Optimal scale sizes in economic efficiency models with integer measures: a case study of foundry industry

Data envelopment analysis (DEA) is a measurement method for estimating the relative efficiency of decision-making units (DMUs) that can calculate economic (i.e., cost and revenue) efficiency levels of DMUs and can move economic activities toward the performance improvement. DEA also determines the optimal scale sizes (OSSs) of economic activities with real-valued measures in the right combination of scale and allocative efficiencies. Due to the presence of integer input–output measures in many applications, in this paper, alternative concepts of average-cost efficiency and average-revenue efficiency with integer measures are proposed. In fact, by considering the known prices of inputs (outputs), two-step models are introduced for numerically calculating the OSSs with integer inputs and outputs. In addition, the proposed methods are used for a twelve-period data set of a foundry company in Iran. The automotive industry and related industries, especially the foundry and parts manufacturing industries are among the industries that play an important role in the growth of a country's economy. Therefore, the provision of an appropriate scale to ensure their economic efficiency (cost and revenue) is necessary. The results of the investigation show that the proposed approach is practical for estimating OSSs in terms of minimizing inputs (maximizing outputs) of companies with integer input–output values.