Research Article Enhanced Two-Step Method via Relaxed Order of α-satisfactory Degrees for Fuzzy Multiobjective Optimization

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1 Hndaw Publshng Corporaton Mathematcal Problems n Engneerng Artcle ID pages Research Artcle Enhanced Two-Step Method va Relaxed Order of α-satsfactory Degrees for Fuzzy Multobjectve Optmzaton Na Wang 2 Chaofang Hu 3 Wux Sh 2 Chunbo Xu 2 and Yme Chen 2 School of Electrcal Engneerng and Automaton Tanjn Polytechnc Unversty Tanjn Chna 2 Key Laboratory of Advanced Electrcal Engneerng and Energy Technology Tanjn Polytechnc Unversty Tanjn Chna 3 School of Electrcal Engneerng and Automaton Tanjn Unversty Tanjn Chna Correspondence should be addressed to Chaofang Hu; cfhu@tjueducn Receved 6 September 203; Revsed 3 December 203; Accepted 4 January 204; Publshed 2 March 204 Academc Edtor: Km-Hua Tan Copyrght 204 Na Wang et al Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense whch permts unrestrcted use dstrbuton and reproducton n any medum provded the orgnal work s properly cted An enhanced two-step method va relaxed order of satsfactory degrees for fuzzy multobjectve optmzaton s proposed n ths paper By ntroducng the concept of fuzzy numbers and α-level set theory fuzzy parameters are taken as varables and all the objectves are transformed nto fuzzy goals nvolvng three fuzzy relatons The order of α-satsfactory degrees whch means the objectves wth hgher prorty achevng hgher satsfactory degree s appled to model preemptve prorty requrement Ths strct order constrant s relaxed by prorty varable to fnd the preferred soluton satsfyng optmzaton and prorty The orgnal optmzaton problem s dvded nto two steps to be solved teratvely The M-α-Pareto optmalty of the soluton s ensured and the satsfactorysoluton can be acqured by regulatng the slack parameterδδ or changng α The numercal examples demonstrate the power of the proposed method Introducton In recent years Multobjectve Optmzaton (MOO) problem has become more and more obvous and mportant n producton and economy where multple objectves are conflctng noncommensurable and mprecse Many researchers are nterested n ts study and development [ 4] In the real world MOO problem often takes place n uncertan envronment [5 6] In the early works research on randomness s devoted to uncertanty [7 8] However there s a qualtatvely dfferent type of mprecson whch cannot be equated wth probablty Therefore t may lead to a fuzzy understandng for Decson Maker (DM) to the optmzaton problem [9 0] For example the goals or parameters of the objectves and constrants are not known precsely n vagueenvronmentethertheperspectvegoalsaregven by mprecse DM s judgment or the parameters cannot be determned accurately Therefore fuzzy set theory s ntroduced to represent uncertanty[0 2] where the targets or parameters are modeled usng membershp functons Such a MOO problem s called Fuzzy Multobjectve Optmzaton (FMOO) problem [3 4] Its research ncludes dfferent types of FMOO and dfferent preferences [5] In MOO problem the fnal soluton s usually dependent upon DM s preference whch refers to the opnon of DM concernng ponts n crteron space Accordng to the way that DM artculates or ncorporates preference s categorzed nto pror posteror and progressve artculaton Wheren mportance s the commonly pror artculaton of preference In real lfe especally for FMOO problem mportance has become the most nterestng preference and has attracted many researchers [6 7] It represents the opnon of DM for dfferent objectves and s gven by DM n advance In order to get the preferred result analyzer needs to model the mportance preference mathematcally Generally weghted method s a tradtonal strategy about mportance where a scalarcrteronsformulatedbasedonp-norm ( p ) [8] As a specal case of mportance preemptve prorty requrement s used frequently n practce It means that all the objectves are classfed nto the dfferent levels accordng to DM s requrement and then they are optmzed step by

