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 inventory routing problem (IRP) involves the distribution of one or more products from a supplier to a set of customers over a discrete planning horizon. The version treated here, the so-called vendor managed IRP (VMIRP), is the IRP arising when the replenishment policy is decided a priori. We consider two replenishment policies. The first is known as order-up (OU): if a client is visited in a period, then the amount shipped to the client must bring the stock level up to the upper bound. The latter is called maximum level (ML): the maximum stock level in each period cannot be exceeded. The objective is to find replenishment decisions minimizing the sum of the storage and distribution costs. VMIRP contains two important subproblems: a lot-sizing problem for each customer and a classical vehicle routing problem for each time period. In this paper we present a priori reformulations of VMIRP-OU and VMIRP-ML derived from the single-item lot-sizing substructure. These reformulations take into account the special nature of the test instances-constant demand for each client over time and stock/production upper bounds that are fixed small multiples of the client's demand. In addition we introduce two new cutting plane families-the cut inequalities-deriving from the interaction between the lot-sizing and the routing substructures. A branch-and-cut algorithm has been implemented to demonstrate the effectiveness of the single-item reformulations. Computational results on benchmark instances with 50 customers and six periods with a single product and a single vehicle are presented.

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.