Description
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.
International Journal of Data Envelopment Analysis and *Operations Research*, 2014, Vol. 1, No. 1, 12-15
Available online athttp://pubs.sciepub.com/ijdeaor/1/1/2
©Science and Education Publishing
DOI:10.12691/ijdeaor-1-1-2
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
*Corresponding author: [email protected]
Received September 26, 2013; Revised January 30, 2014; Accepted February 07, 2014
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
Cite This Article: Roxana Asayesh, 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*, vol. 1, no. 1 (2014): 12-15. doi: 10.12691/ijdeaor-1-1-2.
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].
International Journal of Data Envelopment Analysis and *Operations Research* 13
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 n DMUs, DMU
j
: 1,..., j n = ,
and the performance of each DMU is characterized by a
production process of m inputs ( : 1,...,
ij
x i m = ) to
produce s outputs ( : 1,...,
rj
y r s = ). 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
1
,
s
r ro
r
o
m
i io
i
u y
e
v x
=
=
=
?
?
(1)
Where the weights
r
u and
i
v 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:
0 0 0
1
0
1 1
0
1
0
. . 0, 1,..., ,
1,
, , , .
s
r r
r
s m
r rj i ij
r i
m
i i
i
i r
Max u y u
s t u y v x u j n
v x
v u for all i r and u free in sign
?
? ?
=
= =
=
= ?
? ? ? =
=
? ?
?
? ?
?
(2)
The above formulations assume that , 0 , .
ij rj
x y i j ? ?
All variables in (2) are also constrained to be non-negative
except for
0
u 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 0 ? > 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
0
1 ? = . 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):
0 0 0
1
0
1 1
0
1
0
. . 0, 1,..., , 0.
1,
, , , .
s
r r
r
s m
r rj i ij
r i
m
i i
i
i r
Max u y u
s t u y v x u j n j
v x
v u for all i r and u free in sign
?
? ?
=
= =
=
= ?
? ? ? = ?
=
? ?
?
? ?
?
(3)
Let the optimal objective value to (3) be ?
0
. For an
efficient DMU
o
, ?
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 ?o of the following program:
1
0
1
1
0
1
0
1
1
. . , 1,... ,
, 1,... ,
, 1,... ,
0 , 1,... ,
0, 1
m i
i
o
s
r
r
o
n
i
j ij
j
j
n
j rj r
j
j
i
io
ro r
j
x
m x
Min
y
s y
s t x x i m
y y r s
x x i m
y y r s
j
?
?
?
?
=
=
=
?
=
?
=
? =
? =
? =
? ? =
? =
?
?
?
?
,... . n
(4)
?
o
is a weighted L
1
distance from
0 0
( , ) x y to the
production possibility set spanned by
0 0
( , ), x y
1,..., , 0. j n j = ?
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 (
1
x )(liter), number of nozzles (
2
x ),
number of staff (
3
x ) and area (
4
x )(
2
m ). The output
variables are sold-out products (
1
y )(this money variable
is stated as current Iranian million Rials), Automatic
14 International Journal of Data Envelopment Analysis and *Operations Research*
power generator (
2
y ) and Automated teller machine
(ATM)(
3
y ). 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,
1
1
y is the variable sold-out products for the first
period of the year,
2
1
y is the variable sold-out products for
the second period of the year, number 1 for two variables
2
y and
3
y 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.
Table 1. Input/output data of NIOPDC
Station
1
x
2
x
3
x
4
x
1
1
y
2
1
y
2
y
3
y
1 157000 6 8 2350 699913333 523717500 1 0
2 140000 7 9 1700 1006051250 817430833 1 1
3 61920 12 10 2400 1443842500 1161450000 0 1
4 106000 8 5 1000 833737500 583571250 1 1
5 225000 15 12 2600 1919315416 1521854166 1 0
6 185000 8 3 2000 11157430833 922810833 0 0
7 135000 12 11 1540 677804166 634358333 1 0
8 180000 10 9 2000 796336666 713984166 1 0
9 90000 6 5 1200 421283333 351310000 1 0
10 225000 21 12 1400 867883333 838262500 1 0
11 100000 6 5 500 520480833 442768333 0 0
12 187000 13 11 1348 1093674166 950575000 0 1
13 225000 10 8 1374 892810000 750239166 1 1
14 240000 12 6 1500 908190833 844565833 0 0
15 165000 14 7 2150 2023383333 1913783333 1 1
16 84000 3 4 1400 381770833 289864166 1 0
17 90000 32 12 1270 1014266666 490700000 1 1
18 225000 12 11 2300 1149968333 960133333 0 1
19 48500 7 6 2050 1331900000 1098833333 1 1
20 84000 5 4 300 299524166 256952500 1 0
21 180000 7 6 2150 510232500 377775000 1 0
22 135000 8 3 3800 335839166 305242500 0 0
23 98000 8 5 1300 518133333 435833333 0 0
24 90000 8 8 1300 905000000 777733333 1 1
25 135000 7 6 1200 441935000 433509166 1 0
26 135000 13 11 1978 1071883333 1004216666 0 1
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.
