Applied Mathematics and Computation 162 (2005) 1265–1277
www.elsevier.com/locate/amc
Multi-component efficiency measurement with imprecise data Alireza Amirteimoori a
a,*
, Sohrab Kordrostami
b
Department of Mathematics, Islamic Azad University, Pol-e-Taleshan, Rasht, Iran b Department of Mathematics, Islamic Azad University, Lahijan, Iran
Abstract Data envelopment analysis (DEA) evaluates the efficiency of decision making units with multiple inputs and outputs. In most applications of DEA, presented in literature, the models presented are designed to obtain a single measure of efficiency where all inputs and outputs are known exactly. In many instances, however, the decision making units involved may perform several different functions, or can be separated into different components and some inputs and outputs are unknown decision variables such as bounded data and ordinal data. In such situations, inputs are often shared among those components and all components play an important role in producing some outputs. In this case, the standard DEA model becomes a non-linear program. We develop in this paper, an alternative approach for dealing with imprecise data in multi-component efficiency measurement in DEA that preserves the linearity of DEA model. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Data envelopment analysis; Optimization; Imprecise data
1. Introduction Data envelopment analysis (DEA) is a non-parametric method for evaluating the relative efficiency of decision making units (DMUs) on the basis of multiple inputs and outputs. The original DEA models Charnes et al. [2] assume
*
Corresponding author. E-mail address:
[email protected] (A. Amirteimoori).
0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.03.007
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that inputs and outputs are measured by exact values on a ratio scale. In most applications, a single measure of production efficiency provided by DEA methodology using exact values for all inputs and outputs, has been an adequate and useful means of comparing units and identifying best performance. However, these assumptions may not be true, in the sense that some inputs and outputs may be only known as in forms of bounded data or ordinal data, and in some cases, the DMU involved, may perform several different and clearly identifiable functions. In this case, inputs (known inputs and unknown inputs) are shared among those functions and all functions play an important role in producing some outputs. Hence, we have to determine the performance of DMUs in each component. The idea of measuring efficiency relative to certain components of a DMU is not new. F€ are and Grosskopf [7] look at a multi-stage process wherein intermediate products or outputs at one stage can be both final products and inputs to later stage of production. Another example is due to Cook et al. [4] and involves the evaluation of efficiency of components of a DMU where some inputs are shared among these components. Also, the idea of measuring efficiency with imprecise data is not new. Cooper et al. [10], for example, addressed the problem of imprecise data in DEA in its general form. An alternative approach for dealing with imprecise data in DEA have been proposed by Despotis and Smirils [6]. Afterward, Zhu [9], has shown an efficiency evaluation with imprecise data. These applications of efficiency evaluation with imprecise data, do not involve multi-component DMUs with shared inputs and outputs. The current paper, extends the method of Zhu [9] to deal with imprecise data and multi-component DMUs. We assume that some inputs are shared among all components of DMUs and also, all components play an important role in producing some outputs. In the sections to follow, we present a model for deriving an aggregate performance measure with imprecise data with accompanying component measures that make up that aggregate value. Also, we show herein that our treatment of shared inputs and outputs and imprecise data, leads to a linear than a non-linear model. The paper is structured as follows: A summary of basic DEA model and imprecise data proposed by Zhu [9] is given in Section 2. The next section of the paper, addresses a multicomponent efficiency measurement. Multi-component efficiency measurement with imprecise data is given in Section 4. The paper is then illustrated by a numerical example. Conclusions appear in Section 6.
