Intra-Parametric Analysis of a Fuzzy MOLP

Size: px

Start display at page:

Download "Intra-Parametric Analysis of a Fuzzy MOLP"

Christian Maxwell
5 years ago
Views:

1 Intra-Parametrc Analyss of a Fuzzy MOLP a MIAO-LING WANG a Department of Industral Engneerng and Management a Mnghsn Insttute of Technology and Hsnchu Tawan, ROC b HSIAO-FAN WANG b Insttute of Industral Engneerng and Engneerng Management b Natonal Tsng Hua Unversty Abstract: - Gven an MOLP wth fuzzy coeffcents, ths study proposed a soluton procedure to fnd the eact soluton-m for any -leveled satsfacton, whch s n the form of an nterval-valued problem. By ncorporatng the perturbaton on weghts and the degrees of possblty of the profts and resources, a multparametrc model can be solve by usng ntra-parametrc analyss for the effcent control and managemen A nondomnated set wth the crtcal regon for the weghted problem can be derved. The nondomnated set of a b-obectve lnear program problem wth ts crtcal regon can be derved as a specal case. Theoretcal evdences are provded wth numercal llustraton ey-words: degree of possblty, ntra-parametrc analyss, fuzzy coeffcents, nterval-valued Introducton Consder a multobectve lnear programmng MOLP wth fuzzy coeffcents n the followng form: [,..., ] [,..., Z = z z = c c ] A b where A = [ a ], =,..., m, n s an m n crsp constrant matr; b = [ b ], =,... m and [ c = c ], =,...,, n are fuzzy numbers of rght hand-sdes RHSs and cost coeffcents wth membershp functons, =, m and, =,...,, n respectvely. By -level cut, the fuzzy numbers of c and b becomes crsp nterval That s, we can fnd the feasble sets of c and b satsfyng the followng relatons c and b n the forms { c }, c { c }] [ Mnc c c { b b }, { b b }] [ Mn b b and for each, and respectvely. Furthermore, by the convety of a fuzzy number, the ntervals can be obtaned as [nf { },sup{ }] and [nf{ },sup{ }] for each, and. Therefore, we can obtan an MOLP wth nterval-valued coeffcents defned by: Z = [ z,..., z A b where c = c ], ] = [ c,..., c ] [ [ c, c ] c for =,,, =,,n and, b = [b ], b [ b, b ] for =,,m. When the weghts of the crtera are gven by a decson maer DM denoted by λ, =,...,, λ =, model wll transformed nto a weghted sngle obectve nterval-valued lnear program LP. Z = λ c = A b Many nvestgatons have focused on a lnear system wth the nterval-valued coeffcent and were prmarly concerned wth the bounds of a soluton set but not the eact solutons [,9,0,]. Wth regard to an optmzaton problem, Btran and Steuer developed some algorthms to solve an MOLP problem of whch the cost coeffcents are nterval-valued[,]. By applyng the vector-mamzaton Theory, an nterval-valued MOLP can be transformed nto a constant-valued problem, then a nondomnated set can then be obtaned [5, 6, 4] by the found effcent etreme pont Delgade et al. have nvestgated a mathematcal program wth nterval-valued cost coeffcents from the vewpont of fuzzy set theory and a fuzzy soluton can be obtaned [4].

