## 4 Proposed Metaheuristic Schemes
### DPSO Algorithm  
Because the MCMRS problem defined in Equations (1)-(5) is a discrete optimization problem, traditional PSO algorithm cannot be used and, thus, discrete particle swarm optimization algorithm will be developed for the MCMRS problem. Therefore, the evolutionary process of particle velocity and position represented by (6) and (7) in the original PSO algorithm, respectively, must be modified such that they can span the discrete search domain. For our proposed MCMRS problem, we developed a DPSO algorithm which is an adaptation of the method established in [45] for updating the particle positions. According to [45], the position of the \(i\)th particle at iteration \(t\) can be updated as follows:  
\[X_{i}^{t+1}=c_{2}\bigoplus F_{3}\Big{(}c_{1}\bigoplus F_{2}\Big{(}\omega\ \bigoplus F_{1}\big{(}X_{i}^{t}\big{)},\ P_{i}^{t}\Big{)},\ G^{t}\Big{)} \tag{8}\]  
where \(F_{1}\), \(F_{2}\), and \(F_{3}\) are operations with the probabilities \(\omega\), \(c_{1}\), and \(c_{2}\), respectively. In the DPSO algorithm, the particle velocities and positions are updated by three operations. The first operation is \(\tau_{i}^{t+1}=\omega\bigoplus\ F_{1}\big{(}X_{i}^{t}\big{)}\), which represents the velocity operation for a particle and generates a temporary particle \(\tau_{i}^{t+1}\). \(F_{1}\) denotes the mutation operator, which is applied with a probability of \(\omega\). Generate a random number \(r\in(0,1)\). A new particle \(\tau_{i}^{t+1}=F_{1}\big{(}X_{i}^{t}\big{)}\) is generated by applying the mutation operator to the current particle if \(r<\omega\). Otherwise, set the new particle to be the current particle, i.e., \(\tau_{i}^{t+1}=X_{i}^{t}\). The second operation is \(\delta_{i}^{t+1}=c_{1}\bigoplus F_{2}\big{(}\tau_{i}^{t+1},\ P_{i}^{t}\big{)}\), which is the cognitive part of the particle. \(F_{2}\) represents the crossover operator, which is applied with a probability of \(c_{1}\). The outcome is, thus, either \(\delta_{i}^{t+1}=F_{2}\Big{(}\tau_{i}^{t+1},\ P_{i}^{t}\big{)}\) or \(\delta_{i}^{t+1}=\tau_{i}^{t+1}\), where \(\delta_{i}^{t+1}\) is a temporary particle. The third operation is \(X_{i}^{t+1}=c_{2}\bigoplus F_{3}\Big{(}\delta_{i}^{t+1},\ G^{t}\big{)}\), which is the social part of the particle. \(F_{3}\) represents the crossover operator, which is applied with a probability of \(c_{2}\). The outcome is either \(X_{i}^{t+1}=F_{3}\Big{(}\delta_{i}^{t+1},\ G^{t}\Big{)}\) or \(X_{i}^{t+1}=\delta_{i}^{t+1}\). Mutation operator \(F_{1}\) for position \(X_{i}^{t}\) represents the replacement of the original cover sector for a sensor by a randomly selected cover sector. For example, with \(X_{i}^{t}\) = (1, 3, 0, 2, 3, 1, 4, 0, 2) and mutations occurring in the second, third, and seventh positions, we might obtain \(\tau_{i}^{t+1}\) = (1, 2, 1, 2, 3, 1, 3, 0, 2). Operator \(F_{2}\) represents selecting \(\tau_{i}^{t+1}\) and \(P_{i}^{t}\) to be the first and second parents for a crossover operation, respectively. Each of the cover sectors in \(\tau_{i}^{t+1}\) and \(P_{i}^{t}\) are exchanged, from left to right, if a random number \(r\in(0,1)<c_{1}\). For example, with \(\tau_{i}^{t+1}\) = (1, 3, 0, 2, 3, 1, 4, 0, 2) and \(P_{i}^{t}\) = (1, 3, 1, 2, 4, 1, 3, 0, 1), we obtain \(\delta_{i}^{t+1}\) = (1, 3, 1, 2, 4, 1, 3, 0, 1) if all generated uniform random numbers are less than \(c_{1}\). The crossover operator \(F_{3}\) represents selecting \(\delta_{i}^{t+1}\) and \(G^{t}\) as the first and second parents, respectively, for a crossover operator in a similar manner for that of \(F_{2}\); each of the different cover sectors in \(\delta_{i}^{t+1}\) and \(G^{t}\), from left to right, is exchanged if a random number \(r\in(0,1)<c_{2}\). The pseudocode of the procedure of our proposed DPSO algorithm for the MCMRS problem is shown in Algorithm 2. ```
1: Initialize the parameters: \(P\), \(\omega\), \(c_{1}\), and \(c_{2}\). Initialize the position and velocity for each particle randomly.