PARTICLE SWARM OPTIMIZATION FOR INTERACTIVE FUZZY MULTIOBJECTIVE NONLINEAR PROGRAMMING. T. Matsui, M. Sakawa, K. Kato, T. Uno and K.

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1 Scenae Mahemacae Japoncae Onlne, e-2008, PARTICLE SWARM OPTIMIZATION FOR INTERACTIVE FUZZY MULTIOBJECTIVE NONLINEAR PROGRAMMING T. Masu, M. Sakawa, K. Kao, T. Uno and K. Tamada Receved February 27, 2007; revsed Ocober 22, 2007 Absrac. In recen years, parcle swarm opmzaon (PSO) proposed by Kennedy e al. has been wdely used as a general approxmae soluon mehod for opmzaon problems. The auhors proposed a revsed PSO (rpso) mehod ncorporang he homomorphous mappng and he mulple srechng echnque n order o cope wh shorcomngs of PSO and showed s effcency for nonlnear programmng problems. In hs paper, we consruc an neracve fuzzy sasfcng mehod for mulobjecve nonlnear programmng problems based on he rpso. In order o oban beer soluons n consderaon of he propery of mulobjecve programmng problems, we ncorporae he drecon o nondomnaed soluons no he rpso. Furhermore, he effcency of he proposed mehod (MOrPSO) s shown hrough applcaons o numercal examples. 1 Inroducon A nonlnear programmng problem s called a convex programmng problem when s objecve funcon and s consran regon are convex. For such convex programmng problems, here have been proposed many effcen soluon mehods such as he successve quadrac programmng mehod, he generalzed reduced graden mehod and so forh. Unforunaely, here have no been proposed any decsve soluon mehod for nonconvex programmng problems. As praccal soluon mehods, mea-heursc opmzaon mehods such as he smulaed annealng mehod, he genec algorhm and so on, have been proposed. In recen years, however, more speedy and more accurae opmzaon mehods have been desred because acual problems become more large-sze and more complex. As a new opmzaon mehod, he parcle swarm opmzaon (PSO) mehod was proposed by Kennedy e al. [2]. PSO s a search mehod smulang he socal behavor ha ndvduals (parcles) lke brd or fsh consue a populaon (swarm) and each parcle n he swarm search beer pons usng he knowledge owned by as well as ha owned by he swarm. Snce he orgnal PSO has shorcomngs such as he concenraon of parcles o local soluons and he napplcably o consraned problems, he auhors proposed a revsed PSO (rpso) by ncorporang he homomorphous mappng and he mulple srechng echnque n order o overcome hese shorcomngs [5]. In recen years, wh he dversfcaon of socal requremens, he demand for he programs wh mulple objecve funcons, has been ncreasng raher han a sngle-objecve funcon (e.g. maxmzng he oal prof and mnmzng he amoun of polluon n a producon plannng). Snce here does no always exs a complee opmal soluon whch opmzes all objecves smulaneously for mulobjecve programmng problems, a soluon ha any mprovemen of one objecve funcon can be acheved only a he expense of a leas one of oher objecve funcons s consdered as a reasonable soluon, whch s 2003 Mahemacs Subjec Classfcaon. Prmary 90C29, 90C70; Secondary 90C59, 90C26. Key words and phrases. parcle swarm opmzaon, mulobjecve nonlnear programmng problem, neracve fuzzy sasfcng mehod, nondomnaed soluon.

