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Appled Mathematcal Scences, Vol. 4, 2010, no. 34, The Prorty Method of Interval Fuzzy Preference Relaton 1 HeFeng Jang 2, MeMe Xa 3 and CuPng We 4 2 Tayuan Normal Unversty, Tayuan, , Chna
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Appled Mathematcal Scences, Vol. 4, 2010, no. 34, The Prorty Method of Interval Fuzzy Preference Relaton 1 HeFeng Jang 2, MeMe Xa 3 and CuPng We 4 2 Tayuan Normal Unversty, Tayuan, , Chna 3 School of Economc and Management Southeast Unversty, Nanng, , Chna 4 College of Operatons Research and Management Qufu Normal Unversty, Rzhao , Chma Abstract The prorty method of nterval fuzzy preference s studed. One lnear programmng (LP model based on addtve transtvty s constructed to derve nterval weghts from consstent or nconsstent nterval fuzzy preference relatons. The proposed method requres only one LP model whch can reduce the amount of computaton. Some numercal examples are llustrated to show the potental practcablty of the proposed method. Keywords: Interval fuzzy preference relaton; Consstency; Addtve transtvty 1 Introducton Multplcatve and fuzzy preference relatons are two mportant tools for decson makers to express hs/her preference on decson alternatves or crtera. 1 Ths research was supported by the Proect of Shandong Provnce Hgher Educatonal Scence and Technology Program No. J09LA14. 2 1682 HeFeng Jang, MeMe Xa and CuPng We Multplcatve preference relatons [1-4] have been appled extensvely n many felds,.e., technology transfer, populaton forecast. For fuzzy preference relatons, a number of technques [5-8] have been developed to deal wth ts prorty method. However, due to the complexty and uncertanty nvolved n real-world decson problems, t s sometmes unrealstc or mpossble for decson maker to estmate hs/her preference wth exact numercal value. In such stuatons, an nterval fuzzy preference relaton [9] s very sutable for expressng the decson maker s uncertan preference nformaton. Some authors have pad attenton on the prorty method of nterval fuzzy preference relatons. Gong et al. [10] constructed a nonlnear programmng method to derve the prorty weghts from nterval fuzzy preference relatons. Xu [11] establshed two programmng models to derve the prorty weghts from varous nterval fuzzy preference relatons, but they requre the soluton of (2n+1 lnear programmng (LP models. Hu et al. [12] proposed an egenvector method for fndng prortes of the multplcatve nterval fuzzy preference relatons, but the method s complex. Feng et.al.[13, 14] provded the optmal programmng models for nterval preference relatons by transformng the nterval fuzzy preference nto nterval multplcatve preference relatons, whch also requre the soluton of (2n+1LP models. Our lteratures revew shows that among the methods mentoned above, some of the methods are somewhat complex, or some transform the nterval fuzzy preference relatons nto nterval multplcatve preference relatons, whch can lost decson nformaton. Ths paper s focused on the prorty method of nterval fuzzy preference relatons. To do that, the remander of ths paper s structured as follows. Secton 2 ntroduces some concepts about the fuzzy preference relatons. Secton 3 establshes one lnear programmng model based on addtve transtvty to derve nterval prorty weghts from consstent or nconsstent nterval fuzzy preference relatons. Secton 4 establshes one lnear programmng model based on multplcatve transtvty to derve nterval prorty weghts from consstent or nconsstent nterval fuzzy preference relatons. Secton 5 provdes two numercal examples to show the potental applcatons and valdty of the proposed method. Fnally, n secton 6, we conclude the paper and gve some remarks. 