Performance evaluation: An integrated method using data envelopment analysis and fuzzy preference relations

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European Journal of Operational Research 194 (2009) 227–235 www.elsevier.com/locate/ejor

Decision Support

Performance evaluation: An integrated method using data envelopment analysis and fuzzy preference relations Desheng Dash Wu

*

Joseph L. Rotman School of Management, University of Toronto, 105 St. George Street, Toronto, Canada M5S 3E6 RiskLab, University of Toronto, Canada Received 25 April 2005; accepted 8 October 2007 Available online 13 October 2007

Abstract In a multi-attribute decision-making (MADM) context, the decision maker needs to provide his preferences over a set of decision alternatives and constructs a preference relation and then use the derived priority vector of the preference to rank various alternatives. This paper proposes an integrated approach to rate decision alternatives using data envelopment analysis and preference relations. This proposed approach includes three stages. First, pairwise efficiency scores are computed using two DEA models: the CCR model and the proposed cross-evaluation DEA model. Second, the pairwise efficiency scores are then utilized to construct the fuzzy preference relation and the consistent fuzzy preference relation. Third, by use of the row wise summation technique, we yield a priority vector, which is used for ranking decision-making units (DMUs). For the case of a single output and a single input, the preference relation can be directly obtained from the original sample data. The proposed approach is validated by two numerical examples.  2007 Elsevier B.V. All rights reserved. Keywords: Performance evaluation; Data envelopment analysis (DEA); Preference relations; Cross evaluation

1. Introduction In a multi-attribute decision-making (MADM) situation the decision maker (DM) is faced with the question of which decision making unit (DMU) to adopt from among a set of alternative DMUs that are available to him. To model this problem, one typical method is to ask the DM to provide his preferences over a set of evaluated decision alternatives and construct preference relations using his expressed pairwise comparison information. The DM then solve this MADM problem by ranking all the evaluated DMUs based on the priority vector derived from a consistency judge matrix. There are usually two widely used preference relations: multiplicative preference relation (Saaty,

* Address: Joseph L. Rotman School of Management, University of Toronto, 105 St. George Street, Toronto, Canada M5S 3E6. Tel.: +1 416 946 0052. E-mail address: [email protected]

0377-2217/$ - see front matter  2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.10.009

1980; Herrera et al., 2001; Chiclana et al., 2001) and fuzzy preference relation (FPR) (Orlovsky, 1978; Nurmi, 1981; Kacprzyk, 1986; Tanino, 1990). We present the definitions of two preference relations in Appendix II. FPR is preferred to multiplicative preference relation when MADM with incomplete information is presented to the DM (Chiclana et al., 2007). Both preference relations are based on pairwise comparison and thus incur some common research issues, e.g., the construction of preference relations (Vargas, 1990; Gheorghe et al., 2005) and the consistency problem of preference relations (Herrera-Viedma et al., 2004). Either the multiplicative or the fuzzy preference relation is actually constructed based on a self-rated scheme (indicated by the diagonal elements in the preference matrix) and a cross-rated scheme (indicated by the non-diagonal elements). Classical techniques used to construct a preference relation are based on subjective evaluation, requiring much involvement of expert knowledge and time. An objective technique can greatly reduce the cost incurred

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D.D. Wu / European Journal of Operational Research 194 (2009) 227–235

