An Approach to Data Envelopment Analysis Models with Fuzzy Inputs and Outputs by Application of Fuzzy Mathematical Programming

Data envelopment analysis (DEA) is a widely used mathematical programming approach for comparing the inputs and outputs by evaluating their relative efficiency. It represents a set of linear programming techniques and uses deterministic data (inputs and outputs), in stable conditions. An efficient integration of production and distribution plans into a unified framework is critical to achieving competitive advantage. In the conventional DEA, all the data assume the form of specific numerical values. However, the observed values of the input and output data in real-life problems are sometimes imprecise or vague. This paper finds efficiency measures with fuzzy inputs and outputs via proposed model. However, a real supply chain operates in a highly dynamic and uncertain environment. From the theory point of view, the objective of this study is to develop a simple and effective Fuzzy DEA model. This paper proposes an interactive evaluation process for measuring the relative efficiencies in fuzzy DEA with consideration of the preferences. it becomes hard for a productivity measurement expert to specify the amount of resources and the outputs as exact scalar numbers. The present paper, applies this method in the application to the oil exploration and production companies of Iran to measure and compare productivity performance of transformation processes when numerical data cannot be specified in exact terms. Realizations of the Solution of the Total Schrödinger Equation from the Link Between Relativity and Quantum Mechanics International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page | 12 The Data Envelopment Analysis (DEA) approach has been extensively applied in companies to measure the productive efficiency of production entities. Charnes et al. (1978) developed the DEA methodology for measuring relative efficiencies within a group of decision-making units (DMU’s) which utilize several inputs to produce a set of outputs. DEA constructs a nonparametric frontier over data points so that all observations lie on or below the frontier. A competing method for computing technical efficiency scores is the stochastic frontier approach (SFA) developed by Aigner et al. (1977) and Meeusen and van den Broeck (1977). DEA approach has been favored over the SFA for several seasons. First, it requires no assumption about the distribution of the underlying data and deviation from the estimated frontier is interpreted purely as inefficiency. Second, it does not require specification of a functional form for the frontier just as economic theory does not imply a particular functional form. Third, multiple inputs and outputs can be considered simultaneously, and fourth, inputs and outputs can be quantified using different units ofmeasurement. However, DEA requires detailed data about inputs and outputs. It is based on the assumption that all the input and output data are crisp, i.e., all the observations are considered as feasible with probability one, meaning no noise or measurement error is assumed (Simar 2007, Henderson and Zelenyuk 2007). This assumption may not be realistic in production oil companies where inputs and outputs of a decision making unit (DMU) are ever changing because of weather, seasons, operating states and so on (Guo and Tanaka 2001). The dominance of uncertainty in oil production has seen the flourish of studies of production under risk in oil economics (Just and Pope 2001). Factors used in production oil, such as labor, are sometimes difficult to measure in a precise manner. Input measures are often based on accounting data even though the definition of accounting costs differs from that of economic costs by excluding the opportunity cost (Kuosmanen et al., 2007). Producer data may also be available only in linguistic form such as “high yield”, “low yield”, “labor intensive” or “capital intensive. ” The convectional DEA approach is very sensitive to data measurement errors and changes in data, including outliers and missing data, can change the efficient frontier significantly. The DEA model is deterministic in nature, meaning that it does not account for statistical noise. A number of techniques to account for the deterministic nature have beensuggested in the literature, such as the techniques for detecting possible outliers (Cazals et al., 2002) and the stochastic programming approach (Cooper et al., 1998). Notably, Simar and Wilson (1998, 2000a) introduced bootstrapping into the DEA framework to allow for consistent estimation of the production frontier, corresponding efficiency scores, as well asstandard errors and confidence intervals. However, as observed by Kousmanen et al., (2007), the statistical properties and hypothesis tests suggested by Simar and Wilson (2000a, 2000b ) focus exclusively on the effect of the sampling of plants from the production possibilities set and, hence, the bootstrap approach does not