A generalized model for data envelopment analysis with interval data

Applied Mathematical Modelling 33 (2009) 3237–3244

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A generalized model for data envelopment analysis with interval data G.R. Jahanshahloo, F. Hosseinzadeh Lotfi, M. Rostamy Malkhalifeh *, M. Ahadzadeh Namin Islamic Azad University, Department of Mathematics, Science and Research Branch, Tehran 14515-775, Iran

a r t i c l e

i n f o

Article history: Received 23 January 2008 Received in revised form 18 October 2008 Accepted 22 October 2008 Available online 5 November 2008

Keywords: Data envelopment analysis FDH GDEA Interval data

a b s t r a c t Data envelopment analysis (DEA) is a method to estimate the relative efficiency of decision-making units (DMUs) performing similar tasks in a production system that consumes multiple inputs to produce multiple outputs. So far, a number of DEA models with interval data have been developed. The CCR model with interval data, the BCC model with interval data and the FDH model with interval data are well known as basic DEA models with interval data. In this study, we suggest a model with interval data called interval generalized DEA (IGDEA) model, which can treat the stated basic DEA models with interval data in a unified way. In addition, by establishing the theoretical properties of the relationships among the IGDEA model and those DEA models with interval data, we prove that the IGDEA model makes it possible to calculate the efficiency of DMUs incorporating various preference structures of decision makers. Ó 2008 Published by Elsevier Inc.

1. Introduction Data envelopment analysis (DEA ) is a non-parametric technique for measuring and evaluating the relative efficiencies of a set of entities, called decision-making units (DMUs), with common inputs and outputs [1–4]. Examples include schools, hospitals, libraries and, more recently, the whole economic and social systems, in which outputs and inputs are always multiple in character. In recent years, in different applications of DEA, inputs and outputs have been observed whose values are indefinite [5–7]. Such data are called ‘‘inaccurate”. Inaccurate data can be probabilistic, interval, ordinal, qualitative, or Fuzzy. Therefore, some papers were presented on the theoretical development of DEA with interval data, of which we can name Despotis and Smirlis [8] and Jahanshahloo et al. [9]. Yun et al. [10] proposed a generalized model for DEA, called GDEA model, which can treat basic DEA models, specifically the CCR model, the BCC model and the FDH model in a unified way. In this study, we propose a generalized model with interval data for interval DEA(IDEA), called IGDEA model, which can treat basic IDEA models, specifically the CCR, BCC, FDH models with interval data in a unified way. In addition, we show the theoretical properties of the relationships among the IGDEA and those IDEA models. 2. DEA model We assume that there are n DMUs to be evaluated indexed by j ¼ 1; 2; . . . ; n. And each DMU is assumed to produce s different outputs from m different inputs. Let the observed input and output vectors of DMU j be X j ¼ ðx1j ; x2j ; . . . ; xmj Þ and Y j ¼ ðy1j ; y2j ; . . . ; ysj Þ respectively, where all components of vectors X j and Y j for all DMUs are non-negative, and each DMU has at least one strictly positive input and output. For evaluating the efficiency of DMU o we use some models as follows: * Corresponding author. Tel./fax: +98 2144804172. E-mail address: [email protected] (M.R. Malkhalifeh). 0307-904X/$ - see front matter Ó 2008 Published by Elsevier Inc. doi:10.1016/j.apm.2008.10.030

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Min h  e1Sz ; n X kj xij þ sx ¼ hxio ; s:t:

i ¼ 1; . . . ; m;

j¼1 n X

ð1Þ kj yij  sy ¼ yro ;

r ¼ 1; . . . ; s;

j¼1

kj K; j ¼ 1; 2; . . . ; n; where Sz ¼ ðsx ; sy Þ and K are as follows:

K ¼ KCCR ¼ fkj jkj P 0; j ¼ 1; 2; . . . ; ng; ( ) n X K ¼ KBCC ¼ kj j kj ¼ 1; kj P 0; j ¼ 1; 2; . . . ; n ; (

K ¼ KFDH ¼

kj j

j¼1 n X

) kj ¼ 1; kj f0; 1g; j ¼ 1; 2; . . . ; n ;

j¼1

Definition 1. DMU o is CCR  efficient if and only if h ¼ 1 and Sz ¼ 0 for K equal to KCCR. BCC efficiency and FDH efficiency are defined likewise.

