Gert A. Tijssen

Routing helicopters for crew exchanges on off-shore locations

This paper deals with a vehicle routing problem with split demands, namely the problem of determining a flight schedule for helicopters to off-shore platform locations for exchanging crew people employed on these platforms. The problem is formulated as an LP model and solved by means of a column-generation technique including solving TSP problems. Since the final solution needs to be integral, we have chosen a rounding procedure to obtain an integer solution. Since the LP approach needs a considerable amount of computer time, it is only suitable for long-term planning practices. For the usual short-term planning, we have designed the so-called Cluster-and-Route Heuristic together with a number of improvement heuristics. The Cluster-and-Route procedure constructs a suitable clustering of the platforms and simultaneously forms the routes of the helicopter flights associated with the clusters. This approach is different from the usual heuristics, in which the clusters are constructed first, and the routes for each cluster are made afterwards. Simulations with various data sets show that the new heuristic outperforms the usual heuristics for vehicle routing problems. Even better results are obtained when improvement heuristics are applied. We use four improvement heuristics, including, so-called 1-opt and 2-opt procedures.

Hamiltonian properties of Toeplitz graphs

Conditions are given for the existence of hamiltonian paths and cycles in the so-called Toeplitz graphs, i.e. simple graphs with a symmetric Toeplitz adjacency matrix.

Faces with large diameter on the symmetric traveling salesman polytope

This paper deals with the symmetric traveling salesman polytope and contains three main theorems. The first one gives a new characterization of (non)adjacency. Based on this characterization a new upper bound for the diameter of the symmetric traveling salesman polytope (conjectured to be 2 by M. Grotschel and M.W. Padberg) is given in the third theorem. D. Hausman has proved in 1980 that (non)adjacency of two vertices of the symmetric traveling salesman polytope can be tested on a face, induced by these two vertices. The second theorem shows that some of these faces have a large diameter and therefore cannot be used in proving the above conjecture.

Balinski--Tucker simplex tableaus: Dimensions, degeneracy degrees, and interior points of optimal faces

This paper shows the relationship between degeneracy degrees and multiple solutions in linear programming (LP) models. The usual definition of degeneracy is restricted to vertices of a polyhedron. We introduce degeneracy for nonempty subsets of polyhedra and show that for LP-models for which the feasible region contains at least one vertex it holds that the dimension of the optimal face is equal to the degeneracy degree of the optimal face of the corresponding dual model. This result is obtained by means of the so-called Balinski—Tucker (B—T) Simplex Tableaus. Furthermore, we give a strong polynomial algorithm for constructing such a B—T Simplex Tableau when a solution in the relative interior of the optimal face is known.

Degeneracy degrees of constraint collections

Abstract. This paper presents an unifying approach to the theory of degeneracy of basic feasible solutions, vertices, faces, and all subsets of polyhedra. It is a generalization of the usual concept of degeneracy defined for basic feasible solutions of an LP-problem. We use the concept of degeneracy degree for arbitrary subsets of ℝn with respect to linear constraint collections. We discuss the connection with the usual definitions, and establish the relationship between minimal representations of polyhedra and the degeneracy of their faces. We also consider a number of complexity aspects of the problem of determining degeneracy degrees. In the last section we show how our definition of degeneracy can be used to analyze the convergence of interior point methods when the optimal solutions are degenerate.

Faces of diameter two on the Hamiltonian cycle polytype

In this paper it is shown that faces of the Hamiltonian cycle polytope (also called the symmetric traveling salesman polytope) formed by the edge union of two cycles for which the symmetric difference contains only alternating cycles without common points, have diameter at most two. As a consequence, a logarithmic upper bound for the diameter of the Hamiltonian cycle polytope and the perfect two-matching polytope are derived.