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Journal of the Operational Research Society (2011) 62, 1585–1595
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A new approach for weight derivation using data envelopment analysis in the analytic hierarchy process 1
SM Mirhedayatian and R Farzipoor Saen 1
2 2
Islamic Azad University-Qazvin Branch, Qazvin, Iran; and Islamic Azad University-Karaj Branch, Karaj, Iran
Recently, some researches have been carried out in the context of using data envelopment analysis (DEA) models to generate local weights of alternatives from pairwise comparison matrices used in the analytic hierarchy process (AHP). One of these models is the DEAHP. The main drawback of the DEAHP is that it generates counter-intuitive priority vectors for inconsistent pairwise comparison matrices. To overcome the drawbacks of the DEAHP, this paper proposes a new procedure entitled Revised DEAHP, and it will be shown that this procedure generates logical weights that are consistent with the decision maker’s judgements and is sensitive to changes in data of the pairwise comparison matrices. Through a numerical example, it will be shown that the Revised DEAHP not only produces correct weights for inconsistent matrices but also does not suffer from rank reversal when an irrelevant alternative is added or removed. Journal of the Operational Research Society (2011) 62, 1585–1595. doi:10.1057/jors.2010.105 Published online 11 August 2010 Keywords: data envelopment analysis; analytic hierarchy process; DEAHP; rank reversal
1. Introduction The derivation of priority vectors from pairwise comparison matrices is one of the important issues in the analytic hierarchy process (AHP) literature. After proposing the eigenvalue method (EM) by Saaty (2000), numerous researches were conducted in this area, such as weighted least square method (Chu et al, 1979), the logarithmic least square method (Crawford, 1987), the geometric least square method (Islei and Lockett, 1988), the fuzzy programming method (Mikhailov, 2000), the gradient eigen weight method (Cogger and Yu, 1985), the robust estimation method (Lipovetsky and Conklin, 2002), singular value decomposition approach (Gass and Rapcsak, 2004). Recently, Ramanathan (2006) proposed an approach called the DEAHP, in which data envelopment analysis (DEA) is used for priority derivation of pairwise comparison matrices in the AHP. Although, the DEAHP has the advantage of removing the rank reversal problem when an irrelevant alternative is added or removed, it has some drawbacks. First of all, it only uses some information of pairwise comparison matrices and it is not sensitive to changes in some data of judgement matrices. Second, the generated weights from inconsistent pairwise comparison
Correspondence: R Farzipoor Saen, Department of Industrial Management, Faculty of Management and Accounting, Islamic Azad UniversityKaraj Branch, Karaj, P.O. Box: 31485-313, Iran. E-mail: [email protected]
matrices are irrational and counter-intuitive. In this paper, a new procedure entitled Revised DEAHP is proposed to overcome these drawbacks, and show that this method generates rational weights for inconsistent pairwise comparison matrices, while it retains the DEAHP advantages on the rank reveral problem. The paper is organized as follows. In Section 2, the DEAHP is reviewed and its drawbacks are numerically illustrated. In Section 3, the proposed method for weight derivation is discussed. Numerical examples are provided in Section 4. The paper is concluded in Section 5.
2. DEAHP The DEAHP method uses two similar methods for local weights and final weights derivation from pairwise comparison matrices, in which local weights are the weights of criteria (or alternatives with respect to each criterion) and final weights are the weights of alternatives that exist in the lowest level of the hierarchical structure.
2.1. Obtaining local weights using the DEAHP Let
2 6 6 A ¼ ðaij Þn n ¼ 6 4
a11 a12 a21 a22 .. . an1 an2
3 a1n a2n 7 7 7 5 ann
ð1Þ
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be a pairwise comparison matrix, where aji ¼ 1/aij for jai and W ¼ (w1, . . . , wn)T is its weights vector. In the DEAHP, each row of the matrix is considered as a decision-making unit (DMU) and the columns are viewed as outputs. A dummy input is added to be solved by DEA models. Ramanathan (2006) used the following input-oriented CCR (Charnes et al, 1978) model for the local weights derivation: Maximize wo ¼
n X
aoj vj
j¼1
Subject to
8 u1 ¼ 1 > > > n < P aij vj u1 p0; i ¼ 1; . . . ; n > j¼1 > > : u1 ; vj X0; j ¼ 1; . . . ; n
ð2Þ
1 4
where wo is the efficiency score of DMUo, which is viewed as its local weight. Model (2) is solved for all DMUs to obtain the local weights vector W ¼ (w1, . . . , wn)T of the comparison matrix A. It has been proved that the DEAHP provides correct weights for perfectly consistent comparison matrices (aij ¼ aik akj, for i, j, kA{1, . . . , n}) (Mehrabian et al, 2000).
