A Fuzzy Model for the Multiobjective Emergency Facility Location Problem with A-Distance

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1 The Open Cybernecs and Sysemcs Journal, 007, 1, A Fuzzy Model for he Mulobecve Emergency Facly Locaon Problem wh A-Dsance T. Uno *, H. Kaagr and K. Kao Deparmen of Arfcal Complex Sysems Engneerng, Graduae School of Engneerng, Hroshma Unversy, Hgashhroshma, Hroshma, Japan Absrac: Ths paper proposes a new mulobecve locaon problem for emergency facles, e.g. ambulance servce saons and fre saons. In he problem, one of he obecves s o mnmze he maxmal dsance of he pahs from he locaed emergency facles o hospals va accdens. The oher s o maxmze he frequency of accdens ha he emergency facles can respond quckly. An neracve fuzzy sasfcng mehod wh parcle swarm opmzaon (PSO) mehod s proposed for fndng a sasfcng locaon for he problem. Compuaonal resuls llusrae he mehod wh numercal examples of he mulobecve locaon problem. INTRODUCTION In hs paper, we consder a new emergency facly locaon problem (EFLP), such ha ambulance servce saons [1-3], fre saons [1, 4, 5], ec. Masuom and Ish [] consdered EFLPs wh he suaon ha f an accden occurs, he neares emergency facly sends ambulances o and nured people are conveyed o he neares hospal. We propose a new mulobecve EFLP by exendng Masuom and Ish s EFLP. EFLPs have he followng wo mporan facors. One s dsance (or norm); for deals of he relaon beween facly locaon and norm, he reader can refer o he sudy of Marn [6]. There are wo norms wdely used n sudes abou he EFLPs. One s he Eucldean norm [7, 8], whch s assumed ha can be raveled o any orenaons a any pons. However, hs assumpon does no usually hold for facly locaon n urban areas. The oher s he block norm [9-11], whch s assumed ha can be raveled o gven several allowable orenaons of movemen wh weghs a any pons. Reclnear dsance [8] s regarded as one of he block norms such ha here are wo allowable orenaons whch cross a rgh angles wh he same weghs. The EFLPs wh he reclnear dsance are ofen suded [7, 1]. The A-dsance defned by Wdmayer e al. [13] s also regarded as one of block norms such ha here are several allowable orenaons of movemen wh he same weghs. Masuom and Ish [] consder an EFLP wh he A- dsance. In hs paper, we propose a new EFLP based on he EFLP wh he A-dsance. The oher s creron of opmaly for facly locaon. In general EFLPs [, 4, 7, 14, 15], an obecve for facly locaon s o mnmze he maxmal dsance beween emergency facles and he scenes of accdens. In hs paper, we nroduce anoher new obecve, whch s o maxmze *Address correspondence o hs auhor a he Deparmen of Arfcal Complex Sysems Engneerng, Graduae School of Engneerng, Hroshma Unversy, Hgashhroshma, Hroshma, Japan; E-mal: unoake@hroshma-u.ac.p frequency of accdens ha emergency facles can respond quckly. Then, a new mulobecve EFLP wh he wo obecves s formulaed. Mos mulobecve EFLPs do no have complee opmal soluons. For fndng a sasfcng soluon of he mulobecve EFLP for he decson maker (DM), we apply neracve fuzzy sasfcng mehod proposed by Sakawa and Yano [16]. Kaagr e al. [17] recenly proposed neracve fuzzy sasfcng mehod for mulobecve fuzzy random lnear programmng problems. In hs mehod, we need o fnd an opmal soluon for each of he mnmax problems wh he correspondng reference membershp values. Parcle swarm opmzaon (PSO), whch s proposed by Kennedy and Eberhar [18] and mproved by Masu e al. [19], s one of he effcen soluon mehods for nonlnear programmng problem. We propose o apply he PSO mproved by ulzng some characerscs of he EFLPs. The organzaon of he paper s as follows. In Secon, we gve he defnon of he A-dsance and s properes. In Secon 3, we formulae a mulobecve EFLP wh he A- dsance. For he formulaed EFLP, frs we propose he mehod o compue he obecve values for each locaon n Secon 4. In order o fnd a sasfcng soluon for he DM, we nroduce he neracve fuzzy sasfcng mehod proposed by Sakawa and Yano [16] n Secon 5. In order o solve he mnmax problems n hs mehod, we proposed a PSO mehod consderng characerscs of he EFLP n Secon 6. In secon 7, we show he resuls for applyng he mehod o numercal examples of he mulobecve EFLPs. Fnally, we make menon of conclusons and fuure remarks n Secon 8. A-DISTANCE In hs secon, we descrbe he defnon of A-dsance and s properes. We consder he suaon ha here are a orenaons whch can only move n he plane R. The orenaons are represened as he angles beween he correspondng sragh lnes o orenaons and he Caresan x-axs; for example, orenaon 0 s he x-axs and orenaon / s X/ Benham Scence Publshers Ld.

