International Journal of Mathematical Archive-3(5), 2012, Available online through ISSN

Size: px

Start display at page:

Download "International Journal of Mathematical Archive-3(5), 2012, Available online through ISSN"

Douglas Anthony Allison
5 years ago
Views:

1 Iteratoal Joural of Matheatcal Archve-(5,, Avalable ole through ISSN FULLY FUZZY LINEAR PROGRAMS WITH TRIANGULAR FUZZY NUMERS S. Mohaaselv Departet of Matheatcs, SRM Uversty, Kattaulathur, Chea - 6, INDIA K. Gaesa Departet of Matheatcs, SRM Uversty, Kattaulathur, Chea - 6, INDIA (Receved o: -4-; Accepted o: 4-5- ASTRACT I ths paper we propose a ew ethod for solvg fully fuzzy lear prograg proble whose paraeters are tragular fuzzy uber. We prove soe of the portat theores for the soluto of fuzzy lear prograg proble whch tur helps us to solve the gve fully fuzzy lear prograg proble by applyg the fuzzy verso of Splex algorth wthout covertg to classcal proble. A uercal exaple s provded to llustrate the proposed ethod. Keywords: Tragular fuzzy ubers, Fuzzy Lear prograg, Rag, locato dex uber, fuzzess dex fuctos. AMS Subect Classfcato: 9C5, E7, 9C7.. INTRODUCTION Lear prograg s a oe of the ost portat operatoal research (OR techques appled to solve ay decso ag probles. To solve optzato odels operato research, the decso paraeters of the odel ust be ow precsely. ut real world systes, ay cases, the decso paraeters volved the proble are precse ature due to ay reasos ad have to be terpreted as fuzzy ubers to reflect the real world stuato. A crsp lear prograg proble s a atheatcal prograg proble havg a lear obectve fucto ad lear costrats whose coeffcets are crsp real ubers. If the coeffcets volved the obectve ad costrat fuctos are precse ature ad s terpreted as fuzzy ubers, the the resultg atheatcal proble s referred to as a fuzzy lear prograg proble. Fuzzy lear prograg proble occur ay felds such as Matheatcal odelg, Cotrol theory ad Maageet sceces, etc. The dea of fuzzy set was troduced by Zadeh [] 965. The cocept of a fuzzy decso ag was frst proposed by ella ad Zadeh []. A applcato of fuzzy optzato techques to lear prograg proble wth sgle ad ult-obectve fuctos has bee proposed by Zera []. I lterature several authors have proposed dfferet approaches for solvg fuzzy lear prograg proble such as Iuguch [5], Gaesa ad Veeraa [4], Roelfager [9], Nasser [8], Taaa ad Asa [] ad Male [6] etc. Most of the exstg ethods are based o the cocept of coparso of fuzzy ubers by use of rag fucto. Iuguch et.al [5] deals wth the fuzzy lear progra wth cotuous pecewse lear ebershp fuctos. They proposed a techque to solve the proble usg a stadard lear prograg whe ebershp fuctos are strctly quascocave ad the u operator s adopted for aggregatg fuzzy goals. Gaesa ad Veeraa [4] troduced a ew ethod based o pral splex algorth for solvg lear prograg proble wth syetrc trapezodal fuzzy ubers wthout covertg the to crsp lear prograg probles. Roelfager [8] proposed a ew ethod for solvg stochastc lear prograg probles wth fuzzy paraeters. Nasser et al. [8], have proposed a ew ethod for solvg fuzzy uber lear prograg probles, by use of lear rag fucto. Taaa ad Asa [] also proposed forulato of fuzzy lear prograg wth fuzzy costrats ad proposed a ethod for ts soluto based o equalty relato betwee fuzzy ubers. Male [6] proposed a ethod to ufy soe of the exstg approaches whch are usg dfferet rag fuctos for solvg fuzzy prograg probles. I ths paper we propose a ew ethod for solvg fully fuzzy lear prograg proble whose coeffcets all are represeted by tragular fuzzy uber. We propose a fuzzy verso of Splex ethod to solve the gve fully fuzzy Correspodg author: K. Gaesa Departet of Matheatcs, SRM Uversty, Kattaulathur, Chea - 6, INDIA Iteratoal Joural of Matheatcal Archve- (5, May 88

