Research Article A MOLP Method for Solving Fully Fuzzy Linear Programming with LR Fuzzy Parameters

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1 Mahemaical Problems i Egieerig Aricle ID pages hp://dx.doi.org/ /2014/ Research Aricle A MOLP Mehod for Solvig Fully Fuzzy Liear Programmig wih Fuzzy Parameers Xiao-Peg Yag 12 Xue-Gag Zhou 13 Big-Yua Cao 1 ad S. H. Nasseri 4 1 School of Mahemaics ad Iformaio Sciece Key Laboraory of Mahemaics ad Ierdiscipliary Scieces of Guagdog Higher Educaio Isiues Guagzhou Uiversiy Guagzhou Chia 2 Deparme of Mahemaics ad Saisics Hasha Normal Uiversiy Chaozhou Chia 3 Deparme of Applied Mahemaics Guagdog Uiversiy of Fiace Guagzhou Chia 4 Deparme of Mahemaics Mazadara Uiversiy Babolsar Ira Correspodece should be addressed o Big-Yua Cao; caobigy@163.com Received 22 March 2014; Acceped 15 Sepember 2014; Published 29 Sepember 2014 Academic Edior: Yag Xu Copyrigh 2014 Xiao-Peg Yag e al. This is a ope access aricle disribued uder he Creaive Commos Aribuio Licese which permis uresriced use disribuio ad reproducio i ay medium provided he origial work is properly cied. Kaur ad Kumar 2013 use Mehar s mehod o solve a kid of fully fuzzy liear programmig (FFLP problems wih fuzzy parameers. I his paper a ew kid of FFLP problems is iroduced wih a soluio mehod proposed. The FFLP is covered io a muliobjecive liear programmig (MOLP accordig o he order relaio for comparig he fla fuzzy umbers. Besides he classical fuzzy programmig mehod is modified ad he used o solve he MOLP problem. Based o he compromised opimal soluio o he MOLP problem he compromised opimal soluio o he FFLP problem is obaied. A las a umerical example is give o illusrae he feasibiliy of he proposed mehod. 1. Iroducio The research o fuzzy liear programmig (FLP has rise highly sice Bellma ad Zadeh [1] proposed he cocep of decisio makig i fuzzy evirome. The FLP problem is said o be a fully fuzzy liear programmig (FFLP problem if all he parameers ad variables are cosidered as fuzzy umbers. I rece years some researchers such as Lofi ad Kumar were ieresed i he FFLP problems ad some soluio mehods have bee obaied o he fully fuzzy sysems [2 4] ad he FFLP problems [5 13]. FFLP problems ca be divided i wo caegories: (1 FFLP problems wih iequaliy cosrais; (2 FFLP problems wih equaliy cosrais. If he FFLP problems are classified by he ypes of he fuzzy umbers hey will iclude he ex hree classes: (1 FFLP problems wih all he parameers ad variables represeed by riagular fuzzy umbers; (2 FFLP problems wih all he parameers ad variables represeed by rapezoidal fuzzy umbers; (3 FFLP problems wih all he parameers ad variables expressed by fuzzy umbers (or fla fuzzy umbers. Fuzzy programmig mehod is a classical mehod o solve muliobjecive liear programmig (MOLP [14 15]. I his paper he fuzzy programmig mehod is modified ad he used o obai a compromised opimal soluio of he MOLP. The modified fuzzy programmig mehod is show i Seps 4 10 of he proposed mehod i Secio 3. Dehgha e al. [2 4] employed several mehods o fid soluios of he fully fuzzy liear sysems. Hosseizadeh Lofi e al. [6] used he lexicography mehod o obai he fuzzy approximae soluios of he FFLP problems. Allahviraloo e al. [7]adKumareal.[5 8] solved he FFLP problem by use of a rakig fucio. Fa e al. [12] adopedheα-cu level o deal wih a geeralized fuzzy liear programmig (GFLP probelm. The feasibiliy of fuzzy soluios o he GFLP was ivesigaed ad a sepwise ieracive algorihm based o he idea of desig of experime was advaced o solve he GFLP problem.

