DUALITY IN NONLINEAR FRACTIONAL PROGRAMMING PROBLEM USING FUZZY PROGRAMMING AND GENETIC ALGORITHM

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1 Inernaional Journal of Sof Copuing, Maheaics and Conrol (IJSCMC), Vol. 4, No., February 5 DUALITY IN NONLINEAR FRACTIONAL PROGRAMMING PROBLEM USING FUZZY PROGRAMMING AND GENETIC ALGORITHM Ananya Chakrabory Assisan Professor, Deparen of Maheaics, Veana Insiue of Technology, Korangala 3 rd Block, Bangalore-34. Karnaaka, India ABSTRACT In his paper we have considered nonlinear fracional prograing proble wih uliple consrains. A pair of prial and dual for a special ype of nonlinear fracional prograing has been considered under fuzzy environen. Exponenial ebership funcion has been used o deal wih he fuzziness. Dualiy resuls have been developed for he special ype of nonlinear prograing using exponenial ebership funcion. The ehod has been illusraed wih nuerical exaple. Geneic Algorih as well as Fuzzy prograing approach has been used o solve he proble. KEYWORDS Fuzzy Maheaical Prograing, Nonlinear Fracional Prograing, Exponenial ebership funcion, Decision Analysis, Geneic Algorih.. INTRODUCTION Several facors in he real world iply he increase in use of nonlinear prograing odels. There are several classes of probles where nonlinear prograing had had a grea ipac for exaple oil and perocheical indusries, nonlinear nework probles and econoic planning odels. The area where nonlinear prograing can be used in hese several classes of probles is given in deail in (Ladson e al., 98). The concep of fuzzy prograing in decision aking proble was firs proposed by (Bellann and Zadeh, 97). Many auhors have applied fuzzy prograing approach in differen area of linear prograing (Zierann, 978, Sancu- Minasian e al., 978, Chakrabory and Gupa, ). In (Jienez, 5), auhor has considered a non linear prograing proble wih fuzzy consrains and he soluion has been obained by ui obecive evoluionary Algorihs. An ineracive cuing plane algorih for fuzzy uli obecive nonlinear prograing probles has been presened in (Kanaya, ). In (Jaeel, ), auhor has obained accurae resuls for solving non linear prograing using fuzzy environen by using properies of fuzzy se and fuzzy nuber wih linear ebership funcion. Fracional progras arise in anageen decision aking as well as ouside of i. They also occur soeies indirecly in odeling where iniially no raio is involved. The efficiency of a syse is soeies characerized by a raio of echnical and/or econoical ers. Maxiizing syse efficiency hen leads o a fracional progra. Lis of frequenly occurring obecives are axiizaion of produciviy, axiizaion of reurn on invesen, axiizaion of reurn/risk, DOI :.48/iscc

