Combining ranking scales and selecting variables in the DEA context: The case of industrial branches

Abstract

Data Envelopment Analysis (DEA) has been introduced by Charnes et al. (Charnes, A., Cooper, W. W. and Rhodes, E., Measuring the efficiency of decision making units. Eur. J. Oper. Res., 1978, 2, 429–444) based on the Farrell measurement of productive efficiency ( J. Royal Statist. Soc., 1957, A120, 253–290). Originally DEA was designed to evaluate the efficiency of decision making units in the public sector (e.g. schools, towns, hospitals and nations) based on their given multiple inputs and outputs which are not measured in unified units (e.g. money). Eventually DEA was used in business and industry (e.g. bank branches). The DEA merely classifies the units into two dichotomic groups, efficient and inefficient. The purpose of our paper is to fully rank the units from the most efficient to the least efficient within the DEA context. For this purpose we use here three recent ranking methods developed within the DEA framework. The multi-ranking approach is utilized for validating the ranks and forming new overall ranking by combining the ranks which statistically fit the DEA classification. In a way this approach bridges between the DEA frontiers approach and the statistical/econometric approach of averages. The motivating example here is the case of ranking Israeli Industrial Branches. In order to determine the appropriate labor variables (number of man hours, average wage, vs total labor cost) we run two versions of the model. In this paper we use various recent scale ranking methods in the DEA (Data Envelopment Analysis) context. Two methods are based on multivariate statistical analysis: canonical correlation analysis (CCA) and discriminant analysis of ratios (DR/DEA), while the third is based on the cross efficiency matrix (CE/DEA) derived from the DEA. This multirank approach is necessary for rank validation of the model. Their consistency and goodness of fit with the DEA are tested by various nonparametric statistical tests. Once we had validated the consistency among the ranking methods, we constructed a new overall rank combining all of them. Actually, given the DEA results, we here provide ranks that complement the DEA for a full ranking scale beyond the mere classification to two dichotomic groups. This new combined ranking method does not replace the DEA, but it adds a post-optimality analysis to the DEA results. In this paper, we combine the ranking approach with stochastic DEA: each approach is in the forefront of DEA. This is an attempt to bridge between the DEA frontier Pareto Optimum approach and the average approach used in econometrics. Furthermore, the quality of this bridge is tested statistically and thus depends on the data. We demonstrate this method for fully ranking the Industrial Branches in Israel. In order to delete unmeaningful input and output variables, and to increase the fitness between the DEA and the ranking, we utilize the canonical correlation analysis to select the meaningful variables. Furthermore, we run the ranking methods on two sets of variables to select the proper combination of variables which best represents labor.