Integer programming in the context of Polyhedral combinatorics

Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models.

In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics.

The vehicle routing problem (VRP) is a combinatorial optimization and integer programming problem which asks "What is the optimal set of routes for a fleet of vehicles to traverse in order to deliver to a given set of customers?" The problem first appeared, as the truck dispatching problem, in a paper by George Dantzig and John Ramser in 1959, in which it was applied to petrol deliveries. Often, the context is that of delivering goods located at a central depot to customers who have placed orders for such goods. However, variants of the problem consider, e.g, collection of solid waste and the transport of the elderly and the sick to and from health-care facilities. The standard objective of the VRP is to minimise the total route cost. Other objectives, such as minimising the number of vehicles used or travelled distance are also considered.

The VRP generalises the travelling salesman problem (TSP), which is equivalent to requiring a single route to visit all locations. As the TSP is NP-hard, the VRP is also NP-hard.

In mathematics, a constraint is a condition of an optimization problem that the solution must satisfy. There are several types of constraints—primarily equality constraints, inequality constraints, and integer constraints. The set of candidate solutions that satisfy all constraints is called the feasible set.