The fuzzy weighted average within a generalized means function

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1 Computers and Mathematcs wth Applcatons ) The fuzzy weghted average wthn a generalzed means functon Yuh-Yuan Guh a,, Rung-We Po b, E. Stanley Lee c a Graduate School of Busness Admnstraton, Chung Yuan Chrstan Unversty, ChungL 32023, Tawan, ROC b Insttute of Technology Management, Natonal Tsng Hua Unversty, Hsnchu 30013, Tawan, ROC c Department of Industral and Manufacturng Systems Engneerng, Kansas State Unversty, Manhattan, KS 66506, USA Accepted 26 September 2007 Abstract The fuzzy weghted average s wdely used to solve herarchcal evaluaton problems, ncludng fuzzy consderaton for the operatons of scorng, weghtng and aggregatng. Prevous works consdered the fuzzness of score and weght, and used the addtve functon to aggregate these weghted scores. Ths study consders the aggregaton operator also as a fuzzy varable, and uses a generalzed means functon to fuzzfy the aggregaton operator wthn a fuzzy weghted average. In practce, the proposed model not only consders both the relatve mportant of the crtera and ts acheved performance, but also conveys the nfluence of the DM s Decson Maker s) evaluaton atttude. Thus the proposed model can flexbly reflect any DM s evaluaton atttude, such as open, neutral or rgorous. Thereby, the proposed model can make an objectve evaluaton that approaches a real decson makng stuaton, and thus has the potental to be a useful management tool for mproved resoluton of fuzzy herarchcal evaluaton problems. c 2007 Elsever Ltd. All rghts reserved. Keywords: Fuzzy sets; Fuzzy weghted average FWA); Generalzed mean operator; Evaluaton atttude; Herarchcal evaluaton; Performance 1. Introducton In mult-crtera decson makng, how to establsh an effectve evaluaton model to objectvely evaluate large-scale or complex problems has been a contnung concern for decson scence. Ths knd of evaluaton problem generally ncludes varous crtera, and thus can be represented as a herarchy of goals and means n the shape of a systematc dagram of all the crtera elements, namely the herarchy evaluaton structure. Stllwell et al. [1] found that the herarchcal weghts are steeper than the non-herarchcal weghts, and thus a herarchy evaluaton structure provdes a more powerful method of dentfyng the performance dfferences among a group of evaluated objects. In these problems of herarchy evaluaton there s the lack of precson n assessng the relatve mportance of attrbutes and the performance ratngs of alternatves wth respect to attrbutes. Thus, the tradtonal crsp, determnstc and precse mult-crtera decson makng methods cannot effectvely handle problems contanng such mprecse nformaton. Rbero [2] stated that mprecson may come from a varety of sources, such as unquantfable nformaton, ncomplete nformaton, partal gnorance, and no obtanable nformaton. To solve ths dffculty, fuzzy Correspondng author. Tel.: ; fax: E-mal address: yuhyuan@cycu.edu.tw Y.-Y. Guh) /$ - see front matter c 2007 Elsever Ltd. All rghts reserved. do: /j.camwa

2 2700 Y.-Y. Guh et al. / Computers and Mathematcs wth Applcatons ) Fg. 1. Herarchy evaluaton problem n a vague envronment. set theory, frst ntroduced by Zadeh [3], has been used and s the focus of ths current study. The llustraton of the herarchy evaluaton problem n a vague envronment can be expressed as follows: Generally for a specfc evaluaton problem, once the herarchy s constructed, there are three basc operatons nvolvng n calculatng the aggregated ndex for herarchy evaluaton: 1) scorng the crtera x, 2) weghtng the crtera x and 3) aggregatng the crtera x see Fg. 1). Incorporatng fuzzy set theory nto herarchy evaluaton problems, there have been numerous artcles consderng that the scorng crtera x ) and ther relatve weghtng ) are both fuzzy numbers, and