2 Inernaional Journal of Sof Copuing, Maheaics and Conrol (IJSCMC), Vol.4, No., February 5 iniizaion of cos/ie, axiizaion of oupu/inpu, Non-Econoic Applicaions. There are a nuber of anageen science probles which indirecly give rise o a fracional progra; see e.g. (Schaible, 98, 983, 995). A new ehod has been given by (Borza e al. ) for solving he linear fracional prograing probles wih inerval coefficiens in he obecive funcion. An algorih has been developed by (Saad e al., ) for uli obecive ineger nonlinear fracional prograing proble under fuzziness. In order o defuzzify he proble, he has developed he concep of -level se of he fuzzy nuber is given and for obaining an efficien soluion o he proble (FMOINLFP), a linearizaion echnique is presened o develop he soluion algorih. In he paper (Biswas and Bose, ) auhor presened a fuzzy prograing procedure o solve nonlinear fracional prograing probles in which he paraeers involved in obecive funcion are considered as fuzzy nubers. The dualiy heory for nonlinear uliobecive opiizaion probles in he field of he opiizaion heory has inensively developed during he las decades. In (Rodder and Zierann, 98), a generalizaion of axin and inax probles in a fuzzy environen is presened and hereby a pair of fuzzy dual linear prograing probles is consruced. An econoic inerpreaion of his dualiy in ers of arke and indusry is also discussed in ha paper. In (Becor and Chandra, ), a pair of linear prograing prial-dual proble is inroduced under fuzzy environen and appropriae resuls were proved o esablish he dualiy relaionship beween he. In (Liu, 995) a consrucive approach has been proposed o dualiy for fuzzy uliple crieria and uliple consrain level linear prograing probles. (Biswas and Bose, ) gives a paraeric approach for he dualiy in fuzzy uli crieria and uli consrain level linear prograing proble. In (Gupa and Mehlawa, 9), a sudy of a pair of fuzzy prial dual linear prograing probles has been presened and calculaed dualiy resuls using an aspiraion level approach using exponenial ebership funcion, while a discussion of fuzzy prial dual linear prograing proble wih fuzzy coefficiens has been presened in (Becor and Chandra, ; Liu, 995). In (Becor and Chandra, ), a pair of linear prograing prial-dual proble is inroduced under fuzzy environen and appropriae resuls were proved o esablish he dualiy relaionship beween he. Also in (Chakrabory e al. 4), auhor has presened a pair of linear prial dual prograing using linear and exponenial ebership funcion using fuzzy prograing approach and geneic algorih approach. In his paper we have exended he work done by (Gupa and Mehlawa, 9) in which he has solved dualiy in convex fracional prograing using linear ebership funcion. We have aken he sae kind of proble and proved he dualiy heores considering exponenial ebership funcion for fuzzified obecive funcion and consrains and hen solved using LINGO and Geneic Algorih also and he resuls have been analysed. The paper has been organized as follows: In secion, he nonlinear fracional prograing proble and is dual has been defined. The fuzzified and he crisp forulaion have been defined for he prial and dual. In secion 3, he necessary dualiy resuls have been developed. The nuerical exaple defined in (Gupa and Mehlawa, 9) has been illusraed in secion 4. Finally, analysis of he soluion and concluding rearks has been presened in secion 5.. NON LINEAR FRACTIONAL PROGRAMMING PROBLEM The nonlinear fracional prograing proble (Gupa and Mehlawa, 9) can be defined as:

3 Inernaional Journal of Sof Copuing, Maheaics and Conrol (IJSCMC), Vol.4, No., February 5 ( c x) Max f ( x) = Subec o Ax b x () The dual for of he above proble can be wrien as Min g( u, v) = b u Subec o A u + dv cv u, v () Where A is x n arix, x, c, b are colun vecors wih n coponens and b is a colun vecor of coponens.. Fuzzified Non Linear Fracional Prograing Proble Fuzzified for of nonlinear fracional prograing proble can be defined as Find x R n Subec o ( c x) f ( x) = Z Ax b x (3) Where Z is he aspiraion level of he obecive funcion of he prial proble. and denoes he flexibiliy of he obecive funcion and he uli consrains. Fuzzified for of he dual of he proble can be defined as Find u R Subec o g(u,v) = u b w A u + dv cv u, v (4) Where w is he aspiraion level of he obecive funcion of he dual proble.. Crisp Forulaions using Exponenial Mebership Funcion The advanage of nonlinear ebership funcion over linear ebership funcion is already described in (Gupa and Mehlawa, 9; Chakrabory e al. 4). Linear ebership funcion is os coonly used in fuzzy linear prograing proble because i is siple and i is defined by fixing wo poins. Also, linear ebership funcion has been used in a vas range of nonlinear probles (Becor and Chandra, ; Gupa and Mehlawa, 9; Chakrabory e al. 4). Bu in any pracical siuaions linear ebership funcion is no a suiable

4 Inernaional Journal of Sof Copuing, Maheaics and Conrol (IJSCMC), Vol.4, No., February 5 represenaion. Furherore, if he ebership funcion is inerpreed as he fuzzy uiliy of he decision aker, used for describing levels of indifference, preference owards uncerainy, hen a nonlinear ebership funcion provides a beer represenaion han a linear ebership funcion. Moreover, unlike linear ebership funcion, for nonlinear ebership funcions he arginal rae of increase (or decrease) of ebership values as a funcion of odel paraeers is no consan- a echnique ha reflecs realiy beer han he linear case (Gupa and Mehlawa, 9). Le us consider he following for of exponenial ebership funcion (Gupa and Mehlawa, 9; Chakrabory e al. 4) for he fuzzy nonlinear fracional prograing proble respecively. ( c x) if z ( c x) z / p e e ( c x) µ ( x) = if z p < < z - e ( c ) if x z p if A x b {( b A x) / q } e e µ ( x) = if b < A x < b + q - e if A x b + q (5) =,,. Where and are user defined paraeers which deerine he shape of he ebership funcion. p and q ( =,,..) are subecively chosen consan of adissible violaions such ha p is associaed wih nonlinear fracional obecive funcion and q ' s are associaed wih linear consrains of he proble. Therefore using exponenial ebership funcion he crisp forulaion of he proble can be given by Max Subec o ( c ) x z / p e e λ -e {( b A x)/ q } e e λ =,, -e,, x. (6)