2 The Open Cybernecs and Sysemcs Journal, 007, Volume 1 Uno e al. he y-axs. Le A = { 1,..., a } be a se of orenaons such ha 0 1 a <. A lne, a half lne, or a lne segmen s called A-orened f s orenaon s one of hose n A. Then, he A-dsance beween wo pons p 1 and s represened as follows: p R d, p ), f p 1 and p are n an A-orened lne, d A, p ):= (1) mn d p 3 { R A, p 3 ) + d A (p 3, p )}, oherwse where d (, ) means he Eucldean dsance. Fg. (1) shows an example of he A-dsance beween p 1 and p R. The reclnear dsance s represened as he A-dsance wh A = { 0, / }. Fg. (). An A-crcle ( A = { / 3, /,3 / 4} ). The sdes and verces of he Vorono polygons are called Vorono edges and Vorono pons, respecvely. The se of all Vorono polygons, whch can be regarded as a paron of R, s called Vorono dagram wh he A-dsance. Vorono dagram for Q s denoed by VD A (Q). Fg. (1). An example of he A-dsance. Wdmayer e al. [13] show ha f pons p 1 and p are no n any A-orened lnes, here exss a leas one pon p 3 R such ha d A, p ) = d, p 3 ) + d (p 3, p ) () Theorem 1: A Vorono dagram n he A-dsance for Q can be consruced n O ( n log n) mes usng O (n) space, whch s asympocally opmal n he wors case. Proof: Correcness follows from he consderaon n Secon 3 of he reference [13]. Opmaly s due, for nsance, o a reducon of sorng [0]. Fg. (3) shows an example of he Vorono dagram wh A = { 0, / }. For a pon p and a dsance d > 0, he locus of all pons p ' wh d A ( p, p' ) = d s called he A-crcle wh cener p and radus d. As shown n Fg. (), A-crcle has s boundary of a-gon whose corner pons are he nersecons of he crcle wh cener p and radus d and he A-orened lnes hrough p. For wo pons p 1 and p, he bsecor of p 1 and p wh he A-dsance s defned as follows: B p, p ) = { p d ( p, p ) = d ( p, )}. (3) A( 1 A 1 A p Le Q = { p 1,..., p n } be a se of n pons p 1,..., p n R. Then, he Vorono polygon V A ( p ), = 1,..., n, wh he A- dsance s defned as follows: V A (p ) = p d A (p, p ) d A (p, p ) (4) { } Fg. (3). Vorono dagram wh he A-dsance. FORMULATION OF MULTI-OBJECTIVE EFLP In hs secon, we formulae a mulobecve EFLP wh he A-dsance. Le S R be a closed convex polygon n whch accdens occur and he DM needs o locae emergency facles. We consder he suaon ha f an accden occurs a a pon, he neares emergency facly o he pon sends ambulances o he pon and hen nured people n he accden are conveyed from he pon o he neares hospal.

