AGGREGATION OF FUZZY OPINIONS UNDER GROUP DECISION-MAKING BASED ON SIMILARITY AND DISTANCE

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1 Jrl Syst Sc & Complexty (2006) 19: AGGREGATION OF FUZZY OPINIONS UNDER GROUP DECISION-MAKING BASED ON SIMILARITY AND DISTANCE Chengguo LU Jbn LAN Zhongxng WANG Receved: 6 December 2004 / Revsed: 7 September 2005 Abstract In ths artcle, a new method for aggregatng fuzzy ndvdual opnons nto a group consensus opnon s proposed. To obtan the aggregaton weghts of each ndvdual opnon, a consstency ndex of each expert wth the other experts s ntroduced based on smlarty and dstance. The mportance of each expert s also taken nto consderaton n the process of aggregaton. Fnally, a numercal example s presented to llustrate the effcency of the procedure. Key words Consstency degree, fuzzy ndvdual opnons, fuzzy numbers, group consensus opnon, group decson-makng. 1 Introducton Under the crcumstance that a fuzzy decson-makng problem nvolves multple actors, there may arse stuatons of conflct and agreement among the experts because of dfferent values and nformaton systems. Hence, how to fnd a reasonable method to aggregate each ndvdual opnon nto a group consensus opnon s an mportant ssue n a group decsonmakng envronment. Up to now, some aggregaton methods have been proposed to solve the group decson makng problems n an acceptable way [1 7], but most of them are based on the fuzzy preference relatons [3 6]. Hsu and Chen [8] proposed a smlarty aggregaton method (SAM) to combne the ndvdual opnons. They ntroduced a smlarty ndex to measure the consstency of each expert wth the other experts. However, there are several problems n Hsu and Chen s work. Frst, they assumed that the opnons of all experts represented by fuzzy numbers should have a common ntersecton at some α-level cut, α [0, 1]. Otherwse, t wll not work. For example, let us consder three fuzzy numbers Ã, B, and C rangng from 2 to 3, 4 to 5, and 6 to 8, respectvely. Accordng to Hsu and Chen s method, the degree of smlarty between à and B s zero; the degree of smlarty between à and C s also equal to zero. However, these two smlartes are obvously dfferent. To avod ths dsjont stuaton, Hsu and Chen suggested that Delph s method should be used to modfy each expert s opnon. But t wll dstort the opnons of the experts to some extent. Moreover, f the supports of fuzzy Chengguo LU School of Mathematcs and Informaton Scence, Guangx Unversty, Nannng , Chna. Jbn LAN School of Economcs and Management, South West Jao-tong Unversty, Chengdu , Chna. Zhongxng WANG School of Mathematcs and Informaton Scence, Guangx Unversty, Nannng , Chna. Emal: zxwgx@126.com.

2 64 CHENGGUO LU JIBIN LAN ZHONGXING WANG numbers do not ntersect, we can t conclude that the opnons represented by fuzzy numbers do not ntersect [9]. For example, two experts assgn dfferent supports to an alternatve under a crteron rangng from 0.78 to 0.80 and 0.81 to 0.83, respectvely. The supports are dsjont, but ther opnons are very close and the smlarty between the opnons should not equal zero. The second problem of Hsu and Chen s work s that the degrees of smlarty between fuzzy opnons are determned by the proporton of the consstent area to the total area only, and they dd not consder the supports of the consstent area and the total area (leadng to a loss of nformaton), and ths s dscussed n Secton 3. In ths artcle, we propose a consstency aggregaton method (CAM) to combne fuzzy ndvdual opnons. Ths method s based on smlarty and dstance because they are ndces equally mportant to the comparson of fuzzy opnons. By the method, the sum of weghted consstency among aggregated consensus and the group consensus opnon s obtaned. Then an algorthm s