## 4 Analysis of Single Objective Algorithms Overview
### Learner Performance-Based Behavior
#### 5.3.2 Multi-Objective Differential Evolution

As a general evolutionary algorithm, the MODE method is made up of three primary parts: selection, Pareto-based evaluation, and mutation. This section [16] goes into great detail about these elements.Algorithm 6 describes the steps of MODE

Figure 7: Diagram illustrating a two-step MOO process. ### Mutation Operator

Two types of vectors should be established: the perturbation vector and the differential vector, to duplicate the mutation operator in the differential evolution method that was described in Section 2. For a single-objective problem in the DE, the vector that depicts the difference between the best individual and the individual undergoing operation is known as the differential vector. The individual with the highest value of fitness within the population is usually the best [51]. The Pareto-optimum solution group is the goal of the evolutionary approach when applied to a multi-objective domain. The original objective of finding every Pareto solution is to identify them all, not to direct individuals toward a single option. In every generation of the evolutionary process, the proposed MODE presents a Pareto-based method for choosing the best individual for mutation procedure identification regarding non-dominated solutions (Pareto-optimum solutions) in D across the population. We should determine whether an individual \(\mathrm{P_{i}}\) is dominated before performing the mutation operation on them. We need to look into whether or not that specific individual is being dominated. The \"best\" answer for identifying a dominated individual is to identify the group of non-dominated individuals, \(\mathrm{D_{i}}\), that dominate this individual. A value is chosen at random from the set \(\mathrm{D_{i}}\), called \(\mathrm{P_{best}}\). For the mutation process, the vector that lies between points \(\mathrm{P_{best}}\) and \(\mathrm{P_{i}}\) is the differential vector. \(\mathrm{P_{best}}\) will be the individual itself if they are already non-dominated. In this case, only perturbation vectors are affected and the differential vector equals zero. In contrast to single objective DE, multi-objective differential evolution changes rather than stays the same to reproduce all of the individuals in a population. This is consistent with the main objective of finding the entire Pareto-optimal set. A random selection of individual pairs from the parent population yields perturbation vectors. A structure akin to the single-objective DE could be used for the mutation process once differential vectors as well as perturbation vectors are determined. If one uses the representation of the natural chromosome, each allele is associated with a single choice variable. They conduct the experiments detailed in their study using this form of representation in their implementation of MODE. So, given a mutation probability of \(\mathrm{P_{m}}\), each allele of an individual will undergo a specific process. An individual could be thought of as a vector, and every element of the vector will be subject to the operation, as shown in Equation (11) [16]. \[\mathcal{P}_{i}=\left\{ \begin{aligned} & P_{i}+F*\sum_{k=1}^{k}\left(P_{a}^{k}-P_{b}^{k} \right)\text{ if $P_{i}$ is non-dominated}\\ &\gamma*P_{best}+(1-\gamma)P_{best}+F*\sum_{k=1}^{k}\left(P_{a}^{ k}-P_{b}^{k}\right)\text{ else }\end{aligned} \right. \tag{11}\]

where \(P_{best}\) is the best Pareto-selected individual from the parent population, \(\gamma\in[0,1]\) denotes the operator's greediness, \(K\) is number of perturbation vectors, \(F\) is the factor that determines the perturbation's scale, \(P_{a}^{k}\) and \(P_{b}^{k}\) are mutually distinct individuals from the parent population that are randomly selected, and \(\mathcal{P}_{i}\) is the offspring. The boundary used to determine how many individuals are inside the predefined sharing radius is the radius. An individual's fitness value is penalized more heavily as the number of individuals in their border grows.