Solving Fuzzy Linear Programming Problems with Piecewise Linear Membership Function

Transcription

1 Avalable a hp://pvamu.edu/aam Appl. Appl. Mah. ISSN: Vol., Issue December ), pp. Prevously, Vol., Issue, pp. 6 6) Applcaos ad Appled Mahemacs: A Ieraoal Joural AAM) Solvg Fuzzy Lear Programmg Problems wh Pecewse Lear Membershp Fuco S. Effa Deparme of Mahemacs Ferdows Uversy of Mashhad Mashhad, Ira s-effa@um.ac.r H. Abbasya Deparme of Mahemacs Islamc Azad Uversy Alabadeaool, Ira abbasya@yahoo.com Receved: Jauary, 9; Acceped: Augus, Absrac I hs paper, we cocerae o lear programmg problems whch boh he rgh-had sde ad he echologcal coeffces are fuzzy umbers. We cosder here oly he case of fuzzy umbers wh lear membershp fucos. The symmerc mehod of Bellma ad Zadeh 97) s used for a defuzzfcao of hese problems. The crsp problems obaed afer he defuzzfcao are o-lear ad eve o-cove geeral. We propose here he "modfed subgrade mehod" ad "mehod of feasble drecos" ad uses for solvg hese problems see Bazaraa 99). We also compare he ew proposed mehods wh well ow "fuzzy decsve se mehod". Fally, we gve llusrave eamples ad her umercal soluos. Keywords: Fuzzy lear programmg; fuzzy umber; augmeed Lagraga pealy fuco mehod; feasble drecos of Tops ad Veo; fuzzy decsve se mehod MSC ) No.: 9C, 9C7. Iroduco I fuzzy decso mag problems, he cocep of mamzg decso was proposed by Bellma ad Zadeh 97). Ths cocep was adaped o problems of mahemacal programmg

2 AAM: Ier. J., Vol., Issue December ) [Prevously, Vol., Issue, pp. 6 6] by Taaa e al. 9). Zmmerma 9) preseed a fuzzy approach o mul-obecve lear programmg problems. He also suded he dualy relaos fuzzy lear programmg. Fuzzy lear programmg problem wh fuzzy coeffces was formulaed by Negoa 97) ad called robus programmg. Dubos ad Prade 9) vesgaed lear fuzzy cosras. Taaa ad Asa 9) also proposed a formulao of fuzzy lear programmg wh fuzzy cosras ad gve a mehod for s soluo whch bases o equaly relaos bewee fuzzy umbers. Shaocheg 99) cosdered he fuzzy lear programmg problem wh fuzzy cosras ad defuzzfcaed by frs deermg a upper boud for he obecve fuco. Furher he solved he obaed crsp problem by he fuzzy decsve se mehod roduced by Saawa ad Yaa 9). Guu ad Ya-K 999) proposed a wo-phase approach for solvg he fuzzy lear programmg problems. Also applcaos of fuzzy lear programmg clude lfe cycle assessme [Raymod )], produco plag he ele dusry [Elamvazuh e al. 9)], ad eergy plag [Caz 999)]. We cosder lear programmg problems whch boh echologcal coeffces ad rghhad-sde umbers are fuzzy umbers. Each problem s frs covered o a equvale crsp problem. Ths s a problem of fdg a po whch sasfes he cosras ad he goal wh he mamum degree. The dea of hs approach s due o Bellma ad Zadeh 97). The crsp problems, obaed by such a maer, ca be o-lear eve o-cove), where he oleary arses cosras. For solvg hese problems we use ad compare wo mehods. Oe of hem called he augmeed lagraga pealy mehod. The secod mehod ha we use s he mehod of feasble drecos of Tops ad Veo 99). The paper s ouled as follows. I seco, we sudy he lear programmg problem whch boh echologcal coeffces ad rgh-had-sde are fuzzy umbers. The geeral prcples of he augmeed Lagraga pealy mehod ad mehod of feasble drecos of Tops ad Veo are preseed seco ad, respecvely. The fuzzy decsve se mehod s suded seco. I seco 6, we eame he applcao of hese wo mehods ad he compare wh he fuzzy decsve se mehod by cocree eamples.. Lear Programmg Problems wh Fuzzy Techologcal Coeffces ad Fuzzy Rgh Had-sde Numbers We cosder a lear programmg problem wh fuzzy echologcal coeffces ad fuzzy rgh-had-sde umbers: Mamze c Subec o a~ ~ b, m ),,

