Fuzzy Set Approach to Solve Multi-objective Linear plus Fractional Programming Problem

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1 Internatonal Journal of Oeratons Research Vol.8, o. 3, 5-3 () Internatonal Journal of Oeratons Research Fuzzy Set Aroach to Solve Mult-objectve Lnear lus Fractonal Programmng Problem Sanjay Jan Kalash Lachhwan,ψ Deartment of Mathematcal Scences, Government College, Ajmer- 35, Inda. Deartment of Mathematcs, Government Engneerng College, Bkaner-3344, Inda. Receved January ; Revsed January ; Acceted May Abstract In ths aer we ugraded the Luhjula s method (Fuzzy Sets Systems, 3(), (984), -3) for multle objectve lnear lus fractonal rogrammng roblems (MOL+FPP) wth modfcaton gven by Dutta et al. (Fuzzy Sets System, 5(), (99), 39-45). The am of ths aer s to show new fuzzy set aroach for MOL+FPP by defnng new membersh functon for lnear functon art smlar modfed membersh functon of the goal nduced by the quotent art of the objectve functons choose weghts corresondng to these goal membersh functons. We also rovde condtons on the weghts ndcatng the relatve mortance gven by decson maker, so that certan hyothess verfed. We extend the current roof of theorem for MOL+FPP rove ts valdty n obtanng the effcent soluton. Keywords Fuzzy mathematcal rogrammng, multle objectve lnear lus fractonal rogrammng, lngustc varable, membersh functon.. ITRODUCTIO Mathematcally, lnear fractonal rogrammng roblem nvolves otmzaton of objectve functon n the form of lnear fractonal functons.e objectve functon n the form ( X ). But n ractce fractonal rogrammng deals wth D( X ) stuaton where a relaton between hyscal / or economcal functons. For examle cost / tme, cost / roft or other quanttes that measure the effcency of a system, s mnmzed. The state of art n the theory, methods alcatons of fractonal rogrammng s resented n Stancu Mnasan s book (997). Fractonal rogrammng has been wdely revewed by many authors Craven (998), Horst Pardalos (995), Cabellero Hernadez (4). But many economcal hyscal roblems of fractonal rogrammng nvolves lnear functon wth addton of quotent functon.e. a new tye of ( X) otmzaton roblems where the objectve functon s of the form L( X) +, subject to certan condtons. D( X) In real world decson stuaton, decson makers sometmes may face u wth the decson to otmze Proft + Inventory/Sales, Salary + Outut/Emloyee etc. wth resect to some constrants. Such tyes of roblems wth multle objectves formulate the multobjectve lnear- lus-lnear fractonal rogrammng (MOL+FP) roblems. Mathematcally, Multobjectve lnear-lus-lnear fractonal rogrammng (MO+FP) roblem seeks to otmze more than one objectve g( X) functon n the form of f( X) +.e sum of lnear functon rato of two lnear functons of non negatve varables hx ( ) subject to lnear constrants under the assumton that the set of feasble solutons s a convex olyhedral wth a fnte number of extreme onts the denomnator art of each objectve functon s non zero on the constrant set. In lterature, Hrche (996) nvestgated the facts about behavor of lnear-lus lnear fractonal objectve functons. Recently, Jan Lachhwan (9) develoed an algorthm to solve multobjectve lnear lus fractonal rogram by convertng t nto fuzzy rogrammng roblem. In the case when several objectve functons (conflctng non commensurable) exsts, the otmal soluton for a functon s not necessarly otmal for the other functons, hence one ntroduce the noton of the best comromse soluton, also known as on domnated soluton, effcent soluton, on-nferor soluton, Pareto s otmal soluton. Multobjectve rogrammng roblems nvolve the modelng of nut data whch can also be made by means of the fuzzy set theory. Sgnfcant contrbutons have been made to fuzzy mult-objectve fractonal rogrammng roblem. For an Corresondng author s emal: drjansanjay@gmal.com ψ Emal: kalashclachhwan@yahoo.com 83-73X Coyrght ORSTW

