Solving Interval and Fuzzy Multi Objective. Linear Programming Problem. by Necessarily Efficiency Points

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1 Iteratoal Mathematcal Forum, 3, 2008, o. 3, Solvg Iterval ad Fuzzy Mult Obectve ear Programmg Problem by Necessarly Effcecy Pots Hassa Mshmast Neh ad Marzeh Aleghad Mathematcs Departmet, Faculty of Scece versty of Ssta ad Baluchesta, Zaheda, Ira Abstract I ths paper we focus o a kd of lear programmg wth fuzzy umbers ad multple obectves. Frst by usg α -cuts ad fuzzy rakg,we trasform these problems to mult obectve problem wth fuzzy coeffcets ad crsp costrat the defe ecessarly effcecy pots for ew problem ad for solvg the problem try to fd all of these ecessarly effcecy pots. Keywords: mult obectve fuzzy programmg, ecessarly effcecy pots, terval programmg, Fuzzy rakg.. Itroducto I covetoal mathematcal programmg, coeffcets of problems are usually crsp values ad determed by the experts. But real world, t s a ucerta assumpto that the kowledge ad represetato of a expert are so precse. Hece order to develop good operato research methodology fuzzy, terval ad stochastc approaches are frequetly used to descrbe ad treat mprecse ad ucerta elemets preset a real decso problem. I fuzzy programmg problems the costrats ad goals are vewed as fuzzy sets ad assumed that ther membershp fuctos are kow. But, realty, to a decso maker (DM t s ot always easy to specfy the membershp fucto. At least some of the cases, use a terval coeffcets may serve the purpose better. Though by usg α - cuts, fuzzy umbers ca be trasformed to terval umbers. Moreover, most real word problems are heretly characterzed by multple, coflctg ad commesurate aspects of evaluato.these axes of evaluato are geerally operatoalzed by obectve fuctos to be optmzed framework of multple obectve lear programmg models.

2 00 H. M. Neh ad M. Aleghad I ths paper, we focus o mult obectve lear programmg wth fuzzy coeffcet. By troducg α -cuts ad amk ad amek rakg, we trasforme the problem to a multobectve lear programmg wth terval obectve fuctos ad crsp costrats the defe ecessarly effcet pots ad fd these pots for ew problems wth proposed algorthms. 2. Multobectve lear programmg wth fuzzy coeffcets The mult obectve lear programmg wth fuzzy coeffcets ca be formulated as follow: max maze ~ c k x k =,, p = ~ ~ a x b,, m x 0 = ( where c~ k, a ~ ad b ~ are fuzzy umbers. To fd the ecessarly effcecy pots for problem (, we trasform t to a problem wth terval obectve fucto ad crsp costrats. So we use α -cuts to trasforme coeffcets of obectve fucto to terval, as dcated follow, ad amk ad amek rakg to obta crsp costrats from our fuzzy costrats. 2.. Some deftos The α -level set (α -cut of a fuzzy set M ~ s defed as a ordary set M ~ α for whch the degree of ts membershp fucto exceeds the levelα : M ~ α = { x μ M ~ α }, α [ 0, ]. Observe that the α -level set M ~ ca be defed by the characterstc fucto α μ ( x α C M ~ = α M 0 μ M ( x < Actually, a α -level set s a ordary set whose elemets belog to the correspodg fuzzy set to a certa degree α. A fuzzy umber s a covex ormalzed fuzzy set of the real le whose membershp fucto s pecewse cotuous. From the defto of a fuzzy umber M ~, t s sgfcat to ote that the α -level set M ~ α of a fuzzy umber M ~ ca be represeted by the closed terval whch depeds o terval value of α. Namely, M ~ α = { x μ M ~ α }= [ M,M α α ] Where M α or respectvely. M α represets the left or rght extreme pot of the α -level set α M ~ α,

