Document Type : Research Article
Authors
1 Khozestan
2 University of Zabol
Abstract
Introduction: Several techniques are used to evaluate decision making units in DMUs with a restricted multiplier. DEA is recognized as a methodology widely used to evaluate the relative efficiency of a set of decision-making units (DMUs) involved in a production process. This approach assumes that the input and output data of the different decision making units (DMUs) are measured with precision. Although DEA is a powerful tool to use measure efficiency, there are some restrictions that need to be considered. One important restriction involves the sensitivity of DEA to the specific data under analysis. In this paper, the linear robust optimization framework of Bertsimas and Sim is used to concentrate on the DEA with uncertain data to determine the efficiency of irrigated wheat farms in Neyshabur County.
Materials and Methods: This paper proposes a linear robust data envelopment analysis (LRDEA) model using imprecise data represented by an uncertainty set. The method is based on the robust optimization approach of Bertsimas and Sim to seek maximization of efficiency under uncertainty (as does the original DEA model). In this approach, it is possible to vary the degree of conservatism to allow for a decision maker to understand the tradeoff between a constraint’s protection and its efficiency. The method incorporates the degree of conservatism in the maximum probability bound for constraint violation. The most significant uncertainties for a DEA model are input and output data that arise from errors. Application of the proposed model (LRDEA) to the case study (Neishabour district irrigated wheat farms) demonstrates the reliability and flexibility of the model. Monte Carlo simulation was implemented to examine the quality of the LRDEA model 100 random numbers were generated for each input and output of DMUs.
Results and Discussion: In this section, a case study of Neishabour county irrigated wheat farms is presented to illustrate the use of the methodology in this proposal, which consists of 95 DMUs, one input and five outputs. For the input and output data uncertainty, ten given maximums of a constraint’s violation probability were considered with respect to nominal values: 10%, 20%, up to 100% (i.e. we used Γ = 0.10, 0.20, up to1.00). The results show that the Gamma value decreases as the probability of constraint violation increases. The LRDEA model result shows how efficiency declines as the level of conservatism of the solution increases, that is, as the constraint violation probability decreases. According to the method, if all Gammas equal 0, then robust and original DEA models are the same. The most of the difference between the mean of optimal and actual amount of inputs is related to the two inputs of pesticide and cultivation land in both of the DEA and RDEA models. Accordingly, holding participatory extension classes to train farmers to increase yield and optimal use of existing agricultural land with a cooperative of efficient farmers is recommended. Also, the extinction of integrated pest management (IPM) to increasing non-optimal use of pesticide in the study area is proposed. Monte Carlo simulation was implemented to examine the quality of the LRDEA model 100 random numbers were generated for each input and output of DMUs. In the simulation violation probabilities ranging from 0.1 to 1.0 (at a constant the level of ε), percentages of average conformity are quite high. . However, it declines very rapidly as P approaches 0.7.
Conclusions: Evaluating the performance of many activities by a traditional DEA approach requires a precise input and output data. However, input and output data in real-world problems are often imprecise or vague. To deal with imprecise data, this study uses a robust optimization approach as a way to quantify vague data in DEA models. It is shown that the Bertsimas and Sim approach can be a useful tool in DEA models without introducing additional complexity into the problem (we called linear robust data envelopment analysis (LRDEA)). A case study of Neishabour county irrigated wheat farms is presented to illustrate the reliability and flexibility of the proposed model. The problem was solved for a range of given uncertainty and constraint violation probability levels using the GAMS software. This example suggests that our approach identifies the tradeoff between levels of conservatism and efficiency. As a result, efficiency decreases as the constraint violation probability increased. Additionally the LRDEA approach provides both a deterministic guarantee about the efficiency level of the model, as well as a probabilistic guarantee that is valid for all symmetric distributions.
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