A New Method for Solving Fuzzy Linear. Programming by Solving Linear Programming

Transcription

1 ppled Matheatcal Sceces Vol 008 o New Method for Solvg Fuzzy Lear Prograg by Solvg Lear Prograg S H Nasser a Departet of Matheatcs Faculty of Basc Sceces Mazadara Uversty Babolsar Ira b The Research Ceter o lgebrac Hyperstructures ad Fuzzy Matheatcs Babolsar Ira bstract Sce ay real-world egeerg systes are too cople to be defed precse ters precso s ofte volved ay egeerg desg process Fuzzy lear prograg probles have a essetal role fuzzy odelg whch ca forulate ucertaty actual evroet Oe of the ost practcable subects recet studes s based o LR fuzzy uber whch was defed ad used by Dubos ad Prade wth soe useful ad easy approato arthetc operators o the We use soe vector coputatos o fuzzy vectors where a fuzzy vector appears as a vector of tragular fuzzy ubers Here our a scope s fdg soe oegatve fuzzy vector whch azes the obectve fucto z c so that b where ad b are a real atr ad a fuzzy vector respectvely ad c s a real vector too Keywords: Fuzzy arthetc Fuzzy lear prograg Fuzzy uber Itroducto Fuzzy set theory has bee appled to ay dscples such as cotrol theory ad aageet sceces atheatcal odelg ad dustral applcatos The cocept of fuzzy lear prograg FLP o geeral level was frst proposed by Taaka et al [6] fterwards ay authors cosdered varous types of the FLP probles ad proposed several approaches for solvg these probles I partcular the ost coveet ethods are based o the cocept of coparso of fuzzy ubers by use of rakg fuctos [5] Usually such

2 7 S H Nasser ethods authors defe a crsp odel whch s equvalet to the FLP proble ad the use optal soluto of the odel as the optal soluto of the FLP proble I [] by usg a geeral lear rakg fucto we troduced a dual sple algorth for solvg lear prograg proble wth fuzzy varables ad ts dual fuzzy uber lear prograg proble drectly I ths paper we cosder a lear prograg proble wth tragular fuzzy ubers Our a cotrbuto here s the establshet of a ew ethod for solvg the FLP probles wthout usg ay rakg fucto Moreover we llustrate our ethod wth a eaple Prelary I ths secto we revew soe ecessary backgrouds of the fuzzy theory whch wll be used ths paper Below we gve deftos ad otatos take fro [] Fuzzy ubers Defto fuzzy uber s a cove oralzed fuzzy set o the real le R such that: There ests at least oe 0 R wth μ 0 μ s pecewse cotuous Let us assue that the ebershp fucto of ay fuzzy uber s as follows: α < α μ + β β 0 otherwse where s the ea value of ad α ad β are left ad rght spreads respectvely ad t s tered as tragular fuzzy uber We show ay tragular fuzzy uber by α β Let FR be the set of all tragular fuzzy ubers Defto fuzzy uber { μ R } s oegatve f ad

3 Method for solvg fuzzy lear prograg 75 oly f μ 0 for all < 0 The a tragular fuzzy uber α β s oegatve f 0 α Defto Two tragular fuzzy ubers α β ad B B B B α β are sad to be equal f ad oly f B B B α α adβ β Defto fuzzy uber α β s called syetrc fα β rthetc o tragular fuzzy ubers B B B Let α β ad B α β be two tragular fuzzy ubers the arthetc o the s defed as []: B B B ddto: B + α + α β + β Scalar ultplcato: For ay scalar λ we have λ λα λβ λ λ α β λ λβ λα B B B Subtracto: B α + β β + α f λ 0 f λ 0 Noegatve atr ad oegatve fuzzy vector Defto5 atr s called oegatve ad deoted by 0 f each eleet of be a oegatve uber Defto6 fuzzy vector b b s called oegatve ad deoted by b 0 f each eleet of b be a oegatve fuzzy that s b 0 Fuzzy Lear Syste of Equatos Defto Cosder the where a ] lear syste as: b [ s a oegatve crsp atr ad b b are R oegatve fuzzy vectors ad b F for all s called a fuzzy lear syste wth oegatve tragular ubers Defto We say a oegatve fuzzy vector s the soluto of b

