On the Solution of a Special Type of Large Scale. Linear Fractional Multiple Objective Programming. Problems with Uncertain Data

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1 Appled Mathematcal Sceces, Vol. 4, 200, o. 62, O the Soluto of a Specal Type of Large Scale Lear Fractoal Multple Obectve Programmg Problems wth Ucerta Data Tarek H. M. Abou-El-Ee Departmet of Operatos Research & Decso Support Faculty of Computers & Iformato Caro Uversty 5 Dr. Ahmed Zowel St.- Orma - Postal Code Gza - Egypt t.haafy@fc-cu.edu.eg Omar M. Saad Departmet of Mathematcs, Faculty of Scece Helwa Uversty, A Helwa Postal Code 792, Caro, Egypt omar_saad@scece.helwa.edu.eg Abstract Ths paper s cocered wth solvg large scale lear fractoal multple obectve programmg (LSLFMOP) problems wth chace costrats. Chace costrats volve radom parameters the rght-had sdes. These radom rght-had sdes are cosdered to be statstcally depedet radom varable. The ma features of the proposed soluto procedure are based o chacecostraed techue, Chares ad Cooper trasformato ad the Rose's parttog procedure. A llustratve umercal example s gve to clarfy the ma results developed the paper. Mathematcs Subect Classfcato : 90C05, 90C06, 90C5, 90C29 Keywords: Large Scale Systems; Lear fractoal programmg ; Multple obectve programmg; Block agular structure; Agular ad dual agular structure; Chace-costrats. Correspodg author

2 3096 T. H. M. Abou-El-Ee ad O. M. Saad () Itroducto Fractoal programmg has attracted the atteto of may researchers the past. The ma reaso for terest fractoal programmg stems from the fact that programmg models could better ft the real problems f we cosder optmzato of rato betwee the physcal ad/or ecoomc uattes. Lterature survey reveals wde applcatos of fractoal programmg dfferet areas ragg from egeerg to ecoomcs. I real world decso stuatos, whe formulatg a LSLFMOP problem, some or all of the parameters of the optmzato problem are descrbed by stochastc (or radom or probablstc) varables rather tha by determstc uattes. Most of LSLFMOP problems arsg applcatos have specal structures that ca be exploted. There are may famlar structures for large scale optmzato problems such as: () the block agular structure, ad () agular ad dual- agular structure to the costrats, ad several kds of decomposto methods for lear ad olear programmg problems wth those structures have bee proposed [3, 4]. Recetly a sgfcat umber of studes have deed bee reported o sgle ad multple obectve fractoal lear ad olear programmg problems [, 2, 3, 4, 5, 7, 8, 9,, 5, 6 7, 8, 9, 20, 2, 24, 25, 26]. Ammar [] dscussed Dkelbach s global optmzato approach for fdg the global maxmum of the fractoal programmg problem. Based o ths dea, the author gave several characterzatos of the soluto set of a covex cocave fractoal programs. Caballero ad Herádez [2] troduced a ew method to estmate the weakly effcet set for the multobectve lear fractoal programmg problem. I aother work, they [3] troduced a test to establsh whether a lear fractoal goal programmg problem has solutos that verfy all goals ad, f so, how to fd them by solvg a lear programmg problem. Also, they outle a techue for restorg effcecy based o a mmax phlosophy. A useful ad ew approach to solve fractoal programmg wth absolute-value fuctos (FP-A) has bee proposed by Chag [4]. Later o, Chag [5] troduced a approxmate approach to reachg as close as possble a optmal soluto of the fractoal programmg problem wth absolute value fucto (FP- A). Che [7] cosders two popular vetory models: the cotuous revew ad perodc revew reorder-pot, order-uatty, cotrol systems. Specfcally we preset two procedures whch determe optmal values for the two cotrol parameters (.e., reorder-pot ad order-uatty) whe the holdg-ad-shortage costs are o-uas-covex. The algorthms based o a fractoal programmg method.

