The Necessarily Efficient Point Method for Interval Molp Problems

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1 ISS 6-69 Eglad K Joural of Iformato ad omputg Scece Vol. o. 9 pp. - The ecessarly Effcet Pot Method for Iterval Molp Problems Hassa Mshmast eh ad Marzeh Alezhad + Mathematcs Departmet versty of Ssta ad aluchesta Zaheda IA. (eceved September accepted October 9 Abstract. I the most real world stuatos a obectve fucto s ot satsfed the decso maker s goals ad reduce the effcecy of the models. Also the coeffcets of decso varables are ot exactly kow. Oe way to llustrate the ucertaty s tervals. I ths paper we cosder multobectve lear programmg wth terval coeffcets ad solve t wth respect to ecessarly effcet pots. Keywords: Iterval mult obectve programmg ecessarly effcet pots terval equalty. Itroducto We usually face some dffcultes whe a real world problem s formulated to a mathematcal programmg problem. Oe of the dffcultes s caused by the ucertaty s kowledge formato ad decso maker s preferece. Aother dffculty s that most of the problems are heretly characterzed by multple ad coflctg aspects of evaluato. Multobectve lear programmg wth terval coeffcets s oe of the approaches to tackle the above dffcultes mathematcal programmg models. Ths paper s amed at provdg a approach whch s based o effcet pots to solve a ucertaty mult obectve lear programmg wth equal costrats ad equalty costrats respectvely. osder wth out loss of geeralty the followg MOP wth terval coeffcets Max X s. t AX b (. X Φ Where Φ s a set of matrces whch ts elemets for k p m matrx b s a m vector ad X s a vector. We defe ecessarly effcet soluto ths maer: A soluto s ecessarly effcet to problem (. f ad oly f t s effcet for ay ecessarly effcet set ( s obtaed by p E Where s the effcet soluto set for each E X E ( Φ Φ. ad. A s a Φ. The.. Fdg ecessarly effcet pots To fd the etre set of ecessarly effcet soluto to (. we use the followg algorthms but frst we state some ecessary deftos. et ( g w w g ( g w w g where the colums of the terval matrx ( g w w g are defed as ( g w w g.... g w g w g m. + E-mal address: hmeh@hamoo.usb.ac.r ad m.alezhad@yahoo.com. Publshed by World Academc Press World Academc o

2 Hassa Mshmast eh et al: The ecessarly Effcet Pot Method for Iterval Molp Problems Hassa Mshmast eh g s the tree level. ( g w w g ( g w wg et ad be composed of the lower ad upper bouds of each elemets ( g w w g belogg to the terval matrx respectvely. The operator "sc" s defed as: ( g w w g ( g+ w w g ( g+ w w g sc( { } ad ( g w w g ( g+ w w g ( g+ w w sc( {.. The ecessary effcecy tests We lsted below some ecessarly effcecy tests: The tra ecessarly effcecy test Step. let { ( } S. } g ( g w w g Step. Select oe elemet from S ad check whether t s effcet. (a If t s effcet the remove the elemet from S. ( g w w g ( g w w (b Otherwse add sc( to S.If g ( m w wm the s ot ecessarly effcet. Step. If the set S s empty the s ecessarly effcet. Step: etur to step. If the soluto beg aalyzed s ot ecessarly effcet we use the followg algorthm To test the ecessary effcecy. The Ida ecessarly effcecy test Step. let { } ( S. ( g w wg Step. Select oe elemet from S ad check whether t s effcet. (a If t s ot effcet the (b Otherwse add sc( s ot ecessarly effcet. g w w ( g ( g w w g ( m w.if w to S m Do ot add ay thg to S. Step. If the set S s empty the s ecessarly effcet. Step. etur to step. The herkova s effcecy test Step: ompute. Step: Aalyze colums ad rows of ad proceed as follows: (a If there are ay colums such that the elmate these colums.. (b If there are ay rows such that. the elmate these rows. Step: Aalyze colums ad rows of ad proceed as follow: (a If there s a colum such that the s ot effcet.. (b If there s a row r such that the s effcet.. > (c If there s a row such that ad a row such that > ( the s effcet.. Step: alculate the summato of the colums( ad rows ( of. (aif. Σ the s ot effcet. (b If Σ. > the s effcet.. Iterval MOP wth equal costrats Defto : We defe the set of. A X b That s S s the set of all solutos of.σ Σ. S { X AX b A A b b } AX b for all A A b b as the soluto of terval system of equatos. Ths set s ot a terval vector. ecause S s geerally so complcated shape t s usually mpractcal to use t. Istead t s commo practce to seek the terval vector X cotag S that has the arrowest possble terval compoets. JI emal for cotrbuto: edtor@c.org.uk