2 2 Mahemaical Problems i Egieerig Kaur ad Kumar [9] iroduced Mehar s mehod o he FFLPproblems wih fuzzy parameers. They cosider he followig model: 2. Prelimiaries 2.1. Basic Noaios Maximize (or Miimize subjec o ((p j q j α j β j (x jy j α j β j ((a ij b ij α ij β ij (x j y j α j β j (b ig i γ i δ i i=12...m (1 Defiiio 1 ( fuzzy umber see [2]. A fuzzy umber u is said o be a fuzzy umber if L( m x α x m α>0 u (x = R( x m β x m β>0 where m ishemeavalueof u ad α ad β are lef ad righ spreads respecively ad fucio L( meas he lef shape fucio saisfyig (3 where he parameers ad variables are fla fuzzy umbers ad he order relaio for comparig he umbers is defied as follows. (i u Ṽ if ad oly if R( u R(Ṽ (ii u Ṽ if ad oly if R( u R(Ṽ (iii u Ṽ if ad oly if R( u = R(Ṽ. Here u ad Ṽ are wo arbirary fla fuzzy umbers. I our sudy we cosider a ew kid of FFLP problems wih fla fuzzy parameers as follows: mi (or max z ( x = c 1 x 1 c 2 x 2 c x subjec o a i1 x 1 a i2 x 2 a i x b i x j 0 j = i=12...m where he parameers ad variables are fla fuzzy umbers ad he order relaio show i Defiiio 4 is differe from he oe above. I his paper we modify he classical fuzzy programmig mehod. The FFLP is chaged io a MOLP problem solved by he modified fuzzy programmig mehod. We ge he compromised opimal soluio o he MOLP ad geerae he correspodig compromised opimal soluio o he FFLP. The res of he paper is orgaized as follows. I Secio 2 he basic defiiios ad he FFLP model are iroduced. I Secio 3 we propose a MOLP mehod o solve he FFLP problems. Some resuls are discussed from he soluios obaied by he proposed mehod. I Secio 4aumerical exampleisgiveoillusraehefeasibiliyofheproposed mehod. I Secio 5 we show some shor cocludig remarks. (2 (1 L(x = L( x; (2 L(0 = 1 ad L(1 = 0; (3 L(x is oicreasig o [0. Naurally a righ shape fucio R( is similarly defied as L(. Defiiio 2 ( fla fuzzy umber see [9 16]. A fuzzy umber u deoed as (mαβ issaidobea fla fuzzy umber if is membership fucio u(x is give by L( m x x m α>0 α u (x = R( x x β>0 β 1 m x. Defiiio 3 (see [5 9]. A fla fuzzy umber u = (mαβ is said o be oegaive fla fuzzy umber if m α 0ad is said o be oposiive fla umber if +β 0. We defie u = (m00 as a fuzzy umber wih membership fucio ad deoe (0000 as 0. u (x = 1 m x 0 oherwise (4 (5

