Fuzzy data envelopment analysis: A fuzzy expected value approach

Expert Systems with Applications 38 (2011) 11678–11685

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Fuzzy data envelopment analysis: A fuzzy expected value approach q Ying-Ming Wang a,⇑, Kwai-Sang Chin b a b

School of Public Administration, Fuzhou University, Fuzhou 350002, PR China Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Tong, Hong Kong

a r t i c l e

i n f o

Keywords: Fuzzy data envelopment analysis Fuzzy expected values Double frontier analysis (DFA) Flexible manufacturing system

a b s t r a c t Performance assessment often has to be conducted under uncertainty. This paper proposes a ‘‘fuzzy expected value approach’’ for data envelopment analysis (DEA) in which fuzzy inputs and fuzzy outputs are first weighted, respectively, and their expected values then used to measure the optimistic and pessimistic efficiencies of decision making units (DMUs) in fuzzy environments. The two efficiencies are finally geometrically averaged for the purposes of ranking and identifying the best performing DMU. The proposed fuzzy expected value approach and its resultant models are illustrated with three numerical examples, including the selection of a flexible manufacturing system (FMS). Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Traditional data envelopment analysis (DEA) (Charnes, Cooper, & Rhodes, 1978) requires crisp input and output data, which may not always be available in real word applications. Significant efforts have been made to handle fuzzy input and fuzzy output data in DEA. For example, Sengupta (1992) incorporated fuzziness into DEA by defining tolerance levels for both the objective function and violations of constraints and proposed a fuzzy mathematical programming approach. Triantis and Girod (1998) transformed fuzzy input and fuzzy output data into crisp data using membership function values and suggested a mathematical programming approach in which efficiency scores were computed for different values of membership functions and then averaged. Guo and Tanaka (2001) converted fuzzy constraints such as fuzzy equalities and fuzzy inequalities into crisp constraints by predefining a possibility level and using the comparison rule for fuzzy numbers and presented a fuzzy CCR model. León, Liern, Ruiz, and Sirvent (2003) suggested a fuzzy BCC model based on the same idea. Lertworasirikul, Fang, Joines, and Nuttle (2003) proposed a possibility DEA model for fuzzy DEA. In the special case that fuzzy data are trapezoidal fuzzy numbers, the possibility DEA model became a linear programming (LP) model. They (Lertworasirikul, Fang, Joines, & Nuttle, 2003) also presented a credibility approach as an alternative way for solving fuzzy DEA problems. The possibility and credibility approaches were further extended to fuzzy BCC model in Lertworasirikul, Fang, Nuttle, and Joines (2003) by the

q The work described in this paper was supported by the National Natural Science Foundation of China (NSFC) under the Grant Nos. 70771027 and 70925004 and also partially supported by a grant from CityU (Project No. 7002571). ⇑ Corresponding author. Tel.: +86 591 22866681; fax: +86 591 22866677. E-mail addresses: [email protected], [email protected] (Y.-M. Wang).

0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.03.049

same authors. Wu, Yang, and Liang (2006) applied the possibility DEA model for efficiency analysis of cross-region bank branches in Canada. Garcia, Schirru, and Melo (2005) utilized the possibility DEA model for failure mode and effects analysis (FMEA) and presented a fuzzy DEA approach to determining ranking indices among failure modes. Wen and Li (2009) employed credibility measure to represent fuzzy CCR model as an uncertain programming and solved it with a hybrid intelligent algorithm which integrates fuzzy simulations and genetic algorithms. Kao and Liu (2000a, 2000b, 2003, 2005) transformed fuzzy input and fuzzy output data into intervals by using a-level sets and Zadeh’s extension principle, and built a family of crisp DEA models for the intervals. Based on their crisp DEA models for a-level sets, Liu (2008) and Liu and Chuang (2009) took further into consideration the concept of assurance region (AR) and developed a fuzzy DEA/AR model for the selection of flexible manufacturing systems (FMSs) and the assessment of university libraries, respectively. Saati, Menariani, and Jahanshahloo (2002) defined fuzzy CCR model as a possibilistic-programming problem and transformed it into an interval programming by specifying a a-level set. Their approach was further extended in Saati and Memariani (2005) so that all decision making units (DMUs) could be evaluated with a common set of weights under a given a-level set. Entani, Maeda, and Tanaka (2002) and Wang, Greatbanks, and Yang (2005) also changed fuzzy input and fuzzy output data into intervals by using alevel sets, but suggested two different interval DEA models. Dia (2004) proposed a fuzzy DEA model based upon fuzzy arithmetic operations and fuzzy comparisons between fuzzy numbers. The model requires the decision maker (DM) to specify a fuzzy aspiration level and a safety a-level so that the fuzzy DEA model could be transformed into a crisp DEA model for solution. Wang, Luo, and Liang (2009) constructed two fuzzy DEA models from the perspective of fuzzy arithmetic to deal with fuzziness in input

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and output data in DEA. The two fuzzy DEA models were both formulated as linear programs and could be solved to determine fuzzy efficiencies of DMUs. Triantis (2003) introduced a fuzzy DEA approach to calculate fuzzy non-radial technical efficiencies and implemented the approach in a newspaper preprint insertion manufacturing process. Soleimani-damaneh, Jahanshahloo, and Abbasbandy (2006) addressed some computational and theoretical pitfalls of the fuzzy DEA models developed in Kao and Liu (2000a), León et al. (2003) and Lertworasirikul et al. (2003) and provided a fuzzy DEA model to yield crisp efficiencies for the DMUs with fuzzy input and fuzzy output data. Jahanshahloo, Soleimani-damaneh, and Nasrabadi (2004) extended a slack-based measure (SBM) of efficiency in DEA to fuzzy settings and developed a two-objective nonlinear DEA model for fuzzy DEA. Existing fuzzy DEA models exhibit some drawbacks. For instance, fuzzy DEA models derived from the direct fuzzification of crisp DEA models ignore the fact that a fuzzy fractional program cannot be transformed into an LP model in the traditional way that we do for a crisp fractional program. Fuzzy DEA models built on the basis of a-level sets require the solution of a series of LP models and thus considerable computational efforts. Fuzzy DEA models constructed from the perspective of fuzzy arithmetic demand a rational yet easy-to-use ranking approach for fuzzy efficiencies. To overcome these drawbacks, we propose in this paper a ‘‘fuzzy expected value approach’’ for fuzzy DEA, which first weights fuzzy inputs and fuzzy outputs, respectively, and then utilizes their expected values for measuring the performances of DMUs in fuzzy environments. The paper is organized as follows. Section 2 introduces the measures of fuzzy expected values and develops fuzzy expected value models for fuzzy DEA. Section 3 illustrates the developed fuzzy expected value models with three numerical examples, including the selection of a FMS. Section 4 concludes the paper. 2. Fuzzy expected values and fuzzy DEA models Fuzzy numbers are convex fuzzy sets, characterized by given intervals of real numbers, each interval with a grade of membership between 0 and 1. The most commonly used fuzzy numbers are triangular and trapezoidal fuzzy numbers defined by the following membership functions, respectively:

