On the fusion of multi-granularity linguistic label sets in group decision making

Computers & Industrial Engineering 51 (2006) 526–541 www.elsevier.com/locate/dsw

On the fusion of multi-granularity linguistic label sets in group decision making Zhifeng Chen *, David Ben-Arieh Department of Industrial and Manufacturing Systems Engineering, Kansas State University, Manhattan, KS 66506, USA Available online 27 September 2006

Abstract Group decision-making problem is a common and crucial human activity. Many times due to inherent uncertainty, exact numbers can be either costly or unnecessary to be applied to express experts’ opinions or preferences. The use of linguistic labels makes expert judgment more reliable and informative for decision-making. This paper presents a new fusion approach for multi-granularity linguistic information for managing information assessed in different linguistic term sets (multi-granularity linguistic term sets). The paper also presents the application of this approach to a decision-making problem with multiple information sources, assuming that the linguistic performance values given to the alternatives by the different experts are represented in linguistic term sets with different granularity and/or semantic.  2006 Elsevier Ltd. All rights reserved. Keywords: Group decision-making; Fuzzy set; Linguistic label; Fusion operator

1. Introduction Decision-making is a common and important human activity. Group decision-making (i.e., multi-expert) is a typical one where the inherent complexity and uncertainty necessitates the participation of many experts in the decision-making process. In real world, there are decision situations in which the information cannot be assessed precisely in a quantitative form but may be qualitative, and thus, the use of a linguistic description is necessary. As an example, when attempting to qualify phenomena related to human perception, we are often led to use words in natural language instead of numerical values (Herrera, Herrera-Viedma, & Martinez, 2000). Sometimes, precise quantitative information may not be stated because the cost of its computation is too high, so an ‘‘approximate value’’ may be tolerated (e.g., when describing the speed of a car, linguistic terms like ‘‘fast’’, ‘‘very fast’’, ‘‘slow’’ may be used instead of numerical values). 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 2 E pre-

*

Corresponding author. Tel.: +1 847 700 4801; fax: +1 847 700 5033. E-mail address: [email protected] (Z. Chen).

0360-8352/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2006.08.012

Z. Chen, D. Ben-Arieh / Computers & Industrial Engineering 51 (2006) 526–541

527

sents his/her preference relation on Ai as xik 2 S. Where S is a finite but totally ordered term set of linguistic labels S = {s0, s1, . . . , sT}, with si > sj, if i > j. Usually each label has a membership function and preferably each set S has odd number of elements. Also, we have the importance uk, k = [1, 2, . . . , q] assigned to each expert k. These weights represent the importance or trust that each expert carries. A general procedure for group decision-making in a fuzzy environment requires these steps (Herrera et al., 2000): first, the evaluations from each expert should be unified. The second stage is to aggregate opinions from all group members to a final score for each alternative. This score is usually a fuzzy number or a linguistic label, which is used to order the alternatives. Weights of experts can be assigned or calculated. The third stage is to select preferred alternative(s) based on the ranking of the alternatives considering the experts’ weights. Finally, there is a need to measure the consensus level and the individual contribution in order to assess the validity of the group decision. In some cases, another round of consultation with the experts is performed in order to improve the group consensus (Ben-Arieh & Chen, 2004). This is a general procedure for group decision-making in fuzzy environment. Good reviews on group decision-making can be found in Zimmermann (1987), Hwang and Lin (1987), Hwang and Yoon (1981), Kacprzyk and Fedrizzi (1990), etc. The use of linguistic variables makes decision makers’ evaluations more flexible and reliable, but makes the aggregation of the linguistic labels complicated, especially when considering the weights associated with the experts’ evaluations. Generally, there are two main approaches to aggregate linguistic labels in group decision-making. Most methods use the associated membership functions. Among them, Baas and Kwakernaak’s rating algorithm (1977) which aggregates fuzzy scores and weights at different a-cut levels with their associated membership functions. Chen and Hwang (1989) present a conversion scales approach which transforms the linguistic expressions into fuzzy numbers one attribute at a time. This work describes eight conversion scales and finds a scale from the pool containing all linguistic terms such that the scale is as simple as possible. By assigning crisp scores to fuzzy numbers, they then apply classical MCDM method such as TOPSIS. The introduction of the OWA method by Yager in 1988 opened the door to conducts aggregation by applying OWA and linguistic quantifiers (Kacprzyk, Fedrizzi, & Nurmi, 1992; Yager, 1993, 1994, 1995). The other approach is to calculate linguistic labels directly. Defined in (Herrera & Herrera-Viedma, 1997) and (Herrera, Herrera-Viedma, & Verdegay, 1996), Linguistic OWA (LOWA) is based on the OWA (Yager, 1983) and the convex combination of linguistic labels (Delgado, Verdegay, & Vila, 1993). The idea is that the combination resulting from two linguistic labels should be itself an element in the set S. So, given si, sj 2 S and i, j 2 [0, T], the LOWA method finds an index k in the set S representing a single resulting label. Another FLOWA method also based on OWA is from Ben-Arieh and Chen (2004). This model assigns membership functions to all linguistic labels in S by linearly spreading the weights from the labels to be aggregated. The aggregating result changes from a single label in S to a fuzzy set with membership levels of each label in S. A 2-tuple Fuzzy linguistic representation model based on the symbolic translation is introduced by Herrera and Martinez (2000). Here, a linguistic 2-tuple (s, a) is used where s is a linguistic term and a is a numeric value representing the symbolic translation. A new approach to extend different classical aggregation operators with the 2-tuple linguistic model is developed by Herrera and Martı´nez (2001). When both scores and weights are not crisp numbers, Yager, 1998a, 1998b uses the fuzzy modeling technology termed IOWA to model the importance of the scores as well as their values. Other approaches can be applied to group decision-making under linguistic assessments include Fuzzy Analytical Hierarchy Process (Mikhailov, 2004; Shamsuzzaman, Sharif Ullah, & Bohez Erik, 2003), Fuzzy Delphi (Cheng, 1999; Iggland, 1991; Ishikawa, 1993), Fuzzy Compromising (Prodanovic & Simonovic, 2003). Blin (1974) proposed to represent a relative group preference as a fuzzy preference matrix composed from individual preferences. Fan, Ma, and Zhang’s (2002) weighting construction method tries to minimize the difference between the decision makers’ fuzzy preference information and the group fuzzy preference information calculated from the decision matrices. By solving this optimization model the optimal weights to the criteria are calculated. Then the alternatives are ranked based on the aggregation results. Hsu and Chen (1996) proposed a similarity based aggregation method. The basic idea of the method is that weight of an expert’s opinion should be larger if his opinion is closer to other opinions.

