4 Column generation (CG) 4.1 Basics of column generation. 4.2 Applying CG to the Cutting-Stock Problem. Basic Idea of column generation

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1 4 Colun generaton (CG) here are a lot of probles n nteger prograng where even the proble defnton cannot be effcently bounded Specfcally, the nuber of coluns becoes very large herefore, these probles are hard to tackle by general algorths Consder an optal splex tableau of such a proble: Most varables are nonbasc, so ther value s 0 Only a tny part of the atrx s of nterest at all Basc queston: How can we restrct unnecessary coputatonal te that results fro the consderaton of unused varables? Wrtschaftsnforatk und Operatons Research Bascs of colun generaton Basc Idea of colun generaton Select a sall subset of (prosng) varables Solve the correspondng LP-relaxaton Derve and solve a subproble n order to dentfy whether there exsts an unused varable, whch would prove the obectve value If such a varable exsts: nclude t and resolve the proble If there s none: the proble s already solved to optalty Wrtschaftsnforatk und Operatons Research Applyng CG to the Cuttng-Stock Proble Certan aterals (e.g., paper, etal) are anufactured n standard rolls of large wdth W hs wdth s dentcal for all rolls hese rolls are cut n saller ones (called fnals) =,, wth wdths w such that the nuber of slced rolls s nzed Addtonally, we have a deand of b fnals of wdth w A soluton defnes n detal n whch fnals each roll s cut n order to satsfy all deands and to nze the nuber of consued rolls Clearly, a an proble arses by the fact that the entre soluton space coprses a huge set of varables Wrtschaftsnforatk und Operatons Research 369

2 Preparng the proble defnton In what follows, we ntroduce the proble defnton proposed by Glore and Goory We defne specfc cuttng patterns gven by an nteger vector a=(a,,a ) It defnes a specfc feasble selecton of a roll Altogether, we consder n feasble cuttng patterns wth (,..., ),wth, {,..., } a = a a a w W n = Such a cuttng pattern defnes the segentaton of a roll n the fnals of predeterned wdth Consequently, we obtan the followng lnear nteger progra Wrtschaftsnforatk und Operatons Research 370 he lnear progra and ts dual We obtan the contnuous pral proble P Mnze Z = n s.t. a x b, {,..., } = x 0 {,..., n},wth a w W, {,..., n} and the correspondng contnuous dual D n = x = = Maxze Y = b y s.t. a y, {,..., n} = y 0 Wrtschaftsnforatk und Operatons Research 37 Intractablty of the pral proble Clearly, the nuber of possble coluns ay becoe extreely large, even for sall nstances of the Cuttng-Stock Proble herefore, Glore and Goory (96) proposed colun generaton for solvng ts LP-relaxaton Frst of all, an ntal soluton of the LP-relaxaton s generated by defnng a set of ntalzng coluns We denote B as the atrx of the current coluns B n W d = w = W d = w Wrtschaftsnforatk und Operatons Research 372 2

3 After solvng the proble We have a frst row that tells us that the found soluton n the splex tableau s optal Specfcally, n our case, t s non-negatve,.e., t holds: ( π ) 0 c = c A We substtute the paraeters accordngly, and obtan c = ( y B) = y a 0 y a = = Note that ths apples only to the coluns of atrx B hus, there ay be addtonal coluns n A that lead to negatve entres n the frst row Wrtschaftsnforatk und Operatons Research 373 Most attractve coluns hus, the larger (a ). y s, the ore attractve becoes the colun a to be ntegrated nto the current tableau hs drectly results fro the defnton of the reduced costs n the pral tableau herefore, n order to generate prosng coluns, we consder the followng proble Maxze Z = a y s.t. = { } Obvously, t s a specal Knapsack Proble = a 0, a nteger,,..., a w W Wrtschaftsnforatk und Operatons Research 374 Dervng a lower bound hese cogntons provde us wth a lower bound of the optal obectve functon value Clearly, for a current optal dual soluton y (optal accordng to the current coluns), t holds that ( ) ( ) y A y A, ax ( ) ( ) ax ( y A) ( y A) y A y A, y A, we denote z = ( y A) ( y A) ax Apparently, z s a feasble dual soluton to the LP-relaxaton. herefore, b z provdes us wth a lower bound ax y ax Wrtschaftsnforatk und Operatons Research 375 3

