Bernard Gendron

Parallel Branch-and-Branch Algorithms: Survey and Synthesis

We present a detailed and up-to-date survey of the literature on parallel branch-and-bound algorithms. We synthesize previous work in this area and propose a new classification of parallel branch-and-bound algorithms. This classification is used to analyze the methods proposed in the literature. To facilitate our analysis, we give a new characterization of branch-and-bound algorithms, which consists of isolating the performed operations without specifying any particular order for their execution.

Bundle-based relaxation methods for multicommodity capacitated fixed charge network design

Abstract To efficiently derive bounds for large-scale instances of the capacitated fixed-charge network design problem, Lagrangian relaxations appear promising. This paper presents the results of comprehensive experiments aimed at calibrating and comparing bundle and subgradient methods applied to the optimization of Lagrangian duals arising from two Lagrangian relaxations. This study substantiates the fact that bundle methods appear superior to subgradient approches because they converge faster and are more robust relative to different relaxations, problem characteristics, and selection of the initial parameter values. It also demonstrates that effective lower bounds may be computed efficiently for large-scale instances of the capacitated fixed-charge network design problem. Indeed, in a fraction of the time required by a standard simplex approach to solve the linear programming relaxation, the methods we present attain very high-quality solutions.

Multicommodity Capacitated Network Design

Network design models have wide applications in telecommunications and transportation planning; see, for example, the survey articles by Magnanti and Wong (1984), Minoux (1989), Chapter 16 of the book by Ahuja, Magnanti and Orlin (1993), Section 13 of Ahuja et al. (1995). In particular, Gavish (1991) and Balakrishnan et al. (1991) present reviews of important applications in telecommunications. In many of these applications, it is required to send flows (which may be fractional) to satisfy demands given arcs with existing capacities, or to install, in discrete amounts, additional facilities with fixed capacities. In doing so, one pays a price not only for routing flows, but also for using an arc or installing additional facilities. The objective is then to determine the optimal amounts of flows to be routed and the facilities to be installed.

A Comparison of Mixed-Integer Programming Models for Nonconvex Piecewise Linear Cost Minimization Problems

We study a generic minimization problem with separable nonconvex piecewise linear costs, showing that the linear programming (LP) relaxation of three textbook mixed-integer programming formulations each approximates the cost function by its lower convex envelope. We also show a relationship between this result and classical Lagrangian duality theory.

A mathematical programming approach for scheduling physicians in the emergency room

Preparing a schedule for physicians in the emergency room is a complex task, which requires taking into account a large number of (often conflicting) rules, related to various aspects: limits on the number of consecutive shifts or weekly hours, special rules for night shifts and weekends, seniority rules, vacation periods, individual preferences, ... In this paper, we present a mathematical programming approach to facilitate this task. The approach models the situation in a major hospital of the Montréal region (approximately 20 physicians are members of the working staff). We show that the approach can significantly reduce the time and the effort required to construct a six-month schedule. A human expert, member of the working staff, typically requires a whole dedicated week to perform this task, with the help of a spreadsheet. With our approach, a schedule can be completed in less than one day. Our approach also generates better schedules than those produced by the expert, because it can take into account simultaneously more rules than any human expert can do.

0-1 reformulations of the multicommodity capacitated network design problem

We study 0-1 reformulations of the multicommodity capacitated network design problem, which is usually modeled with general integer variables to represent design decisions on the number of facilities to install on each arc of the network. The reformulations are based on the multiple choice model, a generic approach to represent piecewise linear costs using 0-1 variables. This model is improved by the addition of extended linking inequalities, derived from variable disaggregation techniques. We show that these extended linking inequalities for the 0-1 model are equivalent to the residual capacity inequalities, a class of valid inequalities derived for the model with general integer variables. In this paper, we compare two cutting-plane algorithms to compute the same lower bound on the optimal value of the problem: one based on the generation of residual capacity inequalities within the model with general integer variables, and the other based on the addition of extended linking inequalities to the 0-1 reformulation. To further improve the computational results of the latter approach, we develop a column-and-row generation approach; the resulting algorithm is shown to be competitive with the approach relying on residual capacity inequalities.

Models and Methods for Merge-in-Transit Operations

We develop integer programming formulations and solution methods for addressing operational issues in merge-in-transit distribution systems. The models account for various complex problem features, including the integration of inventory and transportation decisions, the dynamic and multimodal components of the application, and the nonconvex piecewise linear structure of the cost functions. To accurately model the cost functions, we introduce disaggregation techniques that allow us to derive a hierarchy of linear programming relaxations. To solve these relaxations, we propose a cutting-plane procedure that combines constraint and variable generation with rounding and branch-and-bound heuristics. We demonstrate the effectiveness of this approach on a large set of test problems with instances derived from actual data from the computer industry that contain almost 500,000 integer variables.

A Slope Scaling/Lagrangean Perturbation Heuristic with Long-Term Memory for Multicommodity Capacitated Fixed-Charge Network Design

This paper describes a slope scaling heuristic for solving the multicomodity capacitated fixed-charge network design problem. The heuristic integrates a Lagrangean perturbation scheme and intensification/diversification mechanisms based on a long-term memory. Although the impact of the Lagrangean perturbation mechanism on the performance of the method is minor, the intensification/diversification components of the algorithm are essential for the approach to achieve good performance. The computational results on a large set of randomly generated instances from the literature show that the proposed method is competitive with the best known heuristic approaches for the problem. Moreover, it generally provides better solutions on larger, more difficult, instances.

Benders, metric and cutset inequalities for multicommodity capacitated network design

Abstract Solving multicommodity capacitated network design problems is a hard task that requires the use of several strategies like relaxing some constraints and strengthening the model with valid inequalities. In this paper, we compare three sets of inequalities that have been widely used in this context: Benders, metric and cutset inequalities. We show that Benders inequalities associated to extreme rays are metric inequalities. We also show how to strengthen Benders inequalities associated to non-extreme rays to obtain metric inequalities. We show that cutset inequalities are Benders inequalities, but not necessarily metric inequalities. We give a necessary and sufficient condition for a cutset inequality to be a metric inequality. Computational experiments show the effectiveness of strengthening Benders and cutset inequalities to obtain metric inequalities.

Formulations and relaxations for a multi-echelon capacitated location-distribution problem

We consider a multi-echelon location-distribution problem arising from an actual application in fast delivery service. We present and compare two formulations for this problem: an arc-based model and a path-based model. We show that the linear programming (LP) relaxation of the path-based model provides a better bound than the LP relaxation of the arc-based model. We also compare the so-called binary relaxations of the models, which are obtained by relaxing the integrality constraints for the general integer variables, but not for the 0-1 variables. We show that the binary relaxations of the two models always provide the same bound, but that the path-based binary relaxation appears preferable from a computational point of view, since it can be reformulated as an equivalent simple plant location problem (SPLP), for which several efficient algorithms exist. We also show that the LP relaxation of this SPLP reformulation provides a better bound than the LP relaxation of the path-based model.