thus they form a Fuzzy Weghted Average FWA) problem. To generalze the fuzzy weghted average, let A 1, A 2... A n, and W 1, W 2... W n be the fuzzy numbers defned on the unverses X 1, X 2..., X n, and Z 1, Z 2..., Z n respectvely. If f s a functon whch maps from X 1 X 2 X n Z 1 Z 2 Z n to the unverse Y, then the fuzzy weghted average y s defned as follows: ) w1 x 1 + x 2,..., +w n x n y = f x 1, x 2,..., x n,,,..., w n ) = = +,..., +w x 1 + x 2,..., +w n x n) n where for each = 1, 2... n, x X, Z and normalzed weght w = / +,..., +w n ). Essentally, the FWA problem s a nonlnear fractonal programmng problem. On the bass of α-cuts and nterval analyss, Dong and Wong [4], Tee and Bowman [5], Lou and Wang [6], Guh et al. [7], Hon et al. [8], Lee and Park [9], Vanegas and Labb [10], Chang and Hung [11], Lou and Guu [12], Guu [13], Srbladze and Skharuldze [14,15] and Chang et al. [16] used a computatonal algorthm and ts mproved algorthms to obtan a dscrete but exact soluton for the fuzzy weghted average. However, even wth ths above mprovement, the computatonal requrements are stll very hgh, even for moderate szed problems. Thereby Guh et al. [17], Kao and Lu [18] and Chen et al. [19] used Charnes and Cooper s lnear transformaton [20,21] to convert the orgnal nonlnear fractonal programmng problem nto a lnear programmng problem. The advantage of ths approach s that the reducton to a lnear problem s very straghtforward and smple. The above authors consdered the fuzzness of score and weght, usng the addtve type to aggregate these weghted scores, wthout smultaneously consderng the fuzzness of the operator for aggregaton. However, aggregaton s a crtcal factor n a herarchy evaluaton problem snce t reflects the evaluaton atttude of the decson maker n evaluaton. When the DM s open or optmstc n evaluaton, a functon toward maxmum the aggregated ndex wll be adopted. On the other hand, f the evaluaton atttude of the DM s rgorous or pessmstc, a functon that mnmzes the aggregated ndex should be taken. In summary, the operatons of scorng, weghtng and aggregatng have the followng effects: 1) scorng: reflects the performance acheved for each crteron. 2) weghtng: reflects the relatve mportance of each crteron for a cluster of relevant crtera. 3) aggregatng: a connectve operator that reflects the percepton functon of DM n makng fnal decson. Although the functons of these three basc operatons are ndependent, they are related and have the followng domnatng effect on the aggregated ndex: aggregatng > weghtng > scorng. For example, f a crteron s gven a hgher score than other crtera but wth a lower weght, and ts weghted score s stll low, ths means that the nfluence of weghtng domnates the scorng. Agan for example, f the DM adopts an open atttude maxmum functon) to aggregate these weghted scores, the aggregated ndex s determned only by the maxmum weghted score. In ths paper we follow the procedure of Dong and Wong [4] n usng α-cut representaton for calculatng fuzzy weghted averages wthn a generalzed mean operator. For each membershp value α, a par of maxmum and

3 Y.-Y. Guh et al. / Computers and Mathematcs wth Applcatons ) mnmum fractonal programmng problems s formulated. As α vares, the famly of fractonal programs s modeled by usng parametrc programmng technque. Thus the exact membershp functon of the fuzzy weghted average wthn the generalzed mean operator can be derved. Ths study s organzed as follows. Frst, the basc concept of a herarchy evaluaton problem n a vague envronment s ntroduced. Second, the model of fuzzy weghted average wthn a generalzed mean operator s developed, and then converted nto a fractonal lnear programmng. Thrd, the process of usng Charnes and Cooper s lnear transformaton method to convert a fractonal lnear programmng to a lnear programmng model s ntroduced. Fnally, the example dscussed n Dong and Wong [4] s solved analytcally by usng the proposed approach. 