**5. Conclusions**

In this work, we have studied the assessment of multi-objective optimization approaches when trying to optimize single-objective problems. From our point of view, it is interesting to analyze the differences between the solutions provided by these multi-objective techniques (when considering each objective value separately) and those reached by algorithms that are specifically designed to optimize single and independent objectives. For this reason, in this paper, we presented a comparative study between Multi-Objective Evolutionary Algorithms and Single-Objective Evolutionary Algorithms. For the experimental analysis, we focused on two well-known and widely studied optimization problems: the Knapsack Problem and the Travelling Salesman Problem. We considered bi-objective formulations of the aforementioned problems. These bi-objective optimization problems were directly—in a single run—processed using the multi-objective approaches, thus yielding a Pareto front, from which we only are interested in two values: the point optimizing objective 1 and, separately, the point optimizing objective 2. Meanwhile, the single-objective approaches must be executed twice: once to optimize objective 1, defined in the bi-objective formulation of the problem, and again to optimize objective 2.

The computational study carried out allows us to conclude that although MOEAs have to deal with several objectives simultaneously, in some cases they have proven to be more effective than single-objective approaches. In particular, the multi-objective approaches exhibited better behavior when dealing with larger instances or with instances where the objectives are strongly correlated. For those specific cases, the direct application of a multi-objective solver to a multi-objective problem is a better choice in comparison to the transformation of the multi-objective problem into a single-objective one to be solved by means of a single-objective algorithm. This conclusion can be explained by the intrinsic capacity of MOEAs to maintain diversity within a population. MOEAs need conflicting objectives and more time to converge, thus performing a larger exploration of the solution space. The more negatively correlated the objectives, the more they conflict one with each other. Otherwise, if we consider a context with non-conflicting objective functions, the Pareto front converges to a single point. Hence, in these cases, it is better to address the problem by optimizing independently each of the objective functions through a single-objective algorithm.

Considering the above, in the future, further evaluations should be done with a more—representative and independent— set of problems and instances. We could thus further investigate the key factors influencing the improvement of MOEA approaches to single-objective environments. Since the design of MOEAs allows each objective to have a helper-objective effect on the other objective, this property can provide more freedom to maintain the diversity of individuals within a population. Such a feature is not present under the single-objective approaches.

It is important to note that what is sought is useful diversity. A greater diversity does not necessarily imply a proper balance between exploration and exploitation, so a high diversity might be counterproductive. In this work, we did not employ a suitable diversity managemen<sup>t</sup> strategy because our intention was to study the intrinsic capacity of MOEAs to maintain diversity and to analyze how effective these approaches are in single-objective optimization. However, after this initial analysis, it would be worthwhile to design new experiments were the intrinsic and specific features of MOEAs could be evaluated separately in some way.

**Author Contributions:** Conceptualization, G.M., C.L. and E.S.; Formal analysis, M.M., G.M., C.L. and E.S.; Funding acquisition, M.M. and C.L.; Investigation, M.M., G.M., C.L. and E.S.; Methodology, G.M. and C.L.; Software, M.M., G.M. and E.S.; Supervision, G.M., C.L. and E.S.; Validation, G.M., C.L. and E.S.; Visualization, M.M., G.M. and E.S.; Writing of original draft, M.M., G.M., C.L. and E.S.; Writing of review & editing, M.M., G.M., C.L. and E.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 program "I+D+i Orientada a los Retos de la Sociedad" [contract number TIN2016-78410-R]. The work of Mohammed Mahrach was funded by the Canary Government "Agencia Canaria de Investigación Innovación y Sociedad de la Información—ACIISI" [contract number TESIS2018010095].

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