2 Wangs have developed theoretcal propertes of an MOLP wth fuzzy cost coeffcents, by ntroducng the concepts of the membershp functons and the cone polyhedral theory, a soluton procedure s proposed to fnd the fuzzy nondomnated solutons[6]. Turn to the problem, when a RHS b s vared n the nterval b, b ], based on the prncple of the least [ the better n resource consumpton, t can be concluded that b represents the least but the largest possblty of consumpton; conversely, b represents the largest but the least possblty of consumpton. Smlarly, a cost coeffcent s presented by the nterval [ c, c ] wth c beng the least and the largest possblty of proft and c beng the largest and the least possblty of prof Snce the coeffcents can be vared wthn the gven ntervals, let the parameters θ be the degrees of possblty on achevng the addtonal proft of the cost coeffcents and δ be the degree of the possblty n consumng the addtonal resource of the RHSs respectvely. Due to the most effcent control, dfferent parameters are gven for dfferent types of coeffcent Therefore, by ncorporatng the parameters, the nterval-valued cost coeffcents can be represented as c = c c θ,,...,, n and that of the RHSs are b b b δ, =,... m where θ, δ [0,]. Furthermore, by consderng the robustness of the weghts, the senstvty analyss s performed to get the complete nformaton about the optmal soluton. By usng the technque of parametrc programmng, model can be transformed nto a sngle weghted obectve lnear program wth mult-parametrc weghts, cost coeffcents and RHSs as follows: Z = λ = + ρ ς [ c A b δ b b θ c c ] 4 where ρ = [ ρ ], ρ = 0 are the gven vector of the parameterς. Model 4 s a mult-parametrc LP problem, many approaches are focused on the sngle and multple parametrc analyss on the gven coeffcents ndvdually and smultaneously [7,8,,8]. In ths study, the Intra-Parametrc Analyss are proposed to fnd the optmal soluton of model 4 n Secton where the nondomnated sets of the weghted MOLP problem and ts specal case of the un-weghted b-obectve LP problem can be found. In Secton, a b-obectve eample s provded for llustraton Fnally, dscusson and conclusons are drawn n Secton 4. Intra-Parametrc Analyss of a Fuzzy MOLP Intra-parametrc analyss n model 4 of a weghted MOLP s a means for fndng all possble optmal solutons n whch the coeffcents are lnear functons of some parameters as defned n the followng: Defnton : An ntra-parametrc analyss of an MOLP s a parametrc programmng n whch the crtcal regons found by dfferent levels of parameters s defned for the same type of coeffcent In practce, there are many drectons for searchng the optmal bass by usng the parametrc programmng. The leavng varable s determned as the basc varable whose value n the RHS column goes negatve frs Snce all possble optmal bases are needed to be consdered, the order of enterng basc varable wth ts possble searchng path s not what we concern although t may affect the computaton complety. Thus, we shall smply change the optmal bass one by one and dscuss the crtcal regons of the parameters of each bas It s well-nown that when m constrants and n varables are consdered, there are at most m+ n optmal basesetreme ponts m needed to be searched.whle the range of each parameter contnuously changes, all of optmal solutons can be determned. Furthermore, based on the geometrc propertes, the authors have developed the Inter-Parametrc Tolerance Analyss of an MOLP wth global perturbatons, a b-level crteron s proposed to determne comparatvely the senstve parameter [5]. When the crtcal regon of each optmal bass s determned, t means that, whenever the values of parameters are chosen wthn the crtcal regon, the correspondng bass not unque remans optmal. On the ntersecton of crtcal regons, two or more bases may be optmal smultaneously, whch are determned q by combnatons of q etreme Snce any pont whch s a conve combnaton of two adacent optmal etreme ponts s also an optmal soluton, therefore, these ponts construct an optmal soluton set, namely, a nondomnated set, for the weghted problem. Followng ths property, we can derve the crtcal regon of the nondomnated set of a weghted MOLP problem and

