Transport operations can be found at different stages of supply chains, from the collection of raw materials to the distribution of final products. In India, transportation represents 13% to 14% of GDP of the country. Not surprisingly, operations research has been successfully applied to model and optimize different transportation decisions. Among them, node-routing decisions are faced in different settings such as heavy industrial product distribution, soft-drink distribution, sales-force routing, milk collection, dairy distribution, frozen-food distribution, meat distribution, recycling operations, school-bus routing, postal delivery, industrial-waste collection, home healthcare, ready-mixed concrete delivery, maintenance operations in public utilities, raw-materials collection, final-products delivery, and after-sales services, among others. However, they all share the same underlying structure in which a set of mobile servers (usually vehicles) satisfies the demand for a good (or service) of a set of geographically scattered customers. This underlying structure has been captured in the well-known vehicle routing problem (VRP).
Formally, the VRP is defined as an optimization problem in which a set of vehicles of limited capacities based at a depot has to serve the demand of a geographically scattered set of customers. The objective of the VRP is to find a set of routes of minimum total length such that each customer is visited exactly once, all the routes begin and end at the depot and the total demand of the customers visited in each route does not exceed the capacity of the allocated vehicle. Since the seminal paper by Dantzig and Ramser, the VRP has attracted the attention of many researchers in the optimization field. For instance, Eksioglu et al report more than one thousand journal articles addressing VRP-related problems. Moreover, these authors found a significant growth in the number of publications during the last two decades.
As pointed out by Golden et al the VRP is one of the success stories of operational research. A recent survey of vehicle routing software presents 22 products with customers from different industries. Moreover, a study of different real-world applications estimates that the cost reduction achieved with the use of optimization techniques range from 7% to 37%, depending on the characteristics of the routing problem. Currently, the need to reduce green-house emissions gives new reasons to study the VRP and its extensions.
The classical VRP has been extended with side constraints to model several characteristics found in practice. A non-exhaustive list of VRP extensions includes: the distance constrained VRP in which a maximum limit for the length or travel time of any route exists; the VRP with time windows, where each customer must be visited within a pre specified time interval; the VRP with backhauls, where a subset of delivery customers (or line haul customers) must be visited before a subset of collection customers (or backhaul customers), the VRP with pick-ups and deliveries in which a transportation request has a pick-up and a delivery location; heterogeneous fleet VRPs that consider a fleet composed of vehicles with different capacities and routing costs; the site-dependent VRP in which each customer can be served only by a subset of vehicles from a heterogeneous fleet (incompatibility constraints); the open VRP in which the vehicles are not required to return to the depot after completing service; the periodic VRP, where customers must be visited several times over a planning horizon that spans a few days; the split-delivery VRP, where each customer may be visited by several routes; the multi-compartment VRP, where each customer orders several products, the vehicles have several compartments, and each compartment is dedicated to one product; the multi-depot VRP (MDVRP), where vehicles are based at several depots, the truck and trailer routing problem, where a heterogeneous fleet composed of trucks and trailers has to serve a set of customers with incompatibility constraints, among others.
New variants of the VRP also appear when the routing is integrated with other strategic, tactical or operational decisions. For instance, the routing decisions are included when locating depots in the location-routing problem; inventory control and routing decisions have to be made simultaneously in the inventory routing problem; and two and three-dimensional loading vehicle routing problems consider the loading of the vehicles when designing their routes. More recently, the term rich VRP has been coined to denote vehicle routing problems that include several real-world features often ignored in academic research. For other extensions of the VRP arising in practical applications the reader is referred to Hasle and Kloster.
It is clear that a unique vehicle routing problem does not exist, rather there is a wide family of problems with a common structure. Therefore, good vehicle routing methods are those that produce accurate results in short running times, but also they are easy to code and understand, have few parameters, and easily adjust to a wide variety of side constraints found in practice. Traditionally, the solutions methods for vehicle routing problems have been classified into three groups: (i) exact methods, (ii) heuristics, and (iii) meta heuristics.