2. Fuzzy weghted average wthn a generalzed mean operator On the bass of the above demonstraton, ths method s extended to assume that the aggregaton ) as well as the score x ) and normalzed weght w ) wthn a herarchy evaluaton are all fuzzy numbers, and thus forms a problem such as y = w 1 x 1 w 2 x 2,..., w n x n), w = / +,..., +w n ). Here and x are fuzzy numbers, for each α j [0, 1], wth the correspondng ntervals for x denoted by [a, b ], and the correspondng ntervals for denoted by [c, d ], = 1,..., n. Addtonally, the weghted generalzed mean operator, ntally proposed by Dujmovc and revsed by Dyckoff and Pedrycz [22], s employed to fuzzfy the aggregaton operator. Thus the fuzzy weghted average functon ncorporatng the generalzed mean operator can be denoted as follows: y = w 1 x 1 w 2 x 2,..., w n x n) y = f x 1, x 2,..., x n,, w 2,..., w n, p) = x w 2 x p 2,..., +w n x n p ) w1 x p 1 y = f x 1, x 2,..., x n,,,, w n, p) = + x p 2,..., +w nxn p +,..., +w n a x b, c d. The weghted generalzed mean operator s one knd of averagng operator, that s, mnx 1, x 2..., x n ) y maxx 1, x 2..., x n ). It s a monotonc nondecreasng functon. By varyng the parameter value p, the weghted generalzed means operator can produce varous dfferent aggregaton operators. Some of the better known ones are p =, the mnmum operator p = 1, the harmonc mean operator p = 0, the geometrc mean operator p = +1, the arthmetc mean operator p = +, the maxmum operator. On the bass of the dversty of operators produced by weghted generalzed means, we make use of them wth dfferent values of p to enumerate dfferent decson atttudes for the DM. For example, f the DM takes a more open atttude n evaluaton, ths mples that a hgher value p should be adopted for the weghted generalzed means functon, and the functon wll gve a hgher aggregated ndex toward the maxmum value that the functon can produce. In contrast, f the evaluaton atttude of the DM s rgorous or conservatve, a smaller value of p s adopted for the weghted generalzed means functon, and thus the functon s toward a mnmum aggregaton. On the other hand, f the DM takes a neutral atttude n evaluaton, a weghted generalzed means functon wth p value near one s adopted;.e. the arthmetc mean operator s used to aggregate all crtera. In summary, a DM can choose an approprate value for the parameter p representng the approprate evaluaton atttude. Consder the nfluence of the value p on a generalzed mean functon, where p 0, snce a x b and 0, 0, = 1,..., n. We have a p x p b p, = 1,..., n. Takng the summaton on the nequalty, and then dvdng by n, ths can be denoted as follows: a p x p b p a p x p ) b p.

4 2702 Y.-Y. Guh et al. / Computers and Mathematcs wth Applcatons ) Defne the upper and lower bounds of aggregated ndex for each α-cut nterval beng [L, U ] whle p 0, where L = Mn { f L }, U = Max { f U }. L = Mn f L = Mn a U = Max f U = Max b a a Takng the Naperan logarthm for L and U, ln L = Mn ln f 1 L = Mn p ln a c d, = 1, 2,..., n = 1 ln Mn p a ln U = Max ln f 1 U = Max p ln b c d, = 1, 2,..., n = 1 ln Max p b b b a a a a = 1 p ln L c d, = 1, 2,..., n c d, = 1, 2,..., n. b b c d, = 1, 2,..., n b b = 1 p ln U c d, = 1, 2,..., n. Snce ln L and L have the same solutons, ln U and U also do. The problems L and U are reduced to L and U. The problems L and U are fractonal lnear programmng, whch can be converted by means of Charnes and Cooper s lnear transformaton method nto the typcal lnear programmng model. In the case p < 0, snce a x b and 0, = 1,..., n, we have a p > x p > b p, = 1,..., n. Takng the summaton on the nequalty, and then dvdng by n, ths can be denoted as follows: a p > x p > b p a p Takng the Naperan logarthm for L and U, ln L = Mn ln f 1 L = Mn p ln a c d, = 1, 2,..., n x p a a b p.

