However, during the week, hospitals also place last minute orders to reflect an unexpected, increase in demand. They can also lower their orders if demand at their hospital is lower than expected. These last minute adjustments are reflected in this model with noise around the average demand of 770 units. Accurate data was available from Dr. Eric Senaldi at The Blood Center of New Jersey regarding supply of blood. The following histogram and cumulative percentage table are created from 51 weeks of donation data. Figure 4-4: Histogram of Donation Data at the Blood Center of NJ Table 4-1: Cumulative Distribution Data for Donations Supply (units) Cumulative% % 1.92% % % % % % % % % % % % Based on this data, an appropriate supply distribution can be created. Approximately 25% of the time, the supply is lower than 550 and 75% of the time it is greater than 700. Using the maximum and minimum values in addition to these percentiles, three different supply scenarios are simulated, given in Table 4-2. Although information regarding the variance of the demand at the Blood Center of New Jersey was not available, 66 weeks of distribution data from the New York Blood Center

66 Chapter 4: Multi-Period Model Framework was also provided by Dr. Senaldi. This data indicates that 50% of the time, the demand ranges 5% above or below the mean. The lowest demands fall to 20% below the mean while the highest demands rise to approximately 20% of the mean of the distribution. Therefore, three scenarios can also be created for demand data using these percentages along with the mean demand of 770 units. While the assumption that the variation in demand at the New York Blood Center is the same as for The Blood Center of New Jersey might not be completely accurate, it is the best approximation to make with the lack of data. following summarizes the parameters for normal supply and demand distributions at the Blood Center of New Jersey: Table 4-2: Simulation Scenarios for Donation and Demand Frequency of Occurrence Minimum Demand (# units) Maximum Demand (# units) Minimum Supply (# units) Maximum Supply (# units) 25% % % The Given these three scenarios for both supply and demand, uniform random variables are simulated between the maximum and minimum values. Each scenario occurs with the appropriate frequencies given in Table 4-2. Since the scenarios are divided by frequency according to provided data, a uniform random variable simulation within each interval is reasonable. An important note is that high supply and high demand do not have to occur in the same scenario. For each of the supply scenarios, all three of the demand scenarios are possible, each one occurring with its respective frequency of occurrence percentage. Thus, this simulation framework provides an accurate method for creating the exogenous supply and demand for the model. Additionally, the initial age distribution must be set. The National Blood Data Resource Center s Comprehensive Report on Blood Collection and Transfusion in the United States in 2001 provides an average age distribution for blood supply at hospitals [11]. This data is translated to fit the six-week age categories in this analysis. The supply distribution data by type and age used in this model is shown below: Table 4-3: Age Distribution for Initial Blood Inventory Age (# of weeks) Percent of Total Supply 9.5% 39.9% 32.6% 13.6% 3.2% 1.1%

67 Chapter 4: Multi-Period Model Framework The data is based on hospital blood banks and not blood centers, but it is the best available data that could be found to represent an initial age distribution. After the first period, the decisions alter the age distribution and thus this initial data has little effect on the overall model. It is simply necessary to initialize the supply of blood in the first iteration. This data will also give insight later to the results of the model. Since a blood center holds blood before it is shipped to a hospital blood bank, it is expected that the average age of blood in a blood center is slightly younger than blood at a hospital. Therefore, when examining results of the model, this information is useful for a comparison to actual inventories 4.9 Review This chapter has described all necessary variables and equations needed to explain the formulation of a multi-period model. The goal of this model is to investigate the effects of using value function approximations when making decisions at a blood center with the hope that an improvement will be made. Improvement will be measured by examining the behavior of this model compared to results when no value function approximations are included. Although traditional blood banking models examine shortages and outdates as performance metrics, this model will extend the metrics to include a variety of other statistics such as average age of blood by type. The specific performance measures are discussed in more detail in the next chapter.

68 Every time you give blood, you are saving up to three lives [13] Chapter 5 Multi-Period Model Results Successful blood centers maintain minimal levels of total shortages and outdates. Therefore, the performance of the dynamic programming model will be assessed based on the total shortages and outdates incurred. The optimal behavior patterns of the model must also be examined to identify how the model produces the given results. Comparisons to a model in which no value functions are included highlight the utility of the value function approximations. This model with no value functions is structured such that all demand is filled whenever enough inventory is available, even if inventory becomes completel depleted. Since no data regarding actual daily or weekly shortages at blood centers is available, these comparisions are the best availble measures for the success of the dynamic programing model. Before testing the dynamic programming model with random sample realizations of donations and demands, the value functions must be adequately trained based on the given supply and demand distributions. After convergence is established, average behavior patterns, total shortages, and total outdates over a large number of iterations can identify the large-scale differences between the optimal strategy of the dynamic programming model and the strategy of a model with no value functions. To better understand the specific behaviors that bring about differences in the aggregate results, it is helpful to scrutinize specific behaviors over a few iterations. Thus, Section 5.3 will study a three iteration example that reveals particular differences in holding and substitution behaviors critical to the success (as defined by total shortages) of the dynamic programming model. This model should reduce total shortages and outdates through a change in optimal decision-making policies in comparison to a model with no value function approximations. In terms of realistic inventory management, if a blood center has accurate estimates for the

69 Chapter 5: Multi-Period Model Results value of holding each blood type by age for future use, that center should be able to make more intelligent shipping and holding decisions by striking a proper balance between how much blood to hold and how much blood to ship each week. Therefore, this chapter implements the framework established in Chapter 4 to assess the advantages of incorporating value function information into a decision-making model. 5.1 Convergence The base model refers to value functions trained with the supply and demand distributions for The Blood Center of New Jersey. Before the model can be tested, it is essential to train it sufficiently such that the value functions are adequately updated based upon specific supply and demand distributions. When the value function smoothing no longer produces large modifications, the training phase is complete. This model uses the convergence of the contribution function as an indicator of the convergence of the value functions. Figure 5-1 shows the smoothed contribution functions, as calculated by Equation 4.19, over 500 iterations including both a lower and upper bound for a 95% confidence interval (specified by Equation 4.21). Figure 5-1: Smoothed Contribution Function with a 95% Confidence Interval over 500 Iterations.

70 Chapter 5: Multi-Period Model Results Even after 500 iterations, the contribution functions still have some fluctuations, but after approximately 300 iterations, reasonable convergence is reached. The fluctuations are primarily a result of the high uncertainty in supply and demand. Hence, the significant noise generated in both the supply and demand data is responsible for the slight variations in the smoothed contribution function values. Since the amount of variation in iterations 300 to 500 is so much smaller in comparison with the variation in the first 300 iterations, convergence can be assumed in this model to be reached at 500 iterations. Thus, the value function approximations in this dynamic programming model are trained over 500 iterations. The smoothed contribution function values from the dynamic programming model can be compared with contribution function values obtained from a model with no value functions. Using the same sample realization, both models are simulated over 500 iterations. Figure 5-2 reveals the difference in smoothed contribution function values: Figure 5-2: Comparison of Smoothed Contribution Functions with and without use of Value Function Approximations Both models have dramatic fluctuation in the first 100 iterations, but the contribution function converges to a higher value for the dynamic programming model than for the model without value functions. In some problem classes, the contribution function reflects real costs, such as purchasing costs over a period of time. However, the costs in this model are simply set to reflect realistic behaviors. A higher contribution function in this case implies