2 2 Mathematcal Problems n Engneerng step Beng the most conventonal strategy the lexcographc method requres that multple subproblems consstng of dfferent objectves are solved n lexcographc order [9 20] However ths may lead to heavy computaton burden Even degeneratve optmzaton may happen when some objectves wth hgher prorty have obtaned the optmum result but the remaned ones are not optmzed snce the soluton keeps dentcal Thus Chen and Tsa [2] proposetheprncple that the objectves wth hgher prorty should have hgher satsfactory degree to model the preemptve prorty by means of fuzzy theory Although the multstep optmzaton procedure s avoded the satsfactory even feasble soluton may not exst because the prorty order constrants are too strct In order to guarantee the feasblty of optmzaton Aköz and Petrovc [22] use fuzzy bnary relatons to represent mprecse prorty; however they only consder the nequalty fuzzy relatons In addton Hu and L [23]present enhanced nteractve satsfyng optmzaton method va goal programmng and L et al [24] also desgn generalzed varyng-doman optmzaton approach Nevertheless the nonlnearty of the orgnal problem s strengthened and the more optmzaton varables are ntroduced n ther methods Ths ncreases computaton burden Therefore L and Hu [25] propose a two-step satsfactory method for FMOO to acqure the satsfactory soluton Usually FMOO problem ncludes two cases: MOO wth fuzzy goals and MOO wth fuzzy parameters Although the orgnal research s resulted from fuzzy goals [0] actually more and more FMOO problems are manly orented from fuzzy parameters Fuzzy parameters of the constrants wll nfluence the feasble feld and fuzzy parameters of the objectves wll decde the optmzaton result So they produce more complcated optmzaton problem than MOOwthfuzzygoalsThsrequresDMtopartcpaten the optmzaton procedure to provde more nformaton whch brngs about more dffculty to obtan the preferred soluton Therefore more researchers focus on progressve artculaton of preference n MOO wth fuzzy parameters such that dfferent nteractve methods [4 26] are studed Nevertheless fndng the preferred soluton becomes more dffcult when prorty preference s requred The above early works [9 25] about prorty only adapt to MOO wth fuzzy goals and they are not appled drectly to fuzzy parameters For example n [25] we have proposed the two-step satsfactory method to handle preemptve prorty requrement usng relaxed prorty constrants But t only consders the case that the goals are vague The uncertan objectve functons and constrants (e feasble feld) that resulted from fuzzy parameters are not nvolved Ths leads to that the prevous method cannot be used to solve the MOO model wth fuzzy parameters Moreover the optmalty of soluton s not guaranteed for the approach n [25] where there may exst the weak M-Pareto optmal soluton Ths probably leads to the unreasonable optmzaton result In ths paper the MOO problem wth fuzzy parameters and preemptve prorty s concerned where fuzzy parameters have possblstc dstrbutons The two-step satsfactory method n [25] s ntroduced and enhanced Fuzzy parameters are treated as fuzzy numbers Usng the concept of αlevel set fuzzy parameters are defned as varables and all the objectves are regarded as fuzzy goals ncludng three types of fuzzy relatons Then the FMOO problem usng α-level sets s formulated nto an α-fmoo problem For preemptve prorty requrement the relaxed order of αsatsfactory degrees denotng that the objectve wth hgher prorty acheves the hgher α-satsfactory degree s appled Moreover the orgnal FMOO problem s reformulated nto two models that s the prelmnary optmzaton model and theprortymodelthetwomodelsaresolvedteratvely The soluton of the frst model that s α-maxmum overall satsfactory degree s obtaned and then t s decreased n the second model for balance between optmzaton and prorty In addton the value of α also nfluences the fnal result and helps to fnd the optmum value Thus the satsfactory soluton can be acqured by relaxng α-maxmum overall satsfactory degree or changng α where the M-α-Pareto optmaltyofthesolutonsensuredbymeansoftestmodel Wth our method DM can easly get the preferred result In ths paper Secton 2 descrbes MOO problem wth fuzzy parameters and preemptve prorty requrement The enhanced two-step method va relaxed order of α-satsfactory degrees s proposed n Secton 3 Secton 4 demonstrates ts powerbythenumercalexamplesinsecton 5 the concluson s drawn 2 α-fmoo wth Preemptve Prorty 2 MOO wth Fuzzy Parameters In practce the possble values about the parameters of the objectves or constrants are often consdered to be mprecse snce they often nvolve the ambguty of DM s understandng of the real system These parameters are called fuzzy parameters Then the MOO problem wth fuzzy parameters s descrbed n the followng [4 27]: mn (f (x ã )f k (x ã k )) st x G( b) =x g j (x b j ) 0j=m} where f (x) ( = k) are multple objectve functons to be mnmzed and G R n s system constrants ã = ( a a 2 a r ) and b j = ( b j b j2 b jsj ) denote fuzzy parameter vectors Usually they are characterzed as the fuzzy numbers [4 27] It s reasonable to treat a real fuzzy number as a convex contnuous fuzzy subset There are usually varous knds of membershp functons for fuzzy numbers All of them are contnuously mappng monotonously ncreasng or decreasng For example fuzzy number c s shown n Fgure ()

