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Linguistic-Labels Aggregation and Consensus Measure for Autocratic Decision Making Using Group Recommendations David Ben-Arieh and Zhifeng Chen
Abstract—Group decision making is a common and important activity in everyday life. In many cases, due to inherent uncertainty, experts cannot express their score or preference using exact numbers. The use of linguistic labels makes expert judgment more reliable and informative for decision-making. One of the problems of group decision making in fuzzy domains is aggregating experts’ opinions, expressed using linguistic labels, into a group opinion. This aggregation allows the group to select the most “preferred” alternative from a finite set of candidates. The aggregation of individual judgments into a group opinion requires a measured level of consensus. In this paper, by introducing a new linguistic-labels aggregation operation, we present a procedure for handling an autocratic group decision-making process under linguistic assessments. The methodology presented results in two consequent outcomes: a group-based recommendation, and a score for each expert, reflecting the expert’s contribution towards the group recommendation. By changing the weights of the experts based on their contributions, we increase the consensus and reinforce the common decision, without forcing the experts to modify their opinions. This methodology allows an autocratic decision maker to use a diversified group of consultants for a succession of decisions reaching a high level of consensus. Index Terms—Aggregation operators, consensus measure, fuzzy sets, multicriteria decision making.
totally ordered term set of linguistic labels S = {s0 , s1 . . . sT }, with si > sj , for i > j [11], [18]. For example, S = {s0 = none, s1 = very low, s2 = low, s3 = medium, s4 = high, s5 = very high, s6 = perfect}. Usually, the set S has an odd number of elements. Also, each expert is assigned a degree of importance or weight uk , k = [1, 2 . . . q]. Yager [42] introduces the concept of ordered weighted average (OWA), in which weights can represent linguistic aggregation quantifiers. These weights can be viewed in two ways: the importance or trust that each expert carries, or weights derived from a required degree of orness [40], [42]. In a fuzzy environment, the group decision-making problem can be solved in four steps [15]. First, one should unify the evaluations from each expert. The second step is to aggregate the opinions of all group members to a final score for each alternative. This score is usually a fuzzy set or a linguistic label, which is used to order the alternatives. The third step is to rank the linguistic labels or fuzzy sets and select the preferred alternatives based on this order. Finally, the decision manager assesses the consensus level and the individual contribution to the group decision. The following procedure describes these steps in more detail.
I. I NTRODUCTION
D
ECISION making is an important subject in business, manufacturing, and service. Group decision making (i.e., multiexpert) is a typical one, where the inherent complexity and uncertainty necessitates the participation of many experts in the decision-making process. In the real world, the uncertainty, constraints, and even the vague knowledge of the experts imply that decision makers cannot provide exact numbers to express their opinions. The use of linguistic labels makes expert judgment more reliable and consistent. In a fuzzy environment, a group decision-making problem is composed of the following elements: a finite set of alternatives A = {A1 , A2 . . . An }, a finite set of experts E = {E1 , E2 . . . Eq } with each expert ek ∈ E presenting his/her preference relation on Ai as xik ∈ S, where S is a finite, but
A. Uniform Experts’ Evaluations Herrera-Viedma et al. [19] present four ways for group members to express their opinions: preference ordering, utility values, fuzzy preference relations, and multiplicative preference relations. These opinions can be converted into the various representations using different transformations [8], [9]. For instance in [19], the function of transforming the multiplicative preference relations akij into the fuzzy preference relations pkij is given as pkij = (1/2)(1 + log9 akij ). As mentioned before, group members can also produce linguistic opinions, especially when the problem could not be evaluated by exact numbers. In this way, expert Ek can choose sj from the linguistic-label set S as the score xik to alternative Ai . B. Aggregate All Experts’ Opinions
Manuscript received October 21, 2003; revised March 12, 2004 and September 9, 2004. This paper was recommended by Associate Editor J. Lambert. The authors are with the Department of Industrial and Manufacturing Systems Engineering, Kansas State University, Manhattan, KS 66506 USA (e-mail:
[email protected]). Digital Object Identifier 10.1109/TSMCA.2005.853488
In this stage, all experts’ opinions are combined to get a final rating for each alternative. The selection of aggregation function plays an important role in the accuracy of the final solution. Many methods for aggregation can be found in [2], [6], [7], [12], [14], [15], [17], [21], [34], [35], and [41]. Regardless
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BEN-ARIEH AND CHEN: LINGUISTIC-LABELS AGGREGATION AND CONSENSUS MEASURE FOR DECISION MAKING
of whether the weights and scores are linguistic or numeric, the general form of the aggregation function is fi =
q
uk ⊗ xik
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example, the experts’ opinions represent a pairwise comparison of alternatives expressed as linguistic preference relations. The summary, conclusions, and future research are presented in Section VI.
k=1
where fi is the final score for alternative i. The weights of experts uk could have quantitative or qualitative values. The qualitative weights can be rationalized using some algorithms introduced in [3], [14], [15], [37], [39], and [40]. In this paper, we assume that the initial weights of experts are given, either in quantitative or in qualitative terms, or derived from a given linguistic quantifier Q(r), as defined in [22] and [38]. ⊗ represents the general aggregation operator. C. Rank Alternatives The objective is to find one or several best alternatives, which is accomplished by ranking all the alternatives based on the aggregated result from the group members. In Section III, we present a fuzzy ranking method that can be used in ranking both fuzzy sets and groups of linguistic labels.
