# A Learnheuristic Algorithm for the Capacitated Dispersion Problem under Dynamic Conditions
## 4 Solution Approaches for the Dynamic Capacitated Dispersion Problem
### A Constructive Heuristic
## 6 Conclusions and Future Work

This work introduces a learnheuristic algorithm that hybridizes a constructive heuristic with reinforcement learning to tackle a dynamic version of the capacitated dispersion problem, thus offering a robust solution that surpasses existing methods for deterministic static versions. The presented algorithm adopts a constructive approach, where it integrates

Figure 4: Average of nodes in the solution by type of instance. Figure 3: Gaps between static and dynamic approaches for different environments. elements using reinforcement learning techniques, thus continuously learning and adapting to the dynamism inherent in the problem space. The first part of our study explores the advantages of this constructive mechanism. Beginning with an empty set, the algorithm iteratively adds promising components, which are influenced by heuristic knowledge and guided by reinforcement learning feedback, until an efficient solution structure is achieved. This phase demonstrates a remarkable capacity to navigate and adapt to the problem's complexities, thus showcasing superiority over established methods, particularly in deterministic scenarios. The results show a generalized decrease in the number of selected nodes compared to the static version of the algorithm. It does, however, raise the computational time due to its complex learning and adaptation processes. This minor increase in resource is usually made up for by the algorithm's skill in quickly identifying and skipping nodes that probably will not give the best solution, thereby ensuring a more efficient problem-solving process. In the CDP, the objective function is logically connected to the number of selected nodes; this reduction, which was undoubtedly due to the predictive intelligence used by the algorithm to avoid nodes with a low chance of success, led to an improvement in the objective function. Looking forward, we identify several directions for further research. Firstly, future studies could explore problem variants where the element selection is associated with specific costs that are possibly linked to individual element capacities. This expansion would bring a new layer of complexity and realism to the problem, thereby providing possibilities for further optimization and refinement. Secondly, considering scenarios with stochastic inputs, including fluctuating demands or capacities, would be a natural next step: this would extend our algorithm to include simheuristic components [30; 31]. Thus, combining our learnheuristic with simulation could provide robust solutions in fluctuating environments, thus clearing the path for optimization algorithms that are flexible and capable of adapting to dynamic and stochastic scenarios. Additionally, another possible extension refers to the combined use of parallel computing [32] and biased randomization techniques [33] in order to promote 'agile' optimization algorithms. Conceptualization, A.A.J. and J.P.; methodology, J.F.G. and A.R.U. ; software, J.F.G. and A.R.U. ; validation, A.A.J. and J.P.; formal analysis, J.F.G. and A.R.U. ; investigation, J.F.G. and A.R.U. ; writing--original draft preparation, J.F.G. and A.R.U. ; writing--review and editing, A.A.J. and J.P.; supervision, A.A.J. All authors have read and agreed to the published version of the manuscript. This work has been partially funded by the Spanish Ministry of Science and Innovation (PID2022-138860 NB-I00, RED2022-134703-T), by the SUN project of the Horizon Europe program (HORIZON-CL4-2022-HUMAN-01-14-101092612), and by the i4 OPT project of the Generalitat Valenciana (PROMETEO/2021/065). All data employed is publicly available in the given references. The authors declare no conflict of interest. ## References

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