3 6 S. Effa ad H. Abbasya where a leas oe ad a ~ ad b ~ are fuzzy umbers wh he followg lear membershp fucos:, a, a d, a a d, d, a d, a where R ad d for all,, m,,,, ad b, b, b p, b b p, p, b p, where p, for,, m. For defuzzfcao of he problem ), we frs calculae he lower ad upper bouds of he opmal values. The opmal values z l ad z u ca be defed by solvg he followg sadard lear programmg problems, for whch we assume ha all hey he fe opmal value z l Mamze c Subec o a d b, m ), ad z u Mamze c Subec o a b p, m,. )

4 AAM: Ier. J., Vol., Issue December ) [Prevously, Vol., Issue, pp. 6 6] 7 The obecve fuco aes values bewee z l ad z u whle echologcal coeffces ae values bewee a ad a d ad he rgh-had sde umbers ae values bewee b ad b p. The, he fuzzy se of opmal values, G, whch s a subse of, s defed by, c, z l c z l G ), zl c, z u zu zl, c. z u ) The fuzzy se of he cosra, c, whch s a subse of s defed by, b a, b a c ), a ), b a d p d p, b a. d p ) By usg he defo of he fuzzy decso proposed by Bellma ad Zadeh 97) [see also La ad Hwag 9)], we have ) m ),m ))). 6) D G C I hs case, he opmal fuzzy decso s a soluo of he problem ma )) ma m ),m ))). 7) D G C Cosequely, he problem ) rasform o he followg opmzao problem Mamze Subec o G ) ), m C ).

5 S. Effa ad H. Abbasya By usg ) ad ), he problem ) ca be wre as: Mamze Subec o z z ) c zl a l u d ) p b, m,. 9) Noce ha, he cosras problem 9) coag he cross produc erms are o cove. Therefore he soluo of hs problem requres he specal approach adoped for solvg geeral o cove opmzao problems.. The Augmeed Lagraga Pealy Fuco Mehod The approach used s o cover he problem o a equvale ucosraed problem. Ths mehod s called he pealy or he eeror pealy fuco mehod, whch a pealy erm s added o he obecve fuco for ay volao of he cosras. Ths mehod geeraes a sequece of feasble pos, hece s ame, whose lm s a opmal soluo o he orgal problem. The cosras are placed o he obecve fuco va a pealy parameer a way ha pealzes ay volao of he cosras. I hs seco, we prese ad prove a mpora resul ha usfes usg eeror pealy fucos as a meas for solvg cosraed problems. Cosder he followg prmal ad pealy problems: Prmal problem: Mmze Subec o z z l a u ) d ) c z l p,,..., m,,...,,, )

6 AAM: Ier. J., Vol., Issue December ) [Prevously, Vol., Issue, pp. 6 6] 9 Pealy problem: Le be a couous fuco of he form a d ) p b m,...,, ) ) ) zl zu c zl, ) where s couous fuco sasfyg he followg: y, f y ad y), f y. ) The basc pealy fuco approach aemps o solve he followg problem: Mmze ) Subec o, where f{, ) : R, R}. From hs resul, s clear ha we ca ge arbrarly close o he opmal obecve value of he prmal problem by compug µ) for a suffcely large µ. Ths resul s esablshed Theorem.. Theorem.. Cosder he problem ). Suppose ha for each µ, here ess a soluo, ) R o he problem o mmze +µ, ) subec o R ad R, ad ha {,) } s coaed a compac subse of R. The, lm sup{ : R, R, g, ) }, where g g, g,..., g, g,..., g, g, g ) ad m m m m m g, ) g g g m m m g, ) z, ) z a, ), ) l u d ),,..., c z p,,... m l )

7 S. Effa ad H. Abbasya ad f{, ) : R, R} [, ) ]. Furhermore, he lm, ) of ay coverge subsequece of {, ) } s a opmal soluo o he orgal problem, ad [, ) ] as. Proof: For proof, see Bazaraa 99)... Augmeed Lagraga Pealy Fucos A augmeed lagraga pealy fuco for he problem ) s as: F AL m u m u,, u) ma, ), g, ) where u are lagrage mulpler. The followg resul provdes he bass by vrue of whch he AL pealy fuco ca be classfed as a eac pealy fuco. Theorem... Cosder problem P o ), ad le he KKT soluo,,u sasfy he secod-order suffcecy codos for a local mmum. The, here ess a such ha for, he AL pealy fuco F AL..,u ), defed wh u =u, also acheves a src local mmum a,. Proof: For proof, see Bazaraa 99). Algorhm The mehod of mulplers s a approach for solvg olear programmg problems by usg he augmeed lagraga pealy fuco a maer ha combes he algorhmc aspecs of boh Lagraga dualy mehods ad pealy fuco mehods.