II. A N EW FLOWA M ETHOD TO A GGREGATE L INGUISTIC L ABELS There are two main approaches to aggregate linguistic labels. Most methods use the associated membership functions. Such methods include Chen and Hwang’s eight conversion scales [6], Baas and Kwakernaak’s rating and ranking algorithm [2], Cheng’s adjusted fuzzy rating method, fuzzy Delphi method [7], and Yager’s all/and/min-type of aggregation [41]. This approach calculates an average or a center of gravity for the experts’ opinion. The second approach directly computes the linguistic labels. The LOWA method defined in [14] and [15] is one of these approaches. This method is based on the OWA [40], [42] and the convex combination of linguistic labels [12]. The idea is that the combination resulting from two linguistic labels should be itself an element in the set S. So, given si , sj ∈ S and i, j ∈ [0, T ], the LOWA method finds an index k in the set S representing a single resulting label.
D. Measure Consensus After the group decision is created, we evaluate how good it is by checking whether it represents the majority of the group members’ opinions. It could happen that when the group members have conflicting opinions, the solution could be a medium one that no expert in the group likes. The aggregation function, ranking method, and consensus measure are the main problems to be solved for a fuzzy group decision-making problem. One contribution of the methodology presented herein is that the result of this aggregation approach is a collection of linguistic labels with a calculated degree or membership function, presenting a more informative aggregation. Most methodologies reported in the literature aggregate linguistic labels into a single label. Moreover, this paper has a unique approach towards consensus in group decision making. In this case, the experts are allowed not to modify their opinions in order to reach a group decision. The group decision is handled by an autocratic decision maker who combines the experts’ opinions (recommendations) into a single decision. The consensus in this approach is enhanced by modifying the weight of the experts such that dissenting or eccentric experts loose credibility. This point is further discussed in Section IV. The paper is organized as follows. In Section II, we introduce a new linguistic-label aggregation operator based on linguisticOWA (LOWA) [14], [15] operator that we call the fuzzy-LOWA (FLOWA) operator. After comparing several fuzzy ranking approaches, we introduce a fuzzy ranking method with some favorable properties in Section III. In Section IV, we present a new method to measure consensus and the contributions of disagreeing experts. Also, we present the entire process of generating a group decision/recommendation and converging on the consensus measure. Section V presents a numeric example to demonstrate the group-solution procedure. In this
A. Review of the OWA Operator First defined by Yager [42], an OWA operator of dimension n is a mapping F : I n → I (where I ∈ [0, 1]) that has an associated weight vector W = [w1 w2 . . . wn ], such that 1) wi ∈ [0, 1] 2) i wi = 1 and where F (a1 , a2 , . . . , an ) = w1 b1 + w2 b2 + . . . + wn bn where bi is the ith largest element in the collection of {a1 , a2 , . . . an }. The main idea of the OWA operator is the ordering of arguments by their values. An orness α is associated with the degree of optimism of a decision maker. The relationship between the orness level α and the weighting vector W is defined by [40], [42] α=
n n−j wj . n −1 j=1
Given a certain orness level, we could find the corresponding weighting vector W . Yager in [39] also suggested a way to compute the weights of an OWA aggregation operator by using linguistic quantifiers, which, in the case of nondecreasing proportional quantifiers Q(r) is given by wi = Q(i/n) − Q[(i − 1)/n]. By appropriately selecting the weights in W , one can emphasize different arguments based upon their positions in the order. For example, given the linguistic quantifier “most” with four experts (n = 4), using a common accepted membership function of the linguistic quantifier Q(r) = r2 , one can generate the weights W = [0.0625, 0.1875, 0.3125, 0.4375].
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Fig. 1. Concept of FLOWA.
B. Review of the LOWA Operator Let X = {a1 . . . am } be a set of labels to be aggregated, and the LOWA [15] operator ϕ is defined as a recursive function: ϕ{a1 , . . . , am } = W · B T = C m {wk , bk , k = 1, . . . , m}
k = min {8, 1 + round (0.3748 × (5 − 1))}
= w1 ⊗ b1 ⊕ (1 − w1 ) ⊗ C m−1 × {βh , bh , h = 2, . . . , m}
For example, assume that there are nine linguistic labels S = {s0 , s1 , s2 , . . . , s8 }, where T = 8, and we need to combine two linguistic labels s1 and s5 with the weights given as 0.6252 and 0.3748, respectively. Applying the LOWA method, with i = 1, j = 5, wi = 0.6252, and wj = 0.3748 produces
= min {8, 1 + round (1.499)} = 2. (1) So the result is s2 . Now, if we change the weights of wj from 0.3748 to 0.3753, the result is
where W = [w1 , . . . , wm ] is a weighting vector, such that wi ∈ [0, 1] wi = 1.
k = min {8, 1 + round (0.3753 × (5 − 1))} = min {8, 1 + round (1.501)} = 3.
i
m
βh = wh / 2 wk , h = 2, . . . , m, and B = {b1 , . . . , bm } is a vector associated to X, such that (2) B = σ(X) = aσ(1) , . . . , aσ(m)
Thus, the aggregation result is s3 . The example shows that the LOWA method loses some useful information in the aggregation process, in the case where w1 ∗ ( j − i) is around 0.5.