2 2 T. MATSUI, M. SAKAWA, K. KATO, T. UNO AND K. TAMADA called a Pareo opmal soluon or a nondomnaed soluon. For such mulobjecve opmzaon problems, fuzzy programmng approaches (e.g. Zmmermann [12], Rommelfanger [7]), consderng he mprecse naure of he decson maker s judgmens n mulobjecve opmzaon problems, seem o be very applcable and promsng. In he applcaon of he fuzzy se heory no mulobjecve lnear programmng problems sared by Zmmermann [12], has been mplcly assumed ha he fuzzy decson or he mnmum-operaor of Bellman and Zadeh [1] s suable o represen he decson maker s preference for fuzzy goals of objecve funcons. In general, however, he fuzzy decson s no always he bes represenaon of he decson maker s preference for fuzzy goals. Thereby, Sakawa e al. have proposed neracve fuzzy sasfcng mehods for varous mulobjecve programmng problems o derve a sasfcng soluon for he decson maker along wh checkng he local preference of he decson maker hrough neracons [8, 9]. In parcular, hey [10] consdered an neracve fuzzy sasfcng mehod for mulobjecve nonlnear programmng problems. In [10], hey adoped a genec algorhm called RGENOCOPV and showed he feasbly of he proposed mehod, bu here are drawbacks abou calculaon me and precson of soluons. In hs research, focusng on mulobjecve nonlnear programmng problems, we aemp o derve a sasfcng soluon hrough he neracve fuzzy sasfcng mehod wh shor calculaon me and hgh precson. As he soluon mehod used n he neracve fuzzy sasfcng mehod, we adop rpso [5] whch s promsng for nonlnear programmng problems and we propose an exenson of rpso o mulobjecve programmng problems, MOrPSO. 2 Mulobjecve nonlnear programmng problems In hs research, we consder mulobjecve nonlnear programmng problems formulaed as: (1) mnmze subjec o f l (x), l =1, 2,...,k g (x) 0, =1, 2,...,m l j x j u j, j =1, 2,...,n x =(x 1,x 2,...,x n ) T R n where f l ( ), g ( ) are lnear or nonlnear funcons, l j and u j are he lower lm and he upper lm of each decson varable x j. In addon, we denoe he feasble regon of (1) by X. 3 An neracve fuzzy sasfcng mehod In order o consder he mprecse naure of he decson maker s judgmens for each objecve funcon n (1), f we nroduce he fuzzy goals such as f l (x) should be subsanally less han or equal o a ceran value, (1) can be rewren as: (2) maxmze (µ 1 (f 1 (x)),...,µ k (f k (x))) x X where µ l ( ) s he membershp funcon o quanfy he fuzzy goal for he l h objecve funcon n (1). To be more specfc, f he decson maker feels ha f l (x) should be less han or equal o a leas fl 0 and f l (x)) fl 1( f l 0 ) s sasfacory, he shape of a ypcal membershp funcon s shown n Fg. 1. Snce (2) s regarded as a fuzzy mulobjecve decson makng problem, here rarely exs a complee opmal soluon ha smulaneously opmzes all membershp funcons. As a reasonable soluon concep for he fuzzy mulobjecve decson makng problem, Sakawa e al. defned M-Pareo opmaly on he bass of membershp funcon values by drecly exendng he Pareo opmaly n he ordnary mulobjecve programmng problem [8].