4 76 S H Nasser where ad b are defed f satsfes syste Now sce F R ad b F R we ay let α β b b b α b β where The we ay rewrte the syste b as: R α β ad b b α β α β α b b b 0 α b β R I other had b ad are two oegatve fuzzy vectors hece by use of Defto ad arthetc o oegatve tragular fuzzy uber t s eough to solve the followg crsp syste: α α β β b b b Note that f we use fro the syetrc tragular fuzzy ubers the last syste β β α α b s ot ecessary to solved because t s equal to syste b Fuzzy Lear Prograg Defto Cosder the followg lear prograg proble: a s t z c b 0 where the coeffcet atr a ] [ ad the vector c c c are a oegatve crsp atr ad vector respectvely ad b b are R oegatve fuzzy vectors such that b F for all s called a fuzzy lear prograg FLP proble Defto We say that a fuzzy vector s a fuzzy feasble soluto of b 0 where ad b are defed f satsfes syste Now sce F R ad b F R we ay let α β b b b α b β where α β R ad b b α b β R The we ay rewrte the syste b as: α β α β b b b 5 I other had b ad are two oegatve fuzzy vectors hece by use of Defto ad arthetc o oegatve tragular fuzzy ubers t s equvalet to the followg crsp syste:

5 Method for solvg fuzzy lear prograg 77 α α b b β β b α 0 6 Now we defe a operator "a" for a fuzzy lear fucto whch s defed as: z f c c c where c are real ubers ad F R Defto Let T ad T b b be two oegatve fuzzy vectors where ad b b b b F R fuzzy vector azes the lear fucto z f such that b 0 b 7 where ad are oegatve real ubers f ad oly f R azes the below real fucto: c > 0 c< 0 z c + + c + 8 such that b b b Here we gve a llustrate eaple Eaple ssue that a copay akes two products Product P has a 0$ per ut proft ad product P has a 0$ per ut proft Each ut of product P requres twce as ay labor hours as each product P The total avalable labor hours are soewhat close to 500 hours per day ad ay possbly be chaged due to specal arrageets for overte work The supply of ateral s alost 00 uts of both products P ad P per day but ay possbly be chaged accordg to prevous eperece The proble s how ay uts of products P ad P should be ade per day to aze the total proft? Let

6 78 S H Nasser ad deote the uber of uts of products P ad P ade oe day respectvely The the proble ca be forulated as the followg lear prograg wth tragular fuzzy varables proble a s t z 0 The supply of ateral ad the avalable labor hours are close to 00 ad 500 ad hece are odeled as 0055 ad respectvely Now the curret fuzzy lear prograg odel ay be wrtte the stadard for as follows: a s t z where ad are two slack varables Hece the equvalet fuzzy lear prograg proble as follows: a s t z Sce the fuzzy ubers are syetrc therefore t s eough to solve the followg fuzzy lear prograg proble: a s t z Now we ca obta a optal fuzzy soluto for proble 0 by solvg the followg lear prograg:

7 Method for solvg fuzzy lear prograg 79 a s t z The optal soluto of the above lear prograg s: Therefore the optal fuzzy soluto of the proble 0 s: * * * * * 000 ad the optal fuzzy value of the obectve fucto s: z * * 5 Cocluso I ths paper we proposed a ew ethod for solvg the FLP probles by solvg the classcal lear prograg probles where we kow how to solve the The sgfcace of ths paper s provdg a ew ethod for solvg the fuzzy lear prograg wthout usg ay rakg fucto Moreover ths paper wll be useful for future works o fuzzy lear prograg ad wll be useful to study o soe fudaetal cocepts of the fuzzy lear prograg ad partcular vestgatg o dualty results ad ore portat o sestvty aalyss Refereces [] L Capos ad JL Verdegay Lear prograg probles ad rakg of fuzzy ubers Fuzzy Sets ad Systes [] D Dubos ad H Prade Fuzzy Sets ad Systes: theory ad applcatos New York cadec Press 980 [] YJ La ad CL Hwag Fuzzy Matheatcal Prograg Methods ad pplcatos Sprger Berl 99

8 80 S H Nasser [] N Mahdav-r ad SH Nasser Dualty results ad a dual sple ethod for lear prograg probles wth trapezodal fuzzy varables Fuzzy Sets ad Systes [5] HR Malek M Tata ad M Mashch Lear prograg wth fuzzy varables Fuzzy Sets ad Systes [6] H Taaka T Okuda ad K sa O fuzzy atheatcal prograg The Joural of Cyberetcs Receved: February 5 008