Conducting fuzzy division by using linear programming

Transcription

1 WSES TRNSCTIONS on INFORMTION SCIENCE & PPLICTIONS Muat pe Basaan, Cagdas Hakan adag, Cem Kadia Conducting fuzzy division by using inea pogamming MURT LPER BSRN Depatment of Mathematics Nigde Univesity Nigde TURKEY CGDS HKN LDG Depatment of Statistics Hacettepe Univesity Beytepe, nkaa TURKEY CEM KDILR Depatment of Statistics Hacettepe Univesity Beytepe, nkaa TURKEY bstact: Some appoimation methods have been poposed fo fuzzy mutipication and division in the iteatue. Instead of doing aithmetic opeations using fuzzy membeship functions fo fuzzy numbes, paameteized epesentation of fuzzy numbes have been used in aithmetic opeations. The most appied paameteized fuzzy numbes used in many of the eseach papes ae symmetic and asymmetic tiangua and tapezoida fuzzy numbes. In this study, we popose a new appoimation method based on inea pogamming fo fuzzy division. In ode to show the appicabiity of the poposed method, some eampes ae soved using the poposed method and the esuts ae compaed with those geneated by othe methods in the iteatue. The poposed method has poduced bette esuts than those geneated by the othes. Key-wods: ppoimation method; Fuzzy aithmetic; Fuzzy division; Linea pogamming; Tiangua fuzzy numbe 1 Intoduction Instead of using membeship functions in fuzzy aithmetic, paameteized fom of fuzzy numbes have been used fo aithmetic opeations. Despite of the fact that the addition and subtaction of the paameteized fuzzy numbes esut in cosed fom, the same does not hod fo mutipication and division. Howeve, fuzzy mutipication in cosed fom can be done unde the weakest T nom as an eception [4]. Theefoe, some appoimation methods ae poposed fo mutipication and division of the paameteized fuzzy numbes [1,,3,4]. The most appied paameteized fuzzy numbes ae symmetic and asymmetic tiangua and tapezoida fuzzy numbes since they ae simpe and easy to impement fo many puposes [8]. Dubois and Pade [1] fist intoduced aithmetic opeations on paameteized fuzzy numbes. Then, Giachetti and Young [,3] poposed a new method fo fuzzy mutipication and division. nothe method fo fuzzy aithmetic opeation is the weakest T nom [4]. In these studies, eseaches have focused on deceasing the fuzziness of the esuting fuzzy numbe. It is assumed that two paameteized tiangua fuzzy numbes ae mutipied o divided. This computation can be done using one of the methods mentioned above [1,,3,4]. Howeve, the esuts obtained fom these methods have diffeent fuzzy end points o spead vaues ecept cente vaue. Instead of using paameteized fom of fuzzy numbes that consist of ea numbes, the aea and the popotions of cente vaue to eft and to ight end points ae used to do mutipication o division of two fuzzy numbes. We poposed a new method based on inea pogamming fo paameteized fuzzy division. The paameteized fuzzy numbes ae imited to symmetic and asymmetic and tapezoida fuzzy numbes because of thei simpicity and ease of use. In this pape, new fuzzy division method is appied to tiangua fuzzy numbes. The essence of poposed method depends on utiizing geometic featues and definition of tiangua fuzzy numbes [5,7]. In the net section, some peiminay knowedge eated to fuzzy set is given. Section 3 gives bief infomation about eisting appoaches avaiabe in the iteatue fo fuzzy division. The new poposed method is intoduced in Section 4. Section 5 contains the impementation of the poposed method and the compaisons with othe avaiabe methods in the iteatue. Last section is discussion and concusion. Peiminaies fuzzy set [6] of the ea ine R with membeship function : R [ 0,1] μ is caed a fuzzy numbe if 1. is noma, namey, thee eist an eement such that μ ( = 1. is fuzzy conve, that is, μ ( λ1 + (1 λ μ ( 1 μ ( 1, R, λ [0,1] 3. μ is semi continuous 4. supp is bounded whee supp = R : μ ( > 0 { } ISSN: Issue 6, Voume 5, June 008

