THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 10, Number 2/2009, pp

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1 THE PUBLIHING HOUE PROCEEDING OF THE ROMANIAN ACADEMY, eres A, OF THE ROMANIAN ACADEMY Volume 0, Number /009, ON ZALMAI EMIPARAMETRIC DUALITY MODEL FOR MULTIOBJECTIVE FRACTIONAL PROGRAMMING WITH -ET FUNCTION Ioa M. TANCU-MINAIAN, Vasle PREDA, Mrua BELDIMAN, Adreea Mădăla TANCU Isue o Mahemacal ascs ad Aled Mahemacs o he Romaa Academy, Calea 3 eembre r. 3, 0507, Buchares 5, Romaa, E-mal: sacum@csm.ro Uversy o Buchares, Faculy o Mahemacs ad Comuer ceces, r. Academe 4, 0004, Buchares, Romaa New dualy resuls or a semaramerc dualy model are gve or a racoal rogrammg roblem volvg -se ucos. Key words: mulobecve rogrammg, -se uco, dualy, geeralzed covexy.. INTRODUCTION AND PRELIMINARIE We cosder he rame o omzao heory or -se [,5,8]. For ormulag ad rovg varous dualy resuls, we use he class o geeralzed covex -se ucos called (F,b,φ,ρ, θ)-uvex ucos, whch were deed Zalma []. Ul ow, F was assumed o be a sublear uco he hrd argume. I our aroach, we suose ha F s a covex uco he hrd argume, as Preda e al. [7,8] ad Băăorescu e al. []. Le ( X, A, μ ) be a e aomless measure sace wh L ( X, A, μ ) searable, ad le d be he seudomerc o A deed by d( R, ) : = μ ( RΔ) = where R = ( R,, R ), = (,, ) Aⁿ ad Δ sads or symmerc derece. Thus, ( Aⁿ, d) s a seudomerc sace. For h L ( X, A, μ ) ad T A wh dcaor (characersc) uco χt L ( X, A, μ ), he egral hdμ s deoed by h, χ T. Deo.. [4] A uco : A s sad o be dereable a A here exs D ( ) L ( X, A,μ ), called he dervave o a, ad V : A A such ha ( ) = F( ) D ( ), χ χ V (, ) or each A, where V (, ) s od ( (, )), ha s, 0 d(, ) 0 V (, ) lm = 0. d (, ) /

2 Ioa M. TANCU-MINAIAN, Vasle PREDA, Mrua BELDIMAN, Adreea Mădăla TANCU Deo.. [] A uco g : A s sad o have a aral dervave a =(,..., ) Aⁿ wh resec o s -h argume he uco ( ) = g (,,,,,, ) has dervave D( ), = {,,..., }. We dee D g ( ) = D ( ) ad wre D g ( ) = (D g ( ),,D g ( )). Deo.3. [] A uco g: Aⁿ R s sad o be dereable a here exs Dg( ) ad W g : Aⁿ Aⁿ R such ha χ χ G = G ( ) = G ( ) D G ( ), W(, ), where (, ) WG s od ( (, )) or all Aⁿ. Le be he -dmesoal Eucldea sace ad ( ) s osve orha,.e. = { x = x,, x : x 0, =,, }. For ay vecors x ( x,, x ) ad y ( y,, y ) = =, we u x y x y, or each ={,,...,}; x y x y, wh x y; x < y x < y, or each = {,,...}; x y meas he egao o x y. Clearly, x x 0. I hs aer, we cosder he mulobecve racoal subse rogrammg roblem subec o ( ) ( ) ( ) Φ = g ( ) g ( ) g ( ) (P) m ( ),,, { } h ( ) 0, =,,,, A, where Aⁿ s he -old roduc o he σ-algebra A o subses o a gve se X, : Aⁿ, g : Aⁿ, { } =,,...,, ad h : A, deoed by ⁿ, such ha ( ) 0 ( ) P = { Aⁿ : h 0, } he se o all easble soluos o (P). Deo.4. A easble soluo oher easble soluo P such ha g > or each ad all P. We 0 P s sad o be a ece soluo o (P) here exss o g g g g g g ( ) ( ) ( ) ( ) ( ) ( ),,,,,, ( ) ( ) ( ) ( ) ( ) ( ) I he ollowg we cosder F: Aⁿ Aⁿ R R ad a dereable uco : Aⁿ. The deos below uy he coces o ( F, ρ ) covexy, ( F, ρ ) seudocovexy, ( F, ρ ) uascovexy rom Preda [6] ad uvexy, seudouvexy, uasuvexy rom Mshra [3]. Le b : Aⁿ Aⁿ, θ : Aⁿ Aⁿ Aⁿ Aⁿ such ha θ(, ) (0,0), φ : R R, ad a real umber ρ. Deo.5. [] A uco s sad o be (srcly) ( Fbϕρθ,,,, ) uvex a or each A. ( ) ( > ) ( ) ρ ( θ ( )) ϕ( F F( )) F(, ; b, D F( )) d²,

