MULTIOBJECTIVE NONLINEAR FRACTIONAL PROGRAMMING PROBLEMS INVOLVING GENERALIZED d - TYPE-I n -SET FUNCTIONS

Transcription

1 THE PUBLIHING HOUE PROCEEDING OF THE ROMANIAN ACADEMY, eres A OF THE ROMANIAN ACADEMY Volue 8, Nuber /27,.- MULTIOBJECTIVE NONLINEAR FRACTIONAL PROGRAMMING PROBLEM INVOLVING GENERALIZED d - TYPE-I -ET FUNCTION Ioa M. TANCU-MINAIAN The Roaa Acadey, Isttute of Matheatcal tatstcs ad Aled Matheatcs, Calea 3 etebre r. 3, 57, Bucharest 5, Roaa, E-al: stacu_asa@yahoo.co We establsh dualty results uder geeralzed covexty assuto for a ultobectve olear fractoal rograg roble volvg geeralzed d-tye-i -set fuctos. Key words: dualty, ultobectve rograg, fractoal rograg, -set fuctos, geeralzed d- tye-i fuctos.. PRELIMINARIE I ths secto we troduce the otato ad deftos whch wll be used throughout the aer. Let! be the -desoal Eucldea sace ad! ts ostve orthat,.e. { x ( x) x }! = =,, =,,.! for each { 2 } For x = ( x,,x ) ( )!, y = y,, y we ut x y ff x y M =,,, ; x y ff x y for each M, wth x y;x< y ff x<y for each M. We wrte x! ff x. For a arbtrary vector x! ad a subset J of the dex set { 2,,..., }, we deote by x J the vector wth cooets x, J. for Let ( X,Γ, μ) be a fte o-atoc easure sace, ad let d be the seudoetrc o Γ defed by,, T,,T 2 d,t = μ ( T ) = = ( ),T = Γ, where Γ s the -fold roduct of a σ -algebra Γ of subsets of a gve set X, ad deotes the syetrc dfferece. Thus (, d ) s a seudoetrc sace, whch wll serve as the doa for ost of the fuctos that wll be used ths aer. For (,Γ, ) h L X μ, the tegral h dµ wll be deoted by hi, /2 Γ, where I s the dcator (characterstc) fucto of Γ. We ext troduce the oto of dfferetablty for -set fuctos. Ths was orgally troduced by Morrs [4] for set fuctos ad subsequetly exteded by Corley [] to -set fuctos. A fucto ϕ :Γ! s sad to be dfferetable at Γ f there exsts Dϕ( ) L ( X,Γ,µ ), called the dervatve of ϕ at, ad ψ : Γ Γ! such that for each Γ, ( ) D ( ) I ( I ) ϕ = ϕ ϕ, ψ,

2 ψ, s ( ) where ( T) od,, that s Ioa M. TANCU-MINAIAN 2 l ψ, / d, =, ad d s a seudoetrc o Γ [4]. (, ) d A fucto F :Γ!"s sad to have a artal dervatve at arguet,, f the fucto has dervatve ( ) F ϕ =,,,,,, =,, wth resect to ts -th Dϕ ( ), ad we defe DF D ( ). If the DF = DF,, D F. If A fucto F such that :Γ where ψ, s od, for all H :Γ,! H ( H H )! s sad to be dfferetable at DF, all exst, the we ut =,,, we ut DH ( ) DH ( ) f there exst = F = F D F, I I ψ, Γ. DF ad, =. ψ:γ Γ! A vector set fucto f = f,, f :Γ! s dfferetable o Γ f all ts cooet fuctos f,, are dfferetable o Γ. Cosder the ultobectve olear fractoal rograg roble volvg -set fuctos. (P) where = {, 2, ", } ze F F =,,, G G subect to H, M, = (,", ) Γ F F,G, P, ad H, M are dfferetable real valued fuctos defed o Γ wth F ad G >,for all P. The ter ze beg Proble (P) s for fdg effcet, ad wea effcet solutos. Let = { Γ, H } be the set of all feasble solutos to (P), where H = ( H,", H ). A feasble soluto to (P) s sad to be a effcet soluto to roble (P) f there exsts o other F F( ), for all P, wth strct equalty for at least P. feasble soluto to (P) such that A feasble soluto other feasble soluto to (P) such that to (P) s sad to be a wealy effcet soluto to roble (P) f there exsts o F < F( ), for all P. Let ρ,, ρ, ρ,, ρ, ρ, ρ ' be real ubers ad ut ρ = ( ρ,, ρ ) ad ρ = ρ ',, ρ. Also let ' ' ( ) : Γ θ Γ! be a fucto such that θ(, ) for. Alog the les of Jeyauar ad Mod [2] ad uea ad rvastava [7], Preda, tacu-masa ad Koller [5] defed ew classes of -set fuctos, called ( ρ, ρ',d)-tye-i, ( ρρ, ',d )-quas tye-i, ( ρρ, ',d )- seudo tye-i, ( ρρ, ',d )-quas-seudo tye-i, ( ρρ, ',d )-seudo-quas tye-i.