3 Mahemaical Problems i Egieerig Arihmeic Operaios. Le u =(m 1 1 α 1 β 1 ad Ṽ = (m 2 2 α 2 β 2 be wo fla fuzzy umbers k R.The he arihmeic operaios are give as follows [9 16]: u Ṽ =(m 1 +m α 1 +α 2 β 1 +β 2 (1 I is obvious ha u w =(m 1 +m α 1 +α 3 β 1 + β 3 Ṽ σ =(m 2 +m α 2 +α 4 β 2 +β 4.Sice u Ṽ w σwege m 1 m m 1 α 1 m 2 α 2 u Ṽ =(m m 2 α 1 +β 2 α 2 +β 1 k u = (km 1k 1 kα 1 kβ 1 k 0 (k 1 km 1 kβ 1 kα 1 RL k<0 1 +β 1 2 +β 2 m 3 m m 3 α 3 m 4 α 4 3 +β 3 4 +β 4. (8 u Ṽ = (m 1 m m 1 α 2 +α 1 m 2 1 β 2 +β 1 2 u 0 Ṽ 0 (m m 2 α 1 2 m 1 β 2 β 1 m 2 1 α 2 u 0 Ṽ 0 ( 1 m 2 m α 2 β 1 m 2 m 1 β 2 α 1 2 u 0 Ṽ 0 ( 1 2 m 1 m 2 1 β 2 β 1 2 m 1 α 2 α 1 m 2 u 0 Ṽ 0. (6 I is easy o verify ha he operaor saisfies associaive law. Hece he formula u j = u 1 u 2 u is reasoable where u 1 u 2... u are fla fuzzy umbers Order Relaio for Comparig he Fla Fuzzy Numbers. For comparig he fla fuzzy umbers we iroduce he order relaio as follows. Defiiio 4. Le u =(m 1 1 α 1 β 1 ad Ṽ =(m 2 2 α 2 β 2 be ay fla fuzzy umbers. The (i u = Ṽ if ad oly if m 1 = m 2 1 = 2 α 1 = α 2 β 1 =β 2 ; (ii u Ṽ if ad oly if m 1 m m 1 α 1 m 2 α 2 1 +β 1 2 +β 2 ; (iii u Ṽ if ad oly if m 1 m m 1 α 1 m 2 α 2 1 +β 1 2 +β 2. So m 1 +m 3 m 2 +m (m 1 +m 3 (α 1 +α 3 (m 2 +m 4 (α 2 +α 4 ( (β 1 +β 3 ( (β 2 +β 4. This idicaes ha u w Ṽ σ. (2 I is clear ha k u = (km 1 k 1 kα 1 kβ 1 kṽ = (km 2 k 2 kα 2 kβ 2.From u Ṽwege m 1 m m 1 α 1 m 2 α 2 Therefore 1 +β 1 2 +β 2. km 1 km 2 k 1 k 2 km 1 kα 1 km 2 kα 2 for k 0ad k 1 +kβ 1 k 2 +kβ 2 km 1 km 2 k 1 k 2 km 1 kα 1 km 2 kα 2 k 1 +kβ 1 k 2 +kβ 2 (9 (10 (11 (12 Based o he defiiio of order we may obai ha (i u is oegaive if ad oly if u 0; (ii u is oposiive if ad oly if u 0. The followig proposiios are give o show he properies of he order relaio defied above. for k 0.Thisidicaesha k u kṽ k 0 k u kṽ k 0. (13 Proposiio 5. Le u Ṽ w σ be four arbirary fla fuzzy umbers ad k a arbirary real umber. The (1 u Ṽ w σ u w Ṽ σ; (2 u Ṽ k u kṽ k 0 k u kṽ k 0. Proof. Suppose u =(m 1 1 α 1 β 1 Ṽ =(m 2 2 α 2 β 2 w =(m 3 3 α 3 β 3 ad σ =(m 4 4 α 4 β 4. (7 Proposiio 6. Le u Ṽ w be hree arbirary fla fuzzy umbers. The (1 u u; (2 u Ṽ Ṽ u u =Ṽ; (3 u Ṽ Ṽ w u w. Proof. Suppose u =(m 1 1 α 1 β 1 Ṽ =(m 2 2 α 2 β 2 ad w =(m 3 3 α 3 β 3. (1 Obviously u = u; hece wehave u u.