2. DEA and imprecise data To describe the DEA efficiency measurement, we assume there are n DMUs and the performance of each DMU is characterized by a production process of m inputs ðxij ; i ¼ 1; 2; . . . ; mÞ to yield s outputs ðyrj ; r ¼ 1; 2; . . . ; sÞ. To estimate a DEA efficiency score of a specific pth DMU, we solve
A. Amirteimoori, S. Kordrostami / Appl. Math. Comput. 162 (2005) 1265–1277
Max
ep ¼
s X
1267
ur yrp ;
r¼1
subject to :
m X
vi xip ¼ 1;
i¼1 s X
ur yrj
r¼1
m X
ð1Þ vi xij 6 0;
j ¼ 1; . . . ; n;
i¼1
ur P ;
r ¼ 1; . . . ; s;
vi P ;
i ¼ 1; . . . ; m;
where > 0 is a non-archimedean constant to assure strongly efficient solutions. When xij (for some i) and yrj (for some r) are imprecise and unknown decision variables such as bounded and ordinal data, model (1) becomes a nonlinear program and is called Imprecise DEA (IDEA) (see Cooper et al. [5]). The bounded data can be expressed as y rj 6 yrj 6 yrj
and
xij 6 xij 6 xij ;
for r 2 BO; i 2 BI;
ð2Þ
where y rj and xij are lower bounds, yrj and xij are upper bounds, and BO and BI represent the associated sets containing bounded outputs and inputs, respectively. The weak ordinal data can be expressed as yrj 6 yrp
and
xij 6 xip ;
for all j 6¼ p; r 2 WO; i 2 WI;
ð3Þ
or to simplify the representation, yr1 6 yr2 6 6 yrp 6 yrn ; r 2 WO; xi1 6 xi2 6 6 xip 6 xin ; i 2 WI;
ð4Þ
where WO and WI represent the associated sets containing weak ordinal outputs and inputs, respectively. The strong ordinal data can be expressed as yr1 < yr2 < < yrp < yrn ;
r 2 SO;
xi1 < xi2 < < xip < xin ;
i 2 SI;
ð5Þ
where SO and SI represent the associated sets containing strong ordinal outputs and inputs, respectively. If we incorporate (2)–(5) into model (1), we have
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Max
ep ¼
s X
ur yrp ;
r¼1
subject to :
m X
vi xip ¼ 1;
i¼1 s X
ur yrj
r¼1
m X
vi xij 6 0;
i¼1
j ¼ 1; . . . ; n;
ð6Þ
xij 2 Ti ; yrj 2 Trþ ; ur P ;
r ¼ 1; . . . ; s;
vi P ;
i ¼ 1; . . . ; m;
where xij 2 Ti and yrj 2 Trþ represent any of or all of (2)–(5). The following theorem provides the theoretical foundation to the approach developed by Chen et al. [3] where the standard DEA model (1) is used to solve the IDEA model (6). Theorem 1. Suppose Ti and Trþ are given by (2). Then for DMUp , the optimal value to (6) can be achieved at yrp ¼ yrp and xip ¼ xip for DMUp and yrj ¼ y rj and xij ¼ xij for DMUj ðj 6¼ pÞ. Proof. See [8,9].
h
Theorem 1 shows that when DMUp is under evaluation, we can have a set of exact data via setting yrp ¼ yrp and xip ¼ xip for DMUp and yrj ¼ y rj and xij ¼ xij for DMUj ðj 6¼ pÞ. Zhu [9] proposed a procedure to convert the weak and strong ordinal data, into a set of exact data.
3. Multi-component performance With the increased emphasis on the components of a DMU, there is a need to provide a performance measurement tool with component-based information as part of the aggregate efficiency score. For notational purposes, let Ypð1Þ ; Ypð2Þ ; . . . ; YpðbÞ denote the set of each component transactions of DMUp in which ðiÞ ðiÞ ðiÞ YpðiÞ ¼ yp1 ; yp2 ; . . . ; ypJi ; i ¼ 1; . . . ; b:
