**4. Experimental results**

Now we will present the most important results, which proved the superiority of the proposed algorithm MaBAT/R2 over other algorithms using the well-known functions DTLZ (the DTLZ suite of benchmark problems, created by [13], is unlike the majority of multi-objective test problems in that the problems are scalable to any number of objectives), from which we took only nine functions for comparison and with different sizes in terms of directions, number of target functions, and number of repetitions, especially regarding the problems of irregular Pareto Front (PF) patterns.

#### **4.1 Inverted generational distance (IGD)**

Let S denote the search result of a MOEA on a specific MOP. Should R be a set of PF representation points that are equally spaced? [1] Can be used to determine S's IGD value in relation to R.

$$IGD(\mathcal{S}, R) = \frac{\sum\_{r \in R} d(r, \mathcal{S})}{|R|} \tag{3}$$

where |R| is the cardinality of R and d (r, S) is the minimum Euclidean distance between r and the points in S. It is important to note that perhaps the elements in R should really be spread evenly, and |R| should be large enough to ensure that the points in R fairly reflect the PF. This ensures that the IGD value of S may accurately assess the solution set's confluence and diversification. S has a lower IGD value, which indicates that it is of higher quality [14].

A set R of indicative points of the PF must be provided in this section to calculate the IGD value of a result set S of a MOEA executing on a MOP.

#### **4.2 Hypervolume indicator**

The hyperbolic quantity indicator *Ihyp*ð Þ A calculates the volume of a territory H that is composed of a set of points A and a set of reference points N:

$$I\_{\rm hyp}(\mathcal{A}) = \text{volume}\left(\bigcup\_{\forall \mathfrak{a} \in \mathcal{A}; \forall \mathfrak{a} \in \mathcal{N}} \text{hypercube}(\mathbf{a}, \ \mathbf{n})\right) \tag{4}$$

As a result, higher indicative values correspond to better solutions. The S metric and the Lévesque measure are other names for the hyperdensity indicator. It has a number of appealing attributes that have aided in its adoption and success. It is, in example, the only marker with metric features and the only one that is strictly Pareto monotonic [15]. Because of these characteristics, this indicator has been employed in a variety of applications, including measuring performance and evolutionary programming.
