A method for ranking based on common weights and benchmark point with fuzzy data

International Journal of Advanced Information Science and Technology (IJAIST) ISSN: 2319:2682 Vol.21, No.21, January 2014

A method for ranking based on common weights and benchmark point with fuzzy data: an application for ranking university departments B. Rahmani Parchikolaei Department of Mathematics, Nour Branch Islamic Azad University, Nour, Iran.

Bijanrah40@gmail.com

Abstract-

The highest efficiency score is regarded as the common benchmark level for decision making units. Such cases can have more than one DMU with the highest score. This can happen in DEA for the evaluation of DMUs and in methods of common set of weights for the ranking of DMUs, which does not lead to complete ranking. As defined, an ideal DMU (IDMU) has the highest efficiency score. Therefore, IDMU can be regarded as benchmark for all DMUs. This taken into account, we present a linear programming model for obtaining the common set of weights (CSWS). Since the data are fuzzy, we obtain an upper bound and a lower bound base on the best and the worst evaluation respectively for each Îą- cut. This method with upper bound efficiency score of unit less than the highest score, and with lower bound efficiency score of unit more than the least score guarantees a complete ranking of DMUs. This mode is economical, too. A comparative example of data from university department and other methods is presented.

Keywords: DEA, Ideal decision making, common set of weight, benchmark, data fuzzy.

1.INTRODUCTION Data envelopment analysis (DEA) was first offered by charnes et al. in 1978. DEA is a power full tool for measurement of efficiency of congruent decision making units through mathematical programming.

DMUs are divided into efficient and inefficient groups.

In major models of DEA (CCR, BCC, SBM) the efficient units have efficient score of 1, and inefficient units have efficiency score of less than1. Unlike reality, DEA makes no distinction among units with efficiency

A. Payan Department of Mathematics, Zahedan Branch, Islamic Azad University, Zahedan, Iran. a.payan@iauzah.ac.ir score of 1. To resolve this problem, there are a number of distinction methods among efficient units known as ranking methods. Ranking methods are of two kinds.

One includes methods of ranking of vertex efficient DMUs only, offered Andersen and Petersen (1993) and known as super-efficiency method. The other includes methods for ranking of all DMUs; these are divided into three groups: Cross-efficiency methods such as work Doyle and Green, (1994), Multi-criteria decision making methods (MCDM) such as work Li and Reeves (1999), and Periodic DEA methods such as work Wang and Yang (2007); for a complete review see Adler et al.

(2002). Extension the MCDM approach has led to common set of weights (CSWs) methods. In this method, the efficiency of DMUs simultaneously with a fixed set of weights is measured. In this approach, the efficiency of DMUs is simultaneously measured by a fix set of weights. Liu and Peng (2008) presented a linear programming problem model for obtaining CSWs. Chiang et al. (2011) to obtain the CSWs introduced a linear model with separation vector.

Almost, all CSWs approaches consider number one as the highest common benchmark level for the data are definite in existing models of DEA. There are, however, many issues fuzzy in nature. The theory of fuzzy sets was developed to examine the truth values extending from perfectly wrong. The fuzzy sets algebra was introduced by Zadeh Lotfi (1965). The theory of fuzzy sets has turned into a method of determining uncertain quantities and data in DEA models. DEA fuzzy models were first introduced by Sengupta (1992), which shows an increasing interest and growing number of articles in the related literature. Fuzzy DEA models can demonstrate real-world problems more realistically than

1

International Journal of Advanced Information Science and Technology (IJAIST) ISSN: 2319:2682 Vol.21, No.21, January 2014 usual DEA models. DEA fuzzy models are considered fuzzy linear programming models. The application of fuzzy sets theory is classified into four categories: a) the tolerance approach b) The- level base approach c) the fuzzy ranking approach d) possibility approach. The main idea in tolerance approach (Sengupta, 1992) is that uncertainty in DEA models combines with tolerance level on breach of constraints. The method was improved by Kahraman and Tolga (1998). The α- cut method is used more in fuzzy DEA model. The main idea is to convert CCR model into parametric programs pairs in order to find the upper and lower bounds of α of membership functions of efficiency scores. Girod and Triantis (1999), for instance, presented fuzzy programming method based on Carlsson and Korhonen (1986).Kao and Liu (2000) converted fuzzy DEA model into pairs of mathematical parametric performance of decision making units through ranking method. Hatamimarbani and Saati (2009) develop BCC fuzzy model.

