Goal Programming Approach to Solve Multi- Objective Intuitionistic Fuzzy Non- Linear Programming Models

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1 Internatonal Journal o Mathematcs rends and echnoloy IJM Volume Number 7 - January 8 Goal Prorammn Approach to Solve Mult- Objectve Intutonstc Fuzzy Non- Lnear Prorammn Models S.Rukman #, R.Sopha Porchelv * # Assstant Proessor, Department o Mathematcs, S..E.. Women s Collee, Mannarud, amlnadu, Inda * AssocateProessor, Department o Mathematcs, A.D.M. Collee or Women, Naappattnam, amlnadu, Inda Abstract In ths Paper Mult-objectve Intutonstc Fuzzy Non-lnear Prorammn Problem has been solved throuh oal prorammn approach. Solvn non-lnear prorammn problem strahtaway s qute dcult. So, the Mult-objectve Intutonstc Fuzzy Non-lnear Prorammn Problem s transormed nto Mult-objectve Intutonstc Fuzzy Lnear Prorammn Problem by the use o aylor Polynomal Seres. Intutonstc uzzy oal prorammn approach s dscussed by ormn sutable membershp and nonmembershp uncton n whch the aspraton level and the tolerance levels are ed by the decson maker or by ndn the best and worst soluton o each objectve uncton. Usn the membershp and non-membershp uncton o each objectve, the ven problem was ormulated and the solutons are ound by varous approaches and the obtaned solutons are dentcal. Numercal eample wll llustrate the ecency o the proposed approach. Keywords Non-Lnear Prorammn, Goal Prorammn, Intutonstc Fuzzy Set. I. Introducton Lke Lnear Prorammn, Non- Lnear Prorammn s a mathematcal technque or determnn the optmal solutons to many busness problems. Lnear Prorammn problem s characterzed by the presence o lnear relatonshp n decson varables n the objectve uncton as well as lnear constrants. But n the real le systems, nether the objectve uncton nor the constrants are lnear unctons n decson varables,.e., the relatonshps are non-lnear. Samr Dey et al., proposes an ntutonstc uzzy optmzaton approach to solve a non-lnear prorammn problem n the contet o structural applcaton. Mahapatra.G.S et al., solved a ntutonstc mult-objectve nonlnear prorammn problem n the contet o relablty applcaton. Based on these, Multobjectve Intutonstc Fuzzy Non-lnear Prorammn Problem has been solved throuh oal prorammn approach n ths paper. In the rst secton, Mult-objectve Intutonstc Fuzzy Non-lnear Prorammn Problem s dscussed and aylor Polynomal seres s ntroduced n the subsequent secton. he membershp and non-membershp unctons are ormed ordn to the eptance and rejecton level n the thrd secton. Varous ormulatons o the problem are dscussed n the ourth secton. Step-wse soluton procedure s eplaned n the net secton. Fnally, a numercal eample s ven and t s solved by varous ormulatons n two cases and the obtaned solutons are dscussed at the end. II. Prelmnares A. Formulaton o Mult-Objectve Intutonstc Fuzzy Non-Lnear Prorammn Problem Consder the Mult-objectve Intutonstc FuzzyNon-lnear Prorammn Problem, Ma or MnZ X =,,. k the constrants j = b, =,,, m; j =,,, n ISSN: -7 Pae

2 Internatonal Journal o Mathematcs rends and echnoloy IJM Volume Number 7 - January 8 ISSN: -7 Pae 6 and j, j =,,.., n In the above, s are real valued unctons o n varables,,.., n and each,,. k are ntutonstc uzzy non-lnear unctons. o overcome the computatonal dcultes o solvn non-lnear prorammn problems, the ollown aylor olynomal seres s used to transorm the non-lnear uncton nto lnear uncton. B. aylor Polynomal Seres For each objectve uncton, =,,, k the ndvdual optmzed value =,,.., n s to be determned and transorm the non-lnear objectves, =,,, k nto lnear objectves by the use o aylor Polynomal Seres, P = n n n By replacn the non-lnear uncton by P o all objectve unctons, =,,.., k become the lnear uncton. Usn aylor Polynomal seres, the ven non-lnear unctons are transormed nto lnear unctons. Accordn to these lnear unctons, the ollown membershp and non-membershp unctons ormed. C. Formulaton o Membershp and non-membershp unctons Case : he aspraton level and tolerance levels are ed by the decson maker Form the membershp and non-membershp unctons or and or, Here, and are eptance and rejecton tolerance or the th oal. Case : he eptance and rejecton levels are ed by ndn the best and worst solutons o each objectve uncton. he membershp and non-membershp uncton o mamzaton problem s dened as ollows:

3 Internatonal Journal o Mathematcs rends and echnoloy IJM Volume Number 7 - January 8 L U L U U re re L re L L re L U U L U re U re he membershp and non-membershp uncton o mnmzaton problem s dened as: U U L L L U re U re L re L re re L U re re U L re U Here, U = ma {Z } and L = mn {Z } or membershp uncton. Moreover, U re = U ε, L re = L and U re = U, L re = L + ε are non-membershp uncton o mamzaton type and mnmzaton type respectvely, ε, =,., k based on the decson maker s choce. III. Problem Formulaton Rajesh Danwal et al., proposed a method to solve Mult-Objectve Lnear Fractonal Prorammn Problem by vaue set. Here, Mult-objectve Intutonstc Fuzzy Non-lnear Prorammn Problem has been solved n whch the membershp unctons can be chosen as per hs method. Furthermore, the non-membershp should be mnmzed and ν are any one o the our possbltes dened above and each o them are reater than or equal to zero. Moreover, n ntutonstc uzzy sets μ + ν, the membershp and non-membershp are ormed separately but the sum o both values are less than or equal to one another. hereore, the ntutonstc uzzy oal prorammn can be ormulated as: Formulaton I For membershp k Ma λ = = μ μ = / μ = or μ = L U L / μ = U U L μ j = b ISSN: -7 Pae 7

4 Internatonal Journal o Mathematcs rends and echnoloy IJM Volume Number 7 - January 8 and j, μ, =,,, m; j =,,., n For non-membershp k Mn λ = = ν ν = / ν = or ν = U U re re / ν = L re L U re re L ν j = b and j, ν, =,,, m; j =,,., n o et the best achevement, the membershp should be mamzed and non-membershp should be mnmzed and n ntutonstc uzzy sets, μ + ν. hereore, the ven Mult-objectve Intutonstc Fuzzy Non-lnear Prorammn Problem can be reormulated as: Formulaton II k Ma λ = = {μ ν } μ ν μ + ν < μ + ν j = b and j, μ, ν, =,,, m; j =,,., n Snce, the hhest deree o membershp s and the lowest deree o non-membershp s. hereore, n oal prorammn approach, Acheve: { / μ + d d + = & ν Z + d d + = }, j In whch, d, d and d +, d + are the under attanment and over attanment, respectvely o the th oal, the devatonal varables d + and d can be removed rom the oal prorammn because the overachevement o membershp uncton and underachevement o non-membershp uncton are eptable. hereore, the ven problem s ormulated as: Formulaton III Mn λ = d + + d μ + d ν d + j = b and j, d, d +, =,,, k; j =,,., n he ollown soluton procedure s used to solve the problem wthout dculty by the above ormulatons. IV. Soluton Procedure ISSN: -7 Pae 8