3. Interval DEA Let the input and output values of any DMU o be located in a certain interval, where xlij and xuij are the lower and upper bounds of the jth DMU, respectively, and ylrj and yurj are the lower and upper bounds of the rth DMU, respectively; that is to say, xlij 6 xij 6 xuij and ylrj 6 yrj 6 yurj . Such data are called interval data, because they are located in intervals. Note that always xlij 6 xuij and ylrj 6 yurj . If xlij ¼ xuij , then the ith input of the jth DMU has a definite value. The BCC model for evaluating DMU o with interval data is as follows:

Max U t Y o þ uo ; s:t:

U t Y j  V t X j þ uo 6 0 j ¼ 1; 2; . . . ; n; V t X o ¼ 1;

ð2Þ

U P 1e; V P 1e; where Y j ¼ ½y1j ; y2j ; . . . ; ysj , X j ¼ ½x1j ; x2j ; . . . ; xmj , Y o ¼ ½y1o ; y2o ; . . . ; yso  and X o ¼ ½x1o ; x2o ; . . . ; xmo  for each j ¼ 1; 2; . . . ; n. In problem (2), one can see that all parameters of the problem are in intervals and the efficiency of DMU o is also located in an interval. The upper and lower bounds of the relative efficiency of DMU o are obtained by solving the following problems, respectively.

h ¼ Max U t Y u þ uo ; o s:t:

U t Y lj  V t X uj þ uo 6 0 j ¼ 1; 2; . . . ; n; j–o; U t Y uo  V t X lo þ uo 6 0;

ð3Þ

V t X lo ¼ 1; U P 1e;

V P 1e;

and

h ¼ Max U t Y lo þ uo ; s:t:

U t Y uj  V t X lj þ uo 6 0 j ¼ 1; 2; . . . ; n; j–o; U t Y lo  V t X uo þ uo 6 0;

ð4Þ

V t X uo ¼ 1; U P 1e;

V P 1e;

If we suppose uo ¼ 0 in models (2)–(4), we have the CCR models. Considering that the efficiency of any DMU lies in an interval, all DMUs can be divided into one of the three following classes:

G.R. Jahanshahloo et al. / Applied Mathematical Modelling 33 (2009) 3237–3244

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Class 1: Includes all DMUs which are efficient both in their best and worst situation; in other words,

Eþþ ¼ fDMU j jh ¼ h ¼ 1g: Class 2: Consists of all DMUs which are efficient in their best situation, but inefficient in their worst situation; in other words,

Eþ ¼ fDMU j jh < 1; h ¼ 1g; and Class 3: Consists of all DMUs which are inefficient in their best situation. It goes without saying that such DMUs are, also, inefficient in their worst situation; that is to say,

E ¼ fDMU j jh < 1; h < 1g:

4. The GDEA model based on parametric domination structure The GDEA model, was suggested by Yun et al. [10]. They established the relationship between the GDEA model and basic DEA models. The GDEA model can evaluate the efficiency in several basic models in special cases, as follows:

Max D; s:t:

~ þa D6d j

s X

ur ðyro  yrj Þ þ

r¼1 s X

ur þ

m X

r¼1

m X

!

v i ðxio þ xij Þ

j ¼ 1; 2; . . . ; n;

i¼1

ð5Þ

v i ¼ 1;

i¼1

ur P e;

vi P e

i ¼ 1; 2; . . . ; m;

r ¼ 1; 2; . . . ; s;