8 > < ðx  aÞ=ðb  aÞ; a 6 x 6 b; leA ðxÞ ¼ ðd  xÞ=ðd  bÞ; b 6 x 6 d; > 1 : 0; otherwise;

ð1Þ

8 ðx  aÞ=ðb  aÞ; > > > < 1; leA ðxÞ ¼ > 2 ðd  xÞ=ðd  cÞ; > > : 0;

ð2Þ

a 6 x 6 b; b 6 x 6 c; c 6 x 6 d; otherwise:

For brevity, triangular and trapezoidal fuzzy numbers are often denoted as (a, b, d) and (a, b, c, d). It is evident that triangular fuzzy numbers are special cases of trapezoidal fuzzy numbers with b = c. For any two positive trapezoidal fuzzy numbers e ¼ ðaL ; aM ; aN ; aU Þ and B e ¼ ðbL ; bM ; bN ; bU Þ, fuzzy addition and fuzzy A e e are respectively defined as multiplication on A and B eþB eB e ¼ ðaL þ bL ; aM þ bM ; aN þ bN ; aU þ bU Þ e  ðaL bL ; A and A aM bM ; aN bN ; aU bU Þ. Let n be a fuzzy variable with a membership function l: R ! ½0; 1 and r be a real number. The possibility and the necessity of {n P r} are respectively defined by

Posfn P rg ¼ sup lðxÞ; xPr

ð3Þ

Necfn P rg ¼ 1  Posfn < rg ¼ 1  sup lðxÞ;

ð4Þ

x
which show respectively the possibility and the necessity degrees to which n is not smaller than r. Pos and Nec are a pair of dual fuzzy measures in the sense that Pos{A} = 1  Nec{AC} with AC is the complement of A. Based upon the possibility and the necessity measures, credibility measure is defined as

1 ðPosfn P rg þ Necfn P rgÞ: 2

Crfn P rg ¼

ð5Þ

The fuzzy expected value of n can thus be defined as (Liu & Liu, 2002)

E½n ¼

Z

1

Crfn P rgdr  0

Z

0

Crfn 6 rgdr:

ð6Þ

1

It has been shown (Liu & Liu, 2002) that if fuzzy variable n is replaced with a random variable whose probability density function is / and Cr is replaced with the probability measure Pr, then there R1 R0 R þ1 exists 0 Prfn P rgdr  1 Prfn 6 rgdr ¼ 1 x/ðxÞdx, which is exactly the expected value of the random variable n. It is also shown (Liu & Liu, 2002) that if n is a trapezoidal fuzzy variable (r1, r2, r3, r4), then the expected value of n is (1/4)(r1 + r2 + r3 + r4). In particular, if n is a triangular fuzzy variable (r1, r2, r3), then the expected value of n is (1/4)(r1 + 2r2 + r3). Suppose we have n DMUs to be evaluated in terms of m inputs and s outputs. Let xij (i = 1, . . . , m) and yrj (r = 1, . . . , s) be the input and output data of DMUj (j = 1, . . . , n). Without loss of generality, all input and output data xij and yrj are assumed to be uncertain and character  N U and ized by trapezoidal fuzzy numbers ~ xij ¼ xLij ; xM ij ; xij ; xij   L M N U L L ~rj ¼ yrj ; yrj ; yrj ; yrj with xij P 0 and yrj P 0 for i = 1 to m, r = 1 to y s, and j = 1 to n. Crisp data and triangular fuzzy data are treated as ~rj with special cases of trapezoidal fuzzy data ~ xij and y N U L M N U M N M N ¼ x ¼ x ; y ¼ y ¼ y ¼ y , and x ¼ x ; xLij ¼ xM ij ij ij rj rj rj rj ij ij yrj ¼ yrj , respectively. The total fuzzy weighted output (FWO) and the total fuzzy weighted input (FWI) of DMUj are given by

FWOj ¼ FWIj ¼

s X

s X 

~r y ~rj ¼ u

r¼1 m X

r¼1 m X

i¼1

i¼1

v~ i ~xij ¼



  L M N U N U uLr ; uM r ; ur ; ur  yrj ; yrj ; yrj ; yrj ;

v Li ; v Mi ; v Ni ; v Ui



ð7Þ

  N U  xLij ; xM ij ; xij ; xij ;

ð8Þ

    N U N U ~ r ¼ uLr ; uM where u and v~ i ¼ v Li ; v M are fuzzy r ; ur ; ur i ; vi ; vi ~rj , respectively. Accordweights for fuzzy input ~xij and fuzzy output y ing to fuzzy addition and fuzzy multiplication operations on two positive fuzzy numbers, (7) and (8) can be approximately expressed as

FWOj 

s X

uLr yLrj ;

r¼1

FWIj 

m X i¼1

v Li xLij ;

s X

M uM r yrj ;

r¼1 m X i¼1

v Mi xMij ;

s X

uNr yNrj ;

r¼1 m X i¼1

v Ni xNij ;

s X

! uUr yUrj

r¼1 m X

;

ð9Þ

;

ð10Þ

!

v Ui xUij

i¼1

which can be viewed as two trapezoidal fuzzy variables, whose expected values can therefore be determined as

EðFWOj Þ ¼ ¼

s s s s X X X 1 X M uLr yLrj þ uM uNr yNrj þ uUr yUrj r yrj þ 4 r¼1 r¼1 r¼1 r¼1 s   1X M N N U U uLr yLrj þ uM r yrj þ ur yrj þ ur yrj ; 4 r¼1

!

