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Logarithmic least squares method to priority for group decision making with incomplete fuzzy preference relations
Applied Mathematical Modelling 37 (2013) 2139–2152
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Logarithmic least squares method to priority for group decision making with incomplete fuzzy preference relations Yejun Xu a,b,⇑, Ravi Patnayakuni c, Huimin Wang a,b a
State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, PR China Business School, Hohai University, Jiangning, Nanjing, Jiangsu 211100, PR China c Department of Economics and IS, University of Alabama in Huntsville, Huntsville, AL 35763, USA b
a r t i c l e
i n f o
Article history: Received 10 November 2011 Received in revised form 28 April 2012 Accepted 8 May 2012 Available online 22 May 2012 Keywords: Group decision making Incomplete fuzzy preference relation Logarithmic least squares method Multiplicative consistency
a b s t r a c t The aim of this paper is to present a logarithmic least squares method (LLSM) to priority for group decision making with incomplete fuzzy preference relations. We give a reasonable definition of multiplicative consistent for incomplete fuzzy preference relation. We develop the acceptable fuzzy consistency ratio (FCR for short), which is simple and similar to Saaty’s consistency ratio CR for multiplicative fuzzy preference relations. We also extend the LLSM method to the case of individual preference relation with complete information. Finally, some examples are illustrated to show that our method is simple, efficient, and can be performed on computer easily. Crown Copyright 2012 Published by Elsevier Inc. All rights reserved.
1. Introduction Group decision making (GDM) [1–21] is participatory process in which multiple individuals, often experts, together formulate problems, develop alternatives and eventually select among the alternatives to reach a decision. Each expert may have unique motivations or goals and may approach the decision process from a different perspective, but share a common interest in reaching eventual agreement on selecting the best solution (s) to the problem. Fuzzy preference relations [2–4,22,10–12,14–16,23,24] are commonly used to represent decision makers’ preferences over the set of possible alternative solutions, and have received considerable research attention in the past decades. The research primarily focuses on examining fuzzy preference relations with complete information. A complete fuzzy preference relation of order n necessitates the completion of all n(n 1)/2 judgments in its entire top triangular portion. However, sometimes, a decision maker (DM) may develop fuzzy preference relations with incomplete information. Firstly, because of time pressure, lack of knowledge, and/or the DM’s limited expertise related with problem domain [25,26,19,27,20], the DM may not be able to provide a complete set of preference relations. Secondly, when the number of alternatives, n, is large. In such cases it may be practically impossible, or at least unacceptable from the view point of the decision maker, to perform all the n(n 1)/2 required comparisons to complete the pairwise comparison matrices [28]; A third reason should be that it may be convenient or even necessary to skip some direct critical comparisons between alternatives, even if the total number of alternatives is small [28]. Finally, an expert may not be able to efficiently express any kind of preference degree between two or more of the available alternatives. This may be due to an expert not possessing a precise or sufficient level of information for part of the problem, or because that expert is unable to discriminate the degree to which some alternatives are better than others ⇑ Corresponding author at: State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, PR China. Tel.: +86 25 68514612; fax: +86 25 85427972. E-mail address: [email protected] (Y. Xu). 0307-904X/$ - see front matter Crown Copyright 2012 Published by Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2012.05.010
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[29,30,5,6]. There have been seen some theoretical studies with incomplete preference relations. Alonso et al. [30] 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. [6] proposed a new model to deal with GDM problems with the incomplete fuzzy preference relations based on the additive-consistency (AC) property. The new model consists of two steps: estimation of missing preference values and the selection of alternatives. In another paper, they [5] 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 a moderator, using both consensus and consistency criteria. More recently, Alonso et al. [29] put forward a general procedure that attempts to estimate the missing information of incomplete preference relations in any of the above formats: fuzzy, multiplicative, interval valued and linguistic. Xu [19] 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 programing models, based on additive consistent incomplete fuzzy preference relation and multiplicative consistent incomplete fuzzy preference relation respectively, for obtaining the priority vector for incomplete fuzzy preference relation. Later in our analysis (Section 2), we will argue that the concept of multiplicative consistent incomplete fuzzy preference relation proposed in this paper have limitations. In another paper, Xu [27] proposed a procedure for decision making with incomplete fuzzy preference relation based on multiplicative consistency. Xu [20] also developed a simple but practical approach to deriving the ranking of the alternatives from an incomplete reciprocal relation based on additive transitivity. He postulated a correspondence between priority vector and additive consistent incomplete fuzzy preference relation. Shen et al. [31], and Xu [32,33,17] had pointed out that the correspondence may be unreasonable, and deduced a function between the fuzzy preference relation and priority vector. Gong [34] developed a least-square model to obtain the collective priority vector from the incomplete preference relations of multiple decision makers derived with a simple equation. As will be evident in the examples presented in this paper, in some situations, Gong’s simple equation cannot be used to obtain the priority vector because the matrix Q (See Example 5) is a singular matrix, where Q1 does not exist. Xu et al. [35] proposed a weighted least square method and quadratic programing method for developing priorities from incomplete fuzzy preference relation based on multiplicative transitivity. In another paper they [32] presented two priority methods-the least variance priority method (LVM) and quadratic programing method for incomplete fuzzy preference relation based on additive consistency. Liu et al. [9] explored the group decision making problems with incomplete additively consistent fuzzy preference relations. They developed least square completion and inconsistency repair methods to deal with incomplete and inconsistent fuzzy preference relations. The consistency in preference relations (whether multiplicative or fuzzy) [25,36,37] given by DMs has a direct impact on the ranking results of the final decision. The issue of consistency in Analytic Hierarchy Process (AHP) was first addressed by Saaty [38], who originally developed the notions of perfect consistency and acceptable consistency, and the latter was checked by a consistency index CI and consistency ratio CR. Saaty pointed out that a multiplicative preference matrix is of acceptable consistency if CR < 0.1. The method is simple and has been widely used. Studies on the consistency of fuzzy preference relations have also attracted much attention recently. Most of the research is associated with transitivity [39,40,10], such as weak transitivity [41], max–min transitivity, max–max transitivity, restricted max–min transitivity, restricted max–max transitivity, multiplicative consistency and additive consistency. For incomplete fuzzy preference relations, the consistency measure and consistency level have been studied for the additive consistency preference relation [5,6,26]. To our knowledge, there have no studies on the consistency index for incomplete multiplicative fuzzy preference relations. Therefore, we direct our attention to this issue. The aim of this paper is to present a logarithmic least squares method (LLSM) to prioritize among a set of alternatives in the group decision making context when DMs have incomplete fuzzy preference relations. Although LLSM is not new and is well known in AHP [42–46,15], it has not been applied to modeling incomplete fuzzy preference relations in group decision making problems based on multiplicative consistency. In order to demonstrate the method, this paper is structured in the following way. The next section presents the basic concepts of a multiplicative preference relation, fuzzy preference relation and incomplete fuzzy preference relation. We also illustrate with an example to explain that multiplicative consistent incomplete fuzzy preference relation as defined by Xu is not correct. We then offer a reasonable definition of multiplicative consistent for an incomplete fuzzy preference relation. We also give an approach to judge whether an incomplete fuzzy relation is acceptable or not acceptable. Section 3, the LLSM is extended to model incomplete fuzzy preference relations based on multiplicative consistency. The corresponding acceptable fuzzy consistency ratio (FCR for short) is developed, which is simple and similar to Saaty’s consistency ratio CR for multiplicative consistent fuzzy preference relations in Section 4. Section 5, some examples are presented to illustrate that the proposed method is simple, efficient, and can be performed easily on computer. Finally, in Section 6, we draw our conclusions and point out future research.
2. Preliminaries This section describes a multiplicative preference relation and fuzzy preference relation on alternatives, and also some related concepts such as incomplete fuzzy preference relation, multiplicative consistency of incomplete fuzzy preference relation, acceptable incomplete fuzzy preference relation etc., which will be used in the subsequent sections.
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For simplicity, we denote N = {1, 2, . . . , n}. Let X = {x1, x2, . . . , xn} (n P 2) be a finite set of alternatives, where xi denotes the ith alternative. In multi-attribute decision making problems, a decision maker needs to rank the alternatives x1, x2, . . . , xn from the best to the worst according to the preference information. Next, a brief description of a multiplicative preference relation and fuzzy preference relation is provided below. 2.1. Multiplicative preference relation A multiplicative preference relation is a positive preference relation, A X X, A = (aij)nn, where aij denotes the relative weight of alternative xi with respect to xj. The measurement of aij is described using a ratio scale and in particular, as shown by Saaty [38], aij 2 {1/9.1/8, 1/7, . . . , 1, 2, . . . , 9}:aij = 1 denotes the indifference between xi and xj, aij = 9 (or aji = 1/9) denotes that xi is definitely preferred to xj, and aij 2 {2,3, . . . , 8} denotes the intermediate evaluations. It is multiplicative reciprocal, i.e., aijaji = 1, "i, j 2 {2, . . . , n} and in particular, aij = 1 "i 2 {1, 2, . . . , n}. Thus we have the following definition. Let A = (aij)nn be a multiplicative preference relation, then A is called a consistent multiplicative preference relation (or called consistent reciprocal judgment matrix), if aij = aikakj, for all i, j, k. 2.2. Fuzzy preference relation The fuzzy preference relation R is described as follows: R X X, R = (rij)nn, with membership function uR:X X ? [0, 1], where uR(xi, xj) = rij denotes the preference degree of the alternative xi over xj [3,7,10,12]: rij = 0.5 denotes indifference between xi and xj, rij = 1, denotes that xi is definitely preferred to xj, and 0.5 < rij < 1 (or 0 < rji < 0.5) denotes that xi is preferred to xj. Let R = (rij)nn be a preference relation, then R is called a reciprocal fuzzy preference relation if
rij 2 ½0; 1;
rij þ rji ¼ 1;
r ii ¼ 0:5;
for all i; j 2 N:
Tanino [10] proposed the definition of multiplicative consistency on fuzzy preference relations, i.e. R = (rij)nn is perfectly consistent if
ril r lj r ji ¼ rli r jl r ij ;
i; j; l 2 N:
It has been found that a perfectly consistent multiplicative preference relation and a perfectly consistent fuzzy preference P relation can be precisely characterized by a priority vector w = (w1,w2, . . . , wn)T, which satisfies the conditions ni¼1 wi ¼ 1 and wi > 0 for i = 1, 2, . . . , n. That is [16]
wi ; i; j 2 N; wj wi ; i; j 2 N: rij ¼ wi þ wj
aij ¼
ð1Þ ð2Þ
Next, we will cover some related concepts where the preference information is incomplete. Definition 1 [19]. Let C = (cij)nn be a fuzzy preference relation, then C is called an incomplete fuzzy preference relation, if some of its elements cannot be provided by the DM, which we denote the unknown number x, and the others can be provided by the DM, which satisfy cij 2 [0, 1], cij + cji = 1, cii = 0.5. Definition 2 19. Let C = (cij)nn be an incomplete fuzzy preference relation, then C is called a multiplicative consistent incomplete fuzzy preference relation, if all the known elements satisfy multiplicative transitivity i.e. cilcljcji = clicjlcij. Next, we offer proof that this definition is not correct. Proof. Since
cil clj cji ¼ cli cjl cij ;
ð3Þ
that is
cij cil clj ¼ : cji cli cjl
ð4Þ c
c
Let bij ¼ cijji ; bil ¼ ccil ; blj ¼ clj , then Eq. (4) is equivalent to the following equation: li
bij ¼ bil blj :
jl
ð5Þ
We label this as the corresponding incomplete multiplicative preference relation. So Definition 2 is equivalent to the following Definition 3. h Definition 3. Let B = (bij)nn be an incomplete multiplicative preference relation, then B is called a multiplicative consistent incomplete multiplicative preference relation, if all the known elements satisfy the multiplicative transitivity bij = bilbij.
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Herrera-Viedma et al. [47] further gave the following proposition. Proposition 1. For a reciprocal multiplicative preference relation A = (aij)nn, the following statements are equivalent: (i) aij ajk = aik, "i, j, k. (ii) aij = aii+1 aii+2 . . . aj1j, "i, j. Let B = (bij)nn be an incomplete multiplicative preference relation, and let G be its corresponding directed graph, which implies the relationship in matrix B. There are n vertices 1, 2, . . ., n in G, Moreover, if bij – x, there exists a directed arc ! in the directed graph G i;j going from i to j, and its weight is bij. If bij = x, there is no directed arc between i and j, which denotes that there is no comparison information between alternative xi and xj. We will see that Definition 3 is not correct using the following example. Example 1. For an incomplete multiplicative preference relation B which is in the following:
2
1 3 6
61 63 61 B¼6 66 6x 4 1 4
x
4
3
x7 7 7 1 x7 7; 1 2 1 27 5 x x 12 1 1 2 1 1 2
1 2
Fig. 1 is the directed graph of matrix B. We can test and verify, for all i, j, l, bijbjlbli = 1 (where bij – x, for all i, j), which can also be tested and verified in Fig. 1. For all the circuit whose length is 3, the weight is 1. For example, the circuit (1, 2, 3, 1), its weight is b12 b23 b31 ¼ 3 2 16 ¼ 1, but for the circuit (1, 3, 4, 5, 1), its weight is b13 b34 b45 b51 ¼ 6 12 2 14 – 1, so matrix B is not consistent. If we change a15 to 6, then for all the circuit, its weight is 1. Thus, the following result characterizes incomplete reciprocal multiplicative consistency. Theorem 1. Let B = (bij)nn be an incomplete multiplicative preference relation, then B is called a multiplicative consistent incomplete preference relation, if all the known elements satisfy the multiplicative transitivity bi1 i2 bi2 i3 . . . bik1 ik bik i1 ¼ 1, "i, j, bij – x. Example 2. The decision maker gives the following incomplete fuzzy preference relation.
3 3 6 4 x 0:5 4 7 5 7 6 1 2 1 0:5 x 7 6 4 3 2 7 6 1 1 1 C¼6 0:5 x 7 3 3 7: 6 7 6 1 2 2 7 0:5 4 x 2 3 3 5 1 1 x x 0:5 5 3 2
If we use Xu’s Definition 2 to verify, C as a multiplicative consistent incomplete fuzzy preference relation, we will find that for the same reasons, it is not correct. So Definition 2 should be corrected as follows:
Fig. 1. The directed graph of matrix B.
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Definition 4. Let C = (cij)nn be an incomplete fuzzy preference relation, then C is called a multiplicative consistent incomplete fuzzy preference relation, if all the known elements satisfy the multiplicative transitivity ci1 i2 ci2 i3 . . . cik ij ¼ ci2 i1 ci3 i2 . . . cij ik , "i, j, cij – x. Next, we offer a simple method to judge whether an incomplete fuzzy preference is acceptable or not. Definition 5 [35,48]. The elements cij, ckl, of are called adjacent, if {i, j} \ {k, l} – £. For the unknown element cij, cij can be determined indirectly, if there exist a series of known elements cij1 ; cj1 j2 ; . . . ; cjk j . Definition 6 [35,48]. Let C = (cij)nn be an incomplete fuzzy preference relation, if the missing elements of C can be determined by the known elements, then C is called an acceptable incomplete fuzzy preference relation, otherwise, C is not an acceptable incomplete fuzzy preference relation. Theorem 2. Let C = (cij)nn be an incomplete fuzzy preference relation, the necessary condition of acceptable incomplete fuzzy preference relation C is that there exists at least one known element in each row or column of C except for the diagonal elements (cii,i = 1, 2, . . . , n), i.e. there needs at least (n 1) judgments. Proof. Let ckl(k – l) be a missing element of C = (cij)nn, where C is an acceptable incomplete fuzzy preference relation. From Definitions 5 and 6, we know that there must be a series of known elements ckl1 ; cl1 l2 ; . . . ; cl;l , in other words, there is at least one given element ckl1 in the kth row, for k – l1, and at the same time, there is at least one given element cls l in the lth column, for ls – l. So the necessary condition holds for "k, l, and thus, there need to be at least (n 1) judgments in the upper (lower) triangular of the incomplete preference relation C. Herrera-Viedma et al. [6] proved the above result by induction on the number of alternatives. At the same time, their proof is based on the additive reciprocity property of a fuzzy preference relation. Our method is suitable not only for additive reciprocity property but for the multiplicative property of a fuzzy preference relation. Gong [34] also gave an approach to judge whether an incomplete fuzzy preference relation is acceptable or not, as we can see, his method is so complicated and cannot judge easily. h
3. Logarithmic least squares method for deriving priorities from group incomplete fuzzy preference relations For simplicity, we denote M = {1, 2, . . . , m}. Let X = {x1, x2, . . . , xn} (n P 2) be a finite set of alternatives and D = {d1, d2, . . . , dn} (m P 2) be a finite set of decision makers. Suppose that these m decision makers provide their preferences over the set X, and ðkÞ ðk 2 MÞ. Let w = (w1,w2, . . . , wn)T be the priority vector of the give m incomplete fuzzy preference relations C ðkÞ ¼ cij nn ðkÞ group’s incomplete multiplicative consistent fuzzy preference relations C ðkÞ ¼ cij , then these preference relations are nn
given by [19] ðkÞ ðkÞ
ðkÞ
dij cij ¼ dij ðkÞ
where dij
wi ; wi þ wj
i; j 2 N;
ð6Þ
ðkÞ is a zero or one integer variable of the indicator matrix D ¼ dij
ðkÞ
dij ¼
nn
8 < 0; if cðkÞ ¼ x ij
: 1; if cðkÞ – x ij
, which is defined as
i; j 2 N; k 2 M:
ð7Þ
From Eq. (6), we have ðkÞ ðkÞ
ðkÞ ðkÞ
dij cij wj ¼ dij cji wi ; If none of ðkÞ dij
ðkÞ cij
i; j 2 N; k 2 M:
ð8Þ
is zero for "i, j 2 N, k 2 M, then Eq. (8) can be equivalently written as ðkÞ
ðkÞ
ln wi ln wj ln cij þ ln cji
¼ 0:
ð9Þ
However, in the general case, Eq. (9) does not hold. Let
ðkÞ ðkÞ ðkÞ eðkÞ ln wi ln wj ln cij þ ln cji ; ij ¼ dij ðkÞ
ð10Þ
where eij ði; j 2 N; k 2 MÞ are deviation variables, whose absolute values should be kept as small as possible. Thus, we can construct the following logarithmic least squares model (LLSM):
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ðM 1Þ min J ¼
m X n X n X
eijðkÞ
2
¼
k¼1 i¼1 j¼1
s:t
m X n X n 2 X ðkÞ ðkÞ ðkÞ dij ln wi ln wj ln cij þ ln cji
ð11Þ
k¼1 i¼1 j¼1
n X wi ¼ 1;
ð12Þ
i¼1
wi > 0; i 2 N:
ð13Þ ðkÞ
The above model (M-1) is only suitable for cij duce the following model.
ðM 2Þ min J ¼
ðkÞ
– 0 or cij
– 1. In order to avoid this problem, in the general case, we intro-
m X n X n X
2
ðkÞ ðkÞ ðkÞ rðkÞ ln wi ln wj ln cij þ ln cji ; ij dij
ð14Þ
k¼1 i¼1 j¼1
s:t
n X wi ¼ 1;
ð15Þ
i¼1
wi > 0; i 2 N;
r
where
ðkÞ ij
ð16Þ
is a zero or one integer variable defined as
(
ðkÞ ij
r ¼
ðkÞ
0; if cij ¼ 1 or 0 1; otherwise
i; j 2 N; k 2 M;
;
ð17Þ
ðkÞ ðkÞ ðkÞ ðkÞ which is introduced to avoid taking logarithm of zero. rij dij denotes that the element cij is missing cij ¼ x or equivalent ðkÞ ðkÞ ðkÞ ðkÞ to 1 or 0. rij rij ¼ 1 if and only if the element is cij known and satisfies 0 < cij < 1. In order to solve the model (14)–(16), we construct the Lagrange function m X n X n X
r
Lðw; kÞ ¼
ðkÞ ðkÞ ij dij
! n 2 X ðkÞ ðkÞ ln wi ln wj ln cij þ ln cji þ 4k wi 1 ;
k¼1 i¼1 j¼1
ð18Þ
i¼1
where k is the Lagrange multiplier. Differentiating Eq. (18) with respect to wi, i = 1, 2, . . . , n, and setting the partial derivatives equal to zero, we get the following set of equations:
n X m X
1 þ k ¼ 0; wi
rijðkÞ dijðkÞ ln wi ln wj ln cijðkÞ þ ln cjiðkÞ
j¼1 k¼1
i 2 N;
ð19Þ
which is equivalent to
n X m X
rijðkÞ dijðkÞ ln wi ln wj ln cijðkÞ þ ln cjiðkÞ þ kwi ¼ 0; i 2 N:
ð20Þ
j¼1 k¼1
Summing both sides of Eq. (20) with respect to i,i = 1, 2, . . . , n, we have
n X n X m X
n X wi ¼ 0:
ðkÞ ðkÞ ðkÞ rðkÞ ln wi ln wj ln cij þ ln cji þ k ij dij
i¼1 j¼1 k¼1
Since
Pn Pn Pm i¼1
j¼1
k¼1
n X m X
ðkÞ ðkÞ ðkÞ rðkÞ ln wi ln wj ln cij þ ln cji 0 and ij dij
ð21Þ
i¼1
Pn
i¼1 wi
¼ 1, As a result, k = 0, therefore, Eq. (20) becomes
rijðkÞ dijðkÞ ln wi ln wj ln cijðkÞ þ ln cjiðkÞ ¼ 0; i 2 N:
ð22Þ
j¼1 k¼1
That is n X m X
n X m X
n X m X
j¼1 k¼1
j¼1 k¼1
j¼1 k¼1
rijðkÞ dijðkÞ ln wi
ðkÞ rðkÞ ij dij ln wj ¼
ðkÞ ðkÞ rijðkÞ dðkÞ ln cij ln cji ; i 2 N; ij
ð23Þ
which can be equivalently expressed in the form of matrix as
Dn W n ¼ Y n ;
ð24Þ T
where W n ¼ ðln w1 ; ln w2 ; . . . ; ln wn Þ , D and Y are, respectively, the matrix and vector defined as
Y. Xu et al. / Applied Mathematical Modelling 37 (2013) 2139–2152
2X n X m
r1jðkÞ d1jðkÞ
6 6 j¼2 k¼1 6 6 X m 6 ðkÞ ðkÞ 6 r21 d21 6 Dn ¼ 6 k¼1 6 6 .. 6 . 6 6 X m 4 ðkÞ ðkÞ rn1 dn1
m X ðkÞ ðkÞ r12 d12
k¼1
3
7 7 7 7 m X 7 ðkÞ ðkÞ r2n d2n 7 7 k¼1 7; 7 7 .. 7 . 7 7 n m X X ðkÞ 5 rðkÞ d n;j n;j k¼1
n m X X
r2jðkÞ dðkÞ 2j
j¼1;j – 2 k¼1
.. .
m X ðkÞ ðkÞ rn2 dn2
k¼1
m X ðkÞ ðkÞ r1n d1n
2145
ð25Þ
j¼1;j–n k¼1
k¼1
3 ðkÞ ðkÞ r1jðkÞ dðkÞ 1j ln c 1j ln c j1 7 6 7 6 k¼1 j¼1 7 6 6X m X n 7 6 ðkÞ ðkÞ ðkÞ ðkÞ 7 6 r2j d2j ln c2j ln cj2 7 7 6 Y n ¼ 6 k¼1 j¼1 7: 7 6 7 6 . .. 7 6 7 6 6X m X n 7 4 ðkÞ ðkÞ ðkÞ ðkÞ 5 rnj dnj ln cnj ln cjn 2X m X n
ð26Þ
k¼1 j¼1
As can be seen, the sum of each row or column of Dn is zero, that is jDn j ¼ 0, so Dn is a singular matrix. Therefore, there are an infinite number of solutions to Eq. (24), but there is only one solution that satisfies the constraint Eq. (15), letting wn = 1, and solving the first (n 1) equations, we have
Dn1 W n1 ¼ Y n1 ;
ð27Þ
where
Dn1
2 X n X m r1jðkÞ dðkÞ 1j 6 6 j¼2 k¼1 6 6 m X 6 ðkÞ ðkÞ 6 r21 d21 6 ¼6 k¼1 6 6 .. 6 . 6 6 X m 4 ðkÞ ðkÞ rn1;1 dn1;1 k¼1
2
W n1
ln w1 6 ln w 2 6 ¼6 .. 6 4 .
m X ðkÞ ðkÞ r12 d12
k¼1
r2jðkÞ dðkÞ 2j
j¼1;j–2 k¼1
.. .