5 Y.-Y. Guh et al. / Computers and Mathematcs wth Applcatons ) = 1 ln Max p a Fg. 2. Charnes and Cooper s lnear transformaton method. a a = 1 p ln L p < 0) c d, = 1, 2,..., n ln U = Max ln f 1 U = Max p ln b c d, = 1, 2,..., n = 1 ln Mn p b c d, = 1, 2,..., n. b b b b = 1 p ln U p < 0) Snce ln L and L have same solutons, ln U and U also do. The problems L and U are reduced to L and U. The problems L and U are fractonal lnear programmng, whch can be converted by means of Charnes and Cooper s lnear transformaton method nto the typcal lnear programmng model. 3. Charnes and Cooper s lnear transformaton method Charnes and Cooper s lnear transformaton s summarzed as follows, and only the needed equatons are summarzed. For detaled dervatons and the use of the dual problem, the reader s referred to Charnes and Cooper [20, 21]. Consder the followng smple fractonal programmng problem see the left part of Fg. 2): where p and q are two n-dmensonal constant vectors, x s the n-dmensonal varable, A s an m n matrx, and b s an m-dmensonal constant vector. Durng the transformaton process, we assume that q x 0. Multplyng both the objectve functon and the constrants by z and usng the equaton z = 1/q x and zx = y, we obtan a lnear programmng problem lsted on the rght of Fg. 2. Usng the lnear transformaton, the above problem of a fuzzy weghted average wthn a generalzed mean operator can be converted nto the followng lnear programmng problems: 1) In the case where p 0, L and U are transformed to lnear programmng L p a 1 y 1 + a p 2 y 2,..., +an p ) y n U = Max L = Mn and U, p b 1 y 1 + b p 2 y 2,..., +bn p ) y n c z y d z = 1, 2..., n c z y d z = 1, 2..., n y 1 + y 2,..., +y n = 1 y 1 + y 2,..., +y n = 1 z 0 z 0 y 0, = 1, 2..., n y 0, = 1, 2..., n.

6 2704 Y.-Y. Guh et al. / Computers and Mathematcs wth Applcatons ) ) In the case where p < 0, L and U are transformed to lnear programmng L p a 1 y 1 + a p 2 y 2,..., +an p ) y n U = Mn L = Max and U, p b 1 y 1 + b p 2 y 2,..., +bn p ) y n c z y d z = 1, 2..., n c z y d z = 1, 2..., n y 1 + y 2,..., +y n = 1 y 1 + y 2,..., +y n = 1 z 0 z 0 y 0, = 1, 2..., n y 0, = 1, 2..., n. 4. Numercal example To llustrate ths approach, the example used by Dong and Wong [4] s solved usng the lnear transformaton algorthm. Ths problem can be regarded as a sub-herarchy belongng to a herarchy evaluaton problem e.g. performance evaluaton), whch has three crtera. The followng trangular fuzzy numbers were used for the values of the scorng crtera and the weghtng factors: { { x1 0 x u A1 x 1 ) = 1 1 w1 /0.3 0 w u 2 x 1 1 x 1 2 w1 ) = )/ { { x2 2 2 x u A2 x 2 ) = 2 3 w2 0.4)/ w u w2 ) = u A3 x 3 ) = 4 x 2 3 x 2 4 { x3 4 4 x x 3 5 x 3 6 u w3 w 3 ) = 1 )/ { w3 0.6)/ w 3 )/ Frst, we solve ths problem at α = 0.5. The ntervals at ths α value for the scorng crtera and the weghtng factors for the above trangular fuzzy numbers are x 1 = [0.50, 1.50], x 2 = [2.50, 3.50], x 3 = [4.50, 5.50] = [0.15, 0.60], = [0.55, 0.85], w 3 = [0.70, 0.90]. 1) When p 0, the lnear programmng L and U for α = 0.5 are L = Mn 0.5 p y p y p ) y 3 U 1.5 p y p y p ) y z y z 0.15z y z 0.55z y z 0.55z y z 0.70z y z 0.70z y z y 1 + y 2 + y 3 = 1 y 1 + y 2 + y 3 = 1 z 0 z 0. 2) In the case where p < 0, the lnear programmng L and U for α = 0.5 are L = Max 0.5 p y p y p ) y 3 U 1.5 p y p y p ) y z y z 0.15z y z 0.55z y z 0.55z y z 0.70z y z 0.70z y z y 1 + y 2 + y 3 = 1 y 1 + y 2 + y 3 = 1 z 0 z 0. These smple lnear programng problems are solved usng LINDO. The aggregated ndex ntervals [L, U] obtaned for α = 0.5 are lsted n Table 1 as the value p vares from p = to p =. It can be seen that the maxmum or mnmum value for the above lnear programng problems occurs n the boundary ponts of the weght nterval, whch conforms wth the relevant theorems presented n Dong and Wong [4], Lou and Wang [6], Guh et al. [7] and Hon et al. [8]. As shown n Table 1, the larger the values of p, the larger the lower and upper bounds of the aggregated ndex obtaned. Ths ndcates the atttude that a DM takes n evaluaton toward an open standpont. When p approxmates

