A Study on Fuzzy Complex Linear. Programming Problems

Transcription

1 It. J. Cotemp. Math. Scieces, Vol. 7, 212, o. 19, A Study o Fuzzy Complex Liear Programmig Problems Youess, E. A. (1) ad Mekawy, I. M. (2) (1) Departmet of Mathematics, Faculty of Sciece Tata Uiversity, Tata, Egypt eyouess1989@yahoo.com (2) Miistry of Educatio A. R. E., Al-Imam Muhammad Ib Sauid Islamic Uiversity, Scietific Istitute i Jouf Jouf, Saudi Arabia Abstract. I this paper we cosider a certai kid of liear programmig problems. This kid, i which fuzzy complex umber is the coefficiet of the objective fuctio, is called fuzzy complex liear programmig problem. Solutios of this kid of problems will be characterized by derivig its fuzzy Kuh-Tucker coditios. The results iclude a duality relatio betwee α -cut programmig ad its dual are discussed. Also, we itroduce the proof of Kuh-Tucker statioarypoit ecessary optimality theorem for our problem. Keywords: Coe, polyhedral coe, fuzzy set, fuzzy umber, α -cut programmig fuzzy complex umber, complex programmig. 1. Itroductio Sice the cocept of fuzzy complex umbers was first itroduced by J. J. Buckley i 1989 [6], may papers were devoted to studyig the problems of the cocept of fuzzy complex umbers. This ew brach subject will be widely applied i fuzzy system theory, especially i fuzzy mathematical programmig, ad will also be widely applied i complex mathematical programmig. It is well kow that, complex programmig, the extesio to complex variables ad fuctios of mathematical programmig was iitiated by N. Leviso i [9], where a duality theory for complex liear programmig is give ad the basic theorem of liear iequalities are exteded to the complex case. Similar results were previously developed by Bellma ad Fa i [1], for Hermitia matrices. This work of Leviso was cotiued by Haso ad Mod i [3], where it is Correspodig Author, imekawy1971@yahoo.com

2 898 Youess, E. A. ad Mekawy, I. M. cosidered as a extesios to quadratic ad o liear programmig. The fuzzy set theory has bee applied to may disciplies such as cotrol theory ad maagemet scieces, mathematical modelig, operatios research ad may idustrial applicatios. The cocept of fuzzy mathematical programmig o geeral level was first proposed by Taaka et al. [4]. I the framework of the fuzzy decisio of Bellma ad Zadeh [12]. Zimmerma [5] first itroduced fuzzy liear programmig as covetioal liear programmig. He cosidered problem with a fuzzy goal ad fuzzy costraits, used liear membership fuctios ad the mi operator as a aggregator for these fuctios, ad assiged a equivalet problem to fuzzy liear programmig.this study focuses o fuzzy complex liear programmig (FCLP) problems. Hece, first some importat cocepts o fuzzy ad complex mathematical programmig are metioed. This paper orgaized i 5 sectios. I sectio 2, we give some ecessary otatios ad defiitios of fuzzy set theory, ad complex mathematical programmig problems. The cocept of fuzzy complex umbers is itroduced i sectio (3). Formulatio of fuzzy complex liear programmig, ad derivig Kuh-Tucker statioary poit problem are preseted i sectio 4. Fially, sectio 5 for coclusio. 2. Notatios ad Defiitios 2.1. Notatios: - C ( R ) the -dimesioal complex (real) vector space over the field m m - C [ R ] the m complex [real] matrices. m - For A = ( ai j) C : A ( a i j ) the cojugate, T A ( a ji ) the traspose, H A T A the cojugate traspose. - For x = ( xi ) C, y C : H ( x, y ) y x the ier product of x ad y, x ( xi ) R the absolute value, x ( x i ) the cojugate, Re x ( Re xi ) R the real part, of x Im x ( Im xi ) R the imagiary part, of x arg x ( arg x i ) the argumet of x. - For a oempty set S C : S { y C : x S Re( y, x ) } the dual (also polar) of S - R+ { x R : xi )( i = 1,..., )} R x y deoted x y R +, for ( x, y ) R. R.

3 Fuzzy complex liear programmig problems For a aalytic fuctio f : C C ad a poit z C f z f ( z ) ( z ), ( i = 1,..., ) the gradiet of f at z. z i 2.2. Defiitios Defiitio [1]. A o empty set S C is: (a) covex if λ 1 λ S + (1 λ) S S, (b) a coe if λ λ S S, k (c) a polyhedral coe if for some positive iteger k ad A C : k k S = A R+ = { A x : x R+ }, i.e. S is geerated by fiitely may vectors (the colums of A). Defiitio [7]. A fuzzy set a% i R is a set of ordered pairs: a% = ( x, μ ( x )) : x R }, { % } a μ % ( x ) is called the membership fuctio of x i a% a μa ( x ) : R [, 1] if μ a% ( x ) = 1, the fuzzy set a% is called ormal. Defiitio [7]. The support of a fuzzy set a% o R is the crisp set of all x R such that μ a% ( x ) >. Defiitio The α - level set (cut set) of a fuzzy set a% is defied as a ordiary set a α for which the degree of its membership fuctio exceeds the level α : aα = { x : μa% ( x ) α}, α [, 1]. Defiitio [4]. A fuzzy set a% o R is covex if μa% ( λx + (1 λ) y ) mi{ μa% ( x ), μa% ( y )}, x, y R, ad λ [,1]. Note that, a fuzzy set is covex if all α -cuts are covex. Defiitio [5]. A fuzzy umber a% is a covex ormalized fuzzy set o the real %, lie R such that: (1) It exists at least oe x R which μ a% ( x ) = 1, (2) μ a % ( x ) is piecewise cotiuous. From the defiitio of a fuzzy umber a%, it is sigificat to ote that, the α -level set a α of a fuzzy umber a% ca be represeted by the closed iterval which depeds o the value of α. Namely, + aα = { x R : μa% ( x ) α} = [ a ( α ), a ( α )], where a ( α ) or + a ( α ) represets the left or right extreme poit of the α -level set a α, respectively 3. Fuzzy Complex Numbers [2, 6, 8, 13] The cocept of fuzzy complex umbers was first itroduced by J. J. Buckley i 1989, may papers were devoted to studyig the problems of the cocept of fuzzy

4 9 Youess, E. A. ad Mekawy, I. M. complex umbers. Ucertaity of complex valued physical quatities c = a+ i b ca be described by complex fuzzy sets, such sets ca be described by membership fuctios μ ( a, b ) which map the uiverse of discourse (complex plae) ito the iterval [,1]. The problem with this descriptio is that it is difficult to directly traslate ito words from atural laguage. To make this traslatio easier, several authors have proposed to use, istead of a sigle membership fuctio for describig the complex umber, several membership fuctio which describe differet real valued characteristics of this umber, such as its real part, its imagiary part, its absolute value, etc. Thus, a atural idea is to represet a complex fuzzy umber by describig two real fuzzy umbers: a% ad b % characterized by the correspodig membership fuctios μ % ( a), μ % ( b ). I this approach, for every complex value a b a+ i b, i.e., for every pair ( a, b ), the degree μ ( a, b ) with which this complex value is possible ca be defied as μ ( c ) = μ ( a, b ) = mi( μ ( a), μ% ( b )) c% c% a%. b 3.1 Prelimiary Cocepts Let R be a set of real umbers, C = { x + i y : x R, y R, i = 1} a field of complex umbers, a fiite closed iterval X = [ X, X + ] is called a closed iterval umber o R, I ( R ) deotes the set of all closed iterval umbers o R. For arbitrary itervals X [ X, X + + = ], Y = [ Y, Y ] R, Z = X + iy ={x + iy C : x X, y Y, i = 1} is called a closed complex iterval umber, I ( C ) = { Z = X + iy : X, Y I ( R ), i = 1} deotes the set of all closed complex iterval umbers o C, = { +,,, } is a biary operatio o I ( C ), (whe = Z 1 Z2, supp Z2 ). + + Defiitio 3.1. For Zk = X k + iy k = [ X k, X k ] + i [ Y k, Y k ] I ( C ), ( k = 1, 2), + + z1 z2 = { z : ( z1, z2) Z1 Z2, z = z1 z2} ad X = [ X, X ], Y = [ Y, Y ] I ( R ). Specifically, for k R + + = [, + ), let k X = [ k X, k X ], let k Z = ( k X ) + i ( ky ) = + + [ kx, kx ] + i[ ky, ky ], let Z k = X k + iy k ( k = 1, 2) I ( C ), Z1 Z2 X1 X2, Y1 Y2. If X, Y are covex sets o R, Z C ad Z = { x + i y C : x X, y Y }, the Z is a covex set o C. Defiitio 3.2. Let C be a field of complex umber, mappig Z% : C [, 1] is called a fuzzy complex set, Z% ( z ) is called the membership fuctio of fuzzy set Z % for z, F ( C ) = { Z% : Z% : C [, 1]} deotes all fuzzy complex sets o C. Defiitio 3.3. Z % = { z = x + i y C : Z % ( z ) = Z % ( x + i y ) α } is called α -cut set of Z %. α

5 Fuzzy complex liear programmig problems 91 Defiitio 3.4. Z% = supp Z% = { z = x + i y C : Z% ( z ) = Z% ( x + i y ) > } is called support set of Z %. Defiitio 3.5. Z % F ( C ) is called a covex fuzzy complex set o C, if ad oly if for α [,1], Z% α is a covex complex set o C. Defiitio 3.6. Z % F ( C ) is a ormal fuzzy complex set o C, if ad oly if { z C : Z% ( z ) = Z% ( x + i y ) = 1} φ. Defiitio 3.7. A ormal covex fuzzy complex set o C is called a fuzzy complex umber Buckley's Membership Fuctio Descriptio. I order to describe a complex umber C = a+ i b, we must describe two real umbers: its real part a ad its imagiary pat b. Thus, a atural idea is to represet a complex fuzzy umber c% = a% + i b %, by describig two real fuzzy umbers: a% ad b % (see [2]) characterized by the correspodig membership fuctios μ a % ( a ) ad μ ( b ) b %. I this approach, for every complex value a+ i b, i.e., for every pair ( a, b ), the degree μc% ( c ) = μc% ( a, b ) with which this complex value is possible ca be defied as μc% ( c ) = μc% ( a, b ) = mi( μa% ( a), μ ( b )) b%. The, for each α [,1], the α -cut for the real part a% is a iterval [ a ( α ), a + ( α )], the α -cut for the imagiary part is also a iterval [ b ( α ), b + ( α )], ad hece, the α -cut for the resultig 2-D membership fuctio μc% ( c ) = mi( μa% ( a), μ ( b )) b%, + + is a rectagular "box" [ a ( α ), a ( α )] [ b ( α ), b ( α )]. The boudary of this box cosists of two straight lie segmets, which are parallel to the a-axis, ad of two straight lie segmets which are parallel to the b-axis. 4. Fuzzy Complex Liear Programmig Cosider the complex liear programmig problem [1, 3, 9, 1] mi Re ( c, x ) ( P1 ) s.t Ax b T, x S, m m where, A C, c C, ad let S C ad T C be closed covex coes.

6 92 Youess, E. A. ad Mekawy, I. M. The fuzzy complex liear programmig ca be formulated as: mi Re ( c%, x ) ( P% ) s.t Ax b T, x S, where c% = a% + i b % is a fuzzy complex umber, μ % ( c ) = mi( μ% ( a), μ % ( b )), for each c a b α [,1], the α -cut for the real part a% is a iterval [ a ( α ), a + ( α )], ad the α -cut for b % is [ b ( α ), b + ( α )], ad hece the α -cut for the resultig 2-D membership fuctio μ ( c ) = μ ( a, b ) = mi( μ ( a), μ % ( b )) c% c% a% is b + + [ a ( α ), a ( α )] [ b ( α ), b ( α )], ad it ca be writte i the followig form Lα = { c : μc% ( c ) α }, where c = a+ i b, μc% ( c) = mi( μa% ( a), μ ( b)) b%, α = ( α1, α2), μa% ( a ) α1 ad μ ( b ) b% α2. The fuzzy complex liear programmig ( P % ) ca be coverted to the determiistic α - cut programmig as mi Re ( c, x ) s. t ( Pα ) Ax b T, x S, c Lα. Choose ĉ L α ad characterize the solutio of ( P α ) correspodig to ĉ, i.e. we will solve the problem mi Re ( cˆ, x ) ( P ) s.t Ax b T, x S, The dual problem of problem ( P ) is max Re ( b, y ) ( D ) s.t H cˆ A y S, y T where S ad T are dual coes of S ad T respectively. Defiitio 4.1. A vector x C is (i) a feasible solutio of ( P ) is A x b T, x S, (ii) a optimal solutio of ( P ) if x is feasible ad Re ( cˆ, x ) = mi{re ( cˆ, x ); x is feasible }. Defiitio 4.2. The problem ( P ) is (i) cosistet if it has feasible solutios, (ii) ubouded if it is cosistet, ad if it has feasible solutios { x k ; k = 1, 2,...} with Re ( cˆ, x ). k

7 Fuzzy complex liear programmig problems 93 Cosistecy ad boudedess of ( D ) ad feasibility ad optimality of its solutios, are similarly defied. Defiitio 4.3. The Lagragia of the problems ( P ) ad ( D ) is L ( x, y ) = Re{( cˆ, x ) ( y, A x b )} H = Re{( b, y ) + ( cˆ A y, x )} Defiitio 4.4. The poit ( x, y ) S T is a saddle poit of L ( x, y ) with respect to S T if L ( x, y ) L ( x, y ) L ( x, y ) for all x S, y T. A duality relatios betwee ( P ) ad ( D ) ad a characterizatio of there optimal solutios, (if such solutios exist) are give i the followig theorem. Theorem 4.1. [6]. Let S ad T i problems ( P ) ad ( D ) be polyhedral coes, the: (a) If oe of the problems is icosistet the, the other is icosistet or ubouded. (b) Let the two problems be cosistet, ad let x be a feasible solutio of ( P ) ad y be a feasible of ( D ). The Re ( cˆ, x ) Re ( b, y ). (c) If both ( P ) ad ( D ) are cosistet, the they have optimal solutios ad their optimal values are equal. (d) Let x ad y be feasible solutios of ( P ) ad ( D ) respectively. The x ad y are optimal if ad oly if H Re{( Ax b, y ) + ( cˆ A y, x )} =, or equivaletly if ad oly if H Re ( Ax b, y ) = ( cˆ A y, x ) =. (e) The vectors x C ad y C m are optimal solutios of ( P ) ad ( D ) respectively if ad oly if the poit ( x, y ) is a saddle poit of L ( x, y ) with respect to S T, i which case L ( x, y ) = Re( cˆ, x ) = Re( b, y ). Now, let us retur to the problem ( P α ), suppose that x is a solutio, the problem becomes: mi Re ( c, x ) s. t Ax b T, x S, c L α, which is equivalet to mi Re ( c, x ) s. t c L α, where Lα = { c : μc% ( c) α}, α = ( α1, α2 ), c% = a% + ib %, μc% ( c) = mi{ μa% ( a), μ ( b)} b%. The problem becomes

8 94 Youess, E. A. ad Mekawy, I. M. mi Re ( c, x ) s. t α μc ( c ), % where μc% ( c ) = mi( μa% ( a), μ ( b )) b%, α = ( α1, α2 ), μa% ( a ) α1, μ ( b ) b% α2. We ca defie the followig problems: 1- The miimizatio problem ( MP ). Fid c L, if it exists, such that α MP Re ( c, x ) = mi Re ( c, x ), c Lα = { c : α μc% ( c ) }. c Lα 2- The local miimizatio problem ( L MP ) Fid a c i Lα, if it exists, such that for some ope ball Bδ ( c ) LMP aroud c with radius δ > c Bδ ( c ) I L α Re( c, x ) Re( c, x ) 3- The Kuh-Tucker statioary-poit problem ( KTP ): Fid, m c Lα u R, if they exist, such that ψ ( c, u ) = Re( c, x ) + uα u μc% ( c ), cψ ( c, u ) = Re c ( c, x ) u c μc% ( c ) =, KTP u ψ ( c, u ) α μc% ( c ), u u ψ ( c, u ) = u [ α μc% ( c )] =, u I the followig, we itroduce some costrait qualificatios, which, we eed it i our study, see [8] 1- Slater's Costrait Qualificatio. Let L α be a covex set i C, ad μ c % ( c ) be cocave o L α. α μ c % ( c ) is said to satisfy slater's costrait qualificatio if there exists a c L α such that α μ c ( c ) < 2- Karli's Costrait Qualificatio. Let L α be a covex set i C, ad μ c % ( c ) is cocave o L α. α μ c % ( c ) is said to satisfy Karli's costrait qualificatio if there exists o P R, P such that Pα P μc% ( c ) for all c L α. 3- The Strict Costrait Qualificatio. Let L α be a covex set i C, ad μ c % ( c ) is cocave o L α. α μ c % ( c ) is said to satisfy strict costrait qualificatio if α μ c% ( c ) is strictly 1 2 covex at distict poits c, c L α. 4- The Kuh-Tucker Costrait Qualificatio. Let L α be a ope set i C, ad μ c % ( c ) defied o L α, α μ c % ( c ) is said to satisfy the Kuh-Tucker costrait qualificatio at c if α μ c% ( c ) is %. L α

9 Fuzzy complex liear programmig problems 95 differetiable at c ad if y C, c μc% ( c ) Rey implies there exists a - dimesioal vector fuctio e defied o the iterval [, 1] such that (a) e () = c, (b) e ( τ ) L α for τ 1, d (c) e is differetiable at τ = ad ( e ()) = λ y for some λ >. d τ 5- The Arrow-Hurwicz- Uzawa Costrait Qualificatio. Let L α be a ope set i C, ad μ c % ( c ) defied o L α, α μ c% ( c ) is said to satisfy the Arrow-Hurwicz-Uzawa costrait qualificatio at c L α if α μ c% ( c ) is differetiable at c ad if c μc % ( c )Rez has a solutio z C. 6- The Reverse Covex Costrait Qualificatio. Let L α be a ope set i C, ad α μ c% ( c ) defied o L α, α μ c% ( c ) is said to satisfy the Arrow-Hurwicz-Uzawa costrait qualificatio at c L α if α μ c% ( c ) is differetiable at c, ad either α μ c% ( c ) is cocave at c or α μ c% ( c ) is liear i R. Theorem 4.2. Let L α be a ope subset of C, let Re ( c, x ) be covex ad μ c % ( c ) be cocave o L α, c L α solve the problem mi Re ( c, x ) s. t. c Lα = { c : α μc% ( c ) }, ad let Re ( c, x ) ad α μ c% ( c ) be differetiable at c ad let α μ c% ( c ) satisfies oe of the costrait qualificatio 1-6. The, there exists a u R such that ( c, u ) satisfies: Re c ( c, x ) u c μc% ( c ) =, ad u [ α μc% ( c )] =. Proof: Let c solve LMP with δ ˆ = δ. If α μ c % ( c ) <, the, for T y C, y y = 1, we have: α μc% ( c + δ y ) = α μc% ( c ) δ[ μc% ( c ) y + β ( c, δ y )] μc% ( c + δ y ) μc% ( c ) = δ[ c μc% ( c ) y + β ( c, δ y )] Sice α μ c % ( c ) < ad lim β ( c, δ y ) =, δ ˆ < δ < δ < δ, α μc% ( c + δ y ) < ad c + δ y L α. But c solves LMP, so Re( c + δ y, x ) Re( c, x ) = δ [Re c ( c, x ) y + β ( c, δ y ) for < δ < ˆ δ. Hece Re c ( c, x ) y + β ( c, δ y ). Sice lim β ( c, δ y ) =, Re c ( c, x ) y. δ

10 96 Youess, E. A. ad Mekawy, I. M. T Sice y is a arbitrary vector i C satisfyig y y = 1 we coclude from this last i iequality, by takig y =± e, where e i C is a vector with oe i the i th positio ad zero elsewhere, that Re c ( c, x ) =. Hece c ad u = satisfy that Re ( c, x ) u ( c, x ) =, ad c c [ α μc% ( c )] =. u If α μ c % ( c ) =. Let α μ c% ( c ) satisfy the Kuh-Tucker costrait qualificatio at c, ad let y C satisfy c ( c )Rey, there exists a -dimesioal vector fuctio e defied o [, 1] such that e () = c, e ( τ ) L α for τ 1, d ( e ()) e is differetiable at τ =, ad = λ Re y for some λ >. Hece for τ 1 dt dei () ei ( τ ) = ei () + τ + γi (, τ ) for i = 1,...,. d τ where lim γ (, τ ) =. Hece by takig τ small eough, say < τ < ˆ τ < 1, we have i τ that e ( τ ) Bδ ( c ) sice e ( τ ) L α, for τ 1 ad c solves LMP, we have that Re ( e ( τ ), x ) Re ( e ( ), x ) for < τ < ˆ τ. Hece by the chai rule ad the differetiability of Re ( c, x ) at c, we have for < τ < τˆ that Re( e ( τ ), de( ) x ) Re( e (), x ) = Re ( e ( ), x ) τ + β (, τ ) τ, d τ where lim β (, τ ) =. Hece τ de() Re ( e ( ), x ) + β (, τ ) for < τ < ˆ τ. d τ Takig the lim as τ approaches zero gives de() Re ( e ( ), x ). d τ de() Sice e () = c ad = λ Re y for some λ >, we have that: d τ Re ( c ( c, x ) y ). Hece, w have show that if Re ( c μc% ( c ) y ) Re ( c ( c, x ) y ), or that Re ( c ( c, x ) y ) < has o solutio y C. Re ( μ ( c ) y ) c c%

11 Fuzzy complex liear programmig problems 97 Hece by Motzki's theorem of alterative i [7], there exists a r ad r such that r Re c ( c, x ) r c μc% ( c ) = r, or Re c ( c, x ) ( ) c μc% c =, r r, r. Sice r is a real umber, r meas r >, the by defiig r u =, we have that r Re ( c, x ) u μ % ( c ) =, ad u c c c ( α μc% ( c )) =. 5. Coclusios I this work, we ca fid a certai kid of mathematical programmig problems called fuzzy complex liear programmig by usig the cocept of fuzzy complex umber i complex mathematical programmig ad α -cut of a fuzzy complex umber. Also, we itroduce the proof of Kuh-Tucker statioary-poit ecessary optimality theorem for our problem. Refereces [1] A. Berma, Lecture otes i ecoomics ad mathematical systems, Spriger Verlag, Berli, Heidelberg, New York, [ 2 ] A. Kaufma ad M. M Gupta, Itroductio to fuzzy arithmetic, Va Nostrad Reihold Co., N. Y., [3] B.Mod ad M. A Haso, O duality for real ad complex programmig problems, J. Math. Aal. Appl., 24, (1968), [4] H.Taaka, T. Dkuda ad K. Asai, O fuzzy mathematical programmig. The joural of Cyberetics 3, (1974), [5] H. J. Zimmerma, Fuzzy programmig ad liear programmig with several objective fuctios", Fuzzy Sets ad System 1 (1978), [6] J. J Buckley, Fuzzy complex umbers. Fuzzy Sets ad Systems, 33 (1989), [7] M. Sakawa, Fuzzy sets ad iteractive multiobjective optimizatio, pleum press, New York ad Lodo, [8] M. Shegqua ad C. Chu, Fuzzy complex aalysis, the Ethic House, Beijig 21. [ 9 ] N Leviso, Liear programmig i complex space, J. Math. Appl., 14, (1966), [1] O. L Magasaria, No liear programmig, McGraw Hill, New York (1969).

12 98 Youess, E. A. ad Mekawy, I. M. [11] R. E. Bellma ad K. Fa, O system of liear iequalities i Hermitia matrix variables, proceedigs of symposia i pure mathematics, volume VII, Covexity (edited by V. Klee), Amer, Math. Soc., Providece, Rhode Islad, (1963), [12] R. E. Bellma. ad L. A. Zadeh, Decisio makig i a fuzzy eviromet, Maagemet Sci. 17 (197), [13] S. C. Fag ad C. F Hu, Liear programmig with fuzzy coefficiets i costrait, Comput. Math. Appl. 37 (1999), Received: October, 211