A PSO based algorithm with an efficient optimal split procedure for the multiperiod vehicle routing problem with profit

Abstract

The multiperiod vehicle routing problem with profit (mVRPP) is a selective vehicle routing problem where the planning horizon of each vehicle is divided into several periods. The aim of solving mVRPP is to design service itineraries so that the total amount of collected profit is maximized and the travel time limit of each period is respected. This problem arises in many real life applications, as the one encountered in cash-in-transit industry. In this paper, we present a metaheuristic approach based on the particle swarm optimization algorithm (PSO) to solve the mVRPP. Our approach incorporates an efficient optimal split procedure and dedicated local search operators proposed to guarantee high search intensification. Experiments conducted on an mVRPP benchmark show that our algorithm outperforms the state of the art metaheuristic approaches in terms of performance and robustness. Our PSO algorithm determines all the already known optimal solutions within a negligible computational time and finds $88$ strict improvements among the $177$ instances of the benchmark.

A Appendix: Detailed results

In the Tables 5, 6, 7, 8, 9 and 10, we compare our proposed PSO algorithm with the other existing methods in the literature on a per-instance basis. In these tables, the columns Instance, n and m represent the name of the instance, the number of customers and the number of available vehicles, respectively. In column $Z_{best}$, the best score is retained for each instance. This score can either be the best score found by one of the two algorithms (MA of Zhang et al. (2013) or our PSO), or the best/optimal score found in the literature for the case of a single period per route (Boussier et al. 2007; Dang et al. 2013b; El-Hajj et al. 2016; Groër 2008). The two main columns in these tables show the results obtained by MA of Zhang et al. (2013) and our PSO algorithm, where we present the maximum score found for each method, the average one and the average cpu time in seconds obtained among the ten executions, which are denoted by $Z_{max}$, $Z_{avg}$ and CPU, respectively. We also added the RPG columns to compute the gap between the obtained solution and the best-known solution in the literature for each instance, using the following expression: $RPG = 100\times \frac{(Z_{best}-Z_{max})}{Z_{best}}$. At the end of each table, we present the average computational time for the corresponding set of instances and the average gap.

Table 7 Comparison with the literature on the first class ($D=4$)

Table 8 Comparison with the literature on the second class ($D=1$)

Table 9 Comparison with the literature on the second class ($D=2$)

Table 10 Comparison with the literature on the second class ($D=4$)

We mark the value of our $Z_{max}$ with a star each time the optimal solution is reached by our PSO, and in bold when we find an improvement with respect to the literature.