Solving fuzzy linear programming problems with piecewise linear membership functions by the determination of a crisp maximizing decision

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1 Frs Jo Cogress o Fuzzy ad Iellge Sysems Ferdows Uversy of Mashhad Ira 9-3 Aug 7 Iellge Sysems Scefc Socey of Ira Solvg fuzzy lear programmg problems wh pecewse lear membershp fucos by he deermao of a crsp mamzg decso S. Effa ad H. Abbasya e-mal:effa9@yahoo.com e-mal:abbasya58@yahoo.com Absrac: I hs paper we cocerae o lear programmg problems whch boh he rgh-had sde ad he echologcal coeffces are fuzzy umbers.we cosder here oly he case of fuzzy umbers wh lear membershp fuco. The deermao of a crsp mamzg decso [] s used for a defuzzfcao of hese problems. The crsp problems obaed afer he defuzzfcao are o-lear ad o-cove geeral. We propose here he augmeed lagraga pealy fuco mehod ad use for solvg hese problems. We also compare he ew proposed mehod wh well ow fuzzy decsve se mehod. Fally we gve llusrave eample ad hs solve by he ew proposed mehod ad compare he umercal soluo wh he soluo obaed from fuzzy decsve se mehod. Keywords: Fuzzy lear programmg fuzzy umber augmeed lagraga pealy fuco mehod fuzzy decsve se mehod. Iroduco A model whch he obecve fuco s crsp ha s has o be mamzed or mmzed ad whch he cosras are or parally fuzzy s o loger symmercal. Fuzzy lear programmg problem wh fuzzy coeffces was formulaed by Negoa [3] ad called robus programmg. Dubos ad Prade [] vesgaed lear fuzzy cosras. Taaa ad Asa [5] also proposed a formulao of fuzzy lear programmg wh fuzzy cosras ad gave a mehod for s soluo whch bases o equaly relao bewee fuzzy umbers. We cosder lear programmg problems whch boh echologcal coeffces ad rghhad-sde umbers are fuzzy umbers. Each problem s frs covered o a equvale crsp problem. Ths s a problem of fdg a po whch sasfes he cosras ad he goal wh he mamum degree. The crsp problems obaed by such a maer ca be o-lear (eve ocove where he o-leary arses cosras. For solvg hese problems we use ad compare wo mehods. Oe of hem called he fuzzy decsve se mehod as roduced by Saawa ad Yaa [7]. I hs mehoda combao wh he bseco mehod ad phase oe of he smple mehod of lear programmg s used o obaed a feasble soluo. The secod mehod we use s he augmeed lagraga pealy mehod. I hs mehod a combes he algorhmc aspecs of boh Lagraga dualy mehods ad pealy fuco mehods. For hs d of problems we cosder hs mehod s appled o solve cocree eamples. The paper s ouled as follows. I seco we sudy he lear programmg problem whch boh echologcal coeffces ad rgh-had-sde are fuzzy umbers. The geeral prcples of he augmeed lagraga pealy mehod s preseed seco 3. I seco we eame he applcao of hs mehod ad fuzzy decsve se mehod o cocree eample. 386

2 . Lear Programmg Problems wh fuzzy echologcal coeffces ad fuzzy rgh-had-sde umbers We cosder a lear programmg problem wh fuzzy echologcal coeffces ad fuzzy rghhad-sde umbers: Mamze c Subec o a b m ( where a leas oe > ad a ad b are fuzzy umbers wh he followg lear membershp fucos: < a a + d μ a ( a a + d d a + d Where ad d > for all K m K. ad < b b + p μ ( b b b + p p b + p Where p > for K m. The fuzzy se of he h cosra C whch s a subse of s defed by b a μc ( + d p b b < a ( a + d a b < ( a + d + p + p ( The problem we fac s he deermao of a crsp fuco over a fuzzy doma. The approaches are cocevable:. he deermao of he fuzzy se "decso".. he deermao of a crsp "mamzg decso" by aggregag he obecve fuco afer approprae wh he cosras. -. The deermao of a fuzzy se "decso" Orlovs [977] suggess compug for all α - level ses of he soluo space he correspodg opmal values of he obecve fuco ad cosderg as he fuzzy se decso he opmal values of he obecve fucos wh he degree of membershp equal o he correspodg α -level of he soluo space. Defo. [Werers 98] μ α be he α -level { } Le ( α C ses of he soluo space ad N α f sup f y he se { } ( ( ( α y α of opmal soluos for each he se of opmal soluos for each α -level se where f s a obecve fuco. The fuzzy se "decso" s he defed by he membershp fuco sup ( α N α α > N μop ( ( α oherwse For he case of lear programmg he μ ca be obaed by deermao of he ( op paramerc programmg [Chaas 983]. For each α a LP of he followg d would have o be solved: Mamze c Subec o α ( K m (3 μ C X. I he followg we shall cosder a approach ha suggess a crsp soluo depede o he soluo space. -. The deermao of a crsp mamzg decso We shall prese a model ha s parcularly suable for he ype of lear programmg model 386