528

Z. Chen, D. Ben-Arieh / Computers & Industrial Engineering 51 (2006) 526–541

Those approaches work well with homogeneous data type, but have problems with composite data, for example when different linguistic definitions are used. An information fusion operator is necessary before aggregation. In this case, a fusion operation in group decision-making is needed to make the opinions uniform. As discussed above, this is the first step of group decision-making and should be performed before aggregating all experts’ evaluations. To propose a new fusion algorithm, the paper is organized as following. In Section 2 of this paper, we provide a literature review of data fusion approaches used in prior research in group decision-making. We propose a new fusion approach of multi-granularity linguistic information for managing information assessed in different linguistic term sets in Section 3. In Section 4, we apply this approach to a numeric example demonstrating the procedure of solving the fuzzy group decision-making problem. Section 5 provides the summary and conclusions. 2. Prior research on data fusion in group decision-making Data fusion is a technology of combining various forms of data such as sound, image, numerical, and linguistic, and acquiring knowledge by combining these data types. This enables a group decision-making where experts use heterogeneous data types. Currently, most of the research in data fusion is in the fields of sensing and communicating (Chen & Luo, 1999; Goodridge, Luo, & Kay, 1994; Hussien, Bender, & Ismael, 1994a; Hussien, Ismael, & Bender, 1994b; Singh & Bailey, 1997; Valet, Mauris, Bolon, & Keskes, 2003). When there are both linguistic and numeric data in a group decision-making problem, we need a fusion algorithm to transform the experts opinion between linguistic and numeric types in order to aggregate experts’ opinions. In support of data conversion there are many fuzzification and defuzzification approaches such as Maximum Defuzzifier, and the Centroid Defuzzifier (Mendel, 1995). In the following subsection, we review the current researches on fusion models applied in group decision-making. 2.1. Fusion between linguistic and numeric data Akiyama (2000) presents an object-oriented fusion and diffusion algorithms. The fusion algorithm glues a set of linguistically defined object types to create the new object type that transforms crisp information to the linguistic and primitive format. As the opposite operation of the fusion operation, the object diffusion is to unglue a linguistic object type and transforms it from a linguistic format into crisp and primitive formats. Chen and Chen (2002) present a new information fusion algorithm based on a similarity measure. The algorithm can handle heterogeneous fuzzy group decision-making problems in a more flexible and intelligent manner. Delgado, Herrera, Herrera-Viedma, and Martinez (1998) introduces two transformation functions, from linguistic to numerical and from numerical to linguistic based on the fuzzy number characteristic values. They also propose a group decision-making process based on the fusion operator. Grabisch and Saveant (1998) propose a framework for handling uncertainty in data fusion based on possibility theory and also present a linguistic interface to translate possibility distributions into a natural language form. Torrez, Bamber, Goodman, and Nguyen (2002) combine ‘‘Boolean related event algebra’’ and ‘‘one point random set coverage representations of fuzzy sets’’ together in order to integrate fuzzy input into a probability input. Hathaway, Bezdek, and Pedrycz (1996) presents a model to integrate numbers, intervals and linguistic assessments from three types of sensors. Moses, Degani, Teodorescu, Friedman, and Kandel (1999) developed a linguistic coordinate transformation algorithm for complex fuzzy sets. This can be applied to an adaptive control system with linguistic inputs and outputs. 2.2. Linguistic to linguistic data In linguistic approach, an important decision parameter is the ‘‘granularity of uncertainty’’ (Herrera et al., 2000), i.e., the cardinality of the linguistic term set being used to express the information. The linguistic term set is chosen according to the uncertainty degree that an expert has on qualifying a phenomenon. When different experts have different uncertainty degrees on the phenomenon, they use several linguistic term sets with different granularity of uncertainty.

Z. Chen, D. Ben-Arieh / Computers & Industrial Engineering 51 (2006) 526–541

529

The use of more linguistic label sets gives decision makers more flexibility. Similar to using crisp numbers, one expert may choose values from 1 to 5 as his evaluation, while another expert prefers values from 1 to 10 to make a more detailed opinion. The use of linguistic labels has the same quality. Some experts like to use five linguistic labels set such as S1 = {none, low, medium, high, perfect} and others may prefer more linguistic labels like S2 = {none, very low, low, medium, high, very high, perfect}. Using less linguistic labels simplifies the representation of the data while using more linguistic labels improves the accuracy of the decision while increasing the computational complexity of the solution. A fusion algorithm is necessary to transform one set of linguistic labels to another. Suppose two experts use the two linguistic labels sets S1 and S2, respectively. Even both of them choose ‘‘low’’ as their score to an alternative, the same linguistic label ‘‘low’’ may have different definitions, i.e., the same label ‘‘low’’ may have different membership functions. Actually, the ‘‘low’’ in S1 (in the above example) covers the ‘‘low’’ in S2 while the ‘‘low’’ in S2 is a subset of the ‘‘low’’ in S1. So, we cannot use these two ‘‘lows’’ equivalently in the process of the decision-making. A fusion algorithm is used to convert multi-granularity linguistic term sets into a specific linguistic domain, which provides the basic linguistic term set. The basic linguistic term set is chosen so as not to impose useless precision to the original evaluations and in order to allow an appropriate discrimination of the initial performance values. Herrera et al. (2000) propose an algorithm for transformation between linguistic sets. 2.2.1. The fusion method (Herrera et al., 2000) This approach allows experts to use different sets of linguistic labels as their score sources. The method proposed a transformation function to unify the information. After aggregating all experts’ opinions, ranking methods are applied to choose the best alternative(s). Fig. 1 (Herrera et al., 2000) shows an example of layers

Expression

None

Low

Medium

Very Low

None

Low

High

Medium

Perfect

Very High

High

Perfect

Layer 3 Very Low

None

None

Almost None

Very Low

Low

Almost Medium

Medium

Almost High

High

Very High

Low

Almost Medium

Medium

Almost High

High

Very High

Fuzzy Set in (0, 0.5)

Real number in (0, 0.5)

Fuzzy Set approx 0.5

0.5

Perfect

Almost Perfect

Fuzzy Set in (0.5, 1)

Real number in (0.5, 1)

Fig. 1. The layers of the linguistic labels (Herrera et al. (2000)).

