## 2 The Proposed Multi-Objective Majority-Minority Cellular Automata Algorithm (MOMmCAA)
### Basic Concepts of Cellular Automata with Majority Rule  
Cellular Automata (CA) are dynamic systems of cells that initially take a value from a finite set of possible states. The dynamics of CA proceed in discrete steps, making CA discrete systems in time and space. At each step, a cell considers its current state and that of its close neighbors in order to update its state at the next time step. In this way, a mapping from blocks of states to individual states is called an evolution rule. CA can generate chaotic and complex global behaviors depending on the evolution rule that defines their local mapping. Because of this, they have been widely investigated and applied in various engineering and computational problems [51; 52]. One of the rules of evolution extensively studied in recent work is the majority rule. In this rule, each cell takes its new state as the most common state in its local neighborhood. The dynamics of this rule are characterized by patchy patterns that stabilize as the system's evolution progresses. Its counterpart is the minority rule, which takes the least common element of each neighborhood to update the state of a cell in the next generation. The evolution of the minority rule is characterized by the fact that it does not tend quickly to a fixed or periodic state, as a minority state tends to become the majority and vice versa, resulting in oscillating global dynamics. Figure 1 shows various dynamic behaviors of the majority rule and minority rule along with the application of the majority rule with probability in cellular automata of two states and various neighborhood sizes. In these examples, 500 cells and 250 evolutions were used. Evolution generates a periodic pattern for the original majority rule, while the minority rule generates a chaotic pattern of heterogeneous triangular shapes. When the rules are alternated probabilistically, the result is the formation of complex patterns in which non-periodic structures are combined with a stable background. This combination of majority and minority rules was used as inspiration to define the Majority-minority Cellular Automata Algorithm (MmCAA) for single-objective optimization [47]. The inspiration behind the MmCAA is to emulate the dynamic behavior of applying majority and minority rules in cellular automata. The MmCAA starts by generating a random population \(S\) of \(n_{S}\) smart-cells, where each smart-cell is represented as \(\mathbf{s}\in\mathbb{R}^{n}\). The dynamics of each \(\mathbf{s}\) is defined by a set of rules, with one of them chosen randomly to improve the position of the smart-cell. The rules take information from other smart-cells to generate new neighbors, from which the best one is selected to upgrade the smart-cell position. With this mechanism, the positions of all smart-cells in the population are improved and the system evolves iteratively during the optimization process. The rules used by the MmCAA for smart-cell evolution are as follows. The majority (minority) rule applied to a single smart-cell is described in Algorithm 1. The input is a smart-cell \(\mathbf{s}_{i}\) for \(1\leq i\leq n_{S}\) and a weight parameter \(prop\_w\) that defines a limit on the change in the values of \(\mathbf{s}_{i}\). The rule takes the differences \(\mathbf{cm}\) between the values in \(\mathbf{s}_{i}\) and the most repeated element \(el\) in the smart-cell, and \(rand\) generates a random value between 0 and 1. A new solution \(\mathbf{evol}\) is formed by taking the differences between the original smart-cell and \(\mathbf{cm}\) randomly weighted between 0 and \(prop\_w\). This rule helps to bring the values of \(\mathbf{s}_{i}\) closer to the most repeated value \(el\).