A Cooperative Multilevel Tabu Search Algorithm for the ... - CIRRELT

1 downloads 0 Views 209KB Size Report
A Cooperative Multilevel Tabu Search. Algorithm for the Covering Design Problem. Chaoying Dai, (Ben) Pak Ching Li and Michel Toulouse. Department of ...
      "!$#% & ')(  *+, - (. /0 1 (23 -542+768 9:5+4)?A@0BDCFEHGJIK@0EMLONQPSR5GUTWV @0XY>)?ZE[GZI\]E @^GJ_2`E[a?JR5b]cdB^eZb[B^eJfMR gWhˆOiFjn"kmloj5n&y€‰‹ŠhŒpDxl ŽFqj5rdqƒ^s:yzqpun&^|Fiu‰zt^yziFlohvj5k)‘’x}w^|^xloyzhq}p{tux‰zqh}tu|{m~Sh5p{“5”Sy€}xhvko•mtuy‚l„n"ƒ"jqp{rdy‚lo…†q}‡Fj5j0pu•y‚xlojq}‡uj x -–,qŽu—™’yzhv˜xMkošvŽyz›5puhvœu¥^}iuš}¢{‰zžhqmkoŸ yzh}OŽup¤yzm¡hi{’ kohMq}q5k‡{koj‘‰z‰uhn†¢{‰zhh}• mhxvŸ ‰zkodŽuyz‡q5h&hrZOmq5h}j5jhvpkokoxMj‰€ŽYj¢uiujhi{i^puyzlpuj5yzWpuloyzmh&q}hpjyzkoqxMj,rFŽ&n£lmjtuiu‡{‰‚jlot¤xyz‰zhvhmd’hq5jhrJ‰DkoxMmjSŽ2h¦^jjko§xM‰z}Ž¨‹q5©Flokoªoq)yz«loloŽ{ªŽ{¬n h ­ xxqq’’hvhvkokoyzyzpupuYY¢{¢{hhmmyzyzppu&i{yzkop´q‡uh‰zjhxn†Ž´• mŸ hjj5ko‡uxMt®Ž®mmhiFj5j5koxxh}Ž°•d¯Fµ:pF‰zqD¢^x2‘^³±„rHqDkoq0q}¢{n1²loj5Ž{iuq}i^mkoh¶q¥^j5yziun"i^j5koqlo¥^yzqyzpun"³j5loq5h r mmq}hj5‰zt{koloxyzŽ°q}p{m†iFjjxvkohh%• lmkŸ j5Župuh·r¸hM¢^kmƒ^kopFh¢¹j5n&loy€qxv£qloq5ŽurShMxk†qDq‰zhi’hvhk‰zj5lo|yzqkop¹h¢{hvj¯un&puqyzpupu2¶‰zlohŽ{’h%hv‰€xv£qlokmhkopFhm¢^i£qlopFq¶¢{yzkop{hv º ŽFkojq}"t{‡ihhvqDp½q0¢¼Dtuj5y‚iuloh¶i{kohvq¾Z¥^hyzn"xvloj5yz’loh%h»mjq}l,‰zt{¯ulopFyzq¢^puy€p{-¤yzp0kolohvqºQxmqn"nj‡u‰z‰yzpumhj5jlokoyzq}xMŽp{,miFq5j5rx‡uh‰zqD• xMŸ ‘0j"‡{yzt,p½mhmn"j5kojxŽ‰z‰ mmhiFj5jkoxvxhŽ†• miFj5xh  -ŽuyzxŽ†i{koq0y€¢{hWmt{xxhm·r[tu‰Amhj5koxŽ¶¢{y‚kohxMloy€qpu»yzp‰€j5kohvkSmhj5koxŽ ¿ R5CFÀSB^Á_ZfÂ`°eZb€ÃvEHb[R5Ä{R5bu@0b[I^B^ÁvEHÃM?ZŶfL>)BDÄ^R5ÁvE[GZIS_ZRfMEHI{GÆJÁMB{ÇZbHRÅ2L5cd@^ÇZe&fR}@0Áa?