Jahanshahloo et al. (2007) suggested fuzzy L1-norm model with fuzzy trapezoidal data. The fuzzy ranking method has been used a lot in fuzzy data literature, as well. Obtaining fuzzy efficiency scores of decision making units by fuzzy linear programs is the main objective of this method, which requires ranking of fuzzy sets. This method was first presented by Guo and Tanaka (2001). Hosseinzadeh Lotfi and Mansouri (2008) regarded separation analysis DEA method as fuzzy data and changed fuzzy model into a definite model by linear ranking function. The possibility method, In DEA fuzzy models, coefficients can be considered fuzzy variables, and constraints be regarded as fuzzy occurrences. Lertworasirikul et al. (2002) and Lertworasirikul (2002) mentioned two methods to resolve ranking problem in fuzzy DEA models and named them probability and validity methods, respectively. Khdabakhshi et al (2009) offered two random and fuzzy collective models to determine the efficiency to scale in DEA. Saati and Memariani (2005) presented a method to determine the common weights in fuzzy DEA based on α-cut method with triangular fuzzy data. Tavakoli-mogaddam and Mahmoodi (2010) found a way to obtain common weights by fuzzy entropy. A thorough review of the fuzzy methods can be found in Hatami-marbani et al (2011).In most methods base on common weights set number 1 is taken as the highest benchmark of decision making units.

More than one DMU with the highest score can exist in

such cases, which is not achieved complete ranking.

We, therefore, need a criterion the achievement of which is harder than criterion level 1 for efficient decision making units. The main idea is to use a criterion point instead of criterion level in order to obtain common weights. The present article consists of the following parts: Section 2 defines background of DEA. Section 3 briefly introduces fuzzy DEA. In section 4 present our proposed method in section 5 numerical example is given, finally the conclusion present in section 6.

2. BACKGROUND OF DEA Assume

n

decision

making

( DMU j : j  J  {1,..., n}) outputs

y rj , r  1,..., s

units

each producing S from

m

inputs

xij , i  1,..., m . Assume input and output vectors for DMUj are X j  ( x1 j ,..., xmj ) and Y j  ( y1 j ,..., y sj ) ,

respectively.

For

DMUj it

is

assumed

that

X j  0, X j  0 and Y j  0, Y j  0. the efficiency of DMUo (o  J ) in proportion to the rest of units is obtained from the following fractional model: Max s.t.

U t Yo V t Xo U tY j VtX

 1,

j  1,..., n

(1)

j

U  0,

V  0.

Where U and V are weight vectors for the input and output vectors, respectively. This model is converted into the following linear model by Charnes- Cooper transformation, also known as CCR model.

Max U t Yo

S.t. V t X o  1,

(2)

 V t X  U tY  0,

2

International Journal of Advanced Information Science and Technology (IJAIST) ISSN: 2319:2682 Vol.21, No.21, January 2014 U  0, V  0.

development of Zadeh Lotfi and membership function, DMUp efficiency can be stated as:

mn

sn

Where X  R and Y  R are input and output matrices, respectively. When the optimal amount of the target function is equal to 1, the DMUo is said to be efficient or otherwise inefficient.