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1269
Also, let Xpð1Þ ; Xpð2Þ ; . . . ; XpðbÞ denote I1 ; I2 ; . . . ; Ib -dimensional vectors of dedicated inputs to each components and XpðcÞ is a Ic -dimensional vector of shared inputs. All components are involved in producing the Jc -dimensional vector of output YpðcÞ . Some portion ai of the shared input XpðcÞ is allocated to the ithcomponent. Also, the ith component is involved in producing some bi of the shared Pb Pportion b output YpðcÞ . (Note that ai P 0; bi P 0 and i¼1 ai ¼ i¼1 bi ¼ 1.) In proposed model, ai and bi are decision variables which must be determined. By the above mentioned discussion, it is evident that the outputs Ypð1Þ ; Ypð2Þ ; . . . ; YpðbÞ ; YpðcÞ are produced from the inputs Xpð1Þ ; Xpð2Þ ; . . . ; XpðbÞ ; XpðcÞ . To estimate a DEA efficiency score of a specific pth DMU and its components, we solve the following linear program (See [1]): Max
b X
T
lðiÞ YpðiÞ þ
b X
i¼1
subject to :
b X
i¼1 T
V ðiÞ XpðiÞ þ
i¼1 b X
T
ðsi Þ YpðcÞ ; l
b X
T V ðsi Þ XpðcÞ ¼ 1;
i¼1 T
ðiÞ
lðiÞ Yj þ
i¼1
b X
T
ðcÞ
ðsi Þ Yj l
i¼1 b X
b X
T
ðiÞ
V ðiÞ Xj
i¼1
T ðcÞ V ðsi Þ Xj 6 0;
8j;
i¼1 T
ðiÞ
T
T
ðcÞ
ðiÞ
T
ðcÞ
ðsi Þ Yj V ðiÞ Xj V ðsi Þ Xj 6 0; lðiÞ Yj þ l
ð7Þ
i ¼ 1; . . . ; b; 8j; ðsi Þ 2 X1 ; i ¼ 1; . . . ; b; lðiÞ ; l
V ðiÞ ; V ðsi Þ 2 X2 ;
i ¼ 1; . . . ; b;
lðiÞ P ;
i ¼ 1; . . . ; b;
V ðiÞ P ;
i ¼ 1; . . . ; b;
ðsi Þ P bi ; l
i ¼ 1; . . . ; b;
V ðsi Þ P ai ;
i ¼ 1; . . . ; b:
The sets X1 and X2 are assurance regions defined by any restrictions imposed on multipliers (See Thompson et al. [8]). If we incorporate (2)–(5) into model (7), we have (to simplify the representation, we set V and l instead of V ðsi Þ and ðsi Þ ) l
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Max
" b Xh X
ðiÞ lðiÞ r yrp
þ
ðcÞ lðsiÞ r yrp
i
# i Xh ðiÞ ðiÞ ðsiÞ ðcÞ lr yrp þ lr yrp ; þ
r2BO
i¼1
r3BO
"
subject to:
# b i Xh i Xh X ðiÞ ðiÞ ðsiÞ ðcÞ ðiÞ ðiÞ ðsiÞ ðcÞ vt xtp þ vt xtp þ vt xtp þ vt xtp ¼ 1; t2BI
i¼1
t3BI
"
b Xh X i¼1
ðiÞ lðiÞ r yrj
þ
ðcÞ lðsiÞ r yrj
i
i Xh ðiÞ ðsiÞ ðcÞ lðiÞ þ r yrj þ lr yrj
r2BO
#
r3BO
" # b i Xh i Xh X ðiÞ ðiÞ ðsiÞ ðcÞ ðiÞ ðiÞ ðsiÞ ðcÞ vt xtj þ vt xtj þ vt xtj þ vt xtj 6 0; i¼1
t2BI
t3BI
j ¼ 1; . . . ; n; i Xh i Xh ðiÞ ðiÞ ðsiÞ ðcÞ ðsiÞ ðcÞ lðiÞ lðiÞ þ r yrj þ lr yrj r yrj þ lr yrj r2BO
r3BO
i Xh i Xh ðiÞ ðsiÞ ðcÞ ðiÞ ðiÞ ðsiÞ ðcÞ vðiÞ x x v x x þ v þ v 6 0; tj tj tj tj t t t t t2BI
t3BI
j ¼ 1; . . . ; n; i ¼ 1; . . . ; b; xij 2 Ti ; yrj 2 Trþ ;
lðiÞ ; lðsi Þ 2 X1 ;
V ðiÞ ; V ðsi Þ 2 X2 ;
i ¼ 1; . . . ; b; i ¼ 1; . . . ; b; ð8Þ
where xij 2 Ti and yrj 2 Trþ represent any of or all of (2)–(5). Theorem 2 provides the theoretical foundation to the approach developed by Zhu [9] where the DEA model (8) is used to solve the IDEA model. þ Theorem 2. Suppose / i and /r are given by (2). Then, for DMUp , the optimal value to (8) can be achieved at yrp ¼ yrp and xip ¼ xip for DMUp and yrj ¼ y rj and xij ¼ xij for DMUj ðj 6¼ pÞ.
