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Comparison of Bundle and Classical Column Generation

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(1)Comparison of Bundle and Classical Column Generation O. Briant, C. Lemaréchal, Ph. Meurdesoif, S. Michel, N. Perrot, F. Vanderbeck. To cite this version: O. Briant, C. Lemaréchal, Ph. Meurdesoif, S. Michel, N. Perrot, et al.. Comparison of Bundle and Classical Column Generation. [Research Report] RR-5453, INRIA. 2005, pp.31. �inria-00070554�. HAL Id: inria-00070554 https://hal.inria.fr/inria-00070554 Submitted on 19 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Comparison of Bundle and Classical Column Generation O. Briant — C. Lemaréchal — Ph. Meurdesoif — S. Michel — N. Perrot — F. Vanderbeck. N° 5453 Janvier 2005. ISSN 0249-6399. ISRN INRIA/RR--5453--FR+ENG. THÈME 4. apport de recherche.

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(213) O*] \[ ! — 4. ¡ Ó¶Ó g(hjlv|{€†wv¶Ç™}™r±yzl}jlU]`{‚ª rt2Š‹u™ yµw4y]lU†hj}jrx†wv wz]ujkm~€y]rt{}¶©™y]{@›Su™w‚‰w‚}Eyzll+r|} É ˜ EÊ"y]h™w‚y = x"º · zŠI = · xgº · z ÉY l6l |˜fڀ© `ÀË­ “j­Ý’ ¼ {‚y]h™l‰`wS]]ujkm~€y]rt{}¸'w‚‰zlµ~:{Sz]r|§jvtl©jª«{‰'l‡jwk4~™v|lµy]h¸w>y rx'~¸{Sv|„shjl ƒ ‰w‚vìÊË­2g(hjl6}  d ¤ ›S{sl6 µ wS(r|}Ïg(hjk­t˜­ n¸É«rìÊ ¼  d ‰ ¤ rt'y]hjl]{‚¨Á†wv|vtl ƒ Àjl}¸†hjlv ƒ u¸w‚vtr±y‹„žy]hjl6{‰zlkÅw} ƒ yzhjl ‰zl6 y(rx'†vtl6w‰6­ g(hsu™6Š© ¹ {‚ª 6˜S­|˜ rx`‰zl~™vtwS†l ƒ §s„ ©:°'hjrt†h¹†6w‚}ϧ:l{§jyzw‚rt}jl ƒ §E„~j‰z{>•sr|}j›¦{S~€y]rtk¦w‚vtr±y‹„{‚ª r|} É ˜ EÊË©™{S‰`§s„ ƒ r|‰zl6†y]vt„ ]{vt•sr|}j› r|yz[7 wz {€†rxw>yzl ƒ ‹§™r ƒ u¸w‚v ÉM–€˜UÊ7 ­

(214) Žl6†wv|vjy]hjl ƒ lǙ}jr|y]rt{}¹É ˜ EÊ

(215) {‚ª E ¹ · xgº · z È)w2§¸lª«{‰zl© 7SI º · Œ N e N 7 N ª«{‰[w+}sujkP§¸l6‰2{‚ª †{vtujkm}™  u¸†h@y]h¸w>y ­ ¥ h™w‚y(rx(kmrtz]r|}j›4rx )[

(216) "\) ‰& ‰,d† w‚} ƒ wª«{‰]y]rt{‰zr¿†{km~jvtlkml}Eyw‚‰zr±y‹„¿È © uj}™v|lU] : ‚P n #­ X`šwm‰]lU ujx"7 v|y6N ©™8©º y‹°\· z{mI yzhjOrt}jxgº ›S· 'z kPu™ y`§¸lP†hjl6†¯Sl ƒ §¸lª«{‰zlw¦‹yw‚§jrtvt¸ r|Â6l ƒ w‚vt›{S‰]r|y]h™kņw}§:l y]{S~j~¸l ƒ È 6 7“:  7<>x º¹ = z kPu™‹y(§:lµ]k¦w‚vtv:wS[§:lª«{S‰]lS©E§™u€y\yzhjlµ~¸{E r|y]rt•lš~™w‚‰]y\{ª :mkPu™ y"w‚vx]{§:lµ]k¦w‚vtv

