Duality in linear interval equations

Transcription

1 Aville online t Int. J. Industril Mthemtis Vol. 1, No. 1 (2009) Dulity in liner intervl equtions M. Movhedin, S. Slhshour, S. Hji Ghsemi, S. Khezerloo, M. Khezerloo, S. M. Khorsny () Deprtment of Siene, Quhn Brnh, Islmi Azd University, Quhn, Irn. () Deprtment of Mthemtis, Siene nd Reserh Brnh, Islmi Azd University, Tehrn, Irn. () Deprtment of Mthemtis, Krj Brnh, Islmi Azd University, Krj, Irn. { Astrt In this pper we nd solution of liner intervl eqution in dul form sed on the generlized proedure of intervl extension whih is lled intervl extended zero method. Moreover, proposed method hs signint dvntge, is tht it sustntilly dereses the exess width eet. Keywords : Intervl eqution, Intervl extended zero method, Intervl rithmetis - 1 Introdution At the ore of mny engineering prolems is the solution of sets of equtions nd inequlities nd the optimiztion of ost funtion. Unfortuntely, expet in speil ses suh s when set of equtions is liner in its unknowns or when onvex onstrint, the results otined y onventionl numeril methods re only lol nd n e gurnteed. By ontrst, intervl nlysis mkes it possile to otin gurnteed pproximtions of the set of the ll tul solutions of the prolem eing onsidered. So, to investigte intervl nlysis, we should propose intervl rithmeti. Intervl rithmeti is n rithmeti on sets of intervls, rther thn sets of rel numers. Moreover, onsider [x] = [x; x] nd [y] = [y; y] e two risp intervls 2 f+; ; ; =g, then [x]@[y] = fx@y 8x 2 [x]; 8y 2 [y]g (1) As the diret outome of the si denition(1), the following expressions were otined: [x] + [y] = [x + y; x + y]; (2) [x] [y] = [x y; x y] (3) Corresponding uthor. Emil ddress: movhedin 10@yhoo.om(M. Movhedin ). 41

2 42 M. Movhedin et l. / IJIM Vol. 1, No.1 (2009) [x] [y] = [min(xy; xy; xy; xy); mx(xy; xy; xy; xy)] (4) [x]=[y] = [x; x] [1=y; 1=y] (5) One of the importnt pplitions of intervl nlysis is nding solution of intervl eqution. However, the lssil solution too often fils to exist[1, 2, 3, 4, 5, 6]. So, numeril methods re pplied to nd suh equtions. The intervl eqution x + where ; nd e intervls is lled liner intervl eqution in dul form. Reently, Sevstjnov nd Dymov[9] proposed new method for solving liner intervl nd fuzzy equtions in the form, where ; re intervl numers. They shown tht the resulting solution of intervl liner equtions sed on the intervl extended zero method my e nturlly treted s fuzzy numer. Sine there is ertin plurlism when hoosing n pproprite method for solving intervl equtions in dul form, we propose (similr[9]) to turn k to the lssil pproh, ut looking t the prolem from other point of view. We elieve tht the solution of the prolem is tht the equtions x + ; x x = 0 where ; ; re intervl numers, re not equivlent ones. Moreover, the min prolem is tht the onventionl intervl extension of the usul eqution, whih leds to the intervl eqution suh s x x = 0, is not orret proedure, sine, in the left hnd side we hve intervl nd the right hnd side we hve rel numer zero. So, to modied suh shortomings, we develop the method proposed in [8, 9] to solve intervl equtions in dul form. The rest of the pper is set out s follows. In setion 2, we review riey intervl extended zero method for solving intervl eqution in dul form. Finlly, some numeril exmples re given. 2 Intervl extended zero method Here, we riey desrie the intervl extended zero method for solving intervl equtions in dul form. Let us onsider intervl extensions of liner eqution nd its lgerilly equivlent forms: x + (6) (7) x x = 0 (8) for intervls, nd suh tht 0 =2. Let [] = [; ], [] = [; ] nd [] = [; ] e intervls. For the ske of simpliity, let us onsider the se of []; []; [] > 0. Moreover, for desriing of our solution we ssumed > mxf; g nd >. Notie tht lter ssumption is pplied only for simpliity nd 42

3 M. Movhedin et l. / IJIM Vol. 1, No.1 (2009) nother ses ould e esily extended sed on the suh ssumptions. However, the intervl extension of (8) is [; ][x; x] = [; ][x; x] + [; ]: By pplying onventionl intervl rithmeti rule(4), we get: [x; x] = [x; x] + [; ] = [x + ; x + ] Oviously, the equlity is hold i x + nd x +. So we hve: ; nd s onsequene of rule(5), intervl extension of Eq.(8) led to otin ; (9) (10) To illustrte the solution (4) nd (5), onsider three exmples s following: Exmple 2.1 Let =[5,8],=[2,4] nd =[1,2]. Then from Eq.(9) we get 3 1 nd from Eq.(10) 6 1 nd 2. Exmple 2.2 Let =[7,8],=[5,6] nd =[3,3.5]. Then from Eq.(9) we get 3 2, 3:5 2 nd from Eq.(10), 1, 3:5 Exmple 2.3 Let =[6,9], =[5,5] nd =[1,2]. Then from Eq.(9) we get 1, 0:5 nd from Eq.(10) 1 4, 2 We n see tht intervl extension of Eq.() my result in the inverted intervl [x], i.e., x < x, while the extension of Eq.(8) provides orret intervl(x < x). The stndrd intervl extension of Eq.(8) is [x; x] [x; x] [; ] = 0, y pplying the intervl rithmeti we get nlly: ; : It is esy to see tht in ny se x > x, i.e., we get n inverted intervl. Also, we dene the degenerte (usul zero) s the result of the opertion, where is ny rel vlued numer or vrile. Therefore, in similr wy, we n dene the intervl zero s the result of the opertion [] [], where [] is n intervl. So we get [; ] [; ] = [ ; ] = [ ( ); ]: Thus, in ny se the result of the intervl sutrtion [] [] is n intervl symmetril with respet to 0. So, y suh omputtions, the intervl extension of Eq.(8) should e s following: [; ][x; x] [; ][x; x] [; ] = [ y; y] (11) where the right hnd side of Eq.(11) is n intervl entered round zero. So, y using Eq.(11) we get: x x = y x x = y (12) 43

4 44 M. Movhedin et l. / IJIM Vol. 1, No.1 (2009) Summing the expressions in the left nd right hnd sides of Eq.(12) we get: ()x + ()x ( + ) = 0 (13) It is impossile to get single rel vlued solution of (13) s it is n under-determined eqution. Similr [9], Eq.(13) is pplied s the so lled onstrint stisftion prolem[10]. In ddition, y using nother onstrint its intervl solution my e derived. To this end, suh onstrint whih is pplied on the vriles x nd x is the solution of Eq.(13) y setting x. In this degenerte se the solution of Eq.(13) s: x m = + () + () It is esy to see tht x m is the upper ound for x nd the lower ound for x. Also, the lower ound for x nd the upper ound for x should e dened too. So, we dene the nturl lower ound for x nd the upper ound for x s follows: i h i, x m ; : h. Thus, we hve [x] = ; x m nd [x] = It is ler tht the right ound of x nd the left ound of x, i.e., x m n not hnged s they present the degenerte(rel vlue) solution of (13). So, from (13), we get: + + ( ) ; x ( ) Oviously, when x is mximl in the intervl ; x 2 h x m ; x m ; vlue of x, i.e., x min = ( ) + ()(). Similrly, we get the mximl vlue of x, i.e., x m i ; x m, i.e., (14) (15) (16) we get the miniml ( ) + (! )(), when. Generlly, it is possile tht x min < nd x mx >. So, the miniml lower ound o x nd the mximl upper ound of x n e presented s following: x L min = min ; ( ) + (17) ()() x U m mx ; ( ) + (18) ()() Therefore, we get the following intervl solution of Eq.(8): [x] = x L min; ; [x] = + () + () + () + () ; xu mx It is seen tht Eq.(19) dene ll possile solutions of Eq.(8). The mximl intervl solution width w m x U mx x L min orresponds to the mximl vlue of y: y m mx ; (20) nd y min is otined y sustituting the degenerte solution x m in Eq.(13) suh tht y min = min ( )x m + ; ()x m (21) 44 (19)

5 3 Conlusion in this pper we proposed new method to solve liner intervl eqution in dul form. To this end, we turn k to the lssil pproh ut looking t the prolem from the other point of view. An importnt of dvntge of new method is tht sustntilly dereses the exess width eet. Moreover, our pproh gurnteed tht suh new method lwys gives intervl solution not inverted intervl solution while, lssil intervl method hs no suh property every where. Referenes M. Movhedin et l. / IJIM Vol. 1, No.1 (2009) [1] S. Asndy, B. Asdy, Newton's method for solving fuzzy nonliner equtions, Applied Mthemtis nd Computtion, 159(2004) [2] J. J. Bukley, Y. Qu, Solving system of liner fuzzy equtions, Fuzzy Sets nd Systems 43(1991) [3] J. J. Bukley, Y. Qu, Solving liner nd qudrti fuzzy equtions, Fuzzy Sets nd Systems 38(1990) [4] J. J. Bukley, Y. Qu, Solving fuzzy equtions: new onept, Fuzzy Sets nd Systems 39(1991) [5] J. J. Bukley, Solving fuzzy equtions in eonomis nd nne, Fuzzy Sets nd Systems 48(1992) [6] J. J. Bukley, E. Eslmi, Neurl net solutions to to fuzzy prolems: the qudrtieqution, Fuzzy Sets nd Systems 86(1997) [7] B. S. Shieh, Innite fuzzy reltion equtions with ontinuous t-norms, Informtion Sienes, 178(2008) [8] P. Sevstjnov, L. Dymov, A new method for solving intervl nd fuzzy equtions: Liner Cse, iinformtion Sienes, 179(2009) [9] P. Sevstjnov, L. Dymov, Fuzzy solution of intervl liner equtions, in:proeedings of SeventhInterntionl Conferene on Prllel Proessing nd Applied Mthemtis(PPAM'07), Gdnsk, In Press. [10] J. R. Ullmnn, Prtition serh for non-inry onstrint stisftion, Informtion Sienes, 177(2007)