ð11Þ

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m m m m X X X   1 X E FWIj ¼ v Li xLij þ v Mi xMij þ v Ni xNij þ v Ui xUij 4 i¼1 i¼1 i¼1 i¼1

!

m   1X v Li xLij þ v Mi xMij þ v Ni xNij þ v Ui xUij : ¼ 4 i¼1

Minimize hworst ¼ 0 ð12Þ

Subject to

v

Pv Pv

M i

P v P 0; L i

j ¼ 1; . . . ; n;

ð13Þ

ð14Þ

r ¼ 1; . . . ; s; i ¼ 1; . . . ; m:

L uUr P uNr P uM r P ur P 0; r ¼ 1; . . . ; s;

i ¼ 1; . . . ; m: ð15Þ

With the help of the Charnes–Cooper transformation (Charnes & Cooper, 1962), the above two fractional programming models can be converted into the LPs below for solution:

Maximize hbest ¼ 0

s X 

M N N U U uLr yLr0 þ uM r yr0 þ ur yr0 þ ur yr0



r¼1

Subject to

m X 

v Li xLi0 þ v Mi xMi0 þ v Ni xNi0 þ v Ui xUi0



¼ 1;

i¼1 s   X M N N U U uLr yLrj þ uM r yrj þ ur yrj þ ur yrj m  X

v Li xLij þ v Mi xMij þ v Ni xNij þ v Ui xUij

v Ui P v Ni P v Mi P v Li P 0;



6 0;

j ¼ 1; . . . ; n;

¼ 1;

M N N U U uLr yLrj þ uM r yrj þ ur yrj þ ur yrj



m  X

v Li xLij þ v Mi xMij þ v Ni xNij þ v Ui xUij



P 0; j ¼ 1; . . . ; n;

v Ui P v Ni P v Mi P v Li P 0;

i ¼ 1; . . . ; m: ð17Þ

If hbest ¼ 1, we refer to DMU0 as optimistic efficient; otherwise, 0 it is referred to as optimistic inefficient. As such, if hworst ¼ 1, we 0 call DMU0 pessimistic inefficient; otherwise, we refer to it as pessimistic efficient. Since the two efficiencies are measured from different perspectives within different efficiency ranges, they are usually not comparable in magnitude. In other words, the pessimistic efficiency hworst will not be less than the optimistic efficiency 0 hbest 0 . According to Wang, Chin, and Yang (2007), the optimistic and pessimistic efficiencies measure the performances of the n DMUs in two extreme cases, which are either the best or the worst. Theoretically, the two extreme efficiencies should be considered at the same time to give an overall assessment of the performance of each of the n DMUs. This is what we called double frontier analysis (DFA). To this end, Wang et al. (2007) suggested a geometric average efficiency index defined by

hGeometric ¼ j

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hbest  hworst ; j j

j ¼ 1; . . . ; n;

ð18Þ

which is believed to measure the overall performance of DMUj. First, hGeometric is the geometric average value of two efficiencies j and is thus of the meaning of efficiency just like the average cross-efficiency. Next, hGeometric is the integration of optimistic and j pessimistic efficiencies and is thus more comprehensive than either of them. Interested readers may refer to Wang et al. (2007) for theoretical justifications for preferring geometric average efficiency over arithmetic or other alternatives for the purposes of ranking DMUs. In this paper, we adopt the geometric average efficiency to measure the overall performances of the n DMUs in fuzzy environments. We point out that crisp inputs and crisp outputs should be weighted with crisp weights. This requires fuzzy weights     v~ i ¼ v Li ; v Mi ; v Ni ; v Ui and u~r ¼ uLr ; uMr ; uNr ; uUr to meet the conditions L U N M L of v Ui ¼ v Ni ¼ v M i ¼ v i and ur ¼ ur ¼ ur ¼ ur for crisp inputs and L crisp outputs. If we set uUr ¼ uNr ¼ uM and r ¼ ur ¼ ur P 0 U N M L v i ¼ v i ¼ v i ¼ v i ¼ v i P 0 for all r = 1, . . . , s and i = 1, . . . , m, then we have

  Ps N U yLrj þ yM ~rj Þ rj þ yrj þ yrj ur Eðy   ¼ Pr¼1 ; hj ¼ P m m ~ L M N U v Eð x i ij Þ i¼1 i¼1 v i xij þ xij þ xij þ xij

j ¼ 1; . . . ; n; ð19Þ

r ¼ 1; . . . ; s; i ¼ 1; . . . ; m ð16Þ

and



L uUr P uNr P uM r P ur P 0; r ¼ 1; . . . ; s;

r¼1 ur

i¼1 L uUr P uNr P uM r P ur P 0;

v Li xLi0 þ v Mi xMi0 þ v Ni xNi0 þ v Ui xUi0

Ps

r¼1





i¼1

From pessimistic perspective, h0 can be determined with the following model, which measures the worst performance of DMU0 relative to the other DMUs:  Ps  L L M N N U U ur yr0 þ uM r yr0 þ ur yr0 þ ur yr0  Minimize hworst ¼ Pr¼1 m  L L 0 M M N N U U i¼1 v i xi0 þ v i xi0 þ v i xi0 þ v i xi0  Ps  L L M M N N U U r¼1 ur yrj þ ur yrj þ ur yrj þ ur yrj worst  P 1; j ¼ 1; . . . ; n; Subject to hj ¼P  m L L M M N N U U i¼1 v i xij þ v i xij þ v i xij þ v i xij

v Ui P v Ni P v Mi P v Li P 0;

s  X



j ¼ 1; . . . ; n;

