# Pressure Vessel Design Problem Using Improved Gray Wolf Optimizer Based on Cauchy Distribution
## 3 The Proposed Algorithm
### Cauchy Mutation Strategy Based on Dynamic Inertia Weight  
Cauchy mutation, as an update strategy, has been applied to optimization algorithms by many scholars, and it has been proven to be an effective technique for improving algorithms [42]. Cauchy mutation can improve the exploration or development ability of the algorithm [43]. In this paper, the Cauchy nonlinear inertia weight is used as the mutation probability, and a new Cauchy mutation position update strategy combined with the inertia weight is proposed. Compared with the original position update strategy, the improved method can generate a wider range of individual mutations. Compared with the conventional Cauchy mutation strategy, the proposed mutation strategy is more flexible and can adapt to each stage of the search. If the leading wolf is trapped in the local optimal value, other individuals will also be greatly affected. Therefore, the objects of the Cauchy mutation are the three leading wolves \(\alpha\), \(\beta\), and \(\delta\). The Cauchy mutation strategy based on inertia weight is shown in Equation (13):  
\[X^{i+1}_{\alpha,\beta,\delta}(t)=X^{i}_{\alpha,\beta,\delta}(t)+Cauchy(x_{0}, \gamma)\cdot X^{i}_{\alpha,\beta,\delta}(t) \tag{13}\]  
where \(X^{i}_{\alpha,\beta,\delta}(t)\) is the individual before the mutation and \(X^{i+1}_{\alpha,\beta,\delta}(t)\) is the individual after the mutation. The mutation probability, which determines whether an individual needs mutation, is calculated using Equation (14). \[ \begin{array}{ll}P_{s}=\omega=&\omega_{min}+\left(\frac{\omega_{max}-\omega _{min}}{2}\right)\cdot\frac{1}{\ln\left[e^{+\left(k\cdot\tau\right)^{2}} \right]}\\ &+\left(\frac{\omega_{max}-\omega_{min}}{2}\right)\cdot Cauchy(x_{0},\gamma) \end{array}  \tag{14}\]  
In Equation (14), the inertia weight is taken as the mutation probability \(P_{s}\), which is the basis for judging whether the mutation operation is needed. In the early stage, the inertia weight remains at a large value, and the global search ability of the algorithm is strong. These perturbations are then added to the current position to generate new candidate solutions. The Cauchy distribution is a heavy-tailed distribution that does not have finite mean and variance; therefore, greater randomness can be introduced, which helps the algorithm jump out of the local optimum. Due to the heavy-tail nature, the Cauchy mutation may also produce large disturbances. In order to prevent large disturbances from affecting the convergence speed, this paper first compares the fitness values of the mutant and the current optimal individual and then uses the survival of the fittest strategy [44] to determine the final new individual--this ensures that the perturbations generated by the Cauchy distribution are controllable. The specific position update formula is shown in Equation (15):  
\[X_{\alpha,\beta,\delta}(t)=\left\{ \begin{array}{ll}X^{i}_{\alpha,\beta, \delta}(t)&rand<P_{s}\\ X^{i+1}_{\alpha,\beta,\delta}(t)&rand>P_{s}\end{array} \right.\text{ and }f\left(X^{i}_{\alpha,\beta,\delta}(t)\right)>f \left(X^{i+1}_{\alpha,\beta,\delta}(t)\right)\\ X^{i}_{\alpha,\beta,\delta}(t)&rand>P_{s}\end{array}\right.\text{ and }f\left(X^{i}_{\alpha,\beta,\delta}(t)\right)<f \left(X^{i+1}_{\alpha,\beta,\delta}(t)\right) \tag{15}\]  
where \(f\left(X^{i}_{\alpha,\beta,\delta}(t)\right)\) is the fitness function value of the \(\alpha\beta\), and \(\delta\) individual before the mutation, and \(f\left(X^{i+1}_{\alpha,\beta,\delta}(t)\right)\) is the fitness function value of the individual after the mutation. \(P_{s}\) is the mutation probability, that is, the Cauchy inertia weight proposed in this paper, and it is a dynamically changing value.