8 AAM: Ier. J., Vol., Issue December ) [Prevously, Vol., Issue, pp. 6 6] Ialzao Sep: Selec some al Lagraga mulplers u u ad posve Values for,..., m, for he pealy parameers. Le, ) be a ull vecor, ad deoe VIOL, ), where for ay R R ad, VIOL, ) ma{ g, ), I { : g, ) }} s a measure of cosra volaos. Pu = ad proceed o he er loop of he algorhm. Ier Loop: Pealy Fuco Mmzao) Solve he ucosraed problem o Mmze F AL,, u ), ad le, ) deoe he opmal soluo obaed. If VIOL, ), he sop wh, ) as a KKT po, Praccally, oe would ermae f VIOL, ) s lesser ha some olerace. ) Oherwse, f VIOL, ) VIOL, ), proceed o he ouer loop. O he oher had, for each cosra,..., m for whch g, ) VIOL, ), replace he correspodg pealy parameer by, repea hs er loop sep. Ouer Loop: Lagrage Mulpler Updae) Replace u by u ew, where,, u},,..., m. u ) u ma{ g ew Icreme by, ad reur o he er loop.

9 S. Effa ad H. Abbasya. The Modfcao of Tops ad Veo Revsed Feasble Drecos Mehod The frs, we descrbe he mehod of revsed feasble drecos of Tops ad Veo. So we propose a modfcao from hs mehod. A each erao, he mehod geeraes a mprovg feasble dreco ad he opmzes alog ha dreco. We ow cosder he followg problem, where he feasble rego s defed by a sysem of equaly cosras ha are o ecessarly lear: Mmze Subec o z z l a u ) d ) c z p l b,,,..., m,..., ),. Theorem below gves a suffce codo for a vecor d o be a mprovg feasble dreco. Theorem.. Cosder he problem ). Le ˆ, ˆ ) be a feasble soluo, ad le I be he se of bdg cosras, ha s I { : g ˆ, ˆ }, where g ' s are as ). If ) ˆ, ˆ d. e d ) ad g ˆ, ˆ d mprovg feasble dreco. for I, he d s a Proof: For proof see Bazaraa 99). Theorem.. Le ˆ ˆ, ) R be a feasble soluo of ). Le z, d ) be a opmal soluo o he followg dreco fdg problem: Mmze Subec o z g ) ˆ, ˆ d ˆ, ˆ d z g ˆ, ˆ,,..., m d,,...,, 6)

10 AAM: Ier. J., Vol., Issue December ) [Prevously, Vol., Issue, pp. 6 6] f z, he d s a mprovg feasble dreco. Also, ˆ, ˆ ) s a Frz Joh po, f ad oly f z. Afer smplfy, we ca rewre he problem 6) as follows: Mmze z Subec o d z c d z z ) d z g, ) u l a d d p d d z g m ) ), ),,..., 7) d z,,..., d z d,,...,. Ths revsed mehod was proposed by Tops ad Veo 967) ad guaraees covergece o a Frz Joh po. Geerag a Feasble Dreco The problem uder cosderao s Mmze Subec o g, ),,..., m, where g ' s are as ). Gve a feasble po ˆ, ˆ ), a dreco s foud by solvg he dreco-fdg lear programmg problem DF ˆ, ˆ ) o 7). Here, boh bdg ad o bdg cosras play a role deermg he dreco of moveme... Algorhm of Tops ad Veo Revsed Feasble Drecos Mehod A summary of he mehod of feasble drecos of Tops ad Veo for solvg he problem ), s gve below. As wll be show laer, he mehod coverges o a Frz Joh po. Ialzao Sep: Choose a po, ) such ha g, ) for,..., m, where g are as ). Le = ad go o he ma sep.