presented to determne the aggregaton weghts of each ndvdual opnon. Ths artcle s organzed as follows. In Secton 2, Hsu and Chen s smlarty aggregaton method (SAM) s descrbed n bref. In Secton 3, we gve an mproved method to measure the smlarty between fuzzy numbers, and a modfed dstance measure s also defned as another ndex to compare fuzzy numbers. Then we combne the smlarty and dstance measures to obtan the weghts of each opnon. In Secton 4, a numercal example s presented to llustrate our consstency aggregaton method (CAM). In Secton 5, concludng remarks are presented. 2 Prelmnares A fuzzy number s a fuzzy set n the real lne and s completely defned by ts membershp functon (µ(x) : R [0, 1]). For computatonal purposes, the fuzzy number s often strct to both normal and convex. Normalty: sup{µ(x)} = 1, x R. Ths requrement means that there s at least one pont n a real lne wth the maxmum membershp value equal to 1. Convexty: µ{λx 1 + (1 λ)x 2 } mn{µ(x 1 ), µ(x 2 )}, x 1, x 2 R, λ [0, 1]. Ths means the ponts of the real lne wth the hghest membershp values are clustered around a gven nterval (or pont). For convenence, the fuzzy number s often represented by a trapezodal fuzzy number. A trapezodal fuzzy number can be denoted by a 4-tuple (a 1, a 2, a 3, a 4 ) where a 1 a 2 a 3 a 4. When a 2 = a 3 the fuzzy number s called a trangular fuzzy number. The condton that a 1 = a 2 and a 3 = a 4 mples a closed nterval. If a 1 = a 2 = a 3 = a 4, we obtan a crsp real number. Let R = (a, b, c, d ) be a postve trapezodal fuzzy number representng th expert s subjectve estmate to an alternatve under a crteron. Now the key ssue s to construct an aggregaton functon F to combne R ( = 1, 2,, n) nto a group opnon R = F( R 1, R 2,, R n ). Before the descrpton of our method, Hsu s aggregaton method (SAM) s brefly descrbed. In Hsu and Chen s artcle, they calculated the average agreement degree of each expert E ( = 1, 2,, n) by averagng the degrees of smlarty wth respect to other experts: where A(E ) = 1 n 1 S( R, R j ) = j=1,j S( R, R j ), (1) x (mn{µ R (x)})dx x (max{µ R (x)})dx. (2) It s a smlarty measure functon by Zwck et al. [10], whch means the proporton of the consstent area ( x (mn{µ R (x)})dx) to the total area ( x (max{µ R (x)})dx).

3 AGGREGATION OF FUZZY OPINIONS UNDER GROUP DECISION-MAKING 65 Then the aggregaton weght of the th expert s gven by RAD = A(E ). (3) A(E j ) j=1 Wthout consderng the mportance of the th expert, the aggregaton result s therefore defned as R = F( R 1, R 2,, R n ) = RAD R, (4) where s the fuzzy multplcaton operator [11]. 3 New Aggregaton Method Based on Smlarty and Dstance Hsu and Chen s work s based on the dea that the weght of an expert s opnon should be larger f hs opnon s closer to the other opnons. But there are some problems as we have ponted out earler. To solve these problems, Lee [9] proposed an optmal aggregaton method (OAM) on the bass of the dea that the weght of an expert s opnon should be larger f the dstances between hs opnon and other opnons are smaller. In fact, nether Hsu and Chen s work nor Lee s work can be effcent enough because dstance and smlarty are dual mportant ndces for comparng fuzzy ndvdual opnons. Any one of them can reflect only one aspect of the ssue whle leadng to the loss of other nformaton. Therefore, we propose a new consstency aggregaton method (CAM) based on smlarty and dstance. 3.1 Smlarty Between Fuzzy Numbers Hsu and Chen employed Zwck s smlarty measure functon to calculate the smlarty between fuzzy numbers. However, each element n the unverse may have a dfferent mportance. Let us study an example: Example 3.1 There are three fuzzy numbers à = (1, 3, 5, 7), B = (2, 3, 5, 6), C = (1, 4, 4, 7) (see Fg. 1). Fgure 1 Three fuzzy numbers Ã, B, and C Usng Hsu and Chen s smlarty measure, Eq. (2), the degrees of smlarty of à and B, à and C are S(Ã, B) = S(Ã, C) = 3 4. However, one can see from Fg. 1 that these two smlartes should not be equal because the supports of the ntersecton of à and B are larger than the supports of the ntersecton of à and C. To avod ths stuaton we need to consder the weght