4 4 Mahemaical Problems i Egieerig (2 Sice u Ṽ Ṽ uwege m 1 m m 1 α 1 m 2 α 2 1 +β 1 2 +β 2 m 2 m m 2 α 2 m 1 α 1 2 +β 2 1 +β 1. This meas m 1 =m 2 1 = 2 m 1 α 1 =m 2 α 2 1 +β 1 = 2 +β 2. Tha is (14 (15 he fla fuzzy umbers boh i he objecive fucio ad he cosrai iequaliies are as show i Defiiio Proposed Mehod Seps of he proposed mehod are give o solve problem (20 as follows. This mehod is applicable o miimizaio of FFLP problems ad he soluio mehod of maximizaio problems is similar o ha of miimizaio oes. Sep 1. If all he parameers c j b i a ij x j are represeed by fla fuzzy umbers (c j1 c j2 α cj β cj (b i1 b i2 α bi β bi (a ij1 a ij2 α aij β aij ad(x j1 x j2 α xj β xj he he FFLP (20cabewrieas m 1 =m 2 1 = 2 α 1 =α 2 β 1 =β 2. (16 Therefore we have u =Ṽ. (3 From u Ṽ Ṽ wwege m 1 m m 1 α 1 m 2 α 2 mi z ( x = ((c j1 c j2 α cj β cj (x j1 x j2 α xj β xj 1 +β 1 2 +β 2 m 2 m m 2 α 2 m 3 α 3 (17 ((a ij1 a ij2 α aij β aij (x j1 x j2 α xj β xj 2 +β 2 3 +β 3. (b i1 b i2 α bi β bi i=12...m This idicaes m 1 m m 1 α 1 m 3 α 3 Therefore we have u w. 1 +β 1 3 +β 3. (18 From Proposiio 6 we kow ha he order relaio is a parial order o he se of all fuzzy umbers Fully Fuzzy Liear Programmig wih Fuzzy Parameers. I his paper we will cosider he followig model; ha is or mi mi z ( x = c x A x b x 0 z ( x = c 1 x 1 c 2 x 2 c x a i1 x 1 a i2 x 2 a i x b i i=12...m (19 (20 (x j1 x j2 α xj β xj ( (21 Sep 2. Calculae (c j1 c j2 α cj β cj (x j1 x j2 α xj β xj ad (a ij1 a ij2 α aij β aij (x j1 x j2 α xj β xj respecively ad suppose ha (c j1 c j2 α cj β cj (x j1 x j2 α xj β xj = (x j1 x j2 α x j β x j ad (a ij1 a ij2 α aij β aij (x j1 x j2 α xj β xj =(p ij q ij γ ij δ ij ; he he FFLP problem obaied i Sep 1 ca be wrie as mi z ( x =( ( p ij x j1 x j2 α x j q ij γ ij δ ij β x j (b i1 b i2 α bi β bi i=12...m x j (x j1 x j2 α xj β xj ( where c =[ c j ] 1 b =[ b i ] m 1 A =[ a ij ] m ad x =[ x j ] 1 represe fuzzy marices ad vecors ad c j b i a ij ad x j are fla fuzzy umbers. The order relaios for comparig 2... (22

5 Mahemaical Problems i Egieerig 5 Sep 3. Accordig o he order relaio defied above he problem obaied i Sep 2 is equivale o mi x j1 x j2 (x j1 α x j (x j2 +β x j p ij b i1 q ij b i2 i=12...m i=12...m (p ij γ ij (b i1 α bi (q ij +δ ij (b i2 +β bi x j1 x j2 α xj 0 β xj 0 x j1 α xj i=12...m i=12...m (23 We deoe X = (x 11 x 12 α x1 β x1 x 21 x 22 α x2 β x2... x 1 x 2 α x β x T z 1 (X = x j1 z 2(X = x j2 z 3 (X = (x j1 α x j z 4 (X = (x j2 +β x j ad D=X Xsaisfies he cosrais of programmig (23}. Programmig (23 may be wrie as he programmig (24 below for shor as follows: mi z 1 (X z 2 (X z 3 (X z 4 (X X D. (24 Obviously programmig (24 is a crisp muliobjecive liear programmig problem. I fac we have z( x = (z 1 (X z 2 (X z 1 (X z 3 (X z 4 (X z 2 (X. Sep 4. Solve he subproblems where = WefidopimalsoluiosX 1 X 2 X 3 ad X 4 respecively. Ad he correspodig opimal values will be z mi 1 =z 1 (X 1 z mi 2 =z 2 (X 2 z mi 3 =z 3 (X 3 ad z mi 4 =z 4 (X 4. Sep 5. Le z max = maxz (X 1 z (X 2 z (X 3 z (X 4 } = adhemembershipfucioofz (X is give by μ z (z (X = where = z (X <z mi z max z (X z max z mi z mi 0 z (X >z max z (X z max (26 Sep 6. Le I 0 = }; he MOLP problem obaied i Sep 3 cabeequivalelywrieas max λ μ z (z (X λ I 0 (27 Suppose X 1 is oe of he opimal soluios (if here exis oly oe opimal soluio X 1 is he uique oe ad λ 1 is he opimal objecive value (i fac he opimal soluio should be wrie as (X 1 λ 1.Siceλ is a auxiliary variable we deoe (X 1 λ 1 as X 1 for simpliciy. The μ zs1 (X 1 = λ 1 for a leas oe s 1 i I 0.