3 we are cosderg here. Werers [98] suggess he followg defo. Defo. le f : be he obecve fuco a fuzzy rego (soluo space ad S he suppor of hs rego.e. S { μ C ( > }. The mamzg se over he fuzzy rego M ( f s he defed by s membershp fuco ( f ( f ( f ( S μ ( M f sup ( f ( f ( f ( S S f f f f sup ( ( ( S S S f ( < f ( f ( < sup f ( S f ( For defuzzfcao of he problem ( we frs calculae f f c ad S f sup c for whch c S c. I fac f ad f are he lower ad upper bouds of he opmal values respecvely. The opmal values f ad f ca be defed by solvg he followg sadard lear programmg problems for whch we assume ha all hey have he fe opmal values. f Mamze a c Subec o ( + d b m (5 f Mamze c Subec o ( a + d b + p m (6 The obecve fuco aes values bewee f ad f whle echologcal coeffces ae values bewee a ad a + d ad he rgh-had sde umbers ae values bewee b ad b + p. The erseco of hs mamzg se wh he fuzzy se decso could he be used o compue a mamzg decso as he soluo wh he hghes degree of membershp hs fuzzy se (see []. Hece we have he followg lemma: Lemma. s a crsp opmal soluo of he problem ( f ad oly f μ (. op ( μ M ( f If we forgo alerave opmal soluos of he problem (3 accordg o above lemma we have: c f c S α. sup c c S f S By addg hs he cosra o (3 ad α [] we have: Mamze c Subec o c α ( f f f μc ( α m (7 α. By usg ( he problem (6 ca be wre as Mamze c c α ( f f f Subec o ( a + αd + αp b m (8 α. Noce ha he cosras problem (8 coag he cross produc erms α are o cove. Therefore he soluo of hs problem requres he specal approach adoped for solvg geeral o cove opmzao problems. 3. The Augmeed Lagraga Pealy Fuco Mehod The approach used s o cover he problem o a equvale ucosraed problem. Ths mehod s called he pealy or he eeror pealy fuco mehod whcha pealy erm s added o he obecve fuco for ay volao of he cosras. Ths mehod geeraes a sequece of feasble pos hece s ame whose lm s a opmal soluo o he orgal problem. The cosras are placed o he obecve fuco va a pealy parameer a way ha pealzes ay volao of he cosras. I hs seco we prese ad prove a mpora resul ha usfes usg eeror pealy fucos as a meas for solvg cosraed problems. 386