3. FUZZY DEA

min{ X~ij ( xij ), Y~ik ( yik ),

 F~ (t )  Sup p

i, j, k : t  Fp ( x, y )}

x, y

Where Fp ( x, y ) is the as DEA model. To obtain the

 F~

membership function of

~ ~ Assume X ij and Yrj to be inputs and outputs in DMU s

} (6)

we found of high and low

p

 F~

bounds of α-cut of

. Therefore, the following

p

set defined by fuzzy sets with membership functions of

 X~

and

models can be used.

, respectively. Since exact data can be

ij

s

produced by membership functions with only one quantity in their domain, all data are assumed to be fuzzy. Therefore, the fuzzy DEA model can be formulated as follows: s

U

~ F p  max

r

r 1 m

r 1 m

s

U r 1 m

r

i

i 1

pi

)U 

U r 1 m

r

(Ykr ) U 

 Vi ( X ki )L

 1,

k  1,..., n, k  p, (7)

i 1

~

i

(Y pr ) L

s

s

U

pi

~ Ykr

V X

r

V ( X

~

i

i 1

i 1

~ ( F p ) L  max

s.t.

~ Y pr

V X

s.t.

U

r

r 1 m

 1,

k  1,..., n

(Y pr ) L

V ( X

(3)

i

i 1

pj

 1,

)U 

U r , Vi    0,

ki

U r , Vi    0,

r  1,..., s

r  1,..., s

i  1,..., m,

i  1,..., m,

The inputs and outputs can be demonstrated by αcuts at different levels. Therefore, the fuzzy DEA model ~ U changes into a family of exact DEA models with a set( F p )   max of α-cuts of inputs and outputs (0    1) . The set of

s

U r 1 m

r

(Y pr ) U

V ( X

α-cuts are exact intervals capable of being defined as:

i

i 1

L pi 

)

s

~ ( X ij )   [min{xij  S ( X ij ) :  X~ij ( xij )   }, xij

~ max{xij  S ( X ij ) :  X~ij ( xij )   }] xij

U s.t.

(4)

(Ykr ) L

V ( X i 1

i

 1, k  p, (8)

U

ki

)

s

~ (Yik )   [min{ y ij  S (Yik ) :  ~yik ( y ik )   }, yik

~ max{ y ij  S (Yij ) :  ~yik ( y ik )   }] yik

~

~

Where S ( X ij ), S (Yik ) and

r 1 m

r

are the supports of

(5)

U r 1 m

r

(Y pr ) U

 Vi ( X pj ) L

 1,

i 1

U r , Vi    0,

r  1,..., s

i  1,..., m,

~ X ij

~ Yrj , respectively. According to the principle of 3

International Journal of Advanced Information Science and Technology (IJAIST) ISSN: 2319:2682 Vol.21, No.21, January 2014 4. OBTAIN THE COMMON SET OF WEIGHTS BY

We

BENCHMARK POINT

Ej 1

know

IDMU  1

that

and

( j  1,..., n)

An ideal (IDMU) and an anti-ideal decision making unit (ADMU) can be defined as follows:

, therefore IDMU has the highest efficiency score. We use this score as the highest criterion level for DMUj .

Definition 1: A virtual decision making unit is called IDMU if it uses the least input for the most output.

A corresponding virtual DMU is shown by

DMU j

Definition 2: A virtual decision making unit is call ADMU if it uses the most input for the least output.

( vi xij ,  u r y rj ) that (v1 ,..., vm , u1 ,..., u s ) is the

Based on this, if we demonstrate the inputs and

(i  1,..., m)

min

outputs of on ideal unit by

y

max r

(r  1,.., s) min

x y

i max r

x

i

and

, then

 min 1 j  n {xij }  max 1 j n { y rj }

anti-ideal unit by

y

min r

x

(r  1,.., s),

max

i

max

i

(i  1,..., m)

and

i 1

r 1

set of corresponding weight.

In comparison with another DMU, a DMU shows a better performance when its virtual DMU is closer to the benchmark level compared with the other one. A method to minimize the distance between

( j  1,..., n) ,

m

v x i 1

i

r 1

r  1,..., s

We show the IDMU efficiency by E IDMU and can be determine by the following model: s

E IDMU  Max u r y rmax r 1 m

i 1

i

min i

 1,

s

m

r 1

i 1

 u r y rj   vi xig  0, u r    0,

r  1,..., s,

vi    0.

i  1,..., m.

