Long-Haul Vehicle Routing and Scheduling with Working Hour Rules

Transcription

1 Long-Haul Vehicle Routing and Scheduling with Working Hour Rule Marie-Ève Rancourt Jean-Françoi Cordeau Gilbert Laporte October 2010 Bureaux de Montréal : Bureaux de Québec : Univerité de Montréal Univerité Laval C.P. 6128, ucc. Centre-ville 2325, de la Terrae, bureau 2642 Montréal (Québec) Québec (Québec) Canada H3C 3J7 Canada G1V 0A6 Téléphone : Téléphone : Télécopie : Télécopie :

2 Marie-Ève Rancourt 1,2,3,*, Jean-Françoi Cordeau 1,2, Gilbert Laporte 1,3 1 Interuniverity Reearch Centre on Enterprie Network, Logitic and Tranportation (CIRRELT) 2 Canada Reearch Chair in Logitic and Tranportation, HEC Montréal, 3000 Côte-Sainte- Catherine, Montréal, Canada H3T 2A7 3 Canada Reearch Chair in Ditribution Management, HEC Montréal, 3000 Côte-Sainte-Catherine, Montréal, Canada H3T 2A7 Abtract. Long-haul carrier mut comply with variou afety rule which are rarely taken into account in model and algorithm for vehicle routing problem. In thi paper, we conider the rule on truck driver afety during long-haul trip in North America. The problem under tudy ha two dominant feature: a routing component that conit in determining the equence of cutomer viited by each vehicle, and a cheduling component that conit in planning the ret period and the ervice time of each cutomer. We have developed different cheduling algorithm embedded within a tabu earch heuritic. The overall olution method were teted on modified Solomon intance, and the computational reult confirm the benefit of uing a ophiticated cheduling procedure when planning long-haul tranportation. Keyword. Vehicle routing, trip cheduling, multiple time window, driver rule, HOS regulation, enumeration procedure, tabu earch heuritic. Acknowledgement. Thi work wa partly upported by the Natural Science and Engineering Reearch Council of Canada (NSERC) under grant and Thi upport i gratefully acknowledged. We alo thank Groupe Robert for their kind cooperation. Reult and view expreed in thi publication are the ole reponibility of the author and do not necearily reflect thoe of CIRRELT. Le réultat et opinion contenu dan cette publication ne reflètent pa néceairement la poition du CIRRELT et n'engagent pa a reponabilité. * Correponding author: Marie-Ève.Rancourt@cirrelt.ca Dépôt légal Bibliothèque nationale du Québec, Bibliothèque nationale du Canada, 2010 Copyright Rancourt, Cordeau, Laporte and CIRRELT, 2010

3 1 Introduction In long-haul tranportation, the ditance traveled by truck driver during a journey can be coniderable and driver often have to be on the road for everal conecutive day. Thi i in contrat with claical vehicle routing problem (VRP) in which all requirement are typically fulfilled within the ame day. A a reult, truck driver fatigue i often a contributing factor to eriou accident in long-haul tranportation. To reduce fatigue and improve afety, everal wetern countrie have adopted legilation which regulate the amount of time long-haul truck driver can drive without a ret over a given period. Trucking companie are monitored, and non-compliance with the legilation can reult in ubtantial fine. Accordingly, legilative requirement hould be conidered when planning long-haul vehicle route. However, thee requirement are rarely conidered in the vehicle routing literature or in routing and cheduling oftware. The aim of thi paper i to preent, model and olve a rich vehicle routing problem ariing in the le-than-truckload indutry, and which take into account the North American legilative requirement on work and ret hour. We olve the problem by mean of cheduling algorithm embedded within a tabu earch heuritic. Our tudy i motivated by the cae of Groupe Robert Inc., one of the larget and bet known Canadian third-party logitic provider, but our contribution i of general applicability. In addition to the North American legilation, our tudy alo take into account other contraint ometime conidered in VRP, uch a multiple time window and heterogeneou fleet contraint. Multiple time window are aigned to each cutomer ince the problem extend over everal day, but cutomer are viited only once. The North American legilation tipulate for how long commercial vehicle driver may drive without reting during long-haul trip. The legilation generally impoe three retriction. Firt, a driver can only cumulate a certain number of driving hour between two conecutive ret period. Second, a driver may not drive beyond a given number of conecutive hour after reuming duty following a ret. Thee two retriction are meant to enure that a driver doe not work over a precribed limit within a certain time interval. Third, a driver may only drive for a maximum cumulated time during a certain number of conecutive day after being off-duty for more than a precribed time. We note that the American and Canadian regulation are imilar in tructure, but ome of their parameter differ. For intance, it i forbidden to drive after 13 hour of cumulated driving time in Canada, while the limit i 11 hour in the United State (US). Similar legilation are alo enforced in everal other countrie. For example, the European Union (EU) ha adopted regulation No 561/2006 (the EU regulation) which et retriction on driving hour 2

