# Hyper-Heuristic Based on ACO and Local Search for Dynamic Optimization Problems +
## 5 Conclusions  
This paper presented the hyper-heuristic HULK based on ACO and LSOs. We have used one of the best ACO variants named \(\mathcal{MMAS}\) and integrated it with two powerful LSOs (LK and US). This integration enabled the adaptation capabilities of \(\mathcal{MMAS}\) in dynamic environmental changes and the solution improvements of LSO that led to better exploitation and exploration in the search space. HULK has shown better performance than other compared methods in more than 80% of tested instances (66.7% considering symmetric changes and 94.4% for asymmetric changes). The idea of using an adaptive weighted roulette wheel to chose among the LSO available proved to be an important issue since when there was some change in the LSO during the same dynamic environment, the ACO retained the information due to the previous one and this synergy could direct the solution search to unexplored areas in the search space. We also applied both LSOs in sequence when some stagnation was detected. This behavior can be better seen when we analyze each environment and see how HULK improves the method intelligence. The method of adjusting the weights during the execution of HULK always seeks to favor the best LCO at the current moment, using as a guide the average of up to three latest solutions, but never leaving a single LSO no chance of being chosen. The TSP was the base problem used to generate the dynamic test instances; to make the problem more realistic, we included the possibility of blocking arcs, in both senses if the dynamic change was symmetric and if the change was asymmetric in just one sense.  
At the beginning of the experiments, we compared HULK with other ACO with immigrant schemes found in the literature. HULK outperformed them by a minimal of 6.8%. So, we turned our attention to instances of higher dimensions, and to the best of our knowledge, ACO-based methods for DOPs have never been mentioned before in the literature. We also extended the dynamic changes, including arc blocking and asymmetry, making the instances more difficult to solve. Section 4 analyzes in detail the behavior of all methods exploring the results from Tables 1-5, showing the superiority of HULK mainly in asymmetric environments.  
Finally, we can conclude that HULK has had better performance than \(\mathcal{MMAS}+\texttt{LK}\)\(\mathcal{MMAS}+\texttt{LK}\) in solving dynamic symmetric changes and even better performance in dynamic asymmetric changes. In HULK, both LSOs can work together, improving the search for solutions capacity. In future research, we can more widely explore the parameters to choose the most suitable LSO, and the possibility exists to include more LSOs. The idea of HULK is easily adapted to cope with other DOPs such as dynamic routing problems that interact very well with ACO and have a wide variety of LSOs in the literature.  
**Funding:** The work of [PERSON] was financed in part by the Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior--Brasil (CAPES)--Finance Code 2380-14-5. [PERSON] is a Doctor Student partially supported by CAPES.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The instances are available at [[http://comopt.ifi.uni-heidelberg.de/software/TSPLIB95/](http://comopt.ifi.uni-heidelberg.de/software/TSPLIB95/)]([http://comopt.ifi.uni-heidelberg.de/software/TSPLIB95/](http://comopt.ifi.uni-heidelberg.de/software/TSPLIB95/)) (accessed on 22 October 2021) and the computational codes are available by request.  
**Acknowledgments:** The authors thanks the anonymous reviewers for their valuable contributions.  
**Conflicts of Interest:** The authors declare no conflict of interest.