DMU

and

r

y rj increases. To have

the smallest distance possible, the following model can be considered: s

u

Yrmin  min 1 j n { y rj }

v x

u

decreases and

n

i  1,..., m

virtual

s

ij

Then

 m max 1 j n {xij }

s.t.

s

benchmark point is to determine their weight so that

r  1,..., s

And if we demonstrate the inputs and outputs of on

x

m

DMU j

i  1,..., m

( j  1,..., n) , and DMUj by pair of

Max(

j 1 r 1 n

s

r

r rj

m

  vi xij



j 1 i 1

u r 1 m

r

yr

 vi xi

)

i 1

s

u s.t.

r 1 m

r

y rmax

v x i 1

i

 E IDMU ,

(10)

min i

u r    0,

r  1,..., s,

vi    0,

i  1,..., m.

The optimal amount of this problem is cumulative j  1,..., n,DMU (9) efficiency, a DMU whose inputs and outputs are the same as the sum of all DMUs inputs and outputs, respectively. Changing the appropriate variable, we convert the above model into the following:

4

International Journal of Advanced Information Science and Technology (IJAIST) ISSN: 2319:2682 Vol.21, No.21, January 2014 s

Max  u r y r r 1 m

v x

s.t.

i

i 1

s

u r 1

i

 1, m

r

y rmax  E IDMU  vi ximin  0,

(11)

i 1

u r   .0,

r  1,..., s,

vi    0,

s

i

i 1

i

r 1

y

m

i 1

i 1

u

r

min i

 0,

Max(

(12)

j 1 r 1 n

u

model:

s

u

yr

 vi xi

)

i 1

r 1 m

r

y rmin

 vi ximax

 E ADMU ,

(14)

i 1

u r    0,

r  1,..., s,

vi    0,

i  1,..., m.

The optimal amount of this problem is the efficiency of cumulative DMU, a DMU whose inputs and outputs are the sum of all DMUs inputs and outputs, respectively. Changing the appropriate variable, we can convert the above model to the following:

r 1 m

 1,

s.t.

y rj   vi xig  0,

j  1,..., n,

(13)

i 1

u r    0,

r  1,..., s,

vi    0.

i  1,..., m.

i

with

s

u r 1

i

 1, m

r

y rmin  E ADMU  vi ximax  0,

u r   .0, vi    0,

We know that ADMU has the lowest efficiency score.

We use this as the lowest benchmark level for DMUj. We show a virtual DMU corresponding with

( j  1,..., n)

v x i 1

m

r

r

s

r 1

i

r 1 m

Max  u r y r

s

E ADMU  Min  u r y rmin

v x

m



u

s

E ADMU which can be determined by the following

max i

rrj

j 1 i 1

It is evident that there is no set of weights where a virtual DMU complies with the benchmark point unless its corresponding DMU is IDMU. If an IDMU is not an observed DMU in practice, there is no DMU whose virtual DMU can reach the benchmark point for every possible answer (12) or an equivalent model. Therefore, all DMUs can be ranked. In this way, an ADMU can be assumed. The efficiency of ADMU is shown by

DMU j

decreases and

s

r

  vi xij

i  1,..., m.

m

s

u

s.t.

r 1

ij

y rj increases. To have the smallest distance

r  1,..., s,

vi    0,

i 1

i

s

n

 E IDMU  vi x

max r

u r   .0,

s.t.

v x

of weights so that the

 1, m

r

( j  1,..., n) is the

DMU j

between

possible, we can consider the following model:

s

u

r 1

r 1

r 1

v x

i 1

set of corresponding weight. A DMU has a better performance in comparison to another when its virtual DMU is more distant from the benchmark level compared to the other. A method to maximize the virtual DMUj and the benchmark point of determination

Max  u r y r s.t.

s

distance

i  1,..., m.

