M.Jayalakshmi and P. Pandian Department of Mathematics, School of Advanced Sciences, VIT University, Vellore-14, India.

Size: px

Start display at page:

Download "M.Jayalakshmi and P. Pandian Department of Mathematics, School of Advanced Sciences, VIT University, Vellore-14, India."

Roy Hamilton
5 years ago
Views:

1 M.Jayalakshmi, P. Padia / Iteratioal Joural of Egieerig Research ad Applicatios (IJERA) ISSN: Vol., Issue 4, July-August 0, pp A New Method for Fidig a Optimal Fuzzy Solutio For Fully Fuzzy Liear Programmig Problems M.Jayalakshmi ad P. Padia Departmet of Mathematics, School of Advaced Scieces, VIT Uiversity, Vellore-4, Idia. Abstract A ew method amely, boud ad decompositio method is proposed to fid a optimal fuzzy solutio for fully fuzzy liear programmig (FFLP) problems. I the proposed method, the give FFLP problem is decomposed ito three crisp liear programmig (CLP) problems with bouded variables costraits, the three CLP problems are solved separately ad by usig its optimal solutios, the fuzzy optimal solutio to the give FFLP problem is obtaied. Fuzzy rakig fuctios ad additio of oegative variables were ot used ad there is o restrictio o the elemets of coefficiet matri i the proposed method. The boud ad decompositio method is illustrated by umerical eamples. Keywords: Liear programmig, Fuzzy variables, Fuzzy liear programmig, Boud ad Decompositio method, Optimal fuzzy solutio. Itroductio Liear programmig (LP) is oe of the most frequetly applied operatios research techiques. Although, it is ivestigated ad epaded for more tha si decades by may researchers from the various poit of views, it is still useful to develop ew approaches i order to better fit the real world problems withi the framework of liear programmig. I the real world situatios, a liear programmig model ivolves a lot of parameters whose values are assiged by eperts. However, both eperts ad decisio makers frequetly do ot precisely kow the value of those parameters. Therefore, it is useful to cosider the kowledge of eperts about the parameters as fuzzy data [5]. Usig the cocept of decisio makig i fuzzy eviromet give by Bellma ad Zadeh [5], Taaka et al. [4] proposed a method for solvig fuzzy mathematical programmig problems. Zimmerma [6] developed a method for solvig fuzzy LP problems usig multiobective LP techique. Campos ad Verdegay [8] proposed a method to solve LP problems with fuzzy coefficiets i both matri ad right had of the costrait. Iuiguchi et al. [5] used the cocept of cotiuous piecewise liear membership fuctio for FLP problems. Cadeas ad Verdegay [7] solved a LP problem i which all its elemets are defied as fuzzy sets. Fag et al. [] developed a method for solvig LP problems with fuzzy coefficiets i costraits. Buckley ad Feurig [6] proposed a method to fid the solutio for a fully fuzzified liear programmig problem by chagig the obective fuctio ito a multiobective LP problem. Maleki et al. [9] solved the LP problems by the compariso of fuzzy umbers i which all decisio parameters are fuzzy umbers. Liu [7] itroduced a method for solvig FLP problems based o the satisfactio degree of the costraits. Maleki [0] proposed a method for solvig LP problems with vagueess i costraits by usig rakig fuctio. Zhag et al. [7] itroduced a method for solvig FLP problems i which coefficiets of obective fuctio are fuzzy umbers. Nehi et al. [] developed the cocept of optimality for LP problems with fuzzy parameters by trasformig FLP problems ito multiobective LP problems. Ramik [3] proposed the FLP problems based o fuzzy relatios. Gaesa ad Veeramai [3] proposed a approach for solvig FLP problem ivolvig symmetric trapezoidal fuzzy umbers without covertig it ito crisp LP problems. Hashemi et al. [4] itroduced a two phase approach for solvig fully fuzzified liear programmig. Jimeez et al. [6] developed a method usig fuzzy rakig method for solvig LP problems where all the coefficiets are fuzzy umbers. Allahviraloo et al. [] solved fuzzy iteger LP problem by reducig it ito two crisp iteger LP problems. Allahviraloo et al. [] proposed a method based o rakig fuctio for solvig FFLP problems. Nasseri [] solved FLP problems by usig classical liear programmig. Ebrahimead ad Nasseri [0] solved FLP problem with fuzzy parameters by usig the complemetary slackess theorem. Ebrahimead et al. [] proposed a ew primal-dual algorithm for solvig LP problems with fuzzy variables by usig duality theorems. Dehgha et al. [9] proposed a FLP approach for fidig the eact solutio of fully fuzzy liear system of equatios which is applicable oly if all the elemets of the coefficiet matri are o-egative fuzzy umbers. Lotfi et al. [8] proposed a ew 47 P a g e