ðkÞ ðkÞ n1;2 dn1;2
r
ðkÞ ðkÞ r1;n1 d1;n1
k¼1
n m X X
m X
m X
m X
ðkÞ ðkÞ r2;n1 d2;n1
k¼1
.. .
n m X X
k¼1
ðkÞ ðkÞ rn1;j1 dn1;j
3 7 7 7 7 7 7 7 7; 7 7 7 7 7 5
ð28Þ
j¼1;j–n1 k¼1
3 7 7 7; 7 5
ð29Þ
ln wn1 3 m X n X r1jðkÞ d1jðkÞ ln c1jðkÞ ln cj1ðkÞ 7 6 7 6 k¼1 j¼1 7 6 7 6 m X n X 7 6 ðkÞ ðkÞ ðkÞ ðkÞ 7 6 r d2j ln c2j ln cj2 2j 7 6 ¼6 k¼1 j¼1 7: 7 6 7 6 . .. 7 6 7 6 6X m X n 7 4 ðkÞ ðkÞ ðkÞ ðkÞ rn1;j dn1;j ln cn1;j ln cj;n1 5 2
Y n1
ð30Þ
k¼1 j¼1
Because Dn1 is full rank, that is jDn1j – 0, therefore, we have
W n1 ¼ D1 n1 Y n1 :
ð31Þ
wi ¼ expðW i Þ i ¼ 1; 2; . . . ; n 1
ð32Þ
Thus,
and wn = 1. By normalizing the weighting vector, we have
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Y. Xu et al. / Applied Mathematical Modelling 37 (2013) 2139–2152
expðW i Þ wi ¼ Pn1 ; j¼1 expðW i Þ þ 1 1 wi ¼ Pn1 ; expðW iÞ þ 1 j¼1
i ¼ 1; 2; . . . ; n 1;
ð33Þ
i ¼ n:
ð34Þ
Thus, we get the following theorem: T Theorem 3. Let w ¼ w1 ; w2 ; . . . ; wn be the optimal solution to the model (14)–(16), then
expðW i Þ wi ¼ Pn1 ; j¼1 expðW i Þ þ 1 1 wi ¼ Pn1 ; expðW iÞ þ 1 j¼1
i ¼ 1; 2; . . . ; n 1;
ð35Þ
i ¼ n;
ð36Þ
where wi is determined by Eqs. (28)–(31). If we solve Eq. (23) another way, we will get the following result. T Theorem 4. Let w ¼ w1 ; w2 ; . . . ; wn be the optimal solution to the model (14)–(16), then
Qn Qm j¼1
ðkÞ
cij
k¼1
ij ij Pn P m
rðkÞ dðkÞ k¼1 ij ij
j¼1
ðkÞ cji
0
wi ¼
rðkÞ dðkÞ
1;
rðkÞ dðkÞ
Pn BQn Qm B i¼1 @ j¼1 k¼1
ij ij ðkÞ Pn P m
cij
rðkÞ dðkÞ k¼1 ij ij
j¼1
ðkÞ
cji
i 2 N:
ð37Þ
C C A
Proof. Eq. (23) can then be rewritten as
P P n m ln wi ¼
j¼1
P
k¼1
Pn Pm
ðkÞ ln wj rðkÞ ij dij
n Pm j¼1 k¼1
rijðkÞ dðkÞ ij
þ
j¼1
k¼1
ðkÞ ðkÞ ðkÞ rðkÞ ln cij ln cji ij dij
P
n Pm j¼1 k¼1
rijðkÞ dðkÞ ij
i 2 N:
;
ð38Þ
Eq. (38) can be rewritten as
0Pn Pm wi ¼ c exp @
ðkÞ ðkÞ ðkÞ ðkÞ ln cij ln cji k¼1 rij dij
j¼1
P
n j¼1
Pm
k¼1
rijðkÞ dðkÞ ij
1 A¼c
rðkÞ dðkÞ
n Y m Y j¼1 k¼1
ðkÞ cij ðkÞ cji
ij ij !P n P m j¼1
rðkÞ dðkÞ k¼1 ij ij
;
ð39Þ
0P P 1 n m rðkÞ dðkÞ ln wj ij j¼1 k¼1 ij A. where c ¼ exp @ Pn Pm j¼1
k¼1
ðkÞ rðkÞ dij ij
Summing on both sides of Eq. (39) with respect to i, we have: rðkÞ dðkÞ
ðkÞ cij ðkÞ cji
n n n Y m X Y X wi ¼ c i¼1
i¼1
c¼
0
j¼1 k¼1
ij ij !Pn P m
rðkÞ dðkÞ k¼1 ij ij
j¼1
¼ 1:
ð40Þ
Thus
1
Pn BQn Qm B i¼1 @ j¼1 k¼1
Qn Qm j¼1
wi ¼
0
k¼1
rðkÞ dðkÞ
ðkÞ cij ðkÞ cji
ij ij Pn P m j¼1
ð41Þ
C C A
rðkÞ dðkÞ
ðkÞ
cij
ij ij Pn P m
rðkÞ dðkÞ k¼1 ij ij
j¼1
ðkÞ cji
Pn BQn Qm B i¼1 @ j¼1 k¼1
rðkÞ dðkÞ k¼1 ij ij
1;
rðkÞ dðkÞ
ij ij ðkÞ Pn P m
cij
ðkÞ
cji
j¼1
rðkÞ dðkÞ k¼1 ij ij
1: C C A
ð42Þ
Y. Xu et al. / Applied Mathematical Modelling 37 (2013) 2139–2152
2147
Corollary 1. If there is no ckij ¼ 1 or 0 for any i, j 2 N and k 2 M, then the optimal solution to the model (14)–(16) is
j¼1
wi ¼
ðkÞ d
Qn Qm
ij Pn P m
ðkÞ
cij
k¼1
ðkÞ d k¼1 ij
j¼1
ðkÞ
cji
0
d
Pn BQn Qm B i¼1 @ j¼1 k¼1
ðkÞ
ij ðkÞ Pn P m
cij
ðkÞ d k¼1 ij
j¼1
ðkÞ
cji
1:
ð43Þ
C C A
Especially for a single incomplete fuzzy preference relation C = (cij)nn, then d
ij Qn cij Pn
d j¼1 ij
j¼1 cji
wi ¼ Pn
i¼1
0 1: dij Qn cij Pn dij @ j¼1 A
ð44Þ
j¼1 cji
Corollary 2. If there is no ckij ¼ x, for any i, j 2 N and k 2 M, then the optimal solution to the model (14)–(16) is
j¼1
wi ¼
rðkÞ
Qn Qm
ij Pn P m
ðkÞ
cij
k¼1
rðkÞ k¼1 ij
j¼1
ðkÞ
cji
0
rðkÞ
Pn BQn Qm B i¼1 @ j¼1 k¼1
ij ðkÞ Pn P m
cij
ðkÞ
j¼1
cji
rðkÞ k¼1 ij
1;
ð45Þ
C C A
which denotes all the fuzzy preference relations presented by multiple DMs with complete information. So our model can deal with either complete or incomplete information. ðkÞ Furthermore, if there is no cij ¼ 1 or 0 for any i, j 2 N and k 2 M, we have
Qn Qm j¼1
wi ¼
Pn
i¼1
k¼1
Qn Qm j¼1
ðkÞ
cij
1 nm
ðkÞ
cji
k¼1
ðkÞ
cij
!: 1 nm
ð46Þ
ðkÞ
cji
Especially for a single complete fuzzy preference relation, then
Qn cij 1n wi
¼
j¼1 cji
Pn
i¼1
; Qn cij 1n
ð47Þ
j¼1 cji
which is the same as Wang et al. [15] and Fedrizzi et al. [49]. From above analysis, we know that, our model can be used to deal with incomplete or complete fuzzy preference relations, single or group decision making problems, while Wang’s method only can be used to deal with the complete fuzzy preference relations. As a result, our proposed method has broader application.
4. Measurement of multiplicative consistency Consistency test is an important aspect of AHP. Since the DM may be unable to provide perfectly consistent pairwise comparisons, it is required that the pairwise comparison matrix should have an acceptable consistency. Saaty [38] proposed a consistency index of multiplicative preference matrix as:
CI ¼
kmax n ; n1
where kmax is the largest eigenvalue of multiplicative preference matrix A. Using the consistency index Eq. (48), Wang [50] further got the following formula
ð48Þ
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Y. Xu et al. / Applied Mathematical Modelling 37 (2013) 2139–2152
CI ¼
X wj 1 wi aij þ aji 2 ; nðn 1Þ 16i
ð49Þ
where w = (w1,w2, . . . , wn)T is the eigenvector of A. Saaty also proposed a consistency ratio CR = CI/RI, where RI is the mean consistency index of randomly generated multiplicative preference matrix (RI, for short) which is shown in Table 1. If CR 6 0.1, the pairwise comparison matrix is thought to have an acceptable consistency; otherwise, it needs to be revised. By Eqs. (2) and (49), we get the following formula, and call them fuzzy consistency index (FCI) and fuzzy consistency ratio (FCR) respectively, which can be used to check the consistency of a fuzzy preference matrix.
