SOLVING METHOD FOR FUZZY MULTIPLE OBJECTIVE INTEGER OPTIMIZATION

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1 SOLVING METHOD FOR FUZZY MULTIPLE OBJECTIVE INTEGER OPTIMIZATION BOGDANA POP Taslvaa Uvesty of Basov Romaa Abstact Statg fom the dea of Wag ad Lao (00 fo solvg fuzzy o-lea tege ogammg oblem ad tag to accout the multle ctea otmzato fuzzy evomet a solvg method fo fuzzy multle objectve tege otmzato oblem s develoed hee. Theoetcal aalyss o effcet solutos fo multle ctea otmzato oblem ad comutatoal esults ae also eseted. Keywods: fuzzy ogammg tege ogammg multle ctea. INTRODUCTION Statg fom the dea of Wag ad Lao [7] fo solvg fuzzy o-lea tege ogammg oblem ad tag to accout the multle ctea otmzato fuzzy evomet a solvg method fo fuzzy multle objectve tege otmzato oblem s develoed hee. Wag ad Hog [8] oosed a aoach to efom comlete aametc aalyss tege ogammg by cosdeg all of the ossble caddates of fuzzy evomet aametes costats. We use the methods to solvg multle objectve ogammg wth tege vaables ad fuzzy costats. I [] Pegoz et al. focused o multle objectve tege ogammg oblems wth adom vaable coeffcets objectve fuctos ad/o costats. I [5] Saawa ad Kato eseted a teactve fuzzy satsfyg method fo olea tege ogammg oblems though a geetc algothm. I [7] α-paeto otmal solutos wee detemed fo the multle objectve tege olea ogammg oblems havg fuzzy aametes the costats togethe wth the coesodg stablty set of the fst d. Cte [] eseted a method useful solvg a secal class of lage-scale multle objectve tege oblems based uo a combato of the decomosto algothm couled wth the weghtg method togethe wth the bach-ad-boud method. Secto descbe the mathematcal model of a fuzzy multle objectve tege otmzato. A solvg method fo such a model s eseted Secto usg a algothm whch wos o detemstc oblems ad a cocet of cal caddates fo fuzzy evomet aametes. Comutatoal esults ae seted Secto 4. The cosdeed examle solved a multle ctea lea factoal ogammg oblem havg fuzzy costats ad tege decso vaables. Secto 5 s fo shot cocludg emas.. FUZZY MULTIPLE OBJECTIVE INTEGER OPTIMIZATION MODEL Cosde the multle objectve ogammg wth fuzzy costats ad tege vaables " m" { z( x = ( z ( x z ( x... z ( x x X } (. s the feasble set of the oblem A s a m costat matx x s a -dmesoal vecto of decso vaable ad m b R X = { x R Ax b x 0 x Z}

2 Vecto ( z ( x eesets the objectve fuctos whch could be lea lea factoal = o covex fuctos ( ode to mae the ew method woable. The tem "m" used Poblem (. s fo fdg all effcet solutos a mmzato sese tems of the Paeto otmalty. A ossble way to hadle costats mecso s to cosde the followg aametc oblem m" z( x = z ( x z ( x... z ( x x X ( θ { ( } We oted X ( = { x R Ax b + θb' x 0 x Z} " [ 0]. (. θ the feasble set of the oblem θ ad b a gve etubato vecto ([7]. 3. THE SOLVING METHOD I [3] a algothm to solvg multle objectve lea factoal ogammg (MOLFP s eseted. We set below a befly descto of ths algothm. Ste Establsh the oblem costat ad ut = 0. Ste Comute magal values usg ERA ([3]. 0 Ste 3 Choose x as a feasble soluto of MOLFP. Ste 4 If ( z x = 0 the x s the otmal soluto. Othewse go to Ste 5. Ste 5 (Accetablty test If x s a accetable soluto of MOLFP the favoable STOP If x s ot a accetable soluto of MOLFP but a ossble movg exsts go to Ste 6 Othewse ufavoable STOP. Coveet soluto does ot exst fo MOLFP. Ste 6 Seach a decto of mmzato s fo as may objectve fuctos as ossble. If thee s o such a decto s the let s be the decto of mmzato fo the cteo whch ealzed the lagest slac Ste 5. Comute λ ad accodg wth ERA go to Ste 7 wth x + = x + λ max s. + the go to Ste 4 wth = +. Othewse comute x = x + λs fo λ < λmax ad etu to the test of Ste 7. We wll use the above algothm (ude the ame MultObj (θ to solve Poblem (. meag the detemstc oblem wth the feasble set X ( θ. To move the teactvty of the method dffeet values fo θ wll be cosdeed. Wag ad Hog [8] oosed a aoach to efom comlete aametc aalyss tege ogammg by cosdeg all of the ossble caddates of θ. They defed the cal caddates of θ as bee that θ whch maes b + θ ' a tege fo = m. We also wo wth these θ's. I [7] Wag ad Lao oosed a + Ste 7 If S( x S( x b heustc algothm to aalyze the same fuzzy oblem. We wll use the method to solvg multle objectve ogammg wth tege vaables ad fuzzy costats. Fo a fxed value θ we stat defg Poblem (3. as Poblem (. wthout the tegty estcto of vaables. { ( x = ( z ( x z ( x... z ( x Ax b + b' x 0} " m" z θ (3. We obta a effcet soluto fo Poblem (3. usg MultObj (. It s a soluto fo Poblem (. f ad oly f ts comoets ae tege umbes. I ths case t s also soluto fo Poblem (. but wth mmal degee fuzzy evomet. Cosequetly ou ext goals ae to tasfom the soluto to a tege oe ad also to move ts fuzzy degee. These max