3 Mulobecve Emergency Facly Locaon Problem wh A-Dsance The Open Cybernecs and Sysemcs Journal, 007, Volume 1 3 Frs, we show he mnmax creron abou pah from he emergency facles o he hospals va he pons of accdens. Le h 1,..., h m S be ses of m hospals, le y 1,..., y n S be ses of n emergency facles, and le n Y = ( y 1,..., yn ) S. Then, f an accden occurs a a pon p S, he A-dsance for he above pah s represened as follows: u(y, = mn d A (y, + mn d A (p, h ) (5) =1,...,n =1,...,m Because he DM does no know where accdens occur n S beforehand, one of our obecves s nerpreed as copng wh any accden pons n S quckly. Then, he frs obecve funcon s represened as follows: f 1 (Y ):= max u(y, (6) ps Secondly, we show a new creron abou frequency of accdens. We assume ha he DM knows pons where accdens frequenly occur n S, called accden pons. There are k accden pons whose ses are denoed by a 1,...,a k S, and each of her accden pons has a wegh abou frequency of accdens, denoed by w 1,..., w k > 0, respecvely. Le > 0 be an upper lm of he dsance from he emergency facles o he hospals such ha a medcal reamen for nured people can be n me. The oher of our obecves s nerpreed as maxmzng he sum of he weghs of frequency of accden pons ha he emergency facles can cover for a gven. Then, he second obecve funcon s represened as follows: f (Y ):= (7) w I (Y ) where I (Y ):= a u(y, { } (8) Therefore, a mulobecve EFLP s formulaed as follows: mnmze f 1 (Y ) maxmze f (Y ) subec o Y = (y 1,..., y n ) S n COMPUTATION OF THE OBJECTIVE VALUES OF MULTIOBJECTIVE EFLP In order o fnd an opmal soluon of (9), we need o compue he wo obecve values for each soluon. For he second obecve funcon, we can compue s obecve value by measurng he A-dsances from he emergency facles o he hospals va he k accdens pons. In he followng par of hs secon, we propose he mehod o compue he frs obecve value for each locaon. For he case n = 1, Masuom and Ish [] showed he followng heorem. Theorem : If n = 1, p S whch maxmzes u ( Y, s one of he followng pons: (9) Verces of he boundary of S, Inersecons of Vorono edges of each V h ),..., ( h ) and he boundary of S. If he DM locaes wo or more emergency facles, we need o consder whch of he emergency facles s used a any pons n S. From he defnon of Vorono polygon, V A ( y ) s he se of pons whch uses emergency facly. Whle p S whch maxmzes u ( Y, s only on he boundary of S f n = 1, p S whch maxmzes u ( Y, may be n he neror of S and on he Vorono edges of V A ( h 1 ),..., V A ( h m ) and V A ( y 1 ),..., V A ( y m ) f n. Then, Theorem 1 can be exended o he followng corollary: Corollary 1: If n, p S whch maxmzes u ( Y, s one of he followng pons: Verces of he boundary of S, Inersecons of Vorono edges of each V h ),..., ( h ) and he boundary of S, Vorono pons of each V h ),..., ( h ), Vorono pons of each V y ),..., ( y ), Inersecons of Vorono edges of each V y ),..., ( y ) and he boundary of S, Inersecons of Vorono edges of each V h ),..., ( h ) and Vorono edges of each V A ( 1 ),..., V A ( y m y ). The above pons can be found by drawng Vorono dagram for hospals and Vorono dagram for each locaon of emergency facles. Then, we can fnd he frs obecve value by compung he maxmal dsance for pahs from emergency facles o hospals va hese pons. INTERACTIVE FUZZY SATISFICING APPROACH In hs secon, we nroduce he neracve fuzzy sasfcng mehod proposed by Sakawa and Yano [16] n order o fnd a sasfcng soluon of (9) for he DM. For decson makng n real world, he DM usually prefer o make an obecve funcon value more/less han a ceran value raher han o maxmze/mnmze s obecve funcon value. Such an obecve, called a fuzzy obecve, ncludes vagueness based upon udgmen of he DM. In hs paper, we represen he wo obecves of (9) as fuzzy obecves provded by membershp funcons, denoed by μ 1 and μ. Now we nroduce an example of membershp funcons for each obecve funcon. Le d e denoe he dsance such ha he DM s que sasfed f he frs obecve value s less han d e, and d l denoe he dsance such ha she/he s sasfed o a ceran degree f s obecve value s more han d bu less han e