2.2. Obtaining final weights using the DEAHP To aggregate local weights into final weights, Ramanathan (2006) considered two cases: Case 1: the weights of criteria are not considered in the model. Case 2: the weights of criteria are considered in the model. In Case 1, to obtain final weights, the DEAHP uses the following model in order to obtain final weights: Maximize
J X
woj vj
j¼1
Subject to
8 u1 ¼ 1 > > > J < P wij vj u1 p0; i ¼ 1; . . . ; N > j¼1 > > : u1 ; vj X0; j ¼ 1; . . . ; J
J X
6
3
1 3
1
For matrix A, the consistency ratio1 (CR) is CR ¼ 0.229>0.1, which means A is inconsistent. The weight vector obtained by the DEAHP for A is WDEAHP ¼ (1, 1, 1/3, 1, 1, 1), which makes no sense, while considering the first row of A. Obviously, the first criterion or alternative is more important than the others. The DEAHP not only produces irrational weights for highly inconsistent matrices, but also may generate illogical weights for those pairwise comparison matrices with satisfactory CR,2 that is, CRo0.1. For example, consider the following pairwise comparison matrix. 2 3 1 13 13 1 12 6 7 6 3 1 12 12 1 7 6 7 1 7 B¼6 63 2 1 2 17 6 7 41 2 2 1 15 2 1 12 1 1 For matrix B, the consistency ratio is CR ¼ 0.0999o0.1, which means that B has an acceptable CR. The weights vector obtained by the DEAHP is WDEAHP ¼ (1, 1, 1, 1, 1), which makes no sense, because the generated weights are similar for all alternatives or criteria.
3. Revised DEAHP The drawback of the DEAHP in generating irrational weights for inconsistent matrices has two basic features: (1) As a rule of thumb, to use DEA, the following condition must be satisfied (Cooper et al, 2000).
dj woj Þ
nX maxfm s; 3ðm þ sÞg
j¼1
8 J > < v ðP d w Þp1; 1 j ij Subject to j¼1 > : v1 X0
1
ð3Þ
where wij is the local weight of the ith alternative (i ¼ 1, . . . , N) with respect to the jth criterion ( j ¼ 1, . . . , J). In Case 2, the DEAHP uses the following model: Maximize v1 ð
It is claimed that rank reversal does not occur with the DEAHP if an irrelevant alternative, which is DEA inefficient, is added or removed. As it was mentioned in the previous section, the DEAHP has some drawbacks. The main drawback is that the DEAHP generates irrational and counter-intuitive weights for inconsistent pairwise comparison matrices. For example, consider the following pairwise comparison matrix: 3 2 1 4 3 1 3 4 6 1 1 7 3 1 1 7 7 6 4 5 7 6 1 1 6 3 7 1 15 15 16 7 7 6 A¼6 1 1 7 6 1 3 5 1 1 3 7 7 6 1 4 3 5 5 1 1 1 5
i ¼ 1; . . . ; N
ð4Þ
The above model is the same as model (3) with the additional constraint vj ¼ djv1( j ¼ 1, . . . , J; d1 ¼ 1).
where n is the number of DMUs. In addition, m and s are the number of inputs and outputs, respectively. It has been empirically shown that if this condition would not be held, 1
For more details, please see Appendix A. According to Saaty (2000), a consistency ratio of 0.1 or less is considered acceptable. 2
SM Mirhedayatian and R Farzipoor Saen—New approach for weight derivation using DEA
using DEA models causes more units to be efficient (ie, have efficiency scores equal to 1). Consequently, it is not possible to discriminate among efficient DMUs. In the DEAHP, the number of DMUs equals to the number of alternatives or criteria (n) and the number of outputs equals to n, and there is a dummy input with a value of 1 for all DMUs. Therefore, the previous condition would not be satisfied when the DEAHP is used to produce weights for inconsistent pairwise comparison matrices. That is, no maxfn; 3ðn þ 1Þg (2) The DEAHP is based on the CCR model in which the multipliers (vj) are assumed to be positive (vjX0), which permits DMUs to have multipliers with zero values in the optimum solution. In this case, it is possible that some DMUs, because of having a superior input/output, are recognized as efficient DMUs. Here, a new procedure is proposed to overcome the drawbacks of the DEAHP with respect to the aforementioned reasons. To solve the first problem, the superefficiency model for discriminating among efficient DMUs is utilized. To overcome the second problem, an infinitesimal non-Archimedean positive value (e) can be considered as a lower bound for multipliers (vj)3 Therefore, the following proposed super-efficiency model (based on Andersen and Petersen, 1993) can be used to obtain the weight vector for comparison matrices. Maximize wo ¼
n X
aoj vj
j¼1
8 u1 ¼ 1 > > > n < P aij vj u1 p0; i ¼ 1; . . . ; n iao ð5Þ Subject to > j¼1 > > : vj Xe; j ¼ 1; . . . ; n where wo is the efficiency score (weight) for DMUo. Notice that, e is considered as a lower bound of multipliers (vj), because it prevents the multipliers having zero value after solving an ordinary CCR model (with vjX0), but, in our model, e guarantees that all vj will have strict positive values (vjXe) after solving model (5). In addition, note that the objective function of model (5) is to maximize P wo ¼ nj¼q aoj vj , and if one or more vj become zero after solving the model, the pairwise comparison value for alternative ‘o’, according to alternative j (aoj) will not be 3
Epsilon is an infinitesimal non-Archimedean value. In other words, it is a very small positive value. Here, definitions of Archimedean and infinitesimal are presented as below. Archimedean: A number series is Archimedean if, (8x) (8y) ((n) (0oxoy-yonx) well-behaved numbers are Archimedean. Take any two numbers, however far apart they are. Then there is some number that you can multiply the smaller by to give a result greater than the larger. Non-standard analysis introduces infinitesimals of which this is not true. Infinitesimal: A function or variable continuously approaching zero as a limit (a variable with a limit of zero).
