A New Method for Fuzzy Group Decision-Making Based on Interval Linguistic Labels
Shyi-Ming Chen and Li-Wei Lee
Department of Computer Science and Information Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan, R. O. C.
Abstract-This paper presents a new method for dealing with fuzzy group decision-making problems based on interval linguistic labels. We propose the interval linguistic labels ordered weighted average (ILLOWA) operator to aggregate interval linguistic labels and propose likelihood-based comparison relations of interval linguistic labels to develop a new method to deal with fuzzy group decision-making problems. It provides us with a useful way to deal with fuzzy group decision-making problems.
Kqwords-Fuzzy group decision-making, interval linguistic labels, likelihood-based comparison relations
I. INTRODUCTION
Some methods have been presented for fuzzy group decision-making [1]-[15], [17], [19]. In [1], Ben-Arieh and Chen presented a linguistic labels aggregation operator, called the fuzzy linguistic order weighted average (pLOW A) operator, and applied it to deal with fuzzy group decision
making problems. However, Ben-Arieh and Chen's method can not deal with the situation that decision-makers use interval linguistic labels to express their preferences.
In this paper, we propose the interval linguistic labels ordered weighted average (ILLOWA) operator and propose likelihood-based comparison relations of interval linguistic labels to develop a new method to deal with fuzzy group decision-making problems. Although Ben-Arieh and Chen [1]
have presented a method for dealing with fuzzy group decision
making problems based on FLOW A operator, their method has the following drawbacks: 1) It can not deal with fuzzy group decision-making problems in which interval linguistic labels are used to represent decision makers' preference. 2) The FLOW A operator proposed by Ben-Arieh and Chen's has the divided by zero problem. The proposed method can overcome the drawbacks of Ben-Arieh and Chen's method [1]. It provides us with a useful way to deal with fuzzy group decision-making problems.
II. PRELIMINARIES
In [17], Yager proposed the ordered weighted averaging (OW A) operator, shown as follows:
n
OWA(ah a2, . . . , an) = L wjbj,
j=1
978-1-4244-6588-0/10/$25.00 ©201 0 IEEE
(1)
1
where bj is the jth largest of ai, 1 :s i :s n, Wj is the corresponding
n
weight of bj, WjE [0, 1], and L Wj = 1. In [18], Yager also presented a formula to calculate the weights j=1 Wj of bj for the OW A aggregation operator, where 1 :s j :s n, by using the linguistic quantifiers Q(r), shown as follows:
j j-l
w· J =Q(-)-Q(n -n ),
where Q(r) = rG, a � 0 and 1 :Sj:S n.
(2)
Assume that there is a linguistic labels set S = {so, Sh . . . , St}
and assume that there are two interval linguistic labels fi - -
= [sal ' S p.] and 12 = [Sa2 ' S p), where 0 :s a,:s PI :s t and 0
:s a2 :s P2 :s t, then the addition operation between the interval - -
linguistic labels fi and 12 is defined as follows:
fi (37 12 = [sal' s p.] Ee [Sa2 , S p)
= [min(St , max(sal+a2 , SO )), min(St, max(s P.+P2 , So))] (3) Assume that fi = [s q' S p.] is an interval linguistic label, then � is defined as follows:
� = A[Sal' Sp.] = [Asq, Asp'] = [SAal' SA.P.], (4) where 0 :s A. :s 1, 0 :s al:S PI :s t and 0 :s AaI:S A.fJI :s t. Based on the likelihood-based comparison relation of intervals [16], we
"'""W � ... <""<oJ
can define the likelihood p(fi � 12) of I, � 12, shown as
follows:
- - P2 -a,
p(fi � 12) = max(l- max( _ _ , 0), 0), (5)
L(fi ) + L(f2 )
where L(li) = PI -a, and L<l2) = P2 -a2. The likelihood
� � '""-J �
p(fi � 12) of fi � 12 has the following properties:
(1 ) 0 s; ...p(fi � ... 12) S; ...1, ...
(2) p(fi � 12) + P(f2 � fi) = 1,
(3) If x2 S; Yl' then p(fi � 12) = 0,
(4) If xI �Y2, thenp(fi �/2)=1, (5) p(fi � fi) = 0. 5.
If al = PI and a2 = P2 , then the likelihood p(fi � 12) of fi � 12 is defmed as follows:
{I,
if al > a2p(li � 12) = �, if al = a2
0, if al < a2 (6)
Assume that there is a linguistic labels set S = {so, Sh • • • , SI}. Assume that the interval linguistic labels fi, 12' . . . , and In are needed to be ranked and assume that there is a fuzzy target T, where T = [so, sa. The proposed ranking procedure of interval linguistic labels is presented as follows:
- -
Step 1: Calculate the ranking value p(J; � T) of J; , where 1 :s
.<
I_n.
Step 2: Rank fi, 12' . . . , and In according to their ranking values. The larger the value of p(J; � T), the better the ranking of J;, where 1 :s i :s n.