8 X 1 > < FCI ¼ nðn1Þ > :
rij
r ij wj r ji wi
16i6j6n
r
þ rjiij
wi wj
2
ð50Þ
:
FCR ¼ FCI RI
We extend the above measurement of consistency of a fuzzy preference relation to the incomplete fuzzy preference environment, and derive the following formula:
8 X 1 > < FCI ¼ nðn1Þ > :
rij dij
16i6j6n
r ij wj r ji wi
r
þ rjiij
wi wj
2
ð51Þ
;
FCR ¼ FCI RI
where dij is a zero or one integer variable of the indicator matrix D = (dij)nn, which is defined by Eq. (7). rij is defined by Eq. (17). We say that if FCR < 0.1, then the incomplete fuzzy preference relation C is of acceptable consistency; otherwise, we can return the matrix to the DM to reconsider structuring a new matrix according to his/her new judgments and follow this procedure until the matrix with satisfactory consistency is obtained. 5. Numerical examples
Example 3. For a decision-making problem, there are six decision alternatives xi (i = 1, 2, . . . , 6). The DM provides his/her preference over these six decision alternatives, and gives an incomplete fuzzy preference relation as follows (adapted from [19]):
2
0:5 0:4
x
0:3 0:8 0:3
3
6 0:6 0:5 0:6 0:5 x 0:4 7 7 6 7 6 6 x 0:4 0:5 0:3 0:6 x 7 7 C¼6 6 0:7 0:5 0:7 0:5 0:4 0:8 7: 7 6 7 6 4 0:2 x 0:4 0:6 0:5 0:7 5 0:7 0:6
x
0:2 0:3 0:5
Using Eqs. (28) and (30), we have (where n = 6):
2 Dn1
4
0
1 1
3
7 6 6 1 4 1 1 0 7 7 6 7 ¼ D5 ¼ 6 6 0 1 3 1 1 7; 7 6 4 1 1 1 5 1 5 1 0 1 1 4 2
Y n1
1
3
þ ln 0:3 þ ln 0:8 þ ln 0:3 ln 0:4 0:6 0:7 0:2 0:7
7 6 ln 0:6 þ ln 0:6 þ ln 0:5 þ ln 0:4 7 6 0:4 0:4 0:5 0:6 7 6 0:4 0:3 0:6 7: ¼ Y5 ¼ 6 ln þ ln þ ln 0:6 0:7 0:4 7 6 6 0:7 0:5 0:7 0:4 0:8 7 4 ln 0:3 þ ln 0:5 þ ln 0:3 þ ln 0:6 þ ln 0:2 5 ln 0:2 þ ln 0:4 þ ln 0:6 þ ln 0:7 0:8 0:6 0:4 0:3
Table 1 The mean consistency index of randomly generated matrices. n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
RI
0
0
0.52
0.89
1.12
1.26
1.36
1.41
1.46
1.49
1.52
1.54
1.56
1.58
1.59
Y. Xu et al. / Applied Mathematical Modelling 37 (2013) 2139–2152
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Using Eqs. (31)–(34), we have
W n1 ¼ W 5 ¼ ð0:0534; 0:2749; 0:0267; 0:6138; 0:0387ÞT ; T w ¼ w1 ; w2 ; w3 ; w4 ; w5 ; w6 ¼ ð0:1448; 0:1807; 0:1410; 0:2536; 0:1427; 0:1373ÞT : If we use Eq. (51) to compute its consistency index, we get: FCR = 0.1694 > 0.1, so C is not acceptable consistent preference relation. Then we return this matrix to the DM to reconsider structuring new matrix in the following (where c15 = 0.8 ? 0.3, c51 = 0.2 ? 0.7):
2
0:5 0:4
x
0:3 0:3 0:3
3
7 6 6 0:6 0:5 0:6 0:5 x 0:4 7 7 6 7 6 6 x 0:4 0:5 0:3 0:6 x 7 7: 6 C¼6 7 6 0:7 0:5 0:7 0:5 0:4 0:8 7 7 6 6 0:7 x 0:4 0:6 0:5 0:7 7 5 4 0:7 0:6
x
0:2 0:3 0:5
Again, we repeat the above computation, we have:
T w ¼ w1 ; w2 ; w3 ; w4 ; w5 ; w6 ¼ ð0:0878; 0:1599; 0:1551; 0:2464; 0:2208; 0:1301ÞT : Then, using Eq. (51), we have
FCR ¼ 0:0719 < 0:1: Thus, C is of acceptable consistency. Therefore the ranking of the six alternatives is
x4 x5 x2 x3 x6 x1 : Example 4. For a decision making problem, there are six decision alternatives xi (i = 1, 2, . . . , 6). The DM provides his/her preferences over these six decision alternatives, and gives an incomplete fuzzy preference relation as follows (adapted from [27]):
2
0:5
x
x
0:3 0:8 0:3
3
7 6 x x x 7 6 x 0:5 x 7 6 7 6 x 0:5 x x x 7 6 x 7: C¼6 7 6 x 0:5 0:4 0:8 7 6 0:7 x 7 6 6 0:2 x x 0:6 0:5 0:7 7 5 4 0:7
x
x
0:2 0:3 0:5
From Theorem 3, we know that C is not an acceptable incomplete fuzzy preference relation, because there is no know element in the second and third row (or column) of C except for the diagonal elements (c22 = 0.5, c33 = 0.55). Xu [27] solved a series of equations to obtain the weighting vector, but he found that there existed infinite solutions to the equations. As we can see, it is difficult to find that there exists a finite set of solutions to a series equations, and it requires careful analysis or complicated computations. Compared with Xu’s method, our method is simple. Example 5. Suppose that three DMs provide the following incomplete preference relations Ci (i = 1, 2, 3) for a set of three alternatives X = {x1, x2, x3},
2
0:5 0:5 0:6
6 C 1 ¼ 4 0:5 0:5 0:4
x
3
7 x 5;
2
0:5
6 C2 ¼ 4 x
0:5
x
0:6
7 0:5 0:6 5;
0:4 0:4 0:5
Using Eqs. (28) and (30), we have (where n = 3):
" Dn1 ¼ D2
1
2
2
4
"
# ;
Y n1 ¼ Y 2
Then, using Eqs. (31)–(34), we have
3
2 ln 0:6 0:4 2 ln 0:6 0:4
# :
2
0:5 0:5
x
3
6 7 C 3 ¼ 4 0:5 0:5 0:6 5: x
0:4 0:5
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Y. Xu et al. / Applied Mathematical Modelling 37 (2013) 2139–2152
W n1 ¼ W 2 ¼ ð0:4055; 0:4055ÞT ; w ¼ ð0:375; 0:375; 0:25Þ : Next, applying Eq. (51), we have
CR1 ¼ CR2 ¼ CR3 ¼ 0 < 0:1: Which means all the three fuzzy preference relations is perfectly multiplicative consistent and gives the ranking of x1 x2 x3. Where the symbol ‘ ’ means ‘is equal to’, ‘ ’ means ‘is preferred to’. If we use Gong’s [34] least-square method to compute, we have
2
3 0:82 0:5 0:48 6 7 Q ¼ 4 0:5 0:82 0:48 5: 0:48 0:48 1:44 Because Q is a singular matrix, Q1 does not exist, therefore, we cannot use Q1e/eTQ1e [34] to compute the priority vector proposed by Gong. From the above illustration, we demonstrate that our method is more robust. Example 6. For a multi-attribute decision making problem, there are four decision alternatives x1, x2, x3 and x4. The decisionmaker provides his/her preferences over these four decision alternatives, and gives a fuzzy preference relation as follows:
2
0:5 0:7 0:6 0:8
3
6 0:3 0:5 0:4 0:6 7 7 6 C¼6 7: 4 0:4 0:6 0:5 0:7 5 0:2 0:4 0:3 0:5 Using Eqs. (28) and (30), where n = 4, we have:
2
3
1 1
3
6 7 Dn1 ¼ D3 ¼ 4 1 3 1 5; 1 1 3
2
þ ln 0:6 þ ln 0:8 ln 0:7 0:3 0:4 0:2
3
6 7 Y n1 ¼ Y 3 ¼ 4 ln 0:3 : þ ln 0:4 þ ln 0:6 0:7 0:6 0:4 5 ln 0:4 þ ln 0:6 þ ln 0:7 0:6 0:4 0:3
Then, using Eqs. (31)–(34), we have
W n1 ¼ W 3 ¼ ð1:3195; 0:4479; 0:8716ÞT ; w ¼ ð0:4302; 0:1799; 0:2749; 0:1150Þ : Next, applying Eq. (51), we have
FCR ¼ 0:000897 < 0:1: Thus, C is of acceptable consistency, and therefore gives the ranking of x1 x2 x3 x4. Accordingly, the most desirable alternative is x1. If we use Eq. (47) to compute the weight vector, which will get the same result. Our weighting vector is approximate to Xu’s least deviation method and also results in the same ranking. Xu [51] used a convergent iterative algorithm to derive the weighting vector, the algorithm is fairly complicated. With our method, we only need to solve a series of equations, which can be solved easily using software such as Matlab or Lingo etc.
6. Discussion and conclusion In this paper, we have proposed the logarithmic least squares method (LLSM) to rank alternatives in group decision making with incomplete fuzzy preference relations and where entries are not equal to 0 or 1. In order to do this, we first illustrate an example to explain that the concept of multiplicative consistent incomplete fuzzy preference relation defined by Xu [19] is not correct, and propose a reasonable definition of multiplicative consistency for incomplete fuzzy preference relations. We also propose an approach to assess if an incomplete fuzzy relation is acceptable. The method is simple compared to Gong’s [34] method. We also develop an acceptable consistency ratio, which is simple and similar to Saaty’s consistency ratio CR for incomplete multiplicative consistent fuzzy preference relations. As shown, our LLSM method and consistency test method can not only be used to derive the weighting vector for the incomplete fuzzy preference relations, but also for the complete fuzzy preference relations. We further use some examples to illustrate that compared with the existing methods, our proposed approach has distinct characteristics. Compared with Gong’s [34] least-square method, our method can be used to deal with the following two situations: (1) all the incomplete fuzzy preference relations are perfectly multiplicative consistent, (2) at least one of the incomplete fuzzy preference relations is not perfectly multiplicative consistent, while Gong’s method can only be applied in the latter case.
Y. Xu et al. / Applied Mathematical Modelling 37 (2013) 2139–2152
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The proposed method can deal with both the complete and incomplete fuzzy preference relations, while Wang’s [15] method is only suitable to deal with the complete fuzzy preference relations. In other words, Wang’s method would be considered as a special case of our proposed method. For the complete fuzzy preference relations, Xu’s [51] least deviation method can only be used to derive the priority vector for a fuzzy preference relation, and can not be used in the context group decision making. The iterative algorithm of the least deviation method is fairly complicated, as compared to our proposed method which can handle not only one single preference relation, but a set of some fuzzy preference relations, Finally, our method is simple, efficient, and can be solved easily using software such as Matlab or Lingo, etc. If the incomplete fuzzy preference relation does not have acceptable consistency, it needs to be returned to the DM to reconsider and structure a new matrix. As shown in Example 3, by only changing c15 and c51, we were able to recalculate the weighting vector and the consistency radio. Once the incomplete fuzzy preference relation has acceptable consistency we are able to rank the alternatives. In future research we will consider Unusual and False Observations (we call them UFO). In addition in this paper, we only consider the decision makers having equal importance, in the future research, we will pay attention to the GDM problems where DMs are heterogeneous and there are explicit importance degrees associated with the DMs. Acknowledgements We would like to thank the anonymous reviewers for valuable comments that have improved the quality of the paper. This work was partly supported by the National Natural Science Foundation of China (NSFC) under Grants 71101043, 50979024 and 90924027, College Philosophy and Social Science Research Project of Jiangsu Province under Grant 2011SJD630007, the International Academic Cooperation Team Development Foundation of Business School, Hohai University under Grant UAH-2011-181. References [1] T.Y. Chen, Multiple criteria group decision-making with generalized interval-valued fuzzy numbers based on signed distances and incomplete weights, Appl. Math. Model. 36 (2012) 3029–3052. [2] F. Chiclana, F. Herrera, E. Herrera-Viedma, Integrating multiplicative preference relations in a multipurpose decision-making model based on fuzzy preference relations, Fuzzy Sets Syst. 122 (2001) 277–291. [3] F. Chiclana, F. Herrera, E. Herrera-Viedma, Integrating three representation models in fuzzy multipurpose decision making based on fuzzy preference relations, Fuzzy Sets Syst. 97 (1998) 33–48. [4] F. Chiclana, F. 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Contents lists available at SciVerse ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Logarithmic least squares method to priority for group decision making with incomplete fuzzy preference relations Yejun Xu a,b,⇑, Ravi Patnayakuni c, Huimin Wang a,b a
State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, PR China Business School, Hohai University, Jiangning, Nanjing, Jiangsu 211100, PR China c Department of Economics and IS, University of Alabama in Huntsville, Huntsville, AL 35763, USA b
a r t i c l e
i n f o
Article history: Received 10 November 2011 Received in revised form 28 April 2012 Accepted 8 May 2012 Available online 22 May 2012 Keywords: Group decision making Incomplete fuzzy preference relation Logarithmic least squares method Multiplicative consistency
a b s t r a c t The aim of this paper is to present a logarithmic least squares method (LLSM) to priority for group decision making with incomplete fuzzy preference relations. We give a reasonable definition of multiplicative consistent for incomplete fuzzy preference relation. We develop the acceptable fuzzy consistency ratio (FCR for short), which is simple and similar to Saaty’s consistency ratio CR for multiplicative fuzzy preference relations. We also extend the LLSM method to the case of individual preference relation with complete information. Finally, some examples are illustrated to show that our method is simple, efficient, and can be performed on computer easily. Crown Copyright 2012 Published by Elsevier Inc. All rights reserved.