Perfect

Layer 2

Layer 1

530

Z. Chen, D. Ben-Arieh / Computers & Industrial Engineering 51 (2006) 526–541

of the linguistic labels. The set can have 5, 7, 9, 11, or even more linguistic labels. However, regardless of the cardinality of the sets each set covers the real numbers interval [0, 1]. The more linguistic labels used, the smaller the range that each linguistic label covers. For instance, for S1 = {none, low, medium, high, perfect}, there are two linguistic labels ‘‘none’’ and ‘‘low’’ cover the range [0, 0.5). But if we use S2 = {none, almost none, very low, low, almost medium, medium, almost high, high, very high, almost perfect, perfect}, the same range [0, 0.5) is covered by five linguistic labels. For different purposes, different experts can choose different linguistic sets for evaluation pool. Here the linguistic label set Sj is defined by the expert Ej, from Mj + 1 linguistic labels to evaluate the alternatives. The problem now is to transform all sets Sj into a standard linguistic set ST for later aggregation. The idea of the fusion method from Herrera et al. (2000) is to assign a membership value to every linguistic label in the target set for each linguistic label being transformed. The membership is computed by finding the interaction of two linguistic labels, target and source. The multi-granularity transformation function sS T is defined as sS T : S M ! F ðS T Þ sS T ðli Þ ¼

fðsj ; aij Þjj

ð1Þ 2 f0; . . . ; T gg;

8li 2 S M

ð2Þ

aij ¼ max minflli ðyÞ; lsj ðyÞg

ð3Þ

y

where SM = {l0, l1, . . . , lM} and ST = {s0, s1, . . . , sT} are source and target linguistic label sets, respectively, such that T P M. One problem with this approach is selecting the target set ST from all the linguistic label sets. When there is only one term set with the maximum granularity, we choose the set with the maximum granularity as ST, where T = max (M1, M2, . . . , Mq). If we have two or more linguistic label sets with maximum granularity, then ST is chosen depending on the semantics of these linguistic label sets, finding two possible situations to establish ST: If all the linguistic label sets have the same semantics, then ST is any of them. There are some linguistic label sets with different semantics. Then, ST is a basic linguistic label set with a larger number of labels than the number of labels that a person is able to discriminate (Normally 11 or 13). Example 1. This example shows how this fusion approach works. Here, we have M = 4 and T = 6, S4 = {l0, l1, . . . , l4} and S6 = {s0, s1, . . . , s6} with the following membership functions: l0 : ð0; 0; 0:25Þ;

s0 : ð0; 0; 0:16Þ

l1 : ð0; 0:25; 0:5Þ; l2 : ð0:25; 0:5; 0:75Þ;

s1 : ð0; 0:16; 0:34Þ s2 : ð0:16; 0:34; 0:5Þ

l3 : ð0:5; 0:75; 1:0Þ;

s3 : ð034; 0:5; 0:66Þ

l4 : ð0:75; 1:0; 1:0Þ

s4 : ð0:5; 0:66; 0:84Þ s5 : ð0:66; 0:84; 1:0Þ s6 : ð0:84; 1:0; 1:0Þ

s0

s1

l1

s2

s3

s4

s5

s6

Fig. 2. The transformation of l1 by F. Herrera’s method.

Z. Chen, D. Ben-Arieh / Computers & Industrial Engineering 51 (2006) 526–541

531

Applying sS T , for l0 and l1 are: sS T ðl0 Þ ¼ fðs0 ; 1Þ; ðs1 ; 0:58Þ; ðs2 ; 0:18Þ; ðs3 ; 0Þ; ðs4 ; 0Þ; ðs5 ; 0Þ; ðs6 ; 0Þg sS T ðl1 Þ ¼ fðs0 ; 0:39Þ; ðs1 ; 0:85Þ; ðs2 ; 0:85Þ; ðs3 ; 0:39Þ; ðs4 ; 0Þ; ðs5 ; 0Þ; ðs6 ; 0Þg Fig. 2. demonstrates the calculation of sS T ðl1 Þ. 3. A new fusion method The fusion approach mentioned above has several drawbacks. One of the problems of the method is that we need to assign a membership function to each linguistic label. Different membership function will result in different transformation results. A second and more serious limitation is that we can only transfer a small linguistic label set into a larger one, but cannot do the inverse operation, i.e., we can use this method to transform a five linguistic labels set S4 to another linguistic set with cardinality of 7, but we cannot apply this method to map seven labels to five. In this section, we describe a new fusion method which computes the linguistic labels directly and without the limitations mentioned above. 3.1. A new fusion approach The new fusion process assigns membership functions to all linguistic labels in the target set directly. Note that no matter how many linguistic labels a set has, it covers the real numbers interval [0, 1]. Fig. 3 shows the basic idea of the new approach with T = 2, 4, 6, 8, 10, 12, 14. In this case, the interval that each linguistic label covers overlaps with the other intervals from other sets. The ratio of the common interval they cover indicates the membership function that should be assigned. Since usually membership functions of linguistic labels are symmetric, the transformation result depends only on the size of overlap between the various labels. As we defined before, two linguistic label sets ST = {s0, s1, . . . , sT} and SM = {l0, l1, . . . , lM} are the target and source sets, respectively, so we transform SM to ST. In Fig. 4, the linguistic label li in SM covers a small interval of Sj  1, the whole of Sj, and a part of Sj + 1. The corresponding memberships lS i;j1 , lS i;j , lS i;jþ1 of the three linguistic labels Sj  1, Sj, and Sj + 1 in ST (which overlaps with li) are greater than 0. The membership function lS i;j1 ¼ a=w is defined by the ratio of the interval covered by the intersection of linguistic labels Sj  1 and li (the length a in Fig. 4) to w, the width of the labels in ST. Other labels in ST get membership of 0s since they have no common interval with the label li in SM. Then the linguistic label li can be expressed by: li ¼ fðsj ; lS ij ðxÞÞjj 2 ½0; T g We can use the following formulas to compute the membership functions lS ij ðxÞ: sS T : S M ! F ðS T Þ

Fig. 3. The tiers of the linguistic label sets.

532

Z. Chen, D. Ben-Arieh / Computers & Industrial Engineering 51 (2006) 526–541

Fig. 4. The idea of tier method on the linguistic label sets fusion.

sS T ðli Þ ¼ fðsj ; lS ij ðxÞÞjj 2 ½0; T g 8 1 > >   > > > i < Tjþ1  ðT þ 1Þ Mþ1 þ1 lS ij ðxÞ ¼   > j iþ1 > > > Mþ1  T þ1 ðT þ 1Þ > : 0

ð4Þ for jmin < j < jmax for j ¼ jmin

ð5Þ

for j ¼ jmax others

where, jmin and jmax are the indexes of the first and the last linguistic labels with nonzero membership functions in the target set ST. In Fig. 4, jmin = j  1 and jmax = j + 1. For each i, the corresponding jmin and jmax are defined by: jmin i j þ1 6 min ; 6 T þ1 T þ1 M þ1 jmax iþ1 j þ1 6 max ; 6 T þ1 T þ1 M þ1

i ¼ 0; 1; . . . ; M

ð6Þ

i ¼ 0; 1; . . . ; M

ð7Þ

þ1 i i It is easy to see that when j = jmin in Eq. (5), Mþ1 6 jmin , then ðTjþ1  Mþ1 ÞðT þ 1Þ P 0. Similarly, when j = jmax, þ1 T þ1 j iþ1 ðMþ1  T þ1ÞðT þ 1Þ P 0. Then lS ij ðxÞ P 0 is always true. This means the membership function is always nonnegative.

