Tabu Search-Based Algorithm for Large Scale Crew Scheduling Problems 1

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1 Tab Searh-ased Agorthm for arge Sae Crew Shedng robems Maro Caserta* Abstrat n ths paper, the probem of fndng a work shede for arne rew members n a gven tme horzon s taked. Ths probem s known n the teratre as Arne Crew Shedng. The obetve s to defne the mnmm ost shedes where eah rew, assoated to a ombnaton of ommera fghts or egs aed parng, s assgned to one or more fghts ensrng that the whoe set of fghts s overed by rew members. The Crew Shedng robem an be modeed by sng the Set Coverng formaton. Ths paper presents a new agorthm whose enterpee s a prma-to-da sheme amed at nkng any prma soton to the da feasbe vetor that best refets the qaty of the prma soton. Ths new mehansm s sed to ntertwne a tab searh based, prma ntensve, sheme wth a agrangan based, da ntensve, sheme to desgn a prma-da agorthm that progressvey redes the gap between pper and ower bond. The agorthm has been tested on benhmark probems from the teratre. n ths paper, rests on rea-word arne nstanes are presented: ot of sx we-known probems, the agorthm s abe to math the optma soton for for of them whe for the ast two, whose optma soton s not known, a new best known soton s fond. Key words: set overng, tab searh, metaherst, agrangan optmzaton, prma-toda. The athor s gratef to the anonymos referees for ther sef omments, and to Aberto Caprara for provdng the nstanes presented n Tabe. * rofesor nvestgador de nsttto Tenoógo y de Estdos Sperores de Monterrey, amps Cdad de Méxo.Correo eetróno: [email protected]

2 Maro Caserta. ntrodton The Arne Crew Shedng probem s a we know probem where one desres to ompte the set of mnmm parngs overng the whoe set of sheded fghts of an arne. A parng s a seqene of fghts overed by a rew, where the startng pont and the endng pont are the rew member s home base. The typa ength of a parng, or eg, s omprsed between one day and fve or sx days. A eg s an ndvda fght segment. Conseqenty, a parng s made p by a set of egs, where the frst eg departs from the same ty where the ast eg ends. Eah parng orresponds to a possbe fght shede for a snge fght rew. Fnay, a rew shede s a set of parngs that overs a the egs at exaty one (or at east one f deadheadng, whh means fght wthot workng, s aowed.) As expaned by Anb et a. (998), the rew shedng probem s the thrd stage of a for-stage proess. The frst stage of the arne pannng probem s the Shede Creaton, whh onssts of the defnton of a pan of fghts, based on marked demand. For exampe, t s possbe to shede a fght from Mexo Cty to Monterrey every weekday at 9:00 A.M.The fght from Mexo Cty to Monterrey and retrn s aed eg. The seond step of the pannng proess s made p by the Feet Shede. The obetve of ths probem s to hoose a feet for eah fght defned n the frst step. The assgnment of the feet to the fght needs to take nto aont apaty onstrants and fyng haratersts. The obetve at ths stage s to mnmze the operatng osts of the arraft. The thrd stage s the Crew Shedng, where, as mentoned, eah eg, or ndvda fght segment, s assgned to a parng, or rew. The probem s modeed sng the Set Coverng robem, where one ams at fndng the mnmm set of omns (parngs) that an over the whoe set of rows (ommera fghts, or egs) of the probems. Fnay, the forth and ast stage of the pannng probem s the Crew Rosterng, where eah ndvda rew member s assgned to eah parng aordng to dfferent mehansms, sh as senorty, aton, et. Severa fators ontrbte to enhane the dffty assoated to the rew shedng probem, sh as, for exampe: 50