3 Mathematcal Problems n Engneerng 3 Its membershp functon s wrtten as c 2 c c c 2 c c c 2 c μ c (c) = 2 c c 3 c c 3 c c 4 c 3 c c otherwse Then α-level set of fuzzy parameter (ã b) s defned as (ã b) α = (a b) μ ad (a d (b ) αμ bje je ) α =2k;d=2r j=2m;e=2s j where a =(a a 2 a k ) and b =(b b 2 b m ) are taken asthedecsonvarablesaboutα Snce fuzzy parameters wll brng about or ncrease the mprecsenatureofdm sjudgmentaboutthegoalsofthe objectves they wll result n the fuzzy goals under an αlevel set Therefore DM can gve ther mplct targets to the objectves The fuzzy decson s presented as [ ] fnd (x a) such that st f (x a ) f =k x G(b) =x g j (x b j ) 0j=m} (a b) (ã b) α where f s the perspectve goal value for the fuzzy objectve functon f (x ã ) and denotes three types of fuzzy relatons that s (fuzzy mnmzaton) (fuzzy maxmzaton) and = (fuzzy equal) Correspondngly they mean that the th fuzzy objectve s approxmately less than or equal to f approxmately more than or equal to f and n the vcnty of f In ths paper the conventonal lnear trangle-lke membershp functons are adopted to defne fuzzy goals of the objectve functons wth dfferent fuzzy relatons For (a b) (ã b) α the membershp functon under α-level set s called α-membershp functon and ts value s called also αsatsfactory degree of the fuzzy objectve For fuzzy mnmzaton the tolerant nterval for the fuzzy objectve f (x ã ) s regarded as (f f max ) f max s the tolerant lmt Then the α-membershp functon μ f (x a ) s defned as μ f (x a ) f (x a ) f = f (x a ) f f max f f f (x a ) f max 0 f (x a ) f max It s shown n Fgure 2 (2) (3) (4) (5) For fuzzy maxmzaton the tolerant nterval s (f mn f ) Equaton (6) s ts α-membershp functon Fgure 3 llustrates the graph of ths fuzzy relaton: μ f (x a ) f (x a ) f = f f (x a ) f f mn f mn f (x a ) f 0 f (x a ) f mn For = (f mn f max ) s the tolerant nterval (see Fgure 4) The α-membershp functon s μ f (x a ) 0 f (x a ) f max f (x a ) f f max f f = f (x a )=f f (x a ) f f f (x a ) f max f mn f mn 0 f (x a ) f mn f (x a ) f Introducng the α-membershp functons of fuzzy goals nto (4) FMOO problem s reformulated nto the followng α-fmoo problem: max (x a ) (x a 2 )μ fk (x a k ) st x G(b) =x g j (x b j ) 0j=m} (a b) (ã b) α For general MOO or MOO under α-level set there s usually Pareto or α-pareto optmalty of the soluton [4 28] Correspondngly the defnton of M-α-Pareto optmalty for α-fmoo problem (8)sgven Defnton (M-α-Pareto optmal soluton see [27]) A pont x G(b )where(a b ) (ã b) α sm-α-pareto optmal soluton f and only f there does not ext another soluton x G(b) and (a b) (ã b) α suchthatμ f (x a ) μ f (x a ) for all ( = k)and μ fh (x a )>μ fh (x a ) foratleast one h h k} Smlarly we can defne the concept of Weak M-α-Pareto optmal soluton Defnton 2 (weak M-α-Pareto optmal soluton) A pont x G(b )where(a b ) (ã b) α sweakm-α-pareto optmal soluton f and only f there does not ext another soluton x G(b) and (a b) (ã b) α suchthatμ f (x a )> μ f (x a ) for all (=k) 22 Preemptve Prorty In real MOO problem preemptve prorty requrement s generally descrbed usng lngustc form For example suppose the prorty of the objectve f s (x ã s ) s hgher than that of f s (x ã s ) (s s (6) (7) (8)

4 4 Mathematcal Problems n Engneerng μ c (c) μ f (x a ) c =(c c 2 c 3 c 4 ) α c c 2 c 3 c 4 f (x a ) Fgure : Fuzzy number c f mn f μ f (x a ) Fgure 3: α-membershp functon μ f (x a ) for fuzzy relaton μ f (x a ) f f max f (x a ) f (x a ) Fgure 2: α-membershp functon μ f (x a ) for fuzzy relaton f mn f f max k}s = s )Thatsf s (x ã s ) should be optmzed before f s (x ã s ) Thuswecanseethatpreemptveprortyhas strcter requrement than common mportance It usually requres that all the objectves must be optmzed n sequence accordng to ther mportance Ths means that the objectves wth the hghest prorty must be satsfed frstly and then the other objectves havng the lower prorty are consdered based on the obtaned results Therefore all the objectves need to be grouped accordng to ther prorty where there wll be dfferent levels ncludng one or several objectves Consequently the α-fmoo problem wth preemptve prorty can be formulated as follows [25]: max [P (μ f (x a )μ f l (x a l )) st P L (μ f L (x a L )μ f L l L (x a L l L ))] x G(b) =x g j (x b j ) 0j=m} (a b) (ã b) α where P ( = L) s the prorty factor and P >> P + ThsmeanstheobjectvesbelongngtoP have hgher prorty than those of P + f (x ã )f l (x ã l ) represent the objectves belongng to th prortylevel (9) Fgure 4: α-membershp