When products get ready - e.g., in a production process scheduled to be ecient -, they have to be distributed. The cost of product delivery is decreased as the total length of the routes of delivering vehicles becomes shorter. Motivated by this fact, a signicant amount of research in applied optimization is devoted to design strategies of optimal vehicle or vehicle eet operation, where the problem of vehicle routing is of primary importance. Therefore, a large number of papers and a lot of research aim to optimally determine the routes of a vehicle eet serving consumers under dierent conditions. Problems of routing vehicles are classied based on these conditions. Following the taxonomy presented in [91], [92], [93] and [94], Figure 1.6 shows the most signicant and most relevant variants of the problem. It is also worth mentioning the paper of Wang and Wasil [95] that reviews all the signicant VRP-related papers published by the journal Networks in the last50 years.
Figure 1.6. Signicant variants of Vehicle Routing Problem.
Denition 41. Travelling Salesman Problem (TSP): An agent starts from the depot, visits all customers exactly once, and returns to the depot. This route has to be done with minimal cost. The problem is equivalent to nding the minimal Hamiltonian cycle in the graph that describes the road-system. TSP is NP-hard [9698].
Denition 42. Vehicle Routing Problem (VRP): A multi-agent generalization of TSP. Here, the procedure of solving TSP is extended by partitioning the customers into disjoint routes [99].
Denition 43. Capacitated VRP (CVRP): A VRP that contains constraints related to the amount of customer demands and the capacity of the vehicles.
Because of the evident nature of capacity constraints, CVRP is often reected as - simply - a VRP.
Denition 44. VRP with Time Windows (VRP-TW): A VRP, where every cus-tomer has a time window and serving time. If the vehicle arrives at the cuscus-tomer before the lower bound of the customer's time window, it has to wait. When the ar-rival is later than the upper bound of the customer's time window, the vehicle missed the customer. In some cases, the depot has a time window, too [100103].
Denition 45. Multi-Depot VRP (MD-VRP): A VRP with more than one depot.
Each vehicle starts from a depot, and it has to return to the same depot after per-forming the delivery tasks [104].
Denition 46. MD-VRP with Inter-Depot Routes (MD-VRPI): An MD-VRP, where the replenishment of vehicles at intermediate depots is allowed [105].
Denition 47. Periodic VRP (PVRP): A VRP version, where the transfer has to happen at certain times in a well dened time interval (e.g., two times a week) instead of a single delivery [106].
Denition 48. VRP with Pickup and Delivery (VRPPD): In this VRP version, the customers are allowed to send goods by the vehicles besides receiving products [107, 108].
Denition 49. VRP with Multiple Use of Vehicles (VRPM): A VRP, where one vehicle can be used for more than one trip. It often happens when a company has a small cardinality vehicle set, or the average distance between the customers and the depot is small.
Aside from various constraints, dierent objectives of the delivery also generate their VRP variants.
Denition 50. VRP with Route Balancing (VRPRB): A VRP, where the objective is to minimize the dierence between the shortest and the longest route [109].
Extra demands of customers, a technical problem of the vehicle, or a closed road necessitates modifying the predened transports. Recent advances in information technology made it possible to process additional data and do modications while vehicles are on the road.
Denition 51. Dynamic VRP (DVRP): A VRP, where modication of problem-related data during the transport is allowed, and the trips can be adjusted to that [110112].
There are problem variants, in which some data are not determined as static values but are random variables. These data can be customer demand, the travel times, or the service/waiting times, for example.
Denition 52. Stochastic VRP (SVRP): A VRP variant, where some data are stochastic variables [113, 114].
Denition 53. VRP with Stochastic Demands (VRPSD): An SVRP where cus-tomer demands are stochastic [115117].
An important class of time-varying circumstances is when the reason for the change of the problem state is an error during execution, caused by a fault of an element of the system, e.g., the engine of a vehicle or a blocked road. Despite the modied state of the system, the main goal is to avoid service failure, i.e., servicing all the customers. Fault-related topics, like fault detection, diagnosis, or tolerance, are extensively examined, e.g., in the case of control and technical systems of both military [118] and public applications for a long time [119].
In [120], the threat is either an unknown future request or a trac problem. The authors apply a fuzzy fault-tolerant control approach: all customer requests whose future servicing is aected by a problem will be reassigned as new requests. In VRPSD literature, the notion of "route failure" is used when a vehicle does not have enough capacity to service the customers who are assigned to its route. Usual solution methods for this problem have two stages: the rst one species a route for each vehicle, then the second stage deals with the recourse action in case of route failure [111]. It means that an appropriate response may be signicantly delayed.
In [121], the traditional recourse action - when in case of route failure, the vehicle may return to the depot and then perform an extra trip - was completed by a new recourse strategy. In this strategy, a vehicle that has completed its own original route and has remaining capacity can serve customers from the failed route. (However, it needs to be emphasized that contrary to the deterministic demand that is used in our research, Novoa et al. [121] deals with stochastic demands. Moreover, their concept of "route failure" also diers from ours.)
The aid one vehicle oers another is one of the main priorities in my third thesis point - however, the threat to the service considered here is the breakdown of a vehicle before it has served its last customer. Based on the fault classication in [122], this kind of fault belongs to the category of "abrupt faults". Moreover, in this thesis, the routes' execution is already prepared in the route planning phase, which minimizes the fault-related extra costs without changing the set of routes itself.
To achieve fault tolerance, the fault has to be detected. In my thesis, it is assumed that when a fault - e.g., the fault of the engine of a vehicle - causes a vehicle failure - e.g., a vehicle breakdown - and the eet's vehicles communicate with one another, fault detection is trivial. Specic fault detection methods can be found for other VRP-related fault concepts in [120] and [122].
The execution of a VRP program can be made less sensitive to vehicle breakdown by considering the possibility of cooperation among vehicles. Cooperation has a positive eect on transportation costs, even in cases where there are no unforeseen events.
This is demonstrated in [123]. The concept of "cooperation" here means that a vehicle takes over the servicing of a customer or a set of customers originally assigned to another vehicle. In the common case of multiple vehicles servicing customers, two questions need to be considered: selecting which vehicle should take over the task and its costs.