3 Tab Searh-ased Agorthm for arge Sae Crew Shedng robems The feasbty of parng s governed by a ompex set of res, both of governmenta natre (FAA) as we as non-reated. For exampe, typa restrtons are those mposed on the maxmm draton of a parng ength n terms of days, the maxmm dty fght tme, the mnmm tme aowed between onnetng fght and the maxmm nmber of arraft hanges permtted. The aaton of the parng ost s ompex, say takng nto aont severa non near omponents. These nde rew pay as we as hote osts, et. The tota nmber of feasbe pars s very arge, hene nreasng the sze of the orrespondng rew shedng probem. For ths reason, t s not feasbe to generate a the possbe parngs. et N be the set of a feasbe parngs. For eah par, we defne a bnary varabe x whose vae s f the parng x s n the mnmm set and 0 otherwse. et be the ost of parng, whh s gven for ths probem. n addton, et s ndate wth M the whoe set of ommera fghts, or egs, to be overed. The Crew Shedng robem s modeed sng the Set Coverng formaton. The Set Coverng robem (SC) s a 0- nteger probem, defned as (SC) mn z s. t. N N x a x x, M 0,, N where a = f parng serves eg and 0 ve versa. n the foowng, we a a over a vetor x {0,} n that s a feasbe soton of SC, and a prme over s a over wth no redndant omns. Aso, et J ={ N: a =} be the ndex set of omns overng row, and ={ M: a =} the ndex set of rows overed by omn. Typay, Crew Shedng robems are haraterzed by the exstene of mons of omns (parngs) and hndreds, or thosands n the ase of the bggest arrers, of rows (ommera fghts, or egs.) No. 25, enero-abr

4 Maro Caserta SC s N-ompete (e.g., Garey and Johnson 979), hene exat soton proedres are doomed to fa n sovng prata SC probems. Spported by ts appabty, a great dea of effort has been dreted toward the deveopment of approxmate agorthms for SC. As a rest, some agorthms are apabe of sovng SCs wth thosands of rows and mons of omns (e.g., Cera et a. 998, Caprara et a. 999). To smmarze, most approxmate soton proedres for SC are da herst proedres based pon the agrangan reaxaton of SC and sbgradent optmzaton (e.g., aas and Ho 980, Vaasko and Wson 984, Fsher and Keda 990, aas and Carrera 996, Cera et a. 998, Caprara et a. 999). As the da proedres reqre greedy-type prma hersts n order to bd a prma over based pon the da soton obtaned va sbgradent optmzaton, they an aso be vewed as prma-and-da agorthms wth da-to-prma mehansms. n addton, more advaned da proedres for SC n the teratre typay featre some form(s) of probng and varabe fxng shemes that dynamay pdate prma and da nformaton of SC and ad n more effetve soton of SC (e.g., easey 990, aas and Carrera 996, Cera et a. 998, Caprara et a. 999). A maor ontrbton of the paper s the deveopment of a prma-to-da (p2d) mehansm that, for any gven prma soton, onstrts a feasbe da vetor that mnmzes the gap between the pper bond of SC gven by the over and the ower bond by a feasbe da soton wth respet to the sffent optmaty ondtons that we provde n Theorem. The beneft of the prma-to-da mehansm s two-fod: () f the rrent over s optma to SC, t verfes the optmaty and the searh proess an be termnated; () otherwse, t onstrts a da vetor that serves as a new startng vetor for sbgradent optmzaton. f dfferent prme overs are provded, the prma-to-da sheme onstrts dfferent s, aowng sbgradent optmzaton to expore dfferent regons of the da soton spae and onverge to respetve oa sotons theren. Ths, n trn, aows greedytype da-to-prma hersts to onstrt dfferent prme overs for SC. ntegratng effetve da-to-prma mehansms from the teratre and the nove prma-to-da mehansm wth a speazaton of the tab searh metaherst of Caserta and Ryoo (200) for sovng SC, n ths paper we deveop a prma-ntensve, dynam prma-and-da metaherst for arge-sae SC wth m<<n. Comptatona experments wth the proposed metaherst on 6 we known nstanes taken from the arne ndstry ndate that the proposed agorthm 52

