A ranking framework based on interval self and cross-efficiencies in a two-stage DEA system

The evaluation of the performance of a decision-making unit (DMU) can be measured by its own optimistic and pessimistic multipliers, leading to an interval self-efficiency score. While this concept has been thoroughly studied with regard to single-stage systems, there is still a gap when it is extended to two-stage tandem structures, which better correspond to a real-world scenario. In this paper, we argue that in this context, a meaningful ranking of the DMUs is obtained; this outcome simultaneously considers the optimistic and pessimistic viewpoints within the self-appraisal context, and the most favourable and unfavourable weight sets of each of the other DMUs in a peer-appraisal setting. We initially extend the optimistic-pessimistic Data Envelopment Analysis (DEA) models to the specifications of such a two-stage structure. The two opposing self-efficiency measures are merged to a combined self-efficiency measure via the geometric average. Under this framework, the DMUs are further evaluated in a peer setting via the interval cross-efficiency (CE). This methodological tool is applied to evaluate the target DMU in relation to the most favourable and unfavourable weight profiles of each of the other DMUs, while maintaining the combined self-efficiency measure. We, thus, determine an interval individual CE score for each DMU and flow. By treating the interval CE matrix as a multi-criteria decision making problem and by utilising several well-established approaches from the literature, we delineate its remaining elements; we show how these lead us to a meaningful ultimate ranking of the DMUs. A numerical example about the efficiency evaluation of ten bank branches in China illustrates the applicability of our modelling approaches.