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Notes on “Logarithmic least squares method to priority for group decision making with incomplete fuzzy preference relations” Zhen Zhang∗, Chonghui Guo Institute of Systems Engineering, Dalian University of Technology, Dalian 116024, PR China
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 16 July 2014 Revised 25 May 2015 Accepted 25 July 2015 Available online xxx
In this note, we provide some numerical examples and theoretical analysis to demonstrate that a theorem and two corollaries in a recent paper (Xu et al., 2013) are incorrect. © 2015 Elsevier Inc. All rights reserved.
Keywords: Group decision making Incomplete fuzzy preference relation Logarithmic least squares method
1. Introduction Group decision making with fuzzy preference relations is an interesting topic in the decision science research field. In a recent paper, Xu et al. [1] investigated group decision making problems with incomplete fuzzy preference relations and constructed a logarithmic least squares model to deal with this type of problems. Moreover, the authors provided two solutions to the model. However, we noticed that one solution reported in [1] is incorrect. Therefore, in this note we will provide some numerical examples and theoretical analysis about this problem. 2. Problems in Xu et al.’s paper and analysis Consider the following group decision making problem with incomplete fuzzy preference relations [1]. Let X = {x1 , x2 , . . . , xn } be the set of alternatives, where xi is the ith alternative, i ∈ N = {1, 2, . . . , n}. Moreover, let D = {d1 , d2 , . . . , dm } be the set of decision makers, where dk is the kth decision maker, k ∈ M = {1, 2, . . . , m}. The decision makers provide their preference information over the alternatives using incomplete fuzzy preference relations. The fuzzy preference relation of the kth decision maker is de(k) (k) (k) noted as C (k) = (cij )n×n , where cij is the preference degree to which xi is preferred to xj . If cij is provided by the kth decision (k)
(k)
maker, cij ∈ [0, 1], otherwise, let cij = x, i, j ∈ N, k ∈ M. Xu et al. [1] established the following optimization model to deal with this group decision making problem:
min
J=
n n m 2 δij(k) σij(k) ln wi − ln w j − ln cij(k) + ln c(jik) k=1 i=1 j=1
∗
Corresponding author. Tel.: +86 411 84706512; fax: +86 411 84707425. E-mail addresses:
[email protected],
[email protected] (Z. Zhang),
[email protected] (C. Guo).
http://dx.doi.org/10.1016/j.apm.2015.07.024 S0307-904X(15)00539-9/© 2015 Elsevier Inc. All rights reserved.
Please cite this article as: Z. Zhang, C. Guo, Notes on “Logarithmic least squares method to priority for group decision making with incomplete fuzzy preference relations”, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.07.024
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s.t.
n
wi = 1
i=1
wi > 0, i ∈ N, (k)
(k)
where δij and σij
1,
and
σij(k) =
are defined as
if cij(k) = x, , if cij(k) = x,