2 M.Jayalakshmi, P. Padia / Iteratioal Joural of Egieerig Research ad Applicatios (IJERA) ISSN: Vol., Issue 4, July-August 0, pp method to fid the fuzzy optimal solutio of FFLP problems with equality costraits which ca be applied oly if the elemets of the coefficiet matri are symmetric fuzzy umbers ad the obtaied solutios are approimate but ot eact. Amit Kumar et al. [3, 4] proposed a method for solvig the FFLP problems by usig fuzzy rakig fuctio i the fuzzy obective fuctio. I this paper, a ew method amely, boud ad decompositio method is proposed for fidig a optimal fuzzy solutio to FFLP problems. I this method, the FFLP problem is decomposed ito three CLP problems ad the usig its solutios, we obtai a optimal fuzzy solutio to the give FFLP problem. With the help of umerical eamples, the boud ad decompositio method is illustrated. The advatage of the boud ad decompositio method is that there is o restrictio o the elemets of the coefficiet matri, fuzzy rakig fuctios ad oegative variables are ot used, the obtaied results eactly satisfy all the costraits ad the computatio i the proposed method is more easy ad also, simple because of the LP techique ad the level wise computatio. The boud ad decompositio method ca serve maagers by providig a appropriate best solutio to a variety of liear programmig models havig fuzzy umbers ad variables i a simple ad effective maer.. Prelimiaries We eed the followig defiitios of the basic arithmetic operators ad partial orderig relatios o fuzzy triagular umbers based o the fuctio priciple which ca be foud i [, 4, 5,6 ]. Defiitio. A fuzzy umber a is a triagular fuzzy umber deoted by (, a, are real umbers ad its member ship fuctio a ( ) is give below: ( a) /( a a) for a a a ( ) ( a3 ) /( a3 a) for a a3 0 otherwise Defiitio. Let ( a, a, ad ( b, b ) be two triagular fuzzy umbers. The (i) ( a, a, ( b, b ) = ( a b, a b, a3 b3 ). (ii) ( a, a, ( b, b ) = ( a b3, a b, a3 b ). (iii) k ( a, a, = ( ka, ka, ka3 ), for k 0. (iv) k ( a, a, = ( ka 3, ka, ka ), for k 0. ( ab, ab, a3b3 ), a ( b, b ) ( ab3, ab, a3b3 ), ( ab3, ab, a3b ), (v) (, a, Let F(R) be the set of all real triagular fuzzy umbers. Defiitio.3 Let ( a, a, ) (i) (iii) a a 0, a a 0, 3 3 0, 0. A a3 ad B ( b, b ) be i F (R), the A B iff ai b i, i,, 3 ; (ii) A B iff ai b i, i,, 3 A B iff ai b i, i,, 3 ad A 0 iff a i 0, i,, 3. a where a, a ad a3 3. Fully Fuzzy Liear Programmig Problem Cosider the followig fully fuzzy liear programmig problems with m fuzzy iequality/equality costraits ad fuzzy variables may be formulated as follows: (P) Maimize (or Miimize) z ct 48 P a g e