11 S. Effa ad H. Abbasya Ma Sep:. Le z, d ) be a opmal soluo o lear programmg problem 7). If z, sep;, ) s a Frz Joh po. Oherwse, z ad we go o.. Le l be a opmal soluo o he followg le search problem: Mmze ld Subec o l l, ma where l ma sup{ l : g y ld ),,..., m }, ad y, ) ad g, for all,..., m, are as ). Le y y l d. Replace by +, ad reur o sep. Theorem... Cosder he problem ). Suppose ha he sequece {, )} s geeraed by he algorhm of Tops ad Veo. The, ay accumulao po of {, )} s a Frz Joh po... The Modfcao of Algorhm I above algorhm, we eed o oba he grade of he obecve fuco ad also he grade of he cosra fucos. I hs modfcao we do o eed a feasble po. Noe ha we ca forgo from he le search problem of sep he ma sep, sce, obvously, opmal soluo for hs le search problem l. Hece, sep of he ma sep, we have l. s ma l ma Ialzao Sep The mehod of fd a he al feasble po). Se = ad = ad go o.

12 AAM: Ier. J., Vol., Issue December ) [Prevously, Vol., Issue, pp. 6 6]. Tes wheher a feasble se sasfyg he cosras of he problem ) ess or o, usg phase oe of he smple mehod,.e., solvg he problem below: Mmze a Subec o g, ) s a,,..., m, where s s he vecor of slac varables ad s he vecor of arfcal varables. Le, s, a ) be a opmal soluo of hs he problem. If a, ha, ) s a al feasble po for he problem ) ad go o ; oherwse, go o.. Se ad, reur o.. Se Ma Sep:,, = ad go o he ma sep.. Le z, d ) be a opmal soluo o lear programmg problem 7). If z, sep, ) s a Frz Joh po. Oherwse, z ad we go o.. Se l l ma, where l ma sup{ l : g y ld ),,..., m }, y, ), ad g s as ). Le y y l d, replace by +, ad reur o. The algorhm for fdg l ma sup{ l: g yld) }, by employg he bseco mehod. Ths algorhm s as below: Ialzao Sep:. Se l ad =.. If for a leas oe, oba g y ld), he go o, oherwse, se l l, = + ad repea.. Se a l, b l, ad go o he ma sep.

13 6 S. Effa ad H. Abbasya Ma Sep:. Sel a b, f a b where? s a small posve scaler); sop, lma l. Oherwse, go o.. If for a leas oe oba g y ld), he se b l. Oherwse, se a, = + ad repea. l. Numercal Eamples Eample.. Solve he opmzao problem Mamze ~ ~ Subec o ) ~, ~ 6, ~ ~ ~ ~ whch ae fuzzy parameers as L,), L,), L,) ad L, ), as used by Shaocheg 99). Tha s, a ), d ), a d ). For eample, L a, d ) s as:, a, ad a ) d, a a d,,, or,, a ),,,.

14 AAM: Ier. J., Vol., Issue December ) [Prevously, Vol., Issue, pp. 6 6] 7 For solvg hs problem we mus solve he followg wo subproblems: z = mamze ad Subec o 6, z = mamze Subec o 6., Opmal soluos of hese sub problems are,.6. z 6. ad.6.7 z.6, respecvely. By usg hese opmal values, problem ) ca be reduced o he followg equvale o-lear programmg problem: Mamze Subec o ,. Tha s, Mamze Subec o ) ) ) ) ) 6.,

15 S. Effa ad H. Abbasya Le us solve problem 9) by usg he modfcao mehod of feasble drecos of Tops ad Veo. Ialzao Sep: The problem he phase s as: Mmze a a a Subec o a a s a.6.7 ) ) s a ) ) ) s a 6, s, s, s,,,,, a a a where a, a, a are arfcal varables ad s, s, s are slac varables. Se, he, opmal soluo of above problem we have: a.776, a a, ad sce a so he feasble se s empy, he ew value of s red. For hs, he a.77 so he feasble se s empy. The ew value of., he he opmal soluo of he problem ) s as follows: s.76 s.9779 s.69 a a a. Hece, we are sar from he po, ).7),.99976). We frs formulae he problem 9) he form Mmze Subec o g,, ).7.6 g,, ) ) ) g,, ) ) ) 6 g,, ) ) g,, ) g,, ) g 6 7,, ).

