Data envelopment analysis for bi-level systems with multiple followers

This study aims to evaluate the performance of a bi-level system including two kinds of players: a leader positioned at the upper level and multiple followers located at the lower level. The overall system tries to use inputs to produce outputs in a cost-efficient way. However, each player in the bi-level system controls different decision variables and has varying objectives. Based on this, a novel bi-level data envelopment analysis model with multiple followers is developed, in order to evaluate the performance of overall bi-level system as well as each player in detail. Then the extended Kuhn-Tucker conditions are used to solve the proposed model. An illustrative example is presented to validate the proposed approach by evaluating supply chains with a single plant and two distribution centers. A comparative analysis is conducted to demonstrate that the developed model has stronger discriminating power and can provide more managerial insights. Several conclusions can be reached from this study: the lower level subsystem is cost-efficient if and only if every follower is cost-efficient, and the lower level cost efficiency lies between the smallest cost efficiency and the largest cost efficiency among all followers; being cost-efficient at both the upper level and the lower level results in an achievement of overall cost efficiency in a bi-level structure, and the overall cost efficiency falls in between the upper level and the lower level cost efficiencies.