functon μ f (x a ) for fuzzy relaton = 3 Enhanced Two-Step Method va Relaxed Order of α-satsfactory Degrees 3 Relaxed Order of α-satsfactory Degrees For (9) the tradtonal strategy s lexcographc optmzaton [20]whose multstep optmzaton procedure may result n complex computaton even degeneratve optmzaton Accordng to Chen and Tsa [2] the objectve wth hgher prorty should have hgher satsfactory degree by whch prorty can be presented as the order of satsfactory degrees It can convert the multstep optmzaton nto a sngle formulaton Ths strategy not only adapts to general FMOO but also can be used to α-fmoo For the example about prorty n Secton 22 f s (x ã s ) has the hgher prorty than f s (x ã s ) we can get that the former has the hgher α-satsfactory degree than the latter Then the preemptve prorty requrement s modeled nto the followng order of α-satsfactory degrees: μ fs (x a s ) μ f s (x a s ) ss k} s =s (a b) (ã b) α (0)

5 Mathematcal Problems n Engneerng 5 However the order (0) s too strct If t s taken as the constrant the satsfactory even feasble soluton maybe cannot be obtaned In addton the prorty dfference between the objectves cannot be quantfed from (0) Ths means that the prorty requrement may not be reflected accurately when the α-satsfactory degrees are the same Thus the new prorty formulaton n [25] μ fs (x) μ f s (x) γ s adopted where the prorty varable γ ( γ ) s ntroduced to relax strct comparson between the satsfactory degrees For MOO wth fuzzy parameters ths formulaton s mproved to relax the order of α-satsfactory degrees Then the enhanced preemptve prorty s reformulated as follows: μ fs (x a s ) μ f s (x a s ) γ ss k} s =s (a b) (ã b) α () When γ 0 () denotes that the relaxed order of α-satsfactory degrees conforms to the basc preemptve prorty order It s noted that γ=0s a specal case where prorty of all the objectves wll be the same On the contrary the prorty requrement wll be volated f γ > 0By() prorty dfference can be quantfed 32 Enhanced Two-Step Method 32 The Frst Step In MOO problem optmzaton and prorty requrements need to be balanced Therefore the two-step satsfactory method [25] sntroducedhowever fuzzy parameters and three types of fuzzy relatons are consdered n ths paper Ths s sgnfcantly dfferent from [25] whch only focuses on fuzzy goals wth the relaton of fuzzy mnmzaton Therefore the orgnal two-step method needs a great mprovement In ths paper by means of α-level set of fuzzy parameters (3) and the relaxed order of α-satsfactory degrees () α-fmoo problem wth preemptve prorty (9) s dvded nto two models whch consst of the prelmnary optmzaton model and the prorty model They wll be solved teratvely The purpose of the prelmnary optmzaton model s to optmze all the objectves smultaneously as much as possble regardless of prorty Thus the frst step s to construct and solve the prelmnary optmzaton model Based on max-mn decson the prelmnary optmzaton model s presented n the followng: max λ st μ f (x a ) λ =k μ f (x a ) x G(b) =x g j (x b j ) 0j=m} (a b) (ã b) α (2) where λ s called α-overall satsfactory degree It s equal to the mnmum α-satsfactory degree of all the objectves when preference s not consdered Accordng to max-mn decson the optmal soluton of (2) λ s the maxmum value of α-overall satsfactory degree whch denotes that all the objectves are optmzed fully Therefore t s called α-maxmum overall satsfactory degree and t wll be treated as the gven condton of the next step optmzaton 322 The Second Step On bass of the frst step we have to consder preemptve prorty requrement Consequently the prorty model s constructed to balance optmzaton and the prorty order n the second step After solvng (2) the feasble feld of orgnal α-fmoo has become very small In order to acqure more potental solutons satsfyng prorty requrement t s necessary to enlarge the decreased feasble feld by the followng equatons: μ f (x a ) λ Δδ =k (3) where Δδ (Δδ 0) s slack parameter by whch the αmaxmum overall satsfactory degree λ canbedecreased Ths means that there wll be more solutons f the constrant (3) s relaxed Then DM can choose the most satsfactory soluton from the new feld For realzng preemptve prorty requrement the relaxed order of α-satsfactory degrees () stakenasanew constrant Combnng the orgnal system constrants of (4) the prorty model s constructed as follows: mn γ st μ f (x a ) λ Δδ =k μ fs (x a s ) μ f s (x a s ) γ s s k} s=s γ x G(b) =x g j (x b j ) 0j=m} (a b) (ã b) α (4) In (4) the prorty varable γ s mnmzed to enlarge prorty dfference Durng solvng the slack parameter Δδ s regulated n order to acqure the satsfactory soluton of α-fmoo