Dynamic flow problems arising from traffic planning

Abstract

The subject of this work is the solution of four classes of combinatorial optimization problems arising from traffic planning applications. To this end, we use algorithms and techniques from the field of mixed-integer programming, tailoring them to the specific problem classes. The robust shortest path problem is a generalization of the famous shortest path problem to a scenario in which the travel times through a given network are subject to uncertainty. The air-to-air refueling problem arises in the context of commercial trans-continental flights. A special feature of this problem is the nonlinearity with respect to the fuel consumption of the feeder aircraft. We address this issue by containing the nonlinearity in a combinatorial subproblem. The time-dependent TSP is a generalization of the notorious traveling salesman problem to scenarios in which the travel times are subject to change over time, introducing a temporal dynamic. Finally we study the Aircraft Landing Problem, a large-scale optimization problem where a large amount of domain constraints --- scheduling of landing times and consistent runway assignments --- have to be satisfied. Column Generation is a subject of paramount importance for the solution of large-scale problem instances. The technique consists of generating problem variables one after another based on the predicted reduction of the objective function value, producing a provably optimal solution using a fraction of the whole set of problem variables, significantly reducing computational overhead. Modern column generation variants such as dual stabilization methods allow to control the generation of variables in such a way as to significantly increase the speed of convergence. Further techniques include the usage of cutting planes, tightening the outer approximations of the polyhedra of feasible integral points, as well as branching rules and priorities supporting existing codes and primal heuristics used to find feasible solutions. For each problem class we begin by settling its computational complexity, discussing approximation methods where appropriate. We proceed to derive exact solution algorithms based on mixed-integer programming formulations of the problem classes. To evaluate different solution techniques we perform a series of computational experiments with both real-world and synthetic data. Our work increases the scale of tractable problem instances dramatically, decreasing computational time by orders of magnitude.