4 4 The Open Cybernecs and Sysemcs Journal, 007, Volume 1 Uno e al. d l. Then, we use he followng lnear membershp funcon for he former obecve: 1, f f 1 (Y ) < d e, f μ 1 ( f 1 (Y )) := 1 (Y ) d e, f d e f 1 (Y ) < d l, d l d e 0, f d l f 1 (Y ) (10) Nex, one of he smples ways o provde membershp funcon for he laer obecve s as follows: μ ( f (Y )) := f (Y ) (11) k w =1 Then, (9) s ransformed as he followng mulobecve fuzzy programmng problem: mnmze μ 1 ( f 1 (Y )) maxmze μ (f (Y )) subec o Y = (y 1,..., y n ) S n (1) Snce here s generally no complee opmal soluon for mulobecve programmng problem ncludng (1), he concep of he M-Pareo opmal soluon s usually used for mulobecve fuzzy programmng problems. Defnon 1: Soluon Y * s an M-Pareo opmal soluon of (1) f and only f here does no exs any soluons n S Y such ha μ ( f ( Y )) μ ( f ( Y*)) for all = 1, and μ ( f ( Y )) > μ ( f ( Y*)) for a leas one {1, }. The neracve fuzzy sasfcng mehod [16] s o fnd a sasfcng M-Pareo opmal soluon hrough neracon o he DM. Le ( μ 1, μ) be a par of nal reference membershp levels of membershp funcon μ 1 and μ, respecvely. The neracve fuzzy sasfcng mehod for (1) can be descrbed as follows: Algorhm 1: Ineracve Fuzzy Sasfcng Mehod Sep 1. Provde wo membershp funcons μ 1 and μ accordng o (10) and (11). Sep. Se he nal reference membershp levels μ, μ ) (1,1). Sep 3. ( 1 = For he gven par of reference membershp levels ( μ 1, μ), solve he followng correspondng mnmax problem: mnmze max{μ μ ( f (Y )) =1, + (μ μ ( f (Y )))} =1 subec o Y = (y 1,..., y n ) S n where, s a suffcenly small posve number. Sep 1. (13) If he DM s sasfed wh he curren levels of he M-Pareo opmal soluon, STOP. Then he curren M-Pareo opmal soluon s a sasfcng soluon for he DM. Sep. Updae he par of curren reference membershp levels ( μ 1, μ) based on nformaon of preference of he DM, he curren values of he membershp funcons, ec. Reurn o Sep 3. In he neracve fuzzy sasfcng mehod, we need o solve he mnmax problems n Sep 3 effcenly. In he nex secon, we propose an effcen soluon mehod for he mnmax problem. A SOLUTION ALGORITHM FOR MINIMAX PROB- LEM A PSO mehod proposed by Kennedy and Eberhar [18] s based on he socal behavor ha a populaon of ndvduals adaps o s envronmen by reurnng o promsng regons ha were prevously dscovered [1]. Ths adapaon o he envronmen s a sochasc process ha depends upon 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 he followng hree arbues: he poson vecor n he search space, he velocy vecor and he bes poson n s rack, and he bes poson of he swarm. The process can be oulned as follows. Algorhm : Oulne of he PSO Mehod Sep 1. Generae he nal swarm nvolvng N parcles a random. Sep. Calculae he new velocy vecor for each parcle, based on s arbues. Sep 3. Calculae he new poson of each parcle from he curren poson and s new velocy vecor. Sep 4. If he ermnal condon s sasfed, STOP. The bes soluon gven n he searchng hsory s an approxmae opmal soluon. Oherwse, go o Sep. To be more specfc, for he poson and he velocy vecor of he -h parcle a me, denoed by x and v, respecvely, he new velocy vecor of he -h parcle a me + 1 s calculaed by he followng scheme nroduced by Sh and Eberhar []. v + 1 : 1 1 g where me, and = v + c R ( p x ) + c R ( p x ) (14) R 1 and R are random numbers beween 0 and 1 a p s he bes poson of he -h parcle n s rack p g s he bes poson of he swarm. There are hree problem-dependen parameers, he nera of he parcle, and wo rus parameers c 1 and c. Then, he new poson of he -h parcle a me + 1 s calculaed from he followng equaon: + 1 : + 1 x = x + v (15)