4 66 CHENGGUO LU JIBIN LAN ZHONGXING WANG of each element x R so that we have the followng weghted smlarty measures. For fuzzy numbers R and R j, assume the weght of each x R s w(x), 0 w(x) 1. Then the weghted smlarty between R and R j s defned as S w ( R, R j ) = x (w mn(x)mn{µ R (x)})dx x (w max(x)max{µ R (x)})dx, (5) where w mn (x) and w max (x) denote the weght functons of the consstent area and the total area, respectvely. In general, let w mn (x) = mn{µ R (x)}, w max (x) = max{µ R (x), µ Rj (x)} because mn{µ R (x)} and max{µ R (x)} are membershp functons of the consstent area and total area, respectvely. Thus the mproved smlarty s gven as S w ( R, R j ) = x (mn{µ R (x)}) 2 dx x (max{µ R (x)}) 2 dx. (6) It s easy to prove that S w ( R, R j ) satsfes the followng propertes: (SP1) 0 S w ( R, R j ) 1; (SP2) S w ( R, R j ) = 1 f and only f R = R j ; (SP3) S w ( R, R j ) = S w ( R j, R ); (SP4) S w ( R, R j ) S w ( R, R k ) and S w ( R j, R k ) S w ( R, R k ) f R R j R k. Let us agan consder Example 3.1. Now usng our smlarty measure Eq. (6) we obtan The result s far and reasonable. 3.2 Dstance Between Fuzzy Numbers S w (Ã, B) = 4 5 S w(ã, C) = 3 5. Dstance s an mportant concept n fuzzy set theory and s also a sgnfcant ndex of the comparson of fuzzy numbers. Many dstance measure methods have been proposed up to now, a survey of such dstances can be found n papers of Dubos et al. [12], and Helpern [13]. We synthesze the man characterstcs of these methods and classfy them as two categores as: The frst category takes no consderaton of the membershp functon of fuzzy numbers, such as the dstance used by Tong [14] and Hausedorff metrc. The advantages of these methods are the convenence of operatng and ease of understandng. However, t wll lead to the loss of nformaton for the neglgence of the supports of fuzzy numbers. The second category s characterstc of takng the membershp functon nto account. In these methods, the Hammng metrc and the Eucldean metrc are most often used. For any two fuzzy numbers à and B wth membershp functons µã and µ B, respectvely, we have (see [15]) the Hammng dstance d H (Ã, B) d H (Ã, B) = µã(x) µ B(x) dx (7) x and the Eucldean dstance d E (Ã, B) d E (Ã, B) = (µã(x) µ B(x)) 2 dx. (8) x These formulas are straghtforward generalzatons of dstances wth the membershp functons. However, these methods do not always operate. In the case that the ntersecton between two fuzzy numbers s empty, no concluson wll be drawn. Let us see the followng example.

5 AGGREGATION OF FUZZY OPINIONS UNDER GROUP DECISION-MAKING 67 Example 3.2 There are three fuzzy numbers à = (1, 2.5, 3.5, 4), B = (5, 6, 7, 8), C = (9, 10, 11, 12). Usng the Hammng dstance Eq. (7), the dstance between à and B and the dstance between à and C are calculated as follows: d H (Ã, B) = Sà + S B, d H (Ã, C) = Sà + S C, where SÃ, S B, and S C mply the areas of Ã, B, and C, respectvely. Snce S B = S C, t follows: d H (Ã, B) = d H (Ã, C). By analogy, we have d E (Ã, B) = d E (Ã, C). But t s obvous that the dstance between à and B should be smaller than the dstance between à and C. To overcome ths dsadvantage, we ntroduce a dstance between two sets A and B d nf (A, B) = nf{d(a, b), a A, b B}, (9) where d s the usual metrc. For trapezodal fuzzy numbers à = (a 1, a 2, a 3, a 4 ) and B = (b 1, b 2, b 3, b 4 ), d nf (Ã, B) = nf{d(a, b), a [a 1, a 4 ], b [b 1, b 4 ]}. (10) Consderng the Hammng dstance s a lnear dstance and s easy to operate, now we defne a new dstance measure between fuzzy numbers à and B on the bass of the Hammng dstance d H (Ã, B) and the dstance d nf (Ã, B) as follows: D(Ã, B) = 1 ( d H 2 (Ã, B) ) + d nf (Ã, B) = 1 ( ) µã(x) µ B(x) dx + d nf (Ã, B). (11) 2 x Note that ths dstance can be regarded as the Hammng dstance when d nf (Ã, B) = 0, but t does overcome the shortcomng of the Hammng dstance. To llustrate the effcency of ths dstance method, now consder Example 3.2 agan. By our dstance measure Eq. (11), we have D(Ã, B) = 1 2 (S à + S B + 1), D(Ã, C) = 1 2 (S à + S C + 5) > D(Ã, B). Ths concdes wth our ntuton. For the fuzzy numbers R =(a, b, c, d )( = 1, 2,, n) of each expert s opnon, we calculate the dstance d( R, R j ) between each par of R and R j, then select the largest dstance D ( R p, R q ) = max D( R, R j ). We dvde each dstance by D ( R p, R q ) so that we obtan the,j followng normalzed dstance: d( R, R j ) = D( R, R j ) D ( R p, R q ). (12) Ths dstance measure fulflls the followng propertes: (DP1) 0 d( R, R j ) 1; (DP2) d( R, R j ) = 0 f and only f R = R j ; (DP3) d( R, R j ) = d( R j, R ); (DP4) d R, R j ) d( R, R k ) and d( R j, R k ) d( R, R k ) f R R j R k. 3.3 Aggregaton Method Let R 1, R 2,, R n be n fuzzy numbers representng each expert s opnon for an alternatve under a gven crteron. Snce we have obtaned the smlarty S w ( R, R j ) and the dstance

6 68 CHENGGUO LU JIBIN LAN ZHONGXING WANG d( R, R j ) for, j = 1, 2,, n, now we defne a new consstency measure between fuzzy opnons R and R j : r( R, R j ) = βs w ( R, R j ) + (1 β)(1 d( R, R j )), (13) where β [0, 1] s the weght of S w ( R, R j ), whch reflects the relatve mportance degree between the smlarty and the dstance wth respect to the decson maker. 1 β s the weght of d( R, R j ). The basc dea of ths aggregaton method s that the larger S w ( R, R j ) and the smaller d( R, R j ), the larger consstency degree r( R, R j ) between fuzzy opnons R and R j. Snce r( R, R j ) s the lnear combnaton of S w ( R, R j ) and d( R, R j ), t follows (RP1) 0 r( R, R j ) 1; (RP2) r( R, R j ) = 1 f and only f R = R j ; (RP3) r( R, R j ) = r( R j, R ); (RP4) r R, R j ) r( R, R k ) and r( R j, R k ) r( R, R k ) f R R j R k. In practce, group decson-makng s hghly nfluenced by the degrees of the mportance of partcpants. For example, there are some experts such as the managers of a company wth authorty and some experts who are more experenced than the others. So an effectve aggregaton method should consder the relatve mportance weght of each expert. For no loss of generalty, let the degree of mportance of th expert be e (0 e 1), and e = 1. (14) Then the weghted consstency degree of each expert E s gven as C(E ) = Thus the aggregaton weght of each expert E s calculated by The aggregaton result s therefore defned as r( R, R j )e j. (15) j=1 w(e ) = C(E ). (16) C(E j ) j=1 R = F( R 1, R 2,, R n ) = w(e ) R, (17) where s the fuzzy multplcaton operator [11]. Now we summarze our consstency aggregaton method (CAM) and gve an algorthm. 3.4 Algorthm CAM Step 1: For each expert E ( = 1, 2,, n), he/she constructs a fuzzy number R ( = 1, 2,, n) to represent hs/her opnon to the alternatve under a gven crteron. Step 2: Calculate the smlarty S w ( R, R j ) between each par of experts by Eq. (6). Step 3: Calculate the dstance d( R, R j ) between each par of experts by Eq. (12). Step 4: Gven β [0, 1], calculate the consstency degree r( R, R j ) between each par of experts by Eq. (13). Step 5: Select the degree of mportance e of each expert E, then calculate the weghted consstency degree C(E ) of each expert E by Eq. (15).