5 Inernaional Journal of Sof Copuing, Maheaics and Conrol (IJSCMC), Vol.4, No., February 5 or Max ( c ) x Subec o p log{ λ( e ) + e } z q log{ λ( e ) + e } ( b A x), =,,,, x. (7) The exponenial ebership funcion for he dual proble obecive funcion and nonlinear consrains are given by if g( u, v) w {( w g ( u, v))/ r} e e w = w < g u v w + r µ ( ) if (, )< -e if (, ) + g u v w r if A u + d v c v {( A u+ d v c v)/ s } e e µ ( w) = if c v s <A u + d v < c v -e if A u + d v c v s =,,,n (8) Where and ' s are user defined paraeers which deerine he shape of he ebership funcion. r and s ( =,,., n) are subecively chosen consan of adissible violaions such ha r is associaed wih obecive funcion and s ' s are associaed wih n linear consrains of he dual proble. Min (-η) Subec o e η { ( w g ( u, v ))/ r} e -e {( A u+ d v c v)/ s } e e η =,, n -e η, η, u,v. (9) or Min (-η) Subec o r log{ η( e ) + e } ( w g( u, v) ) 3

6 Inernaional Journal of Sof Copuing, Maheaics and Conrol (IJSCMC), Vol.4, No., February 5 () log{ η( ) } ( ) s e + e A u + d v c v, =,, n η, η, u,v. 3. THEOREMS ON DUALITY We shall prove soe dualiy heores. In he heores, Q= (q, q,,q ), S= (s, s,., s n ) are colun vecors Theore 3. (Modified fuzzy weak dualiy heore): Le (x, ) be he feasible soluion of crisp prial proble (7) and (u, v, η ) be feasible soluion of crisp dual proble (). Then n log{ λ( e ) + e } log{ η( e ) + e } ( c x) u Q + S x u b i= i = Proof: Since (x, ) be he feasible soluion of crisp prial proble (7) and (w, η ) be feasible soluion of crisp dual proble (), hen q log{ λ( e ) + e } ( b A x), i=,, i i i i log{ η( ) } ( ) s e + e A u + d v c v, =,, n Or log{ λ( e ) + e } Q b Ax () i= i n log{ η( e ) + e } S A u + dv cv () = Muliplying he equaion () by ranspose of u and aking ranspose of equaion () and uliplying equaion by x, we have i= n = log{ λ( e ) + e } u Q u b u Ax i log{ η( e ) + e } S x u Ax + v v c x Adding he above wo inequaliies, we have i= i = n η log{ λ( e ) + e } log{ ( e ) + e } u Q + S x u b + v v c x n Or λ + η + ( ) u Q + S x u b + v i= i = ( ) log{ ( e ) e } log{ ( e ) e } ( c x) ( c x) 4

7 Inernaional Journal of Sof Copuing, Maheaics and Conrol (IJSCMC), Vol.4, No., February 5 n log{ λ( e ) + e } log{ η( e ) + e } ( c x) ( c x) u Q + S x u b + ( ) v i= i = ( ) n log{ λ( e ) + e } log{ η( e ) + e } ( c x) u Q + S x u b i= i = Reark: When =, η = in he above inequaliy we ge he obecive funcion of prial ( ) c x proble is iprecisely less han he obecive funcion of he dual u b which can be defined as he sandard fuzzified weak dualiy heore in a uli obecive linear prograing proble. Theore 3.: Le ( x, λ ) be feasible soluion of crisp prial proble (7) and ( w, η ) be feasible soluion of crisp dual proble () such ha log{ ( ) } n log{ ( ) } λ e + e ( ) T η e + e T c x w Q + S x = u b i= i = (i) log{ ( e ) e } log{ ( e ) e } ( c x) λ + η + (ii) p + r = u b + ( w z) (iii) The aspiraion levels z and w saisfy ( z w ) Then ( x, λ ) is opial soluion of prial proble and ( w, η ) be opial soluion of dual proble. Proof: Le (x, ) be feasible soluion of crisp prial proble (7) and (w, η ) be feasible soluion of crisp dual proble (). Then fro Theore 3 we have, ( c x) n log{ λ( e ) + e } log{ η( e ) + e } u Q + S x u b i= i = (3) Fro condiion (i) and Equaion (3) we have ( c x ) log{ λ ( e i ) + e i } n log{ η ( e ) + e } u Q + S x u b i= i = n c x log{ λ( e ) + e } log{ η( e ) + e } u Q S x u b + i= i = 5