(217) ÉY {y]h™w‚y 7 rt"w‚~™~j‰]{º‡€rtk¦w>y]l6v|„Pª«l6wS rt§jvtlUÊ­Jp€ll ¶ª«{S‰wm†{}s•Sl‰z›l}¸†lš y]u ƒ „ž{ª ›l6}jl‰w‚v< yzw§jr|vtrtÂl ƒ kmlyzhj{ ƒ 6­ psl6•l6‰zwv¿†h™{rx†l6"h™wº•Sl+§¸l6l}~j‰z{~:{S]l ƒ ª«{S‰ ©j°'hjrx†h°\l+§j‰zr|l ™„ž‰zl•srtl° }™{>° ­ ƒ ”ƒ"  Ó  Ÿ6  Î  X Ǚ‰ y`‹yw‚§jrtv|rtÂrt}j› ƒ l6•Erx†lµª«{S‰z†l6 º rt}w. }

(218) P¨8§:{º‡žw‚‰z{uj} ƒ º ¹ ­2g(h™l‹yw‚§jrtv|rtÂrt}j›m~j‰z{§jvtlkÅrt ÉY–S–Ê 2 K · x"º*z#G%ºA<>nU8 )ºªÂ º¹ ‘T8 rY­ l­ 2 ¼ · xgºzÂm3%x"ºªÂ º*¹ z ¨` rx"y]hjl rt} ƒ rx†w‚y]{‰(ª«u™}™†Ëyzr|{S}{ª ‚©™]lžl À

(219) rt›™­ “j­ 98 ƒ ox"º ·  º*¹ z. 7 =. 0. . ),. . 1D/. .-. >. >. >. . . 1. 0. . . 4. . :. . >. . . . >. . , 0L-. . . . )L K>. . . )L > M .R S. I.          .  .

(220) rt›uj‰zl “jÈ2g(hjl+rt} ƒ rx†w‚y]{‰(ª«u™}™†Ëyzr|{S}{ª g(hjrxJrx

(221) y]hjl'§:{º‡j‹yzl~4kmly]h™{ ƒ {ª ““ 8­)À8yzJ{S‰]rt›rt}™wvswr|k°"wS)}j{‚y2kPu™†hyz{ ‰]l ƒ u™†l\yzhjl'}sujk§:l‰J{ª¿†6w‚vtvt

(222) yz{+y]hjl'{S‰zwS†vtl© §jujy`‰zw‚y]hjl6‰"y]{ml6wS lµy]h™l ‰]lU {Sv|u€yzr|{S}ž{ª(-É ÚSÊË©™‰]l6~jvtwS†rt}j›4r±y§s„ À. , .-. º“ º¹ !  #8|ºA?A@B¼. 2)L > xgºz*8. . g(hjlm yzw‚§™r|vtr±y‹„¹†l}Ey]l6‰S¹ °(wµ†hj{S]l}¹w~j‰zr|{S‰]r ¼ r±y+°"wS`}™{‚yškm{>•Sl ƒ wS`rt}FÉ ˜MÚSÊ`§ju€y+{}™v|„Ïw>ªìyzl‰µy]h™l4w§¸{>•Sl~j‰z{§jvtlk&°(w ª«ujvtvt„  {Sv|•Sl ƒ Éàuj}jvtl6z )º º ·  º¹  ©2rt} ƒ rx†w>yzr|}™›®{>•l6‰zwv|v\{~€yzr|k¦wv|r|y‹„ w‚} ƒ yzl‰zkmr|}™w‚y]rt{} {ª'y]hjl†{Sv|u™k4}F›l6}jl‰w>yzr|{S} ~j‰z{€†lU]ÊË­ g(hjl†{}>Š‹u™›Sw>yzlÉY–—SÊ