5 Tab Searh-ased Agorthm for arge Sae Crew Shedng robems advanes the state-of-art n SC qte sbstantay. n order to prove the robstness of the agorthm, we tested t on the easey s OR brary. Ot of 94 benhmark probems, 2 of them have not been soved to optmaty. For 6 of the 2 probems, or agorthm redes the gap between and best ower and pper bonds: the agorthm fnds new best sotons for 3 probems and mproves the ower bonds for 5 probems. For the 73 benhmarks soved to optmaty, the proposed agorthm fnds the optma sotons. n ths paper, we mt the presentaton of the rests to those probems taken from the arne ndstry, provded by Weden (995). The proposed agorthm s made p of dynamay nteratng and ntertwned (meta)-herst omponents that synergstay ontrbte to the effeny and effay of the proposed agorthm. We present these (meta-)herst omponents of the proposed agorthm n Setons 2 to 7, both from the prma perspetve and from the da perspetve. ater, a ompete presentaton of the overa agorthm s offered n Seton 8. Comptatona experments wth 6 benhmark probems are smmarzed n Seton 9 and ondng remarks are provded n Seton Tab Searh Metaherst The tab searh metaherst of the proposed agorthm s based pon the work of Gover (See Gover 989, Gover 990, and Gover and agna 997) and ses the rest of a speazaton of the metastrategy provded n Caserta and Ryoo (2003) for SC. For reasons of spae, we provde detas for those omponents that are probem-spef n natre for SC. et x k denote the rrent prme over. n the k th teraton of tab searh, we move from x k to a dfferent prme over x k+ kt throgh a seqene of moves to x, t=,,t k, some of whh are overs and the others are not. et x 0 = x k k k and x T = x k+ kt. For smpty, x w be denoted as x t n the remander of ths seton. For reasons of spae, the proedra steps w be presented n a psedo-angage form wth omments provded n brakets. 2. Aowed nfeasbe Regon and Attratve Feasbe Spae One of the bas deas of the proposed agorthm s reated to the defnton of sbareas of the feasbe and nfesbe spaes. We dfferentate between the Aowed nfeasbe Regon and the Attratve Feasbe Spae. The former enhanes the searh spae by ndng a porton of the nfeasbe spae thoght to be attratve. No. 25, enero-abr

6 Maro Caserta On the other hand, the atter redes the searh spae by ttng ot those areas of the feasbe spaes thoght to be non-attratve, hene mtng the searh to a porton of the whoe feasbe spae. For x t {0,} n (t=0,, T k ), denote by the ndex set of omns that take vae n x t. et M 0 = { M : J = }, M = { M : J = }, and M 2 = { M : J = 2 } denote the set of rows that are novered, the set of rows that are nqey overed, and the set of rows that are overed twe by x t, respetvey. Frthermore, et = M 0 for N \ and = M for. et T denote the st of rrent tab moves. Wth the notaton above, the feasbe spae of SC an be defned as X := {x {0,} n : M 0 = 0 }. n ontrast to X, et s defne the aowed nfeasbe spae of SC as X := { x {0,} n : M 0 αm }, where α s a predetermned parameter hosen n [0, ). A key featre of the proposed tab searh metaherst s ts abty to esape from a oay optma soton va an exrson nto an aowed nfeasbe spae. Owng to the monotone dereasng property of the obetve fnton n x, sotons n X are, say, more attratve than the feasbe sotons. Hene, even f x k s a oay domnant prme over, the searh path w be abe to esape from t to a remote, dfferent prme over x k+ throgh a seqene of x t X X. n addton, et s defne the set of attratve overs of SC as X * := { x X : M \ M βm }, where β [0, ) s a predetermned parameter. Denote the set of prme overs by X. et ndate a -neghborhood move that sets the th omponent of x t to f = + (a set overng move) and to 0 f = - (a set reeasng move) and et denote the move n the opposte dreton of. Then, eah omposte move from x t to x t+ s omprsed of a seqene of a fnte nmber of -neghborhood moves, and the hoe of the frst move pays a rta roe n the proposed metastrategy. 2.2 Seeton of Frst Move and Three Types Moves As mentoned n the prevos seton, eah teraton of the agorthm s haraterzed by a omposte move, made p by a frst move, hosen aordng to nformaton 54

7 Tab Searh-ased Agorthm for arge Sae Crew Shedng robems 55 No. 25, enero-abr 2005 reated to searh stats, memory and earnng mehansms, and a seond move, whh s drety reated to the hoe of the frst move. n ght wth ths observaton, t s vta to devse ntegent mehansms amed at hoosng the frst move n the rrent neghborhood. For the seeton of, the frst move, we desgn for -neghborhood move seeton rtera. et s frst dentfy for sets wth varabes, rearranged n the order of for sores that are empoyed n the tab searh mehansm: : 0,, T M M : 0,, T : 0,, r r T N where r, 2 : 0,, M r M r T M N To aow the searh path to devate from foowng a pre-determned traetory gven by the se of the greedy mert fntons, we seet t probabstay. Eah move s, n trn, assfed as ether a regar, dversfed or ntensfed move dependng pon the way t s seeted. Remark. n order to aow for a more rgoros searh of the soton spae, we rer to three dfferent strateges that defne three searh phases of the agorthm, namey the regar, dversfaton, and ntensfaton phases by adaptng the probabst sheme Move Assgnment Sheme The next step, after hoosng the frst move n the rrent neghborhood, s to defne the omposte move, whh aows reahng a new pont n the feasbe