5 Internatonal Journal o Mathematcs rends and echnoloy IJM Volume Number 7 - January 8 Step : Determne the optmal soluton or each objectve uncton, =,,, k n subject to the ven set o constrants. Step : ransorm the non-lnear objectve uncton nto equvalent lnear objectve uncton at the optmal ponts =,,.. n by aylor Polynomal Seres Step : I the decson maker desres to the oal, then the aspraton level, eptance tolerance and rejecton tolerance or each o the obtaned lnear objectve uncton based on the decson maker s choce. I the decson maker may have the decson deadlock to decde the oals, then the eptance and rejecton levels by ndn the best and worst values o each lnear objectve uncton. Step 4 : Construct the membershp and non-membershp uncton or each lnear objectve uncton wth the help o tolerance and rejecton levels n step. Step : Construct the ntutonstc uzzy oal prorammn models I, II and III as presented n secton III. Step 6 : Solve the ntutonstc uzzy oal prorammn models by the use o optmzaton sotware ORA. he ollown numercal eample wll demonstrate the ecency o the above soluton procedure. It has been solved n two proposed cases and the results are dscussed at the end. V. Numercal Eample Ma = + Ma = Ma = + subject to + 4 4, +,, By ndn the optmzed solutons or each objectve uncton subject to the ven constrants, ma, =, ma, = 4, ma, = By usn aylor s seres.., the lnear objectve unctons are, = +, +, = 4 +, +, = +, +, he obtaned lnear unctons are, = +, = + +, = he ntutonstc uzzy oal prorammn s as ollows: Case : I the oals and tolerance values are decded by the decson maker Accordn to the decson maker s choce,, 6 and he eptance and rejecton tolerance or oal G, oal G and oal G are as ollows: G : = and = G : = and = 4 G : = and = 4 he membershp and non-membershp unctons ormed wth respect to the eptance and rejecton levels dened above are: ISSN: -7 Pae 9

6 Internatonal Journal o Mathematcs rends and echnoloy IJM Volume Number 7 - January 8 ISSN: -7 Pae Usn these membershp and non-membershp uncton dened above, the problem wll be ormulated by I, II and III as ollows: Formulaton I For membershp Ma λ = μ + + = ; μ + + = ; μ + = ; μ ; μ ; μ ; + 4 4; + ; And,, μ, μ, μ he optmal soluton s =., =.67, μ =.8, μ =, μ =.44 For non-membershp, Mn λ = ν + + = ; 4ν + + = 4; 4ν + = ; ν ;ν ;ν ; + 4 4; + ; and,, ν, ν, ν he optmal soluton s =., =.67, ν =., ν =, ν =.4 Formulaton II Ma λ = ; + 4; ; + 9; ; 8; 7 7 ; + ; ; + 4 4; + ;

7 Internatonal Journal o Mathematcs rends and echnoloy IJM Volume Number 7 - January 8 and, he optmal solutons s =., =. 67 By substtutn these values n the correspondn membershp and non-membershp unctons, the ollown values obtaned: μ =. 8, μ =, μ =. 446, ν =., ν =, ν =. 4 Formulaton III Mn λ = d + d + d + d + + d d + + d ; + + d d > 4; + + 4d + 4; + + d ; ; + + 4d + ; + 4 4; + ; and,, d, d, d, d +, d +, d + he optmal solutons are =., =.67, d =.7, d =, d =.6, d., d + =, d + =.4 From the membershp and non-membershp unctons dened above, the values are, μ =.8, μ =, μ =.44, ν =., ν =, ν =.4 Case : he oals and tolerance values are ed by ndn the best and worst values o the lnear unctons. he best and worst values o each o the objectve unctons are, Ma = 4.67; Mn = ; Ma = 6; Mn = ; Ma = ; Mn = ; Accordn to these values, the membershp and non-membershp unctons are, ISSN: -7 Pae + =

8 Internatonal Journal o Mathematcs rends and echnoloy IJM Volume Number 7 - January 8 Usn these membershp and non-membershp uncton dened above, the problem wll be ormulated by I, II and III or the values ε =., ε =. and ε = as ollows: Formulaton I For membershp Ma λ = μ = ; 4μ = ; μ + = ; μ ; μ ; μ ; + 4 4; + ; And,, μ, μ, μ he optmal soluton s =., =.67, μ =, μ =, μ =.44 For non-membershp, Mn λ = ν + + = 4.6;.97ν + + =.97; ν + = ; ν ;ν ;ν ; + 4 4; + ; and,, ν, ν, ν he optmal soluton s =., =.67, ν =, ν =, ν =. Formulaton II Ma λ = ; ; ; ; ; ; 4; ; + 4; + 4 4; + ; and, he optmal solutons s =., =.67 By substtutn these values n the correspondn membershp and non-membershp unctons, the ollown values obtaned: μ =, μ =, μ =.4, ν =, ν =, ν =. Formulaton III Mn λ = d + d + d + d + + d d d 4.67; d + 4.6; + + 4d 4; d +.97; + + d ; d + ; + 4 4; + ; and,, d, d, d, d +, d +, d + ISSN: -7 Pae