71 Chapter 5: Multi-Period Model Results that more demand is generally filled and since the dynamic programming model converges to a higher value, it can be initially concluded that this model is able to fill more demand than a model without value function approximations. However, to accurately gauge success of the model, it is necessary to examine tangible results such as shortages, outdates, and substitution behaviors of interest to blood center managers. 5.2 Optimal Policy Analysis Once trained, the model is tested over 200 iterations, representing a period of approximately 4 years since each iteration corresponds to one week. With a same sample realization of exogenous information, two sets of decisions are simultaneously recorded, one for the dynamic programming model and the other for a model with no value function approximations. The following sections investigate the differences in behavior between these two models to thereby assess the advantages of the dynamic programming approach Total Shortages and Outdates In practice, blood center managers are most concerned with performance in terms of shortages and outdates. Therefore, the best measure for success of a model is in the total shortages and outdates expected to occur. Since the model is trained and tested using data from The Blood Center of New Jersey, a net importing center, it is not surprising that the model faces chronic shortages. Regular imports are excluded from this model since the purpose is to investigate the optimal inventory management strategy given only the donations at the center. Conclusions from this analysis can then identify optimal import policies. The model can be trained with supply and demand distributions for a net exporting center as well, in which case outdates would be the primary focus. However, the conclusions in this thesis relate to optimal behaviors for a net importing blood center. The expected total outdates of the model are not shocking given the net importing classification: no outdates are encountered in any of the iterations. This is also true for a model with no value functions. Even though some iterations may have higher supply than demand, the supply and demand distributions are significantly different such that there is never enough inventory build-up to incur outdates. The more interesting statistic to consider is therefore the total shortages. The results of this model are somewhat surprising in that

72 Chapter 5: Multi-Period Model Results total shortages on average are not lower than shortages for a model with no value functions. Table 5-1 illustrates that the inclusion of value function approximations has no effect on average total shortages: Table 5-1: Mean and Standard Deviation of Total Shortage Data for 200 Iterations Model Type Mean Total Shortages Standard Deviation Using Value Function Approximations 96.5 units 13.5 Not Using Value Function Approximations 96.5 units However, it is in the reduction of the variance of total shortages over the 200 iterations that the results can be considered optimal and significant. The standard deviations shown in Table 5-1 highlight this result numerically and Figure 5-3 displays all total shortages for both models. The blue dots represent total shortages for the dynamic programming model. Figure 5-3: Total Shortages for Dynamic Programming Model and a Model without Value Functions The initial hypothesis is that any optimal model will minimize total shortages. However, Figure 5-3 clearly shows that the inclusion of value functions does not produce this result. There is not a single iteration that produces total shortages of zero with the dynamic programming model. However, a critical concern at a blood center is controlling the unpredictability of supply and demand. Blood center managers want to reduce fluctuations in inventory so that they can more consistently fill hospital orders. Therefore this reduction in the variance of total shortages is extremely desirable because a blood center operating

73 Chapter 5: Multi-Period Model Results according to the behavior dictated by the dynamic programming model never has to fear extremely large shortage situations. The following histograms explicitly illustrate the distribution of total shortages for the dynamic programming model and a model with no value functions. Although the model with no value functions does have a much larger proportion of total shortages ranging from zero to fifty, it also has a much larger proportion of shortages above 150 units as can be seen in Figure 5-5. The dynamic programming model, on the other hand, has a highly concentrated range of expected shortages and prevents frequent shortages above 150 units as shown in Figure 5-4. Figure 5-4: Histogram of Total Shortages for the Dynamic Programming Model Figure 5-5: Histogram of Total Shortages without use of Value Functions

74 Chapter 5: Multi-Period Model Results The dynamic programming model thus identifies the unavoidable structural shortage at the blood center. Given the information in Table 5-1, the blood center should import 100 units of blood each week and can then expect total shortages to approach zero when applying the optimal decisions determined by the dynamic programming model. With the average supply of 630 units, the imports would increase the total supply to 730 units. The imports as a percentage of total supply would then be 13.7%. Current practices at The Blood Center of New Jersey require that 23% of total inventory be supplied through imported blood. The implementation of this importing policy of 100 units per week in conjunction with the behaviors supported by the dynamic programming model would thus greatly reduce the current import need and simultaneously eliminate extremely large shortage situations Shipping and Holding Behavior To achieve such consistent total shortage results, the dynamic programming model implements decisions that greatly differ from optimal behaviors determined by a model with no value functions. Each week, blood centers must make decisions about how much blood to keep in inventory, how much blood to ship to hospital blood banks, and how much blood to possibly send to another community blood center. Figures 5-6 and 5-7 illustrate the average allocation of total supply over all 200 iterations for both the dynamic programming model and the model without value function approximations. Using Value Functions Not Using Value Functions Fill Demand 76% Hold 61% Fill Demand 39% Send to CBC 0% Send to CBC 1% Hold 23% Figure 5-6: Average Holding and Shipping Behavior with the Dynamic Programming Model. Figure 5-7: Average Holding and Shipping Behavior without Value Function Approximations

75 Chapter 5: Multi-Period Model Results The pie represents total inventory and thus each piece represents on average the percentage of inventory allocated to each specific behavior. The difference in behavior patterns is dramatic, barring the similarity that neither sends a significant amount of blood to another CBC. Since The Blood Center of New Jersey is in an importing region, it is expected that no blood should be shipped to another CBC. The dynamic programming model on average allocates only 39% of a blood center s inventory each week to ship to hospital blood banks and keeps the remaining 61% in inventory. This 61% is much greater than the 23% of supply held according to a model with no value functions. The dynamic program thus uses a much smaller portion of inventory to fill demand. However, both simulations face an identical supply and demand sample realization so the difference in behavior patterns is in fact significant since it is not a reflection of different supply and demand outcomes. The dynamic programming model holds a larger portion of inventory each week as shown by Figure 5-6 and is thus not able to fill all demand regularly. However, this strategy also prevents any considerable shortages from occurring. Therefore, if The Blood Center of New Jersey implemented the results of the model, their general shipment should be approximately 39 % of inventory and total shortages should fluctuate around 100 units. This percentage will vary from week to week in response to fluctuating demand, but the aggregate results suggest that reducing variance in total shortage outcomes is obtained through increased holding behaviors. However, the optimal results are not simply a function of just holding and shipping frequency. Other statistics must also be investigated to better understand how the dynamic programming model makes decisions regarding blood type substitution and the age distribution of blood maintained in inventory Age Distribution Behavior Monitoring the distribution of ages of blood in inventory is critical to properly discard outdated blood. However, it is also important because a blood center wants to avoid outdates as much as possible since donations are such a valuable resource and supply is unpredictable. Hence, blood centers try to prevent outdates using the FIFO method, or if the blood center is a net exporter, by sending appropriate amounts of blood out to other blood centers. The dynamic programming model encourages holding behavior so it is expected

76 Chapter 5: Multi-Period Model Results that the average age of blood will be older. Yet while encouraging holding, the model is simultaneously discouraging outdates. Therefore, the dynamic programming model finds a proper balance in the age distribution required to optimally manage inventory. Figure 5-8: Average Age by Blood Type of Inventory with the Dynamic Programming Model and a Model with no Value Functions Figure 5-9: Average Age Distribution of Inventory with and without use of Value Functions Figure 5-8 illustrates the average age of each blood type held in inventory. The dynamic programming model contains blood with a higher average age for all blood types when compared to a model with no value functions. The average ages do vary by blood type, specifically inventories of B-positive and AB-positive should have the highest average