8 Maro Caserta spae. n ths seton, we present the proedre sed to defne the omposte move. The presentaton s offered n the form of meta-angage ode. Denote by e a nt vetor whose th omponent s - f = - and + otherwse. proedre Composte Move Assgnment; npt:, x t, T, (ore) probem nstane otpt:, x t+, T begn 0 a Seet Frst Move f x t + e X * X then, go to ne 0 t T T f x t X X then f ese t then : endf esef x t Î X * \ X then f = - then :,, : t, t ese :,, : t, t endf f ese endf 2 t then : 2 f = - then :,, 2 : t, t ese :,, 2 : t, t 56

9 Tab Searh-ased Agorthm for arge Sae Crew Shedng robems x t x t e e : T T end Remark. Note n the above that eah move s seeted n sh a way that a monoton property n the searh path s preserved wth respet to: M \ M, a measrement of redndany n x t, f x t X * ; and M 0, a measrement of the amont of nfeasbty assoated wth x t, otherwse. Remark. Sne the agorthm s espeay desgned to hande arge-sae nstanes of SC, we aways work on a sbset of the omns N C N and we empoy prng tehnqes to add or remove omns to and from N C. Conseqenty, eah orrene of N n the defnton of neghborhoods mst be repaed by N C. 2.4 Steps Tab Searh Metaherst Fnay, pttng together what presented n setons 2- to 2-3, we present the overa tab searh metaherst proedre. proedre Tab Searh Metaherst; npt: x *, U, x 0 (nta over), T, (ore) probem nstanes otpt: x *, U, T begn for phase regar, ntensfaton, dversfaton do k 0 {ont # of exrson n feasbe spae} = - {start wth feasbe spae} whe k < 2 do {fttng rteron} a Composte Move Assgnment {see Seton 2.3 } f x t+ x then {f the soton s feasbe} Note that even a prme over say satsfy M \ M > 0. No. 25, enero-abr

10 Maro Caserta end endf end whe end for f x k+ < U then {pdate prma nformaton} x * x t+ U x t+ endf f x t or = + then {end tab teraton} sove (p2d) {see Seton 5} parta prng {see Seton 6} a agrangan Optmzaton {see Seton 3} endf 3. agrangan Reaxaton and Greedy Hersts The best known prma herst s the greedy one, whh ses the reded ost nformaton nformaton provded by the da phase to onstrt a prme over. aas and Ho (980) presented a st of sores based pon the omn ost per row overed to reate a prme over. Vaasko and Wson (984) seeted a omn to be added to the parta over aordng to the vae of a sore fnton, randomy hosen among a poo of fntons based pon the omn ost per row overed. At eah teraton the prma herst s rn 30 tmes wth randomy hosen sore fntons. easey (990a) proposed a agrangan based prma herst sheme that extended the parta over of the agrangan probem to a prme over. A sore based pon the omn ost per row overed s sed to rank the omns. Fsher and Keda (990) proposed as sore the reded ost ompted sng ony the mtpers of rows eft novered, rather than the ata reded ost. rker and Tehapetvanh (993) stded the effetveness of fve dfferent prma herst sores, based pon the omn ost per row overed and the reded ost per row overed, both the rea and the modfed reded ost. aas and Carrera (996) oped the approah of Vaasko and Wson (984) wth a prma sheme that reates a prme over extendng the parta over of the agrangan phase by hoosng omns based pon ther reded ost. The prma sheme, as a byprodt, prodes an mproved da vetor. et M * denote the set of rows eft novered by x, and denotes the set of omns fxed to n the rrent (parta) over x. et k = M * be the nmber 58