4 Interpretng the lower bound hs lower bound concdes wth the obectve value of the current optal soluton (.e., the optal soluton correspondng to the actve set of coluns) wth the obectve functon value b y dvded by the optal value of the derved specfcally desgned knapsack subproble for all possble coluns ( y A) ax Wrtschaftsnforatk und Operatons Research 376 Qualty of the LP-relaxaton bound he bound provded by the optal soluton of the LPrelaxaton of Glore and Goory s odel s usually very tght (see Aor and Carvalho n Desaulners, G.; Desrosers, J.; Soloon, M.M.: Colun Generaton. p.37) Specfcally, ost of the one densonal cuttng stock nstances have gaps saller than one, and we say that the nstance has the nteger round-up property, but there are nstances wth gaps equal to (Marcotte, 985, 986), and as large as 7/6 (Retz and Schethauer, 2002). It has been conectured that all nstances have gaps saller than 2, a property denoted as the odfed nteger round-up property (Schethauer and erno, 995) Wrtschaftsnforatk und Operatons Research 377 Soluton technque Colun generaton Hence, a pragatc approach would be to solve the correspondng contnuous proble by applyng the Splex Algorth and round the resultng non-nteger results accordngly Consequently, snce we have restrctons n the pral proble, we obtan an optal soluton wth at ost non-zero varables Each roundng can cost us at ost one addtonal roll hus, the resultng cost dfference between optal contnuous soluton and found nteger soluton s upper bounded by Note that ths dfference s extreely unlkely Wrtschaftsnforatk und Operatons Research 378 4

5 General approach I We generate specfc cuttng patterns whch are used as coluns of a atrx B hs atrx defnes the followng lnear progra = If B s not drectly avalable, we ay use the dagonal atrx wth the dagonal values d defned as follows ( ) Mnze x, s.t., B x b, x 0 * B n W d = w = W d = w Wrtschaftsnforatk und Operatons Research 379 General approach II hen, we solve the defned proble optally by applyng the revsed Splex Algorth hs provdes us wth an optal soluton x* However, ts optalty accordng to the orgnal atrx A depends on the defnton of B Clearly, f the choce of cuttng patterns was not approprate, we ay have generated a soluton that wll be outperfored by alternatve constellatons basng on odfed coluns But, n order to check ths, we conduct the followng steps Wrtschaftsnforatk und Operatons Research 380 General approach III Subsequently, we calculate a dual optal vector y wth y B = c y = c B B Next, we try to fnd an nteger vector a=(a,,a ), wth a 0 satsfyng B w a W and y a > = = If such a vector exsts, we replace one colun n B by t (proposton) Otherwse, x defnes an optal soluton to the contnuous proble (proposton) Wrtschaftsnforatk und Operatons Research 38 5

6 Proof of the proposton If we have no nteger vector a=(a,,a ), wth a 0 satsfyng we obvously know that for all a = a,..., a wth w a W, t holds y a ( ) w a W and y a > = = = = hus, we ay conclude that y s a feasble dual soluton and the coplete splex tableau has a postve frst row Wrtschaftsnforatk und Operatons Research 382 hs frst row s defned as Proof of the proposton c c ( A ) y a = π = 0 = and therefore the found soluton s optal We cannot prove the found optal soluton of the reduced proble (*) by ntegratng any addtonal cuttng pattern,.e., by ntegratng any addtonal colun Addtonally, the obectve value of x* provdes a lower bound on the obectve functon value of the best nteger soluton Wrtschaftsnforatk und Operatons Research 383 Concluson We apply the revsed Splex Algorth to a lnear progra hen, t s not necessary that the whole atrx A N of non-basc coluns s avalable Moreover, t s suffcent to store the current base atrx B and to have a procedure at hand whch calculates an enterng colun a (.e., a colun a of A N satsfyng y. a >c (MnProb)), or proves that no such colun exsts hs proble s denoted as the prcng proble and s solved by a prcng procedure Wrtschaftsnforatk und Operatons Research 384 6