3 ear programmg problem 0 ~ Especally f M = ( m, γ, β s a tragular fuzzy umber the M ~ ~ α = [ γ ( α + m,m β( α ] ad f M = ( m, m, γ, β s a trapezodal fuzzy umber the M ~ = [ γ ( α + m,m β( α ]. α 2.2. akg fuzzy umbers Dubos ad prade proposed a method of rakg fuzzy umbers as follow: Defto let M ~ ad N ~ be fuzzy umbers, the we have M ~ N ~ M ~ N ~ = M ~ By usd the defto amk ad ramek suggested the followg lemma: emma f M ~ ad N ~ be a fuzzy umbers, M ~ N ~ = M ~ f ad oly f for every h [0,] we have f { s : μ ( s M ~ h } f { t : μ N ~ ( t h } sup { s : μ ( s M ~ h } sup { t : μ N ~ ( t h } Especally f M ~ = ( m,m, α, β ad N ~ = (,, γ, δ be two trapezodal fuzzy umbers, the above relato s true f ad oly f m * * ( h α ( h β h [0, ] m + * ( h α * + ( h β h [0, ] where * ( h = sup{ z : ( z ( h = sup{ z : ( * * ( h = z sup{ z : T( z h } h } h } ( h = sup{ z : ( z h }. * From the defto of amk ad amek, f M ~ = ( m, α, β ad N ~ = (, γ, δ, be tragular fuzzy umbers, the we have M ~ N ~ m m α γ m + β + δ If M ~ = ( m,m, α, β ad N ~ = (,, γ, δ be two fuzzy umbers, smlarly we ca compare them as follow m M ~ N ~ m m α γ m + β + δ By usg above defto ad wth respect to the kd of fuzzy umbers of the problem, we ca trasform the fuzzy costrats to crsp oes.

4 02 H. M. Neh ad M. Aleghad I ths way, f c ~ k = ( ck, α k, β k, a~ = ( a, γ, δ ad b ~ = ( b, η, ν be tragular fuzzy umbers, problem ( s equvalet to max maze [ α k( α + ck,ck β k( α ] x k =,, p = a x = = ( 2 = b ( a γ x b η ( a + β x b + ν x 0. =,,m =,,m =,,m Smlarly f c ~ k = ( ck,ck, α k, β k, ~ a = ( a, a, γ, δ ad b ~ = ( b,b, η, ν be trapezodal fuzzy umbers the, problem ( s equvalet to max maze [ α k( α + ck,ck β k( α ] x k =,, p ( 3 = a x = a x = = = b b ( a γ x b η ( a + β x b + ν x 0. =,,m =,,m =,,m =,,m 3. Necessarly effcet pots Cosder the followg MOP problem wth terval obectve fuctos: max z ( x = Cx Ax b ( 4

5 ear programmg problem 03 x 0 C Φ. Where Φ s a set of p matrces, wth elemets Ck [ Ck,Ck ] for k =,, p, =,,, A s a m matrx ad b s a m vector. A soluto s ecessarly effcet to problem (4 f ad oly f t s effcet for ay C Φ. The ecessarly effcet soluto set ( N E s obtaed by N I E = X E ( C C Φ Where X E ( C s the effcet soluto set for each C Φ. Btra (980 proposed a mplct eumerato algorthm that uses a subproblem to test effcecy of a gve basc soluto ad a brach ad boud algorthm to solve subproblem. But hs method may result a hgh computatoal burde f the soluto beg aalyzed s ot ecessarly effcet. Ida (999 suggested a exteso of the mplct eumerato algorthm.the proposed method uses two effcecy test. Oe of them checks ecessary effcecy ad the other checks o-ecessary effcecy. et matrx,, C C ( g,w,,wg ( g,w,,wg = CB ( g,w,,wg B ( g, w,, wg B CB. = CB. [ CB., C B are defed as B. ] g, w N C g, w N = = g m., where the colums of the terval I ths cotext, g s the tree level, m s the umber of basc varables wth terval coeffcets, w s ( f the colum elemets of the matrx wth the tree level B. are the lower bouds C (upper boudsc of the tervals ad =,,g. ( g,w,,w ( g,w,,w g g et ad be composed of the lower ad upper bouds of ( g,w,,wg each elemet belogg to the terval matrx, respectvely. The operatos " sc" s defed as ad sc( sc( = {, B. ( g,w,,w ( g+,w,,w, ( g+,w, g g g, w,, wg ( g+, w,, wg, ( g+, w,, wg, = {, }. ( 3.. Necessary effcecy algorthms The ecessary effcecy test s obtaed the followg way: Algorthm Step. et S = { ( 0 }. ( g,w,,wg Step2. Select oe elemet from S ad check whether t s effcet. (a If t s effcet the remove the elemet from S. ( g,w,,wg ( g,wk,w g ( m,wk,w m (b Otherwse, add sc( to S.If ( = (, the s ot ecessarly effcet.,w g, }