3 goals could be atted modfyg values x [ x ] o [ ] + etubato vecto deceases usg bac ( Pocedue bac ( : If = the f Ax b +θ b' the y ( x θ z( x x such that the devato of the ocedue descbed below. = ; etu. If u[]=0 the x[]=[x[]]; u[]=; bac ( + ; u[]=0. If u[]=0 the x[]=[x[]]+; u[]=; bac ( + ; u[]=0. Tag to accout the above emas the solvg algothm could be descbed as follows. Ste Defe the thesholds θ < θ <... < θ usg the cal caddates of aametesθ. Put q=. Ste Fo =q dow to do Comute values x ( x x... x If x Z the favoable STOP wth oblem. Othewse call bac ( y 0 = q = = usg MultObj θ. x the ( ad obta x ( x x'... x' θ fuzzy degee accetable soluto of the = '. Idetfy m such that Ax b m +θ '. The x s the θ fuzzy degee accetable soluto of the oblem. b m Ste 3 Descbe the oblem soluto as ( q m y =. 4. COMPUTATIONAL RESULTS I ode to llustate ou solvg method let us cosde the followg detemstc lea factoal ogam x ( x + 7 "m" z x = (4. x + 50 x + 3 Subject to x x + (4. x 0 x Z. The fuzzy feasble set of Poblem (4. meag (4. s teated as a aamete feasble set (4.3 θ 0 ad cosdeg [ ] x θ 3x + + 6θ (4.3 x 0 x Z. Ste gave q=7 ad θ = ( The solvg algothm gave fuzzy tege solutos cotaed Table 4.. Cosequetly a soluto of Poblem (4. s

4 Y [ y = (( 33 0 ( = (( ( ] θ x f ( f ( x ' Z = y x x x ' Z Table 4. To deal wth lea factoal otmzato we used classc solvg methods whch ae descbed detal [6]. Also to mae basc otmzato calculus we used classc tools. To efom global comutatoal aalyze we mlemet ou algothms (ERA MultObj (θ bac y. ( 5. CONCLUDING REMARKS I ths ae we have oosed a method of solvg fuzzy (costats multle objectve tege (decso vaables otmzato oblems. We woed wth the cocet of cal caddates fo the fuzzy evomet aametes (ad to move the teactvty of the method dffeet values fo aametes θ wee cosdeed ad obtaed a fuzzy soluto meag ossble soluto values vesus fuzzy degee of ths. We have aled a algothm whch solves detemstc classc oblems ad the we tasfomed the soluto to a tege oe (develog ocedue bac ( also movg ts fuzzy degee. BIBLIOGRAPHY [] M.S. Osma O.M. Saad A.G. Hasa (999 Solvg a secal class of lage-scale fuzzy multobjectve tege lea ogammg oblems Fuzzy Sets Syst. 07/ ; [] C. Pegoz K. Kato H. Katag M. Saawa (004 A teactve fuzzy satsfyg method fo multobjectve stochastc tege ogammg oblems though vaace mmzato model Sc. Math. J. 60/ ; [3] Bogdaa Po (004 Algothm to solvg multle ctea lea factoal otmzato} A. Uv. Bucuest Mat. Ifom. 05-4; [4] O.M. Saad (999 O the soluto of fuzzy multobjectve tege olea ogammg oblems wth a aametc study J. Fuzzy Math. 7/ ; [5] M. Saawa K. Kato (003 A teactve fuzzy satsfyg method fo multobjectve olea tege ogammg oblems though geetc algothms Lect. Notes Comut. Sc ; [6] I.M. Stacu-Masa (997 Factoal Pogammg: Theoy Methods ad Alcatos Kluwe Academc Publshes; [7] H.F. Wag Y.C. Lao (00 Fuzzy o-lea tege ogam by aametc ogammg aoach Fuzzy Sets Syst. /3 45-5;

5 [8] H.F. Wag J.S. Hog (996 Stuctual aoach to aametc aalyss of a IP o the case of the ght-had-sde Euo. J. Oe. Res

A COMPARISON OF MULTIVARIATE DISCRIMINATION OF BINARY DATA

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