77 Chapter 5: Multi-Period Model Results age while inventories of AB-negative should have the lowest average age. The significant difference in the age distribution with and without the value function model highlights the different inventory compositions obtained with the inclusion of value function approximations. The distinction of average age by blood type is seen in both cases, with and without the model, thus the model does not greatly affect relative age by type. The important result here is that the average age of blood in inventory is older when value functions are incorporated, but no outdates are consequently encountered. Figure 5-9 concurs with Figure 5-8 in that the inclusion of value functions leads to an older inventory. The extremely large percentage of inventory that is one week old in the case without value functions increases the sensitivity of the blood center to fluctuations in donations. The dynamic programming model greatly reduces this sensitivity by holding blood for more than three weeks and equivalently diversifying the ages of blood in inventory. The presence of six-week old blood in the inventory should be carefully monitored since a drop in demand could lead to outdates. However, such a small proportion of inventory is five and six week-old blood that no outdates are ever incurred. The age distribution of hospital inventory given in Table 4-3 generally matches the results of the age distribution shown in Figure 5-9 if a one-week shift is applied to ages in the blood center. In other words, the percentage of one-week old blood at the blood center according to the dynamic programming model (see Figure 5-9) is similar to the average percentage of two-week old blood at hospital blood banks. Assuming a shipping period that ages the blood by one week, this validates the results here in terms of representing realistic blood inventories. This similarity in age distributions is observed for all ages, thus the model is able to reproduce realistic inventory attributes Substitution Behavior Another interesting behavior contributing to the total shortage results of the dynamic programming model is the use of blood-type substitution at the blood center when filling demand. Since this model assumes that 50% of demand cannot be filled by substitution, and since the mean supply is lower than the mean demand, not very much blood is actually available for substitution. However, the addition of value function approximations encourages additional substitution behavior above the amount of substitution performed in a

78 Chapter 5: Multi-Period Model Results model without value functions. These substitutions maximize the value of remaining inventory at the blood center. They also help to optimally build up inventory for filling demand during shortage periods. This particular behavior will be discussed in greater detail in Section 5.3, but the broad trends are brought out in this analysis. Figure 5-10 shows the percent of total inventory assigned to fill demand of a different blood type. The O-negative Substitution refers to O-negative units of blood assigned to fill demand of another blood-type other than O-negative. The Other Blood-Type Substitution refers to blood of any type (excluding O-negative) assigned to fill a demand of a different blood type, for instance O-positive blood filling an A-positive demand. Figure 5-10: Percent of Total Supply Allocated for Blood-Type Substitution with and without Value Functions. The dynamic programming model clearly performs blood-type substitutions more frequently than a model without value functions. While the actual percentages are quite low, all below 3.5% of total supply, the structural shortage situation prevents a large amount of substitution behavior. For instance, although O-negative is the universal donor, the only blood type that can fill O-negative demand is O-negative itself. Thus, only a small portion of O-negative blood is actually available for substitution. Similarly, O-positive can be used to fill three other demand types, but O-positive demand can only be filled with either O-positive or O-negative blood. Therefore, substitution possibilities are indeed inherently limited by the substitution pattern given in Figure 2-2.

79 Chapter 5: Multi-Period Model Results Figure 5-11 is another representation of substitution frequency measured by the percent of iterations that implement some type of blood type substitution. This statistic is important to examine as an optimal model should use blood-type substitutions whenever possible to satisfy demand optimally. Figure 5-11: Percent of Iterations in which Blood-Type Substitution is Performed In all 200 iterations, O-negative blood type substitution is applied to fill demands in the dynamic programming model and in over 90% of the iterations, substitutions with other blood types take place. A model without value functions does not substitute as often. This is extremely significant as the use of blood type substitution is highly valued in the dynamic programming model in order to produce the stable total shortages. This aggregate statistic is thus a reflection of the importance of blood-type substitution for optimal inventory management. It is clear that a model incorporating approximate value functions recognizes that blood-type substitutions can improve the overall performance by freeing up units of certain blood types in relative shorter supply. The particular blood types valuable for substitution are identified in Section Summary of Optimal Behaviors and Policies The optimal behaviors discussed in this section emphasize the importance of holding blood for the next period. Implementing optimal blood-type substitution when filling demand improves the overall value of blood remaining in inventory. This holding behavior

80 Chapter 5: Multi-Period Model Results leads to an older average age of blood, but can still be performed without incurring any outdates. These behaviors result in significantly more predictable expected shortages- an extremely desirable result for blood center managers. Additionally, extremely high shortage situations can be completely avoided using the optimal behavior from the dynamic programming model. This result also translates to an improvement over current practices at The Blood Center of New Jersey. The policies supported by the model reduce the required imports from 23% of total supply down to 14%. This is a significant drop and has an impact on the demand placed on other blood centers as well. This reduction in inter-regional demand on the aggregate network of blood centers is highly attractive considering the high growth rate of demand for blood in the United States. 5.3 Specific Example of Model Behavior These aggregate results highlight general behavior patterns of the dynamic programming model, but to truly demonstrate and explain the significance of the value function approximations, a specific three-period example is discussed. This section examines the divergent behaviors between the dynamic programming model and a model with no value function approximations. After only three iterations, dramatically disparate inventories emerge. This is incredibly significant because every iteration reflects a single week. Thus, in the short period of three weeks, a blood center not using proper estimates for the value of holding blood can potentially face huge shortage situations whereas with different behavior patterns, these large shortages can be completely avoided. This section seeks to highlight the particular decisions that lead to such different results General Scenario The particular scenario chosen for discussion illustrates the reaction of the value function model to a drop in donations. Using three iterations of random variable simulation based on the given supply and demand distributions in Table 4-2, the following donation and demand behavior is observed:

81 Chapter 5: Multi-Period Model Results Figure 5-12: Total Donation and Demand over the Three Iterations A dramatic drop in donations is observed. This example is particularly significant as donations are extremely low in both the second and third iterations causing an extreme thinning of inventory. Although donations as small as 400 units occur extremely infrequently, this is a scenario generated with the given supply and demand distributions and is thus a possible sequence of donation and demand events. The dynamic programming model handles this scenario much better than a model without value functions. Figure 5-13 illustrates total available inventory in each iteration. Both models begin with identical inventories, but the dynamic programming model maintains higher levels of inventory in both the second and third iteration. Figure 5-13: Total Supply over the Three Iterations for the Dynamic Programming Model and a Model with no Value Functions

82 Chapter 5: Multi-Period Model Results To understand the effects of the differing inventories, Figure 5-14 is included to show the total shortages for all three iterations. Figure 5-14: Total shortages over the Three Iteration for the Dynamic Programming Model and a Model with no Value Functions The model without value functions is able to fill demand in the first and second iterations with limited shortages, but an extremely large shortage (288 units) is seen in the third iteration. The dynamic programming model, however, values holding some blood instead of filling all demand. So, consistent total shortages are incurred across all three iterations. Maintaining higher inventory levels in the second iteration produces shortages, but builds adequate inventory for use in the third iteration where the difference in total shortages is dramatic. This example is thus a clear illustration of the effectiveness of the value function approximations since the huge shortage situation can be completely avoided. The next two sections investigate more specifically the behaviors that contribute to the success of the value function model in this three-iteration example Particular Optimal Behaviors The following tables present the optimal allocations of inventory over the 3 iterations. These tables are included for completeness, but certain data within the tables will be highlighted in the discussion of this section.