3 Joural of Iformato ad omputg Scece (9 pp - Ths terval vector s called hull ad we solve the system whe we fd the hull X. Theorem : For a terval equalty costrat The followg par of equalty costrats A A x A A x b b A x b ad ~ A x b where A ad A defe a covex rego of possblty whch every pot x ~ A A x x could satsfy some legal verso of the orgal terval equalty by a approprate choce of fxed values for the terval coeffcets ad vce versa. ow we cosder the followg MOP wth terval coeffcets ad equal costrats: Max X s. t X b A (. X Φ where A s a m terval matrx b s a m terval vector ad Φ s a set of all p matrces wth. y usg the theorem problem (. s equvalet to the followg problem: Max X s. t A X b (. A X b X Φ ote that the problem (. s a multobectve programmg wth terval coeffcets ad crsp costrats. So we ca fd ts ecessarly effcet soluto by usg the algorthm we metoed the secto.. Iterval MOP wth equalty costrats osder the followg terval MOP wth equalty costrats : Max X s. t X b A (. X Φ where Φ s a set of all p matrces wth A s a m terval matrx ad b s a m terval vector. To fd the ecessarly effcet soluto for the problem (. we try to covert t to a problem smlar to (. ad fd the ecessarly effcet pots as the method whch expla secto. For ths purpose we suggest the followg method: osder the followg costrat wth terval coeffcets: where b b b A A A x. A x b Fdg the feasble rego s a essetal problem for terval programmg problem. Several deftos of feasble rego have bee proposed by researchers 6. Here we use the defto gve by Ishbuch ad Taaaka whch s based o the cocept of degree of equalty holdg true for two tervals6. (. JI emal for subscrpto: publshg@wa.org.uk

4 6 Hassa Mshmast eh et al: The ecessarly Effcet Pot Method for Iterval Molp Problems Hassa Mshmast eh Defto : For a terval A A A ad a real umber x the degree for equalty A x holdg true s gve as follows: Defto : For two tervals A A gve as follows: x A g( A x max{ m{ }} A A A ad the degree for equalty A holdg true s A g( A g( A max{ m{ }} A + A Accordg to defto the feasble rego of terval costrat (. ca be determed by the followg theorem: Theorem : For a gve degree of equalty holdg true q the terval equalty costrat (. ca be trasformed to ( qa x + ( q A x ( q b + qb For more detaled dscusso o the feasble of terval costrat see 6. Therefore by usg theorem for a gve degree of equalty holdg true q the problem (. covert to: Max X s. t ( qa x + ( q A x ( q b + qb (. X Φ umercal example osder the followg MOP wth equalty costrats whch the terval matrces follow: A If q.the by usg theorem the matrces the terval costrats matrces A ad b : 6 6 A b 6 6 ow we wat to use the ecessary effcecy test for the extreme pot So we obta Thus 6 6 T ( b x b A ad b as A trasform to followg JI emal for cotrbuto: edtor@c.org.uk

5 Joural of Iformato ad omputg Scece (9 pp - ( ( Hece ( (. Sce ( ad there s a elemet ( ( ( the ( s effcet ad ( s ot effcet. Therefore by the two effcecy tests t may be ecessary to exame the effcecy of both ( ad (. Thus ( ( ( ( Hece ( (. ( (. Sce ad there s a elemet the s effcet. However s ot effcet sce. O the other had s ot effcet sce. Fally ad are ot effcet because. Therefore by the two effcecy tests we coclude that s ot ecessarly effcet(there s a elemet that s ot effcet ad the soluto ( ( ( ( ( ( (. ( (. ( ( (. S ( T 6 6 ( s ot ecessarly effcet whe. q. JI emal for subscrpto: publshg@wa.org.uk