N i

m X

r¼1

 Ps  L L M M N N U U r¼1 ur yr0 þ ur yr0 þ ur yr0 þ ur yr0   P Maximize hbest ¼ m 0 L L M M N N U U i¼1 v i xi0 þ v i xi0 þ v i xi0 þ v i xi0   Ps uLr yLrj þ uM yM þ uNr yNrj þ uUr yUrj r r¼1 rj  6 1; ¼P  Subject to hbest j m L L M M N N U U i¼1 v i xij þ v i xij þ v i xij þ v i xij

U i



i¼1

which is a crisp function, making the performances of the n DMUs easy to be measured and compared. hj can be measured from different perspectives. For a DMU under evaluation, say DMU0, from the optimistic point of view, h0 can be determined with the following fractional programming model, which measures the best relative performance of DMU0:

L uUr P uNr P uM r P ur P 0;

M N N U U uLr yLr0 þ uM r yr0 þ ur yr0 þ ur yr0

r¼1

Accordingly, we define the efficiency of DMUj in fuzzy environments as

  E FWOj  hj ¼  E FWIj  Ps  L L M M N N U U r¼1 ur yrj þ ur yrj þ ur yrj þ ur yrj ; ¼P  m L L M M N N U U i¼1 v i xij þ v i xij þ v i xij þ v i xij

s X 

~rj Þ are respectively the expected values of fuzzy where Eð~xij Þ and Eðy ~rj , and can be seen as the crisp input and input ~xij and fuzzy output y output values of DMUj. In this case, fuzzy expected value models (16) and (17) become the traditional DEA models, as shown below:

Y.-M. Wang, K.-S. Chin / Expert Systems with Applications 38 (2011) 11678–11685

Maximize hbest ¼ 0

s X

11681

  U ~rj ¼ yLrj ; yM fuzzy output y rj ; yrj , respectively.

~r0 Þ ur Eðy

r¼1

Subject to

m X

3. Numerical examples

v i Eð~xi0 Þ ¼ 1;

ð20Þ

i¼1 s X

~rj Þ  ur Eðy

r¼1

m X

v i Eð~xij Þ 6 0;

In this section, we provide three numerical examples to illustrate the applications of the proposed fuzzy expected value models in performance assessment.

j ¼ 1; . . . ; n;

i¼1

ur ; v i P 0;

r ¼ 1; . . . ; s; i ¼ 1; . . . ; m

and s X

Minimize hworst ¼ 0

~r0 Þ ur Eðy

r¼1

Subject to

m X i¼1 s X

v i Eð~xi0 Þ ¼ 1; ~rj Þ  ur Eðy

r¼1

m X

ð21Þ

v i Eð~xij Þ P 0;

j ¼ 1; . . . ; n;

i¼1

ur ; v i P 0;

r ¼ 1; . . . ; s; i ¼ 1; . . . ; m:

These two LP models have less flexibility and freedom than models (16), (17) in selecting the most favorable and the least favorable weights for each DMU. As a result, the hbest determined by model 0 (20) will not be higher than that by (16). For the same reason, the hworst in model (21) will not be less than that in (17). This fact will 0 be illustrated with numerical examples in the next section. Since triangular fuzzy numbers and crisp numbers are special cases of trapezoidal fuzzy numbers, LP models (16), (17), (20) and (21) are also applicable to crisp input and output data as well as triangular fuzzy input and fuzzy output data. Take triangular fuzzy data for example. LP models (16) and (17) in this situation can be expressed as

Maximize hbest ¼ 0

s X 

M U U uLr yLr0 þ 2uM r yr0 þ ur yr0



r¼1

Subject to

m X 

v Li xLi0 þ 2v Mi xMi0 þ v Ui xUi0

i¼1 s  X

M U U uLr yLrj þ 2uM r yrj þ ur yrj



¼ 1;



r¼1



m  X

v Li xLij þ 2v Mi xMij þ v Ui xUij



6 0;

j ¼ 1; . . . ; n;

i¼1 L uUr P uM r P ur P 0;

v

U i

Pv

M i

P v P 0; L i

r ¼ 1; . . . ; s; i ¼ 1; . . . ; m ð22Þ

and s X 

Minimize hworst ¼ 0

M U U uLr yLr0 þ 2uM r yr0 þ ur yr0



r¼1

Subject to

m X 



i¼1 s  X



v Li xLi0 þ 2v Mi xMi0 þ v Ui xUi0 M U U uLr yLrj þ 2uM r yrj þ ur yrj

¼ 1;

r¼1



m  X

v Li xLij þ 2v Mi xMij þ v Ui xUij



P 0;

j ¼ 1; . . . ; n;

i¼1 L uUr P uM r P ur P 0;

v Ui P v Mi P v Li P 0;

r ¼ 1; . . . ; s; i ¼ 1; . . . ; m; ð23Þ



   U U ~ r ¼ uLr ; uM where u and v~ i ¼ v Li ; v M are triangular fuzzy r ; ur i ; vi   L M weights for triangular fuzzy input ~xij ¼ xij ; xij ; xUij and triangular

