APPLYING TRANSFORMATION CHARACTERISTICS TO SOLVE THE MULTI OBJECTIVE LINEAR FRACTIONAL PROGRAMMING PROBLEMS

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1 Iteratoal Joural of Computer See & Iformato eholog IJCSI Vol 9, No, Aprl 07 APPLYING RANSFORMAION CHARACERISICS O SOLVE HE MULI OBJECIVE LINEAR FRACIONAL PROGRAMMING PROBLEMS We Pe Departmet of Busess Amstrato, Chug Hua Uverst, Hshu, awa, ROC ABSRAC For some maagemet programmg problems, multple objetves to be optmze rather tha a sgle objetve, a objetves a be epresse wth rato equatos suh as retur/vestmet, operatg proft/et-sales, proft/maufaturg ost, et. I ths paper, we propose the trasformato haratersts to solve the mult objetve lear fratoal programmg MOLFP problems. If a MOLFP problem wth both the umerators a the eomators of the objetves are lear futos a some tehal lear restrtos are satsfe, the t s efe as a mult objetve lear fratoal programmg problem MOLFPP ths researh. he trasformato haratersts are llustrate a the soluto proeure a umeral eample are presete. KEYWORDS rasformato Charatersts, MOLFP, MOLFPP. INRODUCION Maagemet programmg problems are base upo estmate values. hese problems have multple objetves to be optmze rather tha a sgle objetve. hus optmal soluto to oe objetve futo s ot eessarl optmal for other objetve futos a hee oe ee a soluto as the ompromse soluto. I the meatme, for some maagemet programmg problems, objetves a be epresse rato equatos suh as retur/vestmet, operatg proft/et-sales, proft/maufaturg ost, et. hese multple objetve fratoal programmg moels were frst stue b Luhajula [6]. Korbluth a Steuer [4] have presete a algorthm for solvg the MOLFP b ombg aspets of multple objetve, sgle objetve fratoal programmg a goal programmg. Valpour et al. [9] suggeste a teratve parametr approah for solvg MOLFP problems whh ol uses lear programmg to obta effet solutos a overges to a soluto. Mshra et al. [7] presete a MOLFP approah for mult objetve lear fuzz goal programmg problem. L et al. [5] propose a two-level lear fratoal water maagemet moel base o teratve fuzz programmg. Saha et al. [8] propose a approah for solvg lear fratoal programg problem b overtg t to a sgle lear programmg problem, whh a be solve b usg a tpe of lear fratoal programmg tehque. mmerma [0,] frst apple fuzz set theor oept wth hoes of membershp futos a erve a fuzz lear program whh s etal to the mamum program. He DOI:0.5/jst

2 Iteratoal Joural of Computer See & Iformato eholog IJCSI Vol 9, No, Aprl 07 showe that solutos obtae b fuzz lear programmg are effet solutos a also gves a optmal ompromse soluto. Luhajula [6] solve MOLFP b applg fuzz approah to overome the omputatoal ffultes of usg ovetoal fratoal programmg approahes to solve multple objetve fratoal programmg problem. Chares a Cooper [] have show that a lear fratoal programmg problem a be optmze b reug t to two lear programs to solve MOLFP. Dutta et al [3] mofe the lgust approah of Luhajula [6] b ostrutg the esrable membershp futos. Charabort a Gupta [] propose a fferet methoolog for solvg MOLFP. he approah state that sutable trasformato shoul have bee apple to formulate a equvalet mult objetve lear programmg a the resultg mult objetve lear programmg oul be solve base o fuzz set theoret approah. I ths researh, base o Charabort a Gupta [], f a MOLFP problem wth both the umerators a the eomators of the objetves are lear futos a some tehal lear restrtos are satsfe, the t s efe as a MOLFPP. We propose the trasformato haratersts to solve the MOLFPP. he trasformato haratersts are llustrate a the soluto proeure a umeral eample presete.. MEHODS.. he rasformato Charatersts of MOLFPP... Fuzz Lear Programmg Fuzz lear programmg s fuzz set theor apple to lear mult rtera eso mag problems. he mult objetve lear fratoal programmg problem a be osere as a vetor optmzg problem. he frst step s to assg two values U a L as upper a lower bous for eah objetve futo : U Hghest aeptable level of ahevemet for objetve L Aspre level of ahevemet for objetve Let U L the egraato allowae for objetve. aes a elemet X that has a egree of membershp the -th objetve, eote b a membershp futo µ X, to trasform the fuzz moel to a rsp sgle objetve lear programmg moel of λ. he rage of the membershp futo s [ 0,]. 78

3 Iteratoal Joural of Computer See & Iformato eholog IJCSI Vol 9, No, Aprl < <. 0,, U f U L f L U L L f X µ hs approah s smlar, ma respets, to the weghte lear goal programmg metho.... Lear Fratoal Programmg he geeral format of a lassal lear fratoal programmg problem Chares a Cooper [] a be state as Ma β α s.t. m R b b A R X, 0,, where R, ; R β α,, X s oempt a boue. We propose the bas trasformato haraterst of the orgal objetve to solve the problem. he followg trasformato s propose: N D M N D M D N Ma...3. Multple Objetves Lear Fratoal Programmg Problem he geeral format of mamzg MOLFPP a be wrtte as Ma β α β α β α

