A SCORING-BASED RANKING SYSTEM AND ITS APPLICATION IN CADAVER KIDNEY ALLOCATION A DISSERTATION SUBMITTED TO THE DEPARTMENT OF MANAGEMENT SCIENCE AND ENGINEERING AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Yichuan Ding August 2012 © 2012 by Yichuan Ding. All Rights Reserved. Re-distributed by Stanford University under license with the author. This work is licensed under a Creative Commons Attribution- Noncommercial-No Derivative Works 3.0 United States License. http://creativecommons.org/licenses/by-nc-nd/3.0/us/ This dissertation is online at: http://purl.stanford.edu/nt830sv6639 ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Stefanos Zenios, Primary Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Peter Glynn I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Lawrence Wein I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Yinyu Ye Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost Graduate Education This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives. iii Abstract This dissertation presents a modeling-based analysis of scoring-based policies for al- locating cadaver kidneys to recipient patients. The policies are categorized into two major groups: donor-dependent and donor-independent, depending on whether or not the quality of a kidney plays a role in calculating the candidate’s score. I develop three different models for evaluating different types of scoring-based policies with re- spect to three major criteria: 1) whether or not patients and kidneys are matched according to expected survival times; 2) whether or not most kidneys will be accepted by the patients; and 3) the amount of freedom patients have in choosing their kidneys. All of the models assume that patients and kidneys are classified into different groups according to their estimated survival time, and that the patients leave the waitlist system either by transplantation or by death. The first model assumes that patients always accept the first-offered kidney, and then investigates the allocation outcome under different types of scoring policies. To capture the scoring-based nature of the allocation scheme, I study the asymptotic behavior of an overloaded queuing system when the abandon rate is much smaller than the traffic throughput. I prove that the waiting time at steady-state converges in probability to a point limit in the asymptotic regime. This limit can be solved through a system of equations. The analysis based on this limit reveals that, under a donor-independent scoring scheme, the quality of kidneys offered to a patient is independent of the patient’s estimated survival years. In contrast, a supermodular, donor-dependent scoring scheme increases the chance of a higher-quality kidney being allocated to a higher-longevity patient. The second model considers the possibility that a patient may reject a kidney iv being offered to receive a better kidney in the future. It has been reported that, in the current system, a high number of rejections occur, which results in delays in transplantation and a decrease in organ quality. To address this issue, I propose a policy called ”partitioning and scoring,” where patients must specify the class of kidneys they are waiting for, and cannot change their choice later. Because a low- quality kidney usually corresponds to a less crowded queue and a shorter waiting time, patients must choose between a better kidney or a shorter waiting time. By modeling the waitlist as a multi-queue system, I show that the queue-length process converges to a diffusion process in the heavy traffic limit regime, and therefore the allocation outcome can be approximately predicted. In particular, the allocation outcome shows that a partitioning and scoring policy can improve the survival matching between a recipient and a donor and reduce the kidney rejection rate. This result supports the idea of keeping a separated waitlist for kidneys from Expanded Criterion Donors (ECD), which has been implemented since 1992. The third model is motivated by the fact that a partitioning-based policy gives pa- tients less freedom in choosing kidneys. An alternative is considered that merges the candidates into a single waitlist; each candidate is under consideration for all kidney offers and can accept or reject a kidney based on individual preference. I propose a model that approximately predicts the outcome of using such a policy, and use simu- lation to verify the approximation. My analysis shows that a donor-dependent scoring scheme (DDSS) constitutes a good compromise between the two conflicting objectives of reducing kidney rejections and allowing patients more freedom in choosing their kidneys. That is, a carefully calibrated donor-dependent scoring policy reduces kid- ney rejections to an affordable level and meanwhile gives patients adequate freedom to choose between a shorter waiting time or a higher-quality organ. I compared the performance of four representative scoring-based allocation policies by a simulation test using kidney-pancreas simulated allocation model (KPSAM), a software devel- oped by the Scientific Registration of Transplant Research (SRTR). The results shows that a DDSS reduces the number of discarded kidneys by about 7% compared with a DISS, although the difference could be even larger if the KPSAM could capture the impact of different policies on patient behavior. v Acknowledgements First, I would like to express my sincere gratitude to my dissertation advisor, Professor Stefanos Zenios. He has introduced me to the area of health care management and given me numerous help with my study and career. He is always my role model as a top rated researcher as well as a successful teacher. Second, I would like to thank my program advisor, Professor Yinyu Ye, for his great supervision on my research in mathematical programming, as well as his en- couragement and support in different aspects when I was in Stanford. Third, I would like to thank Professor Henry Wolkowicz. He was my supervisor when I studied in the University of Waterloo towards a Master degree. He has led me into the world of optimization and continued to be my resource after I graduated from Waterloo. I would like to thank Professor Peter Glynn, for many insightful discussions on my research project, as well as his support to me in the MS&E department. I would like to thank Professor Margaret Brandeau, for her advising on my research in health care as well as her valuable help with my interview and job talks. I would also like to thank Professor Baris Ata, for his valuable feedback on my research on kidney allocation policies. I would thank Professor Lawrence Wein for his very useful class, stochastic networks, and his many helpful suggestions in improving my dissertation. I want to thank Professor Amin Saberi, for his help with my second-year tutorial paper, which later gets published in a peer-reviewed journal. I would like to express my gratitude to my collaborators and colleagues. I have been fortunate to have many excellent collaborators when I was in Stanford, e.g., Shipra Agrawal, Dongdong Ge, David Lowsky, Donald Lee, Zizhuo Wang. Our vi collaboration becomes unforgettable experience for me. I am also grateful to get many great colleagues in Stanford, e.g., Sabina Alistar, Rob Bray, John Carlsson, Su Chen, Danny Greenia, Joel Goh, Kris Iyer, Bora Keskin, Nur Keskin, Anicham Kumarasamy, Hugo Mora, Jessica McCoy, Sechan Oh, and Zheng Wen. I learned a lot from many discussions with those wonderful colleagues, who are my life-long friends and teachers. My friends have given me invaluable supports in the past years. My special thanks go to Zizhuo Wang. He is an amazing human being as well as my best resource when I need a help. I want to thank my good buddies in the MS&E department, e.g., Shi Chen, Dongdong Ge, Yihan Guan, Ruixue Guo, Lei Liu, Shan Liu, Jing Ma, Janbai Li, Wenhao Liu, Chen Peng, Qi Qi, Xi Wang, Yu Wu, Qinqin Zhang, Xiaowei Zhang, Yanchong Zheng, Wugang Zhao. I am also grateful for many friends in the Chinese community at Stanford, e.g., Jing Li, Yuankai Ge, Qicong Hu, Xiaoyu Liu, Yue Shi, Yunyun Song, Yanjing Yin... I cannot forget the days when I were working together in the Chinese Association of Students and Scholars, preparing the drama plays, enjoying the sunset in the Mexico Bay, and singing on the stage of CCTV. Although my student life ends here, I will keep these memories with me forever. Finally, I would like to express my deepest gratitude to my parents, Mei Lin and Bingang Ding, for their great support. Special thanks go to my grandfather, Yongcai Ding, who just passed away several months ago when I was job hunting. This dissertation is dedicated to him. vii Contents Abstract iv Acknowledgements vi 1 Introduction 1 2 Overloaded Systems with Scoring-Based Rankings 7 2.1 Introduction................................ 7 2.2 LiteratureReview............................. 10 2.3 ModelsandAnalysis ........................... 12 2.3.1 Service-Independent Scoring Scheme . 13 2.3.2 Service-Dependent Scoring Scheme .