by the involvement of expert knowledge and time in the evaluation process. DEA provides a tool for objective evaluation (Charnes et al., 1978). DEA is a nonparametric programming technique used to treat problems of multiple inputs and outputs associated with multiple DMUs. DEA is used to establish a best practice group from among a set of observed units and to identify the units that are inefficient when compared to the best practice group. Researchers have discussed the similarity of performance measurement and decision making problems using DEA (Doyle and Green, 1994; Parkan, 2006). The purpose of the present paper is two-fold: (a) to introduce an integrated DEA/FPR method for ranking the DMUs in a decision making situation, and (b) to develop an alternative tool for construction of relation preference and extend the traditional DEA models by proposing a cross-evaluation method for use in performance evaluation. The proposed approach includes three stages. The first stage is to run the CCR model and the proposed cross-evaluation DEA model to yield pairwise efficiency scores. The pairwise efficiency scores are then used to construct the fuzzy preference relation at the second stage. At the last step, we use the row wise summation technique to yield the priority vector for ranking DMUs. Similar technique was used in Sinuany-Stern et al. (2000) to address the relationship between DEA and AHP. Sinuany-Stern et al.’s model may not be practical due to two drawbacks. First, as indicated in their own work, ‘‘. . .we receive many efficient values, especially as the number of inputs and outputs increases. . .’’ (Sinuany-Stern et al., 2000; p. 115), which reflects a weak diagnostic power of their model. This is always a hurdle for their model since their model violates the DEA rule of thumb by allowing only two DMUs to be involved each time they run DEA. The rule of thumb requires in data the number of DMUs is no less than three times of the total number of input and output variables (Cooper et al., 2000). Second, by selecting only two DMUs in each DEA run, a total number of 2n(n  1) linear programs have to be solved, involving n(n  1) programs for the CCR version (self-rated problem) and another n(n  1) programs for the cross-rated problem, which obviously result in a heavy computation task. Our proposed method directly applies n DMUs of interests to CCR DEA and the revised benevolent DEA. Computation complexity is greatly reduced and diagnostic power is improved comparing to Sinuany-Stern et al.’s model. These advantages are to be discussed in both Sections 2 and 5. The rest of this paper unfolds as follows. Section 2 presents the methodology to construct preference relations. An algorithm for alternative evaluation by use of the constructed preference relation is designed in Section 3. Section 4 shows how to construct preference relations directly from the original sample data in the case of single output and single input. Section 5 gives two illustrative examples, and finally, concluding remarks and further consideration are presented in Section 6.

2. Methodology In this section, the preference relation is constructed by implementing a three-stage methodology. Note that we argue in the introduction that a preference relation is actually constructed based on a self-rated scheme and a crossrated scheme, thus we need to establish self-rated and cross-rated problem by use of DEA at first. Hence, of the three stages, first we yield pairwise efficiency scores using two DEA models: the CCR model and the proposed cross-evaluation DEA model. The resulting pairwise efficiency scores are then utilized to construct the fuzzy preference relations at the second stage. At the last stage, by use of the row wise summation technique, the priority vector for ranking DMUs is obtained. 2.1. Paired DEA: CCR and cross-evaluation DEA Suppose there are n DMUs, denoted as DMUl (l = 1, 2, . . . , n) to be evaluated. Each DMUl has m different inputs xil (i = 1, 2, . . . , m) and s different outputs yrl (r = 1, 2, . . . , s). Let the observed input and output vectors of T DMUl be X l ¼ ðx1l ; x2l ; . . . ; xml Þ > 0, l = 1, 2, . . . , n, and T Yl = (y1l, y2l, . . . , ysl) > 0, respectively, where ‘‘T’’ denotes the transpose. DEA determines for each alternative its efficiency value as the maximum of the ratio of its weighted scores for output criteria to weighted scores for input criteria under the constraint that this efficiency is bound from above by unity for all the alternatives of interest. This is known as the CCR DEA as a fractional programming problem (Charnes et al., 1978), which is then transformed into the following linear programming model (1) by use of the Charnes–Cooper transformation (Charnes and Cooper, 1962). CCR DEA (self-rated problem) Þ ¼ Max Edd ¼ ðhCCR d s:t:

lTd Y d

xTd X l  lTd Y l P 0; xTd X d ¼ 1; xTd P 0;

l ¼ 1; 2; . . . ; n;

ð1Þ

lTd P 0;

where DMUd is under evaluation and xd, ld are the associated input and output weight vectors. While DMUd (d = 1, 2, . . . , n) is changed in the above CCR model n T times, each for one DMU, the optimal weights ðxT d ; ld Þ and optimal efficiency Edd given to DMUd (d = 1, 2, . . . , n) are obtained. In the CCR model, each DMU optimizes the most favorable weights and receives its most favorable evaluation relative to any other unit. In other words, each DMU is self-evaluated. DMUd is termed weakly efficient if and only if the optimal objective is equal to 1, i.e., Edd = 1. The cross-efficiency of DMUj using the optimal weight of DMUd is calculated as lT d Yj xT d Xj

ðd; j ¼ 1; 2; . . . ; nÞ:

