Dominance relations and ranking of units by using interval number ordering with cross-efficiency intervals

Abstract

This paper proposes an approach to the cross-efficiency evaluation that considers all the optimal data envelopment analysis (DEA) weights of all the decision-making units (DMUs), thus avoiding the need to make a choice among them according to some alternative secondary goal. To be specific, we develop a couple of models that allow for all the possible weights of all the DMUs simultaneously and yield individual lower and upper bounds for the cross-efficiency scores of the different units. As a result, we have a cross-efficiency interval for the evaluation of each unit. Existing order relations for interval numbers are used to identify dominance relations among DMUs and derive a ranking of units based on the cross-efficiency intervals provided. The approach proposed may also be useful for assessing the stability of the cross-efficiency scores with respect to DEA weights that can be used for their calculation.

Appendix

Proof of proposition

Let be a point in the interval then for some λ∈[0, 1]. Let (v _d ^R, u _d ^R), d=1, …, n, be an optimal solution of (5), and (v _d ^L, u _d ^L), d=1,…,n, an optimal solution of (8). We assume, without loss of generality, both that and Thus, we have that and Consider the following vectors where and Then, for every j=1, …, n, we have

and it is also satisfied

Therefore, are optimal solutions of the CCR model for DMU _d , d=1, …, n, respectively. In addition,