Enhanced Mud Ring Algorithm (EMRA) for Global Optimization

EMRA: an enhanced, adaptive Mud Ring Algorithm with 3-phase exploration-exploitation control for global optimization problems.

The Enhanced Mud Ring Algorithm (EMRA) is a population-based, nature-inspired metaheuristic for continuous global optimization. EMRA extends the original Mud Ring Algorithm (MRA) — inspired by the cooperative mud-ring foraging behavior of bottlenose dolphins — by replacing MRA's single, static exploration-exploitation control rule with a three-phase adaptive schedule. This directly addresses MRA's main known weakness: high sensitivity to its transition parameter K, which can otherwise cause premature convergence or excessive, unproductive exploration.

How EMRA works

At each iteration, the balance between exploration and exploitation is governed by K, computed according to run progress (t / T_max):

- Early exploration phase (progress < 30%) — K uses the standard MRA term plus a small Gaussian perturbation, maximizing search-space coverage.

- Balanced phase (30%-70% progress) — an exponential decay term gradually tempers randomness while preserving diversity.

- Exploitation phase (progress ≥ 70%) — K is scaled down to concentrate the search around the best-known solutions, accelerating convergence.

When |K| ≥ 1, agents take a randomized exploration step. When |K| < 1, agents perform the characteristic "mud ring" encircling move around the current best solution. A quadratic penalty term discourages infeasible (out-of-bounds) candidates.

Key features

- Adaptive, three-phase exploration-exploitation control — the core enhancement over standard MRA.

- Simple, low-overhead update rules — no gradients or problem-specific tuning required.

- Built-in penalty-based constraint handling for box-constrained problems.

- Convergence-curve output for immediate visualization.

- Validated against the classical 23-function benchmark suite, the CEC2019 benchmark suite (10 functions), and three real-world constrained engineering problems (welded beam, pressure vessel, tension spring), outperforming the original MRA and compared against 9 established metaheuristics (CSA, SCA, SSA, HHO, WOA, PSO, TSA, GWO, CGO).

Usage

[Best_score, Best_pos, Convergence_curve] = EMRA(SearchAgents_no, T_max, lb, ub, dim, fobj);

Example:

dim = 30; lb = -100; ub = 100;

fobj = @(x) sum(x.^2);

[Best_score, Best_pos, cg] = EMRA(30, 500, lb, ub, dim, fobj);

figure; semilogy(cg); xlabel('Iteration'); ylabel('Best score');

Note: the exploitation branch calls a helper function mu_inv, not distributed with this submission. If it is not on your MATLAB path, you will see "Undefined function 'mu_inv'" when that branch triggers — supply your own implementation before running EMRA.

Full methodology, derivations, and benchmark results are in the associated publication (see Cite As).