Fixed-charge Capacitated Multicommodity Network Design Research Articles

Node-based Lagrangian relaxations for multicommodity capacitated fixed-charge network design

Classical Lagrangian relaxations for the multicommodity capacitated fixed-charge network design problem are the so-called flow and knapsack relaxations, where the resulting Lagrangian subproblems decompose by commodities and by arcs, respectively. We introduce node-based Lagrangian relaxations, where the resulting Lagrangian subproblem decomposes by nodes. We show that the Lagrangian dual bounds of these relaxations improve upon the linear programming relaxation bound, known to be equal to the Lagrangian dual bounds for the flow and knapsack relaxations. We also develop a Lagrangian matheuristic to compute upper bounds. The computational results on a set of benchmark instances show that the Lagrangian matheuristic is competitive with the state-of-the-art heuristics from the literature.

A Learning-Based Matheuristic for Stochastic Multicommodity Network Design

This paper proposes a solution approach for the multicommodity capacitated fixed-charge network design problem with uncertain demand modeled as a two-stage stochastic program. The proposed learning-based matheuristic combines heuristic search techniques with mathematical programming. It provides a systematic approach to identifying structures of good-quality solutions by gradually considering scenarios and their influences on design decisions. Extensive computational experiments illustrate the efficiency of the proposed matheuristic in obtaining high-quality solutions with limited computational efforts.

MIP Neighborhood Search Heuristics for a Capacitated Fixed-Charge Network Design Problem

The fixed-charge capacitated multicommodity network design problem is a fundamental optimization problem arising in many network configurations. The solution of the problem provides an appropriate network design as well as routes of multicommodity flows aimed at minimizing the total cost, which is the sum of the flow costs and fixed-charge costs over a network with limited arc capacities. In the present paper, we introduce a combined approach with a capacity scaling procedure for finding an initial feasible solution and an MIP neighborhood search for improving the solutions. Besides, we modify the procedure for application to large-scale problems. Computational experiments demonstrate the effectiveness of the proposed approach, and high-quality solutions are obtained for two problem sets from the literature.

Cell-Based Local Search Heuristics for Guide Path Design of Automated Guided Vehicle Systems With Dynamic Multicommodity Flow

This article discusses the guide path design of automated guided vehicle (AGV) systems for which we propose a model that incorporates a dynamic multicommodity flow with capacity constraints from a pickup to a delivery point. The problem is formulated as the selection of a guide path connecting a given set of pickup/delivery points. A cell-based local search that is based on an original neighborhood technique is developed. In the proposed method, the guide path is effectively optimized based on a local search of cell-based neighborhood search preserving connectivity constraints and the subsequent solution of a dynamic multicommodity flow problem. The performance of the proposed method is strengthened by the redundant elimination of arcs and intensification of search according to the concept of the flow-concentrated cell. The effectiveness of the proposed method is demonstrated by comparing it with a general-purpose solver and recent algorithms for fixed-charge capacitated multicommodity network design problems. A real case study is presented to demonstrate the applicability of the proposed method by using simulation software. Note to Practitioners —Due to the recent growth of online shipping, the design of automated warehousing systems is receiving much attention. Most automated warehousing systems introduce multiple automated guided vehicles (AGVs) for transportation. This article presents an efficient optimization of guide path design considering congestions and dynamic multicommodity flow routing. We develop cell-based local search heuristics for the guide path design problem for AGV systems that can be applied to a large-scale transportation model to solve the problem efficiently. Our major finding is that the guide path design with the dynamic multicommodity flow can significantly reduce the delays caused by congestions and improve the efficiency of the warehousing systems. The effectiveness of the derived guide path design is evaluated by simulation software. The feasibility of the guide path is confirmed in several case studies.