2 WSES TRNSCTIONS on INFORMTION SCIENCE & PPLICTIONS Muat pe Basaan, Cagdas Hakan adag, Cem Kadia The α cut of a fuzzy set is a non- fuzzy set defined as α = { R : μ ( α}. ccoding to the definition of a fuzzy numbe, evey α cut of a fuzzy numbe is a cosed inteva. Thus, α = [ L ( α, U ( α ] can be witten, whee L ( α = inf{ R : μ ( α} U ( α = sup{ R : μ ( α} The membeship function fo tiangua fuzzy numbe is [7] defined as 0 fo < a a fo a b b a μ ( = c fo b c c b 0 fo > c Thee ae two ways of denoting paameteized fuzzy numbes. The fist one denotes eft end point, cente vaue and the ight end point. On the othe hand, the second one shows eft spead vaue, cente vaue, and the ight spead vaue. Thoughout the pape, two ways of denoting fuzzy numbes ae adopted in fuzzy aithmetic opeations. 3 Review of Eisting ppoaches The fist way of doing paameteized fuzzy aithmetic was intoduced by Dubois and Pade [1]. The esuting fuzzy numbe is just an appoimation. Howeve, the usage of them is easy and simpe in appications. Let M and N be two tiangua fuzzy numbes denoted by M = ( m, α, β, N = ( n, γ, δ. Dubois and Pade [1] intoduced fuzzy division with the epession given beow. M m m n m n X δ + α γ + β = = (,, N n n n In Tabe 1, thee iustations ae given utiizing the epession given by [1]. Giachetti and Young [,3] intoduced a new appoimation method fo fuzzy mutipication and division. Whie a new appoimation method fo mutipication was intoduced in [], [3] incuded a new appoach fo fuzzy division. The motivation behind those papes is that the esuting fuzzy numbe shoud be obtained with ess eo which means that the fuzziness of the esuting fuzzy numbe is epected to gow ess. the mathematica epessions fo the new appoimation method fo fuzzy division [3] was given as foows: QN( L = DL+ G( α, n τl( n, λ( b a QN( R = DR+ G( α, n τr( n, ρ( c b whee D L is the actua division using α -cut, G( α, n is the coection tem which is witten in tems of α. The inea fit between the geometic mean of the spead atios and the numbe incuded in division is denoted by τ ( n L, λ. ( b a is the eft spead vaue. The same tems can be witten fo the ight side of the esuting fuzzy numbe. Heein, we ae not going into the detaied epanations about how the method poposed by [,3] is appied to tiangua fuzzy numbes. The moe detaied epanations can be found in [,3]. 4 The Poposed Method Instead of using numbes given in paameteized fom of the fuzzy numbes to pefom fuzzy aithmetic, we popose a new method based on inea pogamming utiizing the geometic featues and definition of fuzzy numbes. It is assumed that two asymmetic fuzzy numbes ae divided. The paameteized fom of these fuzzy numbes ae denoted by ( 1,, 1 and (,,, whee 1 and, c 1 and c, and, 1 and denote eft end points, cente points and, ight end points, espectivey. Let = ( 1,, 1 and B = (,, be two fuzzy numbes. The esuting fuzzy numbe obtained fom the division of these fuzzy numbes can be witten as foows. X = (1 B whee X = (,, is a fuzzy numbe. c The constaints and the objective function of the inea pogamming pobem fo the fuzzy division given in (1 ae defined as foows: The fist constaint is based on the aea of fuzzy numbes. The aea of the esuting fuzzy numbe X shoud be equa to o ess than the aea obtained fom the division of two aeas of fuzzy numbes and B. Thus, the mathematica epession fo the fist constaint is witten as foows: ( 1 1 ( ( ( ISSN: Issue 6, Voume 5, June 008

3 WSES TRNSCTIONS on INFORMTION SCIENCE & PPLICTIONS Muat pe Basaan, Cagdas Hakan adag, Cem Kadia whee the aeas of fuzzy numbe and B ae computed ( 1 as 1 ( and espectivey and the aea of fuzzy numbe X ( is computed as. Epession ( can be ewitten as foows: ( 1 1 (3 ( The second and the thid constaints ae defined using the atios of cente vaues to eft and ight speads espectivey. The second constaint is constucted based on the eft speads. The eft spead vaue ove cente vaue fo X shoud be equa to o ess than the division of the same atios cacuated fo and B. Thus, the mathematica epession of the second constaint is witten as foows: ( 1 ( c (4 ( c c The cente vaue of X is equa to the division of the cente vaues of and B since a the avaiabe methods in the iteatue [1,,3] have used this notion. Thus, the epession given in (4 is ewitten as foows: ( 1 (5 ( In a simia manne, the thid constaint is constucted as foows: ( 1 + (6 ( The fouth and fifth constaints can be defined based on the definition of fuzzy numbe. The eft end point of a fuzzy numbe shoud be ess than the cente vaue. Then, the mathematica epession fo the fouth constaint is given beow. < (7 The ight end point of a fuzzy numbe shoud be geate than the cente vaue. Then, the mathematica epession fo the fifth constaint is given beow. > (8 When taking the second (5, thid (6, fouth (7, and the fifth (8 constaints into account, the uppe and owe bounds fo and can be witten as given beow. ( 1 < c ( c c ( 1 < + ( In the objective function, what we ae ooking fo is a fuzzy numbe X which contains the fuzziness esuted fom dividing two fuzzy numbes. Theefoe, the diffeence between the end points and shoud be maimized. Then, the objective function is given beow. f ( = (9 Instead of maimizing the objective function in (9, if the diffeence between and wee tied to be minimized, the vaue of f ( woud take zeo. This means that the esuting fuzzy numbe becomes having zeo speads, that is, it is a degeneated fuzzy numbe. Finay, the defined inea pogamming pobem is given beow. Ma f ( = ( 1 1 ( ( 1 ( ( 1 + (9 ( < > In the fomuation of the pobem, thee ae two unknown vaiabes and. In the esut of soving the inea pogamming pobem, the eft and ight end points of X, whose cente vaue known as in advance, ae obtained. 5 Impementation Let and B be two fuzzy numbes which ae denoted by = ( 1,, 1 and B = (,, whee 1 and, and c, and, 1 and denote eft end points, cente points and, ight end points, espectivey. Fo eampe, et = (70,100,130 and B = (4,10,16 be two asymmetic fuzzy numbes. The poposed method is used in ode to divide these two fuzzy ISSN: Issue 6, Voume 5, June 008