where aσ( j) ≤ aσ(i)
C. Definition of the FLOWA Operator
∀i ≤ j
with σ being a permutation over the set of labels X. C m is the convex combination operator of m labels and if m = 2, then it is defined as C 2 {wi , bi , i = 1, 2} = w1 ⊗ sj ⊕ (1 − w1 ) ⊗ si = sk , si , sj ∈ S, ( j ≥ i)
(3)
such that k = min {T, i + round (w1 · ( j − i))} where round is the usual round operation, and b1 = sj , b2 = si . The LOWA operator does not consider the degree of membership associated with each linguistic label and the combined result is a single element in the linguistic-label set S. Thus, it is easy to order the result. The LOWA has one minor problem in using the round w1 ∗ ( j − i). When this value is around 0.5, varying the experts’ weights even slightly can result in a different linguistic label, thus a different solution.
Based on the LOWA method, we present a new aggregation operator denoted as FLOWA. Fig. 1 shows the basic concept of the FLOWA. Let m linguistic labels X = {si . . . sl . . . sj } (i < l < j), be a set of labels to be aggregated, where si is the smallest label in X and sj is the largest one and X ⊆ S. (A detailed description of the linguistic labels’ common properties is available, for example, in [14] and [15]). The FLOWA operator F is defined as F {si . . . sl . . . sj } = {(sk , µsk ) |sk ∈ S} where µsk is the fuzzy membership assigned to the kth linguistic label sk after aggregating the weights on label set X = {si . . . sj }. It is defined as
µsk =
T l=0
µlsk
(4)
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where, µlsk is the membership function of the kth linguistic label sk , sk ∈ S generated from the weighted linguistic label sl , sl ∈ X. The µlsk is defined by the following. 1) l = i µlsk =
2( j − k) wl . ( j − i)( j − i + 1)
(5)
µlsk =
2(k − i) wl . ( j − i)( j − i + 1)
(6)
2) l = j
Fig. 2. Example involving the aggregation of three linguistic labels.
3) i < l < j µlsk
=
2(k−i) ( j−i)(l−i) wl ,
for I ≤ k ≤ l
2( j−k) ( j−i)( j−l) wl ,
for l ≤ k < j.
(7)
Also, the experts have the weights w5 = 0.5, w1 = 0.125, and w7 = 0.375. As an example, for k = 1 µs1 = µ1s1 + µ5s1 + µ7s1
4) l < i or l > j µlsk = 0.
=
The weighting vector W = [wi . . . wl . . . wj ] (i < l < j) associated with the linguistic labels represents the experts’ weight. Thus, wl represents the weight of the expert who chooses label l as the linguistic representation of his/her preference. And wl ∈ [0, 1] wl = 1. l
Notice that if wi , wj > 0, and wl = 0, for all l = i, j, then (4) has only two parts for µisk and µsjk from (5) and (6), respectively. In the FLOWA approach, the final result should lie between si and sj (including si and sj ). Instead of only choosing the sk in the set of linguistic labels S, we assign membership functions to all the linguistic labels between si and sj . We decreasingly spread the original weight wi on si to the linguistic labels from si to sj . Similarly, we increasingly spread the original weight on sj to the linguistic labels from sj to si . The weight of label l, which lies between labels i and j, is spread linearly, decreasing in both directions. After aggregating m linguistic labels X = {si . . . sj }, any linguistic label between i and j gets a membership value as the aggregated result, while the linguistic labels in S with k < i and k > j get weights of 0. This shows that the aggregated result is not another linguistic label, but a set of labels between si and sj , each with a membership function. This membership value represents the degree of confidence in the label. Example: Assume three experts E1 , E2 , and E3 are evaluating an alternative A. Each one chooses a linguistic label from the set S to express his/her opinion. Let us use the same nine linguistic-labels set defined in [15] as S = {I, EV, VLC, SC, IM, MC, ML, EL, C}, where si < sj , given i < j and si , sj ∈ S. Suppose the labels that the experts choose are X = {s1 , s5 , s7 }. The aggregate value of these three linguistic labels is the score of the alternative under consideration.
2 × (7 − 1) × 0.125 (7 − 1) × (7 − 1 + 1) +
2 × (1 − 1) × 0.5 (7 − 1) × (5 − 1)
+
2 × (1 − 1) × 0.375 (7 − 1) × (7 − 1 + 1)
= 0.0357. The final result shown in Fig. 2 is a fuzzy set {0/s0 , 0.0357/s1 , 0.0893/s2 , 0.1429/s3 ,0.1964/s4 , 0.25/s5 , 0.1706/s6 , 0.1071/s7 , 0/s8 }. We can see that after the aggregation, the linguistic label s5 = MC has the highest possibility as the aggregation result. In contrast, the aggregation result of the LOWA algorithm is simply s5 . D. Properties of the FLOWA Algorithm The FLOWA algorithm has several interesting properties. 1) Property 1: The aggregation result is normalized. (Meaning that the membership functions of all the labels in the aggregate sum is equated to one) T
µsk = 1.
k=0
Proof: We have T k=0
µsk =
T T k=0 l=0
T
µlsk =
l=0
µlsk =
T T l=0 k=0
j l=i
µlsk =
µlsk = wl , then
j T l=0 k=i
µlsk =
T
wl = 1.
l=0
Since all the weights wl , (0 ≤ l ≤ T , sl ∈ X) are distributed among all the linguistic labels that participate in the solution, and Tl=0 wl = 1, sl ∈ X, so all the fuzzy memberships should sum to 1. 2) Property 2: Linearity: The aggregation result is a linearly distributed weight between any two linguistic labels to be aggregated, either increasing, decreasing, or constant.
