Evaluation technique with a modified cross-efficiency in Data Envelopment Analysis

The cross-efficiency measure in DEA can be used as another way of addressing the issue of efficiency, such as characterizing decision-making units (DMUs) and ordering efficient DMUs. In this paper, a theoretical framework to use the cross-efficiency measure properly for evaluating DMUs is discussed and many useful techniques are proposed. First, a new efficiency measure for cross-evaluation, called a modified cross-efficiency (MCE), is proposed in order to solve both arbitrariness and inequality problems related to the cross-efficiency. MCE is defined as one efficiency score in the range, decided uniquely and fairly by using an appropriate criterion, after minimum and maximum values of cross-efficiency are computed. Second, seven ways of setting criteria to decide a specific and unique value are shown, such as (a) a minimum and a maximum value, (b) a satisfaction and an adjusted satisfaction value which degrees of DMUs are equally achieved, (c) a value computed using an average of multipliers as vertices of DEA solutions, (d) an arithmetic and a geometric mean. Moreover, seven techniques to evaluate DMUs using MCE are shown, such as (a) an arithmetic, a geometric, and a conditional weighted mean of specific (uniquely decided) MCE, (b) a satisfaction and an adjusted satisfaction value by which degrees of DMUs are equally achieved, (c) a minimum and a maximum value. The characteristics of MCE are discussed with numerical examples. Moreover, seven criteria to decide a unique score, and seven evaluating techniques are compared and discussed with the same examples. Various criteria and techniques related to MCE and the numerical results can help to evaluate DMUs appropriately in the sense of cross-evaluation.