2 Some concepts Defnton 1 [6]. A fuzzy preference relaton R on the set X { x x x } represented by a complementary matrx ( R = r X X wth n n = 1, 2, L, n s r 0, r + r = 1, r = 0.5, for all, = 1,2, L, n (1 Prorty method of nterval fuzzy preference relaton 1683 where r represents a crsp preference degree of x over x provded by the decson maker. Especally, r = 0 ndcates that x s absolutely preferred to x ; r = 0.5 ndcates ndfference between x and x ; r 0.5 ndcates that x s preferred to x, r = 1 ndcates that x s absolutely preferred to Let T w= ( w1, w2, L, w n be the vector of prorty weghts, where the mportance degree of x. All the w ( 1, 2,, n sum to one,.e. w 0, = 1,2, L, n, Defnton 2 [7]. A fuzzy preference relaton R ( r n = 1 x. w reflects = L are no less than zero and w = 1 (2 = s called an addtve consstent fuzzy preference relaton, f the followng addtve transtvty s satsfed r = rk rk + 0.5, for all,, k = 1,2, L, n (3 and such a fuzzy preference relaton s gven by[15]: α r = ( w w, α n 1, for all, = 1,2, L, n (4 2 where w satsfes the condton(2. Defnton 3 [16]. Let w% = w, w and w% = w, w be two any nterval weghts, where 0 w w 1 and 0 w w 1, then the degree of possblty of w% w% s defned as + max{ 0, w w} max{ 0, w w} p( w% w% =. (5 + w w + w w n n 3 Lnear programmng models based on addtve transtvty Consder a multple crtera decson makng problem wth a fnte set of n crtera, and let X = { x1, x2, L, xn} be the set of crtera. A decson maker compares each par of crtera n X, and provdes hs/her nterval preference degree r = r, r of the crteron x over x, where r ndcates that the crtera x s between r and r + as mportant as the crteron x. All these nterval r, = 1,2 L, n compose an nterval fuzzy preference preference degrees ( relaton ( r + + R = r X X wth r = r, r, r + r = r + r = 1, r r 0, = r = 0.5, for all, = 1,2 L, n. + 1684 HeFeng Jang, MeMe Xa and CuPng We By defnton 2, for the addtve consstent nterval fuzzy preference R, there should exst a normalzed nterval weght vector, W = ( w% 1, w% 2, L, w% n = (, +,,, + w1 w 1 wn w n Τ α L, whch s close to R n the sense that r = ( w% w% α α + = ( w w,0.5 + ( w w 2 2 for, = 1, L, n ;. Accordng to [17], the nterval weght vector W s sad to be normalzed f and only f w + max ( w + w 1 (6 ( w + max w + w 1 (7 whch can be equvalently rewrtten as w + w 1, = 1, 2, L, n (8 w = 1, + w 1, = 1, 2, L, n (9 = 1, As s known, f the nterval preference relaton R s the precse comparson α α + about the nterval weght vectorw, namely, r = r, r ( w w,0.5+ ( w w 2 2, then R must can be wrtten as α α + α α + _ ( w1 w2,0.5 + ( w1 w2 0.5 ( 1,0.5 ( L + w wn + wwn 2 2 α α + α α + _ ( w2 w1,0.5 + ( w2 w ( w2 wn,0.5 + ( w2wn R= 2 2 L 2 2 (10 M M 0.5 M α _ + α + α α ( wn w1,0.5+ ( wn w ( wn w2, ( ww n L whch can be splt nto the followng nonnegatve preference relaton: and R α α ( w1 w2 L ( w1 wn 2 2 α α ( w2 w1 0.5 L ( w2 wn = M M M α α ( wn w ( wn w2 L (11 Prorty method of nterval fuzzy preference relaton 1685 Let α + α ( w1 w2 L 0.5+ ( w1 wn 2 2 α + α ( w2 w1 0.5 L 0.5+ ( w2 wn R = M M M α + α ( wn w ( wn w2 L (12 0 α α α α 0 α α D =, M M L M α α α 0 (13 n n n C = 2 r1 n, 2 r2 n,, 2 rn n L, = 2 = 1, 2 = 1, n (14 n n n C = 2 r1 n, 2 r2 n,, 2 rn n L. = 2 = 1, 2 = 1, n (15 It s easy to prove that α n 1 W + DW = C, (16 ( ( 1 + α n W + DW = C, (17 T where W = ( w 1, w 2, L, w n and W + ( w + 1, w + 2,, w + n = L. Due to the presence of subectvty and uncertanty, the DM s subectve udgments cannot be 100% exact. Therefore, Eq. (13 and Eq. (14 may not hold precsely. Based on such an analyss, we ntroduce the followng devaton vector T T T ( ε1, L, εn, ( ε1, L, εn, ( γ1, L, γn and Γ = ( γ L γ n Ε = Ε = Γ = whch satsfes ( T 1,, T α n 1 W + DW C Ε +Ε = 0, (18 ( + + α n 1 W + DW C Γ +Γ = 0, (19 + where Ε, Ε, Γ, Γ 0, It s most desrable that the devaton varables should be kept as small as possble, whch leads to the followng model to be constructed: 1686 HeFeng Jang, MeMe Xa and CuPng We n ( ε + ε γ + γ e Τ ( + Mnmze J = = Ε +Ε +Γ +Γ ( ( = 1 α n 1 W + DW C Ε +Ε = 0, + + α n 1 W + DW C Γ +Γ = 0, w + w 1, = 1, L, n, = 1, st.. (20 w + w 1, = 1, L, n, = 1, + W W 0, W, W, Ε, Ε, Γ, Γ 0. where the frst two constrants are the Eqs.(14 and (15, the mddle two constrants are the normalzaton constrants on the nterval weght vectorw, and the last three constrants are those on the lower and upper bounds of W and the nonnegatvty of both W and devaton varables. 4. Case llustratons In ths secton, two numercal examples wll be examned to show the applcaton of the proposed method n consstent and nconsstent nterval fuzzy preference relatons. Example 1 [11]. Consder a multple crtera decson makng problem, there are four crtera. A decson maker compares each par of crtera x and x, and provdes hs/her nterval preference degree r of the crteron x over x, and then constructs the followng nterval fuzzy preference relaton: [ 0.5, 0.5] [ 0.3, 0.4] [ 0.5, 0.7] [ 0.4, 0.5] [ 0.6, 0.7] [ 0.5, 0.5] [ 0.6, 0.8] [ 0.2, 0.6] R =. [ 0.3, 0.5] [ 0.2, 0.4] [ 0.5, 0.5] [ 0.4, 0.8] [ 0.5, 0.6] [ 0.4, 0.8] [ 0.2, 0.6] [ 0.5, 0.5] By the model (20, we get W = ([ , ],[ , ],[ , ],[ , ] Usng Defnton 3, we have the rankng w% 2 w% 4 w% 1 w% 3. If we replace the elements r 12 = [ 0.3,0.4] and r 21 = [ 0.6,0.7] of R n Example ' ' 1 wth a par of new elements r 12 = [ 0.1,0.2] and r 21 = [ 0.8,0.9], that s, R n example 1 s revsed as, Prorty method of nterval fuzzy preference relaton 1687 [ 0.5, 0.5] [ 0.1, 0.2] [ 0.5, 0.7] [ 0.4, 0.5] [ 0.8, 0.9] [ 0.5, 0.5] [ 0.6, 0.8] [ 0.2, 0.6] [ 0.3, 0.5] [ 0.2, 0.4] [ 0.5, 0.5] [ 0.4, 0.8] [ 0.5, 0.6] [ 0.4, 0.8] [ 0.2, 0.6] [ 0.5, 0.5] R =. By the model (20, we have W = ([ , ],[ , ],[ , ],[ , ] usng Defnton 3, we have the same rankng w% 2 w% 4 w% 3 w% 1. From Example1, we can conclude that the rankng dvded by the model (20 s smlarly to the rankng n lterature [11], but the proposed methods are much smpler than the methods n lterature[11]. In what follows, we shall utlze a practcal case nvolvng the assessment of a set of agroecologcal regons n Hube Provnce, Chna, to llustrate the developed models. Example 2[11]. Located n Central Chna and the mddle reaches of the Changang(YangtzeRver, Hube Provnce s dstrbuted n