0,
δij(k) =
(M–1)
i, j ∈ I, k ∈ M,
if cij(k) = 1 or 0, , otherwise,
0, 1,
(1)
i, j ∈ I, k ∈ M.
(2)
Based on the logarithmic least squares method, the model (M–1) is solved as follows. Let
⎛
m n δ (k) σ (k)
1j 1j ⎜ ⎜ j=2 k=1 ⎜ m ⎜ ⎜ − δ (k) σ (k) ⎜ 21 21 ⎜ k=1 Dn−1 = ⎜ ⎜ ⎜ .. ⎜ . ⎜ ⎜ ⎜ m (k) (k) ⎝− δn−1,1 σn−1,1
−
⎛
12
···
12
k=1 n m δ (k) σ (k) 2j
···
2j
j=1 k=1 j=2
.. . m − δ (k)
n−1,2
k=1
and
m δ (k) σ (k)
··· (k)
σn−1,2
···
⎞ (k) σ i,n−1 i,n−1 ⎟ ⎟ k=1 ⎟ m ⎟ (k) (k) − δ2,n−1 σ2,n−1 ⎟ ⎟ ⎟ k=1 ⎟ ⎟ ⎟ .. ⎟ . ⎟ ⎟ m n (k) (k) ⎟ δn−1, j σn−1, j ⎠ −
m δ (k)
(3)
j=1 k=1 j=n−1
k=1
⎞
n m δ (k) σ (k) ( ln c(k) − ln c(k) )
⎜ ⎟ 1j 1j 1j j1 ⎜ k=1 j=1 ⎟ ⎜ ⎟ ⎜ ⎟ n m ⎜ (k) (k) (k) (k) δ2 j σ2 j ( ln c2 j − ln c j2 ) ⎟ ⎜ ⎟ Yn−1 = ⎜ ⎟, ⎜ k=1 j=1 ⎟ ⎜ ⎟ ··· ⎜ ⎟ ⎜ ⎟ n m ⎝ (k) (k) (k) (k) δn−1, j σn−1, j ( ln cn−1, j − ln c j,n−1 )⎠
(4)
k=1 j=1
then the solution to the model (M–1) is
⎧ exp (Wi ) ⎪ ⎪ ⎨ n−1 exp (W ) + 1 , j j=1 w∗i = 1 ⎪ ⎪ , ⎩ n−1 j=1 exp (W j ) + 1
i = 1, 2, . . . , n − 1 ,
(5)
i=n
Y . where W = (W1 , W2 , . . . , Wn−1 )T = D−1 n−1 n−1 Xu et al. solved the model (M–1) from another perspective and gave the following theorem [1]. Theorem 1. Let w∗ = (w∗1 , w∗2 , . . . , w∗n ) be the optimal solution to the model (M–1), then
n
m
j=1
w∗i =
k=1
n n i=1
j=1
(k)
(k)
(cij /c ji )
m k=1
δ (k) σ (k) ij ij n m δ (k) σ (k) j=1 k=1 ij ij
(cij(k) /c(jik) )
δ (k) σ (k) ij ij n m δ (k) σ (k) j=1 k=1 ij ij
,
i ∈ N.
(6)
In what follows, we utilize some numerical examples to demonstrate that Theorem 1 is incorrect. Please cite this article as: Z. Zhang, C. Guo, Notes on “Logarithmic least squares method to priority for group decision making with incomplete fuzzy preference relations”, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.07.024
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3
Example 1 [1]. For a decision making problem, a decision maker provided his preference information over six alternatives xi (i = 1, 2, . . . , 6) using the following incomplete fuzzy preference relation (here m = 1):
⎡
0.5 ⎢0.6 ⎢x C=⎢ ⎢0.7 ⎣ 0.7 0.7
0.4 0.5 0.4 0.5 x 0.6
x 0.6 0.5 0.7 0.4 x
0.3 0.5 0.3 0.5 0.6 0.2
⎤
0.3 x 0.6 0.4 0.5 0.3
0.3 0.4⎥ x ⎥ ⎥ 0.8⎥ ⎦ 0.7 0.5
If we use Eqs. (3)–(5) to derive the priority weights, we have
⎛4 ⎜−1 D5 = ⎜ 0 ⎝ −1 −1
−1 4 −1 −1 0
⎞
−1 −1 −1 5 −1
0 −1 3 −1 −1
−1 0⎟ −1⎟, ⎠ −1 4
⎛−2.9474⎞
⎜ 0.4055 ⎟ Y5 = ⎜−0.8473⎟. ⎝ ⎠ 2.6754 1.6946
Therefore, w∗ can be derived as (0.0878, 0.1599, 0.1551, 0.2464, 0.2208, 0.1301)T , which is reported in Xu et al.’s work (see page 2149 of [1]). However, if Eq. (6) is utilized, the following results can be derived: 6
(c1 j /c j1 )
δ1 j σ1 j 6 δ σ j=1 1 j 1 j
= (0.5/0.5)0.2 × (0.4/0.6)0.2 × 1 × (0.3/0.7)0.6 = 0.5546.
j=1
Similarly, 6
(c2 j /c j2 )
δ2 j σ2 j 6 δ σ j=1 2 j 2 j
= 1.0845,
j=1 6
(c3 j /c j3 )
δ3 j σ3 j 6 δ σ j=1 3 j 3 j
= 0.8091,
j=1
(c4 j /c j4 )
δ4 j σ4 j 6 δ σ j=1 4 j 4 j
= 1.5619,
j=1 6
6
6
(c5 j /c j5 )
δ5 j σ5 j 6 δ σ j=1 5 j 5 j
= 1.4034,
j=1
(c6 j /c j6 )
δ6 j σ6 j 6 δ σ j=1 6 j 6 j
= 0.8219.
