In Search of Efficient Networks Using Bilevel Evolutionary Optimization

This study introduces a method for finding input and output values that result in efficiency for all decision making units (DMUs) in all stages of a production network. This method contrasts with the usual application of DEA for measurement of efficiency and comparison of decision making units. Two-stage data envelopment analysis (DEA) networks represent internal production processes, including inputs to each stage, intermediate variables that serve as outputs from one stage and inputs to another, and feedback among stages. A directed acyclic network is usually used for the measure of efficiency, but the presence of feedback requires an undirected network. The static network with feedback is important for analysis of recycling and re-use processes, but the measures in these cases are non-linear with multiple solutions. Here, a two-stage DEA network was defined as a bilevel optimization where the upper level maximized ParetoKoopmans efficiency while efficiency of each DMU in each stage was calculated in the lower level. In an application to published data from a two-stage network with feedback, hypervolumes were observed where every interior point was Pareto-Koopmans efficient. The Pareto-Koopmans hypervolume associated with the feedback network was modeled with a Bayesian network, and simulations from the model used to visualize efficient space. The use of Bayesian networks to generate virtual DMUs provided a computationally inexpensive method to explore the trade-offs among efficient networks to meet stakeholder preferences. Overall, this application of bilevel evolutionary optimization provides a novel method for solving nonlinear DEA problems with feedback