~ ¼ max fur ðy  y Þ; v ðx þ x Þg. where a is appropriately given according to the give problems, and d j i;r i io ij ro rj Note that when j ¼ o, the right-hand side of the inequality constraint in the problem (GDEA) is zero, and hence its optimal value is not greater than zero. Definition 2 (a-Efficiency). For a given positive number a, DMU o is defined to be a-efficient if and only if the optimal value to the problem (GDEA) is equal to zero. Otherwise, DMU o is said to be a-inefficient. Yun et al. [10] showed that (a) DMU o is FDH-efficient if and only if DMU 0 is a-efficient for some sufficiently small positive number a. b) DMU o is BCC-efficient if and only if DMU o is a-efficient for some sufficiently large positive number a. c) DMU o is CCR-efficient if and only if it is a-efficient for some sufficiently large positive number a when condition U t Y o  V t X j ¼ 0 is added to the GDEA model.

5. GDEA with interval data based on parametric domination structure In this section, we formulate the GDEA model with interval data (IGDEA) based on parametric domination structure and define a new efficiency in the GDEA model with interval data. Next, we establish the relationships between the GDEA model with interval data and the IDEA models. The IGDEA model, which can evaluate the efficiency in several basic models with interval data as special case, is the following:

D ¼ Max D; s:t:

~ þa D6d j

s X

ur ðyro  yrj Þ þ

r¼1 s X

ur þ

r¼1

ur P e;

m X

m X

!

v i ðxio þ xij Þ

j ¼ 1; 2; . . . ; n;

i¼1

ð6Þ

v i ¼ 1;

i¼1

v i P ei ¼ 1; 2; . . . ; m;

r ¼ 1; 2; . . . ; s;

where xij ½xlij ; xuij , yij ½ylij ; yuij . To evaluate the efficiency of model (6) we introduce two models as follows:

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Dmax ¼ Max D; s:t:

~0þa D6d j

s X

ur ðyuro



ylrj Þ

þ

r¼1

m X

!

v

l i ðxio

þ

xuij Þ

j ¼ 1; 2; . . . ; n;

j–0;

i¼1

ð7Þ

D 6 0; s m X X ur þ v i ¼ 1; r¼1

i¼1

v i P e; i ¼ 1; 2; . . . ; m;

ur P e;

r ¼ 1; 2; . . . ; s;

~0 ¼ maxi;r fur ðyu  yl Þ; v i ðxl þ xu Þg. where a P 0 is appropriately given according to the given problems, and d ro j rj io ij

Dmin ¼ Max D; s:t:

~00 þ a D6d j

s X

ur ðylro  yurj Þ þ

r¼1

m X

!

v i ðxuio þ xlij Þ

j ¼ 1; 2; . . . ; n;

j–0;

i¼1

ð8Þ

D 6 0; s X

ur þ

r¼1

m X

v i ¼ 1;

i¼1

ur P e;

v i P e; i ¼ 1; 2; . . . ; m;

r ¼ 1; 2; . . . ; s;

~00 ¼ max fur ðyl  yu Þ; v ðxu þ xl Þg. where a P 0 is appropriately given according to the given problems, and d i;r i ro j rj io ij   In the following theorem,we define the relation between D , Dmax , and Dmin : Theorem 1. If D , Dmax , and Dmin , are the optimal values of models (6)–(8) respective then we have D ½Dmin and Dmax . Proof. Let D , u ¼ ðu1 ; u2 ; . . . ; us Þ,

(

v ¼ ðv 1 ; v 2 ; . . . ; v m Þ be the optimal solution for DMUo in model (6). Then

xlio P xio ;

ð9Þ

yuro P yro :

Suppose that zo ¼ ðx1o ; x2o ; . . . ; xmo ; y1o ; y2o ; . . . ; yso Þt and zuo ¼ ðxl1o ; xl2o ; . . . ; xlmo ; yu1o ; yu2o ; . . . ; yuso Þt . That is zuo P zo . Also  xij P xuij and zlo ¼ ðxu1o ; xu2o ; . . . ; xumo ; yl1o ; yl2o ; . . . ; ylso Þt .That is, zo P zlo . So, yrj P ylrj

zo  zj 6 zuo  zlj ;