Example 1. The performances of manufacturing enterprises depend mainly upon their manufacturing costs (MCs), numbers of employees (NOEs), gross output values (GOVs), and product qualities (PQs), where MC and NOE are inputs and GOV and PQ are outputs. Consider eight manufacturing enterprises (DMUs) in China, which are to be evaluated in terms of the two inputs and the two outputs. Data on GOV, MC and PQs are estimated as fuzzy numbers due to unavailability at the time of assessment. The input and output data for the eight manufacturing enterprises are documented in Table 1. For the input and output data in Table 1, since NOE is a crisp input, it is weighted with a crisp weight. Fuzzy weights are only used for input MC and the two outputs. By solving LP models (16) and (17) for each of the eight manufacturing enterprises, we obtain the optimistic and pessimistic efficiencies of the eight manufacturing enterprise, as shown in Table 2 together with their geometric averages. It is seen from the table that enterprises A, B and E are all optimistic efficient, but E is also rated as pessimistic inefficient. So, its overall performance cannot be as good as those of A and B. The geometric average efficiencies in the table show the ranking of the eight manufacturing enterprises as B  A  G  E  C  H  D  F, which is somewhat different from the ranking B  E  A  G  C  H  D  F obtained by Wang et al. (2009) through the solution of a fuzzy DEA model based on fuzzy arithmetic. This is because the fuzzy DEA model based upon fuzzy arithmetic in Wang et al. (2009) considers only the optimistic efficiencies of the eight enterprises while ignores their pessimistic efficiencies. When both the optimistic and pessimistic efficiencies are taken into account, E performs worse than not only A and B, but also G, which is an optimistic inefficient enterprise. In Table 3, we show the efficiencies of the eight manufacturing enterprises obtained by solving LP models (20) and (21) for each manufacturing enterprise. These two models utilize crisp weights for all inputs and outputs. From the table it is seen that the use of crisp weights for fuzzy inputs and fuzzy outputs achieves exactly the same ranking for the eight manufacturing enterprises as the use of fuzzy weights for fuzzy inputs and fuzzy outputs in Table 2, but the efficiencies under crisp weights are slightly different from those under fuzzy weights. In particular, the optimistic efficiencies in Table 3 are less than or equal to those in Table 2, whereas the pessimistic efficiencies in Table 3 are not less than those in Table 2. This is because crisp weights have less freedom and less flexibility than fuzzy weights in determining their values. Example 2. Ten DMUs are to be evaluated in light of three fuzzy inputs and two fuzzy outputs, whose data for the 10 DMUs are shown in Table 4. For the fuzzy data in Table 4, Dia (2004) employed their fuzzy DEA model based upon fuzzy arithmetic operations and fuzzy comparisons between fuzzy numbers to rate the 10 DMUs as DMU7  DMU6  DMU4  DMU10  DMU1  DMU2  DMU5  DMU3  DMU8  DMU9. We now use fuzzy weights and crisp weights to solve LP models (20)–(23) for each of the 10 DMUs, respectively. The results are recorded in Tables 5 and 6. From the two tables, it is seen that the two rankings are different from the ranking obtained by Dia. This is because Dia’s ranking was generated just based on the optimistic efficiencies of the 10 DMUs, whereas

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Table 1 Input and output data for eight manufacturing enterprises (Wang et al., 2009). Enterprises (DMUs)

Inputs

Outputs

MC

NOE

GOV

PQ

A B C D E F G H

(2120, 2170, 2210) (1420, 1460, 1500) (2510, 2570, 2610) (2300, 2350, 2400) (1480, 1520, 1560) (1990, 2030, 2100) (2200, 2260, 2300) (2400, 2460, 2520)

1870 1340 2360 2020 1550 1760 1980 2250

(14500, 14790, 14860) (12470, 12720, 12790) (17900, 18260, 18400) (14970, 15270, 15400) (13980, 14260, 14330) (14030, 14310, 14400) (16540, 16870, 17000) (17600, 17960, 18100)

(3.1, 4.1, 4.9) (1.2, 2.1, 3.0) (3.3, 4.3, 5.0) (2.7, 3.7, 4.6) (1.0, 1.8, 2.7) (1.6, 2.6, 3.6) (2.4, 3.4, 4.4) (2.6, 3.6, 4.6)

Table 2 Efficiencies of the eight manufacturing enterprises under fuzzy weights. Enterprises (DMUs)

  Optimistic efficiency hbest 0

  Pessimistic efficiency hworst 0

  Geometric average efficiency hGeometric 0

Ranking

A B C D E F G H

1.0000 1.0000 0.9606 0.9091 1.0000 0.8852 0.9593 0.9149

1.0423 1.1253 1.0000 1.0000 1.0000 1.0000 1.0673 1.0000

1.0209 1.0608 0.9801 0.9535 1.0000 0.9409 1.0119 0.9565

2 1 5 7 4 8 3 6

Table 3 Efficiencies of the eight manufacturing enterprises under crisp weights. Enterprises (DMUs)

  Optimistic efficiency hbest 0

  Pessimistic efficiency hworst 0

  Geometric average efficiency hGeometric 0

Ranking

A B C D E F G H

1.0000 1.0000 0.9553 0.9076 1.0000 0.8829 0.9582 0.9137

1.0453 1.1255 1.0121 1.0000 1.0000 1.0000 1.0747 1.0054

1.0224 1.0609 0.9833 0.9527 1.0000 0.9396 1.0148 0.9585

2 1 5 7 4 8 3 6

Table 4 Fuzzy input and fuzzy output data for 10 DMUs (Dia, 2004). DMUs

Input 1

Input 2

Input 3

Output 1

Output 2

1 2 3 4 5 6 7 8 9 10

(7, 10, 12) (11, 15, 19) (12, 12, 15) (5, 10, 13) (15, 18, 21) (5, 7, 8) (6, 10, 15) (9, 12, 17) (10, 14, 18) (7, 8, 9)

(0.65, 0.8, 0.95) (0.7, 1.0, 1.25) (1.7, 2.1, 2.4) (0.45, 0.6, 0.82) (0.35, 0.5, 0.7) (0.6, 0.9, 1.35) (0.25, 0.3, 0.35) (1.1, 1.5, 1.75) (0.65, 0.8, 1.15) (0.75, 0.9, 1.27)

(490, 540, 575) (455, 480, 510) (475, 510, 525) (400, 420, 435) (520, 600, 645) (495, 520, 565) (450, 500, 560) (515, 550, 605) (540, 570, 585) (420, 450, 470)

(0.75, 0.9, 1.25) (0.83, 1.0, 1.31) (0.7, 0.8, 0.95) (0.71, 0.9, 1.05) (0.55, 0.7, 0.92) (0.8, 1.0, 1.17) (0.68, 0.8, 0.97) (0.63, 0.75, 0.83) (0.6, 0.65, 0.71) (0.7, 0.85, 0.9)