In addton to Δδ the feasble feld can also be ncreased by decreasng α These ways provde more regulatng freedoms than the orgnal two-step approach n [25] Therefore the preferred soluton can be found by regulatng Δδ or changng α If γ >0 the soluton does not conform to prorty λ needs to be decreased through ncreasng Δδ tll γ 0and DM s satsfed where Δδ s determned by the nteracton between DM and the analyzer If the result stll remans unsatsfactory after regulatng Δδwecandecreaseα 323 Maxmum Stable Relaxaton From (3) we know that the feasblty s enlarged contnually by ncreasng Δδ such that the smaller γ canbeobtanedwhenδδ s ncreased to equal Δδ (3) wll become nactve constrant Ths means γ wll reman dentcal That s prorty dfference wll not

6 6 Mathematcal Problems n Engneerng change when Δδ > Δδ ThusΔδ s called maxmum stable relaxaton μ f = μ f μ f k } sthevectorofalltheαsatsfactory degrees about the optmal soluton Δδ can be acqured by the followng algorthm Step Let Δδ = λ and assgn a value to α andthensolve model (4)toobtantheoptmalsolutonx μ f and γ Step 2 Compare the elements of μ f fnd the mnmum one as μ f m andletδδ =λ μ f m Δδ s taken as a pror knowledge by whch the case of arbtrary enlargng of the feasble feld can be avoded 33 M-α-Pareto Optmalty Test From Secton 32weknow that the prelmnary optmzaton model of the frst step s max-mn decson on -norm and the α-maxmum overall satsfactory degree n the prorty model of the second step s relaxed It cannot be guaranteed that each objectve under preemptve prorty requrement s optmzed as much as possble fnally although the optmal soluton satsfes the prorty order Accordng to Defnton the expected satsfactory soluton must be M-α-Pareto optmal Thus the followng M-α-Pareto optmalty test model s proposed By means of the model the M-α-Pareto optmalty of the solutoncanbeensured: max k ε st μ f (x a ) ε =μ f (x a ) =k If at least one ε h s not zero then x must be the M-α- Pareto optmal soluton Or else there wll exst an M-α- Pareto optmal soluton xand t must be the optmalsoluton of (5) Ths s contrary to the fact that x s the optmal soluton Therefore x must be M-α-Pareto optmal 34 Algorthm The specfc algorthm of enhanced two-step method s summarzed as follows Step (ntalzaton) Formulate the α-level set of fuzzy parameters and calculate the ndvdual f mn and f max of the objectve functon f (x ã ) (=k)under the gven constrants respectvely when α=0and α= Step 2 Determne the desrable targets and the tolerances construct the α-membershp functons of all the objectves andaskdmtoselectthentalvalueofα Step 3 Solve the prelmnary optmzaton model (2)ofthe frst step and get the α-maxmum overall satsfactory degree λ Step 4 Formulate and solve the prorty model (4) ofthe second step accordng to preemptve prorty requrement and test M-Pareto optmalty of the soluton by (5) Step 5 Judge: f γ > 0 go to next step If γ 0butthe soluton s not satsfactory also go to Step 6 Otherwsestop optmzaton and the satsfactory soluton s acqured Step 6 Relax λ by ncreasng Δδ f Δδ Δδ andgoback to Step 4Ordecreaseα and go back to Step 3 μ fs (x a s ) μ f s (x a s ) γ s s k} s=s (5) 35 Analyss of Feasblty By the followng Theorem t s proven that there must be the feasble soluton n our approach x G(b) =x g j (x b j ) 0j=m} (a b) (ã b) α ε 0 where ε ( = k) are the error varables and (x γ ) s the optmzaton soluton of (4) Let x be the optmal soluton of (5) The theorem about M-α-Pareto optmalty test s gven Theorem 3 If ε (=k)are zero then x s M-α-Pareto optmal Otherwse f at least one ε h h k}s not zero x s the M-α-Pareto optmal soluton Proof If x s not M-α-Pareto optmal when all ε are zero then the optmal soluton x must be M-α-Pareto optmal Because μ f (x a ) μ f (x a ) ( = k)must hold and μ fh (x a )>μ fh (x a ) for at least one h h k}ths means there must be one nonzero ε h at least whch s contrary to the fact that all ε are zero So x s M-α-Pareto optmal f ε (=k)are all zero Theorem 4 Forenhancedtwo-stepmethodconsstngofthe prelmnary optmzaton model (2) and the prorty model (4) there must exst an optmal soluton when G( b) s nonempty Proof Frstly the prelmnary optmzaton model (2) of the frst step must be feasble because G( b) s nonempty Secondly for the prorty model (4) of the second step the relaxed order of α-satsfactory degrees () can be rewrtten as (μ fs (x a s ) μ f s (x a s ) γ) = Z where Z s the number of the prorty constrants ncludng actve constrants and nactve constrants Then we have γ=max [(μ (x a fs s ) μ f s (x a s )) ] (6) Thus (4)canbereformulatedas mn max [(μ (x a fs s ) μ f s (x a s )) ] st μ f (x a ) λ Δδ =k max [(μ (x a fs s ) μ f s (x a s )) ]

