The cross-e ciency ranking method is a well-known method in DEA which is frequently used under the constant returns to scale assumption; while various applications exist based on the Variable returns to scale (VRS). This is due to the presence of negative input-oriented VRS cross-e ciencies. In this paper, each cross-e ciency is replaced by an equivalent distance measure as ine ciency measure. Then, the cross-ine ciency method is developed under the VRS assumption.

Data envelopment analysis (DEA) is a non-parametric approach for performance analysis of decision making units (DMUs) which uses a set of inputs to produce a set of outputs without the need to consider internal operations of each unit. In recent years, there have been various studies dealt with two-stage production systems, i.e. systems which consume some inputs in their first stage to produce some intermediate outputs which are used as the inputs of the second stage in producing final outputs. One of these researches done by Kao and Hwang (2008) gives a decomposition of system efficiency score based on the efficiency of its sub-processes in the case of constant returns to scale (CRS) technology. This paper presents an extension of this approach for the technologies with Variable returns to scale (VRS) and explains the results.

We suggest a method for finding the non-dominated points of the production possibility set (PPS) with Variable returns to scale (VRS) technology in data envelopment analysis (DEA). We present a multiobjective linear programming (MOLP) problem whose feasible region is the same as the PPS under Variable returns to scale for generating non-dominated points. We demonstrate that Pareto solutions of the MOLP produce efficient units in DEA, and vice versa. We solve the MOLP problem by using a finite number of weights which are extreme rays of the cone generated by the efficient solutions. We obtain new efficient points by changing weights, and thus the efficient solutions set are produced.

This paper addresses issue of sensitivity of efficiency classification of Variable returns to scale (VRS) technology for enhancing the credibility of data envelopment analysis (DEA) results in practical applications when an additional decision making unit (DMU) needs to be added to the set being considered. It also develops a structured approach to assisting practitioners in making an appropriate selection of variation range for inputs and outputs of additional DMU so that this DMU be efficient and the efficiency classification of VRS technology remains unchanged. This stability region is simply specified by the concept of defining hyper planes of production possibility set of VRS technology and the corresponding halfspaces. Furthermore, this study determines a stability region for the additional DMU within which, in addition to efficiency classification, the efficiency score of a specific inefficient DMU is preserved and also using a simulation method, a region in which some specific efficient DMUs become inefficient is provided.