4 Cosder he followg prmal ad pealy problems: Prmal Problem: Mmze c Subec o c α ( f f f ( a + α d + αp b m (9 α α Pealy problem: le ρ be a couous fuco of he form ρ( K α ( ( a + αd + αp b m φ + + ψ φ( ( c α( f f f + φ( α + φ( α ( Where φ ad ψ are couous fucos sasfyg he followg: φ( y f y ad φ( y > f y > ψ ( y f y ad ψ ( y > f y ( 3.. Augmeed Lagraga Pealy Fucos Augmeed lagraga pealy fucos for he problem (9 s as: F α u v c + AL Where ( m+ + m+ + u μ ma g ( α + μ u + vh( α + ηh ( α μ ( u ad v are are lagrage mulpler. The followg resul provdes he bass by vrue of whch he AL pealy fuco ca be classfed as a eac pealy fuco. Algorhm The mehod of mulplers s a approach for solvg olear programmg problems by usg he augmeed lagraga pealy fuco a maer ha combes he algorhmc aspecs of boh Lagraga dualy mehods ad pealy fuco mehods. Ialzao Sep: Selec some al Lagraga mulplers u u ad v v ad posve values μ for each K m + + ad η for he pealy parameers. Le ( α be a ull vecor ad deoed VIOL ( α where for ay ad α VIOL( α ma{ h( α g ( α I { : g ( α > } s a measure of cosra volaos. Pu ad proceed o he "er loop" of he algorhm. Ier Loop: (pealy fuco mmzao Solve he ucosraed problem o Mmze F AL ( α u v ad le ( α deoe he opmal soluo obaed. If VIOL( α he sop wh ( α as a KKT po (Praccally oe would ermae f VIOL( α s lesser ha some olerace ε >. Oherwse f VIOL( α VIOL( α proceed o he ouer loop. O he oher had f VIOL( α > VIOL( α he for each cosra K m + + for whch g ( ( α > VIOL α ad h( α > ( VIOL α replace he correspodg pealy parameer μ by μ ad η η respecvely ad repea hs er loop sep. Ouer Loop: (Lagrage Mulpler Updae eplace u by u ew where for each K m + + ( uew u + ma{ μ g ( α u } Ad also v by v ew where ( vew v + μh( α. Icreme by ad reur o he er loop.. Numercal Eample Solve he opmzao problem Mamze + 3 Subec o (3 386

5 Whch ae fuzzy parameers as L( L( 3 3 L( 3 ad L( 3 by shaocheg [6]. Tha s as used ( a ( d 3 ( a + d 5. For eample L( s as: < + μ ( + +. For solvg hs problem we mus solve he followg wo subproblems: f Mamze + 3 Subec o Ad f Mamze + 3 Subec o Opmal soluo of hese subproblems are f 6.8 ad 6. By usg hese opmal values problem (3 ca be reduced o he followg equvale o-lear programmg problem: Mamze + 3 Subec o α 3. 6 ( + α + ( + 3α ( 3 + α + ( + 3α 6 ( α Le s solve problem ( by usg he augmeed lagraga pealy fuco mehod.we frs formulae he form g 6 α Selec al Lagraga mulplers ad posve values for he pealy parameers u μ. 6 ad η. v The sarg po s ae as ( ad ε.. sce VIOL ( α 3 > ε we gog o er loop. The augmeed lagraga pealy fuco s as F AL ( α u v [ 3 α ] [( + α + ( + 3α ] + [(3 + α + ( + 3α 6] Wh solvg problem mmze F AL he we obaed ( α ( Ad VIOL ( α > ε ad also VIOL ( α > VIOL( α hece we have μ ew (..... ad η ew. So repea hs er loop sep. 5 6 Sce VIOL ( α < ε so vecor * * ( ( s a soluo o he problem (3 whch has bes membershp grad α * The progress of he algorhm of he mehod of he augmeed lagraga pealy fuco of eample s depced he followg fgure. Mmze 3 Subec o h α g ( + α + ( + 3α g ( 3 + α + ( + 3α 6 g 3 g g 5 α 386

6 Now we solve problem ( by usg he fuzzy decsve se mehod. We oba value of λ a he hry hree erao by usg he fuzzy decsve se mehod. λ Noe ha he opmal value of. foud a he s erao of he augmeed lagraga pealy fuco mehod s appromaely equal o he opmal value of calculaed a he hry hree erao of he fuzzy decsve se mehod. EFEENCES [] M. S. Bazaraa H. D. Sheral ad C. M. Shey No-lear programmg heory ad algorhms Joh Wley ad Sos New Yor ( [] H. J. Zmmerma Fuzzy se heory ad s applcaos 8-3 Kluwer Academc Publshers Boso/Dordrech/Lodo. [3] H. Taaa T. Ouda ad K. Asa O fuzzy mahemacal programmg J. Cyberecs 3 ( [] D. Dubos H. Prade Sysem of lear fuzzy cosras Fuzzy se ad sysems 3 (98 -. [5] H. Taaa K. Asa Fuzzy lear programmg problems wh fuzzy umbers Fuzzy ses ad sysems 3 (98 -. [6] T. Shaocheg Ierval umber ad fuzzy umber lear programmg Fuzzy ses ad sysems 66 ( [7] M. Saawa H. Yaa Ieracve decso mag for mul-obecve lear fracoal programmg problems wh fuzzy parameers Cyberecs Sysems 6 ( Deparme of Mahemacs Teacher Trag Uversy of Sabzevar Sabzevar Ira. 386