5 Mulobecve Emergency Facly Locaon Problem wh A-Dsance The Open Cybernecs and Sysemcs Journal, 007, Volume 1 5 The -h parcle calculaes he nex search drecon vecor v by (14) n consderaon of he curren search drec- +1 on vecor v, he drecon vecor gong from he curren search poson x o he bes poson n s rack p and he drecon vecor gong from he curren search poson x o he bes poson of he swarm p g, moves from he +1 curren poson x o he nex search poson x calculaed by (14). The parameer conrols he amoun of he move o search globally n early sage and o search locally by decreasng gradually. I s defned by follows: poson of he swarm. Moreover, hey proposed he mulplex srechng mehod, whch s he exended verson of he srechng mehod proposed by Parsopoulos and Varahas [3]. In order o sele he second problem, Masu e al. [19] proposed o generae nal parcles n he feasble se by ulzng he homomorphsm proposed by Kozel and Mchalewcz [4]. Moreover, here are ofen cases ha a parcle afer move s no always nfeasble f we use he updang equaon of search poson menoned above. To deal wh such a suaon, Masu e al. [19] dvded he swarm no wo subswarms; one s he move of a parcle o he nfeasble regon s acceped and he oher s no. 0 T max 0 ( ) : = (16) 0.75T max 0 where T max s he number of maxmum eraon mes, Tmax s an nal value a he me eraon, and s he las value. The searchng procedure of PSO s shown n Fg. (4). Fg. (4). Movemen model for PSO. Comparng he evaluaon value of a parcle afer +1 movemen, denoed by f ( x ), wh ha of he bes poson n s rack, denoed by f ( p ), f f ( x ) s beer +1 han ) f ( x f ( p ), hen he bes poson n s rack s updaed as f ( p ). Oherwse, hen he bes poson n he + 1 swarm s updaed as f ( p ) f ( p ). In hs way, a parcle ges nformaon of he bes poson of new oneself and he swarm, and moves accordng o (14) and (15) agan, and searches newly. The summary of he PSO mehod s shown n Fg. (5). Such a PSO echnque ncludes wo drawbacks. One s ha parcles concenrae on he bes search poson of he swarm and hey canno easly escape from he local opmal +1 soluon snce he move drecon vecor v calculaed by (14) always ncludes he drecon vecor o he bes search poson of he swarm. Anoher s ha a parcle afer move s no always feasble for problems wh consrans. In order o sele he frs ssue, Masu e al. [19] proposed he leavng acs for parcles whch are on he bes Fg. (5). Summary of PSO algorhm. IMPROVEMENT OF PSO METHOD In hs sudy, based upon he PSO mehod nroduced n he prevous secon, we proposed o add a new fourh movemen n (14). Le q S be maxmzer of u ( x,, such a pon can be found by usng Theorem 1. In (9) approachng he neares emergency facly o q, obecve value of f 1( Y ) may be mproved. Therefore we nroduce such a movemen o our soluon mehod. Le p a be a poson such ha for x, a se of he neares facly s changed o q and ses of he oher facles are fxed. Then, our proposng new velocy vecor of he -h parcle a me + 1 s represened as follows: v +1 := v + c 1 R 1 (p x ) + c R (p g x ) +c 3 R 3 (p a x ) where c 3 s a rus parameer and beween 0 and 1. NUMERICAL EXPERIMENTS (17) R 3 s a random number In hs secon, we apply neracve fuzzy sasfcng approach wh he PSO mehod o an example of our proposng mulobecve EFLPs. In hs example, we consder an EFLP

