A note on group decision-making procedure based ... - Semantic Scholar
Soft Comput (2011) 15:1289–1300 DOI 10.1007/s00500-010-0662-3
2011,15(7)/Soft Computing
ORIGINAL PAPER
A note on group decision-making procedure based on incomplete reciprocal relations Yejun Xu • Qingli Da • Huimin Wang
Published online: 23 October 2010 Springer-Verlag 2010
Abstract In a very recent paper by Xu and Chen (Soft Comput 12:515–521, 2008), a novel procedure for group decision making with incomplete reciprocal relations was developed. In this note, we examine the function between the fuzzy preference relation and its corresponding priority vector developed by Xu and Chen with a numerical example and show that the function does not hold in general cases. Then, we deduce an exact function between the additive transitivity fuzzy preference relation and its corresponding priority vector. Based on this, we develop a procedure for the decision making with an incomplete reciprocal relation and also develop a procedure for the group decision making with incomplete reciprocal relations. In order to compare the performances of our method with Xu and Chen’s method in fitting the reciprocal relation, we introduce some criteria. Theoretical analysis and numerical examples have shown that the function deduced by us is more reasonable and effective than Xu and Chen’s.
Y. Xu (&) H. Wang Business School, HoHai University, Nanjing 210098, Jiangsu, China e-mail: [email protected] Q. Da School of Economics and Management, Southeast University, Nanjing 210096, Jiangsu, China e-mail: [email protected] H. Wang State Key Laboratory of Hydrology Water Resource and Hydraulic Engineering, HoHai University, Jiangsu 210098, China e-mail: [email protected]
Keywords Group decision making Incomplete reciprocal relation Additive transitivity consistency Auxiliary reciprocal relation
1 Introduction Reciprocal relations [or called fuzzy preference relations (Orlovsky 1978)] have received a great deal of attention from researchers. A complete fuzzy preference relation of order n necessitates the completion of all n(n - 1)/2 judgements in its entire top triangular portion. Sometimes, however, a decision maker (DM) may develop a fuzzy preference relation with incomplete information because of (1) time pressure, lack of knowledge, and the DM’s limited expertise related with problem domain(Chiclana et al. 2008; Lee et al. 2007; Xu 2004, 2005; Xu and Chen 2008); (2) when the number of the alternatives, n, is large it may be practically impossible, or at least unacceptable from the point of view of the decision maker, to perform all the n(n - 1)/2 required comparisons to complete the pairwise comparison matrices (Fedrizzi and Giove 2007); (3) It can be convenient/necessary to skip some direct critical comparison between alternatives, even if the total number of alternatives is small (Fedrizzi and Giove 2007). (4) An expert would not be able to efficiently express any kind of preference degree between two or more of the available options. This may be due to an expert not possessing a precise or sufficient level of knowledge of part of the problem, or because that expert is unable to discriminate the degree to which some options are better than others (Alonso et al. 2004, 2008, 2009; Cabrerizo et al. 2010a, b; Herrera-Viedma et al. 2007a, b). Up to now, some related theory studies with incomplete reciprocal preference relations have been given.
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Alonso et al. (2004) put forward a procedure which attempts to find out the missing information in an expert’s incomplete fuzzy preference relation based on additive consistency, using only the preference values provided by that particular expert. Herrera-Viedma et al. (2007b) presented a new model to deal with GDM problems with the incomplete fuzzy preference relations based on the additive consistency (AC) property. The new model is composed of two steps: the estimation of missing preference values and the selection of alternatives. Fedrizzi and Giove (2007) proposed a new method to calculate the missing elements of an incomplete fuzzy preference relation by maximizing the global consistency of the ‘completed’ matrix. Afterwards, Chiclana et al. (2009) found that the above two methods (the first one by Herrera-Viedma et al. 2007a, b, and the second one by Fedrizzi and Giove 2007) for calculating missing values, although different, are very similar. Both methods derive the same estimated values for the independent-missing-comparison case, while they differ in the dependent-missing-comparison case. Herrera-Viedma et al. (2007a) further presented a consensus model for GDM problems with incomplete fuzzy preference relations. The main novelty of the consensus model is that of being guided by both consensus and consistency measures. Furthermore, the consensus-reaching process is guided automatically, without moderator, through both consensus and consistency criteria. Alonso et al. (2008) put forward a general procedure that attempts to estimate the missing information in any of the above formats of incomplete preference relations: fuzzy, multiplicative, interval valued and linguistic. Xu (2004) defined the concepts of incomplete fuzzy preference relation, additive consistent incomplete fuzzy preference relation and multiplicative consistent incomplete fuzzy preference relation, and then proposed two goal programming models, based on additive consistent incomplete fuzzy preference relation and multiplicative consistent incomplete fuzzy preference relation respectively, for obtaining the priority vector of incomplete fuzzy preference relation. Xu (2005) proposed a procedure for decision making with incomplete fuzzy preference relation based on multiplicative consistency. Gong (2008) developed a least-square model to obtain the collective priority vector of the incomplete preference relations by multiple decision makers. And the priority vector only derived by a simple equation. We (Xu and Da 2008) proposed weighted least-square method and quadratic programming method for priorities of incomplete fuzzy preference relation based on multiplicative transitivity. We (Xu and Da 2009) presented two priority methods: the least variance priority method (LVM) and the quadratic programming method for incomplete fuzzy preference relation based on additive consistency.
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Very recently, Xu and Chen (2008) also developed a simple but practical approach to deriving the ranking of the alternatives from an incomplete reciprocal relation based on additive transitivity. They postulated a correspondence between priority vector and additive consistent incomplete fuzzy preference relation. Shen et al. (2009), we (Xu et al. 2009), respectively, have proved that the correspondence does not always hold. In this paper, we will still investigate the problem which Xu and Chen had investigated. The purpose of this paper is to present that the function between the fuzzy preference and priority vector proposed by us is more reasonable than Xu and Chen’s. In order to do this, the rest of this paper is structured as follows. In Sect. 2, we recall some concepts of incomplete reciprocal relation. In Sect. 3, we first illustrate with a numerical example to point out the correspondence between priority vector and additive consistent incomplete fuzzy preference relation given by Xu and Chen does not always hold in general cases, and then we deduce a simple functional equation between fuzzy preference’ element and priority vector. We prove that our function is more effective to fitting the relationship between the fuzzy preference’s element and the priority vector. In Sect. 4, we present a procedure for group decision making with incomplete reciprocal relations, and propose some new concepts and criteria to measure the performances of our method. In Sect. 5, we examine two numerical examples to present the effectiveness of our methods, and compare the performances between our method and Xu and Chen’s method. Conclusions are offered in Sect. 6. 2 Incomplete reciprocal relation For simplicity, we let N = {1, 2,…, n}. Definition 1 Let R = (rij)n9n be a preference relation, then R is called a fuzzy preference relation (Orlovsky 1978; Tanino 1984) if rij 2 ½0; 1; rij þ rji ¼ 1; rii ¼ 0:5;
for all i; j 2 N:
ð1Þ
Definition 2 Let R = (rij)n9n be a fuzzy preference relation, then R is called an additive transitive fuzzy preference relation, if the following additive transitivity (Tanino 1984) is satisfied: rij ¼ rik rjk þ 0:5;
for all i; j; k 2 N:
ð2Þ
We also called it to be perfectly additive transitive consistent. From Definition 2, we can get the following result easily: Theorem 1
Let R = (rij)n9n be a complete preference
relation, then the sum of all the elements of R is nðn1Þ 2 ; that is
2011,15(7)/Soft Computing
Group decision-making procedure n n X X i¼1 j¼1;j6¼i
rij ¼
n X n X i¼1 j¼1
rij
n X i¼1
rii ¼
nðn 1Þ : 2
ð3Þ
The above Theorem 1 denotes that the sum of the elements of the complete reciprocal relation does not involve the diagonal elements. If the diagonal elements are taken into account, the sum will be n2/2. Definition 3 Xu (2004) Let R = (rij)n9n be a fuzzy preference relation, then R is called an incomplete fuzzy preference relation, if some of its elements cannot be given by the DM, which we denote the unknown number x, and the others can be provided by the DM, which satisfy Eq. (1). Definition 4 Xu (2004) Let R = (rij)n9n be an incomplete fuzzy preference relation, then R is called an additive consistent incomplete fuzzy preference relation, if all the known elements of R satisfy the additive transitivity rij = rik - rjk ? 0.5.
3 An approach to constructing a complete reciprocal relation from an incomplete reciprocal relation
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additive consistent incomplete fuzzy preference relation [i.e., Eq. (4)] does not always hold. In the following, we will deduce the more reasonable relationship between rij and wi - wj. Zhang (2000), we (Xu et al. 2009) have proved the following results: Lemma 1 Let R = (rij)n9n be a fuzzy additive transitive preference relation, w = (w1, w2, …, wn)T be the corresponding weighting vector, where 0 B wi B 1, i = 1, P 2, …, n, ni¼1 wi ¼ 1; then there exists a positive number b, and such a relation can be expressed as follows: rij ¼ 0:5 þ bðwi wj Þ where b [ 0. Theorem 2 If the priority vector of the additive transitive perfectly consistent fuzzy preference relation R is derived by normalizing rank aggregation method, then b ¼ n1 2 : Proof If the priority vector of the additive transitive perfectly consistent fuzzy preference relation R is derived by normalizing rank aggregation method, then Pn Pn rik 0:5 rik 0:5 k¼1 wi ¼ Pn Pn ¼ k¼1nðn1Þ ; i2N ð6Þ i¼1 k¼1;k6¼i rik 2 Pn Pn rjk 0:5 rjk 0:5 k¼1 P P ¼ k¼1nðn1Þ ; i 2 N: ð7Þ wj ¼ n n r i¼1 k¼1;k6¼j jk
As we stated above, suppose that we have a set of alternatives, X = {x1, x2, …, xn}, the expert gives his/her fuzzy preference relation, and construct the judgement matrix R = (rij)n9n,and w1, w2, …, wn be the corresponding ranking vector of each alternative x1, x2, …, xn, where Pn i¼1 wi ¼ 1; wi 0: In our previous work (Xu et al. 2009), we use function f to denote the relationship between the element rij and wi - wj, which is rij = f(wi - wj). Xu (2004; Xu and Chen 2008) inferred the relationship between rij and wi - wj as follows:
since
rij ¼ 0:5ðwi wj þ 1Þ:
rij ¼ rik rjk þ 0:5
ð4Þ
ð5Þ
2
Put (6), (7) into (5), then rij ¼ bðwi wj Þ þ 0:5 Pn ðrik rjk Þ þ 0:5 ¼ b k¼1nðn1Þ 2
Now, we will show the equation does not hold in general cases. For example, 2 3 0:5 1 0:5 R ¼ 4 0 0:5 0 5: 0:5 1 0:5
then
Obviously, the reciprocal relation R is a consistency reciprocal relation according the Definition 2. On the other hand, since r12 = 1, then according to Eq. (4), we have w1 = 1, w2 = 0, and thus w3 = 0. But r32 = 1, we should have w3 = 1, w2 = 0 and w1 = 0. Therefore, the results are conflict between themselves. Shen et al. (2009) proved that the correspondence between priority vector and
Theorem 3 If b ¼ n1 2 ; then the priority vector of the additive transitive perfectly consistent fuzzy preference relation R derived by Eq. (5) is normalizing rank aggregation method, that is Pn Pn j¼1;j6¼i rij j¼1 rij 0:5 wi ¼ : ð9Þ ¼ Pn Pn nðn 1Þ=2 i¼1 j¼1;j6¼i rij
rij ¼ b
Pn
k¼1 ðrij nðn1Þ 2
0:5Þ
nrij n þ 0:5 ¼ b nðn1Þ 2 þ 0:5:
ð8Þ
2
So we can get b ¼ n1 2 ; which completes the proof.
