A unified differential-equation and operations-research framework for benchmarking metaheuristic optimization of engineering systems

Tuning an engineering controller or a structural design normally means running a differential-equation model inside an outer search loop: the objective is available only pointwise, carries no usable gradient, and mixes smooth tracking terms with non-smooth actuator saturation and constraint penalties. The search algorithm is therefore itself a design decision, yet it is almost always fixed once, on a single plant, without evidence that the choice transfers to any other. This paper asks whether it does, and when a metaheuristic is needed at all. We answer both questions with a unified differential-equation (DE) and operations-research (OR) framework in which six representative plants—a separately excited DC servomotor, a mass–spring–damper stage, a stirred heat exchanger, a continuously stirred tank reactor (CSTR), a series RLC circuit, and an inverted pendulum on a cart—are wrapped in a single proportional–integral–derivative (PID) evaluation harness, alongside three constrained mechanical-design benchmarks and the ten-dimensional Rastrigin function. Seven long-established metaheuristics (GA, PSO, DE, SA, GWO, WOA, SCA) are compared over fifteen seeded runs at a budget of exactly 3000 objective evaluations, enforced by an evaluation counter rather than by nominal population and iteration settings, so that trajectory- and population-based solvers are treated identically. Four non-metaheuristic baselines, a hyperparameter sensitivity sweep and two ablation studies test the framework itself. The solver ranking reverses across problem classes: differential evolution attains the best Friedman rank on the PID landscapes (1.52 of 7) where the sine–cosine algorithm is worst (6.44), whereas on Rastrigin that ordering inverts (SCA 1.20, WOA 2.07, DE 5.60), and GWO and DE dominate the constrained designs. The baselines explain why. On the smooth PID surfaces a multi-start Nelder–Mead search matches every metaheuristic to five decimal places, so the metaheuristic buys robustness rather than accuracy; classical Ziegler–Nichols tuning, by contrast, is 8.9 to 19.8 times worse, and no local method remains competitive once constraints or multimodality are present. The framework therefore does more than rank solvers—it identifies which problems justify a metaheuristic—and is released as a fully specified, reproducible pipeline.