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Group consensus algorithms based on preference relations
Information Sciences 181 (2011) 150–162
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Information Sciences journal homepage: www.elsevier.com/locate/ins
Group consensus algorithms based on preference relations Zeshui Xu a,b,⇑, Xiaoqiang Cai b a b
School of Economics and Management, Southeast University, Nanjing, Jiangsu 210096, China Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong
a r t i c l e
i n f o
Article history: Received 12 June 2009 Received in revised form 1 August 2010 Accepted 3 August 2010
Keywords: Group decision making Fuzzy preference relation Multiplicative preference relation Goal programming model Quadratic programming Iterative algorithm
a b s t r a c t In many group decision-making situations, decision makers’ preferences for alternatives are expressed in preference relations (including fuzzy preference relations and multiplicative preference relations). An important step in the process of aggregating preference relations, is to determine the importance weight of each preference relation. In this paper, we develop a number of goal programming models and quadratic programming models based on the idea of maximizing group consensus. Our models can be used to derive the importance weights of fuzzy preference relations and multiplicative preference relations. We further develop iterative algorithms for reaching acceptable levels of consensus in group decision making based on fuzzy preference relations or multiplicative preference relations. Finally, we include an illustrative example. Ó 2010 Published by Elsevier Inc.
1. Introduction Group decision-making problems generally involve finite non-empty sets of alternatives that have been generated through search processes [13] and require a predefined group of decision makers. In the knowledge-based technique for group decision making developed by Wanyama and Far [15], multiple user agents utilize a systematic model that assists Commercial Off-The-Shelf (COS) selection stakeholders to identify conflicts, and to determine and evaluate conflict resolution options. In addition, each user agent analyzes the agreement options in detail before advising its client about which goals to optimize, and which goals to compromise, to reach agreement with the other stakeholders. In many group decision-making situations, such as investment decisions, project evaluations, personnel evaluation, and medical diagnoses, the decision makers usually compare each set of alternatives. They also use predefined evaluation scales to determine which alternatives they prefer. Fuzzy preference relations and multiplicative preference relations are two of the most common and useful tools for representing decision makers’ preferences. To get the group’s opinion in a decision-making process, an appropriate technique is required to aggregate all the individual fuzzy preference relations or all the multiplicative preference relations. Consequently, much of the literature has investigated approaches for information aggregation, and a variety of aggregation operators have been developed [14,20]. Among these, the Ordered Weighted Averaging (OWA) [9,24,26], Ordered Weighted Geometric (OWG) [7,19,23], Weighted Arithmetic Averaging (WAA) [6], and Weighted Geometric Averaging (WGA) [12] are the most common operators for aggregating fuzzy preference relations and multiplicative preference relations. The fundamental feature of the OWA and OWG operators is their reordering step. They reorder all the given arguments in descending order, and then weight the ordered position of an argument but ignore the importance of the argument itself. Chiclana et al. [2] used fuzzy preference relations to make the information uniform for problems where the information about alternatives is represented by preference orderings, utility functions and fuzzy preference relations. Based on the ⇑ Corresponding author at: School of Economics and Management, Southeast University, Nanjing, Jiangsu 210096, China. Tel.: +86 25 84483382. E-mail addresses: [email protected] (Z. Xu), [email protected] (X. Cai). 0020-0255/$ - see front matter Ó 2010 Published by Elsevier Inc. doi:10.1016/j.ins.2010.08.002
Z. Xu, X. Cai / Information Sciences 181 (2011) 150–162
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concept of fuzzy majority, they used the OWA operator, whose weights are calculated by the fuzzy quantifier, to aggregate fuzzy preference relations. Drawing on the Induced Ordered Weighted Averaging (IOWA) operator developed by Yager [25], Chiclana et al. [5] introduced an importance IOWA operator, a consistency IOWA operator and a preference IOWA operator. They then used these operators to aggregate fuzzy preference relations based on the concept of fuzzy majority. Herrera et al. [7] studied problems where the information about alternatives can be presented by means of preference orderings, utility functions and multiplicative preference relations. They used multiplicative preference relations to make the information uniform, and then employed the OWG operator to aggregate multiplicative preference relations based on the concept of fuzzy majority. In another study, Chiclana et al. [4] used the importance IOWG, consistency IOWG and preference IOWG operators to aggregate multiplicative preference relations. However, the aggregated results produced by these operators generally do not satisfy the reciprocity property (i.e., the aggregated results are generally not fuzzy preference relations or multiplicative preference relations). Chiclana et al. [3,8] further used the OWA and OWG operators to study the conditions in which the reciprocity property is maintained when aggregating fuzzy preference relations or multiplicative preference relations. Furthermore, numerous methods have been proposed to determine the OWA weights [17,27]. However, while the weights associated with these operators reflect the ordered positions of the preferences, they do not consider the fuzzy preference relations or multiplicative preference relations. The WAA and WGA operators first weight all the given arguments using a normalized weight vector, and then aggregate the weighted arguments. The underlying characteristic of both the WAA and WGA operators is that they take the importance of the sources of information into account when computing the aggregated arguments [21]. The preference relation of all the fuzzy preference relations aggregated by the WAA operator is a fuzzy preference relation [22]. The preference relation of all the multiplicative preference relations aggregated by the WGA operator is also a multiplicative preference relation [12]. Clearly, the key step in using the WAA (or WGA) operator to aggregate fuzzy preference relations (or multiplicative preference relations) is determining the importance weight of each fuzzy preference relation (or multiplicative preference relation). For the sake of convenience, all fuzzy preference relations (or multiplicative preference relations) are generally assigned equal weight. However, the fuzzy preference relations (or multiplicative preference relations) are provided by the decision makers. A decision maker is usually only able to comment on the part of a problem he/she feels competent to address [16]. In other words, he/she cannot be expected to have sufficient expertise to comment on all aspects of the problem. Thus, in many practical applications the fuzzy preference relations (or multiplicative preference relations) provided by decision makers require different weights, especially in situations involving policy specification [11]. In this paper, we address this issue by developing goal programming models and quadratic programming models to derive the importance weights of fuzzy preference relations and multiplicative preference relations. In addition, we will also introduce iterative algorithms for establishing acceptable levels of consensus in group decision making. The remainder of the paper is organized as follows. Section 2 formulates models to derive the importance weights of fuzzy preference relations from the perspective of maximizing group consensus. Based on our models, we then develop iterative algorithms for group decision making with fuzzy preference relations to make the level of group consensus level as high as possible. Section 3 establishes a number of optimization models for group consensus based on multiplicative preference relations. Section 4 applies our algorithms in the evaluation and selection of suitable locations for shopping centers. Finally, Section 5 concludes the paper. 2. Group consensus algorithms based on fuzzy preference relations Consider a group decision-making problem based on fuzzy preference relations, where there are a collection of n alternatives {x1, x2, . . . , xn}, and a group of m decision makers {e1, e2, . . . , em} participate in the decision making. To express their opinions about alternatives, each decision maker ek compares each pair of alternatives (xi, xj) on a 0–1 scale. He/she provides his/ her preference value pijk, where 0 6 pijk 6 1 denotes the preference degree of the alternative xi over xj. Specifically, 0 6 pijk < 0.5 indicates that xj is preferred to xi, and the smaller the pijk, the greater the preference degree of the alternative xj over xi. In particular, pijk = 0 indicates that xj is absolutely preferred to xi, pijk = 0.5 indicates indifference between xi and xj, 0.5 < pijk 6 1 indicates that xi is preferred to xj, and pijk = 1 indicates that xi is absolutely preferred to xj. All the preference values pijk(i, j = 1, 2, . . . , n) provided by the decision maker ek are contained in a fuzzy preference relation Pk = (pijk)nn, which satisfies
0 6 pijk 6 1;
piik ¼ 0:5;
pijk þ pjik ¼ 1;
i; j ¼ 1; 2; . . . ; n
ð1Þ
The importance weight of each fuzzy preference relation needs to be taken into account in practical applications, because the fuzzy preference relations provided by different decision makers generally have different levels of importance. Thus, to assign the appropriate weights, let w = (w1, w2, . . . , wm)T be the weight vector (which is to be determined) of the fuzzy preference relations Pk = (pijk)nn (k = 1, 2, . . . , m), where
wk P 0;
k ¼ 1; 2; . . . ; m;
m X
wk ¼ 1
ð2Þ
k¼1
To get the group’s opinion, we use the Weighted Arithmetic Averaging (WAA) operator:
pij ¼
m X k¼1
wk pijk ;
i; j ¼ 1; 2; . . . ; n
ð3Þ
152
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to aggregate all the individual fuzzy preference relations Pk = (pijk)nn (k = 1, 2, . . . , m) into the collective preference relation P = (pij)nn. It can be easily shown that P satisfies condition (1), and is thus also a fuzzy preference relation. Clearly, the fundamental role of the WAA operator is to determine the weight vector w. To examine this role, we will now discuss the relationship between each individual opinion and the group’s opinion: Case 1. If the individual fuzzy preference relation Pk is consistent with the collective fuzzy preference relation P, then Pk = P. In this case, each preference value pijk in Pk should be equal to the corresponding preference value pij in P, i.e., pijk = pij, for all i,j = 1, 2, . . . , n. Using (3), we have
pijk ¼
m X
wl pijl ;
for all i; j ¼ 1; 2; . . . ; n
ð4Þ
l¼1
If all the individual fuzzy preference relations Pk (k = 1, 2, . . . , m) are consistent with the collective fuzzy preference relation P, then (4) holds, for all k = 1, 2, . . . , m, i.e.,
pijk ¼
m X
wl pijl ;
for all i; j ¼ 1; 2; . . . ; n; k ¼ 1; 2; . . . ; m
ð5Þ
l¼1
In this case, the group reaches complete consensus. Accordingly, (5) is equivalent to (4), and using (1) and (4) is equivalent to the following:
pijk ¼
m X
wl pijl ;
for all i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n
ð6Þ
l¼1
from which we can derive the weight vector w. Case 2. If the individual fuzzy preference relation Pk is inconsistent with the collective fuzzy preference relation P, then (6) does not always hold. If the group does not reach complete consensus, then (5) does not always hold. In this case, we introduce the deviation variable eijk as follows:
eijk
m X ¼ pijk wl pijl ; l¼1
for all i; j ¼ 1; 2; . . . ; n; k ¼ 1; 2; . . . ; m
ð7Þ
It follows from (1) that (7) is equivalent to the following:
m
eijk ¼ pijk m wl pijl ; for all i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n; k ¼ 1; 2; . . . ; m l¼1
ð8Þ
In practice, there are many group decision-making problems, such as personnel examination, military system efficiency evaluation, and venture capital project evaluation, where some individual decision makers may assign unduly high or low preference values to their preferred or unwanted objects (alternatives). In these situations, it is necessary to employ an appropriate measure to adjust the extreme individual opinions, so that a sensible group decision can be reached. We propose to determine the weights of the individual preferences based on their deviations eijk (i = 1, 2, . . . , n 1; j = i + 1, . . . , n; k = 1, 2, . . . , m) from the group’s opinion. This is because the deviation eijk reflects the degree to which the individual fuzzy preferences depart from the group’s opinion. By adjusting the weights based on the deviations, we try to discount those opinions that are highly inconsistent with the assessments of the majority. Specifically, we aim to minimize eijk (i = 1, 2, . . . , n 1; j = i + 1, . . . , n; k = 1, 2, . . . , m), which leads to the following multiobjective programming model:
ðM-1Þ
min s:t:
eijk
m X ¼ pijk wl pijl ; l¼1
wl P 0;
i ¼ 1; 2; . . . ; n 1;
l ¼ 1; 2; . . . ; m;
m X
j ¼ i þ 1; . . . ; n;
k ¼ 1; 2; . . . ; m
wl ¼ 1
l¼1
A solution to the above minimization problem can be found by solving the following goal programming model:
ðM-2Þ
min
J1 ¼
m X n1 X n X þ sijk dijk þ t ijk dijk k¼1 i¼1 j¼iþ1
s:t:
pijk
m X
þ
wl pijl dijk þ dijk ¼ 0; i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n; k ¼ 1; 2; . . . ; m
l¼1 þ
dijk P 0;
dijk P 0;
i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n; k ¼ 1; 2; . . . ; m m X wl P 0; l ¼ 1; 2; . . . ; m; wl ¼ 1 l¼1
Z. Xu, X. Cai / Information Sciences 181 (2011) 150–162
153
þ
where dijk is the positive deviation from the target of the goal eijk, defined as
(
þ
dijk ¼ max pijk
m X
)
wl pijl ; 0
ð9Þ
l¼1
dijk is the negative deviation from the target of the goal eijk, defined as dijk
¼ max
( m X
) wl pijl pijk ; 0
ð10Þ
l¼1 þ
sijk is the weighting factor corresponding to the positive deviation dijk , and tijk is the weighting factor corresponding to the negative deviation dijk . Considering that all the goal functions eijk (i = 1, 2, . . . , n 1; j = i + 1, . . . , n; k = 1, 2, . . . , m) are fair, we can set sijk = tijk = 1 (i = 1, 2, . . . , n 1; j = i + 1, . . . , n; k = 1, 2, . . . , m). The model (M-2) can then be rewritten as
ðM-3Þ
min
J1 ¼
m X n1 X n X þ dijk þ dijk k¼1 i¼1 j¼iþ1
s:t:
pijk
m X
þ
wl pijl dijk þ dijk ¼ 0;
i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n; k ¼ 1; 2; . . . ; m
l¼1 þ
dijk P 0; wl P 0;
dijk P 0;
i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n; m X l ¼ 1; 2; . . . ; n; wl ¼ 1
k ¼ 1; 2; . . . ; m
l¼1
By solving this model (M-3), we can obtain the optimal weight vector w = (w1, w2, . . . , wm)T of the fuzzy preference relations Pk = (pijk)nn (k = 1, 2, . . . , m). Another approach can also be used to solve the minimization problem (M-1), which transforms the model (M-1) into the following quadratic programming model:
ðM-4Þ
J 01
min
¼
m X n X n X k¼1 i¼1
s:t:
wl P 0;
2 ijk
e ¼
m X n X n X
j¼1
k¼1 i¼1
l ¼ 1; 2; . . . ; m;
pijk
m X
j¼1
m X
!2 wl pijl
l¼1
wl ¼ 1
l¼1
To solve this model, we can construct the Lagrange function:
! 2 m X n X n m X X m Lðw; kÞ ¼ pijk m wl pijl 2k wl 1 k¼1 i¼1
l¼1
j¼1
ð11Þ
l¼1
where k is the Lagrange multiplier. Differentiating (11) with respect to wh (h = 1, 2, . . . , m) and k, and setting these partial derivatives equal to zero, the following set of equations is obtained:
! m X n X n m X X @Lðw; kÞ ¼ 2 pijk wl pijl pijh 2k ¼ 0; @wh k¼1 i¼1 j¼1 l¼1
h ¼ 1; 2; . . . ; m
m @Lðw; kÞ X ¼ wl 1 ¼ 0 @k l¼1
ð12Þ ð13Þ
(12) can be simplified as m X n X n m X X k¼1 i¼1
j¼1
! wl pijl pijh pijk pijh
k ¼ 0;
h ¼ 1; 2; . . . ; m
ð14Þ
l¼1
i.e., n X m X i¼1
l¼1
wl
n X
! mpijl pijh
j¼1
n X n m X X i¼1
j¼1
! pijk pijh k ¼ 0;
h ¼ 1; 2; . . . ; m
ð15Þ
k¼1
which can be rewritten in a matrix form as
Dw p ke ¼ 0
ð16Þ
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Z. Xu, X. Cai / Information Sciences 181 (2011) 150–162
where n X n X m X
p¼
i¼1
pijk pij1 ;
j¼1 k¼1
n X n X m X i¼1
pijk pij2 ; . . . ;
j¼1 k¼1
n X n X m X i¼1
!T pijk pijm
;
e ¼ ð1; 1; . . . ; 1ÞT
ð17Þ
j¼1 k¼1
and
0
n P n P 2 B i¼1 j¼1 mpij1 B B n P n BP B mpij1 pij2 D¼B B i¼1 j¼1 B ... B B n n @P P mpij1 pijm
n P n P
mpij1 pij2
i¼1 j¼1 n P n P i¼1 j¼1
mp2ij2
... n P n P
i¼1 j¼1
mpij2 pijm
i¼1 j¼1
1 mpij1 pijm C i¼1 j¼1 C C n P n C P ... mpij2 pijm C C C i¼1 j¼1 C ... ... C C n n PP A 2 ... mpijm ...