~ d ~0 and d j6 j :

ð10Þ

~ þ aðU; VÞðZ o  Z 6 d ~0 þ aðU; VÞðZ u  Z l Þ, then D 6 Dmax , we can prove D 6 D , likewise. h Therefore, d j min o j 5.1. Relationships between IGDEA and IDEA In this subsection, we establish the theoretical properties of the relationship between IDEA’s efficiency and IGDEA efficiency. þ  Definition 3. We define Eþþ G ,EG and EG as follows: j Eþþ G ¼ fDMU j jDmin ¼ 0g;

EþG ¼ fDMU j jDjmin 6 0; Djmax ¼ 0g; EG ¼ fDMU j jDjmax < 0g: Definition 4. For a given positive number a, DMU 0 is a-efficient in GDEA model with interval data if and only if Dmin ¼ Dmax ¼ 0. Definition 5. Let xij ½xlij ; xuij  and yij ½ylij ; yuij ; i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; n and r ¼ 1; 2; . . . ; s. 1. DMU o is said to be FDH – efficient in the best situation if it does not exist a DMU j in its worst condition that dominates it. 2. DMU o is said to be FDH – efficient in the worst situation if it does not exist a DMU j in its best condition that dominates it. þ  Theorem 2. Let xij ½xLij ; xUij  and yij ½yLij ; yUij ; i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; n and r ¼ 1; 2; . . . ; s. DMU o is in Eþþ G ðEG ; EG Þ for some suf ; E Þ for FDH model. ficiently small positive number a if and only if DMU o is in Eþþ ðEþ G G

G.R. Jahanshahloo et al. / Applied Mathematical Modelling 33 (2009) 3237–3244

3241

Proof (only if part for Eþþ ). By contradiction assume that DMU o is not in Eþþ by the FDH evaluation, then

Ro ¼ max minfxlij =xuio g < 1; j–0

i

that is,

9jDðoÞ ¼ fjjxlij 6 xuio ; yurj P ylro g for some i and j ¼ 1; . . . ; n; thus xlij < xuio , yurj P ylro for some i.That is,

zlo  zuj < 0;

ð11Þ

~00 6 0 for each j. and d j 1 ; u 2 ; . . . ; u  s Þ, V ¼ ðv  1; v 2; . . . ; v  m Þ, we have For any positive U ¼ ðu

~00 þ aðU; VÞðzl  zu Þ < 0: Dmin 6 d j o j

ð12Þ Eþþ G

Dmin

< 0. This contradicts the assumption that DMU o is in for some sufficiently small positive number a. (if part for Eþþ ). By contradiction suppose that DMU 0 is not in Eþþ G for some sufficiently small positive number a. Then for ~00 þ aðU; VÞðzl  zu Þ < 0 for some j–0. any sufficiently small positiveDmin < 0, so d 0 j j Regarding the assumption that DMU o is in Eþþ , we have hmin ¼ 1 and there is not a j such that ~00 ¼ max fur ðyl  yu Þ; v ðxu þ xl Þg < 0 and ur P e, v P e for each r ¼ 1; 2; . . . ; s, i ¼ 1; 2; . . . ; m, then zl  zu < 0. d i;r i i ro o jðj–0Þ rj io ij j This contradicts the assumption that DMU o is in Eþþ . h

So

 < 1, R ¼ maxjDðoÞ mini fxu =xl g < 1, that is, h Proof (Only if part for E ). Suppose that DMU o is in E . Then o ij io u l l u u l l u u l ~0 6 o for some i. 9jDð0Þ ¼ fjjxij 6 xio ; yrj P yro g thus xij < xio , yrj P yro for each i.That is, zo  zj < 0 and d j–0 ~0 þ aðu; v Þðzu  zl Þ < 0. That is, Dmax 6 0 1 ; u 2 ; . . . ; u  s Þ, V ¼ ðv 1; v  2; . . . ; v  m Þ, we have Dmax 6 d Then for any positive U ¼ ðu o j j and DMU o is in E for some sufficiently small positive number a. (if part for E ): Suppose that DMU o is in E for some sufficiently small positive number a. Then for some sufficiently small positive number a, Dmax 6 0, that is,