83 Chapter 5: Multi-Period Model Results Table 5-2: Allocation of Total Supply during the 1 st Iteration Non- Substitutable Demand Total Supply Hold Substitutable Blood Demand Type Filling own blood Filling other blood n=1 Total Total type type Without Without Without With With V With V With V Without V With V Without V V V V V AB AB A A B B O O TOTAL Table 5-3: Allocation of Total Supply during the 2 nd Iteration Non- Substitutable Demand Total Supply Hold Substitutable Blood Demand Type Filling own blood Filling other blood n=2 Total Total type type Without Without Without With With V With V With V Without V With V Without V V V V V AB AB A A B B O O TOTAL

84 Chapter 5: Multi-Period Model Results Table 5-4: Allocation of Total Supply during the 3 rd Iteration Non-Substitutable Substitutable Demand Total Supply Hold Demand Blood Filling own blood Filling other blood Type n=3 Total Total type type Without With Without Without With With V With V Without V With V Without V V V V V V AB AB A A B B O O TOTAL The dynamic programming model (referred to in Tables 5-2, 5-3, and 5-4 as With V ), clearly places more emphasis on holding blood for the next period. Instead of filling all demand, the optimal decisions according to the dynamic programming model dictate that it is preferable to save some blood for future use, particularly blood of types A-positive, O- negative and O-positive. Since 34% of the population in the US has A-positive blood, it is logical that saving this resource is valuable, even though substitution possibilities are limited to just AB-positive. If demand spikes, a reasonable assumption is that demand for A-positive blood will dramatically rise since such a large portion of the population has this blood type. For the same reason, it is logical to save O-positive blood since 38% of the population has O- positive blood. These two blood types are found in 72% of the US population, thus it is reasonable that the model has updated the value of holding these types of blood to encourage additional holding behaviors. Since O-negative is the universal donor, holding this blood type is extremely important since it can be used to fill any demand in the event of a spike in demand or a drop in donations. In the first iteration, the value function model does not fill all demand even though adequate supply exists. The second iteration shows similar behavior as shortages are incurred because blood is instead held in inventory. The following pie charts show inventory allocations over all three iterations.

85 Chapter 5: Multi-Period Model Results Filling own Blood Type % Filling Non- Substitutable Demand % Filling a Different Blood Type % Total Supply: 1098 units Filling own Blood Type % 4% Total Supply: 1098 Hold units % Hold % Filling Non- Substitutable Demand % Filling a Different Blood Type Figure 5-15: 1st Iteration Optimal Decisions with Value Function Figure 5-16: 1 st Iteration Optimal Decisions without Value Functions Filling own Blood Type % Filling a Different Blood Type % Total Supply: 883 units Hold % Filling own Blood Type % Filling a Different Blood Type 0 0% Total Supply: 731 units Hold % Filling Non- Substitutable Demand % Figure 5-17: 2 nd Iteration Optimal Decisions with Value Function Filling Non- Substitutable Demand % Figure 5-18: 2 nd Iteration Optimal Decisions without Value Functions Filling a Different Blood Type Filling own 9% Blood Type % Hold % Total Supply: 653 units Filling own Blood Type % Filling a Different Blood Type 0 0% Hold % Total Supply: 473 units Filling Non- Substitutable Demand % Figure 5-19: 3 rd Iteration Optimal Decisions with Value Function Filling Non- Substitutable Demand % Figure 5-20: 3 rd Iteration Optimal Decisions without Value Functions

86 Chapter 5: Multi-Period Model Results The first column illustrates behaviors of the dynamic programming model and the second column illustrates behaviors without value functions. The label Filling a Different Blood Type refers to blood used to fill demands of different blood types, the label Filling own Blood Type refers to blood filling demand of its own type, and the label Filling Non- Substitutable Demand refers to blood filling demands that do not allow substitution. The first iteration is the most helpful to analyze as the total inventories are identical. The value function model holds 51% of inventory whereas the non-value function model fills all demand and only holds the remaining 27% of blood. Although this incurs a shortage of just over 100 units in the first period for the dynamic programming model, units of O- negative and O-positive blood are held that become critical in iterations two and three. The first iteration also shows a higher proportion of blood allocated for substitution in the dynamic programming model (8% verses 4%). An optimal balance is thus found between filling demands and holding blood for future periods. Figures 5-17 and 5-18 represent optimal decisions for the second period and illustrate similar trends in that 27% of inventory is held in comparison to only 8% without value functions. An important note is that the size of the pie, or the total supply, is different in iterations two and three. Thus, the 23% represents an even larger absolute amount. Figures 5-19 and 5-20 for iteration three show slightly different behaviors as only 4% of inventory is held in the dynamic programming model whereas 7% is held without value functions. This is a response to the dramatic drop in donations. The dynamic programming model uses almost all inventory to fill demand and only incurs a 101 unit shortage. The model without value functions has such low inventory at the beginning of iteration three (437 units) that a significantly large shortage (288 units) is necessarily incurred. Thus the emphasis on holding blood is critical to avoid these shortages. Another important aspect of the dynamic programming model is the illustration that filling all non-substitutable demand can be detrimental. Table 5-2 clearly illustrates this phenomenon. Without value functions, all non-substitutable demand is filled in the first iteration. Therefore, there is inadequate AB-positive, A-positive, and AB-negative supply to fill their respective substitutable demands. Thus, 38 units of O-negative are used to fill A- positive demand and another 7 units of O-negative blood are used to fill the remaining ABand AB+ demand (thus reflecting the 45 units of O-negative blood shown in Table 5-2). The

87 Chapter 5: Multi-Period Model Results value function model, however, does not fill all non-substitutable demands and instead allocates a portion of this blood to fill substitutable demands and then holds the rest in inventory. Additionally, the dynamic programming model does not use all available O-negative blood to fill demands in the first iteration and instead holds some for future use. The model without value functions illustrates the result of using all O-negative to fill demand: not enough O-negative remains in iterations two and three to support the drop in donations. Thus it is clear that the dynamic programming model is able to hold just enough blood of the proper types such that when demand spikes or donations drop, the same total shortages are generally incurred. Tables 5-2,5-3, and 5-4 thus show the divergent behaviors that create changes from the identical initial inventories. The dynamic programming model holds more blood of all types and performs blood-type substitution when possible, but balances the allocation of each blood type according to how useful it is to keep in inventory Analysis of A-positive, O-positive, and O-negative Blood Behaviors Since A-positive, O-negative, and O-positive are most affected by the inclusion of value functions in terms of holding and substitution behaviors, this section investigates the particular behavior of these three blood types. Figures 5-21 and 5-22 illustrate the different behaviors in the first iteration regarding A-positive blood: Filling own Blood Type % Filling a Different Blood Type 0 0% Hold % Filling own Blood Type % Filling a Different Blood Type 0 0% Hold % Filling Non- Substitutable Demand % Filling Non- Substitutable Demand % Figure 5-21: 1 st Iteration Optimal Allocation of A-positive with Value Functions Figure 5-22: 1 st Iteration Optimal Allocation of A-positive without Value Functions

88 Chapter 5: Multi-Period Model Results Given the same initial amount of A-positive blood (307 units) and the same nonsubstitutable A-positive demand (191 units), the value function model fills only 163 units of non-substitutable demand, thereby increasing the number of units held to 97 units. The model without value functions fills all 191 units and thus holds only 70 units in inventory. Thus, this is a specific illustration of the model deciding not to fill all demand even though there is sufficient supply and instead choosing to hold units for future use. The additional units of A-positive remaining in inventory are then used to fill A-positive demands in iteration two whereas the model without value functions must use O-negative to fill these demands. Some A-positive blood is also held in the second iteration whereas none is held in the model without value functions. This blood is critical in iteration three when donations drop and not enough new A-positive blood is donated. Therefore, this regular conservation strategy of A-positive blood employed by the dynamic programming model provides a larger inventory level of A-positive in iteration three, critical in preventing a large shortage given the dramatic drop in donations. This strategy also limits the required amount of O-negative substitution that then allows additional units of O-negative to be saved. Similar behavior is seen for O-positive blood: Filling own Blood Type % Filling a Different Blood Type % Hold % Filling own Blood Type % Filling a Different Blood Type 0 0% Hold % Filling Non- Substitutable Demand % Filling Non- Substitutable Demand % Figure 5-23: 1 st Iteration Optimal Allocation of O-positive with Value Functions Figure 5-24: 1 st Iteration Optimal Allocation of O-positive without Value Functions