We take the optimal reply to the problem as the common weight. Equally constraint can be replaced by inequality constraint in the problem above.

m

m

( vi xij ,  u r y rj ) that (v1 ,..., vm , u1 ,..., u s ) is the

the

pair

the

(15)

i 1

r  1,..., s, i  1,..., m.

We take the optimal answer of the problem as the common weight. The equality constraint in the above problem can be replaced by inequality constraint:

5

International Journal of Advanced Information Science and Technology (IJAIST) ISSN: 2319:2682 Vol.21, No.21, January 2014 s

Table1. Inputs and outputs

Max  u r y r r 1 m

s.t.

v i 1

s

u r 1

r

i

inputs

xi  1,

y rmin  E ADMU

u r   .0, vi    0,

m

v i 1

i

ximax  0,

(16)

r  1,..., s, i  1,..., m.

1.

2.

3.

4.

5.

5. NUMERICAL EXAMPLE

This example includes the evaluation of the departments with fuzzy data, gathered from 17 departments during 2008, 2009 and 2010 academic years. Seven inputs and two outputs have been assumed for each department as illustrated in the following table:

6.

7.

Area of the room Number of faculty members Number of probation students Number of transferred student Number of education officials Number of expelled student Number of computers

outputs 1. Number of students 2. Individuals being admitted to higher educational level

The data of departments have been presented in table 2.

6

International Journal of Advanced Information Science and Technology (IJAIST) ISSN: 2319:2682 Vol.21, No.21, January 2014

Table2. major

Data of department

I1

I2

I3

I4

I5

I6

I7

O1

O2

(6,6,6)

(1,1,1)

(0,0,0)

(0.2,0.2,0.2)

(0, 0.33,1)

(2, 3,4)

(0,0,0)

(11, 14.67,21)

(0, 2.33,4)

(13,13,13)

(5,5,5)

(0,0,0)

(0.25,0.25,0.25)

(1,1,1)

(11, 26.33,52)

(1,1,1)

(33, 42,53)

(12, 15,17)

(6,6,6)

(1,1,1)

(0,0,0)

(0.25,0.25,0.25)

(0,0.67,1)

(15, 24.33,37)

(0,0,0)

(45, 59.67,69)

(0,0,0)

(8‫و‬10.66 ,12)

(7,7,7)

(0,0,0)

(0.17,0.17,0.17)

(1,1,1)

(2, 12.67,22)

(1,1,1)

(1,1,2)

(7, 8,9)

(12,13.33, 16)

(7,7,7)

(3,3,3)

(0.17,0.17,0.17)

(1, 1.33,2)

(37, 46.67,56)

(1,1,1)

(49, 63.33,75)

(7, 9,12)

1

History

2

Geography

3

Accounting A.A

4

Applied mathematics

5

Physics

6

Electronic Engineering

(6,6.67,8)

(3,3,3)

(3,3,3)

(0.17,0.17,0.17)

(3, 3.33,4)

(44, 74.33,112)

(1,1,1)

(67, 67.33,68)

(6, 9.33,14)

7

Power Engineering

(4,5.33, 6)

(2,2,2)

(3,3,3)

(0.2,0.2,0.2)

(4, 5,7)

(33, 71,108)

(0.3, 0.76,1)

(108, 226.33,397)

(0,4,6)

8

Communication Engineering

(6,6.67, 8)

(2,2,2)

(2,2,2)

(0.2,0.2,0.2)

(5, 6,8)

(44, 54.33,73)

(0.3, 0.76,1)

(15, 39.33,53)

(4, 6,9)

9

Mechanic Engineering

(6,6,6)

(1,1,1)

(2,2,2)

(0.25,0.25,0.25)

(8, 9.67,13)

(52, 64,70)

(0, 0.33,1)

(43, 72.33,89)