3 dscuss the property of the nondomnated set wth the gven vectors of each parameter. Especally, when a b-obectve LP problem s consdered, we have the followng result Theorem : [7] Any pont n the feasble regon s an effcent basc soluton for an MOLP problem f and only f t s an optmal soluton to a weghted LP problem. That s, we have followng model 5: Z = w = [ c θ c c A b δ b b where 0 w, w = w. ] 5 It s noted that when we replace the weght w wth λ + ς, model 5 s equvalent to model 4 wth w = λ ς. Therefore, the crtcal regons derved by the above procedure s a parallel transformaton of the crtcal regons of model 4 n w -as wth the quantty λ. We can use the results derved from the weghted problem of whch a feasble bass s optmal wthn the crtcal regons of the parameters θ and δ when w moves wthn ths regon Then, the bass s an effcent bass of the un-weghted problem. That s, the crtcal regons of the un-weghted problem s the proecton of the crtcal regons derved from model 4 wth = on θ, δ -plane. When a nondomnated set of the un-weghted problem has been obtaned, the crtcal regon of a nondomnated set pocesses the followng propertes: Theorem : The crtcal regon of a nondomnated set of an un-weghted b-obectve problem where all nondomnated etreme ponts are effcent s the ntersecton of the crtcal regons of the effcent base proof: Let us prove by contradcton. If there ests some θ, δ that does not belong to the ntersecton. There wll est at least one bass, say B, whose largest tolerance regon does not contan θ, δ. Therefore, f we choose θ and δ as the value of the parameters θ and δ, B s no longer effcen Theorem : The crtcal regon of a nondomnated set of an un-weghted b-obectve problem where at least one nondomnated etreme pont s effcent s the unon of the crtcal regons of the effcent base proof: Any θ, δ that does not belong to the unon mples that the correspondng bass f t ests s not an effcent bass of the orgnal problem and s not a member of the nondomnated se Therefore, the unon s the largest tolerance regon of the nondomnated set contanng at least one nondomnated effcent bas Numercal Analyss Let us use the followng eample to llustrate the above analys Eample : z = c + c + c z = c + c + c + + b + b + + b,, 0 The membershp functons of the fuzzy coeffcents are defned as follows and the graphs are shown n Fgure. c 7 0.5c +.5, c [,4], 0, otherwse; 7 c +, [, 7], = c c 6 6 0, otherwse; 8 0.5c +.5, c [, ], c 0, otherwse; 9 0.5c +, c [,4], c 0, otherwse; 0 0.5c +.5, c [, ], c 0, otherwse; 4 c +, [, 8], = c c 6 0, otherwse; 0.005b +.5, b [00, 600], b 0, otherwse; 0.005b +.5, b [00, 00], b 0, otherwse; 4 b +.5, [900,500], = b b 600 0, otherwse; 5 6

4 For each level, we can obtan the followng parametrc ntervals by solvng the followng system, tang c as an eample: c f andonly f , [0,] 6 Therefore, we have b [00, ], [900, b [00,00 00 ], b ],, c [,5 4 ], c [,7 6 ], c c [, ], c [,4 ], c [,8 6 ]. Fgure Membershp Functons of a c,b c, c c, d c, e c, f c, g b, h b, b When the level s gven as 0.5, then we have the followng problem 6 wth nterval-valued cost coeffcents and RHSs: 4 a c 4 d b 00 z z g = [,] = [,] c [,4] + [,] b e h + [,] + [,5] [00,400] [00,00] [900,00],, 0 6 If a decson maer provdes the weghts of the mportance wth λ = 0. 5 for the frst crteron and λ = 0.5 for the other, the problem s changed nto a weghted problem. When a perturbaton s consdered b 7 c c c 8 f c c b on the weghts wth ρ =,, we have the followng mult-parametrc problem: Z = ς θ,4 θ, θ,, ς θ, θ,5 θ,, =.5θ ςθ st θ + ς + 5ςθ θ ς + ςθ δ 0000δ 00 00δ,, 0, θ, δ [0,], ς R Ths problem has decson varable n= and constrants m=. So, we have 0 bases to consdered. Every bass B determne an optmal smple tableau as stated n Table. Z * B Table Optmal Smple Tableau Orgnal varables Λ n n+ n + m B C B X A A C B Λ RHS C B B C B B b B B b 7 Then, bass s optmal f and only f the feasblty and the optmalty hold. That s, C B B 0, C B B A C 0 and B b 0 smultaneously. The regon determned by the above constrants s what we call the crtcal regon of parameter By usng the proposed nter-parametrc tolerance analyss, the weghtng constrant can be obtaned frstly as 0.5 ς 0.5 and t has been obtaned that the crtcal regon of parameter δ can be determned ndependently, that s, the feasblty condton should be consdered frst to reduce the number of possble optmal base Furthermore, there ests nter-relatons between parameters θ and ς of each bass wth ther relatve senstvtes[5]. Followng the procedure of ntra-parametrc analyss, all the possble bases wth correspondng crtcal regons can be found. Let us summarze the results n Table and the correspondng optmal solutons are lsted n Table. Note that bases of,, 6,,, 5,, 5, 6,,, 5,,,,,,,,,,,,, , 4, 5,,, 4, 5, and are nfeasble under the assumpton δ [0,]. Bases of,,,,,,,,,,,,,, Slac varables C = λ + ςρ c θ c c, b = b + δ b b