3 PSO FOR INTERACTIVE FUZZY MULTIOBJECTIVE NONLINEAR PROGRAMMING 3 l ( ( x) ) f l 1 0 f l 1 f l 0 f l () x Fgure 1: An example of membershp funcons. Inroducng an aggregaon funcon µ D (x) for k membershp funcons n (2), he problem can be rewren as: (3) maxmze µ D (x) x X where he aggregaon funcon µ D ( ) represens he degree of sasfacon or preference of he decson maker for he whole of k fuzzy goals. Followng convenonal fuzzy approaches, as aggregaon funcons, he mnmum operaor and he produc operaor are ofen adoped. However, should be emphaszed here ha such approaches are preferable only when he decson maker feels ha he mnmum operaor or he produc operaor s approprae. In oher words, n general decson suaons, he decson maker does no always use he mnmum operaor or he produc operaor when combnng he fuzzy goals. Probably he mos crucal problem n (3) s he denfcaon of an approprae aggregaon funcon whch well represens he decson maker s fuzzy preferences. If µ D ( ) can be explcly denfed, hen (3) reduces o a sandard mahemacal programmng problem. Unforunaely, however, hs rarely happens. Thereby, as an alernave, an neracon wh he decson maker s necessary for fndng he sasfcng soluon of (2). In he neracve fuzzy sasfcng mehod, n order o generae a canddae for he sasfcng soluon whch s M-Pareo opmal, he decson maker s asked o specfy he aspraon levels of achevemen for all membershp funcons, called he reference membershp levels [8]. For he decson maker s reference membershp levels µ l, l =1, 2,...,k, he correspondng M-Pareo opmal soluon, whch s neares o he requremens n he mnmax sense or beer han ha f he reference membershp levels are aanable, s obaned by solvng he followng augmened mnmax problem (4). (4) mnmze x X max {( µ l µ l (f l (x))) + ρ l=1,...,k k ( µ µ (f (x))) where ρ s a suffcenly small posve real number. We can now consruc he neracve algorhm n order o derve he sasfcng soluon for he decson maker from he M-Pareo opmal soluon se. The procedure of an neracve fuzzy sasfcng mehod s summarzed as follows. Sep 1: Under gven consrans, he mnmal value and he maxmal one of each objecve funcon are calculaed by solvng followng problems. (5) mnmze f l (x), l =1, 2,...,k x X (6) maxmze f l (x), l =1, 2,...,k x X =1

4 4 T. MATSUI, M. SAKAWA, K. KATO, T. UNO AND K. TAMADA Sep 2: In consderaon of he mnmal value and he maxmal one of each objecve funcon, he decson maker subjecvely specfes membershp funcons µ l (f l (x)), l = 1, 2,...,k o quanfy fuzzy goals for objecve funcons. Nex, he decson maker ses nal reference membershp levels µ l, l =1, 2,...,k. Sep 3: We solve he augmened mnmax problem correspondng o curren reference membershp levels (4). Sep 4: If he decson maker s sasfed wh he soluon obaned n sep 3, he neracve procedure s fnshed. Oherwse, he decson maker updaes reference membershp levels µ l, l =1, 2,...,k based on curren membershp funcon values and objecve funcon values, and reurn o sep 3. 4 Parcle swarm opmzaon Parcle swarm opmzaon (PSO) [2] s based on he socal behavor ha a populaon of ndvduals adaps o s envronmen by reurnng o promsng regons ha were prevously dscovered [3]. Ths adapaon o he envronmen s a sochasc process ha depends on boh he memory of each ndvdual, called parcle, and he knowledge ganed by he populaon, called swarm. In he numercal mplemenaon of hs smplfed socal model, each parcle has four arbues: he search pon n he search space, he search drecon vecor and he bes search pon n s rack and he bes search pon of he swarm. The process can be oulned as follows. Sep 1: Sep 2: Generae he nal swarm nvolvng N parcles a random. Calculae he nex search drecon vecor of each parcle, based on s arbues. Sep 3: Calculae he nex search pon of each parcle from he curren search pon and s nex search drecon vecor. Sep 4: If he ermnaon condon s sasfed, sop. Oherwse, go o Sep 2. To be more specfc, he nex search drecon vecor of he -h parcle a me, v +1,s calculaed by he followng scheme nroduced by Sh and Eberhar [11]. (7) v +1 := ω v + c 1 R 1(p x )+c 2 R 2(p g x ) In (7), R1 and R2 are random numbers beween 0 and 1, p s he bes search pon of he -h parcle n s rack and p g s he bes search pon of he swarm. There are hree problem-dependen parameers, he nera of he parcle ω, and wo rus parameers c 1, c 2. Then, he nex search pon of he -h parcle a me, x +1, s calculaed from (8). (8) x +1 := x + v +1 where x s he curren search pon of he -h parcle a me. The -h parcle calculaes he nex search drecon vecor v +1 by (7) n consderaon of he curren search drecon vecor v, he drecon vecor from he curren search pon x o he bes search pon n s rack p and he drecon vecor from he curren search pon x o he bes search pon of he swarm p g, moves from he curren search pon x o he nex search pon x +1 calculaed by (8). The parameer ω conrols he amoun of move o search globally n early sage and o search locally by decreasng ω gradually. The movemen of a parcle n PSO s shown n Fg. 2. Comparng he evaluaon value of a parcle afer move, f(x +1 ), wh ha of he bes search pon n s rack, f(p ), f

5 PSO FOR INTERACTIVE FUZZY MULTIOBJECTIVE NONLINEAR PROGRAMMING 5 p g p x v v +1 x +1 Fgure 2: Movemen of a parcle n PSO. f(x +1 ) s beer han f(p ), hen he bes search pon n s rack s updaed as p := x+1. ) s beer han f(p g ), hen he bes search pon of he swarm s Furhermore, f f(p +1 updaed as p +1 g := p +1. Such a smple PSO [2, 11] ncludes wo problems. One s ha parcles concenrae on he bes search pon of he swarm and hey canno easly escape from he pon (local soluon) snce he search drecon vecor v +1 calculaed by (7) always ncludes he drecon vecor o he bes search pon of he swarm. Anoher s ha a parcle afer move s no always feasble for problems wh consrans. 5 Improvemen of parcle swarm opmzaon In hs sudy, o preven he concenraon and sop of parcles a local opmal soluons n he smple PSO, we nroduce he modfcaon of move schemes of a parcle, he secesson and he mulple srechng echnque. In addon, n order o rea consrans, we dvde he swarm no wo subswarms. In one subswarm, snce he move of a parcle o he nfeasble regon are no acceped, f a parcle becomes nfeasble afer move, s repared o be feasble. In he oher subswarm, he move of a parcle o he nfeasble regon are acceped. 5.1 Move of a parcle Le us consder he move from he curren search pon x of he -h parcle. Frs, f he prevous search pon x 1 s he bes search pon of he parcle n s rack p, he nex search pon x+1 moves near he bes search pon of he swarm p g wh hgh possbly. Thereby, as shown n Fg. 3, we change (7) o deermne he nex search drecon vecor v +1 as (9) v +1 := c 1 R 1 (p x )+c 2R 2 (p k x ) where p k s he bes search pon of he k-h parcle. By he change of he search drecon vecor deermnaon scheme, we can relax concenraon of parcles o p g. Nex, n case ha he curren search pon x s he bes search pon of he parcle n s rack p, he drecon from he prevous search pon o he curren search pon s desrable. Thus, as n Fg.4, we change (7) o deermne he nex search drecon vecor as v +1 (10) v +1 := ω v Oherwse, we use (7) o deermne he nex search drecon vecor v +1.

6 6 T. MATSUI, M. SAKAWA, K. KATO, T. UNO AND K. TAMADA x +1 v +1 p k p g p x v Fgure 3: The new search drecon vecor when he bes search pon of a parcle n s rack s renewed a he prevous search pon. p g x =p v +1 v x +1 Fgure 4: The new search drecon vecor when he bes search pon of a parcle n s rack s renewed a he curren search pon.

