**6. Conclusions**

This paper proposes a memetic multi-objective algorithm based on the well-known MOEA/D, which applies an ILS as the improvement phase, i.e., ILS-MOEA/D, to solve a novel multi-objective constrained formulation of the MPP. The experimental assessment conducted to contrast the diversity in the decision variable space of ILS-MOEA/D in comparison to a single-objective MA yielded interesting conclusions and possible future lines of work.

Firstly, depending on the problem size evaluated, ILS-MOEA/D obtained different Pareto Front shapes. This hinders the application of a priori methods to guide the search in a certain region of the space. Second, the results show that for reduced problem sizes, i.e., *n* = 20, ILS-MOEA/D and MA yield practically the same results in terms of the menu plan cost. However, as the problem size increases, such as *n* = 40, 60, the results provided by ILS-MOEA/D are slightly worse than those of MA in terms of the cost. Third, in the case of ILS-MOEA/D, diversity managemen<sup>t</sup> in the decision variable space is not as crucial when compared to MA. Considering the results, both techniques gradually decrease the level of diversity; however, ILS-MOEA/D preserves lower levels of diversity in the same time even though MA includes an explicit diversity managemen<sup>t</sup> technique. This is due to the fact that ILS-MOEA/D properly manages diversity in the decision variable space implicitly, since it also promotes diversity in the objective function space. Last but not least, the working hypothesis of this work is confirmed by the results provided in Section 5. Even though MA and ILS-MOEA/D yielded similar results in terms of the menu plan cost for the different instances assessed, the latter provided a much lower level of repetition of specific courses and food groups in comparison to the former. As a consequence, the application of ILS-MOEA/D to solve the multi-objective constrained formulation of the MPP presented here, which considers the cost and the level of repetition as the objectives to be optimised, produces not only affordable, but also considerably more balanced, menu plans when compared to the plans obtained by MA when solving the single-objective formulation of the MPP, which only considers the cost.

As we previously mentioned, note that the single-objective MA incorporates a mechanism to explicitly promote diversity in the decision variable space, something that has not been included into ILS-MOEA/D. The above could be the main reason why ILS-MOEA/D obtained slightly worse meal plans in terms of their cost in comparison to the single-objective MA, particularly, for larger instances. Consequently, including techniques in ILS-MOEA/D to explicitly manage the diversity in the decision variable space could improve its performance in terms of the cost of the menu plans provided. Moreover, even though proposing a deep comparison of evolutionary algorithms is not aligned with our working hypothesis, further research may include a comparison with other multi-objective evolutionary algorithms and state-of-art metaheuristics. For example, it would be interesting to compare ILS-MOEA/D to differential evolution or particle swarm optimisation, even though major design changes would have to be considered to apply those algorithms to a combinatorial optimisation problem. As an alternative line of further research, it would be interesting to consider a comparison between ILS-MOEA/D and a single-objective MA intended to optimise the level of repetition of courses and food groups, rather than optimising the menu plan cost. Finally, we should note that our method could be applied to constrained multi-objective problems in other domains by performing a few modifications. First, the perturbation step of the ILS, as well as how neighbours are obtained, should be adapted by considering information of the particular problem at hand. Moreover, other problem-dependent variation operators should be incorporated into ILS-MOEA/D as well.

**Author Contributions:** Conceptualization, E.S., C.L. and C.S.; Formal analysis, A.M., E.S., C.L. and C.S.; Funding acquisition, A.M., E.S., C.L. and C.S.; Investigation, A.M., E.S., C.L. and C.S.; Methodology, E.S., C.L. and C.S.; Software, A.M., E.S. and C.S.; Supervision, E.S., C.L. and C.S.; Validation, E.S., C.L. and C.S.; Visualization, A.M.; Writing of original draft, A.M.; Writing of review & editing, A.M., E.S., C.L. and C.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was partially funded by the Spanish Ministry of Economy, Industry and Competitiveness as part of the programme "I+D+i Orientada a los Retos de la Sociedad" [contract number TIN2016-78410-R]. It was also partially funded by the Spanish Ministry of Science, Innovation and Universities, as well as by the University of La Laguna, as part of the programme "Nuevos Proyectos de Investigación: Iniciación a la Actividad Investigadora" [contract number 1203\_2020]. The work of Alejandro Marrero was funded by the Canary Islands Government "Agencia Canaria de Investigación Innovación y Sociedad de la Información - ACIISI" [contract number TESIS2020010005]. The work was also funded by CONACyT through the "Ciencia Básica" project number 285599. Finally, the authors acknowledge the support from "Laboratorio de Supercómputo del Bajio" through project 300832 from CONACyT.

**Acknowledgments:** The problem formulation and constraints follow the recommendations of the school canteen programme of the Government of the Canary Islands: "Programa de Eco-comedores Escolares de Canarias", which is a joint priority action of the "Instituto Canario de Calidad Agroalimentaria", the "Dirección General de Salud Pública del Gobierno de Canarias" and the "Dirección General de Ordenación, Innovación y Promoción Educativa".

**Conflicts of Interest:** The authors declare no conflicts of interest.
