Irene Hepzibah.R 1 and Vidhya.R 2

Transcription

1 Internatonal Journal of Scentfc & Engneerng Research, Volume 5, Issue 3, March ISSN INTUITIONISTIC FUZZY MULTI-OBJECTIVE LINEAR PROGRAMMING PROBLEM (IFMOLPP) USING TAYLOR SERIES APPROACH Irene Hepzbah.R and Vdhya.R 2 Abstract These nstructons gve you gudelnes for preparng papers for JOURNALS. Use ths document as a template f you are usng Mcrosoft Word 6.0 or later. Otherwse, use ths document as an nstructon set. The electronc fle of your paper wll be formatted further at. Defne all symbols used n the abstract. Do not cte references n the abstract. Do not delete the blan lne mmedately above the abstract; t sets the footnote at the bottom of ths column. Don t use all caps for research paper ttle. Index Terms Intutonstc fuzzy mult-objectve lnear programmng problem (IFMOLPP), membershp and non-membershp functons, Taylor seres. INTRODUCTION In real world, we frequently deal wth vague or mprecse nformaton. Informaton avalable s sometmes vague, sometmes nexact or sometmes nsuffcent. The concept of Fuzzy sets was ntroduced by Zadeh n 965. In the lterature, dfferent approaches appear to solve dfferent models of lnear programmng problem (LPP).Because, programmng solves more effcently the above problems wth respect to the some constrants. The concept of maxmzng decson was ntally proposed by Bellman and Zadeh [3. By adoptng ths concept of fuzzy sets was appled n mathematcal programs frstly by Zmmerman [20. In the past few years, many researchers have come to the realzaton that a varety of real world problems whch have been prevously solved by lnear programmng technques are n fact more complcated. Frequently, these problems have multple goals to be optmzed rather than a sngle objectve. Moreover, many practcal problems cannot be represented by lnear programmng model. Therefore, attempts were made to develop more general mathematcal programmng methods and many sgnfcant advances have been made n the area of mult-objectve lnear programmng. Out of several hgher order fuzzy sets, ntutonstc fuzzy sets (IFS) [,2 have been found to be hghly useful to deal wth vagueness. There are stuatons where due to nsuffcency n the nformaton avalable, the evaluaton of membershp values s not also always possble and consequently there remans a part ndetermnstc on whch hestaton survves. Dr. R. Irene Hepzbah, Professor of Mathematcs, A.V.C. College of Engneerng, Mayladuthura, Inda, PH E-mal: reneraj74@gmal.com R. Vdhya, Assstant Professor of Mathematcs, As-Salam College of Engneerng and Technology, Thrumangalaud, Aduthura, Inda, PH E-mal: vdhya4m@gmal.com 204 Certanly fuzzy sets theory s not approprate to deal wth such problems; rather ntutonstc fuzzy sets (IFS) theory s more sutable. The Intutonstc fuzzy set was ntroduced by Atanassov.K.T [ n 986. For the fuzzy multple crtera decson mang problems, the degree of satsfablty and non- satsfablty of each alternatve wth respect to a set of crtera s often represented by an ntutonstc fuzzy number (IFN), whch s an element of IFS [8,6. Ths Intutonstc fuzzy mathematcs s very lttle studed subject. In a recent revew, Tosar [5 gave a Taylor seres approach to fuzzy mult-objectve fractonal programmng problem. However, many methods of solvng mult-objectve lnear programmng problems are avalable n the lterature. In ths paper, membershp and non-membershp functons, whch are assocated wth each objectve of ntutonstc fuzzy mult-objectve lnear programmng problem (IFMOLPP) are transformed by usng frst order Taylor polynomal seres [5. Then the IFMOLPP can be reduced to a sngle objectve lnear programmng. The paper s organzed as follows: The formulaton of the problem s gven n Secton 2, and Secton 3 deals wth an algorthm for solvng a ntutonstc fuzzy mult-objectve lnear programmng problem. Fnally, n Secton 4, the effectveness of the proposed method s llustrated by means of an example. Some concludng remars are gven n secton 5.