3 Lear fractoal multple obectve programmg 3097 Problems Dutta et al. [8] developed a method for optmzg multobectve lear fractoal programmg problem whch yelds always a effcet soluto. Gómez et al [9] preset a lear fractoal goal programmg model to a tmber harvest schedulg problem order to obta a balaced age class dstrbuto of a forest platato Cuba. Husa ad Jabee [] derved ecessary ad suffcet optmalty codtos for a cotuous -tme fractoal mmax programmg problem. I aother work, Husa et al. [2] troduced ecessary ad suffcet optmalty codtos for a odfferetable fractoal mmax programmg problem. Moreover, suffcet optmalty codtos for a olear multple obectve fractoal programmg problem volvg η-semdfferetable type I-prevex ad related fuctos have bee derved by Mshra et al. [6]. Morta et al. [7] preseted a probablty maxmzato model of a stochastc lear kapsack problem where the radom varables cosst of several groups wth mutually correlated oes. They proposed a soluto algorthm to the euvalet olear fractoal programmg problem wth a smple rakg method. Ths approach s effectvely applcable to oe of the portfolo selecto problems. A goal programmg (GP) procedure for fuzzy multobectve lear fractoal programmg problems has bee suggested by Pal et al. [8]. Preda [9] preseted ecessary ad suffcet optmalty codtos for a olear fractoal multple obectve programmg problem volvg η- semdfferetable fuctos. Rav ad Reddy [20] have modeled chemcal process plat operatos plag a ol refery as fuzzy lear fractoal multple goal programmg problem. A varat of the fuzzy lear fractoal goal programmg model of Dutta et al. [8] has bee developed ad used that study to solve the problem whch has two fuzzy fractoal goals ad 22 crsp costrats. Saad ad Sharf [2] proposed a soluto procedure to solve the chacecostraed teger lear fractoal programmg problem. Also, Saad ad Abou-El-Ee [22] suggested a soluto algorthm for teger lear fractoal multple obectve programmg problems (ILFMOP) wth block agular structure of the costrats. Sgh et al. [23, 24] studed multparametrc sestvty aalyss for programmg problems wth lear-plus-lear fractoal obectve fucto usg the cocept of maxmum volume the tolerace rego. Smelyasky ad Skedzelewsk [25] descrbe the computatoal kerels that are the buldg blocks of the Iteror Pot Method (IPM), ad they expla the dfferet sources of parallelsm sparse parallel lear solvers, the domat computato of IPM. Also, they aalyze the scalablty ad performace of two mportat optmzato workloads for solvg large scale lear ad uadratc programmg problems.

4 3098 T. H. M. Abou-El-Ee ad O. M. Saad Stacu-Masa ad Pop [26] poted out certa shortcomgs the work of Dutta et al. [8] ad gve the correct proof of theorem whch valdates the obtag of the effcet solutos. I the preset paper, a soluto algorthm s suggested for the soluto of LSLFMOP problems wth chace costrats whch has block agular structure of the costrats. The paper s orgazed as follows: I the followg secto, the problem formulato of LSLFMOP wth chace costrats (CHLSLFMOP). Chace costrats volve radom parameters the rght-had sdes. These radom rght-had sdes are cosdered to be statstcally depedet radom varable A algorthm s descrbed fte steps for solvg the problem of cocer s proposed Secto 3. For the sake of llustrato, a umercal example s provded Secto 4. Fally, the paper s cocluded Secto 5. (2) Problem Formulato Cosder the followg CHLSLFMOP problem wth a block agular structure of the costrats as: Maxmze [f (X), f 2 (X),,f k (X)] (-a) subect to P{ = P{ ah o xh o vh o } α h o, h o =,2,,m o, (-b) bh xh vh } α h, h =m - +, m - +2,,m, (-c) x 0, N, =,2,,, > } (-d) where the th obectve fucto ca be wrtte as follows: = ( X ) = f ( X ) = =, =,2,..., k f ( X ) D X + β = f = C X + γ ad γ, a, b ad β are costats, =,2,..., k; k : the umber of obectve fuctos, : the umber of subproblems, m : the umber of costrats, : the umer of varables, : the umber of varables of the th subproblem, =,2,,, >, : the umber of the commo costrats represeted by m o (2)