Table 2. Results for the first period
Station E SE-AP SE-SBM
1 0.6195 - -
2 0.6725 - -
3 0.8613 - -
4 1 2.1614(2) 1.4957(4)
5 1 1.4158(6) 1.0631(5)
6 1 2.1733 (1) 1.9614(2)
7 0.6817 - -
8 0.7114 - -
9 0.4 - -
10 0.6668 - -
11 0.4 - -
12 0.9282 - -
13 0.802 - -
14 0.3444 - -
15 1 1.7143(4) 1.0091(7)
16 0.5 - -
17 1 1.0754(7) 1.0407(6)
18 0.6738 - -
19 1 2.0669(3) 2.4038(1)
20 0.7872 - -
21 0.4838 - -
22 0.6715 - -
23 0.4062 - -
24 0.9973 - -
25 0.453 - -
26 1 1.6209(5) 1.7013(3)
International Journal of Data Envelopment Analysis and *Operations Research* 15
Table 3. Results for the second period
Station E SE-AP SE-SBM
1 0.6311 - -
2 1 1.0444(10) 1.0387(9)
3 0.9278 - -
4 1 1.4975(8) 1.4957(6)
5 1 1.7660(10) 1.2695(7)
6 1 2.2156(3) 1.901(2)
7 0.6117 - -
8 0.9817 - -
9 1 2.6495(1) 1.1769(4)
10 0.2359 - -
11 0.4555 - -
12 1 1.0772(9) 1.1163(8)
13 0.8549 - -
14 0. 4389 - -
15 1 1.8124(4) 1.82(3)
16 0.5354 - -
17 1 1.0076(11) 1.0076(11)
18 0.6853 - -
19 1 2.4042(2) 2.2928 (1)
20 0.5281 - -
21 0.37 - -
22 0.7185 - -
23 0.4542 - -
24 1 1.5090(7) 1.0268(10)
25 0.3782 - -
26 1 1.7830(5) 1.7552(5)
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.
References
[1] Charnes, A., Cooper, W.W., Rhodes, E., Measuring the efficiency
of decision making units, European J ournal of Operational
Research, 2. 429-444. 1987.
[2] Seiford, L.M., Thrall, R.M., Recent developments in DEA:The
mathematical programming approach to frontier analysis, J ournal
of Econometrics, 46. 7-38. 1990.
[3] Cooper, W.W., Seiford, L.M., Tone, T., Introduction to Data
Envelopment Analysis and its Uses: with DEA-SolverSoftware and
References, Springer, New York, 2006.
[4] Amirteimoori, A. An extended shortest path problem: a DEA-
based approach Applied Mathematics Letters, 25 (11). 1839-1843.
November 2012.
[5] Amirteimoori, A., Emrouznejad, A., Flexible measure in
production process: a DEA-based approach. RARIO Operation
Research,45. 63-74. 2011.
[6] Cooper, W.W., Seiford, L.M., Zhu, J., Handbook of Data
Envelpment Analysis. Kluwer Academic Publishers, Norwell, MA,
2004.
[7] Thompson, R.G., Lee, E., Thrall, R.M., DEA/AR efficiency of US
independent oil/gas producers over time, Computers and
Operations Research, 19. 377-391. 1992.
[8] Hawdon, D., Efficiency, performance and regulation of the
international gas industry – a bootstrap DEA approach, Energy
Policy, 31. 1167-1178. 2003.
[9] Boyd, G.A., Pang, J .X., Estimating the linkage between energy
efficiency and productivity, Energy Policy, 28. 289-296. 2000.
[10] Ramanathan, R., A holistic approach to compare energy
efficiencies of different transport models, Energy Policy, 28. 743-
747. 2000.
[11] Dyckhoff, H., Allen, K., Measuring ecological efficiency with
data envelopment analysis (DEA), European J ournal of
Operational Research, 132. 312-325. 2001.
[12] Korhonen, P.J ., Luptacik, M., Eco-efficiency analysis of power
plants: An extension of data envelopment analysis, European
J ournal of Operational Research, 154. 437-446. 2004.