D.D. Wu / European Journal of Operational Research 194 (2009) 227–235

It is recognized that the above CCR DEA can suffer from a weak discriminating power and unrealistic weight distributions (Li and Reeves, 1999), because it only reflects a self-rated scheme and each DMU is evaluated under the best possible light. To deal with these problems, researchers have made a number of efforts. Among these revisions, one route is to impose restrictions on data. For example, it is required in data that the number of DMUs is no less than three times of the total number of input and output variables (Cooper et al., 2000). This is known as the rule of thumb, by which an effective DEA computation can be obtained. Sinuany-Stern et al.’s model actually violates this rule by allowing only two DMUs to be involved each time DEA is run. As a result, their model suffers from a weak diagnostic power, indicated in their own paper as: ‘‘. . .we receive many efficient values, especially as the number of inputs and outputs increases. . .’’ (see, Sinuany-Stern et al., 2000; p. 115). Another well-known route to improve the DEA diagnosing power is the cross-efficiency method (Sexton et al., 1986; Doyle and Green, 1994; Anderson et al., 2002). This method constructs a cross-efficiency matrix, by using both a DMU’s self-rated DEA efficiency, and cross-efficiencies, rated by its peers. The following model (2) is a cross-efficiency model by revising the benevolent form of Doyle and Green (1994). Revised benevolent DEA (Cross-rated Problem) Max uTd

Yl X

s:t:

vTd X l  uTd Y l P 0; l ¼ 1; 2; . . . ; n; Xl X ¼ 1; vTd l–d ad  vTd X d  uTd Y d 6 0; vTd ; uTd P 0; ad 2 ð0; 1Þ is a parameter:

ð2Þ

Model (2) is a revision of the benevolent (Doyle and Green, 1994) formulation by allowing the self-rated efficiency to be deviated from the CCR score in the third constraint. Doyle and Green (1994) set the efficiency value of DMUd to the CCR score by using an equation constraint in their model. Our inequality constraint gives a lower bound to the efficiency value of DMUd, which means that it maximizes the efficiency of other (n  1) DMUs by holding its own efficiency no less than a given parameter value ad. Hence, the revised benevolent DEA (2) can allow the self-rated efficiency to adaptively change in the cross-evaluation setting and thus the cross-evaluation process can be flexible and adaptive to real situations. In the above model (2), while DMUd (d = 1, 2, . . . ,n) is T changed, the optimal weights ðuT d ; vd Þ are obtained. The cross-efficiency of DMUj using the optimal weight optimized for DMUd is calculated by the following formula (3): uT d Yj ; Edj ¼ T vd X j

Þ, the Note that if the parameter ad 6 min16j6n ðhCCR j third inequality constraint will be redundant and can be removed, indicating an invalid cross-evaluation. This is not our expectation and thus the parameter ad is set to Þ; 1Þ. be no less than unity, i.e., ad2 ðmin16j6n ðhCCR j Note also that our proposed self-rated and cross-rated models can have an advantage in computation time compared to Sinuany-Stern et al.’s models. In Sinuany-Stern et al. (2000), by selecting only two DMUs in each DEA run, a total number of 2n(n  1) linear programs have to be solved: n(n  1) programs for the CCR version (selfrated problem) and another n(n  1) programs for the cross-rated problem, resulting in a heavy computation task. Our proposed method directly applies n DMUs of interests to CCR DEA and the revised benevolent DEA. Computation complexity is greatly reduced and the diagnostic power is better than that of Sinuany-Stern et al.’s model. Now, we have established self-rated and cross-evaluated models using DEA and are ready to construct preference relations. 2.2. Construction of preference relations After running the paired models (1) and (2) to obtain the efficiency Ejj, Edd, Ejd and Edj, we construct the pairwise comparison fuzzy preference relation (matrix) R = (rdj)n · n (see Appendix II), so that for every pair of units d and j: rdj ¼

l–d

d–j;

d; j ¼ 1; 2; . . . ; n:

ð3Þ

229

Edd þ Ejd ; Edd þ Ejd þ Ejj þ Edj

rjj ¼ 0:5;

j ¼ 1; 2; . . . ; n: ð4Þ

Note that the pairwise comparison fuzzy preference relation (matrix) R = (rdj)n · n on the diagonal has a value of 0.5 and the elements rdj reflect the evaluation of unit d over unit j. If rdj < 0.5, unit d is preferred to unit j. rdj is constructed from the paired DEA results and satisfied by rdj + rjd = 1. Obviously, A ¼ ðadj Þnn is a fuzzy preference relation (see Appendix II). With the transformation function between the fuzzy preference relation (matrix) and the multiplicative preference relation(see Appendix II), we naturally yield the associated multiplicative preference relation A = (adj)n · n, where Edd þ Ejd ; ajj ¼ 1; j ¼ 1; 2; . . . ; n: ð5Þ adj ¼ Ejj þ Edj The matrices R and A are both objective evaluated, calculated from the pairwise DEA runs, reflecting a crossevaluation scheme. 2.3. Construction of consistency fuzzy preference relation for ranking DMUs Based upon the fuzzy preference relation R, a consistency fuzzy preference relation B = (bdj)n · n can be constructed by use of formula (6) and (7) (see Appendix II)

