BRANCH-AND-PRICE FOR INTEGRATED MULTI-DEPOT VEHICLE AND CREW SCHEDULING PROBLEM. 1. The Integrated Vehicle and Crew Scheduling Problem

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1 Avance OR an AI Methos n Transportaton BRANCH-AND-PRICE FOR INTEGRATED MULTI-DEPOT VEHICLE AND CREW SCHEDULING PROBLEM Marta MESQUITA 1, Ana PAIAS 2, Ana RESPÍCIO 3 Abstract. We propose a branch-an-prce algorthm to solve the ntegrate multepot vehcle an crew scheulng problem. An nteger mathematcal formulaton that combnes a multcommoty network flow moel wth a set partonnng moel s presente. We solve the corresponng lnear relaxaton usng a column generaton scheme. Two branchng strateges are teste over benchmark nstances avalable n the Internet. Computatonal results show the effectveness of our approach. 1. The Integrate Vehcle an Crew Scheulng Problem The vehcle an the crew scheulng problems are mportant combnatoral optmzaton problems that arse n the plannng process of mass transt companes. Usually, vehcles are scheule before the crews. Durng the vehcle scheulng process one has to optmally lnk tmetable trps proucng vehcle blocks, so that each vehcle block forms a feasble scheule for a vehcle. If vehcles are locate at fferent epots, one has to smultaneously construct the vehcle blocks an assgn them to the epots. Then, the problem s NP-Har. On the crew scheulng problem one has to ensure that a set of crews covers the set of vehcle blocks at mnmum cost. Each vehcle block s splt nto peces of work, sequences of tmetable trps an eahea trps, where a change of rver s not allowe. A crew uty s a combnaton of peces of work that respects several constrants such as maxmum an mnmum sprea, maxmum workng tme wthout a break, break uraton an a maxmum number of changeovers. The vehcle blocks characterstcs nfluence the resultng crew utes an the crew uty set may lea to changes on the orgnal vehcle blocks. 1 Unversae Técnca e Lsboa, Insttuto Superor e Agronoma, Centro e Investgação Operaconal, Tapaa a Ajua, Lsboa, Portugal, martaolv@sa.utl.pt 2 Unversae e Lsboa, Faculae e Cêncas, DEIO, Centro e Investgação Operaconal, Bloco C6, pso 4, Cae Unverstára, Lsboa, Portugal, ampaas@fc.ul.pt 3 Unversae e Lsboa, Faculae e Cêncas, DI, Centro e Investgação Operaconal, Bloco C6, pso 3, Cae Unverstára, Lsboa, Portugal, respco@.fc.ul.pt

2 554 M. Mesquta The strong epenency between these two problems suggests that an ntegrate approach may lea to better scheules. Sequental approaches n the lterature solve the vehcle scheulng frstly. Conserng a preefne vehcle scheule mposes constrants that may lea to lack of feasblty an flexblty n the crew scheulng problem. In consequence, costs are potentally hgher. In ths paper, we propose a branch-an-prce algorthm that generates exact solutons for the ntegrate vehcle an crew scheulng problem wth mult-epots. We escrbe the ntegrate problem by an nteger mathematcal formulaton smlar to the one propose by Husman et al. [2], but wth fewer ecson varables an fewer constrants. We solve the corresponng lnear relaxaton, whle Husman et al. consere a lagrangean relaxaton. Both approaches use column generaton schemes, although embee n fferent optmzaton schemes. To our knowlege, ths paper presents the frst exact approach for the ntegrate problem wth mult-epots. 2. Mathematcal Formulaton Let T 1, L,Tn be a set of tmetable trps. Each trp s characterze by a startng tme an locaton, an, an enng tme an locaton. Let D 1, L, Dk be a set of epots wth a known locaton. At epot D, = 1, L, k, there are v vehcles. All vehcles are entcal. An orere par of trps ( T T, j ) s sa to be compatble f the bus release after trp T completon can be assgne to trp T j. The orere par ( T T, j ) efnes a eahea trp, where a bus runs wthout passengers, between the en locaton of T an the start locaton of T j. Pull-out trps correspon to the movement of a bus from a epot to the start locaton of a trp, whle pull-n trps correspon to the movement of a bus from an en locaton of a trp to a epot. We assume that each en locaton of a tmetable trp s a potental relef pont, where a change of rver s allowe. Therefore, each task correspons to a eahea trp followe by a tmetable trp. Consequently, crew utes can start (en) at a epot or at an en locaton of a trp N. In what follows we efne the vehcle scheulng problem on a recte multgraph an propose an nteger mathematcal formulaton for the ntegrate problem that combnes a multcommoty flow moel wth a set parttonng moel. Let N = { 1, L,n} be the set of vertces, where vertex N represents trp T. Denote by D the set of k epots. There s a vertex n +, D corresponng to each epot. For each D we assocate a graph G = ( V, A ), where V = N { n + } an A = I { ( n + ) N} { N ( + n) }. The arc set A contans arcs that correspon to compatble pars of trps, I N N, an arcs relate wth pull-n an pull-out trps to an from epot. Costs c j, c, n + an c n+, j are assocate wth the corresponng arcs. A fxe cost s assgne to each crew. If trps N are orere by ncreasng value of ther startng tme, then the arc set I contans only arcs (, j) wth < j an no crcut contanng only vertces, j N exsts n a graph G. Let L be the set of all feasble utes. Conser the ecson varables,

