Abstract
•Established models can evaluate hierarchical systems with Stackelberg relationships.•Multiple followers are considered at the lower level.•Cost efficiency DEA is integrating into bi-level programming framework.•Relationships among different cost efficiencies are revealed.•Shared and direct inputs and intermediate measures are considered.
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.