Proof. We prove this theorem in following ratio model, which is equivalent to (8).
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1271
hP
h i P h ii Pb ðiÞ ðiÞ ðsiÞ ðcÞ ðiÞ ðiÞ ðsiÞ ðcÞ i¼1 r2BO lr yrp þ lr bi yrp þ r3BO lr yrp þ lr bi yrp hP h i P h ii ; Max eðaÞ p ¼ Pb ðiÞ ðiÞ ðsiÞ ðcÞ ðiÞ ðiÞ ðsiÞ ðcÞ t2BI vt xtp þ vt ai xtp þ t3BI vt xtp þ vt ai xtp i¼1 Pb h P subject to:
i P h ii ðiÞ ðcÞ ðiÞ ðcÞ ðsiÞ lrðiÞ yrj þ lrðsiÞ bi yrj þ r3BO lðiÞ r yrj þ lr bi yrj i P h ii 6 0; Pb h P h ðiÞ ðiÞ ðsiÞ ðcÞ ðiÞ ðiÞ ðsiÞ ðcÞ þ t3BI vt xtj þ vt ai xtj i¼1 t2BI vt xtj þ vt ai xtj i¼1
h
r2BO
j ¼ 1; . . .; n; h i P h i P ðcÞ ðcÞ ðiÞ ðiÞ ðsiÞ ðiÞ ðiÞ ðsiÞ l y þ l b y l y þ l b y þ i rj i rj r rj r r rj r r2BO r3BO i P h i 6 0; P h ðiÞ ðiÞ ðsiÞ ðcÞ ðiÞ ðiÞ ðsiÞ ðcÞ v x þ v a x v x þ v a x þ i tj i tj t t t t tj tj t2BI t3BI i ¼ 1; . .. ; b; j ¼ 1;. . . ; n; b X
ai ¼ 1;
i¼1 b X
bi ¼ 1;
i¼1
lðiÞ ; lðsi Þ 2 X1 ; i ¼ 1; . .. ; b; V ðiÞ ; V ðsi Þ 2 X2 ; i ¼ 1; . . . ; b; ai P0; i ¼ 1; . . . ; b; bi P0; i ¼ 1; . . . ; b: ð9Þ
Now, suppose the optimal value to (9) is achieved at yrj and x ij for DMUj ðj 6¼ pÞ such that y rj < yrj 6 yrj ðr 2 BOÞ and xij 6 x ij < xij ði 2 BIÞ. Pb h P
i P h ii ðcÞ ðiÞ ðiÞ ðsiÞ ðcÞ ðiÞ ðiÞ ðsiÞ l y þ l b y l y þ l b y þ i i rj rj i¼1 r rj r r r r2BO r3BO rj h i P h ii Pb h P ðiÞ ðiÞ ðsiÞ ðcÞ ðiÞ ðiÞ ðsiÞ ðcÞ v þ v a v x þ v a x x x þ i i t t t t tj tj tj tj i¼1 t2BI t3BI h i P h ii Pb h P ðcÞ ðcÞ ðiÞ ðiÞ ðsiÞ ðiÞ ðiÞ ðsiÞ l y þ l b y l y þ l b y þ i rj i rj i¼1 r rj r r rj r r2BO r3BO h i P h ii ; < P hP ðiÞ ðiÞ ðsiÞ ðcÞ ðiÞ ðiÞ ðsiÞ ðcÞ b þ t3BI vt xtj þ vt ai xtj i¼1 t2BI vt xtj þ vt ai xtj j 6¼ p;
h
for i ¼ 1; . . . ; b;
ð10Þ
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we have h i P h i ðcÞ ðiÞ ðiÞ ðsiÞ ðcÞ ðiÞ ðiÞ ðsiÞ l y þ l b y l y þ l b y þ rj rj i i r rj r r r r2BO r3BO rj h i P h i P ðiÞ ðiÞ ðsiÞ ðcÞ ðiÞ ðiÞ ðsiÞ ðcÞ x x v þ v a v x þ v a x þ i i t t t t tj tj tj tj t2BI t3BI
P
h i P h i ðcÞ ðcÞ ðiÞ ðiÞ ðsiÞ ðiÞ ðiÞ ðsiÞ l y þ l b y l y þ l b y þ i rj i rj r rj r r rj r r2BO r3BO h i P h i ; < P ðiÞ ðiÞ ðsiÞ ðcÞ ðiÞ ðiÞ ðsiÞ ðcÞ v x þ v a x v x þ v a x þ i tj i tj t t t t tj tj t2BI t3BI P
j 6¼ p;
ð11Þ