On the crtcal regon of the bass,, 5, both,, 6 and,, 5 are optmal base If a DM choose the degrees of possblty for achevng the addtonal proft of the cost coeffcents and the degree of the possblty n

5 and,, are nfeasble under the assumpton θ [ 0,] and ς [. The crtcal regon of bass,, 6 { θ, ς, δ ς [, θ [0,], δ [0,]}. The correspondng graphs of the crtcal regons for θ and ς of bases,, 4 and,, 5 are shown n Fgure. By ncorporatng the crtcal nterval of δ, the crtcal regons of each bass are obtaned. On the crtcal regon of the bass,, 5, both,, 6 and,, 5 are optmal base If a DM choose the degrees of possblty for achevng the addtonal proft of the cost coeffcents and the degree of the possblty n consumng the addtonal resource of the RHSs n the above regon, any pont whch s the conve combnatons of these etreme ponts,.e. the edge connected by these two ponts, must be an optmal soluton and the obectve values are the same wthn ths area. Then, a nondomnated set can be constructed. Smlar dscusson can be performed to other bases and we can develop the global relatons between the bases and the crtcal regons for a weghted problem. Dfferent choce of the parameters wll derve dfferent soluton-me Table The Crtcal Regons of Dfferent Bases Bass Crtcal Regons of Parameters δ θ, ς,, 6 [0.8,] { θ, ς θ [0,], ς [ },, 5 [0,0.8] θ [0,], ς [ θ, ς 0.5θ + ς 0.5θς 0,, 4 [0,0.8] θ [0,], ς [ θ, ς θ ς + θς 0 6 Table The Optmal Solutons of Dfferent Bases Bass Optmal Soluton,, δ, δ, δ 6,, 5,, δ,800 50δ,00 50δ δ, δ, δ Fgure The Crtcal Regons of θ and ς of bases 5,, and,, ς 0,0.5 4,0.5,0 θ s :the crtcal regon of bass,, 4 :the crtcal regon of bass,, 5 Furthermore, let us focus on the orgnal b-obectve LP problem. By proect the crtcal regons on the θ, δ -plane, the above three bases are also the effcent base The crtcal regons of these three bases are shown n the followng table. Table 4 The Crtcal Regons of Dfferent Bases of the un-weghted problem Bass Crtcal Regons of Parameters δ θ,, 6 [0.8,] [0,],, 5 [0,0.8] [0,],, 4 [0,0.8] [0,] It can be noted that when the degree of possblty n achevng the addtonal proft of the cost coeffcents and the degree of the possblty n consumng the addtonal resource,.e. θ, δ, are chosen n the crtcal regon of bass,,, both,, and 4 4,, 5 are effcent, therefore, the conve combnaton of these bases are the nondomnated se 4 Concluson In ths study, an MOLP wth fuzzy cost coeffcents and fuzzy RHSs s focused. By a gven level cut, the MOLP problem wll be transformed nto an nterval-valued problem. When the weghts are gven by a DM, based on the prncple of the least the better n resource consumpton and the largest the better n proft, t would be desrable that the least consumpton has the largest possblty and the largest consumpton has the least possblty and the least proft has the largest possblty and the largest proft has the least possblty. Snce the coeffcents can be vared wthn the gven ntervals, the parameters θ and δ are ntroduced to be the degrees of possblty n achevng the addtonal proft of the cost coeffcents and the degree of the possblty n consumng the addtonal resource of the RHSs respectvely. When a the robustness of the weghts s consdered for control, a soluton procedure for the transformed MOLP problems have been developed n ths study, an ntra-parametrc analyss s proposed to fnd the complete nondomnated se The eact soluton-mes and the crtcal regons of 0,-0.5