9 Internatonal Journal o Mathematcs rends and echnoloy IJM Volume Number 7 - January 8 he optmal solutons are =., =.67, d =, d =, d =., d + =, d + =, d + =. From the membershp and non-membershp unctons dened above, the values are, μ =, μ =, μ =.4, ν =, ν =, ν =. Comparson o Optmal Solutons or Case able : Approach Soluton Ponts, Achevement Level μ, μ, μ Non-achevement level ν, ν, ν Formulaton I.,.67.8,,.44.,,.4 Formulaton II.,.67.8,,.44.,,.4 Formulaton III.,.67.8,,.44.,,.4 Comparson o Optmal o Solutons or Case able : Approach Soluton Ponts, Achevement Level μ, μ, μ Non-achevement level ν, ν, ν.,.67,,.4,,. Formulaton I Formulaton II.,.67,,.4,,. Formulaton III.,.67,,.4,,. In comparson o the optmal solutons n the three ormulatons under two varous cases, he values are dentcal n all the ormulatons and the possblty o the achevements under case, the oal s 8%, oal s % and oal s 4%. he non-achevement level o oal s %, oal s % and oal s 4%. he values are dentcal n all the ormulatons and the possblty o the achevements under case, the oal s %, oal s % and oal s 4%. he possbltes o non-achevement o the oal s %, oal s % and oal s %..e. the rst two oals are ully acheved and the thrd oal s acheved by 4%. Hence, the solutons are dentcal n all the ormulatons and n addton the second case ves more ecent soluton than the rst one. Moreover, n the second case the epslon ε value can be chaned to et the better soluton. VI. CONCLUSION Mult-objectve Intutonstc Fuzzy Non-lnear prorammn problem has been solved n two varous cases under three ormulatons and the obtaned solutons are compared at the end. aylor s Polynomal Seres s used to convert the non-lnear nto lnear to solve the problem rather easly. he decson maker s achevement level was satsed wth ood percentae n all the objectves. he stepwse soluton procedure wll helpul to the reader to solve the mult-objectve non-lnear prorammn problem wth better soluton. Moreover, the numercal eample clearly eplaned or better understandn o the soluton procedure. REFERENCES [] H.J. Zmmermann 978, Fuzzy prorammn and lnear prorammn wth several objectve unctons, Fuzzy Sets and Systems, vol., 4- ISSN: -7 Pae

10 Internatonal Journal o Mathematcs rends and echnoloy IJM Volume Number 7 - January 8 [] P.P.Anelov 997, Optmzaton n ntutonstc uzzy envronment, Fuzzy Sets and Systems, vol.86, 99-6 [] R. Sopha Porchelv, An Alorthmc approach to Mult-objectve uzzy lnear prorammn problem, Internatonal Journal o Alorthms, Computn & Mathematcs, Eashwar Publcatons, 4,6-66 [4] Rajesh Danwal, aylor seres soluton o mult-objectve lnear ractonal prorammn problem by vaue set, Internatonal Journal o Fuzzy Mathematcs and Systems, Vol., No., 4- [] Mahapatra.G.S, Intutonstc uzzy mult-objectve mathematcal prorammn on relablty optmzaton model, Internatonal Journal o Fuzzy Systems, Vol-, No-, pp.9-66 [6] Samr Dey and apan Kumar Roy, Intutonstc Fuzzy Goal Prorammn echnque or Solvn Non-Lnear Mult-Objectve Structural Problem, Journal o Fuzzy Set Valued Analyss, No., pp.79-9 [7] Pramank.S and Roy..K, An ntutonstc uzzy oal prorammn approach to vector oprmzaton problem, Notes on Intutonstc Fuzzy sets, Vol-, No-, pp.-4. [8] Attanassov.K.. 986, Intutonstc uzzy sets, Fuzzy Sets and Systems, Vol-, pp ISSN: -7 Pae 4