Pierre Chardaire

Upper and lower bounds for the two‐level simple plant location problem

In this paper, we consider a problem relevant to the telecommunications industry. In atwo‐level concentrator access network, each terminal has to be connected to a first‐levelconcentrator, which in turn must be connected to a second‐level concentrator. If no extracomplicating constraints are taken into account, the problem, translated into the language ofdiscrete location theory, amounts to an extension to two levels of facilities of the simpleplant location problem (SPLP). A straightforward formulation can be used, but we proposea more complicated model involving more variables and constraints. We show that the linearprogramming relaxations of both formulations have the same optimal values. However, thesecond formulation can be tightened by using a family of polyhedral cuts that define facetsof the convex hull of integer solutions. We develop a Lagrangian relaxation method tocompute lower bounds on the optimal value of the linear programming formulations andfeasible solutions of the integer programming model. A simulated annealing algorithm isalso designed to improve upon some of the upper bounds returned by the Lagrangian relaxationalgorithm. Experiments show the effectiveness of the formulation incorporating poly‐hedralcuts and of an approach combining a Lagrangian relaxation method and a simulatedannealing algorithm.

Multiscale nonlinear decomposition: the sieve decomposition theorem

Sieves decompose one dimensional bounded functions (d/sub m/)/sub m=1//sup R/ that represent the information in a manner that is analogous to the pyramid of wavelets obtained by linear decomposition. Sieves based on sequences of increasing scale open-closings with flat structuring elements (M and N filters) map f to {d} and the recomposition, consisting of adding up all the granule functions, maps {d} to f. Experiments show that a more general property exists such that {d/spl circ/} maps to f/spl circ/ and back to {d/spl circ/} where the granule functions {d/spl circ/} are obtained from {d} by applying any operator /spl alpha/ consisting of changing the amplitudes of some granules, including zero, without changing their signs. In other words, the set of granule function vectors produced by the decomposition is closed under the operation /spl alpha/. An analytical proof of this property is presented. This property means that filters are useful in the context of feature recognition and, in addition, opens the way for an analysis of the noise resistance of sieves.

Solving a Time-Space Network Formulation for the Convoy Movement Problem

We give a formal specification for a strategic network routing problem known as the convoy movement problem (CMP) and establish that the corresponding feasibility problem is NP-complete. We then introduce an integer programming (IP) model based on the concept of a time-space network and apply a Lagrangian relaxation to this model. We discuss how the dual function may be evaluated using a modified version of Dijkstras algorithm suitable to very large, implicitly defined graphs and show how heuristic solutions to the primal problem may be obtained. We present results for a number of instances of the CMP, most of which are based on real-world problems. The number of convoys in these instances varies between 15--25, and their movement time requires up to several thousand time units in networks ranging in size from a few dozen to several thousand vertices and edges. The most difficult instance tested involves 17 long convoys each taking four times the average link travel time to pass through a point in the network. This instance is solved within 3.3% of optimality in less than 3.5 hours of computing time on a Dell Precision 420 dual processor computer. Every other test instance is solved within 2% of the optimal value in less than 20 minutes of computing time.

Application of GRASP to the Multiconstraint Knapsack Problem

A number of approaches based on GRASP are presented for the Multiconstraint Knapsack Problem. GRASP combines greedy construction of feasible solutions with local search. Results from applying our algorithms to standard test problems are presented and compared with results obtained by Chu and Beasley.

The Radio Link Frequency Assignment Problem: A Case Study Using Genetic Algorithms

As part of a multi-national study to compare exact and heuristic techniques to solve a range of very hard combinatorial problems associated with military applications, we describe the application of genetic algorithms to the solution of the Radio Link Frequency Assignment Problem, (RLFAP). The RLFAP, as described here, is that of assigning frequencies, from a limited set of discrete frequency values, to a number of radio links in such a manner as to simultaneously satisfy a large number of constraints and use as few distinct frequencies as possible. This problem is known to be NP-complete. A range of new genetic operators is described. These were developed to overcome the high level of epistasis that occurs. Dynamically altering the priorities of the search also proved to be effective in improving the performance of the genetic algorithm, and two methods of effecting this are described. In addition, we comment on the ways and effectiveness with which ideas based on dedicated heuristics for this problem can be incorporated into the GA. Finally we describe a hybrid GA for the RLFAP and comment on the performance of the various approaches described. This work is being undertaken as part of the EUCLID (European Cooperation for the Long Term in Defence) CALMA Project — RTP 6.4 (Combinatorial Algorithms for Military Applications [Haje93]).

The core and nucleolus of games: A note on a paper by Göthe-Lundgren et al.

Abstract.In the paper “On the nucleolus of the basic vehicle routing game”, Mathematical Programming 72, 83–100 (1996), Göthe-Lundgren et al. develop a constraint generation method to compute the pre-nucleolus of a game. Their method assumes that constraints that are redundant in the representation of the core can be ignored in the computation of the pre-nucleolus. We provide an example that shows that for a game with an empty core such an assumption is, in general, not valid. Further, we show that a statement made by Göthe-Lundgren et al. about an intuitive interpretation of the pre-nucleolus is misleading.

The Multiscale Morphology Decomposition Theorem

Sieves decompose one dimensional bounded functions, e.g. f to a set of increasing scale granule functions {d r } r=1 R , that that represent the information in a manner that is analogous to the pyramid of wavelets obtained by linear decomposition. Sieves based on sequences of increasing scale open-closings with flat structuring elements (M and N filters) map f to {d r } and the inverse process maps {d r } to f. Experiments show that a more general inverse exists such that \( \left\{ {{{\hat d}_r}} \right\} \) maps to \( \hat f \) and back to \( \left\{ {{{\hat d}_r}} \right\} \), where the granule functions \( \left\{ {{{\hat d}_r}} \right\} \) are a subset of {d r } in which granules may have changed amplitudes, that may include zero but not a change of sign. An analytical proof of this inverse has been obtained. This key property could prove important for feature recognition and opens the way for an analysis of the noise resistance of these sieves. The resulting theorems do not apply to parallel open-closing filters.

The probe metaheuristic and its application to the multiconstraint knapsack problem

A new metaheuristic technique called PROBE is presented. The application of PROBE to the multiconstraint knapsack problem is described. Experimental results obtained using the resulting algorithm are compared with the results obtained by Chu and Beasley using a Genetic Algorithm.

A GRASP-based approach for the Pure Parsimony Haplotype Inference problem

The availability and study of haplotype data is of considerable interest to a wide range of areas including general health care, personalized medicine, and pharmacogenetics. The inner workings of contemporary sequencing techniques however imply that genotype data is generated from a chromosome rather than haplotype data. The reconstruction of the latter from this kind of data lies at the heart of the well studied Pure Parsimony Haplotype Inference problem (PPHI). In this paper, we present a proof of concept that a GRASP-based approach for solving PPHI has the potential of yielding an attractive tool that complements existing approaches. The usage of this strategy for solving PPHI is novel. To assess its suitability, we have implemented it in basic form in the novel and freely available HAPLOGRASP approach which we assessed in terms of simulated and real data. Our findings are highly encouraging.