A three-phase matheuristic for capacitated multi-commodity fixed-cost network design with design-balance constraints

Abstract

This paper proposes a three-phase matheuristic solution strategy for the capacitated multi-commodity fixed-cost network design problem with design-balance constraints. The proposed matheuristic combines exact and neighbourhood-based methods. Tabu search and restricted path relinking meta-heuristics cooperate to generate as many feasible solutions as possible. The two meta-heuristics incorporate new neighbourhoods, and computationally efficient exploration procedures. The feasible solutions generated by the two procedures are then used to identify an appropriate part of the solution space where an exact solver intensifies the search. Computational experiments on benchmark instances show that the proposed algorithm finds good solutions to large-scale problems in a reasonable amount of time.

Appendix

Table 13 explains the column headers used in the subsequent tables.

Table 13 Notation used in tables of the Appendix

Table 14 lists the solutions for the instances used in the calibration phase.

Table 14 Best solutions for instances used in calibration

Tables 15 and 16 display the characteristics of the R and C instances in terms of number of nodes, arcs, and commodities, as well as cost and capacity ratios (see, e.g., Ghamlouche et al. 2003 for more detailed information). For the C instances, a high or low fixed cost relative to the routing cost is signaled by the letter F or V, respectively, while the letters T and L indicate, respectively, if the problem has a tight or loose capacity given the total demand. For the R instances, the fixed cost ratio is computed as $|\mathcal P |\sum _{(i,j)\in \mathcal A }f_{ij}/\sum _{p\in \mathcal P }\sum _{(i,j)\in \mathcal A }c_{ij}^{p},$ and the three values considered are F01 = 0.01, F05 = 0.05, and F10 = 0.10 corresponding to increasing levels of fixed costs compared to routing costs. The capacity ratio is computed as $|\mathcal A |\sum _{p\in \mathcal P }w^{p}/\sum _{(i,j)\in \mathcal A }u_{ij},$ and the values considered are C1 = 1, C2 = 2, and C8 = 8, indicating that the total capacity becomes increasingly tight relative to the total demand.

Table 15 Characteristics of R instances

Table 16 Characteristics of C instances

Tables 17 and 18 display the objective values for each method and the gaps between the proposed algorithm and the other methods for the C instances. Tables 19 and 20 display the same information for the R instances. Negative gaps indicate the proposed method improves over previous ones; some of these values (Columns “TS–PR/P-TS”) are considerable. The values in Columns “TS–PR/CPLEX 1 h” and “TS–PR/CPLEX 5 h” indicate that our algorithm is also competitive with the state-of-the-art solver. The maximum difference between the results of the proposed algorithm and CPLEX 5 h is less than 1.5 %, and many values are less than 0.5 %. This quality is obtained in less computational time than the MIP solver.

Table 17 Best solutions for C instances

Table 18 TS–PR improvement with respect to other methods for C instances

Table 19 Best solutions for R instances

Table 20 TS–PR improvement with respect to other methods for R instances

Tables 21 and 22 compare the performance of the proposed tabu search to that of the other methods. The tabu search phase finds feasible solutions for all instances, and improves over the state-of-the-art tabu search method 22 (of 24) C instances and 47 (of 54) R instances. The largest improvements are obtained for instances with high cost ratios (“F10” for R instances and “F” for C instances) because the feasibility phase always adds the arcs with the smallest total cost when satisfying the design-balance constraints, which has a more important impact when fixed costs are high.

Table 21 Improvement achieved by the tabu search procedure on C instances

Table 22 Improvement achieved by the tabu search procedure on R instances

Tables 23 and 24 give the number of arcs fixed by each type of intensification and CPLEX after 5 h, as well as comparative ratios. Column 5 compares the number of open arcs by the intensification procedure and CPLEX. The high values in this column indicate that the intensification phase yields good results, the solver having to add only a small number of arcs to satisfy all the constraints. Column 6 displays the ratio of the number of arcs fixed to closed and the number of arcs in the instance. When the number of arcs is large, closing unpromising arcs helps reduce the search space and the running time. We note that there are some instances for which the entries in this column column are relatively low (about 10–40 %). These are instances for which the number of arcs in the best solutions found by CPLEX is generally quite large (e.g., instances “R14,F01,C8” and “R15,F01,C8”) compared to the number of arcs in these instances, which indicates that most arcs are needed in the design. The values in the last two columns are the ratios of the open- and closed-intensification schemes compared to the tabu search methods indicating the good behaviour of the procedures.

Table 23 Open and closed arc statistics for C instances

Table 24 Open and closed arc statistics for R instances

Tables 25 and 26 display information relating to the execution of the proposed procedures. The second column gives the system imbalance for the solution obtained by the initialization step, and the third column reports the total running time. The last two columns give the number of solutions found by tabu search and path relinking phases.

Table 25 Statistical information execution C instances

Table 26 Statistical information execution R instances