(223) rxJl6wS „y]{†{Skm~ju€y]l'°'h™l} rt[w‚}¦rt} ƒ rx†w>yz{‰)ª«uj}¸†Ëy]rt{}Èyzhjrt[›rt•lU 3   x : zI"# : Ù ­$X'††{S‰ ƒ r|}j› yz{@Ž'lk¦w‚‰z¯µ˜­ Úw‚} ƒ g(h™k­t˜­ s©jŒ"{º‡j y]l~žy]hjl6‰]lª«{‰zl+~j‰]{ ƒ u¸†l6'w} 7 ]{vt•sr|}™› ÉY–“SÊ 23546 7 › º ¹ : › # : Ù 8;:7<{=~ : 8 x"7”8 : z#? t)v 4‘wyxgE · z ½ @B  Œ"{‚yzh~j‰]{S§jvtlk¦+ÉY––SÊ(w‚} ƒ ÉY–“SÊ"†w}§:l+ª«{‰zku™vtw‚y]l ƒ w(vtrt}jl6w‰'~j‰]{S›‰w‚k¦ ¼ yzhjrt°'r|vtv¶§:l ƒ {}jl+rt} 6˜­ “j­ “P§¸l6v|{>° ­ ƒ ”ƒ 7`Ó + Ð + Î Ô ¡>¢+ [Î ¢+«  Î Ÿ X ]l‰zr|lU"{‚ª

(224)  yzw§jrtv|rtÂrt}j›4y]l6‰]k¦ h™wº•lµ§¸l6l}ρ]uj§™]l6ÃEujl}Eyzv|„@~™‰]{S~¸{E l ƒ ©sr|}°'hjrx†hyzhjl ‹yw‚§jrtvtr|Â6l ƒƒ u™w‚v<‰zl6 y]‰zrx†Ëy]l ƒ k¦wS‹yzl‰ É ˜ SÊ"rx'w4km{‰zl+w} ƒ k4{S‰]l ]{~jh™rt y]rx†w‚y]l ƒÕ ˆ)­™g(h™l†{‰z‰zl6]~¸{S} ƒ rt}j›P yzw§jr|vtrtÂrt}j›4y]l‰zk¦w‚‰zl ƒ l~jrx†Ëyzl ƒ rtS} À

(225) rt›™­ n™­ . . . . 1. . -. +$. . 0. )(  . 

(226) &  . *. *$ *   $ rt›u™‰]l+n™È[psyzw‚§™r|vtr|ÂUw>yzr|{S}§s„+ Ù ~:l}™wv±y‹„. À. ÀÁ} t˜ 8© rt-, ¨8]h™w‚~:l ƒ É«vtlªìyš~™w‚‰]y'{‚ªiÀ

(227) rt›™­ nEÊȍɋ˜ EÊ\h™w"yzhjl+ª«{‰zk B · x"º*z ^ . º ™  º ¹ ™ . 8 rY­ l­ 2 2 ¼ , 0.-. 

(228) '   .  %$ . 0. .  > M .R S. ¨`. U. ™ ؏Ù. )L > M .R S. ¨` ¼. · xgºziÂ.  º“ º¹ Ù U. 8. ÉY–‚nEÊ ë«î ë .

(229) 

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(232) "! #$%&'()*) +%,- "% 0. °'r|y]h~:{Sz rt§jvt„ Å ~:l}™wv±y‹„ž†{sl @†rtl}Eyz ™ w‚} ƒ ™  rt}™ y]l6w ƒ {ªy]h™l rt}j›vtl ­JÀÁ}Eyzl‰zl6 y]rt}j›vt„¦l}j{Suj›h©sy]hjrx'‹yw‚§jrtvtr|ÂUw>y]rt{} rx(y]h™l ƒ u™w‚v¶†{uj}Eyzl‰z~™w‚‰]y"{ª)Œ"{º‡j y]l~È

(233) rt}ªàw†ËyU©  Ù I  w‚} ƒ y]hjl9À™l}™†h™lv¶§jr ƒ u™wv2ÉY–j˜UÊ\yzw¯lU[yzhjl+ª«{‰zk ÉY–S’Ê 2346 7 › º ¹ : 8;:7¡<{=# : 8 : Q8 xg7”8 : zo? tv 4rw}x"E · z ½ @BC8 °'hjrx†hφw}w›Sw‚rt}ž§:l+ª«{‰zku™vtw‚y]l ƒ w'w4vtr|}™l6w‚‰'~™‰]{S›‰w‚k­ g(hjlž~¸l6}™w‚v|y‹„_u¸ l ƒ r|} “S’ \h™wSwÏ~™wr|‰{‚ª§j‰zl6w¯E~:{rt}Eyz¦ÉY†l}Eyz‰zwv[~™w‰ y{ª À rt›™­ nEÊ©Jw‚›Swr|}0°'r|y]h ~:{Sz rt§jvt„×w ƒ w‚~jyzw‚§™v|l ]v|{S~¸lU ©  w‚} ƒ §j‰zl6w‚¯s~:{rt}Eyz  ­)g(hjl' yzw‚§™r|vtr|Â6l‰){‚ª¸y]hjl'‰zrt›hEy