£Å†R5Ãv@DÈ ?ZReZÁME¸fÃME¸a0É Ê Ë{Ì)͍ÎZÏÐÑÒUÍZÓÏ»Ì ÔÖÕO× NQØAÙÚ¡ÙvÛUT,ÜÝ0ޒß5àá‹âuã¤ä^ßåá€ã0â¶E¸fW@ÆA@0E[Á³NQæ°ÙMçTLÀ?ZRÁMRæèE¸f@fR5ÃWB^édfMEHêRØULZa5@^bHb[R_ ëÕ ÈmÝDfá‹eZâJÇAìífåR5ó@0GJB^_îé»æôçèE¸E¸f&f¶a@B{GuaÃvB^@0b[b[E[RGZaR}Ãv_°EHB{E[G G´B^@Dé³Ã&Úub[RÈm@{feJfoÃ"ÇJfÛR5ÃvÇZfb[BFB0aé£XæïfB0L éWa5ç@0b[b[É]R\O_ñRÃ"ðòHÝ}õSÜöUó’NQåØAL-ÙfMÚ¡eJÙ aÕ ? T_Ãv?JRGZ@DöB0ÃvRRÄ^ÃvR5?ZÁvCR ņEHGJEHÅeZÅ*GFeZÅÇARÁ,B^éSÇZb[BaXf³EHGï@^GuC Õ»× N÷ØUÙvÚ¡ÙÛAT&a5BDÄ^RÁME[GZI2_RfMEHI{G]É ÔøÕ)× NQØAÙÚ¡ÙvÛUT aBDÄ{R5ÁvEHGJI2_ZRfMEHI{GE¸f%Ý ë ì„á‹ù†ú0òE€éEHÃ,?J@^fõ ö NQØAÙÚ¡Ù Õ T³ÇJbHBaXfûHü}ý}þ·ÉdcW?JR´ÜÝDÞDßàá‹âFãä{ßåá€ã^â ë_R}àvÝ^fE[ð5I^òHßG°ùÿÆJÁME¸B{fÇZÃvbHR?ZÅ RÆZ?JÁM@^B{fÇZ@0b[R5ÆZÅÆJbHE¸B0a5é,@0ÃM_E[B^R5GJÃMR5f£ÁvEHņG´EHGJbHB^EHGZÏÃvI R5ÁvÃMC2?ZR´_ZRÄDfM@0EHI{b[eZG]R®LU_JB^@DéÃv@Yõ ö aB{NQØAÅÙÆJÚ¡ÁMÙ R}Õ TfMfMÉSEHB{cWG°?ZR@0GJa_®BDÄ{R5R5ÁvÁvÁMEHB{GZÁI È ÃMÁ@0ÆJÆZEHGJI†_ZRaB_E[GZIû þ·É cW?ZRÄD@0b[eZR´õ ö N÷ØUÙvÚ¡Ù Õ T¶a5@^GÇAR®_ZRÃMRÁMņE[GZR_eAfE[GZIï@0GR @{aÃ%fMR@^Áva?î@0b[I^B^ÁvEHÃM?ZÅ2É GZé÷B^ÁMÃMeZGJ@0ÃMRbHC{L^feAa?@^bHI{B^ÁvE€Ãv?ZŶf:@^ÁMRSE[GZR ¡RaÃME[Ä^RWé÷B^Á-@^bHbÇJeÃ»@£é÷R5ÀfR5Ã-B0éAÆA@0Á@0ņRÃvR5Áf5L _eZR†ÃvB2ÃM?ZR¶R UR}aÃf&B0éWaB^ÅÇZE[GJ@DÃvB^ÁvE[@^b:R ÆZbHBufE[B^GOÉ]cW?ZR5ÁvRé÷B{ÁMR{L]fR}@0Áa?®?ZReZÁME¸fÃME¸a5f³Å¶@’C ÇUR"@ÄFE¸@0ÇZb[R&B^ÆZÃME[B^GYé÷B^ÁE[ÅÆJÁMBDÄFE[GZIeJÆZÆARÁ ÇUB^eZGJ_JfB^GõWöANQØAÙÚ¡Ù Õ TÉ GÃv?ZE¸fWÆJ@0ÆUR5Á}LFÀ)REHGuÃvÁMB_eJa5R@aBFB{ÆARÁv@0ÃME[Ä^RÅeZbHÃME[bHRÄ^R5b™fMR@^Áva??JR5eZÁvE[fÃME¸a£ÅR5ÃM?ZB_ ÃMBEHņÆZÁvBDÄ^ReJÆZÆARÁ»ÇUB^eZGA_Zf»B{G%õWöANQØAÙÚ¡Ù Õ TÉ Ô fMfMeZņR ÀSR@0ÁvRb[BuB{XFEHGZI"é÷B^Á)@ ÕJ× NQØAÙÚ¡ÙvÛUT aBDÄ{R5ÁvEHGJI³_ZRfMEHI{G¶B0é SÇZb[BaXFfÉ{\ORà ÇURÃv?ZR£fMRÃSB0é]@0b[bUÚuÈmfMeZÇJfMRÃvf»EHG¶æ @^GJ_†b[Rà "ÇUR"Ãv?ZR,fMB^b[eÃME[B^G°fMÆJ@^a5R³é÷B^Á Õ × NQØAÙÚ¡ÙvÛUTÉ Ô fvfMeZņR Ù Ù ]Ù @0ÁvR,fMeZÇJfMRÃfB0é fMeJa?2ÃM?J@0Ã É »@{a?2fMeZÇJfMRà _R JGZRf³@YfR}@0Áa?°fÆJ@{aR B{G Õ × N÷ØUÙvÚ¡ÙÛATE[G°ÃM?JRfv@0ņR,ÀW@’C2@^f É Ô Ã@0ÇZefR}@0Áa? @0b[I^B{ÁMEHÃM?JÅ R ÆZbHB{ÁMR}f»E[GJ_RÆARGJ_RG{ÃvbHC¶R}@^a?¤fR}@0Áa?fÆJ@{aRÃMB†fRR5X¶fR5ÃvfSB0é ÇZb[BaXFf»ÃM?A@Dà 