7 AGGREGATION OF FUZZY OPINIONS UNDER GROUP DECISION-MAKING 69 Step 6: Calculate the aggregaton weght w(e ) of expert by Eq. (16). Step 7: Aggregate each fuzzy opnon nto a group fuzzy opnon by Eq. (17). Our consstency aggregaton method (CAM) also preserves some mportant propertes as follows. Property 3.1 (Agreement preservaton [1] ) If R = R j for all, j, then R = R j. That s, f all opnons of experts are dentcal the aggregaton result s the common opnon. Property 3.2 (Order ndependence [1] ) The result of CAM would not be nfluenced by the dfferent order wth whch ndvdual opnons are combned. That s, f {(1), (2),, (n)} s a permutaton of {1, 2,, n}, then R = F( R 1, R 2,, R n ) = F( R (1), R (2),, R (n) ). Property 3.1 and Property 3.2 are consstency equpments. Property 3.3 Let the uncertanty measure H( R ) of ndvdual opnon R be defned as the area under ts membershp functon [1] H( R ) = + µ R (x)dx, (18) then the uncertanty measure defned n Eq. (18) satsfes the followng equaton: H( R) = w(e ) H( R ). (19) Ths means that the uncertanty of aggregaton result s between the uncertantes of all experts,.e., mn H( R ) H( R) max H( R ). Referrng to Example 4.1, H( R 1 ) = 2, H( R 2 ) = 2.25, H( R 3 ) = 2.75, so H( R) = s between H( R 1 ) and H( R 3 ). Property 3.4 [8] The common ntersecton of all expert s opnons s concluded n the fnal n aggregaton result. That s, R R. Snce Proof Let λ cut of R be R λ = [aλ, bλ ], then n t follows Notce R λ = a λ = maxa λ, bλ = mn b λ. R = w(e ) R, R λ = [aλ, b λ ], where w(e ) [ n ] Rλ = (w(e ) a λ ), (w(e ) b λ ). (w(e ) a λ ) aλ = max a λ, n (w(e ) b λ ) bλ = mn b λ. We have proved ths property. n Property 3.5 If R =, the consstent group opnon R also can be derved. When the opnons are dsjont, we can measure the dstance d( R, R j ) (for all, j) among them, then calculate the consstency degree r( R, R j ) between each par of experts by Eq. (13) so that aggregaton process can be contnued.

8 70 CHENGGUO LU JIBIN LAN ZHONGXING WANG 4 Numercal Example Example 4.1 [8] Consder a group decson-makng problem wth three experts. The opnons of each expert are gven as three postve trapezodal fuzzy numbers (see Fg. 2): R 1 = (1, 2, 3, 4), R 2 = (1.5, 2.5, 3.5, 5), R 3 = (2, 2.5, 4, 6). Fgure 2 Aggregaton result of fuzzy opnons R 1, R 2, and R 3 Now use our consstency aggregaton method (CAM) to study ths problem. Step 1: It s gven above. Step 2: Calculate the smlarty S w ( R, R j ) for, j = 1, 2, 3 as follows: S w ( R 1, R 1 ) = 1, S w ( R 1, R 2 ) = 1 2, S w( R 1, R 3 ) = 1 3 ; S w ( R 2, R 1 ) = 1 2, S w( R 2, R 2 ) = 1, S w ( R 2, R 3 ) = 2 3 ; S w ( R 3, R 1 ) = 1 3, S w( R 3, R 2 ) = 2 3, S w( R 3, R 3 ) = 1. Step 3: Calculate the dstance d( R, R j ) for, j = 1, 2, 3 as follows: d( R 1, R 1 ) = 0, d( R 1, R 2 ) = 5 9, d( R 1, R 3 ) = 1; d( R 2, R 1 ) = 5 9, d( R 2, R 2 ) = 0, d( R 2, R 3 ) = 4 9 ; d( R 3, R 1 ) = 1, d( R 3, R 2 ) = 4 9, d( R 3, R 3 ) = 0. Step 4: Select β = 1 2, calculate the consstency degree r( R, R j ) for, j = 1, 2, 3 as follows: r( R 1, R 2 ) = 1, r( R 1, R 2 ) = 0.472, r( R 1, R 3 ) = 0.167; r( R 2, R 1 ) = 0.472, r( R 2, R 2 ) = 1, r( R 2, R 3 ) = 0.611; r( R 3, R 1 ) = 0.167, r( R 3, R 2 ) = 0.611, r( R 3, R 3 ) = 1. Step 5: Let the degrees of mportance of three experts be e 1 = 0.42, e 2 = 0.25, e 3 = 0.33, respectvely, and we obtan: C(E 1 ) = 0.593, C(E 2 ) = 0.649, C(E 3 ) = Step 6: Calculate the aggregaton weght w(e ) of expert E as follows: w(e 1 ) = 0.330, w(e 2 ) = 0.361, w(e 3 ) = Step 7: Aggregate each fuzzy opnon nto a group fuzzy opnon as 3 R = w(e ) R = (1.489, 2.334, 3.489, 4.979). Referrng to Fg. 2, we can see the second opnon R 2 s close to the other opnons, so the aggregaton weght of R2 s the largest, and the consensus opnon R s therefore close to R 2. The consstency degree of the opnon R 3 s the smallest of three ones so that R 3 s less mportant, whch s far and reasonable. However, n Hsu and Chen s result, the consensus degree coeffcent of R 2 (CDC 2 = 0.33) s smaller than CDC 1 = 0.34 although R 2 s closer to the consensus opnon R than R 1. Ths does not concde wth people s ntuton. Furthermore,