4 and limit the working hour of driver. Although imilar in objective, the proviion of the North American regulation are different from thoe of Europe. The main difference tem from the more retrictive nature of the EU regulation which require additional break after pecified working time period. In fact, in addition to daily ret period, the EU regulation tipulate that hort break mut be cheduled after certain driving time interval. Other rule and alternative proviion alo apply. 1.1 Literature review Several olution approache have been propoed to deal with working hour retriction and driver break cheduling requirement within the VRP. Rochat and Semet [17] model ret break a fictitiou cutomer, wherea Cordeau et al. [5] treat them a arc in a multi-tage network. Deaulnier et al. [7] have hown that maximum working time contraint can be handled within contrained hortet path algorithm by mean of reource contraint like thoe ued for time and load variable. Campbell and Savelbergh [3] have preented a modified inertion heuritic to handle maximum hift time limit for driver. Savelbergh and Sol [20] incorporate break and daily ret into a branch-and-price algorithm for a general pickup and delivery problem. Dealing with legilation on working hour cannot be treated a baic break cheduling becaue continuou movement on the road generate everal dependent daily chedule over the planning horizon. A number of author have developed algorithm incorporating the EU or US regulation. Goel and Gruhn [11] conider the maximum driving time retriction impoed by the EU regulation within a VRP with time widow (VRPTW) and olve the problem by mean of a large neighbourhood earch algorithm. Goel [10] ubequently improved thi methodology, although hi reearch i focued on the baic proviion of the EU regulation and doe not conider ome of the alternative proviion of thi regulation. The author ha applied a large neighbourhood earch algorithm to modified Solomon [21] tet intance for the VRPTW. In their conideration of the EU regulation within a VRPTW, Kok et al. [13] preent a baic break cheduling method embedded within a dynamic programming framework. Precott-Gagnon, Drexl and Roueau [16] have propoed a large neighbourhood earch algorithm baed on a column generation heuritic to olve the problem. Thi method relie on a tabu earch for generating route (column) and a labeling algorithm to check their feaibility. Kok et al. [13] and Precott-Gagnon, Drexl and Roueau [16] have introduced in their algorithm the alternative EU proviion that were not conidered by Goel [10]. The algorithm developed by Precott-Gagnon, Drexl and Roueau [16] outperform thoe of Goel [10] and Kok et al. [13] on VRP benchmark intance. However, all thee method may not conider ome route within the optimization proce becaue the truck driver cheduling algorithm they ue i ometime unable to find a feaible olution. Goel [9] 3

5 preent a procedure that alway identifie a feaible chedule atifying the EU regulation whenever one exit but, to our knowledge, thi method ha not yet been embedded within an overall algorithm for the combined VRP and truck driver cheduling problem. Zäpfel and Bögl [23] and Bartodziej et al. [2] have worked on vehicle routing problem temming from real cae tudie and incorporating ret contraint pecified by the EU regulation. Zäpfel and Bögl [23] have preented a two-phae heuritic for a complex combined vehicle routing and peronnel aignment problem, including outourcing deciion. A VRP i olved during the firt phae by a tabu earch or by a genetic algorithm. An aignment problem i olved heuritically during the econd phae. Bartodziej et al. [2] have tudied a block planning problem that deal with time window, ret regulation and a heterogeneou fleet. They have combined a column generation heuritic and a large neighbourhood earch with two type of neighbourhood. To our knowledge, the only North Americain regulation conidered in a routing context are thoe of the US Department of Tranportation (DOT). Thee were firt conidered by Powell [15] who preented a hybrid model that take into account forecat demand to perform dynamic routing and driver cheduling. The imulator ued to tet Powell model integrated the verion of the DOT regulation that were then in force. The firt paper that explicitly integrate DOT regulation driver retriction in a VRP i due to Xu et al. [22]. Thee author have propoed a column generation algorithm in which DOT regulation are handled in the ubproblem which i olved by mean of a fat heuritic. More recently, Ceelli, Righini and Salani [4] have worked on a rich problem embodying everal operational difficultie ariing in real-world application, among which they conider a working time limit before a ret. They have developed a column generation algorithm in which the pricing problem i a particular reource-contrained elementary hortet-path problem olved by a bounded bidirectional dynamic program. However, the cheme ued to chedule ret period i rather baic. A ret i inerted only when no more driving time i available, o that the poibility of reting before the allowable driving time i depleted i not conidered. A hown by Archetti and Salvelbergh [1], not uing early ret mean that ome feaible cutomer equence are conidered to be infeaible. More recently, ome author have focued their attention on the difficulty of contructing a feaible chedule for a given equence of cutomer, in conformity with the US legilation. The two following tudie take into account the Federal Motor Carrier Safety Adminitration Hour-of-Service (HOS) regulation for commercial vehicle driver, the latet DOT working hour rule, and concentrate on the trip cheduling problem rather than on the VRP. Archetti and Salvelbergh [1] have conidered the problem of determining how a equence of full truckload tranportation requet, each with a dipatch window at the origin, can be executed by a driver in conformity with the HOS regulation. They have 4

6 developed an algorithm, called SMARTRIP, to chedule the working and driving hour of a driver. It find a feaible chedule in polynomial time, if one exit. Goel and Kok [12] have alo tudied a trip cheduling problem in which each location mut be viited within one of everal time window, and have preented a cheduling method capable of finding a feaible chedule in polynomial time if one exit. They have hown that the complexity of their algorithm for the cae of ingle time window remain the ame in the cae of multiple time window when there i a gap of at leat ten hour between them. 1.2 Scientific contribution The problem conidered in thi paper differ from thoe adreed by the above mentioned author. Firt, the US regulation are different from thoe of the EU regulation. The EU regulation can be viewed a an extenion of the North American regulation ince they are more retrictive and everal alternative proviion can be applied. Thi being aid, in the problem conidered by Goel [10], Kok et al. [13], and Precott-Gagnon, Drexl and Roueau [16], a ingle wide time window i aociated to each cutomer and only one vehicle type i ued. In the preent paper, we conider multiple time window for each cutomer and a heterogeneou fleet of vehicle. Second, we extend the idea of the author who have conidered the HOS regulation, by embedding a trip cheduling module that can create intermediate infeaible olution for the combined vehicle routing and driver cheduling problem. In the cheduling proce, we alo include the poibility for the driver to plit a ret into two horter period pent in the leeper berth, an alternative not yet treated in the paper that have dealt with the HOS regulation. Third, we integrate the labor cot of driver in the routing and cheduling procee to better repreent the cot of operating a vehicle in practice. In the cheduling proce, we integrate optimization by minimizing the duration of the planned trip, rather than only trying to find a feaible chedule a wa previouly done. We alo analyze the impact of conidering the total chedule duration in the objective function a oppoed to only concentrating on the total ditance traveled. The remainder of the paper i organized a follow. In Section 2, we provide a decription of the contraint and objective of the problem. The algorithm we have developed are decribed in Section 3. The computational reult are preented in Section 4, followed by concluion in Section 5. 2 Contraint and objective We conider a VRP with multiple time window (VRPMTW) that combine cheduling ret period for driver in conformity with the North American regulation, a well a other typical VRP contraint. 5

