A two-stage approach for modelling undesirable outputs in DEA with production trade-offs: A case on chain stores

Information about production trade-offs between inputs and outputs can be included in data envelopment analysis (DEA) models. In production processes, undesirable outputs are produced simultaneously with desirable outputs. We propose the production possibility set (PPS) with production trade-offs in the presence of undesirable outputs. This paper presents a two-stage process for measuring the efficiency of production units in the presence of undesirable outputs based on DEA with production trade-offs. In the first stage, the radial targets of the decision-making unit (DMU) under evaluation is calculated. In the second stage, we calculate the maximum amount of inefficiency slack corresponding to the components of inputs, desired outputs, and undesirable outputs. We prove that the targets obtained from the two-step process corresponding to inefficient DMUs are efficient units on the efficiency frontier of PPS. These targets have a minimum level of undesirable outputs. Also, by choosing the right direction in the presented models based on the directional distance function (DDF), we can obtain different efficient targets corresponding to each of the DMUs. We show that by changing the weight restrictions on the inputs and outputs, the efficiency score and the corresponding targets of the DMUs change. These weight restrictions are determined by the decision-maker (DM) in order to consider the importance of inputs and outputs. An application of the presented approach is presented to evaluate a set of chain stores, and at the end we present the results of the paper. ‎ ‎