n P n P
i¼1 j¼1
ð18Þ
mm
Because (13) can be rewritten as
eT w ¼ 1
ð19Þ
solving (16) and (19) we get
k¼
1 eT D1 p
ð20Þ
eT D1 e
and
w¼
D1 eð1 eT D1 pÞ eT D1 e
þ D1 p
ð21Þ
which is also the optimal weight vector w = (w1,cw2, . . . , wm)T of the fuzzy preference relations Pk = (pijk)nn(k = 1, 2, . . . , m). Then, using (4), we get the collective fuzzy preference relation P. In addition, based on (8) and the optimal weight vector w, we can calculate the deviation between the individual fuzzy preference relation Pk and the collective fuzzy preference relation P by
dðP k ; PÞ ¼
n1 X n X
eijk
i¼1 j¼iþ1
n1 X n m X X 2 ¼ wl pijl pijk nðn 1Þ i¼1 j¼iþ1 l¼1
ð22Þ
Accordingly, the weighted sum of all the deviations d(Pk, P) (k = 1, 2, . . . , m) can be defined as
D1 ¼
m X
wk dðPk ; PÞ
ð23Þ
k¼1
From (22) and (23), we know that if d(Pk, P) = 0, then the individual fuzzy preference relation Pk is consistent with the collective fuzzy preference relation P. If D1 = 0, then the group reaches complete consensus. In addition, we suppose that if D1 6 k1, then the group reaches an acceptable level of consensus, where k1 is the threshold of an acceptable level of group consensus of fuzzy preference relations, which can be predefined by the decision makers. Below, we develop iterative algorithms for group decision making with fuzzy preference relations based on the goal programming model (M-3) and the quadratic programming model (M-4). Algorithm 1 Step 1. The decision makers ek (k = 1, 2, . . . , m) provide their preferences over the alternatives xi (i = 1, 2, . . . , n) on a 0–1 scale, and construct fuzzy preference relations Pk = (pijk)nn (k = 1, 2, . . . , m). Let t be the number of iterative times, t* the maximum iteration number, k1 the deadline for acceptable group consensus of fuzzy preference relations ð0Þ ð0Þ Pk ¼ ðpijk Þnn ¼ Pk ¼ ðpijk Þnn ðk ¼ 1; 2; . . . ; mÞ, and t = 0. ðtÞ ðtÞ Step 2. Utilize the WAA operator (3) to aggregate all the individual fuzzy preference relations P k ¼ ðpijk Þnn ðk ¼ 1; 2; . . . ; mÞ ðtÞ into the collective fuzzy preference relation PðtÞ ¼ ðpij Þnn . Step 3. Apply the goal programming model (M-3) or the quadratic programming model (M-4) to determine the optimal ðtÞ ðtÞ ðtÞ ðtÞ ðtÞ weight vector wðtÞ ¼ ðw1 ; w2 ; . . . ; wm ÞT of P k ¼ ðpijk Þnn ðk ¼ 1; 2; . . . ; mÞ, and then get the collective fuzzy prefer(t) ence relation P using (4). ðtÞ Step 4. Calculate the deviations dðPk ; P ðtÞ Þ ðk ¼ 1; 2; . . . ; mÞ and D1(t) using (22) and (23), respectively. If D1(t) 6 k1 or t = t*, ðtÞ ðtÞ then stop. Otherwise, there must exist at least one fuzzy preference relation P k such that dðP k ; PðtÞ Þ > k1 . In this ðtÞ (t) case, return P k together with P to the decision maker ek for revaluation, and construct a new fuzzy preference relation. Let t = t + 1, then go to Step 2.
Z. Xu, X. Cai / Information Sciences 181 (2011) 150–162
155
With Algorithm 1, when the group does not reach an acceptable level of consensus, some decision makers need to revaluate their preferences over the alternatives. Occasionally, this process may require an excessive amount of time to complete. In cases where consensus is urgently required, or the decision makers cannot or are unwilling to revaluate the alternatives, we can use the following algorithm to develop an acceptable level of group consensus: Algorithm 2 Steps 1–3. See Algorithm 1. ðtÞ Step4. Calculate the deviations dðPk ; P ðtÞ Þðk ¼ 1; 2; . . . ; mÞ and D1(t) using (22) and (23), respectively. If D1 (t) 6 k1 or ðtþ1Þ ðtþ1Þ t = t*, then stop. Otherwise, let Pk ¼ ðpijk Þnn ðk ¼ 1; 2; . . . ; mÞ, where ðtþ1Þ
pijk
ðtÞ
ðtÞ
¼ gpijk þ ð1 gÞpij ;
i; j ¼ 1; 2; . . . ; n; k ¼ 1; 2; . . . ; m; 0 < g < 1
ð24Þ
Then return to Step 2. Algorithm 2 is convergent. In fact, similar to [18], we have Theorem 1. Let Pk = (pijk)n n (k = 1, 2, . . . , m) be the fuzzy preference relations provided by the decision makers ek (k = 1, 2, . . . , m), ðtÞ and fP k g ðk ¼ 1; 2; . . . ; mÞ and {P(t)} the fuzzy preference relations sequences generated in Algorithm 2. Then for each t:
D1 ðt þ 1Þ ¼ gD1 ðtÞ < D1 ðtÞ;
0
ð25Þ
Especially, let k1 = 0, then
lim D1 ðtÞ ¼ 0
ð26Þ
t!þ1
3. Group consensus models based on multiplicative preference relations As in Section 2, here we consider a group decision-making problem based on multiplicative preference relations. Each decision maker ek compares each pair of alternatives (xi, xj) using a ratio scale (in particular as Saaty [12] showed, the 1–9 scale). He/she provides his/her preference value aijk, where 1/9 6 aijk 6 9, and aijk denotes the preference intensity of the alternative xi over xj. Specifically, 1/9 6 aijk < 1 means that xj is preferred to xi, and the smaller the aijk, the stronger the preference intensity of the alternative xj over xi. In particular, aijk = 1/9 means that xj is absolutely preferred to xi, aijk = 1 implies indifference between xi and xj, and 1 < aijk 6 9 means that xi is preferred to xj(aijk = 9 indicates that xi is absolutely preferred to xj). All the preference values aijk (i, j = 1, 2, . . . , n) provided by the decision maker ek are contained in a multiplicative preference relation Ak = (aijk)nn, which satisfies
aijk > 0;
aiik ¼ 1;
aijk ajik ¼ 1;
i; j ¼ 1; 2; . . . ; n
ð27Þ
Let v = (v1, v2, . . . , vm)T be the weight vector of the multiplicative preference relations Ak = (aijk)nn (k = 1, 2, . . . , m), where P vk P 0, k = 1, 2, . . . , m, and mk¼1 v k ¼ 1. To get the group’s opinion, we use the Weighted Geometric Averaging (WGA) operator:
aij ¼
m Y
ðaijk Þv k ;
i; j ¼ 1; 2; . . . ; n
ð28Þ
k¼1
to aggregate all the individual multiplicative preference relations Ak = (aijk)nn (k = 1, 2, . . . , m) into the collective preference relation A = (aij)nn. Note that A satisfies condition (28) and is therefore also a multiplicative preference relation. Now we discuss the relationship between each individual multiplicative preference relation and the collective multiplicative preference relation: Case 1. If the individual multiplicative preference relation Ak is consistent with the collective multiplicative preference relation A, then Ak = A. In this case, each preference value aijk in Ak should be equal to the corresponding preference value aij in A, i.e., aijk = aij, for all i, j = 1, 2, . . . , n. Using (28), we have
aijk ¼
m Y
ðaijl Þv l ;
for all i; j ¼ 1; 2; . . . ; n
ð29Þ
l¼1
According to (27) and (29) is equivalent to the following form:
aijk ¼
m Y
ðaijl Þv l ;
for all i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n
ð30Þ
l¼1
If all the individual multiplicative preference relations Ak (k = 1, 2, . . . , m) are consistent with the collective multiplicative preference relation A, i.e., (30) holds for all k = 1, 2, . . . , m, then the group reaches complete consensus. In this case, we transform (30) into the following form by taking common logarithms on both sides:
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Z. Xu, X. Cai / Information Sciences 181 (2011) 150–162
lg aijk ¼ lg
m Y
ðaijl Þv l ¼
l¼1
m X
v l lg aijl ;
for all i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n
ð31Þ
l¼1
with which we can derive the weight vector v. Case 2. If the individual multiplicative preference relation Ak is inconsistent with the collective multiplicative preference relation A, then (31) does not always hold. If the group does not reach complete consensus, then
lg aijk ¼ lg
m Y
ðaijl Þv l ¼
l¼1
m X
v l lg aijl ;
for all i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n; k ¼ 1; 2; . . . ; m
ð32Þ
l¼1
does not always hold. In this case, we introduce the deviation variable fijk as follows:
fijk
m X ¼ lg aijk v l lg aijl ; l¼1
for all i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n; k ¼ 1; 2; . . . ; m
ð33Þ
Similar to the model (M-1), we minimize fijk (i = 1, 2, . . . , n 1; j = i + 1, . . . , n; k = 1, 2, . . . , m), and construct the following multi-objective programming model:
ðM-5Þ
min
m X f ijk ¼ lg aijk v lg a l ijl ; l¼1
v l P 0;
s:t:
l ¼ 1; 2; . . . ; m;
i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n; k ¼ 1; 2; . . . ; m m X
vl ¼ 1
l¼1
The model (M-5) can be transformed into the following goal programming model:
ðM-6Þ
min
J2 ¼
m X n1 X n X þ sijk dijk þ t ijk dijk k¼1 i¼1 j¼iþ1
s:t:
lg aijk
m X
v l lg aijl dþijk þ dijk ¼ 0;
i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n; k ¼ 1; 2; . . . ; m
l¼1 þ
dijk P 0;
v l P 0;
dijk P 0;
i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n; k ¼ 1; 2; . . . ; m m X l ¼ 1; 2; . . . ; m; vl ¼ 1 l¼1
þ
where dijk is the positive deviation from the target of the goal fijk, defined as
( þ dijk
¼ max lg aijk
m X
)
v l lg aijl ; 0
ð34Þ
l¼1
dijk is the negative deviation from the target of the goal fijk, defined as dijk
¼ max
( m X
)
v l lg aijl lg aijk ; 0
ð35Þ
l¼1 þ
sijk is the weighting factor corresponding to the positive deviation dijk , and tijk is the weighting factor corresponding to the negative deviation dijk . Considering that all the goal functions fijk(i = 1, 2, . . . , n 1; j = i + 1, . . . , n; k = 1, 2, . . . , m) are fair, we can set sijk = tijk = 1(i = 1, 2, . . . , n 1; j = i + 1 ,. . . , n; k = 1, 2, . . . , m), and then the model (M-6) can be rewritten as
ðM-7Þ
min
J2 ¼
m X n1 X n X þ dijk þ dijk k¼1 i¼1 j¼iþ1
s:t:
lg aijk
m X
v l lg aijl dþijk þ dijk ¼ 0;
i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n; k ¼ 1; 2; . . . ; m
l¼1 þ
dijk P 0;
v l P 0;
dijk P 0;
i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n; k ¼ 1; 2; . . . ; m m X l ¼ 1; 2; . . . ; m; vl ¼ 1 l¼1
Solving the model (M-7), we can obtain the optimal weight vector v = (v1, v2, . . . , vm)T of the multiplicative preference relations Ak = (aijk)nn (k = 1, 2, . . . , m).