(234) ~¸w‚‰]y

(235) {ª À

(236) r|›¸­ n+rx)~j‰z{~:{S]l ƒ rt} n 8©‚°'hjl6‰]l"yzhjl(kmw}™w‚›Slkml}Ey {žª º ¹ rx'rt}™ ~™r|‰zl ƒ §s„@§¸{º‡j y]l6~¶©€‰zw‚y]hjl6‰(y]h™w}×É M˜ ÚEÊË­ X †{Sk4km{S}¦ª«l6w‚y]uj‰zl`{ª<y]hjlU l+‹yw‚§jrtvtr|ÂUw>y]rt{}¸2rx[yzhjlµ}jl6†l6z r|y‹„4y]{k¦w}™w‚›Sl`†w‰]lª«ujv|vt„4y]hjl6r|‰(~™w‰zwkmly]l6‰z ©  ©sly†‚ ­ À™{‰ l‡€wkm~jv|lSÈ ª À

(237) rt›™­z“j©2w¹vxw‚‰z›l   yzw§jrtv|rtÂlU}j{y]hjrt}j›¸©J°'hjr|vtlw_ k¦wv|v   vt{>° ƒ {>°'}0y]hjlr±yzl‰w>yzl6 ¸ ÀÁ} yzhjlž§:{º‡j‹yzl~  yzw§jrtv|rtÂ6w‚y]rt{} {‚ uj} ƒ ujvt„­ n ©sy]{4ª«{S‰z†l+wm]k¦w‚vtv :@r|}Ïg(hjk­6˜­ ¸ ]ll¦ÉY–S’Ê­ ¸ g(hjlµª«{S‰]k Ù {‚$ª À rt›™­]nP‰zl6ÃEujrt‰zl6 ƒ ”ƒ  7`Ó + Ð + Î Ô ¡>¢+(Ÿ U¢€ + ¢€  Ó   + Î  ¡  ¢+ g(hjlPk¦w‚rt}®kml‰zr±yµ{‚ªJyzhjlP yzw§jr|vtrtÂ6w>yzr|{S}™‰zl•srtl°"l ƒ ]{@ªàw‰`rxyz{w‚vtvt{>° Õ ˆ ª«{S‰]kPujvtw‚y]rt{}™6©™§:{‚yzh®rt}Ïy]h™lP~j‰zrtkmwv w} ƒ¹ƒ u™wv ]~™w†l6 ¸ w°(w`w‚vtv|u ƒ l ƒ y]{žª«{‰ÉM–‚“EÊ'w‚} ƒ ÉM–’SÊË­ Õ ly`u™šk¦w‚¯l l‡€~jv|rx†r|yyzhjlP†{‰z‰]lU ~:{} ƒ r|}j› Õ ˆ ~j‰z{›‰w‚k¦(rt}y]hjl4†wS l ox <rz~I Œ ™B 2 KnU8 . < ™ .  ‘T ­ ؏٠U . . U. U. . . U. , .-. U. , L-. U. U. U 9. . . $ #$ ,. U. . $. -. $#-. $. )L.   

(238)  .     &. . *. .  . *.

(239) .

(240) rt›uj‰zl’€È$X}rtv|vtu™ y]‰w>yzr|}j›¦ yzw§jr|vtrtÂ6w>yzr|{S}×ÉYw‚} ƒ r|yz'†{}>Š‹uj›Ew>yzlUÊ [wS†h{S}jl¨ ƒ rtkml}™]r|{S}™w‚v<†{km~¸{S}jl}Eym{ª É ƒ l~jrx†Ëyzl ƒ {}y]h™l v|lªìy~™w‚‰]y'{‚ªiÀ

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