11 Tab Searh-ased Agorthm for arge Sae Crew Shedng robems of rows rrenty novered that wod be overed by settng x to. Drng the da agrangan phase we se the sore: s r * M n a fashon smar to what presented n Caprara et a. (999). 4. Sbgradent Optmzaton The agrangan reaxaton of SC s defned as ( ) mn n r x m x {0,}, () where r (the reded ost for, =,, n) s defned as and reqres sh that a vetor x mnmzng the agrangan fnton an be ompted by a standard tehnqe: x, 0, f otherwse 0 Most sessf approahes for SC n the teratre sove a seres of agrangan reaxaton of SC and se the sbgradent optmzaton tehnqe to generate the vetor for the agrangan optmzaton. For sbgradent optmzaton, we se the forma of Hed and Karp (97) k U max s ( 2 k s( ) k k ),0, M where U and are the pper and ower bonds of the optmm of SC, ë s the step sze parameter, and J k s (x ) x s the omponent of the sbgradent. As n Caprara et a. (999), 0 s ntazed as No. 25, enero-abr

12 Maro Caserta 0 mn, J M and ë s pdated after every p = 20 teratons, tzng the best and worst ower bonds nformaton obtaned drng the ast p teratons. n addton, f the ower bond mprovement n the ast 4p teratons s beow the threshod mt of % we appy a pertrbaton sheme based pon the prma-to-da sheme of 5 to enfore a drast modfaton of the vetor. Frthermore, whenever the pertrbaton sheme s apped for more than 5 tmes n the same agrangan phase, we ntrode an ampfed modfaton of the vetor. 5. rma-to-da Sheme etx denote a (prme) over for SC. Smary, et x denote an optma soton to the agrangan probem wth respet to the rrent da vetor. Denote by z( ) and (, ) the obetve vae of SC and the vae of the agrangan fnton evaated at, respetvey. et = { N : x } and = { N : x }. emma. Sppose that gven x {0,} n, x {0,} n m, and satsfy ( x ) 0 J, for a M. Then, z ( x ) ( x, ) r r. roof. Aordng to the defnton of the agrangan fnton (see () at Seton 4), we have ( x, ) M r 60

13 Tab Searh-ased Agorthm for arge Sae Crew Shedng robems 6 No. 25, enero-abr 2005 The agrangan fnton an be rewrtten as where the seond eqaty s obtaned va 0 ) ( J x, M. Now, we have z ) (x. Conseqenty, we an wrte J M M x ), ( x r r z ), ( ) ( x x Theorem (Sffent Condtons). Sppose that gven x {0,} n, x {0,} n, and m satsfy: Then, x soves SC. N r r M x 0, ) (, ) ( 0, ) (

14 Maro Caserta roof. mmedate. The sffent optmaty ondtons of x * for SC n Theorem an be expoted n the dervaton of a mehansm that onstrts a feasbe da soton that propery refets the mportane of eah onstrant of SC wth respet to the haratersts of x. Frst, note that Condtons () and () of Theorem, aong wth the reqrement m gve the feasbty of to the da near program of the nearzed SC. Condtons () and (), aong wth x * {0,} n ensre that the prma and da sotons are optma to ther respetve programs. et M 0 := M : x, J M 0 : M M 0 and N C Í N wth N C << N. Consder the foowng near program: ( p2d ) mn g 0 M 0, 0, 0, N M 0 Frst, note that (p2d) s an wth (N C \ ) ( 0 ) rows and M omns.aso, note that the two non-trva onstrants of (p2d) set = 0 for a M 0, and, throgh the mnmzaton proess, (p2d) modfes the remanng omponents of the vetor feasbe to the da of the nearzed SC that satsfes the sffeny ondtons of Theorem as mh as possbe to yed that refets the haratersts of x. The foowng s an obvos onseqene of (p2d) and Theorem : Coroary. f the optmm of (p2d) s eqa to zero, then x soves SC. 62

15 Tab Searh-ased Agorthm for arge Sae Crew Shedng robems The foowng aso hods tre: m Theorem 2. Of a da feasbe, * obtaned from sovng (p2d) * mnmzes the gap z( x ) ( x, ) wth respet to x and x and Condton () of Theorem. roof. The da feasbty of * s mmedate. The formaton of (p2d) and Theorem * easy show that z( x ) ( x, ) s mnmzed by the x and * par. 6. Varabe Fxng wth robng Tehnqes Cera et a. (995) fxed a varabe to zero f ts reded ost s greater than the gap between pper and ower bond. aas and Carrera (996) ompted a fator Ä for every omn N \ defned as the mprovement n the vae of the vetor obtaned by fxng x to one. Sbseqenty, one omn s fxed to zero f + r +Ä U. n ths seton we gve the probng tehqnqes and varabes fxng shemes for SC. When probng x at, not ony the agrangan mtpers of a rows mst be set to 0 bt aso a r q, q J for every mst be reded to propery refet the new mportane of the omns wth respet to settng x =. et: q J : r 0 q The proposed sore s embedded n the foowng rersve fxng-to-zero sheme as n aas and Carrera (996): proedre Fx-to-Zero; npt: N, N C,, U, otpt: N, N C begn for N C \ do f r N C N C \ {} then No. 25, enero-abr