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Analogous to the model (M-4), we can transform the model (M-5) into the following quadratic programming model:
min
ðM-8Þ
J 02
¼
m X n X n X k¼1 i¼1
s:t:
v l P 0;
2 fijk
¼
j¼1
m X n X n X k¼1 i¼1
l ¼ 1; 2; . . . ; m;
lg aijk
j¼1
m X
m X
!2
v l lg aijl
l¼1
vl ¼ 1
l¼1
Solving the model (M-8), we can get the optimal weight vector
v¼
G1 e 1 eT G1 a eT G1 e
v = (v1, v2, . . . , vm)T as follows:
þ G1 a
ð36Þ
where
a¼
n X n X m X i¼1
lg aijk lg aij1 ;
j¼1 k¼1
n X n X m X i¼1
lg aijk lg aij2 ; . . . ;
j¼1 k¼1
n X n X m X i¼1
!T lg aijk lg aijm
ð37Þ
j¼1 k¼1
and G is the matrix, denoted as
0 P n P n mðlg aij1 Þ2 B i¼1 j¼1 B B n P n BP B m lg aij1 lg aij2 G¼B B i¼1 j¼1 B ... B B n n @P P m lg aij1 lg aijm i¼1 j¼1
n P n P
m lg aij1 lg aij2
i¼1 j¼1
mðlg aij2 Þ2
i¼1 j¼1
... i¼1 j¼1
n P n P
m lg aij1 lg aijm
1
C C C n P n C P ... m lg aij2 lg aijm C C C i¼1 j¼1 C ... ... C C n n PP 2 A ... mðlg aijm Þ i¼1 j¼1
n P n P
n P n P
...
m lg aij2 lg aijm
i¼1 j¼1
ð38Þ
mm
Then using (28), we can get the collective multiplicative preference relation A. In addition, based on (33) and the optimal weight vector v, we can calculate the deviation between the individual multiplicative preference relation Ak and the collective multiplicative preference relation A using
dðAk ; AÞ ¼
n1 X n n1 X n m X X X 2 fijk ¼ v lg a lg aijk l ijl nðn 1Þ i¼1 j¼iþ1 i¼1 j¼iþ1 l¼1
ð39Þ
The weighted sum of all the deviations d(Ak, A) (k = 1, 2, . . . , m) can be derived using the WAA operator:
D2 ¼
m X
v k dðAk ; AÞ
ð40Þ
k¼1
From (39) and (40), we know that if d(Ak, A) = 0, then the individual multiplicative preference relation Ak is consistent with the collective multiplicative preference relation A. If D2 = 0, then the group reaches complete consensus. In addition, we suppose that if D2 6 k2, then the group has an acceptable level of consensus, where k2 is the threshold of acceptable group consensus for multiplicative preference relations, which can be predefined by the decision makers. In the case where D2 > k2, we can utilize the idea of Algorithm 1 or Algorithm 2 to improve the group consensus. 4. Illustrative example To illustrate the algorithms developed in this paper, we consider a group decision-making problem that concerns the evaluation and selection of suitable locations for shopping centers (adapted from [10]). An investment company has conducted a feasibility study to determine an appropriate site on which to establish a shopping center in one of the strategic demand areas of Istanbul. Many different objectives must be achieved if the project is to be undertaken. One of the basic objectives is to select a suitable location. After identifying the range of feasible sites, the task is then to select the one most likely to optimize the company’s strategic performance. Hence, detailed interviews were conducted with 15 experts, including professional consultants, academics, and business development managers and staff working for the company, to evaluate the most suitable locations. The six potential locations finally identified were Mecidiyeköy, Levent, Kozyatag˘i, Maltepe, Bakırköy and Beylikdüzü, denoted as xi (i = 1, 2, . . . , 6), respectively. Mecidiyeköy and Levent are the most important financial centres on the European side of Istanbul. Kozyatag˘i is a developing business district on the Asian side of Istanbul that is also close to a residential area. Bakırköy is a densely populated residential district on the European side on the coast of the Marmara Sea, which also contains an important shopping and commercial center close to the Atatürk International Airport. Maltepe is a rapidly growing residential district on the Asian side on the coast of the Marmara Sea, which is also close to the Sabiha Gökçen Airport and the Istanbul Park GP Racing Circuit. Although Beylikdüzü
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is primarily an industrial district on the European side, it has started to develop residentially following the opening of exhibition, congress and amusement centers, and because it has easy access to motorways. Assume that a committee comprising five experts (decision makers) ek (k = 1, 2, . . . , 5) from each strategic decision area has been set up to provide information for assessing the locations xi (i = 1, 2, . . . , 6). After comparing each pair of locations, the experts ek (k = 1, 2, . . . , 5) give their preferences using a 0–1 scale, and then construct the fuzzy preference relations ð0Þ
Pk ¼ P k ¼ ðpijk Þ66 ðk ¼ 1; 2; . . . ; 5Þ:
0
ð0Þ
P1 ¼ P1
ð0Þ
P3 ¼ P3
ð0Þ
P5 ¼ P5
0:5 B 0:6 B B 0:8 ¼B B 0:4 B @ 0:3 0:4 0 0:5 B 0:5 B B 0:4 ¼B B 0:4 B @ 0:3 0:1 0 0:5 B 0:7 B B 0:7 ¼B B 0:3 B @ 0:2 0:5
0:4 0:5 0:6 0:4 0:1 0:3 0:5 0:5 0:7 0:2 0:3 0:2 0:3 0:5 0:8 0:3 0:2 0:4
0:2 0:4 0:5 0:4 0:2 0:0 0:6 0:3 0:5 0:3 0:3 0:2 0:3 0:2 0:5 0:3 0:3 0:2
0:6 0:6 0:6 0:5 0:3 0:4 0:6 0:8 0:7 0:5 0:2 0:4 0:7 0:7 0:7 0:5 0:1 0:3
0:7 0:9 0:8 0:7 0:5 0:7 0:7 0:7 0:7 0:8 0:5 0:8 0:8 0:8 0:7 0:9 0:5 0:6
1 0:6 0:7 C C 1:0 C C; 0:6 C C 0:3 A
0
ð0Þ
P2 ¼ P2
0:5 1 0:9 0:8 C C 0:8 C C; 0:6 C C 0:2 A
ð0Þ
P4 ¼ P4
0:5 1 0:5 0:6 C C 0:8 C C 0:7 C C 0:4 A
0:5 B 0:7 B B 0:7 ¼B B 0:5 B @ 0:2 0:3 0 0:5 B 0:8 B B 0:9 ¼B B 0:5 B @ 0:2 0:1
0:3 0:5 0:6 0:3 0:0 0:2 0:2 0:5 0:8 0:1 0:4 0:0
0:3 0:4 0:5 0:5 0:1 0:1 0:1 0:2 0:5 0:2 0:4 0:4
0:5 0:7 0:5 0:5 0:4 0:3 0:5 0:9 0:8 0:5 0:0 0:2
0:8 1:0 0:9 0:6 0:5 0:6 0:8 0:6 0:6 1:0 0:5 0:6
1 0:7 0:8 C C 0:9 C C 0:7 C C 0:4 A 0:5 1 0:9 1:0 C C 0:6 C C 0:8 C C 0:4 A 0:5
0:5
In [2], Chiclana et al. utilized the OWA operator [24]:
OWAða1 ; a2 ; . . . ; an Þ ¼
n X
xj bj
ð41Þ
j¼1
to aggregate the individual preferences into the group’s opinion, where bj is the jth largest of a collection of arguments ai (i = 1, 2, . . . , n). x = (x1, x2, . . . , xn)T is a weighting vector associated with the OWA operator, such that xj P 0, j = 1, 2, . . . , n, P and nj¼1 xj ¼ 1. They can then be computed by means of a fuzzy linguistic quantifier Q used to represent the fuzzy majority as follows:
xj ¼ Q
j j1 Q ; n n
j ¼ 1; 2; . . . ; n
ð42Þ
where
Q ðrÞ ¼
8 > < 0;
ra ; > ba
:
1;
if r < a if a 6 r 6 b if r > b
ð43Þ
with a, b, r 2 [0, 1]. Some examples of non-decreasing proportional fuzzy linguistic quantifiers with the parameters (a, b) are [2]: ‘‘most” (0.3, 0.8), ‘‘at least half” (0, 0.5), and ‘‘as many as possible” (0.5, 1), respectively. Using (42) and the fuzzy majority criterion with the fuzzy quantifier ‘‘at least half”, with the pair (0, 0.5), we get the weighting vector x = (x1, x2, . . . , x5)T as: x = (0.4, 0.4, 0.2, 0, 0)T. Then using the OWA operator (41), we aggregate all the fuzð0Þ zy preference relations P k ¼ P k (k = 1, 2, . . . , 5) into the collective preference relation:
0
P_ ð0Þ
0:5
0:42 0:42 0:64 0:80 0:86
B 0:74 0:5 0:38 0:82 B B B 0:82 0:72 0:5 0:74 ¼B B 0:48 0:38 0:42 0:5 B B @ 0:28 0:32 0:34 0:32 0:42 0:32 0:28 0:38
1
0:92 0:88 C C C 0:82 0:92 C C 0:92 0:74 C C C 0:5 0:40 A 0:72
0:5
From the foregoing analysis, we know that the OWA operator based on the fuzzy linguistic quantifier can only utilize the preferences of the majority of decision makers, and that it produces serious losses of preference information in the process of aggregation. The collective preference relation derived from the individual fuzzy preference relations using the OWA operator is also generally not a fuzzy preference relation. Additionally, the weights associated with the OWA operator are those of the ordered positions of the considered arguments, rather than the arguments themselves.