7 We begin by decribing the VRP and it contraint. We then elaborate on the work regulation in greater detail. Finally, we introduce two alternative objective function for the problem. 2.1 Vehicle routing and cheduling contraint We firt decribe the baic concept of the VRPMTW. Given a et of vehicle baed at a depot, the problem conit in determining a et of feaible route to erve a et V of cutomer in order to minimize a given objective. The depot i denoted by 0 and we define V 0 = V {0}. Let c ij be the travel ditance between i and j, where i, j V 0, and let d ij be the driving time to reach j from i. The problem i olved over a planning horizon of length H. A unique time window [a 1,0, b 1,0 ] of length H i aociated with the depot, where a 1,0 and b 1,0 repreent the earliet poible departure from the depot and the latet admiible arrival at the depot, repectively. Many companie provide cutomized tranportation ervice to meet their cutomer requirement, which may lead to the ue of a heterogeneou fleet of vehicle uited to variou function (e.g. refrigerated vehicle, tank-truck, livetock tranportation vehicle, etc.). Conequently, we conider a heterogeneou vehicle fleet in thi tudy. Every cutomer i V 0 ha a non-negative demand q i and can be erved only by a ubet of the vehicle fleet. Every vehicle type ha a given load capacity which cannot be exceeded by the total demand it carrie. Moreover, a retriction on on-duty time ha to be impoed to prevent overloaded work chedule for driver. A driver can be on-duty for a maximum of h work hour during the planning horizon and thi retriction tranlate directly into a duration contraint on the vehicle route. An ordered et of time window T i = {[a ti, b ti ], t = 1,..., t i }, with b t 1,i b ti, i aociated with each cutomer i to determine the time interval within which delivery i allowed. Each cutomer i V 0 mut be viited once by a vehicle during w i time unit, without ervice interruption. Conequently, a ingle time window ha to be choen for each cutomer delivery. If a vehicle arrive before the opening of thi time window, it ha to wait. It mut alo arrive before the cloing of the elected time window, for otherwie the driver will have to wait until the opening of the next available time window, if one exit. 2.2 Working hour contraint By law, every driver operating a commercial vehicle during long-haul trip in North America i required to record hi duty tatu for each 24-hour period on a pecific grid in a log book (Figure 1). A change of 6

8 duty tatu ha to be regitered a one of the following four poible tate: driving, on-duty, off-duty, or leeper berth. Thee tate are decribed a follow. Figure 1: Illutration of a completed driver log for one working day [8]. Driving: Driving time i the total time pent at the driving control of a commercial vehicle in operation, even when the vehicle i tuck in a traffic jam. On-duty: On-duty time mean all the time a driver i performing work for any employer or i required to be available for work. Accordingly, thi time i computed from the moment a driver begin work until he i relieved from work. On-duty time include the following activitie: 1) driving time; 2) time pent at a property of a carrier or a hipper, unle the driver ha been relieved from duty; 3) time pent inpecting, ervicing, or conditioning any commercial vehicle; 4) time pent in or upon any commercial vehicle, except time pent reting in the leeper berth; 5) time pent loading, unloading, uperviing or attending a commercial vehicle; 6) time pent handling work paper for hipment; 7) time pent repairing or remaining in attendance upon a diabled commercial vehicle; 8) time pent performing any compenated work for a peron who i not a carrier. 7

9 Off-duty: When off-duty, driver have no obligation to perform any work. They are free to purue any activity and are allowed to leave the place where the vehicle i parked. Sleeper berth: The time the driver i in the leeper berth of a commercial vehicle in conformity with pecific requirement. In thi paper, we conider the HOS regulation for commercial vehicle driver (Part 395 of the Federal Carrier Safety Regulation) which have been in force in the US ince January The HOS retriction on driving period are decribed in Table 1. Table 1: Hour of ervice rule 70-hour on-duty limit 11-hour driving limit 14-hour limit A driver cannot drive after 70 hour on-duty in eight conecutive day. He may retart an eight conecutive day period after 34 or more conecutive hour off-duty. A driver may only drive a maximum of 11 hour after 10 conecutive hour off-duty. A driver cannot drive beyond the 14 th conecutive hour after coming on-duty, following 10 conecutive hour off-duty. Off duty time doe not expand the 14-hour period. Intead of applying the 70-hour on-duty limit rule, a imilar retriction with a limit of 60 working hour during a period of even conecutive day can alo be applied. In thi cae, we ue the 70-hour on-duty limit to be conitent with the rule applied by Groupe Robert, but the idea remain the ame for the 60-hour on-duty limit. A driver can alo ue the leeper berth to extend the 14-hour limit. Any period of at leat eight conecutive hour pent in a leeper berth will not be included in the 14-hour horizon. Thi allow a driver to extend the time during which he can ue the 11 hour of driving a long a the condition decribed in Table 2 are repected. In ummary, a driver can drive at mot h drive hour and can be on-duty at mot h on duty hour before a precribed ret of at leat h ret hour ha to be taken to gain the right to drive again. In fact, a ret period ha to tart at the latet when the driving limit or the on-duty limit i reached. Duty time conit primarily of driving, waiting and ervice time. When a driver make ue of the leeper berth proviion, he ha to take a break of at leat h long break conecutive hour and another one of at leat h hort break conecutive hour. In uch a cae, the driving limit and the on-duty limit remain. According to the HOS regulation, we can et the parameter a follow: h work = 70 hour, the maximal cumulated on-duty hour during eight conecutive day; h ret = 10 hour, the minimal duration of a ret period to regain driving time; 8

