Interntionl Journl of Computer pplictions 097 8887 Volume 8 No 8 Decemer 0 nking of egonl Fuzzy Numers Solving ulti Ojective Fuzzy Liner Progrmming Prolem jrjeswri. P Deprtment of themtics Chikknn Government rts College Tirupur STCT In this pper rnking Procedure sed on egonl Fuzzy numers is pplied to ulti-ojective Liner progrmming prolem OLPP with fuzzy coefficients. y this rnking method ny ultiojective Fuzzy Liner Progrmming prolem OFLPP cn e converted in to crisp vlue prolem to get n optiml solution. This method provides n insight the plnner due to uncertin environment in n orgniztionl Economics. In n orgniztion where numer of lterntives nd vriles such s production inventory finncil mngement costing nd vrious other prmeters re involved this rnking procedure serves s n efficient method wherein numericl emple is lso tken nd the inference is given. Keywords nking egonl fuzzy numers OFLPP Decision mking.. INTODUCTION nking fuzzy numer is used in decision- mking process in n economic environment. In n orgniztion vrious ctivities such s plnning eecution nd other process tkes plce continuously. This requires creful oservtion of vrious prmeters which re ll in uncertin in nture due the competitive usiness environment glolly. In fuzzy environment rnking fuzzy numers is very importnt decision mking procedure. The ide of fuzzy set ws first proposed y ellmn nd Zdeh [] s men of hndling uncertinty tht is due to imprecision rther thn rndomness. The concept of fuzzy liner progrmming FLP ws first introduced y Tnk et l. [ ] Zimmermn [7] introduced fuzzy liner progrmming in fuzzy environment. ulti-ojective liner Progrmming ws introduced y Zeleny []. Li Y.J wng C.L [] considered OLPP with ll prmeters hving tringulr possiility distriution. They used n uiliry model nd it ws solved y OLPP. Zimmermn [8] pplied their pproch to vector mimum prolem y trnsming OFLP prolem to single ojective liner progrmming prolem. Qiu- Peng Gu nd ing-yun Co [7] solved Fuzzy liner progrmming prolems sed on Fuzzy numers distnce. Tong Shocheng [] focused on the fuzzy liner progrmming with intervl numers. Chns [] proposed fuzzy progrmming in multi ojective liner progrmming. Verdegy [] hve proposed three methods solving three models of fuzzy integer liner progrmming sed on the representtion theorem nd on fuzzy numer rnking method. In prticulr the most convenient methods re sed on the concept of comprison of fuzzy numers y the use of rnking functions. Nsseri [] hs proposed new method solving FLP prolems in which he hs used the fuzzy rnking method converting the fuzzy ojective function into crisp ojective function. Shy Sudh. Deprtment of themtics Nirml College Women Coimtore. PELIINIES.Definition: Let X e nonempty set. fuzzy set in X is chrcterized : X 0 where is y its memership function interpreted s the degree of memership of element in fuzzy ech X.. Intervl Numer: Let e the set of rel numers. Then closed intervl [ ] is sid to e n intervl numer where Є.. Distnce etween intervl numers: Let = [ ] = [ ] e two intervl numers. Then the distnce etween denoted y d is defined y d. Fuzzy numer: d fuzzy set of the rel line with memership function : 0 is clled fuzzy numer if i must e norml nd conve fuzzy set; ii the support of must e ounded iii must e closed intervl every 0. Support: The support of fuzzy set such tht X 0 S Ã is the crisp set of ll ЄX. Distnce etween fuzzy numers: Let e two fuzzy numers. Then the distnce etween the fuzzy numers nd is defined y D d d 0. EXGONL FUZZY NUE fuzzy numer is hegonl fuzzy numer denoted y where re rel numers nd its memership function is given elow.