16 Maro Caserta end endf end for N N \ {} To fx a omn permanenty at ompte, for eah, the amont of modfaton that needs to be made to the vae sh that at east one other omn q J, q has a reded ost that s non-postve and row w be overed agan. Ths amont of modfaton reqred for s 0 mn qj r q and x,, an be permanenty to f - r + U. 0 proedre Fx-to-One; npt: N, N C,, U, otpt: N, N C begn for do end endf end for 0 f r U x N C N C \ {} N N \ {} then 7. rng and Core robem Generaton To defne ore probems, we empoy the prng sheme beow that resembes the one presented n Caprara et a. (999). 64

17 Tab Searh-ased Agorthm for arge Sae Crew Shedng robems proedre Defne Core robem; begn NC for N do f r 0. then N C N C {} end for for M do f J 5 and J N C < 5 then add (5 - J N C ) omns n J \ N C wth smaest r vaes to N C endf end for end As shown, we ensre that eah row s overed by at east 5 omns, f possbe, n a ore probem. 8. roposed Agorthm N N C The foowng smmarzes the steps of the proposed metaherst agorthm, whose behavor s strated n Fgre. Ths fgre presents the agorthm s steps from the feasbty pont of vew as we as from the optmaty pont of vew. S. ntazaton: defnton of a vetor va (5), defnton of a ore probem as presented n Seton 7, and defnton of the tota nmber of yes as K = S2. Ca Tab Searh Metaherst as strated n Seton 2. rma ntensve phase, wth searh both n the feasbe and nfeasbe spae. The overa tme ompexty for ths phase s O(r), where r << m s the ardnaty of the soton fond. S3. Sove (p2d), as presented n Seton 5. Fnd a new da vetor, whh best desrbe the rrent prma soton. Ths phase takes p O(p) tme, where p s the nmber of rows n (p2d) and s ess than N. S4. Defne a new Core robem, as presented n Seton 7. The new ore probem s defned sng the da vetor provded as soton of (p2d). Ths phase s exeted n O(n) tme. No. 25, enero-abr

18 Maro Caserta S5. agrangan Optmzaton hase, as desrbed n Seton 4. Da ntensve phase, wth searh of a new da vetor and a new ower, tghter, ower bond. Eah teraton of ths phase reqres the omptaton of the agrangan vetor, whh s obtaned n O(q) tme, where q s the nmber of non-zero entres n the matrx A (oeffent matrx), the omptaton of the sbgradent, done n O(n og n + q), de to omn sortng and other omptaton, and, fnay, the pdate of the mtper vetor, n O(m) tme. Overa, ths phase reqres O(m(n og n + q)) tme. S5. robng Tehnqes Appaton. Fx varabes to zero and to one, sng pdated da nformaton. Ths phase takes p O(q) tme. S6. f the maxmm nmber of teratons has been reahed, STO. Otherwse, go to S2. 9. Comptatona Rests on SC enhmarks n ths seton we present the rests obtaned testng the agorthm on benhmark probems. The agorthm was mpemented n GNU C++. The (p2d) probem s soved sng Cp CON-OR (ogee-hemer, 2003). The omptng patform sed s a nx workstaton wth AMD.GHz proessor and 52 Mb of memory. Tabe reports the rests on the nstanes appeared n Weden (995). A the probems are rea-word probems from the arne ndstry. A omparson wth the performanes of ommera software pakages s nfeasbe, sne, at the moment, no rests have been provded wth respet to these nstanes. However, the rests obtaned wth the proposed agorthm are ompared wth the best rests avaabe, obtaned throgh ad-ho agorthms deveoped by researhers and prattoners of the area. As a genera re, ad-ho agorthms, sh as the one proposed n ths paper, offer the advantage that very arge nstanes of the Arne Crew Shedng probem an be taked n a short omptatona tme. The maor haraterst of these nstanes s that they present rea osts for eah rew assgnment, havng osts n a range of,600 2,,450. Ot of 6 nstanes, for 4 of them the agorthm fnds the optma soton, on the ffth and sxth t fnds a soton that s better than any other soton fond so far. 66