a transtonal belt where physcal condtons and landscapes are on the transton from north to south and from east to west, Thus, Hube Provnce s well known as a land of rce and fsh snce the regon enoys some of the favorable physcal condtons, wth a dversty of natural resources and the sutablty for growng varous crops. At the same tme, however, there are also some restrctve factors for developng agrculture such as a tght man land relaton between a constant degradaton of natural resource sand a growng populaton pressure on land resource reserve. Despte chershng a burnng desre to promote ther standard of lvng, people lvng n the area are frustrated because they have no ablty to enhance ther power to accelerate economc development because of a dramatc declne n quantty and qualty of natural resources and a deteroratng envronment. Based on the dstnctness and dfferences n envronment and natural resources, Hube Provnce can be roughly dvded nto sx agroecologcal regons: x1 Wuhan Ezhou Huang-gang; x2 Northeast of Hube; x 3 Southeast of Hube; x4 Janghan; x5 North of Hube; x 6 West of Hube. In order to prortze these agroecologcal regons x ( = 1, 2, L, 6 wth respect to ther e k = 1, 2, 3 comprehensve functons, a commttee comprsed of three experts k ( Τ (whose weght vector s λ = ( 1/3,1/3,1/3 has been set up to provde assessment nformaton on x( =1, 2,..., 6. The experts compare these sx agroecologcal regons wth respect to ther comprehensve functons and ( k ( k construct, respectvely, the nterval fuzzy preference relatons R = r (, 1688 HeFeng Jang, MeMe Xa and CuPng We ( k = 1, 2, 3 : ( 1 R = ( 2 R = ( 3 R = [ 0.5, 0.5] [ 0.5, 0.7] [ 0.7, 0.8] [ 0.5, 0.6] [ 0.6, 0.7] [ 0.8, 0.1] [ 0.3, 0.5] [ 0.5, 0.5] [ 0.6, 0.7] [ 0.3, 0.4] [ 0.5, 0.6] [ 0.5, 0.9] [ 0.2, 0.3] [ 0.3, 0.4] [ 0.5, 0.5] [ 0.3, 0.5] [ 0.4, 0.5] [ 0.6, 0.8] [ 0.4, 0.5] [ 0.6, 0.7] [ 0.5, 0.7] [ 0.5, 0.5] [ 0.5, 0.7] [ 0.7, 0.8] [ 0.3, 0.4] [ 0.4, 0.5] [ 0.5, 0.6] [ 0.3, 0.5] [ 0.5, 0.5] [ 0.4, 0.8] [ 0.0, 0.2] [ 0.1, 0.5] [ 0.2, 0.4] [ 0.2, 0.3] [ 0.2, 0.6] [ 0.5, 0.5] [ 0.5, 0.5] [ 0.6, 0.7] [ 0.6, 0.9] [ 0.4, 0.7] [ 0.6, 0.8] [ 0.8, 0.9] [ 0.3, 0.4] [ 0.5, 0.5] [ 0.5, 0.8] [ 0.4, 0.5] [ 0.5, 0.7] [ 0.5, 0.9] [ 0.1, 0.4] [ 0.2, 0.5] [ 0.5, 0.5] [ 0.4, 0.5] [ 0.4, 0.6] [ 0.7, 0.8] [ 0.3, 0.6] [ 0.5, 0.6] [ 0.5, 0.6] [ 0.5, 0.5] [ 0.6, 0.7] [ 0.7, 0.9] [ 0.2, 0.4] [ 0.3, 0.5] [ 0.4, 0.6] [ 0.3, 0.4] [ 0.5, 0.5] [ 0.5, 0.8] [ 0.1, 0.2] [ 0.1, 0.5] [ 0.2, 0.3] [ 0.1, 0.3] [ 0.2, 0.5] [ 0.5, 0.5] [ 0.5, 0.5] [ 0.4, 0.6] [ 0.5, 0.7] [ 0.4, 0.7] [ 0.6, 0.8] [ 0.7, 0.8] [ 0.6, 0.4] [ 0.5, 0.5] [ 0.4, 0.6] [ 0.3, 0.5] [ 0.4, 0.6] [ 0.6, 0.9] [ 0.3, 0.5] [ 0.6, 0.4] [ 0.5, 0.5] [ 0.4, 0.6] [ 0.4, 0.7] [ 0.5, 0.8] [ 0.3, 0.6] [ 0.5, 0.7] [ 0.4, 0.6] [ 0.5, 0.5] [ 0.6, 0.8] [ 0.4, 0.7] [ 0.2, 0.4] [ 0.4, 0.6] [ 0.3, 0.6] [ 0.2, 0.4] [ 0.5, 0.5] [ 0.6, 0.7] [ 0.2, 0.3] [ 0.1, 0.4] [ 0.2, 0.5] [ 0.3, 0.6] [ 0.3, 0.4] [ 0.5, 0.5] We frst utlze the fuzzy weghted averagng operator [8]: ( 1 ( 2 ( 3 R = λ R λ R λ R, to aggregate all the ndvdual nterval fuzzy preference relatons k ( k = 1, 2, 3 nto the collectve nterval fuzzy preference relaton where λ = ( λ, λ, λ Τ s the weght vector of the experts ( 1, 2, 3 ( 6 6 k ( 6 6 ( k ( k R = r ( ( R = r, ek k =, Thus, we have [ 0.50, 0.50] [ 0.50, 0.67] [ 0.60, 0.80] [ 0.43, 0.67] [ 