6 Hassa Mshmast eh et al: The ecessarly Effcet Pot Method for Iterval Molp Problems Hassa Mshmast eh. Multobectve lear programmg wth fuzzy coeffcets. I covetoal mathematcal programmg coeffcets of problems are usually determed by the experts ad crsp values. but realty a mprecse ad ucerta evromet 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. ut 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 degeerated 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. I ths secto we focus o multobectve lear programmg wth fuzzy coeffcet. y troducg α -cuts ad amk ad amek rakg we degeerate 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. The multobectve lear programmg wth fuzzy coeffcets ca be formulated as follow: Where c~ A ~ max c~ x s.t A ~ x b ~ x ad b ~ are fuzzy umbers. k p m (. 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 degeerate coeffcets of obectve fucto to terval as dcated follow ad amk ad amek rakg to obta crsp costrats from our fuzzy costrats... Some deftos The α -level set (α -cut of a fuzzy set M ~ s defed as a ordary set membershp fucto exceeds the level α : M ~ α for whch the degree of ts M ~ α { x μ M ~ α } α 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 umber M ~ ca be represeted by the closed terval whch depeds o terval value of α. amely M ~ α { x μ M ~ α } M M α α Where or M α represets the left or rght extreme pot of the α -level set M ~ α respectvely. M α M ~ α of a fuzzy Especally f M ( m γ β s a tragular fuzzy umber the M ~ α γ ( α + mm β( α ad f M ( m m γ β s a trapezodal fuzzy umber the M ~ α γ ( α + m m β( α... akg fuzzy umbers JI emal for cotrbuto: edtor@c.org.uk

7 Joural of Iformato ad omputg Scece (9 pp - 9 Dubos ad prade proposed a method of rakg fuzzy umbers as follow: Defto. et M ~ ad ~ be fuzzy umbers the we have M ~ ~ M ~ ~ M ~. y usg the defto amk ad ramek suggested the followg lemma: emma. If M ~ ad ~ be a fuzzy umbers M ~ ~ M ~ f ad oly f for every h we have f { s : μ ( s M ~ h } f { t : μ ~ ( t h } sup { s : μ ( s M ~ h } sup { t : μ ~ ( t h } Especally f M ~ ( m m α β ad ~ ( γ δ be trapezodal fuzzy umbers the above relato s true f ad oly f m l * ( h α l * ( h β h m r + * ( h α r + * ( h β h Where * ( h sup{ z : ( z h } * ( h * ( h sup{ z : ( z h } sup{ z : T( z h } ( h sup{ z : ( z h }. * From the defto of amk ad amek f M ~ ( m α β ad ~ ( γ δ be tragular fuzzy umbers the we have M ~ ~ m m α γ m + β + δ If M ~ l r ( m m α β ad ~ l r ( γ δ be trapezodal fuzzy umbers smlarly we ca compare them as follow l l m M ~ ~ r r m l l m α γ r r m + β + δ y usg above defto ad wth respect to the kd of fuzzy umbers of the problem we ca trasform the fuzzy costrats to crsp oes. I ths way f c~ ( c α β A ~ ( a γ δ ad b ~ ( b η ν be tragular fuzzy umbers problem (. s equvalet to Max α ( α + c c β ( α x k p s.t a x b ( a γ x b η m m (. JI emal for subscrpto: publshg@wa.org.uk

8 Hassa Mshmast eh et al: The ecessarly Effcet Pot Method for Iterval Molp Problems Hassa Mshmast eh ( a + β x b + ν x. m Smlarly f c~ ( c c α β A ~ ( a a γ δ ad b ~ ( b b η ν be trapezodal fuzzy umbers the problem (. s equvalet to Max α ( α + c c β ( α x k p s.t a x a x b b ( a γ x b η ( a + β x b + ν x. m m (. m m. ocluso I ths paper a MOP wth terval coeffcets s focused. we proposed a procedure to trasform the problem to a Molp wth terval obectve coeffcets ad crsp costrats. The we try to fdg the ecessarly effcet pots for solvg prmary problem. 6. efereces tra G... ear multple obectve problems wth terval coeffcets. Maagemet Scece. 9 6: heck J.W. amada K.. ear programmg wth terval coeffcets. Joural of the operatoal esearch Socety. : 9-. Hase E.. Global optmzato usg terval aalyss. ew York Press. Ida M.. ecessary effcet test terval multobectve lear programmg. I : Proceedg of th teratoal fuzzy system assocato world cogress. pp. -. Ida M.. Effcet soluto geerato for multple obectve lear programmg ad ucerta coeffcets. I: proceedgs of th ellma cotuum. pp Ishbuch H. Taaka H.. Multobectve programmg optmzato of the terval obectve Systems. Pretc - Hall Eglewood lffs J 9. Olvera. Atues.H.. Multple obectve lear programmg models wth terval coeffcets -a llustrated overvew. Europea oural of operatoal esearch. 6 pp. -. oh J.. System of terval lear equatos ad equaltes(rectagular case. Techcal report. (. 9 Steuer P.E.. Multple crtera optmzato: theory computato ad applcato. ew york Joh wley & Sos 96. JI emal for cotrbuto: edtor@c.org.uk