7 Y.-Y. Guh et al. / Computers and Mathematcs wth Applcatons ) Table 1 Aggregated ndex ntervals as p vares from p = to for α = 0.5 w 3 Lower bound of aggregated ndex w 3 Upper bound of aggregated ndex p p = p = p = p = p = p = p = p , the DM s open atttude s extremely strong, domnatng the nfluence of the crtera weght; and thus the lower and upper bounds of the aggregated ndex gnore the crtera weght nfluence and take the maxmum value of these three crtera scores. In contrast, f p has a small value, ths means that the evaluaton atttude of the DM s rgorous. When p approxmates to, ths mples that the DM s atttude s extremely strongly rgorous, so both the lower and upper bounds of the aggregated ndex also gnore the crtera weght and take the mnmum value of these three crtera scores. When p = 1, t ndcates the DM takes a neutral perspectve, and so the lower and upper bounds of the aggregated ndex are completely determned by these three crtera weghts. In summary, the value p flexbly conveys and emulates the evaluaton atttude of the DM. Addtonally, t s notable that the crtera weghts,, w 3 ) for the lower bound of the aggregated ndex for α = 0.5 change from 0.6, 0.85, 0.7) to 0.6, 0.55, 0.7) between p = +1 and p = Newton s method of numercal analyss s used to obtan the exact turnng pont as follows refer to Ypma [23] and Kelley [24]). Let ) 0.5p = ) 0.5p + whch can be rearranged as f p) = ) 2.5p ) 0.6 ) p ) ) 2.5p ) ) ) 4.5p ) 4.5p ) 0.5 p ) ) ) ) ) 4.5 p = 0. Newton s method s used to develop an algorthm and solve p n+1 = p n f p n )/f p n ), and the turnng pont p = s thus obtaned. Smlarly, the crtera weghts,, w 3 ) of the upper bound U) change from 0.15, 0.55, 0.9) to 0.15, 0.85, 0.9) whle p = Concluson The fuzzy weghted average s wdely used n herarchcal evaluaton problems n engneerng and management, and thus varous algorthms whch approxmate the membershp values of the fuzzy weghted average have been studed. But prevous works consdered only the fuzzness of score and weght, and used the addtve type to aggregate these weghted scores. Ths study successfully uses generalzed means to further consder the aggregaton operator beng also a fuzzy varable. In dong so, t s the frst work to treat all factors ncludng scorng, weghtng and aggregatng wth fuzzy consderaton for a herarchcal evaluaton problem. Ths study also dscusses why the proposed fuzzy weghted average wthn the generalzed mean operator model s so mportant for the fuzzy weghted average approach. Ths s because the tradtonal fuzzy weghted average approach never consders the nfluence of the DM s evaluaton atttude, although t s ndeed an mportant factor