2 S. Mohaaselv & K. Gaesa/ FULLY FUZZY LINEAR PROGRAMS WITH TRIANGULAR FUZZY NUMERS / IJMA- (5, May-, Page: lear prograg proble wth tragular fuzzy uber wthout covertg the gve proble to crsp equvalet proble. The rest of the paper s orgazed as follows. I secto, we recall the basc cocepts ad the results of tragular fuzzy ubers ad ther arthetc operatos. I secto, we troduce the fully fuzzy lear prograg proble wth tragular fuzzy ubers ad related results. I secto 4, we prove soe of the portat theores for the soluto of fully fuzzy lear prograg proble wth tragular fuzzy ubers ad apply fuzzy verso of splex algorth ad obtaed the fuzzy optal soluto. A uercal exaple s also provded to llustrate the theory developed ths paper.. PRELIMINARIES Defto.: A fuzzy set a defed o the set of real ubers R s sad to be a fuzzy uber f ts ebershp fucto a : R [,] has the followg characterstcs: (. a s covex, + ( { } (. a s oral.e. there exsts a x R such that (.e. a(λx - λ x a(x, a(x, for all x, x R ad λ [,]. a x = (. a s Pecewse cotuous. Defto.: A fuzzy uber a o R s sad to be a tragular fuzzy uber (TFN or lear fuzzy uber f ts ebershp fucto a : R [,] has the followg characterstcs: ( x-a, for a x a a -a a -x a x =, for a x a a -a, elsewhere We deote ths tragular fuzzy uber by a = (a, a, a. We use F(R to deote the set of all tragular fuzzy ubers. Also f = a represets the odal value or dpot, α= (a a represets the left spread ad β= (a a represets the rght spread of the tragular fuzzy uber a = (a, a, a, the the tragular fuzzy uber a ca be a = α,, β. a=(a,a,a = α,, β. represeted by the trplet (.e. ( Defto.: A tragular fuzzy uber a F(R a ( r ad a ( r for r whch satsfes the followg requreets: (. a( r s a bouded ootoc creasg left cotuous fucto. (. a( r s a bouded ootoc decreasg left cotuous fucto. (. a( r a( r, r ca also be represeted as a par a, a of fuctos, IJMA. All Rghts Reserved 89

3 S. Mohaaselv & K. Gaesa/ FULLY FUZZY LINEAR PROGRAMS WITH TRIANGULAR FUZZY NUMERS / IJMA- (5, May-, Page: a a( + a ( = a,a, the uber a = s sad to be a Defto.4: For a arbtrary tragular fuzzy uber ( locato dex uber of a. The two o-decreasg left cotuous fuctos a = (a a, a = (a a are called the left fuzzess dex fucto ad the rght fuzzess dex fucto respectvely. Hece every tragular fuzzy uber a = (a,a,a a = a,a,a ca also be represeted by (. Rag of tragular Fuzzy Nubers May dfferet approaches for the rag of fuzzy ubers have bee proposed the lterature. Abbasbady ad Haar [] Proposed a ew rag ethod based o the left ad the rght spreads at soe α -levels of fuzzy ubers. a = (a, a, a = a,a,a a = a(r, a(r, we defe For a arbtrary tragular fuzzy uber ( wth paraetrc for ( the agtude of the tragular fuzzy uber a by Mag(a = (a+ a+ af(rdr = ( a + 4a a f (r dr where the fucto f (r s a o-egatve ad creasg fucto o [,] wth f ( =, f ( = ad f (r dr =. The fucto f(r ca be cosdered as a weghtg fucto. I real lfe applcatos, f(r ca be chose by the decso aer accordg to the stuato. I ths paper, for coveece we use f(r = r. a + 4 a a a+ a+ a Hece Mag(a = =. 4 4 The agtude of a tragular fuzzy uber a sythetcally reflects the forato o every ebershp degree, ad eag of ths agtude s vsual ad atural. Mag ( a s used to ra fuzzy ubers. The larger Mag( a, the larger fuzzy uber. For ay two tragular fuzzy ubers a,a,a ad b = ( b,b,b by coparg the Mag ( a ad Mag ( b o R as follows: (. a b f ad oly f Mag( a Mag( b (. a b f ad oly f Mag( a Mag( b (. a b f ad oly f Mag( a = Mag( b Defto.5: A tragular fuzzy uber a,a,a Defto.6: A tragular fuzzy uber a,a,a deoted by a. Further f Mag( a >, the a,a,a a. F(R, we defe the rag of a ad b s sad to be syetrc f ad oly f a =a. Defto.7: Two tragular fuzzy ubers a,a,a ad b = ( b,b,b s sad to be o-egatve f ad oly f Mag( a ad s s sad to be a postve fuzzy uber ad s deoted by F(R are sad to be equvalet f ad oly f Mag( a = Mag( b.that s a b f ad oly f Mag( a = Mag( b.two tragular fuzzy ubers a = a,a,a b = b,b,b F(R are sad to be equal f ad oly f a = b,a = b,a = b. That s a = b f ( ad ( ad oly f a = b,a = b,a = b.. Arthetc operato o tragular Fuzzy Nubers Mg Ma et al. [7] have proposed a ew fuzzy arthetc based upo both locato dex ad fuzzess dex fuctos. The locato dex uber s tae the ordary arthetc, whereas the fuzzess dex fuctos are, IJMA. All Rghts Reserved 84