5 Lear fractoal multple obectve programmg 3099 = ah o xh o vh o m : the umber of depedet costrats of the th subproblem represeted by bh xh vh, =,2,,, > X : a -dmesoal colum vector of varables, X : a -dmesoal colum vector of varables for the th subproblem, =,2,.,, >, C : a -dmesoal row vector for the th subproblem the th obectve fucto, D : a -dmesoal row vector for the th subproblem, K = {,2,.,k}, N = {,2,..,}, I addto, P meas probablty, αh o ad αh are a specfed probablty levels. For the sake of smplcty, cosder that the radom parameters, vh o ad vh are dstrbuted ormally ad depedetly of each other wth kow meas E{vh o } ad E{vh } ad varaces Var{vh o } ad Var{vh }. Furthermore, we assume that D X + β s everywhere postve. = Usg the chace costraed programmg techue [6, 0], the determstc verso of the CHLSMOP problem () ca be wrtte as follows : Maxmze [f (X), f 2 (X),,f k (X)] (3-a) subect to = ah o xh o E{vh o }+ kα o Var{vh o }, h o =,2,,m o, bh xh E{vh } + kα Var{vh }, h =m - +, m - +2,,m, (3-b) (3-c) x 0, N, =,2,,, > }. (3-d) where k α, =0,,2,.,, s the stadard ormal value such that Φ(k α )=- α, =0,,,, ad Φ represets the cumulatve dstrbuto fucto of the stadard ormal dstrbuto.

6 300 T. H. M. Abou-El-Ee ad O. M. Saad Problem (3) ca be treated usg the oegatve weghted sum approach [3] ad wll be coverted to the followg problem wth a sgle-obectve fucto as: Maxmze subect to = bh k w f ( X ) (4-a) ah o xh o E{vh o }+ kα o Var{vh o }, h o =,2,,m o, xh E{vh } + kα Var{vh }, h =m - +, m - +2,,m, (4-b) (4-c) x 0, N, =,2,,, > }. (4-d) where w 0, ( =,2,..., k ) ad w = k =. Coseuetly, usg Chares-Cooper trasformato method [6] by makg the varable chage: μ = (5-a) D X + β = wth the addtoal varable chages Y = X μ =,2,...,, > (5-b) the uder these chages, problem (4) s euvalet to the oe of solvg the followg: k Maxmze [w f (Y, µ)] =[ = k w ( = C Y +γ µ)] (6-a) subect to = A Y ( E{vh o }+ kα o Var{vh o }) µ 0, h o =,2,,m o, = D Y + β µ = (6-b) (6-c) B Y (E{vh } + kα Var{vh } ) µ 0, h =m - +, m - +2,,m (6-d) µ >0, Y 0, =,2,.,, > (6-e) Now, problem (6) has agular ad dual-agular structure [4] ad ca be solved usg Rose's parttog procedure [4] to fd ts optmal soluto Y ad µ.