3 M.Jayalakshmi, P. Padia / Iteratioal Joural of Egieerig Research ad Applicatios (IJERA) ISSN: Vol., Issue 4, July-August 0, pp subect to A {,, } b, () 0 () where T c c, A a i, m ( ) ad b ( bi ) m ad a, c,, b F( R i i ), for all ad i m. Let the parameters a, c, ad b be the triagular fuzzy umber p, q, r ),, y, t ), ( i i i a, b, c ) ad b, g, h ) i ( i i i i ( respectively. The, the problem (P) ca be writte as follows: (P) Maimize (or Miimize) z, z z p, q, r, y, t subect to, 3 a, b, c, y, t {,, } b, g, h i i i, y, t 0 i i i, for all i,,..., m. Now, sice, y, t is a triagular fuzzy umber, the y t, =,,,m. (3) The relatio (3) is called bouded costraits. ( Now, usig the arithmetic operatios ad partial orderig relatios, we decompose the give FLPP as follows: Maimize z = lower value of ( p, q, r ) (, y, t ) Maimize Maimize 3 z = middle value of ( p, q, r ) (, y, t ) z = upper value of ( p, q, r ) (, y, t ) subect to lower value of middle value of upper value of ad all decisio variables are o-egative. ( a i, bi, ci) (, y, t ) {,, } bi ( a i, bi, ci) (, y, t ) {,, } gi ( a i, bi, ci ) (, y, t ) {,, } hi, for all i,,..., m ;, for all i,,..., m ;, for all i,,..., m From the above decompositio problem, we costruct the followig CLP problems amely, middle level problem (MLP), upper level problem (ULP) ad lower level problem (LLP) as follows: 49 P a g e

4 M.Jayalakshmi, P. Padia / Iteratioal Joural of Egieerig Research ad Applicatios (IJERA) ISSN: Vol., Issue 4, July-August 0, pp (MLP) Maimize z = middle value of ( p, q, r ) (, y, t ) subect to Costraits i the decompositio problem i which atleast oe decisio variable of the (MLP) occurs ad all decisio variables are o-egative. (ULP) Maimize 3 subect to ad (LLP) Maimize where z = upper value of ( p, q, r ) (, y, t ) ( p, q, r ) (, y, t ) z ; upper value of Costraits i the decompositio problem i which atleast oe decisio variable of the (ULP) occurs ad are ot used i (MLP) ; all variables i the costraits ad obective fuctio i (ULP) must satisfy the bouded costraits ; replacig all values of the decisio variables which are obtaied i (MLP) ad all decisio variables are o-egative. subect to z = lower value of ( p, q, r ) (, y, t ) ( p, q, r ) (, y, t ) z ; lower value of Costraits i the decompositio costraits i which atleast oe decisio variable of the (LLP) occurs which are ot used i (MLP) ad (ULP); all variables i the costraits ad obective fuctio i (LLP) must satisfy the bouded costraits; replacig all values of the decisio variables which are obtaied i the (MLP) ad (ULP) ad all decisio variables are o-egative, z is the optimal obective value of (MLP). Remark 3.: I the case of LP problem ivolvig trapezoidal fuzzy umbers ad variables, we decompose it ito four CLP problems ad the, we solve the middle level problems ( secod ad third problems) first. The, we solve the upper level ad lower level problems ad the, we obtai the fuzzy optimal solutio to the give FLP problem ivolvig trapezoidal fuzzy umbers ad variables. 4. The Boud ad Decompositio Method Now, we prove the followig theorem which is used i the proposed method amely. Boud ad decompositio method to solve the FFLP problem. Theorem 4.: Let [ M ] {, M } be a optimal solutio of (MLP), [ U ] {, U } be a optimal solutio of ) (ULP ad [ ] {, L} L be a optimal solutio of (LLP) where L, M ad U are sets of decisio variables i the ( LLP ),( MLP) ad ( ULP) respectively. The 3 (,, ),,,..., 3, ad,,,..., is a optimal fuzzy solutio to the give problem (P) where each oe of is a elemet of L,M ad U. 50 P a g e