Interntionl Journl of Computer pplictions 097 8887 Volume 8 No 8 Decemer 0 0 0 X Figure Grphicl representtion of hegonl fuzzy numer. rithmetic Opertions on egonl Fuzzy numers Let nd e two hegonl fuzzy numers then i ii iii.. nking of egonl Fuzzy Numers: numer of pproches hve een proposed the rnking of fuzzy numers. In this pper hegonl fuzzy numer rnking method is devised sed on the following mul. 8 8 Let nd e two hegonl fuzzy numers then. POPOSED NKING LGOIT The ove procedure. cn e used to develop rnking hegonl fuzzy numers. sed on this rnking procedure rnking lgorithm is developed hegonl fuzzy numers. oreover it is pplied to OLPP under constrints with fuzzy coefficients.. lgorithm: Step: Consider the fuzzy numers & of hegonl fuzzy numer Step: Find Supremum = Sups U s where s =Support set of nd s =Support set of. Step: Tke nd s hegonl fuzzy numers Step: Clculte 8 D & 8 D Step: If then D D then D D then D D Step : Stop.. ETOD OF SOLVING ULTI- OJECTIVE FUZZY LINE POGING POLE In this pper we discuss ulti-ojective Fuzzy Liner Progrmming Prolem in constrint conditions with fuzzy coefficients. oreover the ojectives considered in this pper re mied with oth mimiztion nd minimiztion types. We discuss model whose stndrd m is
Interntionl Journl of Computer pplictions 097 8887 Volume 8 No 8 Decemer 0 imize inimize z c z c X X Suject to 0 Where c c c... c is n n- dimensionl crisp ij i i in is n m n fuzzy mtri row vector ij... T m is n m-dimensionl fuzzy line T vector nd X... n is n n-dimensionl decision vrile vector. We now consider i-ojective Fuzzy liner progrmming Prolem with constrints hving fuzzy coefficients is given y imize z c c c c... n n inimize z c c c c... n n Suject to i i i... in n i... n 0 i... m where fuzzy numers re hegonl where i i i i i i i i i i i i i in i i in in in in in in i i i i i y the rnking lgorithm the ove OFLPP is trnsmed into OLPP is s follows: imize inimize i z c c c c... n n z c c c c... n n Suject to i i... in n i i... inn i i... inn i i... inn i i... inn i i... inn i i i i i i n... 0 i... m Using this cn e converted into single ojective prolem suject to the constrints with trnsmed crisp numer coefficients nd hence solved ccordingly. Similrly multi-ojective prolems with more thn two ojectives cn lso e solved using the ove procedure here in the very first stge itself the prolem is trnsmed into crisp prolem nd fterwrds there will e no more fuzziness in the constrints s well s in the prolem.. NUEICL EXPLE The elow mentioned emple is tken from jrjeswri. P nd ShySudh [8] in production plnning process. Suject to Where Z 7 90 imize Z 0 7 inimize 0 8090000000 0000070 8090000000 8090000000 000000000000000000 00000000000000 Z 7 90 imize Suject to 8090000000 0000070 000000000000000000 8090000000 8090000000 00000000000000
Interntionl Journl of Computer pplictions 097 8887 Volume 8 No 8 Decemer 0 0 y this cn e trnsmed into crisp LPP s Suject to imize Z 7 90 80 0 90 0 00 0 00 0 0 0 0 70 000 000 000 000 000 000 80 80 90 90 00 00 00 00 0 0 0 0 00 000 00 00 000 00 Cse i: We consider the prolem with the mimiztion ojective Z 7 90 imize Suject to 00 00 00 00 000 0 000 The solution is.9 8. 7 nd z 07. 8 9 Net we proceed to solve the prolem with minimiztion ojective Z 0 7 inimize Suject to 00 00 00 00 0 000 000 The solution is.9 8. 7 nd in z. 9 Cse ii: Consider the prolem with the mimiztion ojective nd n dditionl constrint Z 7 90 imize Suject to 00 00 00 00 9.9 8.7 0 000 000 07.88 The solution is 9.9 8. 7 nd z 07. 8 Net we proceed to solve the prolem with minimiztion ojective with dditionl constrint Z 0 7 inimize Suject to 00 00 00 00 9.9 8.7 0 000 000 07.88 The solution is 9.9 8. 7 nd in z. Cse iii: The fesiility of the solution will e eplined s follows When ll the fuzzy numers re in lesser priority vlues the OLPP ecomes imize Z 7 90 Z 0 7 inimize Suject to 90 0 80 90 000 9000 0 The solution is.. 8 nd z 89. in z 08. Cse iv: When ll the fuzzy numers re in priority vlues the OLPP ecomes imize Z 7 90 7
