1
General Presentation of Vehicle Routing Problems

Vehicle routing problems (VRPs) represent an important family of problems encountered in the fields of logistics, as well as in many other applications. In general, a number of customers have to be served with a fleet of vehicles. They can be modeled as an integer programming problem, solved by combinatorial optimization tools. However, exact methods cannot solve instances that consider a large set of customers, as encountered in most real cases. It is, therefore, often necessary to resort to approximate paradigms generally carried out through metaheuristics.

This first chapter introduces what the logistics management and the combinatorial optimization are, before giving a formal definition of the CVRP, with notations useful throughout the remainder of the book.

1.1. Logistics management and combinatorial optimization

In the last few decades, a great interest has grown up in the area of logistics among both industry and academia, for different reasons [BRA 98]. First, companies are facing fierce competition in today's global markets. They need to innovate to keep their position, and they realize the savings that can be achieved by a better planning and management of their logistic systems.

Furthermore, the evolution of lifestyles is significant. Modes of consumption are changing and expectations of consumers switch to products with short lifecycles, and the advancement in communications and transportation technologies, such as mobile communication and overnight delivery, motivates continuous development of the management of logistic systems.

These changes attract attention of the academic community, whose approach consists of determining characteristics of the problems and developing solution methodologies, as well as providing specific guarantees of effectiveness.

1.1.1. History of logistics

Logistics is not a recent trend in managing the flow of goods from an origin to a destination, with the aim to meet some requirements. Logistics made an important stride during the construction of the pyramids in ancient Egypt, for example. It played a key role in global sea trade with the invention of rowing vessels around 300 B.C. Logistics was also one of the main factors for the victory of most wars throughout history.

In military context, logistics is responsible for supplying the troops. It deals with the inventory management and transportation. However, this type of requirement also predominates in carriers and wholesalers activities. Thus, it is natural that modern logistics appears in industry.

Figure 1.1. Example of a supply chain

Nowadays, the function extends from production to distribution, leading to the supply chain (Figure 1.1). In this chain, upstream activities take place prior to a particular link, when the latter orders for material to suppliers in the aim to bring its added value. On the contrary, downstream activities involve the sale of a material to other businesses, governments or private individuals. The extreme link in the upstream part usually concerns raw materials, while the extreme downstream link is related to the final customer. However, each other link in the midstream is both customer of predecessor actors and supplier of successors. Midstream can be a manufacturer, a cooperative warehouse, a regional consolidation center, a city hub, local depot, etc.

Most of the freight transport in the chain is carried in containers, although bulk transport is used more for large volumes of durable goods. The reason is that this option is often the most efficient and cost-effective way to supply the products. However, for the smaller quantities generally required at the final destination, the supply chain is often less efficient. This characteristic is known as the "last mile problem", which can represent up to 28% of the total cost to move goods. In addition, if transport plays an important role in economic growth and globalization, it causes air pollution and a large amount of traffic. Hence, a good transport planning is essential to control the costs, as well as the flow and limit nuisances.

Figure 1.2. Possible outline of a reverse logistics chain

In an even more global view, the network also integrates reverse flows. These cover all operations related to the recycling of products and materials thrown away by the public or by industries (obsolete products, mixed waste and even hazardous). The so-called reverse logistics brings together the movements of products from consumers to producers through a distribution chain (Figure 1.2). The growing concern for integrating environmental requirements into green supply chain management concepts and practices makes it even more relevant. The reverse logistics process refers to activities undertaken to reduce, manage and dispose of waste from industrial activities. It meets the need to decommission the products after use and treat the destruction, by transforming or recycling in order to reduce costs, and valuing the recovered products. Several related activities, therefore, involve: collecting waste, the location of recycling points/storage, inventory management and integration of products from the collection at the related industries. It also includes the optimization of the Ecodesign to facilitate future recycling.

Other issues have arisen recently about city logistics which are obviously related to the last mile problem described before. In fact, the freight distribution in urban area has to deal with several aspects. First, traffic may be difficult because of congestion at some rush hours, which makes the travel time dependent on the time of the day. Another particularity is the accessibility constraint. It might be quite complicated to deliver the goods in some areas because of the lack of parking for example, or because of city restrictions on the use of trucks in favor to smaller vehicles. In the same vein, economic and environmental problem concerns might lead to choose alternative types of transport for urban freight distribution (such as electric vehicles), as well as to adopt new commercialization behaviors. For example, the growth of e-commerce brings new questions and some retail companies have studied the use of drones to deliver online purchased goods to consumers.

Hence, many activities are involved in the supply chain, from the network design, to logistics of transportation, passing through warehouse management, international commerce or information systems. Transportation is one of the main parts of logistics. It can be made through several modes such as air, rail, road, water, cable, pipeline and space and may require particular infrastructures (Figure 1.3). These include links in the network (roads, railways, canals or pipelines, for instance) and terminals such as airports, railway stations, warehouses and depots. A wide range of issues emerges in this context, sweeping topics as diverse as the routing, inventory, cross-docking or network structure.

Figure 1.3. Example of transportation modes

1.1.2. Logistics as a science

The logistics function has risen to such an important place that it is now a profession in itself, and even a science. The goal in logistics management is to be efficient and cost-effective across the entire system [BRA 98]. Therefore, the objective is not simply to minimize locally transportation cost or reducing inventories. Every facility that has an impact on system effectiveness must be taken into consideration, from suppliers to retailers through manufacturing facilities, warehouses and distribution centers.

In fact, logistics management encompasses many of the firm's activities, from the strategic level through the tactical to the operational level:

1. - the strategic level deals with decisions that have usually a long-term effect. Concerning logistics, this includes, for instance, decisions regarding the number, location and capacities of warehouses and manufacturing plants;
2. - the tactical level typically includes decisions that are updated anywhere between once every quarter and once every year. This includes purchasing and production decisions, inventory policies and transportation strategies including the frequency with which customers are visited;
3. - the operational level refers to day-to-day decisions such as scheduling, routing and loading trucks.

Therefore, logistics activities obviously deserve to be recognized as a science, and this has begun to be true from the middle of the 20th Century [TAY 07].

1.1.3. Combinatorial optimization

The science of logistics can be seen as the study of the physical flow of products and services through the supply chain. Therefore, the chain can be seen as a network, or a graph, in which a flow has to go through, with some constraints that need to be encounter and an objective, often relative to a cost function, to optimize. Thus, most of the decision-making to manage the logistics can be taken by modeling the problem in terms of a mathematical program to optimize.

Optimization is a branch of mathematics particularly applied in operations research and management science. It consists of finding one or more best (optimal) solutions from all feasible solutions. Optimization problems can be divided into two categories depending on whether the variables are continuous or discrete. The latter case is known as a combinatorial optimization problem. Solving such...