A simple hyper-heuristic approach for a variant of many-to-many hub location-routing problem

Abstract

This paper addresses a variant of the many-to-many hub location-routing problem. Given an undirected edge-weighted complete graph $G = (V, E)$, this problem consists in finding a subset of V designated as hub nodes, partitioning all the nodes of V into cycles such that each cycle has exactly one hub node, and determining a Hamiltonian cycle on the subgraph induced by hub nodes. The objective is to minimize the total cost resulting from all these cycles. This problem is referred to as Many-to-Many p-Location-Hamiltonian Cycle Problem (MMpLHP) in this paper. To solve this problem, one has to deal with aspects of subset selection, grouping, and permutation. The characteristics of MMpLHP change according to the values of its constituent parameters. Hence, this problem can be regarded as a general problem which encompasses a diverse set of problems originating from different combinations of values of its constituent parameters. Such a general problem can be tackled effectively by suitably selecting and combining several different heuristics each of which cater to a different characteristic of the problem. Keeping this in mind, we have developed a simple multi-start hyper-heuristic approach for MMpLHP. Further, we have investigated two different selection mechanisms within the proposed approach. Experimental results and their analysis clearly demonstrate the superiority of our approach over best approaches known so far for this problem.

Appendix A

Tables 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31 and 32 of this appendix provide the instance-by-instance performance of HH_RAND and HH_GREEDY as well as the approaches of Lopes et al. (2016) on 24 different groups of instances. The detailed results of Exact, BRKGA, M-VND and LS approaches are taken from Lopes et al. (2016), where they reported only the best value among all the approaches (and not the individual best results for each approach). Further, %-deviation of Exact, BRKGA, M-VND and LS approaches from the best value is also reported. The results of M-CNS have been obtained from the corresponding author of Lopes et al. (2016) through personal communication, and these results were received in the same format as the reported results of other 4 approaches. From the best result and %-deviation information for each instance, we have calculated the cost for Exact, BRKGA, M-VND, M-CNS and LS approaches. Further, Lopes et al. (2016) did not report the execution times of Exact, BRKGA, M-VND, M-CNS and LS. Instead, time to reach the best solution is reported for each of these methods on each instance. Therefore, for Exact, BRKGA, M-VND, M-CNS and LS approaches, we can report the time till best only. We have decided to report the execution times of HH_RAND and HH_GREEDY as that is the standard practice in OR community and that will make future researchers working on this problem also to follow the standard practice. In Tables 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31 and 32, the first column lists the instance name where the digits at the end represent the number of nodes in that instance. For each approach, the column ‘cost’ reports the objective value obtained by that approach. The column ‘btime’ for Exact, BRKGA, M-VND, M-CNS and LS approaches reports the time in seconds to find best solution for that approach. The column ‘etime’ for HH_RAND and HH_GREEDY reports the execution time in seconds for that approach. All the times are actual times on respective machines without any multiplication factor to account for difference in processing speeds. Please also note that there is a time limit of 3600 s for all the approaches, however this condition is checked only once during an iteration, and hence, reported time can exceed 3600 s in some cases.

Table 9 Results of different approaches on small instances with ST scenario and $\alpha = 0.2$

Table 10 Results of different approaches on small instances with ST scenario and $\alpha = 0.4$

Table 11 Results of different approaches on small instances with ST scenario and $\alpha = 0.6$

Table 12 Results of different approaches on small instances with ST scenario and $\alpha = 0.8$

Table 13 Results of different approaches on small instances with SL scenario and $\alpha = 0.2$

Table 14 Results of different approaches on small instances with SL scenario and $\alpha = 0.4$

Table 15 Results of different approaches on small instances with SL scenario and $\alpha = 0.6$

Table 16 Results of different approaches on small instances with SL scenario and $\alpha = 0.8$

Table 17 Results of different approaches on small instances with SQ scenario and $\alpha = 0.2$

Table 18 Results of different approaches on small instances with SQ scenario and $\alpha = 0.4$

Table 19 Results of different approaches on small instances with SQ scenario and $\alpha = 0.6$

Table 20 Results of different approaches on small instances with SQ scenario and $\alpha = 0.8$

Table 21 Results of different approaches on large instances with ST scenario and $\alpha = 0.2$

Table 22 Results of different approaches on large instances with ST scenario and $\alpha = 0.4$

Table 23 Results of different approaches on large instances with ST scenario and $\alpha = 0.6$

Table 24 Results of different approaches on large instances with ST scenario and $\alpha = 0.8$

Table 25 Results of different approaches on large instances with SL scenario and $\alpha = 0.2$

Table 26 Results of different approaches on large instances with SL scenario and $\alpha = 0.4$

Table 27 Results of different approaches on large instances with SL scenario and $\alpha = 0.6$

Table 28 Results of different approaches on large instances with SL scenario and $\alpha = 0.8$

Table 29 Results of different approaches on large instances with SQ scenario and $\alpha = 0.2$

Table 30 Results of different approaches on large instances with SQ scenario and $\alpha = 0.4$

Table 31 Results of different approaches on large instances with SQ scenario and $\alpha = 0.6$

Table 32 Results of different approaches on large instances with SQ scenario and $\alpha = 0.8$