1. Introduction Group decision making (GDM) [1–21] is participatory process in which multiple individuals, often experts, together formulate problems, develop alternatives and eventually select among the alternatives to reach a decision. Each expert may have unique motivations or goals and may approach the decision process from a different perspective, but share a common interest in reaching eventual agreement on selecting the best solution (s) to the problem. Fuzzy preference relations [2–4,22,10–12,14–16,23,24] are commonly used to represent decision makers’ preferences over the set of possible alternative solutions, and have received considerable research attention in the past decades. The research primarily focuses on examining fuzzy preference relations with complete information. A complete fuzzy preference relation of order n necessitates the completion of all n(n 1)/2 judgments in its entire top triangular portion. However, sometimes, a decision maker (DM) may develop fuzzy preference relations with incomplete information. Firstly, because of time pressure, lack of knowledge, and/or the DM’s limited expertise related with problem domain [25,26,19,27,20], the DM may not be able to provide a complete set of preference relations. Secondly, when the number of alternatives, n, is large. In such cases it may be practically impossible, or at least unacceptable from the view point of the decision maker, to perform all the n(n 1)/2 required comparisons to complete the pairwise comparison matrices [28]; A third reason should be that it may be convenient or even necessary to skip some direct critical comparisons between alternatives, even if the total number of alternatives is small [28]. Finally, an expert may not be able to efficiently express any kind of preference degree between two or more of the available alternatives. This may be due to an expert not possessing a precise or sufficient level of information for part of the problem, or because that expert is unable to discriminate the degree to which some alternatives are better than others ⇑ Corresponding author at: State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, PR China. Tel.: +86 25 68514612; fax: +86 25 85427972. E-mail address: [email protected] (Y. Xu). 0307-904X/$ - see front matter Crown Copyright 2012 Published by Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2012.05.010
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[29,30,5,6]. There have been seen some theoretical studies with incomplete preference relations. Alonso et al. [30] 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. [6] proposed a new model to deal with GDM problems with the incomplete fuzzy preference relations based on the additive-consistency (AC) property. The new model consists of two steps: estimation of missing preference values and the selection of alternatives. In another paper, they [5] 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 a moderator, using both consensus and consistency criteria. More recently, Alonso et al. [29] put forward a general procedure that attempts to estimate the missing information of incomplete preference relations in any of the above formats: fuzzy, multiplicative, interval valued and linguistic. Xu [19] 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 programing models, based on additive consistent incomplete fuzzy preference relation and multiplicative consistent incomplete fuzzy preference relation respectively, for obtaining the priority vector for incomplete fuzzy preference relation. Later in our analysis (Section 2), we will argue that the concept of multiplicative consistent incomplete fuzzy preference relation proposed in this paper have limitations. In another paper, Xu [27] proposed a procedure for decision making with incomplete fuzzy preference relation based on multiplicative consistency. Xu [20] also developed a simple but practical approach to deriving the ranking of the alternatives from an incomplete reciprocal relation based on additive transitivity. He postulated a correspondence between priority vector and additive consistent incomplete fuzzy preference relation. Shen et al. [31], and Xu [32,33,17] had pointed out that the correspondence may be unreasonable, and deduced a function between the fuzzy preference relation and priority vector. Gong [34] developed a least-square model to obtain the collective priority vector from the incomplete preference relations of multiple decision makers derived with a simple equation. As will be evident in the examples presented in this paper, in some situations, Gong’s simple equation cannot be used to obtain the priority vector because the matrix Q (See Example 5) is a singular matrix, where Q1 does not exist. Xu et al. [35] proposed a weighted least square method and quadratic programing method for developing priorities from incomplete fuzzy preference relation based on multiplicative transitivity. In another paper they [32] presented two priority methods-the least variance priority method (LVM) and quadratic programing method for incomplete fuzzy preference relation based on additive consistency. Liu et al. [9] explored the group decision making problems with incomplete additively consistent fuzzy preference relations. They developed least square completion and inconsistency repair methods to deal with incomplete and inconsistent fuzzy preference relations. The consistency in preference relations (whether multiplicative or fuzzy) [25,36,37] given by DMs has a direct impact on the ranking results of the final decision. The issue of consistency in Analytic Hierarchy Process (AHP) was first addressed by Saaty [38], who originally developed the notions of perfect consistency and acceptable consistency, and the latter was checked by a consistency index CI and consistency ratio CR. Saaty pointed out that a multiplicative preference matrix is of acceptable consistency if CR < 0.1. The method is simple and has been widely used. Studies on the consistency of fuzzy preference relations have also attracted much attention recently. Most of the research is associated with transitivity [39,40,10], such as weak transitivity [41], max–min transitivity, max–max transitivity, restricted max–min transitivity, restricted max–max transitivity, multiplicative consistency and additive consistency. For incomplete fuzzy preference relations, the consistency measure and consistency level have been studied for the additive consistency preference relation [5,6,26]. To our knowledge, there have no studies on the consistency index for incomplete multiplicative fuzzy preference relations. Therefore, we direct our attention to this issue. The aim of this paper is to present a logarithmic least squares method (LLSM) to prioritize among a set of alternatives in the group decision making context when DMs have incomplete fuzzy preference relations. Although LLSM is not new and is well known in AHP [42–46,15], it has not been applied to modeling incomplete fuzzy preference relations in group decision making problems based on multiplicative consistency. In order to demonstrate the method, this paper is structured in the following way. The next section presents the basic concepts of a multiplicative preference relation, fuzzy preference relation and incomplete fuzzy preference relation. We also illustrate with an example to explain that multiplicative consistent incomplete fuzzy preference relation as defined by Xu is not correct. We then offer a reasonable definition of multiplicative consistent for an incomplete fuzzy preference relation. We also give an approach to judge whether an incomplete fuzzy relation is acceptable or not acceptable. Section 3, the LLSM is extended to model incomplete fuzzy preference relations based on multiplicative consistency. The corresponding acceptable fuzzy consistency ratio (FCR for short) is developed, which is simple and similar to Saaty’s consistency ratio CR for multiplicative consistent fuzzy preference relations in Section 4. Section 5, some examples are presented to illustrate that the proposed method is simple, efficient, and can be performed easily on computer. Finally, in Section 6, we draw our conclusions and point out future research.
2. Preliminaries This section describes a multiplicative preference relation and fuzzy preference relation on alternatives, and also some related concepts such as incomplete fuzzy preference relation, multiplicative consistency of incomplete fuzzy preference relation, acceptable incomplete fuzzy preference relation etc., which will be used in the subsequent sections.
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For simplicity, we denote N = {1, 2, . . . , n}. Let X = {x1, x2, . . . , xn} (n P 2) be a finite set of alternatives, where xi denotes the ith alternative. In multi-attribute decision making problems, a decision maker needs to rank the alternatives x1, x2, . . . , xn from the best to the worst according to the preference information. Next, a brief description of a multiplicative preference relation and fuzzy preference relation is provided below. 2.1. Multiplicative preference relation A multiplicative preference relation is a positive preference relation, A X X, A = (aij)nn, where aij denotes the relative weight of alternative xi with respect to xj. The measurement of aij is described using a ratio scale and in particular, as shown by Saaty [38], aij 2 {1/9.1/8, 1/7, . . . , 1, 2, . . . , 9}:aij = 1 denotes the indifference between xi and xj, aij = 9 (or aji = 1/9) denotes that xi is definitely preferred to xj, and aij 2 {2,3, . . . , 8} denotes the intermediate evaluations. It is multiplicative reciprocal, i.e., aijaji = 1, "i, j 2 {2, . . . , n} and in particular, aij = 1 "i 2 {1, 2, . . . , n}. Thus we have the following definition. Let A = (aij)nn be a multiplicative preference relation, then A is called a consistent multiplicative preference relation (or called consistent reciprocal judgment matrix), if aij = aikakj, for all i, j, k. 2.2. Fuzzy preference relation The fuzzy preference relation R is described as follows: R X X, R = (rij)nn, with membership function uR:X X ? [0, 1], where uR(xi, xj) = rij denotes the preference degree of the alternative xi over xj [3,7,10,12]: rij = 0.5 denotes indifference between xi and xj, rij = 1, denotes that xi is definitely preferred to xj, and 0.5 < rij < 1 (or 0 < rji < 0.5) denotes that xi is preferred to xj. Let R = (rij)nn be a preference relation, then R is called a reciprocal fuzzy preference relation if
rij 2 ½0; 1;
rij þ rji ¼ 1;
r ii ¼ 0:5;
for all i; j 2 N:
Tanino [10] proposed the definition of multiplicative consistency on fuzzy preference relations, i.e. R = (rij)nn is perfectly consistent if
ril r lj r ji ¼ rli r jl r ij ;
i; j; l 2 N:
It has been found that a perfectly consistent multiplicative preference relation and a perfectly consistent fuzzy preference P relation can be precisely characterized by a priority vector w = (w1,w2, . . . , wn)T, which satisfies the conditions ni¼1 wi ¼ 1 and wi > 0 for i = 1, 2, . . . , n. That is [16]
wi ; i; j 2 N; wj wi ; i; j 2 N: rij ¼ wi þ wj
aij ¼
ð1Þ ð2Þ
Next, we will cover some related concepts where the preference information is incomplete. Definition 1 [19]. Let C = (cij)nn be a fuzzy preference relation, then C is called an incomplete fuzzy preference relation, if some of its elements cannot be provided by the DM, which we denote the unknown number x, and the others can be provided by the DM, which satisfy cij 2 [0, 1], cij + cji = 1, cii = 0.5. Definition 2 19. Let C = (cij)nn be an incomplete fuzzy preference relation, then C is called a multiplicative consistent incomplete fuzzy preference relation, if all the known elements satisfy multiplicative transitivity i.e. cilcljcji = clicjlcij. Next, we offer proof that this definition is not correct. Proof. Since
cil clj cji ¼ cli cjl cij ;
ð3Þ
that is
cij cil clj ¼ : cji cli cjl
ð4Þ c
c
Let bij ¼ cijji ; bil ¼ ccil ; blj ¼ clj , then Eq. (4) is equivalent to the following equation: li
bij ¼ bil blj :
jl
ð5Þ
We label this as the corresponding incomplete multiplicative preference relation. So Definition 2 is equivalent to the following Definition 3. h Definition 3. Let B = (bij)nn be an incomplete multiplicative preference relation, then B is called a multiplicative consistent incomplete multiplicative preference relation, if all the known elements satisfy the multiplicative transitivity bij = bilbij.