10 Table 2: Sleeper berth proviion Sleeper berth partial ret period Driving and duty limit with partial ret period Driver uing the leeper berth proviion mut pend: 1) at leat eight conecutive hour (but le than 10 conecutive hour) in the leeper berth; 2) a eparate block of at leat two conecutive hour (but le than 10 conecutive hour) either in the leeper berth, off-duty, or in any combination of the two. After the econd required ret period i completed, a new calculation point for the 14- hour limit, tarting at the end of the previou ret period, will have to be conidered to determine the available on-duty and driving hour. In thi calculation, only the time block in the leeper berth of at leat eight conecutive hour will not be counted a part of the 14-hour limit; a block of le than eight hour will. The leeper berth proviion can be ued continually until 10 conecutive hour off duty are taken. After 10 conecutive hour off duty, a driver ha 11 hour of driving time and 14 hour of duty time available again. h drive = 11 hour, the maximal cumulated driving hour between two ret period; h on duty = 14 hour, the maximal cumulated on-duty hour after which it i illegal to drive before reting; h long break = eight hour, the minimal duration of a long break period when a ret i plit in conformity with the leeper berth proviion; h hort break = two hour, the minimal duration of a hort break period when a ret i plit in conformity with the leeper berth proviion. 2.3 Objective function In tudie in which driver working hour are conidered (ee Section 1.1), the primary objective i typically to minimize the number of vehicle in the olution, and the econdary objective i to minimize the total ditance traveled. However, minimizing thi objective without conidering driver chedule may yield unacceptable olution uch a long route with ignificant reting time. To avoid uch ituation we alternatively conider the total trip duration a a econdary objective. The problem i olved twice, uing each of the two different objective. The olution obtained can then be compared and the election of the bet compromie olution i left to the deciion maker. To olve the problem of Groupe Robert, we minimize the number of vehicle ued in the olution, and then the real routing cot. Thi cot i in fact a weighted um of the total traveled ditance and total duration. 9

11 2.4 Illutration Figure 2 depict a feaible olution for a VRPMTW with working rule. Each horizontal line repreent either a time line for the depot (the top and the bottom line) or for a cutomer. The demand of each cutomer i indicated on the left-hand ide of their time line and the capacity of each vehicle i equal to 10 unit. The quare bracket and aociated value on a time line repreent the time window aociated with the correponding cutomer. The number in the middle of the double arrow on the right-hand ide how the driving time between location. A vehicle trip i repreented by a path that read from the top left corner to the bottom right corner. More preciely, a path between the two depot time line repreent a driver chedule. An inclined black egment mean that the driver i driving, wherea a horizontal egment repreent working time (dark dotted line) or reting time (light grey line). A dotted light grey line how that the ret period i longer than the precribed reting time h ret. A cutomer i erved by a driver when a dark dotted line appear in a time window. In Figure 2, a firt vehicle erve cutomer 1, 2 and 3, and a econd one erve cutomer 4 and 5. Cutomer Driving time (min) between cutomer i = 0 0 H = i = 1 q 1 = 3, w 1 = i = 2 q 2 = 1, w 2 = i = 3 q 3 = 5, w 3 = i = 4 q 4 = 4, w 4 = i = 5 q 5 = 6, w 5 = 100 i = H = 5300 Time (min) Figure 2: Illutration of a feaible olution for a VRPMTW with working hour rule. An inclined black egment repreent driving time, wherea a horizontal egment repreent working time (dark dotted line) or reting time (light grey line). A cutomer i erved by a driver when a dark dotted line appear in a time window. 10

12 3 Tabu earch heuritic The problem under tudy contain two dominant feature: a routing component coniting of determining the equence of cutomer viited by each vehicle, and a cheduling component coniting of planning the ret period and the ervice time of each cutomer. Therefore, we have developed everal cheduling algorithm embedded within a routing heuritic. Our algorithm for the VRPTW with HOS regulation i baed on an updated verion of the Cordeau, Laporte and Mercier [6] unified tabu earch algorithm (UTSA). Thi heuritic ha been proven to be effective and flexible for a variety of VRP. In the following, we will ummarize the main feature of the tabu earch and detail the cheduling algorithm that can handle the HOS regulation. 3.1 Outline of the tabu earch heuritic The algorithm tart with an initial olution x 0, obtained with a imple contructive procedure, and move at each iteration from a olution x of value f(x) to another olution in the neighbourhood N(x) of x. The neighbourhood N(x) conit of all olution that can be obtained by applying a given type of tranformation to x. An attribute et B(x) = {(i, k) : i V, k = 1,..., m}, i aociated with olution x and indicate on which route k each cutomer i i viited. A neighbour olution i obtained by replacing a pair (i, k) B(x) by another non-tabu pair (i, k ) / B(x). The attribute (i, k) i then declared tabu for a fixed number of iteration. However, an apiration criterion allow the earch proce to accept a olution x containing a tabu attribute (i, k) if thi olution i the bet known olution with thi attribute. The earch can be broadened by accepting intermediate infeaible olution. Thi i achieved through the ue of a penalized objective function with elf-adjuting penaltie. Let c(x) be the cot of the olution x, q(x) the violation of capacity contraint, d(x) the violation of duration contraint and v(x) the violation of time window contraint. The global cot function i: f(x) = c(x) + αq(x) + βd(x) + γv(x), where the parameter α, β and γ are dynamically updated throughout the earch. After each iteration, the value of thee parameter are modified by a factor of 1 + δ, where δ > 0: if the current olution i feaible with repect to a contraint, the value of it aociated parameter i divided by 1 + δ, and it i multiplied by 1 + δ otherwie. Diverification and intenification mechanim are alo ued to improve the earch. The goal of the diverification i to penalize olution containing frequently encountered attribute, wherea intenification i a proce aimed at deepening the earch around good olution. In our problem, multiple time window and the HOS regulation are integrated during route contruction. To evaluate duration and time window violation of a route, a chedule complying with the 11