6 6 The Open Cybernecs and Sysemcs Journal, 007, Volume 1 Uno e al. for wo emergency facles, ha s n =. We represen S as a convex hull ncludng 0 pons gven n [ 0,100] [0,100] randomly. For he A-dsance, we se A = { 0, / 4, /,3 / 4}. For hospals, we se m = 3 and her ses are gven n S randomly. For he frequency of accdens, we se = 15, and for each of 100 accden pons, s se and wegh are randomly gven n S and (0,1], respecvely. We llusrae he neracve fuzzy sasfcng approach for he above example of he mulobecve EFLP. For parameers of PSO, we use he same parameer as he sudy of Masu e al. [19], ha s, s populaon sze s 40, s generaon s 500, and c 1 = c =. Moreover, we se he new parameer c 3 = A Sep 1, n order o represen fuzzness abou wo obecves, we use membershp funcons (10) and (11) n Secon 5 wh seng d = 5 and d = 10. e A Sep 3, we solve mnmax problem (13) for each gven μ, ) by solvng he PSO mehod n secon 6. We se ( 1 μ = In order o verfy effcency of PSO mehod, we apply he soluon mehod GENOCOP [4], whch s a genec algorhm for numercal opmzaon for consraned problem proposed by Kozel and Mchalewcz [4], o mnmax problems. For parameers of GENOCOP, we se he same populaon sze and generaon. Moreover, we se he oher parameers smlarly o he sudy of Kozel and Mchalewcz [4]. Compuaonal resuls for each gven ( μ 1, μ) a 0 mes by PSO and GENOCOP are shown n Tables 1 and, respecvely, where hese resuls are gven by usng DELL Opplex GX60 (CPU: Penum(R) 4.33GHz, RAM: 51MB). Table 1. Compuaonal Resuls by PSO l Table. Compuaonal Resuls by GENOCOP Mnmax Problem 1 3 μ μ Bes Mean Wors Mean CPU Tme (Sec) curren reference membershp levels ( μ 1, μ) by consderng he curren values of he membershp funcons, and resolve mnmax problem o Sep 3. In hs example of EFLP, we assume ha he DM hnks ha he frs obecve s more mporan han he second obecve. Then he DM hopes o mprove he value of μ 1 even f he value of μ s changed for he worse. However, he DM does no hope o go he value of μ oo bad. Then, an example of he neracve fuzzy sasfcng mehods s shown n Table 3. Table 3. Resuls of Ineracve Fuzzy Sasfcng Approach Ieraon 1 3 μ μ μ 1 ( f 1 (Y*)) μ ( f (Y*)) Mnmax Problem 1 3 μ μ Bes Mean Wors Mean CPU Tme (Sec) From Tables 1 and, PSO can fnd beer soluons han GENOCOP by meanngs of boh mean and sably. Ths means ha effcency of PSO for such mnmax problems. A Sep 4, he DM evaluaes wheher he M-Pareo opmal soluon gven by solvng he mnmax problem a Sep 3 s sasfed or no. If s soluon sasfes he DM, hs algorhm s ermnaed. Oherwse, ask he DM o updae he In Table 3, he DM consders as follows: A Ieraon 1, he DM s no sasfed M-Pareo opmal soluon because he frs obecve value s bad. A Ieraon, for mprovng he frs obecve value, he DM decreases μ, whch s reference membershp levels for he second obecve. Then, he M- Pareo opmal soluon gven a Ieraon s mproved for he frs obecve. However, she/he s no sasfed because he second obecve value s oo bad. A Ieraon 3, for mprovng he second obecve value a lle, he DM decreases μ 1, whch s reference membershp levels for he frs obecve. Then, she/he obans a sasfcng soluon for boh wo obecves, so he algorhm s ermnaed. CONCLUSIONS AND FUTURE STUDIES In hs paper, we have proposed a new mulobecve EFLP wh he A-dsance. In order o oban a sasfcng

7 Mulobecve Emergency Facly Locaon Problem wh A-Dsance The Open Cybernecs and Sysemcs Journal, 007, Volume 1 7 soluon for he DM, we have proposed an neracve fuzzy sasfcng mehod whch nvolves he procedure of solvng mnmax problems by he PSO mehod. By applyng an example of mulobecve EFLPs, we showed effcency of PSO and llusraed he neracve fuzzy sasfcng mehod. In he mulobecve EFLPs, we assume ha he regon of facly locaon S s a convex polygon because Theorem and Corollary 1 use he assumpon. However, n order o apply he EFLPs o more general cases, we need o consder varous shapes of S whch are non-convex, non-conneced, ec. To consruc our soluon mehod for general shapes of S s a fuure sudy. Moreover, f S ncludes many hospals and he DM locaes many