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Herrera-Viedma et al. [47] further gave the following proposition. Proposition 1. For a reciprocal multiplicative preference relation A = (aij)nn, the following statements are equivalent: (i) aij ajk = aik, "i, j, k. (ii) aij = aii+1 aii+2 . . . aj1j, "i, j. Let B = (bij)nn be an incomplete multiplicative preference relation, and let G be its corresponding directed graph, which implies the relationship in matrix B. There are n vertices 1, 2, . . ., n in G, Moreover, if bij – x, there exists a directed arc ! in the directed graph G i;j going from i to j, and its weight is bij. If bij = x, there is no directed arc between i and j, which denotes that there is no comparison information between alternative xi and xj. We will see that Definition 3 is not correct using the following example. Example 1. For an incomplete multiplicative preference relation B which is in the following:
2
1 3 6
61 63 61 B¼6 66 6x 4 1 4
x
4
3
x7 7 7 1 x7 7; 1 2 1 27 5 x x 12 1 1 2 1 1 2
1 2
Fig. 1 is the directed graph of matrix B. We can test and verify, for all i, j, l, bijbjlbli = 1 (where bij – x, for all i, j), which can also be tested and verified in Fig. 1. For all the circuit whose length is 3, the weight is 1. For example, the circuit (1, 2, 3, 1), its weight is b12 b23 b31 ¼ 3 2 16 ¼ 1, but for the circuit (1, 3, 4, 5, 1), its weight is b13 b34 b45 b51 ¼ 6 12 2 14 – 1, so matrix B is not consistent. If we change a15 to 6, then for all the circuit, its weight is 1. Thus, the following result characterizes incomplete reciprocal multiplicative consistency. Theorem 1. Let B = (bij)nn be an incomplete multiplicative preference relation, then B is called a multiplicative consistent incomplete preference relation, if all the known elements satisfy the multiplicative transitivity bi1 i2 bi2 i3 . . . bik1 ik bik i1 ¼ 1, "i, j, bij – x. Example 2. The decision maker gives the following incomplete fuzzy preference relation.
3 3 6 4 x 0:5 4 7 5 7 6 1 2 1 0:5 x 7 6 4 3 2 7 6 1 1 1 C¼6 0:5 x 7 3 3 7: 6 7 6 1 2 2 7 0:5 4 x 2 3 3 5 1 1 x x 0:5 5 3 2
If we use Xu’s Definition 2 to verify, C as a multiplicative consistent incomplete fuzzy preference relation, we will find that for the same reasons, it is not correct. So Definition 2 should be corrected as follows:
Fig. 1. The directed graph of matrix B.
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Definition 4. Let C = (cij)nn be an incomplete fuzzy preference relation, then C is called a multiplicative consistent incomplete fuzzy preference relation, if all the known elements satisfy the multiplicative transitivity ci1 i2 ci2 i3 . . . cik ij ¼ ci2 i1 ci3 i2 . . . cij ik , "i, j, cij – x. Next, we offer a simple method to judge whether an incomplete fuzzy preference is acceptable or not. Definition 5 [35,48]. The elements cij, ckl, of are called adjacent, if {i, j} \ {k, l} – £. For the unknown element cij, cij can be determined indirectly, if there exist a series of known elements cij1 ; cj1 j2 ; . . . ; cjk j . Definition 6 [35,48]. Let C = (cij)nn be an incomplete fuzzy preference relation, if the missing elements of C can be determined by the known elements, then C is called an acceptable incomplete fuzzy preference relation, otherwise, C is not an acceptable incomplete fuzzy preference relation. Theorem 2. Let C = (cij)nn be an incomplete fuzzy preference relation, the necessary condition of acceptable incomplete fuzzy preference relation C is that there exists at least one known element in each row or column of C except for the diagonal elements (cii,i = 1, 2, . . . , n), i.e. there needs at least (n 1) judgments. Proof. Let ckl(k – l) be a missing element of C = (cij)nn, where C is an acceptable incomplete fuzzy preference relation. From Definitions 5 and 6, we know that there must be a series of known elements ckl1 ; cl1 l2 ; . . . ; cl;l , in other words, there is at least one given element ckl1 in the kth row, for k – l1, and at the same time, there is at least one given element cls l in the lth column, for ls – l. So the necessary condition holds for "k, l, and thus, there need to be at least (n 1) judgments in the upper (lower) triangular of the incomplete preference relation C. Herrera-Viedma et al. [6] proved the above result by induction on the number of alternatives. At the same time, their proof is based on the additive reciprocity property of a fuzzy preference relation. Our method is suitable not only for additive reciprocity property but for the multiplicative property of a fuzzy preference relation. Gong [34] also gave an approach to judge whether an incomplete fuzzy preference relation is acceptable or not, as we can see, his method is so complicated and cannot judge easily. h
3. Logarithmic least squares method for deriving priorities from group incomplete fuzzy preference relations For simplicity, we denote M = {1, 2, . . . , m}. Let X = {x1, x2, . . . , xn} (n P 2) be a finite set of alternatives and D = {d1, d2, . . . , dn} (m P 2) be a finite set of decision makers. Suppose that these m decision makers provide their preferences over the set X, and ðkÞ ðk 2 MÞ. Let w = (w1,w2, . . . , wn)T be the priority vector of the give m incomplete fuzzy preference relations C ðkÞ ¼ cij nn ðkÞ group’s incomplete multiplicative consistent fuzzy preference relations C ðkÞ ¼ cij , then these preference relations are nn
given by [19] ðkÞ ðkÞ
ðkÞ
dij cij ¼ dij ðkÞ
where dij
wi ; wi þ wj
i; j 2 N;
ð6Þ
ðkÞ is a zero or one integer variable of the indicator matrix D ¼ dij
ðkÞ
dij ¼
nn
8 < 0; if cðkÞ ¼ x ij
: 1; if cðkÞ – x ij
, which is defined as
i; j 2 N; k 2 M:
ð7Þ
From Eq. (6), we have ðkÞ ðkÞ
ðkÞ ðkÞ
dij cij wj ¼ dij cji wi ; If none of ðkÞ dij
ðkÞ cij
i; j 2 N; k 2 M:
ð8Þ
is zero for "i, j 2 N, k 2 M, then Eq. (8) can be equivalently written as ðkÞ
ðkÞ
ln wi ln wj ln cij þ ln cji
¼ 0:
ð9Þ
However, in the general case, Eq. (9) does not hold. Let
ðkÞ ðkÞ ðkÞ eðkÞ ln wi ln wj ln cij þ ln cji ; ij ¼ dij ðkÞ
ð10Þ
where eij ði; j 2 N; k 2 MÞ are deviation variables, whose absolute values should be kept as small as possible. Thus, we can construct the following logarithmic least squares model (LLSM):
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ðM 1Þ min J ¼
m X n X n X
eijðkÞ
2
¼
k¼1 i¼1 j¼1
s:t
m X n X n 2 X ðkÞ ðkÞ ðkÞ dij ln wi ln wj ln cij þ ln cji
ð11Þ
k¼1 i¼1 j¼1
n X wi ¼ 1;
ð12Þ
i¼1
wi > 0; i 2 N:
ð13Þ ðkÞ
The above model (M-1) is only suitable for cij duce the following model.
ðM 2Þ min J ¼
ðkÞ
– 0 or cij
– 1. In order to avoid this problem, in the general case, we intro-
m X n X n X
2
ðkÞ ðkÞ ðkÞ rðkÞ ln wi ln wj ln cij þ ln cji ; ij dij
ð14Þ
k¼1 i¼1 j¼1
s:t
n X wi ¼ 1;
ð15Þ
i¼1
wi > 0; i 2 N;
r
where
ðkÞ ij
ð16Þ
is a zero or one integer variable defined as
(
ðkÞ ij
r ¼
ðkÞ
0; if cij ¼ 1 or 0 1; otherwise
i; j 2 N; k 2 M;
;
ð17Þ
ðkÞ ðkÞ ðkÞ ðkÞ which is introduced to avoid taking logarithm of zero. rij dij denotes that the element cij is missing cij ¼ x or equivalent ðkÞ ðkÞ ðkÞ ðkÞ to 1 or 0. rij rij ¼ 1 if and only if the element is cij known and satisfies 0 < cij < 1. In order to solve the model (14)–(16), we construct the Lagrange function m X n X n X
r
Lðw; kÞ ¼
ðkÞ ðkÞ ij dij
! n 2 X ðkÞ ðkÞ ln wi ln wj ln cij þ ln cji þ 4k wi 1 ;
k¼1 i¼1 j¼1
ð18Þ
i¼1
where k is the Lagrange multiplier. Differentiating Eq. (18) with respect to wi, i = 1, 2, . . . , n, and setting the partial derivatives equal to zero, we get the following set of equations:
n X m X
1 þ k ¼ 0; wi
rijðkÞ dijðkÞ ln wi ln wj ln cijðkÞ þ ln cjiðkÞ
j¼1 k¼1
i 2 N;
ð19Þ
which is equivalent to
n X m X
rijðkÞ dijðkÞ ln wi ln wj ln cijðkÞ þ ln cjiðkÞ þ kwi ¼ 0; i 2 N:
ð20Þ
j¼1 k¼1
Summing both sides of Eq. (20) with respect to i,i = 1, 2, . . . , n, we have
n X n X m X
n X wi ¼ 0:
ðkÞ ðkÞ ðkÞ rðkÞ ln wi ln wj ln cij þ ln cji þ k ij dij
i¼1 j¼1 k¼1
Since
Pn Pn Pm i¼1
j¼1
k¼1
n X m X
ðkÞ ðkÞ ðkÞ rðkÞ ln wi ln wj ln cij þ ln cji 0 and ij dij
ð21Þ
i¼1
Pn
i¼1 wi
¼ 1, As a result, k = 0, therefore, Eq. (20) becomes
rijðkÞ dijðkÞ ln wi ln wj ln cijðkÞ þ ln cjiðkÞ ¼ 0; i 2 N:
ð22Þ
j¼1 k¼1
That is n X m X
n X m X
n X m X
j¼1 k¼1
j¼1 k¼1
j¼1 k¼1
rijðkÞ dijðkÞ ln wi
ðkÞ rðkÞ ij dij ln wj ¼
ðkÞ ðkÞ rijðkÞ dðkÞ ln cij ln cji ; i 2 N; ij
ð23Þ
which can be equivalently expressed in the form of matrix as
Dn W n ¼ Y n ;
ð24Þ T
where W n ¼ ðln w1 ; ln w2 ; . . . ; ln wn Þ , D and Y are, respectively, the matrix and vector defined as
Y. Xu et al. / Applied Mathematical Modelling 37 (2013) 2139–2152
2X n X m
r1jðkÞ d1jðkÞ
6 6 j¼2 k¼1 6 6 X m 6 ðkÞ ðkÞ 6 r21 d21 6 Dn ¼ 6 k¼1 6 6 .. 6 . 6 6 X m 4 ðkÞ ðkÞ rn1 dn1
m X ðkÞ ðkÞ r12 d12
k¼1
3
7 7 7 7 m X 7 ðkÞ ðkÞ r2n d2n 7 7 k¼1 7; 7 7 .. 7 . 7 7 n m X X ðkÞ 5 rðkÞ d n;j n;j k¼1
n m X X
r2jðkÞ dðkÞ 2j
j¼1;j – 2 k¼1
.. .
m X ðkÞ ðkÞ rn2 dn2
k¼1
m X ðkÞ ðkÞ r1n d1n
2145
ð25Þ
j¼1;j–n k¼1
k¼1
3 ðkÞ ðkÞ r1jðkÞ dðkÞ 1j ln c 1j ln c j1 7 6 7 6 k¼1 j¼1 7 6 6X m X n 7 6 ðkÞ ðkÞ ðkÞ ðkÞ 7 6 r2j d2j ln c2j ln cj2 7 7 6 Y n ¼ 6 k¼1 j¼1 7: 7 6 7 6 . .. 7 6 7 6 6X m X n 7 4 ðkÞ ðkÞ ðkÞ ðkÞ 5 rnj dnj ln cnj ln cjn 2X m X n
ð26Þ
k¼1 j¼1
As can be seen, the sum of each row or column of Dn is zero, that is jDn j ¼ 0, so Dn is a singular matrix. Therefore, there are an infinite number of solutions to Eq. (24), but there is only one solution that satisfies the constraint Eq. (15), letting wn = 1, and solving the first (n 1) equations, we have
Dn1 W n1 ¼ Y n1 ;
ð27Þ
where
Dn1
2 X n X m r1jðkÞ dðkÞ 1j 6 6 j¼2 k¼1 6 6 m X 6 ðkÞ ðkÞ 6 r21 d21 6 ¼6 k¼1 6 6 .. 6 . 6 6 X m 4 ðkÞ ðkÞ rn1;1 dn1;1 k¼1
2
W n1
ln w1 6 ln w 2 6 ¼6 .. 6 4 .
m X ðkÞ ðkÞ r12 d12
k¼1
r2jðkÞ dðkÞ 2j
j¼1;j–2 k¼1
.. .