4 Iteratoal Joural of Computer See & Iformato eholog IJCSI Vol 9, No, Aprl 07 s.t. m X R A b, 0, b R, 3 Where, R ; α, β R,,,...,, X s oempt a boue. Smlarl, mmum problem a also be efe as M [,,..., ] Where m s.t. X R A b, 0, b R, 4, R ; α, β R,,,...,, X s oempt a boue, α wth β N. D he geeral format of mmum MOLFPP s as the followg equvalet mult objetve lear programmg problem: M s.t. α t, β t γ A bt 0,, t 0,,,...,. he membershp futos for N a D are as followe: If I, the µ tn t 0 tn t 0 0 If I, the µ td t f f f tn t 0 0 < tn t < tn t 80

5 Iteratoal Joural of Computer See & Iformato eholog IJCSI Vol 9, No, Aprl 07 0 td t 0 0 f f f td t 0 0 < td t < td t he mmerma s [0,] operator s use to trasform the equvalet mult objetve lear programmg problem to the rsp moel as: Ma λ s.t. µ tn t λ for I, µ td t λ for I, td t for I, tn t for I, A t b 0, t > 0, 0,,,...,. I s a set suh that I { : N 0 for some } a I { : N < 0 where I U I {,,..., }. he omputg of, s proeee as f * the mamum asprato level s, a f propose ths paper suggests that wth t 0 I, the I for eah }, the t ma assume. he metho *, b Chares a Cooper [] metho, the problem oul ot be solve, the trasformato haratersts a be use to solve the MOLFPP. 3. RESULS AND DISCUSSION he soluto proeure s state a umeral eamples aopte from Charabort a Gupta [] are use to show the trasformato haratersts. 3.. Soluto Proeure he trasformato haratersts are use to solve MOLFPP whe t 0 from the orgal problems. he followg proeure s evelope: Step. Solve the orgal MOLFPP b Chares a Cooper []. Step. If t 0, the propose methoolog s apple. Step 3. Solve the problem b mmerma s [0,] operator to trasform the equvalet mult objetve lear programmg problem to the rsp moel. 3.. Numeral Eamples Let s oser a MOLFP wth two objetves as follows: 8

6 Iteratoal Joural of Computer See & Iformato eholog IJCSI Vol 9, No, Aprl 07 Ma s.t., 3 5, 3, 0,,. 3, Solve the MOLFP b Chares a Cooper [] approah. f, t 3 Ma f, t 7 s.t. 3t, 5 t, t 0, 3 5t 0, 3t 0,, t 0,,. Where 0, 0, t 0 for f, t, a , 0, t for f,. hus U, L 0, , U, L.3636,0 t Wth mmerma s [0,] approah, the above mult objetve lear programmg problem oul be solve. he soluto of the problem s obtae as λ , , 0.396, a t 0. he orgal problem oul be traslate to the followg MOLFPP: 3, 3 M 5 7 s.t., 3 5, 3, 0,,. 8

7 Iteratoal Joural of Computer See & Iformato eholog IJCSI Vol 9, No, Aprl 07 he equvalet MOLFPP s as followe: M f f, t, t 5 3t t s.t. 3, 7, t 0, 3 5t 0, 3t 0,,,, t 0. he soluto are U, L 0, , U, L ,0, λ , , , a t he soluto of the orgal problem s: 5 3,,, CONCLUSIONS he trasformato haratersts to solve MOLFPP base o fuzz set theoret approah are propose ths researh. he MOLFPP a be trasforme to the equvalet approprate mult objetve lear programmg problem b usg the trasformato haratersts. he resultg mult objetve lear programmg problem s solve usg fuzz set theoret approah b membershp futos. Numeral eample s utlze to llustrate the propose methoolog. REFERENCE [] Charabort, M. a Gupta, S. 00, Fuzz mathematal programmg for mult objetve lear fratoal programmg problem, Fuzz Sets a Sstems, Vol. 5, pp [] Chares, A. a Cooper, W.W. 96, Programmg wth lear fratoals, Naval Researh Logsts Quarterl, Vol. 9, pp [3] Dutta, D., war, R.N. a Rao, J.R. 99, Multple objetve lear fratoal programmg-a fuzz set theoret approah, Fuzz Sets a Sstems, Vol. 5, pp [4] Korbluth, J.S.H. a Steuer, R.E. 98, Multple objetve lear fratoal programmg, Maagemet See, Vol. 7, pp [5] L, M., Guo, P., & Re, C. 05, Water resoures maagemet moels base o two-level lear fratoal programmg metho uer uertat, Joural of Water Resoures Plag a Maagemet, 49, [6] Luhajula, M.K. 984, Fuzz approahes for multple objetve lear fratoal optmzato, Fuzz Sets a Sstems, Vol. 3, pp. -3. [7] Mshra, B., Nsha, A. K., & Sgh, S. R. 04, Fuzz Mult-fratoal Programmg for La Use Plag Agrultural Prouto Sstem, Fuzz Iformato a Egeerg, 6,

8 Iteratoal Joural of Computer See & Iformato eholog IJCSI Vol 9, No, Aprl 07 [8] Saha, S. K., Hossa, M. R., U, M. K., & Moal, R. N. 05, A New Approah of Solvg Lear Fratoal Programmg Problem LFP b Usg Computer Algorthm, Ope Joural of Optmzato, 403, 74. [9] Valpour, E., Yaghoob, M. A., & Mashh, M. 04, A teratve approah to solve mult objetve lear fratoal programmg problems, Apple Mathematal Moellg, 38, [0] mmerma, H.J. 976, Desrpto a optmzato of fuzz sstems, Iteratoal Joural of Geeral Sstems, Vol., pp [] mmerma, H.J. 978, Fuzz programmg a lear programmg wth several objetve futos, Fuzz Sets a Sstems, Vol., pp