A New Method for Solving Integer Linear. Programming Problems with Fuzzy Variables

Transcription

1 Appled Mathematcal Scences, Vl. 4, 00, n. 0, A New Methd fr Slvng Integer Lnear Prgrammng Prblems wth Fuzzy Varables P. Pandan and M. Jayalakshm Department f Mathematcs, Schl f Advanced Scences, VIT Unversty, Vellre-4, Inda. pandan6@redffmal.cm Abstract A new methd namely, decmpstn methd fr slvng nteger lnear prgrammng prblems wth fuzzy varables by usng classcal nteger lnear prgrammng has been prpsed. In the decmpstn methd, rankng functns are nt used. The prpsed methd can serve managers by prvdng the best slutn t a varety f nteger lnear prgrammng prblems wth fuzzy varables n a smple and effectve manner. Wth the help f numercal examples, the decmpstn methd s llustrated. Mathematcs Subect Classfcatns: 90C0, 90C70, 90C90, 65K05 Keywrds: Fuzzy thery, Fuzzy varables, Integer lnear prgrammng, Fuzzy nteger lnear prgrammng, Decmpstn methd Intrductn Lnear prgrammng [] has applcatns n many felds f peratns research. It s cncerned wth the ptmzatn f a lnear functn whle satsfyng a set f lnear equalty and/r nequalty cnstrants r restrctns. In real wrld stuatn the avalable nfrmatn n the system under cnsderatn are nt exact, therefre fuzzy lnear prgrammng (FLP) was ntrduced and studed by many researchers [0, 9, 4,, 6, 7, 8, ]. Fuzzy set thery has been appled t many dscplnes such as cntrl thery and management scences, mathematcal mdelng and ndustral applcatns. The cncept f FLP n a general level was frst prpsed by Tanaka et al. [0]. FLP prblems have an essental rle n fuzzy mdelng, whch can frmulate uncertanty n actual envrnment. Afterwards, many

2 998 P. Pandan and M. Jayalakshm authrs have cnsdered varus types f the FLP prblems and prpsed several appraches fr slvng these prblems. In partcular, the mst cnvenent methds are based n the cncept f cmparsn f fuzzy numbers wth help f rankng functns [,7,8]. Usually n such methds authrs defne a crsp mdel whch s equvalent t the FLP prblem and then use ptmal slutn f the mdel as the ptmal slutn f the FLP prblem. Mahdav-Amr and Nasser [7] ntrduced a dual smplex algrthm fr slvng lnear prgrammng prblem wth fuzzy varables and ts dual by usng a general lnear rankng functn and lnear prgrammng drectly. Recently, Herrera and Verdegay [5] have prpsed three methds fr slvng three mdels f fuzzy nteger lnear prgrammng based n the representatn therem and n fuzzy number rankng methd. Allahvranl et al. [] have prpsed a new methd based n fuzzy number rankng methd fr a FILP prblem va crsp nteger lnear prgrammng (ILP) prblems. Nasser [9] has prpsed a new methd fr slvng the FLP prblems n whch he has used the fuzzy rankng methd fr cnvertng the fuzzy bectve functn nt crsp bectve functn. In ths paper, we have prpsed a new methd namely, decmpstn methd fr slvng a ILP prblem wth fuzzy varables by usng the classcal ILP. The sgnfcance f ths paper s prvdng a new methd fr slvng ILP prblems wth fuzzy varables wthut usng any rankng functns. Ths methd can serve managers by prvdng the best slutn t a varety f nteger lnear prgrammng prblems wth fuzzy varables n a smple and effectve manner. The decmpstn methd s llustrated wth the help f numercal examples. Prelmnares We need the fllwng defntns f the basc arthmetc peratrs n fuzzy trangular numbers based n the functn prncple whch can be fund n [9]. Defntn. A fuzzy number a s a trangular fuzzy number dented by ( a, a, a) where a, a and a are real numbers and ts membershp functn μ a ( x) s gven belw. ( x a) /( a a) fr a x a μ a ( x) = ( a x) /( a a) fr a x a 0 therwse Defntn. Let ( a, a, a) and ( b, b, b ) be tw trangular fuzzy numbers. Then () a, a, ) b, b, ) = a + b, a + b, a + ). ( a ( b ( b