5 M.Jayalakshmi, P. Padia / Iteratioal Joural of Egieerig Research ad Applicatios (IJERA) ISSN: Vol., Issue 4, July-August 0, pp Proof: Let [ y ] { y,,,..., } be a feasible solutio of (P). Clearly, [ y M ],[ yu ]ad [ y L ] are feasible solutios of ( MLP ),( ULP) ad ( LLP) respectively. Now, sice [ M ],[ U ]ad [ L] are optimal solutios of ( ),( ULP) ad ( LLP) Z ([ L]) Z([ yl]); Z([ M ]) Z([ ym ]) ad Z3([ U ]) Z3([ y Z([ ]) Z([ y { 3 (,, ),,,..., U This implies that ]), for all feasible solutio of the problem (P). Therefore, } each oe of 3, ad,,,..., MLP respectively, we have ]) is a optimal fuzzy solutio to the give problem (P) where is a elemet of L, M ad U. Hece the theorem. Now, we propose a ew method amely, boud ad decompositio method for solvig a FFLP problem. The boud ad decompositio method proceeds as follows. Step : Costruct (MLP), (ULP) ad (LLP) problems from the give the FFLP problems. Step : Usig eistig liear programmig techique, solve the (MLP) problem, the the (ULP) problem ad the, the (LLP) problem i the order oly ad obtai the values of all real decisio variables z, z ad z, y ad t ad values of all obectives 3. Let the decisio variables values be, y ad t, =,,,m ad obective values be z, z ad z3. Step 3: A optimal fuzzy solutio to the give FFLP problems is (, y, t ), =,,,m ad the maimum fuzzy obective is ( z, z, z3 ) ( by the Theorem 4..). Now, we illustrate the proposed method usig the followig umerical eamples Eample 4.: Cosider the followig fully fuzzy liear programmig problem (Obective fuctio cotais egative term) z,, 3, 3, 4 Maimize subect to 0,,,, 3 (,0, 4 );,, 3 0,, (, 8, ); ad 0. Let (, y, ), (, y, ) ad z ( z, z, ). t t Now, the decompositio problem of the give FLPP is give below: z t Maimize Maimize z y 3y Maimize z3 3t 4t subect to ; ; 0 0 z3 5 P a g e

6 M.Jayalakshmi, P. Padia / Iteratioal Joural of Egieerig Research ad Applicatios (IJERA) ISSN: Vol., Issue 4, July-August 0, pp y y 0 ; y y 8 ;. t 3t 4 ; 3t t ;, 0, y, y 0, t, t 0. Now, the Middle Level problem is give below: ( P ) : Maimize z y 3y subect to y y 0 ; y y 8 ; y, y 0. Now, solvig the problem ( P ) usig simple method, we obtai the optimal solutio y ; y 4 ad z 6. Now, the Upper Level problem is give below: ( P 3 ) : Maimize z3 3t 4t subect to 3t 4t 6 ; t 3t 4 ; 3t t ; t y ; t y ; t, t 0. Now, solvig the problem ( P3 ) with y ; y 4 usig simple method, we obtai the optimal solutio ad t ; t 6 ad z Now, the Lower Level problem: ( P ) : Maimize z t subect to t 6 ; 0 ; 0 ; y, y ;, 0 Now, solvig the problem ( P ) with t 3; y ; y 4 solutio ; ad z. Therefore, a optimal fuzzy solutio to the give fully fuzzy liear programmig problem is z (,6, 33. (,, 3 ), (, 4, 6 ) ad ). by simple method, we obtai the optimal Remark 4.: The solutio of the Eample 4. is obtaied at the 8 th iteratio with 0 o-egative variables by Amit Kumar et al.[4] method. Eample 4.: Cosider the followig fully fuzzy liear programmig problem: z, 6, 9, 3, 8 Maimize subect to, 3, 4,, 3 ( 6,6, 30 );,,, 3, 4 (,7, 30 ); ad 0. Now, by solvig the boud ad decompositio method, a optimal fuzzy solutio to the give fully fuzzy liear programmig problem is (,, 3 ), ( 4, 5, 6 ) ad z ( 9, 7, 75 ). Remark 4.: The solutio of the Eample 4., obtaied by the boud ad decompositio method is same as i Amit Kumar et al. [4]. Eample 4.3: Cosider the followig fully fuzzy liear programmig problem: z, 6, 9,, 8 Maimize subect to 5 P a g e