(65, 70, 97) (77, 95, 103) (71, 75, 93) (85, 90, 100) (67, 80, 97) (45, 50, 56) (63, 70, 81) (69, 75, 87) (52, 55, 69) (85, 90, 113)

Table 5 Efficiencies of the 10 DMUs under fuzzy weights. DMUs

  Optimistic efficiency hbest 0

  Pessimistic efficiency hworst 0

  Geometric average efficiency hGeometric 0

Ranking

1 2 3 4 5 6 7 8 9 10

1.00000 1.00000 0.76946 1.00000 0.88667 1.00000 1.00000 0.64763 0.54516 1.00000

1.35232 1.35234 1.00000 1.83446 1.00000 1.00000 1.24251 1.00000 1.00000 1.40874

1.16289 1.16290 0.87719 1.35442 0.94163 1.00000 1.11468 0.80476 0.73835 1.18690

4 3 8 1 7 6 5 9 10 2

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Y.-M. Wang, K.-S. Chin / Expert Systems with Applications 38 (2011) 11678–11685 Table 6 Efficiencies of the 10 DMUs under crisp weights. DMUs

  Optimistic efficiency hbest 0

  Pessimistic efficiency hworst 0

  Geometric average efficiency hGeometric 0

Ranking

1 2 3 4 5 6 7 8 9 10

0.90490 1.00000 0.75429 1.00000 0.83106 1.00000 1.00000 0.63321 0.54188 1.00000

1.39424 1.45422 1.00000 1.84443 1.00000 1.00000 1.38697 1.00000 1.00000 1.47091

1.12323 1.20591 0.86850 1.35810 0.91163 1.00000 1.17770 0.79575 0.73612 1.21281

5 3 8 1 7 6 4 9 10 2

our rankings in Tables 5, 6 are given by combining both the optimistic and pessimistic efficiencies of the 10 DMUs. If we take into consideration only the optimistic efficiencies in Tables 5 and 6, it is not difficult to find that the optimistic efficiencies in the two tables both evaluate DMU2, DMU4, DMU6, DMU7, and DMU10 as optimistic efficient while rank DMU3, DMU5, DMU8, and DMU9 at the bottom as DMU5  DMU3  DMU8  DMU9. Such a conclusion is nearly consistent with that drawn from the ranking of Dia. From the two tables, it is also observed that the use of crisp weights for fuzzy inputs and fuzzy outputs and the use of fuzzy weights for fuzzy inputs and fuzzy outputs produce slightly different rankings for the 10 DMUs. The only difference lies in the rankings of DMU1 and DMU7. The use of fuzzy weights for fuzzy inputs and fuzzy outputs evaluates DMU1 to be better than DMU7, whereas the use of crisp weights for fuzzy inputs and fuzzy outputs leads to an opposite ranking and rates DMU7 to be better than DMU1. Such a difference is understandable because fuzzy weights give DMU1 more freedom and more flexibility in selecting its most favorable weights. As a result, DMU1 is rated as performing better than DMU7. In addition, it is noted from Table 6 that DMU1 is no longer optimistic efficient under crisp weights. This shows that crisp weights may evaluate less DMUs as optimistic efficient than fuzzy weights. In this sense, the use of crisp weights for fuzzy inputs and fuzzy outputs may have a better discrimination power than the utilization of fuzzy weights for fuzzy inputs and fuzzy outputs. Example 3. Twelve flexible manufacturing systems are compared in terms of two inputs and four outputs. Capital and operating cost and floor space requirement are the two inputs. Improvements in qualitative benefits, work-in-process (WIP), average number of tardy jobs, and average yield are the four outputs. Table 7 shows the fuzzy input and output data of the 12 FMSs, which were

randomly generated from the crisp data in Shang and Sueyoshi (1995) by keeping the crisp data as the most possible values (i.e. modal values) of triangular fuzzy data. Since the data for input 2 (floor space needed) and output 1 (improvement in qualitative benefit) are crisp values, they can only be weighted with crisp weights. Input 1 and outputs 2–4, however, could be weighted by either triangular fuzzy weights or crisp weights. By solving LP models (16), (17), (20) and (21) for each of the 12 FMSs, respectively, we get the optimistic and pessimistic efficiencies of each FMS under fuzzy and crisp weights. The results are shown in Tables 8 and 9, from which it is seen that both fuzzy and crisp weights evaluate FMS5 as the best flexible manufacturing system. This evaluation is completely consistent with the selections of Shang and Sueyoshi (1995) and Wang and Chin (2009) in crisp environment, but differs from the selection of Liu (2008) under fuzzy environments. Liu (2008) selected FMS7 as the best flexible manufacturing system by imposing the following assurance region (AR) on input and output weights: 0:75=0:25 6 v 1 =v 2 6 0:8333=0:1667; 0:4023 u1 0:4667 0:4023 u1 0:4667 0:4023 u1 0:4667 6 6 ; 6 6 ; 6 6 ; 0:1361 u2 0:0795 0:1850 u3 0:1392 0:3146 u3 0:2766 0:0795 u2 0:1361 0:0795 u2 0:1361 0:1392 u3 0:1850 6 6 ; 6 6 ; 6 6 : 0:1850 u3 0:1392 0:3146 u4 0:2766 0:3146 u4 0:2766

When we append such an AR to LP models (20) and (21) and solve them again for each of the 12 FMSs, we get the new optimistic and new pessimistic efficiencies of the 12 FMSs. The results are provided in Table 10, from which it is seen that FMS7 stands out as the best flexible manufacturing system under the role of the above AR. This evaluation is consistent with the selection of Liu (2008), but our expected value models are much simpler than his fuzzy DEA/AR models based upon a-level sets and Zadeh’s extension principle.