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considered in calculating wo, and therefore in model (5) all vj are strictly positive, which means that all aoj will be considered in calculating efficiency score wo (local weight) for alternative ‘o’. Since choosing arbitrary values for e causes the infeasibility of solution in multiplier form (or unboundedness in envelopment form), the main problem in using model (5) is to determine the e value. To overcome the infeasibility problem of the solution, the following proposed model (based on Mehrabian et al, 2000) can be applied: Maximize e
Subject to
8 u1 ¼ 1 > > >
j¼1 > > : j ¼ 1; . . . ; n vj Xe;
ð6Þ
Since the first constraint u1 ¼ 1 is similar to all DMUs, to determine the optimum value of e, model (6) is solved just one time. To generate local weights of the alternatives or criteria, the following steps should be implemented: Step 1: Obtain the optimum value of e using model (6) and determine the overall assurance interval for e, that is, (0, e ]. Step 2: Compute local weights of alternatives or criteria by solving model (5) for all DMUs. As seen earlier, Ramanathan (2006) considered two cases for aggregating local weights to obtain final weights. The first case, in which local weights of criteria are determined by the DEAHP, is not logical because determining the weights of criteria arbitrarily leads to changes in the final ranking of the alternatives. In the second case, model (4) is used to obtain final weights. However, the optimum value of final weights of the alternatives can be computed directly as follows. m P
wAi
wij wj ( ); ¼ m P max wkj wj k2f1;...;ng j¼1
i ¼ 1; . . . ; n
ð7Þ
j¼1
where WAi is the final weight of alternative Ai. Proof Consider model (4). It is obvious that, u1 1 ¼ Pm ; w d ij j j¼1 j¼1 wij dj
v1 p Pm
i ¼ 1; . . . ; n
ð8Þ
The above constraints can be converted to a single constraint: ( ) 1 1 nP o ð9Þ Pm v1 p min ¼ m w d i2f1;...;ng ij j j¼1 w d max ij j j¼1 i2f1;...;ng
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To make the objective function of model (4) achieve its maximum value, v1 has to take its upper bound value. That is, 1 nP o ð10Þ v1 ¼ m w d max ij j j¼1 i2f1;...;ng
Therefore, the maximum objective function value of model (4) is computed as: m P woj dj m X j¼1 ( ) WAo ¼ v1 ð woj dj Þ ¼ m P j¼1 max wij dj i2f1;:::;ng j¼1
m P
¼
m P
woj ðwj =w1 Þ woj wj j¼1 ( )¼ ( ) m m P P max max wij ðwj =w1 Þ wij wj i2f1;:::;ng i2f1;:::;ng j¼1
j¼1
ð11Þ
j¼1
It can be seen that the final weights obtained by Equation (11) are the same as the weights obtained by simple additive weighting (SAW) method. Then, the SAW method can be used to aggregate local weights (as is shown in Table 1). This new procedure is called Revised DEAHP. The procedure is shown in Figure 1.
4. Numerical examples 4.1. Example 1 Here, the objective is to compute local weights for inconsistent pairwise comparison matrix A, by the Revised DEAHP. First, e is determined by solving model (6). Max e Subject to: u1 ¼ 1
Figure 1 Procedure of the Revised DEAHP for weight derivation in the hierarchical structure.
1 1 v1 þ v2 þ 5v3 þ v4 þ v5 þ v6 p1 3 3 1 v1 þ 5v2 þ 5v3 þ v4 þ v5 þ 3v6 p1 3 1 1 v1 þ v2 þ 6v3 þ 3v4 þ v5 þ v6 p1 4 3
v1 þ 4v2 þ 3v3 þ v4 þ 3v5 þ 4v6 p1 1 1 v1 þ v2 þ 7v3 þ 3v4 þ v5 þ v6 p1 4 5 1 1 1 1 1 v1 þ v2 þ v3 þ v4 þ v5 þ v6 p1 3 7 5 5 6
v1 Xe v2 Xe v3 Xe
Table 1 Aggregating of local weights by SAW method Alternative
Criterion w1
w2
Final weights
v5 Xe v6 Xe
...