7 M.Jayalakshmi, P. Padia / Iteratioal Joural of Egieerig Research ad Applicatios (IJERA) ISSN: Vol., Issue 4, July-August 0, pp ,,,, 3 ( 4, 7,4 );,, 3, 4, 4 ( 4,4, );, 3, 4,, 3 (, 3, 6 ); ad 0. Now, by solvig the boud ad decompositio method, a optimal fuzzy solutio to the give fully fuzzy liear programmig problem is ( 0,, ), (, 3, 4 ) ad z ( 4,, 50 ). Remark 4.3: The solutio of the Eample 4.3, obtaied by the Boud ad Decompositio method is same as i Amit Kumar et al. [3]. Eample 4.4: Cosider the followig fully fuzzy liear programmig problem: z,, 3, 3, 4 Maimize subect to 0,,,, 3 (,0, 7 );,, 3 0,, (,, 8 ); ad 0. Now, by solvig the boud ad decompositio method, a optimal fuzzy solutio to the give fully fuzzy liear programmig problem is (, 4, 6 ), (, 3, 5 ) ad z ( 4,7, 38 ). Remark 4.4: The solutio of the Eample 4.4, obtaied by the boud ad decompositio method is same as i Amit Kumar et al. [3]. Refereces [] T. Allahviraloo, K. H. Shamsolkotabi, N. A. Kiai ad L. Alizadeh, Fuzzy Iteger Liear Programmig Problems, Iteratioal Joural of Cotemporary Mathematical Scieces, (007), [] T. Allahviraloo, F. H. Lotfi, M. K. Kiasary, N. A. Kiai ad L. Alizadeh, Solvig Fully Fuzzy Liear Programmig Problem by the Rakig Fuctio, Applied Matematical Scieces, (008), 9-3. [3] Amit Kumar, Jagdeep Kaur, Pushpider Sigh, Fuzzy optimal solutio of fully fuzzy liear programmig problems with iequality costraits, Iteratioal Joural of Mathematical ad Computer Scieces, 6 (00), [4] Amit Kumar, Jagdeep Kaur, Pushpider Sigh, A ew method for solvig fully fuzzy liear programmig problems, Applied Mathematical Modelig, 35(0), [5] R. E. Bellma ad L. A. Zadeh, Decisio Makig i A Fuzzy Eviromet, Maagemet Sciece, 7(970), [6] J. Buckley ad T. Feurig, Evolutioary Algorithm Solutio to Fuzzy Problems: Fuzzy Liear Programmig, Fuzzy Sets ad Systems, 09(000), [7] J. M. Cadeas ad J. L. Verdegay, Usig Fuzzy Numbers i Liear Programmig, IEEE Trasactios o Systems, Ma ad Cyberetics-Part B: Cyberetics, 7(997), [8] L. Campos ad J. L. Verdegay, Liear Programmig Problems ad Rakig of Fuzzy Numbers, Fuzzy Sets ad Systems, 3( 989), -. [9] M. Dehgha, B. Hashemi, M. Ghatee, Computatioal methods for solvig fully fuzzy liear systems, Appl. Math. Comput., 79 ( 006 ), P a g e