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Proof: Without loss of generality, suppose we are going to aggregate three linguistic labels X = {si , sl , sj } with i < l < j. Then, when k < l, the kth linguistic label in S gets the weight by 2j 2i wi − wl µsk = ( j − i)( j − i + 1) ( j − i)(l − i) 2i wj − ( j − i)( j − i + 1) 2(l − i)(wj − wi ) + 2( j − i + 1)wl + k ( j − i)(l − i)( j − i + 1) whereas, when k > l 2j 2j wi + wl µsk = ( j − i)( j − i + 1) ( j − i)(l − i) 2i wj − ( j − i)( j − i + 1) 2(l − i)(wj − wi ) − 2( j − i + 1)wl + k. ( j − i)(l − i)( j − i + 1) In both cases, after aggregation, the membership function µsk of the kth linguistic label in S is a linear function of k. 3) Property 3: The maximum membership after aggregation could only happen in a linguistic label sk , where sk ∈ X. Proof: From Property 2, it is obvious that the maximum membership never happens between si , sl , and sj . If X = {si , sl , sj }, the maximum membership could only happen in si , sj , or sl with the corresponding value of µsi =
2 wi j−i+1
µsj =
2 wj j−i+1
2 2( j − l)wi + 2(l − i)wj wl + . µsl = j−i ( j − i)( j − i + 1) 4) Property 4: The FLOWA operator is commutative in a limited sense: F (si , . . . , sk , . . . , sl , . . . , sj ) = F (F (si , . . . , sj ), π(sk , . . . , sl )) where π is any permutation over the set of arguments. This property implies that the aggregation process has to start with the two extreme labels. Once this is accomplished, the order of integrating the other labels is immaterial. Proof: From (4)–(6), it is easy to see that once labels i and j (the two extremes) are aggregated the contribution of any label, l and k are independent of the order of aggregation. An example is provided in Fig. 3 and Table I. This example shows that the FLOWA operator is order dependent in this limited sense. When labels are aggregated without the two extreme labels, the results are erroneous as shown. Thus, when
Fig. 3.
Example of property 4.
we aggregate linguistic labels, the min and max labels need to be aggregated first. In this example, only the bottom row presents correct results. Note that F (s5, F (s2, s3, s6)) = F ((s5, s3), F (s2, s6 )) = F (s3, F (s2, s5, s6)). 5) Property 5: The result of aggregating two adjacent labels is the same two labels. Proof: As presented in Fig. 4, if j = i + 1, by formula (5) and (6), µisi = wi , µsjj = wj . III. F UZZY R ANKING Ranking is the last step to select a solution from a set of alternatives. Fuzzy-sets ranking is applied to decision making when each alternative has a set of fuzzy sets associated with the set of linguistic labels. Based on the ranking order of these fuzzy sets, one could rank the alternatives. Many methods for ranking fuzzy sets have been proposed so far. Good summaries of fuzzy-sets ranking methods can be found in books [6], [21] and papers [5], [26], [27]. After summarizing nearly 40 fuzzy-sets ranking-method sources, Chang and Lee proposed the following classification in [5]. 1) α-cut methods. Usually a method developed by this approach is easy and fast to calculate. 2) Methods based on the possibility concept. 3) Method by integration. Measure a fuzzy set by its mean value. 4) Multiple-indexes approach. Rank fuzzy sets using the results of multiple ranking or comparison functions. 5) Linguistic approach. This method is developed mainly due to the desire to maintain the fuzzy characteristics of the problem. Using the third approach, Lee and Li’s fuzzy mean and standard deviation method [26] ranks fuzzy sets based on two different criteria: the fuzzy mean and the fuzzy spread of the fuzzy sets. They pointed out that human intuition would favor a fuzzy set with the following characteristics: higher mean value, and at the same time, lower spread. The fuzzy mean is defined as
xu (A) =
xµA (x)dx
S(A)
S(A)
µA (x)dx
(8)
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TABLE I AN EXAMPLE OF PROPERTY 4
Fig. 4.
Result of aggregating two adjacent labels. TABLE II RANKING RULES FOR FUZZY MEAN AND SPREAD METHOD
where S(A) is the support of fuzzy set A. The standard deviation is calculated as 12 2 x µA (x)dx S(A) σu (A) =
(9) − [xu (A)]2 . µA (x)dx S(A)
Assuming that the mean values and spreads are calculated for two fuzzy sets A and B, the rules for ranking are shown in Table II. The discrete-case equivalent is presented by (10) and (11).