89 Chapter 5: Multi-Period Model Results The value function model fills only 182 units of the total 214 units of nonsubstitutable O-positive demand and fills only 78 units of the total 92 units of O-positive substitutable demand. The model with no value functions, however, fills all O-positive demand. Therefore, the dynamic programming model chooses to experience some shortages in return for holding 169 units of O-positive blood verses the 132 units held if all the demand is filled. The second iteration shows similar holding behaviors that consequently build up the inventory of O-positive for the third iteration. Table 4-4 shows that 51 units of O-positive blood are available for substitution in this shortage situation, whereas the model with no value functions does not have adequate O-positive inventory to fill demand through substitution. The O-positive analysis also highlights another important behavior of the dynamic programming model. In the first iteration, the model chooses not to fill some O-positive demand and instead uses 8 units of O-positive to fill A-positive demand, thereby freeing up some A-positive blood. Without the value function approximations, this substitution is not performed and additional units of A-positive blood are not conserved. Finally, the behavior of O-negative blood is important to analyze: Filling a Different Blood Type % Filling own Blood Type % Hold % Filling Non- Substitutable Demand % Filling a Different Blood Type % Hold 0 0% Filling Non- Substitutable Demand % Filling own Blood Type % Figure 5-25: 1 st Iteration Optimal Allocation of O-negative with Value Functions Figure 5-26: 1 st Iteration Optimal Allocation of O-negative without Value Functions The value function model only fills 34 units of the total 40 units of O-negative nonsubstitutable demand and fills only 14 units of the total 17 units of O-negative substitutable

90 Chapter 5: Multi-Period Model Results demand. The model with no value functions fills all demand and uses all remaining O- negative blood to fill other demand types, therefore this model holds no units of O-negative blood. The dynamic programming model only allocates 17 units for blood type substitution (primarily to fill A-positive demand), and chooses to hold the rest. As can be seen in Table 5-3, the O-negative blood held in the dynamic programming model is helpful in the second iteration to fill demand of other blood types whereas the model with no value functions does not have adequate O-negative supply to perform substitutions. The third iteration (see Table 5-4) also shows the utility of conserving O-negative blood for the purposes of filling substitutable demand. This three iteration example illustrates the balance that the model finds in terms of filling demand, substituting, and holding blood. Deciding not to fill demand even though adequate supply exists is the key to the dynamic programming model in this net importing scenario. The supplies of A-positive, O-negative, and O-positive are particularly important to conserve since they represent a large portion of total demand and also since they have many substitution possibilities. Performing blood type substitutions when possible is a crucial action that frees up inventory. The optimal behaviors in the dynamic programming model generate the consistent total shortages and prevent any large shortage situations from ever arising. 5.4 Review The dynamic programming model provides an attractive strategy for blood centers to manage inventory. The drastic reduction in the fluctuation of total shortages is the most impressive result of the model. Hospital blood banks rely on blood centers for regular shipments each week, and in some cases, on a daily basis. If these regular needs cannot be met, patients at hospitals often face life-threatening situations. Therefore, the ability of the model to successfully fill hospital demands with consistent shortages of approximately 100 units provides valuable reliability for the blood center. This consistent shortage is a structural shortage and can possibly be eliminated through an import strategy, which will be explored in Section 6-1. The behaviors described in Section 5.3 highlight the particular decisions the model makes to prevent such large shortage situations from occurring. With a proper balance

91 Chapter 5: Multi-Period Model Results between filling demand, exercising substitution possibilities, and holding blood in inventory, the blood center can optimally satisfy as much demand as possible in a reliable manner. The highly unpredictable demands and donations are thereby transformed into a fairly predictable pattern of behaviors and outcomes.

92 Donating blood is literally giving the gift of life. [27] Chapter 6 Import Policy Simulation and Robustness The findings from Chapter 5 suggest that a structural shortage exists at The Blood Center of New Jersey. The consistent shortages of 100 units each week when using the dynamic programming model suggest an optimal import quantity of 100 units. Therefore, this chapter investigates a simulation of this import policy to determine if the structural shortages can be eliminated. Other import quantities are also evaluated. In addition to this policy simulation, Chapter 6 also investigates the robustness of the dynamic programming model. Blood banks often face seasonal shifts in supply and demand and thus it is important to understand how sensitive a CBC is to these changes. Section 6.2 examines behaviors during the summer and holiday periods where donations generally drop and Section 6.3 examines behaviors just after emergency situations where demand spikes. The analysis compares the results using the base model (in which value functions are trained according to the original supply and demand distributions) and the results using newly trained value functions according to the shifted supply/demand distributions. The conclusions from this analysis should illustrate how robust the dynamic programming model is as well as provide insight into how often a blood bank needs to reformulate their estimates of the value of holding blood for the future. 6.1 Import Policy Simulation With an appropriate import policy along with optimal behaviors specified by the dynamic programming model, a blood center should be able to eliminate almost all shortages. Therefore, an import policy of ordering 100 units each week is simulated here. The assumption is initially made that all imports enter the blood center with an age of one

93 Chapter 6: Import Policy Simulation and Robustness week, so imports are simply added to the donations. However, this is generally not the age of imports since blood centers often keep blood for themselves for a few weeks before sending it to other centers. Thus, a simulation in which imports arrive as four-week old blood is also studied Import Policy Results The 100 imported units are distributed by type according to the supply distribution given in Table 3-1. When these imports are added to the random donations in each period, total shortages are in fact often zero as can be seen in Figure 6-1. Figure 6-1: Total Shortages Resulting from an Import Policy of 100 units of Blood per Week with an Age of One Week Although shortages above zero do occur, a large proportion of iterations experience no shortages at all. Given that the model values holding blood, some iterations show a situation in which some demand is not met and blood is instead held for future use. This partly explains the shortages above zero. Another explanation is that random spikes in demand will still most likely cause shortages regardless of the import policy. Despite the presence of some shortages, this import policy produces dramatic improvement over results from Chapter 5 in which no imports are considered. The average shortages according to this import policy fall from 96.5 units (with no imports) down to 18.5 units given this sample realization of supply and demand.

94 Chapter 6: Import Policy Simulation and Robustness Since the assumption that imported blood has an age of one week is not entirely valid, another import policy that assumes an age of four weeks is also simulated. The results of a model without value functions are also included. The total shortages for the dynamic programming model are very similar to those of the first import policy in that shortages are often zero, but some random fluctuations in total shortages occur: Figure 6-2: Total Shortages Resulting from an Import Policy of 100 units of Blood per Week with an Age of Four Weeks Figure 6-2 shows that total shortages of approximately 40 to 50 units often are encountered with the dynamic programming model. However, these shortages are incurred so that blood can be held to prevent any large shortages above 120 units. When compared to Figure 6-1, the import policies produce similar results (different sample realizations are used for the two policies). Thus, it can be concluded that even imports arriving with an age of four weeks dramatically improve blood center performance. Figure 6-2 also shows that the dynamic programming model does not have a significant advantage over a model without value functions. The average total shortages are not significantly different, but the strength of the dynamic programming model is again in the reduction of variance, from a standard deviation of 41 without value functions to a standard deviation of 29 with value functions. The dynamic programming model is advantageous as it eliminates any shortages above 100 units whereas the model with no value functions still has some extremely large shortages.