In this paper, based on the concept of OWA operator [17], we propose the interval linguistic labels ordered weighted average (lLLOWA) operator, defined as follows:
ILLOWA([sal' S p.], [sa2 ,S P2]' ... , [San'S Pn ]) =
L n wj[saj ,spj],
j=1 (7)
where 0 :s aj :s Pj :s t, [Sa ' sp ] j j is the jth largest interval linguistic label of [sa; ,S p), 0 :s aj:S Pi :s t, 1 :s i :s n, Wj is the corresponding weight of[saj ,sp), WjE [0,1] and L n Wj = 1.
j=1
III. A NEW METHOD FOR Fuzzy GROUP DECISION-MAKING BASED ON THE PROPOSED ILLOWA OPERATOR AND THE PROPOSED LIKELIHOOD-BASED COMPARISON RELATIONS OF
INTERVAL LINGUISTIC LABELS
In this section, we propose a fuzzy decision-making method based on the proposed interval linguistic labels ordered weighted average (lLLOW A) operator and the proposed likelihood-based comparison relations of interval linguistic labels. Assume that there is a linguistic labels set S = {so, S h • • • , SI}, there are n alternatives A" A2, ... , An and there are q experts
E" E2, • • • , Eq. Assume that u?, ug, . . . , and u� are the initial weights of the experts E" E2, • • • , and Eq, respectively, where u? E [0, 1], ug E [0, 1], . . . , and u� E [0, 1], and assume that there is a fuzzy target T = [so, sa. The interval linguistic preference matrix Pk with respect to expert Ek is shown as follows:
2
(8)
where II = [sa'sp], It = [St-p,St-a], j:F-i, 0 :s a :s P :s t and 1 :s k :s q.
The proposed fuzzy group decision-making method is now presented as follows:
Step 1: For alternative Ai from expert Ek, collect n - 1 interval linguistic labels I;, where 1 :s i:S n, 1 :Sj:S n, j :F-i and 1 :s k:s q. For example, for alternative Al from expert Eh we collect the
. mterva mgulsbc a e s 12, 11· · · 1 b 1 jj-k jj-k 13, . . . , an In. djj-k
Step 2: Based on Eq. (5) and the collected n - 1 interval linguistic labels I; obtained in Step 1, calculate the ranking value P<f; � T), where 1 :s i:S n, 1 :Sj:S n, j:F-i and 1 :s k:s q.
The larger the value of p(J;J � T), the better the ranking order of I; , where 1 :s i :s n, 1 :Sj :s n, j :F-i and 1 :s k :s q. Based on Eq. (7), we can get the aggregating result J;k of alternative Ai
from expert Eh shown as follows:
J;k = ILLOWA({I; 11 :Si:Sn, 1 :Sj:Sn, j:F-i and 1 :Sk:Sq}) n-I
= L wr[sa, ,S p,], (9)
r=1
where [sa, ,sp,] is the rth largest of I; ,Wr is the corresponding weight of [sa , ,sp), WrE [0, 1] and L n-I Wr =1.
r=1
Step 3: Based on Eq. (5), calculate the ranking values
-k -k -k
p(fi � T), P(f2 � T), . . . , and p(fn � T) of the aggregated
-k -k -k
results II , 12 , . . . , and In , respectively. The larger the value of p(J;k � T), the more the preference of alternative Ai from expert Ek, where 1 :s i:S n and 1 :s k:s q.
-1 -2 -q .
Step 4: Aggregate J; , J; , . . . , and J; WIth respect to experts
E" E2, • • • , and Eq to get the aggregated result J; of alternative -0
Ai from all experts, shown as follows:
(10) where Uk denotes the weight of expert Ek at the rth iteration, 1 :s i:S n and r � o.
Step 5: Based on Eq. (5), calculate the ranking values
-G -G -G
p(fi � T), P(f2 � T), . . . , and p(fn � T) of the aggregated
-G -G -G
results fi , 12 , . . . , and In , respectively. The larger the value of p(J;G � T), the more the preference of alternative Ai from all experts, where 1 � i � n.
Step 6: Calculate the consensus level Ci of alternative Ai and the group consensus level CG as follows:
(11)
1 P CG =-LC[il'
P i=' (12)
where Ci denotes the consensus level of alternative Ai, O� denotes the ranking order of alternative Ai from the group, O!k denotes the ranking order of alternative Ai from expert Elo Co denotes the group consensus level and [z1 represents the alternative ranked in the ith position.
Step 7: Calculate the consensus level Clz of alternative Ai without expert Ez, the contribution Diz of expert Ez on alternative Ai and the cumulative contribution Dz of expert Ez on all alternatives as follows:
Clz = L
[
(1-lOG A, _oEk Ai I )xPk ,1
keE\{z) n-l (13)
(14)
(15)
where E = {I, 2, . . . , q}, "ke E\{z} " denotes ke E and k::f:. z, Clz denotes the consensus level of alternative Ai without expert Ez, Diz denotes the contribution of expert Ez on alternative Ai, Dz denotes the cumulative contribution of expert Ez on all alternatives, Pk = i and "ie E\{z}"
ieE\{z} Ui
denotes i E E and i ::f:. z. The larger the value of D., the higher the contribution of expert Ez•
Step 8: IfCG � 0, where 0 is a threshold value and 0 E [0, 1], then Stop. Otherwise, go to Step 9.
