Laurent Houssin

In this paper, we focus on exact methods to solve the preemptive Flexible Job-Shop Scheduling Problem with makespan and total completion time minimisation objective functions. Mathematical and constraint programming models enable the resolution of this problem for small instances. However, as an NP-hard problem, the cost of solving grows rapidly when considering larger instances. In this regard, we propose a logic-based Benders decomposition that relies on an efficient branch-and-bound procedure to solve the subproblem representing a pure (non-flexible) preemptive job-shop scheduling problem. Computational experiments are carried out and show the very good performance of our proposals.

Euclidean norm computations over continuous variables appear naturally in the constraints or in the objective of many problems in the optimization literature, possibly defining non-convex feasible regions or cost functions. When some other variables have discrete domains, it positions the problem in the challenging Mixed Integer Nonlinear Programming (MINLP) class. For any MINLP where the nonlinearity is only present in the form of inequality constraints involving the Euclidean norm, we propose in this article an efficient methodology for linearizing the optimization problem at the cost of entirely controllable approximations. They make it possible to rely fully on Mixed Integer Linear Programming and all its strengths. This methodology is successfully applied to a critical problem in the telecommunication satellite industry: the optimization of the beam layouts in multibeam satellite systems. We provide a proof of the NP-hardness of this very problem along with experiments on a realistic reference scenario.

In this article, models and methods for solving a real-life frequency assignment problem based on scheduling theory are investigated. A realistic frequency assignment problem involving cumulative interference constraints in which the aim is to maximize the number of assigned users is considered. If interferences are assumed to be binary, a multiple carrier frequency assignment problem can be treated as a disjunctive scheduling problem since a user requesting a number of contiguous frequencies can be considered as a non-preemptive task with a processing time, and two interfering users can be modelled through a disjunctive constraint on the corresponding tasks. A binary interference version of the problem is constructed and a disjunctive scheduling model is derived. Based on the binary representation, two models are proposed. The first one relies on an interference matrix and the second one considers maximal cliques. A third, cumulative, model that yields a new class of scheduling problems is also proposed. Computational experiments show that the case-study frequency assignment problem can be solved efficiently with disjunctive scheduling techniques.