Researches On Cross Efficiency Of Data Envelopment Analysis (DEA)

Efficiency evaluation(performance evaluation) is a very common and important work in the organization,while it is also very difficult when there are multiple inputs and multiple outputs to the system.The difficulties are further enhanced when the relationships between the inputs and the outputs are complex and involve unknown tradeoffs.Data Envelopment Analysis(DEA),as an effective non-parametric approach treating problem with multiple inputs and outputs,has been widely used in the evaluation and improvement of decision making unit(DMU). Recently,more and more ranking approaches based on DEA appear,as a typical representative,cross efficiency evaluation method is also developed and widely used. The main idea of cross evaluation is to use DEA in a peer evaluation mode instead of a self-evaluation mode.Cross evaluation can provide for a unique ordering of the DMUs and eliminate unrealistic weight schemes without requiring the elicitation of weight restrictions from application area experts.However,there are still several limitations in cross evaluation.One is the possible diversification of cross efficiencies.The second is the losing association with the weights by averaging among the cross efficiencies,which means that this method cannot clearly supply the weights to help decision maker improve his performance.The third problem lies in the reality that average cross-efficiency measure is not good because it is not Pareto solution,and all the DMUs have no power to accept the results.These problems need a game issue to deal with,and we should combine game theory and cross efficiency evaluation method.The main contributions of this thesis are as follows:(1) Studies on the non-uniqueness of cross efficiency.The optimal solutions of traditional DEA models are always not unique,so the generated cross efficiency is not unique either,which possibly reduces the usefulness of cross efficiency evaluation method.There are no enough remedies to eliminate this drawback,and the existed approaches also have many drawbacks,for example,the proposed models are nonlinear,the problem of how to choose the appropriate model among various models,and so on.In this thesis,three different approaches are proposed to eliminate this drawback and improve the existed approaches.The proposed approaches are all illustrated by numerical examples,and the result are also analyzed and compared. Through exact proves and detailed numerical illustrations,we can find the proposed approaches in this thesis can commendably solve the problem of nonuniqueness,and greatly improve the existed approaches.The relevant results of this research have been published in International Journal of Production Economics(2008,113(2): 1025-1030),Expert Systems with Applications(2008, doi:10.1016/j.eswa.2008.05.042) and Journal of Systems Engineering and Electronics(2008,30(10):1890-1894).(2) Study on game cross efficiency model.After the examination of the cross efficiency evaluation method,the major drawbacks are illustrated,such as the non-uniqueness of cross efficiency,the results of the cross efficiency can not be accepted by all DMUs and so on.This thesis combines the non-cooperative game theory and cross efficiency evaluation method in a creative way,and the concept of game cross efficiency is proposed.In the game cross efficiency model proposed, each DMU is viewed as a player that seeks to maximize its own efficiency under the condition that the cross efficiency of each of the other DMUs does not deteriorate. An algorithm for deriving the game cross efficiency scores is presented and we show that the game cross efficiency scores constitute a Nash equilibrium point.The results of this research have been published in Operations Research(2008,56(5): 1278-1288),which is the flagship journal in the profession of management science and operations research.To our knowledge,it is one of the few researches by Chinese scholars of management published in this journal in the last more than fifty years since it was founded.The results of this research have been attracted by lots of researchers before its publication:Wade D.Cook and Larry M.Seiford,two famous DEA experts in the world,introduced this result detailedly in their review of DEA literatures in memory of DEA's thirty years(Wade D.Cook and Larry M.Seiford (2009).Data envelopment analysis(DEA)-Thirty years on.European Journal of Operational Research 192(1),1-17).The Development of Subject of Management Science and Engineering in 2007-2008 of China,presided by China Association for Science and Technology and edited by Chinese Society of Optimization,Overall Planning and Economical Mathematics,reported this result as the research overlight and breakthrough in the field of management science and engineering.(3) Study on the cross efficiency evaluation method via cooperative game theory.This thesis combines cooperative game theory with cross efficiency evaluation method.To the shortcomings of CCR model for self-appraisal and cross efficiency evaluation model for peer-appraisal in the evaluation,Nash Bargaining game model is proposed to yield a bargaining equilibrium solution between CCR model and cross efficiency evaluation model,and the properties of the bargaining equilibrium solution are also discussed.Then,aiming to the flaws when the ultimate average cross efficiency is used to evaluate DMUs,we eliminate the assumption of average and combine the theory of cooperative games,the DMUs under evaluation are considered as the players of the game,and we use the characteristic function value of coalitions to compute the Shapley value and nucleolus to determine the final weights of every DMU in the cooperative games and the ultimate cross efficiency. The results of this research have been accepted for publication by Expert Systems with Applications(2009,36(1):872-876),(2008,doi:10.1016/j.eswa.2008.05.001), Systems Engineering Theory and Practice(2008,28(5):92-97) and Journal of Systems Engineering.(4) Study on the efficiency of participants at the Summer Olympics.As for the ranking of countries at the Summer Olympics based on the medals won,there is no recognized method.DEA,as an effective tool for evaluating the DMUs with multiple inputs and multiple outputs,has been widely used in the Olympic ranking. This thesis uses the cross efficiency evaluation method and game cross efficiency evaluation method to do detailed researches on the efficiency evaluation of the participants in the Summer Olympic Games,the main content includes:(â…°) Study on the performance of countries at the Summer Olympics using cross efficiency evaluation method.This thesis firstly uses the cross efficiency evaluation method with weight restrictions to measure the performances of the nations participating in the last six Summer Olympic Games.The results of cross efficiency for the last six Summer Olympic Games are analyzed for the performance of each country.Also cluster analysis technique is used to select the more appropriate targets for poorly performing countries to use as benchmarks.The results of this research have been accepted for publication by European Journal of Operational Research (2008,doi:10.1016/j.ejor.2008.06.030) and International Journal of Enterprise Network Management(2008,2(4):377-392).(â…±) Study on the performance of countries at the Summer Olympics using game cross efficiency model.This thesis presents a new and modified DEA game cross efficiency model,which is also valid even if the returns to scales are considered.This modification can also eliminate a great drawback that cross efficiency may be negative in the case of variable returns to scales.Considering the competitions among participants in Olympic Games,the modified game cross efficiency model is applied to the last six Summer Olympic Games.We extend the results of existed literatures to incorporate competition among the DMUs.Because game cross efficiency is unique and constitutes a Nash equilibrium point,the results will be more reliable and beneficial to decision makers. The results of this research have been accepted for publication by Omega:The International Journal of Management Science(2009,37(4):909-918).Finally,in addition to the above achievements,the thesis discusses some challenging topics which deserve further research in future.