230

cd ¼

D.D. Wu / European Journal of Operational Research 194 (2009) 227–235 n X

rdj ¼

j¼1

n X j¼1

Edd þ Ejd Edd þ Ejd þ Ejj þ Edj

ðd ¼ 1; 2; . . . ; nÞ; ð6Þ

cd  cj bdj ¼ þ 0:5: 2ðn  1Þ

ð7Þ

The consistency fuzzy preference relation B provides a ranking order of the alternatives. This is accomplished by using the row wise summation technique in (8). The ranking weight (score) wd given to DMUd is calculated P P n j bdj j bdj þ 2  1 wd ¼ P P : ð8Þ ¼ nðn  1Þ d j bdj Sinuany-Stern et al. (2000) argue that a single hierarchical level AHP can be run based on the pairwise comparison matrices R and A. Our framework is similar to their methods by using DEA/AHP to better rate DMUs. Remark 1. (advantage of row wise summation approach): Other than the proposed row wise summation approach, there are several methods by which we can derive the priority vector, i.e., the ranking weight vector, from the comparison matrices R and A (Bryson, 1995; Blankmeyer, 1987; Saaty, 1980). The eigenvector method (Saaty, 1980) and least squares method (Jensen, 1984; Bryson, 1995), logarithmic least squares method (Crawford and Williams, 1985) are among these methods. Comparing with other methods, the proposed row wise summation approach exhibits many good properties such as being easy for computation and exhibiting strong rank preservation (Xu, 2004). 3. Algorithm Before the algorithm is explored, a proposition is presented regarding the determination of the value of ad as follows. Proposition 1. Set ad in the revised benevolent DEA (2) to P lT Y d average cross-efficiency 1n nj¼1 xjT X (d = 1,2, . . . , n), i.e., we d j have ad ¼ E d ¼

n lT 1X j Yd ; n j¼1 xT j Xd

ð9Þ

T where ðxT d ; ld Þ is the optimal solution to CCR model (1), then there is at least one optimal solution to the revised benevolent DEA (2).

Proof. Please see Appendix I for the proof. Based on the afore-mentioned discussion, an iterative algorithm is designed: Step 1: Solve the CCR DEA program (1) and obtain the optimal weight vectors xd ; ld ðd ¼ 1; 2; . . . ; nÞ and optimal CCR efficiency Edd. Calculate Ej by use of formula (9).

Step 2: Set ad in the revised benevolent DEA (2) to Ed ðd ¼ 1; 2; . . . ; nÞ and solve the revised benevolent DEA (2) to obtain the optimal solution T vT d ; ud ðd ¼ 1; 2; . . . ; nÞ. Calculate Edj by the formula (3) as the solution to cross-rated model. Step 3: Use formula (4) to compute the value of rdj (d,j = 1, 2, . . . , n) and construct the fuzzy preference relation R as R = (rdj)n · n. Step 4: Calculate the consistency fuzzy preference relation B = (bdj)n · n by use of formulas (6) and (7). Step 5: The ranking weight (score) wd given to DMUd is calculated using formula (8). Step 6: Rank the DMUs in the descending order of ranking scores wd (d = 1, 2, . . . , n). The most desirable DMU is the one with the highest score. h

4. The case of single output and single input In this section, we present the construction of preference relations using DEA for the case of single output and single input. Proposition 2. For the case of single output and single input Edd ¼

Y d =X d min ðX l =Y l Þ

ðd ¼ 1; 2; . . . ; nÞ;

Y j =X j min ðX l =Y l Þ

ðd–j; d; j ¼ 1; 2; . . . ; nÞ:

16l6n

Edj ¼

16l6n

Proof. See Appendix III. h Theorem 1. For the case of single output and single input, the constructed fuzzy preference relation R = (rdj)n · n resulted from our approach is a product transitive consistency fuzzy preference relation (see Definition 3 in Appendix II), where Y d =X d ; d; j ¼ 1; 2; . . . ; n: rdj ¼ Y j =X j þ Y d =X d Proof. It is easy to have rijrjkrki = rjirkjrik for all i, j, k = 1, 2, . . . , n. Thus Theorem 1 holds. h Remark 2. Based on Theorem 1, we can easily construct a fuzzy preference relation from raw sample data. Moreover, d Þ is a consistency it is obvious that if A ¼ ðadj ¼ YYdj =X =X j nn multiplicative preference relation (consistency reciprocal matrix) for the case of a single output and a single input (see Appendix II). Also, we can obtain the corresponding additive transitive consistency fuzzy preference relation d Þ. AR = (ardj)n · n, where ardj ¼ 12 ð1 þ log9 YYdj =X =X j 5. Numerical illustration To compare the proposed method with existing methods, we draw two examples from previous study: Fan

