SCUBA DIVER OPTIMIZATION ALGORITHM: for Speed Reducer Design

SCUBA DIVER OPTIMIZATION ALGORITHM: for Speed Reducer Design Optimization A NOVEL METAHEURISTICS ALGORITHM FOR CONSTRAINED ENGINEERING

SDOA Algorithm Implementation for Speed Reducer Design Optimization

Core Algorithm: Scuba Diver Optimization Algorithm (SDOA)

• Nature-inspired, population-based metaheuristic derived from scuba diving dynamics
• Adaptive search behaviour regulated by “oxygen” and “depth-stage” transitions
• Key mechanism: oxygen depletion and replenishment drive a controlled shift from exploration to exploitation

Problem Solved: Speed Reducer Design Optimization

• Objective: Minimize speed reducer weight
• Design variables: 7 variables [x1,…,x7], with mixed-integer structure (x3 is integer: number of teeth)
• Constraints: 11 nonlinear constraints reflecting mechanical feasibility:
• Gear tooth bending stress constraints
• Shaft deflection constraints
• Shaft stress constraints
• Geometric and dimensional constraints

Algorithm Components

1) Diver Representation & Initialization

• Each diver represents a candidate solution vector: 7 design variables
• Initialization ensures broad coverage of the search space using a structured distribution strategy
• Includes random diversification for a subset of the population to strengthen global coverage
• Integer enforcement: x3 is rounded after initialization and after every update step

2) Oxygen & Depth Management

• Oxygen decay model:

Oi​(t)=Oi​(t−1)exp(−αt/tmax​)

• The population is partitioned dynamically into 5 depth stages using oxygen thresholds:

Depth stages (oxygen thresholds):

• D1 (high oxygen): Global exploration (Lévy-flight style updates)
• D2 (moderate oxygen): Moderate exploration (crossover with best + random walk)
• B1 (reduced oxygen): Exploitation (local search with optional elite crossover)
• D3 (low oxygen): Fine-tuning (non-uniform mutation with decreasing intensity)
• Reset (very low oxygen): Diversification (reinitialization, especially if infeasible)

This staged strategy implements a structured exploration-to-exploitation pipeline without requiring manual phase switching.3) Search Operators

• Lévy-flight exploration: heavy-tailed perturbations for escaping local minima
• Crossover operator: blends current diver with best/elite diver to accelerate convergence
• Random walk: bounded stochastic perturbations to enhance neighbourhood exploration
• Local search: deterministic refinement around current candidate (hill-climbing style)
• Non-uniform mutation: progressively shrinking perturbations for late-stage tuning

All moves are followed by:

• bound repair (clipping to [Lb,Ub])
• integer snapping for x3​

4) Constraint Handling

• Penalty-based constraint handling: pfit=f(x)+106∑max(0,gi​(x))
• Acceptance policy prioritises feasibility:
• Improved penalised fitness is accepted
• Feasible solutions are favoured over infeasible ones (feasibility-first bias)
• Constraint verification is included via a dedicated validation routine that reports each constraint value and feasibility status

5) Adaptive Mechanisms

• Elitism: top solutions are preserved and restored each iteration (elite protection)
• Parameter adaptation: mutation and exploration strength decrease with iteration count
• Oxygen replenishment: successful improvements partially restore oxygen to reinforce productive behaviours
• Communication strategy: divers probabilistically move relative to the best diver, strengthening collective convergence

Key Parameters (as provided)

• Population size: 500 divers
• Maximum iterations: 2000
• Runs: 10 independent runs (feasible-first statistics reported)
• Elite size: 15
• Initial oxygen: 200, oxygen decay factor α=0.4\alpha = 0.4α=0.4
• Initial mutation rate: 0.3 (decreases over iterations)
• Initial current strength: 0.3 (decreases faster to focus refinement later)

Results & Visualization

The implementation generates detailed diagnostics and plots, including:

• Best-run convergence history (best vs. feasible-average)
• Constraint-violation history for the best run
• Best design variable bar plot
• Diver distribution across depth stages (area plot)
• Feasible-run percentage (pie chart)
• Cross-run overlays: convergence and constraint histories for all runs
• Final best objective distribution (scatter + boxplot) and feasibility count

Strengths of the Implementation

• Exploration–exploitation balance: controlled by oxygen-driven stage switching
• Robust feasibility management: strong penalty handling with feasibility-first selection pressure
• Mixed-integer compatibility: integer decision handling integrated directly into the update loop
• Convergence stability: elite preservation prevents loss of high-quality solutions
• Rich behavioural tracking: depth-stage population statistics provide interpretability of algorithm dynamics

Algorithm Characteristics

• Computational complexity: O(nDivers×maxIter×nVars)
• Memory usage: moderate (stores histories across iterations and runs)
• Scalability: suitable for constrained, nonlinear engineering design problems with small-to-medium dimension

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