A1
w11
w12
w1m
A2 .. .
w21 .. .
w22 .. .
w2m .. .
An
wn1
wn2
wnm
v4 Xe
Pm
j¼1
w1j wj /
j¼1
w2j wj /
Pm Pm
j¼1
wnj wj /
Pn Pm i¼1
j¼1
wij wj
i¼1
j¼1
wij wj
i¼1
j¼1
wij wj
Pn Pm .. P. n Pm
The optimum value of e is 0.0625. Thus, the overall assurance interval for e is (0, 0.625]. After determining e, model (5) is solved to obtain local weights of the alternatives or criteria. For matrix A, model (6) should
SM Mirhedayatian and R Farzipoor Saen—New approach for weight derivation using DEA
be solved six times for each alternative or criterion. For instance, the model for generating local weights of alternative 1 is as below: Max
w1 ¼ v1 þ 4v2 þ 3v3 þ v4 þ 3v5 þ 4v6
Subject to: 1 1 v1 þ v2 þ 7v3 þ 3v4 þ v5 þ v6 p1 4 5 1 1 1 1 1 v1 þ v2 þ v3 þ v4 þ v5 þ v6 p1 3 7 5 5 6 1 1 v1 þ v2 þ 5v3 þ v4 þ v5 þ v6 p1 3 3 1 v1 þ 5v2 þ 5v3 þ v4 þ v5 þ 3v6 p1 3 1 1 v1 þ v2 þ 6v3 þ 3v4 þ v5 þ v6 p1 4 3 v1 X0:0625 v2 X0:0625 v3 X0:0625 v4 X0:0625 v5 X0:0625 v6 X0:0625 As it can be seen, the first constraint is eliminated when model (5) is used for computing local weight of alternative 1. The computed local weights of the alternatives or criteria using the Revised DEAHP are presented in Table 2.4 Furthermore, to show that it is not possible to choose e arbitrarily, we chose e ¼ 0.07 (which is greater than maximum permitted value (0.0625) determined by solving model (6), and it can be seen that at least for one alternative the model will have infeasible solution). Table 2 shows the ranking of the alternatives or criteria as below. C1 4C5 4C2 4C6 4C4 4C3 This ranking is consistent with the information of pairwise comparison matrix A. With respect to the first row of matrix A, it is obvious that C1 is more important than others, and can be removed from pairwise comparison matrix. In this reduced matrix, C5 is the most important alternative or criterion and should be ranked in the second place. Therefore, it can be eliminated from the matrix for further comparisons. From this reduced pairwise comparison matrix, it is observed that C2 is weakly more important than C6 because it is seven times as important as C3, while C6 is only six times as important as C3. Therefore, C2 and C6 are, respectively, the third and 4
For benefit of the readers, a model with solution for matrix A has been represented in Appendix B.
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Table 2 Local weights generated using Revised DEAHP for matrix A Alternative or criterion 1 2 3 4 5 6
e
Local weights using Revised DEAHP
Rank
0.0625 0.0625 0.0625 0.0625 0.0625 0.0625
1.125 0.778 0.128 0.542 0.958 0.724
1 3 6 5 2 4
fourth most important alternatives or criteria. The left criteria or alternatives, C3 and C4, are very easy to be ranked as C44C3 in terms of their direct comparisons a34 ¼ 15 and a43 ¼ 5. Therefore, the ranking obtained by the Revised DEAHP is consistent with pairwise comparison matrix information. This shows that the Revised DEAHP can produce logical local weights for inconsistent pairwise comparison matrices. The DEAHP does not use all information of pairwise comparison matrices for computing local weights of alternatives or criteria. For example, consider the following pairwise comparison matrices: 2 3 2 3 1 2 5 1 5 5 6 7 6 7 C ¼4 12 1 3 5 D ¼ 4 15 1 3 5 1 1 1 1 1 1 5 3 5 3 2 3 1 9 5 6 7 E ¼4 19 1 3 5 1 1 1 5 3 The local weights vector obtained using the DEAHP for the above pairwise comparison matrices is the same as (1, 0.6, 0.2), which means that the DEAHP is not sensitive to changes in c12 to d12 and e12 (the second element of the first row in matrices C, D, and E, respectively). In fact, the DEAHP only uses the third column data for generating local weights. Table 3 compares local weights using the DEAHP and the Revised DEAHP methods for pairwise comparison matrices C, D, and E. As can be seen from Table 3, the local weights obtained by the Revised DEAHP are changed when c12 varies from 2 to 5 and 9 in matrices D and E, respectively. It is obvious that the Revised DEAHP reflects better the changes of pairwise comparisons in a pairwise comparison matrix than the DEAHP.
