## Evaluation of the Relative Efficiency of Gas Stations by Data Envelopment Analysis

**Roxana Asayesh**^{1,}, **Zahra Faeghi Raad**^{2}

^{1}Young Researchers and Elite Club, Rasht Branch, Rasht, Iran

^{2}Department of management, Islamic Azad University, Rasht, Iran

### Abstract

Performance measurement is an important part of management science and operation research. Data Envelopment Analysis is a powerful analytical tool that has been successfully applied for measuring and benchmarking the relative performance in a wide variety of activities. Data Envelopment Analysis assists decision makers to distinguish efficient and inefficient decision making units in a homogeneous group. Super-efficiency Data Envelopment Analysis models can be used in ranking the performance of efficient decision making units. In this paper, Data Envelopment Analysis is employed to present a mathematical model for evaluating the relative efficiency of gas stations of Iranian Oil products Company. Banker, Charnes and Cooper model is applied to determine the relative efficiency of the stations. Super efficiency model of Andersen and Petersen and Slack Based Measure of Super efficiency ranking method are used to determine the most efficient unit.

**Keywords:** data envelopment analysis, decision making unit, ranking, super efficiency, input/output

*International Journal of Data Envelopment Analysis and *Operations Research**, 2014 1 (1),
pp 12-15.

DOI: 10.12691/ijdeaor-1-1-2

Received September 26, 2013; Revised January 30, 2014; Accepted February 07, 2014

**Copyright:**© 2014 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- Asayesh, Roxana, and Zahra Faeghi Raad. "Evaluation of the Relative Efficiency of Gas Stations by Data Envelopment Analysis."
*International Journal of Data Envelopment Analysis and *Operations Research**1.1 (2014): 12-15.

- Asayesh, R. , & Raad, Z. F. (2014). Evaluation of the Relative Efficiency of Gas Stations by Data Envelopment Analysis.
*International Journal of Data Envelopment Analysis and *Operations Research**,*1*(1), 12-15.

- Asayesh, Roxana, and Zahra Faeghi Raad. "Evaluation of the Relative Efficiency of Gas Stations by Data Envelopment Analysis."
*International Journal of Data Envelopment Analysis and *Operations Research**1, no. 1 (2014): 12-15.

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### 1. Introduction

Today, energy and fuel are prominent elements in the progress of the industrialized countries. In Iran, because of having a lot of oil sources, oil and petroleum products play important and strategic role in the economy of the country. Gas stations are directly involved in distributing petroleum products. The main function of these stations is presenting desirable and high-grade products and services to consumers. So evaluating their performance of them is significant. In NIOPDC (National Iranian Oil Products Distribution Company), gas stations are ranked every six months (twice a year) based on some specific indexes. Most of these indexes are qualitative and they pay attention to the beauty and appearance of the stations. Measurable and quantitative indexes are used less. This method is required to spend a long time and eventually the result is not satisfactory. In this paper, Data Envelopment Analysis (DEA) is employed to present a mathematical model as a precise and assured method for measuring the performance of the stations and also ranking them. In This systematic and comprehensive approach, every gas station is considered as a system with specified and quantitative inputs and outputs and then their efficiencies will be evaluated.

DEA is a well established methodology used to evaluate the relative efficiency of a set of comparable entities called decision making units (DMUs) with multiple inputs and outputs by some specific mathematical programming models ^{[1, 2]}. DEA was introduced in 1978 when Charnes et al. ^{[3]} (CCR approach) demonstrated how to change a fractional linear measurement of efficiency into a linear programming format. Since the first DEA model developed, many other DEA models and applications have been developed and extended (see ^{[4, 5, 6]}). In energy and environmental studies, DEA has been widely applied to estimate the technical efficiency of energy industries ^{[7, 8]}, assessing energy efficiencies of different organizations ^{[9, 10]} and measuring ecological efficiency ^{[11, 12]}. DEA can be used to optimize the performance measure of each DMU. It calculates a maximal performance measure for each DMU relative to all DMUs in the firms under observation ^{[13]}. Assessment of bank branch performance ^{[14]}, examining bank efficiency ^{[15]}, measuring the efficiency of higher education institutions ^{[16]}, solving facility layout design (FLD) problem ^{[17]} and measuring the efficiency of organizational investments in information technology ^{[18]} are examples of using DEA in various areas.

Data Envelopment Analysis assists decision makers to distinguish efficient and inefficient decision making units in a homogeneous group. Standard DEA models cannot provide more information about efficient units. Super-efficiency DEA models can be used in ranking the performance of efficient DMUs and overcome this obstacle ^{[19]}. Super-efficiency DEA model is obtained when a DMU under evaluation is excluded from the reference set of the original DEA model. This model was developed by Banker et al. ^{[20]} and Andersen and Petersen ^{[21]}.

The rest of the paper is organized as follows: Section 2 describes the DEA methodology. Section 3 points out the application of DEA in evaluating 26 gas stations of oil company in two northern cities of Iran. Section 4 contains the conclusion.