D.D. Wu / European Journal of Operational Research 194 (2009) 227–235 Table 1 Raw data of the numerical Example 1

X1(R2) X2(R5) Y1(R1) Y2(R3) Y3(R4)

Table 2 Cross-efficiencies by model (2)

DMU1

DMU2

DMU3

DMU4

5300 0.17 8350 6135 0.82

4952 0.13 7455 6527 0.65

8001 0.15 11,000 9008 0.59

5822 0.12 9624 8892 0.74

et al. (2004) and Sinuany-Stern et al. (2000). We present in Table 1 the first numerical example with the raw data replicated from Fan et al. (2004). In this MADM problem, a bank intends to select a factory from four investment alternatives (DMU1, DMU2, DMU3 and DMU4). When making a decision, the attributes considered include: (1) (2) (3) (4) (5)

R1: output value ($10,000); R2: cost of investment ($10,000); R3: total sales ($10,000); R4: profit proportion of the country; and R5: environmental pollution degree (fuzzy assessment value).

Among the five attributes, R1, R3 and R4 are of the benefit type and R2 and R5 are of the cost type. The decision matrix with five attributes and four alternatives is presented as follows. With the raw data in Table 1, we implement the algorithm proposed in Section 3 from Steps 1 to 6. Step 1: Solve the CCR DEA model (1) and compute the value of average cross-efficiency Ej using formula (9). The value of Edd (d = 1–4) is set to the CCR score documented in the third column of Table 4. The average cross-efficiency given to each DMU P4 lT Y 1 P4 lT Y 2 E1 ¼ 14 j¼1 xjT X ¼ 0:8914, E2 ¼ 14 j¼1 xjT X ¼ 1 2 j j P4 lT j Y3 1 0:8776, E3 ¼ 4 j¼1 xT X ¼ 0:7969, E4 ¼ 3 j P4 lT j Y4 1 j¼1 xT X ¼ 1. 4 j

4

Step 2: Set ad in the revised benevolent DEA (2) to Ed ðd ¼ 1; 2; 3; 4Þ and solve the four associated benevolent DEA programs, each program for one DMU, to obtain the optimal solution T vT d ; ud ; ðd ¼ 1; 2; 3; 4Þ. For example, the benevolent DEA associated with DMU4 is presented as follows: Max uT4 ½8350 þ 7455 þ 11000;6135 þ 6527 þ 9008; 0:82 þ 0:65 þ 0:59 s:t:

vT4 X l  uT4 Y l P 0; l ¼ 1; 2; ... ;4; vT4 ½5300 þ 4952 þ 8001;0:17 þ 0:13 þ 0:15 ¼ 1; 1  vT4 X 4  uT4 Y 4 6 0; vT4 ; uT4 P 0:

231

DMU1 DMU2 DMU3 DMU4

DMU1

DMU2

DMU3

DMU4

ad

/ 0.8197 0.5221 0.8563

0.9531 / 0.8317 1

0.9531 0.9107 / 1

0.9545 0.9114 0.8303 /

0.8914 0.8776 0.7969 1

The optimal solution to the above program is ðvT d ; 5 5 Þ ¼ ð5:47856  10 ; 0; 0; 3  10 ; 0; 0:00235Þ: uT d The cross-efficiency of DMU1, DMU2 and DMU3 using the optimal weights optimized for DMU4 is uT Y

1 4 ¼ 0:9545; calculated as: E41 ¼ vT X

uT Y

4

1

uT Y

2 4 E42 ¼ vT ¼ X 4

2

3 4 0:9114; E43 ¼ vT ¼ 0:8303. These scores are docX3 4 umented in the fifth column of Table 2. The CCR score and the revised benevolent DEA score are combined in matrix E, with values presented in the Table 3. Now we are ready to construct the preference relations using the square matrix E. Step 3: The formula (4) is used to compute the value of rdj (d, j = 1,2, . . . , n) and construct the fuzzy preference relation R. For example, the element in the first row and second column of R is computed as: r12 = (1 + 0.9531)/(1 + 0.9531 + 0.9524 + 0.8197) = 0.5243. Table 4 presents the result of the consistency fuzzy preference relation and fuzzy preference relation. Step 4: Calculate the consistency fuzzy preference relation B = (bdj)n · n by use of formula (6) and (7). For example, the element in the first row and second column of the consistency fuzzy preference relation B is computed as

