Modified goat optimization algorithm with a population-dispersion-based adaptive exploration coefficient

The goat optimization algorithm (GOA) is a novel nature-inspired optimization technique that has demonstrated promising performance across diverse numerical optimization problems. However, its reliance on a fixed exploration coefficient during iterations often leads to premature convergence to local optima and diminished solution accuracy. To address this limitation, we introduced the GOA-PD, a lightweight population-dispersion-driven adaptive strategy that adjusts the exploration coefficient online by integrating normalized dispersion feedback, iteration-dependent decay, and exponential moving average smoothing. Empirical evaluations on the 20-dimensional CEC2021 benchmark suite, together with statistical tests, ablation analysis, and parameter sensitivity analysis, indicate that the proposed method improves the baseline GOA in terms of average ranking, convergence behavior, and stability while maintaining competitive overall performance. In addition, a case studied on welded beam design further suggests that the GOA-PD can obtain feasible and competitive solutions to constrained engineering optimization problems with limited additional computational cost.