3 Branch-an-prce for ntegrate mult-epot vehcle f trp j follows rectly after trp by a vehcle locate at = x j (, j) I, x, n 1 f after thecompleton of trp the bus returns to epot + = D, N, D, 1 f epot rectly supples a bus for trp xn+, = N, D, to escrbe the vehcle scheulng problem, an the ecson varables 1 f crew uty l s n theoptmalsoluton yl= l L, to represent the crew scheulng problem. Let s l be the cost assocate to uty y l an L, j = utesl coverng the eahea trp from trp to trp j an coverng trp j, ( ) { } ( j) = { utesl coverng the eahea trpfrom a epot to trp j an coverng trp j } ( ) = { utesl coverng the eahea trp from trp to a epot }, DL LD Then, the VCSP can be formulate as follows (SP-VCSP): mn c j xj +, n+, n+ n+, n+, D (, j) I D N ( c x + c x ) + sl yl (1) l L s. t. x j + x, n+ = 1 N (2) D j: (, j) I D j x j + x, n + x xn+, = 0 N, D (3) j: (, j ) I j: ( j, ) I x, v D (4) N l l n+ y DL( j ) l x n +, j =0 j N D (5) yl x j = 0 (, j) I (6) L(, j ) D l y LD( ) j {0, 1} l x n+ N D, =0 (7) x ( j) I, D, (8) x x {0,1} N, D (9), n+, n+, y l {0,1} l L (10) The objectve s to mnmze a lnear combnaton of vehcle an crew costs. Flow contons (2) state that each tmetable trp s performe exactly once. The flow conservaton constrants (3) guarantee that each vehcle block s assgne to a vehcle that

4 556 M. Mesquta returns to the source epot. Constrants (4) are epot capacty constrants. The couplng constrants (5), (6) an (7) relate the vehcle scheulng varables wth the crew varables requrng that a eahea trp s covere by a vehcle f an only f t s covere by a crew. Although (5), (6) an (7) are state just for arcs corresponng to eahea trps, they are suffcent to guarantee that all eahea trps an all tmetable trps n a vehcle block are covere by crew utes. Ths statement follows from our task efnton where the eahea trp from the en of a trp (or from the epot) to the start of trp j an trp j are performe by the same rver. Consequently, the mathematcal formulaton (SP-VCSP) has fewer ecson varables an fewer constrants than the one propose by Husman et al, [2]. These authors conser a secon nex on the uty varables to ncate the epot to whch the crew belongs. In our moel, each crew s assgne to the corresponng vehcle epot. 3. Branch-an-Prce The ILP formulaton for the VCSP nvolves a huge number of uty varables (crew varables) an t s neffcent to hanle these varables explctly. Therefore, the moel s tackle usng a branch-an-prce technque. At each noe of the branchng tree we solve the lnear relaxaton of a restrcte master problem usng a column generaton scheme. We conser explctly all the vehcles scheulng varables an we conser mplctly all the crew scheulng varables. On a prevous work [3], Mesquta an Paas prove that an nteger optmal soluton to the SP-VCSP was obtane when (8) an (9) are relaxe to x 0 an x x 0, respectvely. Ths result suggests to branch only on the uty crew varables. j, n+, n+, 3.1. Root noe At the root noe, an ntal set of columns s neee to start the prcng problem. Therefore, we solve a mult-epot vehcle scheulng problem, wthout requrng that each vehcle returns to the source epot. Ths problem can be vewe as an assgnment problem or a mnmum cost flow problem, an can be solve n polynomal tme. The optmal soluton gves a set of vehcle blocks coverng all tmetable trps. Frstly, each vehcle block s assgne to exactly one epot. Seconly, vehcle blocks are splt nto utes satsfyng a subset of the crew feasblty constrants. The resultng set of utes s checke for the remanng constrants an a bg cost s assgne to non-feasble utes Solvng the prcng problem The column generator evelopment was nspre on the one propose by Desrochers an Soums [1] for solvng the lnear relaxaton of a set coverng moel for the crew scheulng problem. Each feasble crew uty can be seen as an aequate path n a network that takes nto account specal features of the ntegrate problem. The feasblty s establshe by usng resources that are consume along the network. Imposng tme wnows on each

