Evaluation techniques with a modified cross-efficiency in DEA

The cross-efficiency measure in DEA can be used as another way of addressing the issue of efficiency, such as characterizing 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. The 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-efficiencies are computed. Second, seven ways of setting criteria to decide a specific and a 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 the 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 which degrees of DMUs are equally achieved, (c) a minimum and a maximum value. The characteristics of the MCE is discussed with numerical examples. Moreover, seven criteria to decide a unique score, and seven evaluation techniques are compared and discussed with the same examples. Various criteria and techniques related to the MCE and the numerical results can help to evaluate DMUs appropriately in the sense of cross-evaluation.