8 Inernaional Journal of Sof Copuing, Maheaics and Conrol (IJSCMC), Vol.4, No., February 5 Thus, ( x, λ, w, η) is opial o he proble whose axiu obecive value is. ( ) log{ λ ( e i ) + e i } n log{ η ( e ) + e } c x Max u Q + S x u b i= i = Subec o ( c x) p log{ λ( e ) + e } Z q log{ λ( e ) + e } ( b A x) i i i i r log{ η( e ) + e } ( w u b) log{ η( ) + } ( + ) s e e A u d v c v,, x, η, u, v, η. (i =,,, ) ( =,,.., n ) Adding condiion (i) and (ii), we have ( c x ) log{ λ ( e i ) + e i } n log{ η ( e ) + e } u Q + S x u b i= i = log{ ( e ) e } log{ ( e ) e } ( c x) λ + η + p + r u b ( w z) = Or log{ λ ( e i ) + e i } n log{ η ( e ) + e } u Q + S x + i= i = log{ λ( e ) + e } log{ η( e ) + e } p + r + ( z w ) = Each er in he above su is non posiive as λ, η. log{ λ( e ) + e } Therefore, u Q = i= n x = i log{ η( e ) + e } S x = log{ λ( e ) + e } p = + 6

9 Inernaional Journal of Sof Copuing, Maheaics and Conrol (IJSCMC), Vol.4, No., February 5 log{ η( e ) + e } r = z w = Since log{ ( e ) λ + e } p and log{ ( ) η e + e } r, because λ, η, we ge log{ λ( e ) + e } log{ λ( e ) + e } p p log{ η( e ) + e } log{ η( e ) + e } r r or λ λ and η η Thus, ( x, λ ) is opial soluion of (7) and ( w, η ) be opial soluion of (). 4. NUMERICAL EXAMPLE Le us consider he sae nuerical exaple which is defined in []. The prial dual nonlinear fracional prograing proble can be defined as Min ( x + x) f ( x) = x + x Subec o x + x 6 x + 3x 8 x, x Max g( u, v) = 6u + 8u Subec o u + u + v 4v u u v v u, u, v The fuzzified nonlinear fracional prograing proble can be wrien as: Find x R n Min Subec o ( x + x ) x + x ( ) = f x x + x 6 7

10 Inernaional Journal of Sof Copuing, Maheaics and Conrol (IJSCMC), Vol.4, No., February 5 Max x + 3x 8 x, x (4) g( u, v) = 6u + 8u Subec o u + u + v 4v u u v v u, u, v (5) Le us ake p =, q =, q =, =, =, = for prial proble. Using exponenial ebership funcion, he crisp odel of (4) can be wrien as: Min λ Subec o 4x + x + 4 xx ( x + x ) log(.865λ +.353) 4x + x log(.865λ +.353) x + 3x 8 log(.865λ +.353), x, x. (6) Le us ake r =, s =, s =, =, =, = for dual proble (5). Using exponenial ebership funcion, he crisp odel can be wrien as: Max η Subec o log(.865η +.353) u + 6u log(.865η +.353) 4u u v + 8v log(.865η +.353) 3 + u u v v η, u, u, v. (7) 5. SOLUTION AND ANALYSIS Using LINGO sofware, he soluion of he prial proble (6) is given by Local opial soluion found. Obecive value: -. Infeasibiliies:. Exended solver seps: 5 Toal solver ieraions: 5 Model Class: NLP Toal variables: 3 Nonlinear variables: 3 Ineger variables: 8