b12 ¼ ðð0:5 þ 0:5234 þ 0:5762 þ 0:5129Þ  ð0:4757 þ 0:5 þ 0:5162 þ 0:4824ÞÞ=6 þ 0:5 ¼ 0:5243: The value of the computed consistency fuzzy preference relation is presented in Table 4. Step 5: The ranking weight (score) wd given to DMUd is calculated using formula (8). The result is documented in Table 5. Table 5 also gives different scores by Fan et al. (2004)’s approach and CCR DEA. Step 6: By the DEA/FPR method, the ranking order of four DMUs is DMU1  DMU4  DMU2  DMU3. The most desirable DMU is selected as DMU1 with the highest score 0.2594. However, Table 3 Cross-rated and self rated efficiency score (the value of E)

Ed1 Ed2 Ed3 Ed4

E1j

E2j

E3j

E4j

1 0.8197 0.5221 0.8563

0.9531 0.9524 0.8317 1

0.9531 0.9107 0.9144 1

0.9545 0.9114 0.8303 1

232

D.D. Wu / European Journal of Operational Research 194 (2009) 227–235

Table 4 Consistency fuzzy preference relation and fuzzy preference relation in Example 1

Table 7 Cross-rated and self rated efficiency score in Example 2

Consistency fuzzy preference relation

Fuzzy preference relation

0.5 0.47682 0.46002 0.4876

0.5 0.4757 0.4238 0.4871

DMU1 DMU2 DMU3 DMU4

0.52318 0.5 0.4832 0.5108

0.53998 0.5168 0.5 0.5276

0.5124 0.4892 0.4724 0.5

0.5243 0.5 0.4838 0.5176

0.5762 0.5162 0.5 0.5341

0.5129 0.4824 0.4659 0.5

DMU1 DMU2 DMU3 DMU4

Fan et al.’ ranking

CCR

0.2344 0.2344 0.7033 0.8918

3 3 2 1

1.00000 0.95244 0.91438 1.00000

DMU2

DMU3

DMU4

ad

1 1 0.2109 0.7898

1 1 0.276 0.8953

1 0.1558 0.8508 1

1 0.9755 0.2205 1

0.8053 0.8334 0.5567 0.8053

Table 8 Consistency fuzzy preference relation and fuzzy preference relation in Example 2

Table 5 Scores by different approach in Example 1 Fan et al.’ score

DMU1

CCR ranking

DEA/ FPR

DEA/FPR ranking

1 3 4 1

0.2594 0.2479 0.2395 0.2532

1 3 4 2

Fan et al. (2004) yields a ranking order DMU4  DMU3  DMU2  DMU1 based on the score in Table 5. Our ranking result greatly differs from that of Fan et al. (2004) since Fan et al. based their computing on a decision maker’s (e.g., a bank manger) subjective fuzzy preference relation, while the current analysis directly comes from an objective fuzzy preference relation generated from the original data by DEA. Moreover, comparing with CCR model, our DEA/FPR approach obviously exhibits a more powerful diagnosing power since it succeeds in ranking DMU1 and DMU4, while the CCR model fails to do so. The second example is drawn from Sinuany-Stern et al. (2000) and the data is presented in Table 6, where we have both the raw data and the optimal weights and the DEA score given to each DMU. By implementing the proposed algorithm in Section 3, we yield the computed result of the second example. Table 7 gives the cross-rated and self rated efficiency scores and Table 8 documents the consistency fuzzy preference relation and fuzzy preference relation. Scores by different approaches are presented in Table 9. Table 9 indicates that our DEA/FPR method yields the ranking order for four DMUs as: DMU1  DMU4  DMU2  DMU3. The most desirable DMU is selected as DMU1 with the highest score 0.2651. This result is somewhat consistent with that by Sinuany-Stern et al.’s model