j=1
It follows that
0.5546 = 0.0889, 0.5546 + 1.0845 + 0.8091 + 1.5619 + 1.4034 + 0.8219 1.0845 = 0.1739, w∗2 = 0.5546 + 1.0845 + 0.8091 + 1.5619 + 1.4034 + 0.8219 0.8091 w∗3 = = 0.1298, 0.5546 + 1.0845 + 0.8091 + 1.5619 + 1.4034 + 0.8219 1.5619 w∗4 = = 0.2505, 0.5546 + 1.0845 + 0.8091 + 1.5619 + 1.4034 + 0.8219 1.4304 = 0.2251, w∗5 = 0.5546 + 1.0845 + 0.8091 + 1.5619 + 1.4034 + 0.8219 0.8219 w∗6 = = 0.1318. 0.5546 + 1.0845 + 0.8091 + 1.5619 + 1.4034 + 0.8219
w∗1 =
It is obvious that the results derived by Eq. (6) are different from those obtained by Eqs. (3)–(5). Example 2 [1]. For a group decision making problem with three alternatives, three decision makers provided the following incomplete fuzzy preference relations:
C (1) =
0.5 0.5 0.4
0.5 0.5 x
0.6 x , 0.5
C (2) =
0.5 x 0.4
x 0.5 0.4
0.6 0.6 , 0.5
C (3) =
0.5 0.5 x
0.5 0.5 0.4
x 0.6 . 0.5
By Eqs. (3)–(5), the priority weight vector can be derived as w∗ = (0.375, 0.375, 0.25)T [1]. However, if Eq. (6) is utilized, we can obtain 3 3
(k)
(k)
(c1 j /c j1 )
δ (k) σ (k) 1j 1j 3 3 δ (k) σ (k) j=1 k=1 1 j 1 j
= 1.1228,
j=1 k=1
Please cite this article as: Z. Zhang, C. Guo, Notes on “Logarithmic least squares method to priority for group decision making with incomplete fuzzy preference relations”, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.07.024
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3 3
(k)
(k)
(c2 j /c j2 )
δ (k) σ (k) 2j 2j 3 3 δ (k) σ (k) j=1 k=1 2 j 2 j
= 1.1228,
j=1 k=1 3 3
(c3(kj) /c(j3k) )
δ (k) σ (k) 3j 3j 3 3 δ (k) σ (k) j=1 k=1 3 j 3 j
= 0.7932.
j=1 k=1
Therefore,
1.1228 = 0.3695, 1.1228 + 1.1228 + 0.7932 1.1228 = 0.3695, w∗2 = 1.1228 + 1.1228 + 0.7932 0.7932 w∗3 = = 0.2610. 1.1228 + 1.1228 + 0.7932
w∗1 =
By contrast, the results derived by Eq. (6) are also different from those obtained by Eqs. (3)–(5). In what follows, we will make some analysis to demonstrate that Theorem 1 is incorrect, i.e. Eq. (6) cannot be used to derive priority weights from incomplete fuzzy preference relations. In [1], the authors proved Theorem 1 using the Lagrange method and constructed the following Lagrange function:
n n m n 2 min J = δij(k) σij(k) ln wi − ln w j − ln cij(k) + ln c(jik) + 4λ wi − 1 . k=1 i=1 j=1
(7)
i=1
Through proof, we have λ = 0 (see the proof in [1]). Therefore, m n δ (k) σ (k) ln (w /w ij
i
ij
j
)=
j=1 k=1
m n δ (k) σ (k) ln c(k) /c(k) , ij
ij
ij
ji
i ∈ N.
(8)
j=1 k=1
By Eq. (8), the following equation can be derived:
1
ln wi = n
m
j=1
k=1
δij(k) σij(k)
m m n n δij(k) σij(k) ln w j + δij(k) σij(k) ln cij(k) /c(jik) , j=1 k=1
i ∈ N.
(9)
j=1 k=1
Xu et al. [1] wrote Eq. (9) as δ (k) σ (k)
m n n ijm ijδ(k) σ (k) wi = c · cij(k) /c(jik) j=1 k=1 ij ij ,
(10)
j=1 k=1
where
n m c = exp
j=1
n j=1
k=1
δij(k) σij(k) ln w j
m
k=1
δij(k) σij(k)
.