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Although the experts generally have equal weight in deciding group preferences, there are many group decision-making situations that require different weights [11]. Bodily [1] suggested an interesting way of determining the weights of the experts by designating an expert’s voting weight to a delegation subcommittee made up of the other experts of the group. Assume that the weight vector of the experts ek (k = 1, 2, . . . , 5) given by this technique is w = (0.1, 0.3, 0.2, 0.1, 0.3)T. Then, using ð0Þ (4) we can aggregate all the fuzzy preference relations P k ¼ P k ðk ¼ 1; 2; . . . ; 5Þ into the collective preference relation:
0
P€ð0Þ
0:5 B 0:66 B B 0:67 B ¼B B 0:41 B @ 0:28 0:31
0:34 0:5 0:70 0:27 0:17 0:25
0:33 0:30 0:5 0:36 0:24 0:17
0:59 0:73 0:64 0:5 0:22 0:32
0:77 0:83 0:76 0:78 0:5 0:65
1 0:69 0:75 C C 0:83 C C C 0:68 C C 0:35 A 0:5
which is also a fuzzy preference relation. In many practical problems, experts may come from different places and possess different professional backgrounds. As they may be unfamiliar with each other, they may be unable (or unwilling) to designate voting weights to the others. Consequently, Bodily’s [1] approach cannot deal with these cases. However, the algorithms we have developed in this paper are suitable for solving this problem. Not only can they alleviate the influence of unduly high or low preferences by assigning small weights to the fuzzy preference relations with large deviations, they are also capable of achieving group opinions that have high or acceptable levels of consensus. In the following, we utilize Algorithm 1 to obtain the solution to the problem. We first apply the model (M-3) to determine the optimal weight vector:
wð0Þ ¼ ð0:05; 0:29; 0:15; 0:19; 0:32ÞT From (4), we can derive the collective fuzzy preference relation:
0
Pð0Þ
0:5 B 0:68 B B 0:70 B ¼B B 0:42 B @ 0:22 0:30
0:32 0:5 0:72 0:25 0:19 0:23
0:30 0:28 0:5 0:34 0:26 0:20
0:58 0:75 0:66 0:5 0:19 0:30
0:78 0:81 0:74 0:81 0:5 0:63
1 0:70 0:77 C C 0:80 C C C 0:70 C C 0:37 A 0:5
Using (22) and (23), we calculate the deviations: ð0Þ
dðP2 ; Pð0Þ Þ ¼ 0:0747;
ð0Þ
dðP5 ; Pð0Þ Þ ¼ 0:0580;
dðP1 ; Pð0Þ Þ ¼ 0:0940; dðP4 ; Pð0Þ Þ ¼ 0:1393;
ð0Þ
dðP3 ; Pð0Þ Þ ¼ 0:0893
ð0Þ
ð0Þ
D1 ð0Þ ¼ 0:0848
Suppose that the experts go through a negotiation process and predefine the threshold of acceptable group consensus as ð0Þ k1 = 0.05. As a result, the group has an unacceptable level of consensus. Because all the deviations dðPk ; P ð0Þ Þ ðk ¼ 1; 2; . . . ; 5Þ ð0Þ are greater than 0.05, we need to return Pk ðk ¼ 1; 2; . . . ; 5Þ together with P(0) (as a reference) to the experts ek (k = 1,2,. . .,5), respectively, for revaluation. They then construct the new fuzzy preference relations:
0
ð1Þ
P1
ð1Þ
P3
ð1Þ
P5
0:5 B 0:6 B B 0:7 B ¼B B 0:4 B @ 0:2 0:4 0 0:5 B 0:6 B B 0:6 B ¼B B 0:4 B @ 0:3 0:2 0 0:5 B 0:7 B B 0:7 B ¼B B 0:4 B @ 0:2 0:4
0:4 0:5 0:7 0:3 0:2 0:3 0:4 0:5 0:7 0:2 0:2 0:2 0:3 0:5 0:7 0:3 0:2 0:2
0:3 0:3 0:5 0:4 0:2 0:2 0:4 0:3 0:5 0:3 0:3 0:2 0:3 0:3 0:5 0:3 0:3 0:2
0:6 0:7 0:6 0:5 0:2 0:4 0:6 0:8 0:7 0:5 0:2 0:4 0:6 0:7 0:7 0:5 0:1 0:3
0:8 0:8 0:8 0:8 0:5 0:7 0:7 0:8 0:7 0:8 0:5 0:7 0:8 0:8 0:7 0:9 0:5 0:6
1 0:6 0:7 C C 0:8 C C C; 0:6 C C 0:3 A 0:5 1 0:8 0:8 C C 0:8 C C C; 0:7 C C 0:3 A 0:5 1 0:6 0:8 C C 0:8 C C C 0:7 C C 0:4 A 0:5
0
ð1Þ
P2
ð1Þ
P4
0:5 B 0:7 B B 0:7 B ¼B B 0:5 B @ 0:2 0:3 0 0:5 B 0:7 B B 0:7 B ¼B B 0:5 B @ 0:2 0:3
0:3 0:5 0:7 0:3 0:1 0:3 0:3 0:5 0:8 0:2 0:3 0:2
0:3 0:3 0:5 0:4 0:2 0:1 0:3 0:2 0:5 0:3 0:2 0:2
0:5 0:7 0:6 0:5 0:2 0:3 0:5 0:8 0:7 0:5 0:2 0:3
0:8 0:9 0:8 0:8 0:5 0:6 0:8 0:7 0:8 0:8 0:5 0:6
1 0:7 0:7 C C 0:9 C C C 0:7 C C 0:4 A 0:5 1 0:7 0:8 C C 0:8 C C C 0:7 C C 0:4 A 0:5
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Then, based on the fuzzy preference relations Pk ðk ¼ 1; 2; . . . ; 5Þ, we can apply the quadratic programming model (M-4) (or (21)) to determine the optimal weight vector:
wð1Þ ¼ ð0:20; 0:20; 0:20; 0:20; 0:20ÞT After that, we can utilize (4) to get the collective fuzzy preference relation:
0
Pð1Þ
0:5
0:34 0:32 0:56 0:78 0:68
1
B 0:66 0:5 0:28 0:74 0:80 0:76 C C B C B B 0:68 0:72 0:5 0:66 0:76 0:82 C C ¼B B 0:44 0:26 0:32 0:5 0:82 0:68 C C B C B @ 0:22 0:20 0:28 0:19 0:5 0:36 A 0:32 0:24 0:18 0:30 0:64
0:5
Using (22) and (23), we calculate the deviations: ð1Þ
dðP2 ; Pð1Þ Þ ¼ 0:0427;
ð1Þ
dðP5 ; Pð1Þ Þ ¼ 0:0360;
dðP1 ; Pð1Þ Þ ¼ 0:0413; dðP4 ; Pð1Þ Þ ¼ 0:0413;
ð1Þ
dðP3 ; Pð1Þ Þ ¼ 0:0480
ð1Þ
ð1Þ
D1 ð1Þ ¼ 0:0419
Then D1(1) 6 0.05, which means that the group reaches an acceptable level of consensus. In cases where consensus must be urgently obtained, or the experts cannot or are unwilling to revaluate the alternatives, we can use Step 4 of Algorithm 2 (without loss of generality, set g = 0.3) to obtain the new preference relations:
0
ð1Þ
P1
0:5 B 0:66 B B B 0:73 ¼B B 0:41 B B @ 0:24
0:34 0:27 0:59 0:76 0:67
1
0:5 0:32 0:71 0:84 0:75 C C C 0:68 0:5 0:64 0:76 0:86 C C; 0:29 0:36 0:5 0:78 0:67 C C C 0:16 0:24 0:22 0:5 0:35 A
0:33 0:25 0:14 0:33 0:65 0:5 1 0:5 0:37 0:39 0:59 0:76 0:76 B 0:63 0:5 0:29 0:77 0:78 0:78 C C B C B B 0:61 0:71 0:5 0:67 0:73 0:80 C C; B ¼B C B 0:41 0:23 0:33 0:5 0:81 0:67 C C B @ 0:24 0:22 0:27 0:19 0:5 0:32 A
0
ð1Þ
P2
0
ð1Þ
P3
0:5 B 0:69 B B B 0:70 ¼B B 0:44 B B @ 0:21
ð1Þ
P4
ð1Þ
0:32 0:74 0:87 0:78 C C C 0:68 0:5 0:61 0:79 0:83 C C 0:26 0:39 0:5 0:75 0:70 C C C 0:13 0:21 0:25 0:5 0:38 A
0:30 0:22 0:17 0:30 0:62 0:5 1 0:5 0:28 0:24 0:56 0:79 0:76 B 0:72 0:5 0:26 0:80 0:75 0:84 C C B C B B 0:76 0:74 0:5 0:70 0:70 0:74 C C B ¼B C B 0:44 0:20 0:30 0:5 0:87 0:73 C C B @ 0:21 0:25 0:30 0:13 0:5 0:38 A
0:24 0:22 0:20 0:33 0:68 0:5 1 0:5 0:31 0:30 0:62 0:79 0:64 B 0:69 0:5 0:26 0:74 0:81 0:72 C C B C B B 0:70 0:74 0:5 0:67 0:73 0:80 C C B ¼B C B 0:38 0:26 0:33 0:5 0:84 0:70 C C B @ 0:21 0:19 0:27 0:16 0:5 0:38 A 0:36 0:28 0:20 0:30 0:62
0:24 0:16 0:26 0:27 0:62
0:5
and then apply the model (M-4) to derive the optimal weight vector:
wð1Þ ¼ ð0:203; 0:198; 0:200; 0:200; 0:199ÞT Using (4), we can get the collective fuzzy preference relation:
0
Pð1Þ
0:5
0:32 0:30 0:58 0:78 0:71
1
B 0:68 0:5 0:29 0:75 0:81 0:77 C C B C B B 0:70 0:71 0:5 0:66 0:74 0:81 C C ¼B B 0:42 0:25 0:34 0:5 0:81 0:69 C C B C B @ 0:22 0:19 0:26 0:19 0:5 0:36 A 0:29 0:23 0:19 0:31 0:64
and based on
ð1Þ Pk
0:5
ðk ¼ 1; 2; 3Þ and P(1), we can calculate the deviations using (22) and (23):
ð1Þ
ð1Þ
ð1Þ
dðP1 ; Pð1Þ Þ ¼ 0:0260;
dðP2 ; Pð1Þ Þ ¼ 0:0247;
dðP3 ; Pð1Þ Þ ¼ 0:0247
ð1Þ dðP4 ; Pð1Þ Þ
ð1Þ dðP5 ; Pð1Þ Þ
D1 ð1Þ ¼ 0:0280
¼ 0:0440;
¼ 0:0207;
1
0:5
0
0
P5
0:31 0:30 0:56 0:79 0:70
which indicates that the group has also reached an acceptable level of consensus.
0:5
Z. Xu, X. Cai / Information Sciences 181 (2011) 150–162
161
5. Conclusions In group decision making with preference relations, the preference relations (such as fuzzy preference relations and multiplicative preference relations) provided by decision makers generally have different importance weights. Accordingly, the question of how these importance weights should be determined presents an interesting research topic. In this paper, we have established several goal programming models and quadratic programming models based on minimizing deviation variables from the viewpoint of maximizing group consensus. These models allow us to determine the importance weights of fuzzy preference relations. We have further developed two iterative algorithms for group decision making with fuzzy preference relations. In cases where the group does not reach an acceptable level of consensus, Algorithm 1 interacts with the decision makers until the group achieves an acceptable level of consensus or the number of the interactions reaches a predefined maximum number. If the threshold of acceptable group consensus is reduced, the number of interactions generally increases. The algorithm is very suitable for solving group decision-making problems, such as vendor selection in supply chain management, personnel examination, military system efficiency evaluation, and venture capital project evaluation. Algorithm 2 is a convergent iterative algorithm, which can automatically modify the diverging individual fuzzy preference relations so that the group’s opinion achieves an acceptable level of consensus. These two algorithms have their own characteristics, and are complementary in practical applications. The main advantages and functions of our algorithms are as follows: (1) If the weights of the fuzzy reference relations in group decision-making problems cannot be predefined, then using the goal of maximizing group consensus, we can apply our models to determine the optimal weights of these fuzzy preference relations. The models can alleviate the influence of the unduly high or low preferences of some decision makers on the decision result, by assigning appropriately small weights to the fuzzy preference relations with large deviations. Using the well-known WAA operator, we fuse all the individual fuzzy preference relations into the collective fuzzy preference relation. Next, the interactive Algorithm 1 can be used to implement an interactive revaluation process, to reach a high or acceptable level of group consensus. This process may sometimes require an excessive amount of time to complete. In cases where consensus is urgently required, or decision makers cannot or are unwilling to revaluate the alternatives, we can use the iterative Algorithm 2 to automatically modify the diverging individual fuzzy preference relations to make the level of group consensus as high as possible. (2) If the information about the weights of fuzzy reference relations is known, then we can also use the WAA operator to fuse all the individual fuzzy preference relations into the collective fuzzy preference relation, and then utilize Algorithm 1 or Algorithm 2 to reach an acceptable level of group consensus. In addition, we have also established goal programming models and quadratic programming models to derive the importance weights of multiplicative preference relations. Algorithms 1 and 2 can also be used to establish group consensus with multiplicative preference relations. An interesting topic for future study would be to extend our models and algorithms to group decision-making problems based on interval-valued fuzzy preference relations or interval-valued multiplicative preference relations. Acknowledgments The work was partly supported by the National Science Fund for Distinguished Young Scholars of China (No. 70625005), the National Natural Science Foundation of China (Nos. 71071161 and 70932005), and the Research Grants Council of Hong Kong under General Research Fund Project No. 410208. The authors are very grateful to the anonymous referees for their constructive comments and suggestions that have led to an improved version of this paper. References [1] S.E. Bodily, A delegation process for combining individual utility functions, Management Science 25 (1979) 1035–1041. [2] F. Chiclana, F. Herrera, E. Herrera-Viedma, Integrating three representation models in fuzzy multipurpose decision making based on fuzzy preference relations, Fuzzy Sets and Systems 97 (1998) 33–48. [3] F. Chiclana, F. Herrera, E. Herrera-Viedma, L. Martı´nez, A note on the reciprocity in the aggregation of fuzzy preference relations using OWA operators, Fuzzy Sets and Systems 137 (2003) 71–83. [4] F. Chiclana, E. Herrera-Viedma, F. Herrera, S. Alonso, Induced ordered weighted geometric operators and their use in the aggregation of multiplicative preference relations, International Journal of Intelligent Systems 19 (2004) 233–255. [5] F. Chiclana, E. Herrera-Viedma, F. Herrera, S. Alonso, Some induced ordered weighted averaging operators and their use for solving group decisionmaking problems based on fuzzy preference relations, European Journal of Operational Research 182 (2007) 383–399. [6] J.C. Harsanyi, Cardinal, welfare, individualistic ethics, and interpersonal comparisons of utility, Journal of Political Economy 63 (1955) 309–321. [7] F. Herrera, E. Herrera-Viedma, F. Chiclana, Multi-person decision-making based on multiplicative preference relations, European Journal Operational Research 129 (2001) 372–385. [8] F. Herrera, E. Herrera-Viedma, F. Chiclana, A study of the origin and uses of the ordered weighted geometric operator in multicriteria decision making, International Journal of Intelligent Systems 18 (2003) 689–707. [9] J.M. Merigó, A.M. Gil-Lafuente, The induced generalized OWA operator, Information Sciences 179 (2009) 729–741. [10] S. Önüt, T. Efendigil, S.S. Kara, A combined fuzzy MCDM approach for selecting shopping center site: an example from Istanbul, Turkey, Expert Systems with Applications 37 (2010) 1973–1980.