7 Mathematcal Problems n Engneerng 7 Table : Fuzzy parameters n Example c (c c 2 c 3 c 4 ) a ( 2 0) a 2 (2334) a 2 ( 0 0 ) x G(b) =x g j (x b j ) 0j=m} (a b) (ã b) α (7) Therefore the prorty model (4)sequvalentto(7) Snce λ satsfes the constrants of the prelmnary optmzaton model (2) (3) must hold Thus there must be optmal soluton for (7) That s the optmal result also exsts n the model (4) Then both of the frst step and the second step arefeasblesotheoptmalsolutoncanbeobtanedbythe enhanced two-step method when G( b) s nonempty 4 Numercal Examples We demonstrate the effectveness of the proposed optmzaton method by the followng numercal examples Example (see [25]) The example n [25] s consdered where some parameters of the objectves are mprecse: mn f (x ã )= a x +3x 2 + a 2 x 3 mn f 2 (x ã 2 )=(x ) 2 +2(x 2 3) 2 + a 2 (x 3 2) 2 st x +x 2 +x x x 2 x 3 0 (8) ã =(ã ã 2 ) s fuzzy parameter vector where ã =( a a 2 ) and ã 2 =( a 2 ) Ther dstrbutons are gven n Table Preemptve prorty requrement s as follows: ( 20 40) and ( 4503)respectvelyThecorrespondngαmembershp functons are presented n the followng (x a )= (40 a x 3x 2 a 2 x 3 ) 60 (x a 2 )= (03 (x ) 2 2(x 2 3) 2 a 3 (x 3 2) 2 ) 48 (9) (B) The prelmnary model of the frst step s wrtten n the followng expresson: max λ st μ f (x a ) λ μ f (x a ) = 2 (x a )= (40 a x 3x 2 a 2 x 3 ) 60 (x a 2 ) = (03 (x ) 2 2(x 2 3) 2 x +x 2 +x x x 2 x 3 0 a (ã) α a 2 (x 3 2) 2 ) (48) (20) (C) Accordng to the gven prorty the relaxed order of α-satsfactory degrees s constructed: (x a ) (x a 2 ) γ (2) Consequently the second step model s formulated as level : f 2 (x ã 2 ) level 2: f (x ã ) Frstly optmzaton smulaton s mplemented for the gven prorty and varous results wth regard to dfferent Δδ and α are presented Then maxmum stable relaxaton Δδ s computed Fnally senstvty analyss s consdered by changng the prorty order to valdate the senstveness of our approach ()OptmzatonforGvenPrortyOn bass of the proposed method Example s solved step by step (A) The membershp functons of ã are formulated by (2) accordngtotable and the correspondng constrants about α-level set are constructed as a (ã) α usng (3) The four optmum values of each objectve functon are computed when α = 0 and α = Thus the aspraton values and the tolerant lmts of the two objectves are determned as mn γ st μ f (x a ) λ Δδ = 2 (x a ) (x a 2 ) γ (x a )= (40 a x 3x 2 a 2 x 3 ) 60 (x a 2 ) = (03 (x ) 2 2(x 2 3) 2 x +x 2 +x x x 2 x 3 0 γ a (ã) α a 2 (x 3 2) 2 ) (48) (22)

8 8 Mathematcal Problems n Engneerng Table 2: Optmzaton results of Example for gven prorty α λ Δδ γ f (x ã ) f 2 (x ã 2 ) (x a ) (x a 2 ) (D) The M-Pareto optmalty test model s wrtten n the followng max ε +ε 2 st μ f (x a ) ε =μ f (x a ) =2 (x a ) (x a 2 ) γ (x a )= (40 a x 3x 2 a 2 x 3 ) 60 (x a 2 ) = (03 (x ) 2 2(x 2 3) 2 x +x 2 +x x x 2 x 3 0 ε 0 a (ã) α a 2 (x 3 2) 2 ) (48) (23) Accordng to the algorthm the above models are solved teratvely The correspondng optmzaton results are gven ntable 2 Comparng the results lsted n Table 2tsknownthat the values of γ arelessthan0andthechangngofγ conforms to that of Δδ Thsmeansthatthepreemptve prorty requrement s reasonable and all the results satsfy t Then DM can choose the preferred soluton from them accordng to hs requrement (2) Computaton of Maxmum Stable Relaxaton Δδ From Secton 323weknowthatthesolutonwllkeepunchanged when Δδ > Δδ Therefore there must exst maxmum stable relaxaton Δδ durng solvng Example Fordfferentα Δδ s correspondngly dfferent For example when α = 085 and Δδ = λ = μ f = ( ) are obtaned by solvng (4) where μ f s mnmum Therefore Δδ = whch means that the result wll reman unchanged when Δδ > (3) Senstvty Analyss For demonstratng the senstveness of the proposed method senstvty analyss s constructed by changng the orgnal prorty order nto the new one f (x ã ) s hgher than f 2 (x ã 2 ) Then the abovemodelsareupdatedusngthenewprortyconstrant (x a 2 ) (x a ) γ They are solved teratvely The correspondngoptmzatonresultsarepresentedntable 3 From Table 3t s seen that the order of α-satsfactory degrees changes wth alternaton of prorty requrement That denotes that our method s senstve to the prorty requrement Example 2 (see [26 27]) In order to verfy the effectveness of our method further the example wth more objectves from [26 27]susedItsfuzzyparametersconsstntheobjectves and the constrants whch makes t more complcated than Example : mn f (x ã )=(x +5) 2 + a x (x 3 a 2 ) 2 mn f 2 (x ã 2 )= a 2 (x 45) 2 mn f 3 (x ã 3 )= a 3 (x + 20) 2 +(x 2 + 5) 2 +3(x 3 + a 22 ) 2 + a 32 (x 2 45) 2 +(x 3 + 5) 2 st g(x b )= b x 2 + b 2 x b 3 x x x 2 x 3 0 (24) ã = (ã ã 2 ã 3 ) and b are fuzzy parameter vectors and ã = ( a a 2 ) ã 2 = ( a 2 a 22 ) ã 3 = ( a 3 a 32 )and b = ( b b 2 b 3 ) Ther characterstcs are presented n Table 4 Preemptve prorty requrement s as follows: level : f (x ã ) and f 3 (x ã 3 ) level 2: f 2 (x ã 2 ) (A) Frstly the fuzzy parameters ã and b are formulated nto membershp functons about α-level set by (3) Then the aspraton values and the tolerant lmts of the three objectves are determned as ( ) ( ) and( )respectvelybymeansofthemnmum and maxmum values at α = 0 and α = Therαmembershp functons are defned as