### 2. DEA Methodology

DEA is based on a linear programming. This method measures the relative efficiency of operational units with multiple inputs and outputs. The principal advantage of the DEA technique is that it does not require the specification of a particular functional form for the technology. This non-parametric approach solves a linear programming (LP) formulation per DMU and the weights assigned to each DMU are the results of the corresponding LP. The original model developed by Charnes, Cooper and Rhodes (CCR model) was applicable when characterized by constant returns to scale(CRS). Imperfect competition may cause a DMU not to operate at optimal scale. Banker, Charnes and Cooper (BCC model, 1984) extended the CCR model to account for technologies that show variable returns to scale(VRS). The technical efficiency score (in both CRS and VRS models) equal one implies full efficiency. On the other hand, if the score is less than one it indicates technical inefficiency.

Suppose that there are DMUs, DMU_{j} : , and the performance of each DMU is characterized by a production process of inputs () to produce outputs (). Relative efficiency is defined as the ratio of weighted sum of outputs to the weighted sum of inputs. The efficiency measure for DMU_{o} is defined as

(1) |

Where the weights and are non-negative.

The efficiency of a specific DMU_{0}_{ }Can be evaluated by the BCC model of DEA which is presented in multiplier form as follows:

(2) |

The above formulations assume that All variables in (2) are also constrained to be non-negative except for which may be positive, negative or zero with consequences that make it possible to use optimal values of this variable to identify RTS. The term in the constraints of (2) is not a real number. It is, instead, a non-Archimedean infinitesimal which is smaller than any positive real number. The entire frontier DMUs (efficient DMUs) has . In order to discriminate the performance of efficient DMUs, Andersen and Petersen ^{[21]} developed a procedure for ranking efficient units. Their methodology enables an extreme efficient unit o* *to achieve an efficiency score greater than one by removing the constraint corresponding to DMU_{o}* *in (2) as shown in model (3):

(3) |

Let the optimal objective value to (3) be *φ*_{0}. For an efficient *DMU*_{o}*, φ*_{0}* *is not less than unity and this value indicates *super-efficiency *of *DMU*_{o}.

Tone ^{[24]} has defined the slack based measure of super efficiency of *DMU*_{o}* *as the optimal objective function value *δo *of the following program:

(4) |

*δ*_{o}* *is a weighted L_{1} distance from to the production possibility set spanned by

### 3. Application of DEA in Gas Stations

In this section**, **DEA method is applied to evaluate the efficiency of 26 gas stations of two cities in the north of Iran. Data of the model have been derived from available documents in NIOPDC (National Iranian Oil Products Distribution Company). Seven variables from the data set as inputs and outputs have been used. Inputs include capacity of the tanks ()(liter), number of nozzles (), number of staff () and area ()(). The output variables are sold-out products ()(this money variable is stated as current Iranian million Rials), Automatic power generator () and Automated teller machine (ATM)(). The chosen input-output data used in the application are available over first and second periods of solar year, 1388 and they are displayed in Table 1. In this table, is the variable sold-out products for the first period of the year, is the variable sold-out products for the second period of the year, number 1 for two variables and shows the existence of the technology and number 0 shows non-existence. The problem is solved by using a BCC model and the super efficiency models of Andersen and Petersen ^{[22]}, and SBM. The results are reported in Table 2 and Table 3.

In Table 2 and Table 3, the 2nd and 3rd columns report the optimal value to models (2) and (3). The BCC model indicates that 7 stations #4, #5, #6, #15, #17, #19, and #26 are full efficient in the first period and 11 stations #2, #4, #5, #6, #9, #12, #15, #17, #19, #24, and #26 are full efficient in the second period (see column 2 in Table 2 and Table 3). The forth column of each Table 2 and Table 3 reports the super-SBM measure of efficiency defined in (4). By the super efficiencies of the stations, in the 1st period, station #19 is the top-ranked station and the other 6 stations are ranked as 6> 26> 4> 5> 17> 15 and in the 2nd period, station #19 is the top-ranked station and the other 10 stations are ranked as 6> 15> 9> 26> 4> 5> 12> 2> 24>17. It is to be noted that based on the results reported in the third column in model (3) station #6 is the top-ranked followed by 4> 19> 15> 26> 5> 17 in Table 2 and station #9 is the top-ranked station in Table 3 followed by 19> 6> 15> 26> 5> 24> 4> 12> 2> 17. Consider a specific station, Say station #6. The super efficiency measures AP* *and SBM* *to this station are respectively 2.1733 and 1.9614. This station is the top-ranked station using the super efficiency model (3) proposed by Andersen and Petersen ^{[22]}, whereas the top-ranked station in SBM methodology is station #19.

### 4. Conclusion

There is a method in NIOPDC (National Oil Products Distribution Company) to evaluate the performance of gas stations and determine the efficient stations. This evaluation is performed every six month. In this paper, data envelopment analysis method has been applied to evaluate the relative efficiency of 26 gas stations of two Northern cities of Iran. Data of the model have been derived from available documents in NIOPDC. BCC model was used for evaluating the relative efficiency. In This approach, each gas station was considered as a system with specified and quantitative inputs and outputs. By using this method the efficient station and also inefficient stations have been identified. Super efficiency (AP-model) and slack based measure of super efficiency were used for ranking gas stations. In both first and second period, station #19 is the top-ranked station by SBM methodology whereas the top-ranked station in AP methodology is station #6 in the first period and station #9 in the second period. Both SBM and AP methodologies are applicable but as matter of fact, comparing with data in NIOPDC, SBM methodology is more accurate and reliable.

### Acknowledgments

The authors are grateful for comments and suggestions made by anonymous referees and to the editor Emma Taylor** **for ensuring a timely review process.

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