Mathematical Modeling for Data Envelopment Analysis with Fuzzy Restrictions on Weights
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Data envelopment analysis (DEA) is a relative technical efficiency measurement tool, which uses operations research techniques to automatically calculate the weights assigned to the inputs and outputs of the production units being assessed. The actual input/output data values are then multiplied with the calculated weights to determine the efficiency scores. Recent variants of the DEA model impose upper and lower bounds on the weights to eliminate certain drawbacks associated with unrestricted weights. These variants are called weight restriction DEA models. Most weight restriction DEA models suffer from a drawback that the weight bound values are uncertain because they are determined based on either incomplete information or the subjective opinion of the decision-makers. Since the efficiency scores calculated by the DEA model are sensitive to the values of the bounds, the uncertainty of the bounds gets passed onto the efficiency scores. The uncertainty in the efficiency scores becomes unacceptable when we consider the fact that the DEA results are used for making important decisions like allocating funds and taking action against inefficient units. In order to minimize the effect of the uncertainty in bound values on the decision-making process, we propose to explicitly incorporate the uncertainty in the modeling process using the concepts of fuzzy set theory. Modeling the imprecision involves replacing the bound values by fuzzy numbers because fuzzy numbers can capture the intuitive conception of approximate numbers very well. Amongst the numerous types of weight restriction DEA models developed in the research, two are more commonly used in real-life applications compared to the others. Therefore, in this research, we focus on these two types of models for modeling the uncertainty in bound values. These are the absolute weight restriction DEA models and the Assurance Region (AR) DEA models. After developing the fuzzy models, we provide implementation roadmaps for illustrating the development and solution methodology of those models. We apply the fuzzy weight restriction models to the same data sets as those used by the corresponding crisp weight restriction models in the literature and compare the results using the two-sample paired t-test for means. We also use the fuzzy AR model developed in the research to measure the performance of a newspaper preprint insertion line.
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