7 Lear fractoal multple obectve programmg 30 (3) Soluto Method I ths secto we propose a soluto method for solvg the CHLSLFMOP problem (). The proposed algorthm ths paper ca be summarzed fte steps as follows: Soluto Algorthm: Step. Formulate CHLSLFMOP problem () whch have fractoal lear obectve fuctos as E. (2). Step 2. Use the stadard ormal dstrbuto table [0] to fd kα, =0,,2,.,, Step 3. Trasform problem () to the form of problem (3) Step 4. Use the oegatve weghted sum approach [3 ] to covert problem (3) to problem (4). Step 5. Use the Chares-Cooper trasformato [6] by makg the varable chage μ wth the addtoal varable chage = = D X + β Y = X μ =,2,...,, > to rewrte problem (4) the form of problem (6). Step 6. () Choose w = w 0, =,2,..., k ad w =. () Use Rose's parttog procedure [4] to obta a optmal soluto (, μ Y ) from whch the optmal soluto X ca be obtaed drectly, where X = Y / μ. Step7. Let X be the optmal soluto of problem (4) such that : (): If w > 0 for all, the X s a effcet soluto of problem () ad go to step 8. (): If w 0 for all, ad X s a uue soluto of problem (4), the X s a effcet soluto of problem () ad the Go to step 8. (): If w 0 for all, ad there are alteratve teger solutos of problem (4), use the o-ferorty test [3] to detfy whch soluto s effcet of problem (), the go to step 8. Step 8. Stop. k = (4) A Illustratve Example I what follows, we provde a umercal example to llustrate the soluto algorthm descrbed the prevous secto. For ths purpose, let us cosder the followg CHLSFMOP problem whch has the agular structure: Maxmze (f (X), f 2 (X))

8 302 T. H. M. Abou-El-Ee ad O. M. Saad subect to P{x + x 2 b o } , P{x b } 0.5, P{ 2x 2 b 2 } 0.403, x 0, x 2 0 where x + x2 x + 2 x2 f ( x) =, ad f 2 ( x) = 4 x + 3 x x + 3 x2 + 3 By usg problem (3), we ca have Maxmze (f (X), f 2 (X)) subet to x + x 2 5, x 2, 2x 2 8, x 0, x 2 0 Assume that μ = 4x + 3 x2 + 3 ad cosder w =.0.5 ad w2 = 0.5, ad make the varable chages: y = xμ ad y2 = x2μ Thus, the above problem ca be rewrtte as follows: Mmze F(y, y 2 ) = y +.5y 2 subect to 4y + 3 y 2 + µ = y + y 2-5 µ 0, y -2 µ 0, 2 y 2-8 µ 0, y, y 2 0 ad µ>0 Ths problem has agular ad dual-agular structure. the by usg Rose's Parttog procedure [4], the optmal soluto y = 0 y = 0.3, μ = 0.08 ad f 0.46 ca be obtaed ad thus x 2 = = 0, x2 = (5) Cocludg Remarks I ths paper we have proposed a soluto procedure for solvg a specal type of CHLSFMOP problems.these problems have block agular structure of the costrats. A llustratve umercal example has bee provded to clarfy the theory ad the proposed algorthm.

9 Lear fractoal multple obectve programmg 303 However, there are some ope pots of research whch should be explored ad studed the feld of fuzzy teger fractoal optmzato problems. Some of these pots are : () A soluto algorthm s eeded for treatg fuzzy bcrtero ad multple obectve teger lear ad olear fractoal programs wth fuzzy parameters the costrats ad the obectve fuctos. () A soluto procedure s reured for treatg fuzzy large- scale teger lear ad olear fractoal multple obectve programmg problems. () A soluto method should be carred out for solvg stochastc largse- scale multple obetve teger lear ad olear fractoal programmg problmes wth block agular structure of the costrats. Refereces [] E.E. Ammar, O solutos ad dualty of olear osmooth fractoal, programs, Appled Mathematcs ad Computato,72 (2) (2006) [2] R. Caballero ad M. Herádez, The cotrolled estmato method the multobectve lear fractoal problem, Computers & Operatos Research, 3 () ( 2004) [3] R. Caballero ad M. Herádez, Restorato of effcecy a goal programmg problem wth lear fractoal crtera, Europea Joural of Operatoal Research,72()(2006)3-39 [4] C. Chag, A approxmate approach for fractoal programmg wth absolute-value fuctos, Appled Mathematcs ad Computato, 6() (2005) [5] C. Chag, Fractoal programmg wth absolute value fuctos: a fuzzy goal programmg approach, Appled Mathematcs ad Computato, 67 () (2005) [6] A. Chares ad W. W. Cooper, Programmg wth lear fractoal crtera, Naval Res. Logst Quart., 9 (962) [7] Y. F. Che, Fractoal programmg approach to two stochastc vetory problems, Europea Joural of Operatoal Research, 60() ( 2005) 63-7 [8] D. Dutta, R. N. Twar ad J. R. Rao, Multple obectve lear fractoal programmg - A fuzzy set theoretc approach, Fuzzy Sets ad Systems 52 () (992) [9] T. Gómez, M. Herádez, M.A. Leó ad R. Caballero, A forest plag problem solved va a lear fractoal goal programmg model, Forest Ecology admaagemet, 227(-2)(2006) [0] W.K. Grassma, Stochastc systems for maagemet, North Hollad, New York, 98.

