Michel Minoux

Networks synthesis and optimum network design problems: Models, solution methods and applications

This paper is intended as a survey in the area of network synthesis and optimum network design, which, in view of the importance and variety of the underlying applications, has attraced, since the early 1960s, much interest in the Operations Research community. Indeed, if the first models were studied in connection with telecommunication networks, the range of applications kept on getting broader and broader, including transportation networks, computer and teleprocessing networks, energy transport systems, water distribution networks, etc. However, beyond the apparent diversity of practical situations involved, most of these applications can be accounted for (modulo possibly a few minor adaptations), by a rather limited number of basic models. One of the main purposes of this paper is to provide the reader with a relevant classification of the area which will help him identify the fundamental structure of the problem (if any) he has to cope with, and relate it to already published work. In order to obtain a fairly good coverage of the matter, we have thus been led to identify three basic models aroung which the whole paper is organized: a general model using minimum cost multicommodity flows (Section 2); models in terms of tree-like networks (Section 3); models using nonsmultaneous single-commodity or multicommodity flows (Section 4). In each bcase the most important variants of the basic models have been surveyed with the purpose of providing as much information as possible concerning (a) the various contexts of applications from which the problem arose; (b) the main computational methods proposed in the literature for solving it, with emphasis on those techniques which appear at present, to be most efficient or promising.

A new approach for crew pairing problems by column generation with an application to air transportation

We propose a new approach to crew-pairing problems arising in the context of airline companies. The problem is first formulated as a large scale set covering problem with many colums, each column representing a valid crew-pairing. We then suggest a solution procedure for the continuous relaxation of this large scale problem, based on generalized linear programming, in which the column generation subproblem is shown to be equivalent to a shortest path problem in an associated graph. Computational results obtained on a series of real problems (involving up to 329 flight segments) are reported, confirming both computational efficiency and practical applicability of the new approach. Indeed not only were the resulting solutions observed to be integral for most test problems, but average savings of about 4 to 5% over the best available hand-built solutions were shown to be obtained.

LTUR: a simplified linear-time unit resolution algorithm for Horn formulae and computer implementation

Abstract Testing for the satisfiability of a Horn expression in propositional calculus is a fundamental problem in the area of logic programming for many reasons. Among these is the fact that the basic solution techniques for propositional Horn formulae have been shown to be central to the design of efficient interpreters for some predicate logic-based languages such as Hornlog (Gallier and Raatz, 1987). The present paper proposes a simplified way of deriving a linear-time algorithm avoiding many of the intricacies of previously known descriptions. In addition, a full, ready-to-use computer code is provided at the end of the paper together with a detailed analysis of the necessary data structures.

Graphs, dioids and semirings : new models and algorithms

Pre-Semirings, Semirings and Dioids.- Combinatorial Properties of (Pre)-Semirings.- Topology on Ordered Sets: Topological Dioids.- Solving Linear Systems in Dioids.- Linear Dependence and Independence in Semi-Modules and Moduloids.- Eigenvalues and Eigenvectors of Endomorphisms.- Dioids and Nonlinear Analysis.- Collected Examples of Monoids, (Pre)-Semirings and Dioids.

Exact solution of multicommodity network optimization problems with general step cost functions

We describe an exact solution procedure, based on the use of standard LP software, for multicommodity network optimization problems with general discontinuous step-increasing cost functions. This class of problems includes the so-called single-facility and multiple-facility capacitated network loading problems as special cases. The proposed procedure may be viewed as a specialization of the well-known BENDERS partitioning procedure, leading to iteratively solving an integer 0-1 linear programming relaxed subproblem which is progressively augmented through constraint generation. We propose an improved implementation of the constraint generation principle where, at each step, several (O(N)) new constraints are included into the current problem, thanks to which the total number of iterations is greatly reduced (never exceeding 15 in all the test problems treated). We report on systematic computational experiments for networks up to 20 nodes, 37 links and cost functions with an average six steps per link.

Accelerating Benders method using covering cut bundle generation

Over the years, various techniques have been proposed to speed up the classical Benders decomposition algorithm. The work presented in the literature has focused mainly on either reducing the number of iterations of the algorithm or on restricting the solution space of the decomposed problems. In this article, a new strategy for Benders algorithm is proposed and applied to two case studies in order to evaluate its efficiency. This strategy, referred to as covering cut bundle (CCB) generation, is shown to implement in a novel way the multiple constraints generation idea. The CCB generation is applied to mixed integer problems arising from two types of applications: the scheduling of crude oil and the scheduling problem for multi-product, multi-purpose batch plants. In both cases, CCB significantly decreases the number of iterations of the Benders method, leading to improved resolution times.

An optimal column-generation-with-ranking algorithm for very large scale set partitioning problems in traffic assignment

Abstract We show in this paper how branch-and-bound and column generation techniques can be conbined very efficiently to solve to optimality some very large scale set partitioning problems with special structure, such as the matrix decomposition problem arising in the context of satellite communication system optimization. The main contribution of our approach is the use of column generation during the tree search procedure, combined with a ranking procedure which ensures that the exact optimal integer solution is obtained. Extensive computational experiments are reported for the matrix decomposition problem encountered in the search for optimal schedules in satellite switching systems, showing the effectiveness of this approach.

Scheduling of loading and unloading of crude oil in a refinery using event-based discrete time formulation

One of the most critical activities in a refinery is the scheduling of loading and unloading of crude oil. Better analysis of this activity gives rise to better use of a systems resources, as well as control of the entire supply chain. It is important that the crude oil is loaded and unloaded contiguously, primarily for security reasons (e.g. possibility of system failures) but also to reduce the setup costs incurred when flow between a dock and a tank or between a tank and a crude distillation unit is reinitialized. The aim of the present paper is to develop an exact solution approach, widely applicable to most refineries where several modes of blending and several recipe preparation alternatives are used. A novel time formulation is proposed for the scheduling of the system under study called event-based time representation where the intervals are now based on events instead of hours.

Discrete Cost Multicommodity Network Optimization Problems and Exact Solution Methods

We first introduce a generic model for discrete cost multicommodity network optimization, together with several variants relevant to telecommunication networks such as: the case where discrete node cost functions (accounting for switching equipment) have to be included in the objective; the case where survivability constraints with respect to single-link and/or single-node failure have to be taken into account. An overview of existing exact solution methods is presented, both for special cases (such as the so-called single-facility and two-facility network loading problems) and for the general case where arbitrary step-increasing link cost-functions are considered. The basic discrete cost multicommodity flow problem (DCMCF) as well as its variant with survivability constraints (DCSMCF) are addressed. Several possible directions for improvement or future investigations are mentioned in the concluding section.

Efficient Greedy Heuristics For Steiner Tree Problems Using Reolptimization And Super Modularity

AbstractThe optimum Steiner Tree problem in a (nondirected) graph is known to belong to the class of NP-hard problems. However large scale instances (typically hundreds or thousands of nodes) arise in such important applications as optimal communication network design, or VLSI routing. There is therefore a strong need for efficient heuristics.We discuss in this paper efficient implementations of greedy-type procedures for approximate solution of s)Trimetric Steiner Tree problems with arbitrary nonnegative lengths on the edges. A special subclass of problems (meeting the so-called “nonadjacency condition”) is also introduced and studied. Various ways of improving the computational efficiency of the basic greedy algorithm are examined, among which the use of reoptimization procedures and proper exploitation of a supermodularity property of an associated set function; thanks to the latter the total number of minimum spanning tree computations can be significantly reduced according to the “accelerated greedy ...