8 2706 Y.-Y. Guh et al. / Computers and Mathematcs wth Applcatons ) affectng the fnal evaluaton result. Ths paper demonstrates that the generalzed mean operator s an approprate functon for fuzzfyng the aggregaton operator for a herarchcal evaluaton problem. By varyng the parameter p of a generalzed mean operator, t can flexbly smulate any DM s evaluaton atttude, such as open, neutral or rgorous. Suffcent evdence from the numercal example suggests that the proposed model can accurately reflect the nfluence of a DM s evaluaton atttude. Ths study concludes that the proposed model s theoretcally sound, readly understood, easly mplemented, and capable of producng results consstent wth expectatons. In summary, the proposed model has the potental to be a useful management tool for mproved resoluton of fuzzy herarchcal evaluaton problems. References [1] W.G. Stllwell, D.V. Wnterfeldt, R.S. John, Comparng herarchcal and nonherarchcal weghtng methods for elctng multattrbute value models, Management Scence ) [2] R.A. Rbero, Fuzzy multple attrbute decson makng: A revew and new preference elctaton technques, Fuzzy Sets and Systems ) [3] L.A. Zadeh, Fuzzy sets, Informaton and Control ) [4] W.M. Dong, F.S. Wang, Fuzzy weghted average and mplementaton of the extenson prncple, Fuzzy Sets and Systems ) [5] A.B. Tee, M.D. Bowman, Brdge condton assessment usng fuzzy weghted averages, Cvl Engneerng Systems ) [6] T.S. Lou, M.J. Wang, Fuzzy weghted average: An mproved algorthm, Fuzzy Sets and Systems ) [7] Y.Y. Guh, C.C. Hon, K.M. Wang, E.S. Lee, Fuzzy weghted average: A max mn pared elmnaton method, Computers and Mathematcs wth Applcatons ) [8] C.C. Hon, Y.Y. Guh, K.M. Wang, E.S. Lee, Fuzzy multple attrbutes and multple herarchcal decson makng, Computers & Mathematcs wth Applcatons ) [9] D.H. Lee, D. Park, An effcent algorthm for fuzzy weghted average, Fuzzy Sets and Systems ) [10] L.V. Vanegas, A.W. Labb, Applcaton of new fuzzy-weghted average NFWA) method to engneerng desgn evaluaton, Internatonal Journal of Producton Research ) [11] P.T. Chang, K.C. Hung, Applyng the fuzzy-weghted-average approach to evaluate network securty systems, Computers & Mathematcs wth Applcatons ) [12] Y.C. Lou, S.M. Guu, Lnear-tme algorthm for the fuzzy weghted average method, Journal of the Chnese Insttute of Industral Engneers ) [13] S.M. Guu, Fuzzy weghted averages revsted, Fuzzy Sets and Systems ) [14] G. Srbladze, A. Skharuldze, Weghted fuzzy averages n fuzzy envronment part I. Insuffcent expert data and fuzzy averages, Internatonal Journal of Uncertanty, Fuzzness and Knowledge-Based Systems ) [15] G. Srbladze, A. Skharuldze, Weghted fuzzy averages n fuzzy envronment part II. Generalzed weghted fuzzy expected values n fuzzy envronment, Internatonal Journal of Uncertanty, Fuzzness and Knowledge-Based Systems ) [16] P.T. Chang, K.C. Hung, K.P. Ln, C.H. Chang, A comparson of dscrete algorthms for fuzzy weghted average, IEEE Transactons on Fuzzy Systems ) [17] Y.Y. Guh, C.C. Hon, E.S. Lee, Fuzzy weghted average: The lnear programmng approach va Charnes and Cooper s rule, Fuzzy Sets and Systems ) [18] C. Kao, S.T. Lu, Fractonal programmng approach to fuzzy weghted average, Fuzzy Sets and Systems ) [19] Y. Chen, Y.K. Fung, J. Tang, Ratng techncal attrbutes n fuzzy QFD by ntegratng fuzzy weghted average method and fuzzy expected value operator, European Journal of Operatonal Research ) [20] A.A. Charnes, W.W. Cooper, Programmng wth lnear fractonal functonals, Naval Research Logstcs Quarterly ) [21] A.A. Charnes, W.W. Cooper, An explct general soluton n lnear fractonal programmng, Naval Research Logstcs Quarterly ) [22] H. Dykhoff, W. Pedrycz, Generalzed means as a model of compensaton connectves, Fuzzy Sets and Systems ) [23] T.J. Ypma, Hstorcal development of the Newton Raphson method, SIAM Revew ) [24] C.T. Kelley, Solvng Nonlnear Equatons wth Newton s Method, n: Fundamentals of Algorthms, vol. 1, SIAM, New York, 2003.