ðkÞ ðkÞ n1;2 dn1;2
r
ðkÞ ðkÞ r1;n1 d1;n1
k¼1
n m X X
m X
m X
m X
ðkÞ ðkÞ r2;n1 d2;n1
k¼1
.. .
n m X X
k¼1
ðkÞ ðkÞ rn1;j1 dn1;j
3 7 7 7 7 7 7 7 7; 7 7 7 7 7 5
ð28Þ
j¼1;j–n1 k¼1
3 7 7 7; 7 5
ð29Þ
ln wn1 3 m X n X r1jðkÞ d1jðkÞ ln c1jðkÞ ln cj1ðkÞ 7 6 7 6 k¼1 j¼1 7 6 7 6 m X n X 7 6 ðkÞ ðkÞ ðkÞ ðkÞ 7 6 r d2j ln c2j ln cj2 2j 7 6 ¼6 k¼1 j¼1 7: 7 6 7 6 . .. 7 6 7 6 6X m X n 7 4 ðkÞ ðkÞ ðkÞ ðkÞ rn1;j dn1;j ln cn1;j ln cj;n1 5 2
Y n1
ð30Þ
k¼1 j¼1
Because Dn1 is full rank, that is jDn1j – 0, therefore, we have
W n1 ¼ D1 n1 Y n1 :
ð31Þ
wi ¼ expðW i Þ i ¼ 1; 2; . . . ; n 1
ð32Þ
Thus,
and wn = 1. By normalizing the weighting vector, we have
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expðW i Þ wi ¼ Pn1 ; j¼1 expðW i Þ þ 1 1 wi ¼ Pn1 ; expðW iÞ þ 1 j¼1
i ¼ 1; 2; . . . ; n 1;
ð33Þ
i ¼ n:
ð34Þ
Thus, we get the following theorem: T Theorem 3. Let w ¼ w1 ; w2 ; . . . ; wn be the optimal solution to the model (14)–(16), then
expðW i Þ wi ¼ Pn1 ; j¼1 expðW i Þ þ 1 1 wi ¼ Pn1 ; expðW iÞ þ 1 j¼1
i ¼ 1; 2; . . . ; n 1;
ð35Þ
i ¼ n;
ð36Þ
where wi is determined by Eqs. (28)–(31). If we solve Eq. (23) another way, we will get the following result. T Theorem 4. Let w ¼ w1 ; w2 ; . . . ; wn be the optimal solution to the model (14)–(16), then
Qn Qm j¼1
ðkÞ
cij
k¼1
ij ij Pn P m
rðkÞ dðkÞ k¼1 ij ij
j¼1
ðkÞ cji
0
wi ¼
rðkÞ dðkÞ
1;
rðkÞ dðkÞ
Pn BQn Qm B i¼1 @ j¼1 k¼1
ij ij ðkÞ Pn P m
cij
rðkÞ dðkÞ k¼1 ij ij
j¼1
ðkÞ
cji
i 2 N:
ð37Þ
C C A
Proof. Eq. (23) can then be rewritten as
P P n m ln wi ¼
j¼1
P
k¼1
Pn Pm
ðkÞ ln wj rðkÞ ij dij
n Pm j¼1 k¼1
rijðkÞ dðkÞ ij
þ
j¼1
k¼1
ðkÞ ðkÞ ðkÞ rðkÞ ln cij ln cji ij dij
P
n Pm j¼1 k¼1
rijðkÞ dðkÞ ij
i 2 N:
;
ð38Þ
Eq. (38) can be rewritten as
0Pn Pm wi ¼ c exp @
ðkÞ ðkÞ ðkÞ ðkÞ ln cij ln cji k¼1 rij dij
j¼1
P
n j¼1
Pm
k¼1
rijðkÞ dðkÞ ij
1 A¼c
rðkÞ dðkÞ
n Y m Y j¼1 k¼1
ðkÞ cij ðkÞ cji
ij ij !P n P m j¼1
rðkÞ dðkÞ k¼1 ij ij
;
ð39Þ
0P P 1 n m rðkÞ dðkÞ ln wj ij j¼1 k¼1 ij A. where c ¼ exp @ Pn Pm j¼1
k¼1
ðkÞ rðkÞ dij ij
Summing on both sides of Eq. (39) with respect to i, we have: rðkÞ dðkÞ
ðkÞ cij ðkÞ cji
n n n Y m X Y X wi ¼ c i¼1
i¼1
c¼
0
j¼1 k¼1
ij ij !Pn P m
rðkÞ dðkÞ k¼1 ij ij
j¼1
¼ 1:
ð40Þ
Thus
1
Pn BQn Qm B i¼1 @ j¼1 k¼1
Qn Qm j¼1
wi ¼
0
k¼1
rðkÞ dðkÞ
ðkÞ cij ðkÞ cji
ij ij Pn P m j¼1
ð41Þ
C C A
rðkÞ dðkÞ
ðkÞ
cij
ij ij Pn P m
rðkÞ dðkÞ k¼1 ij ij
j¼1
ðkÞ cji
Pn BQn Qm B i¼1 @ j¼1 k¼1
rðkÞ dðkÞ k¼1 ij ij
1;
rðkÞ dðkÞ
ij ij ðkÞ Pn P m
cij
ðkÞ
cji
j¼1
rðkÞ dðkÞ k¼1 ij ij
1: C C A
ð42Þ
Y. Xu et al. / Applied Mathematical Modelling 37 (2013) 2139–2152
2147
Corollary 1. If there is no ckij ¼ 1 or 0 for any i, j 2 N and k 2 M, then the optimal solution to the model (14)–(16) is
j¼1
wi ¼
ðkÞ d
Qn Qm
ij Pn P m
ðkÞ
cij
k¼1
ðkÞ d k¼1 ij
j¼1
ðkÞ
cji
0
d
Pn BQn Qm B i¼1 @ j¼1 k¼1
ðkÞ
ij ðkÞ Pn P m
cij
ðkÞ d k¼1 ij
j¼1
ðkÞ
cji
1:
ð43Þ
C C A
Especially for a single incomplete fuzzy preference relation C = (cij)nn, then d
ij Qn cij Pn
d j¼1 ij
j¼1 cji
wi ¼ Pn
i¼1
0 1: dij Qn cij Pn dij @ j¼1 A
ð44Þ
j¼1 cji
Corollary 2. If there is no ckij ¼ x, for any i, j 2 N and k 2 M, then the optimal solution to the model (14)–(16) is
j¼1
wi ¼
rðkÞ
Qn Qm
ij Pn P m
ðkÞ
cij
k¼1
rðkÞ k¼1 ij
j¼1
ðkÞ
cji
0
rðkÞ
Pn BQn Qm B i¼1 @ j¼1 k¼1
ij ðkÞ Pn P m
cij
ðkÞ
j¼1
cji
rðkÞ k¼1 ij
1;
ð45Þ
C C A
which denotes all the fuzzy preference relations presented by multiple DMs with complete information. So our model can deal with either complete or incomplete information. ðkÞ Furthermore, if there is no cij ¼ 1 or 0 for any i, j 2 N and k 2 M, we have
Qn Qm j¼1
wi ¼
Pn
i¼1
k¼1
Qn Qm j¼1
ðkÞ
cij
1 nm
ðkÞ
cji
k¼1
ðkÞ
cij
!: 1 nm
ð46Þ
ðkÞ
cji
Especially for a single complete fuzzy preference relation, then
Qn cij 1n wi
¼
j¼1 cji
Pn
i¼1
; Qn cij 1n
ð47Þ
j¼1 cji
which is the same as Wang et al. [15] and Fedrizzi et al. [49]. From above analysis, we know that, our model can be used to deal with incomplete or complete fuzzy preference relations, single or group decision making problems, while Wang’s method only can be used to deal with the complete fuzzy preference relations. As a result, our proposed method has broader application.
4. Measurement of multiplicative consistency Consistency test is an important aspect of AHP. Since the DM may be unable to provide perfectly consistent pairwise comparisons, it is required that the pairwise comparison matrix should have an acceptable consistency. Saaty [38] proposed a consistency index of multiplicative preference matrix as:
CI ¼
kmax n ; n1
where kmax is the largest eigenvalue of multiplicative preference matrix A. Using the consistency index Eq. (48), Wang [50] further got the following formula
ð48Þ
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Y. Xu et al. / Applied Mathematical Modelling 37 (2013) 2139–2152
CI ¼
X wj 1 wi aij þ aji 2 ; nðn 1Þ 16i
ð49Þ
where w = (w1,w2, . . . , wn)T is the eigenvector of A. Saaty also proposed a consistency ratio CR = CI/RI, where RI is the mean consistency index of randomly generated multiplicative preference matrix (RI, for short) which is shown in Table 1. If CR 6 0.1, the pairwise comparison matrix is thought to have an acceptable consistency; otherwise, it needs to be revised. By Eqs. (2) and (49), we get the following formula, and call them fuzzy consistency index (FCI) and fuzzy consistency ratio (FCR) respectively, which can be used to check the consistency of a fuzzy preference matrix.