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Proof wi ¼
If b ¼ n1 2 ; by (5), we have
rij 0:5 n1 2
þ wj :
ð10Þ
Summing on both sides of Eq. (10) with respect to j, then Pn n X Xn j¼1 ðrij 0:5Þ w ¼ þ wj ð11Þ n1 j¼1 i 2
i.e.,
Pn
nwi ¼
j¼1 rij n1 2
0:5n
j¼1
þ 1:
Therefore, Pn Pn 1 j¼1 rij 0:5n j¼1 rij 0:5 þ ¼ wi ¼ n nðn nðn 1Þ=2 Pn 1Þ=2 j¼1;j6¼i rij ¼ Pn Pn i¼1 j¼1;j6¼i rij
We call the method is normalizing rank aggregation method. For an incomplete reciprocal relation R = (rij)n9n, we can replace the unknown element ‘‘x’’ with n1 2 ðwi wj Þ þ 1 ; and then construct an auxiliary reciprocal relation 2 R ¼ ðr ij Þnn (we also call R the fitting relation of R), where rij 6¼ x rij ; ð12Þ r ij ¼ n1 1 ðw w Þ þ ; rij ¼ x i j 2 2 Example 1 For a decision-making problem, there are four decision alternatives xi(i = 1, 2, 3, 4). The DM provides his/her preference over these four decision alternatives, and gives an incomplete reciprocal relation as follows (the example has been examined by Xu and Chen 2008): 2 3 0:5 x 0:4 0:8 6 x 0:5 0:3 0:7 7 7 R¼6 4 0:6 0:7 0:5 x 5: 0:2 0:3 x 0:5 Let w = (w1, w2, w3, w4)T be the weight vector of R, then we construct an auxiliary reciprocal relation R ¼ ðr ij Þ44 (where n = 4):
0:5 6 1:5ðw2 w1 Þ þ 0:5 R¼6 4 0:6 0:2
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1:5ðw1 w2 Þ þ 0:5 0:5 0:7 0:3
Then, we use the normalizing rank aggregation method to get the weighting vector. It is in the following: P4 1:5ðw1 w2 Þ þ 0:5 þ 0:4 þ 0:8 j¼2 r 1j w1 ¼ P4 P4 ¼ 6 i¼1 j¼1;j6¼i r ij P4 1:5ðw2 w1 Þ þ 0:5 þ 0:3 þ 0:7 j¼1;j6¼2 r 2j ¼ w2 ¼ P4 P4 6 r i¼1 j¼1;j6¼i ij P4 0:6 þ 0:7 þ 1:5ðw3 w4 Þ þ 0:5 j¼1;j6¼3 r 3j ¼ w3 ¼ P4 P4 6 i¼1 j¼1;j6¼i r ij P4 0:2 þ 0:3 þ 1:5ðw4 w3 Þ þ 0:5 j¼1;j6¼4 r 3j ¼ w4 ¼ P4 P4 6 i¼1 j¼1;j6¼i r ij i. e., 2 4:5 1:5 6 1:5 4:5 6 4 0 0 0 0
which completes the proof.
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32 3 2 3 0 0 w1 1:7 6 7 6 7 0 0 7 76 w2 7 ¼ 6 1:5 7: 4:5 1:5 54 w3 5 4 1:8 5 w4 1:5 4:5 1
Solving the linear system of equations, we get: w1 = 0.3, w2 = 0.233, w3 = 0.3667, w4 = 0.1, and the priority vector is w = (0.3, 0.233, 0.3667, 0.1)T. And we also can get the unknown elements as follows: r 12 ¼ 1:5ðw1 w2 Þ þ 0:5 ¼ 0:6; r 34 ¼ 1:5ðw3 w4 Þ þ 0:5 ¼ 0:9: And thus, we can relation R : 2 0:5 0:6 0:4 6 0:4 0:5 0:3 R¼6 4 0:6 0:7 0:5 0:2 0:3 0:1
get a complete reciprocal fitting 3 0:8 0:7 7 7: 0:9 5 0:5
Obviously, we can verify that R is an additive transitivity reciprocal relation according to Definition 2. If we take b = 1/2, which computed by Xu (Xu and Chen 2008) and we have w ¼ ð0:2863; 0:2467; 0:3137; 0:1533ÞT ; 1 r 12 ¼ ðw1 w3 þ 1Þ ¼ 0:5198; 2 r 34 ¼ 0:5ðw3 w1 þ 1Þ ¼ 0:5802:
0:4 0:3 0:5 1:5ðw4 w3 Þ þ 0:5
3 0:8 7 0:7 7: 1:5ðw3 w4 Þ þ 0:5 5 0:5
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Then 2
0:5 6 0:4802 R¼6 4 0:6 0:2
0:5198 0:5 0:7 0:3
0:4 0:3 0:5 0:4198
Theorem 4 R ¼ ðr ij Þnn is an complete additive consistency reciprocal relation if and only if R ¼ R :
3
0:8 0:7 7 7: 0:5802 5 0:5
Proof Necessary condition. If R ¼ ðr ij Þnn is an additive consistency reciprocal relation, then, for all i, j, k, r ij ¼ r ik r jk þ 0:5; and based on Eq. 9, we have
Obviously, R is not an additive consistent reciprocal relation. Therefore, we can see that it is more reasonable to take b = (n - 1)/2 than b = 1/2. Based on the above procedure, we develop a procedure for the decision making with an incomplete reciprocal relation in the following:
n1 ðwi wj Þ þ 0:5 2 0P 1 n n P r ik 0:5 r jk 0:5 C n 1B Bk¼1 C þ 0:5 k¼1 ¼ @ 2 nðn 1Þ=2 nðn 1Þ=2 A
rij ¼
Algorithm I Let w = (w1, w2, …, wn)T be the weighting vector of the incomplete reciprocal relation R = (rij)n9n. Step 1. Replace the unknown element rij in R with n1 2 ðwi wj Þ þ 0:5; and construct the auxiliary relation R ¼ ðr ij Þnn ; r ij satisfy Eq. 12. Step 2 Utilize the normalizing rank aggregation method to obtain the weighting vector w. First, calculate the collective preference degree pi(w) of the alternative xi over all the other alternatives: n X r ij ; i 2 N ð13Þ pi ðwÞ ¼ j¼1;j6¼i
And establish the following linear system of equations: pi ðwÞ pi ðwÞ wi ¼ P ; i2N ¼ n nðn 1Þ=2 pj ðwÞ
ð14Þ
j¼1
which can be further rewritten as the following form: Aw ¼ b
ð15Þ
where b is a positive vector, A is a real symmetrical matrix, whose ith main diagonal element is nðn1Þ n1 2 ai ; here, ai 2 is the count of the missing elements in the ith line in the matrix R. Step 3 Solving Eq. 15, we can get the priority vector w = (w1, w2, …, wn)T. Step 4 Rank all the alternatives and select the best one(s) in accordance with the values of wi (i [ N). Step 5 End. To further compare the performances of b = (n - 1)/2 and b = 1/2 in fitting the reciprocal relation, we introduce the following definition and criteria: Definition 5 Let w = (w1, w2, …, wn)T be the weighting vector which computed by the above Algorithm I, then we call R ¼ ðrij Þnn ¼ ðn1 2 ðwi wj Þ þ 0:5Þnn is the characteristic matrix of R:
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0P n
¼
1
C n 1B Bk¼1 C þ 0:5 2 @ nðn 1Þ=2 A n P
¼
ðr ik r jk Þ
ðr ij 0:5Þ
k¼1
n
þ 0:5
¼ r ij : Thus, R ¼ R : Sufficient condition. If R ¼ R ; then for all i,j,k [ N, we have r ik r jk ¼ rik rjk n1 ¼ ðwi wk Þ þ 0:5 2 n1 ðwj wk Þ þ 0:5 2 n1 ðwi wj Þ ¼ 2 ¼ rij 0:5 ¼ r ij 0:5 i.e., r ik r jk ¼ r ij 0:5; thus, R is an additive consistency reciprocal relation, which completes the proof. Also, from the above proof process of Theorem 4, we can prove that if b = 1/2 [i.e., r*ij = 0.5(wi - wj ? 1)], the above Theorem 4 will not hold. This means that, if R ¼ ðr ij Þnn is an additive consistency reciprocal relation and the weighting vector is computed by Algorithm I in which the unknown element r ij in R is instead by 0.5(wi - wj ? 1), and then the characteristic matrix R* is computed by r*ij = 0.5(wi - wj ? 1), it will be R 6¼ R : Thus, it is more reasonable to take b = (n - 1)/2 than b = 0.5. Definition 6 Let D = (dij)n9n be the difference matrix between R and R*, where dij ¼ r ij rij :
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Obviously, dij = -dji. From Theorem 4, we know that, R is an additive consistency reciprocal relation if and only if D = 0. Thus, the value of D is the smaller the better. That is n X n n X n X X dij ¼ ð16Þ TD ¼ r ij rij i¼1 j¼1
i¼1 j¼1
where dij ¼ r ij rij is the fitting error for the element r ij of the collective auxiliary reciprocal relation R ¼ ðr ij Þnn ; It is easy to find that |dij| : |dji|. Obviously, the smaller TD, the better fitting performance of the weighting vector w. Again, for the Example 1, as we compute in the above, if we take b = (n - 1)/2, we get T
w ¼ ð0:3; 0:233; 0:3667; 0:1Þ : And thus, the characteristic matrix R* computed by rij ¼ 3 2 ðwi wj Þ þ 0:5; we get 2 3 0:5 0:6 0:4 0:8 6 0:4 0:5 0:3 0:7 7 7 R ¼ 6 4 0:6 0:7 0:5 0:9 5: 0:2 0:3 0:1 0:5 Therefore, P P P P TD ¼ ni¼1 nj¼1 jdij j ¼ ni¼1 nj¼1 jr ij rij j ¼ 0:
2
0:5 6 1:5ðw2 w1 Þ þ 0:5 R ¼ 6 4 1:5ðw3 w1 Þ þ 0:5 1:5ðw4 w1 Þ þ 0:5
1:5ðw1 w2 Þ þ 0:5 0:5 1:5ðw3 w2 Þ þ 0:5 1:5ðw4 w2 Þ þ 0:5
0:2
0:3
0:4198
0:5
And thus, the characteristic matrix R* computed by rij ¼ 12 ðwi wj Þ þ 0:5; we get 2 3 0:5 0:5198 0:4863 0:5665 6 0:4802 0:5 0:4665 0:5467 7 6 7 R ¼ 6 7: 4 0:5137 0:5335 0:5 0:5802 5 0:4335
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0:4533
0:4198
0:5
Therefore, 2 0 6 0 6 D¼6 4 0:0863
3
0
0:0863
0:2335
0 0:1655
0:1665 0
0:1533 7 7 7: 0 5
0:2335 0:1533
0
0
And P P P P TD ¼ ni¼1 nj¼1 jdij j ¼ ni¼1 nj¼1 jr ij rij j ¼ 1:2792: As is known, smaller deviations means better performance. Obviously, from the above results, we know that it is more reasonable to take b = (n - 1)/2 than b = 0.5. Example 2 If the DM provides no preferences over these four decision alternatives, then the incomplete reciprocal relation R is reduced to the following: 2 3 0:5 x x x 6 x 0:5 x x 7 7: R¼6 4 x x 0:5 x 5 x x x 0:5 In this case, if we take b = 1.5, and then we can construct the following auxiliary reciprocal relation R ¼ ðr ij Þ44 :
1:5ðw1 w3 Þ þ 0:5 1:5ðw2 w3 Þ þ 0:5 0:5 1:5ðw4 w3 Þ þ 0:5
If we take b = 1/2, we have w = (0.2863, 0.2467, 0.3137, 0.1533)T, and 2 3 0:5 0:5198 0:4 0:8 6 0:4802 0:5 0:3 0:7 7 6 7 R¼6 7: 4 0:6 0:7 0:5 0:5802 5
Y. Xu et al.
3 1:5ðw1 w4 Þ þ 0:5 1:5ðw2 w4 Þ þ 0:5 7 7: 1:5ðw3 w4 Þ þ 0:5 5 0:5
Then, we use the normalizing rank aggregation method to get the weighting vector. It is in the following: P j6¼i 1:5ðwi wj Þ þ 0:5 wi ¼ 6 i.e., 2 1:5 6 1:5 6 4 1:5 1:5
1:5 1:5 1:5 1:5 1:5 1:5 1:5 1:5
3 32 3 2 1:5 1:5 w1 6 7 6 7 1:5 7 76 w2 7 ¼ 6 1:5 7: 4 5 4 5 w3 1:5 5 1:5 w4 1:5 1:5
ð17Þ
Obviously, there are infinite solutions for the above P equation, i.e., for any wi [ [0, 1], and 4i¼1 wi ¼ 1; Eq. 17 always holds. But if we take b = 0.5 (Xu and Chen 2008), we will get w = (0.25, 0.25, 0.25, 0.25)T. Xu interpreted
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that, in the situations where no any preference information is provided, each alternative should be assigned with the same weight. And with the weighting vector, the auxiliary reciprocal relation R ¼ ðr ij Þ44 should be: 2 3 0:5 0:5 0:5 0:5 6 0:5 0:5 0:5 0:5 7 7 R¼6 4 0:5 0:5 0:5 0:5 5: 0:5 0:5 0:5 0:5 That is we only can construct the unique reciprocal relation. But in our view, R is an complete unknown reciprocal relation, in this circumstance, we can construct numerous auxiliary reciprocal relation only need to satisfy P the condition wi [ [0, 1], and 4i¼1 wi ¼ 1: Again, this also denotes that it is more reasonable to take b = (n - 1)/2 than b = 0.5. In the following, we will further investigate the problem in the group decision making.
