# Efficient Task Allocation in Multi-Agent Systems Using Reinforcement Learning and Genetic Algorithm
## 4 A Learning Algorithm
### Genetic Algorithm-Enhanced Proximal Policy Optimization
\(P_{i}\) is evaluated in the environment to obtain fitness score \(F_{i}\) using  
\[F_{i}=\alpha\cdot\frac{R_{i}}{R_{max}}+\beta\cdot\frac{H_{i}}{H_{max}}-\gamma \cdot\frac{E_{i}}{E_{max}}, \tag{12}\]  
where \(R_{i}\) is the cumulative reward obtained by the individual \(P_{i}\), \(H_{i}\) is the policy entropy encouraging exploration, and \(E_{i}\) is the computational resource cost. The factors \(\alpha,\beta,\) and \(\gamma\) are weights balancing these components. To ensure the stability of the training process, PPO employs a clipped objective function for policy updates, which is expressed as  
\[ \begin{split}& L_{1}=\frac{\pi_{\theta}(a_{t}|s_{t})}{\pi_{\theta _{\text{old}}}(a_{t}|s_{t})}\hat{A}_{t},\\ & L_{2}=\text{clip}\bigg{(}\frac{\pi_{\theta}(a_{t}|s_{t})}{\pi_{ \theta_{\text{old}}}(a_{t}|s_{t})},1-\epsilon,1+\epsilon\bigg{)}\hat{A}_{t}, \\ & L(\theta)=\text{min}(L_{1},L_{2}),\end{split}  \tag{13}\]  
where \(r_{t}(\theta)=\frac{\pi_{\theta}(a_{t}|s_{t})}{\pi_{\theta_{\text{old}}}(a_{t }|s_{t})}\) denotes the ratio of the new and old policies, and \(\epsilon\) controls the permissible range of this ratio. The clip function \(\text{clip}(\cdot)\) is expressed as  
\[\text{clip}(x,1-\epsilon,1+\epsilon)= \begin{cases}1-\epsilon&\text{if }x<1- \epsilon\\ x&\text{if }1-\epsilon\leq x\leq 1+\epsilon\\ 1+\epsilon&\text{if }x>1+\epsilon\end{cases} . \tag{14}\]  
This objective function (14) restricts the magnitude of policy updates, thereby preventing significant shifts in the policy and enhancing training stability. Furthermore, incorporating a GA can optimize policy exploration. In each generation, the GA evaluates the fitness of multiple strategies, selecting the best-performing individuals for crossover and mutation, which generates a new set of strategies. This algorithm effectively expands the policy space and enhances the diversity and global optimality of the search. Additionally, the value loss \(L_{V}\) is calculated as  
\[L_{V}=\frac{1}{2}(R_{t}-V(s_{t}))^{2}. \tag{15}\]  
The total loss \(L\) is computed by combining the surrogate loss and the value loss, and the network parameters given as \(\theta\) are updated through gradient descent. As the generations progress, our proposed GAPPO algorithm evaluates each individual in the population to obtain fitness scores \(F_{i}\). Based on those scores, a tournament selection strategy is employed to choose individuals for reproduction. A new population \(P^{\prime}\) is generated through crossover and mutation, and the original population \(P\) is updated accordingly. This evolutionary process continues until the maximum number of generations is reached. The enhanced PPO agent with the genetic algorithm enhances task allocation in multi-agent systems by effectively balancing exploration and exploitation while ensuring diversity within the agent population. This algorithm demonstrates robustness and convergence, even in scenarios with varying numbers of agents. The operational flow of the algorithm is encapsulated in the procedure outlined in Algorithm 1.