(s 1 is a arbirary eleme i he se J=j μ zj (z j (X 1 = λ 1 }. Sep 7. Le I 1 = I 0 s 1 }adsolvehefollowigcrisp programmig: max λ μ z (z (X λ I 1 μ zs1 (X =λ 1 (28 If X 2 is oe of he opimal soluios ad λ 2 is he opimal objecive value he μ zs2 (X 2 = λ 2 foraleasoes 2 i I 1. Sep 8. Le I 2 = I 0 s 1 s 2 }adsolvehefollowigcrisp programmig: max λ μ z (z (X λ I 2 μ zs1 (X =λ 1 (29 mi z (X X D (25 μ zs2 (X =λ 2

6 6 Mahemaical Problems i Egieerig Suppose X 3 is oe of he opimal soluios ad λ 3 is he opimal objecive value. The μ zs3 (X 3 = λ 3 for a leas oe s 3 i I 2. Sep 9. Le I 3 =I 0 s 1 s 2 s 3 }adsolvehefollowigcrisp programmig: max λ μ z (z (X λ I 3 μ zs1 (X =λ 1 μ zs2 (X =λ 2 μ zs3 (X =λ 3 (30 Suppose X 4 is oe of he opimal soluios ad λ 4 is he opimal objecive value. The μ zs4 (z s4 (X 4 = λ 4 wih s 4 i I 3. Proof. (1 From he resuls of Seps 6 9 iisobviouslyclear ha λ 1 =μ zs1 (X 1 λ 2 =μ zs2 (X 2 λ 3 =μ zs3 (X 3 λ 4 =μ zs4 (z s4 (X 4 (34 ad μ zs4 (z s4 (X = λ 4 wih X = X 4.SiceX = X 4 is a opimal soluio o programmig (30 we kow ha X saisfies he cosrais of programmig (30 ad so μ zs1 (X = λ 1 μ zs2 (X = λ 2 adμ zs3 (X = λ 3. (2 I fac (X 1 λ 1 is a opimal soluio o programmig (27; herefore i is a feasible soluio. We have Sep 10. Take X =X 4 as he compromised opimal soluio o programmig (23 ad geerae he compromised opimal soluio x o programmig (21byX. Assumig μ z (z (X 1 λ 1 I 1 I 0 X 1 D (35 X = (x 11 x 12 α x 1 β x 1 x 21 x 22 α x 2 β x 2...x 1 x 2 α x β x T we may obai x =( x 1 x 2... x T = ((x 11 x 12 α x 1 β x 1 (x 21 x 22 α x 2 β x 2 (31 (32 ad i is obvious ha μ zs1 (X = λ 1 from he resul of Sep 6.Hece(X 1 λ 1 is a feasible soluio o programmig (28. The objecive value of (X 1 λ 1 is λ 1 adheopimal objecive value of programmig (28 isλ 2 ;sowegeλ 1 λ 2.Iissimilaroproveλ 2 λ 3 ad λ 3 λ Numerical Example...(x 1 x 2 α x β x T ad he correspodig objecive value z =z( x. Remark 7 (s 1 s 2 s 3 s 4 }=I=1234}. Some properies of he soluios obaied i Seps 6 10 are show i he followig proposiio. Proposiio 8. Suppose μ zsj (z sj (X X j λ j (j = ad X are he oaios described i Seps 1 10; he I his secio we prese a umerical example o illusrae he feasibiliy of he soluio mehod proposed i Secio 3. We aim o fid he compromised opimal soluio ad correspodig objecive value of he followig fully fuzzy liear programmig problem: max z ( x =z( x 1 x 2 = ( x 1 ( x 2 ( x 1 ( x 2 (1 μ zs1 (X = λ 1 =μ zs1 (X 1 ( (36 μ zs2 (X = λ 2 =μ zs2 (X 2 μ zs3 (X = λ 3 =μ zs3 (X 3 μ zs4 (z s4 (X = λ 4 =μ zs4 (z s4 (X 4 (2 λ 1 λ 2 λ 3 λ 4. (33 ( x 1 ( x 2 ( x 1 0 x 2 0 where x 1 =(x 11 x 12 α 1 β 1 ad x 2 =(x 21 x 22 α 2 β 2.