13 HOS regulation i firt determined. Before calculating the cot of a olution x N(x), a ret cheduling algorithm generate a chedule for the given modified equence of node in x in order to ae the cot impact of the modification. In addition, operator favouring the minimization of the number of vehicle in the olution have alo been implemented within the tabu earch. Every time a feaible olution with m vehicle i found and thi value i maller than the leat known number of vehicle in any feaible olution, the upper bound on the number of vehicle available i fixed to m + 1. Thi way, the poibility of moving cutomer to an empty trip remain, which enure a certain flexibility in the earch. A trip detruction operator ha alo been added to the tabu earch procedure, but only when the econdary objective i to minimize the total ditance. Every 100 iteration within the firt 500 iteration, and every 500 iteration thereafter, the trip with the hortet duration i detroyed and it cutomer are moved equentially into another route. Thi procedure i not applied when the econdary objective i to minimize the total duration becaue detroying the trip of horter duration conflict with thi objective. 3.2 Adaptation of a procedure developed by Goel and Kok Goel and Kok [12] have preented a earch cheme for an algorithm that generate feaible driver chedule for an ordered equence of λ cutomer. Thi O(λ 2 ) time algorithm wa not implemented but the author uggeted that uch a procedure could be ued in ome imulation experiment and could be ueful for carrier to avoid precariou delivery chedule. We have adapted thi procedure to our problem and we have embedded it within our tabu earch heuritic. The main difference between the uggetion made by Goel and Kok and our implementation i that we conider infeaible olution during the earch proce a well a extended ret duration during the cheduling proce. In the following ection, we will explain thi modified procedure in more detail. The trip cheduling algorithm we have developed and embedded within the tabu earch heuritic will then be decribed Truck driver cheduling problem The driver cheduling problem conit in deciding when a driver will drive, work and ret in order to comply with the HOS regulation, in uch a way that all cutomer are viited within one of their time window. Conider a route (i 0 = 0, i 1,..., i p, i p+1,..., i λ = 0). The driving time from i p to i p+1 i denoted by d ip,ip+1, and the ervice duration at cutomer i p i equal to w ip. We will deignate by drive any period of continuou driving, by ret an off-duty period of at leat h ret hour, by work any period of 12

14 ervice time at a cutomer and by idle the waiting time at a cutomer. Work and idle time are on-duty period not regarded a driving time. A chedule = (a 0, a 1,..., a u,..., a κ ) i a equence of activitie performed by the driver along the route. Let A := {a u = (a type u, a length u ) a type u {drive,work,ret,idle} and a length u driver activitie. A partial chedule from a i to a j i denoted by ij. 0} denote the et of To preent the algorithm that contruct chedule in compliance with HOS regulation, we define the following value: tarting time of chedule : l tart := a 1,0 + a length 0, if a type 0 = idle ; a 1,0, otherwie ; completion time of chedule : l end := 1 u κ a length u ; index of the lat full ret period activity in chedule : u r := max {u a type u = ret or u = 0} ; cumulated driving time ince the lat ret period in chedule : l drive := u r u κ a type u =drive completion time of the lat ret period in chedule : a length u ; lat ret l := max {l tart, l end 0,u r } ; cumulated lack time of partial chedule ij : l lack ij := i u j a type u =idle a length u. In the claical VRPTW, in order to reduce a partial route duration and unneceary waiting time, it may be advantageou to delay departure from the depot and the beginning of ervice at a cutomer. Similarly, in the context of long-haul tranportation, it may alo be beneficial to expand the length of a ret period in order to reduce waiting time and to increae driver flexibility by delaying the end of the 13

15 14-hour limit. Goel and Kok [12] apply thee idea by taking into account the time by which the lat ret period may be potponed in a chedule. Thi time i denoted by l potpone. We have adapted the computation of thi value ince our problem i characterized by multiple time widow, and infeaible olution are allowed during the earch. The forward time lack i defined by Savelbergh [19] a the larget margin by which one can potpone the beginning of ervice at a cutomer without cauing any time window violation. For a given route, thi value i recurively evaluated tarting from the depot up to the lat cutomer. In our problem, which deal with everal ret and allow temporary time window violation, we compute the forward time lack ince the lat ret, which repreent the latet time at which ervice can begin at a cutomer without increaing time window violation and i recurively evaluated tarting from the previou ret to the following ret. In thi evaluation, for every cutomer i of a route, we conider the time window [a t i, b t i] that allow the earliet poible ervice or generate the leat delay when there i no admiible time window. Let σ(u), with a type u = work, be the index of the cutomer erved during activity u. Thu, the value t that determine the elected time window for a cutomer i i the value of t yielding min {b ti l end u r,κ 1 b ti or b ti = b t i i}, with σ(u) = i. The forward time lack ince lat ret in chedule i defined by F := min u r u κ a type u =work {l lack u r,u + (b t σ(u) max {l end 0,u 1, a t σ(u)}) + }, where (y) + = max{0, y}. Then, the time by which the end of the lat ret period in the work plan may be potponed correpond to l potpone := min {F, l lack u r,κ }. Moreover, if two ret period have to be included to reach the following cutomer tarting from the lat erved cutomer in, then the lat ret period cannot be potponed: l potpone = 0. A ret potponement i implemented whenever a new ret i inerted and when the depot i reached at the end of a route. To thi end, the departure from the depot will be delayed when the newly inerted ret i the firt one of the chedule, and the previou ret i lengthened otherwie. The value by which the departure from the depot or the lat ret (activity a u r) will be brought forward or extended i equal to l potpone. In fact, a length u r will be increaed by a value equal to lpotpone whenever a new ret i inerted. 14

16 3.2.2 Trip cheduling procedure Thi ection preent our extenion of the method developed by Goel and Kok [12] for determining driver chedule. The main idea of thi procedure i to take a partial chedule and complete it by equentially adding activitie, uch a driving and ret period, until the following cutomer i reached. Given a partial route in which the lat cutomer i i p, the remaining driving time required to reach the next cutomer i p+1 in the trip i denoted by with u w = max {u a type u δ := d ip,i p+1 u w u κ a type u =drive a length u, = work}. From a partial chedule, the trip cheduling method illutrated in Algorithm 1 can be ued to determine the equence of activitie to be performed by a driver from cutomer i p to cutomer i p+1. Firt, the trip cheduling method compute, the maximal legal driving time until either the next cutomer i reached or it i neceary to plan a ret period. A driving activity of length i then added to the current chedule. At thi point, if the next cutomer i not yet reached, a full ret period i included after extending the lat ret period or by delaying the departure from the depot. Thi proce continue until the next cutomer i p+1 i reached, at which point two time window are determined: [ at r,i p+1, b t r,i p+1 ] allow the earliet direct ervice and [ at r,i p+1, b t r,i p+1 ] allow the earliet ervice after a full ret period. A firt potential chedule r i then obtained by directly erving the cutomer with a minimum idle time, and another chedule r i defined by reting before erving the cutomer. The et S ip+1 of all identified chedule that comply with HOS regulation i then augmented. By applying thi proce equentially from the depot to the ucceeding node in a route, multiple chedule are generated to reach each cutomer. The bet chedule i the one that complete a trip whitin the hortet time. To reduce the number of poibilitie and to peed up the proce, dominated chedule are pruned during the enumeration. The dominance rule work a follow. Conider two poible chedule and obtained after erving the lat cutomer of a equence. The remaining driving time before reting in chedule correpond to = min {h drive l drive, l lat ret + l potpone When chedule of horter duration are preferable, chedule dominate chedule if l end l tart l end ltart and. + h on duty l end }. Thee condition mean that in chedule the driver pend le time to perform the ame dutie a in chedule and till ha more allowable driving time. Moreover, chedule dominate chedule if l end + h ret l end. Thi i the cae ince the driver in chedule can take a complete ret, enabling a maximum driving time, while till allowing the completion of all activitie earlier than in chedule. 15

17 Algorithm 1 Trip cheduling procedure δ d ip,i p+1 if i p = 0 then a 0 = (idle,0) end if 1. Schedule driving and ret period on the route from cutomer i p to i p+1 : while δ > 0 do min { δ,h drive l drive δ δ (,(drive, )) if δ > 0 then,l lat ret + l potpone Apply foward time lack to the lat ret: a length u r alength u r Include a ret: (,(ret,h ret )) end if end while + lpotpone 2. Schedule ervice without reting: } + h on duty l end Determine the index t r of the time window allowing the earliet ervice at cutomer i p+1 without reting: { } t r the value of t yielding min b t,ip+1 l end 0κ b t,ip+1 or b t,ip+1 = b t ip+1,i p+1. Concatenate the current chedule to create r : if l end a t r,i p+1 then r = (,(work,w ip+1 ) ) ele end if r = (,(idle,a t r,i p+1 l end ),(work,w ip+1 ) ) 3. Schedule ervice with a previou ret: Determine the index t r of the time window allowing the earliet ervice at cutomer i p+1 including a previou ret: t r the value of t yielding min Concatenate the current chedule to create r : if l end + h ret a tr,i p+1 then r = (,(ret,h ret ),(work,w ip+1 ) ) ele end if { b t,ip+1 l end 0κ + h ret b t,ip+1 or b t,ip+1 = b t ip+1,i p+1 }. r = (,(ret,h ret ),(idle,a tr,i p+1 l end h ret ),(work,w ip+1 ) } 4. Update et of poible chedule to reach cutomer i p+1 : S ip+1 S ip+1 { r, r } 16

18 To effectively implement the enumeration procedure, we have ued a breadth-firt earch algorithm which can briefly be decribed a follow. Conider the route (i 0 = 0, i 1,..., i p, i p+1,..., i λ = 0) and let S ip be a et of feaible chedule obtained at node i p. To determine the et S ip+1, the chedule in S ip that are completed the earliet are ucceively conidered and extended within the trip cheduling algorithm. Then, prior to extending the next chedule in S ip, all dominated chedule in S ip+1 are removed. When all chedule of S ip have been generated, the ame tep are repeated for S ip+1. We can determine a et of feaible chedule for the route tarting at i 0 and apply the ame proce up to i λ. In our cae, the feaible chedule retained for the trip, S iλ, will alway be the one with l end for all S iλ. lend Incluion of the leeper berth proviion in the trip cheduling procedure We now decribe how to include the poibility of plitting a ret into two period in compliance with the leeper berth proviion. We firt etablih a new driver tate, called leeper berth, which repreent a period pent in the leeper berth in accordance with the proviion decribed in Table 2. When ret are plit in partial ret period, the computation of the cumulated driving time and the 14-hour limit are modified. Accordingly, ome value linked to a chedule have to be defined or adjuted: index of the ret activity which generate the lat calculation point of the 14-hour and driving limit in chedule : u cp := max {u a type u = leeper berth and u > u : a type u = leeper berth or a type u = ret or u = 0} ; index of the lat activity of type leeper berth in chedule : u b max {u a type u = leeper berth}, if u κ : a type u = leeper berth := 1, otherwie ; index of the lat ret period in chedule : u rp := max {u cp, u b } ; cumulated driving time ince the lat calculation point in chedule : l drive := u cp u κ a type u =drive a length u ; 17