Table 7 Input and output data for 12 flexible manufacturing systems (Liu, 2008). FMSs

1 2 3 4 5 6 7 8 9 10 11 12

Inputs

Outputs (improvements in)

Capital & Operating cost ($100,000)

Floor space needed (1000 ft2)

(16.17, 17.02, 17.87) (15.64, 16.46, 17.28) (11.17, 11.76, 12.35) (9.99, 10.52, 11.05) (9.03, 9.50, 9.98) (4.55, 4.79, 5.03) (5.90, 6.21, 6.52) (10.56, 11.12, 11.68) (3.49, 3.67, 3.85) (8.48, 8.93, 9.38) (16.85, 17.74, 18.63) (14.11, 14.85, 15.59)

5 4.5 6 4 3.8 5.4 6.2 6 8 7 7.1 6.2

Qualitative (%) 42 39 26 22 21 10 14 25 4 16 43 27

WIP (10)

No. of Tardiness (%)

Yield (100)

(43.0, 45.3, 47.6) (38.1, 40.1, 42.1) (37.6, 39.6, 41.6) (34.2, 36.0, 37.8) (32.5, 34.2, 35.9) (19.1, 20.1, 21.1) (25.2, 26.5, 27.8) (34.1, 35.9, 37.7) (16.5, 17.4, 18.3) (32.6, 34.3, 36.0) (43.3, 45.6, 47.9) (36.8, 38.7, 40.6)

(13.5, 14.2, 14.9) (12.4, 13.0, 13.7) (13.1, 13.8, 14.5) (10.7, 11.3, 11.9) (11.4, 12.0, 12.6) (4.8, 5.0, 5.3) (6.7, 7.0, 7.4) (8.6, 9.0, 9.5) (0.1, 0.1, 0.1) (6.2, 6.5, 6.8) (13.3, 14.0, 14.7) (13.1, 13.8, 14.5)

(28.6, 30.1, 31.6) (28.3, 29.8, 31.3) (23.3, 24.5, 25.7) (23.8, 25.0, 26.3) (19.4, 20.4, 21.4) (15.7, 16.5, 17.3) (18.7, 19.7, 20.7) (23.5, 24.7, 25.9) (17.2, 18.1, 19.0) (19.6, 20.6, 21.6) (29.5, 31.1, 32.7) (24.1, 25.4, 26.7)

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Table 8 Efficiencies of the 12 FMSs under fuzzy weights. FMSs

  Optimistic efficiency hbest 0

  Pessimistic efficiency hworst 0

  Geometric average efficiency hGeometric 0

Ranking

1 2 3 4 5 6 7 8 9 10 11 12

1.00000 1.00000 0.98273 1.00000 1.00000 1.00000 1.00000 0.96148 1.00000 0.95389 0.98329 0.80204

1.01321 1.00000 1.11769 1.19153 1.22148 1.14990 1.15861 1.07477 1.00000 1.00000 1.00000 1.00000

1.00658 1.00000 1.04804 1.09157 1.10521 1.07233 1.07639 1.01655 1.00000 0.97667 0.99161 0.89556

7 8 5 2 1 4 3 6 8 11 10 12

Table 9 Efficiencies of the 12 FMSs under crisp weights. FMSs

  Optimistic efficiency hbest 0

  Pessimistic efficiency hworst 0

  Geometric average efficiency hGeometric 0

Ranking

1 2 3 4 5 6 7 8 9 10 11 12

1.00000 1.00000 0.98256 1.00000 1.00000 1.00000 1.00000 0.96144 1.00000 0.95362 0.98314 0.80129

1.01462 1.00000 1.11927 1.19238 1.22242 1.15150 1.15875 1.07598 1.00000 1.00000 1.00000 1.00000

1.00729 1.00000 1.04869 1.09196 1.10563 1.07308 1.07645 1.01710 1.00000 0.97654 0.99154 0.89515

7 8 5 2 1 4 3 6 8 11 10 12

Table 10 Efficiencies of the 12 FMSs under crisp weights with an assurance region (AR). FMSs

  Optimistic efficiency hbest 0

  Pessimistic efficiency hworst 0

  Geometric average efficiency hGeometric 0

Ranking

1 2 3 4 5 6 7 8 9 10 11 12

0.97360 0.95071 0.94860 0.99573 1.00000 0.97957 1.00000 0.93860 0.83412 0.83019 0.92481 0.79071

1.17696 1.14434 1.20020 1.24611 1.26263 1.25136 1.29223 1.18233 1.00000 1.04865 1.12962 1.00000

1.07046 1.04304 1.06701 1.11391 1.12367 1.10716 1.13676 1.05344 0.91330 0.93305 1.02210 0.88922

5 8 6 3 2 4 1 7 11 10 9 12

In conclusion, the fuzzy expected value approach provides a simple, effective and practical way of assessing the performances of DMUs in fuzzy environments, and the double frontier analysis offers an approach to measuring the overall performances of DMUs and an effective way of identifying the best performing DMU in practice. 4. Conclusions In view of the fact that precise input and output data may not always be available in real world performance assessments due to the existence of uncertainty, we have proposed in this paper a fuzzy expected value approach for fuzzy DEA to conduct performance assessments in fuzzy environments from different perspectives. The fuzzy expected value approach transforms fuzzy input and fuzzy output data into two total expected values for inputs and outputs, respectively, based on which two pairs of fuzzy

expected value models have been constructed to measure the optimistic and the pessimistic efficiencies of DMUs by using fuzzy or crisp weights. The two extreme efficiencies have then been integrated through their geometric average to measure the overall performances of the DMUs and identify the best performing DMU. The fuzzy expected value approach and the resultant models have finally been tested with three numerical examples including the selection of a FMS. In comparison with existing fuzzy DEA models or approaches, the fuzzy expected value approach and the resultant fuzzy DEA models are much easier to solve and implement. This is because we only have to solve one pair of LP models for each DMU to determine its optimistic and pessimistic efficiencies without the need of solving any a-level set or possibility based LP models. The fuzzy expected value approach also provides DMUs with more choice and more flexibility to select their most favorable weights that could be either crisp or fuzzy. Most importantly, DMUs are assessed from

Y.-M. Wang, K.-S. Chin / Expert Systems with Applications 38 (2011) 11678–11685

different perspectives to give an overall assessment, which makes assessment conclusions more comprehensive and more convincing than those from just the optimistic point of view. In particular, the identification or selection of the best performing DMU becomes very easy by the overall assessment.