3 Methd fr slvng nteger lnear prgrammng prblems 999 () ( a, a, a) Θ ( b, b, b ) = ( a b, a b, a b ). () k ( a, a, a) = ( ka, ka, ka), fr k 0. (v) k ( a, a, a) = ( ka, ka, ka ), fr k < 0. Let F(R) be the set f all real trangular fuzzy numbers. Defntn. Let A = ( a, a, a ) and B = ( b, b, b ) be n F (R).Then, () A = B a = b, fr all fr = t and () A B a b, fr all fr = t. Defntn 4. Let A = ( a, a, a ) be n F (R). Then, ( ) A s sad t be pstve f a 0, fr all fr = t ; () A s sad t be nteger f a 0, = t are ntegers and () A s sad t be symmetrc f a a = a a. Defntn 5. A real fuzzy vectr b = ( b ) m s called nnnegatve and dented by b 0, f each element f b s a nnnegatve real fuzzy number, that s, b 0,,,,m. Cnsder the fllwng m n fuzzy lnear system wth nnnegatve real trangular fuzzy numbers: A x b, () where A = ( a ) s a nnnegatve crsp matrx and x = ( x ), ( ) m n b = b are nnnegatve fuzzy vectrs and x, b F( R ), fr all n and m. Defntn 6. A nnnegatve fuzzy vectr x s sad t be a slutn f the fuzzy lnear system () f x satsfes equatn (). Usng the defntns and 6 and the arthmetc peratns n trangular fuzzy number, we btan the fllwng therem. Therem. Let Ax b be an m n fuzzy lnear system where A = ( a ) s a m n nnnegatve crsp matrx, ( x = ) b = ( ) are nnnegatve real trangular fuzzy x x b ( b,, b b vectrs and x = ( x, x, ) and b = ) F(R), fr all n nx m. If x = ( x ) s a slutn f the system Ax b, x 0 where x = and b = b ), x = x ) s a slutn f the system ( x) nx ( mx ( nx Ax b, x 0, x x 0 where x = ( x ) nx and b = b ) and x = ( x ) nx s ( mx and

4 000 P. Pandan and M. Jayalakshm a slutn f the system Ax b, x x 0 where x = ( x) mx and b = ( b ) mx, then ( x = ) s a slutn f the system Ax b where (,, x = x x x ). x Nte. If x = ( x, x, x) and b = ( b, b, b ) are symmetrc, we can btan x ( frm the relatn x = x + x x ) wthut slvng the last system Ax = b, x x 0. Fuzzy nteger lnear prgrammng Cnsder the fllwng nteger lnear prgrammng prblem wth fuzzy varables : (P) Maxmze z = c x subect t Ax b, () x 0 and are ntegers, () where the ceffcent matrx A = ( a ) s a nnnegatve real crsp matrx, the cst m n vectr c = ( c,..., c n ) s nnnegatve crsp vectr and x = ( x ) nx and b = ( b ) mx are nnnegatve real fuzzy vectrs such that x, b F( R ) fr all n and m. Defntn 7. A fuzzy vectr x s sad t be a feasble slutn f the prblem (P) f x satsfes () and (). Defntn 8. A feasble slutn x f the prblem (P) s sad t be an ptmal slutn f the prblem (P) f there exsts n feasble u = ( u ) nx f (P) such that c u > c x. Usng the Therem. and the arthmetc peratns f fuzzy numbers, we can btan the fllwng result. Therem. A fuzzy vectr x = ( x, x, x ) s an ptmal slutn f the prblem (P) ff x, x and x are ptmal slutns f the fllwng crsp nteger lnear prgrammng prblems ( P ), ( P ) and ( P ) respectvely where ( P ) Maxmze z = cx subect t Ax b, x 0 are ntegers ; P ) Maxmze z = cx ( subect t Ax b, x 0, x and are ntegers x

5 Methd fr slvng nteger lnear prgrammng prblems 00 and ( P ) Maxmze z = cx subect t Ax b, x 0, x x and are ntegers. Prf: Suppse that x = ( x, x, x ) s an ptmal slutn f the prblem (P). Let x = ( x, x, x) be a feasble slutn f the prblem (P). Ths mples that x cx ; cx cx ; cx cx ; b ; Ax b; Ax b, x, x, x c Ax 0. (4) Let z = ( z, z, z ) be the bectve functn f the prblem (P). Nw, frm (4) we have, x x x Max. z = c ; Max. z = c ; Max. z = c (5) Nw, frm (4) and (5), we can cnclude that x, x and x are ptmal slutns f the crsp nteger lnear prgrammng prblems ( P ), ( P ) and ( P ). Suppse that x, x and x are ptmal slutns f the crsp nteger lnear prgrammng prblems ( P ), ( P ) and ( P ) wth ptmal values z, z and z respectvely. Ths mples that x = ( x, x, x ) s an ptmal slutn f the prblem (P) wth ptmal value z = ( z, z, z ). Hence the therem. 4 Numercal Examples The prpsed methd s llustrated by the fllwng examples. Example. Cnsder the fllwng nteger lnear prgrammng prblem wth fuzzy varables: (P) Maxmze z = 0 x + 0 x subect t 6 x 8 + x (46,48,60) x + x (7,,0) x, x 0 and are ntegers. Let z = ( z, z, ), x = ( y, x, ) and x = ( y, x, ). z t Nw, the prblem ( P ) s gven belw: ( P ) Maxmze z = 0x + 0x t