4 ad for all =,,... S. Mohaaselv & K. Gaesa/ FULLY FUZZY LINEAR PROGRAMS WITH TRIANGULAR FUZZY NUMERS / IJMA- (5, May-, Page: cosdered to follow the lattce rule whch s least upper boud the lattce L. That s for a, b L we defe a b = ax a, b ad a b = a, b. { } { } ad { } For arbtrary tragular fuzzy ubers a,a,a ad b = ( b,b,b operatos o the tragular fuzzy ubers are defed by = ( I partcular for ay two tragular fuzzy ubers a,a,a ad b = ( b,b,b (. Addto: a + b,a,a + ( b,b,b + b,ax{ a,b},ax{ a,b } (. Subtracto: a b,a,a ( b,b,b b,ax{ a,b},ax{ a,b }. (. Multplcato: a b,a,a ( b,b,b b,ax{ a,b},ax{ a,b }. (v. Dvso: a b,a,a ( b,b,b = a b,ax{ a,b},ax{ a,b }.. FUZZY LINEAR PROGRAMMING PROLEM a b a b,a b,a b. ( = +,,,, the arthetc, we defe Let F(R be set of all tragular fuzzy ubers. A fuzzy lear prograg proble wth tragular fuzzy ubers s defed as follows: ax Z cx = = subect to a b for all =,,..., where = a x b for all = +,... ( ad for all =,,... a, c, x, b F(R, =,,,. ad =,,,.,. If a,c,x ad b are represeted by locato dex uber, left fuzzess dex fucto ad rght fuzzess dex fucto respectvely, the the above proble ca be rewrtte as follows: = ax Z (c, (c,(c (x,(x,(x = (a, (a,(a (x,(x,(x (b,(b,(b forall =,,..., = (a,(a,(a (x,(x,(x (b,(b,(b forall = +,... (x,(x,(x for all =,,... ( Defto.: Ay vector x = ( x T ( (, x,..., F R (.e. x { (x,(x,(x, (x,(x,(x,..., (x,(x,(x } F( R = s sad to be a fuzzy feasble soluto, f for each x ( ( = (x,(x,(x F R satsfes the costrats ad o-egatvty restrctos of (. Defto.: A fuzzy feasble soluto s sad to be a fuzzy optal soluto f t optzes the obectve fucto of the fuzzy lear prograg proble. Defto.: Suppose the costrats Ax b where ( A F(R, x ( F(R ad b ( F(R. Let x = ( x, x,..., solves Ax b.if all x = (x,(x,(x for soe (x, the s sad to be a fuzzy basc soluto. If x (x,(x,(x for (x, the has soe o-zero copoets, say, x, x,...,,., IJMA. All Rghts Reserved 84

5 S. Mohaaselv & K. Gaesa/ FULLY FUZZY LINEAR PROGRAMS WITH TRIANGULAR FUZZY NUMERS / IJMA- (5, May-, Page: The Ax b ca be wrtte as a x + a x a x + a + x a x b. If the colus a, a,, a correspodg to these o-zero copoets x, x,..., are learly depedet, the s sad to be a fuzzy basc soluto. Let be ay ( atrx fored by '' learly depedet colu of A. The x b. Now = ( x, x,..., x,,,..., s a fuzzy basc soluto. We represet the fuzzy basc soluto by. Defto.4: A fuzzy basc soluto whch satsfes the o-egatvty costrats s called Fuzzy basc feasble soluto. Defto.5: A stadard for of fuzzy lear prograg proble s defed as follows: C = ax Z.e ax Z Cx subect to A b ad x ( A FR, Cx, where ( ( F(R ad ( F(R b ( 4. IMPROVING FUZZY ASIC FEASILE SOLUTION Cosder the stadard for of fuzzy lear prograg proble, ax Z Cx subect to Ax b ad. Here we assue that ra of A=. a..e [ a a a ].Let be a ( Let the colus of A be gve by A=,,..., o-sgular atrx whose colus are learly depedet colu of A ad = [ b, b,..., b ]. Let be a fuzzy basc soluto..e. = x, x,..., where = b.correspodg to ay such,we defe C,called reduced fuzzy cost vector, cotag the prces of the basc varables.e. C = c,c,...,c.the value of the obectve fucto s T gve by Z=C. To prove the obectve fucto Z, we have to fd other fuzzy basc feasble soluto by replacg oe of the colus of bass atrx. Theore 4.: Let = b s a fuzzy basc soluto of (.If for ay colu a A whch s ot, the codto ( Z C hold ad y for soe =,,..., ad the t s possble to obta a ew fuzzy basc feasble soluto by replacg oe of the colus by a. Proof: Suppose that x = b be a fuzzy basc feasble soluto wth postve copoets such that = x, x,..., x where {,,..., } x = (x,(x,(x ad (x > for =,,..., ad (x = for = +,...,. Now the equato x = bbecoes b +,(x,(x b ,(x,(x b b = + + +,(x,(x = = + b + b b (4 The for ay colu a A whch s ot, we wrte a = y = y + y y r r y y = b b b b b We ow that f the bass vector b r for whch b, b,..., b, a, b,..., b stll for a bass. ( r r+ y s replaced by a A, the the ew set of vectors r, IJMA. All Rghts Reserved 84