11 Inernaional Journal of Sof Copuing, Maheaics and Conrol (IJSCMC), Vol.4, No., February 5 Toal consrains: 5 Nonlinear consrains: 3 Toal nonzeros: Nonlinear nonzeros: 5 Variable Value LAMBDA. X X.E+8 Miniu value of obecive funcion is -. The soluion of he dual proble (7) is given by Local opial soluion found. Obecive value:. Infeasibiliies:. Exended solver seps: 5 Toal solver ieraions: 37 Model Class: NLP Toal variables: 4 Nonlinear variables: Ineger variables: Toal consrains: 5 Nonlinear consrains: 3 Toal nonzeros: 3 Nonlinear nonzeros: 5 Variable Value ETA. U.373 U E- V Maxiu value of he obecive funcion is. Geneic Algorih gives global opial soluion. The NSGA used here is a real paraeer GA ha works direcly wih he paraeer values. NSGA (Nondoinaed Soring Geneic Algorih) uses niching, as well as nondoinaed soring of he soluions in every generaion o ensure ha he good soluions ge preference in selecion for procreaion. The non-doinaed soring GA uses a ranking selecion ehod o ephasize good soluions and hen a niche building procedure o ainain a sable sub populaion of good soluions. Since uli obecive GAs can find uliple pareo opial soluions in one single run, he proposed echnique is capable of finding uliple soluions o he probles. The paraeers used o solve he proble (6) in Geneic Algorih are as follows: Nuber of obecive funcions : 9

12 Inernaional Journal of Sof Copuing, Maheaics and Conrol (IJSCMC), Vol.4, No., February 5 Obecive funcion # : Miniize CROSSOVER TYPE : Binary GA (Single-p) STRATEGY : cross - sie wih swapping Populaion size : Toal no. of generaions : Cross over probabiliy :.9 Muaion probabiliy :. Sring lengh : 6 Nuber of variables Binary : 3 Ineger : Enueraed : Coninuous : TOTAL : 3 Epsilon for closeness : Siga-share value :.3 Sharing Sraegy : sharing on Paraeer Space Lower and Upper bounds :. <= <=.. <= x <=.. <= x <=. Table gives a se of soluion of proble (6). Table : Prial Soluion x x The paraeers used o solve he proble (7) in Geneic Algorih are as follows: Nuber of obecive funcions : Obecive funcion # : Maxiize CROSSOVER TYPE : Binary GA (Single-p) STRATEGY : cross - sie wih swapping Populaion size : Toal no. of generaions : Cross over probabiliy :.9 Muaion probabiliy :. Sring lengh : 4 Nuber of variables Binary : 4 3

13 Inernaional Journal of Sof Copuing, Maheaics and Conrol (IJSCMC), Vol.4, No., February 5 Ineger : Enueraed : Coninuous : TOTAL : 4 Epsilon for closeness : Siga-share value :.8 Sharing Sraegy : sharing on Paraeer Space Lower and Upper bounds :. <= u <=.. <= u <=.. <= v <=.. <= η <=. Using Geneic algorih, a se of soluions of dual proble (7) is given in he for of able in Table. Table : Dual Soluion Η u u V Fro he above able, we can clearly see ha he soluion obained saisfies he fuzzy consrains as well as he fuzzy obecive in boh prial and dual proble. Also, i has been observed ha difference beween he olerance lii and aspiraion level is less. In oher words, he decision akers have been provided wih enough flexibiliy o choose saisfying soluions ha axiize or iniize heir uiliy funcions. Geneic Algorih gives uliple pareo opial soluions in one single run. Therefore, i provides se of soluions o he decision aker. The following graph clearly shows coparison beween he prial and dual obecive funcion of differen ses of soluions. 3

14 Inernaional Journal of Sof Copuing, Maheaics and Conrol (IJSCMC), Vol.4, No., February 5 λ/µ Soluion Nuber Prial Dual Figure : Prial and Dual obecive funcion of differen soluion using GA 6. CONCLUSIONS In his paper, an approach has been presened for a specific kind of fuzzy nonlinear fracional prograing proble by consrucing a pair of fuzzy non linear fracional prial and dual probles. Crisp for of he above prial and dual probles has been obained by using exponenial ebership funcion. Dualiy resuls have been esablished o prove he dualiy relaionship beween above prial and dual proble and are illusraed by an exaple. The dualiy resuls fully saisfy he aspiraion levels or he olerance levels of he obecive funcions and he syse consrains ade by he decision aker. The difference beween he achieved level and he allowable lii of he saisfying soluions of he decision aker is very less. The nuerical exaple has also been solved by Geneic Algorih Approach. Geneic Algorih gives uliple pareo opial soluion. The decision aker can choose any opial soluion according o he convenience. The resuls of he presen paper encourage us o apply dualiy resuls in variey nuber of fields of opiizaion proble. REFERENCES [] Bellan, R. E., Zadeh, L. A., (97) Decision Making in a Fuzzy Environen, Manageen Science, 7, [] Becor, C. R. and Chandra, S., () On dualiy in linear prograing under fuzzy environen, Fuzzy Ses and Syses, 5, [3] Biswas, A. and Bose, K., (), Applicaion of Fuzzy Prograing Mehod for Solving Nonlinear Fracional Prograing Probles wih Fuzzy Paraeers, Maheaical Modelling and Scienific Copuaion Counicaions in Copuer and Inforaion Science [4] Borza, M., Rabely, A. S. and Sara, M., () Solving linear fracional prograing probles wih inerval coefficiens in he obecive funcion. A New Approach, Applied Maheaical Sciences, 6, 69, [5] Chakrabory, M., Gupa, S., () Fuzzy Maheaical Prograing for Muli-obecive Linear Fracional Prograing Proble, Fuzzy Ses and Syses, 5,