Matheuristics based on iterative linear programming and slope scaling for multicommodity capacitated fixed charge network design

We present new matheuristics for the multicommodity capacitated fixed-charge network design problem (MCND). The matheuristics are based on combining iterative linear programming (ILP) methods and slope scaling (SS) heuristics. Each iteration alternates between solving a linear program obtained by adding pseudo-cuts and a restricted mixed-integer programming (MIP) model. The SS heuristic is used as a warm start to a state-of-the-art generic method that solves the restricted MIP model. The resulting ILP/SS matheuristics are compared against state-of-the-art heuristics for the MCND on a set of large-scale difficult instances. The computational results show that the approach is competitive: when performed for a time limit of 1 hour, it finds more best solutions than any other heuristic, using comparable running times; when performed for a time limit of 5 hours, it identifies an optimal solution for each instance for which an optimal solution is known and it is able to find new best solutions for some very hard instances.

Commodity Representations and Cut-Set-Based Inequalities for Multicommodity Capacitated Fixed-Charge Network Design

We improve the mixed-integer programming formulation of the multicommodity capacitated fixed-charge network design problem by incorporating valid inequalities into a cutting-plane algorithm. We use five classes of known valid inequalities: the strong, cover, minimum cardinality, flow cover, and flow pack inequalities. The first class is particularly useful when a disaggregated representation of the commodities is chosen, and the last four are expressed in terms of network cut sets. We develop efficient separation and lifting procedures for these classes of inequalities. We present computational results on a large set of instances of various characteristics, allowing us to measure the impact of the different classes of valid inequalities on the quality of the lower bounds, in particular with respect to the representation of the commodities.

A cycle-based evolutionary algorithm for the fixed-charge capacitated multi-commodity network design problem

This paper presents an evolutionary algorithm for the fixed-charge multicommodity network design problem (MCNDP), which concerns routing multiple commodities from origins to destinations by designing a network through selecting arcs, with an objective of minimizing the fixed costs of the selected arcs plus the variable costs of the flows on each arc. The proposed algorithm evolves a pool of solutions using principles of scatter search, interlinked with an iterated local search as an improvement method. New cycle-based neighborhood operators are presented which enable complete or partial re-routing of multiple commodities. An efficient perturbation strategy, inspired by ejection chains, is introduced to perform local compound cycle-based moves to explore different parts of the solution space. The algorithm also allows infeasible solutions violating arc capacities while performing the “ejection cycles”, and subsequently restores feasibility by systematically applying correction moves. Computational experiments on benchmark MCNDP instances show that the proposed solution method consistently produces high-quality solutions in reasonable computational times.

A Genetic Algorithm Based on Relaxation Induced Neighborhood Search in a Local Branching Framework for Capacitated Multicommodity Network Design

The fixed-charge Capacitated Multicommodity Network Design (CMND) is a well-known problem of both practical and theoretical significance. This article proposes the Genetic Algorithm (GA) cooperative Relaxation Induced Neighborhood Search (RINS) in a Local Branching (LB) framework for CMND problem. GA algorithm is started by initial population which is made by two parents obtain from hybrid LB and RINS algorithms. The basic idea of the proposed solution method is to use the GA algorithm to explore the search space and the hybrid LB and RINS methods to move from current solution to neighbor solution. Adapting the metaheuristic algorithm with RINS method to fit within an LB framework represents an interesting challenge. To evaluate the proposed algorithm, the standard problems with different sizes are used. The parameters of the algorithm are tuned by design of experiments. In order to prove the efficiency and effectiveness of the proposed algorithm, the results are compared with the best results available in the literature. The statistical analysis shows high performance of the proposed algorithm.

A hybrid solution method for fuzzy train formation planning

The train formation plan (TFP) determines the routing and frequency of trains, and assigns the demands to trains. In this paper, in order to consider the real-life condition of railways, a mathematical model with fuzzy costs is proposed for train formation planning in Iranian railway. In this fuzzy model, the costs are considered in three scenarios, namely optimistic, normal and pessimistic. The model is formulated based on the fixed-charge capacitated multicommodity network design problem. Since the TFP problem is NP-hard, an efficient hybrid algorithm combining local branching and relaxation induced neighborhood search methods is presented. A three-step method is applied for parameter tuning using design of experiments approach. To evaluate the efficiency and effectiveness of the proposed algorithm, the results are compared with those of the state-of-the-art optimization software.