Exact methods work over different mathematical formulations of the VRP. For instance, two-index vehicle flow formulations use integer variables to indicate the number of times that a given edge is traversed in the solution. Edges between pairs of customers only take binary values, while edges between the depot and the customers may also take the value of 2 representing back-and-forth trips from the depot to the customers. Since the number of capacity constraints in this formulation is exponential, solution methods based on it usually use branch-and-cut approaches. Moreover, several families of valid inequalities have been developed to strengthen its linear relaxation. Solution methods based on two-index vehicle flow formulations include the earlier work by Laporte et al and the more recent branch-and-cut algorithm by Lysgaard et al. Three-index vehicle flow formulations in which the vehicle that traverses the edge is specified have not been as successful as the two-index formulation. Currently, branch-and-cut algorithms based on two-index vehicle flow formulations are the best methods to solve VRPs in which the capacity of the vehicle is large with respect to the demands of the customers. Using this formulation Augerat et al and Lysgaard et al solved a VRP with 135 customers, that is the largest non-trivial instance solved by an exact method to date. Likewise, commodity-flow formulations use binary variables to indicate if an edge is used in the solution and continuous variables to represent the load of the vehicle when traversing each edge. Valid inequalities for the two-index formulations are also valid for commodity-flow formulations, and can be used to strengthen its linear relaxation.
Since the VRP is an NP- Hard problem , exact methods solve consistently instances with up to one hundred customers. Therefore, heuristics and meta heuristics are used in most practical applications where several hundreds of customers are visited daily. An overview of these approximate methods follows.
Most of the early methods for vehicle routing were simple heuristics intended to find good quality solutions in short times. Laporte and Semet classify them into three groups: (i) constructive methods, (ii) two-phase methods and (iii) improvement heuristics.
Constructive methods merge existing routes using a saving criterion as in the popular Clarke and Wright heuristic, or sequentially add customers to routes using an insertion cost as in the Mole and Jameson heuristic.
Two-phase heuristics decompose the VRP in the assignment of customers to routes and the sequencing of customers within routes. Cluster-first, route-second methods first group the customers in routes (clusters) that can be served by a single vehicle and then solve a traveling-salesman problem (TSP) for each route. There are different variants of this approach depending on the method used in the clustering phase. For instance, the sweep heuristic by Gillet and Miller use intuitive geometric procedures to group customers, while the Fisher and Jaikumar heuristic solves a generalized assignment problem in the clustering phase. On the other hand, route-first, cluster-second heuristics, first construct a TSP tour visiting all the customers and then (optimally) break it into VRP feasible routes using a tour splitting procedure.
Improvement heuristics apply simple local search procedures to explore the neighbourhood of a VRP solution. These methods operate over single or multiple routes. In the former case, any TSP improvement heuristic can be used, for example, the classical Or?? opt, 2-opt and 3-opt procedures. The latter case comprises several edge- and node-exchange procedures; Kindervater and Savelsbergh classified them into node relocation, node exchange and edge crossover. More recently, Funke et al have re?? viewed most of the local search operators for vehicle routing problems and have proposed a unified representation that allows the modeling of problems with complex constraints. Local-search heuristics for the VRP evolved in meta heuristics that achieve much better results in somewhat longer, yet reasonable running times.
Vehicle routing problems exhibit an impressive record of successful metaheuristic implementations. We understand by metaheuristic, a high level heuristic procedure de?? signed to guide other methods or processes towards achieving reasonable solutions to difficult mathematical optimization problems. Meta heuristics are particularly concerned with not getting trapped at a local optimum (when multiple local optima exist) and/or judiciously reducing the search space. Meta heuristics include genetic algorithms, simulated annealing, tabu search, variable neighbourhood search (VNS), iterated local search (ILS), evolutionary strategies, greedy randomized adaptive search procedures (GRASP), scatter search, ant colony optimization, among others. Currently, hybrid meta heuristics combining components and principles from different meta heuristics provide very effective solution methods for several combinatorial optimization problems.
There is a huge number of meta heuristics for the solution of the classical VRP. Among them, the most successful ones are based on evolutionary strategies memetic algorithms, adaptive memory programming, ant colony optimization, tabu search and adaptive large neighbourhood search. Gendreau et al provide an excellent categorized survey of meta heuristics for the VRP and several of its extensions. More recently, matheuristic approaches combining meta heuristics and exact methods in a cooperative way have emerged as a promising alternative for the solution of different VRP variants, such as the classical VRP, the split delivery VRP and the location-routing problem, among others. For a recent survey of matheuristics to solve different VRPs the reader is referred to Doerner and Schmid.
