The Generalized Assignment Problem is a well-known NP-hard combinatorial optimization problem which consists of minimizing the assignment costs of a set of jobs to a set of machines satisfying capacity constraints. Most of the existing algorithms are of a Branch-and-Price type, with lower bounds computed through Dantzig-Wolfe reformulation and column generation.In this paper we propose a cutting plane algorithm working in the space of the variables of the basic formulation, whose core is an exact separation procedure for the knapsack polytopes induced by the capacity constraints. We show that an efficient implementation of the exact separation procedure allows to deal with large-scale instances and to solve to optimality several previously unsolved instances.

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The problem of sequencing and scheduling airplanes landing and taking off on a runway is a major challenge for air traffic management. This difficult real-time task is still carried out by human controllers, with little help from automatic tools. Several methods have been proposed in the literature, including Mixed Integer Programming (MIP) based approaches. However, in a recent survey (Bennell et al. (2011)) MIP is claimed to be unattractive for real-time applications, since computation times are likely to grow too large. In this paper we reverse this claim, by developing a MIP approach able to solve to optimality real-life instances from congested airports in the stringent times allowed by the application. In order to achieve this it was mandatory to identify new classes of strong valid inequalities, along with developing effective fixing and lifting procedures.