Rich vehicle routing in theory and practice

Michael Drexl

doi:10.1007/s12159-012-0080-2

Rich vehicle routing in theory and practice

Drexl, Michael 2012-07-03 00:00:00 Logist. Res. (2012) 5:47–63 DOI 10.1007/s12159-012-0080-2 OR IGINAL PAPER Michael Drexl Received: 6 September 2011 / Accepted: 14 June 2012 / Published online: 3 July 2012 Springer-Verlag 2012 Abstract The contribution of this paper is a comparison can also be found in intra-plant logistics, that is, local of the state of the art of scientiﬁc research on and com- transport within a factory or warehouse building or on mercial software for modelling and solving vehicle rout- company premises. ing problems. To this end, the paper presents a compact Beside the considerable importance of effective and review of vehicle routing literature and an overview of the efﬁcient vehicle routing for the enterprises themselves, the results of a recent study of commercial vehicle routing macroeconomic relevance of vehicle routing must not be software systems with respect to the problem features overlooked: the avoidance of unnecessary or unnecessarily these systems are able to handle and the solution methods long routes with low capacity utilization removes pressure the systems use for automatic generation of vehicle from road infrastructure, improves trafﬁc ﬂow for freight as routes. In this way, existing application and research gaps well as passenger transport, and, by reducing emissions, are identiﬁed. makes a sustained contribution to decrease the harmful effects of transportation. Keywords Rich vehicle routing Commercial vehicle For operational research (OR), vehicle routing consti- routing software Heuristics tutes one of its great success stories. Vehicle routing problems (VRPs) in their many variants have been the subject of intensive study for more than half a century now. This has led to the publication of thousands of scientiﬁc 1 Introduction papers and to the foundation of numerous software com- panies worldwide selling commercial vehicle routing Vehicle routing is a central task in a large number of pri- software (CVRS). This development is certainly due to the vate and public corporations. Routes have to be planned in intellectual challenge VRPs pose as well as to their prac- very diverse sectors of the economy, not only in the tical relevance in logistics and transport. Research on VRPs logistics and transport business, but in virtually all indus- is incessantly ongoing, stimulated by unsolved theoretical trial sectors producing physical goods. In addition to problems and continuous input from logistics practice. transport on public roads, applications of vehicle routing The contribution of this paper is a comparison of sci- entiﬁc research on VRPs and commercial software for modelling and solving VRPs. To this end, the paper pre- sents an overview of the results of a recent study of CVRS M. Drexl (&) with respect to the problem features these systems are able Chair of Logistics Management, Gutenberg School to handle and the solution methods the systems use for of Management and Economics, Johannes Gutenberg University, Mainz, Germany automatic generation of vehicle routes (Drexl [31]). These e-mail: michaeldrexl@web.de ﬁndings are contrasted with the state of the art of scientiﬁc VRP research. In this way, existing application and M. Drexl research gaps are identiﬁed. This should be of interest for Fraunhofer Centre for Applied Research on Supply Chain VRP researchers, and also logistics practitioners using or Services SCS, Nuremberg, Germany 123 48 Logist. Res. (2012) 5:47–63 planning to use CVRS should beneﬁt from this paper, by considered planning horizon. An overview of these learning about the potential of modern CVRS. dimensions of richness in real-world VRPs is given in Throughout the paper, the following deﬁnitions apply. Fig. 1, and a discussion of these characteristics follows. The fundamental activity to be planned in vehicle routing (An orthogonal characterization is by application area. is called a request. A request may be a transport order, such Industry sectors where scientiﬁc VRP research is particu- as the delivery of a shipment from a central depot to a larly widespread are discussed in Sect. 3.2; industry sectors recipient, the pickup of a shipment from a consignor and and ﬁelds of application where the use of CVRS is com- the transfer to a central depot, the pickup of a shipment at mon are presented in Sect. 4.4.2.) Characteristics of rich some location and the transport to some other location, or a VRPs are also discussed in Hasle and Kloster [51], Sect. 3, visit at a location to perform a service there, without and So ¨ rensen et al. [79], who state (p. 241): ‘Although picking up or delivering a physical good. Vehicle routing there is an increasing scientiﬁc focus on so-called ‘‘rich’’ means to group requests into clusters performed by one VRPs (that incorporate more complex constraints and vehicle each, and to determine, for each cluster, a complete objectives), they have not in any way caught up with the sequence of the resulting locations to be visited. This whole complexity of real-life routing problems’. To a large process can be performed manually by a human planner, extent, this point is also supported by the results described automatically by a computer program executing an algo- in the present paper. rithm, or by a combination of both. The goal of vehicle routing is the optimization of an objective function. This 2.1 Requests will regularly be the minimization of a cost function, of the number of used vehicles, of the total distance travelled, etc. There are a large number of different aspects of requests. The rest of the paper is structured as follows. The next First of all, time windows are central properties of requests. section describes the decisive aspects by which the Time windows can be caused by the request itself (e.g. numerous variants of VRPs can be distinguished. Section 3 earliest ready-time of a manufactured good to be picked up then gives a brief overview of the state of the art of sci- or latest delivery time of a component needed at the des- entiﬁc VRP research. Section 4 presents the results of a tination) or by the location where the request is to be comprehensive study of the German CVRS market, performed (opening hours). There can be one or more focussing on modelling and algorithmic aspects for the disjoint time window(s) for a request (opening hours in the automatic solution of VRPs. Section 5 discusses the gaps morning and in the afternoon). Moreover, time windows between theory and practice, and Sect. 6 gives a conclusion can be vehicle-dependent (e.g. large delivery vehicles and an outlook. having more restrictive access to customers in inner-city zones than small ones). Another important aspect is pairing and precedence:ifa 2 An overview of VRPs: dimensions of richness request consists in the transport of a good from a pickup location to a delivery location, then, if no transshipments The archetypal, fundamental VRP, the capacitated vehi- are allowed, one and the same vehicle must visit both cle routing problem (CVRP),isasfollows.Given area locations. Moreover, it is obvious that the pickup must be set of identical vehicles stationed at one depot and performed before the delivery. There are also complex equipped with a limited loading capacity, and a set of requests, which consist of more than one pickup and one geographically dispersed customers with a certain delivery location. Often, a nested execution (so-called demand for a homogeneous good. The task is to deter- LIFO loading: pickup request A, pickup request B, deliver mine an optimal (with respect to an objective function) B, deliver A) is required (e.g. if vehicles can be loaded and route plan, that is, a set of vehicle routes, specifying unloaded only from behind). which customers are visitedbywhich vehicleinwhich Vehicle-driver-request compatibilities are the third sequence, such that each customer is visited exactly once, fundamental aspect. Depending on the vehicle character- the complete demand of each customer is satisﬁed, and istics and the driver qualiﬁcations, not all requests can or the loading capacity of the vehicles is maintained on each may be performed by all vehicles and drivers, even if the route. The objective is to minimize overall cost or trav- request locations are accessible to the vehicle and the elled distance. driver. As mentioned, there are a huge number of variants, Further important types of requests are optional extensions, and generalizations of the CVRP. VRPs can be requests (requests that need not be assigned to a route, but categorized according to their properties with respect to whose execution brings a bonus), periodic requests the requests to be fulﬁlled, the ﬂeet available for doing so, (requests that have to be executed several times within a the desired route structure, the objectives pursued, and the planning horizon, mostly according to visitation patterns, 123 Logist. Res. (2012) 5:47–63 49 Characteristics of VRPs Requests Fleet Route structure Objectives Scope of planning Time windows Costs Closed/open One-dimensional Time horizon � � � � � – Single – Fixed Multiple routes per vehicle – Min. number vehicles – Tactical – Multiple – Variable Fixed route zones – Min. cost – Operational Pairing and precedence con- – Penalty Visually attractive routes – Min. distance – Real-time � � straints – Tariffs � Interdependent routes with – Max. proﬁt � Data availability and accur- – Pickup before delivery � Capacity constraints synchronization � Multi-dimensional acy – LIFO loading – Weight – Between autonomous and – Weighted sum of single – Deterministic � Vehicle-driver-request- – Volume non-autonomous vehicles objectives – Stochastic compatibilities – Loading metres – Between elementary and – Hierarchical ordering of – Uncertain or unknown Special types of requests – Pallet places composite vehicles objectives – Optional Driving speed – To perform transshipments – Multi-criteria objective – Periodic – Same for all vehicles Inter-route resource con- searching for Pareto- – Expected – Different for different straints optimal solutions – Incompatible vehicles � Balanced routes, i.e., all � Hardness of constraints � Different ways of performing – Constant routes should be similar with – Hard a request – Time-dependent respect to – Soft (with penalty in ob- – Request splitting – Load-dependent – Capacity utilization jective function) – Alternative delivery loca- Temporal availability – Number of stops tions Initial and ﬁnal vehicle posi- – Duration Request-dependent vehicle tions – Costs etc. itineraries � Type – Lorry – Trailer – Train – Ship – Aircraft etc. Technical equipment Number of vehicles of each type – Limited – (Virtually) unlimited Drivers – Qualiﬁcations – Social legislation Fig. 1 Dimensions of richness in VRPs for example, twice a week, but not on consecutive days), his home address after six o’clock or on Saturdays. This expected requests (requests that have not yet been issued raises the additional question of how to choose a way of by the customer, but will probably be), incompatible performing the request (where to perform a meet-and-turn requests (parallel incompatibility: do not transport the operation, how to split a request into sub-requests, when to requests at the same time with one vehicle; sequential deliver a package). incompatibility: do not transport request B on a route that Finally, there is the aspect of request-dependent vehicle has transported request A before), and indirect requests itineraries. This refers to situations where the transport (e.g. automatic generation of empty container balancing links a vehicle is able or allowed to use depend on the requests). requests it is carrying, which means that requests determine A further very difﬁcult aspect is when there are different the distance and the travel time between locations. For ways of performing a request. This refers to the possibility example, when a vehicle for transporting bulky goods is to split up the fulﬁlment of a request between several empty, it may be able to pass through a low undergrade vehicles, or to the possibility to perform a request by dif- crossing, but when the vehicle is loaded, it may be too high ferent operations altogether. An example for the ﬁrst case to pass and may thus have to make a detour. Similarly, if a is multi-modal transport, where a request to bring a con- tank vehicle for oil or fuel delivery is empty, it is allowed signment from A to B can be performed by a direct trans- to travel through a water protection area; if such a vehicle port from A to B, via a meet-and-turn operation, or via one is loaded, it must take an alternative, longer way. or several hubs. Another is that the request to deliver x units of a good to a customer can be fulﬁlled by one 2.2 Fleet delivery of x units by one vehicle or by several deliveries by several vehicles. An example for the second case is The term ‘ﬂeet’ refers to the resources available for ful- parcel delivery, where a package must be delivered to the ﬁlling the requests. These resources comprise vehicles of recipient’s ofﬁce address from nine to ﬁve o’clock, and to different types as well as drivers operating them. 123 50 Logist. Res. (2012) 5:47–63 2.2.1 Vehicles board, dangerous-goods equipment, etc. The dimensions of a vehicle also inﬂuence which transport links can be used Vehicles may differ with respect to several criteria, the (large lorries cannot use small inner-city roads, super- most important ones being costs, capacities, driving speed, tankers cannot use the Panama canal). Moreover, depend- temporal availability, actual and desired position at the ing on its weight or emission level, a vehicle may be unable beginning and the end of a planning process, type and or not allowed to use certain roads. technical equipment, and the (in)ability to visit certain In tank vehicles, there are often several compartments locations and use certain transport links. that can be ﬁlled separately to allow the simultaneous Relevant cost categories are ﬁxed costs for using a transport of different goods or products or requests. When vehicle, and distance-, time-, and stop-dependent variable there are n compartments, n different products can be costs. Distance-dependent costs may include road tolls. transported. When all compartments are used, no request Time-dependent costs may be linear or nonlinear and may can be executed that requires the transport of another good, include overtime pay or daily allowances for drivers. even when none of the capacity constraints listed above Moreover, costs may be calculated based on tariffs. Tariffs would be violated. used to be mandatory in Germany until the end of the The number of vehicles of each type and class is twentieth century and were dependent on goods types, important, too. In reality, the number of vehicles is of weight, distance, time, etc.; although the numeric values course always limited. For tactical planning of the ﬂeet size have decreased sharply, the calculation formulas are still and mix, it may, however, be interesting to allow an common in practice. For planning purposes, penalty costs unlimited number of vehicles of each type and class. are often used to consider soft constraints or undesired properties of solutions, or to allow infeasible solutions 2.2.2 Drivers during the solution process. The most common capacity constraints in goods trans- As far as drivers are concerned, restrictions regarding port are weight or payload, volume, loading metres, and qualiﬁcations limit the compatibility between drivers and number of pallet places. Several of these may be relevant at vehicles as well as between drivers and requests. Such the same time. qualiﬁcations may be the type of driving licence a driver The driving speed may be the same for all vehicles or possesses, whether or not a training for the transport of differ between vehicles or vehicle types. It may also dangerous goods was completed, or knowledge of cus- depend on the load a vehicle is carrying (the fuller, the tomer or region speciﬁcs. slower). Moreover, the driving speed may be constant for Another matter of utmost importance in real-world lorry all vehicles throughout the complete planning horizon, or road transport are driver rules. In the European Union and be time-dependent. This is particularly important for short- in other parts of the world, there is extensive social legis- distance and inner-city transport, where travel times are lation on driving, working, break and rest times for drivers; signiﬁcantly higher during rush-hours compared to off- see Humphreys [56] for an overview. The automatic peak times. tachograph introduced in the European Union nowadays Also, the temporal availability of a vehicle may be allows for much tighter supervision of compliance with limited, for example, due to scheduled maintenance or social legislation for drivers, and the road transport Sunday driving bans for heavy lorries, but not for smaller industry in Europe is acknowledging that today, it has to vans. comply with the regulations very exactly. It is important to In operational, short-term planning, initial vehicle note that an algorithm cannot determine a ‘legal schedule’ positions (depots) are given, whereas in tactical, mid-term for a route, because the term ‘legal schedule’ has no legally planning, it is often part of the planning task to determine binding mathematical deﬁnition; it is a purely juristic appropriate locations for the vehicles. In operational as concept. In an unlucky attempt to provide ﬂexibility in well as tactical planning, locations at the end of the plan- practice, the European Union has introduced an intractably ning horizon are arbitrary if open routes are allowed. complex set of optional rules along with the mandatory Concerning type and technical equipment, there are ones. These rules leave a lot of room for interpretation, and different criteria that determine whether or not a vehicle is a dispute about the legality of a schedule will eventually in principle able to perform a request, disregarding the have to be settled in court. For practitioners, this means that current point in time, location, or capacity utilization. trying to exploit the optional rules is dangerous. For Among these criteria are the vehicle type (lorry, train, ship, algorithm developers, the optional rules mean a lot of etc.), the vehicle class (swap-body vehicle, tank vehicle, tedious work: on the one hand, for considering them, on the etc.), vehicle dimensions and weight, and technical other hand, for ensuring that the overall algorithm is not equipment such as a ﬁxed installed tail-lift, a fork-lift on slowed down too much. 123 Logist. Res. (2012) 5:47–63 51 2.3 Route structure itinerary. A very similar case is the planning of separate routes or rotations for vehicles and drivers, where, during Some aspects of the route structure concern each route the planning horizon, a vehicle may be operated by dif- individually, others lead to interdependencies between ferent drivers, and a driver may drive several vehicles. This routes. improves the temporal capacity utilization of vehicles, since these can essentially be used 24 h a day, whereas 2.3.1 Individual routes drivers need regular breaks and rests and have to obey the above-mentioned driver rules. The standard case is a closed route (loop), starting and Furthermore, allowing transshipments of load between ending at the same location (depot). Nevertheless, also vehicles leads to interdependent routes. Transshipments open routes, where a vehicle may be at any location at the occur in the form of meet-and-turn routes with exchange of end of its route, are relevant in many situations. For complete swap-body platforms or as partial exchanges of example, in long-distance road transport, vehicles are en single consignments or one-way transfer of load from one route for a complete week, but routes are planned only for vehicle to another. Multi-modal transport, by deﬁnition, the next day, so the routes for Monday to Thursday end at requires transshipments of load. the last customer scheduled for the respective day. The Also inter-route resource constraints such as processing converse, that is, the planning of multiple closed routes for capacities at depots, a maximum number of vehicles one vehicle, is also possible, for example, in local delivery arriving at a depot per time unit due to limited number of applications, where vehicles return to the depot more than ramps or conveyor belt capacities, etc., make synchroni- once during a day to reload. zation between routes necessary. Further possible types are routes with special geographic Finally, there is the requirement of balanced routes. properties, such as the consideration of ﬁxed route zones in This refers to the stipulation that all routes of a plan be tactical planning, and routes with limits on total duration or similar with respect to covered distance, duration, number waiting time. of requests, costs, etc. 2.3.2 Interdependent routes 2.4 Objectives The usual assumption in almost all VRPs is that the only Objective functions may be one- or multidimensional. coupling or linking or joint constraints between the routes Potential one-dimensional objectives pursued are the min- of different vehicles are related to request covering, to imization of the number of vehicles used, of the overall ensure that each request is performed exactly once. The distance covered by all vehicles, and of the total cost of all preceding aspects leave routes independent of one another vehicles. If not all requests are mandatory, the objective in this sense. The feasibility of one route does not affect the may be the maximization of the difference between the feasibility of another. However, there are also requirements proﬁt obtained from the fulﬁlled requests and the costs that lead to route interdependencies, to routes that must be incurred for fulﬁlling them. synchronized. In such cases, the feasibility of one route When the objective comprises several dimensions, it is may depend on the feasibility of one or more other routes. possible to consider a weighted sum of one-dimensional Multiple synchronization of vehicles or routes may be objectives, to have a hierarchical (lexicographic) ordering relevant with respect to space, time, load, or common of the dimensions (e.g. the minimization of the number of scarce resources. vehicles used as the most important criterion and the One example of such a requirement is that of a ‘visually minimization of cost as the second one, as the tie-breaker attractive’ route plan, which often means intersection-free in the case of two route plans using the same number of routes. Synchronization requirements also occur when vehicles), or a multi-criteria objective searching for Pareto- there are different types of elementary vehicles that may or optimal solutions. must join and form a composite vehicle to be able to move An important aspect that must be mentioned here is that in space or to perform a request. An example is the plan- many constraints and requirements discussed above can be ning of separate routes for lorries and trailers (or tractors considered as hard or soft constraints. Any violation of a and semi-trailers). Each lorry and each trailer is an ele- hard constraint is strictly forbidden and invalidates a route mentary vehicle and can be used to perform requests, and a plan. Constraints such as technical or logical restrictions route is computed for each lorry and each trailer that is (e.g. pickup before delivery) or legal obligations (e.g. actually used. Naturally, the route of a trailer must be working and rest times of drivers) are always hard. A synchronized with the routes of one or more lorries that violation of a soft constraint does not directly invalidate a must pull the trailer on the whole or on a part of its route plan, but is undesired and thus considered with a 123 52 Logist. Res. (2012) 5:47–63 penalty in the objective function. The penalty usually number of benchmark instances for different types of VRP increases with increasing degree of violation, and if the can be found at http://people.brunel.ac.uk/*mastjjb/jeb/ degree of violation exceeds a certain threshold, the con- info.html.) For the latter class, the term ‘rich vehicle straint becomes hard and invalidates the route plan. Time routing’ has been coined rather recently to denote models windows, for example, are sometimes considered as soft and solution approaches for problems that feature several constraints. or all aspects of a real-world application. Most papers belonging to the latter class focus on one new or particu- 2.5 Planning horizon and data availability larly interesting or difﬁcult aspect. A number of important such aspects were queried in the CVRS study mentioned Depending on the frequency of planning and the duration above and was discussed in detail in the previous section. during which plans remain valid, or, put differently, the range and scope of the decisions taken, there are tactical 3.1 Problem variants (medium-term), operational (short-term), and real-time (dynamic) VRPs. Tactical decisions encompass the size Researchers have devoted a lot of effort to the study of a and composition of the ﬂeet (number of vehicles of each rather small number of abstract, generic, and well-deﬁned type, size and technical conﬁguration, assignment to extensions of the CVRP, and rightly so: although these depots) and the preparation of ‘framework routes’ based on scientiﬁc variants of the CVRP hardly ever appear in average data for application areas with periodic supply or practice in their pure form, their study is worthwhile, demand variation (e.g. seasonal route plans for raw milk because the results and insights obtained can serve as a collection at farms with high volume in spring and low basis for tackling the numerous detailed and speciﬁc real- volume in winter or different routes for postal distribution world routing problems. The most important such theo- on different weekdays). The resulting plans may cover retical CVRP extensions are brieﬂy described in the fol- multiple periods. Operational vehicle routing is concerned lowing, by pointing out in which respect these problems with the planning of routes for the next day(s), based on extend the CVRP. The cited references are surveys of the concrete data. Real-time routing takes into account new or respective problem. changing data (such as, for example, incoming requests, In the VRP with time windows (VRPTW, Bra ¨ysy and vehicle breakdowns, trafﬁc congestions) and adapts plans Gendreau [10, 11], Cordeau et al. [20]), the service at each while these are being executed. customer must start within a given single hard time win- A related aspect is the availability and accuracy of the dow. In the split-delivery VRP (SDVRP, (Archetti and data on requests, vehicles, relevant locations, and trafﬁc Speranza [4]), customers may be visited more than once by more than one vehicle. Each vehicle may deliver a fraction links. There are three cases: the deterministic case, where all data are known in advance, the stochastic case, where of a customer’s demand. some data are known in advance only in the form of In Pickup-and-delivery problems (PDP, Parragh et al. probability distributions, and the case of uncertainty, where [68, 69]), the tasks consist in the transport of shipments some data are unknown and become known only during from one location to another, that is, not only the planning or during execution of a route plan. delivery locations are all different, but also the pickup locations. Dial-a-ride problems (DARP, cf. ib.) consider the transport of persons and, in contrast to PDPs, usually 3 Scientiﬁc VRP research feature constraints restricting passenger inconvenience, for example, by limiting the maximum ride duration. It As stated in the introduction, over the last half century, must be noted that there are different sub-types of PDPs, there have been thousands of scientiﬁc publications on such as the VRP with backhauls or the VRP with vehicle routing, starting with the famous paper by Dantzig simultaneous delivery and pickup. The reader is referred and Ramser [25]. Therefore, the following elaborations to the above surveys for a complete taxonomy of PDPs only give a very rough overview, a ‘survey of surveys’, and and DARPs. necessarily refer the reader to the literature for details. The In Periodic VRPs (PVRP, Francis et al. [37]), several existing VRP literature can be divided into theoretical visits are required to serve a customer during a multi-per- papers studying models or methods for idealized or stan- iod planning horizon. These visits must take place in dif- dardized problems and problem-oriented case studies ferent periods. An interesting variant is the consistent VRP dealing with concrete real-world applications. The former (Groe ¨r et al. [47]). This is a periodic VRP where each class considers exact as well as heuristic solution approa- customer must always be visited by the same vehicle in the ches and uses theoretical benchmark instances to measure different periods and each customer must be visited at the effectiveness of the devised algorithms. (A large ‘roughly the same time’ on each visit. 123 Logist. Res. (2012) 5:47–63 53 Heterogeneous ﬂeet VRPs (HVRP, Hoff et al. [54]), as There is also an increasing number of publications on the name implies, consider the case that not all vehicles are algorithms for considering VRPs with driver rules (see, e.g. identical. The ﬂeet size and mix VRP (FSMVRP, cf. ib.) is Archetti and Savelsbergh [3], Goel [43], Drexl and Pres- the tactical variant of the HVRP and considers (different) cott-Gagnon [33], Goel [44], Kok et al. [61], Prescott- ﬁxed costs for using (different types of) vehicles. Gagnon et al. [73]). The capacitated arc routing problem (CARP, Corbera ´n In addition to the above-mentioned surveys, Toth and and Prins [18]) is a variant of the CVRP where the tasks are Vigo [82] and Golden et al. [46] are recent monographs on not to visit customers to perform a service, but where the VRPs and their variants. All of these references contain service is performed while travelling along the links of a results on exact as well as heuristic methods. (road) network. Location-routing problems (LRP, Nagy and Salhi [66]) 3.2 Application-oriented research combine routing and locational decisions. The task is to determine a set of vehicle routes and, for each route, the There are some application areas where OR methods have location where it starts and ends. Using a location by sta- a long-standing tradition (not only in the context of vehicle tioning a vehicle there incurs ﬁxed costs. routing), and where there is a particularly large number of In stochastic VRPs (Flatberg et al. [35], Cordeau et al. application-oriented papers. Such niches of applied VRP [20]), information on occurrence and volume of customer research can be found in the airline industry (Klabjan [60], demand or travel times between customers is given by Ball et al. [7]), public transport (Desaulniers and Hickman probability distributions. In Dynamic VRPs (Powell et al. [29], Hickman et al. [53]), ship routing (Christiansen et al. [72]), as already outlined in Sect. 2.5, the planner is forced [17], Hennig [52]), rail transport (Cordeau et al. [21], to make decisions before all relevant information becomes Caprara et al. [14]), and letter mail or parcel delivery available; decisions must then be modiﬁed as new infor- (Bodin and Levy [9], Wong [84]). mation is received. Essentially, planning is performed Seminal case studies describing the successful solution of parallel to plan execution. rich real-world VRPs are listed in Table 1, sorted chrono- The inventory routing problem (IRP, Moin and Salhi logically. Note that this list is necessarily incomplete. [65], Andersson et al. [2]) is a very special type of VRP. In IRPs, there are no customer demands. Instead, each cus- 3.3 Solution methods tomer has a given consumption rate of a good, a given initial stock and a given storage capacity. The depot has to VRPs are usually modelled using graphs or networks and perform zero or more deliveries to each customer during a formulated as mixed-integer programs (MIPs). As regards multi-period planning horizon to ensure that no customer solution methods, there are two fundamental approaches: runs out of stock. The objective is to plan routes of minimal Mathematical-programming-based algorithms on the one cost for the deliveries. hand, and heuristics and meta-heuristics on the other. Table 1 Selected case studies on rich VRPs Paper Application Savelsbergh and Sol [77] Dynamic, multi-period pickup-and-delivery with complex requests Xu et al. [85] Pickup-and-delivery with complex cost functions and LIFO loading Hollis et al. [55] Simultaneous and interdependent vehicle and crew routing and scheduling Cheung et al. [16] Synchronized routing of lorries and trailers Irnich [58] Arc routing with turn and street crossing restrictions, cluster constraints, and alternative service modes Za ¨pfel and Bo ¨ gl [86] Simultaneous and interdependent vehicle and crew routing and scheduling with outsourcing options and working time regulations Ceselli et al. [15] Heterogeneous ﬂeet, multi-depot, split-delivery VRPTW with open routes and request incompatibilities Bock [8] PDP with time windows and transshipment options Oppen et al. [67] IRP with route duration and precedence constraints using heterogeneous vehicles with compartments Rieck and Zimmermann Simultaneous delivery and pickup with synchronization constraints at loading docks [74] Schmid et al. [78] VRP with splitting of loads and synchronization of different vehicles at customer sites Derigs et al. [26] VRP with multiple use of tractors and trailers 123 54 Logist. Res. (2012) 5:47–63 3.3.1 Exact approaches Method-oriented surveys or tutorials on heuristics and meta-heuristics are given by Funke et al. [38] (local Mathematical programming algorithms are based on MIP search), Røpke [75] (large neighbourhood search), Ahuja models and, in theory, guarantee to ﬁnd an optimal et al. [1] (very large-scale neighbourhood search), Powell solution if one exists. The most successful exact algo- [71] (adaptive dynamic programming), Cotta et al. [22] rithms for VRPs are branch-and-cut-and-price methods, (metaheuristics), and Gendreau and Potvin [41] (meta- which combine cut and column generation with branch- heuristics). Gendreau and Potvin [39] develop an inte- and-bound. However, mathematical programming algo- grating and unifying overview of metaheuristics, and rithms typically require too much time and memory for Gendreau et al. [40] present a categorized bibliography of large instances. Moreover, the computation times for metaheuristics for several types of VRP. instances of the same size and structure often vary to a Hasle and Kloster [51], Sect. 4, give a description of a large degree. At the time of this writing, CVRP and commercial software for solving rich VRPs. In particular, VRPTW instances of more than 200 customers cannot the conceptual approach for modelling and representing be consistently solved to optimality. Rich real-world rich VRPs in a software tool (as opposed to in a mathe- instances with many complicating constraints and a real- matical model) is described in detail. Moreover, the istic number of requests are still untractable with exact implementation of the solution algorithms used is methods. explained. Groe ¨r et al. [48] describe a publicly available A milestone in the ﬁeld of exact methods for VRPs is the programming framework for solving VRPs and give paper by Desaulniers et al. [27], which discusses issues detailed explanations of the framework’s design. Both arising in the modelling and solution of time-constrained codes were not part of the survey presented in the next vehicle routing and scheduling problems using mixed- section. Both papers treat aspects of vehicle routing soft- integer programming and column generation. Desaulniers ware that could not be queried in the survey. et al. [28] present a monograph on column generation and branch-and-price, Røpke [75] describes several exact 3.4 Trends in VRP research algorithms for VRPs and PDPs, Spoorendonk [80] treats issues related to cut and column generation, and Baldacci 3.4.1 Richness and robustness et al. [5] describe an exact solution framework for different types of VRP that outperforms all other exact methods With respect to models, there is a clear trend towards published so far and solves several previously unsolved considering ever ‘richer’ problems (Hartl et al. [50]), and benchmark instances. Finally, Baldacci et al. [6] provide an towards developing generic, uniﬁed modelling frameworks up-to-date review of the state-of-the-art exact algorithms (Irnich [59]) for representing these rich problems. With for the CVRP and the VRPTW. respect to methods, considerable progress has been made concerning the development of exact as well as heuristic 3.3.2 Heuristics and metaheuristics solution algorithms that are robust, that is, work well for a broad range of problems both in terms of running time and Heuristics and metaheuristics do not offer an optimality solution quality (Pisinger and Røpke [70], Baldacci et al. guarantee, but they overcome the limitations of exact [5]). algorithms and are able to ﬁnd close-to-optimal solutions in short time, even for very large instances. Heuristic methods 3.4.2 Self-adaptation and hyperheuristics can be divided into constructive procedures, which are used to compute an initial feasible solution, and improvement A related aspect is the trend towards self-calibrating and/or procedures, which iteratively try to improve a given solu- self-adapting algorithms: Many of the metaheuristics tion by systematically modifying it. Metaheuristics are developed over the years are highly sophisticated and superordinate procedures that control the search processes contain a large number of parameters for which sensible performed by constructive and improvement heuristics. values must be set to obtain good solutions. Different Section 4.6 contains an extensive list of constructive and problems or instances with different data characteristics improvement heuristics as well as metaheuristics. Best- require different parameter settings. Similarly, the recently known heuristic solutions to benchmark instances for the proposed so-called hyperheuristics adapt the search space problems described in Sect. 3.1 have been computed with continuously in the course of the computations, based on many different methods, so there is deﬁnitely no silver the previous solution progress. The monograph edited by bullet. However, it must be noted that most successful Cotta et al. [22] describes several approaches in the ﬁelds heuristic approaches are so-called hybrid procedures of self-calibrating and self-adapting meta- and hyperheu- combining several ‘classical’ ones. ristic methods (see also Burke et al. [13]). 123 Logist. Res. (2012) 5:47–63 55 3.4.3 Matheuristics for details). Lastly, a statistics module serves to compute key performance indicators and to create reports. As mentioned, the most successful heuristic algorithms are Usually, but not exclusively, CVRS is used for planning nearly always hybrid methods combining different construc- routes of motor vehicles on public roads. In this case, a tive and improvement procedures, and sometimes also dif- CVRS is often embedded into a transport management ferent meta-heuristics. A new class of hybrid meta-heuristics system (TMS). A TMS contains components for data entry, has emerged only recently: it was stated in the preceding planning, administration, execution, control, and billing of section that there are two fundamental types of solution transport services. TMSs and other software systems for method, mathematical programming and (meta-)heuristics. logistics and transport are thoroughly discussed in Crainic Due to advances in mathematical programming theory as well et al. [24]. as computer hardware, these two worlds have begun to merge. Hybrid methods that use mathematical programming models 4.2 Reasons for using CVRS and algorithms as subroutines are now commonly subsumed under the term matheuristics. A pertinent monograph is VRPs are, in essence, highly complex mathematical opti- Maniezzo et al. [64]. The paper by Doerner and Schmid [30] mization problems. Because of this complexity, software gives a survey on matheuristics for rich VRPs. for supporting human planners and decision-makers has been widely used for years. There is a considerable number 3.4.4 Parallel algorithms of manufacturers of CVRS, and many of these manufac- turers have been in the business for decades. Reasons for On the technical side, one of the more recent advances in the use of CVRS, as speciﬁed by users, follow. The most computer hardware is multi-core processors, allowing real important reason is surely that CVRS helps to reduce the multi-threaded processing on single, standard personal costs of executing and planning routes, and to increase computers. This has lead to a renewed and increased efﬁciency. Moreover, through automatic planning, dis- interest in parallel algorithms. Both exact and heuristic patchers are relieved from routine jobs. Telematics and methods can beneﬁt from multi-threaded implementations. statistics functionality of CVRS improve the possibilities Crainic [23] presents a survey of parallel solution methods for transport monitoring and surveillance as well as for for VRPs. statistics and controlling, so that the quality and the transparency of the overall planning process is improved. An important point often raised by senior executives is that 4 CVRS: a comprehensive study of the German market the dependency of the company on single persons (expe- rienced dispatchers) and their knowledge is reduced. Also, A CVRS is a computer program that allows to (1) read in the work of the sales department (of freight forwarders) is and display data on vehicle depots, customers, distances, simpliﬁed, because faster and more precise pricing of ad and travel times between locations, on requests, vehicles, hoc customer requests becomes possible. Finally, work and drivers, (2) construct, save, and display vehicle routes, processes in general are uniﬁed and streamlined. The and (3) determine a complete route plan for a given data set beneﬁt of the practical use of CVRS is substantiated by (a problem instance) by executing construction and several scientiﬁc studies. See, for example, the literature improvement algorithms, possibly after entering a set of survey in Eibl [34], p. 45 ff. parameters, without further user interaction. 4.3 Structure of the study 4.1 Components of CVRS For the compilation of the study, a thorough search for A CVRS typically consists of the following ﬁve compo- CVRS manufacturers active on the German market was nents: an interface to a database or enterprise resource performed, and no less than 50 ﬁrms could be found. All of planning (ERP) system allows reading in the relevant data them were asked to ﬁll in a detailed questionnaire con- and writing back the solution. A geographical information taining more than 500 pieces of information on relevant system (GIS) is necessary for geocoding address data, aspects of the company and the CVRS in nine categories: computing distance and travel time matrices, and visual- izing data and solutions in digital maps. A planning mod- 1. Company 2. Product ule, the heart of the system, supports automatic, manual, and interactive planning. A telematics module allows a data 3. Information technology and software engineering 4. User interface exchange between vehicles and the dispatching ofﬁce as well as the tracking and tracing of vehicles (see Goel [42] 5. Geographical information system 123 56 Logist. Res. (2012) 5:47–63 6. Telematics one university. This is mostly done in the form of master’s 7. Models and algorithms for automatic vehicle routing and Ph.D. theses. Twelve ﬁrms offer a free test or demo 8. Reporting, key performance indicators, and statistics version, and several ﬁrms provide low-priced or free li- 9. Prices cences for use in teaching. A basic, single-user licence for commercial use costs The questions posed were, to a large extent, either of the 15,000 Euros on average. This does not include custom- multiple-choice or the yes–no type. The obtained infor- izing and preparatory training of users. mation was evaluated on an aggregate level, by summing or averaging over all questionnaires. No information on 4.4.2 Industry sectors using CVRS single manufacturers or systems is given. The answers in the returned questionnaires were checked for plausibility; As for the industry sectors using CVRS and the respective nevertheless, correctness could of course not be veriﬁed. number of sold licences, no reliable data could be gathered. However, the published results being aggregated, no ven- Some ﬁrms provided detailed data, others did not or gave dor had anything to gain from exaggerating the capabilities only aggregated information. However, a quintessential of his product. With respect to content, aspects that were ﬁnding is that numerous CVRS ﬁrms have customers in the considered relevant by the study author, based on his own following sectors: Industrial ﬁrms producing physical professional experience, were queried. goods use CVRS to plan the sourcing of raw materials and OR/MS Today, the journal edited by the Institute for the transport of (semi-)ﬁnished goods between plants and Operations Research and the Management Sciences warehouses or to wholesalers. Concrete applications are as (INFORMS), features, in a 2-year cycle, a survey on CVRS diverse as milk collection at farmyards or wood transport for the North American market. The latest one, from from forests to mills on the supply side, and delivery of February 2012, comprises 12 vendors (see http://www. ﬁnished cars or ready-made concrete garages on the dis- orms-today.org/surveys/Vehicle_Routing/vrss.html). The tribution side. In the wholesale and retail trade, the dis- results of both studies are hard to compare, because dif- tribution of consumer goods such as drinks, frozen food, ferent information was gathered and organized differently. furniture, or heating oil, to name but a few, is planned with Only one company appeared in both surveys. CVRS. Freight forwarders in the less-than-truckload as well as the full-truckload business determine routes for 4.4 General results local feeding and distribution as well as linehaul and long- distance tramp transports with CVRS. The same holds for Twenty-eight companies sent back a ﬁlled-in question- parcel delivery companies and letter mail services. Also naire. This is a return rate of 56 %, which is acceptable. many reverse logistics and waste collection ﬁrms plan routes with the help of CVRS. An important ﬁeld across 4.4.1 Company structure and size sector boundaries is the solution of service technician, salesman, and other staff dispatching problems. Finally, in Most CVRS companies have their headquarters in Ger- intra-plant logistics of industrial ﬁrms, routes of automated many. The number of employees is 36 on average. The ﬁrst guided vehicles used to fulﬁl transport orders between manufacturer of ‘software for logistics’ was founded in warehouses and production sites are planned with CVRS. 1961. The ﬁrst CVRS, that is, vehicle routing software featuring an automatic, algorithm-based planning compo- 4.5 Models and algorithms for automatic vehicle nent, was offered in 1979. routing in CVRS All ﬁrms offer launching and roll-out support as well as user-speciﬁc adaptation and customization of their soft- Obviously, a central part of a CVRS and the most inter- ware. (This shows that CVRS is (still) not a standard, off- esting one from an OR perspective is the automatic plan- the-shelf product.) In addition, most companies use their ning component. Therefore, detailed information on this own software for project work and consulting services. aspect was queried. This part of the questionnaire was Most ﬁrms, but not all, consider the algorithms used in answered by 27 participants. their systems a core competence. Only four companies do not possess the source code of the algorithms and do not 4.5.1 General features of the automatic planning hold exclusive rights on the code. These companies spe- component cialize in transport management systems (TMS) and use third-party components for automatic planning. Most systems possess the following functionality: they are Cooperation with academia is common. More than three capable of solving general PDP as described above with up to quarters of all ﬁrms stated that they cooperate with at least 10,000 requests, including problems with a heterogeneous 123 Logist. Res. (2012) 5:47–63 57 ﬂeet, a multi-period planning horizon, and multiple use of consider heterogeneous vehicles, at least with respect to vehicles, providing a feasible solution within ﬁve min- costs and capacity, is an absolute must for a CVRS. The utes when distance and travel time matrices are given. surveyed systems generally support different ﬁxed, dis- Most systems can be used to determine only one part of tance- and time-dependent costs. More than half of all the solution of a VRP, namely the clustering of the systems also offer different stop-dependent costs and the requests into groups to be performed by one vehicle use of tariffs; penalty costs for considering soft constraints (whereas the other part, the sequencing of requests, is left are prevalent, too. The simultaneous consideration of dif- to the dispatcher, or, even more often, to the drivers), and ferent capacity constraints is standard as well. Addition- allow the automatic assignment of vehicles and drivers to ally, almost two-thirds of the systems offer support for such groups. Moreover, a re-optimization component multi-compartment vehicles. capable of computing a new feasible solution from an An issue pointed out by several participants of the study existing one after small changes to the instance, such as is that the available geographical data for lorry routing still the arrival of some new requests, is a standard feature. do not cover all relevant attributes (passage heights, barred Another is the possibility to limit the duration of an roads, etc.) in a truly reliable fashion even in Western optimization run by specifying a maximal number of Europe. This is because the commercial providers of geo- iterations and a maximal running time. Lastly, a batch graphical data have concentrated on the much larger car mode for automatic execution of optimization runs with navigation system market. This situation is about to speciﬁc parameter settings is also usual. Interestingly, change, though, and many systems already consider these only eight ﬁrms claim to be able to solve arc routing and data in the preprocessing phase, when distance and travel postman problems. Moreover, although there is a con- time matrices are computed. siderable amount of scientiﬁc literature on the topic (see Different driving speeds per vehicle and time- or load- the survey by Wa ¨ scher et al. [83]), only six systems dependent travel times are rarely supported. This is contain a module for optimizing storage space utilization probably because the necessary data are still difﬁcult to (using 3D packing algorithms). obtain in a sufﬁciently high quality. Tactical ﬂeet plan- ning by directly specifying an unlimited number of 4.5.2 Modelling features vehicles of each type is offered byhalfofall systems; in the other systems, a sufﬁciently large number of vehicles 4.5.2.1 Request-related features Most systems can han- must be speciﬁed by hand if the ﬂeet size and mix is to be dle single as well as multiple time windows for requests determined. and locations, whereas vehicle-dependent time windows Although the concrete type and class of a vehicle are irrelevant for a solution algorithm, the surrounding soft- are only seldom considered. The obvious ‘pickup-before-delivery’ precedence con- ware must be able to manage these attributes in order to straint can be handled by all systems. The requirement of provide sensible feedback to the user. In this respect, most request precedence within a route is also commonly cov- systems are capable of considering lorries/tractors, trailers/ ered. A nested execution is supported by half of all sys- semi-trailers, and cars, but only few systems can also tems, but only few systems can deal with precedence manage pedestrians, bicycles (both of which are relevant, constraints of requests on different routes. for example, in mail delivery), trains, ships, or aircraft. The large majority of the systems allows the consid- Similarly, the technical equipment of a vehicle is only eration of vehicle-driver-request compatibilities, and relevant to determine vehicle-request compatibility, and more than two-thirds of all systems areabletohandle the dimensions and weight of a vehicle are only relevant to optional requests and parallel incompatibility between determine vehicle-location compatibility. The consider- requests. About half of all systems can deal with periodic ation of vehicle class, type, and technical equipment requests and with complex requests consisting of more attributes for specifying vehicle-request-location compati- than one pickup and one delivery location. This indicates bilities is possible with more than four-ﬁfths of the that these request types are commonly encountered in systems. practice. On the other hand, only few systems can con- All CVRS manufacturers claim that their systems pos- sider expected or indirect requests, sequential request sess the feature of considering the European Union social incompatibility, and different ways of performing a legislation on driving, break, and rest times for drivers. request. No system supports request-dependent vehicle During the study, however, the author has gained the itineraries. impression that many systems consider these regulations only incompletely. In particular, only few systems, 4.5.2.2 Fleet-related features A homogeneous ﬂeet is according to the study, contain the rules for double-manned rarely found in the real world. Therefore, the ability to vehicles. 123 58 Logist. Res. (2012) 5:47–63 4.5.2.3 Route-related features Most systems allow the existing plan proposed by the algorithm, such as ﬁx planning of closed and open routes as well as multiple assignments of requests to routes, ﬁx the sequence of routes for one vehicle. Further types of route supported by partial routes, manually assign a certain vehicle to a route, most systems are routes for lorries and trailers, where the etc. Both features, dynamic and interactive planning, trailer can be uncoupled and left behind at parking places, require the capability to re-optimize an existing plan after but with a ﬁxed assignment of lorry and trailer (so that only small changes, without changing the ﬁxed parts. a route for the lorry is computed, and the route of the trailer Only three companies state that their algorithms are is a part of its lorry’s route), the consideration of ﬁxed capable of handling stochastic customers (where the route zones from tactical planning, and routes with a necessity to visit a location is stochastic) or stochastic maximum waiting time. Almost half of all systems support demand/supply (where the amount of demand/supply is the computation of balanced routes with similar capacity stochastic). utilization, number of stops, duration, and cost. The necessity to compute interdependent routes with 4.6 Algorithmic features synchronization requirements between autonomous and non-autonomous vehicles such as lorries and trailers, Whereas participating in the study was seen as a marketing between elementary and composite ‘vehicles’ such as lor- measure by most CVRS manufacturers, several ﬁrms were ries and drivers, or between arbitrary vehicles to perform reluctant to specify details about the algorithms used in transshipments is frequent in practice. However, these their software. However, the questions in this part of the features are supported by few systems only. Mostly, these questionnaire were still answered by 21 ﬁrms, so that also requirements are left to manual planning. The same holds these results may be considered representative. for inter-route resource constraints. Of course, any algorithm used for vehicle routing in practice is necessarily a heuristic. Therefore, the questions 4.5.3 Objective functions on algorithms asked which construction and improvement procedures and which metaheuristics were used. With respect to objective functions, almost everything that is reported in the literature is also available in all or most 4.6.1 Constructive procedures systems: it is possible to minimize the number of vehicles used, the overall distance covered by all vehicles, and the The ranking of constructive procedures is as follows total cost of all vehicles. In addition, about half of all (number of mentions in parentheses): systems support a weighted sum of one-dimensional objective functions, hierarchical, or multi-criteria objective 1. Parallel savings (16) functions. 2. Insertion (11) 3. Cluster ﬁrst, route second (10) 4.5.4 Planning modes Nearest Neighbour (10) 5. Proprietary (7) The classical CVRP corresponds to an operational, single- Sequential savings (7) period, static, deterministic planning situation. Given that 7. Dynamic programming (6) reality is neither static nor deterministic, it is not surprising 8. GENI intra-route (5) that CVRS supports different planning modes. 9. Regret (4) With respect to the frequency of planning, tactical Route ﬁrst, cluster second (4) planning of standard or base routes using aggregate, aver- age data as well as operational, day-by-day planning is 4.6.2 Improvement procedures supported by all systems. Some systems use different algorithms for these two modes, taking into account that The ranking of improvement procedures is as follows: for tactical planning, running time is not critical. Multi- period or rolling horizon planning is also supported by 1. Relocate (move a request to another route) (16) most systems. 2. k-opt (15) In addition, dynamic or real-time planning is possible 3. Swap/exchange (exchange two requests between two with most systems (changing assignments of requests to different routes) (13) vehicles while the latter are already en route, caused by 4. String-relocate (move a route segment to another events such as new requests or breakdowns of vehicles). route) (12) 5. Cross/string-exchange (exchange two route segments Interactive planning is also supported in most cases. This means that the user can make small changes to an between two different routes) (11) 123 Logist. Res. (2012) 5:47–63 59 3. Column generation (4) 6. Or-opt (8) * 4. Branch-and-price (3) 7. k-opt (generalization of k-opt to capacitated prob- 5. Benders decomposition (1) lems) (7) Lagrangian relaxation (1) 8. Lin-Kernighan (6) 9. GENI inter-route (5) k-interchange (exchange at most k requests between 4.6.5 Components, libraries, and benchmarks used two routes) (5) (Very) Large-scale neighbourhood search ((V)LSNS, The ranking of solvers, programming frameworks, and exponential-size neighbourhoods) (5) algorithm libraries is as follows: 12. Double-bridge move (3) 1. CPLEX (3) Ejection chains/cyclic transfers (move a ﬁxed number 2. Boost (2) of requests from route 1 to route 2, then the same COIN (2) number from route 2 to route 3 etc.) (3) LEDA (2) 14. Proprietary (2) XPRESS (2) 6. BCP (1) 4.6.3 Metaheuristics CBC (1) Gurobi (1) The ranking of metaheuristics is as follows: lp_solve (1) SCIP (1) 1. Tabu search (10) SoPlex (1) 2. Genetic algorithms (8) 3. Threshold accepting (7) Eight ﬁrms stated that they tested their algorithms with 4. Proprietary (6) the Solomon VRPTW benchmarks: six have used the Ruin-and-recreate/ﬁx-and-optimize/ripup-and-reroute (6) Gehring/Homberger VRPTW problems and two the Li/Lim Simulated annealing (6) PDPTW instances. No ﬁrm was willing to tell anything 7. Adaptive large neighbourhood search (5) about the results. 8. Ant colony systems (4) Guided local search (4) Variable neighbourhood descent (4) 5 The gaps between theory and practice Variable neighbourhood search (4) 12. Greedy randomized adaptive search procedure (3) It goes without saying that any CVRS is a commercial Memetic algorithms (3) product for end-users without programming and OR 14. Adaptive guided evolution strategies (2) skills. Therefore, it must offer an up-to-date graphical Attribute-based hill climber (2) user interface with adaptable look-and-feel as well as a Backbone search (2) help system. Moreover, a comfortable interface to com- Great deluge (2) mon TMS or ERP systems is also an essential feature. Indirect search (decoder) (2) (Together with the usual GIS, telematics, and statistics Neural networks (2) modules, this means that, typically, less than 10 % of the Scatter search (2) code of a CVRS is for the VRP solution algorithms.) 21. Adaptive/approximate dynamic programming (1) Moreover, aspects such as versatility, genericity, and Adaptive memory programming (1) maintainability are fundamental for any commercial Artiﬁcial immune systems (1) software and must be ensured through adequate design, Particle swarm optimization (1) thorough testing, and extensive documentation. This Record-to-record travel (1) slows down the development process considerably. A scientiﬁc code, on the other hand, is often written as a prototype, a proof-of-concept, to be used only by the 4.6.4 Mathematical-programming-based approaches developers themselves. CVRS must be able to handle many different problems. The ranking of mathematical-programming-based approa- It is not an option to develop and implement a specialized ches is as follows: algorithm for each new customer. Therefore, algorithms 1. Branch-and-cut (5) used in CVRS must necessarily be generic and easily Constraint programming (5) extendable to new problem features. Summing up the 123 60 Logist. Res. (2012) 5:47–63 previous sections, an algorithm for use in a state-of-the-art improvement procedures, which corresponds to an average CVRS supports the following features: of more than ﬁve. • Pickup-and-delivery requests 5.1 Application gaps • Compatibility between locations, requests, vehicles and drivers Aspects that are rather well studied in theory but have not • Multiple time windows for locations and requests yet found widespread use in CVRS and where, conse- • Consideration of service times quently, there is an application gap are stochastic vehicle • Heterogeneous ﬂeet with respect to cost, capacity, start routing (Flatberg et al. [35], Cordeau et al. [20]), time- and end depots dependent travel times (Fleischmann et al. [36], Taniguchi • Fixed, distance-, time-, stop-dependent, penalty costs, and Shimamoto [81], Haghani and Jung [49]), and math- tariffs ematical-programming-based approaches (Maniezzo et al. • Multiple capacity constraints [64]). As far as stochastic VRPs are concerned, this is • Multiple use of vehicles probably because in most cases, it is extremely difﬁcult, if • Driver rules not impossible, to provide sufﬁcient data in sufﬁcient • Weighted and hierarchical cost functions quality to derive useful probability distributions for cus- • Dynamic planning over a one-week planning horizon tomer demands/supplies, and it is doubtful whether this is with event- or time-based rolling horizon planning going to change soon. The consideration of variable travel • Re-optimization options times (driving speeds) depending on the time of day also • Interactive planning requires reliable data. Here, the outlook is more optimistic. Few algorithms described in the literature, if any, are Detailed information will be available in the foreseeable able to deal with all these features. Research algorithms future, at least for large urban regions, which is where peak usually work for special, mostly idealized, types of VRP and off-peak times are most pronounced anyway. Mathe- only. matical-programming-based approaches (matheuristics) are Practitioners need robust, fast, extensible, and simple still a rather new ﬁeld of research, but more and more (parameter-free) algorithms capable of solving instances pertinent publications appear. CVRS manufacturers will with thousands of requests. The last 0.1 % in solution not ignore this trend. quality to be gained from an additional complex algorith- mic device are insigniﬁcant, since the data available in 5.2 Research gaps practice are never 100 % accurate. On the other hand, when using exact methods, the scientiﬁc world strives to ﬁnd On the other hand, a research gap is apparent with respect ‘provably optimal’ solutions for small, idealized problems; to the following aspects: there are only very few papers on using heuristics, researchers are, to a large extent, focussed problems with optional, expected, indirect, or complex on improving best-known results for benchmark instances requests. The existing literature is mostly concerned with or solving concrete real-world problems with prototypical deterministic problems where all requests must be fulﬁlled implementations of specialized algorithms. This point is and consist of single visits or one pickup and one delivery. further elaborated in Cordeau et al. [19] and Pisinger and Exceptions to this rule are Savelsbergh and Sol [76], Røpke Røpke [70]. [75] and Goel and Gruhn [45]. What is more, practitioners A central argument in the above-mentioned paper by often ﬁnd it difﬁcult to give a clear-cut deﬁnition of their So ¨ rensen et al. [79] is that, according to these authors’ problem’s objectives and constraints. Therefore, soft con- experience, commercial CVRS uses quite a large number straints such as visual attractiveness of routes (Lu and of improvement heuristics to improve initial solutions Dessouky [62]) or a preferred assignment of certain drivers determined by constructive procedures. This is in contrast or vehicles to customers (Groe ¨r et al. [47]) are quite to scientiﬁc codes, which tend to use few, but rather important in real-world applications. For the same reason, a complex and sophisticated techniques. The reasons for this ‘fair’ and balanced sharing of the workload between dif- are, according to So ¨ rensen et al., that (1) an approach using ferent routes is important in practice, but seldom consid- many improvement procedures can overcome the greedy ered in the literature (an exception is the paper by behaviour of an approach that uses only a single one and Bredstro ¨ m and Ro ¨ nnqvist [12]). Tariffs and complex cost (2) supplying a large arsenal of diverse search strategies functions are often used in practice, but rarely considered allows a ﬂexible adaptation of the software to the speciﬁc in the literature. See Ceselli et al. [15] for a striking requirements of each customer. The results of the study exception. described in the present paper support this observation: as Most importantly, models and algorithms for integrated can be seen in Sect. 4.6.2, 21 routing tools use 111 different and synchronized vehicle routing are still scarce: in almost 123 Logist. Res. (2012) 5:47–63 61 all vehicle routing models and algorithms, the routes of the powerful decision support tools for integrated and syn- different vehicles are assumed to be independent of one chronized vehicle routing in practice. another, so that modifying one route does not have any effects on other routes. However, in a surprisingly high number of cases, this assumption does not hold. Examples 6 Conclusions and outlook for practical applications requiring a spacial, temporal, and in some cases also load-related synchronization or The exact solution of even the basic variants of VRPs is coordination of routes are the planning of inter- and multi- still impossible for instances of realistic size. An exact modal transports, the planning of meet-and-turn routes, solution of real-world problems with many additional side transports over hubs or cross-docking locations, simulta- constraints will remain impossible in the short and medium neous planning of routes for lorries and trailers, if trailers term. However, close-to-optimal solutions of more and may be pulled by different lorries, simultaneous vehicle more complex and integrated problems, increasingly based and driver routing, if drivers may change vehicles, and on incomplete optimization approaches and mathematical- automatic planning of multiple types of resources (driv- programming-based heuristics, are possible, and this is ers, lorries/tractors, trailers/semi-trailers, swap-bodies/ sufﬁcient to provide useful decision support in practice. containers). Nevertheless, as has already been alluded, in some areas The following quote taken from Irnich [57], p. 9, still there are gaps between industrial needs and the state-of- holds: ‘While research on integrated models and solution the-art CVRS of today. A detailed discussion of these methods for combined vehicle and crew scheduling has issues is, however, beyond the scope of the present paper, made some remarkable advances ..., the literature on but constitutes an interesting topic for further research. integrated vehicle routing still mainly focuses on location For the foreseeable future, CVRS will remain a decision routing problems and inventory routing problems. Litera- support system in almost all application areas. Essentially, ture on other forms of integration is scarce. There is a need fully automatic planning is possible only in some special for new and improved techniques to attack integrated cases, most notably in intra-plant logistics. In road and planning problems. As far as we can see, there is no con- inter-modal transport, interactive planning with a human vincing concept for dealing with VRPs with load transfer at dispatcher having the ﬁnal say is and will remain the rule. hubs or consolidation points, especially in the context of A modern CVRS, however, can considerably facilitate the bimodal or multimodal trafﬁc. The same is true for long- daily routine work for human decision-makers. The sys- haul goods trafﬁc, which requires the coordination between tems have become so mature and user-friendly that, now- feeder processes, linehaul, and distribution’. adays, after introducing a CVRS, nobody wants to return to purely manual planning any more. The concerns often More speciﬁcally, Macharis and Bontekoning [63], p. 400, state in their survey of inter-modal freight transport voiced by many dispatchers, CVRS would invalidate that ‘intermodal freight transportation research is emerging their knowledge and experience or even make them as a new transportation research application ﬁeld, that it lose their jobs, are unfounded. This has become evident in still is in a preparadigmatic phase, and that it needs a dif- many discussions with manufacturers as well as users of ferent type of models than those applicated to uni-modal CVRS. transport’. Summing up, CVRS constitutes a ﬁxed and indispens- In short, a general, unifying modelling and solution able component of logistics planning in practice. Just like concept for integrated and synchronized vehicle routing is any other product, CVRS has to adapt to ever-changing still missing; science has to catch up in this respect. This customer needs and expectations. This requires constant statement is further supported by the fact that the survey further development, both with respect to information article by Gendreau et al. [40], which presents an extensive technology and to OR models and algorithms. Conse- literature list on VRPs, does not mention a reference on quently, even more than half a century after the ﬁrst OR VRPs with multiple synchronization constraints. Moreover, paper in this ﬁeld, the practice of vehicle routing will the recent monograph by Golden et al. [46] does not con- continue to provide interesting and challenging problems tain a paper on this topic. This does not mean that there are for OR researchers. no such papers at all. Rather, it shows that no systematic study of this problem class has yet been performed. Such research is now beginning to emerge (see the survey by References Drexl [32]). On the other hand, the results of the study show that some CVRS systems contain solutions to con- 1. 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Geometric Modelling, Numerical Simulation, and Optimization - Applied Mathematics at SINTEF
Arild Hoff, H. Andersson, Marielle Christiansen, G. Hasle, A. Løkketangen (2010)
Industrial aspects and literature survey: Fleet composition and routing
Comput. Oper. Res., 37
C. Archetti, M. Speranza (2008)
The Split Delivery Vehicle Routing Problem: A Survey
C. Cotta, M. Sevaux, Kenneth Srensen (2008)
Adaptive and Multilevel Metaheuristics (Studies in Computational Intelligence)

Publisher: Springer Journals
Copyright: Copyright © 2012 by Springer-Verlag
Subject: Engineering; Engineering Economics, Organization, Logistics, Marketing; Logistics; Industrial and Production Engineering; Simulation and Modeling; Operation Research/Decision Theory
ISSN: 1865-035X
eISSN: 1865-0368
DOI: 10.1007/s12159-012-0080-2
Publisher site: See Article on Publisher Site

Abstract

Logist. Res. (2012) 5:47–63 DOI 10.1007/s12159-012-0080-2 OR IGINAL PAPER Michael Drexl Received: 6 September 2011 / Accepted: 14 June 2012 / Published online: 3 July 2012 Springer-Verlag 2012 Abstract The contribution of this paper is a comparison can also be found in intra-plant logistics, that is, local of the state of the art of scientiﬁc research on and com- transport within a factory or warehouse building or on mercial software for modelling and solving vehicle rout- company premises. ing problems. To this end, the paper presents a compact Beside the considerable importance of effective and review of vehicle routing literature and an overview of the efﬁcient vehicle routing for the enterprises themselves, the results of a recent study of commercial vehicle routing macroeconomic relevance of vehicle routing must not be software systems with respect to the problem features overlooked: the avoidance of unnecessary or unnecessarily these systems are able to handle and the solution methods long routes with low capacity utilization removes pressure the systems use for automatic generation of vehicle from road infrastructure, improves trafﬁc ﬂow for freight as routes. In this way, existing application and research gaps well as passenger transport, and, by reducing emissions, are identiﬁed. makes a sustained contribution to decrease the harmful effects of transportation. Keywords Rich vehicle routing Commercial vehicle For operational research (OR), vehicle routing consti- routing software Heuristics tutes one of its great success stories. Vehicle routing problems (VRPs) in their many variants have been the subject of intensive study for more than half a century now. This has led to the publication of thousands of scientiﬁc 1 Introduction papers and to the foundation of numerous software com- panies worldwide selling commercial vehicle routing Vehicle routing is a central task in a large number of pri- software (CVRS). This development is certainly due to the vate and public corporations. Routes have to be planned in intellectual challenge VRPs pose as well as to their prac- very diverse sectors of the economy, not only in the tical relevance in logistics and transport. Research on VRPs logistics and transport business, but in virtually all indus- is incessantly ongoing, stimulated by unsolved theoretical trial sectors producing physical goods. In addition to problems and continuous input from logistics practice. transport on public roads, applications of vehicle routing The contribution of this paper is a comparison of sci- entiﬁc research on VRPs and commercial software for modelling and solving VRPs. To this end, the paper pre- sents an overview of the results of a recent study of CVRS M. Drexl (&) with respect to the problem features these systems are able Chair of Logistics Management, Gutenberg School to handle and the solution methods the systems use for of Management and Economics, Johannes Gutenberg University, Mainz, Germany automatic generation of vehicle routes (Drexl [31]). These e-mail: michaeldrexl@web.de ﬁndings are contrasted with the state of the art of scientiﬁc VRP research. In this way, existing application and M. Drexl research gaps are identiﬁed. This should be of interest for Fraunhofer Centre for Applied Research on Supply Chain VRP researchers, and also logistics practitioners using or Services SCS, Nuremberg, Germany 123 48 Logist. Res. (2012) 5:47–63 planning to use CVRS should beneﬁt from this paper, by considered planning horizon. An overview of these learning about the potential of modern CVRS. dimensions of richness in real-world VRPs is given in Throughout the paper, the following deﬁnitions apply. Fig. 1, and a discussion of these characteristics follows. The fundamental activity to be planned in vehicle routing (An orthogonal characterization is by application area. is called a request. A request may be a transport order, such Industry sectors where scientiﬁc VRP research is particu- as the delivery of a shipment from a central depot to a larly widespread are discussed in Sect. 3.2; industry sectors recipient, the pickup of a shipment from a consignor and and ﬁelds of application where the use of CVRS is com- the transfer to a central depot, the pickup of a shipment at mon are presented in Sect. 4.4.2.) Characteristics of rich some location and the transport to some other location, or a VRPs are also discussed in Hasle and Kloster [51], Sect. 3, visit at a location to perform a service there, without and So ¨ rensen et al. [79], who state (p. 241): ‘Although picking up or delivering a physical good. Vehicle routing there is an increasing scientiﬁc focus on so-called ‘‘rich’’ means to group requests into clusters performed by one VRPs (that incorporate more complex constraints and vehicle each, and to determine, for each cluster, a complete objectives), they have not in any way caught up with the sequence of the resulting locations to be visited. This whole complexity of real-life routing problems’. To a large process can be performed manually by a human planner, extent, this point is also supported by the results described automatically by a computer program executing an algo- in the present paper. rithm, or by a combination of both. The goal of vehicle routing is the optimization of an objective function. This 2.1 Requests will regularly be the minimization of a cost function, of the number of used vehicles, of the total distance travelled, etc. There are a large number of different aspects of requests. The rest of the paper is structured as follows. The next First of all, time windows are central properties of requests. section describes the decisive aspects by which the Time windows can be caused by the request itself (e.g. numerous variants of VRPs can be distinguished. Section 3 earliest ready-time of a manufactured good to be picked up then gives a brief overview of the state of the art of sci- or latest delivery time of a component needed at the des- entiﬁc VRP research. Section 4 presents the results of a tination) or by the location where the request is to be comprehensive study of the German CVRS market, performed (opening hours). There can be one or more focussing on modelling and algorithmic aspects for the disjoint time window(s) for a request (opening hours in the automatic solution of VRPs. Section 5 discusses the gaps morning and in the afternoon). Moreover, time windows between theory and practice, and Sect. 6 gives a conclusion can be vehicle-dependent (e.g. large delivery vehicles and an outlook. having more restrictive access to customers in inner-city zones than small ones). Another important aspect is pairing and precedence:ifa 2 An overview of VRPs: dimensions of richness request consists in the transport of a good from a pickup location to a delivery location, then, if no transshipments The archetypal, fundamental VRP, the capacitated vehi- are allowed, one and the same vehicle must visit both cle routing problem (CVRP),isasfollows.Given area locations. Moreover, it is obvious that the pickup must be set of identical vehicles stationed at one depot and performed before the delivery. There are also complex equipped with a limited loading capacity, and a set of requests, which consist of more than one pickup and one geographically dispersed customers with a certain delivery location. Often, a nested execution (so-called demand for a homogeneous good. The task is to deter- LIFO loading: pickup request A, pickup request B, deliver mine an optimal (with respect to an objective function) B, deliver A) is required (e.g. if vehicles can be loaded and route plan, that is, a set of vehicle routes, specifying unloaded only from behind). which customers are visitedbywhich vehicleinwhich Vehicle-driver-request compatibilities are the third sequence, such that each customer is visited exactly once, fundamental aspect. Depending on the vehicle character- the complete demand of each customer is satisﬁed, and istics and the driver qualiﬁcations, not all requests can or the loading capacity of the vehicles is maintained on each may be performed by all vehicles and drivers, even if the route. The objective is to minimize overall cost or trav- request locations are accessible to the vehicle and the elled distance. driver. As mentioned, there are a huge number of variants, Further important types of requests are optional extensions, and generalizations of the CVRP. VRPs can be requests (requests that need not be assigned to a route, but categorized according to their properties with respect to whose execution brings a bonus), periodic requests the requests to be fulﬁlled, the ﬂeet available for doing so, (requests that have to be executed several times within a the desired route structure, the objectives pursued, and the planning horizon, mostly according to visitation patterns, 123 Logist. Res. (2012) 5:47–63 49 Characteristics of VRPs Requests Fleet Route structure Objectives Scope of planning Time windows Costs Closed/open One-dimensional Time horizon � � � � � – Single – Fixed Multiple routes per vehicle – Min. number vehicles – Tactical – Multiple – Variable Fixed route zones – Min. cost – Operational Pairing and precedence con- – Penalty Visually attractive routes – Min. distance – Real-time � � straints – Tariffs � Interdependent routes with – Max. proﬁt � Data availability and accur- – Pickup before delivery � Capacity constraints synchronization � Multi-dimensional acy – LIFO loading – Weight – Between autonomous and – Weighted sum of single – Deterministic � Vehicle-driver-request- – Volume non-autonomous vehicles objectives – Stochastic compatibilities – Loading metres – Between elementary and – Hierarchical ordering of – Uncertain or unknown Special types of requests – Pallet places composite vehicles objectives – Optional Driving speed – To perform transshipments – Multi-criteria objective – Periodic – Same for all vehicles Inter-route resource con- searching for Pareto- – Expected – Different for different straints optimal solutions – Incompatible vehicles � Balanced routes, i.e., all � Hardness of constraints � Different ways of performing – Constant routes should be similar with – Hard a request – Time-dependent respect to – Soft (with penalty in ob- – Request splitting – Load-dependent – Capacity utilization jective function) – Alternative delivery loca- Temporal availability – Number of stops tions Initial and ﬁnal vehicle posi- – Duration Request-dependent vehicle tions – Costs etc. itineraries � Type – Lorry – Trailer – Train – Ship – Aircraft etc. Technical equipment Number of vehicles of each type – Limited – (Virtually) unlimited Drivers – Qualiﬁcations – Social legislation Fig. 1 Dimensions of richness in VRPs for example, twice a week, but not on consecutive days), his home address after six o’clock or on Saturdays. This expected requests (requests that have not yet been issued raises the additional question of how to choose a way of by the customer, but will probably be), incompatible performing the request (where to perform a meet-and-turn requests (parallel incompatibility: do not transport the operation, how to split a request into sub-requests, when to requests at the same time with one vehicle; sequential deliver a package). incompatibility: do not transport request B on a route that Finally, there is the aspect of request-dependent vehicle has transported request A before), and indirect requests itineraries. This refers to situations where the transport (e.g. automatic generation of empty container balancing links a vehicle is able or allowed to use depend on the requests). requests it is carrying, which means that requests determine A further very difﬁcult aspect is when there are different the distance and the travel time between locations. For ways of performing a request. This refers to the possibility example, when a vehicle for transporting bulky goods is to split up the fulﬁlment of a request between several empty, it may be able to pass through a low undergrade vehicles, or to the possibility to perform a request by dif- crossing, but when the vehicle is loaded, it may be too high ferent operations altogether. An example for the ﬁrst case to pass and may thus have to make a detour. Similarly, if a is multi-modal transport, where a request to bring a con- tank vehicle for oil or fuel delivery is empty, it is allowed signment from A to B can be performed by a direct trans- to travel through a water protection area; if such a vehicle port from A to B, via a meet-and-turn operation, or via one is loaded, it must take an alternative, longer way. or several hubs. Another is that the request to deliver x units of a good to a customer can be fulﬁlled by one 2.2 Fleet delivery of x units by one vehicle or by several deliveries by several vehicles. An example for the second case is The term ‘ﬂeet’ refers to the resources available for ful- parcel delivery, where a package must be delivered to the ﬁlling the requests. These resources comprise vehicles of recipient’s ofﬁce address from nine to ﬁve o’clock, and to different types as well as drivers operating them. 123 50 Logist. Res. (2012) 5:47–63 2.2.1 Vehicles board, dangerous-goods equipment, etc. The dimensions of a vehicle also inﬂuence which transport links can be used Vehicles may differ with respect to several criteria, the (large lorries cannot use small inner-city roads, super- most important ones being costs, capacities, driving speed, tankers cannot use the Panama canal). Moreover, depend- temporal availability, actual and desired position at the ing on its weight or emission level, a vehicle may be unable beginning and the end of a planning process, type and or not allowed to use certain roads. technical equipment, and the (in)ability to visit certain In tank vehicles, there are often several compartments locations and use certain transport links. that can be ﬁlled separately to allow the simultaneous Relevant cost categories are ﬁxed costs for using a transport of different goods or products or requests. When vehicle, and distance-, time-, and stop-dependent variable there are n compartments, n different products can be costs. Distance-dependent costs may include road tolls. transported. When all compartments are used, no request Time-dependent costs may be linear or nonlinear and may can be executed that requires the transport of another good, include overtime pay or daily allowances for drivers. even when none of the capacity constraints listed above Moreover, costs may be calculated based on tariffs. Tariffs would be violated. used to be mandatory in Germany until the end of the The number of vehicles of each type and class is twentieth century and were dependent on goods types, important, too. In reality, the number of vehicles is of weight, distance, time, etc.; although the numeric values course always limited. For tactical planning of the ﬂeet size have decreased sharply, the calculation formulas are still and mix, it may, however, be interesting to allow an common in practice. For planning purposes, penalty costs unlimited number of vehicles of each type and class. are often used to consider soft constraints or undesired properties of solutions, or to allow infeasible solutions 2.2.2 Drivers during the solution process. The most common capacity constraints in goods trans- As far as drivers are concerned, restrictions regarding port are weight or payload, volume, loading metres, and qualiﬁcations limit the compatibility between drivers and number of pallet places. Several of these may be relevant at vehicles as well as between drivers and requests. Such the same time. qualiﬁcations may be the type of driving licence a driver The driving speed may be the same for all vehicles or possesses, whether or not a training for the transport of differ between vehicles or vehicle types. It may also dangerous goods was completed, or knowledge of cus- depend on the load a vehicle is carrying (the fuller, the tomer or region speciﬁcs. slower). Moreover, the driving speed may be constant for Another matter of utmost importance in real-world lorry all vehicles throughout the complete planning horizon, or road transport are driver rules. In the European Union and be time-dependent. This is particularly important for short- in other parts of the world, there is extensive social legis- distance and inner-city transport, where travel times are lation on driving, working, break and rest times for drivers; signiﬁcantly higher during rush-hours compared to off- see Humphreys [56] for an overview. The automatic peak times. tachograph introduced in the European Union nowadays Also, the temporal availability of a vehicle may be allows for much tighter supervision of compliance with limited, for example, due to scheduled maintenance or social legislation for drivers, and the road transport Sunday driving bans for heavy lorries, but not for smaller industry in Europe is acknowledging that today, it has to vans. comply with the regulations very exactly. It is important to In operational, short-term planning, initial vehicle note that an algorithm cannot determine a ‘legal schedule’ positions (depots) are given, whereas in tactical, mid-term for a route, because the term ‘legal schedule’ has no legally planning, it is often part of the planning task to determine binding mathematical deﬁnition; it is a purely juristic appropriate locations for the vehicles. In operational as concept. In an unlucky attempt to provide ﬂexibility in well as tactical planning, locations at the end of the plan- practice, the European Union has introduced an intractably ning horizon are arbitrary if open routes are allowed. complex set of optional rules along with the mandatory Concerning type and technical equipment, there are ones. These rules leave a lot of room for interpretation, and different criteria that determine whether or not a vehicle is a dispute about the legality of a schedule will eventually in principle able to perform a request, disregarding the have to be settled in court. For practitioners, this means that current point in time, location, or capacity utilization. trying to exploit the optional rules is dangerous. For Among these criteria are the vehicle type (lorry, train, ship, algorithm developers, the optional rules mean a lot of etc.), the vehicle class (swap-body vehicle, tank vehicle, tedious work: on the one hand, for considering them, on the etc.), vehicle dimensions and weight, and technical other hand, for ensuring that the overall algorithm is not equipment such as a ﬁxed installed tail-lift, a fork-lift on slowed down too much. 123 Logist. Res. (2012) 5:47–63 51 2.3 Route structure itinerary. A very similar case is the planning of separate routes or rotations for vehicles and drivers, where, during Some aspects of the route structure concern each route the planning horizon, a vehicle may be operated by dif- individually, others lead to interdependencies between ferent drivers, and a driver may drive several vehicles. This routes. improves the temporal capacity utilization of vehicles, since these can essentially be used 24 h a day, whereas 2.3.1 Individual routes drivers need regular breaks and rests and have to obey the above-mentioned driver rules. The standard case is a closed route (loop), starting and Furthermore, allowing transshipments of load between ending at the same location (depot). Nevertheless, also vehicles leads to interdependent routes. Transshipments open routes, where a vehicle may be at any location at the occur in the form of meet-and-turn routes with exchange of end of its route, are relevant in many situations. For complete swap-body platforms or as partial exchanges of example, in long-distance road transport, vehicles are en single consignments or one-way transfer of load from one route for a complete week, but routes are planned only for vehicle to another. Multi-modal transport, by deﬁnition, the next day, so the routes for Monday to Thursday end at requires transshipments of load. the last customer scheduled for the respective day. The Also inter-route resource constraints such as processing converse, that is, the planning of multiple closed routes for capacities at depots, a maximum number of vehicles one vehicle, is also possible, for example, in local delivery arriving at a depot per time unit due to limited number of applications, where vehicles return to the depot more than ramps or conveyor belt capacities, etc., make synchroni- once during a day to reload. zation between routes necessary. Further possible types are routes with special geographic Finally, there is the requirement of balanced routes. properties, such as the consideration of ﬁxed route zones in This refers to the stipulation that all routes of a plan be tactical planning, and routes with limits on total duration or similar with respect to covered distance, duration, number waiting time. of requests, costs, etc. 2.3.2 Interdependent routes 2.4 Objectives The usual assumption in almost all VRPs is that the only Objective functions may be one- or multidimensional. coupling or linking or joint constraints between the routes Potential one-dimensional objectives pursued are the min- of different vehicles are related to request covering, to imization of the number of vehicles used, of the overall ensure that each request is performed exactly once. The distance covered by all vehicles, and of the total cost of all preceding aspects leave routes independent of one another vehicles. If not all requests are mandatory, the objective in this sense. The feasibility of one route does not affect the may be the maximization of the difference between the feasibility of another. However, there are also requirements proﬁt obtained from the fulﬁlled requests and the costs that lead to route interdependencies, to routes that must be incurred for fulﬁlling them. synchronized. In such cases, the feasibility of one route When the objective comprises several dimensions, it is may depend on the feasibility of one or more other routes. possible to consider a weighted sum of one-dimensional Multiple synchronization of vehicles or routes may be objectives, to have a hierarchical (lexicographic) ordering relevant with respect to space, time, load, or common of the dimensions (e.g. the minimization of the number of scarce resources. vehicles used as the most important criterion and the One example of such a requirement is that of a ‘visually minimization of cost as the second one, as the tie-breaker attractive’ route plan, which often means intersection-free in the case of two route plans using the same number of routes. Synchronization requirements also occur when vehicles), or a multi-criteria objective searching for Pareto- there are different types of elementary vehicles that may or optimal solutions. must join and form a composite vehicle to be able to move An important aspect that must be mentioned here is that in space or to perform a request. An example is the plan- many constraints and requirements discussed above can be ning of separate routes for lorries and trailers (or tractors considered as hard or soft constraints. Any violation of a and semi-trailers). Each lorry and each trailer is an ele- hard constraint is strictly forbidden and invalidates a route mentary vehicle and can be used to perform requests, and a plan. Constraints such as technical or logical restrictions route is computed for each lorry and each trailer that is (e.g. pickup before delivery) or legal obligations (e.g. actually used. Naturally, the route of a trailer must be working and rest times of drivers) are always hard. A synchronized with the routes of one or more lorries that violation of a soft constraint does not directly invalidate a must pull the trailer on the whole or on a part of its route plan, but is undesired and thus considered with a 123 52 Logist. Res. (2012) 5:47–63 penalty in the objective function. The penalty usually number of benchmark instances for different types of VRP increases with increasing degree of violation, and if the can be found at http://people.brunel.ac.uk/*mastjjb/jeb/ degree of violation exceeds a certain threshold, the con- info.html.) For the latter class, the term ‘rich vehicle straint becomes hard and invalidates the route plan. Time routing’ has been coined rather recently to denote models windows, for example, are sometimes considered as soft and solution approaches for problems that feature several constraints. or all aspects of a real-world application. Most papers belonging to the latter class focus on one new or particu- 2.5 Planning horizon and data availability larly interesting or difﬁcult aspect. A number of important such aspects were queried in the CVRS study mentioned Depending on the frequency of planning and the duration above and was discussed in detail in the previous section. during which plans remain valid, or, put differently, the range and scope of the decisions taken, there are tactical 3.1 Problem variants (medium-term), operational (short-term), and real-time (dynamic) VRPs. Tactical decisions encompass the size Researchers have devoted a lot of effort to the study of a and composition of the ﬂeet (number of vehicles of each rather small number of abstract, generic, and well-deﬁned type, size and technical conﬁguration, assignment to extensions of the CVRP, and rightly so: although these depots) and the preparation of ‘framework routes’ based on scientiﬁc variants of the CVRP hardly ever appear in average data for application areas with periodic supply or practice in their pure form, their study is worthwhile, demand variation (e.g. seasonal route plans for raw milk because the results and insights obtained can serve as a collection at farms with high volume in spring and low basis for tackling the numerous detailed and speciﬁc real- volume in winter or different routes for postal distribution world routing problems. The most important such theo- on different weekdays). The resulting plans may cover retical CVRP extensions are brieﬂy described in the fol- multiple periods. Operational vehicle routing is concerned lowing, by pointing out in which respect these problems with the planning of routes for the next day(s), based on extend the CVRP. The cited references are surveys of the concrete data. Real-time routing takes into account new or respective problem. changing data (such as, for example, incoming requests, In the VRP with time windows (VRPTW, Bra ¨ysy and vehicle breakdowns, trafﬁc congestions) and adapts plans Gendreau [10, 11], Cordeau et al. [20]), the service at each while these are being executed. customer must start within a given single hard time win- A related aspect is the availability and accuracy of the dow. In the split-delivery VRP (SDVRP, (Archetti and data on requests, vehicles, relevant locations, and trafﬁc Speranza [4]), customers may be visited more than once by more than one vehicle. Each vehicle may deliver a fraction links. There are three cases: the deterministic case, where all data are known in advance, the stochastic case, where of a customer’s demand. some data are known in advance only in the form of In Pickup-and-delivery problems (PDP, Parragh et al. probability distributions, and the case of uncertainty, where [68, 69]), the tasks consist in the transport of shipments some data are unknown and become known only during from one location to another, that is, not only the planning or during execution of a route plan. delivery locations are all different, but also the pickup locations. Dial-a-ride problems (DARP, cf. ib.) consider the transport of persons and, in contrast to PDPs, usually 3 Scientiﬁc VRP research feature constraints restricting passenger inconvenience, for example, by limiting the maximum ride duration. It As stated in the introduction, over the last half century, must be noted that there are different sub-types of PDPs, there have been thousands of scientiﬁc publications on such as the VRP with backhauls or the VRP with vehicle routing, starting with the famous paper by Dantzig simultaneous delivery and pickup. The reader is referred and Ramser [25]. Therefore, the following elaborations to the above surveys for a complete taxonomy of PDPs only give a very rough overview, a ‘survey of surveys’, and and DARPs. necessarily refer the reader to the literature for details. The In Periodic VRPs (PVRP, Francis et al. [37]), several existing VRP literature can be divided into theoretical visits are required to serve a customer during a multi-per- papers studying models or methods for idealized or stan- iod planning horizon. These visits must take place in dif- dardized problems and problem-oriented case studies ferent periods. An interesting variant is the consistent VRP dealing with concrete real-world applications. The former (Groe ¨r et al. [47]). This is a periodic VRP where each class considers exact as well as heuristic solution approa- customer must always be visited by the same vehicle in the ches and uses theoretical benchmark instances to measure different periods and each customer must be visited at the effectiveness of the devised algorithms. (A large ‘roughly the same time’ on each visit. 123 Logist. Res. (2012) 5:47–63 53 Heterogeneous ﬂeet VRPs (HVRP, Hoff et al. [54]), as There is also an increasing number of publications on the name implies, consider the case that not all vehicles are algorithms for considering VRPs with driver rules (see, e.g. identical. The ﬂeet size and mix VRP (FSMVRP, cf. ib.) is Archetti and Savelsbergh [3], Goel [43], Drexl and Pres- the tactical variant of the HVRP and considers (different) cott-Gagnon [33], Goel [44], Kok et al. [61], Prescott- ﬁxed costs for using (different types of) vehicles. Gagnon et al. [73]). The capacitated arc routing problem (CARP, Corbera ´n In addition to the above-mentioned surveys, Toth and and Prins [18]) is a variant of the CVRP where the tasks are Vigo [82] and Golden et al. [46] are recent monographs on not to visit customers to perform a service, but where the VRPs and their variants. All of these references contain service is performed while travelling along the links of a results on exact as well as heuristic methods. (road) network. Location-routing problems (LRP, Nagy and Salhi [66]) 3.2 Application-oriented research combine routing and locational decisions. The task is to determine a set of vehicle routes and, for each route, the There are some application areas where OR methods have location where it starts and ends. Using a location by sta- a long-standing tradition (not only in the context of vehicle tioning a vehicle there incurs ﬁxed costs. routing), and where there is a particularly large number of In stochastic VRPs (Flatberg et al. [35], Cordeau et al. application-oriented papers. Such niches of applied VRP [20]), information on occurrence and volume of customer research can be found in the airline industry (Klabjan [60], demand or travel times between customers is given by Ball et al. [7]), public transport (Desaulniers and Hickman probability distributions. In Dynamic VRPs (Powell et al. [29], Hickman et al. [53]), ship routing (Christiansen et al. [72]), as already outlined in Sect. 2.5, the planner is forced [17], Hennig [52]), rail transport (Cordeau et al. [21], to make decisions before all relevant information becomes Caprara et al. [14]), and letter mail or parcel delivery available; decisions must then be modiﬁed as new infor- (Bodin and Levy [9], Wong [84]). mation is received. Essentially, planning is performed Seminal case studies describing the successful solution of parallel to plan execution. rich real-world VRPs are listed in Table 1, sorted chrono- The inventory routing problem (IRP, Moin and Salhi logically. Note that this list is necessarily incomplete. [65], Andersson et al. [2]) is a very special type of VRP. In IRPs, there are no customer demands. Instead, each cus- 3.3 Solution methods tomer has a given consumption rate of a good, a given initial stock and a given storage capacity. The depot has to VRPs are usually modelled using graphs or networks and perform zero or more deliveries to each customer during a formulated as mixed-integer programs (MIPs). As regards multi-period planning horizon to ensure that no customer solution methods, there are two fundamental approaches: runs out of stock. The objective is to plan routes of minimal Mathematical-programming-based algorithms on the one cost for the deliveries. hand, and heuristics and meta-heuristics on the other. Table 1 Selected case studies on rich VRPs Paper Application Savelsbergh and Sol [77] Dynamic, multi-period pickup-and-delivery with complex requests Xu et al. [85] Pickup-and-delivery with complex cost functions and LIFO loading Hollis et al. [55] Simultaneous and interdependent vehicle and crew routing and scheduling Cheung et al. [16] Synchronized routing of lorries and trailers Irnich [58] Arc routing with turn and street crossing restrictions, cluster constraints, and alternative service modes Za ¨pfel and Bo ¨ gl [86] Simultaneous and interdependent vehicle and crew routing and scheduling with outsourcing options and working time regulations Ceselli et al. [15] Heterogeneous ﬂeet, multi-depot, split-delivery VRPTW with open routes and request incompatibilities Bock [8] PDP with time windows and transshipment options Oppen et al. [67] IRP with route duration and precedence constraints using heterogeneous vehicles with compartments Rieck and Zimmermann Simultaneous delivery and pickup with synchronization constraints at loading docks [74] Schmid et al. [78] VRP with splitting of loads and synchronization of different vehicles at customer sites Derigs et al. [26] VRP with multiple use of tractors and trailers 123 54 Logist. Res. (2012) 5:47–63 3.3.1 Exact approaches Method-oriented surveys or tutorials on heuristics and meta-heuristics are given by Funke et al. [38] (local Mathematical programming algorithms are based on MIP search), Røpke [75] (large neighbourhood search), Ahuja models and, in theory, guarantee to ﬁnd an optimal et al. [1] (very large-scale neighbourhood search), Powell solution if one exists. The most successful exact algo- [71] (adaptive dynamic programming), Cotta et al. [22] rithms for VRPs are branch-and-cut-and-price methods, (metaheuristics), and Gendreau and Potvin [41] (meta- which combine cut and column generation with branch- heuristics). Gendreau and Potvin [39] develop an inte- and-bound. However, mathematical programming algo- grating and unifying overview of metaheuristics, and rithms typically require too much time and memory for Gendreau et al. [40] present a categorized bibliography of large instances. Moreover, the computation times for metaheuristics for several types of VRP. instances of the same size and structure often vary to a Hasle and Kloster [51], Sect. 4, give a description of a large degree. At the time of this writing, CVRP and commercial software for solving rich VRPs. In particular, VRPTW instances of more than 200 customers cannot the conceptual approach for modelling and representing be consistently solved to optimality. Rich real-world rich VRPs in a software tool (as opposed to in a mathe- instances with many complicating constraints and a real- matical model) is described in detail. Moreover, the istic number of requests are still untractable with exact implementation of the solution algorithms used is methods. explained. Groe ¨r et al. [48] describe a publicly available A milestone in the ﬁeld of exact methods for VRPs is the programming framework for solving VRPs and give paper by Desaulniers et al. [27], which discusses issues detailed explanations of the framework’s design. Both arising in the modelling and solution of time-constrained codes were not part of the survey presented in the next vehicle routing and scheduling problems using mixed- section. Both papers treat aspects of vehicle routing soft- integer programming and column generation. Desaulniers ware that could not be queried in the survey. et al. [28] present a monograph on column generation and branch-and-price, Røpke [75] describes several exact 3.4 Trends in VRP research algorithms for VRPs and PDPs, Spoorendonk [80] treats issues related to cut and column generation, and Baldacci 3.4.1 Richness and robustness et al. [5] describe an exact solution framework for different types of VRP that outperforms all other exact methods With respect to models, there is a clear trend towards published so far and solves several previously unsolved considering ever ‘richer’ problems (Hartl et al. [50]), and benchmark instances. Finally, Baldacci et al. [6] provide an towards developing generic, uniﬁed modelling frameworks up-to-date review of the state-of-the-art exact algorithms (Irnich [59]) for representing these rich problems. With for the CVRP and the VRPTW. respect to methods, considerable progress has been made concerning the development of exact as well as heuristic 3.3.2 Heuristics and metaheuristics solution algorithms that are robust, that is, work well for a broad range of problems both in terms of running time and Heuristics and metaheuristics do not offer an optimality solution quality (Pisinger and Røpke [70], Baldacci et al. guarantee, but they overcome the limitations of exact [5]). algorithms and are able to ﬁnd close-to-optimal solutions in short time, even for very large instances. Heuristic methods 3.4.2 Self-adaptation and hyperheuristics can be divided into constructive procedures, which are used to compute an initial feasible solution, and improvement A related aspect is the trend towards self-calibrating and/or procedures, which iteratively try to improve a given solu- self-adapting algorithms: Many of the metaheuristics tion by systematically modifying it. Metaheuristics are developed over the years are highly sophisticated and superordinate procedures that control the search processes contain a large number of parameters for which sensible performed by constructive and improvement heuristics. values must be set to obtain good solutions. Different Section 4.6 contains an extensive list of constructive and problems or instances with different data characteristics improvement heuristics as well as metaheuristics. Best- require different parameter settings. Similarly, the recently known heuristic solutions to benchmark instances for the proposed so-called hyperheuristics adapt the search space problems described in Sect. 3.1 have been computed with continuously in the course of the computations, based on many different methods, so there is deﬁnitely no silver the previous solution progress. The monograph edited by bullet. However, it must be noted that most successful Cotta et al. [22] describes several approaches in the ﬁelds heuristic approaches are so-called hybrid procedures of self-calibrating and self-adapting meta- and hyperheu- combining several ‘classical’ ones. ristic methods (see also Burke et al. [13]). 123 Logist. Res. (2012) 5:47–63 55 3.4.3 Matheuristics for details). Lastly, a statistics module serves to compute key performance indicators and to create reports. As mentioned, the most successful heuristic algorithms are Usually, but not exclusively, CVRS is used for planning nearly always hybrid methods combining different construc- routes of motor vehicles on public roads. In this case, a tive and improvement procedures, and sometimes also dif- CVRS is often embedded into a transport management ferent meta-heuristics. A new class of hybrid meta-heuristics system (TMS). A TMS contains components for data entry, has emerged only recently: it was stated in the preceding planning, administration, execution, control, and billing of section that there are two fundamental types of solution transport services. TMSs and other software systems for method, mathematical programming and (meta-)heuristics. logistics and transport are thoroughly discussed in Crainic Due to advances in mathematical programming theory as well et al. [24]. as computer hardware, these two worlds have begun to merge. Hybrid methods that use mathematical programming models 4.2 Reasons for using CVRS and algorithms as subroutines are now commonly subsumed under the term matheuristics. A pertinent monograph is VRPs are, in essence, highly complex mathematical opti- Maniezzo et al. [64]. The paper by Doerner and Schmid [30] mization problems. Because of this complexity, software gives a survey on matheuristics for rich VRPs. for supporting human planners and decision-makers has been widely used for years. There is a considerable number 3.4.4 Parallel algorithms of manufacturers of CVRS, and many of these manufac- turers have been in the business for decades. Reasons for On the technical side, one of the more recent advances in the use of CVRS, as speciﬁed by users, follow. The most computer hardware is multi-core processors, allowing real important reason is surely that CVRS helps to reduce the multi-threaded processing on single, standard personal costs of executing and planning routes, and to increase computers. This has lead to a renewed and increased efﬁciency. Moreover, through automatic planning, dis- interest in parallel algorithms. Both exact and heuristic patchers are relieved from routine jobs. Telematics and methods can beneﬁt from multi-threaded implementations. statistics functionality of CVRS improve the possibilities Crainic [23] presents a survey of parallel solution methods for transport monitoring and surveillance as well as for for VRPs. statistics and controlling, so that the quality and the transparency of the overall planning process is improved. An important point often raised by senior executives is that 4 CVRS: a comprehensive study of the German market the dependency of the company on single persons (expe- rienced dispatchers) and their knowledge is reduced. Also, A CVRS is a computer program that allows to (1) read in the work of the sales department (of freight forwarders) is and display data on vehicle depots, customers, distances, simpliﬁed, because faster and more precise pricing of ad and travel times between locations, on requests, vehicles, hoc customer requests becomes possible. Finally, work and drivers, (2) construct, save, and display vehicle routes, processes in general are uniﬁed and streamlined. The and (3) determine a complete route plan for a given data set beneﬁt of the practical use of CVRS is substantiated by (a problem instance) by executing construction and several scientiﬁc studies. See, for example, the literature improvement algorithms, possibly after entering a set of survey in Eibl [34], p. 45 ff. parameters, without further user interaction. 4.3 Structure of the study 4.1 Components of CVRS For the compilation of the study, a thorough search for A CVRS typically consists of the following ﬁve compo- CVRS manufacturers active on the German market was nents: an interface to a database or enterprise resource performed, and no less than 50 ﬁrms could be found. All of planning (ERP) system allows reading in the relevant data them were asked to ﬁll in a detailed questionnaire con- and writing back the solution. A geographical information taining more than 500 pieces of information on relevant system (GIS) is necessary for geocoding address data, aspects of the company and the CVRS in nine categories: computing distance and travel time matrices, and visual- izing data and solutions in digital maps. A planning mod- 1. Company 2. Product ule, the heart of the system, supports automatic, manual, and interactive planning. A telematics module allows a data 3. Information technology and software engineering 4. User interface exchange between vehicles and the dispatching ofﬁce as well as the tracking and tracing of vehicles (see Goel [42] 5. Geographical information system 123 56 Logist. Res. (2012) 5:47–63 6. Telematics one university. This is mostly done in the form of master’s 7. Models and algorithms for automatic vehicle routing and Ph.D. theses. Twelve ﬁrms offer a free test or demo 8. Reporting, key performance indicators, and statistics version, and several ﬁrms provide low-priced or free li- 9. Prices cences for use in teaching. A basic, single-user licence for commercial use costs The questions posed were, to a large extent, either of the 15,000 Euros on average. This does not include custom- multiple-choice or the yes–no type. The obtained infor- izing and preparatory training of users. mation was evaluated on an aggregate level, by summing or averaging over all questionnaires. No information on 4.4.2 Industry sectors using CVRS single manufacturers or systems is given. The answers in the returned questionnaires were checked for plausibility; As for the industry sectors using CVRS and the respective nevertheless, correctness could of course not be veriﬁed. number of sold licences, no reliable data could be gathered. However, the published results being aggregated, no ven- Some ﬁrms provided detailed data, others did not or gave dor had anything to gain from exaggerating the capabilities only aggregated information. However, a quintessential of his product. With respect to content, aspects that were ﬁnding is that numerous CVRS ﬁrms have customers in the considered relevant by the study author, based on his own following sectors: Industrial ﬁrms producing physical professional experience, were queried. goods use CVRS to plan the sourcing of raw materials and OR/MS Today, the journal edited by the Institute for the transport of (semi-)ﬁnished goods between plants and Operations Research and the Management Sciences warehouses or to wholesalers. Concrete applications are as (INFORMS), features, in a 2-year cycle, a survey on CVRS diverse as milk collection at farmyards or wood transport for the North American market. The latest one, from from forests to mills on the supply side, and delivery of February 2012, comprises 12 vendors (see http://www. ﬁnished cars or ready-made concrete garages on the dis- orms-today.org/surveys/Vehicle_Routing/vrss.html). The tribution side. In the wholesale and retail trade, the dis- results of both studies are hard to compare, because dif- tribution of consumer goods such as drinks, frozen food, ferent information was gathered and organized differently. furniture, or heating oil, to name but a few, is planned with Only one company appeared in both surveys. CVRS. Freight forwarders in the less-than-truckload as well as the full-truckload business determine routes for 4.4 General results local feeding and distribution as well as linehaul and long- distance tramp transports with CVRS. The same holds for Twenty-eight companies sent back a ﬁlled-in question- parcel delivery companies and letter mail services. Also naire. This is a return rate of 56 %, which is acceptable. many reverse logistics and waste collection ﬁrms plan routes with the help of CVRS. An important ﬁeld across 4.4.1 Company structure and size sector boundaries is the solution of service technician, salesman, and other staff dispatching problems. Finally, in Most CVRS companies have their headquarters in Ger- intra-plant logistics of industrial ﬁrms, routes of automated many. The number of employees is 36 on average. The ﬁrst guided vehicles used to fulﬁl transport orders between manufacturer of ‘software for logistics’ was founded in warehouses and production sites are planned with CVRS. 1961. The ﬁrst CVRS, that is, vehicle routing software featuring an automatic, algorithm-based planning compo- 4.5 Models and algorithms for automatic vehicle nent, was offered in 1979. routing in CVRS All ﬁrms offer launching and roll-out support as well as user-speciﬁc adaptation and customization of their soft- Obviously, a central part of a CVRS and the most inter- ware. (This shows that CVRS is (still) not a standard, off- esting one from an OR perspective is the automatic plan- the-shelf product.) In addition, most companies use their ning component. Therefore, detailed information on this own software for project work and consulting services. aspect was queried. This part of the questionnaire was Most ﬁrms, but not all, consider the algorithms used in answered by 27 participants. their systems a core competence. Only four companies do not possess the source code of the algorithms and do not 4.5.1 General features of the automatic planning hold exclusive rights on the code. These companies spe- component cialize in transport management systems (TMS) and use third-party components for automatic planning. Most systems possess the following functionality: they are Cooperation with academia is common. More than three capable of solving general PDP as described above with up to quarters of all ﬁrms stated that they cooperate with at least 10,000 requests, including problems with a heterogeneous 123 Logist. Res. (2012) 5:47–63 57 ﬂeet, a multi-period planning horizon, and multiple use of consider heterogeneous vehicles, at least with respect to vehicles, providing a feasible solution within ﬁve min- costs and capacity, is an absolute must for a CVRS. The utes when distance and travel time matrices are given. surveyed systems generally support different ﬁxed, dis- Most systems can be used to determine only one part of tance- and time-dependent costs. More than half of all the solution of a VRP, namely the clustering of the systems also offer different stop-dependent costs and the requests into groups to be performed by one vehicle use of tariffs; penalty costs for considering soft constraints (whereas the other part, the sequencing of requests, is left are prevalent, too. The simultaneous consideration of dif- to the dispatcher, or, even more often, to the drivers), and ferent capacity constraints is standard as well. Addition- allow the automatic assignment of vehicles and drivers to ally, almost two-thirds of the systems offer support for such groups. Moreover, a re-optimization component multi-compartment vehicles. capable of computing a new feasible solution from an An issue pointed out by several participants of the study existing one after small changes to the instance, such as is that the available geographical data for lorry routing still the arrival of some new requests, is a standard feature. do not cover all relevant attributes (passage heights, barred Another is the possibility to limit the duration of an roads, etc.) in a truly reliable fashion even in Western optimization run by specifying a maximal number of Europe. This is because the commercial providers of geo- iterations and a maximal running time. Lastly, a batch graphical data have concentrated on the much larger car mode for automatic execution of optimization runs with navigation system market. This situation is about to speciﬁc parameter settings is also usual. Interestingly, change, though, and many systems already consider these only eight ﬁrms claim to be able to solve arc routing and data in the preprocessing phase, when distance and travel postman problems. Moreover, although there is a con- time matrices are computed. siderable amount of scientiﬁc literature on the topic (see Different driving speeds per vehicle and time- or load- the survey by Wa ¨ scher et al. [83]), only six systems dependent travel times are rarely supported. This is contain a module for optimizing storage space utilization probably because the necessary data are still difﬁcult to (using 3D packing algorithms). obtain in a sufﬁciently high quality. Tactical ﬂeet plan- ning by directly specifying an unlimited number of 4.5.2 Modelling features vehicles of each type is offered byhalfofall systems; in the other systems, a sufﬁciently large number of vehicles 4.5.2.1 Request-related features Most systems can han- must be speciﬁed by hand if the ﬂeet size and mix is to be dle single as well as multiple time windows for requests determined. and locations, whereas vehicle-dependent time windows Although the concrete type and class of a vehicle are irrelevant for a solution algorithm, the surrounding soft- are only seldom considered. The obvious ‘pickup-before-delivery’ precedence con- ware must be able to manage these attributes in order to straint can be handled by all systems. The requirement of provide sensible feedback to the user. In this respect, most request precedence within a route is also commonly cov- systems are capable of considering lorries/tractors, trailers/ ered. A nested execution is supported by half of all sys- semi-trailers, and cars, but only few systems can also tems, but only few systems can deal with precedence manage pedestrians, bicycles (both of which are relevant, constraints of requests on different routes. for example, in mail delivery), trains, ships, or aircraft. The large majority of the systems allows the consid- Similarly, the technical equipment of a vehicle is only eration of vehicle-driver-request compatibilities, and relevant to determine vehicle-request compatibility, and more than two-thirds of all systems areabletohandle the dimensions and weight of a vehicle are only relevant to optional requests and parallel incompatibility between determine vehicle-location compatibility. The consider- requests. About half of all systems can deal with periodic ation of vehicle class, type, and technical equipment requests and with complex requests consisting of more attributes for specifying vehicle-request-location compati- than one pickup and one delivery location. This indicates bilities is possible with more than four-ﬁfths of the that these request types are commonly encountered in systems. practice. On the other hand, only few systems can con- All CVRS manufacturers claim that their systems pos- sider expected or indirect requests, sequential request sess the feature of considering the European Union social incompatibility, and different ways of performing a legislation on driving, break, and rest times for drivers. request. No system supports request-dependent vehicle During the study, however, the author has gained the itineraries. impression that many systems consider these regulations only incompletely. In particular, only few systems, 4.5.2.2 Fleet-related features A homogeneous ﬂeet is according to the study, contain the rules for double-manned rarely found in the real world. Therefore, the ability to vehicles. 123 58 Logist. Res. (2012) 5:47–63 4.5.2.3 Route-related features Most systems allow the existing plan proposed by the algorithm, such as ﬁx planning of closed and open routes as well as multiple assignments of requests to routes, ﬁx the sequence of routes for one vehicle. Further types of route supported by partial routes, manually assign a certain vehicle to a route, most systems are routes for lorries and trailers, where the etc. Both features, dynamic and interactive planning, trailer can be uncoupled and left behind at parking places, require the capability to re-optimize an existing plan after but with a ﬁxed assignment of lorry and trailer (so that only small changes, without changing the ﬁxed parts. a route for the lorry is computed, and the route of the trailer Only three companies state that their algorithms are is a part of its lorry’s route), the consideration of ﬁxed capable of handling stochastic customers (where the route zones from tactical planning, and routes with a necessity to visit a location is stochastic) or stochastic maximum waiting time. Almost half of all systems support demand/supply (where the amount of demand/supply is the computation of balanced routes with similar capacity stochastic). utilization, number of stops, duration, and cost. The necessity to compute interdependent routes with 4.6 Algorithmic features synchronization requirements between autonomous and non-autonomous vehicles such as lorries and trailers, Whereas participating in the study was seen as a marketing between elementary and composite ‘vehicles’ such as lor- measure by most CVRS manufacturers, several ﬁrms were ries and drivers, or between arbitrary vehicles to perform reluctant to specify details about the algorithms used in transshipments is frequent in practice. However, these their software. However, the questions in this part of the features are supported by few systems only. Mostly, these questionnaire were still answered by 21 ﬁrms, so that also requirements are left to manual planning. The same holds these results may be considered representative. for inter-route resource constraints. Of course, any algorithm used for vehicle routing in practice is necessarily a heuristic. Therefore, the questions 4.5.3 Objective functions on algorithms asked which construction and improvement procedures and which metaheuristics were used. With respect to objective functions, almost everything that is reported in the literature is also available in all or most 4.6.1 Constructive procedures systems: it is possible to minimize the number of vehicles used, the overall distance covered by all vehicles, and the The ranking of constructive procedures is as follows total cost of all vehicles. In addition, about half of all (number of mentions in parentheses): systems support a weighted sum of one-dimensional objective functions, hierarchical, or multi-criteria objective 1. Parallel savings (16) functions. 2. Insertion (11) 3. Cluster ﬁrst, route second (10) 4.5.4 Planning modes Nearest Neighbour (10) 5. Proprietary (7) The classical CVRP corresponds to an operational, single- Sequential savings (7) period, static, deterministic planning situation. Given that 7. Dynamic programming (6) reality is neither static nor deterministic, it is not surprising 8. GENI intra-route (5) that CVRS supports different planning modes. 9. Regret (4) With respect to the frequency of planning, tactical Route ﬁrst, cluster second (4) planning of standard or base routes using aggregate, aver- age data as well as operational, day-by-day planning is 4.6.2 Improvement procedures supported by all systems. Some systems use different algorithms for these two modes, taking into account that The ranking of improvement procedures is as follows: for tactical planning, running time is not critical. Multi- period or rolling horizon planning is also supported by 1. Relocate (move a request to another route) (16) most systems. 2. k-opt (15) In addition, dynamic or real-time planning is possible 3. Swap/exchange (exchange two requests between two with most systems (changing assignments of requests to different routes) (13) vehicles while the latter are already en route, caused by 4. String-relocate (move a route segment to another events such as new requests or breakdowns of vehicles). route) (12) 5. Cross/string-exchange (exchange two route segments Interactive planning is also supported in most cases. This means that the user can make small changes to an between two different routes) (11) 123 Logist. Res. (2012) 5:47–63 59 3. Column generation (4) 6. Or-opt (8) * 4. Branch-and-price (3) 7. k-opt (generalization of k-opt to capacitated prob- 5. Benders decomposition (1) lems) (7) Lagrangian relaxation (1) 8. Lin-Kernighan (6) 9. GENI inter-route (5) k-interchange (exchange at most k requests between 4.6.5 Components, libraries, and benchmarks used two routes) (5) (Very) Large-scale neighbourhood search ((V)LSNS, The ranking of solvers, programming frameworks, and exponential-size neighbourhoods) (5) algorithm libraries is as follows: 12. Double-bridge move (3) 1. CPLEX (3) Ejection chains/cyclic transfers (move a ﬁxed number 2. Boost (2) of requests from route 1 to route 2, then the same COIN (2) number from route 2 to route 3 etc.) (3) LEDA (2) 14. Proprietary (2) XPRESS (2) 6. BCP (1) 4.6.3 Metaheuristics CBC (1) Gurobi (1) The ranking of metaheuristics is as follows: lp_solve (1) SCIP (1) 1. Tabu search (10) SoPlex (1) 2. Genetic algorithms (8) 3. Threshold accepting (7) Eight ﬁrms stated that they tested their algorithms with 4. Proprietary (6) the Solomon VRPTW benchmarks: six have used the Ruin-and-recreate/ﬁx-and-optimize/ripup-and-reroute (6) Gehring/Homberger VRPTW problems and two the Li/Lim Simulated annealing (6) PDPTW instances. No ﬁrm was willing to tell anything 7. Adaptive large neighbourhood search (5) about the results. 8. Ant colony systems (4) Guided local search (4) Variable neighbourhood descent (4) 5 The gaps between theory and practice Variable neighbourhood search (4) 12. Greedy randomized adaptive search procedure (3) It goes without saying that any CVRS is a commercial Memetic algorithms (3) product for end-users without programming and OR 14. Adaptive guided evolution strategies (2) skills. Therefore, it must offer an up-to-date graphical Attribute-based hill climber (2) user interface with adaptable look-and-feel as well as a Backbone search (2) help system. Moreover, a comfortable interface to com- Great deluge (2) mon TMS or ERP systems is also an essential feature. Indirect search (decoder) (2) (Together with the usual GIS, telematics, and statistics Neural networks (2) modules, this means that, typically, less than 10 % of the Scatter search (2) code of a CVRS is for the VRP solution algorithms.) 21. Adaptive/approximate dynamic programming (1) Moreover, aspects such as versatility, genericity, and Adaptive memory programming (1) maintainability are fundamental for any commercial Artiﬁcial immune systems (1) software and must be ensured through adequate design, Particle swarm optimization (1) thorough testing, and extensive documentation. This Record-to-record travel (1) slows down the development process considerably. A scientiﬁc code, on the other hand, is often written as a prototype, a proof-of-concept, to be used only by the 4.6.4 Mathematical-programming-based approaches developers themselves. CVRS must be able to handle many different problems. The ranking of mathematical-programming-based approa- It is not an option to develop and implement a specialized ches is as follows: algorithm for each new customer. Therefore, algorithms 1. Branch-and-cut (5) used in CVRS must necessarily be generic and easily Constraint programming (5) extendable to new problem features. Summing up the 123 60 Logist. Res. (2012) 5:47–63 previous sections, an algorithm for use in a state-of-the-art improvement procedures, which corresponds to an average CVRS supports the following features: of more than ﬁve. • Pickup-and-delivery requests 5.1 Application gaps • Compatibility between locations, requests, vehicles and drivers Aspects that are rather well studied in theory but have not • Multiple time windows for locations and requests yet found widespread use in CVRS and where, conse- • Consideration of service times quently, there is an application gap are stochastic vehicle • Heterogeneous ﬂeet with respect to cost, capacity, start routing (Flatberg et al. [35], Cordeau et al. [20]), time- and end depots dependent travel times (Fleischmann et al. [36], Taniguchi • Fixed, distance-, time-, stop-dependent, penalty costs, and Shimamoto [81], Haghani and Jung [49]), and math- tariffs ematical-programming-based approaches (Maniezzo et al. • Multiple capacity constraints [64]). As far as stochastic VRPs are concerned, this is • Multiple use of vehicles probably because in most cases, it is extremely difﬁcult, if • Driver rules not impossible, to provide sufﬁcient data in sufﬁcient • Weighted and hierarchical cost functions quality to derive useful probability distributions for cus- • Dynamic planning over a one-week planning horizon tomer demands/supplies, and it is doubtful whether this is with event- or time-based rolling horizon planning going to change soon. The consideration of variable travel • Re-optimization options times (driving speeds) depending on the time of day also • Interactive planning requires reliable data. Here, the outlook is more optimistic. Few algorithms described in the literature, if any, are Detailed information will be available in the foreseeable able to deal with all these features. Research algorithms future, at least for large urban regions, which is where peak usually work for special, mostly idealized, types of VRP and off-peak times are most pronounced anyway. Mathe- only. matical-programming-based approaches (matheuristics) are Practitioners need robust, fast, extensible, and simple still a rather new ﬁeld of research, but more and more (parameter-free) algorithms capable of solving instances pertinent publications appear. CVRS manufacturers will with thousands of requests. The last 0.1 % in solution not ignore this trend. quality to be gained from an additional complex algorith- mic device are insigniﬁcant, since the data available in 5.2 Research gaps practice are never 100 % accurate. On the other hand, when using exact methods, the scientiﬁc world strives to ﬁnd On the other hand, a research gap is apparent with respect ‘provably optimal’ solutions for small, idealized problems; to the following aspects: there are only very few papers on using heuristics, researchers are, to a large extent, focussed problems with optional, expected, indirect, or complex on improving best-known results for benchmark instances requests. The existing literature is mostly concerned with or solving concrete real-world problems with prototypical deterministic problems where all requests must be fulﬁlled implementations of specialized algorithms. This point is and consist of single visits or one pickup and one delivery. further elaborated in Cordeau et al. [19] and Pisinger and Exceptions to this rule are Savelsbergh and Sol [76], Røpke Røpke [70]. [75] and Goel and Gruhn [45]. What is more, practitioners A central argument in the above-mentioned paper by often ﬁnd it difﬁcult to give a clear-cut deﬁnition of their So ¨ rensen et al. [79] is that, according to these authors’ problem’s objectives and constraints. Therefore, soft con- experience, commercial CVRS uses quite a large number straints such as visual attractiveness of routes (Lu and of improvement heuristics to improve initial solutions Dessouky [62]) or a preferred assignment of certain drivers determined by constructive procedures. This is in contrast or vehicles to customers (Groe ¨r et al. [47]) are quite to scientiﬁc codes, which tend to use few, but rather important in real-world applications. For the same reason, a complex and sophisticated techniques. The reasons for this ‘fair’ and balanced sharing of the workload between dif- are, according to So ¨ rensen et al., that (1) an approach using ferent routes is important in practice, but seldom consid- many improvement procedures can overcome the greedy ered in the literature (an exception is the paper by behaviour of an approach that uses only a single one and Bredstro ¨ m and Ro ¨ nnqvist [12]). Tariffs and complex cost (2) supplying a large arsenal of diverse search strategies functions are often used in practice, but rarely considered allows a ﬂexible adaptation of the software to the speciﬁc in the literature. See Ceselli et al. [15] for a striking requirements of each customer. The results of the study exception. described in the present paper support this observation: as Most importantly, models and algorithms for integrated can be seen in Sect. 4.6.2, 21 routing tools use 111 different and synchronized vehicle routing are still scarce: in almost 123 Logist. Res. (2012) 5:47–63 61 all vehicle routing models and algorithms, the routes of the powerful decision support tools for integrated and syn- different vehicles are assumed to be independent of one chronized vehicle routing in practice. another, so that modifying one route does not have any effects on other routes. However, in a surprisingly high number of cases, this assumption does not hold. Examples 6 Conclusions and outlook for practical applications requiring a spacial, temporal, and in some cases also load-related synchronization or The exact solution of even the basic variants of VRPs is coordination of routes are the planning of inter- and multi- still impossible for instances of realistic size. An exact modal transports, the planning of meet-and-turn routes, solution of real-world problems with many additional side transports over hubs or cross-docking locations, simulta- constraints will remain impossible in the short and medium neous planning of routes for lorries and trailers, if trailers term. However, close-to-optimal solutions of more and may be pulled by different lorries, simultaneous vehicle more complex and integrated problems, increasingly based and driver routing, if drivers may change vehicles, and on incomplete optimization approaches and mathematical- automatic planning of multiple types of resources (driv- programming-based heuristics, are possible, and this is ers, lorries/tractors, trailers/semi-trailers, swap-bodies/ sufﬁcient to provide useful decision support in practice. containers). Nevertheless, as has already been alluded, in some areas The following quote taken from Irnich [57], p. 9, still there are gaps between industrial needs and the state-of- holds: ‘While research on integrated models and solution the-art CVRS of today. A detailed discussion of these methods for combined vehicle and crew scheduling has issues is, however, beyond the scope of the present paper, made some remarkable advances ..., the literature on but constitutes an interesting topic for further research. integrated vehicle routing still mainly focuses on location For the foreseeable future, CVRS will remain a decision routing problems and inventory routing problems. Litera- support system in almost all application areas. Essentially, ture on other forms of integration is scarce. There is a need fully automatic planning is possible only in some special for new and improved techniques to attack integrated cases, most notably in intra-plant logistics. In road and planning problems. As far as we can see, there is no con- inter-modal transport, interactive planning with a human vincing concept for dealing with VRPs with load transfer at dispatcher having the ﬁnal say is and will remain the rule. hubs or consolidation points, especially in the context of A modern CVRS, however, can considerably facilitate the bimodal or multimodal trafﬁc. The same is true for long- daily routine work for human decision-makers. The sys- haul goods trafﬁc, which requires the coordination between tems have become so mature and user-friendly that, now- feeder processes, linehaul, and distribution’. adays, after introducing a CVRS, nobody wants to return to purely manual planning any more. The concerns often More speciﬁcally, Macharis and Bontekoning [63], p. 400, state in their survey of inter-modal freight transport voiced by many dispatchers, CVRS would invalidate that ‘intermodal freight transportation research is emerging their knowledge and experience or even make them as a new transportation research application ﬁeld, that it lose their jobs, are unfounded. This has become evident in still is in a preparadigmatic phase, and that it needs a dif- many discussions with manufacturers as well as users of ferent type of models than those applicated to uni-modal CVRS. transport’. Summing up, CVRS constitutes a ﬁxed and indispens- In short, a general, unifying modelling and solution able component of logistics planning in practice. Just like concept for integrated and synchronized vehicle routing is any other product, CVRS has to adapt to ever-changing still missing; science has to catch up in this respect. This customer needs and expectations. This requires constant statement is further supported by the fact that the survey further development, both with respect to information article by Gendreau et al. [40], which presents an extensive technology and to OR models and algorithms. Conse- literature list on VRPs, does not mention a reference on quently, even more than half a century after the ﬁrst OR VRPs with multiple synchronization constraints. Moreover, paper in this ﬁeld, the practice of vehicle routing will the recent monograph by Golden et al. [46] does not con- continue to provide interesting and challenging problems tain a paper on this topic. This does not mean that there are for OR researchers. no such papers at all. Rather, it shows that no systematic study of this problem class has yet been performed. Such research is now beginning to emerge (see the survey by References Drexl [32]). On the other hand, the results of the study show that some CVRS systems contain solutions to con- 1. 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Rich vehicle routing in theory and practice

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