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To aggregate all the auxiliary reciprocal relations R ðkÞ ðrij Þnn
Let M = {1, 2, …, m}. Now, we consider a group decision-making problem. Let E = {e1, e2, …, em} be a set of DMs, and let k = (k1, k2, …, km)T be the weighting Pm vector of DMs, with k¼1 kk ¼ 1; kk C 0, k [ M. Suppose that these m DMs provide their preferences over a set of n decision alternatives X = {x1, x2, …, xn}, and give m incomplete reciprocal relations R(k) = (r(k) ij )n9n (k [ M). Similar to the procedure in Sect. 3 which is for the decision making with an incomplete reciprocal relation, in the following, we develop a procedure for group decision making with incomplete reciprocal relations. Algorithm II Let v = (v1, v2, …, vn)T be the collective weighting vector of the incomplete reciprocal relations R(k) = (r(k) ij )n9n(k [ M). (k) Step 1 Replace the unknown element r(k) with ij in R vj Þ þ 0:5; and construct the auxiliary relation
n1 2 ðvi ðkÞ
R
ðkÞ rij
ðkÞ
¼ ðrij Þnn ; where ( ðkÞ rij ; ¼ n1 1 2 ðvi vj Þ þ 2 ;
ðkÞ
rij 6¼ x ðkÞ
rij ¼ x
ð18Þ
Step 2 Utilize the additive weighted averaging (AWA) operator m X ð1Þ ð2Þ ðmÞ ðkÞ r ij ¼ AWAk ðr ij ; r ij ; . . .; r ij Þ ¼ kk r ij ; i; j 2 N: k¼1
ð19Þ
¼
into a collective auxiliary reciprocal relation
R ¼ ðr ij Þnn : Step 3 Utilize the normalizing rank aggregation method to obtain the weighting vector v. First, calculate the collective preference degree pi(v) of the alternative xi over all the other alternatives: n X r ij ; i 2 N ð20Þ pi ðvÞ ¼ j¼1;j6¼i
And establish the following linear system of equations: pi ðvÞ pi ðvÞ ; i 2 N: vi ¼ P ¼ n nðn 1Þ=2 pj ðvÞ
ð21Þ
j¼1
which can be further rewritten as the following form: Av ¼ b
4 A procedure for group decision making with incomplete reciprocal relations
ðkÞ
ð22Þ
where b is a positive vector, A is a real symmetrical matrix. Step 4 Solving Eq. 22, we can get the priority vector v = (v1, v2, …, vn)T. Step 5 Rank all the alternatives and select the best one(s) in accordance with the values of vi (i [ N). Step 6 End. The above algorithm is similar to Xu and Chen’s method, the difference of the algorithm between Xu and Chen and this paper is that in Step 1. In this note, we replace the unknown (k) with n1 element r(k) ij in R 2 ðvi vj Þ þ 0:5; while Xu and Chen utilized 0.5(vi - vj ? 1) to replace the unknown (k) element r(k) ij in R , and then both to construct the auxiliary reciprocal relations. Based on the idea in Sect. 3, we introduce the following definition and criteria: Definition 7 Let v = (v1, v2, …, vn)T be the weighting vector which computed by the above Algorithm II, then we call R ¼ ðrij Þnn ¼ ðn1 2 ðvi vj Þ þ 0:5Þnn is the collective characteristic matrix of R: Definition 8 Let D = (dij)n9n be the difference matrix between R and R*, where dij ¼ r ij rij : Obviously, dij = -dji. From Theorem 4, we know that, R is an additive consistency reciprocal relation if and only if D = 0. Thus, the value of D is the smaller the better. To further compare the performances of the Algorithm II of this paper with Xu’s algorithm for group decision making with incomplete reciprocal relations, we introduce the following performance evaluation criteria:
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•
•
decision makers provide three incomplete reciprocal relations as follows: 2 3 0:5 0:6 0:7 0:3 0:6 6 0:4 0:5 x x x 7 6 7 6 7 7; 0:3 x 0:5 x x Rð1Þ ¼ 6 6 7 6 7 4 0:7 x x 0:5 x 5
Total deviation (TD) between the collective auxiliary reciprocal relation and the collective characteristic matrix. n X n n X n X X dij ¼ ð23Þ TD ¼ r ij rij : i¼1 j¼1
i¼1 j¼1
Total deviation (TD) between each auxiliary reciprocal relation and the collective characteristic matrix. m X n X n m n n X ðkÞ X X X ðkÞ TD1 ¼ dij ¼ rij rij k¼1 i¼1 j¼1
k¼1 i¼1 j¼1
Rð2Þ
where dij ¼ r ij rij is the fitting error for the element r ij of the collective auxiliary fitting reciprocal relation R ¼ ðr ij Þnn ; and
¼
ðkÞ rij
rij
the fitting error for
the fuzzy preference relation R ðkÞ
ðkÞ
ðkÞ rij
ðkÞ
¼ ðrij Þnn : It is easy Rð3Þ
ðkÞ
R
R
R
ð1Þ
ð2Þ
ð3Þ
6 6 0:4 6 ¼6 6 0:3 6 4 0:7 0:4 2
x
0:5
0:5
x
x
x
x
6 x 6 6 ¼6 6 x 6 4 x
0:5
0:6
0:2
0:4
0:5
x
0:8
x
0:5
x
0:5
x
x
0:5
0:5
x
x
x
0:6
0:5
x
x
x
0:5
x
x
x
0:5
0:5 7 7 7 0:4 7 7: 7 0:8 5
0:5
0:6
0:2
0:5
6 x 6 6 ¼6 6 x 6 4 x 0:4
3
0:5 7 7 7 x 7 7; 7 x 5 3
Let v = (v1, v2, …, v5)T be the collective weight vector of the incomplete reciprocal relations R(k) = (r(k) ij )595 n1 (k = 1, 2, 3) (where 2 ¼ 2). (k) with Step 1 Replace each unknown element r(k) ij in R 2(vi - vj) ? 0.5, and construct the auxiliary reciprocal relations:
Example 3 Consider a group decision making problem, there are three decision makers ek (k = 1, 2, 3), and let k = (0.5, 0.3, 0.2)T be the weighting vector of DMs. The
0:6
x
of
5 Numerical examples
0:5
x
2
to find that dij j jdji j; jdij j jdji j: Obviously, the more smaller TD and TD1, the better fitting performance of the weighting vector v.
2
0:4 2
ð24Þ
ðkÞ dij
Y. Xu et al.
0:7
0:3
0:6
3
7 2ðv2 v5 Þ þ 0:5 7 7 2ðv3 v2 Þ þ 0:5 0:5 2ðv3 v4 Þ þ 0:5 2ðv3 v5 Þ þ 0:5 7 7 7 0:5 2ðv4 v5 Þ þ 0:5 5 2ðv4 v2 Þ þ 0:5 2ðv4 v3 Þ þ 0:5 2ðv5 v2 Þ þ 0:5 2ðv5 v3 Þ þ 0:5 2ðv5 v4 Þ þ 0:5 0:5 3 0:5 2ðv1 v2 Þ þ 0:5 2ðv1 v3 Þ þ 0:5 2ðv1 v4 Þ þ 0:5 2ðv1 v5 Þ þ 0:5 6 7 0:5 0:6 0:2 0:5 6 2ðv2 v1 Þ þ 0:5 7 6 7 7 ¼6 v Þ þ 0:5 0:4 0:5 2ðv v Þ þ 0:5 2ðv v Þ þ 0:5 2ðv 3 1 3 4 3 5 6 7 6 7 0:8 2ðv4 v3 Þ þ 0:5 0:5 2ðv4 v5 Þ þ 0:5 5 4 2ðv4 v1 Þ þ 0:5 0:5 2ðv5 v3 Þ þ 0:5 2ðv5 v4 Þ þ 0:5 0:5 2ðv5 v1 Þ þ 0:5 2 3 0:5 2ðv1 v2 Þ þ 0:5 2ðv1 v3 Þ þ 0:5 2ðv1 v4 Þ þ 0:5 0:6 6 7 v Þ þ 0:5 0:5 2ðv2 v3 Þ þ 0:5 2ðv2 v4 Þ þ 0:5 0:5 7 2ðv 6 2 1 6 7 ¼6 0:5 2ðv3 v4 Þ þ 0:5 0:4 7 6 2ðv3 v1 Þ þ 0:5 2ðv3 v2 Þ þ 0:5 7 6 7 0:5 0:8 5 4 2ðv4 v1 Þ þ 0:5 2ðv4 v2 Þ þ 0:5 2ðv4 v3 Þ þ 0:5 0:4 0:5 0:6 0:2 0:5
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0:5
2ðv2 v3 Þ þ 0:5 2ðv2 v4 Þ þ 0:5
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Group decision-making procedure
1297
Step 2 Utilize Eq. 19 to aggregate all the auxiliary ðkÞ
ðkÞ
reciprocal relations R ¼ ðrij Þ55 (k = 1, 2, 3) into the collective auxiliary reciprocal relation:
2
0:5 6 v2 v1 þ 0:45 6 R¼6 6 v3 v1 þ 0:4 4 v4 v1 þ 0:6 0:6ðv5 v1 Þ þ 0:43
v1 v2 þ 0:55 0:5 1:4ðv3 v2 Þ þ 0:47 1:4ðv4 v2 Þ þ 0:59 v5 v2 þ 0:5
v1 v3 þ 0:6 1:4ðv2 v3 Þ þ 0:53 0:5 2ðv4 v3 Þ þ 0:5 1:6ðv5 v3 Þ þ 0:52
v1 v4 þ 0:4 1:4ðv2 v4 Þ þ 0:41 2ðv3 v4 Þ þ 0:5 0:5 1:6ðv5 v4 Þ þ 0:44
3 0:6ðv1 v5 Þ þ 0:57 v2 v5 þ 0:5 7 7 1:6ðv3 v5 Þ þ 0:48 7 7 1:6ðv4 v5 Þ þ 0:56 5 0:5
Step 3 Utilize the normalizing rank aggregation method to obtain the weighting vector v. We can establish the following linear system of equations:
8 > > > v1 > > > > > > > > > > v2 > > < v3 > > > > > > > v4 > > > > > > > > : v5
2
v1 v2 þ 0:55 þ v1 v3 þ 0:6 þ v1 v4 þ 0:4 þ 0:6ðv1 v5 Þ þ 0:57 10 v2 v1 þ 0:45 þ 1:4ðv2 v3 Þ þ 0:53 þ 1:4ðv2 v4 Þ þ 0:41 þ v2 v5 þ 0:5 ¼ 10 v3 v1 þ 0:4 þ 1:4ðv3 v2 Þ þ 0:47 þ 2ðv3 v4 Þ þ 0:5 þ 1:6ðv3 v5 Þ þ 0:48 ¼ 10 v4 v1 þ 0:6 þ 1:4ðv4 v2 Þ þ 0:59 þ 2ðv4 v3 Þ þ 0:5 þ 1:6ðv4 v5 Þ þ 0:56 ¼ 10 0:6ðv5 v1 Þ þ 0:43 þ v5 v2 þ 0:5 þ 1:6ðv5 v3 Þ þ 0:52 þ 1:6ðv5 v4 Þ þ 0:44 ¼ 10 ¼
i.e.,
6:4 1 6 1 5:2 6 6 1 1:4 6 4 1 1:4 0:6 1
1 1 1:4 1:4 4 2 2 4 1:6 1:6
32
3
2
3
2:12 0:6 v1 7 6 7 6 1 7 76 v2 7 6 1:89 7 6 v3 7 ¼ 6 1:85 7 1:6 7 7 76 7 6 1:6 54 v4 5 4 2:25 5 1:89 v5 5:2
Step 4 Solve the above system of equations, thus we have v ¼ ð0:22; 0:17; 0:12; 0:32; 0:17ÞT Step 5 Rank all the alternatives xi(i = 1, 2, 3, 4, 5) in accordance with the values of vi (i = 1, 2, 3, 4, 5). x4 x1 x2 x5 x3 thus the best alternative is x4. ðkÞ
Furthermore, we can get R (k = 1, 2, 3), the collective auxiliary reciprocal relation R; and the characteristic matrix R* are all the same, i.e.,
R
ð1Þ
ð2Þ
ð3Þ
¼ R ¼ R ¼ R 0:5 0:6 0:7 0:3 6 0:4 0:5 0:6 0:2 6 ¼6 6 0:3 0:4 0:5 0:1 4 0:7 0:8 0:9 0:5 0:4 0:5 0:6 0:2 ¼R 2
3 0:6 0:5 7 7 0:4 7 7 0:8 5 0:5
And TD and TD1 computed by Eqs. 23, 24, respectively, we have TD ¼ 0; TD1 ¼ 0: ðkÞ
The above results shows that all the matrices (R (k = 1, 2, 3), R and R*) are the same, and also we can verify that they are all the additive consistency reciprocal relations according to Definition 2. If we take Xu and Chen’s method (i.e, take b = 0.5) to get the weighting vector, we have the following steps: (k) with Step 1 Replace each unknown element r(k) ij in R 0.5(vi - vj ? 1), and construct the auxiliary reciprocal relations:
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2
R
R
R
ð1Þ
ð2Þ
ð3Þ
Y. Xu et al.
3 0:6 7 0:5ðv2 v5 þ 1Þ 7 7 0:5ðv3 v2 þ 1Þ 0:5 0:5ðv3 v4 þ 1Þ 0:5ðv3 v5 þ 1Þ 7 7 7 0:5 0:5ðv4 v5 þ 1Þ 5 0:5ðv4 v2 þ 1Þ 0:5ðv4 v3 þ 1Þ 0:4 0:5ðv5 v2 þ 1Þ 0:5ðv5 v3 þ 1Þ 0:5ðv5 v4 þ 1Þ 0:5 2 3 0:5 0:5ðv1 v2 þ 1Þ 0:5ðv1 v3 þ 1Þ 0:5ðv1 v4 þ 1Þ 0:5ðv1 v5 þ 1Þ 6 7 0:5 0:6 0:2 0:5 6 0:5ðv2 v1 þ 1Þ 7 6 7 6 ¼ 6 0:5ðv3 v1 þ 1Þ 0:4 0:5 0:5ðv3 v4 þ 1Þ 0:5ðv3 v5 þ 1Þ 7 7 6 7 0:8 0:5ðv4 v3 þ 1Þ 0:5 0:5ðv4 v5 þ 1Þ 5 4 0:5ðv4 v1 þ 1Þ 0:5 0:5ðv5 v3 þ 1Þ 0:5ðv5 v4 þ 1Þ 0:5 0:5ðv5 v1 þ 1Þ 2 3 0:5 0:5ðv1 v2 þ 1Þ 0:5ðv1 v3 þ 1Þ 0:5ðv1 v4 þ 1Þ 0:6 6 7 0:5 0:5ðv2 v3 þ 1Þ 0:5ðv2 v4 þ 1Þ 0:5 7 6 0:5ðv2 v1 þ 1Þ 6 7 ¼6 0:5 0:5ðv3 v4 þ 1Þ 0:4 7 6 0:5ðv3 v1 þ 1Þ 0:5ðv3 v2 þ 1Þ 7 6 7 0:5 0:8 5 4 0:5ðv4 v1 þ 1Þ 0:5ðv4 v2 þ 1Þ 0:5ðv4 v3 þ 1Þ 0:4 0:5 0:6 0:2 0:5 0:5 6 6 0:4 6 ¼6 6 0:3 6 4 0:7
0:6 0:5
0:7 0:3 0:5ðv2 v3 þ 1Þ 0:5ðv2 v4 þ 1Þ
Step 2 Utilize Eq. 19 to aggregate all the auxiliary ðkÞ
ðkÞ
reciprocal relations R ¼ ðrij Þ55 (k = 1, 2, 3) into the collective auxiliary reciprocal relation: 2
0:5 6 0:25ðv2 v1 Þ þ 0:45 6 R¼6 6 0:25ðv3 v1 Þ þ 0:4 4 0:25ðv4 v1 Þ þ 0:6 0:15ðv5 v1 Þ þ 0:43
0:25ðv1 v2 Þ þ 0:55 0:5 0:35ðv3 v2 Þ þ 0:47 0:35ðv4 v2 Þ þ 0:59 0:25ðv5 v2 Þ þ 0:5
0:25ðv1 v3 Þ þ 0:6 0:35ðv2 v3 Þ þ 0:53 0:5 0:5ðv4 v3 Þ þ 0:5 0:4ðv5 v3 Þ þ 0:52
0:25ðv1 v4 Þ þ 0:4 0:35ðv2 v4 Þ þ 0:41 0:5ðv3 v4 Þ þ 0:5 0:5 0:4ðv5 v4 Þ þ 0:44
Step 3 Utilize the normalizing rank aggregation method to obtain the weighting vector v. We can establish the following linear system of equations:
8 > > > v1 > > > > > > > > v2 > > > > < v3 > > > > > > > v4 > > > > > > > > : v5
0:25ðv1 v2 Þ þ 0:55 þ 0:25ðv1 v3 Þ þ 0:6 þ 0:25ðv1 v4 Þ þ 0:4 þ 0:15ðv1 v5 Þ þ 0:57 10 0:25ðv2 v1 Þ þ 0:45 þ 0:35ðv2 v3 Þ þ 0:53 þ 0:35ðv2 v4 Þ þ 0:41 þ 0:25ðv2 v5 Þ þ 0:5 ¼ 10 0:25ðv3 v1 Þ þ 0:4 þ 0:35ðv3 v2 Þ þ 0:47 þ 0:5ðv3 v4 Þ þ 0:5 þ 0:4ðv3 v5 Þ þ 0:48 ¼ 10 0:25ðv4 v1 Þ þ 0:6 þ 0:35ðv4 v2 Þ þ 0:59 þ 0:5ðv4 v3 Þ þ 0:5 þ 0:4ðv4 v5 Þ þ 0:56 ¼ 10 0:15ðv5 v1 Þ þ 0:43 þ 0:25ðv5 v2 Þ þ 0:5 þ 0:4ðv5 v3 Þ þ 0:52 þ 0:4ðv5 v4 Þ þ 0:44 ¼ 10 ¼
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3 0:15ðv1 v5 Þ þ 0:57 0:25ðv2 v5 Þ þ 0:5 7 7 0:4ðv3 v5 Þ þ 0:48 7 7 0:4ðv4 v5 Þ þ 0:56 5 0:5
2011,15(7)/Soft Computing
Group decision-making procedure
i.e., 2 9:1 6 0:25 6 6 0:25 6 4 0:25 0:15
0:25 0:25 8:8 0:35 0:35 8:5 0:35 0:5 0:25 0:4
32
3
2
3
2:12 0:25 0:15 v1 6 v2 7 6 1:89 7 0:35 0:25 7 7 76 7 6 7 6 7 6 0:5 0:4 7 76 v3 7 ¼ 6 1:85 7 8:5 0:4 54 v4 5 4 2:25 5 1:89 v5 0:4 8:8
Table 2 Performance evaluation for Example 4 Criteria
b=2
b = 0.5
TD
1.4515
2.0163
TD1
5.1572
6.1471
2
Step 4 Solve the above system of equations, thus we have Rð1Þ
v ¼ ð0:2134; 0:187; 0:1813; 0:2312; 0:1871ÞT Step 5 Rank all the alternatives xi(i = 1, 2, 3, 4, 5) in accordance with the values of vi (i = 1, 2, 3, 4, 5). x4 x 1 x 5 x 2 x 3 thus the best alternative is x4.
ð2Þ
R
Furthermore, by the weighting vector v, we can get ðkÞ
R (k = 1, 2, 3), the collective auxiliary reciprocal relation R; and the characteristic matrix R*[in this case, r*ij = 0.5(vi - vj) ? 0.5)] easily, and we can verify that they are not the same. At the same time, we can further get TD and TD1 by Eqs. 23, 24, respectively. Table 1 shows the performances of the two methods. From the above results, we see that the ranking order is somewhat different between Xu and Chen’s method and ours. In our method, the weights of x2 and x5 are the same, which means that x2 and x5 are same important. But in Xu and Chen’s method, x5 is preferred to x2. From the original incomplete preference relations R(2) and R(3), we see that the two decision makers think that x2 and x5 are same important. Table 1 shows the performances of the two methods. As is known, smaller deviation means better performances. Obviously, our method has the better performance in the two criteria. This shows the advantages of our method. Evidently, Xu and Chen’s method suffer from rank reversal phenomenon, and their method will distort the decision maker’s information in some time. Example 4 Let us suppose that a company wants to renew its cars. There exist five models of car available, X = {x1, x2, …, x5}, and three DMs ek (k = 1, 2, 3), and let k = (0.5, 0.3, 0.2)T be the weighting vector of DMs. The decision makers provide three incomplete reciprocal relations as follows (the example has been examined by Xu and Chen 2008):
Table 1 Performance evaluation for Example 3 Criteria
b=2
b = 0.5
TD
0
0.9583
TD1
0
2.7721
1299
Rð3Þ
0:5 6 6 0:4 6 ¼6 6 0:3 6 4 0:7 0:4 2 0:5 6 6 0:3 6 ¼6 6 0:6 6 4 x
0:6 0:5
0:7 x
0:3 0:8
x
0:5
0:3
0:2 0:7 0:4 0:5 0:7 0:4
0:5 0:2 x
0:5 0:7
0:3 0:5
0:6 0:4
0:4
0:6
0:5
0:4 0:5 x 0:4 2 0:5 x 0:8 0:4 6 x 6 x 0:5 x 6 ¼6 0:2 x 0:5 0:3 6 6 4 0:6 x 0:7 0:5 0:5 x 0:4 0:3
3 0:6 7 0:6 7 7 0:5 7 7; 7 0:8 5 0:5 3 0:6 7 0:5 7 7 x 7 7; 7 0:6 5 0:5 3 0:5 7 x 7 7 0:6 7 7: 7 0:7 5 0:5
Xu and Chen get the priority vector is v = (0.2193, 0.2095, 0.1766, 0.2327, 0.1618)T. But after our carefully computation, the result of Xu and Chen’s linear systems (see Xu and Chen, Eq. 17) is v = (0.2199, 0.2071, 0.1772, 0.2335, 0.1623)T. By using our method, we get v = (0.2194, 0.2124, 0.1733, 0.2348, 0.1601)T. Both the priority vectors lead to the ranking of x4 x1 x2 x3 x5. The fitting performances of the two methods are assessed in terms of the criteria defined by Eqs. 23 and 24. The results are show in Table 2. From which, it is clear that our method still has smaller deviation, that is our method still performs better than Xu and Chen’s method.
6 Concluding remarks In this paper, we study a procedure for group decision making with incomplete reciprocal relations developed by Xu and Chen. We first examine the function between the fuzzy preference relation and its corresponding priority vector developed by Xu and Chen with a numerical example and show that the function does not hold in general cases. Then, we deduce an exact function between the additive transitivity fuzzy preference relation and its corresponding priority vector. Theoretical analysis proved that the function deduced by us is more reasonable than Xu and Chen’s. Based on this, we develop a procedure for the
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decision making with an incomplete reciprocal relation and also develop a procedure for the group decision making with incomplete reciprocal relations. In order to compare the performances of our method with Xu and Chen’s method in fitting the reciprocal relation, we introduce some criteria. Our method turns out to be more reasonable than Xu and Chen’s. It can construct a perfect additive consistency reciprocal relation when the initial incomplete reciprocal relation is additive, while Xu and Chen’s method will distort its information. This advantage also can be found for group decision making with incomplete reciprocal relations. And also, in some cases, Xu and Chen’s method will lead to rank reversal phenomenon. And even for not consistent incomplete reciprocal relations, the performance of our method is still better than Xu and Chen’s method. Acknowledgments The author is very grateful to the two anonymous referees for their insightful and constructive comments and suggestions that have led to an improved version of this paper. This work was supported by Hohai University ‘‘the Fundamental Research Funds for the Central Universities (2009B04514)’’ and the National Natural Science Foundation of China(No. 50979024,No. 90924027).