3 3 Mulobecve rogrammg roblems wh geeralzed V-ye uvexy Deo.6. [] A uco s sad o be (srcly) (F,b,φ,ρ,θ)-seudouvex a ( ) ρ ( θ( )) ϕ ( ) ( > ) F b d (, ;, D ( )) ², ( ( )) 0 or each A,. Deo.7. [] A uco s sad o be (resrcly) (F,b,φ,ρ,θ)-uasuvex a ( ) ( < ) ( ) ρ ( θ ( )) ϕ( ( )) 0 F(, ; b, D ( )) d², or each A. For roblem (P), Zalma [0] gave he ecessary codos or ececy below. Theorem.. Assume ha, g,, ad h,, are dereable a A, ad ha or each here exss ˆ A, such ha ad or each l \ { } we have ( ) D h( ), χ ˆ χ = < 0, h, = g g χ χ ( )D l( ) ( )D l( ), ˆ < 0. I s a ece soluo o (P), he here exss u U = { u : u > 0, u = } ad such ha u g g v h χ χ = = = [ ( )D ( ) ( )D ( )] D ( ), 0 = v >, () or all Aⁿ, vh( ) = 0,. We shall reer o a ece soluo some,, as a ormal ece soluo. ˆ o (P) sasyg he rs wo codos Theorem. or.the DUALITY MODEL AND DUALITY REULT I hs seco we rese a geeral dualy model or (P). Here we use wo aros o he dex ses ad, resecvely. J, J,..., J m a aro o he dex se such I, I,..., I be a aro o he dex se ad { } Le { } 0 ha K={0,,...,} M={0,,...,m}, ad, or xed, u ad v, ad K le he uco Ω ( ;, u, v): Aⁿ be deed by 0 ( Tuv,,, ) u[ ( ) g ( T ) ( T ) g ( ) ] v h ( T ) Ω = I We assocae wh roblem (P) he dual roblem

4 Ioa M. TANCU-MINAIAN, Vasle PREDA, Mrua BELDIMAN, Adreea Mădăla TANCU 4 subec o ( T) ( T) ( T ) max δ,, =,,, g ( T ) g ( T ) g ( T ) ( D ) ( Tuv) F, T; b(, T) u [ G ( T) DF ( T) F ( T) DG ( T) ] v Dh ( T) x 0 A ⁿ = = J ( ) vh T 0, M T A, u U, v I he ollowg we cosder a covex uco F(, T; ) : L ( X, A, μ) Λ v A (, ):, Λ = ( T, v ) J vh( T), M.. ad The resul below esablshes several versos o wea dualy relaed o roblems (P) ad (D). Theorem. (Wea dualy). Le ad (T,u,v) be arbrary easble soluos o (P) ad (D), resecvely, ad assume ha ay oe o he ollowg ses o hyoheses s sased: (a) () Ω (, T, u, v s srcly ) seudouvex a T, ϕ s creasg, ad ϕ ( 0) = 0 or each K ; () ( ) Λ (, v s ) uasuvex a T, ϕ s creasg, ad ϕ ( 0) = 0 or each M\K; ρ () ρ K M\K (b) () Ω (, T, u, v s resrcly ) uasuvex a T, ϕ s creasg, ad ϕ ( 0) = 0 or each K ; () ( ) Λ (, v) s srcly ( Fbϕ,,, ρ, θ ) seudouvex a T, ϕ s crea sg, ad ϕ ( 0) = 0 or each M\K; ρ () ρ K M\K (c) () Ω (, T, u, v s resrcly ) ( Fbϕ,,, ρ, θ ) uasuvex a T, ϕ s creasg, ad ϕ ( 0) = 0 or each K ; () ( ) Λ (, v s ) uasuvex a T, ϕ s creasg, ad ϕ ( 0) = 0 or each M\K; ρ () ρ K M\K (d) () 3 Ω (, T, u, v s srcly ) seudouvex a T or each K, ϕ s creasg, ad ϕ ( 0) = 0 or each K, 3 Ω (, T, u, v ) s resrcly ( Fbϕ,,, ρ, θ ) uasuvex a T or each K, ϕ s creasg, ad ϕ ( 0) = 0 or each K, where { } s a aro o K, wh K,K K, K,, ; = K = K () 3 ( ) Λ (, v s ) uasuvex a T, ϕ s creasg, ad ϕ ( 0) = 0 or each M\K;

