Detecting violations of the monotonicity condition under the assumption of weak disposability in eco-DEA models

Abstract Data envelopment analysis (DEA) was originally developed to evaluate entities (firms, regions, countries, etc.) that carry out production processes in which multiple inputs are converted into multiple outputs and for which prices as a valuation system are (partially) unavailable. Subsequent developed DEA models also consider the undesirable outputs (“bads”) of production. The most prevalent approach that incorporates bads into these models is the assumption of their weak disposability. One drawback of this approach is the potential violation of the monotonicity condition, a violation that could lead to possible misclassifications of efficient and inefficient units and incorrect efficiency values. Unlike with other constraints in DEA, equations represent weak disposability. As is generally known, the shadow prices in the dual program can accept any sign and can also be negative. If the bads increase, the eco-efficiency value also increases, with negative associated shadow prices. This implausible phenomenon is examined, including the (economic) conditions that facilitate its occurrence; the data structure that depends on the relations between inputs, outputs, and bads; and its implications for the presentation of eco-DEA models and their outcomes. Furthermore, it is demonstrated that these models are related to the more-for-less paradox in linear programming in specific instances. Sufficient conditions for identifying units for which this phenomenon occurs are presented. This study further demonstrates that, under the weak disposability assumption, the efficiency frontier changes with the orientation of the DEA model, in contrast to the standard DEA models; this result is also applicable to detecting such units.