Maurizio Boccia

Solving a fuel delivery problem by heuristic and exact approaches

Abstract In this paper we study the case of a company that delivers different types of fuel to a set of fuel pumps. The company has one warehouse and supplies the pumps by a fleet of trucks with several tanks of differing capacities. The company’s aim is to satisfy the orders using the available resources (trucks and drivers) with the minimum total travel cost for delivery. The problem has been formulated as a Set Partitioning model, solved by a Branch-and-Price algorithm. A fast combinatorial heuristic was adopted both to find a good feasible solution very quickly and to provide an initial set of columns for the Branch-and-Price algorithm. Computational results are reported. The exact approach shows low computation time for the real instances provided by the company.

A metaheuristic for a two echelon location-routing problem

In this paper we consider the design problem of a two-echelon freight distribution system. The aim is to define the structure of a system optimizing the location and the number of two different kinds of facilities, the size of two different vehicle fleets and the related routes on each echelon. The problem has been modeled as a two-echelon location-routing problem (2E-LRP). A tabu-search heuristic efficiently combining the composing subproblems is presented. Results on small, medium and large size instances are reported.

An effective heuristic for large-scale capacitated facility location problems

The Capacitated Facility Location Problem (CFLP) consists of locating a set of facilities with capacity constraints to satisfy the demands of a set of clients at the minimum cost. In this paper we propose a simple and effective heuristic for large-scale instances of CFLP. The heuristic is based on a Lagrangean relaxation which is used to select a subset of “promising” variables forming the core problem and on a Branch-and-Cut algorithm that solves the core problem. Computational results on very large scale instances (up to 4 million variables) are reported.

A computational study of exact knapsack separation for the generalized assignment problem

The Generalized Assignment Problem is a well-known NP-hard combinatorial optimization problem which consists of minimizing the assignment costs of a set of jobs to a set of machines satisfying capacity constraints. Most of the existing algorithms are of a Branch-and-Price type, with lower bounds computed through Dantzig–Wolfe reformulation and column generation.In this paper we propose a cutting plane algorithm working in the space of the variables of the basic formulation, whose core is an exact separation procedure for the knapsack polytopes induced by the capacity constraints. We show that an efficient implementation of the exact separation procedure allows to deal with large-scale instances and to solve to optimality several previously unsolved instances.

A Cut and Branch Approach for the Capacitated p-Median Problem Based on Fenchel Cutting Planes

The capacitated p-median problem (CPMP) consists of finding p nodes (the median nodes) minimizing the total distance to the other nodes of the graph, with the constraint that the total demand of the nodes assigned to each median does not exceed its given capacity. In this paper we propose a cutting plane algorithm, based on Fenchel cuts, which allows us to considerably reduce the integrality gap of hard CPMP instances. The formulation strengthened with Fenchel cuts is solved by a commercial MIP solver. Computational results show that this approach is effective in solving hard instances or considerably reducing their integrality gap.

A cutting plane algorithm for the capacitated facility location problem

Abstract The Capacitated Facility Location Problem (CFLP) is to locate a set of facilities with capacity constraints, to satisfy at the minimum cost the order-demands of a set of clients. A multi-source version of the problem is considered in which each client can be served by more than one facility. In this paper we present a reformulation of the CFLP based on Mixed Dicut Inequalities, a family of minimum knapsack inequalities of a mixed type, containing both binary and continuous (flow) variables. By aggregating flow variables, any Mixed Dicut Inequality turns into a binary minimum knapsack inequality with a single continuous variable. We will refer to the convex hull of the feasible solutions of this minimum knapsack problem as the Mixed Dicut polytope. We observe that the Mixed Dicut polytope is a rich source of valid inequalities for the CFLP: basic families of valid CFLP inequalities, like Variable Upper Bounds, Cover, Flow Cover and Effective Capacity Inequalities, are valid for the Mixed Dicut polytope. Furthermore we observe that new families of valid inequalities for the CFLP can be derived by the lifting procedures studied for the minimum knapsack problem with a single continuous variable. To deal with large-scale instances, we have developed a Branch-and-Cut-and-Price algorithm, where the separation algorithm consists of the complete enumeration of the facets of the Mixed Dicut polytope for a set of candidate Mixed Dicut Inequalities. We observe that our procedure returns inequalities that dominate most of the known classes of inequalities presented in the literature. We report on computational experience with instances up to 1000 facilities and 1000 clients to validate the approach.

A penalty function heuristic for the resource constrained shortest path problem

Abstract The resource constrained shortest path problem (RCSP) consists of finding the shortest path between two nodes of an assigned network, with the constraint that traversing an arc of the network implies the consumption of certain limited resources. In this paper we propose a new heuristic for the solution of the RCSP problem in medium and large scale networks. It is based on the extension to the discrete case of the penalty function heuristic approach for the fast e -approximate solution of difficult large-scale continuous linear programming problems. Computational experience on test instances has shown that the proposed penalty function heuristic (PFH) is very effective in the solution of medium and large scale RCSP instances. For all the tests reported it provides very good upper bounds (in many cases the optimal solution) in less than 26 iterations, where each iteration requires only the computation of a shortest path.

Near-Optimal Solutions of Large-Scale Single-Machine Scheduling Problems

The single-machine scheduling problem (SMSP) with release dates concerns the optimal allocation of a set of jobs on a single machine that is not able to process more than one job at a time. Each job is ready to be processed at a release date and without interruption. The goal is to minimize the total weighted completion time of the jobs. In this paper the time-indexed formulation is considered and a new lagrangean heuristic is proposed, based on the observation that lagrangean relaxation of job constraints leads to a weighted stable set problem on an interval graph. The relaxed problem is polynomially solvable by a dynamic-programming algorithm. We report computational experience, showing that instances up to 400 jobs and maximum processing time 50 (around 5,000,000 variables) are solved in less than 40 minutes on a personal computer, yielding duality gaps never exceeding 3%. We also test a set of hard instances, built to produce bad performances where we yield duality gaps less than 5%.

Resource Constrained Shortest Path Problems in Path Planning for Fleet Management

In the management and control of a vehicle fleet on a road network, the problem arises of finding the best route in relation to the mission of the fleet and to the typology of freight or users. Sometimes the route has to be adapted not only to current traffic conditions, but also to the physical, geometric and functional attributes of the roads, related to their urban location and environmental characteristics.This problem is approached here through an extension of the classic Shortest Path problem, named Resource Constrained Shortest Path problem (RCSP), where the resources are related to the road link attributes, assumed as relevant to the specific planning problem. A classification scheme of these attributes is first proposed and RCSP is described and reviewed. Next, a General Resource Constrained Shortest Path problem (GRCSP) is defined, which can be found in all cases where it is necessary to plan, statically or dynamically, the path of a vehicle and to respect a set of constraints expressed in terms of parameters and indices associated with the roads on the network. For this general problem a model has been formulated and a Branch and Cut solution approach is proposed. Computational results obtained on test and real networks during the experimentation of a fleet with low emission vehicles are also given.

A decomposition approach for a very large scale optimal diversity management problem

Abstract.This paper focuses on the solution of the optimal diversity management problem formulated as a p-Median problem. The problem is solved for very large scale real instances arising in the car industry and defined on a graph with several tens of thousands of nodes and with several millions of arcs. The particularity is that the graph can consist of several non connected components. This property is used to decompose the problem into a series of p-Median subproblems of a smaller dimension. We use a greedy heuristic and a Lagrangian heuristic for each subproblem. The solution of the whole problem is obtained by solving a suitable assignment problem using a Branch-and-Bound algorithm.