7 Mahemaical Problems i Egieerig 7 Table 1: The opimal values ad soluios of he four subproblems. (a (b (c (d The opimal objecive value The opimal soluio z max 1 =70 X 1 = ( T z max 2 = X 2 = ( T z max 3 =50 X 3 = ( T z max 4 = X 4 = ( T Accordig o Seps 1 ad 2 i he proposed mehod we obai he followig programmig: Programmig (38 ca be abbreviaed o he followig programmig: max z=(6x 11 +7x 21 7x 12 +9x 22 6α 1 +7α 2 +x 11 +2x 21 7β 1 +9β 2 +2x 12 +x 22 max z 1 (X z 2 (X (9x 11 +x 21 10x 12 +x 22 9α 1 +α 2 +2x 11 z 3 (X (39 +x 21 10β 1 +β 2 +x 12 +x 22 z 4 (X ( (37 X D (2x 11 +4x 21 3x 12 +5x 22 2α 1 +4α 2 +x 11 +x 21 3β 1 +5β 2 +x 12 +2x 22 ( where X=(x 11 x 12 α 1 β 1 x 21 x 22 α 2 β 2 T. Solve he followig subproblems: (x 11 x 12 α 1 β 1 ( (a max z 1 (X (x 21 x 22 α 2 β 2 ( X D By Sep 3 he programmig above is rasformed io he followig programmig: (b max z 2 (X X D (40 max z 1 =6x 11 +7x 21 z 2 =7x 12 +9x 22 z 3 =5x 11 +5x 21 6α 1 7α 2 z 4 =9x x 22 +7β 1 +9β 2 (c max (d max z 3 (X X D z 4 (X X D (41 9x 11 +x x 12 +x x 11 9α 1 α x 12 +2x β 1 +β x 11 +4x x 12 +5x x 11 +3x 21 2α 1 4α x 12 +7x 22 +3β 1 +5β 2 75 x 11 α 1 0 x 21 α 2 0 x 11 x 12 x 21 x 22 α 1 α 2 β 1 β 2 0. (38 respecively ad we obai he opimal objecive value ad oe of he opimal soluios as show i Table 1. Accordig o z mi = miz (X 1 z (X 2 z (X 3 z (X 4 } we acquire he lower objecive values z mi 1 = z mi 2 = z mi 3 = ad z mi 4 = wih correspodig membership fucios give below. Cosider μ z1 (z 1 (X = 1 z 1 (X > 70; z 1 (X z 1 (X 70; 0 z 1 (X <

8 8 Mahemaical Problems i Egieerig 1 z 2 (X > ; z 2 (X μ z2 (z 2 (X = z 2 (X ; 0 z 2 (X < μ z3 (z 3 (X = 1 z 3 (X > 50; z 3 (X z 3 (X 50; 0 z 3 (X < z 4 (X > ; z 4 (X (z 4 (X = z 4 (X ; 0 z 4 (X < By Seps 4 6wege max λ μ z1 (z 1 (X = 6x 11 +7x λ μ z2 (z 2 (X = 7x 12 +9x λ μ z3 (z 3 (X = 5x 11 +5x 21 6α 1 7α λ (z 4 (X = 9x x 22 +7β 1 +9β λ (42 (43 The opimal objecive value is λ 1 = adoe of he opimal soluios is X 1 = ( T. Calculae he value of he membership fucio of z (X ( = ax = X 1 adwegeμ z1 (z 1 (X 1 = μ z2 (z 2 (X 1 = μ z3 (z 3 (X 1 = ad (z 4 (X 1 = max Solve he followig problem: λ μ z1 (z 1 (X = 6x 11 +7x λ μ z2 (z 2 (X = 7x 12 +9x λ μ z3 (z 3 (X = 5x 11 +5x 21 6α 1 7α λ (z 4 (X = 9x x 22 +7β 1 +9β = (44 The opimal objecive value is λ 2 = adoeof he opimal soluios is X 2 = ( T. Calculae he value of he membership fucio of z (X ( = ax = X 2 adwegeμ z1 (z 1 (X 2 = μ z2 (z 2 (X 2 = μ z3 (z 3 (X 2 = ad (z 4 (X 2 = Solve he followig problem: max λ μ z1 (z 1 (X = 6x 11 +7x μ z3 (z 3 (X λ = 5x 11 +5x 21 6α 1 7α λ μ z2 (z 2 (X = 7x 12 +9x = (z 4 (X = 9x x 22 +7β 1 +9β = (45 The opimal objecive value is λ 3 = adoeof he opimal soluios is X 3 = ( T. Calculae he value of he membership fucio of z (X ( = a X=X 3 adwegeμ z1 (z 1 (X 3 = μ z2 (z 2 (X 3 = μ z3 (z 3 (X 3 = ad (z 4 (X 3 =