7 Prcng procedure Usually, a prcng procedure does not calculate only a sngle colun but a set of coluns whch possbly ay enter the bass n the followng teratons hus, we have always a so-called workng set of actve coluns If, fnally, the prcng procedure states that no enterng colun exsts, the current basc soluton s optal and the algorth ternates Wrtschaftsnforatk und Operatons Research 385. Intalze Colun generaton algorth 2. WHILE Calculate Coluns produces new coluns DO 3. END Insert and Delete Coluns Optze Note that only Intalze and Calculate Coluns have to be pleented proble-specfcally Wrtschaftsnforatk und Operatons Research 386 Exaple We ntroduce the followng sple exaple We have rolls of sze W=00 and need 97 fnals of wdth fnals of wdth fnals of wdth 3 2 fnals of wdth 4 Addtonally, we ay apply the followng cuttng patterns a =, a =, a =,and a = Wrtschaftsnforatk und Operatons Research 387 7

8 Exaple hus, we get the syste x x2 60 B x x x4 2 hs syste has the optal contnuous soluton 48,5 0,5 05,5 0,5 x = x = 0, = 5,..., n and the dual soluton y =. 00,75 0,25 97,5 0 Wrtschaftsnforatk und Operatons Research 388 Dual soluton y We consder all dual solutons y It s defned by y ( y2 + y4 ) y B c ( y y2 y3 y4 ) y y2 + 2 y3 Wrtschaftsnforatk und Operatons Research 389 hus, we obtan for the correspondng optal dual soluton cb AB = y = ( 0,5 0,5 0,25 0) Let us now consder y a y a = 0,5 a + 0,5 a + 0,25 a We have to show that t holds: y a y a = 0,5 a + 0,5 a + 0,25 a hus, we obtan 50 a + 50 a + 25 a In what follows, we analyze feasble cuttng patterns for a Wrtschaftsnforatk und Operatons Research 390 8

9 Restrctons A feasble cuttng pattern a ust fulfll the followng restrctons Frst, t ust not consue ore wdths than the roll contans,.e., 45 a + 36 a + 3 a + 4 a 00 ( ) In addton, our found subset of cuttng patterns s optal f t holds for all cuttng patterns that 50 a + 50 a + 25 a hus, we assue to the contrary that t holds: 50 a + 50 a + 25 a > 00 ( 2) 2 3 Wrtschaftsnforatk und Operatons Research 39 Consequences If a exsts, then we know at frst that a 3 =0 Why? If a 3 >3, we have a drect contradcton to () If a 3 =3, we have a =a 2 =0 due to (), but ths contradcts (2) snce 75>00 s obvously not correct If a 3 =2, we have a =0 and a 2 due to (), but ths contradcts (2) snce 00>00 s obvously not correct If a 3 =, we have a +a 2 due to (), but ths contradcts (2) snce 75>00 s obvously not correct Consequently, we obtan a 3 =0 as claed Wrtschaftsnforatk und Operatons Research 392 Consequences hus, we have a 3 =0 and thus we ay wrte now a odfed syste and 2 4 Consequently, snce () we ay conclude that a +a 2 2, and therefore we agan have a contradcton to (2) Consequently, a does not exst at all hus, x* =(48,5;05,5;00,75; 97.5) s an optal soluton for the contnuous relaxaton of our proble P and has the obectve functon value 452,25 ( ) 45 a + 36 a + 4 a 00 2 ( ) 50 a + 50 a > 00 2 Wrtschaftsnforatk und Operatons Research 393 9