2 Internatonal Journal of Scentfc & Engneerng Research, Volume 5, Issue 3, March ISSN , f z (x) g 2. FORMATION OF THE PROBLEM The mult-objectve lnear programmng problem and the mult- objectve fuzzy lnear programmng problem are descrbed n ths secton. 2.. Mult-objectve lnear programmng problem (MOLPP) A lnear mult-objectve optmzaton problem s stated as Maxmze or Mnmze: [z (x), z 2(x),, z (x) Subject to Ax = b, x 0 where zj(x) j =, 2, n s an N vector of cost coeffcents, A an m x N Coeffcents matrx of constrants and b an m vector of demand (resource) avalablty Intutonstc fuzzy mult-objectvelnear programmng problem (IFMOLPP) If an mprecse aspraton level s ntroduced to each of the objectves of MOLPP, then these fuzzy objectves are termed as fuzzy goals. Let g I be the aspraton level assgned to the th objectve z (x).then the fuzzy objectves appear as () Z (x) > I ~ g (for maxmzng Z (x)); () Z (x) < I ~ g (for mnmzng Z (x)); > < Where ~ and ~ ndcate the fuzzness of the aspraton levels, and s to be understood as essentally more than and essentally less than n the sense of Zmmerman[8. Hence, the ntutonstc fuzzy mult-objectve lnear programmng problem can be stated as follows: Fnd X so as to satsfy Z (x) < I ~ g, =,2,... l, Z (x) > I ~ g, = l, l2,, Subject to x X, X 0. Now, n the feld of ntutonstc fuzzy programmng, the ntutonstc fuzzy objectves are characterzed by ther assocated membershp functons and non-membershp functons [0. They can be expressed as follows: If Z (x) > I ~ g, µ I (x) = z (x) t, f z (x) g g t, f t z (x) g 0, f z (x) t and 0, f z (x) g ν I z (x) g (x) =, f t t g z (x) g, f z (x) t If Z (x) < I ~ g, µ I (x) = ν I (x) = t z (x) t g g z (x) t g, f g z (x) t and 0, f z (x) t 0, f z (x) g, f t z (x) g, f z (x) t where t and t are the upper tolerance lmt and the lower tolerance lmt respectvely, for the th ntutonstc fuzzy objectve. Now, n a ntutonstc fuzzy decson envronment, the achevement of the objectve goals to ther aspred levels to the extent possble are actually represented by the possble achevement of ther respectve membershp values and non-membershp values to the hghest degree. The relatonshp between constrants and the objectve functons n the ntutonstc fuzzy envronment s fully symmetrc, that s, there s no longer a dfference between the former and the latter [5. Ths guarantees the maxmzaton of both objectves membershp values and non-membershp values smultaneously. 3. ALGORITHM FOR INTUITIONISTIC FUZZY MULTI- OBJECTIVE LINEAR PROGRAMMING PROBLEM Accordng to Tosar [5,n the ntutonstc fuzzy mult-objectve lnear programmng problem, membershp functons and non-membershp functons assocated wth each objectve are transformed by usng Taylor seres at frst and then a satsfactory value(s) for the varable(s) of the model s obtaned by solvng the fuzzy model, whch has a sngle objectve functon. Based on ths dea, an algorthm for solvng ntutonstc fuzzy mult-objectve lnear programmng problem s developed here. Step.Determne x =(x, x 2,, x n ), that s used to maxmze or mnmze the th membershp functon µ I (x) and non-membershp functon ν I (x) (=,2,.) where n s the number of varables. Step 2.Transform membershp and non-membershp functons by usng frst-order Taylor polynomal seres µ I (x) µ I ı (x)= I µ (x ) [(x x ) µ I (x ) x (x 2 x 2 ) µ I (x )..(x x n x n ) µ 2 µ I (x) µ I ı (x)= I µ (x n ) (x j x j ) µ I (x ) j= ν I (x) ν I ı (x)= I ν (x ) [(x x ) ν I (x ) x (x 2 x 2 ) ν I (x )..(x n x n ) ν I (x ) x n I (x ) x n 204

