*4.1. Mathematical Formulation of MOOPs in the Multi-Robot Exploration*

The process of searching uncertainties by a team of robots can be considered a multi-tasking system. Each robot receives the sensor reading data and upgrades the probabilities of the grid occupancy in the map. One robot should have the same task as another robot in the multi-robot system, wherein the task can be any of the following: scanning the environment using sensors, avoiding obstacles and collisions with other robots, seeking to explore new terrain, and increasing the accuracy of the map. It means that together, each single robot should provide good implementation as a multi-robot system satisfying the multi-objective functions for obtaining the best solutions.

In this paper, we formulated the objective functions of the exploration as follows:

$$\text{Maximize}: f\_1 \to \text{number of explored cells},\tag{6}$$

$$\text{Minimize}: f\_2 \to \text{probability values } P(\text{acc}\_{x,y}). \tag{7}$$

Subject to:

$$\text{If } i \ge \text{(total number of cells - total number of obstacle cells)} / \text{number of robots,} \tag{8}$$

$$wwp\_w \ge \text{number of robots}^3,\tag{9}$$

$$wp\_{w} \le \text{total number of cells} - \text{total number of obstacle cells},\tag{10}$$

where,

*i* − *number o f iterations*,
