Efficiency analyses are crucial to managerial competency for evaluating the degree to which resources are consumed in the production process of gaining desired services or products. Among the vast available literature on performance analysis, Data Envelopment Analysis (DEA) has become a popular and practical approach for assessing the relative efficiency of Decision-Making Units (DMUs) which employ multiple inputs to produce multiple outputs. However, in addition to inputs and outputs, some situations might include certain factors to simultaneously play the role of both inputs and outputs. Contrary to conventional DEA models which account for precise values for inputs, outputs and dual-role factors, we develop a methodology for quantitatively handling imprecision and uncertainty where a degree of imprecision is not trivial to be ignored in efficiency analysis. In this regard, we first construct a pair of interval DEA models based on the pessimistic and optimistic standpoints to measure the interval efficiencies where some or all observed inputs, outputs and dual-role factors are assumed to be characterized by interval measures. The optimal multipliers associated with the dual-role factors are then used to determine whether a factor is designated as an output, an input, or is in equilibrium even though the status of the dual-role factors may not be unique based upon the pessimistic and optimistic standpoints. To deal with the problem, we present a new model which integrates both pessimistic and optimistic models. The integrated model enables us to identify a unique status of each imprecise dual-role factor as well as to develop a structure for calculating an optimal reallocation model of each dual-role factor among the DMUs. As another method to investigate the role for dual-role factors, we introduce a fuzzy decision-making model which evaluates all DMUs simultaneously. We finally present an application to a data set of 20 banks to showcase the applicability and efficacy of the proposed procedures and algorithm.
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