2 Sanjay Kalash: Fuzzy Set Aroach to Solve Mult-objectve Lnear lus Fractonal Programmng Problem 6 IJOR Vol. 8, o. 3, 53 () extensve account on fuzzy on fuzzy fractonal rogrammng roblem wth a sngle or multle objectve functons, we can see Stancu- Mnasan Po s revew book (), Stancu-Mnasan (997), Stancu- Mnasan Po (3) etc.. In recent ast, hybrds of the stochastc aroach fuzzy aroach have been develoed. For nstance, Wang Qao (993) consdered mathematcal rogrammng roblems wth fuzzy rom varables. Ths aer deals wth multobjectve lnear lus fractonal rogrammng roblem (MOL+FPP).e. (x) (x) (x) Maxmze Z(x) = ( L(x) +, L(x) +,..., L (x) + ) x X () D(x) D(x) D (x) X= x R n Ax bx, s a convex bounded set. Where () { } m () A s an m n constrant matrx, X s an n- dmensonal vector of decson varable b R. () (v) L (x) ( ), ( ) ( ), ( ) ( = l x+ α x = c + d D x = e ) x+ f, =,,... n (v) l, c, e R, α, d, f R, =,,..., (v) ( e ) x+ f > =,,..., x X The term Maxmze beng used n roblem () s for fndng all weakly strongly effcent solutons n a maxmzaton sense n terms of followng defntons. Defnton. A ont x X s sad to be weakly effcent for roblem () f only f there s no x X such that (x) L(x) + > L + =,,..., D (x) D Defnton. A ont x X s sad to be strongly effcent soluton for roblem () f only f there s no x X such that (x) L(x) + L + =,,..., D (x) D r (x) o ro Lr (x) + > L o r + o Dr (x) D o ro for at least a r o.. Luhjula (984) used a lngustc aroach to solve MOL+FPP. Dutta et al (99) modfed the lngustc aroach of Luhjula such as to obtan effcent soluton of MOL+FPP. Then Stancu- Mnasan Po (3) onted out certan shortcomng n the work of Dutta et al. (99) gave correct roof of theorem whch valdates the obtanng of the effcent solutons under certan hyothess. The am of ths aer s to show a new fuzzy set aroach for MOL+FPP by defnng new membersh functon for lnear functon art smlar modfed membersh functon for the goal nduced by the quotent art of the objectve functon choose weghts corresondng to these goal membersh functons resectvely. We also rovde condtons on weghts ndcatng the relatve mortance of lnear art quotent art of the objectve functons gven by decson maker so that certan hyothess verfed. It can also be notced that the method resented as a general one does only work effcently f certan hyothess are satsfed. The aer s organzed as follows: In secton, we roose method to solve MOL+FPP wth correct roof of the theorem attestng that, as a result of alyng fuzzy method, an effcent ont s a soluton of roblem (). We also rovde condtons on weghts n ths secton. In secton 3, we consder an examle whch llustrates our roosed method. In secton 4, we gve a comaratve analyss of roosed methodology wth earler dfferent aroach wth consdered numercal examle.. THE PROPOSED METHODOLOGY In order to roose soluton methodology of MOL+FPP, we defne the goal membersh functons begnnng wth the concet of (Z, ε) roxmty used n the larger frame work of the lngustc varable doman as: f ( x) < ( ) x C (x) = f ( ),,..., x = f ( x) > ()

3 Sanjay Kalash: Fuzzy Set Aroach to Solve Mult-objectve Lnear lus Fractonal Programmng Problem 7 IJOR Vol. 8, o. 3, 53 () f D( x) > s s D ( x) C = f D D x s = D (x) ( ),,..., s D f D( x) < D f L( x) < l L ( x) l C = f l L x L = L (x) ( ),,..., L l f L( x) > L (3) (4) Where L, D ( =,,..., ) reresents the maxmal value of lnear functon L (x) numerator (x) the mnmal value of denomnator D (x) on the set X resectvely where, s, l are the thresholds begnnng wth whch values of (x), D (x) L (x) are accetable. Here choce of goal membersh functons s motvated by Klr Yuan (995) wth the followng reasons: ( x). Snce n ractce, t s not convenent to calculate the threshold values of each L ( x) + =,,..., as t D ( x) deend on the threshold values of (x), D (x) L (x) searately. Therefore ts ndvdual threshold values of (x), D (x) L (x) are consdered.. Accetable threshold values for (x), L (x) are obtaned by consderng ther resectve maxmum values for D (x) by ther mnmum values n the constrant regon. (x) As a membersh functon of the goal nduced by the objectve functon L (x) +, we choose the functon D (x) { } µ (x) = w mn C L ( x), C ( x) + wc D ( x) =,,..., Where w an w are the weghts ndcatng the relatve mortance gven by decson maker to the crtera verfyng the condton ( w w) = + = Condtons on weghts We emhasze that the membersh functon µ ( x) =,,..., used n the followng verfy the hyothess: x,x X f L + > L + D D then µ > µ =,,..., (5) Hyothess (5) s used, however, to rove the effcency of the soluton obtaned by solvng the roblem D max V( µ ) = { wµ + wµ } = Subject to mn { (x), (x)} L C C D D µ =, µ = C (x), D,, =,,...,, µ µ Ax b, x ( w + w) = (6) ow we rovde condtons on the weghts w, w so that µ (x) verfes the hyothess. ( x,x X ) ( L + > L + µ < µ ) D D So, µ µ mn, L D w C C wc < w mn C, C L + wc D < { } + { } =