indicating that yrj ¼ y rj and xij ¼ xij are also feasible to (9). Thus, the optimal value to (8) can be achieved at yrj ¼ y rj and xij ¼ xij for DMUj ; ðj 6¼ pÞ. Next, suppose for DMUp , optimality is achieved at y rp 6 yrp < yrp ðr 2 BOÞ and xip < xip 6 xip ði 2 BIÞ with Pb h P
h
i P h ii ðiÞ ðiÞ ðsiÞ ðcÞ ðiÞ ðiÞ ðsiÞ ðcÞ l y þ l b y l y þ l b y þ i rp i rp r rp r r rp r i¼1 r2BO r3BO i P h ii : e ðaÞ ¼ P hP h p ðiÞ ðiÞ ðsiÞ ðcÞ ðiÞ ðiÞ ðsiÞ ðcÞ b v x þ v a x v x þ v a x þ i i t tp t tp t tp t tp i¼1 t2BI t3BI ð12Þ Obviously, when yrp ¼ yrp and xip ¼ xip , we have h i P h ii Pb h P ðiÞ ðiÞ ðsiÞ ðcÞ ðiÞ ðiÞ ðsiÞ ðcÞ l y þ l b y l y þ l b y þ i i rp rp r rp r rp r r r2BO r3BO i¼1 i P h ii eðaÞ ¼ P hP h p ðiÞ ðiÞ ðsiÞ ðcÞ ðiÞ ðiÞ ðsiÞ ðcÞ b v x þ v a x v x þ v a x þ i i t tp t tp t t tp tp t2BI t3BI i¼1 h i P h ii Pb h P ðiÞ ðiÞ ðsiÞ ðcÞ ðiÞ ðiÞ ðsiÞ ðcÞ l þ l b l y þ l b y y y þ i i r r r r rp rp rp rp i¼1 r2BO r3BO i P h ii : < P hP h ðiÞ ðiÞ ðsiÞ ðcÞ ðiÞ ðiÞ ðsiÞ ðcÞ b v x þ v a x v x þ v a x þ i i t t t t tp tp tp tp i¼1 t2BI t3BI ð13Þ A contradiction. This completes the proof.
h
4. A multi-component performance model with imprecis data Theorem 2 shows that when DMUp is under consideration, we can have a set of exact data. In this case we have the following model:
A. Amirteimoori, S. Kordrostami / Appl. Math. Comput. 162 (2005) 1265–1277
"
Max
b Xh X r2BO
i¼1
subject to:
i Xh i ðiÞ ðcÞ ðiÞ ðsiÞ ðcÞ lðiÞ lðiÞ yrp yrp þ lðsiÞ þ r r r yrp þ lr yrp
1273
#
r3BO
" # b i Xh i X Xh ðiÞ ðsiÞ ðcÞ ðiÞ ðsiÞ ðcÞ vðiÞ vðiÞ þ ¼ 1; t xtp þ vt xtp t xtp þ vt xtp i¼1 b X
"
i¼1
t2BI
t3BI
Xh
ðiÞ lðiÞ yrp r
þ
ðcÞ yrp lðsiÞ r
i
i Xh ðiÞ ðsiÞ ðcÞ lðiÞ y y þ þ l r rp r rp
r2BO
#
r3BO
" # b i Xh i X Xh ðiÞ ðiÞ ðsiÞ ðcÞ ðiÞ ðiÞ ðsiÞ ðcÞ vt xtp þ vt xtp þ vt xtp þ vt xtp 6 0; i¼1
t2BI
" b Xh X i¼1
t3BI
ðiÞ lðiÞ r y rj
þ
ðcÞ lðsiÞ r y rj
i
i Xh ðiÞ ðsiÞ ðcÞ lðiÞ þ r yrj þ lr yrj
r2BO
#
r3BO
" # b i Xh i X Xh ðiÞ ðcÞ ðiÞ ðcÞ ðsiÞ vðiÞ vðiÞ xtj þ vðsiÞ xtj þ 6 0; t t t xtj þ vt xtj i¼1
t2BI
t3BI
j ¼ 1; . . . ; n; j 6¼ p; i Xh i Xh ðiÞ ðcÞ ðiÞ ðsiÞ ðcÞ lðiÞ lðiÞ yrp yrp þ lðsiÞ þ r r r yrp þ lr yrp r2BO
r3BO
i Xh i Xh ðiÞ ðsiÞ ðcÞ ðiÞ ðiÞ ðsiÞ ðcÞ vðiÞ x x v x x v v t tp t tp t tp t tp 6 0; t2BI
t3BI