4.2. Example 2 Since the Revised DEAHP is used for weight derivation in the AHP, one can apply it in decision-making problems with hierarchy structures to rank alternatives. Here, the
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Table 3 Local weights using DEAHP and Revised DEAHP methods for pairwise comparison matrices C, D, and E Matrix
Objective function
Local weights DEAHP
Revised DEAHP
Non-normalized
Normalized
Non-normalized
Normalized
C
w1 w2 w3
1 0.6 0.2
0.555 0.333 0.111
1.875 0.562 0.192
0.713 0.214 0.073
D
w1 w2 w3
1 0.6 0.2
0.555 0.333 0.111
4.091 0.382 0.139
0.887 0.083 0.030
E
w1 w2 w3
1 0.6 0.2
0.555 0.333 0.111
7.548 0.277 0.103
0.952 0.035 0.013
Selecting the best president for a university
Academic & functional knowledge (C1)
Professional experience (C2)
Candidate 1
Commitment (C3)
Candidate 2
Personal attribute (C4)
Candidate 3
Figure 2 Hierarchy for selecting the best president for a university.
Revised DEAHP is used to select the best president of a university. The data set of this example has been taken from Wang and Chin (2009). The selection criteria considered by the search committee include academic and functional knowledge (C1), professional experience (C2), commitment (C3), and personal attributes (C4). The ideal candidate should possess as much of the above attributes as possible. After a long period of searching, three potential candidates were shortlisted for committee interview. Figure 2 shows the hierarchical structure of this decision-making problem. Table 4 presents the local weights obtained using the EM, the DEAHP, and the Revised DEAHP methods. To compare the results of the Revised DEAHP with the EM, the weights, dividing to summation, are normalized. As illustrated in Table 4, the local weights for inconsistent matrices A, B, C, and E obtained by the DEAHP are illogical. From pairwise comparison matrix A, it is
obvious that the first criterion (C1) is the most important one, but the results of the DEAHP cause the criteria C1, C3, and C4 to be as important as each other. From pairwise comparison matrix B, it can be seen that the first candidate is more important than candidate 3 (b13 ¼ 2), while the DEAHP generates the same weights for these candidates. The local weights generated for pairwise comparison matrices C and E are also illogical and counter-intuitive for the same reason. From Table 4, it can also be seen that the ranking of criteria or alternatives with respect to each criterion using the Revised DEAHP is the same as the ranking obtained from the EM, and that changing this ranking leads to changes of the final ranking of alternatives (after aggregation of local weights). Therefore, the final weights obtained by the Revised DEAHP are logical and consistent with the decision maker’s judgements. Computing the local weights of the criteria and alternatives with respect to each criterion, the final weights are obtained by the SAW method. The final weights generated by the EM, the DEAHP, and the Revised DEAHP are presented in Table 5. In the AHP method, to generate final weights of alternatives, the local weights obtained for each level are used in aggregation of local weights. Any miscalculation of the local weights influences the value of the final weights, and as a result the ranking of the alternatives. Therefore, because of incorrect calculation of local weights, it is reasonable that the final weights generated by the DEAHP be illogical. As was seen in Table 4, in the DEAHP, the criteria C3 and C4 were considered as important as C1 incorrectly. On the other hand, the relative importance of candidate 2 with respect to criteria C2, and C4 have also been overestimated by the DEAHP and cannot be correct either. These miscalculations make the DEAHP wrongly select candidate 2 as the best candidate for the presidency, as shown in Table 5. The final weights generated by
C1 C2
C3
E: Pairwise comparisons of three candidates with respect to criterion C4 Candidate 1 1 2 2 Candidate 2 1/2 1 2 Candidate 3 1/2 1/2 1 Consistency ratio (CR)=0.0462
1 1 0.5
0.25 1 0.25
three candidates with respect to criterion C3 1 1/4 1 4 1 4 1 1/4 1
D: Pairwise comparisons of Candidate 1 Candidate 2 Candidate 3 Consistency ratio (CR)=0
DEAHP
0.5 1 1
Candidate 3
C: Pairwise comparisons of three candidates with respect to criterion C2 Candidate 1 1 1/2 1/2 Candidate 2 2 1 1/2 Candidate 3 2 2 1 Consistency ratio (CR)=0.0462
Candidate 2
1 0.3333 1
Candidate 1
2 1/5 1/3 1
C4