95 Chapter 6: Import Policy Simulation and Robustness The results are somewhat disappointing, though, as the average total shortage is still above zero. Even when the model is retrained with the imports, almost identical results are seen. A weakness of the model is the extreme emphasis on holding blood such that imports greater than 100 units are required to eliminate shortages. So, a blood center implementing this import policy should expect a reduction of total shortages, but should still expect some small but periodic shortage situations. The next section examines the required import policy needed to eliminate all shortages since 100 units are actually not sufficient Other Import Policies In this analysis, the import quantity is varied from 0 units to 250 units in increments of 25 units. Each import policy is simulated over 200 iterations. The average total shortage of each import policy is recorded to produce the following graph: Figure 6-3: Import Quantity as a Function of Total Shortages. With this model, total shortages cannot be completely eliminated unless imports rise dramatically since the model emphasizes holding blood so much. However, there is a significant drop in the marginal value of additional import units beyond 150 units. Thus, if the blood center is extremely risk averse, the optimal policy should require 150 units per week. The general shape of the function appears to be convex, barring the convexity violation between 50 and 100 units that is most likely a function of random simulation noise. The important conclusion of these findings is that dramatic improvement in terms of total

96 Chapter 6: Import Policy Simulation and Robustness shortages is experienced at about 100 imported units, but additional improvements can be made with larger import quantities. 6.2 Holidays and Summer Policies Even with optimal import policies, blood centers usually face a seasonal drop in donations during holidays and the summer. The value functions are re-trained with the shifted supply distribution to identify optimal decisions given the new supply situation. These results are then compared to the results found if the base model (value functions trained with original supply distribution) is tested with the shifted supply distribution. This comparison provides insight into the robustness of the base model solution and can illuminate how sensitive the model is to fluctuations in underlying supply distributions. The new supply distribution reflects a drop in the average number of donations received at the blood center. No change to the demand distribution is assumed since demand for blood remains reasonably consistent throughout the summer and holiday periods. Thus, using the same absolute ranges as in Table 4-2. the frequency of occurrence for the supply distribution is shifted to the distribution shown in Table 6-1. This distribution represents the seasonal drop in supply during the summer and during holidays when donors are harder to attract and repeat donors tend not to donate. Thus, donations ranging from 300 to 550 units are more frequent. Frequency of Occurrence Table 6-1: Supply and Demand Distribution during Summer and Holiday Periods Minimum Maximum Frequency of Minimum Demand (# Demand (# Occurrence Supply (# units) units) units) 25% % % % % % General Behavior Patterns Maximum Supply (# units) When the model trained with the original supply and demand distributions (from now on referred to as the base model) is used to make decisions based on simulations from this shifted supply distribution, the behavior is similar to behaviors of a model trained with the shifted supply distribution. The purpose for comparing these two sets of results (shown on page 90) is to investigate how sensitive a blood center needs to be to changes in the supply. More specifically, this section should identify whether a blood center needs to significantly

97 Chapter 6: Import Policy Simulation and Robustness Fill Demand 92% Fill Demand 76% Send to CBC 0% Hold 8% Send to CBC 1% Hold 23% Figure 6-4: Optimal Decisions with no Value Functions and Shifted Supply Distribution Figure 6-7: Optimal Decisions with no Value Functions and Original Supply Distribution Fill Demand 79% Hold 61% Fill Demand 39% Send to CBC 0% Hold 21% Send to CBC 0% Figure 6-5: Optimal Decisions with Original Value Functions and Shifted Supply Distribution Figure 6-8: Optimal Decisions with Original Value Functions and Original Supply Distribution Fill Demand 81% Send to CBC 0% Hold 19% Figure 6-6: Optimal Decisions with Retrained Value Functions and Shifted Supply Distribution

98 Chapter 6: Import Policy Simulation and Robustness change its inventory management policies during the summer and holidays. Figures 6-4, 6-5, and 6-6 break down decisions as a percent of total supply using three different decisionmaking models: the base model, the newly trained model, and a model in which no value function approximations are used. Figures 5-6 and 5-7 from Chapter 5 are included as Figures 6-7 and 6-8 for comparison purposes to illustrate the large difference in optimal behaviors given this shift in supply. Thus, the right column of charts represents behaviors with the original supply distribution and the left column reflects behaviors with the shifted supply distribution. The first row shows results for a model with no value functions, the second row shows results for the base model (trained with the original supply distribution), and the last row shows results for a model trained with the shifted supply distribution. Figures 6-5 and 6-6 show very similar behaviors. This verifies the robustness of the model since a similar set of behaviors is found to be optimal based on both the original value function approximations and the retrained value function approximations. Both of these figures are dramatically different from figure 6-4 in which no value functions are used. Both the base and newly trained model with the shifted supply distribution suggest optimal behavior of holding approximately 20% of inventory each week and sending the remaining 80% to hospitals to fill demand. No blood is sent to another community blood center because this net importing region faces a scenario of even lower supply. When this optimal behavior is compared to the behaviors shown in Figure 6-8, it is apparent that a drop in supply should be accompanied by a significant decrease in the amount of blood held in inventory (as shown by the drop from 60% to 20%). Thus, The Blood Center of New Jersey should alter its inventory management policy in the summer months and holiday periods such that they greatly reduce their amount of inventory. However, they do not need to re-evaluate the value of holding blood since the model has proven to be robust in this situation. If the blood center does not use value function approximations to make decisions during summer and holiday periods, about 92% of inventory is shipped out. Figure 6-4 illustrates the breakdown of behaviors as a percent of total supply in the blood center according to this no value function model. This policy of maintaining very low inventory in the blood center is extremely dangerous as the center is highly dependent upon donations.

99 Chapter 6: Import Policy Simulation and Robustness Therefore, both the base model and the newly trained model provide the blood center with an optimal inventory management policy for summer and holidays that is less sensitive to fluctuations in donations. The robustness of the model solution has now been confirmed, but this does not substantiate the success of the model during summer and holiday periods. More specific measurements of blood bank performance need to be assessed, particularly total shortages and outdates Total Shortages and Outdates The dynamic programming model suggests holding 20% of inventory each week during periods of low supply verses only 8% suggested without value function approximations. In all three simulations (the base model, the newly trained model, and a no value function simulation), no outdates are encountered, as expected. However, average shortages do increase in this period of low supply. Table 6-2 provides information regarding the mean and standard deviation of the total shortages in all three models over 200 iterations with the shifted supply distribution: Table 6-2: Mean and Standard Deviation of Total Shortages for all Three Models Model Average Total Shortages Standard Deviation Base Model units 83.9 units Newly Trained Model units 83.4 units No Value Functions units units Table 6-2 highlights the ability of the model to reduce variation in total shortages even though the average shortages do not decrease. The similarity in the total shortages and more importantly in the standard deviation of total shortages between the base model and the newly trained model emphasizes the incredible robustness of the model. Thus, the base model provides the same reduction of variation in total shortages as does a newly trained model so it would not be necessary for a blood center to re-train any sort of model they use for inventory planning. The following histograms reiterate these results:

100 Chapter 6: Import Policy Simulation and Robustness Figure 6-9: Histogram of Total Shortages when Base Model is Used to Make Decisions Figure 6-10: Histogram of Total Shortages when Newly Trained Model is Used to Make Decisions Figure 6-11: Histogram of Total Shortages when no Value Functions are Used

101 Chapter 6: Import Policy Simulation and Robustness The similarity in the histograms in Figures 6-9 and 6-10 again emphasizes that the same distribution of total shortages is observed with the original value functions and with the newly trained value functions. Both of these models reduce the occurrence of extremely high shortages as compared with the distribution of shortages observed when no value function approximations are used (Figure 6-11). Thus, blood center managers can simply use the results of the base model when simulated with the shifted supply distribution to make optimal import decisions. According to the model, a blood center should import approximately 165 units of blood each week to meet demand. The reasonably high variance in total shortages even with the dynamic programming model means that shortages still may be encountered, but not as frequently as when decisions are made without any value function information Summary The discussion of optimal behavior in a low supply situation demonstrates the general robustness of the model solution as well as the change in optimal importing behavior for a blood bank. This analysis shows that it is not necessary to retrain the model with the updated supply distribution. The shift portrayed here is an accurate representation of reality as a 10% shift supply distribution is reasonable for summer and holidays. Therefore, the model solution is robust to shifts of this magnitude but it is important to realize that if a large-scale change in supply distribution occurs, it may be necessary to then retrain the value function approximations. 6.3 Disaster Situation Policies Much like the analysis in Section 6.2, this section seeks to assess the robustness of the model. Spikes in demand are often faced by blood centers when natural disasters occur or when large-scale emergency situations arise. Although natural disasters are sometimes accompanied by an increase in supply with public awareness of the tragedy, this model assumes that the supply distribution remains constant. Thus, the following table gives the supply and demand distributions used for this analysis.