16 AAM: Ier. J., Vol., Issue December ) [Prevously, Vol., Issue, pp. 6 6] 9 Ierao : Search Dreco: The dreco fdg problem s as follows: Mmze z Subec o d z d d.7d z.77.d.7d.696d z.9 d z.7 d z.9976 d z. d z.7,,,. The opmal soluo o he above problem s d, z ).667,.77,.66,.66). d Le Search: The mamum value l such ha, ) s feasble s gve by ld l ma.796. Hece l ma. 796 s opmal soluo. We he have, ), ) l d.96,.779,.796). The process s he repeaed. The, we have:, ).96,.776,.9669), ).76,.776,.977), ).76,.77,.97996), ).7,.7699,.976) 6, 6 ).7,.769,.976) 7, 7 ).779,.76996,.977). The opmal soluo for he ma problem ) s as whch has he bes membershp grad *.977. ) * *, ).779,76996, The progress of he algorhm of he mehod of feasble drecos of Tops ad Veo of Eample s depced Fgure.

17 S. Effa ad H. Abbasya Fgure. Appromae soluo.),.),.). Now, we solve hs problem ) wh he augmeed lagraga pealy fuco mehod. We cover he problem ) o ). Selec al Lagraga mulplers ad posve values for he pealy parameers u,., `,...,7. The sarg po s ae as, ),, ) ad.. Sce VIPOL, ), we choose he er loop. The augmeed Lagraga pealy fuco s as F AL,, u) 7 [ ] [ ) ] ) [ ) ) 6]. Solvg problem mmze F AL,, u ), we oba, ).976,.6,.6976), VIOL, ). 77 ad VIOL, ) VIOL, ).7.

18 AAM: Ier. J., Vol., Issue December ) [Prevously, Vol., Issue, pp. 6 6] Hece, we go o ouer loop sep. The ew Lagraga mulplers are as u ew.76,.99,.,,,,). Se =, ad we go o he er loop sep. The process s he repeaed. The, we VIOL VIOL VIOL VIOL 6, ).7,.7769,.9667), ).679,.79,.966), ).7966,.796,.9776), ).7,.77,.979), ).77,.77,.97) 6 VIOL 6, )., ).6, )., )., ).7. 6 The opmal soluo for he ma problem ) s he po * *, ).77,.77), * whch has he bes membershp grad =.97. The progress of he algorhm of he mehod of he augmeed Lagraga pealy fuco of Eample s depced Fgure. Fgure. Appromae soluo.),.),.).

19 S. Effa ad H. Abbasya Le us solve problem 9) by usg he fuzzy decsve se mehod. For =, he problem ca be wre as 6, 6. L R ad sce he feasble se s empy, by ag ad, he ew value of s red. For., he problem ca be wre as 7,.99 6 L R ad sce he feasble se s empy, by ag ad, he ew value of s red. For., he problem ca be wre as 7 7,.99 6 L R ad sce he feasble se s oempy, by ag ad, he ew value of / / s red. For.7, he problem ca be wre as

20 AAM: Ier. J., Vol., Issue December ) [Prevously, Vol., Issue, pp. 6 6] 7,.6 6 L R ad sce he feasble se s oempy, by ag ad, he ew value of / / s red. For.7, he problem ca be wre as 7,.6 6 L R ad sce he feasble se s oempy, by ag ad, he ew value of // 7 s red. 6 For 7.7, he problem ca be wre as , L R 7 ad sce he feasble se s empy, by ag ad 6, he ew value of / 7 /6 s red. The followg values of are obaed he e wey s eraos:

21 S. Effa ad H. Abbasya Cosequely, we oba he opmal value of a he hry secod erao by usg he fuzzy decsve se mehod. Noe ha, he opmal value of foud a he seve erao of he mehod of feasble dreco of Tops ad Veo ad a he sh erao of he augmeed Lagraga pealy fuco mehod s appromaely equal o he opmal value of calculaed a he wey frs erao of he fuzzy decsve se mehod. Eample 6.. Solve he opmzao problem Mamze Subec o ~ ~ ~ ) ~ ~ ~,, whch ae fuzzy parameers as: ~ ~ ~ ~ L,), L,), L,), L,), ~ ~ b,), b,), L as used by Shaocheg 99). Tha s, L a ), d ), a d ) b ), p ), b p ). 7