Consistency fuzzy preference relation

Fuzzy preference relation

0.5 0.4726 0.4181 0.4887

0.5 0.5 0.347 0.472

0.5274 0.5 0.4454 0.5161

0.5819 0.5546 0.5 0.5706

0.5113 0.4839 0.4294 0.5

0.5 0.5 0.4937 0.4896

0.6532 0.5063 0.5 0.6512

0.5277 0.5104 0.3488 0.5

Table 9 Scores by different approach in Example 2

DMU1 DMU2 DMU3 DMU4

SinuanyStern et al.’s score

SinuanyStern et al. ranking

CCR

CCR ranking

DEA/ FPR

DEA/ FPR ranking

0.4994 0.4994 0.4800 0.5204

2 2 4 1

1.00 1.00 0.85 1.00

1 1 4 1

0.2651 0.2514 0.2241 0.2594

1 3 4 2

and the CCR model in identifying the most undesirable DMU, i.e., DMU3. However, our ranking generally differs from that by these two models. First, the CCR model fails to distinguish among three efficient DMUs, i.e., DMU1, DMU2 and DMU4. This reflects CCR DEA’s weak discriminating power as mentioned in Section 2.1. Second, Sinuany-Stern et al.’s model yields a ranking order DMU4  DMU1  DMU2  DMU3 based on the score in Table 9 and cannot decide the ranking between DMU1 and DMU2. This reflects that Sinuany-Stern et al.’s model suffers from a weak discriminating power, as we already revealed in both the introduction part and Section 2.1. Comparatively, our DEA/FPR allows all the DMUs to be involved in the each self evaluation and cross-evaluation, instead of only two DMUs each time. This obviously enables our proposed method to exhibit a more powerful diagnosing power. As a result, our proposed DEA/FPR can completely discriminate these four DMUs, with rankings shown in Table 9.

Table 6 Data of Example 2 from Sinuany-Stern et al. (2000)

X1 X2 Y1 Y2

DMU1

DMU2

DMU3

DMU4

CCR score

x1

x2

l1

l2

50 55 50 55

130 60 12 78

68 96 45 9

45 30 35 18

1.00 1.00 0.85 1.00

0.02000 0.00274 0.01471 0.02222

0.00000 0.01073 0.00000 0.00000

0.01820 0.00665 0.01891 0.02022

0.01461 0.01180 0.00000 0.01623

D.D. Wu / European Journal of Operational Research 194 (2009) 227–235

Finally we have

6. Conclusions and further consideration In this paper we have discussed how to construct a preference relation using DEA, and derive the priority vector of the preference by a row wise summation technique in a multiattribute decision-making context, and then use the derived priority vector to better rate DMUs. A three-stage DEA/FPR ranking method is developed for performance evaluation. At the first stage, we compute pairwise efficiency scores by use of two DEA models: the CCR model and the proposed cross-evaluation DEA model. The paper has extended the DEA approach (Doyle and Green, 1994). The resulting pairwise efficiency scores are then utilized to construct the fuzzy preference relation and the consistent fuzzy preference relation in the second stage. At the last stage, by use of a row wise summation technique, we yield the priority vector for ranking DMUs. The proposed method not only use DEA to construct preference relations but DEA extracted preference relations are used to enhance DEA itself which is known to be problematic on some known situations. Comparing with previous method (Sinuany-Stern et al., 2000), our models in the DEA computation stage has two advantages: strong diagnosing power and efficient computation power. Specifically, we present the construction method of the preference relation for the case of a single output and a single input. In this case, the preference relation can be directly constructed from the original sample data with this approach. Finally, two numerical examples are demonstrated to validate this approach. Further work can be done based on this research. One direction is the simulation study associated with the parameter in Model (2) (Olson and Wu, 2006). This is important since it provide a robustness validation of the proposed approach. Another direction is to develop new techniques (e.g., eigenvector method) to derive the priority vector based on the constructed objective preference relation. Similar techniques can also be extended to construct other preference relations such as linguistic preference relation (Zadeh, 1975; Kacprzyk and Fedrizzi, 1988). Appendix I Proof of Proposition 1. Let us consider the following Psuperefficiency DEA model where a virtual DMU ð l–d X l , P Y Þ is evaluated. l–d l X Max uT Yl l–d

vT X l  uT Y l P 0; X X l ¼ 1; vT

s:t:

l ¼ 1; 2; . . . ; n

ð10Þ

l–d T

v ;

uT P 0:

The above program (10) is solved to obtain the optimal uT Yd vT   T T d d P solution uT wTdl ¼ X l ,gdl ¼ d ; vd and hd ðhd ¼ vT X d Þ, let  T T T ud gdl Y d ud Yvdd d  l–d P , d5l, d, l = 1, 2, . . . , n, then ¼ ¼ h > Ed . Xl d vT X d T wT X vd

l–d

dl

d

233

d

wTdl X l  gTdl Y l P 0; X X l ¼ 1; wTdl

l ¼ 1; 2; . . . ; n;

ð11Þ ð12Þ

l–d

Ed  wTdl X d  gTdl Y d < 0;

ð13Þ

wTdl ; gTdl

ð14Þ

P 0:

(11)–(14) indicates that ðwTdl ; gTdl Þ is the feasible solution to program (9) and then Proposition 1 holds. h Appendix II Fuzzy preference relation fundamental (Nurmi, 1981; Fan et al., 2004; Xu and Da, 2003; Saaty, 1980; Yager and Kacprzyk, 1997; Chiclana et al., 2001). Definition 1. Let R = (rrm ij)n · n, be a preference relation (matrix), then R is called a fuzzy preference relation, if rij 2 [0, 1]; rij + rji = 1; rii = 0.5 for all i, j 2 [1, . . . , n]. A value of 0.5 for rij or rji indicates an indifference between alternative i and j and a value of 1 for rij or a value of 0 for rji indicates that alternative i is unanimously preferred to j. Similarly, a value between 0.5 and 1 for rij or a value between 0 and 0.5 for rji stands for that alternative i is preferred to j.

Definition 2 (Tanino, 1984). Let R = (rij)n · n be a fuzzy preference relation (matrix), then R is called a additive transitive consistency fuzzy preference relation, if rij 2 [0, 1]; rij = rik  rjk + 0.5 for all i, j 2 [1, . . . , n]. Definition 3 (Tanino, 1984). Let R = (rij)n · n be a fuzzy preference relation (matrix), then R is called a product transitive consistency fuzzy preference relation, if rij 2 [0, 1]; rijrjkrki = rjirkjrik for all i, j, k 2 [1, . . . , n]. Definition 4. Let S = (sij)n · n, be a preference relation (matrix), then S is called a multiplicative preference relation, if sij 2 R+; sij = 1/sji; sii = 1 for all i, j 2 [1, . . . , n]. Definition 5. Let S = (sij)n · n be a multiplicative preference relation (matrix), then S is called a consistency multiplicative preference relation (or called consistent reciprocal judgement matrix (Saaty, 1980), if sij = sik · skj for all i, j 2 [1, . . . , n]. A fuzzy preference relation R = (rij)n · n can be transformed to a additive transitive consistency fuzzy preference relation B = (bij)n · n by the following two formulas: n X rij ði ¼ 1; 2; . . . ; nÞ; ri ¼ j¼1

bij ¼

r i  rj þ 0:5: 2ðn  1Þ

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D.D. Wu / European Journal of Operational Research 194 (2009) 227–235

By applying the row-wised summation technique to the consistency fuzzy preference relation B, we can obtain the priority vector W = (wi), where P ai aj P ai þ n2  1 j ð2ðn1Þ þ 0:5Þ j bij ¼ ¼ wi ¼ P P n2 =2 nðn  1Þ i j bij P n j bij þ 2  1 ¼ : nðn  1Þ A multiplicative preference relation S = (sij)n · n can be obtained from a fuzzy preference relation R = (rij)n · n by the transitive formula: sij = rij/rji. Chiclana et al. (2001) also establish a relationship between multiplicative preference relation and fuzzy preference relation as follows: Theorem 2. (Chiclana et al., 2001). Let S = (sij)n · n be a multiplicative preference relation (matrix), Then the corresponding fuzzy preference relation R = (rij)n · n associated with S is given below: 1 rij ¼ ð1 þ log9 sij Þ: 2 Based upon Theorem 2, we can easily proof that if S = (sij)n · n is a consistency multiplicative preference relation, then the corresponding fuzzy preference relation R = (rij)n · n is a additive transitive consistency fuzzy preference relation, and vice versa. Appendix III Proof of Proposition 2. For the case of single output and single input, CCR model (1) is reduced to model (15) and (16). Edd ¼ Max s:t:

ð15Þ

ld Y d

ld 6 xd X l =Y l ;

l ¼ 1; 2; . . . ; n;

xd ¼ 1=X d ;

Max

xd P 0; X Yl ud

ld P 0; ð16Þ

l–d

s:t:

ud 6 vd X l =Y l ; l ¼ 1; 2; . . . ; n; X X l; vd ¼ 1= l–d

ud P ad  vd X d =Y d ; vd ; ud P 0: Then the optimal solution to the above model (15) is Y d =X d 1 x¼ d 1=X d , ld ¼ X d min16l6n ðX l =Y l Þ and E dd ¼ min16l6n ðX l =Y l Þ. The optimal solution to the above model (16) is P l uT Yj 1 P d , ud ¼ ðmin ¼ vd ¼ 1= Xl–d and Edj ¼ vT Xj X Þ l 16l6n X l =Y l Þð d l–d Y j =X j min16l6n ðX l =Y l Þ. h

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