Meinolf Sellmann

Constraint Programming Based Column Generation for Crew Assignment

Airline crew assignment problems are large-scale optimization problems which can be adequately solved by column generation. The subproblem is typically a so-called constrained shortest path problem and solved by dynamic programming. However, complex airline regulations arising frequently in European airlines cannot be expressed entirely in this framework and limit the use of pure column generation. In this paper, we formulate the subproblem as a constraint satisfaction problem, thus gaining high expressiveness. Each airline regulation is encoded by one or several constraints. An additional constraint which encapsulates a shortest path algorithm for generating columns with negative reduced costs is introduced. This constraint reduces the search space of the subproblem significantly. Resulting domain reductions are propagated to the other constraints which additionally reduces the search space. Numerical results based on data of a large European airline are presented and demonstrate the potential of our approach.

A Framework for Constraint Programming Based Column Generation

Column generation is a state-of-the-art method for optimally solving difficult large-scale optimization problems such as airline crew assignment. We show how to apply column generation even if those problems have complex constraints that are beyond the scope of pure OR methods. We achieve this by formulating the subproblem as a constraint satisfaction problem (CSP). We also show how to efficiently treat the special case of shortest path problems by introducing an efficient path constraint that exploits dual values from the master problem to exclude nodes that will not lead to paths with negative reduced costs. We demonstrate that this propagation significantly reduces the time needed to solve crew assignment problems.

Crew Assignment via Constraint Programming: Integrating Column Generation and Heuristic Tree Search

The Airline Crew Assignment Problem (ACA) consists of assigning lines of work to a set of crew members such that a set of activities is partitioned and the costs for that assignment are minimized. Especially for European airline companies, complex constraints defining the feasibility of a line of work have to be respected. We developed two different algorithms to tackle the large scale optimization problem of Airline Crew Assignment. The first is an application of the Constraint Programming (CP) based Column Generation Framework. The second approach performs a CP based heuristic tree search. We present how both algorithms can be coupled to overcome their inherent weaknesses by integrating methods from Constraint Programming and Operations Research. Numerical results show the superiority of the hybrid algorithm in comparison to CP based tree search and column generation alone.

Constraint Programming Based Lagrangian Relaxation for the Automatic Recording Problem

Whereas CP methods are strong with respect to the detection of local infeasibilities, OR approaches have powerful optimization abilities that ground on tight global bounds on the objective. An integration of propagation ideas from CP and Lagrangian relaxation techniques from OR combines the merits of both approaches. We introduce a general way of how linear optimization constraints can strengthen their propagation abilities via Lagrangian relaxation. The method is evaluated on a set of benchmark problems stemming from a multimedia application. The experiments show the superiority of the combined method compared with a pure OR approach and an algorithm based on two independent optimization constraints.

Cost Based Filtering for the Constrained Knapsack Problem

We present cost based filtering methods for Knapsack Problems (KPs). Cost based filtering aims at fixing variables with respect to the objective function. It is an important technique when solving complex problems such as Quadratic Knapsack Problems, or KPs with additional constraints (Constrained Knapsack Problems (CKPs)). They evolve, e.g., when Constraint Based Column Generation is applied to appropriate optimization problems. We develop new reduction algorithms for KP. They are used as propagation routines for the CKP with Θ(nlogu2009n) preprocessing time and Θ(n) time per call. This sums up to an amortized time Θ(n) for Ω(logu2009n) incremental calls where the subsequent problems may differ with respect to arbitrary sets of necessarily included and excluded items.

Lagrangian Cardinality Cuts and Variable Fixing for Capacitated Network Design

We present a branch-and-bound approach for the Capacitated Network Design Problem. We focus on tightening strategies such as variable fixing and local cuts that can be applied in every search node. Different variable fixing algorithms based on Lagrangian relaxations are evaluated solitarily and in combined versions. Moreover, we develop cardinality cuts for the problem and evaluate their usefulness empirically by numerous tests.

Heuristic Constraint Propagation

For NP-hard constraint satisfaction problems the existence of a feasible solution cannot be decided efficiently. Applying a tree search often results in the exploration of parts of the search space that do not contain feasible solutions at all. Redundant constraints can help to detect inconsistencies of partial assignments higher up in the search tree. Using the social golfer problem as an example we show how complex redundant constraints can be propagated incompletely using local search heuristics.

An Arc-Consistency Algorithm for the Minimum Weight All Different Constraint

Historically, discrete minimization problems in constrained logical programming were modeled with the help of an isolated bounding constraint on the objective that is to be decreased. To overcome this frequently inefficient way of searching for improving solutions, the notion of optimization constraints was introduced. Optimization constraints can be viewed as global constraints that link the objective with other constraints of the problem at hand. We present an arc-consistency (actually: hyper-arc-consistency) algorithm for the minimum weight all different constraint which is an optimization constraint that consists in the combination of a linear objective with an all different constraint.

Coupling Variable Fixing Algorithms for the Automatic Recording Problem

Variable fixing is an important technique when solving combinatorial optimization problems. Unique profitable variable values are detected with respect to the objective function and to the constraint structure of the problem. Relying on that specific structure, effective variable fixing algorithms (VFAs) are only suited for the problems they have been designed for. Frequently, new combinatorial optimization problems evolve as a combination of simpler structured problems. For such combinations, we show how VFAs for linear optimization problems can be coupled via Lagrangian relaxation. The method is applied on a multimedia problem incorporating a knapsack and a maximum weighted stable set problem.