5 Branch-an-prce for ntegrate mult-epot vehcle 557 (some) vertex of that network restrcts the resource consumpton. The column generaton subproblem s a mnmum cost path problem wth resource constrants n the uty network. We solve t, exactly or heurstcally, by ynamc programmng Branchng Rules When the LP soluton of a noe s not nteger, we choose an arc corresponng to a eahea trp such that the sum of all the crew varables coverng ths arc s not nteger. Two new noes are create, one fxng that sum to one an the other fxng t to zero. Note that, fxng the sum to one means that there s one crew coverng the fxe arc an, consequently by constrants (5), (6) an (7), there s a vehcle coverng t. The branchng constrant s explctly ae to the current restrcte master problem n a way that the structure of the prcng problem remans unchange, contrary to the usual scenaro. We teste two branchng strateges. In the frst one, we start to branch conserng arcs, j where an j correspon to tmetable trps an afterwars, f necessary, we look ( ) for arcs ( j), where or j are epots. In the secon one, we frst look for arcs relate wth pull-out an pull-n trps, an after to arcs corresponng to compatble pars of trps. 4. Computatonal Results The branch-an-prce algorthm s coe n C, usng ILOG CPLEX 9.0 Callable Lbrary Routnes to optmze master LP s. Some results have been obtane for ranomly ata problems wth 80 trps an 4 epots avalable at For these test nstances, computatonal results are compare wth the ones obtane by Husman et al [2], although both approaches ffer on some basc assumptons. We consere two types of utes, namely a trpper type an a normal type. A trpper type uty has a sprea between 30 an 300 mnutes. The normal type uty has a break wth mnmum length of 45 mnutes an a sprea between 30 an 585 mnutes. For such type, the maxmum work tme s 540 mnutes an the maxmum uraton allowe before a break occurs s 300 mnutes. The prcng problem s solve exactly conserng smultaneously both types of utes. Husman et al. [2] splt the normal uty type nto four types an solve the prcng problem nepenently for each one of them. In these prelmnary tests, changeovers are not allowe. Our cost structure may also be fferent from the one use n [2] (whch s partally unclose). A fxe cost of 5000 s assgne to each crew n the soluton. To mnmze the number of vehcles, each pulln/pull-out trp has a penalty of 5000 ae to ts cost. Ths correspons to conser a vehcle penalty of The frst upper boun s obtane by ang the fxe cost of a uty crew to the lower boun obtane at the root noe. Table 1 reports average results. The column root gap presents, for each problem type, the gap between the LP optmal value obtane at the root noe an the nteger optmal value. For each branchng strategy, we present the number of noes ( #n ) an the number of calls of the LP solver ( #LP ). Columns #vc an #cr splay the number of vehcles

6 558 M. Mesquta an crews, respectvely, whle tot s the sum of these values. The followng columns splay the values obtane n [2] for the same measures. Prob Root gap Strategy 1 Strategy 2 Husman results type #n #LP #n #LP #vc #cr tot #vch #crh toth 80A 2.1E B 5.5E Table 1. Computatonal Results The gaps obtane at the root noe of the branchng tree were very small, showng that our LP relaxaton s very tght. Regarng results n [2], our approach acheve a smaller number of crews but a greater number of vehcles for both types of problems. We obtane better values for the sum of the two quanttes (number of crews an number of vehcles). However, a fferent cost strbuton may explan these varatons. Concernng the two branchng strateges we teste, we are not able, by now, to conclue whch one performs better n general. For type A problems, strategy 1 outperforms strategy 2, whle for type B, strategy 2 s better than strategy Conclusons an Further Research In ths paper, we propose a branch-an-prce approach to solve exactly the ntegrate multepot vehcle an crew scheulng problem. Prelmnary computatonal results seem to be promsng. For the test nstances, the number of noes n the branchng tree s small. Ths can be explane by the tghtness of the LP relaxaton as well as the effectveness of the branchng rules. The choce of the ntal set of utes for the prcng problem s aequate, as we have always obtane feasble utes an solve the LP relaxaton n a small number of teratons. We are now testng other branchng strateges an lookng for better upper bouns. We are also comparng the behavor of fferent approaches that heurstcally/exactly solve the prcng problem. References [1] M. Desrochers an F. Soums. A Column Generaton Approach to the Urban Transt Crew Scheulng Problem, Transportaton Scence, 23(1):1-12, February [2] D. Husman D, R. Frelng an A.P.M. Wagelmans. Multple-Depot Integrate Vehcle an Crew Scheulng. Econometrc Insttute Report EI , Erasmus Unversty, Rotteram, The Netherlans, Avalable as ns.htm. [3] M. Mesquta an A. Paas. Solvng Vehcle an Crew Scheulng Problems Smultaneously - Workng paper nº 6, Centro e Investgação Operaconal a Faculae e Cêncas a Unversae e Lsboa, (