    !"$#

&%

,+-.,+0/(%1"22*2,.&%34  

 

3 6

7



(' *))*)

,+

(6

  5 8 

9

a5BDÄ^R5Á@{f¶Å¶@0GFC Õ ÈmfMeZÇJfMRÃvf%@^f¶ÆUB{fvfE[ÇZb[R^ÉScW?ZR®aBFB{ÆARÁv@0ÃME[Ä^R2Å,eZbHÃME[b[R5Ä^RbfMR@0Áa? foÃvÁv@0ÃÈ RI^C½aB{GJfE¸fÃvfB0éfMeZÇJfÃMEHÃMeÃvEHGZI¹fMB^ņR%ÇZb[BaXf,B^é ÃM?ZR2fMņ@^bHb[RfÃfMR@^Áva?ïfMÆJ@{aRfÇFC½fR5Ãvf B^é I^BFB_½ÇZb[BFaXf_E¸fMa5BDÄ^R5ÁvR_¹ÇFCÃv@^ÇZe fMR@^Áva?OÉdcW?ZE¸fR Za?J@0GJI^RB^éÇZb[BFaXfR5Ä{R5GuÃMeJ@^bHb[C ÇZÁvE[GZI{f a5B^ÅÇZEHGA@DÃME[B^GAfB0éI^BFB_2ÇZb[BFaXfE[G°fMņ@^bHbdfR}@0Áa?¤fMÆJ@^a5RfÀ?JR5ÁvR&Ãv@^ÇZe°fR}@0Áa?YE¸f eZEHÃMR R UR}aÃvEHÄ{R JGJ_E[GZI,a5BDÄ^RÁME[GZI£é÷B^Á»b¸@0ÁvI^RWGFeZÅÇARÁ-B0é Õ ÈmfeZÇAfR5ÃvfÉ R ?J@’Ä^RSÃvRfÃMR_B^eZÁ @^bHI{B^ÁvE€Ãv?ZÅ B{GaBDÄ^RÁME[GZI&_R}fE[I^G¶ÆZÁvB^ÇJbHRņf»ÀE€Ãv?¶ÃME[I^?uûIu@0ÆJf-ÇAR5ÃoÀ)RR5Gb[BDÀ)RÁ-@0GJ_eZÆZÆUR5Á ÇUB^eZGA_Zf¶û ZL Dþ·É]cW?ZB{fMR@^ÁMR,ÃM?ZR¶Å†BufoÃ&_E ¶a5eZbHÃ"eZÆZÆUR5Á&ÇUB^eZGJ_JfÃMB¤EHņÆZÁvBDÄ^R{É R†À)RÁMR @^ÇZbHRÃMB JGJ_XFGZBDÀG%eZÆZÆUR5ÁWÇAB{eZGJ_Zf)é÷B{ÁS@^bHbAÃM?ZRÆZÁvB^ÇZb[R5ÅE[GJfÃv@0GAaRf»ÃMR}foÃvR_Y@0GJ_†é÷B{eZGJ_ GZRÀ eZÆJÆARÁ ÇAB{eZGJ_Zfé÷B^Á fMR5Ä^RÁv@^b¡B0éOÃv?ZR5Å2É cW?ZR®fMeZÇJfMR eZRGuÃ%fR}aÃME[B^GAf¶B0éÃM?ZE¸fÆA@0ÆUR5Á%@^ÁMRB{ÁMIu@0GZE[ê5R}_ @^f¶é÷B^b[bHBDÀ f É FR}aÃME[B^G ý ÆZÁvBDÄFE[_ZRf ÇA@^aXFI^ÁvB^eZGA_YE[Gé÷B{ÁMŶ@DÃvEHB{GB^GÃv?ZR†aBDÄ^RÁME[GZI_R}fE[I^G®ÆZÁMB{ÇZb[R5Å2É G´fR}aÃME[B^ G ZL ÀSR"_RfvaÁvE[ÇAR³Ãv?ZR&Å,eZbHÃME[b[R5Ä^RbOÆJ@^Áv@{_E[I^Å2 É FRaÃME[B^G fMeZņņ@^ÁME[ê5R}fSÃM?ZR,EHņÆZb[R5ņR5GuÃ@DÃME[B^G B^é)B^eJÁ³a5BFB^ÆUR5Á@DÃME[Ä^RÅ,eZbHÃME[b[R5Ä^Rb @0b[I^B^ÁvEHÃM?ZÅ2É FRaÃME[B^G ÁvR5ÆUB^ÁMÃvfR FÆUR5ÁvE[ÅRGuÃv@0bÁvRfMeZbHÃvf @^GJ_%ÀSR&aB{GJab[eJ_R&E[G FRaÃME[B^G ZÉ    Ò -ÎZϻѣÌÐ G¹Ãv?ZE[ffR}aÃvEHB{G½À)RÆJÁMBDÄFE¸_R@°fM?ZB^ÁMÃ"ÇJ@^aXFI{ÁMB{eZGJ_´B^GïaBDÄ{R5ÁvEHGJI2_ZRfMEHI{GJf"@0GA_½fR}@0Áa? ?ZReZÁvE[fÃME¸a5fSé÷B{Á aBDÄ^RÁME[GZI_RfMEHI{GJfÉ 