Example: Given three fuzzy sets, as shown in Fig. 5
T
kµsk (A)
xu (A) = k=0 T
= µsk (A)
T
kµsk (A)
(10)
k=0
12
T
C = {0/s0 , 0/s1 , 0/s2 , 0.25/s3 , 0.208/s4 , 0.167/s5 , 0.125/s6 , 0.0833/s7 , 0.0417/s8 , 0/s9 }.
k=0
=
T
k=0
12 k µsk (A) − [xu (A)] 2
2
0.1875/s5 , 0.1250/s6 , 0.0625/s7 , 0/s8 , 0/s9 } 0.2500/s5 , 0.1667/s6 , 0.0833/s7 , 0/s8 , 0/s9 }
2
k µsk (A) 2 σu (A) = k=0 (A)] − [x u T µsk (A)
A = {0/s0 , 0/s1 , 0.0833/s2 , 0.1667/s3 , 0.2500/s4 , B = {0/s0 , 0/s1 , 0.0625/s2 , 0.1250/s3 , 0.1875/s4 ,
k=0
Fig. 5. Membership functions of three fuzzy sets A, B, and C.
(11)
The results of the comparison are presented in Table III. The ranking results show that since x(UB ) = x(UC ) > x(UA ), therefore B > A, C > A. While x(UB ) = x(UC ),
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TABLE III RESULTS FOR RANKING EXAMPLE
but σ(UB ) < σ(UC ), thus B > C. Finally, we conclude that, B > C > A. IV. C ONSENSUS M EASURE AND C ONTRIBUTION OF G ROUP M EMBERS Consensus is traditionally meant as a strict and unanimous agreement of all the experts regarding all possible alternatives. Ness and Hoffman define consensus in [29] as “Consensus is a decision that has been reached when most members of the team agree on a clear option and the few who oppose it think they have had a reasonable opportunity to influence that choice. All team members agree to support the decision.” It is very rare when all individuals in a group share the same opinion about the alternatives, since a diversity of opinions commonly exists. Consensus makes it possible for a group to reach a final decision that all group members can support among these differing opinions. Any group decision-making process is basically aimed at reaching a “consensus.” Consensus has become a major area of research in group decision making [3], [16], [19], [24]. Generally, the approaches towards consensus in the literature can be divided into two groups. The first treats consensus as a “mathematical aggregated consensus” [30]. This type of consensus requires some kind of a binding arbitration, so the contributing experts do not need to converge in their opinions. In most cases, the consensus is achieved by changing the weights of the experts (e.g., [25]). In the other type, the experts are encouraged to modify their opinion to reach a closer agreement in opinions (e.g., [20]). The consensus model of Herrera-Viedma, et al. [19] compares the positions of the alternatives based on the individual solutions and the group solution. Based on the consensus level and the offset of individual solutions, the model gives feedback suggesting the direction in which the individual experts should change their opinion. In effect, the experts compromise their opinion for the sake of consensus. Another drawback of this consensus measure is the lack of weighting of the experts’ opinion. A good review of using linguistic quantifiers for a group decision is found in [23]. This early paper describes finding the consensus level as an average of the degree of agreement between all pairs of experts, where the experts are weighted based on their importance. A group consensus based on a threshold model is presented in [28]. This model calculates a group fuzzy preference relation from individual preferences. The preference relation is then used to calculate a consensus measure using a direct or indirect approach. This consensus is based on a two-stage pairwise comparison process; first, between experts, and then, between alternatives. A similar approach is presented in [4], where a group consensus is estimated
based on similarity between preference vectors of the experts. This similarity is used to generate measures of group agreement and group disagreement using threshold parameters. The decision model presented below assumes that experts do not have to agree in order to reach a consensus (as defined above). This assumption is well grounded in research; many of the early decision theorists argue that agreement between experts is a necessary condition for expertise [13]. However, experimentation consistently refuted this hypothesis. In his report, Einhorn found significant differences in diagnoses by three medical pathologists. In another example, four professional livestock judges were asked to evaluate overall breeding quality of swine [31]. The consensus agreement among the four experts was very low (r = 0.5), in spite of the high experts consistency (r = 0.96). An excellent review of this phenomenon of expert disagreement in different domains can be found in [33]. An additional example for such an expert decision is judging figure skating. In this case, the judges, which are carefully trained experts, evaluate very well-defined performance guidelines using uniform criteria. In such a judging, there are no expectation that all experts will eventually converge to an agreement. On the contrary—the experts are expected to produce diversified opinion and the usual procedure is to eliminate the high and low extreme opinions (assign a weight of zero) and average the rest (assign a weight of n − 2). There are several explanations that allow for experts not to converge to a uniform opinion. It is well accepted that experts are not necessarily the decision makers, but provide an advice [32]. (Weiss and Shanteau [36] also describe five structural and five functional factors that explain this necessary lack of agreement.) In the model presented, the degree of importance of each expert is being considered in calculating the consensus. Moreover, once the consensus is calculated, the experts with more extreme opinion will lose some of their weight (credibility, influence, etc.). The experts, however, need not modify their opinions to achieve consensus. This model supports the case in which an autocratic decision maker reaches a decision based on the recommendations of a diverse group of experts, which embody a variety of experience and opinions. A good example for such a case is the decision regarding the actual date of the D-Day invasion, as reported in [1]. The decision maker aggregates the experts’ recommendations to reach a conclusion. However, the more eccentric experts lose some credibility (weight). Subsequent decisions use the updated weight of the experts and reach again a single conclusion (the D-Date decision was made twice by Eisenhower using the same team of experts—once to postpone and once to invade a day later). Adjusting the weight of experts as a result of their opinion is also reported in Davies [10], who states that “holding unequal influence may not necessarily be counter productive to the group.” Similarly, Hsu and Chen [20] uses the approach that “the weight of specific expert’s opinion in aggregation is proportional to the degree of average agreement of that expert.” The consensus model presented below differs from the one in [19], with the added consideration of the degree of importance
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A. Description of the Group-Decision Process 1) Calculate the current consensus level CG and compare it with the desired consensus level δ. If CG > δ, we accept the group’s solution. 2) If not, we measure the contributions of group members Dj using the following equations: G Ek − O O Ai Ai × βk 1 − Ci j = n−1 k∈E\{j} uk where βk = (14) ui I∈E\{j}
Dij = Ci − Ci j n Dj = Dij .