5 5 Mulobecve rogrammg roblems wh geeralzed V-ye uvexy ρ () ρ ρ K K M\K (e) () 3 Ω (, T, u, v) s resrcly ( F, b, ϕ, ρ, θ ) uasuvex a T, ϕ s creasg, ad ϕ ( 0) = 0 or each K ; () 3( m ) Λ (, v ) s srcly ( Fb,, ϕ, ρ, θ ) seudouvex a T or each (M \ K), ϕ s creasg, ad ϕ ( 0) = 0 or each (M \ K), 3 ( m ) Λ (, v ) s ( F, b, ϕ, ρ, θ ) uasuvex a T or each (M \ K), ϕ s creasg, ad ϕ ( 0) = 0 or each (M \ K), where {( M\K ),( M \K) } s a aro o M\K, wh ( M\K), m = ( M\K ), ( M\ K ), = ( M\ ) () ρ ρ m m ρ K (M\K) (M\K) 4 (, ) ( F,,,, ) 4 Ω (, T, u, v ) s resrcly (,,,, ) () () Ω T, u, v s srcly or each ( ) K, K ; m bϕ ρ θ seudouvex a T, ϕ s s ϕ 0 = 0 Fbϕ ρ θ uasuvex a T, ϕ s creasg ad, crea g, ad ( ) ϕ 0 = 0 or each K, where { K,K } s a aro o K, wh K, K, = K, = K ; () 4( m ) Λ (, v ) s srcly ( Fb,, ϕ, ρ, θ ) ϕ ( 0) = 0 or each (M \ K ), 4 ( m ) Λ (, v) s ( Fb,,,, ) creasg, ad ϕ ( 0) = 0 or each (M \ K), where { ( ) ( ) ( M\K ), m = ( M\K), ( M\ K ), m = ( M\ K ) ; ρ ρ () ρ ρ m m K K (M\K) (M\K) (v) K or or ( M\K) ρ ρ ρ ρ > K K (M\K) m (M\K) m The Φ / δ T, u, v. ( ) ( ) Theorem.. (rog dualy). Le D ( ), = seudouvex a T, ϕ s cre asg, ad ϕ ρ θ ϕ s M\ K, M\K } s a aro o M\K, wh P be a ormal ece soluo o (P), le F (, ;D ( )) = χ χ or ay dereable uco : Aⁿ ad Aⁿ, ad assume ha ay oe o he ses o hyoheses seced Theorem.. holds or all easble soluos o (D). The here exs u U ad v such ha (, u, v ) s a ece soluo o (D) ad Φ ( ) = δ (, u, v ). Remar.. Usg Theorems. ad., ad ehues rom [5] ad [0], we ca also oba a src coverse dualy resul. For a dealed reseao o hese resuls, he reader s reerred o [9]. 3. CONCLUION We have obaed dualy resuls or a dual model o Zalma [0], relacg he assumo o subleary by ha o covexy. mlar resuls ca be obaed or he oher dual models rom [0]. Also, almos all resuls o hs ye rese he leraure ca be exeded o he case where F s o ecessarly sublear.

6 Ioa M. TANCU-MINAIAN, Vasle PREDA, Mrua BELDIMAN, Adreea Mădăla TANCU 6 ACKNOWLEDGEMENT Ths wor was arally suored by Gra PN II IDEI o. /007. REFERENCE. BĂTĂTORECU, A., PREDA, V., BELDIMAN, M., O hgher order dualy or mulobecve rogrammg volvg geeralzed (F,ρ,γ,b)-covexy, Mah. Re. (Bucur.) 9(59) (007), CORLEY, H. W., Omzao heory or -se ucos, J. Mah. Aal. Al. 7 (987), MIHRA,.K., Dualy or mulle obecve racoal subse rogrammg wh geeralzed (F,ρ,σ,θ)-V-ye-I ucos, J. Global Om. 36 (006)4, MORRI, R. J. T., Omal cosraed seleco o a measurable subse, J. Mah. Aal. Al. 70(979), PREDA, V., O mmax rogrammg roblems coag -se ucos, Omzao (99)4, PREDA, V., O ececy ad dualy or mulobecve rogramms, J. Mah. Aal. Al. 66 (99), PREDA, V., TANCU-MINAIAN, I.M., BELDIMAN, M., TANCU, A., Omaly ad dualy or mulobecve rogrammg wh geeralzed V-ye I uvexy ad relaed -se ucos, Proc. Rom. Acad. er. A Mah. Phys. Tech. c. I. c. 6(005)3, PREDA, V., TANCU-MINAIAN, I.M., BELDIMAN, M., TANCU, A., Geeralzed V-uvexy ye-i or mulobecve rogrammg roblems wh - se uco, o aear J. Global Om., PREDA, V., TANCU-MINAIAN, I.M., BELDIMAN, M., TANCU, A., Dualy or mulobecve racoal rogrammg wh geeralzed (F,b,φ,ρ,θ)-uvex -se ucos, submed. 0. ZALMAI, G. J., Ececy codos ad dualy models or mulobecve racoal subse rogrammg roblems wh geeralzed (F,α,ρ,θ)-V-covex ucos, Com. Mah. Al. 43 (00), ZALMAI, G. J., Geeralzed (F,b,φ,ρ,θ)-uvex -se ucos ad global semaramerc suce ececy codos mulobecve racoal subse rogrammg, I. J. Mah. Mah. c. 6(005), Receved February 3, 009