19 time pat in the leeper berth during the lat ret period of chedule : leeper berth l := a length u b, if u b 1 0, otherwie ; completion time of the ret period where the lat calculation point tart in chedule : lat ret l := max {l tart, l end 0u cp } ; minimal reting time required to regain the right to drive when the on-duty limit or the driving limit will be reached after the end of chedule : ret required l := hlong break, hhort break, h ret, otherwie ; leeper berth if l > 0 and h hort break a length leeper berth if l > 0 and a length h u b u b long break < hlong break number of partial ret period during the lat ue of the leeper berth proviion in chedule : { } a plit length u a type u = leeper berth and u r u u b, if l leeper berth > 0 l := 0, otherwie ; cumulated lack time of partial chedule ij : l potpone := min {F, l lack u rp,κ, h ret l ret required } leeper berth, if l > 0 min {F, l lack u rp,κ }, otherwie. Before decribing our econd cheduling algorithm, the trip cheduling procedure that exploit the leeper berth proviion, we preent the algorithm ued to generate the chedule repreenting the four trategie that could be adopted before erving a cutomer. The firt poibility, decribed by Procedure 1, i to execute the ervice directly without reting. The econd poibility i to take a full ret before the ervice, which i decribed by Procedure 2. Procedure 3 and Procedure 4 outline the two poibilitie of partial ret period according to the leeper berth proviion: taking a long break or a hort break. In each of thee procedure, the time window elected i the one that allow the earliet ervice. The idle time i then added to the chedule if neceary. Algorithm 2 i imilar to Algorithm 1, but more ret period poibilitie are conidered a it make ue of the leeper berth proviion. Firt, the maximum legal driving time i determined. Then, a driving activity of length i added to the current chedule. If the next cutomer i not yet reached, a ret period will be included after extending the previou ret, or by delaying the departure from the 18

20 Procedure 1 Schedule ervice without reting Determine the index t r of the time window allowing the earliet ervice at cutomer i p+1 without reting: n o t r the value of t yielding min b t,ip+1 l end 0κ b t,ip+1 or b t,ip+1 = b t ip+1,i p+1. Concatenate the current chedule to create r : if l end a t r,i p+1 then r = `, (work, w ip+1 ) ele end if r =, (idle, a t r,i p+1 l end ), (work, w ip+1 ) Procedure 2 Schedule a full ret before ervice Determine the index t r of the time window allowing the earliet ervice at cutomer i p+1 with a previou ret: n o t r the value of t yielding min b t,ip+1 l end 0κ + h ret b t,ip+1 or b t,ip+1 = b t ip+1,i p+1. Concatenate the current chedule to create r : if l end + h ret a t r,i p+1 then r = `, (ret, h ret ), (work, w ip+1 ) ele end if r = `, (ret, h ret ), (idle, a t r,i p+1 l end h ret ), (work, w ip+1 ) Procedure 3 Schedule a long break before ervice Determine the index t lb of the time window allowing the earliet ervice at cutomer i p+1 with a previou long break: n o t lb the value of t yielding min b t,ip+1 l end 0κ + h long break b t,ip+1 or b t,ip+1 = b t ip+1,i p+1. Concatenate the current chedule to create lb : if l end + h long break a t r,i p+1 then r = `, (ret, h long break ), (work, w ip+1 ) ele end if r = `, (ret, h long break ), (idle, a t r,i p+1 l end h long break ), (work, w ip+1 ) Procedure 4 Schedule a hort break before ervice Determine the index t b of the time window allowing the earliet ervice at cutomer i p+1 with a previou hort break: n o t b the value of t yielding min b t,ip+1 l end 0κ + h hort break b t,ip+1 or b t,ip+1 = b t ip+1,i p+1. Concatenate the current chedule to create b : if l end + h hort break a t r,i p+1 then b = `, (ret, h hort break ), (work, w ip+1 ) ele end if b = `, (ret, h hort break ), (idle, a t r,i p+1 l end h hort break ), (work, w ip+1 ) 19

21 depot. Depending on the tate of the driver, a ret plit in the leeper berth will be continued only if a complete ret cannot be included. Thi proce i repeated until the next cutomer i reached, at which point the poibility of a direct ervice i conidered in a firt chedule and all eligible ret period are then conidered in different chedule. Accordingly, if the driver i already making ue of the leeper berth proviion, the chedule with the appropriate break time will be created. Another chedule that end the ret plit in the leeper berth (including a complete ret) will alo be created if it i allowed. When the driver i not making ue of the leeper berth proviion, then all the ret poibilitie are conidered by creating the correponding chedule. Finally, the et S ip+1 of chedule that comply with the regulation i augmented. In the tabu earch heuritic, thi procedure i implemented in the ame way a Algorithm 1, but the dominance criterion i modified. In thi cae, chedule dominate chedule if l end l tart l end ltart, and l ret required ret required l. The firt two condition mean that in chedule the driver ha pent le time to perform the ame dutie of chedule and till ha more allowable driving time to continue hi journey without reting. The lat condition mean that the reting time required to regain the right to drive, when the on-duty limit or the driving limit will be reached, i not more in chedule than in chedule. Moreover, chedule dominate chedule if the driver in chedule i not making ue of the leeper berth proviion or the plit length ue of the leeper berth proviion can be ended by including a complete ret (l = 0 mod 2), and l end + h ret l end Baic trip cheduling procedure To evaluate the benefit of a ophiticated trip cheduling procedure, we compare in Section the Algorithm 1 with the baic trip cheduling procedure ued by Xu et al. [22] and Ceelli, Righini and Salani [4] to olve rich VRP by column generation. In thi procedure, a ret i alway inerted a late a poible, i.e. when the driving or the on-duty limit ha been reached. The departure time from the depot correpond to the earliet poible time that will not create any waiting time before erving the firt cutomer in a route, while departing a early a poible. Algorithm 3 determine the equence of activitie to be fulfilled by a driver from cutomer i p to cutomer i p+1 according to the baic trip cheduling procedure. In thi procedure, the current chedule i equentially extended o that a unique chedule i etablihed for any equence of cutomer. 20