B: Pairwise comparisons of three candidates with respect to criterion C1 Candidate 1 1 3 2 Candidate 2 1/3 1 1/3 Candidate 3 1/2 3 1 Consistency ratio (CR)=0.0462
Candidates
Consistency ratio (CR)=0.0797
A: Pairwise comparisons of four selection criteria with respect to decision goal C1 1 5 3 C2 1/5 1 1/5 C3 1/3 5 1 C4 1/2 5 3
Criteria
1 1 2
2 1 2
2 1 1
1 2 1
Rank
1 0.2 1 1
DEAHP 0.4500 0.0586 0.1709 0.3205
EM
0.4934 0.3108 0.1958
0.1667 0.6667 0.1667
0.1958 0.3108 0.4934
0.5278 0.1396 0.3325
EM
1 2 3
2 1 2
3 2 1
1 3 2
Rank
Local priorities
1 2 1 1
Rank
0.4410 0.0500 0.2100 0.2990
0.5920 0.2590 0.1480
0.0555 0.8889 0.0555
0.1480 0.2590 0.5920
0.5930 0.1100 0.2970
1 2 3
2 1 2
3 2 1
1 3 2
Rank
Revised DEAHP
Revised DEAHP
1 4 3 2
Rank
Local priorities
Table 4 Pairwise comparison matrices for four selection criteria and three candidates and their best local priorities determined by different methods
1 4 3 2
Rank
SM Mirhedayatian and R Farzipoor Saen—New approach for weight derivation using DEA
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Table 5 Aggregation of the best local priorities in Table 4 Criterion 1
Criterion 2
Criterion 3
Criterion 4
Global priorities
Rank
1 1 1 0.5
0.928 1.000 0.770
2 1 3
0.3205 0.4934 0.3108 0.1958
0.4356 0.2946 0.2698
1 2 3
0.457 0.324 0.216
1 2 3
Global priorities of three candidates with respect to decision goal by the DEAHP Local weights of criteria 1 0.2 1 Candidate 1 1 0.5 0.25 Candidate 2 0.3333 1 1 Candidate 3 1 1 0.25 Global priorities of three candidates with respect to decision goal by the EM Local weights of criteria 0.4500 0.0586 0.1709 Candidate 1 0.5278 0.1958 0.1667 Candidate 2 0.1396 0.3108 0.6667 Candidate 3 0.3325 0.4934 0.1667
Global priorities of three candidates with respect to decision goal by the Revised DEAHP Local weights of criteria 0.441 0.050 0.210 0.299 Candidate 1 0.593 0.148 0.055 0.592 Candidate 2 0.144 0.259 0.888 0.259 Candidate 3 0.333 0.592 0.055 0.148 Bold type value shows the most preferred candidates.
the Revised DEAHP show that candidate 1 is correctly selected as the best candidate for the president selection problem, because it performs better than candidate 2, in terms of two important criteria C1 and C4. As it was stated earlier, in the DEAHP, the multipliers (vj) are assumed to be positive (vjX0), and this permits these multipliers to have zero values in the optimum solution and this, in turn, causes more candidates to be efficient. However, in the Revised DEAHP, an infinitesimal non-Archimedean positive value is considered as a lower bound for multipliers (vjXe). The values of multipliers for pairwise comparison matrices B, C, D, and E using the DEAHP and the Revised DEAHP are presented in Table 6. As can be seen from Table 6, the values of multipliers in the Revised DEAHP are strictly positive and this means that all pairwise comparisons are considered in the weight derivation, and this in turn prevents the candidates to be wrongly efficient. To examine the rank reversal problem, the pairwise comparison matrix A is used for this end. As it can be seen from Table 4, the current ranking is as below.
Table 6 Values of multipliers obtained by DEAHP and Revised DEAHP Matrix
B
w1 w2 w3 w1 w2 w3 w1 w2 w3 w1 w2 w3
C D E
DEAHP
Revised DEAHP
v1
v2
v3
v1
v2
v3
0 1 0 0.5 0.5 0.5 0.25 0.25 0.25 0 0 0
0 0 0.333 0 0 0 0 0 0 0 0 0
0.5 0 0 0 0 0 0 0 0 0.5 0.5 0.5
0.111 0.111 0.111 0.2 0.2 0.2 0.111 0.111 0.111 0.2 0.2 0.2
0.111 0.111 0.111 0.2 0.2 0.2 0.111 0.111 0.111 0.2 0.2 0.2
0.111 0.861 0.111 0.5 0.2 0.2 0.111 0.861 0.111 0.5 0.2 0.2
Table 7 Examining rank reversal problem in the Revised DEAHP Local weights by Revised DEAHP (non-normalized)
Rank
Removed criterion
Final ranking of criteria
1 3 4
1.273 0.606 0.864
1 3 2
2
C14C44C3
1 2 4
1.273 0.145 0.863
1 3 2
3
C14C44C2
1 2 3
3.364 0.145 0.606
1 3 2
4
C14C34C3
Criterion
C1 4C4 4C3 4C2 According to the results of the Revised DEAHP, the criteria C2, C3, and C4 are inefficient (ie, their efficiency scores are less than 1). Therefore, these criteria can be eliminated from the pairwise comparison matrix and then the local weights can be obtained to achieve the new ranking of the criteria. The results are presented in Table 7. It is found that, after eliminating inefficient criteria, the ranking results are not changed and this shows that the Revised DEAHP not only produces correct weights for inconsistent matrices but also retains the advantage of the DEAHP on the rank reversal problem.
