# A Swarm Intelligence Solution for the Multi-Vehicle Profitable Pickup and Delivery Problem
## 4 Proposed Method
### ABC without Clustering (ABC)
In addition, as seen in the highlighted rows in Table 7, we noticed that the second strategy for scout bees outperforms the first one in many instances because the first strategy includes selecting either the best customer pairs or random pairs, and the random selection option may affect the quality of the solution, while the second strategy selects customer pairs randomly from the RCL, which contains the best customer pairs in the CSS. ## 6 Conclusions

In this paper, we present eight new metaheuristics to improve the initial solutions for the multi-vehicle profitable pickup and delivery problem (MVPPDP). We proposed two versions of the ABC (Artificial Bee Colony) to improve the initial solutions: the ABC with GRASP and the ABC (C) with GRASP and clustering, i.e., K-means-GRASP (C), modified K-means-GRASP (C), and ACO-GRASP (C). We also have two separate strategies for the scout bee phase. The first strategy, S1, constructs the solution by randomly selecting one of two strategies: greedy and randomised insertion or just greedy insertion. The second strategy, S2, is based on the GRASP. Thus, we have eight different proposed methods. When compared to the state-of-the-art methods in terms of processing time, all eight methods have preference in all instances (except for the 1000-customer instances). Moreover, all proposed algorithms performed well for small-sized instances in terms of profits by achieving thirteen

Figure 3: Comparison the proposed algorithms based on the average of best profits for each category. new best solutions and five equal solutions to the best-known solutions. However, the algorithms slightly lag behind in medium- and large-sized instances due to the greedy randomised strategy and the GRASP that have been used in the scout bee phase. Moreover, our algorithms prioritise minimal solutions and iterations for rapid processing times in daily m-commerce apps, while reducing iteration counts and population sizes reduces the likelihood of obtaining good solution quality. For future research, more neighbourhood operators will be carefully investigated in the scout bee phase to be applicable within the constraints of the MVPPDP, thus increasing the chances of producing high-quality solutions. Moreover, we will try other population-based metaheuristics, such as GAs, evolutionary algorithms, or other swarm-based metaheuristics that may contribute to new solutions for the MVPPDP. Conceptualization, A.I.A. ; methodology, A.I.A. ; formal analysis, A.I.A. ; data curation, A.I.A. ; Writing--original draft, A.I.A. ; funding acquisition, A.I.A. ; Supervision, M.I.H. ; writing--review and editing, M.I.H. All authors have read and agreed to the published version of the manuscript. This research received no external funding. [[http://prolog.univie.ac.at/research/MVPPDP/instances.zip](http://prolog.univie.ac.at/research/MVPPDP/instances.zip)]([http://prolog.univie.ac.at/research/MVPPDP/instances.zip](http://prolog.univie.ac.at/research/MVPPDP/instances.zip)) (accessed on 17 July 2024). The Researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC-2024-9/1). The authors declare no conflict of interest. ## References

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