10 304 T. H. M. Abou-El-Ee ad O. M. Saad [] I. Husa ad Z. Jabee, Cotuous-tme fractoal mmax programmg, Mathematcal ad Computer Modellg, 42 (5-6) (2005) [2] I. Husa, M. A. Haso ad Z. Jabee, O odfferetable fractoal mmax programmg, Europea Joural of Operatoal Research,60 () (2005) [3] C.L. Hwag, & A.S. M. Masud,, Multple Obectve Decso Makg Methods ad Applcatos, Sprger-Verlag, New York, USA, (979). [4] L. S. Lasdo, Optmzato theory for large systems, Macmlla, New York, U.S.A., (970). [5] R. K. Mart, Large scale lear ad teger optmzato, Sprger, U.S.A., (999). [6] S.K. Mshra, S.Y. Wag ad K.K. La, Multple obectve fractoal programmg volvg semlocally type I-prevex ad related fuctos, Joural of Mathematcal Aalyss ad Applcatos, 30 (2) (2005) [7] H. Morta, H. Ish ad T.Nshda, Stochastc lear kapsack programmg problem ad ts applcato to a portfolo selecto problem, Europea Joural of Operatoal Research, 40 (3) (989) [8] B. B. Pal, B. N. Motra ad U. Maulk, A goal programmg procedure for fuzzy multobectve lear fractoal programmg problem, Fuzzy Sets ad Systems, 39 (2) (2003) [9] V. Preda, Optmalty ad dualty fractoal multple obectve programmg volvg semlocally prevex ad related fuctos, Joural of Mathematcal Aalyss ad Applcatos, 288 (2) (2003) [20] V. Rav ad P. J. Reddy, Fuzzy lear fractoal goal programmg appled to refery operatos plag, Fuzzy Sets ad Systems, 96 (2) (998) [2] O. M. Saad ad W.H. Sharf, O the soluto of teger lear fractoal programs wth ucerta data, Joural of the Mathematcs & Computer Sceces, 2 (2) (200) 69-73, [22] O. M. Saad ad T. H. M. Abou-El-Ee, O the Soluto of a Specal Type of Iteger Lear Fractoal Multple Obectve Programmg Problems, Advaces Modellg ad Aalyss, (D), Vol. 3, No., pp.42-55, [23] S.Sgh, P.Gupta ad D.Bhata, Multparametrc sestvty aalyss programmg problem wth lear-plus-lear fractoal obectve fucto, Europea Joural of Operatoal Research, 60 () (2005) [24] S. Sgh, P. Gupta ad D.Bhata, Mult - parametrc sestvty aalyss of the costrat matrx lear- plus- lear fractoal programmg problem, Appled Mathematcs ad Computato, 70 (2) (2005) [25] M. Smelyasky ad S. Skedzelewsk, Parallel computg for large-scale optmzato problems: Challeges ad Solutos, Itel Techology Joural, 9(2) (2005) 5-64.

11 Lear fractoal multple obectve programmg 305 [26] I. M. Stacu-Masa ad B. Pop, O a fuzzy set approach to solvg multple obectve lear fractoal programmg problem, Fuzzy Sets ad Systems, 34 (3) ( 2003) Receved: May, 200