4 WSES TRNSCTIONS on INFORMTION SCIENCE & PPLICTIONS Muat pe Basaan, Cagdas Hakan adag, Cem Kadia numbes. Then, the inea pogamming pobem is given beow. Ma f ( = < 10 > 10 If the pobem is ewitten, the new fomuation is given beow. Ma f ( = < 10 > 10 When the pobem is soved, the soution gives = 15 and = 5. Thus, the esut of division is X = (5,10,15. When the fuzzy numbes having diffeent featues such as geate cente vaues o wide spead vaues ae divided, esuts ae given in ode to show the appicabiity of the poposed method. Fo compaison, the esuts of the soved eampes based on the poposed method and othe methods avaiabe in the iteatue ae summaized in Tabe 1. The ea numbe unde the esuting fuzzy numbe denotes the diffeence between the end points. Fo eampe, when = (1,5,7 is divided by B = (5,0,3, using the poposed method gives X = ( ,0.5, Then the diffeence between end points of this esut is When the esuts ae eamined, the poposed method has esuting fuzzy numbe whose fuzziness is ess than the fuzziness of the method poposed by Dubois and Pade [1]. The fist two eampes poduce appoimatey the same fuzziness fo the methods poposed by Giachetti and Young [3] and the poposed method. Howeve, when the thid eampe is eaimed, the poposed method has poduced much naowe fuzziness than the method poposed by Giachetti and Young [3] since the paametes used in thei method ae out of the ange specified by Giachetti and Young [3]. Theefoe, the poposed method shoud be pefeed to the Giachetti and Young s method since the poposed method does not impose any imitation on any paametes. Tabe 1. The esuts of the diffeent fuzzy division methods (1,5,7 / (5,0,3 esuts ange The poposed method (0.0,0.5, Dubois and Pade (-0.05,0.5, Giachetti and Young (0.01,0.5, (70,110,130 / (4,10,16 esuts ange The poposed method (5,10,15 10 Dubois and Pade (1,10,19 18 Giachetti and Young (6.7,10, (18,147,178 / (3,4,87 esuts ange The poposed method (0.19,3.500, Dubois and Pade (-3.33,3.5, Giachetti and Young (0.765,3.5, It is a known fact that the vaiabes in inea pogamming take positive vaues. Howeve, the components of the esuting fuzzy numbe can take positive and negative vaues at the same time. In ode to sove this pobem in fuzzy division, the inea pogamming pobem is modified. This modification enabes the poposed method to geneate negative end points just as doing aithmetic opeations in ea ines. Theefoe, the imitation of the inea pogamming can be eiminated. When the cente vaue of the esuting fuzzy numbe neas to zeo o the diffeence between eft spead vaues of dividend fuzzy numbe and of diviso fuzzy numbe is age, the eft end point of the esuting fuzzy numbe can be a negative numbe. The constaint given beow can point that might have negative vaue. ( 1 ( Thus, when the condition ( 1 < (10 ( is eaized, the domain of contains negative vaues. This is the mathematica epession of the veba epanation given above. Howeve, the esuts obtained fom soving the inea pogamming pobem fo fuzzy division can not be negative. In ode to ovecome this issue, the vaiabe can be edefined as foows: = 1 whee 1 and ae positive numbes. When the new epession given above is witten in the pobem ISSN: Issue 6, Voume 5, June 008