6 S. Mohaaselv & K. Gaesa/ FULLY FUZZY LINEAR PROGRAMS WITH TRIANGULAR FUZZY NUMERS / IJMA- (5, May-, Page: a Now for yr y a y y ad r, we have, = b = r b b b. y = y y = y = + y r r r r r r r The (4 becoes y y y a x x a + b = +,(x r,(x b b b b b = yr = yr yr = yr = + y r = + r r r b + a + b b r r r y,(x,(x y = y r yr = + y r r = r r r r y b + yr a b whch gves a ew fuzzy basc feasble soluto to Ax b. yr x.e. xˆ ˆ + x x r r b a r b wherexˆ = y ˆ ad x =. r = y y r r Theore 4.: Let be a fuzzy basc feasble soluto to a fuzzy lear prograg proble wth correspodg T obectve value Ζ ˆ =C x.if (Z C for every colu a A, the s optal. Proof: For a gve ay feasble soluto x, we have ( Z C T Z = Cx Zx = ( C = C y = yx C for all =,,..., T ad by Ζ ˆ =C x, we have = = = = = = = Z yx C. That s = yx =, {,,...,} = = = Sce x s a feasble soluto, Ax b ad by y = a. b = ( a = y = yb = b = x. Sce s o-sgular ad b. Therefore =. Hece = = = = = Z x C = Z ˆ for all x the feasble soluto. = 5. A NUMERICAL EXAMPLE Cosder a exaple dscussed At Kuar et al [], ax (,, + (,, 4 such that (,, + (,, (,, 7 (,, + (,, (,, 8 x, Represetg the tragular fuzzy ubers ters of left ad rght dex fucto, we have ( + ( ( + ( ( (, r, r + (, r, r (,9 9r,7 7r ax, r, r x, r, r x such that, r, r x, r, r x,9 9r,7 7r Soluto: Ital terato: C y (, 9 9r,7 7r (, r, r (, r, r (, 9 9r,7 7r (, r, r (, r, r Z C, r, r, r, r ( 5, 9 9r,7 7r (, 9 9r,7 7r ( (, IJMA. All Rghts Reserved 84