9 Mathematcal Problems n Engneerng 9 Table 3: Optmzaton results ofexample for senstvty analyss α Δδ γ f (x ã ) f 2 (x ã 2 ) (x a ) (x a 2 ) Table 4: Fuzzy parameters n Example 2 c (c c 2 c 3 c 4 ) c (c c 2 c 3 c 4 ) a ( ) a 32 ( ) a 2 ( ) b (09) a 2 (852222) b2 (08 2) a 22 ( ) b3 (085 5) a 3 (293335) (C) For prorty requrement the relaxed formulaton s wrtten as (x a 2 ) (x a ) γ (x a 2 ) μ f3 (x a 3 ) γ The second step model s formulated as (27) mn γ (x a )= (59324 (x +5) 2 a x 2 2 2(x 3 a 2 ) 2 ) (x a 2 ) = (79974 a 2 (x 45) 2 (x 2 + 5) 2 3(x 3 +a 22 ) 2 ) (4524) μ f3 (x a 3 ) = (4008 a 3 (x + 20) 2 a 32 (x 2 45) 2 st μ f (x a ) λ Δδ = 2 3 (x a 2 ) (x a ) γ (x a 2 ) μ f3 (x a 3 ) γ (x a ) = (59324 (x +5) 2 a x 2 2 2(x 3 a 2 ) 2 ) (29429) (x 3 + 5) 2 ) (68655) (B) The prelmnary model of the frst step s max λ st μ f (x a ) λ μ f (x a ) = 2 3 (25) (x a 2 ) = (79974 a 2 (x 45) 2 (x 2 + 5) 2 3(x 3 +a 22 ) 2 ) (4524) μ f3 (x a 3 ) = (4008 a 3 (x + 20) 2 a 32 (x 2 45) 2 (x 3 + 5) 2 ) (68655) (x a ) = (59324 (x +5) 2 a x 2 2 2(x 3 a 2 ) 2 ) (29429) (x a 2 ) = (79974 a 2 (x 45) 2 (x 2 + 5) 2 3(x 3 +a 22 ) 2 ) (4524) b x 2 +b 2x 2 2 +b 3x x x 2 x 3 0 γ (a b ) (ã b ) α (28) μ f3 (x a 3 ) = (4008 a 3 (x + 20) 2 a 32 (x 2 45) 2 (x 3 + 5) 2 ) (68655) (D) The correspondng M-Pareto optmalty test model s gven n the followng b x 2 +b 2x 2 2 +b 3x x x 2 x 3 0 (a b ) (ã b ) α (26) max ε +ε 2 +ε 3 st μ f (x a ) ε =μ f (x a ) = 2 3 (x a 2 ) (x a ) γ (x a 2 ) μ f3 (x a 3 ) γ

10 0 Mathematcal Problems n Engneerng Table 5: Optmzaton results of Example 2 α λ Δδ γ f (x ã ) f 2 (x ã 2 ) f 3 (x ã 3 ) (x a ) (x a 2 ) μ f3 (x a 3 ) (x a ) = (59324 (x +5) 2 a x 2 2 2(x 3 a 2 ) 2 ) (29429) (x a 2 ) = (79974 a 2 (x 45) 2 (x 2 + 5) 2 3(x 3 +a 22 ) 2 ) (4524) μ f3 (x a 3 ) = (4008 a 3 (x + 20) 2 a 32 (x 2 45) 2 (x 3 + 5) 2 ) (68655) b x 2 +b 2x 2 2 +b 3x x x 2 x 3 0 ε 0 (a b ) (ã b ) α (29) When α = 09 and α = 08 the models n (B) (C) and (D) are solved teratvely The optmzaton results are shown n Table 5 From Table 5 we know that all the results satsfy the preemptveprortyrequrementdmcanchoosethepreferred soluton from them For example when α = 08 and Δδ = 02 thesatsfactorydegreesare and07049andthe values of the objectves are and 9682 Ths may be regarded as the most reasonable result Remark 5 The prorty varable γ wll decrease wth the ncrement of Δδ How to change Δδ can be determned accordng to the real optmzaton problem Usually t s proper to change Δδ about 0 every tme 5 Concluson Ths paper addresses the problem of FMOO that resulted from fuzzy parameters Although the approach n [25] can work wth preemptve prorty requrement t only focuses on fuzzy goals but not fuzzy parameters Therefore the orgnal two-step method s greatly mproved for fuzzy parameters andprortyinthsworktheconceptoffuzzynumber and α-level set theory are ntroduced to adapt to fuzzy parameters Only nequalty fuzzy relatons are consdered n [25] However the enhanced method s extended to three relatons nvolvng fuzzy mnmzaton fuzzy maxmzaton and fuzzy equal Based on the prevous approach [25] the strct preemptve prorty structure s modeled by the relaxed order of α-satsfactory degrees Moreover there s the possblty of exstence of the weak M-Pareto optmal soluton n the orgnal approach Nevertheless the M-Pareto optmalty can be guaranteed by ths work The examples show that regulatng Δδ and α provdes more optmzaton freedoms than the prevous work durng solvng the two-step models teratvely Thus balance between optmzaton and prortycanberealzedwhenfuzzyparametersarenvolved However we stll need to note that the results of some FMOO problems may be so nsenstve to the change of Δδ or α that t s dffcult to seek the satsfactory soluton Therefore the senstvty research on regulatng Δδ or α wll be studed n the future Conflct of Interests The authors declare that there s no conflct of nterests regardng the publcaton of ths paper Acknowledgments Ths work was supported by the Natonal Nature Scence Foundaton of Chna under Grant