References Alonso S, Chiclana F, Herrera F, Herrera-Viedma E (2004) A learning procedure to estimate missing values in fuzzy preference relations based on additive consistency. MDAI2004, LNAI vol. 3131, pp 227–238 Alonso S, Chiclana F, Herrera F, Herrera-Viedma E (2008) A consistency-based procedure to estimate missing pairwise preference values. Int J Intell Syst 23:155–175 Alonso S, Cabrerizo FJ, Chiclana F, Herrera F, Herrera-Viedma E (2009) Group decision making with incomplete fuzzy linguistic preference relations. Int J Intell Syst 24(2):201–222 Cabrerizo FJ, Heradio R, Pe´rez IJ, Herrera-Viedma E (2010a) A selection process based on additive consistency to deal with incomplete fuzzy linguistic information. J Univers Comput Sci 16(1):62–81 Cabrerizo FJ, Pe´rez IJ, Herrera-Viedma E (2010b) Managing the consensus in group decision making in an unbalanced fuzzy linguistic context with incomplete information. Knowl Based Syst 23(2):169–181
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Chiclana F, Herrera-Viedma E, Alonso S and Herrera F (2008) A note on the estimation of missing pairwise preference values: A U-consistency based method. Int J Uncertainty, Fuzzy Knowl Based Syst 16(2):19–32 Chiclana F, Herrera-Viedma E, Alonso S (2009) A note on two methods for estimating missing pairwise preference values. IEEE Trans Syst Man Cybern B Cybern 39(6):1628–1633 Fedrizzi M, Giove S (2007) Incomplete pairwise comparison and consistency optimization. Eur J Oper Res 183:303–313 Gong ZW (2008) Least-square method to priority of the fuzzy preference relations with incomplete information. Int J Approx Reason 47:258–264 Herrera-Viedma E, Alonso S, Chiclana F, Herrera F (2007a) A consensus model for group decision making with incomplete fuzzy preference relations. IEEE Trans Fuzzy Syst 15(5):863–877 Herrera-Viedma E, Chiclana F, Herrera F, Alonso S (2007b) Group decision-making model with incomplete fuzzy preference relations based on additive consistency. IEEE Trans Syst Man Cybern B Cybern 37(1):176–189 Lee HS, Chou MT, Fang HH, Tseng WK and Yeh CH (2007) Estimating missing values in incomplete additive fuzzy preference relations. KES 2007/WIRN 2007, LNAI, vol. 4693, pp 1307–1314 Orlovsky SA (1978) Decision-making with a fuzzy preference relation. Fuzzy Sets Syst 1:155–167 Shen PD, Chyr WL, Lee HS, and Lin K (2009) Correspondence between incomplete fuzzy preference relation and its priority vector. KES 2009, Part II, LNAI vol. 5712, pp 745–751 Tanino T (1984) Fuzzy preference orderings in group decision making. Fuzzy Sets Syst 12:117–131 Xu ZS (2004) Goal programming models for obtaining the priority vector of incomplete fuzzy preference relation. Int J Approx Reason 36:261–270 Xu ZS (2005) A procedure for decision making based on incomplete fuzzy preference relation. Fuzzy Optim Decis Mak 4:175–189 Xu ZS, Chen J (2008) Group decision-making procedure based on incomplete reciprocal relations. Soft Comput 12:515–521 Xu YJ, Da QL (2008) Weighted least-square method and its improvement for priority of incomplete complementary judgement matrix. Syst Eng Electron 31(7):1273–1276 Xu YJ, Da QL (2009) Methods for priority of incomplete complementary judgement matrices. Syst Eng Electron 31(1):95–99 Xu YJ, Da QL, Liu LH (2009) Normalizing rank aggregation method for priority of a fuzzy preference relation and its effectiveness. Int J Approx Reason 50(8):1287–1297 Zhang JJ (2000) Fuzzy analytical hierarchy process. Fuzzy Syst Math 14(2):80–88
2011,15(7)/Soft Computing
ORIGINAL PAPER
A note on group decision-making procedure based on incomplete reciprocal relations Yejun Xu • Qingli Da • Huimin Wang
Published online: 23 October 2010 Springer-Verlag 2010
Abstract In a very recent paper by Xu and Chen (Soft Comput 12:515–521, 2008), a novel procedure for group decision making with incomplete reciprocal relations was developed. In this note, we examine the function between the fuzzy preference relation and its corresponding priority vector developed by Xu and Chen with a numerical example and show that the function does not hold in general cases. Then, we deduce an exact function between the additive transitivity fuzzy preference relation and its corresponding priority vector. Based on this, we develop a procedure for the decision making with an incomplete reciprocal relation and also develop a procedure for the group decision making with incomplete reciprocal relations. In order to compare the performances of our method with Xu and Chen’s method in fitting the reciprocal relation, we introduce some criteria. Theoretical analysis and numerical examples have shown that the function deduced by us is more reasonable and effective than Xu and Chen’s.
Y. Xu (&) H. Wang Business School, HoHai University, Nanjing 210098, Jiangsu, China e-mail: [email protected] Q. Da School of Economics and Management, Southeast University, Nanjing 210096, Jiangsu, China e-mail: [email protected] H. Wang State Key Laboratory of Hydrology Water Resource and Hydraulic Engineering, HoHai University, Jiangsu 210098, China e-mail: [email protected]
Keywords Group decision making Incomplete reciprocal relation Additive transitivity consistency Auxiliary reciprocal relation
1 Introduction Reciprocal relations [or called fuzzy preference relations (Orlovsky 1978)] have received a great deal of attention from researchers. A complete fuzzy preference relation of order n necessitates the completion of all n(n - 1)/2 judgements in its entire top triangular portion. Sometimes, however, a decision maker (DM) may develop a fuzzy preference relation with incomplete information because of (1) time pressure, lack of knowledge, and the DM’s limited expertise related with problem domain(Chiclana et al. 2008; Lee et al. 2007; Xu 2004, 2005; Xu and Chen 2008); (2) when the number of the alternatives, n, is large it may be practically impossible, or at least unacceptable from the point of view of the decision maker, to perform all the n(n - 1)/2 required comparisons to complete the pairwise comparison matrices (Fedrizzi and Giove 2007); (3) It can be convenient/necessary to skip some direct critical comparison between alternatives, even if the total number of alternatives is small (Fedrizzi and Giove 2007). (4) An expert would not be able to efficiently express any kind of preference degree between two or more of the available options. This may be due to an expert not possessing a precise or sufficient level of knowledge of part of the problem, or because that expert is unable to discriminate the degree to which some options are better than others (Alonso et al. 2004, 2008, 2009; Cabrerizo et al. 2010a, b; Herrera-Viedma et al. 2007a, b). Up to now, some related theory studies with incomplete reciprocal preference relations have been given.
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Alonso et al. (2004) put forward a procedure which attempts to find out the missing information in an expert’s incomplete fuzzy preference relation based on additive consistency, using only the preference values provided by that particular expert. Herrera-Viedma et al. (2007b) presented a new model to deal with GDM problems with the incomplete fuzzy preference relations based on the additive consistency (AC) property. The new model is composed of two steps: the estimation of missing preference values and the selection of alternatives. Fedrizzi and Giove (2007) proposed a new method to calculate the missing elements of an incomplete fuzzy preference relation by maximizing the global consistency of the ‘completed’ matrix. Afterwards, Chiclana et al. (2009) found that the above two methods (the first one by Herrera-Viedma et al. 2007a, b, and the second one by Fedrizzi and Giove 2007) for calculating missing values, although different, are very similar. Both methods derive the same estimated values for the independent-missing-comparison case, while they differ in the dependent-missing-comparison case. Herrera-Viedma et al. (2007a) further presented a consensus model for GDM problems with incomplete fuzzy preference relations. The main novelty of the consensus model is that of being guided by both consensus and consistency measures. Furthermore, the consensus-reaching process is guided automatically, without moderator, through both consensus and consistency criteria. Alonso et al. (2008) put forward a general procedure that attempts to estimate the missing information in any of the above formats of incomplete preference relations: fuzzy, multiplicative, interval valued and linguistic. Xu (2004) defined the concepts of incomplete fuzzy preference relation, additive consistent incomplete fuzzy preference relation and multiplicative consistent incomplete fuzzy preference relation, and then proposed two goal programming models, based on additive consistent incomplete fuzzy preference relation and multiplicative consistent incomplete fuzzy preference relation respectively, for obtaining the priority vector of incomplete fuzzy preference relation. Xu (2005) proposed a procedure for decision making with incomplete fuzzy preference relation based on multiplicative consistency. Gong (2008) developed a least-square model to obtain the collective priority vector of the incomplete preference relations by multiple decision makers. And the priority vector only derived by a simple equation. We (Xu and Da 2008) proposed weighted least-square method and quadratic programming method for priorities of incomplete fuzzy preference relation based on multiplicative transitivity. We (Xu and Da 2009) presented two priority methods: the least variance priority method (LVM) and the quadratic programming method for incomplete fuzzy preference relation based on additive consistency.
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Very recently, Xu and Chen (2008) also developed a simple but practical approach to deriving the ranking of the alternatives from an incomplete reciprocal relation based on additive transitivity. They postulated a correspondence between priority vector and additive consistent incomplete fuzzy preference relation. Shen et al. (2009), we (Xu et al. 2009), respectively, have proved that the correspondence does not always hold. In this paper, we will still investigate the problem which Xu and Chen had investigated. The purpose of this paper is to present that the function between the fuzzy preference and priority vector proposed by us is more reasonable than Xu and Chen’s. In order to do this, the rest of this paper is structured as follows. In Sect. 2, we recall some concepts of incomplete reciprocal relation. In Sect. 3, we first illustrate with a numerical example to point out the correspondence between priority vector and additive consistent incomplete fuzzy preference relation given by Xu and Chen does not always hold in general cases, and then we deduce a simple functional equation between fuzzy preference’ element and priority vector. We prove that our function is more effective to fitting the relationship between the fuzzy preference’s element and the priority vector. In Sect. 4, we present a procedure for group decision making with incomplete reciprocal relations, and propose some new concepts and criteria to measure the performances of our method. In Sect. 5, we examine two numerical examples to present the effectiveness of our methods, and compare the performances between our method and Xu and Chen’s method. Conclusions are offered in Sect. 6. 2 Incomplete reciprocal relation For simplicity, we let N = {1, 2,…, n}. Definition 1 Let R = (rij)n9n be a preference relation, then R is called a fuzzy preference relation (Orlovsky 1978; Tanino 1984) if rij 2 ½0; 1; rij þ rji ¼ 1; rii ¼ 0:5;
for all i; j 2 N:
ð1Þ
Definition 2 Let R = (rij)n9n be a fuzzy preference relation, then R is called an additive transitive fuzzy preference relation, if the following additive transitivity (Tanino 1984) is satisfied: rij ¼ rik rjk þ 0:5;
for all i; j; k 2 N:
ð2Þ
We also called it to be perfectly additive transitive consistent. From Definition 2, we can get the following result easily: Theorem 1
Let R = (rij)n9n be a complete preference
relation, then the sum of all the elements of R is nðn1Þ 2 ; that is
2011,15(7)/Soft Computing
Group decision-making procedure n n X X i¼1 j¼1;j6¼i
rij ¼
n X n X i¼1 j¼1
rij
n X i¼1
rii ¼
nðn 1Þ : 2
ð3Þ
The above Theorem 1 denotes that the sum of the elements of the complete reciprocal relation does not involve the diagonal elements. If the diagonal elements are taken into account, the sum will be n2/2. Definition 3 Xu (2004) Let R = (rij)n9n be a fuzzy preference relation, then R is called an incomplete fuzzy preference relation, if some of its elements cannot be given by the DM, which we denote the unknown number x, and the others can be provided by the DM, which satisfy Eq. (1). Definition 4 Xu (2004) Let R = (rij)n9n be an incomplete fuzzy preference relation, then R is called an additive consistent incomplete fuzzy preference relation, if all the known elements of R satisfy the additive transitivity rij = rik - rjk ? 0.5.
3 An approach to constructing a complete reciprocal relation from an incomplete reciprocal relation
1291
additive consistent incomplete fuzzy preference relation [i.e., Eq. (4)] does not always hold. In the following, we will deduce the more reasonable relationship between rij and wi - wj. Zhang (2000), we (Xu et al. 2009) have proved the following results: Lemma 1 Let R = (rij)n9n be a fuzzy additive transitive preference relation, w = (w1, w2, …, wn)T be the corresponding weighting vector, where 0 B wi B 1, i = 1, P 2, …, n, ni¼1 wi ¼ 1; then there exists a positive number b, and such a relation can be expressed as follows: rij ¼ 0:5 þ bðwi wj Þ where b [ 0. Theorem 2 If the priority vector of the additive transitive perfectly consistent fuzzy preference relation R is derived by normalizing rank aggregation method, then b ¼ n1 2 : Proof If the priority vector of the additive transitive perfectly consistent fuzzy preference relation R is derived by normalizing rank aggregation method, then Pn Pn rik 0:5 rik 0:5 k¼1 wi ¼ Pn Pn ¼ k¼1nðn1Þ ; i2N ð6Þ i¼1 k¼1;k6¼i rik 2 Pn Pn rjk 0:5 rjk 0:5 k¼1 P P ¼ k¼1nðn1Þ ; i 2 N: ð7Þ wj ¼ n n r i¼1 k¼1;k6¼j jk
As we stated above, suppose that we have a set of alternatives, X = {x1, x2, …, xn}, the expert gives his/her fuzzy preference relation, and construct the judgement matrix R = (rij)n9n,and w1, w2, …, wn be the corresponding ranking vector of each alternative x1, x2, …, xn, where Pn i¼1 wi ¼ 1; wi 0: In our previous work (Xu et al. 2009), we use function f to denote the relationship between the element rij and wi - wj, which is rij = f(wi - wj). Xu (2004; Xu and Chen 2008) inferred the relationship between rij and wi - wj as follows:
since
rij ¼ 0:5ðwi wj þ 1Þ:
rij ¼ rik rjk þ 0:5
ð4Þ
ð5Þ
2
Put (6), (7) into (5), then rij ¼ bðwi wj Þ þ 0:5 Pn ðrik rjk Þ þ 0:5 ¼ b k¼1nðn1Þ 2
Now, we will show the equation does not hold in general cases. For example, 2 3 0:5 1 0:5 R ¼ 4 0 0:5 0 5: 0:5 1 0:5
then
Obviously, the reciprocal relation R is a consistency reciprocal relation according the Definition 2. On the other hand, since r12 = 1, then according to Eq. (4), we have w1 = 1, w2 = 0, and thus w3 = 0. But r32 = 1, we should have w3 = 1, w2 = 0 and w1 = 0. Therefore, the results are conflict between themselves. Shen et al. (2009) proved that the correspondence between priority vector and
Theorem 3 If b ¼ n1 2 ; then the priority vector of the additive transitive perfectly consistent fuzzy preference relation R derived by Eq. (5) is normalizing rank aggregation method, that is Pn Pn j¼1;j6¼i rij j¼1 rij 0:5 wi ¼ : ð9Þ ¼ Pn Pn nðn 1Þ=2 i¼1 j¼1;j6¼i rij
rij ¼ b
Pn
k¼1 ðrij nðn1Þ 2
0:5Þ
nrij n þ 0:5 ¼ b nðn1Þ 2 þ 0:5:
ð8Þ
2
So we can get b ¼ n1 2 ; which completes the proof.
