Solving Large Scale Crew Pairing Problems

Crew pairing is one of the most critical processes in airline management operations. Taking a timetable as input, the objective of this process is to find an optimal way to partition flights of the timetable without breaking rules and regulations which are enforced by an airline. The problem has attracted many scientists in recent decades. The main challenge is that there is no general method to work well with all kinds of non-linear cost functions and rules. In order to overcome the non-linearity, the thesis follows a main idea to transfer this combinatorial optimization problem to a set partitioning problem which is one of the most popular \np-hard problems. Although this problem has been studied throughout decades, it becomes more complicated with the increasing size of the input. The complication is induced not only in the transformation process, but also in the methods to solve the resulting set partitioning problem. Finding quickly a good and robust solution for large scale problems is more and more critical to airlines. They are the main targets which are studied by the thesis. The thesis presents exact methods which are usually based on a branch-and-bound scheme. A branch-and-cut approach applies preprocessing techniques, cutting plane generation methods, and heuristics which are suitable for crew pairing problems. The implementation can solve small and medium sized problems. However, for large problems, a branch-and-price approach is necessary to cope with huge constraint matrices. The thesis improves the weakness of standard column generation methods by applying stabilized column generation variants with sophisticated parameter control schemes into this approach. The computation time is reduced significantly by a factor of three. Moreover, the work also focuses on the extensibility of the methods. This is quite important for large scale problems. Then, we easily obtain a heuristic solution method by controlling running parameters of the presented approaches or combining them together. Utilizing the available computing resources to deal with large scale crew pairing problems as effective as possible is also a target of the thesis. A new parallel branch-and-bound library is developed to support scientists to solve combinatorial optimization problems. With little effort, they can migrate their sequential codes to run on a parallel computer. The library contains several load balancing methods and control parameters in order to work well with specific problems. The sequential branch-and-cut code to solve set partitioning problems is parallelized by the library and introduces a good speedup for most crew pairing test problems. Parallel computing is also used to solve a so-called pricing subproblem, which is the most difficult problem in the branch-and-price approach, with a nearly linear speedup. The implementation solves large scale crew pairing problems to optimality within minutes, whereas previous methods ended up in the range of hours or more

Standard Bundle Methods: Untrusted Models and Duality

We review the basic ideas underlying the vast family of algorithms for nonsmooth convex optimization known as "bundle methods|. In a nutshell, these approaches are based on constructing models of the function, but lack of continuity of first-order information implies that these models cannot be trusted, not even close to an optimum. Therefore, many different forms of stabilization have been proposed to try to avoid being led to areas where the model is so inaccurate as to result in almost useless steps. In the development of these methods, duality arguments are useful, if not outright necessary, to better analyze the behaviour of the algorithms. Also, in many relevant applications the function at hand is itself a dual one, so that duality allows to map back algorithmic concepts and results into a "primal space" where they can be exploited; in turn, structure in that space can be exploited to improve the algorithms' behaviour, e.g. by developing better models. We present an updated picture of the many developments around the basic idea along at least three different axes: form of the stabilization, form of the model, and approximate evaluation of the function

A Branch-Price-and-Cut Procedure for the Discrete Ordered Median Problem

International audienceThe Discrete Ordered Median Problem (DOMP) is formulated as a set partitioning problem using an exponential number of variables. Each variable corresponds to a set of demand points allocated to the same facility with the information of the sorting position of their corresponding costs. We develop a column generation approach to solve the continuous relaxation of this model. Then, we apply a branch-price-and-cut algorithm to solve small to large sized instances of DOMP in competitive computational time