Laura Bahiense

The volume algorithm revisited: relation with bundle methods

Abstract. We revise the Volume Algorithm (VA) for linear programming and relate it to bundle methods. When first introduced, VA was presented as a subgradient-like method for solving the original problem in its dual form. In a way similar to the serious/null steps philosophy of bundle methods, VA produces green, yellow or red steps. In order to give convergence results, we introduce in VA a precise measure for the improvement needed to declare a green or serious step. This addition yields a revised formulation (RVA) that is halfway between VA and a specific bundle method, that we call BVA. We analyze the convergence properties of both RVA and BVA. Finally, we compare the performance of the modified algorithms versus VA on a set of Rectilinear Steiner problems of various sizes and increasing complexity, derived from real world VLSI design instances.

Solving Steiner Tree Problems in Graphs with Lagrangian Relaxation

This paper presents an algorithm to obtain near optimal solutions for the Steiner tree problem in graphs. It is based on a Lagrangian relaxation of a multi-commodity flow formulation of the problem. An extension of the subgradient algorithm, the volume algorithm, has been used to obtain lower bounds and to estimate primal solutions. It was possible to solve several difficult instances from the literature to proven optimality without branching. Computational results are reported for problems drawn from the SteinLib library.

A branch-and-cut algorithm for the equitable coloring problem using a formulation by representatives

An equitable k-coloring of a graph is defined by a partition of its vertices into k disjoint stable subsets, such that the difference between the cardinalities of any two subsets is at most one. The equitable coloring problem consists of finding the minimum value of k such that a given graph can be equitably k-colored. We present two new integer programming formulations based on representatives for the equitable coloring problem. We propose a primal constructive heuristic, branching strategies, and the first branch-and-cut algorithm in the literature of the equitable coloring problem. The computational experiments were carried out on randomly generated graphs, DIMACS graphs, and other graphs from the literature.

Scheduling a multi-product pipeline network

Abstract Daily some millions barrels of oil are moved around the world in imports and exports and domestically within countries. While ships are the main mode for intercontinental transport, pipelines are the chief form of transcontinental transport, while regional and local transports is performed by trains and trucks. Despite high installation costs, pipelines are considered highly efficient as a mode for transporting large amounts of oil and oil products over long distances, because they offer lower operation costs, higher reliability rates, lower product loss rates, less environmental impact, and less susceptibility to adverse weather conditions than other modes. This study deals with a multi-product pipeline system that transports a set of oil products (diesel, gasoline and kerosene, for example), which have to be moved from points (operating areas) where they are produced or stored (refineries, terminals) to points where they are needed (other refineries, distribution centers, terminals, ports, customers) through a pipeline or set of pipelines. The present study contributes primarily by offering an efficient tool for the problem of scheduling multi-product pipeline networks. The methodology proposed takes the approach of discretizing both pipelines and planning horizon and combines an efficient MILP model with a post-processing heuristic. When compared with previous models, we propose a more efficient one in which the set of volumetric constraints is modeled in the form of knapsack cascading constraints and constraints on products in pipeline sections, which made for significantly improved performance in the experiments that were conducted. The proposed methodology thus constitutes an advance in terms of modeling the problem, making it feasible to solve problems increasingly close to the realities confronting oil industry operators.

The dynamic space allocation problem

This work is devoted to the Dynamic Space Allocation Problem, where project duration is divided into a number of consecutive periods, each of them associated with a number of activities. The resources required by the activities have to be available in the corresponding workspaces and those sitting idle during a period have to be stored. This problem contains the Quadratic Assignment Problem (QAP) as a particular case, which puts it in the NP-hard class. In this context, the difficulty of identifying optimal solutions, even for instances of medium size, justifies the use of heuristic techniques. This work proposes a construction and a hybrid algorithm (HGT) based on the GRASP and Tabu search metaheuristics. Comparisons are presented for values obtained by HGT, pure GRASP versions, Tabu search and literature results. Computational results show the proposed methods to be competitive in relation to instances in the literature and to existing techniques.

Logistics network planning for offshore air transport of oil rig crews

Abstract Oil discoveries of recent years, especially in the pre-salt Santos Basin, reflect a large increase in petroleum exploration and production in Brazil. Accordingly, drilling rig and production platform crew transport demands will increase. This transport will also become more complex as average distance between fields and Brazil’s coast increases. The helicopter, the modal most used for this purpose, is the most efficient means of transport in terms of speed and safety, but also entails high costs. Optimizing the crew transport logistics network thus becomes an economically significant issue. The study presents an optimization model for crew transport logistics network planning. That model aims to provide managers with accurate information to assist their decision making in logistics infrastructure planning. Such decisions involve airfield locations, distribution of demand among airfields and fleet profile. Since composing the fleet involves considerable expenditures, and once made, this composition is not easily changed, we built several scenarios varying in demand and fleet costs to evaluate the behavior of the model we are proposing as regards processing time and quality of the solution. We have obtained good results, despite the increasing complexity of the scenarios.

The maximum common edge subgraph problem: A polyhedral investigation

In the Maximum Common Edge Subgraph Problem (MCES), given two graphs G and H with the same number of vertices, one has to find a common subgraph of G and H (not necessarily induced) with the maximum number of edges. This problem arises in parallel programming environments, and was first defined in Bokhari (1981) [2]. This paper presents a new integer programming formulation for the MCES and a polyhedral study of this model. Several classes of valid inequalities are identified, most of which are shown to define facets. These findings were incorporated into a branch&cut algorithm we implemented. Experimental results with this algorithm are reported.

A New Lagrangian Based Branch and Bound Algorithm for the 0-1 Knapsack Problem

Abstract This paper describes a new Branch and Bound algorithm for the 0-1 Knapsack Problem (KP). The algorithm is based on the use of a Lagrangean Relax-and-Cut procedure that allows exponentially many Fractional Gomory Cuts and Extended Cover Inequalities to be candidates to Lagrangean dualization. In doing so, the upper bounds thus obtained are stronger than the standard Linear Programming relaxation bound for KP. The algorithm is aimed at solving instances with coefficients as large as 1015, a class of KP instance for which existing solution algorithms might not be directly applicable.

A Branch-and-Cut Algorithm for Equitable Coloring based on a Formulation by Representatives

Let G = (V,E) be an undirected graph, where V is the set of vertices and E is that of edges. An equitable k-coloring of G is a partition of V into k disjoint stable subsets such that the difference on the cardinalities of any two subsets is at most one. Each subset is associated with a color and called a color set. The Equitable Coloring Problem (ECP) consists of finding the minimum value of k such that there is an equitable k-coloring of G. This number is said to be the equitable chromatic number of G and it is denoted by χ=(G). The equitable coloring problem was first introduced in [7], motivated by an application to municipal garbage collection [9]. It was proved to be NPhard in [5]. A branch-and-cut algorithm based on an integer programming

Long-term integrated surgery room optimization and recovery ward planning, with a case study in the Brazilian National Institute of Traumatology and Orthopedics (INTO)

Abstract This paper proposes an integrated approach for the long-term planning and surgery allocation problem with downstream constraints. It is motivated by a case study in the Brazilian National Institute of Traumatology and Orthopedics, which provides elective high complexity surgeries for patients from the Brazilian public health system. We introduce an optimization problem that designs a periodic surgery allocation schedule as well as a recovery ward utilization plan, with a view at balancing patient arrivals and releases in the long term, in such a way that all surgeries are performed in a timely manner.