7 PSO FOR INTERACTIVE FUZZY MULTIOBJECTIVE NONLINEAR PROGRAMMING 7 The bes search pon of he swarm Fgure 5: The concenraon of parcles around he bes search pon of he swarm. 5.2 Dvson of he swarm no wo subswarms In applcaon of PSO o opmzaon problems wh consrans, a parcle afer move s no always feasble f we use he move schemes menoned above. To deal wh such a suaon, we dvde he swarm no wo subswarms. In one subswarm, snce he move of a parcle o he nfeasble regon are no acceped, f a parcle becomes nfeasble afer move, s repared o be feasble. To be more specfc, wh respec o nfeasble parcles whch volae consrans afer move, we repar s search pon o be feasble by he bsecon mehod on he drecon from he search pon before move, x, o ha afer move, x+1. In he oher subswarm, he move of a parcle o he nfeasble regon are acceped. 5.3 Secesson Snce parcles end o concenrae on he bes search pon of he swarm as he search goes forward n PSO, as shown n Fg. 5, he global search becomes dffcul. Thus, we nroduce he followng secesson of a parcle (Fg. 6). (1) [Secesson I] A parcle moves a random o a pon n he feasble regon. (2) [Secesson II] A parcle moves a random o a pon on he boundary of he feasble regon. (3) [Secesson III] A parcle moves a random o a pon n a drecon of some coordnae axs. 5.4 Mulple srechng echnque Srechng echnque s suggesed o preven he sop a a local opmal soluon of parcles n PSO by Parsopoulos e al. [6]. I enables parcles o escape from he curren local opmal soluon and no o approach he same local opmal soluon agan by changng orgnal evaluaon funcon f(x) o oher evaluaon funcon H(x) defned as: ( G(x) =f(x)+γ 1 x x sgn ( f(x) f( x) ) ) (11) +1 (12) sgn ( f(x) f( x) ) +1 H(x) =G(x)+γ 2 anh (µ ( G(x) G( x) ))

8 8 T. MATSUI, M. SAKAWA, K. KATO, T. UNO AND K. TAMADA Secesson I Secesson II Secesson III The bes search pon of he swarm Fgure 6: The secesson. where x s he curren local opmal soluon, and γ 1, γ 2, µ are parameers. In addon, sgn( ) s defned as follows. 1, x > 0 (13) sgn(x) = 0, x =0 1, x < 0 In G(x), he second erm s he penaly dependng on he dsance beween a search pon x and he curren local opmal soluon x. The erm s equal o 0 for x whose objecve funcon value f(x) s beer han f( x), whle akes a value dependng on he dsance beween x and x for x whose f(x) s worse han f( x). Nex, H(x) s defned usng G(x) expressed n (11). The value of H(x) for a search pon x s equal o he objecve funcon value f(x) ofx f f(x) ofx s beer han ha of x, whle akes a very large value f f(x) ofx s worse han ha of x. Usng H(x) as he new evaluaon funcon, parcles can escape from he curren local opmal soluon and search a new regon whch may nclude beer soluons han he curren local opmal soluon. Alhough he srechng echnque [6] enables parcles o escape from he curren local opmal soluon, hey may sop a he same local opmal soluon agan when we apply he srechng echnque o he nex local opmal soluon. Thus, we use he mulple srechng echnque correspondng o plural local opmal soluons. To be concree, we consder he followng funcons for Q local opmal soluons x q, q =1, 2,...,Q. ( G q (x) =f(x)+γ 1 x x q sgn ( f(x) f( x mn ) ) ) (14) +1 (15) sgn ( f(x) f( x mn ) ) +1 H q (x) =G q (x)+γ 2 anh (µ ( G q (x) G q ( x q ) )) (16) S(x) = Q H q (x)/q q=1 Here, x mn s he bes among Q local opmal soluons. The value of S(x) for a search pon x s equal o he objecve funcon value f(x) ofx f f(x) s beer han ha of x mn, whle akes a very large value f he dsance beween x and he neares local opmal soluon s less han a ceran value. Oherwse, akes a value dependng on he dsance.