(15) (16)
i=1
Fig. 6.
Flowchart of the group-decision process.
of each expert. In this model, the consensus of each alternative (Ci ) is calculated, as well as the group consensus (CG ). The group decision accommodates the top p alternatives as well as a single selected one. G E q OAi − OAik × uk 1 − Ci = n−1
Here, Ci j is the group-consensus level on alternative i, without expert j. Dij is the contribution of the jth expert on the ith alternative, which is the difference between the group consensus on alternative i with and without expert j. Dj is the cumulative contribution of the jth expert to all alternatives. The higher Dj it is, the higher the contribution that the jth expert makes to the group. 3) Update the weights of the experts. If the current consensus level is lower than a specified threshold, which means that there is enough discrepancy between the experts’ opinions, we need to update the weights of the experts and recalculate the group-decision solution. The following equations show how to update the weights. = urk · (1 + Dk )β tr+1 k tr+1 ur+1 = k r+1 k tk
(12)
k=1
CG =
1 p
P
C[i]
(17) (18)
k
(13)
i=1
where [i] represents the alternative ranked in the ith position. As before uk is the importance of the kth expert’s opinion. OAGi and OAEik are the ranking orders of the ith alternative from the group and the kth expert, respectively. p is the number of alternatives chosen in the solution set. The consensus measure is used to assess the coherence of the group decision. Consensus is a measure of proximity that evaluates the similarity between the individual preferences and the group preference. It should not be enforced or obtained through compromising. The process of reaching a group decision with high consensus level is depicted in Fig. 6. In this process, we measure the contribution of the individual members to the group decision. Experts who contribute more to the group decision improve their importance, while individuals that are contrary to the group lose some of their weight. The process continues and calculates a new group decision with a new consensus level. The process continues until the desired consensus level is reached.
urk
Here is the importance of expert k in the rth iteration. Parameter β represents the influence of the contribution of the expert on his/her weight. The higher the value of the parameter β, the faster the process converges to the desired consensus level. 4) After updating the weights, we recalculate the group solution and consensus level. The process continues until the desired consensus level is reached. V. A N UMERIC E XAMPLE In this section, we demonstrate the entire process of generating a group decision and converging towards a consensus. The example is adopted from [15], with the purpose of showing how to apply the FLOWA aggregation method and the procedure of reaching the desired consensus level. Let the nine linguistic labels set S to be S = {I, SW, WO, SI, EQ, SB, SU, SS, CS}, where s8 = CS Certainly Superior s7 = SS Significantly Superior Superior s6 = SU s5 = SB Somewhat Better
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TABLE IV AGGREGATION RESULT FOR EXPERT E1
TABLE V AGGREGATED RESULT FROM THE GROUP
s4 = EQ Equivalent Somewhat Inferior s3 = SI s2 = W O Worse s1 = SW Significantly Worse Incomparable. s0 = I Given a set of four alternatives A = {A1 , A2 , A3 , A4 }, as well as a set of four experts E = {E1 , E2 , E3 , E4 } whose opinions are expressed by the following four linguistic preferencerelation matrices. Here Pijk is the degree of preference of the ith alternative Ai over jth the alternative Aj derived from expert k. For example, expert E1 thinks that alternative A1 has a small chance to be better than alternative A2 . Thus, he/she chooses s3 1 as his preference, therefore, the element P12 is s3 . To keep the k k matrices consistent they satisfy Pij + Pji = sT .
− s5 1 P = s3 s7 − s3 3 P = s1 s5
s3 − s1 s5 s5 − s2 s6
s5 s7 − s8 s7 s6 − s8
s1 s3 s0 − s3 s2 s0 −
− s7 2 P = s5 s5 − s2 4 P = s1 s3
s1 − s2 s3 s6 − s4 s5
s3 s6 − s5 s7 s4 − s8
s3 s5 s3 − s5 s3 s0 −
A. Aggregate Each Expert’s Opinion We apply an FLOWA operator guided by a linguistic quantifier Q1 = “As many as possible” to aggregate each expert’s evaluation. The associated weights are calculated as W = [0, 1/3, 2/3]. For alternative A1 from expert E1 , we aggregate {s3 , s5 , s1 }, or ordered linguistic labels {s5 , s3 , s1 }, with associated weights [0, 1/3, 2/3]. Then, we apply formulas (4)–(7) to calculate the membership of the linguistic labels. The aggregating-result matrix for expert E1 is shown in Table IV.