19 Tab Searh-ased Agorthm for arge Sae Crew Shedng robems Tabe : Rests on nstanes from Weden (995) Tme measred n CU seonds est roposed Agorthm Name Sze Densty Range U Tme U Tme b727srath 29x %,600-, * , ataa 8x,65 3.0% 2,200-2,0, * ,258, a320 99x6, %,600-2,, * ,620, a320o 235x8,753.90%,900-,82, , ,495, sasmp 742x0, % 4,720-55, * ,339, sasd9mp2,366x25, % 3,860-35, , ,262,40, Consons We have presented a new dynam sheme for arge sae set overng probems. The bakbone of the agorthm s a new prma-to-da mehansm that, gven any prme over, t onstrts the da feasbe vetor that better refets the qaty of the prme over. Usng ths new mehansm, the agorthm onstanty pdates the stats of the searh n the da spae any tme a new prme over s fond and ve versa, dynamay nkng the prma ntensve phase wth the da ntensve phase. Owng to the ntensve se of prma-based shemes, the agorthm s espeay sted for those nstanes of SC wth a nmber of rows mh arger than the nmber of omns. Some new appatons of SC, sh as probe seeton probem for hybrdzaton experment as we as attrbtes dentfaton and patterns seeton n oga anayss of data, an be better taked wth a prma ntensve approah rather that va the tradtona agrangan based approah. No. 25, enero-abr

20 Maro Caserta Fgre : agorthm behavor (a) feasbty, and (b) optmaty. 68

21 Tab Searh-ased Agorthm for arge Sae Crew Shedng robems Referenes Anb, R., Forrest, J. and eybank, W Comn Generaton and the Arne Crew arng robem. Domenta Mathemata, Extra Vome CM,, aas, E., M. C. Carrera A dynam sbgradent-based branh-and-bond proedre for set overng probem. Operatons Researh, 44, aas, E., A. Ho Set overng agorthms sng ttng panes, hersts and sbgradent optmzaton: a omptatona stdy. Mathemata rogrammng Stdy, 2, easey, J. E. 990a. A agrangan herst for set overng probem. Nava Researh ogsts, 37, easey, J. E. 990b. OR-brary: dstrbtng test probems by eetron ma. Jorna of Operatons Researh Soety, 4, easey, J. E.,. C. Ch A genet agorthm for the set overng probem. Eropean Jorna of Operatona Researh, 94, rker, D., K. Tehapetvanh nvestgaton of agrangan hersts for set overng probems. Tehna Report, Department of ndstra Engneerng, The Unversty of owa, owa Cty, A. Caprara, A., M. Fshett,. Toth A herst method for set overng probem. Operatons Researh, 47, Caserta, M., H. S. Ryoo. Effent Tab Searh-based roedre for Optma Redndany Aoaton n Compex System Reabty. ro. 5th nt Conferene on Optmzaton: Tehnqes and Appatons, vo. 2, , De. 0. Cera, S.,. Nob, A. Sassano A agrangan based herst for arge-sae set overng probems. Mathemata rogrammng, Seres 8, No. 25, enero-abr

22 Maro Caserta Fsher, M..,. Keda Optma soton of set overng/parttonng probems sng da hersts. Management Sene, 36, Garey, M. R., D. S. Johnson Compters and ntratabty: a Gde to the Theory of N-Competeness. Freeman. Gover, F Tab Searh - art. ORSA Jorna on Comptng,, Gover, F Tab Searh - art. ORSA Jorna on Comptng, 2, Gover, F., agna, M Tab Searh. Kwer Aadem bshers. Hed, M., R. M. Karp. 97. The traveng saesman probem and mnmm spannng tree: part. Mathemata rogrammng,, ogee-hemer, R The Common Optmzaton Nterfae for Operatons Researh. M Jorna of Researh and Deveopment, 47(), Vaasko, F. J., G. R. Wson An effent herst for arge set overng probem. Nava Researh ogsts, 3, Weden, D An agorthm for arge Sae 0- nteger rogrammng wth Appatons to Arne Crew Shedng. Annas of Operatona Researh, 57,