6 00 P. Pandan and M. Jayalakshm subect t 6x + 8x 48 ; x + x x, x 0 and are ntegers. Nw, usng an algrthm fr ILP prblem, the slutn f the prblem ( P ) s x = 5, x = and z = 90. Nw, the prblem ( P ) s gven belw: ( P ) Maxmze z = 0y + 0y subect t 6y + 8y 46; y + y 7 ; y 5 ; y y, y 0 and are ntegers. Nw, usng an algrthm fr ILP prblem, the slutn f the prblem ( P ) s y = 4, y = and z = 60. Nw, the prblem ( P ) s gven belw: ( P ) Maxmze z = 0t + 0t subect t 6t + 8t 60 ; t + t 0 ; t 5 ; t t, t 0 and are ntegers. Nw, usng an algrthm fr ILP prblem, the slutn f the prblem ( P ) s t = 6, t = and z = 0. Therefre, the slutn fr the gven fuzzy nteger lnear prgrammng prblem s x = ( y, x, t ) (4,5,6), x = ( y, x, t ) (,, ) and z = (60,90,0). = = Example. Cnsder the fllwng nteger lnear prgrammng prblem wth fuzzy varables: (P) Maxmze z = 4 x + x subect t x + x (4,8,) x + x (6,9,) x, x 0 and are ntegers. Let z = ( z, z, z), x = ( y, x, t) and x = ( y, x, t) and als, x and x be symmetrc. Nw, the prblem ( P ) s gven belw: ( P ) Maxmze z = 4x + x subect t x + x 8 ; x + x 9 ; x, x 0 and are ntegers. Nw, by usng an algrthm fr ILP prblem, the slutn f the prblem ( P ) s z = 9, x = 4 and x =.

7 Methd fr slvng nteger lnear prgrammng prblems 00 Nw, the prblem ( P ) s gven belw: ( P ) Maxmze z = 4y + y subect t y + y 4 ; y + y 6 ; y 4 ; y y, y 0 and are ntegers. Nw, usng an algrthm fr ILP prblem, slutn f the prblem ( P ) s z =, y = and y = 0. Nw, snce x = ( y, x, t) and x = ( y, x, t) are symmetrc, we have t = 5, t = and z = 6. Therefre, the slutn fr the gven fuzzy nteger lnear prgrammng prblem s x = ( y, x, t ) (,4,5) ; x = ( y, x, t ) (0,, ) and z = (,9,6). = = 5 Cnclusn The decmpstn methd prvdes an ptmal slutn t FILP prblems wthut usng rankng functns and applyng classcal nteger lnear prgrammng. Ths methd can serve managers by prvdng the best slutn t a varety f nteger lnear prgrammng prblems wth fuzzy varables n a smple and effectve manner. REFERENCES [] T. Allahvranl, KH. Shamslktab, N. A. Kan and L. Alzadeh, Fuzzy nteger lnear prgrammng prblems, Int. J. Cntemp. Math. Scences, (007), [] M.S. Bazaraa, J.J. Jarvs, and H.D. Sheral, Lnear prgrammng and netwrk flws, Jhn Wley and Sns, New Yrk, 990. [] L. Camps and J.L. Verdegay, Lnear prgrammng prblems and rankng f fuzzy numbers, Fuzzy Sets and Systems, (989), -. [4] M.Delgad, J.L.Verdegay and M.A.Vla, A general mdel fr fuzzy lnear prgrammng, Fuzzy Sets and Systems, 9 (989), -9. [5] F.Herrera and J.L.Verdegay, Three mdels f fuzzy nteger lnear prgrammng, Eurpean Jurnal f Operatnal Research, 8 (995), [6] Y.J.La and C.L.Hwang, Fuzzy mathematcal prgrammng methds and applcatns, Sprnger, Berln,99.

8 004 P. Pandan and M. Jayalakshm [7] N.Mahdav-Amr, and S.H.Nasser, Dualty results and a dual smplex methd fr lnear prgrammng prblems wth trapezdal fuzzy varables, Fuzzy Sets and Systems, 58 (007), [8] H.R.Malek, M.Tata and M.Mashnch, Lnear prgrammng wth fuzzy varables, Fuzzy Sets and Systems, 09 (000), -. [9] S.H. Nasser, A new methd fr slvng fuzzy lnear prgrammng by slvng lnear prgrammng, Appled Mathematcal Scences, (008), [0] H.Tanaka, T.Okuda and K.Asa, On fuzzy mathematcal prgrammng, The Jurnal f Cybernetcs, (974), [] J.L.Verdegay, A dual apprach t slve the fuzzy lnear prgrammng prblem, Fuzzy Sets and Systems, 4 (984), -4. Receved: Octber, 009