9 AGGREGATION OF FUZZY OPINIONS UNDER GROUP DECISION-MAKING 71 the uncertanty of our aggregaton result (H( R) = 2.322) s smaller than the uncertanty of Hsu and Chen s aggregaton result n case 1 (H( R) = 2.341) and case 2 (H( R) = 2.33). The wdth of our aggregaton result (wdth( R) = 3.489) s also smaller than the wdth of Hsu and Chen s aggregaton result n case 1 (wdth( R) = 3.519) and case 2 (wdth( R) = 3.495). 5 Conclusons In ths artcle, we defne a smlarty measure, and a dstance measure s also gven to study the consstency of one expert to the other experts. To deal wth the stuaton when opnons are dsjont, we ntroduce a consstency ndex of each ndvdual opnon on the bass of smlarty and dstance. The basc dea of our aggregaton method s that the aggregaton weght of the opnon should be larger f the degrees of smlarty between the expert s opnon and the other opnons are larger and the degrees of dstance between hs opnon and the other opnons are smaller. The mportance of each expert s also consdered n the procedure of our method. Fnally, a numercal example shows that our method s rather effcent. References [1] A. Bardossy, L. Ducksten and I. Bogarad, Combnaton of fuzzy numbers representng expert opnons, Fuzzy Sets and Systems, 1993, 57: [2] A. Ishkawa, M. Amagasa, T. Shga, G. Tomzawa, R. Tatsuta and H. Meno, The max-mn Delph method and fuzzy Delph method va fuzzy ntegraton, Fuzzy Sets and Systems, 1993, 55: [3] H. Nurm, Approaches to collectve decson makng wth fuzzy preference relatons, Fuzzy Sets and Systems, 1981, 6: [4] J. Kacprzyk and M. Fedrzz, A soft measure of consensus n the settng of partal (fuzzy) preferences, European Journal of Operatonal Research, 1988, 34: [5] J. Kacprzyk, M. Fedrzz and H. Nurm, Group decson makng and consensus under fuzzy preferences and fuzzy majorty, Fuzzy Sets and Systems, 1992, 49: [6] M. Fedrzz and J. Kacprzyk, On measurng consensus n the settng of fuzzy preference relatons, n Non-conventonal Preference Relatons n Decson Makng (ed. by J. Kacprzyk and M. Roubens), Sprnger, Berln, 1988, [7] R. N. Xu and X. Y. Zha, Extensons of the analytc herarchy process n fuzzy envronment, Fuzzy Sets and Systems, 1992, 52: [8] H. M. Hsu and C. T. Chen, Aggregaton of fuzzy opnons under group decson makng, Fuzzy Sets and Systems, 1996, 79: [9] H. S. Lee, Optmal consensus of fuzzy opnons under group decson makng envronment, Fuzzy Sets and Systems, 2002, 132: [10] R. Zwck, E. Carlsten and D. V. Budescu, Measures of smlarty among fuzzy concepts: A comparatve analyss, Internat, J. Approxmate Reasonng, 1987, 1: [11] A. Kauffman and M. M. Gupta, Introducton to Fuzzy Arthmetc: Theory and Applcatons, Van Nostrand Renhold, New York, [12] D. Dubos, E. Kerre and R. Mesar, Fuzzy nterval analyss, n Fundamentals of Fuzzy Sets (ed. by D. Dubos and H. Prade), The Handbooks of Fuzzy Sets Seres, Kluwer Academc Publshers, Boston, 2000, [13] S. Helpern, Representaton and applcaton of fuzzy numbers, Fuzzy Sets and Systems, 1997, 91: [14] R. M. Tong and P. P. Bonssone, A lngustc approach to decson makng wth fuzzy sets, IEEE Trans. Systems Man Cybernet, 1980, 10: [15] J. Kacprzyk, Multstage Fuzzy Control, Wley Chchester, 1997.