0.60, 0.77] [ 0.77, 0.90] [ 0.33, 0.50] [ 0.50, 0.50] [ 0.50, 0.70] [ 0.33, 0.47] [ 0.47, 0.63] [ 0.53, 0.90] [ 0.20, 0.40] [ 0.30, 0.50] [ 0.50, 0.50] [ 0.37, 0.53] [ 0.40, 0.60] [ 0.60, 0.80] R =. [ 0.33, 0.57] [ 0.53, 0.67] [ 0.47, 0.63] [ 0.50, 0.50] [ 0.57, 0.73] [ 0.60, 0.80] [ 0.23, 0.40] [ 0.37, 0.53] [ 0.40, 0.60] [ 0.27, 0.43] [ 0.50, 0.50] [ 0.50, 0.77] [ 0.10, 0.23] [ 0.10, 0.47] [ 0.20, 0.40] [ 0.20, 0.40] [ 0.23, 0.50] [ 0.50, 0.50] Prorty method of nterval fuzzy preference relaton 1689 By the model (20, we get W= ([ , ],[ , ],[ , ],[ , ],[ , ],[ ,0.1163] By Defnton 3, we can have that both models get the same rankng: w% w% w% w% w% w% The result s the same as that n lterature [11].,. 5. Concludng remarks In ths paper, one lnear programmng model s constructed to derve nterval weghts from consstent or nconsstent nterval fuzzy preference relatons. Several numercal examples are llustrated to show that the proposed methods can reduce the amount of computaton. References [1] A.Arbel, Approxmate artculaton of preference and prorty dervaton. European Journal of Operatonal Research 43(1989, [2] S.J. Chen, C.L. Hwang, Fuzzy Multple Attrbute Decson Makng: Methods and Applcatons. Sprnger, New York (1992. [3] L. Mkhalov, Dervng prortes from fuzzy parwse comparson udgments. Fuzzy Sets and Systems 134(2003, [4] P.J.M. Van Laarhoven, W. Pedrycz, A fuzzy extenson of Saaty s prorty theory. Fuzzy Sets and Systems 11(1983, [5] H. Nurm, Approaches to collectve decson makng wth fuzzy preference relatons. Fuzzy Sets and Systems 6(1981, [6] S.A. Orlovsk, Decson-makng wth a fuzzy preference relaton. Fuzzy Sets and Systems 1(1978, [7] T. Tanno, Fuzzy preference orderngs n group decson makng. Fuzzy Sets and Systems 12(1984, 1690 HeFeng Jang, MeMe Xa and CuPng We [8] Z.S. Xu, Q.L. Da, An approach to mprovng consstency of fuzzy preference matrx. Fuzzy Optmzaton and Decson Makng 2(2003, [9] Z.S. Xu, On compatblty of nterval fuzzy preference matrces. Fuzzy Optmzaton and Decson Makng 3(2004, [10] Z.W. Gong, S.F. Lu, Research on Consstency and Prorty of Interval Number Complementary Judgment Matrx. Chnese Journal of Management Scence 14(2006, [11] Z.S. Xu, J.Chen, Some models for dervng the prorty weghts from nterval fuzzy preference relatons, European Journal of Operatonal Research 184(2008, [12] G. Hu, X.Q. Feng, Z.Z. L, Research on the consstency and the prorty method of nterval number complementary udgment matrx, Chna. Journal of Northwest Normal Unversty Natural Scence 44(2007, [13] X.Q. Feng, C.P. We, Z.Z. L, G.Hu, On Consstency and Prorty Method of Interval Complementary Judgment Matrx, Chna. Mathematcs n practcal and theory. 37(2007, [14] X.Q. Feng, G. Hu, J. Zhu, Research on the consstency and the prorty method of nterval number complementary udgment matrx, Chna. Statstc and Decson 7(2008, [15] J. J. Zhang, Fuzzy analytcal herarchy process[j]. Fuzzy Systerm and Mathematcs, 16(2002, [16] Y.M. Wang, J.B. Yang,D.L. Xu, A two-stage logarthmc goal programmng method for generatng weghts from nterval comparson matrces. Fuzzy Sets and Systems 152(2005, [17] K. Sughara, H. Ish, H. Tanaka, Interval prortes n AHP by nterval regresson analyss, European Journal of Operatonal Research 158(2004, Receved: Novemver, 2009
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