4 Sanjay Kalash: Fuzzy Set Aroach to Solve Mult-objectve Lnear lus Fractonal Programmng Problem 8 IJOR Vol. 8, o. 3, 53 () { } { } w mn, L mn, L D D C C C C < w C C (7) Case I. when mn { C, } C L = C { } So, µ µ Or, uttng We obtan < It follows that w w< ka mn C, C L = C D D wc + wc < wc + wc D D w C C < w C C s D s D w w < s D s D D D w w < s D s D k = w > k ff D > D w D D w < k ff D < D w D D w w> ka where A = mn D,, x, x < D < X D D D D A = max D,, x, x > D < X D D D D Thus, f ka< w w< ka then, hyothess (5) s verfed. Ths s same as gven by Stancu- Mnasan Po (3). Case II. When mn { C, } C L = C L { } Or, uttng We obtan It follows that w w< kb mn C, C L C L L D L D L L D D = So, wc + wc < wc + wc w C C < w C C L l L l s D s D w w < L l L l s D s D L L D D w w < L l s D s D = L l k k k w L L > w D D w L L < w D D w w> kb where ff D > D ff D < D µ µ < L L B = mn D,, x, x < D L + < L + X D D D D

5 Sanjay Kalash: Fuzzy Set Aroach to Solve Mult-objectve Lnear lus Fractonal Programmng Problem 9 IJOR Vol. 8, o. 3, 53 () And Thus, f L L B = max D,, x, x > D L + < L + X D D D D kb< w w< kb then, hyothess (5) s verfed. Case III. When mn { C, } C L = C { } Or, uttng We obtan mn C, C L C L = So, µ µ < D L D wc + wc < wc + wc L D D w C C < w C C L l s D s D w w < L l s D s D L l D D w < w L D { } { L L } l D D w < w L D w { (x ) } { (x ) } L L l D > ff D > D w L D D w { (x ) } { (x ) } L L l D < ff D < D w L D D D k = L { (x ) } { L (x ) l } w L > k ff D > D w D D w { (x ) } { (x ) } L L l < k ff D < D w D D mn, L L C C C mn C, C L = C Smlarly for the case { } = { } Condton wll be { L (x ) l } { (x ) } w D L > w L D D w { L (x ) } { (x ) } l D L < w L D D So that hyothess (5) s verfed. ow we can resent the followng roostons for MOL+FPP. ff ff D > D D < D Prooston. Assume that hyothess (5) holds. If x ot s an otmal soluton for roblem (6), then x ot s weakly effcent soluton for the roblem (). Proof: Let x ot be otmal soluton for roblem (6) assume that x ot s not weakly effcent soluton for roblem (). Hence, there s a vector x X such that ot (x) ot L(x) + > L + =,,..., ot D (x) D ot From hyothess (5), t follows that µ (x) > µ =,,..., mn (x), L (x) D ot ot ot w C C wc (x) > w mn C, C L + wc D { } + { } D Thus { wµ (x) + wµ (x)} > { (x ot ) D w (x ot µ + wµ )} = =

