Models and Solutions of Resource Allocation Problems based on Integer Linear and Nonlinear Programming

In this thesis we deal with two problems of resource allocation solved through a Mixed-Integer Linear Programming approach and a Mixed-Integer Nonlinear Chance Constraint Programming approach. In the first part we propose a framework to model general guillotine restrictions in two dimensional cutting problems formulated as Mixed-Integer Linear Programs (MILP). The modeling framework requires a pseudo-polynomial number of variables and constraints, which can be effectively enumerated for medium-size instances. Our modeling of general guillotine cuts is the first one that, once it is implemented within a state of-the-art MIP solver, can tackle instances of challenging size. Our objective is to propose a way of modeling general guillotine cuts via Mixed Integer Linear Programs (MILP), i.e., we do not limit the number of stages (restriction (ii)), nor impose the cuts to be restricted (restriction (iii)). We only ask the cuts to be guillotine ones (restriction (i)). We mainly concentrate our analysis on the Guillotine Two Dimensional Knapsack Problem (G2KP), for which a model, and an exact procedure able to significantly improve the computational performance, are given. In the second part we present a Branch-and-Cut algorithm for a class of Nonlinear Chance Constrained Mathematical Optimization Problems with a finite number of scenarios. This class corresponds to the problems that can be reformulated as Deterministic Convex Mixed-Integer Nonlinear Programming problems, but the size of the reformulation is large and quickly becomes impractical as the number of scenarios grows. We apply the Branch-and-Cut algorithm to the Mid-Term Hydro Scheduling Problem, for which we propose a chance-constrained formulation. A computational study using data from ten hydro plants in Greece shows that the proposed methodology solves instances orders of magnitude faster than applying a general-purpose solver for Convex Mixed-Integer Nonlinear Problems to the deterministic reformulation, and scales much better with the number of scenarios.