Incorporating Ratios in DEA—Applications to Real Data written by Sanaz Sigaroudi Thesis advisor: Dr. Joseph C. Paradi A thesis submitted in conformity with the requirements for the degree of Master of Applied Science Graduate Department of Mechanical and Industrial Engineering University of Toronto Copyright (cid:13)c 2009 by Sanaz Sigaroudi Abstract Incorporating Ratios in DEA—Applications to Real Data Sanaz Sigaroudi Master of Applied Science Graduate Department of Mechanical and Industrial Engineering University of Toronto 2009 In the standard Data Envelopment Analysis (DEA), the strong disposability and convex- ity axioms along with the variable/constant return to scale assumption provide a good estimation of the production possibility set and the efficient frontier. However, when data contains some or all measures represented by ratios, the standard DEA fails to generate an accurate efficient frontier. This problem has been addressed by a number of researchers and models have been proposed to solve the problem. This thesis proposes a “Maximized Slack Model” as a second stage to an existing model. This work implements a two phase modified model in MATLAB (since no existing DEA software can handle ratios) and with this new tool, compares the results of our proposed model against the results from two other standard DEA models for a real example with ratio and non-ratio measures. Then we propose different approaches to get a close approximation of the convex hull of the production possibility set as well as the frontier when ratio variables are present on the side of the desired orientation. ii Acknowledgements First and foremost, I would like to express my sincere gratitude to my supervisor, Pro- fessor Joseph C. Paradi for giving me the opportunity to be part of his team at CMTE. I am indebted to him for his life lessons, continuous support and patience. His passion for having an active social and scientific life, even after retirement, has always been a source of inspiration to me. I am grateful to my committee members Professor Y. Lawryshyn, Professor R. Kwon and Professor B. Balcioglu for providing me with constructive comments and insightful feedback. My appreciation goes to Dr. Emrouznejad for providing access to his paper and the proposed model therein, which became a motivation for this work. DuringmytimeatCMTEIlearnedalotfromdiscussionsonDEAandrelatedmathe- matical modeling with my colleagues. In particular I am thankful to Dr. Barak Edelstein for teaching me the DEA concept and Dr. Haiyan Zu for answering my questions and sharing her expertise with me. Dalia Sheriff ’s knowledge on ratios and Leili Javanmardi knowledge on MATLAB were also very helpful at the early stages of this work. I owe an immeasurable debt of gratitude to Dr. Judy Farvolden for her warm welcome when I first arrived in Canada and her ongoing scientific and non-scientific support and advice throughout my research career. My defense presentation would not be successful without Judy and Haiyan’s help. My friends at CMTE Angela Tran, Pulkit Gupta, Laleh Kobari, Sukrit Ganguly, Steve Frensch, Susuan Mohammadzadeh, Erin Kim, Colin Powel, Joe Pun, and Justin Toupin have made my student life a joyful experience. I also thank Sau-Yan Lee for his help whenever I had a problem with computer and network . My appreciation goes to Professor Balcioglu for trusting me and giving me the opportunity to gain invaluable teachingexperience. IwouldalsoliketothankthestaffattheMIEgraduateoffice,mostly Brenda Fung , Donna Liu, and Joe Nazal for helping me out through the administrative parts of the process. iii I have been always blessed with amazingly nice friends. I am especially grateful to Ghazal and Niayesh for being the best neighbors and friends one can ever ask for. My kind relatives, Makhmal and Anoosh, and their kids, Maryam and Mona, have always welcomed me warmly into their house in Toronto at many late nights when commuting to Waterloo was hard. Yaser and Faranak, my cousin Solmaz, Iman and Golnaz, Sara and Ali, as well as Afsaneh and Reza have made the life experience in Waterloo and Toronto, far from my home country, Iran, a wonderful, adventurous and joyful one. I like to say “merci” to my dearest friend, Mahnaz and her family, for being so kind to me and my husband no matter what we do or where we go. I am indebted to my parents, Habibeh Ghaznavi and Dr. Issa Sigaroudi, and my brother, Sina, for loving me and believing in me and giving me the full freedom to explore the world of unknowns in my own way. I am also very lucky to have caring and loving parents-in-law, Zahra Alemzadeh and Dr. Asadollah Razavi who keep me in their good prayers. My sisters-in-law Maryam, Razieh, and Marziyeh and their families have been always there to cheer for me and support me. I would also like to thank my aunt, Alieh, my cousin Mahnaz, and my husband’s uncle, M. Ali, and his family for their kindness and support during our stay in the United States and Canada. Iwouldnothavebeenabletocontinuemystudieswithoutthelove,support,assistance andencouragementofmybelovedhusband, MohsenRazavi, throughoutthesesyears. His love and kind heart are the most precious gifts I have ever got in this world. This thesis is dedicated to him. iv Contents 1 Introduction 1 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 RA and DEA 4 2.1 RA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.1 How RA is measured? . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.2 RA and linear programming representation . . . . . . . . . . . . . 5 2.2 Data Envelopment Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.2 DEA Basic assumptions and Models . . . . . . . . . . . . . . . . 8 2.2.3 Production Possibility Set . . . . . . . . . . . . . . . . . . . . . . 8 2.2.4 Frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.5 Efficiency Definition . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.6 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2.7 Additive model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Linkage between Data Envelopment Analysis and Ratio Analysis . . . . . 12 2.3.1 Comparing DEA and RA . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.2 Combining DEA and RA . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Advantages of incorporating ratios in DEA . . . . . . . . . . . . . . . . . 16 v 3 Frontier Assessment 17 3.1 Deterministic Frontier Estimators . . . . . . . . . . . . . . . . . . . . . . 18 3.1.1 Free Disposal Hull Frontier . . . . . . . . . . . . . . . . . . . . . 18 3.1.2 Various Returns to Scale Frontiers . . . . . . . . . . . . . . . . . . 18 3.1.3 Constant Returns to Scale Frontier . . . . . . . . . . . . . . . . . 19 3.2 Probabilistic Frontier Estimators . . . . . . . . . . . . . . . . . . . . . . 19 3.2.1 Partial m Frontier . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2.2 Quantile Frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2.3 Practical Frontiers . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4 Incorporating ratios in DEA 24 4.1 Modified DEA models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.1.1 Model I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.1.2 Model II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.2 Motivating Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.3 Nonlinearity and Orientation . . . . . . . . . . . . . . . . . . . . . . . . . 38 5 Comparing the Conventional and the Modified DEA Solver 41 5.1 Use of DEA in Banking Industry . . . . . . . . . . . . . . . . . . . . . . 41 5.2 Evaluation Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 5.2.1 Profitability model . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.3 Data and summary results . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.3.1 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.3.2 “Non-ratio” Profitability Model, DEA solver results . . . . . . . . 48 5.3.3 “Conventional Ratio” Profitability Model, DEA solver results . . 50 5.3.4 “Modified Ratio” Profitability Model, our MATLAB coded DEA solver results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.4 Comparing the Efficiency Scores . . . . . . . . . . . . . . . . . . . . . . . 52 vi 5.4.1 National PPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.4.2 Local PPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.4.3 Targets and Projection . . . . . . . . . . . . . . . . . . . . . . . . 56 5.4.4 Efficient Branches . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.4.5 Practical improvements . . . . . . . . . . . . . . . . . . . . . . . . 60 6 Analysis and Discussion of the Results 62 6.1 Evaluation Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 6.2 National Efficiency Score of Branches in Regions . . . . . . . . . . . . . . 63 6.3 Benchmark Branches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.4 Actual cost savings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.5 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.6 Comparing the results with the Bank’s own metrics . . . . . . . . . . . . 70 7 Approximating the Nonlinear Frontier 71 7.1 Maximum Slack Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 7.2 Additive model for DEA with ratio variables . . . . . . . . . . . . . . . . 72 7.3 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 7.3.1 “for” Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 7.4 Partial PPS Improvement . . . . . . . . . . . . . . . . . . . . . . . . . . 77 8 Conclusions 80 8.1 Summary and Contribution . . . . . . . . . . . . . . . . . . . . . . . . . 80 8.2 Ratio variables on the opposite side of orientation . . . . . . . . . . . . . 80 8.3 Ratio variables in both input and output sides . . . . . . . . . . . . . . . 82 8.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 8.5 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Glossary 84 vii References 88 viii List of Tables 4.1 Lasik Equipment Information . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2 Comprehensive Lasik Equipment Information . . . . . . . . . . . . . . . 31 4.3 Hypothetical DMU Information . . . . . . . . . . . . . . . . . . . . . . . 32 5.1 Choice of input and output for the profitability model . . . . . . . . . . . 45 5.2 Summary of the Input correlations . . . . . . . . . . . . . . . . . . . . . 47 5.3 Summary of the Output correlations in non-ratio and modified model . . 49 5.4 Locally compared . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.5 Input and Output Projections for DMU 58. Note: ROR = Rate of Return 56 5.6 Distance between DMU 58 and the target . . . . . . . . . . . . . . . . . 58 5.7 Detailed road map to achieve ratio improvements for DMU 58 . . . . . . 59 6.1 Areas differentiating the top ten efficient branches from average . . . . . 66 6.2 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 ix List of Figures 2.1 DEA basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.1 Shapes of conventional frontiers:FDH, BCC, and CCR . . . . . . . . . . 19 4.1 Shapes of projections of conventional(red) and modified(black) frontiers on the output plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.2 Shapes of conventional(red) and modified(black) 2D frontiers . . . . . . . 34 4.3 Feasible DMUs generated by conventional(red) and modified(black) PPS estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.4 PPS boundaries in 3D: conventional (red), modified (black) . . . . . . . . 36 4.5 Forming the efficient frontier, output oriented. DEA conventional(red), DEA modified(black). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.6 Forming the efficient frontier, output oriented. DEA conventional(red), DEA modified(black). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.7 Frontier facets, input orientation. . . . . . . . . . . . . . . . . . . . . . . 37 5.1 Nation wide comparison of scores . . . . . . . . . . . . . . . . . . . . . . 54 6.1 Average Input Levels of Regions Branches . . . . . . . . . . . . . . . . . 63 6.2 Average Output Levels of Regions Branches . . . . . . . . . . . . . . . . 64 6.3 Average expenses of top ten branches . . . . . . . . . . . . . . . . . . . . 66 6.4 Average rate of returns of top ten branches . . . . . . . . . . . . . . . . . 67 6.5 Correlation between Modified scores and Bank’s Internal Metrics . . . . . 70 x