7 S. Mohaaselv & K. Gaesa/ FULLY FUZZY LINEAR PROGRAMS WITH TRIANGULAR FUZZY NUMERS / IJMA- (5, May-, Page: Frst terato: C y (, r, r ( 5, 9 9r,7 7r (, r, r (, r, r (, r, r (, r, r (, 9 9r,7 7r ( 6, 9 9r,7 7r Z C ( 5, r, r (, r, r (, r, r (, r, r (, r, r ( 4, 9 9r,7 7r (, r, r (, r, r (, r, r (, r, r Secod terato: C y (, r, r (, 9 9r,7 7r (, r, r (, r, r (, r, r ( 4, 9 9r,7 7r (, r, r (, r, r Z C ( 7, 9 9r,7 7r (, r, r (, r, r (, r, r (, r, r (, r, r (, r, r 8 (, r, r (, r, r Sce all (Z C, the curret fuzzy basc feasble soluto s fuzzy optal. The fuzzy optal soluto s = ( 4, 9 9r,7 7r, = (, 9 9r,7 7r wth Z = ( 7, 9 9r,7 7r. For r =, we have the fuzzy optal soluto ters of locato dex ad fuzzess dex as = ( 4, 9,7 ; = (, 9,7 ; Z = ( 7,9,7. Hece the fuzzy optal soluto the geeral for, ters of a=(a,a,a for r = s = ( 5,4,, = ( 6,, wth Z = ( 8,7, 4. Table 5.: Coparso of Fuzzy Optal Soluto obtaed by our ethod ad by At uar et al ethod. Fuzzy Optal Soluto For r Our Method ax Z = (,, + (,, 4 = 7, 9 9r,7 7r At Kuar et al Method = ( + ( ( = ax Z = ( 8,7,4 ax Z = ( 4,7,8 = ax Z = (.5,7, = ax Z = (.5,7, = ax Z = ( 4.75,7, For r.5 For r.5 ax Z,,,, 4 For r.75 For r = ax Z = It s see that the Decso Maer have the flexblty of choosg r [,] depedg upo the stuato ad hs wsh by applyg the proposed ethod ths paper whereas t s ot possble by applyg At uar et al ethod. 6. CONCLUSION I ths paper, we proposed a ew ethod for solvg fully fuzzy lear prograg proble wth tragular fuzzy uber. The tragular fuzzy ubers are represeted ters of locato dex uber, left fuzzess dex fucto ad rght fuzzess dex fucto respectvely. We appled fuzzy verso of splex algorth for the fuzzy optal soluto of the fully fuzzy lear prograg proble. A uercal exaple dscussed by At uar et al [] s solved usg the proposed ethod wthout covertg the gve proble to crsp equvalet proble. It s to be oted, IJMA. All Rghts Reserved 844

8 S. Mohaaselv & K. Gaesa/ FULLY FUZZY LINEAR PROGRAMS WITH TRIANGULAR FUZZY NUMERS / IJMA- (5, May-, Page: that the Decso Maer have the flexblty of choosg r [,] depedg upo the stuato ad hs wsh by applyg the proposed ethod. ACKNOWLEDGEMENT The authors are grateful to the aoyous referees ad the edtors for ther costructve coets ad valuable suggestos whch helped the authors to prove the presetato of ths paper. 7. REFERENCES [] Abbasbady.S ad Haar. T, A ew approach for rag of trapezodal fuzzy ubers, Coputers ad Matheatcs wth Applcatos, 57 (9, [] At Kuar, Jagdeep Kaur ad Pushpder Sgh., Fuzzy Optal Soluto of Fully Fuzzy Lear Prograg Probles wth Iequalty Costrats, Iteratoal Joural of Matheatcal ad Coputer Sceces, 6( (, 7-4. [] ella, R.E ad Zadeh, L.A., Decso-ag fuzzy evroet, Maageet sceces, 7(4 (97, [4] Gaesa, K., & Veeraa, P., Fuzzy lear progras wth trapezodal fuzzy ubers. Aals Of Operato Research, 4(6, 5-5. [5] Iuguch.M, Ichhash. H ad Kue Y., A soluto algorth for fuzzy lear prograg wth pecewse lear ebershp fucto, Fuzzy Sets ad Systes, 4(99, 5. [6] Male. H. R, Rag fuctos ad ther applcatos to fuzzy lear prograg, Far East J. Appl. Math., 4(, 8. [7] Mg Ma, Meahe Freda, Abraha adel, A ew fuzzy arthetc, Fuzzy sets ad systes, 8(999, 8-9. [8] Nasser, S. H., Ardl, E., Yazda, A., ad Zaefara.R, Splex ethod for solvg lear prograg probles wth fuzzy ubers. Proceedgs of World Acadey of Scece, Egeerg ad Techology, (5, [9] Roelfager, H., Hauschec, R., ad Wolf, J.,Lear prograg wth fuzzy obectve. Fuzzy Sets ad Systes, 9(989, -48. [] Taaa.H, Ouda.T, ad Asa. K, O fuzzy atheatcal prograg, J. Cyberetcs, (984, [] Zadeh.L.A, Fuzzy sets, Iforato Cotrol, 8(965,8-5, [] Zera H. J., Fuzzy prograg ad lear prograg wth several obectve fuctos, Fuzzy sets ad systes, (978,45-55 Source of support: Nl, Coflct of terest: Noe Declared, IJMA. All Rghts Reserved 845

A New Method for Solving Fuzzy Linear. Programming by Solving Linear Programming

A New Method for Solving Fuzzy Linear. Programming by Solving Linear Programming ppled Matheatcal Sceces Vol 008 o 50 7-80 New Method for Solvg Fuzzy Lear Prograg by Solvg Lear Prograg S H Nasser a Departet of Matheatcs Faculty of Basc Sceces Mazadara Uversty Babolsar Ira b The Research