9 PSO FOR INTERACTIVE FUZZY MULTIOBJECTIVE NONLINEAR PROGRAMMING 9 µ 2 1 _ µ Opmal soluon Swarm 0 1 µ 1 Fgure 7: Applcaon of rpso o he augemened mnmax problem. µ 2 1 _ µ Opmal soluon Swarm 0 1 µ 1 Fgure 8: Applcaon of he proposed PSO o he augemened mnmax problem. 5.5 Incorporaon of he drecon o nondomnaed parcles n he swarm The shape of he objecve funcon of he augmened mnmax problem (4), solved n he neracve fuzzy sasfcng mehod for mulobjecve nonlnear programmng problems, ofen becomes complex. Therefore, here does no always exs he opmal soluon n he search drecon of he swarm, as shown n Fg. 7. Thus, consderng he fac ha he opmal soluon o (4) s one of M-Pareo opmal soluons o (2), we nroduce he drecon o a nondomnaed parcle (an approxmae M-Pareo opmal soluon) o carry ou search wh respec o boh M-Pareo opmaly and he opmaly of he objecve funcon of (4), as shown n Fg. 8. In order o ncorporae he new search drecon, we revse (7) o deermne he nex search drecon vecor as: (17) v +1 = ω v + c 3R 3 (x K x ) where x K s he search pon of a ceran nondomnaed parcle K n he curren swarm, c 3 s a parameer and R3 s he unform random number n he nerval [0, 1]. Moreover, when he curren search pon s he bes search pon of a parcle, s consdered ha he curren search pon and he curren search drecon vecor are promsng snce he bes pon of a parcle s jus updaed. Therefore, he nex search drecon vecor v +1 s calculaed by he ranslaon equaon (10) lke Morhara [5]. By he swarm nvolvng parcles whch move n he manner menoned above, he search wh respec o M-Pareo opmaly s carred ou. In rpso [5], he nex search pon s decded by a vecor o he k-h parcle nsead of he bes search pon of he swarm n order o resran he concenraon of parcles on he bes search pon of he swarm when he bes search pon of a parcle s updaed

10 10 T. MATSUI, M. SAKAWA, K. KATO, T. UNO AND K. TAMADA jus before ha. Unforunaely, does no seem suable n order o solve he augmened mnmax problem snce he drecon o he k-h parcle does no seem good f he k-h parcle s domnaed. Thus, we deermne he nex search drecon vecor v +1 by he followng equaon (18) usng he drecon o he curren search pon of a nondomnaed parcle K n s rack nsead of he bes search pon of he k-h parcle. (18) v +1 = c 1 R 1 (p x )+c 2R 2 (x K x ) Ths modfcaon leads o move o he drecon o he nondomnaed soluon as well as he resran of he concenraon on local opmal soluons. 5.6 Algorhm of he proposed PSO mehod The algorhm of he proposed PSO mehod based on rpso for mulobjecve nonlnear programmng problems (MOrPSO) s summarzed as follows. Sep 1: Fnd one feasble soluon by rpso n consderaon of he degree of volaon of consrans, and se a basc pon soluon of he homomorphous mappng r. Se := 0, and go o sep 2. Sep 2: Generae he nal search pon x 0 of each parcle whch s feasble or sasfes all consrans, usng he homomorphous mappng [4]. To be more concree, generae N pons n a hypercube [ 1, 1] n randomly, and map hem no he feasble regon usng he homomorphous mappng. Le hese pons obaned by he mappng be nal search pons x 0, =1, 2,...,N and le p0 := x 0, =1, 2,...,N. Then, fnd he bes search pon among p 0, =1, 2,...,N, and use as he bes search pon of he nal swarm p 0 g. Go o sep 3. Sep 3: Deermne he parameer value ω. If he condons of usng he nformaon of nondomnaed parcles are sasfed, calculae he nex search drecon vecor v +1 and he nex search pon x +1 of each parcle usng he move scheme dependng on s suaon based on (17) and (18). Oherwse, calculae v +1 and x +1 of each parcle usng he usual move scheme dependng on s suaon based on (7), (9) and (10). Go o sep 4. Sep 4: For parcles of he subswarm lmed o he feasble regon, f here exs nfeasble ones, hey are repared o be feasble by he bsecon mehod. Go o sep 5. Sep 5: Check wheher all condons for he execuon of he mulple srechng echnque are sasfed or no. If sasfed, go o sep 6. Oherwse, go o sep 7. Sep 6: Calculae S(x +1 ), =1, 2,...,N where S( ) s he evaluaon funcon n he mulple srechng echnque are sasfed and use hem as objecve funcon values nsead of f(x +1 ). Go o sep 7. Sep 7: Calculae f(x +1 ), =1, 2,...,N and use hem as objecve funcon values as usual. Go o sep 8. Sep 8: For each parcle, f x +1 s no domnaed by any curren nondomnaed parcles n he exernal archve, add x +1 o he exernal archve. Then, remove soluons domnaed by x +1 from he exernal archve. Moreover, f x +1 s beer han p n he sense of he objecve funcon f( ) ors( ), updae he bes search pon of he parcle n s rack as p +1 := x +1. Oherwse, le p +1 := p and go o sep 9.