Remark 1. The new fusion approach assumes that all linguistic labels in a set evenly cover the interval of [0, 1]. Also, there are no overlaps between any two adjacent linguistic labels. Remark 2. The new fusion approach requires M < 2T + 1. This means the number of linguistic labels to be transformed (M) should not be larger than 2 times of the number of the linguistic labels will be presented (T). This can be easily proved by: 1 1 <2 T þ1 M þ1 This inequality comes from the requirement that no two source linguistic labels can be covered by the same linguistic label from the target set. If not, we have M P 2T + 1, from the formula (4) at least    T þ1 sS T ðl0 Þ ¼ sS T ðl1 Þ ¼ s0 ; ; ðs1 ; 0Þ; ðs2 ; 0Þ; . . . ; ðsT ; 0Þ M þ1 In this way, we cannot distinguish the two linguistic labels l0 and l1 after the fusion operation. For instance, T = 4, M = 12, M > 2 · 4 + 1 = 9 does not satisfy the restriction, then l0 and l1 have the same number as    4þ1 sS 4 ðl0 Þ ¼ sS 4 ðl1 Þ ¼ s0 ; ; ðs1 ; 0Þ; ðs2 ; 0Þ; ðs3 ; 0Þ; ðs4 ; 0Þ ¼ fðs0 ; 0:38Þ; ðs1 ; 0Þ; ðs2 ; 0Þ; ðs3 ; 0Þ; ðs4 ; 0Þg 12 þ 1 So, this rule guarantees that we have different linguistic label sets after the fusion operation. Example 2. This example shows how to apply the new fusion algorithm to transform a larger linguistic labels set with cardinality M + 1 = 11 to a small linguistic labels set with T = 8. i 4 iþ1 5 For i = 4, Mþ1 ¼ 11 , Mþ1 ¼ 11

Z. Chen, D. Ben-Arieh / Computers & Industrial Engineering 51 (2006) 526–541

533

4 4 5 4þ1 6 10þ1 6 3þ1 8þ1 and 8þ1 6 10þ1 6 8þ1, we can see that jmin = 3 and jmax = 4     4 4 5 4  ð8 þ 1Þ ¼ 0:73, lS 44 ðxÞ ¼ 10þ1  ð8 þ 1Þ ¼ 0:09  10þ1  8þ1 So, lS 43 ðxÞ ¼ 8þ1

By

3 8þ1

lS 4j ðxÞ ¼ 0;

for all j ¼ 0; 1; . . . ; 8 and j 6¼ 3; 4

Then we have the transformation result as sS 8 ðl4 Þ ¼ fðs0 ; 0Þ; ðs1 ; 0Þ; ðs2 ; 0Þ; ðs3 ; 0:73Þ; ðs4 ; 0:09Þ; ðs5 ; 0Þ; ðs6 ; 0Þ; ðs7 ; 0Þ; ðs8 ; 0Þg Example 3. This example shows how to apply the new fusion algorithm to transform a small linguistic labels set with cardinality M + 1 = 9 to a larger linguistic labels set with T + 1 = 11. i iþ1 For i = 4, Mþ1 ¼ 49, Mþ1 ¼ 59 4 4 4þ1 6 5 6þ1 By 10þ1 6 8þ1 6 10þ1 and 10þ1 6 8þ1 6 10þ1 , we have jmin = 4 and jmax = 6 5 4 5 6 So, lS 44 ðxÞ ¼ ð10þ1  8þ1Þ  ð10 þ 1Þ ¼ 0:11, lS 45 ðxÞ ¼ 1, lS 46 ðxÞ ¼ ð8þ1  10þ1 Þ  ð10 þ 1Þ ¼ 0:11

lS 4j ðxÞ ¼ 0;

for all j ¼ 0; 1; . . . ; 10 and j 6¼ 4; 5; 6

Then we have the transformation result as sS 8 ðl4 Þ ¼ fðs0 ; 0Þ; ðs1 ; 0Þ; ðs2 ; 0Þ; ðs3 ; 0Þ; ðs4 ; 0:11Þ; ðs5 ; 1Þ; ðs6 ; 0:11Þ; ðs7 ; 0Þ; ðs8 ; 0Þ; ðs9 ; 0Þ; ðs10 ; 0Þg 3.2. Extension of the new fusion approach So far the fusion method discussed requires that there are no overlaps between the intervals represented by two linguistic labels. This is not always the case. One of the capabilities of fuzzy set theory is to represents sets with no sharp differences in the boundaries (Figs. 5 and 6). Thus, the former assumption is not always feasible, and there can be overlaps between membership functions of two linguistic labels as shown in Fig. 7. Here we introduce a new parameter a to measure the degree of overlapping. a is defined as the increasing rate of the width of the interval the linguistic label represents without overlapping to the width of the interval the linguistic label covers with overlap. That means, if the width of the linguistic labels is w0 without overlapping, the new width is w1 = (1 + a) Æ w0. For instance, the width without overlap (a = 0) of 5 linguistic label set is w0 = 1.0/5 = 0.2. If a = 0.5, the new width a linguistic label covers is 0.3, so the interval of one of the linguistic labels in this set will change from [0.2, 0.4] to [0.15, 0.45]. Here the increased width is expended to both directions. For the first and the last linguistic labels in linguistic sets, we just expend to one direction, i.e., the first linguistic label covers the interval of [0, 0.25] instead of [0, 0.2] with a = 0.5.

2/9 S0 l0

S1 l1

3/9 S2

l2

4/9 S3

S4

l4

l3 3/11

5/9 S5

l5

4/11

5/11

l6

S6 l7

S7 l8

S8 l9

l10

6/11

Fig. 5. The transformation of large linguistic label set (M = 10) to a small set (T = 8).

4/11 S0 l0

S1

S2 l1

S3 l2

5/11 S4

S5

l3 3/9

6/11

S6

l4 4/9

7/11 S7 l5

5/9

S8 l6

S9 l7

S10 l8

6/9

Fig. 6. The transformation of small linguistic label set (M = 8) to a large set (T = 10).

534

Z. Chen, D. Ben-Arieh / Computers & Industrial Engineering 51 (2006) 526–541 Sj-1

Sj

S j+1 β

li-1

li

α

li+1

Fig. 7. The tier method on the linguistic label sets fusion.