102 Chapter 6: Import Policy Simulation and Robustness Frequency of Occurrence Table 6-3: Supply and Demand Distribution Used for Disaster Situation Analysis Minimum Demand (# units) Maximum Demand (# units) Frequency of Occurrence Minimum Supply (# units) 15% % % % % % Maximum Supply (# units) Unlike the shift in the supply distribution in Section 5.2, the shift in the demand distribution is quite drastic to reflect realistic emergency situations in which the frequency of high demand levels is significant. Although a major emergency may cause demand levels to rise above 925 units, this analysis assumes that the frequency of occurrence shifts, not the minimum or maximum demands General Behavior Patterns This section identifies the similarities and differences in the shipping and holding behaviors when decisions are made with the base model, with model trained with the shifted demand distribution, and with a model in which no value functions are used. Using the same analysis approach as in Section 6.2.1, five pie charts included on page 96 highlight the following: the robustness of the model, the optimal policy under the new demand distribution, and the distinction between results from the dynamic programming model and a model with no value function approximations. In comparison to the results found in Section 6.2, this model is not as robust to the upward shift in demand. There is a more distinct difference between outcomes of the newly trained model and the base model behavior (see Figures 6-13 and 6-14). The newly trained model yields optimal behavior of holding 58% of supply in inventory each week whereas if decisions are made using the original value function approximations, only 47% of supply will be held in inventory. These are significantly different policies and therefore to achieve the most optimal behaviors, the model should be re-trained in these disaster scenarios in which a dramatic shift in demand occurs. However, even with this discrepancy in optimal behavior, both models that include value function approximations encourage more holding behavior than does a model with no value function information. Figure 6-12 illustrates the breakdown of behaviors without the use of value functions. The different behaviors shown in this figure emphasize the

103 Chapter 6: Import Policy Simulation and Robustness Fill Demand 78% Fill Demand 76% Send to CBC 1% Hold 21% Send to CBC 1% Hold 23% Figure 6-12: Optimal Decisions with no Value Functions and Shifted Demand Distribution Figure 6-15: Optimal Decisions with no Value Functions and Original Demand Distribution Fill Demand 53% Hold 61% Hold 47% Fill Demand 39% Send to CBC 0% Figure 6-13: Optimal Decisions with Original Value Functions and Shifted Demand Distribution Send to CBC 0% Figure 6-16: Optimal Decisions with Original Value Functions and Original Demand Distribution Fill Demand 42% Hold 58% Send to CBC 0% Figure 6-14: Optimal Decisions with Retrained Value Functions and Shifted Demand Distribution

104 Chapter 6: Import Policy Simulation and Robustness modifications value function approximations have on behavior patterns as only 21% of inventory is held at the CBC. Thus, there is some sense of robustness in the model since including value function information always encourages holding blood. However, the significant conclusion here is that the original model solution is not as robust to a dramatic shift in the demand distribution Total Shortages and Outdates Similar to the low supply scenario in Section 6.2, high demand is accompanied with zero outdates in all three cases (base model, retrained model, and no value functions). This is expected since outdates are not of significant interest at a net importing blood center. However, the difference in optimal behavior patterns between the base model and the retrained model discussed in Section translate to different total shortage results. The total shortage information for all three models is shown in Table 6-4. Table 6-4: Mean and Standard Deviation of Total Shortages for Models in a High Demand Scenario Model Average Total Shortages Standard Deviation Base Model Newly Trained Model No Value Functions This table verifies the conclusion from Section that the model solution is not as robust in this increased demand scenario. The newly trained model does a better job at reducing variance and it actually also reduces total shortages. Therefore, The Blood Center of New Jersey should have separate estimates for the value of holding blood during emergency situations. Despite this lack of robustness, both the base model and the newly trained model provide significant advantages over a decision-making process blind to value function information. This is seen in the reduction of the standard deviation of total shortages in the value function models when compared to a model with no value functions. The following histograms illustrate this behavior in more detail:

105 Chapter 6: Import Policy Simulation and Robustness Figure 6-17: Histogram of Total shortages using Base Model Figure 6-18: Histogram of Total Shortages using Newly Trained Model Figure 6-19: Histogram of Total Shortages when no value functions are used

106 Chapter 6: Import Policy Simulation and Robustness Figure 6-18 shows the least variance in total shortages and only experiences very few shortages over 150 units. Figure 6-17 is somewhat similar to Figure 6-18 in the distribution of total shortages, but does show a slightly larger variation as total shortages above 150 units are slightly more frequent. The difference between these two histograms illustrates the sensitivity of the blood bank to large shifts in demand and thus the corresponding lack of robustness in the model solution. However, the use of any value function information provides an improvement as is shown by Figure The distribution of total shortages is much wider when no value function approximations are used and many more shortages over 150 units are encountered. Therefore, the advantage of value function information is clearly established in the reduced variance of total shortage outcomes over time Summary The analysis in this section highlights the increased sensitivity of a blood bank to high demand situations. The model is not as robust to such a dramatic shift in demand and thus the most optimal method of handling disaster situations is to retrain the model with an estimate of what the demand distribution will be during an emergency. However, it might not be possible for a blood center to reformulate their estimates for the value of holding each blood type in inventory. This analysis shows that the original model can still provide an improved solution over a model with no value functions. In these disaster situations, it is critical to supply adequate blood since many lives are most likely at stake. Therefore, blood centers should have emergency plans that can be enacted quickly to meet the needs of the many trauma patients after earthquakes, fires, large traffic accidents, or any other large-scale emergency in which significant quantities of blood are needed. 6.4 Review This chapter concludes the analysis of the dynamic programming model with an investigation into the effects of various import policies as well as an assessment of the robustness of the model. Although an import policy of 100 units per week does not eliminate all shortages, it does dramatically decrease expected shortages. Since the model places such a high value on holding blood, it will continue to hold some blood instead of filling demand such that 100 units of imported blood will still not eliminate all shortages. However, the

107 Chapter 6: Import Policy Simulation and Robustness model solution is robust to shifts in supply common during summer and holiday months but is not as robust to spikes in demand. Thus, blood centers can use the same value function approximations year-round for inventory management decisions, but if a large-scale emergency situation arises, the blood center should have a different set of values for holding blood in inventory.