8 M.Jayalakshmi, P. Padia / Iteratioal Joural of Egieerig Research ad Applicatios (IJERA) ISSN: Vol., Issue 4, July-August 0, pp [0] A. Ebrahimead ad S. H. Nasseri, Usig Complemetary Slackess Property to Solve Liear Programmig with Fuzzy Parameters, Fuzzy Iformatio ad Egieerig, (009), [] A. Ebrahimead, S. H. Nasseri, F. H. Lotfi ad M. Soltaifar, A Primal- Dual Method for Liear Programmig Problems with Fuzzy Variables, Europea Joural of Idustrial Egieerig, 4 ( 00 ), [] S. C. Fag, C. F. Hu, H. F. Wag ad S. Y. Wu, Liear Programmig with Fuzzy coefficiets i Costraits, Computers ad Mathematics with Applicatios, 37(999), [3] K. Gaesa ad P. Veeramai, Fuzzy Liear Programs with Trapezoidal Fuzzy Numbers, Aals of Operatios Research, 43 ( 006 ), [4] S. M. Hashemi, M. Modarres, E. Nasrabadi ad M. M. Nasrabadi, Fully Fuzzified Liear Programmig, solutio ad duality, Joural of Itelliget ad Fuzzy Systems, 7(006), [5] M. Iuiguchi, H. Ichihashi ad Y. Kume, A Solutio Algorithm for Fuzzy Liear Programmig with Piecewise Liear Membership Fuctio, Fuzzy Sets ad Systems, 34 (990 ), 5-3. [6] M. Jimeez, M. Areas, A. Bilbao ad M. V. Rodrguez, Liear Programmig with Fuzzy Parameters: A Iteractive Method Resolutio, Europea Joural of Operatioal Research, 77 ( 007 ), [7] X. Liu, Measurig the Satisfactio of Costraits i Fuzzy Liear Programmig, Fuzzy Sets ad Systems, ( 00 ), [8] F. H. Lotfi, T. Allahviraloo, M. A. Jodabeha ad L. Alizadeh, Solvig A Fully Fuzzy Liear Programmig Usig Leicography Method ad Fuzzy Approimate Solutio, Applied Mathematical Modellig, 33 ( 009 ), [9] H. R. Maleki M. Tata ad M. Mashichi, Liear Programmig with Fuzzy Variables, Fuzzy Sets ad Systems, 09 ( 000 ), -33. [0] H. R. Maleki, Rakig Fuctios ad Their Applicatios to Fuzzy Liear Programmig, Far East Joural of Mathematical Scieces, 4 ( 00 ), [] S. H. Nasseri, A New Method for Solvig Fuzzy Liear Programmig by Solvig Liear Programmig, Applied Mathematical Scieces, ( 008 ), [] H. M. Nehi, H. R. Maleki ad M. Mashichi, Solvig Fuzzy Number Liear Programmig Problem by Leicographic Rakig Fuctio, Italia Joural of Pure ad Applied Mathematics, 5 ( 004 ), 9-0. [3] J. Ramik, Duality i Fuzzy Liear Programmig: Some New Cocepts ad Results, Fuzzy Optimizatio ad Decisio Makig, 4 ( 005 ), [4] H. Taaka, T, Okuda ad K. Asai, O Fuzzy Mathematical Programmig, Joural of Cyberetics ad Systems, 3(973), [5] L. A. Zadeh, Fuzzy Sets, Iformatio ad Cotrol, 8 (965), [6] H. J. Zimmerma, Fuzzy Programmig ad Liear Programmig with Several Obective Fuctios, Fuzzy Sets ad Systems, ( 978 ), [7] G. Zhag, Y. H. Wu, M. Remias ad J. Lu, Formulatio of Fuzzy Liear Programmig Problems as Four- Obective Costraied Optimizatio Problems, Applied Mathematics ad Computatio, 39( 003), P a g e

Scholars Journal of Physics, Mathematics and Statistics

Scholars Journal of Physics, Mathematics and Statistics Jaalakshmi M. Sch. J. Phs. Math. Stat., 015 Vol- Issue-A Mar-Ma pp-144-150 Scholars Joural of Phsics, Mathematics ad Statistics Sch. J. Phs. Math. Stat. 015 A:144-150 Scholars Academic ad Scietific Publishers