87

7











 "!#%$'&)(+*-,.#"/0&)*1(./

Wc ?ZR¶fÃMeJ_ZC°B0éWaBDÄ^RÁME[GZI%_ZRfMEHI{GJf³ÇARI{@^G´@0ÁvB^eJGJ_Ãv?ZR†R5GJ_´B0é)ÃM?ZRü2 3 4 fÉ¡cOeZÁ@05 GîNQfMR5R û þ÷TSÀS@{fWB^GZR³B0édÃM?ZR AÁvfÃÁMR}fR}@0Áa?ZR5Áf ÃvB†fÃMeA_CYa5BDÄ^R5ÁvE[GZI_R}fE[I^GAf5É FE[GJa5R£Ãv?ZR5GOLZŶ@0GFC ÁvRfMR@^Áva?ZRÁvfU?J@’Ä{R-foÃveJ_E[R_&a5BDÄ^R5ÁvE[GZI_RfMEHI{GJf™é÷ÁvB^ÅÖÄD@0ÁvEHB{eJf™_E[ÁMR}aÃvEHB{GJf5É76£GZR»fMeJa?&_E[ÁMR}aÈ ÃvEHB{GE¸f Ãv?ZR_R5ÃMRÁMņE[GJ@DÃvEHB{GB^éOõ ö N÷ØUÙvÚ™Ù Õ TÇFCÅR}@0GJf-B0é™aB^ņÆZeZÃMR5Á»ÆZÁMB{I^Á@0ŶfÉ0P)R}a5@^eJfR Ãv?ZR&R Z@^aÃĒ@^bHeJRB^éõWöUN÷ØUÙvÚ¡Ù Õ TS?J@^fWÇUR5RGaB^ņÆZeZÃMR_¤B{GZbHCé÷B^Á fMņ@^bHb]fMRà B^édÄD@0b[eZR}fSé÷B^Á ØULJÚ¡L Õ @^GJ_YÛOLÅ†B{fÃSÁvRfMR@^Áva?B{GYa5BDÄ^R5ÁvE[GZI,_R}fE[I^GJfS?J@{f)é÷BFa5eJfMR_YB^GY_ZRÃMRÁMņE[GZE[GZI,Ãv?ZR eZÆJÆARÁ @0GJ_YbHBDÀSR5ÁSÇUB^eZGJ_JfSé÷B^ÁõWöUN÷ØUÙvÚ¡Ù Õ TÉ GYÃM?ZE¸f fMRaÃvEHB{G]LFÀ)R³ÇZÁvE[9R 8JC%_ZRfvaÁvEHÇURfMB^ņR E[ņÆAB{ÁÃ@0GuÃÁMR}feZbHÃvf@^ÇAB{eÃÃM?JR&bHBDÀSR5ÁWÇUB^eZGJ_Jf@0GJ_¤eZÆZÆUR5ÁÇAB{eZGJ_ZfSé÷B{ÁõSöUNQØAÙÚ¡Ù Õ TÉ cW?Z:R a