108 Concluding Remarks The current trends in the growth of supply and demand for blood will soon lead to widespread shortage situations. Thus, learning how to better manage available inventory is an important endeavor for blood centers and hospitals. Chapter 3 highlights the benefits of creating a universal blood substitute. Even though synthetic blood substitutes will most likely have limited uses since the functionality of real blood is quite complex, these substitutes still will have dramatic effects on the reduction of expected shortages. Thus, resources should be allocated to the development of synthetic blood substitutes because these products would generate a large, reliable source of inventory for blood banks. However, safe and effective blood substitutes will not be available for medical use in the near future. Therefore, different inventory management behaviors can provide immediate changes to blood banking strategies. The dynamic programming approach developed in Chapter 4 illustrates the advantage of incorporating value functions into decision-making processes. The dynamic programming model in effect understands that filling all demand in a given period is actually not optimal because this behavior can greatly reduce inventory, leading to major shortage problems in the future. The value functions also encourage substitution behaviors to maintain an optimal balance of inventory, particularly recognizing the value of O-positive, O-negative, and A-positive blood. Additionally, even if enough O-negative is available to fill demand, blood centers should not send all of it out because maintaining an adequate supply of the universal donor helps to prevent large-scale shortages. These optimal behavior patterns are then able to generate reliable shipments to hospitals each week. The structural shortages identified by the dynamic programming model can then be relieved with adequate importing policies. Although an import quantity of 100 units per week does not eliminate all shortages, it dramatically reduces the average shortage incurred. If a blood center wants to further reduce shortages, increasing the import quantity will further reduce the total shortages, but with a decreasing marginal return. The strength of the dynamic programming model is in the balance achieved between holding, shipping, and substitution behaviors. If blood centers implement the optimal decisions of the model, a

109 Concluding Remarks much more stable system of imports and exports can be established, reducing the total demand on the system as a whole. The results of this dynamic programming model in terms of total shortages are dramatic and a successful inventory management model is created for a single blood center. However, the model has made many simplifying assumptions. Extending the model to include a more extensive representation of actual blood banking practice would be an easy transition and would also provide additional insight into this problem. The following discussion outlines additions and modifications that can be made to the model to improve its reflection of realistic blood banking behaviors. The model assumes that all RBC products are identical, such as washed RBC s and leukocyte-depleted RBC s. However, in reality these are different products and hospitals place orders for each kind individually. The model can be easily translated to accommodate these products through an added attribute to the resource variable specifying the given product type. No other change would be required. Additionally, frozen blood was not included in this analysis. Again, an additional attribute to the resource state can be added to distinguish between frozen blood and fresh blood. Since frozen blood must be thawed and then used within 24 hours, the decision to thaw and send frozen blood would have to occur on a daily basis instead of a weekly basis. The model could then make all decisions daily, which would reflect many blood centers in urban areas with high daily demands, or it could implement two sets of decisions, some made weekly and some made daily. The other components of blood discussed in Chapter 1 can also be modeled with this dynamic programming structure. Each component has specific shelf-life specifications that can be incorporated into a modification of the red blood cell model, simply through a change in the oldest allowable age. These products do not have the substitution patterns of red blood cells, but the formulation of maximizing a one period contribution plus the value of the future state of inventory can be implemented for these components. The management of platelets is particularly interesting as their shelf-life is only five days. Thus, inventory turnover is extremely high and allocation decisions must be made quickly to prevent outdates of the platelets in inventory. Another simplification is that the supply and demand distributions by type are fixed in this model. However, the actual blood-type distributions of donations and demands are not

110 Concluding Remarks constant. Thus, the model can be modified to accommodate either shifted blood-type distributions or could include some level of noise in the distributions. This would provide a more realistic set of outcomes and could also support studies on different regions of the United States with varied ethnic representation. Another simple modification would be to change the percent of demand that allows substitution. This model assumes that half of demand permits blood-type substitution, but depending on physician preferences, this allowable percentage varies. Thus, different analyses can be performed through the modification of this percent. Similarly, the percent of elective demands can be easily altered to assess the results of the model as the amount of elective surgeries does vary from region to region. These modifications can tailor the model to the characteristics of a particular region. Thus, the same model structure can be applied with different inputs to find optimal behaviors for various regions. The model can also be applied to single hospital locations. The exogenous information would then be transfusion demand and the inventory would be replenished through shipments from a CBC. This hospital modeling could also include donations at the hospital since some hospitals do indeed collect blood. Additionally, the application of the model to a hospital location requires that the crossmatch process be included. This process would affect the transition function, requiring that additional units be taken out of available inventory, specified by the crossmatch-transfusion ratio, when filling demand. The excess units assigned but not used would then be put back into available inventory after the specified crossmatch release time. Since situations arise where the excess blood is in fact used, the return of blood to available inventory should occur with some probability. With these modifications, the model can fully describe behaviors at hospital locations. The dynamic programming model can then be used to model a multiple location network for a given region. The hospital blood banks make ordering decisions that are given to their CBC. The CBC sees these orders as their demand and then makes optimal shipment decisions. An allocation method would have to be devised to divide blood between all hospital locations. One possible method would divide blood evenly or another possibility is to make sure each blood center receives the same ratio with respect to their initial request in the event that total orders cannot be met. Thus, the hospital inventories are replenished through

111 Concluding Remarks these shipments from the CBC. This structure would allow the model to be individually applied to each location to then optimize flows over the network in a given region. The model can also be extended to a multiple region network. With multiple community blood centers in the system, optimal import and export decisions can be formulated through information sharing. Each CBC determines its optimal import or export quantity and then the needs are matched over all CBC s. Total desired imports will most likely not match total desired exports, thus a system or method would need to be established to determine how to divide imports and exports to optimally satisfy the needs of all regions. This large-scale application of the model should be able to reduce inefficiencies in the whole system and help each blood center and hospital more effectively manage inventory of all blood products. The dynamic programming model created in this thesis provides a solid base for many additional analyses of blood inventory management. As demand for blood approaches the supply, new solutions must be constructed, whether they involve synthetic blood or improved inventory management systems. Medical procedures will continue to expand and the population will continue growing. Therefore, the need for blood will never cease. Understanding how to manage this vital resource is of utmost importance as you or someone you know will most likely need blood at some point in your lifetime.

112 Appendix A The data included in this appendix represents the portion of data used in this thesis from Dr. Eric Senaldi at the Blood Center of New Jersey. The first table contains 51 weeks of donation data for the Blood Center of New Jersey. The second table contains 66 weeks of distribution data (shipments to hospitals) from the New York Blood Center. Table A-1: Donation Data from The Blood Center of New Jersey WEEK Start End Donation (#units) 1 12/28/08 01/03/ /04/09 01/10/ /11/09 01/17/ /18/09 01/24/ /25/09 01/31/ /01/09 02/07/ /08/09 02/14/ /15/09 02/21/ /22/09 02/28/ /01/09 03/07/ /08/09 03/14/ /15/09 03/21/ /22/09 03/28/ /29/09 04/04/ /05/09 04/11/ /12/09 04/18/ /19/09 04/25/ /26/09 05/02/ /03/09 05/09/ /10/09 05/16/ /17/09 05/23/ /24/09 05/30/ /31/09 06/06/ /07/09 06/13/ /14/09 06/20/ /21/09 06/27/ /28/09 07/04/ /05/09 07/11/ /12/09 07/18/ /19/09 07/25/ /26/09 08/01/ /02/09 08/08/ /09/09 08/15/ /16/09 08/22/ /23/09 08/29/ /30/09 09/05/ /06/09 09/12/ /13/09 09/19/09 741

113 Appendix A WEEK Start End Donation (#units) 39 09/20/09 09/26/ /27/09 10/03/ /04/09 10/10/ /11/09 10/17/ /18/09 10/24/ /25/09 10/31/ /01/09 11/07/ /08/09 11/14/ /15/09 11/21/ /22/09 11/28/ /29/09 12/05/ /06/09 12/12/ /13/09 12/19/ Table A-2: Distribution Data from the New York Blood Center WEEK Date Distribution (#units) 1 07/05/ /12/ /19/ /26/ /02/ /09/ /16/ /23/ /30/ /06/ /13/ /20/ /27/ /04/ /11/ /18/ /25/ /01/ /08/ /15/ /22/ /29/ /06/ /13/ /20/ /27/ /03/ /10/ /17/ /24/ /31/ /07/ /14/ /21/ /28/ /06/ /13/

114 Appendix A WEEK Date Distribution (#units) 38 03/20/ /27/ /03/ /10/ /17/ /24/ /01/ /08/ /15/ /22/ /29/ /05/ /12/ /19/ /26/ /03/ /10/ /17/ /24/ /31/ /07/ /14/ /21/ /28/ /04/ /11/ /18/ /25/ /02/

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