11 PSO FOR INTERACTIVE FUZZY MULTIOBJECTIVE NONLINEAR PROGRAMMING 11 Sep 9: Fnd he bes p +1 bes among p+1, =1, 2,...,N. If p +1 bes s beer han p g n he sense of he objecve funcon f( ) ors( ), updae he bes search pon of he parcle n s rack as p +1 g := p +1 bes. Oherwse, le p+1 g := p g and go o sep 10. Sep 10: For each parcle, apply he secesson operaon. Le := + 1 and go o sep 11. Sep 11: If > T max (he maxmal search me), he search procedure s ermnaed. Oherwse, go o sep 3. Here, he ermnaon condon means ha he search me couner reached o gven maxmal search me T max. An applcaon condon of mulple srechng echnque means ha he bes soluon of he swarm has no been updaed durng a gven perod. 6 Numercal example In order o show he effcency of he proposed PSO (MOrPSO), we consder he followng mulobjecve nonlnear programmng problem. mnmze f 1 (x) =7x 2 1 x2 2 + x 1x 2 14x 1 16x 2 +8(x 3 10) 2 +4(x 4 5) 2 +(x 5 3) 2 +2(x 6 1) 2 +5x (x 8 11) 2 +2(x 9 10) 2 + x mnmze f 2 (x) =(x 1 5) 2 +5(x 2 12) x (x 4 11) x x x4 7 4x 6x 7 10x 6 8x 7 +x (x 9 5) 2 +(x 10 5) 2 mnmze f 3 (x) =x 3 1 +(x 2 5) 2 +3(x 3 9) 2 12x 3 +2x x 2 5 +(x 6 5) 2 +6x (x 7 2)x 2 8 x 9 x 10 +4x x 1 8x 1 x 7 subjec o 3(x 1 2) 2 4(x 2 3) 2 2x x 4 2x 5 x 6 x x 2 1 8x 2 (x 3 6) 2 +2x x 2 1 2(x 2 2) 2 +2x 1 x 2 14x 5 6x 5 x (x 1 8) 2 2(x 2 4) 2 3x x 5x x 1 6x 2 12(x 9 8) 2 +7x x 1 +5x 2 3x 7 +9x x 1 8x 2 17x 7 +2x 8 0 8x 1 +2x 2 +5x 9 17x x j 10.0, j =1,...,10 We apply he orgnal rpso [5], RGENOCOP V [10] and he proposed PSO (MOrPSO) o mnmax problems solved n he neracve fuzzy sasfcng mehod for he above problem. The resuls obaned by hese hree mehods are shown n Table 1. In hese expermens, we se he swarm (populaon) sze N = 70, he maxmal search generaon number T max = 5000, he number of rals s 10. In addon, we use he followng membershp funcons: µ f1 (x) = (1500 f 1 (x))/1420, µ f2 (x) = (3500 f 2 (x))/3300, µ f3 (x) = (3100 f 3 (x))/3050. From Table 1, n he applcaon of rpso [5], we can ge beer soluons n he sense of bes, average and wors han hose obaned by RGENOCOP V [10]. Ths ndcaes ha rpso s beer han RGENOCOP V abou he accuracy of soluons. However, as for he dfference beween bes and wors, ha for rpso s larger han ha for RGENOCOP V,.e., rpso s worse han RGENOCOP V wh respec o he precson of soluons.

