Fixed-charge Multicommodity Network Design Research Articles

Directed fixed charge multicommodity network design: A cutting plane approach using polar duality

We present an efficient cutting-plane based approach to exactly solve a directed fixed charge network design (DFCND) problem, wherein the valid inequalities to the problem are generated using the polar duality approach. The biggest challenge in using this approach arises in constructing the polar dual of the problem. This would require enumerating all the extreme points of the convex hull of DFCND, which is computationally impractical for any instance of a reasonable size. Moreover, the resulting polar dual would be too large to solve efficiently, which is required at every iteration of the cutting-plane algorithm. The novelty of our solution approach lies in suggesting a way to circumvent this challenge by instead generating the violated facets, using polar duality, of the smaller substructures involving only a small subset of constraints and variables, obtained from 2-, 3-and 4-partitions of the underlying graph. For problem instances based on sparse graphs with zero flow costs, addition of these inequalities closes more than 20% of the optimality gap remaining after the addition of the knapsack cover inequalities used in the literature. This allows us to solve the problem instances in less than 400 s, on average, which otherwise take around 1000 s with the addition of only the knapsack cover inequalities, and around 4 hours for the Cplex MIP solver at its default setting.

The transit time constrained fixed charge multi-commodity network design problem

This paper introduces the transit time constrained fixed charge multi-commodity network design problem. Transit times are origin-to-destination time limits for the commodities, which appear for example in transport systems with perishable goods. We discuss how to model the problem and present three different formulations of it. The first formulation is an exponential size path formulation, which we solve with a branch-and-price algorithm. Several speed up techniques from the literature on fixed charge multi-commodity network design problems are implemented, such as lifted cover inequalities and the recently proposed deep dual-optimal inequalities. In an extensive set of computational experiments, we show that these inequalities significantly improve the performance of the algorithm. The other two formulations are of polynomial size: one uses path indices and the other uses time indices. While the branch-and-price algorithm outperforms solving the compact formulations with a general-purpose mixed-integer programming solver, the study of compact models helps better understand the problem, and we can use them as benchmarks. A detailed sensitivity analysis of the branch-and-price algorithm shows that longer transit times and an increased ratio of fixed charge to flow cost increase the difficulty of solving the problem whereas the arc capacity has less impact. We further discuss in-depth implementational details.

Profit-oriented fixed-charge network design with elastic demand

This paper extends classic fixed-charge multicommodity network design by explicitly considering demand elasticity with respect to routing cost in a profit maximization context with service commitments. Demand quantity is determined by a spatial interaction model that accounts for routing costs, thus capturing the trade-off between infrastructure investment, efficient routing, and increased revenue. A numerical example is presented to demonstrate the added value of incorporating demand elasticity in profit-oriented network design problems. An arc-based and a path-based formulation, both with the flexibility of incorporating O/D pair selection by means of network and data transformations, are presented. The arc-based formulation is solved using state-of-the-art global optimization software while the path-based formulation serves as the basis for a hybrid matheuristic that combines a slope scaling metaheuristic and column generation. Computational experience shows the hybrid matheuristic to be superior in terms of solution quality and computation time.

Application of Benders decomposition method in solution of a fixed-charge multicommodity network design problem avoiding congestion

In this paper, a new variable partitioning strategy in Benders decomposition method is applied that enables us to deal with a wide class of mixed-integer nonlinear programming problems including fixed-charge multicommodity network design (FMND) problems under congestion effects. It is proved that the proposed technique for an FMND problem leads to a simple branch-and-bound algorithm such that each node of the branching tree includes a single conic quadratic representable problem consisting of only continuous (flow) variables. Preliminary numerical results are reported.

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.

The congested multicommodity network design problem

This paper studies a version of the fixed-charge multicommodity network design problem where in addition to the traditional costs of flow and design, congestion at nodes is explicitly considered. The problem is initially modeled as a nonlinear integer programming formulation and two solution approaches are proposed: (i) a reformulation of the problem as a mixed integer second order cone program to optimally solve the problem for small to medium scale problem instances, and (ii) an evolutionary algorithm using elements of iterated local search and scatter search to provide upper bounds. Extensive computational results on new benchmark problem instances and on real case data are presented.

A Simplex-based simulated annealing algorithm for node-arc capacitated multicommodity network design

The network design is a well-known problem, both of practical and theoretical significance. Network design models are extensively used to represent a wide range of planning and operations management issues in transportation, telecommunications, logistics, production and distribution. This paper presents a solution method for node-arc formulation of capacitated fixed-charge multicommodity network design problems. The proposed method is a hybrid algorithm of Simplex method and simulated annealing metaheuristic. The basic idea of the proposed algorithm is to use a simulated annealing algorithm to explore the solution space, where the revised Simplex method is used to evaluate, select and implement the moves. In the proposed algorithm, the neighborhood structure is pivoting rules of the Simplex method that provide an efficient way to reach the neighbors of current solution. To evaluate the proposed algorithm, the standard problems with different sizes are used. The algorithm parameters are tuned by design of experiments approach and the most appropriate values for the parameters are adjusted. The performance of the proposed algorithm is evaluated by statistical analysis. The results show high efficiency and effectiveness of the proposed algorithm.