Measurement of the worst practice of decision-making units: Incorporating both undesirable outputs and non-discretionary inputs into imprecise DEA

There are two difficulties in implementing an objective evaluation of the performance of decision-making units (DMUs). The first one is how to treat undesirable outputs jointly produced with the desirable outputs, and the second one is how to treat uncontrollable variables, which often capture the impact of the operating environment. Given difficulties in both model construction and data availability, very few published papers simultaneously consider the above two problems. The objective of the present paper is to propose a novel pair of data envelopment analysis (DEA) models for measurement of relative efficiencies of DMUs in the presence of non-discretionary factors, undesirable factors, and imprecise data. Compared to traditional DEA, the proposed DEA approach measures the efficiency of each DMU relative to the worst practice frontier, also called the input frontier, and is called the worst relative efficiency or pessimistic efficiency. The pair of proposed DEA models simultaneously takes into account the crisp data, ordinal preference information, and interval data, as well as undesirable factors and non-discretionary factors, for measurement of relative efficiencies of DMUs. The results of this study are not only useful for the performance evaluation method, but also have policy implications for industrial and academic researchers. A numeric example has been provided to illustrate the applicability of the DEA models