10 An optal nteger soluton We transfor the contnuous soluton x* =(48,5; 05,5; 00,75; 97,5) to x =(48, 05, 00, 97) hus, we apply the followng constellaton 48 tes cuttng pattern (2,0,0,0), 05 tes cuttng pattern (0,2,0,2), 00 tes cuttng pattern (0,2,0,0), and 97 tes cuttng pattern (0,,2,0) Result 96 fnals of wdth 45 (Deand 97) 607 fnals of wdth 36 (Deand 60) 394 fnals of wdth 3 (Deand 395) 20 fnals of wdth 4 (Deand 2) Wrtschaftsnforatk und Operatons Research 394 An optal nteger soluton Hence, we have altogether 450 rolls of wdth W hs, however, s an nfeasble constellaton, but we ay add three addtonal patterns of the fors (0,2,0,0), (,0,,), and (0,,0,0) Consequently, we obtan a feasble soluton that consues altogether 453 rolls Snce the optal contnuous soluton has an obectve functon value of 452,25, the nteger soluton x s optal Wrtschaftsnforatk und Operatons Research 395 Fndng nteger soluton n general Colun generaton s a technque that enables us to effcently solve probles wth a large nuber of coluns If we have optally solved the correspondng contnuous proble wthout fndng an nteger soluton but a fractonal one, we ay branch by fx a non-nteger varable and repeat the soluton process (Branch&Prce) round varables accordngly apply a specfcally desgned etaheurstc cobne soe of the ethods depcted above n a sophstcated way Clearly, t depends on the applcaton how useful a found optal soluton of the LP-relaxaton s In case of the Cuttng-Stock Proble, these solutons drectly provde us wth tght bounds Bascally, the subproble s the Knapsack Proble that, despte ts NP- Copleteness, can be solved qute effcently herefore, n ths case, the subproble step works qute soothly Wrtschaftsnforatk und Operatons Research 396 0

11 References of Secton 4 Aor, H.B.; Valéro de Carvalho, J: Cuttng Stock Probles. In Desaulners, G.; Desrosers, J.; Soloon, M.M. (eds.): Colun Generaton. Sprnger, pp. 3-62, Danna, E.; Pape, C.L.: Branch-and-Prce Heurstcs: A Case Study on the Vehcle Routng Proble wth e Wndows. In Desaulners, G.; Desrosers, J.; Soloon, M.M. (eds.): Colun Generaton. Sprnger, pp , Desrosers, J.; Lübbecke, M.E.: A Prer n Colun Generaton. In Desaulners, G.; Desrosers, J.; Soloon, M.M. (eds.): Colun Generaton. Sprnger, pp. -32, Glore, P. and Goory, R.: A lnear prograng approach to the cuttng stock proble. Operatons Research 9: , 96. Wrtschaftsnforatk und Operatons Research 397 References of Secton 4 Glore, P. and Goory, R.: A lnear prograng approach to the cuttng stock proble-part 2. Operatons Research : , 963. Gouhs, C.: Optal solutons to the cuttng stock proble. European Journal of Operatonal Research 44:97-208, 990. Kallehauge, B.; Larsen, J.; Madsen, B.G.; Soloon, M.M.: Vehcle Routng Proble wth e Wndows. In Desaulners, G.; Desrosers, J.; Soloon, M.M. (eds.): Colun Generaton. Sprnger, pp , Retz, J. and Schethauer, G.: ghter bounds for the gap and non- IRUP constructons n the one-densonal cuttng stock proble. Optzaton, 5(6): , Wrtschaftsnforatk und Operatons Research 398 References of Secton 4 Schethauer, G. and erno, J.: he odfed nteger roundup property of the one-densonal cuttng stock proble. European Journal of Operatonal Research, 84:562-57, 995. Wrtschaftsnforatk und Operatons Research 399