6 04 H. M. Neh ad M. Aleghad Step3. If the set S s empty, the s ecessarly effcet. Step4. etur to step 2. Ths algorthm s based o the followg theorem: Theorem If all of the elemets of S are effcet, the s ecessarly effcet.(ida, 999 If the soluto beg aalyzed s ot ecessarly effcet, hs mplct eumerato algorthm may be out of acceptable computatoal lmts due to the brachg requred. Therefore, Ida proposed aother algorthm the followg way. Algorthm 2 Step. et S = { ( 0 }. ( g,w,,wg Step2. Select oe elemet from S ad check whether t s effcet. (a If t s ot effcet, the s ot ecessarly effcet. ( g,w,,wg ( g,wk,w g ( m,wk,w m (b Otherwse, add sc( to S.If ( = (, Do ot add ay thg to S. Step3. If the set S s empty, the s ecessarly effcet. Step4. etur to step 2. Ths algorthm s based o the followg theorem (Ida, 999: Theorem If there s a elemet of S that t s ot effcet, the s ot effcet. The effcecy s based o cherkova's algorthm (cherkova, 999 ad proceeds as follows: Algorthm 3 Step. Compute. Step2. Aalyze colum ad rows of ad proceed as follow:. (a If there are ay colums such that. 0, the elmate these colums. (b If there are ay rows such that. 0, the elmate these rows.. Step3. Aalyze colums ad rows of ad proceed as follow: (a If there s a colum such that. 0, the s ot effcet. (b If there s arrow such that. > 0, the s effcet. (c If there s a row such that. 0 ad a row such that > 0 ( = 0,the s effcet. Step4. Calculate the summato of the colums(. Σ ad rows( Σ. of. (a If. Σ 0, the s ot effcet. (b If Σ. > 0, the s effcet. Step5. et D = ad process the rows by usg the extreme ray geerato method T (cherkova,965. et. smultaeously, D = ad process the row parallel by usg extreme ray geerato method ( D s defed cherkova(965 ad Ida (2000b. Step6. etur to step 2.

7 ear programmg problem Numercal Example Cosder the followg MOP wth fuzzy coeffcets: max C ~ x A ~ x b ~ x 0 where ( 5, (0,0.7 = ( 2,0.4 ( 4, ( 2, b ~ = ( C ~, [( 2,0.7 ( 3,0.5 ] et α = 0. 7 A ~ =,, The we have the followg terval for obectve coeffcets. ( 5, (7,0.7 ( 2,0.4 ( 3, = [ 4.7,5.3] = [6.79 7,.2] = [.88,2.2] = [ 2.7,3.3] If we wat to test the ecessary effcecy of the extreme pot (,0,,0,, the we obta B = , N = 2.5 ad C N = Thus, Hece, ( 0 (0 ( 0 Sce. 0 effcet = C ( 0 B 0.08 = [ 4.7,5.3] 0 0 B N C N = B N CN [.88,2.2] 0 0 [.79, 0.08] [ , ] = [ , ] [ , ] ad ( = , the s ot effcet (there s a elemet of S whch s ot

8 06 H. M. Neh ad M. Aleghad Cocluso A ew soluto to a lear multobectve programmg problems wth fuzzy coeffcets proposed based o the ecessary effcecy. The method to fd the ecessarly effcecy pots s dcated. Although we use amk ad amek defto to compare fuzzy umbers, we ca uses the others lear rakg methods, lke lous ad wag dex, ad compare the results. A example s gve to demostrate the proposed soluto. efereces [] Btra G.., lear multple obectve problems wth terval coeffcets, Maagemet Scece 26, pp , 980. [2] Chercova N.V., Algorthm for fdg a geeral formula for oegatve solutos of a system of lear equaltes, SS. Computatoal Mathematcs ad Mathematcal Physcs 5, pp , 965. [3] Ida M., Necessary effcet test terval mult obectve lear programmg, I: proceedgs of 8 th teratoal fuzzy systems assocato world cogress, pp , [4] Ida M., Effcet soluto geerato for multple obectve lear programmg ad ucerta coeffcets, I: proceedgs of 8 th Bellma cotuum, pp.32-36, [5] Ida M., Iterval multobectve programmg ad moble robot path plag, I: Mohammada M. (Eds., New Froters computatoal Itellgece ad ts applcatos. ISO press, pp , [6] Ishbuch H., Taaka H., Multobectve programmg optmzato of the terval obectve Systems, Pretc Hall,Eglewood Clffs, NJ, 982. [7] Olvera C., Atues C.H., ultple obectve lear programmg modeles wth terval coeffcets a llustrated overvew, Europea oural of operatoal esearch, pp.-30, [8] Sakawa M., Fuzzy sets ad teractve multobectve optmzato, Pleum press, New York,993. [9] Steuer P.E., Multple crtera optmzato:theory,computato ad applcato, Joh wley & Sos, New york, 986. eceved: September 2, 2007