Interntionl Journl of Computer pplictions 097 8887 Volume 8 No 8 Decemer 0 Z 0 7 inimize Suject to 00 00 000 00 000 00000 0 The solution is 9.9 8. 7 nd z 07. 8 in z. Cse v: When ll the fuzzy numers re in the higher priority vlues the OLPP ecomes imize Z 7 90 Z 0 7 inimize Suject to 070 70 0 070 77000 000 0 The solution is.0. 0 nd z 0. in z 9. Tle.: Comprison of results otined y using eisting nd proposed method. 7. ESULTS ND DISCUSSIONS s per the ove Tle. we hve otined the sme results from oth the eisting s well s the proposed method nd the fesiility of the solution is lso from the lowest to higher intervls. In the ove prolem the profit mimiztion is from [89. 0.] nd the cost of production rnges from [08. 9.].This new method reduces the miguity in the solution. This ms n optiml solution mnger to tke decision whether to produce prticulr product or not or else ny lternte chnges cn e done.this method cn e used where prticulr prolem cnnot e solved y tringulr or trpezoidl method. This method lso cn e used even if the numer of vriles nd prmeters re incresed this method is fr more efficient nd esy when compred to the erlier method.this cn e etended to ny numer of input prmeters. 8. EFEENCES [] ellmn.e nd.zdeh L. Decision mking in fuzzy environment ngement science 7970 - [] Chns D. Fuzzy progrmming in multi ojective liner progrmming- prmetric pproch Fuzzy set nd system9 989 0-. [] George J.klir oyun Fuzzy sets nd Fuzzy logic Theory nd pplictions-prentice-ll Inc 99 7p [] Ishiuchi; Tnk ulti ojective progrmming in optimiztion of the intervl ojective function Europen journl of Opertionl eserch8 990 9- [] Li Y.J wng C.L Fuzzy mthemticl progrmming lecture notes in Economics nd themticl systems Springer-Verlg 99 [] S. Nsseri new method solving fuzzy liner progrmming y solving liner progrmming.pplied themticl Sciences 008 7- [7] Qiu-Peng Gu. ing-yun Co pproch to liner progrmming with fuzzy coefficients sed on Fuzzy numers distnce IEEE Trnsctions 7-0. 00 [8] jerjeswri.p nd Shy Sudh ultiojective Fuzzy Optimiztion Techniques in Production Plnning Process Proc.of the eer Interntionl conference on pplictions of themtics nd Sttistics TiruchirpplliIndi070-7 [9] jerjeswri.p nd Shy Sudh nd Krthik. new Opertions on hegonl fuzzy numer Interntionl Journl of Fuzzy Logic Systems IJFLS Vol. No July0 [0] Sophiy Porchelvi. Ngoorgni.. Irene epzih. n lgorthimic pproch to ultiojective Fuzzy Liner Progrmming Prolem [] Tnk..si K. Fuzzy liner progrmming prolems with fuzzy numers Fuzzy Sets nd Systems 98-0. [] Tnk..Okud.T nd si K. Fuzzy themticl Progrmming Journl of Cyernetics nd systems 97 7- [] Tong Shocheng Intervl numer nd fuzzy numer liner progrmming Fuzzy Sets nd systems 99 0-0. [] Verdegy J.L. dul pproch to solve the fuzzy liner progrmming prolem Fuzzy sets nd Systems 98 - [] Zdeh L. 9. Fuzzy sets. Inf.control 88-8
Interntionl Journl of Computer pplictions 097 8887 Volume 8 No 8 Decemer 0 [] Zeleny. ultiple criteri decision mking. New York: cgrw-ill ook Compny 98 [7] Zimmermnn.J Fuzzy progrmming nd liner progrmming with severl ojective functions Fuzzy sets nd system 978 -. [8] Zimmermnn.J Fuzzy mthemticl progrmming Computer Science &opertions eserch Vol.0 No 989-98. [9] Zimmermnn.J 98.ppliction of Fuzzy set theory to themticl Progrmming IJC T : www.ijconline.org 9