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[11] R. Ramanathan, L.S. Ganesh, Group preference aggregation methods employed in AHP: an evaluation and an intrinsic process for deriving members’ weightages, European Journal of Operational Research 79 (1994) 249–265. [12] T.L. Saaty, The Analytic Hierarchy Process, McGraw-Hill, New York, 1980. [13] H.A. Simon, Models of Bounded Rationality: Empirically Grounded Economic Reason, vol. 3, MIT Press, Cambridge, 1997. [14] V. Torra, Y. Narukawa, Modeling Decisions: Information Fusion and Aggregation Operators, Springer, Heidelberg, 2007. [15] T. Wanyama, B.H. Far, A multi-agent framework for conflict analysis and negotiation: case of COTS selection, IEICE Transactions on Information and Systems: Special Section on Software Agent and Its Applications 88 (2005) 2047–2058. [16] E.N. Weiss, V.R. Rao, AHP design issues for large scale systems, Decision Sciences 18 (1987) 43–61. [17] Z.S. Xu, An overview of methods for determining OWA weights, International Journal of Intelligent Systems 20 (2005) 843–865. [18] Z.S. Xu, An automatic approach to reaching consensus in multiple attribute group decision making, Computers & Industrial Engineering 56 (2009) 1369–1374. [19] Z.S. Xu, Q.L. Da, The ordered weighted geometric averaging operators, International Journal of Intelligent Systems 17 (2002) 709–716. [20] Z.S. Xu, Q.L. Da, An overview of operators for aggregating information, International Journal of Intelligent Systems 18 (2003) 953–969. [21] Z.S. Xu, J. Chen, An interactive method for fuzzy multiple attribute group decision making, Information Sciences 177 (2007) 248–263. [22] Z.S. Xu, J. Chen, Group decision-making procedures based on incomplete reciprocal relations, Soft Computing 12 (2008) 515–521. [23] Z.S. Xu, Choquet integrals of weighted intuitionistic fuzzy information, Information Sciences 180 (2010) 726–736. [24] R.R. Yager, On ordered weighted averaging aggregation operators in multi-criteria decision making, IEEE Transactions on Systems, Man and Cybernetics 18 (1988) 183–190. [25] R.R. Yager, The induced fuzzy integral aggregation operator, International Journal of Intelligent Systems 17 (2002) 1049–1065. [26] R.R. Yager, Using trapezoids for representing granular objects: applications to learning and OWA aggregation, Information Sciences 178 (2008) 363– 380. [27] R.R. Yager, On the dispersion measure of OWA operators, Information Sciences 179 (2009) 3908–3919.
Contents lists available at ScienceDirect
Information Sciences journal homepage: www.elsevier.com/locate/ins
Group consensus algorithms based on preference relations Zeshui Xu a,b,⇑, Xiaoqiang Cai b a b
School of Economics and Management, Southeast University, Nanjing, Jiangsu 210096, China Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong
a r t i c l e
i n f o
Article history: Received 12 June 2009 Received in revised form 1 August 2010 Accepted 3 August 2010
Keywords: Group decision making Fuzzy preference relation Multiplicative preference relation Goal programming model Quadratic programming Iterative algorithm
a b s t r a c t In many group decision-making situations, decision makers’ preferences for alternatives are expressed in preference relations (including fuzzy preference relations and multiplicative preference relations). An important step in the process of aggregating preference relations, is to determine the importance weight of each preference relation. In this paper, we develop a number of goal programming models and quadratic programming models based on the idea of maximizing group consensus. Our models can be used to derive the importance weights of fuzzy preference relations and multiplicative preference relations. We further develop iterative algorithms for reaching acceptable levels of consensus in group decision making based on fuzzy preference relations or multiplicative preference relations. Finally, we include an illustrative example. Ó 2010 Published by Elsevier Inc.
1. Introduction Group decision-making problems generally involve finite non-empty sets of alternatives that have been generated through search processes [13] and require a predefined group of decision makers. In the knowledge-based technique for group decision making developed by Wanyama and Far [15], multiple user agents utilize a systematic model that assists Commercial Off-The-Shelf (COS) selection stakeholders to identify conflicts, and to determine and evaluate conflict resolution options. In addition, each user agent analyzes the agreement options in detail before advising its client about which goals to optimize, and which goals to compromise, to reach agreement with the other stakeholders. In many group decision-making situations, such as investment decisions, project evaluations, personnel evaluation, and medical diagnoses, the decision makers usually compare each set of alternatives. They also use predefined evaluation scales to determine which alternatives they prefer. Fuzzy preference relations and multiplicative preference relations are two of the most common and useful tools for representing decision makers’ preferences. To get the group’s opinion in a decision-making process, an appropriate technique is required to aggregate all the individual fuzzy preference relations or all the multiplicative preference relations. Consequently, much of the literature has investigated approaches for information aggregation, and a variety of aggregation operators have been developed [14,20]. Among these, the Ordered Weighted Averaging (OWA) [9,24,26], Ordered Weighted Geometric (OWG) [7,19,23], Weighted Arithmetic Averaging (WAA) [6], and Weighted Geometric Averaging (WGA) [12] are the most common operators for aggregating fuzzy preference relations and multiplicative preference relations. The fundamental feature of the OWA and OWG operators is their reordering step. They reorder all the given arguments in descending order, and then weight the ordered position of an argument but ignore the importance of the argument itself. Chiclana et al. [2] used fuzzy preference relations to make the information uniform for problems where the information about alternatives is represented by preference orderings, utility functions and fuzzy preference relations. Based on the ⇑ Corresponding author at: School of Economics and Management, Southeast University, Nanjing, Jiangsu 210096, China. Tel.: +86 25 84483382. E-mail addresses: [email protected] (Z. Xu), [email protected] (X. Cai). 0020-0255/$ - see front matter Ó 2010 Published by Elsevier Inc. doi:10.1016/j.ins.2010.08.002
Z. Xu, X. Cai / Information Sciences 181 (2011) 150–162
151
concept of fuzzy majority, they used the OWA operator, whose weights are calculated by the fuzzy quantifier, to aggregate fuzzy preference relations. Drawing on the Induced Ordered Weighted Averaging (IOWA) operator developed by Yager [25], Chiclana et al. [5] introduced an importance IOWA operator, a consistency IOWA operator and a preference IOWA operator. They then used these operators to aggregate fuzzy preference relations based on the concept of fuzzy majority. Herrera et al. [7] studied problems where the information about alternatives can be presented by means of preference orderings, utility functions and multiplicative preference relations. They used multiplicative preference relations to make the information uniform, and then employed the OWG operator to aggregate multiplicative preference relations based on the concept of fuzzy majority. In another study, Chiclana et al. [4] used the importance IOWG, consistency IOWG and preference IOWG operators to aggregate multiplicative preference relations. However, the aggregated results produced by these operators generally do not satisfy the reciprocity property (i.e., the aggregated results are generally not fuzzy preference relations or multiplicative preference relations). Chiclana et al. [3,8] further used the OWA and OWG operators to study the conditions in which the reciprocity property is maintained when aggregating fuzzy preference relations or multiplicative preference relations. Furthermore, numerous methods have been proposed to determine the OWA weights [17,27]. However, while the weights associated with these operators reflect the ordered positions of the preferences, they do not consider the fuzzy preference relations or multiplicative preference relations. The WAA and WGA operators first weight all the given arguments using a normalized weight vector, and then aggregate the weighted arguments. The underlying characteristic of both the WAA and WGA operators is that they take the importance of the sources of information into account when computing the aggregated arguments [21]. The preference relation of all the fuzzy preference relations aggregated by the WAA operator is a fuzzy preference relation [22]. The preference relation of all the multiplicative preference relations aggregated by the WGA operator is also a multiplicative preference relation [12]. Clearly, the key step in using the WAA (or WGA) operator to aggregate fuzzy preference relations (or multiplicative preference relations) is determining the importance weight of each fuzzy preference relation (or multiplicative preference relation). For the sake of convenience, all fuzzy preference relations (or multiplicative preference relations) are generally assigned equal weight. However, the fuzzy preference relations (or multiplicative preference relations) are provided by the decision makers. A decision maker is usually only able to comment on the part of a problem he/she feels competent to address [16]. In other words, he/she cannot be expected to have sufficient expertise to comment on all aspects of the problem. Thus, in many practical applications the fuzzy preference relations (or multiplicative preference relations) provided by decision makers require different weights, especially in situations involving policy specification [11]. In this paper, we address this issue by developing goal programming models and quadratic programming models to derive the importance weights of fuzzy preference relations and multiplicative preference relations. In addition, we will also introduce iterative algorithms for establishing acceptable levels of consensus in group decision making. The remainder of the paper is organized as follows. Section 2 formulates models to derive the importance weights of fuzzy preference relations from the perspective of maximizing group consensus. Based on our models, we then develop iterative algorithms for group decision making with fuzzy preference relations to make the level of group consensus level as high as possible. Section 3 establishes a number of optimization models for group consensus based on multiplicative preference relations. Section 4 applies our algorithms in the evaluation and selection of suitable locations for shopping centers. Finally, Section 5 concludes the paper. 2. Group consensus algorithms based on fuzzy preference relations Consider a group decision-making problem based on fuzzy preference relations, where there are a collection of n alternatives {x1, x2, . . . , xn}, and a group of m decision makers {e1, e2, . . . , em} participate in the decision making. To express their opinions about alternatives, each decision maker ek compares each pair of alternatives (xi, xj) on a 0–1 scale. He/she provides his/ her preference value pijk, where 0 6 pijk 6 1 denotes the preference degree of the alternative xi over xj. Specifically, 0 6 pijk < 0.5 indicates that xj is preferred to xi, and the smaller the pijk, the greater the preference degree of the alternative xj over xi. In particular, pijk = 0 indicates that xj is absolutely preferred to xi, pijk = 0.5 indicates indifference between xi and xj, 0.5 < pijk 6 1 indicates that xi is preferred to xj, and pijk = 1 indicates that xi is absolutely preferred to xj. All the preference values pijk(i, j = 1, 2, . . . , n) provided by the decision maker ek are contained in a fuzzy preference relation Pk = (pijk)nn, which satisfies
0 6 pijk 6 1;
piik ¼ 0:5;
pijk þ pjik ¼ 1;
i; j ¼ 1; 2; . . . ; n
ð1Þ
The importance weight of each fuzzy preference relation needs to be taken into account in practical applications, because the fuzzy preference relations provided by different decision makers generally have different levels of importance. Thus, to assign the appropriate weights, let w = (w1, w2, . . . , wm)T be the weight vector (which is to be determined) of the fuzzy preference relations Pk = (pijk)nn (k = 1, 2, . . . , m), where
wk P 0;
k ¼ 1; 2; . . . ; m;
m X
wk ¼ 1
ð2Þ
k¼1
To get the group’s opinion, we use the Weighted Arithmetic Averaging (WAA) operator:
pij ¼
m X k¼1
wk pijk ;
i; j ¼ 1; 2; . . . ; n
ð3Þ
152
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to aggregate all the individual fuzzy preference relations Pk = (pijk)nn (k = 1, 2, . . . , m) into the collective preference relation P = (pij)nn. It can be easily shown that P satisfies condition (1), and is thus also a fuzzy preference relation. Clearly, the fundamental role of the WAA operator is to determine the weight vector w. To examine this role, we will now discuss the relationship between each individual opinion and the group’s opinion: Case 1. If the individual fuzzy preference relation Pk is consistent with the collective fuzzy preference relation P, then Pk = P. In this case, each preference value pijk in Pk should be equal to the corresponding preference value pij in P, i.e., pijk = pij, for all i,j = 1, 2, . . . , n. Using (3), we have
pijk ¼
m X
wl pijl ;
for all i; j ¼ 1; 2; . . . ; n
ð4Þ
l¼1
If all the individual fuzzy preference relations Pk (k = 1, 2, . . . , m) are consistent with the collective fuzzy preference relation P, then (4) holds, for all k = 1, 2, . . . , m, i.e.,
pijk ¼
m X
wl pijl ;
for all i; j ¼ 1; 2; . . . ; n; k ¼ 1; 2; . . . ; m
ð5Þ
l¼1
In this case, the group reaches complete consensus. Accordingly, (5) is equivalent to (4), and using (1) and (4) is equivalent to the following:
pijk ¼
m X
wl pijl ;
for all i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n
ð6Þ
l¼1
from which we can derive the weight vector w. Case 2. If the individual fuzzy preference relation Pk is inconsistent with the collective fuzzy preference relation P, then (6) does not always hold. If the group does not reach complete consensus, then (5) does not always hold. In this case, we introduce the deviation variable eijk as follows:
eijk
m X ¼ pijk wl pijl ; l¼1
for all i; j ¼ 1; 2; . . . ; n; k ¼ 1; 2; . . . ; m
ð7Þ
It follows from (1) that (7) is equivalent to the following:
m
eijk ¼ pijk m wl pijl ; for all i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n; k ¼ 1; 2; . . . ; m l¼1
ð8Þ
In practice, there are many group decision-making problems, such as personnel examination, military system efficiency evaluation, and venture capital project evaluation, where some individual decision makers may assign unduly high or low preference values to their preferred or unwanted objects (alternatives). In these situations, it is necessary to employ an appropriate measure to adjust the extreme individual opinions, so that a sensible group decision can be reached. We propose to determine the weights of the individual preferences based on their deviations eijk (i = 1, 2, . . . , n 1; j = i + 1, . . . , n; k = 1, 2, . . . , m) from the group’s opinion. This is because the deviation eijk reflects the degree to which the individual fuzzy preferences depart from the group’s opinion. By adjusting the weights based on the deviations, we try to discount those opinions that are highly inconsistent with the assessments of the majority. Specifically, we aim to minimize eijk (i = 1, 2, . . . , n 1; j = i + 1, . . . , n; k = 1, 2, . . . , m), which leads to the following multiobjective programming model:
ðM-1Þ
min s:t:
eijk
m X ¼ pijk wl pijl ; l¼1
wl P 0;
i ¼ 1; 2; . . . ; n 1;
l ¼ 1; 2; . . . ; m;
m X
j ¼ i þ 1; . . . ; n;
k ¼ 1; 2; . . . ; m
wl ¼ 1
l¼1
A solution to the above minimization problem can be found by solving the following goal programming model:
ðM-2Þ
min
J1 ¼
m X n1 X n X þ sijk dijk þ t ijk dijk k¼1 i¼1 j¼iþ1
s:t:
pijk
m X
þ
wl pijl dijk þ dijk ¼ 0; i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n; k ¼ 1; 2; . . . ; m
l¼1 þ
dijk P 0;
dijk P 0;
i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n; k ¼ 1; 2; . . . ; m m X wl P 0; l ¼ 1; 2; . . . ; m; wl ¼ 1 l¼1
Z. Xu, X. Cai / Information Sciences 181 (2011) 150–162
153
þ
where dijk is the positive deviation from the target of the goal eijk, defined as
(
þ
dijk ¼ max pijk
m X
)
wl pijl ; 0
ð9Þ
l¼1
dijk is the negative deviation from the target of the goal eijk, defined as dijk
¼ max
( m X
) wl pijl pijk ; 0
ð10Þ
l¼1 þ
sijk is the weighting factor corresponding to the positive deviation dijk , and tijk is the weighting factor corresponding to the negative deviation dijk . Considering that all the goal functions eijk (i = 1, 2, . . . , n 1; j = i + 1, . . . , n; k = 1, 2, . . . , m) are fair, we can set sijk = tijk = 1 (i = 1, 2, . . . , n 1; j = i + 1, . . . , n; k = 1, 2, . . . , m). The model (M-2) can then be rewritten as
ðM-3Þ
min
J1 ¼
m X n1 X n X þ dijk þ dijk k¼1 i¼1 j¼iþ1
s:t:
pijk
m X
þ
wl pijl dijk þ dijk ¼ 0;
i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n; k ¼ 1; 2; . . . ; m
l¼1 þ
dijk P 0; wl P 0;
dijk P 0;
i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n; m X l ¼ 1; 2; . . . ; n; wl ¼ 1
k ¼ 1; 2; . . . ; m
l¼1
By solving this model (M-3), we can obtain the optimal weight vector w = (w1, w2, . . . , wm)T of the fuzzy preference relations Pk = (pijk)nn (k = 1, 2, . . . , m). Another approach can also be used to solve the minimization problem (M-1), which transforms the model (M-1) into the following quadratic programming model:
ðM-4Þ
J 01
min
¼
m X n X n X k¼1 i¼1
s:t:
wl P 0;
2 ijk
e ¼
m X n X n X
j¼1
k¼1 i¼1
l ¼ 1; 2; . . . ; m;
pijk
m X
j¼1
m X
!2 wl pijl
l¼1
wl ¼ 1
l¼1
To solve this model, we can construct the Lagrange function:
! 2 m X n X n m X X m Lðw; kÞ ¼ pijk m wl pijl 2k wl 1 k¼1 i¼1
l¼1
j¼1
ð11Þ
l¼1
where k is the Lagrange multiplier. Differentiating (11) with respect to wh (h = 1, 2, . . . , m) and k, and setting these partial derivatives equal to zero, the following set of equations is obtained:
! m X n X n m X X @Lðw; kÞ ¼ 2 pijk wl pijl pijh 2k ¼ 0; @wh k¼1 i¼1 j¼1 l¼1
h ¼ 1; 2; . . . ; m
m @Lðw; kÞ X ¼ wl 1 ¼ 0 @k l¼1
ð12Þ ð13Þ
(12) can be simplified as m X n X n m X X k¼1 i¼1
j¼1
! wl pijl pijh pijk pijh
k ¼ 0;
h ¼ 1; 2; . . . ; m
ð14Þ
l¼1
i.e., n X m X i¼1
l¼1
wl
n X
! mpijl pijh
j¼1
n X n m X X i¼1
j¼1
! pijk pijh k ¼ 0;
h ¼ 1; 2; . . . ; m
ð15Þ
k¼1
which can be rewritten in a matrix form as
Dw p ke ¼ 0
ð16Þ
154
Z. Xu, X. Cai / Information Sciences 181 (2011) 150–162
where n X n X m X
p¼
i¼1
pijk pij1 ;
j¼1 k¼1
n X n X m X i¼1
pijk pij2 ; . . . ;
j¼1 k¼1
n X n X m X i¼1
!T pijk pijm
;
e ¼ ð1; 1; . . . ; 1ÞT
ð17Þ
j¼1 k¼1
and
0
n P n P 2 B i¼1 j¼1 mpij1 B B n P n BP B mpij1 pij2 D¼B B i¼1 j¼1 B ... B B n n @P P mpij1 pijm
n P n P
mpij1 pij2
i¼1 j¼1 n P n P i¼1 j¼1
mp2ij2
... n P n P
i¼1 j¼1
mpij2 pijm
i¼1 j¼1
1 mpij1 pijm C i¼1 j¼1 C C n P n C P ... mpij2 pijm C C C i¼1 j¼1 C ... ... C C n n PP A 2 ... mpijm ...
n P n P
i¼1 j¼1
ð18Þ
mm
Because (13) can be rewritten as
eT w ¼ 1
ð19Þ
solving (16) and (19) we get
k¼
1 eT D1 p
ð20Þ
eT D1 e
and
w¼
D1 eð1 eT D1 pÞ eT D1 e
þ D1 p
ð21Þ
which is also the optimal weight vector w = (w1,cw2, . . . , wm)T of the fuzzy preference relations Pk = (pijk)nn(k = 1, 2, . . . , m). Then, using (4), we get the collective fuzzy preference relation P. In addition, based on (8) and the optimal weight vector w, we can calculate the deviation between the individual fuzzy preference relation Pk and the collective fuzzy preference relation P by
dðP k ; PÞ ¼
n1 X n X
eijk
i¼1 j¼iþ1
n1 X n m X X 2 ¼ wl pijl pijk nðn 1Þ i¼1 j¼iþ1 l¼1
ð22Þ
Accordingly, the weighted sum of all the deviations d(Pk, P) (k = 1, 2, . . . , m) can be defined as
D1 ¼
m X
wk dðPk ; PÞ
ð23Þ
k¼1
From (22) and (23), we know that if d(Pk, P) = 0, then the individual fuzzy preference relation Pk is consistent with the collective fuzzy preference relation P. If D1 = 0, then the group reaches complete consensus. In addition, we suppose that if D1 6 k1, then the group reaches an acceptable level of consensus, where k1 is the threshold of an acceptable level of group consensus of fuzzy preference relations, which can be predefined by the decision makers. Below, we develop iterative algorithms for group decision making with fuzzy preference relations based on the goal programming model (M-3) and the quadratic programming model (M-4). Algorithm 1 Step 1. The decision makers ek (k = 1, 2, . . . , m) provide their preferences over the alternatives xi (i = 1, 2, . . . , n) on a 0–1 scale, and construct fuzzy preference relations Pk = (pijk)nn (k = 1, 2, . . . , m). Let t be the number of iterative times, t* the maximum iteration number, k1 the deadline for acceptable group consensus of fuzzy preference relations ð0Þ ð0Þ Pk ¼ ðpijk Þnn ¼ Pk ¼ ðpijk Þnn ðk ¼ 1; 2; . . . ; mÞ, and t = 0. ðtÞ ðtÞ Step 2. Utilize the WAA operator (3) to aggregate all the individual fuzzy preference relations P k ¼ ðpijk Þnn ðk ¼ 1; 2; . . . ; mÞ ðtÞ into the collective fuzzy preference relation PðtÞ ¼ ðpij Þnn . Step 3. Apply the goal programming model (M-3) or the quadratic programming model (M-4) to determine the optimal ðtÞ ðtÞ ðtÞ ðtÞ ðtÞ weight vector wðtÞ ¼ ðw1 ; w2 ; . . . ; wm ÞT of P k ¼ ðpijk Þnn ðk ¼ 1; 2; . . . ; mÞ, and then get the collective fuzzy prefer(t) ence relation P using (4). ðtÞ Step 4. Calculate the deviations dðPk ; P ðtÞ Þ ðk ¼ 1; 2; . . . ; mÞ and D1(t) using (22) and (23), respectively. If D1(t) 6 k1 or t = t*, ðtÞ ðtÞ then stop. Otherwise, there must exist at least one fuzzy preference relation P k such that dðP k ; PðtÞ Þ > k1 . In this ðtÞ (t) case, return P k together with P to the decision maker ek for revaluation, and construct a new fuzzy preference relation. Let t = t + 1, then go to Step 2.
Z. Xu, X. Cai / Information Sciences 181 (2011) 150–162
155
With Algorithm 1, when the group does not reach an acceptable level of consensus, some decision makers need to revaluate their preferences over the alternatives. Occasionally, this process may require an excessive amount of time to complete. In cases where consensus is urgently required, or the decision makers cannot or are unwilling to revaluate the alternatives, we can use the following algorithm to develop an acceptable level of group consensus: Algorithm 2 Steps 1–3. See Algorithm 1. ðtÞ Step4. Calculate the deviations dðPk ; P ðtÞ Þðk ¼ 1; 2; . . . ; mÞ and D1(t) using (22) and (23), respectively. If D1 (t) 6 k1 or ðtþ1Þ ðtþ1Þ t = t*, then stop. Otherwise, let Pk ¼ ðpijk Þnn ðk ¼ 1; 2; . . . ; mÞ, where ðtþ1Þ
pijk
ðtÞ
ðtÞ
¼ gpijk þ ð1 gÞpij ;
i; j ¼ 1; 2; . . . ; n; k ¼ 1; 2; . . . ; m; 0 < g < 1
ð24Þ
Then return to Step 2. Algorithm 2 is convergent. In fact, similar to [18], we have Theorem 1. Let Pk = (pijk)n n (k = 1, 2, . . . , m) be the fuzzy preference relations provided by the decision makers ek (k = 1, 2, . . . , m), ðtÞ and fP k g ðk ¼ 1; 2; . . . ; mÞ and {P(t)} the fuzzy preference relations sequences generated in Algorithm 2. Then for each t:
D1 ðt þ 1Þ ¼ gD1 ðtÞ < D1 ðtÞ;
0
ð25Þ
Especially, let k1 = 0, then
lim D1 ðtÞ ¼ 0
ð26Þ
t!þ1
3. Group consensus models based on multiplicative preference relations As in Section 2, here we consider a group decision-making problem based on multiplicative preference relations. Each decision maker ek compares each pair of alternatives (xi, xj) using a ratio scale (in particular as Saaty [12] showed, the 1–9 scale). He/she provides his/her preference value aijk, where 1/9 6 aijk 6 9, and aijk denotes the preference intensity of the alternative xi over xj. Specifically, 1/9 6 aijk < 1 means that xj is preferred to xi, and the smaller the aijk, the stronger the preference intensity of the alternative xj over xi. In particular, aijk = 1/9 means that xj is absolutely preferred to xi, aijk = 1 implies indifference between xi and xj, and 1 < aijk 6 9 means that xi is preferred to xj(aijk = 9 indicates that xi is absolutely preferred to xj). All the preference values aijk (i, j = 1, 2, . . . , n) provided by the decision maker ek are contained in a multiplicative preference relation Ak = (aijk)nn, which satisfies
aijk > 0;
aiik ¼ 1;
aijk ajik ¼ 1;
i; j ¼ 1; 2; . . . ; n
ð27Þ
Let v = (v1, v2, . . . , vm)T be the weight vector of the multiplicative preference relations Ak = (aijk)nn (k = 1, 2, . . . , m), where P vk P 0, k = 1, 2, . . . , m, and mk¼1 v k ¼ 1. To get the group’s opinion, we use the Weighted Geometric Averaging (WGA) operator:
aij ¼
m Y
ðaijk Þv k ;
i; j ¼ 1; 2; . . . ; n
ð28Þ
k¼1
to aggregate all the individual multiplicative preference relations Ak = (aijk)nn (k = 1, 2, . . . , m) into the collective preference relation A = (aij)nn. Note that A satisfies condition (28) and is therefore also a multiplicative preference relation. Now we discuss the relationship between each individual multiplicative preference relation and the collective multiplicative preference relation: Case 1. If the individual multiplicative preference relation Ak is consistent with the collective multiplicative preference relation A, then Ak = A. In this case, each preference value aijk in Ak should be equal to the corresponding preference value aij in A, i.e., aijk = aij, for all i, j = 1, 2, . . . , n. Using (28), we have
aijk ¼
m Y
ðaijl Þv l ;
for all i; j ¼ 1; 2; . . . ; n
ð29Þ
l¼1
According to (27) and (29) is equivalent to the following form:
aijk ¼
m Y
ðaijl Þv l ;
for all i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n
ð30Þ
l¼1
If all the individual multiplicative preference relations Ak (k = 1, 2, . . . , m) are consistent with the collective multiplicative preference relation A, i.e., (30) holds for all k = 1, 2, . . . , m, then the group reaches complete consensus. In this case, we transform (30) into the following form by taking common logarithms on both sides:
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Z. Xu, X. Cai / Information Sciences 181 (2011) 150–162
lg aijk ¼ lg
m Y
ðaijl Þv l ¼
l¼1
m X
v l lg aijl ;
for all i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n
ð31Þ
l¼1
with which we can derive the weight vector v. Case 2. If the individual multiplicative preference relation Ak is inconsistent with the collective multiplicative preference relation A, then (31) does not always hold. If the group does not reach complete consensus, then
lg aijk ¼ lg
m Y
ðaijl Þv l ¼
l¼1
m X
v l lg aijl ;
for all i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n; k ¼ 1; 2; . . . ; m
ð32Þ
l¼1
does not always hold. In this case, we introduce the deviation variable fijk as follows:
fijk
m X ¼ lg aijk v l lg aijl ; l¼1
for all i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n; k ¼ 1; 2; . . . ; m
ð33Þ
Similar to the model (M-1), we minimize fijk (i = 1, 2, . . . , n 1; j = i + 1, . . . , n; k = 1, 2, . . . , m), and construct the following multi-objective programming model:
ðM-5Þ
min
m X f ijk ¼ lg aijk v lg a l ijl ; l¼1
v l P 0;
s:t:
l ¼ 1; 2; . . . ; m;
i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n; k ¼ 1; 2; . . . ; m m X
vl ¼ 1
l¼1
The model (M-5) can be transformed into the following goal programming model:
ðM-6Þ
min
J2 ¼
m X n1 X n X þ sijk dijk þ t ijk dijk k¼1 i¼1 j¼iþ1
s:t:
lg aijk
m X
v l lg aijl dþijk þ dijk ¼ 0;
i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n; k ¼ 1; 2; . . . ; m
l¼1 þ
dijk P 0;
v l P 0;
dijk P 0;
i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n; k ¼ 1; 2; . . . ; m m X l ¼ 1; 2; . . . ; m; vl ¼ 1 l¼1
þ
where dijk is the positive deviation from the target of the goal fijk, defined as
( þ dijk
¼ max lg aijk
m X
)
v l lg aijl ; 0
ð34Þ
l¼1
dijk is the negative deviation from the target of the goal fijk, defined as dijk
¼ max
( m X
)
v l lg aijl lg aijk ; 0
ð35Þ
l¼1 þ
sijk is the weighting factor corresponding to the positive deviation dijk , and tijk is the weighting factor corresponding to the negative deviation dijk . Considering that all the goal functions fijk(i = 1, 2, . . . , n 1; j = i + 1, . . . , n; k = 1, 2, . . . , m) are fair, we can set sijk = tijk = 1(i = 1, 2, . . . , n 1; j = i + 1 ,. . . , n; k = 1, 2, . . . , m), and then the model (M-6) can be rewritten as
ðM-7Þ
min
J2 ¼
m X n1 X n X þ dijk þ dijk k¼1 i¼1 j¼iþ1
s:t:
lg aijk
m X
v l lg aijl dþijk þ dijk ¼ 0;
i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n; k ¼ 1; 2; . . . ; m
l¼1 þ
dijk P 0;
v l P 0;
dijk P 0;
i ¼ 1; 2; . . . ; n 1; j ¼ i þ 1; . . . ; n; k ¼ 1; 2; . . . ; m m X l ¼ 1; 2; . . . ; m; vl ¼ 1 l¼1
Solving the model (M-7), we can obtain the optimal weight vector v = (v1, v2, . . . , vm)T of the multiplicative preference relations Ak = (aijk)nn (k = 1, 2, . . . , m).