A Cutting-Plane Neighborhood Structure for Fixed-Charge Capacitated Multicommodity Network Design Problem

In this paper, a cutting-plane neighborhood structure is proposed for the fixed-charge capacitated multicommodity network design (CMND) problem. In the proposed structure, different strategies are used to select an open arc in the incumbent solution to be closed. Then a linear programming (LP) model is generated on the basis of the modified incumbent solution by relaxing binary variables and adding new constraints. The generated LP solution is improved using different cutting-plane inequalities. Subsequently, a new sub-mixed integer programming (MIP) model is created by fixing a number of variables in the generated LP solution. Then the local branching algorithm is used to solve the sub-MIP model and its solution is considered as a neighboring solution. A tabu search algorithm is used to evaluate the proposed neighborhood structure. To tune the parameters of the tabu search algorithm, we have used the design of experiments method. Standard problems with different sizes are employed to evaluate the proposed tabu search algorithm. Results show the efficiency and effectiveness of the tabu search algorithm compared to the best methods found in the literature.

Formulations for the nonbifurcated hop-constrained multicommodity capacitated fixed-charge network design problem

This paper addresses the multicommodity capacitated fixed-charge network design problem with nonbifurcated flows and hop constraints. We present and compare mathematical programming formulations for this problem and we study different relaxations: Lagrangean relaxations, linear programming relaxations, and partial relaxations of the integrality constraints. In particular, we show that the Lagrangean bound obtained by relaxing the flow conservation equations is tighter than the linear programming relaxation bound. We present computational results on a large set of randomly generated instances.

Branch-and-price-and-cut for large-scale multicommodity capacitated fixed-charge network design

We present a branch-and-price-and-cut algorithm for solving large-scale instances of the multicommodity capacitated fixed-charge network design problem. We assume good feasible solutions are already known and we focus on an efficient algorithm for proving the optimality of the solutions. The restricted master problem solved at each column generation iteration is obtained directly from the compact arc-based model by considering only a subset of the commodity flow variables. The pricing subproblem corresponds to a Lagrangian relaxation of the flow conservation and capacity constraints, leaving in the Lagrangian subproblem only the strong inequalities. The column generation procedure is completed by a cut generation step based on strong inequalities. The resulting column-and-row generation procedure is embedded within an enumerative scheme. Computational experiments on a large set of randomly generated instances are presented and analyzed.

Path Relinking, Cycle-Based Neighbourhoods and Capacitated Multicommodity Network Design

In this paper, we propose a path relinking procedure for the fixed-charge capacitated multicommodity network design problem. Cycle-based neighbourhoods are used both to move along paths between elite solutions and to generate the elite candidate set by a tabu-like local search procedure. Several variants of the method are implemented and compared. Extensive computational experiments indicate that the path relinking procedure offers excellent results. It systematically outperforms the cycle-based tabu search method in both solution quality and computational effort and offers the best current meta-heuristic for this difficult class of problems.

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.

Cycle-Based Neighbourhoods for Fixed-Charge Capacitated Multicommodity Network Design

We propose new cycle-based neighbourhood structures for metaheuristics aimed at the fixed-charge capacitated multicommodity network design formulation. The neighbourhood defines moves that explicitly take into account the impact on the total design cost of potential modifications to the flow distribution of several commodities simultaneously. Moves are identified through a shortest-pathlike network optimization procedure and proceed by redirecting flow around cycles and closing and opening design arcs accordingly. These neighbourhoods are evaluated and tested within a simple tabu search algorithm. Experimental results show that the proposed approach is quite powerful and outperforms existing methods reported in the literature.

Revue Des Inégalités Valides Pertinentes Aux Problèmes Des Conception De Réseaux

The objective of this paper is to present a survey of relevant valid inequalities for the multi-commodity capacitated fixed-charge network design problem. We present first a review of the relevant literature on relaxations and heuristic approaches. We observe that these approaches alone are not sufficient to solve large-scale instances of the problem. However, a combination or these approaches and polyhedral methods appears very promising. Then, we present three classes of relevant valid inequalities for the problem that have been studied for related problems. We present as well experimental results on the general implementations of some of these inequalities in a state-of-the-art commercial software. These experiments point towards the necessity of profound polyhedral studies in order to solve efficiently large-scale instances of the problem.