Again, we assume that the membership functions are symmetric. This is a reasonable assumption, since for any given linguistic variable, most people use symmetric evaluation function. Given the desired parameter a and the original width of the interval the linguistic label covers, we can compute the new width we need for the linguistic label. Table 1 shows an example of a 5-linguistic label set: With the new parameter a, we should justify Eqs. (5)–(7). 1þa The new width of the linguistic label are Mþ1 for all i = 0, . . . , M. The interval of the ith label in the set SM i iþ1 2ia 2iþaþ2 1þa Ma changes from ½Mþ1 ; Mþ1 to ½2ðMþ1Þ ; 2ðMþ1Þ, for i = 1, . . . , (M  1), with ½0; Mþ1  when i = 0 and ½ðMþ1Þ ; 1 for L R i = M. To simplify the expressions, we introduce the definitions of ci;M;a and ci;M;a to represent the two boundaries of the intervals that a linguistic label covers Tables 2–4. 8 8 2iþ2aþ2 0; i¼0 > > < < 2ðMþ1Þ ; i ¼ 0 2ia L R 2iþaþ2 ; 0 < i < M ; ci;M;a ¼ 2ðMþ1Þ ci;M;a ¼ 2ðMþ1Þ ; 0 > : 2i2a : ; i ¼ M 1; i¼M 2ðMþ1Þ Then the new functions to compute jmin and jmax with overlap of a in the sources, and b in the target sets:

Table 1 Comparison of the two linguistic sets with different a level for M = 4 a

Original width

New width

s0

s1

s2

s3

s4

0 0.4 0.5

0.20 0.20 0.20

0.20 0.28 0.30

[0, 0.20] [0, 0.24] [0, 0.25]

[0.20, 0.40] [0.16, 0.44] [0.15, 0.45]

[0.40, 0.60] [0.36, 0.64] [0.35, 0.65]

[0.60, 0.80] [0.56, 0.84] [0.55, 0.85]

[0.80, 1] [0.76, 1] [0.75, 1]

Table 2 Properties of the linguistic sets from each expert Expert Ek

Linguistic set Sk

Mk

Overlapping degree a

E1 E2 E3 E4

S1 = {a0, a1, a2, a3, a4, a5, a6, a7, a8} S2 = {b0, b1, b2, b3, b4, b5, b6} S3 = {c0, c1, c2, c3, c4} S4 = {d0, d1, d2, d3, d4, d5, d6, d7, d8}

8 6 4 8

0.5 0.5 0.5 0.3

Table 3 Evaluations from experts

E1 E2 E3 E4

A1

A2

A3

A4

a4 b3 c2 d4

a6 b4 c3 d5

a3 b3 c2 d3

a5 b5 c1 d5

Z. Chen, D. Ben-Arieh / Computers & Industrial Engineering 51 (2006) 526–541

535

Table 4 Mean and variance values to all alternatives by fuzzy outranking

Mean, xðAi Þ Variance, r(Ai)

A1

A2

A3

A4

4.00 0.47

5.37 0.65

3.18 0.39

5.00 0.00

2jmin  b 2i  a 2j þ b þ 2 6 6 min ; 2ðT þ 1Þ 2ðM þ 1Þ 2ðT þ 1Þ

i ¼ 0; 1; . . . ; M;

2jmax  b 2i þ a þ 2 2jmax þ b þ 2 6 6 ; 2ðT þ 1Þ 2ðM þ 1Þ 2ðT þ 1Þ

j ¼ 0; 1; . . . ; T

i ¼ 0; 1; . . . ; M;

ð8Þ

j ¼ 0; 1; . . . ; T

ð9Þ

It could happen that the left side of li lies between the overlapping area of two linguistic labels sj  1 and sj (Fig. 8). Actually, there are three cases for the location of the left sides of li. as well as the right side (Figs. 2jb 2ia 1 lies between the left side of S j1 : 2ðT  T þ1 and the left side 9 and 10) Case one is the left side of li; : 2ðMþ1Þ þ1Þ 2jþb 2ia of Sj. Another case is that the left side of li 2ðMþ1Þ lies between the right side of S j1 : 2ðT and the right side of þ1Þ 2jþb 2jb 2jþb 1 S j : 2ðT þ T þ1 . The third case is it locates between the left side of S j : 2ðT and the right side of S j1 : 2ðT . þ1Þ þ1Þ þ1Þ We can still use the formula (8) and (9) to define jmin and jmax, but now the jmin and jmax are not single numbers, but sets of indexes of the linguistic labels. And they are the indexes of the linguistic labels with the membership functions greater than 0 and less than 1.

Sj-1

S k-1

Sj

Sk β

…

li α

Fig. 8. The jmin and jmax on the linguistic label sets fusion.

0.528

T=8

S0

M=6

l0

S1

S2

S3

l1

S4

l2

0.639 0.694

S5 l3

S6

0.806

S7 l5

l4 0.679

S8

β =0.5

l6

α =0.5

0.893

Fig. 9. The transformation of small linguistic label set (M = 6) to a large set (T = 8) with extension.

0.417

0.528

S4

T=8

S0

S1

S2

S3

M=8

l0

l1

l2

l3

l4 0.539

0.583

0.691

0.806

S5

S6

l5

l6

S7 l7

S8

β =0.3

l8

α=0.5

0.683

Fig. 10. The transformation of two linguistic label sets with same cardinality (M = T = 8).

536

Z. Chen, D. Ben-Arieh / Computers & Industrial Engineering 51 (2006) 526–541

Then there are more linguistic labels whose membership functions lie in the interval (0, 1). For example, in Fig. 8, the membership functions assigned to sj  1 and sj as well as sk  1 and sk should be less than 1 and greater than 0. The membership function should be rewrite as: 8 1 for maxfjmin g < j < minfjmax g > > > >    > > þ1 > > cRj;T ;b  cLi;M;a T1þb for j 2 jmin n fjmin \ jmax g > > > > <   þ1 ð10Þ lS ij ðxÞ ¼ cRi;M;a  cLj;T ;b T1þb for j 2 jmax n fjmin \ jmax g > > >    > > > 1þa T þ1 > for j 2 fjmin \ jmax g > Mþ1 1þb > > > > : 0 others Example 4. This example shows how to apply the new fusion algorithm with the parameter a = 0.5 and b = 0.5 to transform a smaller linguistic labels set with cardinality M + 1 = 7 to a large linguistic labels set with T = 8. For i = 5, a = b = 0.50 By (8) and (9), we can define jmin = {5, 6} and jmax = {7, 8} From (10), jmin \ jmax = / þ1 For j = 8, j 62 {jmin \ jmax} then lS 58 ðxÞ ¼ ðcRi;M;a  cLj;T ;b ÞðT1þb Þ ¼ 0:359, given i = 5, j = 8, M = 6 T = 8 and a = b = 0.50. Similarly, we have the transformation result as sS 8 ðl5 Þ ¼ fðs0 ; 0Þ; ðs1 ; 0Þ; ðs2 ; 0Þ; ðs3 ; 0Þ; ðs4 ; 0Þ; ðs5 ; 0:090Þ; ðs6 ; 0:760Þ; ðs7 ; 0:856Þ; ðs8 ; 0:359Þg Example 5. This example shows the case when jmin \ jmax 5 /.