(4, 6.33,8)

11

Computer Engineering

(0,2.67 ,8)

(1,1,1)

(5,5,5)

(0.2,0.2,0.2)

(0, 1.33,4)

(20, 26.67,35)

(0,0,0)

(60, 65.67,69)

(3, 5.67,9)

11

Rangeland Engineering

(6,6.67, 8)

(3,3,3)

(0,0,0)

(0.2,0.2,0.2)

(0,0,0)

(8, 11.67,16)

(0,0,0)

(0, 0.67,2)

(0, 2,4)

12

)Accounting(B.A)

(4,5.33, 6)

(1,1,1)

(0,0,0)

(0.2,0.2,0.2)

(0, 0.33,1)

(0,0,0)

(0, 0.33,1)

(20,(86,151)

(0, 0.67,2)

13

Mechanic (A.A)

(12,13.33, 16)

(5,5,5)

(1,1,1)

(0.2,0.2,0.2)

(2, 2.33,3)

(27,32,42)

(1,1,1)

(51, 53.67,57)

(0, 1,3)

14

Civil Engineering

(0,2, 6)

(1,1,1)

(2,2,2)

(0.2,0.2,0.2)

(0, 1.33,2)

(0, 1,3)

(0, 0.33,1)

(68, 123.33,179)

(3, 10.33,23)

15

Food sciences

(0,1.33, 4)

(2,2,2)

(0,0,0)

(0.25,0.25,0.25)

(0,0,0)

(0, 1,3)

(0, 0.33,1)

(0, 21.33,46)

(1,2.33,4)

16

Electronic(A.A)

(6, 6.67,8)

(3,3,3)

(0,0,0)

(0.17,0.17,0.17)

(0,0,0)

(14, 17.33,21)

(0.3,0.76 ,1)

(10, 14.67,19)

(0,0,0)

17

Architectural Engineering

(12, 13.33,16)

(5,5,5)

(15,15,15)

(0.2,0.2,0.2)

(5, 9,17)

(23, 42.33,71)

(1,1,1)

(139, 188.37,269)

(12, 17,27)

Using the models (7) and (8) for

  0.0, 0.2, 0.4, 0.6, o.8, 1.0

, we

obtain table 3.

7

International Journal of Advanced Information Science and Technology (IJAIST) ISSN: 2319:2682 Vol.21, No.21, January 2014 Table3. Upper and lower bounds for DMUs

 =0

 =0.2

 =0.4

 =0.6

 =0.8

 =1

DMU1

[0.073, 1]

[1.000, 1]

[1.000, 1]

[1.000, 1]

[1.000, 1]

[1.000, 1]

DMU2

[1 .000, 1]

[1 .000, 1]

[1 .000, 1]

[1 .000, 1]

[1 .000, 1]

[1 .000, 1]

DMU3

[0.298, 1]

[1 .000, 1]

[1.000, 1]

[1.000, 1]

[1.000, 1]

[1.000, 1]

DMU4

[0.606, 1]

[0.638, 1]

[0.672, 1]

[0.707, 1]

[0.818, 1]

[1.000, 1]

DMU5

[0.352, 1]

[0.415, 1]

[0.495, 1]

[0.601, 1]

[0.753, 1]

[0.921,0.928]

DMU6

[0.357, 1]

[0.346, 1]

[0.490, 1]

[0.591, 1]

[0.748, 1]

[0.933,0.943]

DMU7

[0.603, 1]

[0.789, 1]

[1.000, 1]

[1.000, 1]

[1.000, 1]

[1.000, 1]

DMU8

[0.174, 1]

[0.215, 1]

[0.268, 1]

[0.338,0.872]

[0.432,0.667]

[0.547,0.547]

DMU9

[0.186, 1]

[0.285, 1]

[0.345, 1]

[0.415, 1]

[0.494, 1]

[0.613,0.613]

DMU10

[0.335, 1]

[1.000, 1]

[1.000, 1]