22 AAM: Ier. J., Vol., Issue December ) [Prevously, Vol., Issue, pp. 6 6] To solve hs problem, we mus solve he followg wo subproblems z Mamze l ad Subec o, z Mamze u Subec o 7., Opmal soluos of hese subproblems are as follows: z l ad. z u., respecvely. By usg hese opmal values, problem ) ca be reduced o he followg equvale o-lear programmg problem: Mamze Subec o. ) ) ) ) )., Le s solve problem ) by usg he modfcao mehod of feasble drecos of Tops ad Veo. Ialzao Sep: The problem he phase s as follows: Mmze a a a

23 6 S. Effa ad H. Abbasya Subec o. s a a ) ) s ) a ) ) s a,, s, s, s, a, a, where a, a, a are arfcal varables, s, s, s are slac varables ad s fed scaler. Se. The, a. ad sce a so he feasble se s empy, he ew value of s red. The we have. a a. a a a. Hece, we are sar from he po, ).766,.,.). We frs formulae he problem 9) he form Mmze Subec o g,, ). g,, ) ) ) g,, ) ) ),, ) g,, ) g 6,, ) 7,, ) g ) g. The dreco fdg problem for each he arbrary cosa po,, ) s as follows: Mmze z Subec o d z d d.d z g,, ) ) d ) d ) d z g,, ) ) d ) d ) d z g,, ) d z d z d z d, d, d.

24 AAM: Ier. J., Vol., Issue December ) [Prevously, Vol., Issue, pp. 6 6] 7 Ierao Search Dreco: For he al po fdg problem s as follows:, ).766,.,.) he dreco Mmze z Subec o d z d d.d z.67.d.d.6d z.676.d.d.d z.66 The opmal soluo o he above problem s d z.77 d z. d z. d z.7 d,,,. d, z).99,.6,.779,.779 ). Le Search: The mamum value of l such ha ld s feasble s gve by l ma Hece l We he have, ), ) l d.997, ,.797). The process s he repeaed. The, we have: 6 7, ).697,.,.977 ), ).799,.7,.96 ), ).6,.,.79 ), ).,.6,.6 ), ).96,.,. ) 6, ).99,.9,. ) 7, ).9,.,.9 ). The opmal soluo for he ma problem ) s * *, ).9,., )

25 S. Effa ad H. Abbasya * whch has he bes membershp grad =.9. The progress of he algorhm of he mehod of feasble drecos of Tops ad Veo of Eample s depced Fgure. Fgure. Appromae soluo.),.),.). Now, we solve he problem ) wh he augmeed Lagraga pealy fuco mehod. We cover he problem ) o ). Selec al Lagraga mulplers ad posve values for he pealy parameers u,.,,..., 7. The sarg po s ae as, ),, ) ad.. Sce VIOL, ) we gog o er loop. The augmeed Lagraga pealy fuco s as: F AL,, u ) [ ]. [ ) ) ] [ ) ) ], wh solvg problem mmze F AL,, u ) we oba, ).777,.,.69 ), ad VIOL, ). 669 ad also VIOL, ) VIOL, ).

26 AAM: Ier. J., Vol., Issue December ) [Prevously, Vol., Issue, pp. 6 6] 9 Hece, we go o ouer loop sep. The ew Lagraga mulplers are as u.9,,.696,,,,). ew Se =, ad we go o he er loop sep. The process s he repeaed. The, we have:, ).9,.,.799 ) VIOL VIOL VIOL 6 VIOL 7 VIOL VIOL 9 VIOL 9 6 7, ), ), ) 6, ) 7, ). ), ) 9.6.6, )., ).79, ) 6.6, ) 7., )..67, ) ,.96,.76,.9,.,.,. The opmal soluo for he ma problem ) s ) whch has he bes membershp grad * *, ).67,.,,.99,.7767,.9,.9,.,. * =.. The progress of he algorhm of he mehod of he augmeed Lagraga pealy fuco of Eample s depced Fgure. ) ) ) ) ) ) Fgure. Appromae soluo.),.),.).