3 3 Multobectve olear fractoal rograg robles volvg geeralzed d-tye-i -set fuctos,β Defto.. [5] We say that :! {} ad F,H s of ( ρρ, ), d - tye- I at Γ Γ \, P, M, such that for all, we have We say that (,H ) s of ( ρρ, ) (2) s a strct equalty. Now, we troduce = Γ f there exst fuctos F F, D F, I I ρ θ,, P (2) = H ( ) β, D H, I I ρ θ,, M. (3), d -sestrctly tye-i at Defto.2. [8] A feasble soluto ˆ Γ such that subect to f the above defto we have ad to (P) s sad to be a regular feasble soluto f there exsts H D H, I I <, M. ˆ = Now, for each = (,", )! we cosder the araetrc roble ( " ) ( P ) F G F G ze,,. H M, = Γ, (,", ) It s well ow that ( P ) s closely related to roble (P). The followg lea s well ow fractoal rograg. Lea.3. A = = " s a effcet soluto to (P) f ad oly f s a effcet soluto to ( ) P wth F( )/ G ( ),,2,,. I ths aer, the roofs of the dualty results for Proble (P) wll voe the followg ecessary otalty codtos (see Zala [8], Theores 3. ad 3.2 ad Corley [], Theore 3.7.)] Theore.4. Let be a regular effcet (or wealy effcet) soluto to (P) ad assue that F, G, P, ad H, M, are dfferetable at! such that = = = u! u = v! ad.the there exst,, = u ( D F ( ) D G ( )) v D H ( ), I I for all Γ, (4) u F( ) G( ), P, (5) vh =, M. (6)

4 Ioa M. TANCU-MINAIAN 4 2. DUALITY I ths secto, the dfferetable case, based o the equvalece of (P) ad P a dual for P s defed ad soe dualty results ( ρρ',,d)-tye-i assutos are stated. Wth P we assocate a dual stated as subect to ( D) axze (, ", ) u D F T D G T, I I v D H T, I I, Γ, (7) = = = = ( ),, u F T G T P (8),. vh T M (9) u!, u =, v!,!. () = Let D be the set of feasble solutos to (D). that Theore 2.. (Wea dualty). Let ( Tuv,,,) be a feasble soluto to roble (D) ad assue ( ) for each P ad,( () (), () ) M F G H s of We also assue that ay of the followg codtos hold: ρ,ρ ', d -tye-i at T. ( 2 ) u > ' u ρ ρ, v for ay P ad for soe P ad M ; (,T ) β (,T) = = ( 3 ) ( F() G (),H() ) s of ( ) uρ (, T) β (, T) = = ' v ρ >. ρρ,, d - sestrctly tye-i at T; The for ay oe caot have F( )/ G( ) for ay P, F( )/ G ( ) for soe P Corollary 2.2. Let u v be feasble solutos to (P ) ad (D), resectvely. If the hyotheses of Theore 2. are satsfed, the s a effcet soluto to (P ) ad (, u, v, ) s a effcet soluto to (D). Theore 2.3. (trog dualty). Let be a regular effcet soluto to (P). The there exst,,! =!!, u, v, s a feasble soluto to (D). ad (,,, ) u u v ad, such that =

5 5 Multobectve olear fractoal rograg robles volvg geeralzed d-tye-i -set fuctos Further, f the codtos of the wea dualty Theore 2. also hold, the (,,, ) effcet soluto to (D). Now we gve a strct coverse dualty theore of Magasara tye [3] for (P ) ad (D). Theore 2.4. (trct coverse dualty). Let ad (,,, ) (D), resectvely. Assue that ( ) ( ) ( ) u F( ) G( ) u F( ) G( ) ; = = for ay P ad M,( F () G (), H ()) s of ( ρ,ρ ', d ) 2 u v s a u v be effcet solutos to (P ) ad - sestrctly tye I at T; ( 3 ) uρ (,T ) β (,T ) = = ' v ρ >. * The =. The roofs wll aear [6]. REFERENCE. CORLEY, H.W., Otzato theory for -set fuctos, J. Math. Aal. Al., 27 (987), JEYAKUMAR, V., MOND, B., O geeralzed covex atheatcal rograg, J. Austral. Math. oc. er. B, 34 (992), MANGAARIAN, O.L., Nolear Prograg, McGraw-Hll, New Yor, MORRI, R.J.T., Otal costraed selecto of a easurable subset, J. Math. Aal. Al., 7 (979) 2, PREDA, V., TANCU-MINAIAN, I.M., KOLLER, E., O otalty ad dualty for ultobectve rograg robles volvg geeralzed d-tye-i ad related -set fuctos. J.Math.Aal.Al. 283(23), TANCU-MINAIAN, I.M., Dualty for ultobectve fractoal rograg robles volvg geeralzed d-tye- I -set (ubtted). 7. UNEJA,.K., RIVATAVA, M.K., Otalty ad dualty odfferetable ultobectve otzato volvg d-tye I ad related fuctos, J.Math.Aal.Al., 26(997)2, ZALMAI, G.J., Otalty codtos ad dualty for ultobectve easurable subset selecto robles, Otzato, 22 (99) 2, Receved February, 27