12 12 T. MATSUI, M. SAKAWA, K. KATO, T. UNO AND K. TAMADA Table 1: Resuls of he applcaon of hree mehods o augmened mnmax problems. Ineracon 1s 2nd Tme ( µ 1, µ 2, µ 3 ) (1.0, 1.0, 1.0) (0.8, 1.0, 1.0) (sec) Bes MOrPSO (proposed) Average Wors Bes rpso [5] Average Wors Bes RGENOCOP V [10] Average Wors Table 2: Resuls of applcaon of wo mehods o augmened mnmax problems. Ineracon 1s 2nd Tme ( µ 1, µ 2, ) (1.0, 1.0) (0.8, 1.0) (sec) Bes MOrPSO (proposed) Average Wors Bes rpso [5] Average Wors On he oher hand, he resuls obaned by he proposed MOrPSO are beer han hose by rpso n he sense of bes, average, wors, he dfference beween bes and wors. In addon, we appled rpso and MOrPSO o anoher problem whch has 2 objecve funcons, 100 decson varables and 55 consrans. Table 2 shows resuls for he problem. As he resuls for he prevous problem, MOrPSO can oban beer soluons han hose by rpso. Therefore, MOrPSO proposed n hs paper s he bes soluon mehod among hese mehods wh respec o boh he accuracy of soluons and he precson of soluons. 7 Concluson In hs paper, we focused on mulobjecve nonlnear programmng problems and proposed a new PSO echnque whch s effcen for n applyng he neracve fuzzy sasfcng mehod. In parcular, consderng he feaures of augmened mnmax problems solved n he neracve fuzzy sasfcng mehod, we ncorporaed he new search drecon vecor deermnaon scheme based on he nformaon of nondomnaed parcles no rpso. Fnally, we showed he effcency of he proposed MOrPSO by applyng o numercal examples. Acknowledgmen Ths research was parally suppored by The 21s Cenury COE Program on Hyper Human Technology oward he 21s Cenury Indusral Revoluon and he Mnsry of Educaon, Scence, Spors and Culure, Gran-n-Ad for Scenfc Research (C), , 2006.

13 PSO FOR INTERACTIVE FUZZY MULTIOBJECTIVE NONLINEAR PROGRAMMING 13 References [1] R.E. Bellman, L.A. Zadeh, Decson makng n a fuzzy envronmen, Managemen Scence, vol. 17, pp , [2] J. Kennedy and R.C. Eberhar, Parcle swarm opmzaon, Proceedngs of IEEE Inernaonal Conference on Neural Neworks, pp , [3] J. Kennedy, W.M. Spears, Machng algorhms o problems: an expermenal es of he parcle swarm and some genec algorhms on he mulmodal problem generaor, Proceedngs of IEEE Inernaonal Conference on Evoluonary Compuaon, pp , [4] S. Kozel, Z. Mchalewcz, Evoluonary algorhms, homomorphous mappngs, and consraned parameer opmzaon, Evoluonary Compuaon, vol. 7, no. 1, pp , [5] T. Masu, K. Kao, M. Sakawa, T. Uno, K. Morhara Parcle swarm opmzaon based heurscs for nonlnear programmng problems, Proceedngs of Inernaonal MulConference of Engneers and Compuer Scenss 2007, pp , [6] K.E. Parsopoulos, M.N. Varahas, Recen approaches o global opmzaon problems hrough parcle swarm opmzaon, Naural Compung, no. 1, pp , [7] H. Rommelfanger, Fuzzy lnear programmng and applcaons, European Journal of Operaonal Research, vol. 92, pp , [8] M. Sakawa, Fuzzy Ses and Ineracve Mulobjecve Opmzaon, Plenum Press, New York, [9] M. Sakawa, Genec Algorhms and Fuzzy Mulobjecve Opmzaon, Kluwer Academc Publshers, [10] M. Sakawa, K. Kao, T. Suzuk, An neracve fuzzy sasfcng mehod for mulobjecve nonconvex programmng problems hrough genec algorhms, Proceedngs of 8h Japan Socey for Fuzzy Theory and Sysems Chugoku / Shkoku Branch Offce Meeng, pp , [11] Y.H. Sh, R.C. Eberhar, A modfed parcle swarm opmzer, Proceedngs of IEEE Inernaonal Conference on Evoluonary Compuaon, pp , [12] H.-J. Zmmermann, Fuzzy programmng and lnear programmng wh several objecve funcons, Fuzzy Ses and Sysems, vol. 1 pp , Graduae School of Engneerng, Hroshma Unversy 1-4-1, Kagamyama, Hgash-Hroshma, Japan Kosuke Kao TEL: FAX: E-mal: kosuke-kao@hroshma-u.ac.jp