Torsten Fahle

A hybrid setup for a hybrid scenario: combining heuristics for the home health care problem

Home health care, i.e. visiting and nursing patients in their homes, is a growing sector in the medical service business. From a staff rostering point of view, the problem is to find a feasible working plan for all nurses that has to respect a variety of hard and soft constraints, and preferences. Additionally, home health care problems contain a routing component: a nurse must be able to visit her patients in a given roster using a car or public transport. It is desired to design rosters that consider both, the staff rostering and vehicle routing components while minimizing transportation costs and maximizing satisfaction of patients and nurses.In this paper we present the core optimization components of the PARPAP software. In the optimization kernel, a combination of linear programming, constraint programming, and (meta-)heuristics for the home health care problem is used, and we show how to apply these different heuristics efficiently to solve home health care problems. The overall concept is able to adapt to various changes in the constraint structure, thus providing the flexibility needed in a generic tool for real-world settings.

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 parallel algorithm for the vehicle routing problem with time window constraints

In this paper, we describe a new parallel tabu search heuristic for the vehicle routingproblem with time window constraints (VRPTW). The neighborhood structure we proposeis based on simple customer shifts and allows us to consider infeasible interim‐solutions.Similarly to the column generation approach used in exact algorithms, all routes generatedby the tabu search heuristic are collected in a pool. To obtain a new initial solution forthe tabu search heuristic, a fast set covering heuristic is periodically applied to the routes inthe pool. The parallel heuristic has been implemented on a Multiple‐Instruction Multiple‐Datacomputer architecture with eight nodes. Computational results for Solomons benchmarkproblems demonstrate that our parallel heuristic can produce high‐quality solutions.

Simple and Fast: Improving a Branch-And-Bound Algorithm for Maximum Clique

We consider a branch-and-bound algorithm for maximum clique problems. We introduce cost based filtering techniques for the so-called candidate set (i.e. a set of nodes that can possibly extend the clique in the current choice point).Additionally, we present a taxonomy of upper bounds for maximum clique. Analytical results show that our cost based filtering is in a sense as tight as most of these well-known bounds for the maximum clique problem.Experiments demonstrate that the combination of cost based filtering and vertex coloring bounds outperforms the old approach as well as approaches that only apply either of these techniques. Furthermore, the new algorithm is competitive with other recent algorithms for maximum clique.

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(nlog n) preprocessing time and Θ(n) time per call. This sums up to an amortized time Θ(n) for Ω(log n) incremental calls where the subsequent problems may differ with respect to arbitrary sets of necessarily included and excluded items.

The Aircraft Sequencing Problem

In this paper we present different exact and heuristic optimization methods for scheduling planes which want to land (and start) at an airport-the Aircraft Sequencing Problem (ASP). We compare two known integer programming formulations with four new exact and heuristic solution methods regarding quality, speed and flexibility.

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.