Nonetheless, many of the most successful vehicle routing meta heuristics are over fitted to solve efficiently an specific variant, yet they have many components and/or a lot of parameters. This over engineering phenomenon comes at the price of a loss of simplicity and flexibility. Consequently, commercial vehicle routing packages do not include most of the rather standard components of the academic vehicle routing meta heuristics such as memory structures, mutation and crossover operators. Therefore, there is a need for flexible vehicle routing methods capable of solving different VRP variants without many modifications, even if this comes at the price of a reasonable loss in solution quality.
Vehicle routing problems with trailers
In this thesis we studied vehicle routing problems with trailers, that is, a vehicle routing problem in which the capacity of the truck is increased by a trailer. Despite its benefits, trailers cause incompatibility constraints at some customers with limited manoeuvring space or accessible through narrow streets. These customers can be served only by truck (after detaching the trailer).
Real-world applications of this type of problems appear in distribution and collection operations in rural areas and crowded cities. For instance, in several European countries milk collection is performed by a small tanker with a removable tank trailer of larger capacity. Some farms are not reachable by big vehicles, so the tank trailer needs to be detached on main roads before visiting them. Gerdessen reported two applications of vehicle routing problems with trailers in the Netherlands, the first one for the distribution of com pound animal feed in rural regions and the second one for the distribution of dairy products. Semet and Taillard described a vehicle routing problem with trailers, time windows, site dependencies and heterogeneous fleet arising in the distribution operations of a chain of grocery stores in Switzerland.
The arc-routing equivalent of vehicle routing problems with trailers arises in the design of park-and-loop routes for postal delivery, where the postman drives a vehicle from the postal facility to parking locations, loads his sack, and delivers mails by walking the streets; in this case the postman corresponds to the truck and his vehicle to the trailer. Waste collection in small cities and towns also has a similar structure. For instance, in Due Carrare (Italy) small collection vehicles serve narrow streets and big compactors collect waste on streets without accessibility restrictions. Since the disposal facility is far from the town, small collection vehicles meet big compactors in the middle of the routes to dump their contents avoiding long empty trips.
In spite of its wide applicability, vehicle routing problems with trailers are seldomly studied. For instance, the literature review by Gendreau et al only reports three papers with meta heuristics addressing this type of problems. Moreover, apart from the Lagrangian relaxation method by Semet and the branch-and-price by Drexl, we are not aware of other exact algorithms for vehicle routing problems with trailers. Therefore, in this thesis we studied two vehicle routing problems with trailers, namely, the single truck and trailer routing problem with satellite depots (STTRPSD) and the truck and trailer routing problem (TTRP).
In the STTRPSD, a truck with a detachable trailer based at a main depot has to serve the demand of a set of customers accessible only by truck. Therefore, before serving the customers, it is necessary to detach the trailer in designated parking places (called trailer points or satellite depots) where goods are transferred between the truck and the trailer. A solution of the STTRPSD, is composed of a first-level trip departing from the main depot (performed by the truck with the trailer) visiting a subset of trailer points; and several second-level trips (performed by the truck), rooted at trailer points visited in the first-level trip. The multi-depot VRP can be seen as a special case of the STTRPSD where the distance between any two depots is null. Likewise, the STTRPSD can be seen as a simplified version of the two-echelon capacitated location-routing problem (2E-LRP) with only one vehicle in the first echelon and without fixed cost at the depots.
On the other hand, in the TTRP a heterogeneous fixed fleet of trucks and trailers serve the demand of a set customers. The customers are partitioned into truck customers and vehicle customers. Truck customers have incompatibility constraints, being accessible only by truck. In contrast, vehicle customers do not have accessibility restrictions, and their locations can be used to park the trailer before serving truck customers. In the TTRP there are three types of routes: pure truck routes performed by a truck; pure vehicle routes performed by a truck with a trailer serving only vehicle customers; and vehicle routes with sub tours performed by a truck with a trailer and serving both vehicle customers and truck customers. Vehicle routes with sub tours have a STTRPSD-like structure, then the TTRP generalizes in some sense the STTRPSD. The TTRP is also related with the heterogeneous fixed fleet VRP because there are two types of vehicles with different capacities (the truck and the truck-with-the-trailer). In the same way, the TTRP is related with the site-dependent VRP because of the incompatibility constraints of truck customers.