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Proof wi ¼
If b ¼ n1 2 ; by (5), we have
rij 0:5 n1 2
þ wj :
ð10Þ
Summing on both sides of Eq. (10) with respect to j, then Pn n X Xn j¼1 ðrij 0:5Þ w ¼ þ wj ð11Þ n1 j¼1 i 2
i.e.,
Pn
nwi ¼
j¼1 rij n1 2
0:5n
j¼1
þ 1:
Therefore, Pn Pn 1 j¼1 rij 0:5n j¼1 rij 0:5 þ ¼ wi ¼ n nðn nðn 1Þ=2 Pn 1Þ=2 j¼1;j6¼i rij ¼ Pn Pn i¼1 j¼1;j6¼i rij
We call the method is normalizing rank aggregation method. For an incomplete reciprocal relation R = (rij)n9n, we can replace the unknown element ‘‘x’’ with n1 2 ðwi wj Þ þ 1 ; and then construct an auxiliary reciprocal relation 2 R ¼ ðr ij Þnn (we also call R the fitting relation of R), where rij 6¼ x rij ; ð12Þ r ij ¼ n1 1 ðw w Þ þ ; rij ¼ x i j 2 2 Example 1 For a decision-making problem, there are four decision alternatives xi(i = 1, 2, 3, 4). The DM provides his/her preference over these four decision alternatives, and gives an incomplete reciprocal relation as follows (the example has been examined by Xu and Chen 2008): 2 3 0:5 x 0:4 0:8 6 x 0:5 0:3 0:7 7 7 R¼6 4 0:6 0:7 0:5 x 5: 0:2 0:3 x 0:5 Let w = (w1, w2, w3, w4)T be the weight vector of R, then we construct an auxiliary reciprocal relation R ¼ ðr ij Þ44 (where n = 4):
0:5 6 1:5ðw2 w1 Þ þ 0:5 R¼6 4 0:6 0:2
123
1:5ðw1 w2 Þ þ 0:5 0:5 0:7 0:3
Then, we use the normalizing rank aggregation method to get the weighting vector. It is in the following: P4 1:5ðw1 w2 Þ þ 0:5 þ 0:4 þ 0:8 j¼2 r 1j w1 ¼ P4 P4 ¼ 6 i¼1 j¼1;j6¼i r ij P4 1:5ðw2 w1 Þ þ 0:5 þ 0:3 þ 0:7 j¼1;j6¼2 r 2j ¼ w2 ¼ P4 P4 6 r i¼1 j¼1;j6¼i ij P4 0:6 þ 0:7 þ 1:5ðw3 w4 Þ þ 0:5 j¼1;j6¼3 r 3j ¼ w3 ¼ P4 P4 6 i¼1 j¼1;j6¼i r ij P4 0:2 þ 0:3 þ 1:5ðw4 w3 Þ þ 0:5 j¼1;j6¼4 r 3j ¼ w4 ¼ P4 P4 6 i¼1 j¼1;j6¼i r ij i. e., 2 4:5 1:5 6 1:5 4:5 6 4 0 0 0 0
which completes the proof.
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Y. Xu et al.
32 3 2 3 0 0 w1 1:7 6 7 6 7 0 0 7 76 w2 7 ¼ 6 1:5 7: 4:5 1:5 54 w3 5 4 1:8 5 w4 1:5 4:5 1
Solving the linear system of equations, we get: w1 = 0.3, w2 = 0.233, w3 = 0.3667, w4 = 0.1, and the priority vector is w = (0.3, 0.233, 0.3667, 0.1)T. And we also can get the unknown elements as follows: r 12 ¼ 1:5ðw1 w2 Þ þ 0:5 ¼ 0:6; r 34 ¼ 1:5ðw3 w4 Þ þ 0:5 ¼ 0:9: And thus, we can relation R : 2 0:5 0:6 0:4 6 0:4 0:5 0:3 R¼6 4 0:6 0:7 0:5 0:2 0:3 0:1
get a complete reciprocal fitting 3 0:8 0:7 7 7: 0:9 5 0:5
Obviously, we can verify that R is an additive transitivity reciprocal relation according to Definition 2. If we take b = 1/2, which computed by Xu (Xu and Chen 2008) and we have w ¼ ð0:2863; 0:2467; 0:3137; 0:1533ÞT ; 1 r 12 ¼ ðw1 w3 þ 1Þ ¼ 0:5198; 2 r 34 ¼ 0:5ðw3 w1 þ 1Þ ¼ 0:5802:
0:4 0:3 0:5 1:5ðw4 w3 Þ þ 0:5
3 0:8 7 0:7 7: 1:5ðw3 w4 Þ þ 0:5 5 0:5
2011,15(7)/Soft Computing
Group decision-making procedure
Then 2
0:5 6 0:4802 R¼6 4 0:6 0:2
0:5198 0:5 0:7 0:3
0:4 0:3 0:5 0:4198
Theorem 4 R ¼ ðr ij Þnn is an complete additive consistency reciprocal relation if and only if R ¼ R :
3
0:8 0:7 7 7: 0:5802 5 0:5
Proof Necessary condition. If R ¼ ðr ij Þnn is an additive consistency reciprocal relation, then, for all i, j, k, r ij ¼ r ik r jk þ 0:5; and based on Eq. 9, we have
Obviously, R is not an additive consistent reciprocal relation. Therefore, we can see that it is more reasonable to take b = (n - 1)/2 than b = 1/2. Based on the above procedure, we develop a procedure for the decision making with an incomplete reciprocal relation in the following:
n1 ðwi wj Þ þ 0:5 2 0P 1 n n P r ik 0:5 r jk 0:5 C n 1B Bk¼1 C þ 0:5 k¼1 ¼ @ 2 nðn 1Þ=2 nðn 1Þ=2 A
rij ¼
Algorithm I Let w = (w1, w2, …, wn)T be the weighting vector of the incomplete reciprocal relation R = (rij)n9n. Step 1. Replace the unknown element rij in R with n1 2 ðwi wj Þ þ 0:5; and construct the auxiliary relation R ¼ ðr ij Þnn ; r ij satisfy Eq. 12. Step 2 Utilize the normalizing rank aggregation method to obtain the weighting vector w. First, calculate the collective preference degree pi(w) of the alternative xi over all the other alternatives: n X r ij ; i 2 N ð13Þ pi ðwÞ ¼ j¼1;j6¼i
And establish the following linear system of equations: pi ðwÞ pi ðwÞ wi ¼ P ; i2N ¼ n nðn 1Þ=2 pj ðwÞ
ð14Þ
j¼1
which can be further rewritten as the following form: Aw ¼ b
ð15Þ
where b is a positive vector, A is a real symmetrical matrix, whose ith main diagonal element is nðn1Þ n1 2 ai ; here, ai 2 is the count of the missing elements in the ith line in the matrix R. Step 3 Solving Eq. 15, we can get the priority vector w = (w1, w2, …, wn)T. Step 4 Rank all the alternatives and select the best one(s) in accordance with the values of wi (i [ N). Step 5 End. To further compare the performances of b = (n - 1)/2 and b = 1/2 in fitting the reciprocal relation, we introduce the following definition and criteria: Definition 5 Let w = (w1, w2, …, wn)T be the weighting vector which computed by the above Algorithm I, then we call R ¼ ðrij Þnn ¼ ðn1 2 ðwi wj Þ þ 0:5Þnn is the characteristic matrix of R:
1293
0P n
¼
1
C n 1B Bk¼1 C þ 0:5 2 @ nðn 1Þ=2 A n P
¼
ðr ik r jk Þ
ðr ij 0:5Þ
k¼1
n
þ 0:5
¼ r ij : Thus, R ¼ R : Sufficient condition. If R ¼ R ; then for all i,j,k [ N, we have r ik r jk ¼ rik rjk n1 ¼ ðwi wk Þ þ 0:5 2 n1 ðwj wk Þ þ 0:5 2 n1 ðwi wj Þ ¼ 2 ¼ rij 0:5 ¼ r ij 0:5 i.e., r ik r jk ¼ r ij 0:5; thus, R is an additive consistency reciprocal relation, which completes the proof. Also, from the above proof process of Theorem 4, we can prove that if b = 1/2 [i.e., r*ij = 0.5(wi - wj ? 1)], the above Theorem 4 will not hold. This means that, if R ¼ ðr ij Þnn is an additive consistency reciprocal relation and the weighting vector is computed by Algorithm I in which the unknown element r ij in R is instead by 0.5(wi - wj ? 1), and then the characteristic matrix R* is computed by r*ij = 0.5(wi - wj ? 1), it will be R 6¼ R : Thus, it is more reasonable to take b = (n - 1)/2 than b = 0.5. Definition 6 Let D = (dij)n9n be the difference matrix between R and R*, where dij ¼ r ij rij :
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Obviously, dij = -dji. From Theorem 4, we know that, R is an additive consistency reciprocal relation if and only if D = 0. Thus, the value of D is the smaller the better. That is n X n n X n X X dij ¼ ð16Þ TD ¼ r ij rij i¼1 j¼1
i¼1 j¼1
where dij ¼ r ij rij is the fitting error for the element r ij of the collective auxiliary reciprocal relation R ¼ ðr ij Þnn ; It is easy to find that |dij| : |dji|. Obviously, the smaller TD, the better fitting performance of the weighting vector w. Again, for the Example 1, as we compute in the above, if we take b = (n - 1)/2, we get T
w ¼ ð0:3; 0:233; 0:3667; 0:1Þ : And thus, the characteristic matrix R* computed by rij ¼ 3 2 ðwi wj Þ þ 0:5; we get 2 3 0:5 0:6 0:4 0:8 6 0:4 0:5 0:3 0:7 7 7 R ¼ 6 4 0:6 0:7 0:5 0:9 5: 0:2 0:3 0:1 0:5 Therefore, P P P P TD ¼ ni¼1 nj¼1 jdij j ¼ ni¼1 nj¼1 jr ij rij j ¼ 0:
2
0:5 6 1:5ðw2 w1 Þ þ 0:5 R ¼ 6 4 1:5ðw3 w1 Þ þ 0:5 1:5ðw4 w1 Þ þ 0:5
1:5ðw1 w2 Þ þ 0:5 0:5 1:5ðw3 w2 Þ þ 0:5 1:5ðw4 w2 Þ þ 0:5
0:2
0:3
0:4198
0:5
And thus, the characteristic matrix R* computed by rij ¼ 12 ðwi wj Þ þ 0:5; we get 2 3 0:5 0:5198 0:4863 0:5665 6 0:4802 0:5 0:4665 0:5467 7 6 7 R ¼ 6 7: 4 0:5137 0:5335 0:5 0:5802 5 0:4335
123
0:4533
0:4198
0:5
Therefore, 2 0 6 0 6 D¼6 4 0:0863
3
0
0:0863
0:2335
0 0:1655
0:1665 0
0:1533 7 7 7: 0 5
0:2335 0:1533
0
0
And P P P P TD ¼ ni¼1 nj¼1 jdij j ¼ ni¼1 nj¼1 jr ij rij j ¼ 1:2792: As is known, smaller deviations means better performance. Obviously, from the above results, we know that it is more reasonable to take b = (n - 1)/2 than b = 0.5. Example 2 If the DM provides no preferences over these four decision alternatives, then the incomplete reciprocal relation R is reduced to the following: 2 3 0:5 x x x 6 x 0:5 x x 7 7: R¼6 4 x x 0:5 x 5 x x x 0:5 In this case, if we take b = 1.5, and then we can construct the following auxiliary reciprocal relation R ¼ ðr ij Þ44 :
1:5ðw1 w3 Þ þ 0:5 1:5ðw2 w3 Þ þ 0:5 0:5 1:5ðw4 w3 Þ þ 0:5
If we take b = 1/2, we have w = (0.2863, 0.2467, 0.3137, 0.1533)T, and 2 3 0:5 0:5198 0:4 0:8 6 0:4802 0:5 0:3 0:7 7 6 7 R¼6 7: 4 0:6 0:7 0:5 0:5802 5
Y. Xu et al.
3 1:5ðw1 w4 Þ þ 0:5 1:5ðw2 w4 Þ þ 0:5 7 7: 1:5ðw3 w4 Þ þ 0:5 5 0:5
Then, we use the normalizing rank aggregation method to get the weighting vector. It is in the following: P j6¼i 1:5ðwi wj Þ þ 0:5 wi ¼ 6 i.e., 2 1:5 6 1:5 6 4 1:5 1:5
1:5 1:5 1:5 1:5 1:5 1:5 1:5 1:5
3 32 3 2 1:5 1:5 w1 6 7 6 7 1:5 7 76 w2 7 ¼ 6 1:5 7: 4 5 4 5 w3 1:5 5 1:5 w4 1:5 1:5
ð17Þ
Obviously, there are infinite solutions for the above P equation, i.e., for any wi [ [0, 1], and 4i¼1 wi ¼ 1; Eq. 17 always holds. But if we take b = 0.5 (Xu and Chen 2008), we will get w = (0.25, 0.25, 0.25, 0.25)T. Xu interpreted
2011,15(7)/Soft Computing
Group decision-making procedure
that, in the situations where no any preference information is provided, each alternative should be assigned with the same weight. And with the weighting vector, the auxiliary reciprocal relation R ¼ ðr ij Þ44 should be: 2 3 0:5 0:5 0:5 0:5 6 0:5 0:5 0:5 0:5 7 7 R¼6 4 0:5 0:5 0:5 0:5 5: 0:5 0:5 0:5 0:5 That is we only can construct the unique reciprocal relation. But in our view, R is an complete unknown reciprocal relation, in this circumstance, we can construct numerous auxiliary reciprocal relation only need to satisfy P the condition wi [ [0, 1], and 4i¼1 wi ¼ 1: Again, this also denotes that it is more reasonable to take b = (n - 1)/2 than b = 0.5. In the following, we will further investigate the problem in the group decision making.