6 all possble optmal bases can be obtaned to provde the complete nformaton. On the ntersecton of the crtcal regons of the possble bases, they all reach the condton. Therefore, a nondomnated soluton set can be found by the conve combnatons of the properly selected adacent etreme ponts correspondng to these base A specal case of a b-obectve problem s dscussed of whch the crtcal regon of each possble bass of an un-weghted problem has been found to smply proect the one of the perturbed problem onto the θ, δ -plane. Snce n general, the crtcal regons of a nondonated set that contans all or at least one bases of the nondomnated set are shown to be the respectve ntersecton and the unon of the crtcal regons of the effcent bases, therefore, for the specal case of a b-obectve LP, these propertes can be employed to fnd the crtcal regon of a nondomnated se Dfferent solutons have been derved for dfferent values of the parameter Theoretcal evdence s also provded wth numercal llustratons to support the above procedure Acnowledgement The authors gratefully acnowledge the fnancal support from the Natonal Scence Councl, Republc of Chna wth proect No, NSC89--E References [] G.R. Btran, Lnear multple obectve problems wth nterval coeffcents, Management Scence Vol.6,980, pp [] J.L. Cohon, Multobectve Programmng and Plannng, Academc Press, 978. [] J.E. Cope and B.W. Rust, Bounds on solutons of lnear systems wth naccurate data, SIAM Journal of Numercal Analyss, Vol.6,979, pp [4] M. Delgado, J.L. Berdegay, and M.A. Ula, Imprecse costs n mathematcal programmng problems, Control and Cybertcs, Vol.6,987, pp-. [5] J.G. Ecer, and I.A. ouada, fndng all effcent etreme ponts for multple obectve lnear programs, Mathematcal Programng, Vol.4, 978, pp49-6. [6] J.G. Ecer, N.S. Hegner, and I.A. ouada,, Generatng all mamal effcent faces for multple obectve lnear programs, Journal of Optmzaton Theory and Applcatons, Vol.0, 980, pp5-8. [7] T. Gal, RIM multparametrc lnear programmng, Management Scence, Vol., 975, pp [8] T. Gal, Postoptmal Analyss, Parametrc Programmng and Related Topcs, McGraw-Hll, New Yor, 979. [9] E.Hasen, On the soluton of lnear algebrac equatons wth nterval coeffcents, Lnear Algebra and Its Applcatons, Vol., 969, pp5-65. [0] W. Oettl, and W. Prager, Compatblty of appromate soluton of lnear equatons wth gven errors bounds for coeffcents and rght-hand sdes, Numercal Mathematcs, Vol.6, 964, pp [] W. Oettl, On the soluton set of a lnear system wth naccurate coeffcents, SIAM Journal of Numercal Analyss Ser. B, Vol., 965, pp5-8. [] T.L. Saaty and S.I. Gass, Parametrc Obectve functon part, Operatons Research, Vol., 954, pp6-9. [] R.E. Steuer, Algorthms for lnear programmng problems wth nterval obectve functon coeffcents, Mathematcs of Operaton Research, Vol.6, 98, pp-48. [4] R.E. Steuer, Multple Crtera Optmzaton: Theory, Computaton and Applcaton, Wely, 986. [5] H.F. Wang, and M.L. Wang, Inter-parametrc Tolerance Analyss of a multobectve lnear programmng, Journal of Mathematcal Analyss and Applcatons, Vol.79, 99, pp [6] H.F. Wang, and M.L. Wang, A fuzzy multobectve lnear programmng, Fuzzy Sets and Systems, Vol.86, 997, pp6-7. [7] R.E. Wendell, A Revew of a Tolerance Approach to Senstvty Analyss n lnear programmng, Dscrete Mathematcs, Vol.8, 98, pp-4. [8] R.E. Wendell, Usng bounds on the data on lnear programmng: the Tolerance approach to senstvty analyss, Mathematcal Programmng, Vol.9, 984, pp0-.

Type-2 Fuzzy Non-uniform Rational B-spline Model with Type-2 Fuzzy Data

Type-2 Fuzzy Non-uniform Rational B-spline Model with Type-2 Fuzzy Data Malaysan Journal of Mathematcal Scences 11(S) Aprl : 35 46 (2017) Specal Issue: The 2nd Internatonal Conference and Workshop on Mathematcal Analyss (ICWOMA 2016) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES