Vehicle Routing Problem

TheVRPrepresents a classical optimisation problem typically dealt with in the area of opera-tions research: a number of customers have a demand with respect to goods or services and a

set of vehicles exists to perform one or multiple tours to satisfy these requests. In its simplest form solving a VRPintends not only to identify a feasible solution, but finding the solution with lowest cost - often defined by the length of the overall route. A typical restriction is a limitation of visits, so that a customer can only be served once. VRPsare a generalisation to theTravelling Salesman Problem (TSP), a classical combinatorial optimisation problem (Ahuja, Magnanti, and Orlin1993), i.e., theTSPcan be found as a subproblem in aVRP.

In theTSPa salesman has to visit a number of customers, each of which resides in a different city. The optimisation problem consists of finding the optimal route, i.e., lowest cost tour, to serve all customers under the constraint that the salesman visits each city (customer) only once. TheTSP deals with finding one route only, but in aVRPmultiple vehicles have to be considered which can deliver their goods to customers. Therefore, an initial assignment of vehicles to customers is required to subsequently solve a number ofTSPs. A graphG= (V , E) can serve as representation for aVRP, i.e., by defining a network flow (see (Ahuja, Magnanti, and Orlin1993)) where a vertexv∈V represents a customer with a particular demandq_v, and an edgee∈E represents the transition of a vehicle from one customer to another. The graph can be directed when transport cost between two vertices are asymmetric, or undirected if they are symmetric. Additional constraints can be applied to either vertices or edges, e.g., limiting or requiring an in and out-flow of a vertex, or limiting the flow capacity for an edge to encode the maximum transport capacity of a vehicle.

Variants of theVRPallow for a detailed, often application specific formalisation and descrip-tion of the problem, e.g., in order to cover practical issues such as time-based delivery or the loading constraints of the vehicles. While some of these variants support a systematic analysis of solution approaches, they come with a reduced applicability, e.g., when considering a fully homogeneous vehicle fleet. Toth and Vigo (2014) provide a broad overview. Meanwhile, the following section details the relevant variants for this thesis.

Problem variants and models

TheCapacitated VRP (CVRP)is one of the most popularVRPvariants. It accounts for a limited transport capacity of individual vehicles. A fleet of homogeneous vehicles is assumed, i.e., all vehicles have the same capacity and operating cost. All vehicles start and end at a single depot.

To solve theCVRPa variety of representations has been developed, where each representation comes with limitations for its applicability: directed and undirected two-index vehicle-flow formulations (referred to as VRP1 and VRP2 by Toth and Vigo (2014)) use only single edges between any two vertices - the directed formulation use one edge in either direction. As a major disadvantage, these formulations cannot embed vehicle specific solution information and thus cannot explicitly take vehicle characteristics into account. However, these two-index vehicle flow formulations avoid redundant solutions which result from vehicle permutation.

The alternative is a three-index vehicle-flow formulation (referred to as VRP3 by Toth and Vigo (2014)) . The third, vehicle-specific index permits the direct identification of a vehicle which is routed along a particular edge. As a drawback VRP3 leads to the computation of highly redundant solutions.

The problem of vehicle identification is similarly encountered inTemPl. Section 4.3.3refers to the problem in the context of the identification of atomic agent instances. The problem is tackled with a complementary use of symmetry breaking and an explicit multi-commodity min-cost flow formulation (see Section4.3.5).

Another so-called extensive formulation focuses on identification of the best combination of actually feasible routes. Instead of defining the vehicle-flow and edge variables explicitly, the extensive formulation uses variables to represent complete routes. Additionally, it associates each route variable with a binary variable. This binary variable defines whether a route is part of the solution or not. The main benefit of this representation lies in lower computational bounds and the possibility to embed complex feasibility constraints for each route. The ex-tensive formulation increases, however, the combinatorial challenge. A full enumeration of all feasible routes and combinations thereof is only feasible for very small problem sizes. Hence, an implicit handling of routes is required and routes for candidate solutions have to be dynam-ically generated as part of the optimisation process, e.g., using column generation (Desrosiers and Lübbecke2005). TemPl uses a similar dynamic route generation approach. It generates the (full) routes for all mobile agents and partial routes for immobile agents using a constraint-based solution approach. The local search then tries to identify a valid solution including the full routes for immobile agents.

TheVRP with Time Windows (VRPTW)is an extension of theCVRPwhich adds time windows to define when a customer requires its demand of goods to be fulfilled. Time windows can be defined as either hard or soft constraints. The violation of a soft constraint does not invalidate a solution, but involves a penality and increases the solution cost, e.g., when a good is delivered early or late. In effect, any hard constraint is a soft constraint with infinite cost penalty.

TheVRP with Pick-up and Delivery (VRPPD)is also known as Dial-a-Ride Problem (DARP) and represents a public transportation problem: passengers request a ride from one location to another while the pickup and delivery interval can also be constrained. Most of the clas-sical problem instances come with significant limitations for their practical use, e.g., due to assuming a single depot or homogeneous vehicles. Problem-richer variants of higher practical relevance such asMulti-depot heterogeneous fleet VRP with time windows,VRPTTand VRP with multiple synchronization constraints (VRPMSs)exists, but are rarely handled in research.

However, these problem rich variants are of particular interest for this thesis. They comprise main characteristics encountered in an application of reconfigurable multi-robot systems in-cluding, although not limited to the following constraints: (a) multiple depots, (b) heteroge-neous vehicles, (c) time windows, (d) vehicle capacities, (e) vehicle synchronisation, (f) pre-emptive drop-off, (g) and pick-up. The fleet of vehicles and its partitioning into same type vehicles directly corresponds to agent types in the organisation model (see Chapter3). The organisation model focuses on agent properties such as nominal speed, power consumption, and capacity for transporting other agents. Additional fleet characteristics can involve the locomotion capabilities of agents which can be combined with route characteristics such as traversability.

TheVRPTTdeals with trucks and trailers, which can be linked to each other in order to form a transport unit. In contrast to classicalVRPsdefinitions, the problem embeds location acces-sibility constraints, e.g., by accounting for limited manoeuvring space for vehicles or similar.

Though this type of constraint is not explicitly handled inTemPl, Section4.2shows how these constraints can be introduced or accounted for; showing the increased generality of the prob-lem formalisation. The specialisedVRPTTand the more generalisedVRPMSs(Drexl2013) de-scribe problems that deal with vehicles’ interdependence. This interdependency arises through the need for synchronisation of tasks or operations which involve more than one vehicle. Since this is a typical characteristic of any coordinated multi-agent operation,VRPMSsare strongly relevant for reconfigurable multi-robot system operation. Due to the interdependence of

vehi-cle operations, individual routes cannot be optimised in an isolated way, but require so-called inter-route constraints. To tackle the problem, Drexl (2013) focuses on the modelling of the problem using a graph-based approach, while initially leaving the development of appropriate solution approaches open. Parragh and Cordeau (2017), Tilk et al. (2018) and Rothenbächer, Drexl, and Irnich (2018) solve the problem using a combination out of branch, price and cut algorithms.

Drexl identifies orthogonal categories for vehicles: the first category comprises autonomous and non-autonomous, and the second category task and support vehicles. The latter are due to the particular application scenario of milk collection, where customer visits are permitted only for a limited set of vehicles. The remaining vehicles can still be used to support the transport, e.g., by creating intermediate transshipment hubs. Drexl further differentiates between active and passive vehicles. Passive vehicles such as trailers provide additional load capacities. For a location change they depend, however, on an active vehicle such as a lorry which can pull trailers. The general modelling approach remains domain specific in parts due to this categori-sation which is narrowed to trucks (here lorries) and trailers. Thus, it leaves room for further generalisation particularly in the context of reconfigurable multi-robot systems.

Drexl collects constraints forVRPTTsandVRPMSswhich can broadly categorised into: (1) lo-cation access constraints: constrain the parallel or overall access to a lolo-cation by vehicle num-ber and/or by vehicle type, (2) location transfer constraints: constrain the transshipment of commodities by participating vehicles, by the current and overall amount per location, and (3) time constraints: intervals within which the transshipment of goods must be completed.

Drexl splits customers into lorry customers and trailer customers. These subcategories encode a permission of a lorry to perform a visit with or without a trailer - this could also be modelled using location access constraints. TheVRPMSshas no limitation on the starting depots for the vehicles, and no constraint exists on the lorry trailer combination which returns to a depot. Au-tonomous and non-auAu-tonomous vehicles are considered in this problem formulation. A final solution has to synchronise the operation of these vehicles.

Inter- and Intra-Route Constraints In basic VRP variants vehicles can operate indepen-dently from each other. Each vehicle’s route has only to be consistent with existing so-called intra-route constraints. Intra-route constraints can be seen as local constraints. Hence, a solu-tion can be quickly invalidated when a local, i.e. route-based, consistency check fails. Typical intra-route constraints as mentioned by Toth and Vigo (2014) are collected in Table 4.1. In contrast to intra-route constraints, inter-route constraints are global, and can therefore only be verified when all routes for a potential solution have been constructed. Typical inter-route constraints as mentioned by Toth and Vigo (2014) and Drexl (2013) are illustrated in Table4.2.