A

Combinatorial optimisation problems occur naturally in manufacturing processes, logistics, service location, activity planning, etc. They are characterised by wanting to optimise one or more targets on a large number of solutions that grow exponentially depending on the size of the problem. It is necessary for the decision maker to have efficient algorithms to help in the decisions. The exact methods only provide the optimal solution in reasonable time for a few combinatorial problems, while for most of them, we must resort to approximate methods that provide a range of solutions close to the optimal one. Our group has extensive experience in both approximate and exact solving of different combinatorial problems using Integer Linear Programming and metaheuristic techniques. Any combinatorial problem would require a previous analysis and, depending on its size and complexity, the selection and coding of the most appropriate technique.

 B

Metaheuristic procedures are a class of approximate methods that are designed to solve difficult optimisation problems in which classical heuristics are not effective. Metaheuristics provide a general framework to create new hybrid algorithms combining different concepts derived from: artificial intelligence (tabu search), biological evolution (evolutionary algorithms) and statistical mechanisms (simulated annealing). We have worked in many optimisation problems, both in the academic field and in industrial applications, for which we have designed efficient solution methods based on these methods.

Artificial neural networks abstract some of the most important features of biological neural networks and implement them in simple mathematical models in order to imitate their behavior. In recent years there has been a growth in the study and development of neural networks, especially those based on the multilayer model feed-forward. The most common applications are the "classification or pattern recognition" and "function approximation or prediction." In recent years we have worked on the design of these models and in the methods of calibration or adjustment of their parameters.

 C

A Combinatorial Optimisation problem involves finding the minimum cost solution among a finite (or countably infinite) set of possible solutions. Obviously, for these optimisation problems standard techniques of differential calculus of equating derivatives to zero cannot be used because it is not about choosing between the infinite points of a continuum.

An alternative is Linear Programming, which can be defined as the problem of finding the vertex of a specific polyhedron that minimises some linear function, as the number of vertices of a polyhedron is also finite. Polyhedral Combinatorics consist on defining a polyhedron whose vertices correspond one to one to possible solutions to the problem and finding the specific inequalities describing the polyhedron to apply the methods of Linear Programming (a polyhedron can be described by a system of equations and inequalities so that each inequality induces one of its sides).

Note that it is not possible to generate ALL of these inequalities because: (1) it is highly unlikely to get to know the full description of the polyhedron if the original optimisation problem is NP-hard and (2) even if it was known, the number of such inequalities is extremely great. However, it is not needed to generate all the inequalities to solve the problem: it is sufficient with just a limited number of inequalities of the area of a polyhedron ‘close' to the optimal vertex.

This is accomplished by using an iterative algorithm that starts with a small number of inequalities and generates new inequalities when necessary.

The core of this algorithm is the so-called Separation Problem: given an ‘x’ point which does not belong to the polyhedron, find an inequality that induces a side of the polyhedron which is not satisfied by ‘x’. Thus, a significant subset of inequalities describing the polyhedron associated with a Combinatorial Optimisation problem can be enough to find the optimal solution. And when the above iterative algorithm does not reach the optimal solution, we can apply ‘Branch and Bound’ methods to the linear system formed with the inequalities found, which represents an adjusted relaxation of the Combinatorial Optimisation problem.

 D

Vehicle routing problems are important by themselves within the distribution and transport problems, both from a theoretical point of view and for their many applications. They are modelled as routing problems in graphs (mixed in the most general case) which should serve, under certain conditions, lines, vertices or both.

Our group has been working for several years in the theoretical study of different models and, from them, in the design and implementation of algorithms, both exact and approximate, to solve these problems.

 E

All aspects concerning the problems of distribution and transport are of great economic importance. Moreover, the number and size of such problems is growing due to the increase of trade and travel at all levels. Even e-commerce will increase the magnitude of these problems, since the items purchased online must be delivered. Also related to these issues are the problems of network design. Networks that appear in transport logistics and telecommunications planning. Our work in this field consists on the analysis and development of quantitative and computer methods applied to the planning, management and exploitation of distribution networks and transportation of passengers and goods. More specifically, our group is dedicated to the design and development of methods of assessment, analysis, simulation and optimisation required for the study and resolution of real problems. Its purpose is to make the process of analysis and decision making involved in the planning and management of distribution and transportation systems more efficient and of higher quality.