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Analogous to the model (M-4), we can transform the model (M-5) into the following quadratic programming model:
min
ðM-8Þ
J 02
¼
m X n X n X k¼1 i¼1
s:t:
v l P 0;
2 fijk
¼
j¼1
m X n X n X k¼1 i¼1
l ¼ 1; 2; . . . ; m;
lg aijk
j¼1
m X
m X
!2
v l lg aijl
l¼1
vl ¼ 1
l¼1
Solving the model (M-8), we can get the optimal weight vector
v¼
G1 e 1 eT G1 a eT G1 e
v = (v1, v2, . . . , vm)T as follows:
þ G1 a
ð36Þ
where
a¼
n X n X m X i¼1
lg aijk lg aij1 ;
j¼1 k¼1
n X n X m X i¼1
lg aijk lg aij2 ; . . . ;
j¼1 k¼1
n X n X m X i¼1
!T lg aijk lg aijm
ð37Þ
j¼1 k¼1
and G is the matrix, denoted as
0 P n P n mðlg aij1 Þ2 B i¼1 j¼1 B B n P n BP B m lg aij1 lg aij2 G¼B B i¼1 j¼1 B ... B B n n @P P m lg aij1 lg aijm i¼1 j¼1
n P n P
m lg aij1 lg aij2
i¼1 j¼1
mðlg aij2 Þ2
i¼1 j¼1
... i¼1 j¼1
n P n P
m lg aij1 lg aijm
1
C C C n P n C P ... m lg aij2 lg aijm C C C i¼1 j¼1 C ... ... C C n n PP 2 A ... mðlg aijm Þ i¼1 j¼1
n P n P
n P n P
...
m lg aij2 lg aijm
i¼1 j¼1
ð38Þ
mm
Then using (28), we can get the collective multiplicative preference relation A. In addition, based on (33) and the optimal weight vector v, we can calculate the deviation between the individual multiplicative preference relation Ak and the collective multiplicative preference relation A using
dðAk ; AÞ ¼
n1 X n n1 X n m X X X 2 fijk ¼ v lg a lg aijk l ijl nðn 1Þ i¼1 j¼iþ1 i¼1 j¼iþ1 l¼1
ð39Þ
The weighted sum of all the deviations d(Ak, A) (k = 1, 2, . . . , m) can be derived using the WAA operator:
D2 ¼
m X
v k dðAk ; AÞ
ð40Þ
k¼1
From (39) and (40), we know that if d(Ak, A) = 0, then the individual multiplicative preference relation Ak is consistent with the collective multiplicative preference relation A. If D2 = 0, then the group reaches complete consensus. In addition, we suppose that if D2 6 k2, then the group has an acceptable level of consensus, where k2 is the threshold of acceptable group consensus for multiplicative preference relations, which can be predefined by the decision makers. In the case where D2 > k2, we can utilize the idea of Algorithm 1 or Algorithm 2 to improve the group consensus. 4. Illustrative example To illustrate the algorithms developed in this paper, we consider a group decision-making problem that concerns the evaluation and selection of suitable locations for shopping centers (adapted from [10]). An investment company has conducted a feasibility study to determine an appropriate site on which to establish a shopping center in one of the strategic demand areas of Istanbul. Many different objectives must be achieved if the project is to be undertaken. One of the basic objectives is to select a suitable location. After identifying the range of feasible sites, the task is then to select the one most likely to optimize the company’s strategic performance. Hence, detailed interviews were conducted with 15 experts, including professional consultants, academics, and business development managers and staff working for the company, to evaluate the most suitable locations. The six potential locations finally identified were Mecidiyeköy, Levent, Kozyatag˘i, Maltepe, Bakırköy and Beylikdüzü, denoted as xi (i = 1, 2, . . . , 6), respectively. Mecidiyeköy and Levent are the most important financial centres on the European side of Istanbul. Kozyatag˘i is a developing business district on the Asian side of Istanbul that is also close to a residential area. Bakırköy is a densely populated residential district on the European side on the coast of the Marmara Sea, which also contains an important shopping and commercial center close to the Atatürk International Airport. Maltepe is a rapidly growing residential district on the Asian side on the coast of the Marmara Sea, which is also close to the Sabiha Gökçen Airport and the Istanbul Park GP Racing Circuit. Although Beylikdüzü
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is primarily an industrial district on the European side, it has started to develop residentially following the opening of exhibition, congress and amusement centers, and because it has easy access to motorways. Assume that a committee comprising five experts (decision makers) ek (k = 1, 2, . . . , 5) from each strategic decision area has been set up to provide information for assessing the locations xi (i = 1, 2, . . . , 6). After comparing each pair of locations, the experts ek (k = 1, 2, . . . , 5) give their preferences using a 0–1 scale, and then construct the fuzzy preference relations ð0Þ
Pk ¼ P k ¼ ðpijk Þ66 ðk ¼ 1; 2; . . . ; 5Þ:
0
ð0Þ
P1 ¼ P1
ð0Þ
P3 ¼ P3
ð0Þ
P5 ¼ P5
0:5 B 0:6 B B 0:8 ¼B B 0:4 B @ 0:3 0:4 0 0:5 B 0:5 B B 0:4 ¼B B 0:4 B @ 0:3 0:1 0 0:5 B 0:7 B B 0:7 ¼B B 0:3 B @ 0:2 0:5
0:4 0:5 0:6 0:4 0:1 0:3 0:5 0:5 0:7 0:2 0:3 0:2 0:3 0:5 0:8 0:3 0:2 0:4
0:2 0:4 0:5 0:4 0:2 0:0 0:6 0:3 0:5 0:3 0:3 0:2 0:3 0:2 0:5 0:3 0:3 0:2
0:6 0:6 0:6 0:5 0:3 0:4 0:6 0:8 0:7 0:5 0:2 0:4 0:7 0:7 0:7 0:5 0:1 0:3
0:7 0:9 0:8 0:7 0:5 0:7 0:7 0:7 0:7 0:8 0:5 0:8 0:8 0:8 0:7 0:9 0:5 0:6
1 0:6 0:7 C C 1:0 C C; 0:6 C C 0:3 A
0
ð0Þ
P2 ¼ P2
0:5 1 0:9 0:8 C C 0:8 C C; 0:6 C C 0:2 A
ð0Þ
P4 ¼ P4
0:5 1 0:5 0:6 C C 0:8 C C 0:7 C C 0:4 A
0:5 B 0:7 B B 0:7 ¼B B 0:5 B @ 0:2 0:3 0 0:5 B 0:8 B B 0:9 ¼B B 0:5 B @ 0:2 0:1
0:3 0:5 0:6 0:3 0:0 0:2 0:2 0:5 0:8 0:1 0:4 0:0
0:3 0:4 0:5 0:5 0:1 0:1 0:1 0:2 0:5 0:2 0:4 0:4
0:5 0:7 0:5 0:5 0:4 0:3 0:5 0:9 0:8 0:5 0:0 0:2
0:8 1:0 0:9 0:6 0:5 0:6 0:8 0:6 0:6 1:0 0:5 0:6
1 0:7 0:8 C C 0:9 C C 0:7 C C 0:4 A 0:5 1 0:9 1:0 C C 0:6 C C 0:8 C C 0:4 A 0:5
0:5
In [2], Chiclana et al. utilized the OWA operator [24]:
OWAða1 ; a2 ; . . . ; an Þ ¼
n X
xj bj
ð41Þ
j¼1
to aggregate the individual preferences into the group’s opinion, where bj is the jth largest of a collection of arguments ai (i = 1, 2, . . . , n). x = (x1, x2, . . . , xn)T is a weighting vector associated with the OWA operator, such that xj P 0, j = 1, 2, . . . , n, P and nj¼1 xj ¼ 1. They can then be computed by means of a fuzzy linguistic quantifier Q used to represent the fuzzy majority as follows:
xj ¼ Q
j j1 Q ; n n
j ¼ 1; 2; . . . ; n
ð42Þ
where
Q ðrÞ ¼
8 > < 0;
ra ; > ba
:
1;
if r < a if a 6 r 6 b if r > b
ð43Þ
with a, b, r 2 [0, 1]. Some examples of non-decreasing proportional fuzzy linguistic quantifiers with the parameters (a, b) are [2]: ‘‘most” (0.3, 0.8), ‘‘at least half” (0, 0.5), and ‘‘as many as possible” (0.5, 1), respectively. Using (42) and the fuzzy majority criterion with the fuzzy quantifier ‘‘at least half”, with the pair (0, 0.5), we get the weighting vector x = (x1, x2, . . . , x5)T as: x = (0.4, 0.4, 0.2, 0, 0)T. Then using the OWA operator (41), we aggregate all the fuzð0Þ zy preference relations P k ¼ P k (k = 1, 2, . . . , 5) into the collective preference relation:
0
P_ ð0Þ
0:5
0:42 0:42 0:64 0:80 0:86
B 0:74 0:5 0:38 0:82 B B B 0:82 0:72 0:5 0:74 ¼B B 0:48 0:38 0:42 0:5 B B @ 0:28 0:32 0:34 0:32 0:42 0:32 0:28 0:38
1
0:92 0:88 C C C 0:82 0:92 C C 0:92 0:74 C C C 0:5 0:40 A 0:72
0:5
From the foregoing analysis, we know that the OWA operator based on the fuzzy linguistic quantifier can only utilize the preferences of the majority of decision makers, and that it produces serious losses of preference information in the process of aggregation. The collective preference relation derived from the individual fuzzy preference relations using the OWA operator is also generally not a fuzzy preference relation. Additionally, the weights associated with the OWA operator are those of the ordered positions of the considered arguments, rather than the arguments themselves.
