Francesco Mason

Properties of the k-centra in a tree network

The k-centrum of a graph G - (V, E)is the set of vertices each minimizing the sum of the distances from it to the corresponding k farthest vertices of G. In this article properties of k-centrum of trees, where edges have positive real lengths, are investigated.

k-eccentricity and absolute k-centrum of a probabilistic tree

Abstract The k-eccentricity evaluated at a point x of a graph G is the sum of the (weighted) distances from x to the k vertices farthest from it. The k-centrum is the set of vertices for which the k-eccentricity is a minimum. The concept of k-centrum includes, as a particular case, that of center and that of centroid (or median) of a graph. The absolute k-centrum is the set of points (not necessarily vertices) for which the k-eccentricity is a minimum. In this paper it will be proven that, for a weighted tree, both deterministic and probabilistic, the k-eccentricity is a convex function and that the absolute k-centrum is a connected set and is contained in an elementary path. Hints will be given for the construction of an algorithm to find the k-centrum and the absolute k-centrum.

A note on “A perfect forward procedure for a single facility dynamic location/relocation problem”

In a recent paper, M. Bastian and M. Volkmer (Oper. Res. Lett. 12, 11-16) [1] proposed a perfect forward algorithm for the solution of a single facility dynamic location/relocation problem. Here, we first provide a numerical example to demonstrate that this problem does not always have a finite forecast horizon. Secondly, we restate the original problem in terms of a shortest path problem in an acyclic network and give an obvious condition (which is both necessary and sufficient) for the existence of a finite forecast horizon for obtaining an optimal initial decision. Then a simple perfect forward algorithm for obtaining an optimal initial decision (when a finite forecast horizon exists) is presented. This algorithm can be considered as a version of Dijkstras algorithm. It is our opinion that the formulation proposed here is substantially simpler than the one presented in [1] and helps a bit more to understand the real nature of the problem.

Genetic Algorithm with Redundancies for the Vehicle Scheduling Problem

A real vehicle scheduling problem concerning the urban public transportation system of the city of Mestre (Venice) has been approached by Genetic Algorithm enhanced using redundancies. Redundant alleles fix the string at cross-over positions in order to improve solution feasibility. The scheduling problem has been studied both as a single and as a multiple objective optimisation problem. A significant reduction of resources as compared to the currently used solution has been achieved.

Heuristic algorithms for the 2-period balanced Travelling Salesman Problem in Euclidean graphs

In the 2-period Travelling Salesman Problem some nodes, called double nodes, are visited in both of two periods while the remaining ones, called single nodes, are visited in either one of the periods. In this paper we study the case in which a balance constraint is also introduced. We require that the difference between the number of visited nodes in the two periods must be below a fixed threshold. Moreover, we suppose that distances between nodes are Euclidean. The problem is NP-hard, and exact methods, now available, appear inadequate. Here, we propose three heuristics. Computational experiences and a comparison between the algorithms are also given.

Path covering problems and testing of printed circuits

This work deals with a problem, arising in the context of printed circuit testing, that will be formulated as a path covering problem on trees. For this problem, which consists in minimizing the number of continuity testing, an exact algorithm of linear complexity is here proposed for the first time. A general classification of path covering problems is also presented and a brief review of the literature is given.

A model for the assignment of candidates to constituencies in a mixed election system

Abstract A model of a particular mixed election system is presented. Its formulation gives rise to a combinatorial problem which is shown to be strongly NP-complete. An implicit enumeration algorithm is proposed and some computational experience is reported. A relaxation of the problem is presented such that it is reduced to a flow problem.