[1.000, 1]

[1.000, 1]

[1.000, 1]

DMU11

[0.075, 1]

[1.000, 1]

[1.000, 1]

[1.000, 1]

[1.000, 1]

[1.000, 1]

DMU12

[0.869, 1]

[1.000, 1]

[1.000, 1]

[1.000, 1]

[1.000, 1]

[1.000, 1]

DMU13

[0.219, 1]

[0.242,0.912]

[0.336,0.938]

[0.304, 1]

[0.350,0.486]

[0.413,0.429]

DMU14

[0.792, 1]

[1.000, 1]

[1.000, 1]

[1.000, 1]

[1.000, 1]

[1.000, 1]

DMU15

[0.375, 1]

[1.000, 1]

[1.000, 1]

[1.000, 1]

[1.000, 1]

[1.000, 1]

DMU16

[0.002, 1]

[0.014,0.425]

[0.020,0.157]

[0.099,0.283]

[0.037, 1]

[0.052,0.053]

DMU17

[0.522, 1]

[0.635, 1]

[1.000, 1]

[0.974, 1]

[1.000, 1]

[1.000, 1]

After obtain the efficiency of all DMUs, their rank must be determined. There are different methods. We have used Chen and Klein (1997) method. Their ranking index is as follows: n

Ij 

 (( F i 0

n

j

is correct when n number of α-cuts is infinite but Chen and Klein believe that if n=3 or n=4, it is enough to use the formula to distinguish the differences. The information can be found in table 4.

)U  i  c) n

[ (( F j )  i  c)   (( F j ) Li  d )

(17)

U

i 0

Where c

i 0

 min i , j {(Fij )Li } and

d  max i , j {(Fij )Ui } . Theoretically, the formula 8

International Journal of Advanced Information Science and Technology (IJAIST) ISSN: 2319:2682 Vol.21, No.21, January 2014 Table4. The calculated amounts of Ij I1 0.86 6 I10

I2 1.00 0 I11

I3 0.89 5 I12

I4 0.79 3 I13

I5 0.70 7 I14

I6 0.70 0 I15

I7 0.90 8 I16

I8 0.55 7 I17

0.90 0

0.86 6

0.97 9

0.51 1

0.96 6

0.90 6

0.24 3

0.87 3

I9 0.60 5

Based on table above, we observe that

~ ~ ~ ~ ~ ~ ~ ~ ~ F2  F12  F14  F7  F15  F10  F3  F17  F11 ~ ~ ~ ~ ~ ~ ~ ~  F1  F4  F5  F6  F9  F8  F13  F16 Now, we illustrate the use of the proposed procedure with above example. Using the model (16) for   0.0, 0.2, 0.4, 0.6, o.8, 1.0 ,

we

obtain table 5.

Table5. Upper and lower bounds for DMUs based on new method

 =0

 =0.2

 =0.4

 =0.6

 =0.8

 =1

DMU1

[0.002, 0.119]

[0.017, 0.119]

[0.034,0.118]

[0.055,0.117]

[0.081,0.115]

[0.113,0.113]

DMU2

[0.280, 0.399]

[0.320,0.423]

[0.366,0.451]

[0.422,0.485]

[0.049,0.525]

[0.574,0.574]

DMU3

[0.006, 0.010]

[0.007,0.009]

[0.0070.009]

[0.008,0.009]

[0.008,0.009]

[0.008,0.008]

DMU4

[0.236, 0.305]

[0.264,0.325]

[0.298,0.347]

[0.338,0.374]

[0.386,0.406]

[0.446,0.446]

DMU5

[0.243, 0.418]

[0.280,0.432]

[0.322,0.447]

[0.373,0.464]

[0.435,0.486]

[0.511,0.511]

DMU6

[0.212, 0.484]

[0.254,0.491]

[0.304,0.498]

[0.364,0.507]

[0.438,0.516]

[0.527,0.527]

DMU7

[0.019, 0.241]

