Combinatorial Efficiency Evaluation: The Knapsack Problem in Data Envelopment Analysis

The traditional data envelopment analysis (DEA) literatures generally concentrated on the efficiency evaluation of single decision making unit (DMU). However, in many practical problems, the decision makers are required to choose a number of DMUs instead of a single one from the DMUs set. Therefore, it is necessary to study the combinatorial efficiency evaluation problem which can be illustrated as a knapsack problem naturally. It is indicated that the basic model proposed by Cook and Green may have some drawbacks and a modified model, which is combined with the super efficiency model, is proposed in this paper. What is more, our proposed model is more persuasive to the decision makers because it is able to provide a unique best combination of DMUs. An adapted local search algorithm is developed as a solver of this problem. Finally, numerical examples are provided to examine the validity of our proposed model and the adapted local search algorithm.