مدل های برنامه ریزی خطی جایگزین عدد صحیح مختلط برای شناسایی کارآمدترین واحد تصمیم گیری در تحلیل پوششی داده ها

A mixed integer linear model for selecting the best decision making unit (DMU) in data envelopment analysis (DEA) has recently been proposed by Foroughi [Foroughi, A. A. (2011a). A new mixed integer linear model for selecting the best decision making units in data envelopment analysis. Computers and Industrial Engineering, 60(4), 550–554], which involves many unnecessary constraints and requires specifying an assurance region (AR) for input weights and output weights, respectively. Its selection of the best DMU is easy to be affected by outliers and may sometimes be incorrect. To avoid these drawbacks, this paper proposes three alternative mixed integer linear programming (MILP) models for identifying the most efficient DMU under different returns to scales, which contain only essential constraints and decision variables and are much simpler and more succinct than Foroughi’s. The proposed alternative MILP models can make full use of input and output information without the need of specifying any assurance regions for input and output weights to avoid zero weights, can make correct selections without being affected by outliers, and are of significant importance to the decision makers whose concerns are not DMU ranking, but the correct selection of the most efficient DMU. The potential applications of the proposed alternative MILP models and their effectiveness are illustrated with four numerical examples.

Data envelopment analysis (DEA), developed by Charnes, Cooper, and Rhodes (1978), is a useful yet practical methodology for efficiency assessment based on multiple inputs and multiple outputs. Through efficiency measurement, DEA can identify efficient and inefficient decision making units (DMUs). Efficient DMUs are those with efficiency of one, while inefficient DMUs are those with their efficiencies being less than one. Efficient DMUs can be further discriminated through a ranking approach like super-efficiency procedure (Andersen & Petersen, 1993) or cross-efficiency evaluation (Doyle and Green, 1994 and Sexton et al., 1986) to produce a full ranking or select the most efficient DMU. Sometimes, DMU ranking is not a main concern. For example, in DEA applications such as robot selection (Baker and Talluri, 1997 and Khouja, 1995), flexible manufacturing system (FMS) selection (Shang & Sueyoshi, 1995), and computer numerical control (CNC) machine selection (Sun, 2002), what the decision maker (DM) is concerned about is the selection of the most efficient DMU, rather than DMU ranking. So, in these situations, there is no need to measure the performance of every DMU and a very practical way is to develop a model to find the most efficient DMU directly without assessing the performances of the other DMUs. There have been several attempts in DEA literature to develop DEA models for finding the most efficient DMU. For example, Amin and Toloo (2007) proposed an integrated DEA model, which is in nature a mixed integer linear programming, to find the most efficient DMU in two steps. The first step is to find a maximum epsilon value for input and output weight variables and then the second step is to solve the integrated DEA model with the predetermined maximum epsilon value. This integrated model was later applied to find the most efficient association rule in data mining by Toloo, Sohrabi, and Nalchigar (2009) and extended to the situation of variable returns to scale (VRS) by Toloo and Nalchigar (2009). Amin (2009) found the integrated model of Amin and Toloo (2007) flawed, which might produce more than one efficient DMU, and he thus further suggested a mixed integer nonlinear model. This mixed integer nonlinear model, however, was found infeasible in some cases by Foroughi (2011a). To resolve the infeasibility problem of the mixed integer nonlinear model, Foroughi (2011a) proposed a mixed integer linear model to find the most efficient DMU from the perspective of super efficiency. This mixed integer linear model, however, is found involving too many unnecessary constraints and requiring the specification of assurance regions (ARs) for input and output weights to avoid zero weights. It is also found that this mixed integer linear model proposed by Foroughi (2011a) is easy to be affected by outliers, leading to its selection of the best DMU being incorrect. To eliminate these drawbacks and provide more methodological and model options for the decision maker, we propose in this paper three alternative mixed integer linear programming (MILP) models for finding the most efficient DMU under different returns to scales. In comparison with the existing DEA models for finding the most efficient DMU, the proposed alternative MILP models are more succinct, more practical and more reliable and contain only essential constraints and decision variables. More importantly, they can make the best use of input and output information without the need of specifying any assurance regions for input and output weights to avoid zero weights, can make correct selections without being affected by outliers, and are of significant importance to the decision makers whose concerns are not DMU ranking, but the correct selection of the most efficient DMU. Moreover, the proposed alternative MILP models also enhance the theory of DEA. The rest of the paper is organized as follows: Section 2 briefly reviews existing models for finding the most efficient DMU and points out their drawbacks. Section 3 proposes the alternative MILP models under different returns to scales. Section 4 examines four numerical examples to show the potential applications of the proposed alternative MILP models and their effectiveness in finding the most efficient DMU. The paper concludes in Section 5.

The identification of the most efficient DMU is sometimes the main concern of decision makers. This is particularly true in multiple criteria decision analysis. Although the traditional DEA methods can also do this, but require tedious efficiency calculations. In this paper, we have proposed three alternative mixed integer linear programming models for identifying the most efficient DMU under different returns to scales without the need of assessing the performance of every DMU. They can pick up the most efficient DMU in just one step and are thus more practical and more efficient. The proposed MILP models overcome the drawbacks of existing DEA models in finding the most efficient DMU such as infeasibility and making wrong selections in some situations, involving too many unnecessary constraints, and requiring the decision maker to specify assurance regions for input and output weights. The effectiveness and validity of the proposed MILP models have been tested and verified with numerical examples. Numerical examinations have clearly revealed that the use of the proposed alternative MILP models for identifying the most efficient DMU is not only feasible and effective, but also has some significant merits such as making the best use of input and output information without the need of specifying any assurance regions for input and output weights to avoid zero weights, making correct selections in just one step without being affected by outliers, and requiring the least computational effort. It is expected that the proposed MILP models can be applied in a variety of applications such as advanced manufacturing technology (AMT) selection, supplier or supply chain selection, multiple criteria decision analysis, and the like.