[13] J ian cheng guan, Richard C.M.Yam, Chiu kammok, Ning Ma., A
study of the relationship between competitivess and technological
innovation capability based on DEA models, European journal of
operational research, 170. 971-986. 2006.
[14] A.S. Camanho, R.G. Dyson, Cost efficiency measurement with
price uncertainty: a DEA application to bank branch assessments,
European J ournal of Operational Research, 161. 432-446. 2005.
[15] X. Chen, M. Skully, K. Brown, Banking efficiency in China:
application of DEA to pre- and post-deregulation eras: 1993-2000,
China Econ. Rev. 16. 229-245. 2005.
[16] J . J ohnes, Measuring teaching efficiency in higher education: an
application of data envelopment analysis to economics graduates
from UK Universities, European journal of operational research,
174. 443-456. 1993. 2006.
[17] T. Ertay, D. Ruan, U.R. Tuzkaya, Integrating data envelopment
analysis and analytic hierarchy for the facility layout design in
manufacturing systems, Inform. Sci, 176. 237-262. 2006.
[18] S.M. Shafer, T.A. Byrd, A framework for measuring the efficiency
of organizational investments in information technology using
data envelopment analysis, Omega, 28. 125-141. 2008.
[19] Amirteimoori, A., S. Kordrostami., A distance-based measure of
super efficiency in data envelopment analysis: an application to
gas companies, J Glob Optim, 54. 117-128. 2012.
[20] Banker, R.D., Das, S., Datar, S.M.: Analysis of cost variances for
management control in hospitals. Res. Gov, Nonprofit Account, 5.
268-291. 1989.
[21] Andersen, P., Petersen, N.C.:Aprocedure for ranking efficient
units in data envelopment analysis, Manag.Sci, 39, 1261–1264.
1993.
[22] R.D. Banker, A. Charnes, W.W. Cooper, Some models for
estimating technical and scale inefficiency in data envelopment
analysis, Manage. Sci. 30. 1078-1092. 2006.
[23] M. Toloo,S. Nalchigar, A new integrated DEA model for finding
most BCC-efficient DMU, the international journal of management
science. 2008.
[24] Tone, K.: A slack-based measure of super-efficiency in DEA,
European journal of operational research, 143. 32-41. 2002.
doc_900385952.pdf
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.
International Journal of Data Envelopment Analysis and *Operations Research*, 2014, Vol. 1, No. 1, 12-15
Available online athttp://pubs.sciepub.com/ijdeaor/1/1/2
©Science and Education Publishing
DOI:10.12691/ijdeaor-1-1-2
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
*Corresponding author: [email protected]
Received September 26, 2013; Revised January 30, 2014; Accepted February 07, 2014
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
Cite This Article: Roxana Asayesh, 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*, vol. 1, no. 1 (2014): 12-15. doi: 10.12691/ijdeaor-1-1-2.
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].
International Journal of Data Envelopment Analysis and *Operations Research* 13
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 n DMUs, DMU
j
: 1,..., j n = ,
and the performance of each DMU is characterized by a
production process of m inputs ( : 1,...,
ij
x i m = ) to
produce s outputs ( : 1,...,
rj
y r s = ). 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
1
,
s
r ro
r
o
m
i io
i
u y
e
v x
=
=
=
?
?
(1)
Where the weights
r
u and
i
v 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:
0 0 0
1
0
1 1
0
1
0
. . 0, 1,..., ,
1,
, , , .
s
r r
r
s m
r rj i ij
r i
m
i i
i
i r
Max u y u
s t u y v x u j n
v x
v u for all i r and u free in sign
?
? ?
=
= =
=
= ?
? ? ? =
=
? ?
?
? ?
?
(2)
The above formulations assume that , 0 , .
ij rj
x y i j ? ?
All variables in (2) are also constrained to be non-negative
except for
0
u 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 0 ? > 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
0
1 ? = . 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):
0 0 0
1
0
1 1
0
1
0
. . 0, 1,..., , 0.
1,
, , , .
s
r r
r
s m
r rj i ij
r i
m
i i
i
i r
Max u y u
s t u y v x u j n j
v x
v u for all i r and u free in sign
?
? ?
=
= =
=
= ?
? ? ? = ?
=
? ?
?
? ?
?
(3)
Let the optimal objective value to (3) be ?
0
. For an
efficient DMU
o
, ?