~0 þ aðu; v Þðzu  zl Þ < 0: Dmax 6 d j o j

ð13Þ

Then the necessary and sufficient condition so that the above inequality (13) holds for some sufficiently small positive a is that

~0 þ aðu; v Þðzu  zl Þ < 0 ðfor each j–0Þ; d j o j

ð14Þ

and the necessary and sufficient condition so that the above inequality (14) holds for some sufficiently small positive a is that

zuo  zlj < 0;

ð15Þ

ðu; v Þðzuo  zlj Þ < 0;

ð16Þ

~0 < 0: d j

ð17Þ

so,

and

Then Ro ¼ maxjDðoÞ mini fxuij =xlio g < 1, that is Ro < 1 and DMU o is in E by the FDH evaluation. The case for Eþ can be proved likewise. h Theorem 3. Let xij ½xLij ; xUij  and yij ½yLij ; yUij  for each i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; n and r ¼ 1; 2; . . . ; s. DMU o is in Eþþ ðE ; Eþ Þ for some sufficiently large positive number a if and only if DMU o is in Eþþ ðE ; Eþ Þ by the BCC evaluation. Proof (only if part for Eþþ ). Let DMU o in Eþþ of BCC model and ðU  ; V  ; uo Þ is optimal solution, that is

ðU  ; V  Þzuj þ uo 6 0 and ðU  ; V  Þzlo þ uo ¼ 0;

ð18Þ

so,

ðU  ; V  Þðzlo  zuj Þ P 0;

~ 00 P 0; zlo  zuj P 0 and d j

ð19Þ

~00 þ aðU  ; V  Þðzl  zu Þ for some sufficiently small positive a so DMU o in Eþþ for some sufficiently small pothat is, Dmin ¼ 0 6 d G o j j for some sufficiently small positive number a, so for optimal solution sitive a. (if part for Eþþ ). Let DMU o is in Eþþ G ðU  ; V  ; Dmin Þ,

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G.R. Jahanshahloo et al. / Applied Mathematical Modelling 33 (2009) 3237–3244

~00 þ ðU  ; V  Þðzl  zu Þ P 0; d j o j

ð20Þ

and that necessary and sufficient condition so that the above inequality (20) holds for some sufficiently small positive a is that,

zl0  zuj P 0: t

Let V X o ¼ b, so 

ð21Þ

t

V b





X o ¼ 1 and let uo ¼ ðU ; V Þzlo . So,



ðU ; V Þðzlo  zuj Þ P 0;

ð22Þ

that is t

t

U  u V  l uo y  x þ 6 0; b j b j b

ð23Þ

and t

so



t

U  l V  u uo y  x þ ¼ 0; b o b o b t

U b

t

; Vb ;

u0 b



ð24Þ

is feasible solution of BCC model and the value of objective function equal 1.

h

for some suffiTheorem 4. If xij ½xlij ; xuij  and yij ½yLij ; yUij  for each i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; n and r ¼ 1; 2; . . . ; s. DMUo is in Eþþ G þ ciently large positive number a if and only if DMU o is in Eþþ for CCR model. And in E G , EG too consider the problem in which the constraint U t yro ¼ V t xio is added to the model (6) and U t yUro ¼ V t xLio is added to the model (7) and U t yLro ¼ V t xUio is added to the model (8). The proof of this theorem is similar to that of Theorem 3, hence is omitted.