emergency facles n S, we need o fnd an opmal soluon more effcenly for he mnmax problems n he neracve fuzzy sasfcng mehods. To consder an effcen soluon mehod for large-scale mulobecve EFLPs s also a fuure sudy. REFERENCES [1] C. Araz, H. Selm, and I. Ozkarahan, A fuzzy mul-obecve coverng-based vehcle locaon model for emergency servces, Compu. Opera. Res., vol. 34, no. 3, pp , March 007. [] T. Masuom and H. Ish, Mnmax locaon problem wh A- dsance, J. Opl. Res. Soc., vol. 41, pp , June [3] H. K. Raagopalan, C. Saydam, and J. Xao, A mulperod se coverng locaon model for dynamc redeploymen of ambulances, Compu. Opera. Res., vol. 35, no. 3, pp , March 008. [4] D. R. Plane and T. E. Hendrc, Mahemacal programmng and he locaon of fre companes for he Denver fre deparmen, Opns. Res., vol. 5, pp , July-Augus [5] L. Yang, B. F. Jones, and S. H. Yang, A fuzzy mul-obecve programmng for opmzaon of fre saon locaons hrough genec algorhms, Eur. J. Oper. Res., vol. 181, no., pp , Sepember 007. [6] H. Marn and A. Schöbel, Medan hyperplanes n normed spaces - a survey, Dscree Appl. Mah., vol. 89, pp , December [7] J. Elznga and D. W. Hearn, Geomercal soluons for some mnmax locaon problems, Trans. Sc., vol. 6, pp , November 197. [8] S. Özen and S. Güne, Effec of feaure-ype n selecng dsance measure for an arfcal mmune sysem as a paern recognzer Dgal Sgnal Processng (In Press.) [9] B. Pelegrn and F. R. Fernandez, Deermnaon of effcen pons n mulple-obecve locaon problems, Navel Res. Log., vol. 35, pp , December [10] J. F. Thsse, J. E. Hendrc, and R. E. Wendell, Some properes of locaon problems wh block and round norm, Opns. Res., vol. 3, pp , November-December [11] J. E. Ward and R. E. Wendell, Usng block norm for locaon modelng, Opns. Res., vol. 33, pp , Sepember-Ocober [1] G. O. Wesolowsky, Recangular dsance locaon under he mnmax opmaly creron, Trans. Sc., vol. 6, pp , February 197. [13] P. Wdmayer, Y. F. Wu, and C. K. Wong, On some dsance problems n fxed orenaons, SIAM J. Compu., vol. 16, pp , Augus [14] R. L. Francs, A geomercal soluon procedure for a reclnear mnmax locaon problem, AIIE Trans., vol. 4, pp , December 197. [15] D. K. Kulshresha, A mn-max locaon problem wh demand pons arbrarly dsrbued n a compac conneced space, J. Opl. Res. Soc., vol. 38, pp , May [16] M. Sakawa and H. Yano, An neracve fuzzy sasfcng mehod usng augmened mnmax problems and s applcaon o envronmenal sysems, IEEE Trans. Sys. Man Cybern., vol. SMC-15, pp , November-December [17] H. Kaagr, M. Sakawa, K. Kao, and I. Nshzak, Ineracve mulobecve fuzzy random lnear programmng: Maxmzaon of possbly and probably, Eur. J. Oper. Res., (In Press.) [18] J. Kennedy and R. C. Eberhar, Parcle swarm opmzaon, n Proc. of IEEE In. Conf. Neural Neworks, Pscaaway, NJ, pp , [19] T. Masu, M. Sakawa, T. Uno, K. Kao, M. Hgashmor, and M. Kaneko, Parcle swarm opmzaon for ump hegh maxmzaon of a seral lnk robo, Journal of Advanced Compuaonal Inellgence and Inellgen Informacs, vol. 11, no. 8, pp , Ocorber 007. [0] F. Hwang, An O(n log n) algorhm for reclnear mnmal spannng rees, J. Assoc. Compu. Mach., vol. 6, pp , Aprl [1] 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, n Proc. of IEEE In. Conf. Evoluonary Compuaon, Anchorage, Alaska, [] Y. Sh and R. C. Eberhar, A modfed parcle swarm opmzer, n Proc. of IEEE In. Conf. Evoluonary Compuaon, Anchorage, Alaska, [3] K. E. Parsopoulos and M. N. Varahas, Recen approaches o global opmzaon problems hrough Parcle Swarm Opmzaon, Na. Compu., vol. 1, pp , June 00. [4] S. Kozel and Z. Mchalewcz, Evoluonary algorhms, homomorphous mappngs, and consraned parameer opmzaon, Evol. Compu., vol. 7, no. 1, pp , Sprng Receved: Ocober 5, 007 Revsed: Ocober 9, 007 Acceped: November 8, 007