TABLE VI RANKING RESULT FROM EXPERT E1
TABLE VII RANKING RESULT FROM THE WHOLE GROUP
TABLE VIII ALTERNATIVE ORDERS FROM INDIVIDUAL EXPERTS AND THE W HOLE G ROUP
aggregated membership from all experts. Table V shows the calculation result in this example. C. Ranking At this point, the decision of each expert and the group are represented as fuzzy sets, thus, we can apply (10) and (11) to rank these fuzzy sets by their fuzzy means and standard deviations. Tables VI and VII present the ranking results from expert E1 and the whole group, respectively. By comparing the fuzzy means and the standard deviations, we derive the order of the alternatives from each expert and the group. The orders are shown in Table VIII. Thus, Table VIII shows that the group preferred alternative A4 , followed by A1 .
B. Aggregate All Experts’ Opinions In this step, we aggregate the four experts’ opinions to find the final score for each alternative. We assume that all experts have the same importance, meaning that u = [0.25, 0.25, 0.25, the classic aggregation method q 0.25]. EApplying Ek k u µ , where µ µG k ij = ij ij is the membership of the jth k=1 linguistic label on alternative Ai from expert Ek and µG ij is the
D. Consensus Measure Based on the order of the alternatives from each expert and the group, we can apply (12) and (13) to measure the consensus levels. Table IX shows the group consensus level for a single alternative solution, i.e., p = 1, and for p = 2 (the decision maker needs to choose two best alternatives).
BEN-ARIEH AND CHEN: LINGUISTIC-LABELS AGGREGATION AND CONSENSUS MEASURE FOR DECISION MAKING
TABLE IX CONSENSUS LEVEL TO ALTERNATIVES (r = 0)
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TABLE XII CONTRIBUTION OF THE EXPERTS
TABLE XIII WEIGHTS OF EXPERTS
TABLE X NEW GROUP ORDERS WITHOUT A PARTICULAR EXPERT
TABLE XI NEW ALTERNATIVE CONSENSUS LEVEL WITHOUT ONE PARTICULAR EXPERT
To measure the contribution of individual group members, we aggregate the new group without one individual expert. By (14)–(16), we can find the new consensus level for the alternatives without a particular expert. Tables X–XII show the new group orders excluding each expert, the various partialconsensus levels, and the contributions of the four experts towards the group decision. E. Update Weights to Experts We assign the original weighting vector of the experts (r = 0) to be u = [0.25, 0.25, 0.25, 0.25]. After calculating the contribution of each individual expert Di , we apply (17) and (18) to update the weights. Table XIII shows the change of weights to experts during the first two iterations. It indicates that in this case, the weights to experts E1 and E3 increase, while the weights to experts E2 and E4 decrease. This is the result of experts E1 and E3 contributing more to the final group decision. With the change of the weights to experts, the consensus level increases. When the process continues gradually, the weights of the experts change, while the consensus level increases. The process is depicted in Fig. 7. The figure shows the change of the weights of the four experts as the consensus changes from 0.833 to 0.99. The figure shows that to increase consensus initially, the importance of experts E1 and E3 increase. Ultimately, to reach a perfect consensus with experts having differing opinions, eventually all experts but one are completely discounted. One important thing in this procedure is that the alternative order from the four-member group is always A4 , A1 , A2 , and A3 , regardless of the weights to experts changed in this case.
Fig. 7. Group-consensus levels with the changing of weights of experts.
VI. C ONCLUSION This paper introduces a general procedure for a fuzzy groupdecision-making problem that includes four stages: uniforming, aggregating, ranking, and consensus and contribution measure. The paper presents the concepts of the OWA aggregation method and LOWA. The paper summarizes the current use of linguistic labels in decision making and presents a new linguistic-labels aggregation operator FLOWA for the fuzzy group-decision-making problem. The FLOWA method is more detailed and includes more information about the aggregate than existing direct methods. The paper presents a suitable fuzzy ranking method in support of selecting the best alternative. The paper continues and presents an improved consensus measure (among disagreeing experts) that uses a measure of the contributions of the group members. This consensus measurement includes the importance of experts. The paper then presents an entire process that combines all the elements and allows the group to reach a group decision. This decision can be evaluated based on the degree of consensus in the group. In order to improve the consensus, the paper presents a method to modify the weight of the experts based on their support of the group opinion. By updating the importance of the group members, we increase the consensus level. This allows the experts to maintain their professional opinion and not to compromise in order to arrive at a desired consensus level. Additional research is currently underway to determine good stopping rules for the consensus-converging process. This will allow the process to stop at a comfortable level of consensus,