9 Mahemaical Problems i Egieerig 9 Table 2: Values of he four membership fucios a X j. μ z1 (z 1 (X μ z2 (z 2 (X μ z3 (z 3 (X (z 4 (X X=X X=X X=X X=X x = x 1 x = x 2 x = x 3 x = x 4 max Table 3: Values of he objecive fucio z( x a x j. Solve he followig problem: λ z( x ( ( ( ( μ z3 (z 3 (X = 5x 11 +5x 21 6α 1 7α μ z1 (z 1 (X = 6x 11 +7x = μ z2 (z 2 (X = 7x 12 +9x = (z 4 (X = 9x x 22 +7β 1 +9β λ = (46 The opimal objecive value is λ 4 = adoeof he opimal soluios is X 4 = ( T. Calculae he value of he membership fucio of z (X ( = ax = X 4 adwegeμ z1 (z 1 (X 4 = μ z2 (z 2 (X 4 = μ z3 (z 3 (X 4 = ad (z 4 (X 4 = Geerae x j by X j (j = followig Sep 10 ad calculae he value of z( x j.asshowitables2 ad 3 he soluio x j+1 (or X j+1 isbeerha x j (or X j j = Followig Sep 10we fid X =X 4 = ( T. (47 Therefore x = x 4 =(( ( T (48 serves as he compromised opimal soluio wih correspodig objecive value z =z( x = ( (49 5. Cocludig Remarks To he ed we show he followig cocludig remarks. (1Ihispaperweproposedaewmehodofid he compromised opimal soluio o he fully fuzzy liear programmig problems wih all he parameers ad variables represeed by fla fuzzy umbers. The soluio is also a fla fuzzy umber. I his sese we ge a exac soluio which may give more help o he decisio makers. (2 The order relaio i he objecive fucio is he same as ha i he cosrai iequaliies. Based o he defiiio of he order relaio he FFLP ca be equivalely rasformed io a MOLP which is a crisp programmig ha is easy o be solved. (3 Cosiderig he MOLP problem classical fuzzy programmig mehod is modified for obaiig he compromised opimal soluio. Coflic of Ieress The auhors declare ha here is o coflic of ieress regardig he publicaio of his paper. Ackowledgmes The auhors would like o hak he edior ad he aoymous reviewers for heir valuable commes which have bee very helpful i improvig he paper. This work is suppored by he PhD Sar-up Fud of Naural Sciece Foudaio of Guagdog Provice Chia (S ad he Chia Posdocoral Sciece Foudaio Fuded Projec (2014M Refereces [1] R. E. Bellma ad L. A. Zadeh Decisio-makig i a fuzzy evirome Maageme Sciecevol.17pp.B141 B [2]M.DehghaB.HashemiadM.Ghaee Compuaioal mehods for solvig fully fuzzy liear sysems Applied Mahemaics ad Compuaiovol.179o.1pp [3] M. Dehgha ad B. Hashemi Soluio of he fully fuzzy liear sysems usig he decomposiio procedure Applied