Objective function
Bold type shows the inefficient criteria that have been eliminated from pairwise comparison matrix.
SM Mirhedayatian and R Farzipoor Saen—New approach for weight derivation using DEA
5. Concluding remarks In this paper, the drawbacks of the DEAHP were examined and it was shown that the DEAHP generates counterintuitive weights for inconsistent pairwise comparison matrices. To overcome the drawbacks of the DEAHP, a new procedure called Revised DEAHP was proposed. Through a few numerical examples, it was shown that the Revised DEAHP produces rational weights for inconsistent matrices, and it was found that the Revised DEAHP is sensitive to changes in pairwise comparison matrices while the DEAHP is not. Then, through a numerical example, selecting the best president of a university, it was found that the Revised DEAHP not only produces logical weights for inconsistent pairwise comparison matrices, but also retains the advantage of the DEAHP on the rank reversal problem. Because of the simple and understandable nature of the Revised DEAHP, it is expected that the Revised DEAHP be used for many applications. In this research, it was assumed that the data of the pairwise comparisons were deterministic. Future researches can be conducted by using the Revised DEAHP with fuzzy and stochastic data.
Acknowledgements —The authors thank the anonymous reviewer for valuable suggestions and comments.
References Andersen P and Petersen NC (1993). A procedure for ranking efficient units in data envelopment analysis. Mngt Sci 39: 1261–1264. Charnes A, Cooper WW and Rhodes EL (1978). Measuring the efficiency of decision making units. Eur J Opl Res 2: 429–444. Chu ATW, Kalaba RE and Spingarn K (1979). A comparison of two methods for determining the weights of belonging to fuzzy sets. J Optimiz Theory App 27: 531–538. Cogger KO and Yu PL (1985). Eigenweight vectors and leastdistance approximation for revealed preference in pairwise weight ratios. J Optimiz Theory App 46: 483–491. Cooper W, Seiford L and Tone K (2000). Data Envelopment Analysis: A Comprehensive Text with Models, Applications, References. Kluwer Academic Publishers: Boston. Crawford GB (1987). The geometric mean procedure for estimating the scale of a judgment matrix. Math Modelling 9: 327–334. Gass SI and Rapcsak T (2004). Singular value decomposition in AHP. Eur J Opl Res 154: 573–584. Islei G and Lockett AG (1988). Judgemental modelling based on geometric least square. Eur J Opl Res 36: 27–35. Lipovetsky S and Conklin WM (2002). Robust estimation of priorities in the AHP. Eur J Opl Res 137: 110–122. Mehrabian S, Jahanshahloo GR, Alirezaee MR and Amin GR (2000). An assurance interval for the non-Archimedean epsilon in DEA models. Opns Res 48: 344–347. Mikhailov L (2000). A fuzzy programming method for deriving priorities in the analytic hierarchy process. J Opl Res Soc 51: 341–349. Ramanathan R (2006). Data envelopment analysis for weight derivation and aggregation in the analytic hierarchy process. Comput Opl Res 33: 1289–1307.
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Saaty TL (1980). The Analytic Hierarchy Process. McGraw-Hill: New York. Saaty TL (2000). Fundamentals of Decision Making and Priority Theory with the Analytic Hierarchy Process. RWS Publications: Pittsburgh. Wang YM and Chin KS (2009). A new data envelopment analysis method for priority determination and group decision making in the analytic hierarchy process. Eur J Opl Res 195: 239–250.
Appendix A Consistency ratio In the classical AHP, the multiple criteria decision problem is structured hierarchically at different levels, which contain finite number of criteria or alternatives. Applying reciprocal comparisons, the relative importance of the decision elements is determined through exact reciprocal comparison ratios. Then, a reciprocal comparison matrix is obtained and used to aggregate into the weights of alternatives. However, the obtained weights are reliable only if the judgements of a decision maker are perfectly consistent. In most cases, it is unrealistic to expect a reciprocal comparison matrix to be perfectly consistent for the reason that the decision maker would be affected by many uncertainties. Consequently, a definition of acceptably consistent reciprocal comparison matrices was further developed by Saaty (1980) to allow a certain level of acceptable deviation. In what follows, let us first recall the knowledge of consistent and acceptably consistent reciprocal comparison matrices. Let A ¼ (aij)n n denote a reciprocal comparison matrix, where aij ¼ 1/aji, aijAR þ , for all i, j ¼ 1, 2, . . . , n. Definition (Saaty, 1980): A reciprocal comparison matrix A ¼ (aij)n n is consistent, if aij ¼ aik akj for all i, j, k ¼ 1, . . . , n. When A is inconsistent, Saaty (1980) has proposed a consistency index (CI) and a CR to measure the level of inconsistency, that is lmax n ; CR ¼ CI=RI; n1 where lmax and n are the largest eigenvalue and the order of A, respectively. RI is a random index, which is the average CI of a large number of randomly generated reciprocal comparison matrices, and dependent on the orders of the matrices given in Table A1. When CRo0.1, CI ¼
Table A1 The mean consistency index of randomly generated matrices N 1 2
3
4
5
6
7
8
9
10
11
12
RI 0 0 0.52 0.89 1.12 1.26 1.36 1.41 1.46 1.49 1.52 1.54
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the reciprocal comparison matrix A is considered to be acceptably consistent. While CRX0.1, the matrix A is said to be of unacceptable consistency, which should be adjusted to that with acceptable consistency to ensure the rationality of decisions.