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 ?o of the following program:
1
0
1
1
0
1
0
1
1
. . , 1,... ,
, 1,... ,
, 1,... ,
0 , 1,... ,
0, 1
m i
i
o
s
r
r
o
n
i
j ij
j
j
n
j rj r
j
j
i
io
ro r
j
x
m x
Min
y
s y
s t x x i m
y y r s
x x i m
y y r s
j
?
?
?
?
=
=
=
?
=
?
=
? =
? =
? =
? ? =
? =
?
?
?
?
,... . n
(4)
?
o
is a weighted L
1
distance from
0 0
( , ) x y to the
production possibility set spanned by
0 0
( , ), x y
1,..., , 0. j n j = ?
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 (
1
x )(liter), number of nozzles (
2
x ),
number of staff (
3
x ) and area (
4
x )(
2
m ). The output
variables are sold-out products (
1
y )(this money variable
is stated as current Iranian million Rials), Automatic
14 International Journal of Data Envelopment Analysis and *Operations Research*
power generator (
2
y ) and Automated teller machine
(ATM)(
3
y ). 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,
1
1
y is the variable sold-out products for the first
period of the year,
2
1
y is the variable sold-out products for
the second period of the year, number 1 for two variables
2
y and
3
y 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.
Table 1. Input/output data of NIOPDC
Station
1
x
2
x
3
x
4
x
1
1
y
2
1
y
2
y
3
y
1 157000 6 8 2350 699913333 523717500 1 0
2 140000 7 9 1700 1006051250 817430833 1 1
3 61920 12 10 2400 1443842500 1161450000 0 1
4 106000 8 5 1000 833737500 583571250 1 1
5 225000 15 12 2600 1919315416 1521854166 1 0
6 185000 8 3 2000 11157430833 922810833 0 0
7 135000 12 11 1540 677804166 634358333 1 0
8 180000 10 9 2000 796336666 713984166 1 0
9 90000 6 5 1200 421283333 351310000 1 0
10 225000 21 12 1400 867883333 838262500 1 0
11 100000 6 5 500 520480833 442768333 0 0
12 187000 13 11 1348 1093674166 950575000 0 1
13 225000 10 8 1374 892810000 750239166 1 1
14 240000 12 6 1500 908190833 844565833 0 0
15 165000 14 7 2150 2023383333 1913783333 1 1
16 84000 3 4 1400 381770833 289864166 1 0
17 90000 32 12 1270 1014266666 490700000 1 1
18 225000 12 11 2300 1149968333 960133333 0 1
19 48500 7 6 2050 1331900000 1098833333 1 1
20 84000 5 4 300 299524166 256952500 1 0
21 180000 7 6 2150 510232500 377775000 1 0
22 135000 8 3 3800 335839166 305242500 0 0
23 98000 8 5 1300 518133333 435833333 0 0
24 90000 8 8 1300 905000000 777733333 1 1
25 135000 7 6 1200 441935000 433509166 1 0
26 135000 13 11 1978 1071883333 1004216666 0 1
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.
Table 2. Results for the first period
Station E SE-AP SE-SBM
1 0.6195 - -
2 0.6725 - -
3 0.8613 - -
4 1 2.1614(2) 1.4957(4)
5 1 1.4158(6) 1.0631(5)
6 1 2.1733 (1) 1.9614(2)
7 0.6817 - -
8 0.7114 - -
9 0.4 - -
10 0.6668 - -
11 0.4 - -
12 0.9282 - -
13 0.802 - -
14 0.3444 - -
15 1 1.7143(4) 1.0091(7)
16 0.5 - -
17 1 1.0754(7) 1.0407(6)
18 0.6738 - -
19 1 2.0669(3) 2.4038(1)
20 0.7872 - -
21 0.4838 - -
22 0.6715 - -
23 0.4062 - -
24 0.9973 - -
25 0.453 - -
26 1 1.6209(5) 1.7013(3)
International Journal of Data Envelopment Analysis and *Operations Research* 15
Table 3. Results for the second period
Station E SE-AP SE-SBM
1 0.6311 - -
2 1 1.0444(10) 1.0387(9)
3 0.9278 - -
4 1 1.4975(8) 1.4957(6)
5 1 1.7660(10) 1.2695(7)
6 1 2.2156(3) 1.901(2)
7 0.6117 - -
8 0.9817 - -
9 1 2.6495(1) 1.1769(4)
10 0.2359 - -
11 0.4555 - -
12 1 1.0772(9) 1.1163(8)
13 0.8549 - -
14 0. 4389 - -
15 1 1.8124(4) 1.82(3)
16 0.5354 - -
17 1 1.0076(11) 1.0076(11)
18 0.6853 - -
19 1 2.4042(2) 2.2928 (1)
20 0.5281 - -
21 0.37 - -
22 0.7185 - -
23 0.4542 - -
24 1 1.5090(7) 1.0268(10)
25 0.3782 - -
26 1 1.7830(5) 1.7552(5)
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.