i ¼ 1; . . . ; b; i Xh i Xh ðiÞ ðiÞ ðsiÞ ðcÞ ðsiÞ ðcÞ lðiÞ lðiÞ þ r y rj þ lr y rj r yrj þ lr yrj r2BO
r3BO
i Xh i Xh ðiÞ ðsiÞ ðcÞ ðiÞ ðiÞ ðsiÞ ðcÞ vðiÞ v x x x x v v 6 0; tj tj tj tj t t t t t2BI
t3BI
j ¼ 1; . . . ; n; j 6¼ p; i ¼ 1; . . . ; b; ðiÞ ðs Þ
1 ; i ¼ 1; . . . ; b; l ;l i 2 X ðiÞ ðsi Þ
2 ; i ¼ 1; . . . ; b; 2X V ;V ð14Þ where yrj ðr 3 BOÞ and xtj ðt 3 BIÞ are exact data. 4.1. Converting weak ordinal data into exact data Consider DMUp . Suppose we solve model (8), when Ti and Trþ are in forms (4), we have yr1 6 yr2 6 6 yrp 6 yrn
r 2 WO;
x i1 6 x i2 6 6 x ip 6 x in
i 2 WI;
ð15Þ
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where yrj and x ij are a set of optimal solutions. Note that qyrj ; ðr 2 WOÞ and qx ij ; ði 2 WIÞ are also optimal for DMUp where q is a positive constant. Therefore we can always set yrj ¼ x ij ¼ 1: Then, without lose of generality, we have a set of optimal solutions on weak ordinal outputs and inputs as 0 6 yr1 6 6 yrp1 6 yrp ¼ 1 6 yrpþ1 6 yrn 6M
r 2 WO;
0 6 x i1
r 2 WI;
6
6 x ip1
6 x ip
¼
1 6 x ipþ1
6 x in
6M
ð16Þ
where M is sufficiently large. As Zhu [9] proposed, we can set up the following intervals yrj 2 ½0; 1 and xij 2 ½0; 1 ; for DMUj ðj ¼ 1; . . . ; p 1Þ; yrj 2 ½1; M and xij 2 ½1; M ; for DMUj ðj ¼ p þ 1; . . . ; nÞ:
ð17Þ
Based upon Theorem 2, we know that for r 2 WO and i 2 WI, a set of optimal solutions to (14) can be achieved at yrp ¼ xip ¼ 1 for DMUp and yrj ¼ 0 (lower bound, y rj Þ and xij ¼ 1 (upper bound xij ) for DMUj ðj ¼ 1; . . . ; p 1Þ and yrj ¼ 1 (lower bound, y rj ) and xij ¼ M (upper bound xij ) for DMUj ðj ¼ p þ 1; . . . ; nÞ. 4.2. Converting strong ordinal data into exact data Suppose (5) is included in (7) as before, we have a set of optimal solutions on strong ordinal inputs and outputs as yr1 < < yrp1 < yrp ¼ 1 < yrpþ1 < < yrn
r 2 SO;
x i1
r 2 SI:
< <
x ip1
<
x ip
¼1<
x ipþ1
< <
x in
ð18Þ
Cooper et al. [5] proposed that in order to solve model (1) with (5), use the following modification: yrp yrp1 P g
and
xip xip1 P g;
ð19Þ
where g is a positive constant. Zhu [9] showed that adding (19) into model (1) is equivalent to adding yrp yrp1 P
and
xip xip1 P ;
ð20Þ
where 0: It can be shown that it is also true if (20) is added into (8), because of units invariant property.
