Solution of fuzzy multi-objective nonlinear programming problem using interval arithmetic based alpha-cut

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1 Intentionl Jounl of Sttistics nd Applied Mthemtics 016; 1(3): 1-5 ISSN: Mths 016; 1(3): Stts & Mths Received: Accepted: C Lognthn Dept of Mthemtics Mh Ats nd Science College Coimbtoe Tmilndu Indi M Llith Deptment of Mthemtics Kongu Ats nd Science College Eode Tmilndu Indi Solution of fuzz multi-obective nonline pogmming poblem using intevl ithmetic bsed lph-cut C Lognthn nd M Llith Abstct In this ppe fuzz multi-obective nonline pogmming poblem is pesented. All the coefficients of the nonline multi-obective functions nd the constints e fuzz numbes. Hee we find the solution of the nonline pogmming poblem b using Intevl ithmetic bsed on Alph-cut. A numeicl emple is pesented. Kewods: Fuzz multi-obective nonline pogmming (FMONLP) Tingul fuzz numbe Intevl ithmetic Optiml solution - cut. 1. Intoduction The concept of fuzz non line pogmming ws poposed b zimmemnn [1]. Fuzz non line pogmming poblem is useful in solving poblems which e difficult impossible to solve to the impecise subective ntue of the poblem fomultion o hve n ccute solution. Bellmn nd Zdeh [1] poposed the concept of decision mking in fuzz envionment. A new method fo solving line pogmming poblem with vgueness in constints b using nking function ws intoduced b Mleki [8]. Tnk.et.l [11] dopted this concept fo solving mthemticl pogmming poblems. Pndin nd Jlkshmi [9] poposed new method fo solving full fuzz line pogmming poblem with fuzz vibles. Compos nd Vdeg [] consideed line pogmming poblems with fuzz constints nd fuzz coefficients in both left nd ight hnd of the constints set. Kitiwnt nd Tni [5] intoduced new ppoch fo finding n optiml fuzz solution fo fuzz non line pogmming poblem. In this ppe we popose the bsic definitions of fuzz set nd Intevl ithmetic opetions on fuzz numbes bsed on - cut to solve fuzz multi obective non line pogmming poblem.. Peliminies.1 Definition: [4] Let A be clssicl set. A * with the function 1]}. The function () be el vlued function defined fom R [0 1]. A fuzz set () is defined b A * = { ( () ) ; ϵ A nd () is known s the membeship function of A *. () ϵ [ 0 Coespondence: C Lognthn Dept of Mthemtics Mh Ats nd Science College Coimbtoe Tmilndu Indi. Definition: [3] A fuzz numbe is conve nomlized fuzz set of the el line R whose membeship function is piecewise continuous..3 Definition: [1] A fuzz numbe A ~ in R is sid to be tingul fuzz numbe if its membeship function ~ 1 ~

2 Intentionl Jounl of Sttistics nd Applied Mthemtics ~ ( A X ) p p 0 p othewise.4 Definition: [4] Given fuzz set A defined on X nd n numbe ɛ[01] the - cut A = { / A() }..5 Definition: [9] Given fuzz set A defined on X nd n numbe ɛ[01] the stong - cut A = { / A() }. A is the cisp set A is the cisp set 3. Intevl ithmetic [10] Intevl ithmetic intevl mthemtics intevl nlsis o intevl ws developed b vious mthemticins in between 1950 nd This is n ppoch to putting bounds on ounding eos nd mesuement of eos in mthemticl computtion nd thus developing numeicl methods tht ield elible esults. This epesents ech vlue s nge of possibilities. The following e the bsic opetions of intevl ithmetic fo two vibles [p ] nd [ s] tht e subsets of the el line (- ) (i) Addition: [p ] + [ s] = [p + + s] (ii) Subtction: [p ] - [ s] = [p - - s] (iii) Multipliction: [p ]. [ s] = [min (p ps s) m(p ps s)] p p p p p (iv) Division: s = min s s m s s when 0 is not in[c d] 4. Aithmetic opetions of fuzz numbes using - cut method In this section we conside ddition subtction multipliction nd division of fuzz numbes using - cut method [10]. 4.1 Addition of Fuzz Numbes Let X = [p ] nd Y = [ b c] be two fuzz numbes whose membeship functions e defined b ( X ) p p 0 p othewise ( X ) b c c b 0 b b c othewise = [( - p) + p - ( - ) ] nd = [(b - ) + c - (c - b) ] e the -cuts of the fuzz numbes Then X nd Y espectivel. To clculte the ddition of fuzz numbes X nd Y using intevl ithmetic tht is + = [( - p) + p - ( - ) ] + [(b - ) + c - (c - b) ] = [p + + ( p + b - ) + c - ( + c - b). ~ ~