References
[1] Charnes, A., Cooper, W.W., Rhodes, E., Measuring the efficiency
of decision making units, European J ournal of Operational
Research, 2. 429-444. 1987.
[2] Seiford, L.M., Thrall, R.M., Recent developments in DEA:The
mathematical programming approach to frontier analysis, J ournal
of Econometrics, 46. 7-38. 1990.
[3] Cooper, W.W., Seiford, L.M., Tone, T., Introduction to Data
Envelopment Analysis and its Uses: with DEA-SolverSoftware and
References, Springer, New York, 2006.
[4] Amirteimoori, A. An extended shortest path problem: a DEA-
based approach Applied Mathematics Letters, 25 (11). 1839-1843.
November 2012.
[5] Amirteimoori, A., Emrouznejad, A., Flexible measure in
production process: a DEA-based approach. RARIO Operation
Research,45. 63-74. 2011.
[6] Cooper, W.W., Seiford, L.M., Zhu, J., Handbook of Data
Envelpment Analysis. Kluwer Academic Publishers, Norwell, MA,
2004.
[7] Thompson, R.G., Lee, E., Thrall, R.M., DEA/AR efficiency of US
independent oil/gas producers over time, Computers and
Operations Research, 19. 377-391. 1992.
[8] Hawdon, D., Efficiency, performance and regulation of the
international gas industry – a bootstrap DEA approach, Energy
Policy, 31. 1167-1178. 2003.
[9] Boyd, G.A., Pang, J .X., Estimating the linkage between energy
efficiency and productivity, Energy Policy, 28. 289-296. 2000.
[10] Ramanathan, R., A holistic approach to compare energy
efficiencies of different transport models, Energy Policy, 28. 743-
747. 2000.
[11] Dyckhoff, H., Allen, K., Measuring ecological efficiency with
data envelopment analysis (DEA), European J ournal of
Operational Research, 132. 312-325. 2001.
[12] Korhonen, P.J ., Luptacik, M., Eco-efficiency analysis of power
plants: An extension of data envelopment analysis, European
J ournal of Operational Research, 154. 437-446. 2004.
[13] J ian cheng guan, Richard C.M.Yam, Chiu kammok, Ning Ma., A
study of the relationship between competitivess and technological
innovation capability based on DEA models, European journal of
operational research, 170. 971-986. 2006.
[14] A.S. Camanho, R.G. Dyson, Cost efficiency measurement with
price uncertainty: a DEA application to bank branch assessments,
European J ournal of Operational Research, 161. 432-446. 2005.
[15] X. Chen, M. Skully, K. Brown, Banking efficiency in China:
application of DEA to pre- and post-deregulation eras: 1993-2000,
China Econ. Rev. 16. 229-245. 2005.
[16] J . J ohnes, Measuring teaching efficiency in higher education: an
application of data envelopment analysis to economics graduates
from UK Universities, European journal of operational research,
174. 443-456. 1993. 2006.
[17] T. Ertay, D. Ruan, U.R. Tuzkaya, Integrating data envelopment
analysis and analytic hierarchy for the facility layout design in
manufacturing systems, Inform. Sci, 176. 237-262. 2006.
[18] S.M. Shafer, T.A. Byrd, A framework for measuring the efficiency
of organizational investments in information technology using
data envelopment analysis, Omega, 28. 125-141. 2008.
[19] Amirteimoori, A., S. Kordrostami., A distance-based measure of
super efficiency in data envelopment analysis: an application to
gas companies, J Glob Optim, 54. 117-128. 2012.
[20] Banker, R.D., Das, S., Datar, S.M.: Analysis of cost variances for
management control in hospitals. Res. Gov, Nonprofit Account, 5.
268-291. 1989.
[21] Andersen, P., Petersen, N.C.:Aprocedure for ranking efficient
units in data envelopment analysis, Manag.Sci, 39, 1261–1264.
1993.
[22] R.D. Banker, A. Charnes, W.W. Cooper, Some models for
estimating technical and scale inefficiency in data envelopment
analysis, Manage. Sci. 30. 1078-1092. 2006.
[23] M. Toloo,S. Nalchigar, A new integrated DEA model for finding
most BCC-efficient DMU, the international journal of management
science. 2008.
[24] Tone, K.: A slack-based measure of super-efficiency in DEA,
European journal of operational research, 143. 32-41. 2002.
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