22 Algorithm 2 Trip cheduling procedure including leeper berth proviion δ d ip,i p+1 if i p = 0 then a 0 = (idle, 0) end if 1. Schedule driving and ret period on the route from cutomer i p to i p+1 while δ > 0 do leeper berth if l > 0 and a length n u b min δ,h drive l drive,l ele min end if δ δ (, (drive, )) if δ > 0 then n δ, h drive l drive h long break then lat ret, llat ret Apply foward time lack to the lat ret: alength u rp + lpotpone Include a ret or a break period: a length u rp + l potpone + l potpone leeper berth plit length if l > 0 and l 0 mod 2 then ret required, (leeper berth, l ) ret required l h ret ret required l ele ret required, (ret, l ) ret required l h ret end if end if end while 2. Schedule ervice without reting: Execute Procedure 1 3. Schedule ret period before ervice leeper berth if l > 0 then ret required if l = h long break then ele ele Schedule a long break before ervice: Execute Procedure 3 b + h on duty + a length u b Schedule a hort break before ervice: Execute Procedure 4 lb end if plit length if l 0 mod 2 then ele r Schedule a complete ret before ervice: Execute Procedure 2 end if Schedule a complete ret before ervice: Execute Procedure 2 Schedule a long break before ervice: Execute Procedure 3 Schedule a hort break before ervice: Execute Procedure 4 end if 4. Update et of poible chedule to reach cutomer i p+1 S ip+1 S ip+1 r, r, lb, b o + h on duty l end o l end 21

23 Algorithm 3 Baic trip cheduling procedure δ d ip,i p+1 if i p = 0 then a 0 = (idle,0) end if 1. Schedule driving and ret period on the route from cutomer i p to i p+1 : while δ > 0 do min { δ,h drive l drive δ δ (,(drive, )) if δ > 0 then Include a ret: (,(ret,h ret )) end if end while 2. Schedule ervice:,l lat ret } + h on duty l end Determine the index t r of the time window allowing the earliet ervice at cutomer i p+1 : { } t r the value of t yielding min b t,ip+1 l end 0κ b t,ip+1 or b t,ip+1 = b t ip+1,i p+1. Concatenate the current chedule to create r : if i p = 0 and l end > a t r,i p+1 then a 0 (idle,a t r,i p+1 l end ) end if if l end a t r,i p+1 then (,(work,w ip+1 ) ) ele end if (,(idle,a t r,i p+1 l end ),(work,w ip+1 ) ) 22

24 4 Computational experiment Becaue our problem i new, no benchmark intance are available for it. We have firt created tet intance from known VRPTW benchmark problem and we have ued different objective function to compare their impact on the olution. We have alo olved a real intance provided by Groupe Robert for a typical week. We firt decribe the tet intance and we then preent our computational reult. 4.1 Artificial intance We have firt adapted the benchmark intance of Solomon [21] for the VRPTW. Thee intance contain 100 cutomer each and are divided into ix clae that differ by the geographical ditribution of the cutomer and their time window tightne. The cutomer are clutered in the C1 and C2 intance, and uniformly ditributed in the R1 and R2 intance. In the RC1 and RC2 intance ome cutomer are clutered while other are uniformly ditributed. The C2, R2 and RC2 intance have wider time window and a larger load capacity per vehicle than the C1, R1 and RC1 intance, which make them harder to olve ince the number of cutomer per route increae coniderably. For each intance, the travel time matrix and the time window aociated with each cutomer were modified to better uit our context. The other parameter, i.e. the geographic coordinate of cutomer and the depot, the ditance matrice, a well a the vehicle capacitie remain the ame. The time interval during which the vehicle are available in the Solomon intance were firt interpreted a period of 24 hour. The time window of each cutomer have been caled accordingly and then replicated for eight day. The available time of all vehicle wa et to 192 hour (H = 8 day). A propoed by Goel [10] for the EU problem, the travel time matrice have been adjuted o that the traveling peed of a vehicle i et to five ditance unit per hour, intead of 60 a in the original Solomon intance. To illutrate, conider a cutomer i that ha to be viited by a vehicle during the interval [0, 15] and with [3, 9] a the aociated time window in the original Solomon intance. In our cae, the depot time window will be [0, 192] to repreent the eight-day planning horizon and the et of time window for {[ ( ) ) ] } that cutomer will be T i = (t 1),9( (t 1), t = 1,...,8. Furthermore, in order to adjut the travel time matrix of the cutomer viited during an eight-day horizon, each ditance value i multiplied by Real-life Groupe Robert intance We now decribe the intance of Groupe Robert which concern the ditribution of good in the US during a typical week. The geographical ditribution of the 162 cutomer i depicted in Figure 3, which 23

25 wa created uing the MapPoint commercial oftware. A et of time window are alo aociated with each cutomer depending on it delivery chedule and a ervice quality level guaranteed by the carrier. There are three type of good with mutual incompatibilitie. To deal with thi contraint we conider two vehicle type, and a et of compatible vehicle i aociated to each cutomer. The Groupe Robert intance i comparable to the RC1 intance ince ome cutomer are clutered while other are more cattered, and not many cutomer (between one and even) can be erved in the ame route. Moreover, thi i an open vehicle routing problem (Sarikli and Powell [18]) ince the route doe not include the trip back to the depot o a to include further on-line requet. Accordingly, all ditance and traveling time from any cutomer to the depot are et to zero (ee Piinger and Ropke [14]). Figure 3: Cutomer location in the Groupe Robert intance. 4.3 Reult and analyi We have implemented our tabu earch heuritic in C++ and have run all experiment on a Dual Core AMD Opteron 275, 2.19 GHz CPU with 8.0 GB of RAM. We have firt teted the tabu earch heuritic incorporating the trip cheduling algorithm on the modified Solomon intance jut decribed and then on the Groupe Robert cae. We compare the baic trip cheduling procedure with Algorithm 1, and then Algorithm 1 with Algorithm 2. We alo report the mot relevant reult obtained on the real-life intance. 24