Bees Traplining Metaphors for the Vehicle Routing Problem Using a Decomposition Approach

Abstract

In this study, the bees traplining metaphor was adopted for the Bees Algorithm (BA) and the Combinatorial Bees Algorithm (BA_C) and applied to solve the vehicle routing problem. The two-parameter Continuous and Combinatorial Bees Algorithms (BA₂ and BA_C2), equipped with a traplining metaphor intensifier, Bees Routing Optimiser (BRO), were used to solve the capacitated vehicle routing problem with a decomposition approach. In the first phase of the proposed method, the two-parameter Bees Algorithm (BA₂) was employed to solve the capacitated facility location problem, resulting in clusters of customers that did not violate the vehicles’ capacity. Then, BA_C2 combined with BRO was used to produce the routing plan for each cluster. BA₂ and BA_C2 implement the traplining foraging technique of bees, which integrates their exploratory and exploitative search mechanisms, to simplify parameter setting and use their threat avoidance tactics to intensify the solution. The results of comparisons with other BA versions indicate that the proposed algorithm improves the accuracy of the basic version by at least 4% while speeding it up fourfold.

Appendices

Appendix A: The Complete Routing Plan

The results of the decomposition methods presented in this chapter can be found in Tables 3, 4, 5 and 6.

Table 3 Routing plan result for A101 (BA₂ + BA_C2)

Table 4 Routing plan result for A101 (BA₂ + BA_C2 + BRO)

Table 5 Routing plan result for B101 (BA₂ + BA_C2)

Table 6 Routing plan result for B101 (BA₂ + BA_C2 + BRO)

Appendix B: Details of the Clustering Method

The pseudocode for solving the clustering problem with zero capacity violation is shown in Algorithm 2. It has been found that placing the initial cluster centroids in a circular band around the depot (Fig. 15) produced a more robust segmentation of customers than achieved through positioning the initial centroids completely randomly (Fig. 14).

Fig. 14

Clustering with random points initialisation

Fig. 15

Clustering with circular initial centroids (small “o” = initial; large “O” = final centroids)

The existence of clusters means that customers can reasonably be put into groups for which the total demand is less than or equal to the delivery vehicle’s capacity. The initial solutions obtained with customer clustering are better than those produced by random permutation of the whole set of customers.

Figure 16 shows examples of initial solutions with and without clustering. The initial path generated by random permutation is completely random and appears ‘chaotic’, whereas the clustering process limits the chaos within the individual clusters. The random permutation generator created initial solutions with approximately four times the errors achieved with the clustering technique. Due to the existence of good initial solutions, the algorithm was able to reach a near-optimal feasible solution to the routing plan more quickly than it would have done without them. Because the BA₂ for clustering was utilised inside the main BA_C2 initialisation, there will be n initial sets of clustered customers (Fig. 16).

Fig. 16

Initial routes a without and b with clustering on EilA101

Appendix C: Acronyms and Symbols

Tables 7 and 8 define the acronyms and symbols used in this chapter.

Table 7 Acronyms

Table 8 Symbols