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Lagrangian relaxation is usually considered in the combinatorial optimization community as a mere technique, sometimes useful to compute bounds. It is actually a very general method, inevitable as soon as one bounds optimal values, relaxes constraints, convexifies sets, generates columns etc. In this paper we review this method, from both points of view of theory (to dualize a given problem) and algorithms...
Algebraic modelling languages allow models to be implemented in such a way that they can easily be understood and modified. They are therefore a working environment commonly used by practitioners in Operations Research. Having once developed models, they need to be integrated inside the company information system. This step often involves embedding a model into a programming language environment:...
The p-median problem has been widely studied in combinatorial optimisation, but its generalisation to the capacitated case has not. We propose a branch and price algorithm, comparing it with a standard MIP solver and a branch and bound algorithm based on Lagrangean relaxation. We present computational experience, using test instances drawn from the literature and new instances with higher ratio between...
This text summarizes the PhD thesis defended by the author in January 2006 under the supervision of Professor Erik Demeulemeester at the Katholieke Universiteit Leuven. The thesis is written in English and is available from the author’s website (http://www.econ.kuleuven.be/jeroen.belien). In this research we propose a number of exact and heuristic algorithms for various scheduling problems encountered...
We present an overview of the author’s Ph.D. thesis, supervised by P. Dejax and N. Bostel, which was defended in February 2006 at École des Mines de Nantes, France. The thesis is written in French, and is available at http://www.tel.ccsd.cnrs.fr/tel-00078905/ . It was conducted in the context of a research contract with a water distribution company. In a first section, we define multiperiod routing...
Optimising routing of vehicles constitutes a major logistic stake in many industrial contexts. We are interested here in the optimal resolution of special cases of vehicle routing problems, known as team orienteering problems. In these problems, vehicles are guided by a reward that can be collected from customers, while the length of routes is limited. The main difference with classical vehicle routing...
We propose heuristic and exact algorithms for the (periodic and non-periodic) train timetabling problem on a corridor that are based on the solution of the LP relaxation of an ILP formulation in which each variable corresponds to a full timetable for a train. This is in contrast with previous approaches to the same problem, which were based on ILP formulations in which each variable is associated...
This paper surveys recent applications and advances of the constraint programming-based column generation framework, where the master subproblem is solved by traditional OR techniques, while the pricing subproblem is solved by constraint programming (CP). This framework has been introduced to solve crew assignment problems, where complex regulations make the pricing subproblem demanding for traditional...
This is a summary of the author’s Ph.D. thesis supervised by Federico Malucelli and defended on 15 May 2008 at the Politecnico di Milano. The thesis is written in English and is available from the author upon request. This work presents new methods for enhancing the Column Generation approach based on Constraint Programming when it is used for solving combinatorial optimization problems. The methods...
This is a summary of the author’s PhD thesis supervised by Edoardo Amaldi and defended on 3 April 2009 at the Politecnico di Milano. The thesis is written in English and is available from the author upon request. In this work, we extensively study two challenging variants of the general problem of clustering a given set of data points with respect to hyperplanes so as to extract collinearity between...
We survey the main results obtained by the author in his PhD dissertation supervised by Anass Nagih and Lucas Létocart. It was defended in December 2008 at The Computer Science lab of the Paris-Nord University (L.I.P.N.), Villetaneuse, France. The thesis is written in French and is available from http://www-lipn.univ-paris13.fr/~touati/TM_Thesis.pdf . Column generation algorithms are instrumental...
This paper provides a tutorial on column generation and branch-and-price for vehicle routing problems. The main principles and the basic theory of the methods are first outlined. Some additional issues, including reinforcement of the relaxation or stabilization, complete the paper. For the sake of simplicity, this material is illustrated with the case of the vehicle routing problem with time windows.
The Vehicle Routing Problem with Time Windows consists of computing a minimum cost set of routes for a fleet of vehicles of limited capacity visiting a given set of customers with known demand, with the additional constraint that each customer must be visited in a specified time window. We consider the case in which time window constraints are relaxed into “soft” constraints, that is penalty terms...
This article tackles the multi-trip vehicle routing problem with time windows and limited duration. A trip is a timed route such that a succession of trips can be assigned to one vehicle. We provide an exact two-phase algorithm to solve it. The first phase enumerates possible ordered lists of clients which match the maximum trip duration criterion. The second phase uses a Branch and Price scheme to...
The routing and wavelength assignment (RWA) problem typically occurs in wavelength division multiplexing optical networks. Given a number of available wavelengths, we consider here the problem of maximising the number of accepted connections with respect to the clash and continuity constraints. We first propose a new strategy which combines two existing models. This leads to an improved column generation...
In this paper we propose the integration of column generation in the revised normal boundary intersection (RNBI) approach to compute a representative set of non-dominated points for multi-objective linear programmes (MOLPs). The RNBI approach solves single objective linear programmes, the RNBI subproblems, to project a set of evenly distributed reference points to the non-dominated set of an MOLP...
A point-interval $$(I_v, p_v)$$ ( I v , p v ) is a pair constituted by an interval $$I_v$$ I v of $${\mathbb {R}}$$ R and a point $$p_v \in I_v$$ p v ∈ I v . A graph $$G=(V,E)$$ G = ( V , E ) is a Max-Point-Tolerance (MPT) graph if each vertex $$v\in V$$ v ∈ V can be mapped to a point-interval in such a way that (u, v) is an edge of G iff $$I_u...
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