اوزان بهینه در مدل های تحلیل پوششی داده ها با محدودیت های وزن Optimal weights in DEA models with weight restrictions

۱٫ Introduction Data envelopment analysis (DEA) is a nonparametric approach to the assessment of efficiency and productivity of organizational units (Cooper, Seiford, & Tone, 2007; Thanassoulis, Portela, & Despic, ´ ۲۰۰۸). The latter are conventionally referred to as decision making units (DMUs). Standard DEA models are based on the assumption that the underlying production technology is characterized by either constant (CRS) or variable (VRS) returns to scale. Both CRS and VRS models can be stated as two mutually dual linear programs referred to as the envelopment and multiplier models. The optimal value of these two programs is interpreted as the input or output radial efficiency of DMUo under the assessment, depending on the orientation in which the models are solved (Banker, Charnes, & Cooper, 1984; Charnes, Cooper, & Rhodes, 1978). In particular, in the envelopment model, DMUo is benchmarked against the boundary of the CRS or VRS technology, and the radial efficiency of DMUo is interpreted as the utmost proportional improvement factor to its input or output vector possible in the technology. The multiplier models are stated in terms of variable input and output weights (multipliers). The CRS multiplier model can be shown to maximize the ratio of the total weighted output to the total weighted input (efficiency ratio) of DMUo, provided no such ratio across all observed DMUs can exceed the value of 1. The VRS multiplier model has an additional dual variable inter- pretable in terms of returns to scale and scale elasticity (Banker et al., 1984; Podinovski, Chambers, Atici, & Deineko, 2016; Podinovski & Førsund, 2010; Podinovski, Førsund, & Krivonozhko, 2009; Sahoo & Tone, 2015). As pointed by Charnes et al. (1978), the optimal input and output weights are the most favorable to DMUo and show it in the best light in comparison to all observed DMUs. ۱٫۱٫ Weight restrictions Weight restrictions usually represent value judgments incorporated in the form of additional constraints on the input and output weights in the multiplier model. These constraints reduce the flexibility of weights and typically improve the discrimination of the DEA model (see, e.g., Allen, Athanassopoulos, Dyson, & Thanassoulis, 1997; Cook & Zhu, 2008; Joro & Korhonen, 2015; Thanassoulis et al., 2008). The use of weight restrictions generally changes the interpretation of efficiency in both the envelopment and multiplier models. From the technology perspective, the incorporation of weight restrictions results in the expansion of the model of technology (Charnes, Cooper, Wei, & Huang, 1989; Halme & Korhonen, 2000; Roll, Cook, & Golany, 1991). Podinovski (2004a) shows that this expansion is caused by the dual terms in the envelopment model generated by weight restrictions, and that DMUo is projected on the boundary of the expanded technology. Therefore, DMUo is benchmarked against all units in the technology (including those generated by the weight restrictions), and not only against the observed units. The interpretation of efficiency in terms of the multiplier model with weight restrictions is somewhat less obvious and currently incomplete. This can be summarized as follows. If all weight restrictions are homogeneous and not linked (see Section 2 for a formal definition), the multiplier model correctly identifies the optimal weights (within the specified weight restrictions) that represent DMUo in the best light in comparison to all observed DMUs (Podinovski, 2001a). However, a problem with the interpretation arises if at least one weight restriction is nonhomogeneous or is linked. In this case the optimal weights do not generally represent DMUo in the best light in comparison to all observed DMUs. Consequently, the optimal value of the multiplier model with such weight restriction generally underestimates the relative efficiency of DMUo. Examples illustrating this point are given by Podinovski (1999, 2001a); Podinovski and Athanassopoulos (1998) and, recently, by Khalili, Camanho, Portela, and Alirezaee (2010). ۱٫۲٫ Contribution In this paper we show that, for any weight restrictions, the optimal weights of the multiplier model show DMUo in the best light in comparison to all DMUs in the expanded technology generated by the weight restrictions. This result is true if we search among all nonnegative input and output weights, or only among those that satisfy the weight restrictions. Our results also overcome the discrepancy between the interpretation of the envelopment and multiplier models with weight restrictions. Indeed, as pointed above, the envelopment model benchmarks DMUo against all DMUs in the technology expanded by the weight restrictions. However, the conventional interpretation of the multiplier model assumes that DMUo should be benchmarked against the observed DMUs only. As noted, this conventional assumption does not lead to a meaningful interpretation of some types of weight restrictions. Our results show that the multiplier model does exactly the same as the envelopment model—it benchmarks DMUo against all DMUs in the expanded technology, for all types of weight restrictions. From a practical perspective, this new interpretation can be used to justify the incorporation of any types of weight restrictions in the multiplier model, and explain the meaning of the resulting optimal weights and efficiency scores. This includes absolute weight bounds and linked weight restrictions, whose meaning has so far remained unclear.