Step 9: Update the weight uk of expert Ek at the rth iteration, where 1 � k � q, shown as follows:
(16)
3
ut' = tk+' (17)
Ltk+' q
k='
where Uk denotes the weight of expert Ek at the rth iteration and P denotes the influence of the contribution of the experts.
The higher the value of P , the faster the process converges to the desired consensus level; go to Step 4.
Example 3.1: Assume that there is a nine linguistic labels set S = {so, Sh S2, S3, S4, S5, S6, S7, Sg}, where So =
"Incomparable", SI = "Significantly Worse", S2 = "Worse", S3 =
"Somewhat Inferior", S4 = "Equivalent", S5 = "Somewhat Better", S6 = "Superior", S7 = "Significantly Superior" and Sg =
"Certainly Superior". Assume that there are four alternatives Ah A2, A3, A4 and assume that there are four experts Eh E2, E3, E4• Assume that the initial weights u�, ug, u� and u� of the experts Eh E2, E3 and E4 are 0.25, 0.25, 0.25 and 0.25, respectively. Assume that there is a fuzzy target T = [so, Sg].
Assume that 0 = 0.998 and P = 0. 5. Assume that the linguistic opinion of expert Ej is expressed by the interval linguistic preference matrix Pj' where
I
�j � 4, shown as follows:A, A2 A3 A4
�[ -
[s3,s4] [s5,s7] [s"sd A2 [S5,S5] [S7,s8] [s3,s3]li= A3 [s3,s4] [s"s2] [sO,s2]
A4 [S7,S7] [S5,s6] [s8,s8]
A, A2 A3 A4
Al
[ -
[s"s3] [s3,s4] [S"S41]
A2 [S7,S7] [s6,S7] [S5,S5]
P2 =
A3 [s5,s6] [s2,s3] [s3,s3]
A4 [S5,S7] [s3,s3] [S5,s6]
A, A2 A3 A4
A'
[
- [s5,s5] [s7,s8] [S"S41]
A2 [s3,s4] [s6,s8] [s2,s3]
P3 =
A3 [s"sd [s2,s3] [so,sd A4 [S5,S6] [s6,s6] [s8,s8]
A, A2 A3 A4
AI
[
- [s6,s7 ] [s7,s7] Is, 'S71]
A2 [s2,s3] [s4,s5] [s3,s3]
P4 = A3 [s"s2] [s4,s4] [sO,s2]
A4 [s3,s3] [s5,s7] [s8,s8]
Figure 1 shows the group consensus with the changing of weights of the experts.
0.8 f---::...-==---
-+- U,
0.6 f--- -='.=--- U2 0.4 f---- -c=--'=---
O': e:=��
0.8333 0.8539 0.8745 0.8951 0.9157 0.9363 0.9568 0.9774 0.998
Figure I. Group consensus with the changing of weights of experts.
Table I shows the ranking orders of the alternatives from individual experts and the whole group. From Table I, we can see that the proposed method performs better than Ben-Arieh and Chen's method [I] due to the fact that the evaluating values of the proposed method can either be represented by linguistic labels or interval linguistic labels, whereas Ben-Arieh and Chen's method [I] has the drawback that it cannot deal with fuzzy group decision-making problems in which the evaluating values are represented by interval linguistic labels.
TABLE I.
Methods
Ben-Arieh and Chen's method
[[I))
The proposed
method
RANKING ORDERS OF ALTERNATIVES FROM INDIVIDUAL EXPERTS AND THE WHOLE GROUP
Experts Order OAt OA2 OA3 °A.
E. N/A N/A N/A N/A N/A
E2 N/A N/A N/A N/A N/A
E3 N/A N/A N/A N/A N/A
E, N/A N/A N/A N/A N/A
Group N/A N/A N/A N/A N/A
E. A4,A2,A1,A3 3 2 4 I
E2 A2,�,A3,AI 4 I 3 2
E3 A4,A1,A2,A3 2 3 4 I
E, A,,�,A2,A3 I 3 4 2
Group A4,A2,A"A3 3 2 4 I
(Note: "N/A" denotes that Ben-Arleh and Chen's method [I]
cannot deal with the ranking order of alternatives.) IV. CONCLUSIONS
In this paper, we have presented a new method for fuzzy group decision-making based on interval linguistic labels. We have proposed the interval linguistic labels ordered weighted average (lLLOWA) operator to aggregate interval linguistic labels and have proposed likelihood-based comparison relations of interval linguistic labels to deal with fuzzy group decision-making problems. The proposed method can overcome the drawbacks of Ben-Arieh and Chen's method [I].
It provides us with a useful way to deal with fuzzy group decision-making problems.
ACKNOWLEDGMENT
This work was supported in part by the National Science Council, Republic of China, under Grant NSC 97-2221-E-OII- 107-MY3.
4
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