6 Sanjay Kalash: Fuzzy Set Aroach to Solve Mult-objectve Lnear lus Fractonal Programmng Problem IJOR Vol. 8, o. 3, 53 () whch contradct the assumton that x ot s an otmal soluton for roblem (6). The roof s comlete. We can smlarly rove even the strong effcency of soluton x ot for roblem () wthout the further hyothess that x ot beng a unque soluton of roblem (6) beng necessary, as assumed by Dutta et al. (99). Prooston. Assume that hyothess (5) holds. If x ot s an otmal soluton for roblem (6), then x ot s strongly effcent soluton for roblem (). Proof: Let x ot be otmal soluton for roblem (6) assume that x ot s not strongly effcent soluton to roblem (). Hence, a x X exsts such that ot (x) ot L(x) + L + =,,..., ot D (x) D And, for at least an ndex j, we have ot (x) ot L(x) + > L + ot D (x) D ot ot From hyothess (5) t results that µ (x) µ for all =,,..., for the ndex j, µ (x) > µ, multlyng these relatons by w w resectvely, summng after all, yelds D { wµ (x) + wµ (x)} > { (x ot ) D w (x ot µ + wµ )} And ths s agan n contradcton wth the otmalty of x ot for roblem (6). 3. COMPUTATIOAL RESULTS = In ths secton, we consder the followng examle to exlan our argument. x + x x + x max f(x) = x +, f(x) = x + x + x + 7 x + 4 Subject to x + 3x x 6 x, x (9) s aroached by dentfyng the concrete form (6) as: Here we used weghts w =.35, w =.465, We assume that { } = D max V( µ ) = { wµ + wµ } = Subject to mn { (x), (x)} L C C D D µ =, µ = C (x) D,, =,,...,, µ µ I Ax b, x ( w + w) = () w = =.475, w =.5 so that t satsfes ( w + w) =. µ mn (x), L (x) L = C C = C (x), =, L x D D x x L x + D D x For = µ = C (x) =, µ = C (x) =, µ = C (x) =, µ = C (x) =. 6 6 Here the threshold values of ndvdual membersh functons are assumed are: l =, L = 6, l =, L =, s = 7, D =, s = 6, D = 4, Substtutng these values n the roblem (), we get max V( µ ) =.8333x.475x +.5 Subject to x + 3x x 6 x, x Solvng ths lnear rogrammng roblem by smlex method, we get soluton x = 6, x = wth f (x) =, f (x) =.5. Thus x ot = ( 6, ) s the sngle effcent ont of roblem (9) because both the objectve functons reach n ths ont ther otmum, ndeendently one from another on the same feasble regon. Ths effcent ont (6, ) can be obtaned not as a = j j

7 Sanjay Kalash: Fuzzy Set Aroach to Solve Mult-objectve Lnear lus Fractonal Programmng Problem IJOR Vol. 8, o. 3, 53 () soluton of a roblem (6) for any choce of the weghts ( w, w, w, w ) but for a choce of ( w, w, w, w ) from a regon ncluded n OXYZ. The effcent soluton deends on the choce of resectve weghts ( w, w, w, w ) gven to lnear - numerator art denomnator art of objectve functon satsfyng the certan condtons descrbed. Here the weghts assumed satsfy the condtons as dscussed n case II.e. { } µ mn (x), L (x) L = C C = C (x), =, also common condton ( w w) = + =. We can easly dentfy the feasble regon of weghts w for whch the effcent ont (6, ) of roblem (9) can be obtaned as a soluton of roblem (6). Ths regon s rescrbed n fgure by the olyhedral set ABCOD. The nteror of the olyhedral set XOYZ s the feasble regon of values w, w, w, w because there s a relaton w + w + w + w = w, w, w, w between them. 4. COMPARATIVE AALYSIS umerous methods for solvng multobjectve lnear lnear fractonal rogrammng roblems have been suggested n the lterature. Some of them are Guta Chakraborty (997, ) Jan Lachhwan (9) etc. many other researchers used modfed the concet of multobjectve decson makng roblems dscussed dfferent aroaches to tackle these roblems. Here we comare the roosed methodology wth the earler aroach gven by Jan Lachhwan (9) n the context of above numercal examle. Usng the methodology gven by Jan Lachhwan (9), the roblem () can be reduced to Max λ subject to L(x) D(x) (x) + ZD (x) ( λ + D ) (x) =,,..., Ax b x, λ (8) where Z s the maxmum value of Z( X ), dstance functon d wth unt weght as d( X) = Z Z( X) = su { d} =,,3,..., k. Problem (8) s a non-lnear rogrammng roblem can be solved by non-lnear technques. Usng the above methodology to the gven examle, the reduced roblem wll be max λ subject to, x xx 6.5x + 6x.5xλ+ 5xλ+ 7.5λ 6.5 x 5x x +.5xλ+ λ x + 3x x 6 x, x Solvng t usng non lnear rogrammng technques or software ackage lke LIGO (tral verson) as shown n fgure (a) (b), the otmal soluton of the roblem s obtaned as:

8 Sanjay Kalash: Fuzzy Set Aroach to Solve Mult-objectve Lnear lus Fractonal Programmng Problem IJOR Vol. 8, o. 3, 53 () x = 6, x = λ = wth f(x) =, f(x) =.5 Whch s same as obtaned n our roosed aroach. Ths also verfes the roosed aroach. Fgure (a): Descrton of examle n LIGO (tral verson) Fgure (b): Soluton of examle usng LIGO (tral verson) 5. SPECIAL CASE Our roosed methodology for MOL+FPP can be deduced to the Luhjula s (984) method for solvng mult objectve lnear fractonal rogrammng roblem (MOLFPP) by takng lnear functon art L (x) =, =,,..., n each objectve functon whch also justfes our roosed method. 6. COCLUSIO An effort has been made to descrbe fuzzy set aroach to solve MOL+FPP rovde condtons on weghts gven by decson maker so that certan hyothess verfed. The roosed methodology can be useful new aroach to hle MOL+FPP rovded certan hyotheses on weghts are verfed. REFERECES. Craven, B. D. (998). Fractonal rogrammng, In: sgma seres n aled mathematcs, 4, Berln, Heldermann Verlag.. Caballero, R. Hernez, M. (4). The controlled estmaton method n multobjectve lnear fractonal roblem, Comuters Oeratons Research, 3(): Chakraborty, M. Guta, S. (997). Multobjectve lnear rogrammng: A fuzzy rogrammng aroach, Internatonal Journal of Management Systems, 3(): 7-4.

9 Sanjay Kalash: Fuzzy Set Aroach to Solve Mult-objectve Lnear lus Fractonal Programmng Problem 3 IJOR Vol. 8, o. 3, 53 () 4. Chakraborty, M. Guta, San. (). Fuzzy mathematcal rogrammng for multobjectve lnear fractonal rogrammng roblem, Fuzzy Sets Systems, 5(3): Dutta, D. Twar, R.. Rao, R. (99). Multle objectve lnear fractonal rogrammng roblem- A fuzzy set theoretc aroach, Fuzzy Sets Systems, 5(): Horst, R. Pardalos, P. M. (995). Edtor, Hbook of global otmzaton, Kluwer Academc Publshers, Dordrecht. 7. Hrche, J. (996). A note on rogrammng roblems wth lnear lus lnear fractonal objectve functons, Euroean Journal of Oeratonal Research, 89(): Jan, S. Lachhwan, K. (9). An algorthm for the soluton of multobjectve lnear lus fractonal rogram, Bulletn of Australan Socety of Oeratons Research, 8(): Klr, G. J. Yuan, B. (995). Fuzzy sets fuzzy logc: Theory alcatons, Prentce-Hall, PTR, USA.. Luhjula, M. K. (984). Fuzzy aroaches for multle objectve lnear fractonal otmzaton, Fuzzy Sets Systems, 3: -3.. Stancu- Mnasan, I. M. (997). Fractonal rogrammng, theory, methods alcatons, Kluwer Academc Publshers, Dordrecht, Boston, London.. Stancu- Mnasan, I. M. Po, B. (). Fuzzy fractonal rogrammng: Some recent results, aer resented at nd Internatonal Conference on symmetry antsymmetry n mathematcs, Formal languages Comuter scences, Brasov, Romana. 3. Stancu- Mnasan, I. M. Po, B. (3). On a fuzzy set aroach to solvng multle objectve lnear fractonal rogrammng roblem, Fuzzy Sets Systems, 34: Wang, G. Y. Qao, Z. (993). Fuzzy rogrammng wth fuzzy rom varable coeffcents, Fuzzy Sets Systems, 57: 95-3.