8 X 1 > < FCI ¼ nðn1Þ > :
rij
r ij wj r ji wi
16i6j6n
r
þ rjiij
wi wj
2
ð50Þ
:
FCR ¼ FCI RI
We extend the above measurement of consistency of a fuzzy preference relation to the incomplete fuzzy preference environment, and derive the following formula:
8 X 1 > < FCI ¼ nðn1Þ > :
rij dij
16i6j6n
r ij wj r ji wi
r
þ rjiij
wi wj
2
ð51Þ
;
FCR ¼ FCI RI
where dij is a zero or one integer variable of the indicator matrix D = (dij)nn, which is defined by Eq. (7). rij is defined by Eq. (17). We say that if FCR < 0.1, then the incomplete fuzzy preference relation C is of acceptable consistency; otherwise, we can return the matrix to the DM to reconsider structuring a new matrix according to his/her new judgments and follow this procedure until the matrix with satisfactory consistency is obtained. 5. Numerical examples
Example 3. For a decision-making problem, there are six decision alternatives xi (i = 1, 2, . . . , 6). The DM provides his/her preference over these six decision alternatives, and gives an incomplete fuzzy preference relation as follows (adapted from [19]):
2
0:5 0:4
x
0:3 0:8 0:3
3
6 0:6 0:5 0:6 0:5 x 0:4 7 7 6 7 6 6 x 0:4 0:5 0:3 0:6 x 7 7 C¼6 6 0:7 0:5 0:7 0:5 0:4 0:8 7: 7 6 7 6 4 0:2 x 0:4 0:6 0:5 0:7 5 0:7 0:6
x
0:2 0:3 0:5
Using Eqs. (28) and (30), we have (where n = 6):
2 Dn1
4
0
1 1
3
7 6 6 1 4 1 1 0 7 7 6 7 ¼ D5 ¼ 6 6 0 1 3 1 1 7; 7 6 4 1 1 1 5 1 5 1 0 1 1 4 2
Y n1
1
3
þ ln 0:3 þ ln 0:8 þ ln 0:3 ln 0:4 0:6 0:7 0:2 0:7
7 6 ln 0:6 þ ln 0:6 þ ln 0:5 þ ln 0:4 7 6 0:4 0:4 0:5 0:6 7 6 0:4 0:3 0:6 7: ¼ Y5 ¼ 6 ln þ ln þ ln 0:6 0:7 0:4 7 6 6 0:7 0:5 0:7 0:4 0:8 7 4 ln 0:3 þ ln 0:5 þ ln 0:3 þ ln 0:6 þ ln 0:2 5 ln 0:2 þ ln 0:4 þ ln 0:6 þ ln 0:7 0:8 0:6 0:4 0:3
Table 1 The mean consistency index of randomly generated matrices. n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
RI
0
0
0.52
0.89
1.12
1.26
1.36
1.41
1.46
1.49
1.52
1.54
1.56
1.58
1.59
Y. Xu et al. / Applied Mathematical Modelling 37 (2013) 2139–2152
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Using Eqs. (31)–(34), we have
W n1 ¼ W 5 ¼ ð0:0534; 0:2749; 0:0267; 0:6138; 0:0387ÞT ; T w ¼ w1 ; w2 ; w3 ; w4 ; w5 ; w6 ¼ ð0:1448; 0:1807; 0:1410; 0:2536; 0:1427; 0:1373ÞT : If we use Eq. (51) to compute its consistency index, we get: FCR = 0.1694 > 0.1, so C is not acceptable consistent preference relation. Then we return this matrix to the DM to reconsider structuring new matrix in the following (where c15 = 0.8 ? 0.3, c51 = 0.2 ? 0.7):
2
0:5 0:4
x
0:3 0:3 0:3
3
7 6 6 0:6 0:5 0:6 0:5 x 0:4 7 7 6 7 6 6 x 0:4 0:5 0:3 0:6 x 7 7: 6 C¼6 7 6 0:7 0:5 0:7 0:5 0:4 0:8 7 7 6 6 0:7 x 0:4 0:6 0:5 0:7 7 5 4 0:7 0:6
x
0:2 0:3 0:5
Again, we repeat the above computation, we have:
T w ¼ w1 ; w2 ; w3 ; w4 ; w5 ; w6 ¼ ð0:0878; 0:1599; 0:1551; 0:2464; 0:2208; 0:1301ÞT : Then, using Eq. (51), we have
FCR ¼ 0:0719 < 0:1: Thus, C is of acceptable consistency. Therefore the ranking of the six alternatives is
x4 x5 x2 x3 x6 x1 : Example 4. For a decision making problem, there are six decision alternatives xi (i = 1, 2, . . . , 6). The DM provides his/her preferences over these six decision alternatives, and gives an incomplete fuzzy preference relation as follows (adapted from [27]):
2
0:5
x
x
0:3 0:8 0:3
3
7 6 x x x 7 6 x 0:5 x 7 6 7 6 x 0:5 x x x 7 6 x 7: C¼6 7 6 x 0:5 0:4 0:8 7 6 0:7 x 7 6 6 0:2 x x 0:6 0:5 0:7 7 5 4 0:7
x
x
0:2 0:3 0:5
From Theorem 3, we know that C is not an acceptable incomplete fuzzy preference relation, because there is no know element in the second and third row (or column) of C except for the diagonal elements (c22 = 0.5, c33 = 0.55). Xu [27] solved a series of equations to obtain the weighting vector, but he found that there existed infinite solutions to the equations. As we can see, it is difficult to find that there exists a finite set of solutions to a series equations, and it requires careful analysis or complicated computations. Compared with Xu’s method, our method is simple. Example 5. Suppose that three DMs provide the following incomplete preference relations Ci (i = 1, 2, 3) for a set of three alternatives X = {x1, x2, x3},
2
0:5 0:5 0:6
6 C 1 ¼ 4 0:5 0:5 0:4
x
3
7 x 5;
2
0:5
6 C2 ¼ 4 x
0:5
x
0:6
7 0:5 0:6 5;
0:4 0:4 0:5
Using Eqs. (28) and (30), we have (where n = 3):
" Dn1 ¼ D2
1
2
2
4
"
# ;
Y n1 ¼ Y 2
Then, using Eqs. (31)–(34), we have
3
2 ln 0:6 0:4 2 ln 0:6 0:4
# :
2
0:5 0:5
x
3
6 7 C 3 ¼ 4 0:5 0:5 0:6 5: x
0:4 0:5
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Y. Xu et al. / Applied Mathematical Modelling 37 (2013) 2139–2152
W n1 ¼ W 2 ¼ ð0:4055; 0:4055ÞT ; w ¼ ð0:375; 0:375; 0:25Þ : Next, applying Eq. (51), we have
CR1 ¼ CR2 ¼ CR3 ¼ 0 < 0:1: Which means all the three fuzzy preference relations is perfectly multiplicative consistent and gives the ranking of x1 x2 x3. Where the symbol ‘ ’ means ‘is equal to’, ‘ ’ means ‘is preferred to’. If we use Gong’s [34] least-square method to compute, we have
2
3 0:82 0:5 0:48 6 7 Q ¼ 4 0:5 0:82 0:48 5: 0:48 0:48 1:44 Because Q is a singular matrix, Q1 does not exist, therefore, we cannot use Q1e/eTQ1e [34] to compute the priority vector proposed by Gong. From the above illustration, we demonstrate that our method is more robust. Example 6. For a multi-attribute decision making problem, there are four decision alternatives x1, x2, x3 and x4. The decisionmaker provides his/her preferences over these four decision alternatives, and gives a fuzzy preference relation as follows:
2
0:5 0:7 0:6 0:8
3
6 0:3 0:5 0:4 0:6 7 7 6 C¼6 7: 4 0:4 0:6 0:5 0:7 5 0:2 0:4 0:3 0:5 Using Eqs. (28) and (30), where n = 4, we have:
2
3
1 1
3
6 7 Dn1 ¼ D3 ¼ 4 1 3 1 5; 1 1 3
2
þ ln 0:6 þ ln 0:8 ln 0:7 0:3 0:4 0:2
3
6 7 Y n1 ¼ Y 3 ¼ 4 ln 0:3 : þ ln 0:4 þ ln 0:6 0:7 0:6 0:4 5 ln 0:4 þ ln 0:6 þ ln 0:7 0:6 0:4 0:3
Then, using Eqs. (31)–(34), we have
W n1 ¼ W 3 ¼ ð1:3195; 0:4479; 0:8716ÞT ; w ¼ ð0:4302; 0:1799; 0:2749; 0:1150Þ : Next, applying Eq. (51), we have
FCR ¼ 0:000897 < 0:1: Thus, C is of acceptable consistency, and therefore gives the ranking of x1 x2 x3 x4. Accordingly, the most desirable alternative is x1. If we use Eq. (47) to compute the weight vector, which will get the same result. Our weighting vector is approximate to Xu’s least deviation method and also results in the same ranking. Xu [51] used a convergent iterative algorithm to derive the weighting vector, the algorithm is fairly complicated. With our method, we only need to solve a series of equations, which can be solved easily using software such as Matlab or Lingo etc.
6. Discussion and conclusion In this paper, we have proposed the logarithmic least squares method (LLSM) to rank alternatives in group decision making with incomplete fuzzy preference relations and where entries are not equal to 0 or 1. In order to do this, we first illustrate an example to explain that the concept of multiplicative consistent incomplete fuzzy preference relation defined by Xu [19] is not correct, and propose a reasonable definition of multiplicative consistency for incomplete fuzzy preference relations. We also propose an approach to assess if an incomplete fuzzy relation is acceptable. The method is simple compared to Gong’s [34] method. We also develop an acceptable consistency ratio, which is simple and similar to Saaty’s consistency ratio CR for incomplete multiplicative consistent fuzzy preference relations. As shown, our LLSM method and consistency test method can not only be used to derive the weighting vector for the incomplete fuzzy preference relations, but also for the complete fuzzy preference relations. We further use some examples to illustrate that compared with the existing methods, our proposed approach has distinct characteristics. Compared with Gong’s [34] least-square method, our method can be used to deal with the following two situations: (1) all the incomplete fuzzy preference relations are perfectly multiplicative consistent, (2) at least one of the incomplete fuzzy preference relations is not perfectly multiplicative consistent, while Gong’s method can only be applied in the latter case.
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The proposed method can deal with both the complete and incomplete fuzzy preference relations, while Wang’s [15] method is only suitable to deal with the complete fuzzy preference relations. In other words, Wang’s method would be considered as a special case of our proposed method. For the complete fuzzy preference relations, Xu’s [51] least deviation method can only be used to derive the priority vector for a fuzzy preference relation, and can not be used in the context group decision making. The iterative algorithm of the least deviation method is fairly complicated, as compared to our proposed method which can handle not only one single preference relation, but a set of some fuzzy preference relations, Finally, our method is simple, efficient, and can be solved easily using software such as Matlab or Lingo, etc. If the incomplete fuzzy preference relation does not have acceptable consistency, it needs to be returned to the DM to reconsider and structure a new matrix. As shown in Example 3, by only changing c15 and c51, we were able to recalculate the weighting vector and the consistency radio. Once the incomplete fuzzy preference relation has acceptable consistency we are able to rank the alternatives. In future research we will consider Unusual and False Observations (we call them UFO). In addition in this paper, we only consider the decision makers having equal importance, in the future research, we will pay attention to the GDM problems where DMs are heterogeneous and there are explicit importance degrees associated with the DMs. Acknowledgements We would like to thank the anonymous reviewers for valuable comments that have improved the quality of the paper. This work was partly supported by the National Natural Science Foundation of China (NSFC) under Grants 71101043, 50979024 and 90924027, College Philosophy and Social Science Research Project of Jiangsu Province under Grant 2011SJD630007, the International Academic Cooperation Team Development Foundation of Business School, Hohai University under Grant UAH-2011-181. References [1] T.Y. Chen, Multiple criteria group decision-making with generalized interval-valued fuzzy numbers based on signed distances and incomplete weights, Appl. Math. Model. 36 (2012) 3029–3052. [2] F. Chiclana, F. Herrera, E. Herrera-Viedma, Integrating multiplicative preference relations in a multipurpose decision-making model based on fuzzy preference relations, Fuzzy Sets Syst. 122 (2001) 277–291. [3] F. Chiclana, F. Herrera, E. Herrera-Viedma, Integrating three representation models in fuzzy multipurpose decision making based on fuzzy preference relations, Fuzzy Sets Syst. 97 (1998) 33–48. [4] F. Chiclana, F. 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