A Review of Ranking Models in Data Envelopment Analysis <s> (15) <s> In this paper, the cross efficiency evaluation method, regarded as a DEA extension tool, is firstly reviewed for its utilization in identifying the Decision Making Unit (DMU) with the best practice and ranking the DMUs by their respective cross-efficiency scores. However, we then point out that the main drawback of the method lies in non-uniqueness of cross-efficiency scores resulted from the presence of alternate optima in traditional DEA models, obviously making it become less effective. Aiming at the research gap, a weight-balanced DEA model is proposed to lessen large differences in weighted data (weighted inputs and weighted outputs) and to effectively reduce the number of zero weights for inputs and outputs. Finally, we use two examples of the literature to illustrate the performance of this approach and discuss some issues of interest regarding the choosing of weights in cross-efficiency evaluations. <s> BIB001 </s> A Review of Ranking Models in Data Envelopment Analysis <s> (15) <s> Abstract Data Envelopment Analysis (DEA) is a mathematical model that evaluates the relative efficiency of Decision Making Units (DMUs) with multiple input and output. In some applications of DEA, ranking of the DMUs are important. For this purpose, a number of approaches have been introduced. Among them is the cross-efficiency method. The method utilizes the result of the cross-efficiency matrix and averages the cross-efficiency scores of each DMU. Ranking is then performed based on the average efficiency scores. In this paper, we proposed a new way of handling the information from the cross-efficiency matrix. Based on the notion that the ranking order is more important than individual efficiency score, the cross-efficiency matrix is converted to a cross-ranking matrix . A cross-ranking matrix is basically a cross-efficiency matrix with the efficiency score of each element being replaced with the ranking order of that efficiency score with respect to the other efficiency scores in a column. By so doing, each DMU assume the role of a decision maker and how they voted or ranked the other DMUs are reflected in their respective column of the cross-ranking matrix. These votes are then aggregated using a preference aggregation method to determine the overall ranking of the DMUs. Comparison with an existing cross-efficiency method indicates a relatively better result through usage of the proposed method. <s> BIB002 </s> A Review of Ranking Models in Data Envelopment Analysis <s> (15) <s> Data envelopment analysis (DEA) has been developed as a method to evaluate efficiency of Decision Making Unit (DMU). In order to analyze DMU in detail, each DEA model is formulated as a mathematical programming problem utilizing the values of inputs and outputs of all DMUs as coefficients. Each DMU is evaluated by a different weight. Then, the efficiency score of each DMU is determined by using an advantageous weight for itself. In general, the efficiency score is obtained by selecting the most advantage weight. In some real cases, seeking the best ranking is sometimes more important than maximizing the efficiency score. In this paper, we propose a model called rank-based measure (RBM) to evaluate DMU from a different standpoint. We suggest a method to obtain a weight which gives the best ranking, and calculates a weight between maximizing the efficiency score and keeping the best ranking. In order to calculate an efficiency score and the best ranking, we repeatedly solve linear programming problems. Moreover, we apply RBM model to the cross efficiency evaluation. Furthermore, a numerical experiment is shown to compare the rankings and scores with traditional evaluations. <s> BIB003
As it is obvious, nonuniqueness of optimal weight may occur. Wu et al. BIB001 proposed a weight balanced DEA model to reduce differences in weights data and zero weights Therefore, the cross-efficiency score of DMU is the average of these cross-efficiencies: where As it is obvious, nonuniqueness of optimal weight may occur. Zerafat Angiz et al. BIB002 introduced a cross-efficiency matrix based on this idea that ranking order is much more significant than individual efficiency score. Thus, they have provided the following procedure. Step 1. Considering CCR model, calculate the efficiency score of all DMUs and consider * as the efficiency score of DMU . Step 2. Now the cross-efficiency matrix can be constructed by ( * ) × . Note that * is used as the diagonal elements of . Step 3. Convert the cross-efficiency matrix into a crossranking matrix as ( ) × , in which is the ranking order of * in column of matrix . Step 4. Construct the preference matrix as ( ) × considering matrix where is the number of time that DMU is placed in rank . Step 5. Construct matrix Ω as (̂) × in whicĥis calculated by summing the efficiency scores in matrix Z, corresponds to DMU , being placed in rank . Step 6. Obtain a common set of weight for final ranking of DMUs using the following modified method: wherêobtained from Step 5 and * is the optimal solution of the following model. Finally, the DMUs are ranked based on their * = ∑ =1 * ̂v alues. Washio and Yamada BIB003 discussed that in real cases finding the best ranking is more significant than acquiring the most advantage weight and maximizing the efficiency. Thus they presented a model called rank-based measure (RBM) for evaluating units from different viewpoint. Thus, they suggested a method for acquiring those weight resulted from the best ranking as long as calculating those weight that maximizes the efficiency score. Finally they applied the presented model to the cross-efficiency assessment.