Simultaneous Vehicle and Crew Routing and Scheduling for ...

BuR -- Business ResearchOfficial Open Access Journal of VHBGerman Academic Association for Business Research (VHB)��

Simultaneous Vehicle and Crew Routing andScheduling for Partial- and Full-LoadLong-Distance Road TransportMichael Drexl, Researcher at Chair of Logistics Management, Gutenberg School of Management and Economics, Johannes Gutenberg UniversityMainz and Fraunhofer Centre for Applied Research on Supply Chain Services SCS, Nuremberg, Germany, E-mail: [email protected] Rieck, Operations Research Group, Institute of Management and Economics, Clausthal University of Technology, Germany,E-mail: [email protected] Sigl, Fraunhofer Centre for Applied Research on Supply Chain Services SCS, Nuremberg, Germany,E-mail: [email protected] Press, Fraunhofer Centre for Applied Research on Supply Chain Services SCS, Nuremberg, Germany,E-mail: [email protected]

AbstractThis paper studies a simultaneous vehicle and crew routing and scheduling problem arising inlong-distance road transport in Europe: Pickup-and-delivery requests have to be fulfilled over amulti-period planning horizon by a heterogeneous fleet of trucks and drivers. Typically, in the vehiclerouting literature, a fixed assignment of a driver to a truck is assumed. In our approach, we abandonthis assumption and allow truck/driver changes at geographically dispersed relay stations. Thisoffers greater planning flexibility and allows a better utilization of trucks, but also creates intricateinterdependencies between trucks and drivers and requires the synchronization of their routes. Asolution heuristic based on a two-stage decomposition of the problem is developed, taking into accountEuropean Union social legislation for drivers, and computational experiments using real-world dataprovided by a major German forwarder are presented and analyzed. The obtained results suggestthat for the vehicle and driver cost structure prevalent in Western Europe and for transport requeststhat are not systematically acquired to complement one another, no cost savings are possible throughsimultaneous vehicle and crew routing and scheduling, although no formal proof of this fact is possible.

JEL classification: C65, M29

Keywords: vehicle routing, crew scheduling, synchronization, full truckload transportation

Manuscript received October 2, 2012, accepted by Karl Inderfurth (Operations and InformationSystems) June 12, 2013.

1 IntroductionVehicle routing problems (VRPs) are among themost studied mathematical optimization prob-lems; see, for example, the monographs Toth andVigo (2002) and Golden, Raghavan, and Wasil(2008). This interest results from the importanceof vehicle routing in everyday logistics practice aswell as the intellectual challenge of finding feasi-ble or optimal VRP solutions; VRPs belong to theclass of NP-hard optimization problems (Lenstraand Rinnooy Kan 1981). The widespread practicaluse ofmathematical models and algorithms for thesolution of VRPs, as reflected by the large numberof vendors of commercial vehicle routing software,

shows that the efforts of science in this field havehad a significant impact on business operations.One application area, however, where there is alarge gap between practical requirements and per-tinent scientific work is simultaneous vehicle andcrew routing and scheduling. This term denotesthe situation where the required transports haveno given timetable/no fixed schedule, and where adriver-vehicle combination is no longer consideredan inseparable unit. Thus, routes to perform therequired transports have to be planned for bothvehicles and drivers. This is an important issuein practice, because drivers regularly need breaksand rests and have to obey social legislation or

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trade union rules regarding driving, break, andrest times, whereas vehicles can be used twenty-four hours a day. Thus, the planning of separatevehicle and driver routes offers greater planningflexibility and allows a better temporal utilizationof vehicles. In the present paper, we consider theEuropean Union social legislation for drivers.The large majority of papers on vehicle routing,nonetheless, does not distinguish between a vehi-cle and its driver. In their monograph on VRPs,for example, Toth and Vigo (2002) state thatthroughout, ‘‘the constraints imposed on driversare imbedded in those associated with the corre-sponding vehicles’’. Hence, the contribution of thepresent paper is to study how separate, but syn-chronized, routes for drivers and vehicles can becomputed for transports without a given timetable.The remainder of the paper is structured as fol-lows. In the next section, we give a description ofthe proposed simultaneous vehicle and crew rout-ing and scheduling problem (SVCRSP). Then, inSection3, the relevant literature is briefly surveyed.Section 4 points out the distinguishing features ofthe present problem. Section 5 gives an overviewof how the European social legislation on driverdriving and working time is taken into accountin this paper. In Section 6, a two-stage heuristicalgorithm for solving real-world instances of theSVCRSP is described. Computational experimentswith the algorithm are reported in Section 7. Thepaper closes with a summary and a research out-look in Section 8.

2 Problem descriptionIn order to narrow the above-mentioned gap, westudied the following simultaneous vehicle andcrew routing and scheduling problem. Partial andcomplete loads (henceforth referred to as requests)have to be transported by truck from pickup to de-livery locations. There is a time window for thepickup and for the delivery of each request. Forthe transport of the requests, a fleet of trucks anda set of drivers are available. Each driver and eachtruck has a fixed home depot, where he/it is po-sitioned at the beginning of the planning horizonand to where he/it must return at the end of theplanning horizon. The planning horizon spans oneworkingweek, that is, six days fromMonday to Sat-urday. Thus, a driver/truck route starts and endsat an assigned depot and may cover several days.

The truck fleet is heterogeneous; there are severaltypes of trucks thatmay differ with respect to costs,capacity, and ability to perform certain requests.Thedrivers arehomogeneous in the sense that eachdriver is able to drive each truck and that all driversreceive the same wage. In addition to the home de-pots and the pickup and delivery locations, thereare so-called intermediate depots or relay stations.It is possible that a location acts as both a homedepot and a relay station. At relay stations, andonly there, drivers can change trucks. Hence, thereis a free assignment of trucks to drivers and viceversa, that is, a driver may change trucks arbitrar-ily often within the planning horizon, respectively,a truck may be driven by arbitrarily many driverswithin the planning horizon. The fact that thereare multiple depots, where drivers and trucks maybe stationed, and relay stations, where a changeof truck or driver is possible, makes the synchro-nization of drivers and trucks a non-trivial issue.In addition to the requirements already stated, thedrivers have to fulfill the European Union sociallegislation on driver driving and working hours onsingle-manned vehicles. It is assumed that a drivercan take a break or rest anywhere en route. How-ever, a driver must visit a relay station and take adaily rest there, or he must finish his route at hishome depot, upon expiry of a given fixed period af-ter leavinghis homedepot or ending adaily rest at aprevious relay station. Note that the relay stationshave no capacity restrictions, that is, the overallnumber of driver-vehicle changes at a relay stationas well as the number of simultaneous changes isunlimited. Furthermore, we consider the option ofdriver shuttle transports, where small shuttle vansare used to transport truck drivers between relaystations. In doing so, a truck driver is able to leavea truck at a relay station and subsequently drivea truck starting from a different relay station. Theshuttle transports are not counted as driving orworking time for the truck drivers. The possibilityof shuttle transports increases the flexibility formatching trucks and drivers, but of course shuttletransports also induce costs. A typical objective forthe SVCRSP is to minimize the overall operatingcosts, which consist of truck, driver, and shuttlecosts.To fulfill a request, two different objects (a driverand a truck) must be synchronized in space andtime, since both are non-autonomous objects un-able to move in space on their own. From these

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synchronization requirements, the central addi-tional difficulty of the SVCRSP compared to clas-sical VRPs results, the so-called interdependenceproblem:The interdependence problem refers to the factthat a change in one route may have effects on thefeasibility of other routes.This is in marked contrast to classical VRPs with-out synchronization, where any operation mod-ifying a subset of all existing routes leaves theremaining routes unaffected. Indeed, most VRPsolution algorithms, notably local search proce-duresandapproachesbasedoncolumngeneration,implicitly assume independence of routes. Conse-quently, standard VRP solution procedures knownfrom the literature are not directly applicable tosolving SVCRSPs. Hence, it is necessary to usemechanisms for dealing with the synchronizationrequirements and the resulting interdependenceproblem.The SVCRSP is a special case of the broader class ofvehicle routing problems with multiple synchro-nization constraints (VRPMSs). These are VRPswhere the routes of different ‘vehicles’ are inter-dependent. For a classification of VRPMSs, thereader is referred to Drexl (2012). This paper alsoprovides a comprehensive overview of the existingliterature on synchronization in vehicle routing. Inthe following section, the most important papersinfluencing our research are briefly discussed.

3 Literature reviewThe works to be mentioned here may be parti-tioned into two classes: First, papers consideringcases where motor vehicles and drivers are syn-chronized at a central depot and where a changeof driver/vehicle is possible only at the depot. Sec-ond, papers where different types of elementaryautonomous and/or non-autonomous objects arerequired to fulfill tasks, and where these objectsmay join and separate at several different loca-tions. Compared to the first class, the locationalflexibility of the second class adds an additionaldegree of freedom. Hence, problems in the secondclass turn out to be significantly harder to solvethan those in the first class.Synchronization at a central depot. Laurent andHao (2007) consider the problem of simultane-ously scheduling vehicles and drivers for a limou-sine rental company. The required transports are

pickup-and-delivery trips with given time win-dows. The authors use a two-stage solution ap-proach that aims to find a feasible crew and vehicleschedule by assigning a driver-limousine pair toeach trip. First, an initial feasible solution is con-structed by means of a greedy heuristic similar tothe well-known best-fit-decreasing strategy for thebin packing problem, using constraint program-ming techniques for domain reduction. Second,an improvement procedure based on local searchembedded in a simulated annealing metaheuristicis performed.Prescott-Gagnon, Desaulniers, and Rousseau(2010) study the problem of planning oil deliveriesto customers by truck. To solve the problem,the authors develop three metaheuristics, a tabusearch (TS) algorithm, a large neighborhoodsearch (LNS) heuristic based on this TS algorithmand another LNS heuristic based on a columngeneration (CG) heuristic that uses the TSalgorithm to generate columns. In addition, agreedy construction heuristic is considered thatsequentially builds up routes for driver-truckpairs by inserting the temporally closest customer.In the destruction phase of the LNS algorithm,different heuristics similar to those presented inRopke and Pisinger (2006) are used to determinethe vertices to be removed from the routes of thecurrent solution. The reconstruction phase applieseither TS or CG. Among other move types, the TSprocedure employs a driver switch move that triesto switch a pair of drivers, that is, have one driverdrive the other driver’s route and vice versa.Xiang, Chu, and Chen (2006) describe a staticdial-a-ride problem that involves the schedulingof heterogeneous vehicles and a group of driverswith different qualifications. The solution proce-dure is composed of a construction phase to obtainan initial solution, an improvement phase, and anintensification phase to fine-tune the solution. Theimportant aspect of the procedure is that, initially,‘abstract’ routes with a fixed schedule are deter-mined, and only in the last stage, concrete vehiclesand drivers are assigned to the routes.Zäpfel and Bögl (2008) consider an applicationof local letter mail distribution. Pickup routesand delivery routes (but no combined pickup-and-delivery routes) have to be planned within a plan-ning horizon of one week. In pickup routes, out-bound shipments are transported from local postoffices to a letter mail distribution center. Con-

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versely, in delivery routes, shipments are trans-ported from the distribution center to post offices.Schedules are planned for both drivers and vehi-cles, taking into account European Union sociallegislation. The problem is solved heuristically, bydecomposing it into a generalized VRP with timewindows (GVRPTW) and a ‘personnel assignmentproblem’ (PAP). First, a feasible solution to theGVRPTW is computed via a modification of the I1heuristic by Solomon (1987). Then, the PAP tries tofind a feasible driver assignment for the GVRPTWsolution. The assignment is achieved by creatinga table with all feasible combinations of driversand routes. Each table entry represents the costsresulting from the driver performing the route. Acomplete personal assignment is computed, usingthree different strategies, among them a greedyand a random procedure. After that, an improve-ment procedure embedded into a metaheuristicfollows.Synchronization en route. Kim, Koo, and Park(2010) study a combined vehicle routing and staffscheduling problem where a certain number oftasks has to be fulfilled in a fixed sequence atcustomers. Among the tasks, an end-to-start rela-tionship is assumed. In order to fulfill the tasks,different teams ofworkers are available. Each teamis qualified to performone specific type of task. Theteams cannot move by themselves; instead, a setof vehicles is used to transport the teams. Thereis no fixed assignment of a vehicle to a team, andeach vehicle may carry at most one team at a time.The authors develop an astonishingly simple pro-cedure, in which the vehicles, the teams, and thenext tasks for each customer are stored in threelists, along with the relevant information on timesand locations. In each iteration, a triplet (vehicle,team, task) is selected from the lists, using a best-fit criterion. Then, the lists are updated to reflectthe situation resulting when the selected vehicletransports the selected team to the location of theselected task.Hollis, Forbes, and Douglas (2006) describe a si-multaneous vehicle and crew routing and schedul-ing application for urban letter mail distribution atAustralia Post. The authors are the first to considera problem with multiple depots, where vehiclesand drivers may be stationed and interchanged.In order to solve the problem, a two-stage ap-proach is used. In the first stage, the authorsdetermine ‘abstract’ vehicle routes by solving a

pickup-and-delivery problem with time windows,multiple depots, a heterogeneous fleet of vehiclesas well as several working time restrictions fordrivers. A path-basedmixed-integer programming(MIP) model is presented and solved by heuristiccolumn generation. In the second stage, concretevehicle and crew schedules are determined tak-ing an integrated vehicle and crew scheduling ap-proach. This is again done by solving an MIP withheuristic column generation. In the second-stageMIP, the tasks to be performed correspond to thevehicle routes computed in the first stage.It is noteworthy that, for synchronization at a cen-tral depot as well as synchronization en route,although the objects to be synchronized in thepapers described are vehicles and drivers, the con-crete application contexts are almost all different(ranging from limousine rental to mail distribu-tion). As regards solution approaches, and, in par-ticular, the consideration of the interdependenceproblem, Zäpfel and Bögl (2008) perform, in prin-ciple, a decomposition of the problem by objecttype, exploiting the fact that drivers and vehiclesmay join and separate at one location only. On theother hand, Laurent and Hao (2007) and Prescott-Gagnon, Desaulniers, and Rousseau (2010) takethe opposite way and compute routes for predeter-mined driver-vehicle pairs. Xiang, Chu, and Chen(2006) andHollis, Forbes, andDouglas (2006) de-termine abstract routes first, and assign concretevehicles and drivers afterwards. Kim, Koo, andPark (2010) use a heuristic in which a direct selec-tion of vehicles, teams, and tasks is performed.A recurring idea in most solution procedures de-scribed above is to decompose the problem intoseveral stages and, in early stages, partially takeaspects into account that are important to be ableto obtain feasible solutions in later stages. Oursolution approach, described in Section 6, alsofollows this principle.As described in Section 2, the problem under con-sideration here is an SVCRSP in which EuropeanUnion social legislation on drivers’ driving andworking times (driver rules for short) are consid-ered. As mentioned, the relevant rules are brieflydescribed in Section 5; detailed information onthese regulations is given in the Appendix. Pro-cedures for considering them in VRP algorithmsare described, for example, in Goel (2009), Drexland Prescott-Gagnon (2010), Goel (2010), Kok,Meyer, Kopfer, and Schutten (2010), and Prescott-

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Gagnon, Desaulniers, Drexl, and Rousseau (2010).Our SVCRSP algorithm uses the procedure de-scribed in the last reference. Driver rules are inforce also in other countries or regions throughoutthe world. As we focus here on the situation in Eu-rope, considering rules other than the Europeanones is beyond the scope of this paper.A related and well studied field of application isintegrated vehicle and crew scheduling, for ex-ample in the airline business or in the publictransport sector. Pertinent surveys are given inKlabjan (2005), Caprara, Kroon, Monaci, Peeters,and Toth (2007), and Desaulniers and Hickman(2007). However, the models and solution ap-proachesused for solving suchproblemsare alwaysbased on the fundamental assumption of a giventimetable for the requests/tasks. Since we assumehere that the required transports do not have agiven timetable, the SVCRSP requires different orat least modified solution methods.

4 Important features of theproblem

The important and distinguishing features of thepresent problem compared to the ones presentedin the references discussed above lie in the consid-eration of the following criteria:• Practical and real-life assumptions:–Shipments/requests may originate and endanywhere, not only at truck or driver depots.

–A route segment between two relay stationsmay be driven by any driver.

–Atruckneednotbe emptywhenadriver changeis performed at a relay station.

–Driver shuttle transports to and from relaystations are possible.

–European Union social legislation on driverdriving and working times is considered (seenext section).

• Components of the solution approach:–The algorithm is based on a heuristic decom-position of the overall problem into two stages.However, compared to the highly sophisticatedapproaches taken in some of the above papers,our algorithm is rather simple and straightfor-ward and primarily relies on an appropriatenetwork representation of the problem.

–Both stages can be solved with essentially thesame algorithm.

–The large neighborhood search heuristic we

use can very easily be replaced by any othermetaheuristic.

5 Driver rulesThe current EU legislation specifies a complex setof rules on driver driving and working times (fora detailed discussion see the Appendix). For ourwork, we consider the following rules:• A break of at least 45 minutes is required–after 4.5 hours of driving, or–after 6 hours of work.

• A daily rest of at least 11 hours is required–after 9 hours of driving, or–after 9.6 hours of work, or–13 hours of wall-clock time after the end of thelast daily rest.

• A weekly rest of at least 24 hours must be taken–after 45 hours of driving, or–after 48 hours of working, or–144 hours (6 days) of wall-clock time after theend of the last weekly rest.

‘Work’ encompasses driving as well as comple-mentary activities such as loading or unloading,paperwork etc.Driver schedules that comply with these rules maybe considered legal. The EU rules provide addi-tional degrees of freedom and allow splitting ofbreaks or rests, shortening daily rests, and extend-ing daily or weekly driving and working times.We do not consider these options for two reasons:First, taking all these options into accountwouldbetoo computationally costly for the large instanceswe attempt to solve. Second, not considering op-tional rules makes the resulting route plans andschedules more robust and thus more suitable forpractice: In the case of unexpected events, such asdetours, traffic jams, vehicle breakdowns etc., thepossibility to shorten a rest or increase driving orworking time may help to stick to an original planand fulfill all requests in time.Thus, the relevant resources concerning driverrules are the times since the end of the last weeklyanddaily rest, theweekly, daily and interval drivingtimes, and the weekly, daily and interval workingtimes. At this, the daily driving (working) timeis the total accumulated driving (working) timebetween two daily rests and the interval driving(working) time is the total accumulated driving(working) time between two breaks. At the begin-ning of the planning horizon, each truck is located

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at its home depot, and all drivers are consideredrested, that is, the aforementioned time resourcesare all set to zero.Note that the law makes no requirements withrespect to the location where breaks or daily orweekly rests must be taken; these may be takenanywhere en route, such as on customer premises,at public parking places, or even at the roadside.All the references on driver rules cited in the lit-erature review above assume that breaks or restscan be taken directly when scheduled and withoutincurring any detour to or from a suitable location.That is, it is essentially assumed that the driversimply stops at the roadside to take a break or restif he is away fromadepot or a relay station or a cus-tomer location. By allowing breaks or daily reststo be taken anywhere, specifying a maximum timeperiod for two subsequent visits at a relay station(henceforth called inter-visit time), and requiringthat a daily rest must be taken before leaving a re-lay station with a truck, we follow this establishedapproach. (In the computational experiments, dif-ferent values for the inter-visit time were chosen,see Section 7).

6 Solution approachAs mentioned, the basic idea of our solution al-gorithm is to take a two-stage approach. In thefirst stage, routes for trucks are determined, takinginto account somedriver rules. In the second stage,routes for drivers are computed, based on the truckroutes from stage 1 and taking into account the re-maining driver rules to ensure feasibility. We thusadopt some ideas fromHollis, Forbes, andDouglas(2006), but in our first stage, routes are computedfor concrete trucks.More precisely, the first stage consists in the so-lution of a pickup-and-delivery problem with timewindows, relay stations, and additional constraints(PDPTWRS). The vehicles in the PDPTWRS cor-respond to the trucks of the underlying SVCRSP.There are multiple depots, namely, the home de-pots of the trucks. Each truck performs atmost oneroute, and the maximum route duration is set tothe length of the planning horizon. For each truck,driver rules are considered in a manner describedin detail below. At the end of the planning horizon,each truck must return to its home depot.The second stage consists in the solution of avehicle routing problem with time windows and

multiple depots (VRPTWMD). In this problem, thepartial truck routes or segments starting and/orending at a home depot or a relay station, asdetermined in the first stage, are the customers.The drivers are routed according to the solutionof the second-stage problem, that is, the driversact as the vehicles in the VRPTWMD. Each driverdrives at most one route, and the maximum routeduration is again set to the length of the planninghorizon. Figure 1 shows the relationship betweenthe two stages based on an example instance.We consider an instance with three depots (1, 2, 3),three relay stations (r1, r2, r3), and four pickup-and-delivery requests (pi → di, i = 1, . . . , 4). Apossible solution of stage 1, depicted on the left sideof the upper part of the figure, consists of two truckroutes. The route of truck 1 starts at the first depot,picksup the tworequests 1 and2, visits relay stationr1, delivers the two requests 1 and 2, and returns tothe depot. Consequently, the route is comprised oftwo segments. The first one, segment 1.1, goes fromdepot 1 to relay station r1, and yields customer c1for stage 2 of the solution procedure. The secondsegment goes from relay station r1 back to thedepot, and yields customer c2 for stage 2. Similarly,the route of truck 2 consists of three segmentsand, hence, yields three customers for stage 2.A possible solution of stage 2, depicted on theright side of the upper part of the figure, containsthree driver routes. Driver 1 starts at depot 1,‘visits’ the two ‘customers’ c1 and c5, and returnsto the depot. Driver 2 starts at depot 2, visitscustomers c3 and c2, and returns. Finally, driver 3starts at depot 3, visits customer c4, and returns.Each shape on the left-hand side corresponds toa physical location, and the depicted routes showthe exact movements of the two trucks. On theright-hand side, only the triangles correspond toa physical location. The rectangles represent, incompact form, a movement in space. The lowerpart of the figure illustrates the exactmovements ofthe drivers, that is, ‘extracts’ the route informationhidden in the stage 2 customers. There, again, allshapes correspond to physical locations.Note that the idea to model route segments ascustomers in a VRPTW was previously used in thealready mentioned paper by Hollis, Forbes, andDouglas (2006) as well as in Fügenschuh (2006).In the latter paper, however, there is no simultane-ous vehicle and driver routing and scheduling andthus no stage 1.

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Figure 1: Relationship between stages 1 and 2

Stage 1 solution (truck movements)

Truck 1:

Segment 1.1

=̂ Stage 2 customer c1

Segment 1.2


1 p1 p2 r1 d1 d2 1

Segment 2.1

=̂ c3

Segment 2.2

=̂ c4

Segment 2.3

=̂ c5

Truck 2: 2 p3 r2 p4 d3 r3 d4 2

i Depot pi Pickup

di Deliveryri Relay station

Stage 2 solution

Driver 1: 1 c1 c5 1

Driver 2: 2 c3 c2 2

Driver 3: 3 c4 3

i Depot

ci Customer

Driver movements according to stage 2 solution

Driver 1: 1 p1 p2 r1 r3 d4 2 1Shuttle Shuttle

Driver 2: 2 p3 r2 r1 d1 d2 1 2Shuttle Shuttle

Driver 3: 3 r2 p4 d3 r3 3Shuttle Shuttle

In the following, themain features of the problemsin the two stages are summed up.

Stage 1 problem:Pickup-and-delivery problem with timewindows and relay stations (PDPTWRS):• Pickup-and-delivery requests with given capac-ity requirements (weight, pallet places) and load-ing and unloading times

• Hard, single time windows at locations• Vehicles stationed at different depots, and dif-fering with respect to capacities and ability toperform requests

• Each vehicle performs atmost one route startingand ending at specified depots (start and enddepot may be different)

• Maximum route duration equal to the length ofthe planning horizon (6 days)

• Maximum time between two visits at a relaystation

• Driver rules:–Break of 45 minutes after 4.5 hours of driving–Break of 45 minutes after 6 hours of working

(driving, loading, unloading)–Daily rest of 11 hours after∗9 hours of driving or∗9.6 hours of working or∗13 hours of wall-clock time after the end ofthe last daily rest

• Objective function: Minimize sum of fixed andvariable costs

Stage 2 problem:Vehicle routingproblemwith timewindowsand multiple depots (VRPTWMD):• Customers with given time windows and servicetimes, no capacity requirements

• Vehicles stationed at different depots, no restric-tions regarding ability to visit customers

• Each vehicle performs atmost one route startingand ending at the same specified depot

• Driver rules:–Time between two consecutive daily rests lessthan or equal to 13 hours

–Route duration less than or equal to the

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maximum time between two weekly rests(144 hours)

–Overall driving time less than or equal to45 hours

–Overall working time less than or equal to48 hours

• Objective function: Minimize sum of fixed andvariable costs for drivers and shuttles

Note that the maximum route duration in bothproblems is the same. This is because we assumea planning horizon of one working week as thisis common in practice. Theoretically, the plan-ning horizon for the stage 1 problem, where thetruck routes are computed, could be chosen arbi-trarily, whereas in the stage 2 problem, where thedriver routes are determined, the timebetween twoweekly rests is postulated by the law. Note furtherthat it is necessary to consider a period of 13 hoursbetween two daily rests in both problems to en-sure that feasible driver routes can be computed instage 2. This is because in stage 1, the 13-hour ruleis only maintained on each truck route segmentbetween two relay stations, whereas in stage 2,the 13-hour rule must also be ensured if a driverchanges trucks. Otherwise, it might happen that adriver is assigned two truck route segments of lessthan 13 hours’ duration each without scheduleddaily rest. If the two segments have a cumulativeduration of more than 13 hours, then not consid-ering the 13-hour rule in stage 2 would lead to anillegal driver route.In both stages, we apply an implementation ofa large neighborhood search heuristic, see Shaw(1997), Ropke and Pisinger (2006), Pisinger andRopke (2007). Essentially, in such methods,elements of a solution are alternately removed(destruction step) and reinserted (reconstructionstep) in order to improve a given solution. In thecontext of PDPs/VRPs, given a complete routeplan, a subset of requests/customers is removedfrom their respective routes in the destruction stepand reinserted into the resulting partial routes inthe reconstruction step. The basic steps of LNS ingeneral and our implementation in particular arethe following:

Large Neighborhood SearchConstruct an initial feasible solution x and save xas the current best solution x*, i.e., set x* := x

RepeatSelect a destruction and a repair operatorCreate a neighboring solution x′ from x usingthe procedures corresponding to the selecteddestruction and repair operatorsImprove the new solution x′ by local searchIf solutionx′ canbeaccepted, i.e., if anacceptancecriterion is fulfilled, set x := x′

If x is better (has a lower objective function value)than x*, set x* := x

Until a termination criterion is reachedReturn x*

For the construction of an initial feasible solutionas well as for reconstruction purposes, we use aparallel best insertion procedure: In each itera-tion, for each remaining unplanned request, thebest (cheapest) possible insertion positions for thepickup and the delivery location into each existingroute are determined. The request whose best pos-sible insertion causes the smallest increase in thecosts of the current route plan is then included. Anew vehicle is used only if an insertion into existingroutes is impossible. Since our SVCRSP considersa heterogeneous fleet, whenever a new route is tobe created, all unused vehiclesmust be checked forsuitability. After the recreate step, we use a swapoperator, that is, the exchange of two requests be-tween two routes, as a local search improvementstep.Weuse the following three destruction procedures:• A request is selected at random and the routethat contains this request is destroyed, that is,all requests that are on the same route as thespecified request are also selected. This is re-peated until a specified percentage of requestshas been determined.

• A route ρ1 is selected at random. This route andthe one closest to it, say, route ρ2, are destroyed.(A route ρ2 is closest to ρ1 if the overall pairwisedistance of all locations visited in ρ1 to the lo-cations visited in ρ2 is minimal over all routes.)This is repeated until a specified percentage ofrequests is determined.

• A subset S of requests is selected at random. (Therequests need not necessarily be on the sameroute.) Then, further requests whose pickup ordelivery locations are closest to any pickup ordelivery location of the requests in S are selecteduntil a specified percentage of requests has beendetermined.

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The selected requests are then removed from theirroutes, and vehicles serving now empty routes areconsidered unused. In each iteration, one of thethree destruction operators is chosen randomly.The algorithm accepts every new solution that ex-ceeds the old objective function value, c(x), by lessthan a given factor δ ≥ 1; that is, a new solution x′ isaccepted if c(x′) < δc(x). The algorithm terminatesafter a specifiednumberof iterations (cf. Section7).Our implementation is described in more detail inSigl (2009). The design of the code is generic andcan easily be adapted, or, rather, instantiated, tosolve PDPTWRSs as well as VRPTWMDs.In our algorithm, there are data structures thatstore and update the resources of each individualdriver throughout theplanninghorizon to correctlyaccount for each individual driver’s driving andworking history from the beginning of the plan-ning horizon. Similarly, there are data structuresfor each individual vehicle, storing all relevant in-formation (costs, capacities, homedepot, itinerary,schedule).

6.1 Stage 1: Determination of truckroutes

The determination of truck routes in the courseof the PDPTWRS in the first stage is based on anappropriate network representation. In addition,the consideration of driver rules and relay stationsrequires modeling efforts.

6.1.1 Construction of the networkWe consider a network D1 = (V1,A1), where V1

is the set of vertices and A1 is the set of arcs.The vertex set V1 = S ∪· E ∪· P ∪· D∪· R consists ofvertices for start and end depots in subsets S andE respectively, vertices for pickup and delivery insubsets P and D, and vertices for relay stations insubset R. (‘A∪· B’ denotes the union of the disjointsets A and B.) This means that if a physical de-pot location may be used as a relay station, thislocation is represented by three different verticesin the network (one in S, one in E, and one in R).The start of (un)loading at any vertex i ∈ V mustbe within a prescribed time window [ai, bi]. De-pot and relay station vertices have time windowsstarting Monday, 0:00 hours, and ending Satur-day, 24:00 hours. For each request, a pickup and adelivery vertex with associated time windows and(un)loading times are given. The (un)loading must

begin, but need not be finished, within the timewindow. The arc set A1 is composed as follows:(i) From each start depot vertex i, there is one

arc to the corresponding end depot vertex (forunused trucks), to each relay station vertex j ∈R (since it may be sensible or necessary tovisit a relay station before picking up the firstrequest), and to each pickup vertex.

(ii) There is an arc between each pair of relaystation vertices, from each relay station ver-tex i ∈ R to every pickup and every deliveryvertex, and to each end depot vertex j ∈ E.

(iii) Finally, there is an arc between each pair ofrequest vertices (except for an arc from thedelivery vertex of a request to the correspond-ing pickup vertex), an arc from each requestvertex to each relay station vertex, and thereis an arc from each delivery vertex to each enddepot vertex.

Each arc (i, j) has an associated distance dij ≥ 0

and travel time tij ≥ 0. These weights correspondto the real-world distance and travel time by truckbetween the respective locations and are assumedto be equal for each truck type. The travel timeindicates the pure driving time, disregarding anybreaks or rests. Of course, only arcs (i, j) withai + tij ≤ bj are introduced.An example network is depicted in Figure 2. Thenetwork represents an instance with one depotlocation, three relay station locations, and twopickup-and-delivery requests. Consequently, thenetwork contains one start depot vertex, one enddepot vertex, three relay station vertices, and twopickup and twodelivery vertices. To keep the figureclear and concise, there is only one arc for each arctype described in items (i)-(iii). For example, theabsence of an arc from vertex p1 to vertex d1 doesnot mean that there is no such arc in the network.

6.1.2 Consideration of driver rules in stage 1In the PDPTWRS, that is, in the first stage of oursolution approach, routes are determined for thetrucks. The routes start at the depot vertices andvisit pickup and delivery vertices of compatiblerequests, maintaining the capacity constraints ofthe trucks, the time windows of the requests, thetime period until the next visit at a relay station,and the following driver rules resources: time sincethe end of the last daily rest, daily and intervaldriving time, daily and interval working time. Avisit at a relay station resets all these driver rules

250


Figure 2: Example network for stage 1

Start or end depot vertex

pi Pickup vertex

di Delivery vertex

ri Relay station vertex

s

p1

p2

r2

r1

r3

d1

d2

e

resources; the other resources are considered inthe second stage of our solution procedure.An important feature of the approach is that thetrucks need not be empty when visiting a relaystation. Usually, at relay stations, a driver changeis performed. For that reason, a truck may leave arelay station after a short constant period of timeafter having reached the station, a so-called relaytime. In the computational experiments, this timewas set to 15 minutes.Within our insertion heuristics, insertions are per-formed with the objective of minimizing costs. Fora route, that is, for a given sequence of vertices,determined according to this objective, a scheduleconsidering the above-mentioned driver rules iscomputed that has a minimum overall route dura-tion under the condition to arrive at the end depotvertex as early as possible. This is sensible whenone and the same driver drives the entire routeof a truck. For each route, a complete scheduledsequence of activities (drive, wait, (un)load, break,rest) is determined, indicating the earliest sensiblebeginning of each activity. This means that a con-crete departure time from the start vertex of eachsegment is known.

6.1.3 Procedure for considering relay stationsThe procedure for considering relay stations is astraightforward extension of the existing proce-dure for the PDPTW. In each iteration, after routeshave been modified by adding, shifting, or remov-ing requests in the (re)construction or destructionstep, a concrete schedule is computed for all modi-fied routes, taking into account driver rules. Theseroutes are then checked for whether they violatethe maximum time period between two visits ata relay station (but are otherwise feasible with

respect to time windows, driver rules, and truckcapacity).For a route/vertex sequence (s, i1, i2, . . . , im,im+1, . . . , in, e), this check works as follows: Theroute is traversed backward from the end depotvertex e ∈ E. If, according to the route’s schedule,the time period between the arrival at e and thedeparture from the last relay station visited (or thestart depot, if no relay station is visited at all) isgreater than allowed, a relay station is inserted.Assume, for example, that, in the above route,im ∈ S∪· P∪· D∪· R is the first vertex encounteredsuch that there is no relay station between im ande and that the time between the departure from imand the arrival at e exceeds the allowed time. Thena relay station vertex r ∈ R is inserted betweenim and e. (The important question of which ver-tex r ∈ R to insert is discussed below.) An attemptismade to insert rdirectly after im. If this is not pos-sible (due to time window constraints), insertionis attempted directly after im+1 etc. The resultingroute must be feasible with respect to time win-dows, driver rules and time period between theinserted relay station vertex r and the end depotvertex e. To verify this, it is necessary to computea new schedule for the complete route. If the in-sertion was successful, the procedure is repeated,now moving backward from the last inserted relaystation vertex r toward the start depot vertex s.These steps are iterated until a feasible route anda corresponding schedule are determined or theroute is discarded for infeasibility.If a request is removed from a route in the de-struction step of the large neighborhood search,all relay stations visited after the pickup vertex ofthe request are removed, too. This is motivated bythe fact that it may be useful to reduce the number

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of relay stations, but to keep some relay stationsin a partial route that was shown to be feasible.In the computational experiments, also attemptsweremade to remove all relay stations froma routeif at least one request was removed from this route.This increased the running times, but did not leadto better results.As mentioned, an important issue is the selectionof the relay station vertex to be inserted betweentwo vertices. Here, we opted for the station caus-ing the smallest detour. Seen globally, though,this might be suboptimal, since there might be nodriver for changing at the relay station selected inthis way. There might be more drivers availablefor continuing the truck route at other, more dis-tant, relay stations. Choosing such a station mighthelp to avoid shuttle transports. Hence, storingthe number of drivers available at different relaystations in different time intervals might be sensi-ble. However, such a more sophisticated strategyis highly unlikely to be beneficial: Visiting a moredistant relay station to find a suitable truck/driverfor continuing the route will obviously increase thetruck distance and the driving time compared tovisiting the closest station. This means that shut-tle distance and travel time are replaced by truckdistance and travel time, but truck kilometers aremore expensive than shuttle kilometers, and truckhours count as driving time for the driver, whereasshuttle time does not.

6.2 Stage 2: Determination of driverroutes

In the second stage, a driver must be found foreach truck route segment determined in stage 1.Recall that the truck routes consist of one or moreroute segments starting and ending at a depot or arelay station. Each segmentmust by assumption bedriven by one and the same driver with one and thesame truck. To obtain feasible routes for drivers,the following points must be ensured (since theyhave not already been ensured in stage 1):(a)The time between two consecutive daily rests

on a driver route must be less than or equal to13 hours.

(b)The overall duration of a driver route must beless than or equal to 144 hours.

(c)The overall driving (working) time on a driverroute must be less than or equal to 45 hours(48 hours).

To achieve this, an appropriate network represen-tation is required.

6.2.1 Construction of the networkThe network D2 = (V2,A2) for the VRPTWMD isset up as follows: V2 consists of one start and oneend depot vertex for each home depot of a driver,and of one customer vertex for each segment ofa truck route computed in stage 1. For a stage 2customer vertex i, let si ∈ S∪·R and ei ∈ R∪·E denotethe start and end vertices of its correspondingstage 1 segment respectively. That is, the stage 1segment corresponding to stage 2 vertex i startsat location lsi and ends at location lei . All such lsiand lei are either depots or relay stations. Again, thedepot verticeshave timewindowsstartingMonday,0:00hours, andendingSaturday, 24:00hours.Thetime window [ai, bi] of each customer vertex i isequal to the start timewindowof the correspondingsegment. Thedeterminationof these timewindowsis explained in detail below.If no shuttle transports are allowed, the arc set A2

is comprised of the following arcs (where li denotesthe physical location corresponding to vertex i ∈V2):(i) From each start depot vertex i, there is one

arc to the corresponding end depot vertex (forunused drivers) and to each customer vertex jwith li = lsj .

(ii) There is an arc between each pair (i, j) ofcustomer vertices with lei = lsj .

(iii) Finally, there is an arc from each customervertex i to each end depot vertex j with lei = lj.

If shuttle transports are allowed, the li = lsj , lei = lsj ,and lei = lj restrictions are abandoned. That is, iftime windows allow, there are arcs from each startdepot vertex to each customer vertex, between eachpair of customer vertices, and from each customervertex to each end depot vertex. Figure 3 depictsan example of how a network of a stage 2 instanceis constructed from a stage 1 solution.On the left-hand side of Figure 3, a solution tostage 1 is described that contains two truck routes.The first route starts and ends at depot 1 and con-sists of two segments. The second route starts andends at depot 2 and involves one segment. Theresulting stage 2 network is depicted on the right-hand side. It is assumed that drivers are availableat the same two depots where the trucks are sta-tioned. Therefore, the stage 2 network possessestwo start and two end depot vertices. Moreover,

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Figure 3: Example network for stage 2

Stage 1 solution

(truck movements in stage 1 network)

Truck 1:

Segment 1.1


Segment 1.2


s1 p1 p2 r1 d1 d2 e1

Segment 2.1


Truck 2: s2 p3 p4 d3 d4 e2

Start or end depot vertex pi Pickup vertex

di Delivery vertexri Relay station vertex

Resulting stage 2 network

s1

c1

e1

c2

s2

c3

e2

ci Customer vertex

Shuttle between different locations

in accordance with the routes from stage 1, thenetwork comprises three customer vertices, onefor each route segment. The solid arcs representpossible movements of drivers between verticescorresponding to the same physical location, thatis,wherenodriver shuttle transports arenecessary.For example, there is a solid arc from start depotvertex s1 to customer vertex c1, since c1 corre-sponds to stage 1 route segment 1.1, which starts atvertex s1. The dashed arcs represent driver move-ments between different physical locations, thatis, movements requiring a shuttle transport. Thedepicted stage 2 network is complete in the sensethat all possible arcs resulting from the stage 1 so-lution are shown. For example, there is no arc fromvertex c2 to c1, because using such an arc wouldimply that segment 1.2, which corresponds to c2, isperformed before segment 1.1, which correspondsto c1. This is impossible, since the requests deliv-ered on segment 1.2 are picked up on segment 1.1.Moreover, depending on the stage 2 time windows(see below), also some of the depicted arcs may beimpossible to use.Each arc (i, j) ∈ A2 has an associated dis-tance dshuttle

ij ≥ 0 and travel time tshuttleij ≥ 0. Inthe stage 2 network, these weights correspond tothe real-world distance and travel time by shuttlevan between the respective locations. The weightsdshuttleij are needed to determine the shuttle costs,

which are assumed to be distance-dependent. For

arcs (i, j) linking vertices corresponding to thesame physical location, dshuttle

ij := tshuttleij := 0. Thismeans that, if no shuttle transports are allowed,both dshuttle

ij and tshuttleij are zero for all arcs, sincein this case, there are only arcs linking verticescorresponding to the same physical location. Thetravel time tshuttleij indicates the pure driving time,disregarding any breaks or rests, because driverrules are not relevant for drivers of small shuttlevans. However, working time rules are relevant.To completely account for these, it would benecessary to introduce a third algorithmic stageand compute explicit routes and schedules forshuttle vans and their drivers. This is beyond thescope of the present paper and left as a topic forfuture research, see the Conclusions. Instead,we assume that there is a sufficient number ofshuttle vans with rested shuttle van drivers so that(i) a truck driver never has to wait for a shuttleand (ii) it can be assumed that each shuttle tripbetween two relay stations requires at most onebreak of 45 minutes for the shuttle driver, anddoes so only in the rare case that the pure drivingtime of the shuttle transport exceeds 6 hours.As shuttle vans have no speed limit on GermanAutobahns, this break time can be compensatedfor by going faster than the average speed used bya standard GIS system to compute driving times.In order to meet the above-mentioned three re-quirements (a)-(c) concerning driver rules, three

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additional weights, wdij ≥ 0, ww

ij ≥ 0, and tdriverij ≥0, are associated with each arc (i, j) ∈ A2. Theweightswd

ij andwwij respectively measure the driv-

ing andworking timeaccruedon theweeklydrivingand working time account of the driver who usesarc (i, j). The weight tdriverij measures the wall-clocktime that will have elapsed after arrival at ver-tex i or after the beginning of i’s time window,whichever is greater, until a driver can reach j viathe arc (i, j).To illustrate this, the following explanations areappropriate: If a driver uses an arc (i, j) emanatingfrom a customer vertex i and leading to a cus-tomer or end depot vertex j, this indicates that thedriver drives the stage 1 segment corresponding tocustomer i. In other words, a ‘visit’ at a stage 2customer induces a ‘service time’ at the customervertex. This service time is equal to the time forexecuting the corresponding stage 1 segment. Asmentioned above, this time is known from theunequivocal duration determined for each stage 1route and comprises driving, (un)loading, waiting,break, and rest times along the segment. Driv-ing and (un)loading time count as working time.Moreover, a change of vehicle incurs a relay time,which also counts as working time. (Entering thefirst vehicle at the beginning of a route and leavingthe last vehicle at the end of a route also takes time,but this time is not counted as working time.) Inaddition, a driver is assumed to take a daily restat a relay station, which increases overall routeduration.When shuttle transports are allowed, it must beconsidered that the time a truck driver spends be-ing driven in a shuttle van, although not countedas working time,must also not be counted as breakor rest time. Therefore, when a driver reaches arelay station after a shuttle ride, the time sincethe end of his last daily rest will be positive. Toensure that such a driver can still perform the cor-responding segment i ∈ V2, we take into accountthe time tbeforei that elapses before the beginning ofthe first daily rest, and the time tafteri that elapsesafter the end of the last daily rest on i. Both valuesare known from the computations performed instage 1. If no daily rest is taken on a segment, weassume that a daily rest is necessary directly beforeand directly after the segment. In order to ensurethat two segments/customers i and j requiring ashuttle transport can be performed consecutively

by any driver (at least as far as the 13-hour rule isconcerned), we distinguish three cases:(i) If tafteri + tshuttleij ≤ 13, the shuttle is carried out

directly after performing the segment corre-sponding to i, and a daily rest is taken at lsj .(In this case, it is assumed, without loss ofgenerality, that the daily rest lasts at least untilthe beginning of the time window at j.)

(ii) If tafteri + tshuttleij > 13 and tshuttleij + tbeforej ≤13, a daily rest is taken at lei directly afterperforming the segment corresponding to i;subsequently, the shuttle is executed and nodaily rest is taken at lsj . (In this case, it isassumed that the daily rest lasts long enoughso that there is no waiting time at j.)

(iii) If tafteri + tshuttleij > 13 and tshuttleij + tbeforej >13, a daily rest is taken directly before anddirectly after the shuttle transport. (Again, it isassumed that the daily rest at lsj lasts at leastuntil the beginning of the time window at j.)

Similar considerations can be applied to potentialarcs from a depot vertex to a customer vertex.Now, let tdrivei and twork

i respectively denote thedriving time and working time along the segmentcorresponding to i, and let toveralli denote the ‘ser-vice’ time for execution of the segment correspond-ing to i as explained above. The values of tdrivei andtworki can easily be determined from the scheduledsequence of activities computed in stage 1 for eachtruck route. Furthermore, let tdaily_rest denote thetime for the daily rest(s) between two vertices (11or 22 hours, depending on which of the abovecases (i)-(iii) holds), and let trelay denote the timefor handing over a truck to another driver or beinghanded over a truck from another driver. Then, inview of the above elaborations and the describednetwork structure, wd

ij, wwij , and tdriverij are set as

follows:• wd

ij := wwij := tdriverij := 0, if i is a start depot vertex

and j is a customer vertex with li = lsj .• wd

ij := wwij := 0, and tdriverij := tshuttleij + tdaily_rest, if

i is a start depot vertex and j is a customer vertexwith li ≠ lsj .

• wdij := tdrivei , ww

ij := tworki , and tdriverij := toveralli +

tdaily_rest, if i and j are customer vertices andconsecutive segments on one stage 1 route.In this case, no handover time is incurred, sincethe driver stays on the same vehicle.


ij := tworki + 2trelay, and tdriverij :=

toveralli + tdaily_rest + 2trelay + tshuttleij , if i and j are

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customer vertices and not consecutive segmentson one stage 1 route.Here, the first relay time is incurred for handingover the truckk to thedriverwhowill performthenext segment on k’s route. The second relay timeis incurred for being handed over the truck k′

from thedriverwhohasperformed theprecedingsegment on k′’s route.


ij := tworki , and tdriverij := toveralli , if i

is a customer vertex and j is an end depot vertexwith lei = lj.


ij := tworki , and tdriverij := toveralli +

tdaily_rest + tshuttleij , if i is a customer vertex and j isan end depot vertex with lei ≠ lj.Note that the end depot vertex corresponds tothe home depot location of the driver, not thetruck. The truck is left at location lei .

Using these values, only arcs (i, j) with ai + tdriverij ≤bj are introduced in order to satisfy the time win-dow constraints.The observance of 13 hours at most between twodaily rests is now ensured by the fact that a driveralways takes a daily rest between visiting two cus-tomers: At the relay station where the segmentcorresponding to the first customer ends, at therelay station where the segment corresponding tothe second customer starts, or at both locations.Using an arc (i, j) increases the route duration by awall-clock time of tdriverij . Depending on the actualarrival time at j, waiting timemay occur. However,as explained above, it is assumed that this waitingtime is added to the daily rest(s) taken between iand j.The observance of the time since the end of the lastweekly rest is ensured by the network structure.Each driver performs at most one route, he isassumed rested upon leaving his start depot vertex(that is, all driver rules resources are reset), and thetime windows at the start and end depot verticesprohibit routes longer than 144 hours.The only checks concerning driver rules that haveto be performedwithin the algorithm used to solvethe stage 2 problem, no matter which VRPTWMDalgorithm is actually used, are the checks whethertheaccumulatedweeklydrivingandworking times,that is, the sum of the wd

ij and the sum of the wwij

of all arcs on a driver’s route, are less than orequal to 45 and 48 hours respectively. Such checksare essentially equivalent to checking the vehiclecapacity constraint in a standard VRP, because the

capacity consumption in a standard VRP, thoughusually associated with a customer vertex, can ofcourse be associated with the arcs emanating fromthe vertex. For any insertion of a customer intoa route, this check can be performed in constanttime, independent of the number of customers onthe route, cf. Irnich (2008).

6.2.2 Considering time windows and dealingwith the interdependence problem

Even after separating the determination of truckand driver routes, the interdependence problemremains. Recall that in the stage 1 routine for de-termining a schedule considering driver rules, aschedule is computed specifying a concrete depar-ture time from the start vertex of each segment.Now, depending on the request time windows, thedriver rules, and the limited time between two vis-its at a relay station, it may on the one hand bepossible to arrive at a relay station vertex soonerthan the stage 1 schedule indicates, and it may onthe other hand be possible to depart from a relaystation vertex later than the stage 1 schedule de-notes. In other words, there may be slack withina segment/route. Moreover, the amount of slackmaydiffer between the segments of the same route.As an example, consider Figure 4 and assume, forsimplicity, zero (un)loading and relay times and amaximum time between two visits at a relay stationof 130 time units (arbitrarily chosen). Driver rulesare irrelevant in this example, because the overallroute duration is so short that no breaks or restsare necessary or sensible. T denotes the length ofthe planning horizon.The route depicted in Figure 4 performs twopickup-and-delivery requests (p1 → d1 and p2 →d2) and consists of two segments. Segment 1 startsat a start depot, picks up the two requests, andends at a relay station. Segment 2 starts at thisrelay station, delivers the two requests, and endsat an end depot. With respect to the time windowsof the requests, we distinguish three cases. In eachcase, the line ‘Schedule tominimize route duration’indicates the departure time at each vertex so thatthe overall route duration is minimized under thecondition to arrive at the end depot vertex as earlyas possible. The line ‘Overall route slack’ specifiesby how much the scheduled departure time fromthe start depot may be shifted without violatingany time windows. The line ‘Segment slacks’ pro-vides this information for each segment. The line

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Figure 4: Example for stage 2 time windows

s p1 p2 r d1 d2 e30 30 10 10 30 30Driving time:

Segment 1


Segment 2


Case 1:Original time windows: [0, T] [0, T] [75, 90] [0, T] [110, 135] [0, T] [0, T]

Schedule to minimize route duration: 30 60 90 100 110 140 170

Overall route slack: 0Segment slacks: 15 25

Time windows for stage 2 customers: [15, 30] [100, 125]

Case 2:Original time windows: [0, T] [0, T] [75, 90] [0, T] [100, 135] [0, T] [0, T]

Schedule to minimize route duration: 20 50 80 90 100 130 160Overall route slack: 10Segment slacks: 15 35

Time windows for stage 2 customers: [15, 30] [90, 125]Case 3:Original time windows: [0, T] [0, T] [75, 120] [0, T] [100, 125] [0, T] [0, T]

Schedule to minimize route duration: 20 50 80 90 100 130 160Overall route slack: 25

Segment slacks: 45 25

Time windows for stage 2 customers: [15, 60] [90, 115]

‘Time windows for stage 2 customers’ indicatesthe longest possible time windows for the stage 2customers corresponding to the stage 1 segments,based on the minimum duration schedule and thesegment slacks. The durations of the two segments,or, in other words, the service times for the stage 2customers, are the same in all three cases: Theyare simply the sum of the segment driving times,that is, 70 time units (since there is no unavoid-able waiting time, and since there are no stage 1(un)loading or relay times).In case 1, it is easy to see that the time windows forthe customers in the stage 2 VRPTWMD induce nointerdependencies: If customer c1 is visited at time30, that is, as late as possible, then customer c2can still be visited at time 100, that is, as earlyas possible. This is because it is ensured that thestage 1 segment corresponding to c1will be finishedearly enough (30+70 = 100), so that the truck usedto perform both segments will be at the relaystation no later than at time 100.In case 2, if customer c1 is visited later than attime 20, customer c2 is affected, and thus thestage 2 routes visiting the customers are interde-pendent. For example, let c2 be on route ρ2 andsuppose that, due to subsequent customers, c2must be visited no later than at time 99. Further

assume that the route visiting customer c1, say,ρ1, is changed so that c1 can be visited no earlierthan at time 30. Then, the truck used to performboth segments will not have arrived at the relaystation at time 99, and thus ρ2 becomes infeasible.This, in turn, implies that in stage 2, route changessuch as additions, removals, or repositionings ofcustomers may have side effects on other routes,and these effects must be controlled in some wayto be able to determine feasible solutions.In case 3, the situation is still worse. Even if thetime windows for the stage 2 customers are notreduced due to insertion into routes, any routevisiting c2 becomes infeasible if c1 is visited laterthan at time 45.To overcome this interdependence problem, thereare several possibilities. The simplest option, ofcourse, is to use the given concrete schedule foreach route (that is, a timewindowof length zero foreach stage 2 request) and essentially solve a purevehicle scheduling problem. This greatly reducesthe optimization potential for stage 2.The other extreme is to use the slack as describedabove and, in stage 2, consider the route inter-dependencies and devise a specific algorithm forsolving the VRPTWMD. This preserves the com-plete optimization potential (as given by the stage 1

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solution). It is doubtful, though, whether this sec-ond approach is promising for the large instanceswe try to solve.We therefore chose a compromise. Using the flex-ibility given by a stage 1 solution, we set the timewindows of the stage 2 requests in such a way thatall requests, also those corresponding to segmentsbelonging to the same stage 1 route, are indepen-dent of one another. More precisely, we define theslack for each segment as the maximal time bywhich the departure time from the start vertex of asegment canbe deferredwithout violating any timewindow within the current segment and such thatthe actual departure of the subsequent segment isnot delayed. In this way, no request interdepen-dencies arise in stage 2, but still some flexibilitywith respect to the time windows of the requestsis achieved. For the three cases in Figure 4, theprocedure works as follows: In case 1, as statedabove, wemay set the time windows for the stage 2requests to [15, 30] and [100, 125] respectively. Incase 2, we may determine the time windows to[15, 20] and [90, 125], and, finally, in case 3, to[15, 20] and [90, 115].

6.3 Comments on the solution approachThe consequence of disregarding the optionaldriver rules is that our algorithm sometimescannot find a legal schedule for a route, althoughsuch a schedule exists. It is guaranteed, though,that our algorithm only returns driver routesfor which a legal schedule has been found. If nolegal schedule is found for a route (no matterwhether this is because there is no legal scheduleor because the only legal schedule(s) require(s)exploiting optional rules), the route is discarded.It is not possible to reverse the two stages andcompute driver routes in the first stage and truckroutes in the second, unless it is assumed thatonly empty trucks may reach relay stations. This isbecause when a driver route is computed in stage 1,it is unknown which locations the driver will haveto visit after leaving a relay station, because it isunknown which truck he will use and thus whichrequests he will have to deliver.In tramp transportation in practice, the planningsituation is often a dynamic one. Not all requestsmay be known in advance, and it may therefore benecessary to perform rolling horizon planning. Insuch a situation, only one or a few segments must

be computed in advance for a truck. The procedurein stage 1 remains the same, but the number ofrequests decreases. Stage 2 becomes the problemof assigning exactly one driver to each segmentand at most one segment to each driver. Thus, theoverall problem becomes easier.

7 Computational experimentsThe present work was motivated by a researchproject studying the potential of an adoption ofthe advanced truckload business model to Ger-many and Europe. This business model originatedin North America as a reaction to the deregulationof the road transport market and has provideda competitive edge for companies that success-fully implemented it, the so-called advanced truck-load firms (ATLFs). Details on the ATLF businessmodel can be found in Walther (2010). One pillarof ATLFs is the increased temporal utilization oftrucks obtained by allowing drivers to switch vehi-cles (commonly referred to as slipseating). There-fore, to assess the potential of slipseating for long-distance tramp transport in Germany by truck,an extensive real-world data set was provided byour practice partner, a major German freight for-warder. The data set comprises 2,800 pickup-and-delivery requests between 1,975 locations and dur-ing a planning horizon of six days from Mondayto Saturday, 1,645 trucks and drivers stationedat 43 depots, each of which can also function asrelay station, and 157 additional relay station lo-cations. All locations are dispersed over an areaof approximately 350,000 km2. The fleet consistsof two types of trucks differing with respect tocapacity and ability to perform certain requests,but with the same time- and distance-dependentcosts. Every driver is able to drive any truck, andall drivers receive the same wage. An unlimitednumber of identical shuttle vans is available at anytime at any location. The shuttle vans are assumedto incur distance-dependent costs only. Table 1specifies some further indicators.We have no data on driver resources such as driv-ing time during the last week etc. Therefore, we as-sume that at the beginning of the planning horizon,all drivers have the same average driver rules re-sources, as specified in Section6. For our purposes,that is, to identify savings potential by switchingdrivers compared to a fixed truck-driver assign-ment, this is a valid approach, because for both

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Table 1: Data set indicatorsIndicator Min. Avg. Max.

Length of pickup time window [hh:mm] 00:00 06:39 13:20Length of delivery time window [hh:mm] 00:00 18:00 30:25Time between earliest pickup and latest delivery [hh:mm] 02:15 28:10 52:00Distance between pickup and delivery location [truck-kms] 1 391 1,117Request size [Tons / Loading meters] 0.1 / 0.4 10.8 / 9.4 25.0 / 14.6Vehicle capacity type 1 [Tons / Loading meters] 22.0 / 15.3 22.0 / 15.3 22.0 / 15.3Vehicle capacity type 2 [Tons / Loading meters] 25.0 / 13.6 25.0 / 13.6 25.0 / 13.6

situations, free as well as fixed truck-driver as-signment, the same assumptions regarding driverresources are made.To test and evaluate our algorithm, we imple-mented its different modules in Delphi/ObjectPascal as an add-on to our LNS code describedin Section 6. We used the following settings andparameters: In each destruction step, 5% of re-quests are removed. Due to the large number of re-quests, a higher percentage is too time consuming.Initially, only improving solutions are accepted.After 500 iterations without improvement, the δvalue for accepting new solutions is set to 1.5and is then gradually decreased to 1 every 500 it-erations without improvement. 12,500 iterationsof the large neighborhood search algorithm, thatis, 12,500 destruction and 12,500 reconstructionsteps, are performed in each stage.We examined the following 14 scenarios:• Scenario 1: A fixed truck-driver assignmentwith-out the need to visit a relay station. Thiswas usedas the baseline scenario, since it reflects currentpractice.

• Scenarios 2-7: A fixed truck-driver assignmentwhere visiting one of 43, 100, 200 relay stationsis required every 37 or 48 hours.

• Scenario 8: A free truck-driver assignment with-out the need to visit a relay station and with theassumption that a daily rest takes only 15 min-utes. This is equivalent to assuming that thereare infinitely many drivers available at any pointin space and time without incurring any shuttlecosts. This scenario yields an upper bound onthe possible savings.

• Scenarios 9-14: A free truck-driver assignmentwhere visiting one of 43, 100, 200 relay stationsis required every 37 or 48 hours, and where thepossibility of driver shuttle transports exists.

The values of 37 and 48 hours for the inter-visit

time (the time between two consecutive visits ata relay station) were chosen for the following rea-sons. The 37-hour period is motivated by the ex-isting legislation: As mentioned, 13 hours after theend of a daily rest, the next daily rest (of 11 hours)must start. Thus, by ensuring that a truck visits arelay station after 37 hours, it is guaranteed thata driver spends at least every other daily rest at arelay station. A period of 48 hours allows at leasttwo visits per week at a relay station.The objective was to minimize the sum of over-all operating costs, which are comprised of fourcomponents:• Fixed costs for–each truck that leaves its home depot(150.06 EUR) and

–each truck driver who leaves his home depot(175.40 EUR).

According to current practice in long-distanceroad transport in Germany, these costs arecharged in daily rates for each started 24-hourperiod. For example, if a truck route has aduration of 25 hours, counted from the point intime when the truck leaves the start depot untilit returns to the end depot, two daily rates arecharged for this truck.

• Distance-dependent costs for–each truck-km (0.54 EUR) and–each shuttle-km (0.12 EUR).

The first scenario was computed with a basic vari-ant of the stage 1 procedure,whereno relay stationsare considered. This variant was also used for sce-nario 8, where, in addition, the daily rest durationwas set to 15 minutes. Scenarios 2-7 were tackledusing only the stage 1 part of our algorithm; scenar-ios 9-14 were solved using both described stages.Tables 2 and 3 show the computational results. Inthe tables, the lines ‘No. days’, ‘No. truck days’, and‘No. driver days’ indicate the number of daily rates

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Table 2: Computational results for fixed truck-driver assignment

Scenario 1 2 3 4 5 6 7

No. relay stations − 43 43 100 100 200 200

Inter-visit time − 37 48 37 48 37 48

Absolute values:

No. routes 729 767 746 768 752 756 758

No. days 3,419 3,413 3,424 3,456 3,429 3,410 3,457

Kilometers 1,697,000 1,786,000 1,752,000 1,778,000 1,754,000 1,752,000 1,755,000

Costs 2,029,000 2,075,000 2,060,000 2,085,000 2,063,000 2,056,000 2,073,000

Relative changes to baseline scenario 1 in percent:

No. routes 0.00 5.21 2.33 5.35 3.16 3.70 3.98

No. days 0.00 −0.17 0.15 1.08 0.30 −0.28 1.10

Kilometers 0.00 5.21 3.19 4.77 3.33 3.20 3.39

Costs 0.00 2.27 1.53 2.76 1.68 1.33 2.17

charged for trucks and/or drivers respectively.First and foremost, the computational experimentsdemonstrated that our algorithm is capable of solv-ing large real-world instances and is able to achieveconsistent relevant results. Moreover, the follow-ing observations can be made in Tables 2 and 3:• For fixed truck-driver assignment, thenumber ofroutes and the overall distance traveled increaseconsiderably when relay stations have to be vis-ited, compared to the baseline scenario wherethis is not required. The number of truck days,in contrast, changes only slightly. This impliesthat the average route duration decreases.

• For free truck-driver assignment, the numberof truck routes and truck days decreases sig-nificantly in each scenario s = 8, . . . , 14, bothcompared to the baseline scenario 1 as well asto the corresponding scenario with fixed truck-driver assignment (scenario s − 7). This impliesan increase in the average truck route duration,that is, a better temporal utilization of the trucks,which is the result desired. The overall distancetraveled also increases compared to the baselinescenario, but changes only slightly compared tothe corresponding scenarios with fixed truck-driver assignment. On the other hand, however,for scenarios 9-14, the numbers of driver routesand driver days rise sharply.

• For free truck-driver assignment, increasing thetime between two visits at a relay station from 37to 48 hours leads to a clear reduction of all indi-

cators, truck and driver routes and days as wellas truck and shuttle kilometers, independent ofthe number of relay stations used.

• As previously mentioned, the stage 1 algorithmalways inserts the relay station with the smallestdetour between vertices of a truck route. For freetruck-driver assignment, it might be promisingto use a more sophisticated approach for choos-ing a relay station, where the chances are higherto find a rested driver who may readily be avail-able to take over the truck. The usefulness ofsuch an approach, however, should have be-come visible comparing the scenarios with 200and 43 relay stations, but the results showed nosuch effects.

• The impressive savings obtainedwith scenario 8,albeit a theoretical value, demonstrate that therest requirements for drivers are the bottleneckfor an efficient use of trucks: The distance sav-ings compared to the baseline scenario 1 aremoderate, but if no daily rests are necessary, thenumber of trucks and the number of truck dayscan be reduced dramatically.

In addition, the computational experimentsshowed the following results:• An increasednumber of relay stations showednoeffects. The potential benefit of reduced detoursto reach one of a larger number of relay stationsis compensated by the reduced number of routesvisiting the same relay station. Put differently,a network of 43 relay stations may already be

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Table 3: Computational results for free truck-driver assignment

Scenario 8 9 10 11 12 13 14

No. relay stations ∞ 43 43 100 100 200 200

Inter-visit time 13 37 48 37 48 37 48

Absolute values:

No. truck routes 486 705 669 712 678 694 679

No. driver routes 486 968 962 990 970 977 961

No. truck days 2,323 3,229 3,178 3,240 3,187 3,192 3,205

No. driver days 2,323 4,068 3,988 4,156 4,000 4,120 4,006

Truck kilometers 1,652,000 1,793,000 1,756,000 1,777,000 1,761,000 1,771,000 1,770,000

Shuttle kilometers 0 291,000 238,000 351,000 271,000 355,000 274,000

Costs 1,648,000 2,200,000 2,152,000 2,215,000 2,162,000 2,199,000 2,171,000

Relative changes to baseline scenario 1 in percent:

No. truck routes −33.33 −3.29 −8.23 −2.33 −7.00 −4.80 −6.86

No. driver routes −33.33 32.78 31.96 35.80 33.06 34.02 31.82

No. truck days −32.05 −5.56 −7.06 −5.25 −6.80 −6.63 −6.26

No. driver days −32.05 18.99 16.65 21.56 16.99 20.50 17.17

Truck kilometers −2.69 5.63 3.47 4.69 3.73 4.32 4.27

Costs −18.78 8.43 6.06 9.17 6.55 8.38 7.00

Relative changes to corresponding scenario with fixed truck-driver assignment in percent:

No. truck routes −33.33 −8.08 −10.32 −7.29 −9.84 −8.20 −10.42

No. driver routes −33.33 26.21 28.95 28.91 28.99 29.23 26.78

No. truck days −32.05 −5.40 −7.20 −6.26 −7.07 −6.37 −7.28

No. driver days −32.05 19.19 16.47 20.25 16.65 20.84 15.90

Truck kilometers −2.69 0.40 0.27 −0.08 0.39 1.09 0.85

Costs −18.78 6.02 4.47 6.24 4.80 6.96 4.73

considered sufficiently dense for the consideredarea.

• Whereas, as stated above, removing the needfor daily rests leads to an enormous increasein productivity, increasing the temporal capacityand availability of drivers by extending theirweekly driving andworking time onlymarginallyimproved the results for the scenarios with freetruck-driver assignment.

• With free truck-driver assignment, no feasiblesolutions could be computed when no shuttletransports are allowed: In stage 2, only very fewroutes were found that ensured the return ofdrivers and trucks to their home depots at theend of the planning horizon.

• We also tried an inter-visit time of 13 hours,

meaning that every daily rest must be taken ata relay station. This did not work well: A largepart of the requests could not be fulfilled. Forthese requests, the time between earliest pickupand latest delivery is not sufficient to cover thedistance between pickup and delivery locationplus the detour to and from a relay station.

Whether or not monetary savings are possible de-pends on the relative values of the fixed costsfor trucks and drivers and the distance-dependentcosts for trucks and shuttle vans. For the observedsharp increase in the number of necessary drivers,it is clear that no monetary savings are to be ex-pected for firms operating in the notoriously high-wage country Germany, where the daily truck rateis less than the daily driver rate. For the given data

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set, the truck fixed costs would have to rise by anorder of magnitude to yield overall cost savings.Since the number of available drivers is more thantwice the number of truck routes, this sharp in-crease in the number of necessary drivers came asa surprise. After analyzing the data, we identifiedtwo reasons:(i) On the one hand, for the given requests, the

drivers are based at the wrong depots, so thatan allocation of free drivers to truck routesegments requires a lot of shuttle transports.(It is interesting to note that, given the truckroutes fromstage 1, it is possible in scenarios 9-14 to assign the 1,645 drivers to the 43, 100or 200 relay stations so that shuttle transportsare necessary only at the beginning and atthe end of each driver’s working week. Then,monetary savings are obtained also for ourproject partner’s actual cost structure, whichmay be considered representative for long-distance transport in Central and NorthernEurope. Unfortunately, of course, this is onlyof theoretical relevance, because the driverslive where they live and cannot be resettled.)

(ii) On the other hand, given the depot and relaystation locations, the requests do not possessa suitable structure with respect to their posi-tion in space and time. It is too seldom that atruck is available at a relay station for a driverdirectly after the end of his daily rest, or, re-spectively, that a rested driver is available totake over a truckdirectly after its arrival at a re-lay station. The main cause for this mismatchis that the requests in our testbed were notacquired in a systematic manner by a centraldepartment, as it is done in ATLF firms. Theresults obtained by Taylor, Meinert, Killian,and Whicker (1999) support this conclusion:For North American long-distance road trans-port, the authors point out the importance of‘‘regularly scheduled delivery capacity in theform of delivery lanes, hubs and zones whichregularize driver tours while providing perfor-mance benefits for the carrier.’’

To relativize the above results, the following re-marks must be made. The above results are noproof that no savings are possible through slipseat-ing, and itmust be pointed out that a third possiblereason for not obtainingmonetary savings throughslipseating compared to a fixed truck-driver as-signment could be a low solution quality due to

insufficient performance of the heuristic. It is con-ceivable that the observed sharp increase in thenumber of necessary drivers is partly caused bythe solution procedure and is not entirely unavoid-able for the data used. Moreover, a more powerfulheuristic might also be able to further reduce thenumber of truck routes in stage 1.

8 Summary and researchoutlook

This paper has studied a simultaneous vehicle andcrew (truck and driver) routing and schedulingproblem arising in long-distance road transport. Acentral aspect of the problem is that the usual as-sumption in the vehicle routing literature of a fixedassignment of a driver to a truck is abandoned.Instead, truck/driver changes are allowed at ge-ographically dispersed relay stations, and drivershuttle transports between these relay stations areconsidered. This leads to interdependencies be-tween trucks and drivers and requires the syn-chronization of their routes.A heuristic solution algorithm based on largeneighborhood searchhasbeendescribed. The algo-rithmproceeds in two stages, computing routes fortrucks in stage 1 and routes for drivers in stage 2,where the truck routes from stage 1 constitutethe customers/requests to perform in the stage 2problem. The interdependencies between the truckand the driver routes are mitigated by specifyinglimited time windows for the truck routes.Extensive computational experiments have beenperformed with real-world data provided by a ma-jor freight forwarder. The results show the validityof our algorithm. Surprisingly, for typical requeststructures in non-timetabled long-distance trans-port in Central Europe, truck/driver changes seemto offer no savings potential. Under the given cir-cumstances, a fixed truck-driver assignment seemsto be the right set-up. The lack of savings potentialthrough slipseating is at least partially caused bythe lack of appropriate follow-up requests that areadequately situated in space and time, althoughit cannot be excluded that an insufficient perfor-mance of the heuristic algorithm with respect tosolution quality is one reason for these results. Inany case, it must be noted that the results are onlyvalid for the considered business field. The devel-oped algorithm for simultaneous vehicle and crewrouting and schedulingmaywell lead to very differ-ent results with other data or in other application

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areas, so that its further study is justified.A possible improvement of the algorithm is theaddition of a third stage, inwhich routes for shuttlevans are computed. Planning these transports canbe done independently of stages 1 and 2, sincethe stage 2 solution can be used as an input forsuch a stage 3 procedure in the same manner asstage 1 yields the input for stage 2 in the presentedalgorithm.Moreover, as discussed above, it would be highlyinteresting, though highly difficult, to fathom thepotential offered by considering the complete tem-poral flexibility of the routes computed in stage 1of the algorithm and develop a procedure capableof dealing with interdependent requests in stage 2.An even more involved extension of our SVCRSPis to allow that a change of driver/truck at a relaystation is performed without the driver taking adaily rest before switching to another truck. In thiscase, the interdependence of drivers and trucksbecomes still closer, and the described decompo-sition approach no longer works, which leads to ayet more complicated situation. Bürckert, Fischer,and Vierke (2000), Drexl (2007), and Cheung, Shi,Powell, and Simão (2008), in the context of long-distance road transport, short-distance collectiontraffic, and seaport container drayage respectively,consider such problems: Composite objects con-sisting of two or more types of elementary au-tonomous and/or non-autonomous objects are re-quired to fulfill tasks, and the elementary objectsmay join and separate on the fly at many differentlocations. In addition, whereas in our SVCRSP aswell as in the references discussed in Section 3, itwas always clear and fixed that two types of objectsmust visit a location simultaneously, the latterthree papers offer different options as to whichcombinations of object type are used to fulfill atask.Finally, it is sometimes allowed inpractice to trans-ship partial and complete loads from one vehicleto another at specified transfer locations, cf. Bock(2010). Studying a problem where drivers are al-lowed to switch vehicles and where requests maybe transshipped between vehicles constitutes achallenging area for future research.

Acknowledgement. Michael Drexl was fundedby the Deutsche Forschungsgemeinschaft (DFG)under grant no. IR 122/5-1. The authors are grate-ful to the department editor and two anonymous

referees for their detailed and helpful commentsand suggestions.

Appendix: Details on driver rulesRegulation (EC) No. 561/2006 of the EuropeanParliament and of the Council of 15 March 2006on the harmonisation of certain social legislationrelating to road transport and amending Coun-cil Regulations (EEC) No. 3821/85 and (EC) No.2135/98, and repealing Council Regulation (EEC)No. 3820/85 specifies the following rules as quotedfrom Drexl and Prescott-Gagnon (2010):• Interval driving timeA driver must not drive for more than 4 hoursand 30 minutes consecutively.

• Daily driving timeThe driving time between two daily rest periods(see below) must not exceed 10 hours. At mosttwice in a calendar week, the driving time be-tween two daily rest periodsmay exceed 9 hours.

• Weekly driving timeThe driving time within a calendar week mustnot exceed 56 hours.

• Fortnightly driving timeThe driving time in two consecutive calendarweeks must not exceed 90 hours.

• BreaksAfter 4 hours and 30 minutes of driving, thedriver must take a break of at least 45 minutes.The break may be split into two interruptions ofat least 15 and at least 30 minutes in this order.NoteTheabove ruledoesnotmean that there canbe at most 4 hours and 30 minutes of driving ineach interval of 5 hours and 15minutes. Considerthe following example. It is legal for a driver tostart his working day by driving for 1 hour and30 minutes, then take a break of 15 minutes,drive for an additional 3 hours, take a break of30 minutes, and then drive for another 4 hoursand 30 minutes. At the end of these 4 hoursand 30 minutes of driving, the driver has beendriving for 7 hours and 30 minutes in the last8 hours.

• Daily rest periodsThere are two different rules determining whena daily rest period is necessary:–Within 24 hours after the end of the last dailyor weekly rest period, theremust be a daily restperiod of at least 9 hours.

–Additionally, when the daily driving time is

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exhausted, a daily rest period must be takenbefore the driver may continue driving.

Between twoweekly rest periods, theremay be atmost threedaily rest periodsof less than11hours.A daily rest period may be split into two periodsof at least 3 hours and at least 9 hours in thisorder.

• Weekly rest periodsThere are also two different rules determiningwhen a weekly rest period is necessary:–After atmost 144hours after the endof aweeklyrest period, there must be an uninterruptedweekly rest period of at least 24 hours. If theduration of the weekly rest period is less than45 hours, the difference between 45 hours andthe actual duration of the weekly rest periodmust be added as an additional uninterruptedrest period to an uninterrupted rest period of atleast 9 hours within three calendar weeks afterthe calendar week where the thus shortenedweekly rest period ends.

–If the weekly or fortnightly driving time isexhausted, a rest must be taken until the end ofthe current calendar week, even if this is longerthan 45 hours.

In addition, there are rules for vehicles manned bytwo drivers.Directive 2002/15/EC of the European Parliamentand of the Council of 11 March 2002 on the organ-isation of the working time of persons performingmobile road transport activities specifies the fol-lowing rules:• Interval working timeAny mobile worker or self-employed driver per-forming mobile road transport activities mustnot work for more than 6 hours consecutively.

• Daily working timeIf, within a 24-hour period, work lasting at least4 hours is to be performed between 00:00 hoursand 07:00 hours, the working time within this24-hour period must not exceed 10 hours.

• Weekly working timeThe working time within a calendar week mustnot exceed 60 hours. This is allowed only if,over fourmonths, the working time per calendarweek does not exceed 48 hours on average.

• BreaksAfter 6 hours of working, the worker must take abreak of at least 30 minutes. If the daily workingtime exceeds 9 hours, the break must be at least45 minutes. The break may be split into periods

of at least 15 minutes each.These lists may be incomplete. Other national leg-islation in Germany or legislation in other coun-tries need not correspond completely to these rulesand might be stricter. In addition, other interna-tional rules and regulations may apply.In their paper on driver rules, Drexl and Prescott-Gagnon (2010) make the following statement:‘‘This is an OR paper, not a juristic text. It is explic-itly stated that none of the algorithms presentedin this paper is guaranteed to determine a ‘legalschedule’ for a vehicle route, because there is noprecise mathematical definition of the term ‘legalschedule’. The determination of a legal schedulefor a vehicle route is not a question of mathemat-ics or computer science, but solely a juristic one.The pertinent regulations give abundant room forinterpretation, so that any dispute concerning thelegality of a route will eventually have to be settledin court.’’ We agree.

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Biographies

Michael Drexl works in the field of applied operational re-search in logistics and transport and is currently engaged ina project on synchronization in vehicle routing at the Chairof Logistics Management at Johannes Gutenberg University inMainz.

Julia Rieck studied Applied Mathematics at the Universityof Göttingen and the University of Hamburg. She obtainedher PhD in 2008 and now holds the position of a researcherand teaching assistant at Clausthal University of Technology.Her research interests include vehicle routing and locationproblems, distribution network design, and project schedulingproblems.

Thomas Sigl studied Computer Sciences at the Friedrich-Alexander-University Erlangen-Nürnberg. He worked for 10years at the Fraunhofer Centre for Applied Research on SupplyChain Services SCS, Department Networks. His research in-terests include vehicle routing problems and logistics networkdesign.

Bettina Press studied Mathematics at the University ofErlangen-Nürnberg. Since 2010 she works at the FraunhoferCentre for Applied Research on Supply Chain Services SCS, andsince 2011 she is head of the Optimization group at SCS. Herresearch interests include network design and vehicle routing.

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