SCUBA DIVER OPTIMIZATION ALGORITHM: Tension/Compression Spri

Scuba Diver Optimization Algorithm (SDOA) for solving Tension/Compression Spring Design Optimization

SDOA Algorithm Implementation for Tension/Compression Spring Design Optimization

Core Algorithm: Scuba Diver Optimization Algorithm (SDOA)

Nature-inspired metaheuristic based on scuba diving behavior

Population-based approach with adaptive search strategies

Key mechanism: Oxygen levels regulate transition from exploration to exploitation

Problem Solved: Tension/Compression Spring Design Optimization

Objective: Minimize spring weight (mass) while satisfying 4 mechanical constraints

3 design variables: Wire diameter (d), Mean coil diameter (D), Number of active coils (P)

4 nonlinear constraints: Shear stress, Surge frequency, Deflection limits, Diameter relationship

Variable bounds: d ∈ [0.05, 2.0], D ∈ [0.25, 1.3], P ∈ [2.0, 15.0]

Penalty parameter: 10⁹ for constraint violations

Algorithm Components

1. Diver Representation & Initialization

• Each diver = a candidate solution vector (3 variables: d, D, P)
• Structured initialization: Variables initialized across their ranges with segment-based distribution
• Random diversification: 1/3 of population randomly initialized for enhanced diversity
• Bound enforcement: Strict adherence to variable bounds throughout optimization

2. Oxygen & Depth Management

• Oxygen decay: O_i(t) = O_i(t-1) × exp(-α × t / t_max) with α = 0.5
• 5 depth stages based on oxygen thresholds:
• D1 (>75): Global exploration using Levy flight with Cauchy-like distribution
• D2 (>55): Moderate exploration with 70% crossover and 30% random walk
• B1 (>35): Exploitation using local hill-climbing search plus elite crossover (60% probability)
• D3 (>15): Fine-tuning with non-uniform mutation (rate decreases with iteration)
• Reset (≤15): Diversification via reinitialization (40% probability, triggered for constraint violations)

3. Search Operators (Adapted for 3D Spring Space)

• Levy flight: Heavy-tailed exploration using Cauchy distribution: step ∝ tan(π×(rand-0.5))
• Crossover: Linear blending: child = α×parent1 + (1-α)×parent2 with α∼U(0,1)
• Local search: Hill-climbing in ±2% neighborhood of each variable
• Non-uniform mutation: Non-linear perturbation: δ = 1-rand^((1-iter/maxIter)⁵)
• Random walk: Uniform perturbation within [-strength×(Ub-Lb), strength×(Ub-Lb)]

4. Constraint Handling for Spring Design

• Exterior penalty function: fitness = weight + 10⁹ × Σ(max(0, gᵢ))
• Constraint evaluation:

1. g₁ = 1 - (D³×P)/(71785×d⁴) ≤ 0 (Shear stress)
2. g₂ = (4D²-dD)/(12566(d³D-d⁴)) + 1/(5108d²) - 1 ≤ 0 (Surge frequency)
3. g₃ = 1 - (140.45×d)/(D²×P) ≤ 0 (Deflection)
4. g₄ = (d+D)/1.5 - 1 ≤ 0 (Diameter)

• Acceptance criteria: Accept better penalized fitness OR feasible solutions over infeasible ones
• Violation tracking: Count of violated constraints (0-4)

5. Adaptive Mechanisms

• Elite preservation: Top 20 solutions preserved each iteration (based on penalized fitness)
• Parameter adaptation:
• Mutation rate: decays linearly from 0.8 to 0
• Current strength: decays quadratically from 0.3 to 0
• Oxygen replenishment: +5 oxygen units for successful moves
• Communication mechanism: 60% probability to pull toward best solution (scaled by oxygen level)
• Iteration-dependent scaling: Levy flight step size reduces linearly with iterations

Key Parameters

• Population size: 100 divers
• Maximum iterations: 400
• Initial oxygen: 100 units
• Oxygen decay factor (α): 0.5
• Elite size: 20 best spring designs
• Initial mutation rate: 0.8 (decays to 0)
• Initial current strength: 0.3 (decays to 0)
• Constraint penalty: 1,000,000,000 per violation

Algorithm Flow

1. Initialization: Generate 100 divers within bounds with structured distribution
2. Evaluation: Calculate weight and constraint violations for all divers
3. Main Loop (400 iterations):

• Update oxygen levels and determine depth stages
• Preserve elite divers
• Apply movement operators based on depth stage
• Evaluate and accept/reject new positions
• Update best solution
• Record convergence metrics

1. Termination: Output best solution after 400 iterations

Performance Characteristics

• Computational complexity: O(100×400×3) = ~120,000 function evaluations
• Memory usage: Stores position (100×3), oxygen (100×1), depth (100×1), fitness arrays
• Convergence monitoring: Tracks best, average (feasible), and maximum fitness
• Feasibility tracking: Percentage of feasible solutions each iteration
• Depth distribution: Count of divers in each depth stage per iteration

Visualization & Analysis Tools

1. Convergence plot: Best and average feasible fitness over iterations
2. Constraint violation history: Best solution's constraint violation count
3. Optimal variables display: Bar chart of best d, D, P values
4. Depth distribution: Stacked area chart of divers in each depth stage
5. Feasibility pie chart: Final population feasibility percentage
6. Parameter summary: Display of algorithm parameters and final results
7. Constraint verification: Detailed check of all 4 constraints for best solution

Strengths of This Implementation

1. Problem-specific adaptation: Operators tailored for 3-variable spring design space
2. Robust constraint handling: High penalty ensures feasibility drive
3. Balanced search: Five depth stages provide smooth exploration-exploitation transition
4. Adaptive behavior: Parameters decay appropriately across search phases
5. Elitism protection: Preserves best designs throughout optimization
6. Comprehensive monitoring: Full tracking of algorithm state and performance
7. Clear visualization: Six-panel figure provides complete optimization insight

Expected Performance Metrics

MetricExpected RangeNotesBest weight0.0126 - 0.0135Literature optimum ~0.012665Feasibility rate (final)85-100%Depends on penalty parameterConstraint satisfactionAll 4 satisfiedFor best solutionConvergence iterations150-250To reach near-optimal regionFunction evaluations~120,000100 divers × 400 iterations

Implementation Notes for the Provided Code

• Variable naming: Uses P instead of N for number of active coils (consistent with code)
• Bound correction: Includes explicit bound correction in fobj function
• Progress display: Shows iteration progress with key metrics
• Constraint verification: Separate function for detailed constraint analysis
• Initialization method: Segment-based with random diversification
• Termination condition: Fixed iteration count (400)

This implementation demonstrates SDOA's effectiveness on a classic mechanical design problem with a small but highly constrained search space, maintaining the algorithm's biological inspiration while achieving practical engineering optimization results.