15 Inernaional Journal of Sof Copuing, Maheaics and Conrol (IJSCMC), Vol.4, No., February 5 [6] Chakrabory, A., Tiwari, S. P., Chaopadhyay, A. and Chaeree, K. (4), Dualiy in Fuzzy Muli obecive linear prograing wih uli consrain, Inernaional Journal of Maheaics in Operaions Research, Vol. 6, No. 3, pp [7] Gupa, P. and Mehlawa, M. K., (9) Dualiy for a convex fracional prograing under fuzzy environen, Inernaional Journal of Opiizaion: Theory Mehods and Applicaions,, 3, 9-3. [8] Gupa, P. and Mehlawa, M. K., (9), Becor- Chandra ype dualiy in fuzzy linear prograing wih exponenial ebership funcions, Fuzzy Ses and Syses, 6,, [9] Jaeel, A. F., Sadhegi, A., (), Solving Nonlinear Prograing Proble in Fuzzy Environen, In. J. Conep. Mah. Sciences 7, 4, [] Jienez, F., Sanchez, G., Cadenas, J. M., Goez-Skarea, A. F., Verdegay, J. L., (5), Copuaional Inelligence, Theory and Applicaions, Advances in Sof Copuing, 33, [] Kanaya, Z. A. () An Ineracive Mehod for Fuzzy Muliobecive Nonlinear Prograing Probles, JKAU: Sci. 3-; DOI:.497 / Sci [] Lasdon, L. S, Waren, A. D., (98), Feaure Aricle: Survey of Nonlinear Prograing Applicaions, Operaions Research 8(5): [3] Liu, Y. J., Shi, Y. and Liu, Y. H. (995) Dualiy of fuzzy MC linear prograing: a consrucive approach, J. Mah. Anal. Appl., 94, [4] Rodder, W. and Zierann, H. J. (98), Dualiy in fuzzy linear prograing. In: (A. V. Fiacco, K. O. Korane K Eds.). Exernal ehods and Syse Analysis. Berlin, New York, [5] Saad, O. M., M. S. Bilagy and T. B. Farag, An algorih for Muli obecive Ineger nonlinear fracional prograing proble under fuzziness, Gen. Mah. Noes, () -7. [6] Schaible, S., (98) Fracional prograing: applicaions and algorihs, European Journal of Operaional Research, 7,, -. [7] Schaible, S., (995), Fracional prograing, in R. Hors and P.M. Pardalos (eds.), Handbook of Global Opiizaion, Kluwer Acadeic Publishers, Dordrech-Boson-London, [8] Schaible, S. and Ibaraki, T., (983), Fracional prograing, European Journal of Operaional Research, 3, [9] Sancu-Minasian, I. M., Pop, B., (978) On a Fuzzy Se Approach o Solving Muliple Obecive Linear Fracional Prograing Proble, Fuzzy Ses and Syses, 34, [] Zierann, H. J., (978) Fuzzy prograing and linear prograing wih several obecive funcions, Fuzzy Ses and Syses,, Auhors Ananya Chakrabory is Assisan Professor in he Deparen of Maheaics of Veana Insiue of Technology, Bangalore, India. She has done her Ph.D fro Indian School of Mines, Dhanbad, Jharkhand, India. She is having alos years of research experience. She has published any papers in naional/ inernaional ournals. She has presened any papers in differen conferences. Her curren research ineres includes Fuzzy prograing, Operaions Research. She is also a eber of Inernaional associaion of copuer science and inforaion echnology 33