allow for data errors of any kind. Therefore, there is need for a model that can adequately represent the stochastic nature of production data at a micro-level. This paper introduces fuzzy DEA, an approach advanced in the field of mathematical, to measure technical efficiency where data is imprecise. A group of 34 plants producers in iran is used to illustrate how to empirically compute fuzzy technical efficiency scores. The approach incorporates fuzzy set theory and the DEA mathematical programming techniques to compute technical efficiency indices under natural uncertainty inherent in the production processes. Unlike the convectional DEA model, with a fuzzy DEA model the decision maker can consider different degrees of measurement errors (possibilities) when estimating technical efficiency. Expert judgment expressed in linguistic variables can also be incorporated into the fuzzy DEA models (Guo and Tanaka, 2001). Fuzzy DEA models are rare in the economics or oil economics literature. A search for “fuzzy DEA” in the oil, AgEcon Search, and EconLit databases returned no items. Other applications of fuzzy set theory in oil economics include van Kooten et al (2001) who proposed a fuzzy contingent valuation approach to measure uncertain preferences for nonmarket goods. Duval and Featherstone (2002) compared compromise programming and fuzzy programming to a traditional meanvariance approach, and Krcmar and Van Kooten (2008) developed a compromise-fuzzy programming framework to analyze trade-offs of economic development prospects of forest dependent aboriginal communities. Realizations of the Solution of the Total Schrödinger Equation from the Link Between Relativity and Quantum Mechanics International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Page | 13 Analysis of technical efficiency using fuzzy DEA models is very useful to the decision maker and presents several advantages. First, uncertainty in measurement can be incorporated in DEA model at different degrees. Second, linguistic variables can beincorporated into the DEA model, e.g., expert judgment and environmental variables. Third, fuzzy DEA can be used to deal with missing data, and fourth, the decision makercan trace how the efficiency scores vary at different levels of uncertainty. In what follows, the convectional DEA model is presented followed by the basic concepts of fuzzy set theory and how those concepts are integrated into the DEA framework. Then, a literature review of numerical and empirical fuzzy DEA models is presented. The data set is discussed next followed by an application of the fuzzy DEA model to that data and discussion of the results. Then, the article concludes. 2. LITERATURE REVIEW Sengupta (1992) was the first to propose a mathematical programming approach where fuzziness was incorporated into DEA by allowing the objective function and the constraints to be fuzzy. The stochastic DEA model was to be solved using chanceconstrained programming and required the analyst to supply information on expected values of variables, the variancecovariance matrices of all variables, and the probability levels at which the feasibility constraints are to be satisfied. This method was difficult to implement due to those data requirements. Triantis and Girod (1998) suggested a mathematical programming approach that transforms fuzzy inputs and outputs into crisp data using membership function values. Efficiency scores would then be computed for different membership functions and averaged. Hougaard (1999) suggested an approach that allows the decision maker to include other sources of information such as expert opinion in technical efficiencies computation. Kao and Liu (2000a) suggested the use of -cut level sets to transform fuzzy data into interval data so that the fuzzy model becomes a family of convectional crisp DEA models. This approach was much similar to Guo and Tanaka (2001) who proposed a fuzzy CCR model in which fuzzy constraints, including fuzzy equalities and fuzzy inequalities, were all converted to crisp constraints by predefining different possibility levels. Lertworasirikul et al (2003) proposed a possibility approach in which fuzzy constraints are treated as fuzzy events and fuzzy DEA model is transformed into possibility DEA model by using possibility measures on fuzzy events. Saati (2002) adopted the -cut level approach, defined the fuzzy CCR model as a possibilityprogramming problem, and transformed it into an interval programming problem. This model could be solved as a crisp LP model and produce crisp efficiency score for each DMU and for each given -cut level. All the above authors used numerical examples to illustrate the application of the proposed fuzzy DEA approach. application of fuzzy DEA models is still in the infancy stage with only one application in oil economics. Hadi-Vencheh and Matin (2011) used an imprecise DEA (IDEA) model to compute the technical efficiency of 15 Iranian wheat producing provinces. Four inputs (acreage, water, wages and number of tra