[0.047,0.237]

[0.081,0.234]

[0.121,0.231]

[0.168,0.229]

[0.227,0.227]

DMU8

[0.116, 0.268]

[0.140,0.270]

[0.167,0.274]

[0.200,0.278]

[0.240,0.283]

[0.289,0.289]

DMU9

[0.097, 0.196]

[0.118,0.203]

[0.142,0.211]

[0.171,0.221]

[0.206,0.234]

[0.249,0.249]

DMU10

[0.096, 0.271]

[0.1210.273]

[0.150,0.275]

[0.185,0.276]

[0.227,0.227]

[0.280,0.280]

DMU11

[0.000, 0.116]

[0.013,0.113]

[0.027,0.110]

[0.046,0.106]

[0.068,0.068]

[0.095,0.095]

DMU12

[0.004, 0.084]

[0.010,0.079]

[0.017,0.072]

[0.026,0.653]

[0.035,0.035]

[0.047,0.047]

DMU13

[0.009, 0.096]

[0.015,0.091]

[0.023,0.085]

[0.032,0.077]

[0.043,0.043]

[0.057,0.057]

DMU14

[0.098, 0.696]

[0.154,0.673]

[0.220,0.645]

[0.299,0.611]

[0.394,0.394]

[0.514,0.514]

DMU15

[0.023, 0.099]

[0.032,0.098]

[0.433,0.097]

[0.056,0.096]

[0.072,0.072]

[0.092,0.092]

DMU16

[0.002, 0.004]

[0.002,0.004]

[0.244,0.004]

[0.003,0.003]

[0.003,.003]

[0.003,0.003]

DMU17

[0.363, 0.819]

[0.426,0.821]

[0.501,0.824]

[0.590,0.826]

[0.697,0.697]

[0.831,0.831]

9

International Journal of Advanced Information Science and Technology (IJAIST) ISSN: 2319:2682 Vol.21, No.21, January 2014 After obtain the efficiency of all DMUs, They are ranked in the same manner. The information can be found in table 6.

Table6. The calculated amounts of Ij based on new method

I1 0.13 9 I10 0.29 6

I2 0.51 7 I11 0.11 9

I3 0.17 1 I12 0.09 3

I4 0.50 9 I13 0.09 0

I5 0.49 7 I14 0.52 9

I6 0.48 6 I15 0.23 6

I7 0.22 4 I16 0.00 4

I8 0.30 5 I17 0.75 8

I9 0.21 7

Based on table above, we observe that

~ ~ ~ ~ ~ ~ ~ ~ ~ F17  F14  F2  F4  F6  F5  F8  F10  F7 ~ ~ ~ ~ ~ ~ ~ ~  F15  F9  F3  F1  F11  F12  F13  F16 Based on the results obtained, we observe that DMU17 ranks first if our method is used, whereas it ranked eighth in the previous method. DMU2 ranked first in the previous method but it ranks third in our

method. The ranking of other DMUs, too, have changed in our model.

6. CONCLUSION

In the present article, IDMU was employed as a pattern for efficient DMUs. A model of finding CSWs was obtained. The method offered in this article has the following characteristics. First, it obtains CSWs model from an LP. Second, a critical factor in solving an LP is the number of constraints.

The number of constraints is as small as possible in this model; therefore, it is more economical. Third, a complete ranking of DMUs is obtained by this model.

For control weights in the proposed method, we will consider weight restrictions in the proposed method.

Since the data are fuzzy, we obtain an upper bound and a lower bound base on the best and the worst evaluation respectively for each α- cut. This method with upper bound efficiency score of unit less than the highest score, and with lower bound efficiency score of unit more than the least score guarantees a complete ranking of DMUs.

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Author Profile Bijan RahmaniParchikolaei is Assistant Professor of Applied Mathematics at the Nour Branch Islamic Azad University, Iran.

His research interests include, Data Envelopment Analysis, Linear Programming,

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