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To aggregate all the auxiliary reciprocal relations R ðkÞ ðrij Þnn
Let M = {1, 2, …, m}. Now, we consider a group decision-making problem. Let E = {e1, e2, …, em} be a set of DMs, and let k = (k1, k2, …, km)T be the weighting Pm vector of DMs, with k¼1 kk ¼ 1; kk C 0, k [ M. Suppose that these m DMs provide their preferences over a set of n decision alternatives X = {x1, x2, …, xn}, and give m incomplete reciprocal relations R(k) = (r(k) ij )n9n (k [ M). Similar to the procedure in Sect. 3 which is for the decision making with an incomplete reciprocal relation, in the following, we develop a procedure for group decision making with incomplete reciprocal relations. Algorithm II Let v = (v1, v2, …, vn)T be the collective weighting vector of the incomplete reciprocal relations R(k) = (r(k) ij )n9n(k [ M). (k) Step 1 Replace the unknown element r(k) with ij in R vj Þ þ 0:5; and construct the auxiliary relation
n1 2 ðvi ðkÞ
R
ðkÞ rij
ðkÞ
¼ ðrij Þnn ; where ( ðkÞ rij ; ¼ n1 1 2 ðvi vj Þ þ 2 ;
ðkÞ
rij 6¼ x ðkÞ
rij ¼ x
ð18Þ
Step 2 Utilize the additive weighted averaging (AWA) operator m X ð1Þ ð2Þ ðmÞ ðkÞ r ij ¼ AWAk ðr ij ; r ij ; . . .; r ij Þ ¼ kk r ij ; i; j 2 N: k¼1
ð19Þ
¼
into a collective auxiliary reciprocal relation
R ¼ ðr ij Þnn : Step 3 Utilize the normalizing rank aggregation method to obtain the weighting vector v. First, calculate the collective preference degree pi(v) of the alternative xi over all the other alternatives: n X r ij ; i 2 N ð20Þ pi ðvÞ ¼ j¼1;j6¼i
And establish the following linear system of equations: pi ðvÞ pi ðvÞ ; i 2 N: vi ¼ P ¼ n nðn 1Þ=2 pj ðvÞ
ð21Þ
j¼1
which can be further rewritten as the following form: Av ¼ b
4 A procedure for group decision making with incomplete reciprocal relations
ðkÞ
ð22Þ
where b is a positive vector, A is a real symmetrical matrix. Step 4 Solving Eq. 22, we can get the priority vector v = (v1, v2, …, vn)T. Step 5 Rank all the alternatives and select the best one(s) in accordance with the values of vi (i [ N). Step 6 End. The above algorithm is similar to Xu and Chen’s method, the difference of the algorithm between Xu and Chen and this paper is that in Step 1. In this note, we replace the unknown (k) with n1 element r(k) ij in R 2 ðvi vj Þ þ 0:5; while Xu and Chen utilized 0.5(vi - vj ? 1) to replace the unknown (k) element r(k) ij in R , and then both to construct the auxiliary reciprocal relations. Based on the idea in Sect. 3, we introduce the following definition and criteria: Definition 7 Let v = (v1, v2, …, vn)T be the weighting vector which computed by the above Algorithm II, then we call R ¼ ðrij Þnn ¼ ðn1 2 ðvi vj Þ þ 0:5Þnn is the collective characteristic matrix of R: Definition 8 Let D = (dij)n9n be the difference matrix between R and R*, where dij ¼ r ij rij : Obviously, dij = -dji. From Theorem 4, we know that, R is an additive consistency reciprocal relation if and only if D = 0. Thus, the value of D is the smaller the better. To further compare the performances of the Algorithm II of this paper with Xu’s algorithm for group decision making with incomplete reciprocal relations, we introduce the following performance evaluation criteria:
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•
•
decision makers provide three incomplete reciprocal relations as follows: 2 3 0:5 0:6 0:7 0:3 0:6 6 0:4 0:5 x x x 7 6 7 6 7 7; 0:3 x 0:5 x x Rð1Þ ¼ 6 6 7 6 7 4 0:7 x x 0:5 x 5
Total deviation (TD) between the collective auxiliary reciprocal relation and the collective characteristic matrix. n X n n X n X X dij ¼ ð23Þ TD ¼ r ij rij : i¼1 j¼1
i¼1 j¼1
Total deviation (TD) between each auxiliary reciprocal relation and the collective characteristic matrix. m X n X n m n n X ðkÞ X X X ðkÞ TD1 ¼ dij ¼ rij rij k¼1 i¼1 j¼1
k¼1 i¼1 j¼1
Rð2Þ
where dij ¼ r ij rij is the fitting error for the element r ij of the collective auxiliary fitting reciprocal relation R ¼ ðr ij Þnn ; and
¼
ðkÞ rij
rij
the fitting error for
the fuzzy preference relation R ðkÞ
ðkÞ
ðkÞ rij
ðkÞ
¼ ðrij Þnn : It is easy Rð3Þ
ðkÞ
R
R
R
ð1Þ
ð2Þ
ð3Þ
6 6 0:4 6 ¼6 6 0:3 6 4 0:7 0:4 2
x
0:5
0:5
x
x
x
x
6 x 6 6 ¼6 6 x 6 4 x
0:5
0:6
0:2
0:4
0:5
x
0:8
x
0:5
x
0:5
x
x
0:5
0:5
x
x
x
0:6
0:5
x
x
x
0:5
x
x
x
0:5
0:5 7 7 7 0:4 7 7: 7 0:8 5
0:5
0:6
0:2
0:5
6 x 6 6 ¼6 6 x 6 4 x 0:4
3
0:5 7 7 7 x 7 7; 7 x 5 3
Let v = (v1, v2, …, v5)T be the collective weight vector of the incomplete reciprocal relations R(k) = (r(k) ij )595 n1 (k = 1, 2, 3) (where 2 ¼ 2). (k) with Step 1 Replace each unknown element r(k) ij in R 2(vi - vj) ? 0.5, and construct the auxiliary reciprocal relations:
Example 3 Consider a group decision making problem, there are three decision makers ek (k = 1, 2, 3), and let k = (0.5, 0.3, 0.2)T be the weighting vector of DMs. The
0:6
x
of
5 Numerical examples
0:5
x
2
to find that dij j jdji j; jdij j jdji j: Obviously, the more smaller TD and TD1, the better fitting performance of the weighting vector v.
2
0:4 2
ð24Þ
ðkÞ dij
Y. Xu et al.
0:7
0:3
0:6
3
7 2ðv2 v5 Þ þ 0:5 7 7 2ðv3 v2 Þ þ 0:5 0:5 2ðv3 v4 Þ þ 0:5 2ðv3 v5 Þ þ 0:5 7 7 7 0:5 2ðv4 v5 Þ þ 0:5 5 2ðv4 v2 Þ þ 0:5 2ðv4 v3 Þ þ 0:5 2ðv5 v2 Þ þ 0:5 2ðv5 v3 Þ þ 0:5 2ðv5 v4 Þ þ 0:5 0:5 3 0:5 2ðv1 v2 Þ þ 0:5 2ðv1 v3 Þ þ 0:5 2ðv1 v4 Þ þ 0:5 2ðv1 v5 Þ þ 0:5 6 7 0:5 0:6 0:2 0:5 6 2ðv2 v1 Þ þ 0:5 7 6 7 7 ¼6 v Þ þ 0:5 0:4 0:5 2ðv v Þ þ 0:5 2ðv v Þ þ 0:5 2ðv 3 1 3 4 3 5 6 7 6 7 0:8 2ðv4 v3 Þ þ 0:5 0:5 2ðv4 v5 Þ þ 0:5 5 4 2ðv4 v1 Þ þ 0:5 0:5 2ðv5 v3 Þ þ 0:5 2ðv5 v4 Þ þ 0:5 0:5 2ðv5 v1 Þ þ 0:5 2 3 0:5 2ðv1 v2 Þ þ 0:5 2ðv1 v3 Þ þ 0:5 2ðv1 v4 Þ þ 0:5 0:6 6 7 v Þ þ 0:5 0:5 2ðv2 v3 Þ þ 0:5 2ðv2 v4 Þ þ 0:5 0:5 7 2ðv 6 2 1 6 7 ¼6 0:5 2ðv3 v4 Þ þ 0:5 0:4 7 6 2ðv3 v1 Þ þ 0:5 2ðv3 v2 Þ þ 0:5 7 6 7 0:5 0:8 5 4 2ðv4 v1 Þ þ 0:5 2ðv4 v2 Þ þ 0:5 2ðv4 v3 Þ þ 0:5 0:4 0:5 0:6 0:2 0:5
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0:5
2ðv2 v3 Þ þ 0:5 2ðv2 v4 Þ þ 0:5
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Group decision-making procedure
1297
Step 2 Utilize Eq. 19 to aggregate all the auxiliary ðkÞ
ðkÞ
reciprocal relations R ¼ ðrij Þ55 (k = 1, 2, 3) into the collective auxiliary reciprocal relation:
2
0:5 6 v2 v1 þ 0:45 6 R¼6 6 v3 v1 þ 0:4 4 v4 v1 þ 0:6 0:6ðv5 v1 Þ þ 0:43
v1 v2 þ 0:55 0:5 1:4ðv3 v2 Þ þ 0:47 1:4ðv4 v2 Þ þ 0:59 v5 v2 þ 0:5
v1 v3 þ 0:6 1:4ðv2 v3 Þ þ 0:53 0:5 2ðv4 v3 Þ þ 0:5 1:6ðv5 v3 Þ þ 0:52
v1 v4 þ 0:4 1:4ðv2 v4 Þ þ 0:41 2ðv3 v4 Þ þ 0:5 0:5 1:6ðv5 v4 Þ þ 0:44
3 0:6ðv1 v5 Þ þ 0:57 v2 v5 þ 0:5 7 7 1:6ðv3 v5 Þ þ 0:48 7 7 1:6ðv4 v5 Þ þ 0:56 5 0:5
Step 3 Utilize the normalizing rank aggregation method to obtain the weighting vector v. We can establish the following linear system of equations:
8 > > > v1 > > > > > > > > > > v2 > > < v3 > > > > > > > v4 > > > > > > > > : v5
2
v1 v2 þ 0:55 þ v1 v3 þ 0:6 þ v1 v4 þ 0:4 þ 0:6ðv1 v5 Þ þ 0:57 10 v2 v1 þ 0:45 þ 1:4ðv2 v3 Þ þ 0:53 þ 1:4ðv2 v4 Þ þ 0:41 þ v2 v5 þ 0:5 ¼ 10 v3 v1 þ 0:4 þ 1:4ðv3 v2 Þ þ 0:47 þ 2ðv3 v4 Þ þ 0:5 þ 1:6ðv3 v5 Þ þ 0:48 ¼ 10 v4 v1 þ 0:6 þ 1:4ðv4 v2 Þ þ 0:59 þ 2ðv4 v3 Þ þ 0:5 þ 1:6ðv4 v5 Þ þ 0:56 ¼ 10 0:6ðv5 v1 Þ þ 0:43 þ v5 v2 þ 0:5 þ 1:6ðv5 v3 Þ þ 0:52 þ 1:6ðv5 v4 Þ þ 0:44 ¼ 10 ¼
i.e.,
6:4 1 6 1 5:2 6 6 1 1:4 6 4 1 1:4 0:6 1
1 1 1:4 1:4 4 2 2 4 1:6 1:6
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3
2
3
2:12 0:6 v1 7 6 7 6 1 7 76 v2 7 6 1:89 7 6 v3 7 ¼ 6 1:85 7 1:6 7 7 76 7 6 1:6 54 v4 5 4 2:25 5 1:89 v5 5:2
Step 4 Solve the above system of equations, thus we have v ¼ ð0:22; 0:17; 0:12; 0:32; 0:17ÞT Step 5 Rank all the alternatives xi(i = 1, 2, 3, 4, 5) in accordance with the values of vi (i = 1, 2, 3, 4, 5). x4 x1 x2 x5 x3 thus the best alternative is x4. ðkÞ
Furthermore, we can get R (k = 1, 2, 3), the collective auxiliary reciprocal relation R; and the characteristic matrix R* are all the same, i.e.,
R
ð1Þ
ð2Þ
ð3Þ
¼ R ¼ R ¼ R 0:5 0:6 0:7 0:3 6 0:4 0:5 0:6 0:2 6 ¼6 6 0:3 0:4 0:5 0:1 4 0:7 0:8 0:9 0:5 0:4 0:5 0:6 0:2 ¼R 2
3 0:6 0:5 7 7 0:4 7 7 0:8 5 0:5
And TD and TD1 computed by Eqs. 23, 24, respectively, we have TD ¼ 0; TD1 ¼ 0: ðkÞ
The above results shows that all the matrices (R (k = 1, 2, 3), R and R*) are the same, and also we can verify that they are all the additive consistency reciprocal relations according to Definition 2. If we take Xu and Chen’s method (i.e, take b = 0.5) to get the weighting vector, we have the following steps: (k) with Step 1 Replace each unknown element r(k) ij in R 0.5(vi - vj ? 1), and construct the auxiliary reciprocal relations:
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2011,15(7)/Soft Computing
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2
R
R
R
ð1Þ
ð2Þ
ð3Þ
Y. Xu et al.
3 0:6 7 0:5ðv2 v5 þ 1Þ 7 7 0:5ðv3 v2 þ 1Þ 0:5 0:5ðv3 v4 þ 1Þ 0:5ðv3 v5 þ 1Þ 7 7 7 0:5 0:5ðv4 v5 þ 1Þ 5 0:5ðv4 v2 þ 1Þ 0:5ðv4 v3 þ 1Þ 0:4 0:5ðv5 v2 þ 1Þ 0:5ðv5 v3 þ 1Þ 0:5ðv5 v4 þ 1Þ 0:5 2 3 0:5 0:5ðv1 v2 þ 1Þ 0:5ðv1 v3 þ 1Þ 0:5ðv1 v4 þ 1Þ 0:5ðv1 v5 þ 1Þ 6 7 0:5 0:6 0:2 0:5 6 0:5ðv2 v1 þ 1Þ 7 6 7 6 ¼ 6 0:5ðv3 v1 þ 1Þ 0:4 0:5 0:5ðv3 v4 þ 1Þ 0:5ðv3 v5 þ 1Þ 7 7 6 7 0:8 0:5ðv4 v3 þ 1Þ 0:5 0:5ðv4 v5 þ 1Þ 5 4 0:5ðv4 v1 þ 1Þ 0:5 0:5ðv5 v3 þ 1Þ 0:5ðv5 v4 þ 1Þ 0:5 0:5ðv5 v1 þ 1Þ 2 3 0:5 0:5ðv1 v2 þ 1Þ 0:5ðv1 v3 þ 1Þ 0:5ðv1 v4 þ 1Þ 0:6 6 7 0:5 0:5ðv2 v3 þ 1Þ 0:5ðv2 v4 þ 1Þ 0:5 7 6 0:5ðv2 v1 þ 1Þ 6 7 ¼6 0:5 0:5ðv3 v4 þ 1Þ 0:4 7 6 0:5ðv3 v1 þ 1Þ 0:5ðv3 v2 þ 1Þ 7 6 7 0:5 0:8 5 4 0:5ðv4 v1 þ 1Þ 0:5ðv4 v2 þ 1Þ 0:5ðv4 v3 þ 1Þ 0:4 0:5 0:6 0:2 0:5 0:5 6 6 0:4 6 ¼6 6 0:3 6 4 0:7
0:6 0:5
0:7 0:3 0:5ðv2 v3 þ 1Þ 0:5ðv2 v4 þ 1Þ
Step 2 Utilize Eq. 19 to aggregate all the auxiliary ðkÞ
ðkÞ
reciprocal relations R ¼ ðrij Þ55 (k = 1, 2, 3) into the collective auxiliary reciprocal relation: 2
0:5 6 0:25ðv2 v1 Þ þ 0:45 6 R¼6 6 0:25ðv3 v1 Þ þ 0:4 4 0:25ðv4 v1 Þ þ 0:6 0:15ðv5 v1 Þ þ 0:43
0:25ðv1 v2 Þ þ 0:55 0:5 0:35ðv3 v2 Þ þ 0:47 0:35ðv4 v2 Þ þ 0:59 0:25ðv5 v2 Þ þ 0:5
0:25ðv1 v3 Þ þ 0:6 0:35ðv2 v3 Þ þ 0:53 0:5 0:5ðv4 v3 Þ þ 0:5 0:4ðv5 v3 Þ þ 0:52
0:25ðv1 v4 Þ þ 0:4 0:35ðv2 v4 Þ þ 0:41 0:5ðv3 v4 Þ þ 0:5 0:5 0:4ðv5 v4 Þ þ 0:44
Step 3 Utilize the normalizing rank aggregation method to obtain the weighting vector v. We can establish the following linear system of equations:
8 > > > v1 > > > > > > > > v2 > > > > < v3 > > > > > > > v4 > > > > > > > > : v5
0:25ðv1 v2 Þ þ 0:55 þ 0:25ðv1 v3 Þ þ 0:6 þ 0:25ðv1 v4 Þ þ 0:4 þ 0:15ðv1 v5 Þ þ 0:57 10 0:25ðv2 v1 Þ þ 0:45 þ 0:35ðv2 v3 Þ þ 0:53 þ 0:35ðv2 v4 Þ þ 0:41 þ 0:25ðv2 v5 Þ þ 0:5 ¼ 10 0:25ðv3 v1 Þ þ 0:4 þ 0:35ðv3 v2 Þ þ 0:47 þ 0:5ðv3 v4 Þ þ 0:5 þ 0:4ðv3 v5 Þ þ 0:48 ¼ 10 0:25ðv4 v1 Þ þ 0:6 þ 0:35ðv4 v2 Þ þ 0:59 þ 0:5ðv4 v3 Þ þ 0:5 þ 0:4ðv4 v5 Þ þ 0:56 ¼ 10 0:15ðv5 v1 Þ þ 0:43 þ 0:25ðv5 v2 Þ þ 0:5 þ 0:4ðv5 v3 Þ þ 0:52 þ 0:4ðv5 v4 Þ þ 0:44 ¼ 10 ¼
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3 0:15ðv1 v5 Þ þ 0:57 0:25ðv2 v5 Þ þ 0:5 7 7 0:4ðv3 v5 Þ þ 0:48 7 7 0:4ðv4 v5 Þ þ 0:56 5 0:5
2011,15(7)/Soft Computing
Group decision-making procedure
i.e., 2 9:1 6 0:25 6 6 0:25 6 4 0:25 0:15
0:25 0:25 8:8 0:35 0:35 8:5 0:35 0:5 0:25 0:4
32
3
2
3
2:12 0:25 0:15 v1 6 v2 7 6 1:89 7 0:35 0:25 7 7 76 7 6 7 6 7 6 0:5 0:4 7 76 v3 7 ¼ 6 1:85 7 8:5 0:4 54 v4 5 4 2:25 5 1:89 v5 0:4 8:8
Table 2 Performance evaluation for Example 4 Criteria
b=2
b = 0.5
TD
1.4515
2.0163
TD1
5.1572
6.1471
2
Step 4 Solve the above system of equations, thus we have Rð1Þ
v ¼ ð0:2134; 0:187; 0:1813; 0:2312; 0:1871ÞT Step 5 Rank all the alternatives xi(i = 1, 2, 3, 4, 5) in accordance with the values of vi (i = 1, 2, 3, 4, 5). x4 x 1 x 5 x 2 x 3 thus the best alternative is x4.
ð2Þ
R
Furthermore, by the weighting vector v, we can get ðkÞ
R (k = 1, 2, 3), the collective auxiliary reciprocal relation R; and the characteristic matrix R*[in this case, r*ij = 0.5(vi - vj) ? 0.5)] easily, and we can verify that they are not the same. At the same time, we can further get TD and TD1 by Eqs. 23, 24, respectively. Table 1 shows the performances of the two methods. From the above results, we see that the ranking order is somewhat different between Xu and Chen’s method and ours. In our method, the weights of x2 and x5 are the same, which means that x2 and x5 are same important. But in Xu and Chen’s method, x5 is preferred to x2. From the original incomplete preference relations R(2) and R(3), we see that the two decision makers think that x2 and x5 are same important. Table 1 shows the performances of the two methods. As is known, smaller deviation means better performances. Obviously, our method has the better performance in the two criteria. This shows the advantages of our method. Evidently, Xu and Chen’s method suffer from rank reversal phenomenon, and their method will distort the decision maker’s information in some time. Example 4 Let us suppose that a company wants to renew its cars. There exist five models of car available, X = {x1, x2, …, x5}, and three DMs ek (k = 1, 2, 3), and let k = (0.5, 0.3, 0.2)T be the weighting vector of DMs. The decision makers provide three incomplete reciprocal relations as follows (the example has been examined by Xu and Chen 2008):
Table 1 Performance evaluation for Example 3 Criteria
b=2
b = 0.5
TD
0
0.9583
TD1
0
2.7721
1299
Rð3Þ
0:5 6 6 0:4 6 ¼6 6 0:3 6 4 0:7 0:4 2 0:5 6 6 0:3 6 ¼6 6 0:6 6 4 x
0:6 0:5
0:7 x
0:3 0:8
x
0:5
0:3
0:2 0:7 0:4 0:5 0:7 0:4
0:5 0:2 x
0:5 0:7
0:3 0:5
0:6 0:4
0:4
0:6
0:5
0:4 0:5 x 0:4 2 0:5 x 0:8 0:4 6 x 6 x 0:5 x 6 ¼6 0:2 x 0:5 0:3 6 6 4 0:6 x 0:7 0:5 0:5 x 0:4 0:3
3 0:6 7 0:6 7 7 0:5 7 7; 7 0:8 5 0:5 3 0:6 7 0:5 7 7 x 7 7; 7 0:6 5 0:5 3 0:5 7 x 7 7 0:6 7 7: 7 0:7 5 0:5
Xu and Chen get the priority vector is v = (0.2193, 0.2095, 0.1766, 0.2327, 0.1618)T. But after our carefully computation, the result of Xu and Chen’s linear systems (see Xu and Chen, Eq. 17) is v = (0.2199, 0.2071, 0.1772, 0.2335, 0.1623)T. By using our method, we get v = (0.2194, 0.2124, 0.1733, 0.2348, 0.1601)T. Both the priority vectors lead to the ranking of x4 x1 x2 x3 x5. The fitting performances of the two methods are assessed in terms of the criteria defined by Eqs. 23 and 24. The results are show in Table 2. From which, it is clear that our method still has smaller deviation, that is our method still performs better than Xu and Chen’s method.
6 Concluding remarks In this paper, we study a procedure for group decision making with incomplete reciprocal relations developed by Xu and Chen. We first examine the function between the fuzzy preference relation and its corresponding priority vector developed by Xu and Chen with a numerical example and show that the function does not hold in general cases. Then, we deduce an exact function between the additive transitivity fuzzy preference relation and its corresponding priority vector. Theoretical analysis proved that the function deduced by us is more reasonable than Xu and Chen’s. Based on this, we develop a procedure for the
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decision making with an incomplete reciprocal relation and also develop a procedure for the group decision making with incomplete reciprocal relations. In order to compare the performances of our method with Xu and Chen’s method in fitting the reciprocal relation, we introduce some criteria. Our method turns out to be more reasonable than Xu and Chen’s. It can construct a perfect additive consistency reciprocal relation when the initial incomplete reciprocal relation is additive, while Xu and Chen’s method will distort its information. This advantage also can be found for group decision making with incomplete reciprocal relations. And also, in some cases, Xu and Chen’s method will lead to rank reversal phenomenon. And even for not consistent incomplete reciprocal relations, the performance of our method is still better than Xu and Chen’s method. Acknowledgments The author is very grateful to the two anonymous referees for their insightful and constructive comments and suggestions that have led to an improved version of this paper. This work was supported by Hohai University ‘‘the Fundamental Research Funds for the Central Universities (2009B04514)’’ and the National Natural Science Foundation of China(No. 50979024,No. 90924027).
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