27 S. Effa ad H. Abbasya Le us solve he problem ) by usg he fuzzy decsve se mehod. For, he problem ca be wre as., L R ad sce he feasble se s empy, by ag ad, he ew value of s red. For., he problem ca be wre as 6., L R ad sce he feasble se s empy, by ag ad /, he ew value of s red. For., he problem ca be wre as 9 7,.6 L R ad sce he feasble se s empy, by ag ad /, he ew value of s red. For., he problem ca be wre as 9 9 7,. 9 L R ad sce he feasble se s oempy, by ag ad, he ew value of // s red. 6

28 AAM: Ier. J., Vol., Issue December ) [Prevously, Vol., Issue, pp. 6 6] The followg values of are obaed he e wey oe eraos: Cosequely, we oba he opmal value of a he wey ffh erao of he fuzzy decsve se mehod. Noe ha, he opmal value of foud a he secod erao of he mehod of feasble dreco of Tops ad Veo ad a he sh erao of he augmeed Lagraga pealy fuco mehod s appromaely equal o he opmal value of calculaed a he wey ffh erao of he fuzzy decsve se mehod. 7. Coclusos Ths paper preses a mehod for solvg fuzzy lear programmg problems whch boh he rgh-had sde ad he echologcal coeffces are fuzzy umbers. Afer he defuzzfcao usg mehod of Bellma ad Zadeh, he crsp problems are o-lear ad eve o-cove geeral. We use here he "modfed subgrade mehod" ad "mehod of feasble drecos for solvg hese problems. We also compare he ew proposed mehods wh well ow "fuzzy decsve se mehod". Numercal resuls show he applcably ad accuracy of hs mehod. Ths mehod ca be appled for solvg ay fuzzy lear programmg problems wh fuzzy coeffces cosras ad fuzzy rgh had sde values. REFERENCES Bazaraa, M. S., Sheral, H. D. ad Shey, C. M. 99). No-lear programmg heory ad algo-rhms, Joh Wley ad Sos, New Yor, 6-. Bellma, R. E. ad Zadeh, L. A. 97). Decso-mag a fuzzy evrome, Maageme Scece 7, B-B6. Caz, T. 999). Fuzzy lear programmg for DSS eergy plag, I. J. of global eergy ssues, -.

29 S. Effa ad H. Abbasya Dubos, D ad Prade, H. 9). Sysem of lear fuzzy cosras, Fuzzy se ad sysems, -. Elamvazuh, I., Gaesa, T. Vasa, P. ad Webb, F. J. 9). Applcao of a fuzzy programmg echque o produco plag he ele dusry, I. J. of compuer scece ad formao secury 6, -. Guu, Sy-Mg ad Ya-K, Wu 999). Two-phase approach for solvg he fuzzy he fuzzy lear programmg problems, Fuzzy se ad sysems 7, 9-9. Negoa, C. V. 97). Fuzzess maageme, OPSA/TIMS, Mam. Saawa, M ad Yaa, H. 9). Ieracve decso mag for mul-obecve lear fracoal programmg problem wh fuzzy parameers, Cyberecs Sysems 6, Shaocheg, T. 99). Ierval umber ad fuzzy umber lear programmg, Fuzzy ses ad sysems 66, -6. Taaa, H ad Asa, K. 9). Fuzzy lear programmg problems wh fuzzy umbers, Fuzzy ses ad sysems,. Taaa, H, Ouda, T. ad Asa, K. 9). O fuzzy mahemacal programmg, J. Cyberecs, 9-9. Ta, Raymod R. ). Applcao of symmerc fuzzy lear programmg lfe cycle assessme, Evromeal modelg & sofware, -6. Zmmerma, H. J. 9). Fuzzy mahemacal programmg, Compu. Ops. Res. Vol. No., 9-9. APPENDIX The Algorhm of he Fuzzy Decsve Se Mehod Ths mehod s based o he dea ha, for a fed value of, he problem 9) s lear programmg problems. Obag he opmal soluo * o he problem 9) s equvale o deermg he mamum value of so ha he feasble se s oempy. The algorhm of hs mehod for he problem 9) s preseed below. Algorhm Sep. Se ad es wheher a feasble se sasfyg he cosras of he problem 9) ess or o, usg phase oe of he Smple mehod. If a feasble se ess, se, L R oherwse, se ad ad o o he e sep. L R Sep. For he value of, updae he value of follows: L, f feasble se s oempy for, R, f feasble se s empy for. L ad R usg he bseco mehod as

30 AAM: Ier. J., Vol., Issue December ) [Prevously, Vol., Issue, pp. 6 6] Cosequely, for each, es wheher a feasble se of he problem 9) ess or o usg phase oe of he Smple mehod ad deerme he mamum value * sasfyg he cosras of he problem 9).