3 Internatonal Journal of Scentfc & Engneerng Research, Volume 5, Issue 3, March ISSN are as obtaned as follows: µ I (x) = ν I (x) ν (x)= I ı ν I (x n ) (x j x j ) ν I (x ) 0, f z (x) c j= c z (x) Step 3.Fnd satsfactory x =(x, x 2,, x n ) by solvng the reduced problem to a sngle objectve for membershp functon and non-membershp functon respectvely. p(x)= µ I (x n = ) (x j x j ) µ I (x ) j= and q(x)= ν I (x n = ) (x j x j ) ν I (x ) j=. Thus FMOLPP s converted nto a new mathematcal model. The model s as follows: Maxmze or Mnmze µ I (x n = ) (x j x j ) µ (x ) j= and Maxmze or Mnmze ν I (x n = ) (x j x j ) ν I (x ) j=, where µ I (x) = z (x) t g t f z (x) g, f t z (x) g 0, f z (x) t and, f b c b z (x) c z (x) a, f a b a z (x) b 0, f z (x) a = µ I 2 (x) = 0, f z (x) 5, f 3.5 z (x) 5 5 ( x x 2 ) ( x x 2 ) , f 2 z (x) 3.5 0, f z (x) 2 8 ( 3x 2x 2 ) 8 9 ( 3x 2x 2 ) ( 0.0) 9 ( 0.0) 0, f z If µ I (x) = max(mn( 5 ( x x 2 ) (x) g ν I z (x) g (x) =, f t t g z (x) g µ I 2 (x) = max(mn( 8 ( 3x 2x 2 ) 8 9, f z (x) t, f z (x) g and µ I t z (x) (x) =, f g t g z (x) t and µ I (.7, 2.57) and µ I 2 (3, 0). 0, f z (x) t 0, f z (x) g ν I g z (x) (x) =, f t t g z (x) g, f z (x) t respectvely. 0, f z 2 (x) 8, f 9 z 2 (x) 8, f 0.0 z 2 (x) 9 0, f z 2 (x) 0.0, ( x x 2 ) 2 ), 0) and 3.5 2, ( 3x 2x 2 ) ( 0.0) ), 0) then 9 ( 0.0) The membershp and non-membershp functons are transformed by usng frst-order Taylor polynomal seres µ I (x) µ I (x)= µ I (.7,2.57) [(x.7) µ I (.7,2.57) x (x ) µ I (.7,2.57) µ I (x) = 0.667x x (4.2) 4. ILLUSTRATIVE EXAMPLE Consder the followng MOLPP: Maxmze z (x)=x x 2 Maxmze z 2(x)=3x -2x 2 (4.) Subject to the constrants 2x x 2 6 x 4x 2 2 a) The membershp and non-membershp functons were consdered to be ntutonstc trangular (see fgure ) when they depend on three scalar parameters (a,b,c ). z depends on three scalar parameters (2,3.5,5) when z 2 depends on three scalar parameters (-0.0,9,8). The membershp and non-membershp functons of the goals µ I 2 (x) µ I 2 (x)= µ I 2 (3,0) [(x 3) µ 2 I (3,0) x (x 2 0) µ 2 I (3,0) µ I 2 (x) = 0.333x x (4.3) Then the objectve of the FMOLPP s obtaned by addng (4.2) and (4.3), that s p(x) = µ I (x) µ2 I (x) = x-0.889x Subject to the constrants 2x x 2 6 x 4x 2 2` The problem s solved and the soluton s obtaned s as follows: x =3; x 2 = 0; z (x)= 3; z 2(x)= 9 and the membershp values are µ = 0.67 and µ 2 =.The membershp fucton values show that both goals z and z 2 are satsfed wth 67% and 00% respectvely for the obtaned soluton whch s x =3; x 2 =

4 Internatonal Journal of Scentfc & Engneerng Research, Volume 5, Issue 3, March ISSN The problem s solved and the soluton s obtaned s as follows: x =3; x 2 = 0; z (x)= 3; z 2(x)= 9 and the nonmembershp values are ν = 0.33 and ν 2 = 0.The nonmembershp functon values show that both goals z and z 2 are satsfed wth 33% and 0% respectvely for the obtaned soluton whch s x =3; x 2 = 0. b) The membershp and non-membershp functons were consdered to be ntutonstc trapezodal (see fgure 2) when they depend on four scalar parameters (a,b,c,d ).z depends on four scalar parameters (-, 2, 4, 4.5) when z 2 depends on four scalar parameters (7, 0,, The non-membershp functons of the goals are as 2). The membershp and non-membershp functons of the obtaned as follows: goals are as obtaned as follows:, f z (x) c 0, f z z (x) b (x) d, f b ν I c (x) = b z (x) c d z (x), f c d b z (x) c z (x) d, f a b a z (x) b µ I (x) =, f b z (x) c z, f z (x) a (x) a, f a, f z (x) 5 b a z (x) b 0, f z ( x x 2 ) 3.5 (x) a, f 3.5 z (x) 5 = 3.5 ( x x 2 ), f 2 z (x) 3.5 0, f z (x) 4.5, f z 4.5 (x x 2 ) (x) 2, f 4 z (x) 4.5 =, f 2 z (x) 4 (x x 2 ) ( ), f z, f z 2 (x) 8 2 ( ) (x) 2 ( 3x 2x 2 ) 9 0, f z (x), f 9 z ν I (x) 8 2 (x) = 9 ( 3x 2x 2 ), f 0.0 z 9 ( 0.0) 2 (x) 9 0, f z, f z 2 (x) 0.0 (x) 2 2 (3x 2x 2 ), f z If ν I ((x) = max (mn( ( x x 2 ) 3.5, 3.5 ( x 2 (x) 2 x 2 ) ), ) and µ I 2 (x) =, f 0 z (x) (3x 2x 2 ) 7, f 7 z ν I 2 (x) = max (mn( ( 3x 2x 2 ) 9, 9 ( 3x 0 7 (x) 0 2x 2 ) ), ) then 0, f z ( 0.0) (x) 7 ν I (.7,2.57) and ν I 2 (3,0). The non-membershp functons are transformed by usng frst-order Taylor polynomal seres ν I (x) ν I (x)= I ν (.7,2.57) [(x.7) ν I (.7,2.57) x (x ) ν I (.7,2.57) ν I (x) = 0.667x x (4.4) I (3,0) ν I 2 (x) ν I 2 (x)= ν2 (3,0) [(x 3) ν 2 I (3,0) (x 0) ν 2 ν I 2 (x) = 0.333x x (4.5) Then the objectve of the FMOLPP s obtaned by addng (4.4) and (4.5), that s q(x) =ν I (x) ν2 I (x) = x-0.889x Subject to the constrants 2x x 2 6 x 4x If µ I (x) = max(mn( 4.5 (x x 2 ), (x x 2 ) ( ) ), 0) and 2 ( ) µ I 2 (x) = max(mn( 2 (3x 2x 2 ), (3x 2x 2 ) 7 ), 0) thenµ I (.7,2.57) and µ I 2 (3,0). The membershp and non-membershp functons are transformed by usng frst-order Taylor polynomal seres µ I (x) µ I (x)= µ (.7,2.57) [(x.7) µ I (.7,2.57) x (x ) µ I (.7,2.57) µ I (x) = 0.333x -2x 25.0 (4.6) µ I 2 (x) µ I 2 (x)= µ2 (3,0) [(x 3) µ 2 I (3,0) x (x 2 0) µ 2 I (3,0) µ I 2 (x) = x x (4.7)

5 Internatonal Journal of Scentfc & Engneerng Research, Volume 5, Issue 3, March ISSN Then the objectve of the FMOLPP s obtaned by addng (4.6) and (4.7), that s p(x) =µ I (x) µ I 2 (x) =.333 x x Subject to the constrants 2x x 2 6 x 4x 2 2 The problem s solved and the soluton s obtaned s as follows: x =3; x 2 = 0; z(x)= 3; z2(x)= 9 and the membershp values are µ = and µ 2 = 0.67.The membershp functon values show that both goals z and z2 are satsfed wth 00% and 67% respectvely for the obtaned soluton whch s x =3; x 2 = 0. ν I 2 (x) = max (mn( (3x 2x 2 ) 2 ν I (.7,2.57) and ν I 2 (3,0)., 0 (3x 2x 2 ) ), ) then 0 7 The non-membershp functons are transformed by usng frst-order Taylor polynomal seres ν I (x) ν I (x)= ν I (.7,2.57) [(x.7) ν I (.7,2.57) x (x ) ν I (.7,2.57) ν I (x) = 2x x (4.8) ν I 2 (x) ν I 2 (x)= ν I 2 (3,0) [(x 3) ν 2 I (3,0) (x 0) ν 2 ν I 2 (x) = 3x -2x 2-9 (4.9) Then the objectve of the FMOLPP s obtaned by addng (4.8) and (4.9), that s q(x) = ν I (x) ν I 2 (x)) = x x Subject to the constrants 2x x 2 6 x 4x 2 2 I (3,0) The non-membershp functons of the goals are as obtaned as follows: 0, f z (x) d z (x) c, f c d c z (x) d ν I (x) =, f b z (x) c b z (x), f a b a z (x) b 0, f z (x) a = ν I 2 (x) = (x x 2 ) 4 2 (x x 2 ) 2 ( ) (3x 2x 2 ) 2 0 (3x 2x 2 ) 0 7 If ν I (x) = max (mn( (x x 2 ) 4, 2 (x x 2 ) 2 ( ) 204 The problem s solved and the soluton s obtaned s as follows: x =3; x 2 = 0; z (x)= 3; z 2(x)= 9 and the nonmembershp values are ν = 0 and ν 2 = 0.33.The nonmembershp functon values show that both goals z and z2 are satsfed wth 33% and 0% respectvely for the obtaned soluton whch s x =3; x 2 = 0. 