5 WSES TRNSCTIONS on INFORMTION SCIENCE & PPLICTIONS Muat pe Basaan, Cagdas Hakan adag, Cem Kadia constucted fo fuzzy division, the new inea pogamming pobem is obtained as foows: Ma f ( = 1 + ( ( ( 1 1 ( ( 1 + (11 ( 1 < > The fomuation constucted above shoud be soved in ode to obtain the paametes of the esuting fuzzy numbe fo fuzzy division. Thee ae thee unknown vaiabes, and 1.Then, the eft and ight end points of X, whose cente vaue is known as in advance, ae obtained. The eft end point of the esuting fuzzy numbe is obtained by subtacting fom 1. Fo eampe, et = (1,5,7 and B = (5,0,3 be two asymmetic fuzzy numbes. When the vaues ae witten as in (10, the inequaity given beow is obtained. 5 (5 1 < 0 (0 5 Then 0.50 < 0.67 is obtained. It is seen that the condition is satisfied so the eft end point of the esuting fuzzy numbe wi be a negative numbe. Theefoe, the poposed method is used in ode to divide these two fuzzy numbes. Then, the modified inea pogamming pobem is given beow. Ma f ( = < 0.50 > 0.50 When the pobem is soved, the soution gives = and = Thus, the esut of division is X = ( 0.017, 0.50, It shoud be noted that checking the condition given in (9 is not necessay when fuzzy division is conducted since the modified vesion of inea pogamming fomuation eiminates finding just positive numbes. Even though the condition given in (9 is not satisfied, esuts obtained fom the soution of the fomuation (9 ae same as those obtained fom the fomuation (11. Theefoe, the fomuation (11 can be used evey time without checking the condition given in (9. In shot, the fomuation given in (9 is a specia type of the fomuation given in (11 that can poduce positive and negative end points. 6 Concusion Thee have been vaious methods avaiabe in the iteatue in ode to do fuzzy paameteized aithmetic since using membeship functions in fuzzy aithmetic is a compe pocess. Howeve, utiizing paameteized fuzzy aithmetic just poduces appoimated esuts and using paameteized fuzzy aithmetic epeatedy causes to incease the fuzziness of the esuting fuzzy numbe, which is obtained at end of the fuzzy opeations. In this pape, we poposed a new method based on inea pogamming fo fuzzy division of tiangua fuzzy numbes. When the constaints of the inea pogamming pobem ae constucted, we epoit the geometic featues and definition of a fuzzy numbe. The poposed method cacuates the cente vaue of the esuting fuzzy numbe just as the othe methods [1,,3] avaiabe in the iteatue. Thus, the eft and ight end points of the esuting fuzzy numbe ae unknowns. Then, these vaues ae caed decision vaiabes fo the inea pogamming pobem. The esuting fuzzy numbe is obtained by soving this inea pogamming pobem consisting of five constaints. In ode to show the appicabiity of the poposed method, vaious eampes ae soved. The esuts of the poposed method ae compaed with the esuts of the othe methods in the iteatue [1,,3]. It is ceay seen that the poposed method poduces much naowe fuzziness than those poduced by the othe methods. nothe advantage of the poposed method is no dependencies fo any paametes. Fo instance, Giachetti and Young [,3] caimed that thei method woks we fo paametes in specified anges. They caed these paametes as eft and ight side atios. Refeences: [1] D. Dubois and H. Pade, Fuzzy Sets and Systems: Theoy and ppications, cademic Pess, New Yok, [] R.E. Giachetti and R.E. Young, naysis of the eo in the standad appoimation used fo mutipication ISSN: Issue 6, Voume 5, June 008

6 WSES TRNSCTIONS on INFORMTION SCIENCE & PPLICTIONS Muat pe Basaan, Cagdas Hakan adag, Cem Kadia of tiangua and tapezoida fuzzy numbes and the deveopment of a new appoimation, Fuzzy Sets and Systems, Vo. 91, 1997, pp [3] R.E. Giachetti and R.E. Young, paametic epesentation of fuzzy numbes and thei aithmetic opeatos, Fuzzy Sets and Systems, Vo. 91, 1997, pp [4] D.H. Hong, Shape peseving mutipications of fuzzy numbes, Fuzzy Sets and Systems, Vo. 13, 001, pp [5] D. Dubois and H. Pade, Possibiity Theoy n ppoach to Computeized Pocessing of Uncetainity, Penun Pess, [6] Fuzzy Sets and ppications: Seected Papes by L.. Zadeh, Edited by R.R Yage, S. Ovchinnikov, R.M Tong, H.T Nguyen, [7] D. Dubois, H. Pade, Opeations on fuzzy numbes, ntenat. J. Systems Science, Vo. 9(6, 1978, pp [8] W. Pedycz, Why tiangua membeship functions?, Fuzzy Sets and Systems, Vo. 64, 1994, pp ISSN: Issue 6, Voume 5, June 008