is:

For i = 5, a = 0.50, a = b = 0.30 By (8) and (9), we can define jmin = {4, 5} and jmax = {5, 6} From (10), jmin \ jmax = {5} 1þa T þ1 8þ1 When j = 5, j 2 {jmin \ jmax}, then lS 55 ðxÞ ¼ Mþ1  1þb ¼ 1þ0:5 8þ1  1þ0:3 ¼ 0:862. And the transformation result sS 8 ðl5 Þ ¼ fðs0 ; 0Þ; ðs1 ; 0Þ; ðs2 ; 0Þ; ðs3 ; 0Þ; ðs4 ; 0:263Þ; ðs5 ; 0:862Þ; ðs6 ; 0:263Þ; ðs7 ; 0Þ; ðs8 ; 0Þg

Remark 1. The overlapping degree a and b reflect the ‘‘semantics’’ of the linguistic labels. It is similar to the different definitions of membership functions from different experts. Remark 2. With extension, the restriction of M < 2T + 1 should be changed too. By the same philosophy that no two source linguistic labels can be covered by the same linguistic label from the target set, we have the condition: 1þb T þ1

1þa < 2  Mþ1 , we obtain the new restriction

M<

2ðT þ 1Þð1 þ aÞ 1 ð1 þ bÞ

3.3. Properties of the new fusion approach The transformation result from Herrera et al.’s method (2000a) depends on linguistic membership functions. Different membership function will result in different transformation result which will affect the choice of alternative(s). Also the approach can only transform smaller linguistic set to a larger one. The new approach works in both directions, thus suitable to different applications: one can transforms a small set of linguistic labels into a larger set or transform a large set into a smaller one. Both directions have advantages and disadvantages. The transformation from a large set into a smaller one avoids useless precision,

Z. Chen, D. Ben-Arieh / Computers & Industrial Engineering 51 (2006) 526–541

537

and reduces the process time that in turn reduces the total decision cost. The transformation from a small set into a larger one allows for higher accuracy in future decisions. For those applications that do not require high accuracy, we can choose to transform from a large linguistic set to a smaller set to save time and costs. The new approach is easy to calculate. Since we do not compute the membership functions, it simplifies the calculations. And new fusion approach has the following interesting properties: Property 1. The linguistic label set after transformation is ordered. We have li < lj, with i < j Proof. Since both linguistic sets ST and LM are ordered and cover the same increasing crisp number range [0, 1], the transformation function will not change the order of linguistic labels. h Property 2. (Summation) For each linguistic label li after transformation, its membership function satisfies T X

lS ij ðxÞ ¼

j¼0

T þ1 1þa b  þ pþD M þ1 1þb 1þb

where p = min{jmax}  max{jmin},     2 minfjmin g þ b þ 2 2i  a 2i þ a þ 2 2 maxfjmax g  b   D ¼ max 0; þ max 0; 2ðT þ 1Þ 2ðM þ 1Þ 2ðM þ 1Þ 2ðT þ 1Þ Property 3. If we substitute a = 0, b = 0 into formula (10), (11) we can get exactly (5)–(7) Property 4. When M = T and a 5 b, the two linguistic sets have the same cardinality, but different overlapping degree (semantics). This could happen when two experts use the same set of linguistic labels, but have different definitions of membership functions. With M = T, the formula (8), (9) will be,

lS ij ðxÞ ¼

8 1 > > > > > > < ðaþbÞþ2ð1iþjÞ 1þb > ðaþbÞþ2ð1þijÞ > > > 1þb > > : 0

for jmin < j < jmax for j ¼ jmin

ð11Þ

for j ¼ jmax others

Property 5. When M = T and a = b, the two linguistic sets have the same cardinality, and the same overlapping degree (semantics). Or we can say these two linguistic sets are identical. Formula (10) and (11) will give us the same value as jmin = jmax and it is not hard to test that lS ij ðxÞ ¼ 1 for j = jmin. 3.4. The decision-making procedure with fusion In this context, we propose a decision procedure with the objective of finding the best alternative (or a set of alternatives): Each decision maker provides an opinion regarding all alternatives. They choose from their own linguistic set and give the cardinality of the set as well as the parameter a. The fusion of the multi-granularity linguistic performance values is carried out in order to obtain collective performance evaluations. In this step, the multi-granularity linguistic information is made uniform using a linguistic term set as the uniform representation base, the basic linguistic term set. An OWA based aggregation is applied. Finally, the choice of the best alternative(s) from the collective performance evaluations is performed. To do that, a fuzzy preference relation is computed from the collective performance evaluations using a ranking method of pairs of fuzzy sets in the setting of Possibility Theory, applied to fuzzy sets on the basic

538

Z. Chen, D. Ben-Arieh / Computers & Industrial Engineering 51 (2006) 526–541

linguistic term set. Then, a consensus level may be set on the preference relation in order to rank the alternatives.

4. A numeric example In this section, we use the same investment example from Herrera et al. (2000) to demonstrate the group decision process with fusion. 4.1. Data collection In the example, we have an investment company, which wants to invest a sum of money in the best option. There is a panel with four possible alternatives A = {A1, A2, A3, A4} of investment possibilities. A1 is a car industry, A2 is a food company, A3 is a computer company, and A4 is an arms industry. The investment company chooses four experts from four consultancy departments: risk analysis, growth analysis, social-political analysis, and environmental impact analysis departments respectively, to construct a decision group. These experts use different linguistic term sets to provide their preferences over the alternative set as followings: It should be noticed here that the original linguistic sets provide membership functions for each linguistic label in the original set. We estimate the overlapping degrees to each set based on the given membership functions. Then after an in depth study, each expert provides the following preference values. 4.2. Experts’ evaluation unification Following the decision procedure from Section 3.4, we unify experts’ opinions by the proposed fusion algorithm. Since min{Mi, i = 1, 2, 3, 4} = 4 and max{Mi, i = 1, 2, 3, 4} = 8, we have M < 2T + 1, no matter which set is chosen as the target set (BLTS). Notice that s1 and s4 have the same cardinality (M1 = M4 = 8), but different semantics (a1 = 0.5, a4 = 0.3). In the original paper, a special set of 15 labels is used as the target set. Here we objectively choose s1 as our target set simply because a1 = a2 = a3 = 0.5 and M1 = M4 = 8 will simplify the calculation. Then, we obtain the following results: E 1: A 1: A 2: A 3: A 4:

(0, (0, (0, (0,

0, 0, 0, 0,

0, 0, 0, 0,

0, 0, 1, 0,

E 2: A 1: A 2: A 3: A 4:

(0, (0, (0, (0,

0, 0, 0, 0,

0, 0, 0, 0,

0.476, 1, 0.476, 0, 0, 0.286, 1, 0.667, 0.476, 1, 0.476, 0, 0, 0, 0.095, 0.762,

E 3: A 1: A 2: A 3: A 4:

(0, 0, 0.067, 0.733, 1, 0.733, 0.067, 0, 0) (0, 0, 0, 0, 0.2, 0.867, 1, 0.6, 0.102) (0, 0, 0.067, 0.733, 1, 0.733, 0.067, 0, 0) (0.102, 0.6, 1, 0.867, 0.2, 0, 0, 0, 0)

1, 0, 0, 0,

0, 0, 0, 1,

0, 1, 0, 0,

0, 0, 0, 0,

0) 0) 0) 0)

0, 0) 0, 0) 0, 0) 0.857, 0.359)

Z. Chen, D. Ben-Arieh / Computers & Industrial Engineering 51 (2006) 526–541

E4: A1: A2: A3: A4:

(0, (0, (0, (0,

0, 0, 0, 0,

0, 0.267, 0.862, 0.267, 0, 0, 0, 0.267, 0.862, 0.267, 0.267, 0.862, 0.267, 0, 0, 0, 0, 0.267, 0.862, 0.267,

0, 0, 0, 0,

539

0) 0) 0) 0)

4.3. Experts’ opinions aggregation We aggregate four experts’ opinions by OWA operator FQ, guided by the same fuzzy linguistic quantifier Q = ‘‘as many as possible’’ (Yager, 1995), with parameter (0.5, 1). The weighting vector from linguistic quantifier can be calculated as w = [0, 0, 0.5, 0.5]. Then the aggregated performance values for each alternative are: A1: (0, 0, 0, 0.134, 0.931, 0.134, 0, 0, 0) A2: (0, 0, 0, 0, 0.100, 0.434, 0.467, 0, 0) A3: (0, 0, 0, 0.605, 0.134, 0, 0, 0, 0) A4: (0, 0, 0, 0, 0, 0.048, 0, 0, 0) 4.4. Alternatives selection The fuzzy preference relation B and the strict preference relation BS 1 0 0  0 0:471  0:134 0:605 0:048 C B B  0:467 0:048 C B 0:333  0:367 B 0:467 C; B s ¼ B B¼B C B0 B 0:134 0:100 0   0 A @ @ 0:048 0:048

0:048



0

0

0:048

are: 0

1

C 0 C C 0 C A 

Where bij ¼ max minflGAi ðsl Þ; lGAj ðsh Þg, sl,sh 2 ST indicates the degree of possibility of dominance of Ai over Aj sl

sh 6sl

and lGAi ðsj Þ is the aggregated group membership degree to the sj in the linguistic label set for the alternative Ai. bsji ¼ maxfbji  bij ; 0g represents the degree to which Ai is dominated by Aj. Then the non-dominate choice degree (NDD) of each alternative is obtained by NDDi ¼ minf1  bsji ; j 6¼ ig: Aj {NDD1 = 0.667 NDD2 = 1 NDD3 = 0.529 NDD4 = 1} Finally, maximum solution set of alternatives is AND = {A2, A4} To be comparable, we apply the same method in (Herrera et al., 2000) to select the alternatives. Actually, from the aggregated group evaluations, any fuzzy sets outranking method can be applied to rank alternatives, since all linguistic labels are ordered from the properties of linguistic set si < sj, for i < j (Yager, 1998a, 1998b). For example, if we use the fuzzy sets outranking approach in (Lee & Li, 1998), the mean and variance values can be obtained: Then we can directly choose the best alternative as AND = {A2} 5. Conclusion This paper proposed a general procedure for transforming the granular expert preferences in a multi-expert decision-making scenario from fine to coarse or from coarse to fine granule sets. Compared to the other methods, it is more flexible in the sense that one can transform expert opinion from fine granular form to coarse granular form and from coarse granular form to fine granular form. The procedure is free from so-called subjectivity, in the sense that the information fusion is carried out by using algebraic expressions rather than using subjectively defined membership functions and the algebraic expressions are formulated in such a way that the main feature of granular information (overlaps among granules) is well-preserved. The method presented allows using non-overlapping linguistic labels as well as labels with overlap in both source and target sets. The methods presented are demonstrated using several numerical examples.