References Charnes, A., & Cooper, W. W. (1962). Programming with linear fractional functionals. Naval Research Logistics Quarterly, 9, 181–185. Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring the efficiency of decision making units. European Journal of Operational Research, 2, 429–444. Dia, M. (2004). A model of fuzzy data envelopment analysis. INFOR, 42, 267–279. Entani, T., Maeda, Y., & Tanaka, H. (2002). Dual models of interval DEA and its extension to interval data. European Journal of Operational Research, 136, 32–45. Garcia, P. A. A., Schirru, R., & Melo, P. F. F. E. (2005). A fuzzy data envelopment analysis approach for FMEA. Progress in Nuclear Energy, 46, 359–373. Guo, P., & Tanaka, H. (2001). Fuzzy DEA: A perceptual evaluation method. Fuzzy Sets and Systems, 119, 149–160. Jahanshahloo, G. R., Soleimani-damaneh, M., & Nasrabadi, E. (2004). Measure of efficiency in DEA with fuzzy input–output levels: a methodology for assessing, ranking and imposing of weights restrictions. Applied Mathematics and Computation, 156, 175–187. Kao, C., & Liu, S. T. (2000a). Fuzzy efficiency measures in data envelopment analysis. Fuzzy Sets and Systems, 113, 427–437. Kao, C., & Liu, S. T. (2000b). Data envelopment analysis with missing data: An application to University libraries in Taiwan. Journal of the Operational Research Society, 51, 897–905. Kao, C., & Liu, S. T. (2003). A mathematical programming approach to fuzzy efficiency ranking. International Journal of Production Economics, 86, 45–154. Kao, C., & Liu, S. T. (2005). Data envelopment analysis with imprecise data: An application of Taiwan machinery firms. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 13(2), 225–240. León, T., Liern, V., Ruiz, J. L., & Sirvent, I. (2003). A fuzzy mathematical programming approach to the assessment of efficiency with DEA models. Fuzzy Sets and Systems, 139, 407–419. Lertworasirikul, S., Fang, S. C., Joines, J. A., & Nuttle, H. L. W. (2003). Fuzzy data envelopment analysis (DEA): A possibility approach. Fuzzy Sets and Systems, 139, 379–394. Lertworasirikul, S., Fang, S. C., Joines, J. A., & Nuttle, H. L. W. (2003). Fuzzy data envelopment analysis: a credibility approach. In J. L. Verdegay (Ed.), Fuzzy sets based heuristics for optimization (pp. 141–158). Berlin Heidelberg: SpringerVerlag.

11685

Lertworasirikul, S., Fang, S. C., Nuttle, H. L. W., & Joines, J. A. (2003). Fuzzy BCC model for data envelopment analysis. Fuzzy Optimization and Decision Making, 2(4), 337–358. Liu, S. T. (2008). A fuzzy DEA/AR approach to the selection of flexible manufacturing systems. Computers & Industrial Engineering, 54(1), 66–76. Liu, S. T., & Chuang, M. (2009). Fuzzy efficiency measures in fuzzy DEA/AR with application to university libraries. Expert Systems with Applications, 36(2P1), 1105–1113. Liu, B., & Liu, Y. K. (2002). Expected value of fuzzy variable and fuzzy expected value models. IEEE Transactions on Fuzzy Systems, 10(4), 445–450. Saati, S., & Memariani, A. (2005). Reducing weight flexibility in fuzzy DEA. Applied Mathematics and Computation, 161, 611–622. Saati, S., Menariani, A., & Jahanshahloo, G. R. (2002). Efficiency analysis and ranking of DMUs with fuzzy data. Fuzzy Optimization and Decision Making, 1, 255–267. Sengupta, J. K. (1992). A fuzzy systems approach in data envelopment analysis. Computers & Mathematics with Applications, 24, 259–266. Shang, J., & Sueyoshi, T. (1995). A unified framework for the selection of flexible manufacturing system. European Journal of Operational Research, 85(2), 297–315. Soleimani-damaneh, M., Jahanshahloo, G. R., & Abbasbandy, S. (2006). Computational and theoretical pitfalls in some current performance measurement techniques and a new approach. Applied Mathematics and Computation, 181(2), 1199–1207. Triantis, K. (2003). Fuzzy non-radial data envelopment analysis (DEA) measures of technical efficiency in support of an integrated performance measurement system. International Journal of Automotive Technology and Management, 3, 328–353. Triantis, K., & Girod, O. (1998). A mathematical programming approach for measuring technical efficiency in a fuzzy environment. Journal of Productivity Analysis, 10, 85–102. Wang, Y. M., & Chin, K. S. (2009). A new approach for the selection of advanced manufacturing technologies: DEA with double frontiers. International Journal of Production Research, 223(2), 872–878. Wang, Y. M., Chin, K. S., & Yang, J. B. (2007). Measuring the performances of decision making units using geometric average efficiency. Journal of the Operational Research Society, 58(7), 929–937. Wang, Y. M., Greatbanks, R., & Yang, J. B. (2005). Interval efficiency assessment using data envelopment analysis. Fuzzy Sets and Systems, 153(3), 347–370. Wang, Y. M., Luo, Y., & Liang, L. (2009). Fuzzy data envelopment analysis based upon fuzzy arithmetic with an application to performance assessment of manufacturing enterprises. Expert Systems with Applications, 36(3), 5205–5211. Wen, M., & Li, H. (2009). Fuzzy data envelopment analysis (DEA): Model and ranking method. Journal of Computational and Applied Mathematics, 223(2), 872–878. Wu, D., Yang, Z., & Liang, L. (2006). Efficiency analysis of cross-region bank branches using fuzzy data envelopment analysis. Applied Mathematics and Computation, 181(1), 271–281.