Table 1 Inputs and outputs. Inputs

Outputs

Payable interest Personnel Non-performing loans

The total sum of four main deposits Other deposits Loans granted Received interest Fee

Table 2 Input-data for the 20 bank branches. DMU j

xL1j

xU2j

xL2j

xU2j

xL3j

xU3j

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

5007.37 2926.81 8732.7 945.93 8487.07 13759.35 587.69 4646.39 1554.29 17528.31 2444.34 7303.27 9852.15 4540.75 3039.58 6585.81 4209.18 1015.52 5800.38 1445.68

9613.37 5961.55 17752.5 1966.39 17521.66 27359.36 1205.47 9559.61 3427.89 36297.54 4955.78 14178.11 19742.89 9312.24 6304.01 13453.58 8603.79 2037.82 11875.39 2922.15

36.29 18.8 25.74 20.81 14.16 19.46 27.29 24.52 20.47 14.84 20.42 22.87 18.47 22.83 39.32 25.57 27.59 13.63 27.12 28.96

36.86 2016 27.17 22.54 14.8 19.46 27.48 25.07 21.59 15.05 20.54 23.19 21.83 23.96 39.86 26.52 27.95 13.93 27.26 28.96

87243 9945 47575 19292 3428 13929 27827 9070 412036 8638 500 16148 17163 17918 51582 20975 41960 18641 19500 31700

87243 12120 50013 19753 3911 15657 29005 9983 413902 10229 937 21353 17290 17964 55136 23992 43103 19354 19569 32061

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G.R. Jahanshahloo et al. / Applied Mathematical Modelling 33 (2009) 3237–3244 Table 3 Output-data for the 20 bank branches. DMU j

yL1j

yU1j

yL2j

yU2j

yL3j

yU3j

yL4j

yU4j

yL5j

yU5j

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

2696995 340377 1027546 1145235 390902 988115 144906 408163 335070 700842 641680 453170 553167 309670 286149 321435 618105 248125 640890 119948

3126798 440355 1061260 1213541 395241 1087392 165818 416416 410427 768593 696338 481943 574989 342598 317186 347848 835839 320974 679916 120208

263643 95978 37911 229646 4924 74133 180530 405396 337971 14378 114183 27196 21298 20168 149183 66169 244250 3063 490508 14943

382545 117659 503089 268460 12136 111324 180617 486431 449336 15192 241081 29553 23043 26172 270708 80453 404579 6330 684372 17495

1675519 377309 1233548 468520 129751 507502 288513 1044221 1584722 2290745 1579961 245726 425886 124188 787959 360880 9136507 26687 2946797 297674

1853365 390203 1822028 542101 142873 574355 323721 1071812 1802942 2573512 2285079 275717 431815 126930 810088 379488 9136507 29173 3985900 308012

108634.76 32396.65 96842.33 32362.8 12662.71 53591.3 40507.97 56260.09 176436.81 662725.21 17527.58 35757.83 45652.24 8143.79 106798.63 89971.47 33036.79 9525.6 66097.16 21991.53

125740.28 37836.56 108080.01 39273.37 14165.44 72257.28 45847.48 73948.09 189006.12 791463.08 20773.91 42790.14 50255.75 11948.04 111962.3 165524.22 41826.51 10877.78 95329.87 27934.19

965.97 304.67 2285.03 207.98 63.32 480.16 176.58 4654.71 560.26 58.89 1070.81 375.07 438.43 936.62 1203.79 200.36 2781.24 240.04 961.56 282.73

6957.33 749.4 3174 510.93 92.3 869.52 370.81 5882.53 2506.67 86.86 2283.08 559.85 836.82 1468.45 4335.24 399.8 4555.42 274.7 1914.25 471.22

Table 4 CCR-efficiencies of DMUs. DMU j 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

hccr 1 1 1 1 0.76 1 1 1 1 1 1 0.49 0.70 0.72 1 1 1 0.95 1 1

hccr 1 0.371 0.52 1 0.61 0.917 0.72 1 1 1 1 0.32 0.45 0.26 0.41 0.22 1 0.26 0.99 0.18