10 10 Mahemaical Problems i Egieerig Mahemaics ad Compuaio vol.182o.2pp [4] M.DehghaB.HashemiadM.Ghaee Soluioofhefully fuzzy liear sysems usig ieraive echiques Chaos Solios &Fracals vol. 34 o. 2 pp [5] A. Kumar J. Kaur ad P. Sigh A ew mehod for solvig fully fuzzy liear programmig problems Applied Mahemaical Modelligvol.35o.2pp [6]F.HosseizadehLofiT.AllahviralooM.AlimardaiJodabeh ad L. Alizadeh Solvig a full fuzzy liear programmig usig lexicography mehod ad fuzzy approximae soluio Applied Mahemaical Modellig vol.33o.7pp [7] T.AllahviralooF.H.LofiM.K.KiasaryN.A.KiaiadL. Alizadeh Solvig fully fuzzy liear programmig problem by he rakig fucio Applied Mahemaical Sciecesvol.2o. 1 4 pp [8] J. Kaur ad A. Kumar Exac fuzzy opimal soluio of fully fuzzy liear programmig problems wih uresriced fuzzy variables Applied Ielligecevol.37 o.1pp [9] J. Kaur ad A. Kumar Mehar s mehod for solvig fully fuzzy liear programmig problems wih L-R fuzzy parameers Applied Mahemaical Modellig vol.37o.12-13pp [10] J. J. Buckley ad T. Feurig Evoluioary algorihm soluio o fuzzy problems: fuzzy liear programmig Fuzzy Ses ad Sysemsvol.109o.1pp [11] S. M. Hashemi M. Modarres E. Nasrabadi ad M. M. Nasrabadi Fully fuzzified liear programmig soluio ad dualiy Joural of Iellige ad Fuzzy Sysems vol. 17 o. 3 pp [12] Y. R. Fa G. H. Huag ad A. L. Yag Geeralized fuzzy liear programmig for decisio makig uder uceraiy: feasibiliy of fuzzy soluios ad solvig approach Iformaio Sciecesvol.241pp [13] S. H. Nasseri ad F. Zahmakesh Huag mehod for solvig fully fuzzy liear sysem of equaios JouralofMahemaics ad Compuer Sciecevol.1pp [14] H.-J. Zimmerma Fuzzy programmig ad liear programmig wih several objecive fucios Fuzzy Ses ad Sysems vol. 1 o. 1 pp [15] B.-Y. Cao Fuzzy Geomeric Programmig Kluwer Academic Publishers Boso Mass USA [16] D. Dubois ad H. Prade Fuzzy Ses ad Sysems Theory ad Applicaiosvol.144ofMahemaics i Sciece ad Egieerig Academic Press New York NY USA 1980.

11 Advaces i Operaios Research Advaces i Decisio Scieces Joural of Applied Mahemaics Algebra Joural of Probabiliy ad Saisics The Scieific World Joural Ieraioal Joural of Differeial Equaios Submi your mauscrips a Ieraioal Joural of Advaces i Combiaorics Mahemaical Physics Joural of Complex Aalysis Ieraioal Joural of Mahemaics ad Mahemaical Scieces Mahemaical Problems i Egieerig Joural of Mahemaics Discree Mahemaics Joural of Discree Dyamics i Naure ad Sociey Joural of Fucio Spaces Absrac ad Applied Aalysis Ieraioal Joural of Joural of Sochasic Aalysis Opimizaio