3 Intentionl Jounl of Sttistics nd Applied Mthemtics 4. Subtction of Fuzz Numbes Let X = [p ] nd Y = [ b c] be two fuzz numbes. = [( - p) + p - ( - ) ] nd = [(b - ) + c - (c - b) ] e the - cuts of the fuzz numbes X nd Y Then espectivel. To clculte the subtction of fuzz numbes X nd Y using intevl ithmetic tht is - = [( - p) + p - ( - ) ] - [(b - ) + c - (c - b) ] = [(p - c) + ( p + c b) ( - ) - ( + b - ) ] 4.3 Multipliction of Fuzz Numbes Let X = [p ] nd Y = [ b c] be two fuzz numbes. Then = [ ( - p) + p - ( - ) ] nd = [(b - ) + c - (c - b) ] e the - cuts of the fuzz numbes X nd Y espectivel. To clculte the multipliction of fuzz numbes X nd Y using intevl ithmetic tht is * = [( - p) + p - ( - ) ] * [(b - ) + c - (c - b) ] = [(( - p) + p)* (b - ) + )) ( - ( - ) )*(c - (c - b) )] 4.4 Division of Fuzz Numbes Let X = [p ] nd Y = [ b c] be two fuzz numbes. Then = [ ( - p) + p - ( - ) ] nd = [(b - ) + c - (c - b) ] e the - cuts of the fuzz numbes X nd Y espectivel. To clculte the division of fuzz numbes X nd Y using intevl ithmetic tht is = ( p) p = c ( c b) ( ) ( b ) 5. Multiobective nonline pogmming (monlp) fomultion c Mimum Z = Subect to the constints n i b i 1 (i = 1. m) 0 ( = 1. n) The function to be mimized is clled n obective function. This is denoted b Z. c = (c 1 c.c n) is clled cost vecto. The mti [ i] is clled constint mti nd the vecto b i = (b 1 b. b m) is clled ight hnd side vecto. 6. Pocedue The ithmetic opetions of fuzz numbes using -cut opetions discussed in the elie sections e used below to solve the fuzz nonline pogmming poblem. Step 1: Find the vlue of X nd Y Step : Add the - cuts of X nd Y using intevl ithmetic Step 3: The vlues obtined in step1 nd step is conveted into cisp MONLPP. Step 4: B solving the bove nonline pogmming poblem we obtin the optiml solution. 7. Numeicl poblem [5] Using the pocedue n intevl ithmetic nonline pogmming poblem with tingul fuzz numbes is consideed. Mimum Z = [(5 6 7) + (6 7 8)] 1 + [(3 4 5) + (1 3)] (1) Mimum Z = [(6 7 8) + (8 9 10)] 1 + [(4 5 6) + (6 7 8)] () Subect to the constints [(0 1 ) + ( 3 4)] 1 + [(3 4 5) + (1 3)] [(6 8 10) + ( )] [(0 1 ) + ( 3 4)] 1 + [( 3 4) + (4 5 6)] [(4 6 8) + (8 10 1)] (3) Step 1: Detemine X nd Y fo n obective function (1) The - cut of the fuzz numbe (5 6 7) is ~ 3 ~

4 Intentionl Jounl of Sttistics nd Applied Mthemtics 5 7 = 1 1 X = ( ) The - cut of the fuzz numbe (6 7 8) is 6 8 = 1 1 X = ( ) Step : Adding the - cuts of X nd Y using intevl ithmetic we obtin + = ( ) + ( ) = 8 Simill the emining fuzz numbes of obective functions e [(3 4 5) + (1 3)] = 1 [(6 7 8) + (8 9 10)] = 3 [(4 5 6) + (6 7 8)] = 4 Simill the constint mti 11 = [(0 1 ) + ( 3 4)] = 8 1 = [(3 4 5) + (1 3)] = 1 i nd the ight hnd side numbe bi e 1 = [(0 1 ) + ( 3 4)] = 8 = [( 3 4) + (4 5 6)] = 16 b 1 = [(6 8 10) + ( )] = 40 b = [(4 6 8) + (8 10 1)] = 3 Step 3: Now the given poblem becomes the cisp MONLPP s Mimum Z = (4) Mimum Z = (5) Subect to the constints (6) Step 4: Solving the nonline eution (4) with (6). We obtin the optiml solution. M Z = 48 1 = 0 = 1.8 Solving the non line eution (5) with (6) we obtin the optiml solution. M Z = = 0 = Conclusion Fuzz multi-obective nonline pogmming poblem hs been solved b using intevl ithmetic bsed on - cut opetion without conveting them into clssicl NLPP. This is n es ppoch fo solving FNLPP b using intevl ithmetic when comped to the elie ppoches.this is lso cn be pplied in tpezoidl hegonl octgonl fuzz numbes. 9. Refeences 1. Bellmn RE Zdeh LA. Decision mking in fuzz envionment Mngement Science 1970; Cmpose L Vedeg JL. Line Pogmming Poblems nd Rnking of Fuzz Numbes Fuzz Sets nd Sstems Dubois D Pde H. Fundmentls of fuzz sets Kluwe cdemic publishes Boston. 4. Kufmn A Gupt M. Intoduction to fuzz ithmetic theo nd pplictions Vn nostnd einhold co.inc Wokinghm Bekshie Kitiwnt Ghdle P Tni Pw S. New ppoch fo wolfe s modified simple method to solve udtic pogmming poblems Intentionl Jounl of Resech in Engineeing nd Technolog Kiuthig M D Lognthn C. Solving Fuzz Multi-obective Nonline Fctionl Pogmming Poblem Using Fuzz Pogmming Model Intentionl Jounl of Contempo Resech in Compute Science nd Technolog. 015; 1: D. Lognthn C Kiuthig M. Solution of Fuzz Nonline Pogmming Poblem Using Rnking Function Intentionl Jounl of Recent Tends in Engineeing & Resech. 016; : Mleki HR Tt Mshinchi M. Line pogmming with fuzz vibles Fuzz sets nd sstems ~ 4 ~

5 Intentionl Jounl of Sttistics nd Applied Mthemtics 9. Pndin Jlkskmi P. A new method fo solving Intege line pogmming poblems with fuzz vible Applied Mthemtics Sciences Plsh Dutt Hishikesh Bouh Tzid Ali. Fuzz ithmetic with nd without using cut method Intentionl ounl of ltest tends in computing Tnk H Okud T Asi K. On fuzz mthemticl pogmming Jounl of Cbemetics nd Sstems (1973) pp: Zimmemnn HJ. Fuzz pogmming nd Line Pogmming with Sevel Obective Functions Fuzz Sets nd Sstems ~ 5 ~