(11)
Eqs. (10) and (11) mean that c is a constant. However, we observe that the value of c depends on the value of i, i.e. different values of i can lead to different values of c. Thereby, c is not a constant. Xu et al. [1] proved Theorem 1 based on Eqs. (10) and (11), as a result, incorrect conclusions are obtained. It can be observed that the main difference between the two resolution methods is how to solve Eq. (9). We can further check whether the two priority weight vectors derived by the two different methods are solutions to Eq. (9). Now we reconsider the two priority weight vectors in Example 1. If w = (0.0889, 0.1739, 0.1298, 0.2505, 0.2251, 0.1318)T , then the two sides of Eq. (9) are derived as (−2.4197, −1.7492, −2.0421, −1.3843, −1.4913, −2.0264)T and (−2.4037, −1.8432, −1.8785, −1.4063, −1.5339, −2.0104)T , respectively. Clearly, the two sides are not equal, which means that w = (0.0889, 0.1739, 0.1298, 0.2505, 0.2251, 0.1318)T is not a solution to Eq. (9). However, if w = (0.0878, 0.1599, 0.1551, 0.2464, 0.2208, 0.1301)T , then both sides of Eq. (9) are derived as (−2.4329, −1.8330, −1.8640, −1.4009, −1.5107, −2.0396)T , which demonstrates that w = (0.0878, 0.1599, 0.1551, 0.2464, 0.2208, 0.1301)T is a solution to Eq. (9). From the above analysis, it can be concluded that w = (0.0889, 0.1739, 0.1298, 0.2505, 0.2251, 0.1318)T derived by Eq. (6) cannot be regarded as the priority weight vector of the preference relation in Example 1. Similar results can also be obtained for the preference relations in Example 2, which are omitted. On the basis of the above analysis, we can conclude that Eq. (6) is unsuitable for deriving priority weight vector from incomplete preference relations, i.e. Theorem 1 may not hold for all the cases. Furthermore, based on Theorem 1, Xu et al. [1] also gave the following two corollaries which considering the mentioned analysis are incorrect. Please cite this article as: Z. Zhang, C. Guo, Notes on “Logarithmic least squares method to priority for group decision making with incomplete fuzzy preference relations”, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.07.024
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5
(k)
Corollary 1. If there is no cij =1 or 0 for all i, j ∈ N, k ∈ M, then the optimal solution to the model (M–1) is
n
m
j=1
w∗i =
m
j=1
(k)
(cij /c ji )
k=1
n n i=1
(k)
k=1
δ (k) ij n m δ (k) j=1 k=1 ij
(cij(k) /c(jik) )
δ (k) ij n m δ (k) j=1 k=1 ij
,
i ∈ N.
(12)
(k)
Corollary 2. If there is no cij = ϕ for all i, j ∈ N, k ∈ M, then the optimal solution to the model (M–1) is
n
m
j=1
w∗i =
k=1
n n i=1
j=1
(k)
(k)
(cij /c ji )
m k=1
σ (k) ij m n σ (k) j=1 k=1 ij σ (k) ij m σ (k) j=1 k=1 ij
(k) n
cij(k) /c ji
,
i ∈ N.
(k)
(13)
(k)
However, Theorem 1 will hold when there is no cij =1 or 0 and no cij = ϕ for all i, j ∈ N, k ∈ M. In particular, when m = 1, Eq. (6) is reduced to the formulae which is used to derive a priority weight vector from a complete preference relation with no elements of 0 and 1 (see [2,3]). In these cases, c is a constant for Eq. (11). To summarize, we conclude that Eqs. (3)–(5) can be used to derive a priority weight vector from incomplete fuzzy preference relations, while Eq. (6) should not be used. Acknowledgements The authors would like to thank the Editor-in-Chief and the three anonymous referees for their insightful and constructive comments and suggestions that have led to an improved version of this note. This work was partly supported by the National Natural Science Foundation of China (Nos. 71501023, 71171030), the Funds for Creative Research Groups of China (No. 71421001), the China Postdoctoral Science Foundation (2015M570248) and the Fundamental Research Funds for the Central Universities (DUT15RC(3)003). References [1] Y. Xu, R. Patnayakuni, H. Wang, Logarithmic least squares method to priority for group decision making with incomplete fuzzy preference relations, Appl. Math. Model. 37 (4) (2013) 2139–2152. [2] M. Fedrizzi, M. Brunelli, On the priority vector associated with a reciprocal relation and a pairwise comparison matrix, Soft Comput. 14 (6) (2010) 639–645. [3] Y.-M. Wang, Z.-P. Fan, Group decision analysis based on fuzzy preference relations: logarithmic and geometric least squares methods, Appl. Math. Comput. 194 (1) (2007) 108–119.
Please cite this article as: Z. Zhang, C. Guo, Notes on “Logarithmic least squares method to priority for group decision making with incomplete fuzzy preference relations”, Applied Mathematical Modelling (2015), http://dx.doi.org/10.1016/j.apm.2015.07.024