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without discounting all the experts but one. Additionally, we are examining other functions that can distribute the weight in the FLOWA algorithm (in addition to the linear spread). Finally, another research effort underway is the examination of the methodology for a large number of experts. R EFERENCES [1] S. E. Ambrose, The Supreme Commander: The War Years of General Dwight D. Eisenhower. Garden City, NY: Doubleday, 1970. [2] S. M. Baas and H. Kwakernaak, “Rating and ranking of multiple-aspect alternatives using fuzzy sets,” Automatica, vol. 13, no. 1, pp. 47–58, Jan. 1977. [3] G. Bordogna, M. Fedrizzi, and G. Pasi, “A linguistic modeling of consensus in group decision making based on OWA operators,” IEEE Trans. Syst., Man, Cybern. A, Syst., Humans, vol. 27, no. 1, pp. 126–152, Jan. 1997. [4] N. Bryson, “Group decision-making and the analytic hierarchy process: Exploring the consensus-relevant information content,” Comput. Oper. Res., vol. 23, no. 1, pp. 27–35, 1996. [5] P.-T. Chang and E. S. Lee, “Ranking of fuzzy sets based on the concept of existence,” Computers Math. Appl., vol. 27, no. 9/10, pp. 1–21, 1994. [6] S. Chen and C. L. Hwang, Fuzzy Multiple Attribute Decision-Making. New York: Springer-Verlag, 1992. [7] C. Cheng, “Simple fuzzy group decision making method,” in Proc. IEEE Int. Conf. Fuzzy Systems, Seoul, Korea, 1999, vol. 2, pp. II-910–II-915. [8] F. Chiclana, F. Herrera, and E. Herrera-Viedma, “Integrating three representation, models in fuzzy multipurpose decision making based on fuzzy preference relations,” Fuzzy Sets Syst., vol. 97, no. 1, pp. 33–48, 1998. [9] ——, “Integrating multiplicative preference relations in a multipurpose decision-making model based on Fuzzy preference relations,” Fuzzy Sets Syst., vol. 122, no. 2, pp. 277–291, 2001. [10] M. A. P. Davies, “Multicriteria decision model application for managing group decisions,” J. Oper. Res. Soc., vol. 45, no. 1, pp. 47–58, Jan. 1994. [11] M. Delgado, F. Herrera, E. Herrera-Viedma, and L. Martinez, “Combining numerical and linguistic information in group decision making,” Inf. Sci., vol. 107, no. 1–4, pp. 177 194, Jun. 1998. [12] M. Delgado, J. L. Verdegay, and M. A. Vila, “On aggregation operations of linguistic labels,” Int. J. Intell. Syst., vol. 8, no. 3, pp. 351–370, Mar. 1993. [13] J. Einhorn, “Expert judgment: Some necessary conditions and an example,” J. Appl. Psychol., vol. 59, no. 5, pp. 562–571, 1974. [14] F. Herrera and E. Herrera-Viedma, “Aggregation operators for linguistic weighted information,” IEEE Trans. Syst., Man, Cybern. A, Syst., Humans, vol. 27, no. 5, pp. 646–656, Sep. 1997. [15] F. Herrera, E. Herrera-Viedma, and J. L. Verdegay, “Direct approach processes in group decision making using linguistic OWA operators,” Fuzzy Sets Syst., vol. 79, no. 2, pp. 175–190, Apr. 22, 1996. [16] ——, “Rational consensus model in group decision making using linguistic assessments,” Fuzzy Sets Syst., vol. 88, no. 1, pp. 31–49, May 16, 1997. [17] ——, “Sequential selection process in group decision making with a linguistic assessment approach,” Inf. Sci., vol. 85, no. 4, pp. 223–239, Jul. 1995. [18] F. Herrera and L. Martinez, “A model based on linguistic 2-tuples for dealing with multigranular hierarchical linguistic contexts in multi-expert decision-making,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 31, no. 2, pp. 227–234, Apr. 2001. [19] E. Herrera-Viedma, F. Herrera, and F. Chiclana, “A consensus model for multiperson decision making with different preference structures,” IEEE Trans. Syst., Man, Cybern. A, vol. 32, no. 3, pp. 394–402, May 2002. [20] H. M. Hsu and C. T. Chen, “Aggregation of fuzzy opinions under group decision making,” Fuzzy Sets Syst., vol. 79, no. 3, pp. 279–285, 1996. [21] C. L. Hwang and M. Lin, Group Decision Making Under Multiple Criteria: Methods and Applications. Berlin, Germany: Springer-Verlag, 1987. [22] J. Kacprzyk, “Fuzzy linguistic quantifiers in decision making and control,” in Fuzzy Engineering Toward Human Friendly Systems, T. Terano, M. Sugeno, M. Mukaido, and K. Shigemasu, Eds. Amsterdam, The Netherlands: IOS Press, 1992, p. 800. [23] J. Kacprzyk, M. Fedrizzi, and H. Nurmi, “Group decision making and consensus under fuzzy preferences and fuzzy majority,” Fuzzy Sets Syst., vol. 49, no. 1, pp. 21–31, 1992. [24] Consensus Under Fuzziness. J. Kacprzyk, H. Nurmi, and M. Fedrizzi, Eds. Boston, MA: Kluwer, 1997.
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David Ben-Arieh is currently a Professor of Industrial Engineering at Kansas State University, Manhattan. Prior to joining Kansas State University, he taught in the Department of Industrial Engineering and Management and also served as the Head of the Paul Ivanier Center for Robotics and Production Management, Ben-Gurion University, Beer Sheva, Israel. His industrial experience includes working for AT&T Bell Laboratories, and consulting for aerospace industry and health care organizations. He has also received fellowship appointments with the Boeing Company and NASA. He concentrates mainly on applications of decision theory and artificial intelligence applications in manufacturing, and holds one patent in this area.
Zhifeng Chen received the B.S. and M.S. degrees in mechanical engineering from Tongji University, China, in 1997 and 2000, respectively, and the Ph.D. degree in industrial engineering from Kansas State University, Manhattan, in 2005. He currently works at the Enterprise Optimization Group at United Airlines, Chicago, IL. His research interests include deterministic/stochastic optimization, forecasting, scheduling, heuristic optimization, and data mining, with applications in crew planning and supply chain management.