B.2. Weight derivation using model (5) Alternative 1: Model: max ¼ u1 þ 4 u2 þ 3 u3 þ u4 þ 3 u5 þ 4 u6; ð1=4Þ u1 þ u2 þ 7 u3 þ 3 u4 þ ð1=5Þ u5 þ u6o ¼ 1;
Appendix B
ð1=3Þ u1 þ ð1=7Þ u2 þ u3 Models and solution for matrix A using Lingo 8 software
þ ð1=5Þ u4 þ ð1=5Þ u5 þ ð1=6Þ u6o ¼ 1;
B.1. Epsilon determination using model (6)
u1 þ ð1=3Þ u2 þ 5 u3 þ u4
Model:
þ u5 þ ð1=3Þ u6o ¼ 1;
max ¼ e;
ð1=3Þ u1 þ 5 u2 þ 5 u3 þ u4
u1 þ 4 u2 þ 3 u3 þ u4 þ 3 u5 þ 4 u6o ¼ 1;
þ u5 þ 3 u6o ¼ 1;
ð1=4Þ u1 þ u2 þ 7 u3 þ 3 u4
ð1=4Þ u1 þ u2 þ 6 u3 þ 3 u4
þð1=5Þ u5 þ u6o ¼ 1;
þ ð1=3Þ u5 þ u6o ¼ 1;
ð1=3Þ u1 þ ð1=7Þ u2 þ u3 þ ð1=5Þ u4
u14 ¼ 0:0625;
þð1=5Þ u5 þ ð1=6Þ u6o ¼ 1;
u24 ¼ 0:0625;
u1 þ ð1=3Þ u2 þ 5 u3 þ u4 þ u5
u34 ¼ 0:0625; u44 ¼ 0:0625;
þð1=3Þ u6o ¼ 1;
u54 ¼ 0:0625;
ð1=3Þ u1 þ 5 u2 þ 5 u3 þ u4 þ u5 þ 3 u6o ¼ 1;
u64 ¼ 0:0625;
ð1=4Þ u1 þ u2 þ 6 u3 þ 3 u4 þð1=3Þ u5 þ u6o ¼ 1;
solution: Global optimal solution found at iteration: 0 Objective value: 1.125
u14 ¼ e; u24 ¼ e; u34 ¼ e;
Variable
u44 ¼ e;
u1 u2 u3 u4 u5 u6
u54 ¼ e; u64 ¼ e; solution: Global optimal solution found at iteration: 14 Objective value: 0.6250000E-01 Variable e u1 u2 u3 u4 u5 u6
Value
Reduced cost
0.6250000E-01 0.6250000E-01 0.6250000E-01 0.6250000E-01 0.6250000E-01 0.6250000E-01 0.6250000E-01
0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
Value
Reduced cost
0.1875000 0.6250000E-01 0.6250000E-01 0.6250000E-01 0.6250000E-01 0.6250000E-01
0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
If we choose e ¼ 0.07 (which is greater than 0.0625), then at least for one alternative the model will have infeasible solution. For example, for alternative 1, the model will be changed as below: Alternative 1: Model: max ¼ u1 þ 4 u2 þ 3 u3 þ u4 þ 3 u5 þ 4 u6; ð1=4Þ u1 þ u2 þ 7 u3 þ 3 u4 þ ð1=5Þ u5 þ u6o ¼ 1;
SM Mirhedayatian and R Farzipoor Saen—New approach for weight derivation using DEA
ð1=3Þ u1 þ ð1=7Þ u2 þ u3 þ ð1=5Þ u4 þ ð1=5Þ u5 þ ð1=6Þ u6o ¼ 1; u1 þ ð1=3Þ u2 þ 5 u3 þ u4 þ u5 þ ð1=3Þ u6o ¼ 1; ð1=3Þ u1 þ 5 u2 þ 5 u3 þ u4 þ u5 þ 3 u6o ¼ 1; ð1=4Þ u1 þ u2 þ 6 u3 þ 3 u4 þ ð1=3Þ u5 þ u6o ¼ 1; u14 ¼ 0:07; u24 ¼ 0:07; u34 ¼ 0:07; u44 ¼ 0:07; u54 ¼ 0:07; u64 ¼ 0:07; solution:
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As it can be seen, by choosing arbitrary value for epsilon, the solution for model (5) becomes infeasible.
Received March 2009; revised April 2010 after two revisions