5. Example The model proposed in this paper, is applied to a real data set on 20 commercial bank branches. The data for this analysis are derived from oper-
A. Amirteimoori, S. Kordrostami / Appl. Math. Comput. 162 (2005) 1265–1277
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ations during the first six months of 2001 and include a number of cost and revenue figures. For narrow purposes of constructing the production model of banking performance, we use six variables from the data set as inputs and outputs. Also, because of other purposes, we consider only two components that involving deposits and services, which are called hereafter component 1 and component 2, respectively. Inputs include number of staff, number of computer terminals, and square meters of premises; outputs are amount of deposits, amount of loans, and customer satisfaction. (All monetary variables are stated in ten thousands of current Iranian Rial.) It is evident that all inputs
Table 1 Input and output measures used in application Inputs
Outputs
Component 1
Staff
Computer terminals
Space m2
Deposits
Customer satisfaction
Component 2
Staff
Computer terminals
Space m2
Loans
Customer satisfaction
Table 2 The raw data set DMUj
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Exact
Bounded data
Staff
Computer terminals
Space m2
Deposits
Loans
Customer satisfaction yj yj
10 9 8 8.5 11 9.5 5 10 7 7.5 8.5 9 11 12 16.5 4.5 10 15 9 6.5
12.61 11.29 7.66 8.81 7.43 9.88 10.33 13.82 15.10 9.74 11.31 12.28 20.55 15.78 22.69 21.34 14.78 15.52 23.45 20.11
31.10 34.46 21.26 31.72 41.00 44.26 36.78 40.54 65.46 61.08 24.58 23.04 80.38 38.54 90.82 85.04 42.90 60.34 98.02 70.00
700 600 300 500 400 650 950 1000 850 700 450 900 1000 800 700 550 700 1500 600 900
60 40 50 30 250 470 860 900 850 410 820 630 950 470 530 620 430 850 670 430
3 2 5 8 10 11 12 7 11 9 3 6 4 5 10 11 7 4 9 15
8 6 6 10 11 11 15 10 14 12 7 9 4 8 12 4 5 7 11 15
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Table 3 The results DMUj
Efficiency Component 1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0.6491 0.5851 0.6893 0.7784 1.0000 0.7641 1.0000 0.8473 0.6654 0.8452 0.6986 1.0000 0.5261 0.6355 0.3625 0.6427 0.5582 1.0000 0.4040 0.7679
Component 2 0.6170 0.3863 0.6871 0.7717 1.0000 0.7604 1.0000 0.8185 0.6990 0.8385 1.0000 1.0000 0.5524 0.4141 0.3616 0.8004 0.3628 0.6573 0.4283 0.7608
Aggregate 0.6394 0.5254 0.6879 0.7746 1.0000 0.7630 1.0000 0.8387 0.6889 0.8432 0.9096 1.0000 0.5445 0.5691 0.3619 0.7531 0.4996 0.8972 0.4210 0.7641
are shared among components 1 and 2. Also, The deposits are produced by component 1 and the loans are produced by component 2. Also, both components play an important role for the customer satisfaction. The chosen input and output measures that are used in the application, are summarized in Table 1. The relevant data are displayed Table 2. If the method discussed in Section 4 is applied to this set of data, the results are summarized in Table 3.
6. Conclusions The issue of dealing with imprecise data in DEA is an important topic. The existing IDEA approaches, have been focused on a single measure of efficiency. We developed in this paper, an approach for dealing with multi-component efficiency measurement with imprecise data in DEA. As in IDEA, we transform a non-linear DEA model to a linear programming equivalent. The DEA model presented here, can be used for the analysis of any real situations where a significant number of precise and imprecise inputs and outputs are included, but where management views the DMU as a multi-component DMU. In the current paper, we developed the method proposed by Zhu [9] to one that
A. Amirteimoori, S. Kordrostami / Appl. Math. Comput. 162 (2005) 1265–1277
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determines a multi-component efficiency measurement with precise and imprecise data.
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