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Although the experts generally have equal weight in deciding group preferences, there are many group decision-making situations that require different weights [11]. Bodily [1] suggested an interesting way of determining the weights of the experts by designating an expert’s voting weight to a delegation subcommittee made up of the other experts of the group. Assume that the weight vector of the experts ek (k = 1, 2, . . . , 5) given by this technique is w = (0.1, 0.3, 0.2, 0.1, 0.3)T. Then, using ð0Þ (4) we can aggregate all the fuzzy preference relations P k ¼ P k ðk ¼ 1; 2; . . . ; 5Þ into the collective preference relation:
0
P€ð0Þ
0:5 B 0:66 B B 0:67 B ¼B B 0:41 B @ 0:28 0:31
0:34 0:5 0:70 0:27 0:17 0:25
0:33 0:30 0:5 0:36 0:24 0:17
0:59 0:73 0:64 0:5 0:22 0:32
0:77 0:83 0:76 0:78 0:5 0:65
1 0:69 0:75 C C 0:83 C C C 0:68 C C 0:35 A 0:5
which is also a fuzzy preference relation. In many practical problems, experts may come from different places and possess different professional backgrounds. As they may be unfamiliar with each other, they may be unable (or unwilling) to designate voting weights to the others. Consequently, Bodily’s [1] approach cannot deal with these cases. However, the algorithms we have developed in this paper are suitable for solving this problem. Not only can they alleviate the influence of unduly high or low preferences by assigning small weights to the fuzzy preference relations with large deviations, they are also capable of achieving group opinions that have high or acceptable levels of consensus. In the following, we utilize Algorithm 1 to obtain the solution to the problem. We first apply the model (M-3) to determine the optimal weight vector:
wð0Þ ¼ ð0:05; 0:29; 0:15; 0:19; 0:32ÞT From (4), we can derive the collective fuzzy preference relation:
0
Pð0Þ
0:5 B 0:68 B B 0:70 B ¼B B 0:42 B @ 0:22 0:30
0:32 0:5 0:72 0:25 0:19 0:23
0:30 0:28 0:5 0:34 0:26 0:20
0:58 0:75 0:66 0:5 0:19 0:30
0:78 0:81 0:74 0:81 0:5 0:63
1 0:70 0:77 C C 0:80 C C C 0:70 C C 0:37 A 0:5
Using (22) and (23), we calculate the deviations: ð0Þ
dðP2 ; Pð0Þ Þ ¼ 0:0747;
ð0Þ
dðP5 ; Pð0Þ Þ ¼ 0:0580;
dðP1 ; Pð0Þ Þ ¼ 0:0940; dðP4 ; Pð0Þ Þ ¼ 0:1393;
ð0Þ
dðP3 ; Pð0Þ Þ ¼ 0:0893
ð0Þ
ð0Þ
D1 ð0Þ ¼ 0:0848
Suppose that the experts go through a negotiation process and predefine the threshold of acceptable group consensus as ð0Þ k1 = 0.05. As a result, the group has an unacceptable level of consensus. Because all the deviations dðPk ; P ð0Þ Þ ðk ¼ 1; 2; . . . ; 5Þ ð0Þ are greater than 0.05, we need to return Pk ðk ¼ 1; 2; . . . ; 5Þ together with P(0) (as a reference) to the experts ek (k = 1,2,. . .,5), respectively, for revaluation. They then construct the new fuzzy preference relations:
0
ð1Þ
P1
ð1Þ
P3
ð1Þ
P5
0:5 B 0:6 B B 0:7 B ¼B B 0:4 B @ 0:2 0:4 0 0:5 B 0:6 B B 0:6 B ¼B B 0:4 B @ 0:3 0:2 0 0:5 B 0:7 B B 0:7 B ¼B B 0:4 B @ 0:2 0:4
0:4 0:5 0:7 0:3 0:2 0:3 0:4 0:5 0:7 0:2 0:2 0:2 0:3 0:5 0:7 0:3 0:2 0:2
0:3 0:3 0:5 0:4 0:2 0:2 0:4 0:3 0:5 0:3 0:3 0:2 0:3 0:3 0:5 0:3 0:3 0:2
0:6 0:7 0:6 0:5 0:2 0:4 0:6 0:8 0:7 0:5 0:2 0:4 0:6 0:7 0:7 0:5 0:1 0:3
0:8 0:8 0:8 0:8 0:5 0:7 0:7 0:8 0:7 0:8 0:5 0:7 0:8 0:8 0:7 0:9 0:5 0:6
1 0:6 0:7 C C 0:8 C C C; 0:6 C C 0:3 A 0:5 1 0:8 0:8 C C 0:8 C C C; 0:7 C C 0:3 A 0:5 1 0:6 0:8 C C 0:8 C C C 0:7 C C 0:4 A 0:5
0
ð1Þ
P2
ð1Þ
P4
0:5 B 0:7 B B 0:7 B ¼B B 0:5 B @ 0:2 0:3 0 0:5 B 0:7 B B 0:7 B ¼B B 0:5 B @ 0:2 0:3
0:3 0:5 0:7 0:3 0:1 0:3 0:3 0:5 0:8 0:2 0:3 0:2
0:3 0:3 0:5 0:4 0:2 0:1 0:3 0:2 0:5 0:3 0:2 0:2
0:5 0:7 0:6 0:5 0:2 0:3 0:5 0:8 0:7 0:5 0:2 0:3
0:8 0:9 0:8 0:8 0:5 0:6 0:8 0:7 0:8 0:8 0:5 0:6
1 0:7 0:7 C C 0:9 C C C 0:7 C C 0:4 A 0:5 1 0:7 0:8 C C 0:8 C C C 0:7 C C 0:4 A 0:5
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Then, based on the fuzzy preference relations Pk ðk ¼ 1; 2; . . . ; 5Þ, we can apply the quadratic programming model (M-4) (or (21)) to determine the optimal weight vector:
wð1Þ ¼ ð0:20; 0:20; 0:20; 0:20; 0:20ÞT After that, we can utilize (4) to get the collective fuzzy preference relation:
0
Pð1Þ
0:5
0:34 0:32 0:56 0:78 0:68
1
B 0:66 0:5 0:28 0:74 0:80 0:76 C C B C B B 0:68 0:72 0:5 0:66 0:76 0:82 C C ¼B B 0:44 0:26 0:32 0:5 0:82 0:68 C C B C B @ 0:22 0:20 0:28 0:19 0:5 0:36 A 0:32 0:24 0:18 0:30 0:64
0:5
Using (22) and (23), we calculate the deviations: ð1Þ
dðP2 ; Pð1Þ Þ ¼ 0:0427;
ð1Þ
dðP5 ; Pð1Þ Þ ¼ 0:0360;
dðP1 ; Pð1Þ Þ ¼ 0:0413; dðP4 ; Pð1Þ Þ ¼ 0:0413;
ð1Þ
dðP3 ; Pð1Þ Þ ¼ 0:0480
ð1Þ
ð1Þ
D1 ð1Þ ¼ 0:0419
Then D1(1) 6 0.05, which means that the group reaches an acceptable level of consensus. In cases where consensus must be urgently obtained, or the experts cannot or are unwilling to revaluate the alternatives, we can use Step 4 of Algorithm 2 (without loss of generality, set g = 0.3) to obtain the new preference relations:
0
ð1Þ
P1
0:5 B 0:66 B B B 0:73 ¼B B 0:41 B B @ 0:24
0:34 0:27 0:59 0:76 0:67
1
0:5 0:32 0:71 0:84 0:75 C C C 0:68 0:5 0:64 0:76 0:86 C C; 0:29 0:36 0:5 0:78 0:67 C C C 0:16 0:24 0:22 0:5 0:35 A
0:33 0:25 0:14 0:33 0:65 0:5 1 0:5 0:37 0:39 0:59 0:76 0:76 B 0:63 0:5 0:29 0:77 0:78 0:78 C C B C B B 0:61 0:71 0:5 0:67 0:73 0:80 C C; B ¼B C B 0:41 0:23 0:33 0:5 0:81 0:67 C C B @ 0:24 0:22 0:27 0:19 0:5 0:32 A
0
ð1Þ
P2
0
ð1Þ
P3
0:5 B 0:69 B B B 0:70 ¼B B 0:44 B B @ 0:21
ð1Þ
P4
ð1Þ
0:32 0:74 0:87 0:78 C C C 0:68 0:5 0:61 0:79 0:83 C C 0:26 0:39 0:5 0:75 0:70 C C C 0:13 0:21 0:25 0:5 0:38 A
0:30 0:22 0:17 0:30 0:62 0:5 1 0:5 0:28 0:24 0:56 0:79 0:76 B 0:72 0:5 0:26 0:80 0:75 0:84 C C B C B B 0:76 0:74 0:5 0:70 0:70 0:74 C C B ¼B C B 0:44 0:20 0:30 0:5 0:87 0:73 C C B @ 0:21 0:25 0:30 0:13 0:5 0:38 A
0:24 0:22 0:20 0:33 0:68 0:5 1 0:5 0:31 0:30 0:62 0:79 0:64 B 0:69 0:5 0:26 0:74 0:81 0:72 C C B C B B 0:70 0:74 0:5 0:67 0:73 0:80 C C B ¼B C B 0:38 0:26 0:33 0:5 0:84 0:70 C C B @ 0:21 0:19 0:27 0:16 0:5 0:38 A 0:36 0:28 0:20 0:30 0:62
0:24 0:16 0:26 0:27 0:62
0:5
and then apply the model (M-4) to derive the optimal weight vector:
wð1Þ ¼ ð0:203; 0:198; 0:200; 0:200; 0:199ÞT Using (4), we can get the collective fuzzy preference relation:
0
Pð1Þ
0:5
0:32 0:30 0:58 0:78 0:71
1
B 0:68 0:5 0:29 0:75 0:81 0:77 C C B C B B 0:70 0:71 0:5 0:66 0:74 0:81 C C ¼B B 0:42 0:25 0:34 0:5 0:81 0:69 C C B C B @ 0:22 0:19 0:26 0:19 0:5 0:36 A 0:29 0:23 0:19 0:31 0:64
and based on
ð1Þ Pk
0:5
ðk ¼ 1; 2; 3Þ and P(1), we can calculate the deviations using (22) and (23):
ð1Þ
ð1Þ
ð1Þ
dðP1 ; Pð1Þ Þ ¼ 0:0260;
dðP2 ; Pð1Þ Þ ¼ 0:0247;
dðP3 ; Pð1Þ Þ ¼ 0:0247
ð1Þ dðP4 ; Pð1Þ Þ
ð1Þ dðP5 ; Pð1Þ Þ
D1 ð1Þ ¼ 0:0280
¼ 0:0440;
¼ 0:0207;
1
0:5
0
0
P5
0:31 0:30 0:56 0:79 0:70
which indicates that the group has also reached an acceptable level of consensus.
0:5
Z. Xu, X. Cai / Information Sciences 181 (2011) 150–162
161
5. Conclusions In group decision making with preference relations, the preference relations (such as fuzzy preference relations and multiplicative preference relations) provided by decision makers generally have different importance weights. Accordingly, the question of how these importance weights should be determined presents an interesting research topic. In this paper, we have established several goal programming models and quadratic programming models based on minimizing deviation variables from the viewpoint of maximizing group consensus. These models allow us to determine the importance weights of fuzzy preference relations. We have further developed two iterative algorithms for group decision making with fuzzy preference relations. In cases where the group does not reach an acceptable level of consensus, Algorithm 1 interacts with the decision makers until the group achieves an acceptable level of consensus or the number of the interactions reaches a predefined maximum number. If the threshold of acceptable group consensus is reduced, the number of interactions generally increases. The algorithm is very suitable for solving group decision-making problems, such as vendor selection in supply chain management, personnel examination, military system efficiency evaluation, and venture capital project evaluation. Algorithm 2 is a convergent iterative algorithm, which can automatically modify the diverging individual fuzzy preference relations so that the group’s opinion achieves an acceptable level of consensus. These two algorithms have their own characteristics, and are complementary in practical applications. The main advantages and functions of our algorithms are as follows: (1) If the weights of the fuzzy reference relations in group decision-making problems cannot be predefined, then using the goal of maximizing group consensus, we can apply our models to determine the optimal weights of these fuzzy preference relations. The models can alleviate the influence of the unduly high or low preferences of some decision makers on the decision result, by assigning appropriately small weights to the fuzzy preference relations with large deviations. Using the well-known WAA operator, we fuse all the individual fuzzy preference relations into the collective fuzzy preference relation. Next, the interactive Algorithm 1 can be used to implement an interactive revaluation process, to reach a high or acceptable level of group consensus. This process may sometimes require an excessive amount of time to complete. In cases where consensus is urgently required, or decision makers cannot or are unwilling to revaluate the alternatives, we can use the iterative Algorithm 2 to automatically modify the diverging individual fuzzy preference relations to make the level of group consensus as high as possible. (2) If the information about the weights of fuzzy reference relations is known, then we can also use the WAA operator to fuse all the individual fuzzy preference relations into the collective fuzzy preference relation, and then utilize Algorithm 1 or Algorithm 2 to reach an acceptable level of group consensus. In addition, we have also established goal programming models and quadratic programming models to derive the importance weights of multiplicative preference relations. Algorithms 1 and 2 can also be used to establish group consensus with multiplicative preference relations. An interesting topic for future study would be to extend our models and algorithms to group decision-making problems based on interval-valued fuzzy preference relations or interval-valued multiplicative preference relations. Acknowledgments The work was partly supported by the National Science Fund for Distinguished Young Scholars of China (No. 70625005), the National Natural Science Foundation of China (Nos. 71071161 and 70932005), and the Research Grants Council of Hong Kong under General Research Fund Project No. 410208. The authors are very grateful to the anonymous referees for their constructive comments and suggestions that have led to an improved version of this paper. References [1] S.E. Bodily, A delegation process for combining individual utility functions, Management Science 25 (1979) 1035–1041. [2] F. Chiclana, F. Herrera, E. Herrera-Viedma, Integrating three representation models in fuzzy multipurpose decision making based on fuzzy preference relations, Fuzzy Sets and Systems 97 (1998) 33–48. [3] F. Chiclana, F. Herrera, E. Herrera-Viedma, L. Martı´nez, A note on the reciprocity in the aggregation of fuzzy preference relations using OWA operators, Fuzzy Sets and Systems 137 (2003) 71–83. 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