540

Z. Chen, D. Ben-Arieh / Computers & Industrial Engineering 51 (2006) 526–541

References Akiyama, Y. (2000). Object-oriented fusion–diffusion mechanism to handle crisp and linguistic information for better human-system interface. Systems, Man, and Cybernetics, 2000 IEEE International Conference, 2, 1313–1318. Baas, S. M., & Kwakernaak, H. (1977). Rating and ranking of multiple-aspect alternatives using fuzzy sets. Automatica, 13(1), 47–58. Ben-Arieh, D., & Chen, Z. (2004). A new linguistic labels aggregation and Consensus in group decision making. In IIE Annual Conference (IERC 2004), Houston, Texas, USA, May 15–19, 2004. Blin, J. M. (1974). Fuzzy relations in group decision theory. Journal of Cybernetics, 4(2), 17–22. Chen, S. -J, & Chen, S. -M. (2002). A new information fusion algorithm for handling heterogeneous group decision-making problems, Fuzzy Systems, 2002. FUZZ-IEEE’02. In Proceedings of the 2002 IEEE International Conference, 1, pp. 390–395. Chen, S.-J., & Hwang, C.-L. (1989). Fuzzy multiple attribute decision making. Springer. Chen, T. M., & Luo, R. C. (1999). A generalized look-ahead method for adaptive multiple sequential data fusion and decision making; Multisensor Fusion and Integration for Intelligent Systems, 1999. In Proceedings. 1999 IEEE/SICE/RSJ International Conference, pp. 199–204. Cheng, C.-H. (1999). A simple fuzzy group decision making method. IEEE International Conference on Fuzzy Systems, 2, II-910–II-915. Delgado, M., Herrera, F., Herrera-Viedma, E., & Martinez, L. (1998). Combining numerical and linguistic information in group decision making. Journal of Information Sciences, 107, 177–194. Delgado, M., Verdegay, J. L., & Vila, M. A. (1993). On aggregation operations of linguistic labels. International Journal of Intelligent Systems, 8(3), 351–370. Fan, Z.-P., Ma, J., & Zhang, Q. (2002). An approach to multiple attribute decision making based on fuzzy preference information on alternatives. Fuzzy Sets and Systems, 131(1), 101–106. Goodridge, S. G., Luo, R. C., & Kay, M. G. (1994). Multi-layered fuzzy behavior fusion for real-time control of systems with many sensors, Multisensor Fusion and Integration for Intelligent Systems, 1994. In IEEE International Conference on MFI ’94, pp. 272–279. Grabisch, M., & Saveant, P. (1998). Uncertainty modeling and its linguistic expression in data fusion systems. In Fuzzy Systems Proceedings, 1998. IEEE World Congress on Computational Intelligence, The 1998 IEEE International Conference on, 2, pp. 921–926. Hathaway, R. J., Bezdek, J. C., & Pedrycz, W. (1996). A parametric model for fusing heterogeneous fuzzy data. Fuzzy Systems, IEEE Transactions, 4(3), 270–281. Herrera, F., Herrera-Viedma, E., & Verdegay, J. L. (1996). Direct approach processes in group decision making using linguistic OWA operators. Fuzzy Sets and Systems, 79(2), 175–190. Herrera, F., & Herrera-Viedma, E. (1997). Aggregation operators for linguistic weighted information. IEEE Transactions on Systems, Man, and Cybernetics. Part A: Systems and Humans, 27(5), 646–656. Herrera, F., Herrera-Viedma, E., & Martinez, L. (2000). A fusion approach for managing multi-granularity linguistic term sets in decision making. Fuzzy Sets and Systems, 114(1), 43–58. Herrera, F., & Martinez, L. (2000). A 2-tuple fuzzy linguistic representation model for computing with words. Fuzzy Systems, IEEE Transactions, 8(6), 746–752. Herrera, F., & Martı´nez, L. (2001). A model based on linguistic 2-tuples for dealing with multigranular hierarchical linguistic contexts in multi-expert decision-making. IEEE Transactions on Systems, Man and Cybernetics. Part B: Cybernetics, 31(2), 227–234. Hsu, H. M., & Chen, C. T. (1996). Aggregation of fuzzy opinions under group decision making. Fuzzy Sets and Systems, 79, 279–285. Hussien, B., Bender, M., & Ismael, A. (1994a). A method for dynamic, multi-sensor, evidence combination using fuzzy linguistic terms, Intelligent Control, 1994. In Proceedings of the 1994 IEEE International Symposium, pp. 178–183. Hussien, B., Ismael, F., & Bender, M. (1994b). Evidence combination using fuzzy linguistic terms in a dynamic, multisensor environment, Multisensor Fusion and Integration for Intelligent Systems, 1994. In IEEE International Conference on MFI ’94, pp. 387–394. Hwang, C.-L., & Lin, M.-J. (1987). Group decision making under multiple criteria: methods and applications. Berlin, New York: Springer. Hwang, C.-L., & Yoon, K. (1981). Multiple attribute decision making: methods and applications: a state-of-the-art survey. Berlin, New York: Springer. Iggland, B. (1991). Coupling of customer preferences and production cost information. Technology Management: The New International Language, 250–253, October, 27–31. Ishikawa, A. (1993). The new fuzzy Delphi methods: economization of GDS (group decision support), System Sciences, 1993. In Proceeding of the Twenty-Sixth Hawaii International Conference, 4, pp. 255–264. Kacprzyk, J., & Fedrizzi, M. (1990). Multiperson decision making models using fuzzy sets and possibility theory. Dordrecht, Boston: Kluwer Academic Publishers. Kacprzyk, J., Fedrizzi, M., & Nurmi, H. (1992). Group decision making and consensus under fuzzy preferences and fuzzy majority. Fuzzy Sets and Systems, 49, 21–31. Lee, E. S., & Li, R.-J. (1998). Comparison of fuzzy numbers based on the probability measure of fuzzy events. Computers & Mathematics with Applications, 15(10), 887–896. Mendel, J. M. (1995). Fuzzy logic systems for engineering: a tutorial. Proceedings of the IEEE, 83(3), 345–377. Mikhailov, L. (2004). Group prioritization in the AHP by fuzzy preference programming method. Computers & Operations Research, 31, 293–301. Moses, D., Degani, O., Teodorescu, H. -N., Friedman, M. M., & Kandel, A. (1999). Linguistic coordinate transformations for complex fuzzy sets. In Fuzzy Systems Conference Proceedings, 1999. FUZZ-IEEE ’99. 1999 IEEE International, August 22–25, 3, pp. 1340– 1345.

Z. Chen, D. Ben-Arieh / Computers & Industrial Engineering 51 (2006) 526–541

541

Prodanovic, P., & Simonovic, S. P. (2003). Fuzzy compromise programming for Group decision making. Systems, Man and Cybernetics. Part A: IEEE Transactions, 33(3), 358–365. Shamsuzzaman, M., Sharif Ullah, A. M. M., & Bohez Erik, L. J. (2003). Applying linguistic criteria in FMS selection Fuzzy-set-AHP approach. Integrated Manufacturing Systems, 14(3), 247–254. Singh, R.-N. P., & Bailey, W. H. (1997). Fuzzy logic applications to multisensor–multitarget correlation. Aerospace and Electronic Systems, IEEE Transactions, 33(3), 752–769. Torrez, W. C., Bamber, D., Goodman, I. R., & Nguyen, H. T. (2002). A new method for representing linguistic quantifications by random sets with applications to tracking and data fusion, Information Fusion, 2002. In Proceedings of the Fifth International Conference, 2, pp. 1308–1315. Valet, L., Mauris, G., Bolon, P., & Keskes, N. (2003). A fuzzy linguistic-based software tool for seismic image interpretation. Instrumentation and Measurement, IEEE Transactions, 52(3), 675–680. Yager, R. R. (1983). Quantified propositions in a linguistic logic. International Journal of Man Machine Studies, 19, 195–227. Yager, R. R. (1993). Non-numeric multi-criteria multi-person decision making. Group Decision and Negotiation, 2, 81–93. Yager, R. R. (1994). Interpreting linguistically quantified propositions. International Journal of Intelligent Systems, 9(6), 541–569. Yager, R. R. (1995). Multicriteria decision making using fuzzy quantifiers, IEEE/IAFE Conference on Computational Intelligence for Financial Engineering. In Proceedings (CIFEr), pp. 42–46. Yager, R. R. (1998a). On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE Transactions System, Man, Cybernetics A, 18, 183–190. Yager, R. R. (1998b). Including importances in OWA aggregations using fuzzy systems modeling. Fuzzy Systems, IEEE Transactions, 6(2), 286–294. Zimmermann, H.-J. (1987). Fuzzy sets, decision making, and expert systems. Boston: Kluwer Academic Publishers.