IDEA classification þþ

E Eþ Eþ Eþþ E Eþ Eþ Eþþ Eþþ Eþþ Eþþ E E E Eþ Eþ Eþþ E Eþ Eþ

Dmax

Dmin

IGDEA classification

324 845 1000 0 2288 2570 556 0 1372 0 0 4946 4424 6954 762 0 0 2729 0 2922

5630 8282 8190 2715 3448 3969 6717 3097 6556 2065 0 8627 5907 9387 9029 8311 899 9802 2981 9748

E G E G E G Eþ G E G E G E G Eþ G E G Eþ G Eþþ G E G E G E G E G Eþ G Eþ G E G Eþ G E G

6. The efficiency of bank branches We now apply our approach to some commercial bank branches in Iran. There are 20 branches in this district. Each branch uses three inputs to produce five outputs. Table 1 shows these inputs and outputs. In Tables 2 and 3 the interval inputs and interval outputs for these DMUs are given. Also in Table 4 the CCR-efficiencies of these DMUs by IDEA, IGDEA models and their classification are presented. When branches of bank have been CCR evaluated by IDEA, 15 DMUs are efficient in the best situation and also in the worst situation, seven DMUs are efficient, i.e. seven DMUs have been lied in Eþþ . Wherever, in CCR-evaluation by IGDEA, þþ there are seven efficient DMUs in the best situation and an efficient DMU in the worst situation and we have Eþþ G #E þ and Eþ # E . It can be seen that in evaluating by IGDEA, the number of efficient DMUs are less than the other model, and G it shows that IGDEA method improves the discriminating power towards the most efficient units. 7. Conclusions In this paper, we suggested the IGDEA model based on parametric domination structure, and defined a-efficiency in the IGDEA model. In addition, we investigated the theoretical properties of the relationships between the GDEA model and exist-

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ing DEA models, specifically the BCC model,the CCR model, and the FDH model. Then, it was proved that the IGDEA model makes it possible to evaluate the efficiencies of several IDEA models in a unified way, and to incorporate various preference structures of decision makers. Such that, it is shown in numerical example in evaluating DMU by IGDEA the number of efficient DMU have reduced and it is one of the advantage of the proposed model since overcome some difficulties in ranking and also help us to rank efficient and inefficient DMUs. References [1] R.D. Banker, A. Charnes, W.W. Cooper, Some models for estimating technical and scale efficiencies in data envelopment analysis, Manage. Sci. 30 (1984) 1078–1092. [2] R.D. Banker, I. Bardhan, W.W. Cooper, A note on returns to scale in DEA, Euro. J. Operat. Res. 88 (1996) 583–585. [3] R.D. Banker, H. Chang, W.W. Cooper, Equivalence and implementation of alternative method for determining returns to scale in data envelopment analysis, Euro. J. Operat. Res. 89 (1996) 473–481. [4] A. Charnes, W.W. Cooper, Q.L. Qei, Z.M. Huang, Cone ratio data envelopment analysis and multi-objective programming, Int. J. Syst. Sci. 20 (7) (1989) 1099118. [5] R.G. Dyson, E. Thanassoulis, Reducing weight flexibility in data envelopment analysis, J. Operat. Res. Soc. 39 (1988) 563–576. [6] T. Joro, P. Korhonen, J. Wallenius, Structural comparison of data envelopment analysis and multiple objective linear programming, Manage. Sci. 44 (1998) 962–970. [7] T. J Stewart, Relationships between data envelopments analysis and multiple criteria decision analysis, J. Operat. Res. Soc. 47 (1996) 654–665. [8] D.K. Despotis, Y.G. Smirlis, Data envelopment analysis with imprecise data, Euro. J. Operat. Res. 140 (2002) 24–36. [9] G.R. Jahanshahloo, F. Hosseinzadeh Lotfi, M. Moradi, Sensitivity and stability analysis in DEA with interval data, Appl. Math. Comput. 156 (2004) 463– 477. [10] Y.B. Yun, H. Nakayama, T. Tanino, A generalized model for data envelopment analysis, Euro. J. Operat. Res. 157 (1) (2004) 87–105.