nos and Nature Scence Foundaton of Tanjn 2JCZDJC30300 and JCYBJC07000 and the Key Laboratory of Advanced Electrcal Engneerng and Energy Technology Tanjn Polytechnc Unversty The authors are grateful to the anonymous revewers for ther helpful comments and constructve suggestons wth regard to ths paper References [] Y J La and C L Hwang Fuzzy Multple Objectve Decson Makng: Methods and Applcatons Sprnger Berln Germany 994 [2] C T Bornsten N Maculan M Pascoal and L L Pnto Multobjectve combnatoral optmzaton problems wth a cost and several bottleneck objectve functons: an algorthm wth reoptmzaton Computers & Operatons Researchvol39 no9pp [3] S-J Wu C-T Wu and J-Y Chang Fuzzy-based selfnteractve multobjectve evoluton optmzaton for reverse engneerng of bologcal networks IEEE Transactons on Fuzzy Systemsvol20no5pp

11 Mathematcal Problems n Engneerng [4] C J Carmona P González M J del Jesus and F Herrera NMEEF-SD: non-domnated multobjectve evolutonary algorthm for extractng fuzzy rules n subgroup dscovery IEEE Transactons on Fuzzy Systems vol 8 no 5 pp [5] O Crespo J E Bergez and F Garca Multobjectve optmzaton subject to uncertanty: applcaton to rrgaton strategy management Computers and Electroncs n Agrculturevol74 no pp [6] A Detz A Agular-Lasserre C Azzaro-Pantel L Pbouleau and S Domenech A fuzzy multobjectve algorthm for multproduct batch plant: applcaton to proten producton Computers and Chemcal Engneerng vol 32 no -2 pp [7] A Adeyefa and M Luhandjula Multobjectve stochastc lnear programmng: an overvew Amercan Operatons Research vol no 4 pp [8] F B Abdelazz and H Masr A compromse soluton for the multobjectve stochastc lnear programmng under partal uncertanty EuropeanJournalofOperatonalResearchvol202 no pp [9] M K Luhandjula and M J Rangoaga An approach for solvng a fuzzy multobjectve programmng problem European Journal of Operatonal Researchvol232no2pp [0] R E Bellman and L A Zadeh Decson-makng n a fuzzy envronment Management Scence vol 7 no 4 pp [] H-J Zmmermann Fuzzy programmng and lnear programmng wth several objectve functons Fuzzy Sets and Systems vol no pp [2]HTanakaTOkudaandKAsa Onfuzzymathematcal programmng Cybernetcs vol3no4pp [3] Y Gao G Q Zhang J Ma and J Lu A bmλ-cut and goal-programmng-based algorthm for fuzzy-lnear multpleobjectve blevel optmzaton IEEE Transactons on Fuzzy Systemsvol8nopp 3200 [4] S L and C Hu An nteractve satsfyng method based on alternatve tolerance for multple objectve optmzaton wth fuzzy parameters IEEE Transactons on Fuzzy Systems vol 6 no 5 pp [5] R T Marler and J S Arora Survey of mult-objectve optmzaton methods for engneerng Structural and Multdscplnary Optmzatonvol26no6pp [6] C-C Ln A weghted max-mn model for fuzzy goal programmng Fuzzy Sets and Systems vol 42 no 3 pp [7] S L and C Hu Satsfyng optmzaton method based on goal programmng for fuzzy multple objectve optmzaton problem EuropeanJournalofOperatonalResearchvol97no 2 pp [8] J-B Yang Mnmax reference pont approach and ts applcaton for multobjectve optmzaton European Operatonal Research vol 26 no 3 pp [9] R N Twar S Dharmar and J R Rao Fuzzy goal programmng an addtve model Fuzzy Sets and Systemsvol 24nopp [20] B B Pal and B N Motra A goal programmng procedure for solvng problems wth multple fuzzy goals usng dynamc programmng European Operatonal Research vol 44no3pp [2] L-H Chen and F-C Tsa Fuzzy goal programmng wth dfferent mportance and prortes European Operatonal Researchvol33no3pp [22] O Aköz and D Petrovc A fuzzy goal programmng method wth mprecse goal herarchy European Operatonal Researchvol8no3pp [23] C Hu and S L Enhanced nteractve satsfyng optmzaton approach to multple objectve optmzaton wth preemptve prortes Internatonal Informaton Technology & Decson Makngvol5nopp [24] S L Y Yang and C Teng Fuzzy goal programmng wth multple prortes va generalzed varyng-doman optmzaton method IEEE Transactons on Fuzzy Systems vol2no 5pp [25] S L and C Hu Two-step nteractve satsfactory method for fuzzy multple objectve optmzaton wth preemptve prortes IEEE Transactons on Fuzzy Systems vol 5 no 3 pp [26] C Mohan and H T Nguyen Reference drecton nteractve method for solvng multobjectve fuzzy programmng problems European Operatonal Researchvol07no3 pp [27] M Sakawa and H Yano An nteractve fuzzy satsfcng method for multobjectve nonlnear programmng problems wth fuzzy parameters Fuzzy Sets and Systems vol 30 no 3 pp [28] M Sakawa H Yano and T Yumne An nteractve fuzzy satsfcng method for multobjectve lnear-programmng problems and ts applcaton IEEE Transactons on Systems Man and Cybernetcsvol7no4pp

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