5.CONCLUSION In ths paper, a powerful and robust method whch s based on Taylor seres s proposed to solve ntutonstc fuzzy mult-objectve lnear programmng problems (IFMOLPP). Membershp functon and non-membershp functons assocated wth each objectve of the problem are transformed usng Taylor seres respectvely. Actually, the ntutonstc fuzzy mult-objectve lnear programmng problem (IFMOLFP) s reduced to an, f z (x) 4.5 equvalent mult-objectve lnear programmng problem, f 4 z (x) 4.5 (MOLPP) by usng the frst order Taylor polynomal seres. 0, f 2 z (x) 4 The obtaned MOLLP problem s solved assumng that the, f z (x) 2 weghts of the objectves are equal. The proposed soluton, f z (x) method wll be appled to more varables and objectves. Thus, the performance of the proposed soluton method was tested when the number of objectves and varables ncreased and when membershp functons and nonmembershp, f z (x) 2 functons were defned nto dstnct types., f z (x) 2 Then, the proposed soluton method was appled to two 0, f 0 z (x) numercal examples to test the effect on the performance of, f 7 z (x) 0 the proposed soluton method wth respect to dstnct, f z (x) 7 defntons of objectves and changes of the parameters. The proposed method facltates computaton to reduce the ), ) and complexty n problem solvng.

6 Internatonal Journal of Scentfc & Engneerng Research, Volume 5, Issue 3, March ISSN REFERENCES [Atanassov.K.T (986), Intutonstc fuzzy sets, Fuzzy Sets and Systems, 20, [2Atanassov.K.T (989), More on ntutonstc fuzzy sets, Fuzzy sets and systems, 33, [3Bellman, R.E.,Zadeh, L.A., Decson mang n a fuzzy envronment, Management Scence 7 (970) [4Dubos, D., and H. Prade, H., Fuzzy sets and systems, Theory and Applcatons (Academc Press, New Yor, 980). [5Duran Tosar, M., Taylor seres approach to ntutonstc fuzzy mult-objectve lnear fractonal programmng, Informaton Scences 78 (2008), [6Dwyer, P.S.,Lnear Computaton, (New Yor, 95). [7 Edward. L. Hannan, Lnear programmng wth multple fuzzy goals, Fuzzy Sets and Systems 6 (98), [8 Lu H.W. Synthetc decson based on ntutonstc fuzzy relatons. Journal of Shandong Unversty of Technology, 2003, 33(5), [9 G.S.Mahapatra and T.K.Roy(2009), Relablty Evaluaton usng Trangular Intutonstc Fuzzy numbers Arthmetc operatons, World Academy of scence, Engneerng and Technology 50, [0 A.Nagoorgan and K.Ponnalagu (202), Solvng Lnear Programmng Problem n an Intutonstc Envronment, Proceedngs of the Heber nternatonal conference on Applcatons of Mathematcs and Statstcs, HICAMS 5-7. [ Nahmas, S., Fuzzy varables, Fuzzy sets and systems (2) (977)97-0. [2 Nguyen,H.T.,A Note on extenson prncple for fuzzy sets,j.math.anal.appl. 64(978) [3 Pal,B. B., Motra, B.N and Maul, U., A goal programmng procedure for ntutonstc fuzzy multobjectve lnear fractonal programmng problem, Fuzzy Sets and Systems 39 (2003), [4 Rardn, R.L.(2003),Optmzaton n Operatons Research, Pearson Educaton(Sngapore) Pvt.Ltd, Delh. [5 Twar, R.N., Dharmar,S and Rao,J.R., Fuzzy goal programmng - An addtve model, Fuzzy Sets and Systems 24 (987), [6 Xu Z.S. Intutonstc preference relatons and ther applcaton n group decson mang, Informaton Scences, 2007, 77: [7 Zadeh,L.A.,The concept of a Lngustc varable and ts applcatons to approxmate Reasonng parts I, II and III, Inform.Sc.8 (975)99-249; 8, ;9(976) [8 Zadeh, L.A., Fuzzy sets, Informaton and Control,No.8, pp , 965. [9 Zadeh,L.A., Fuzzy sets as a bass for a theory of possblty, Fuzzy sets and systems, No., pp.3-28, 978. [20 Zmmermann. H.J.,Fuzzy programmng and lnear programmng wth several objectve functons, Fuzzy Sets and Systems I (978),