Multi-Objective Optimization for Space Exploration Mission Design

A Pareto Frontier Approach

Space exploration missions require careful balancing of competing objectives: minimizing cost and risk while maximizing scientific return. This is a classic multi-objective optimization problem where we seek Pareto-optimal solutions - configurations where improving one objective necessarily worsens another.

Today, we’ll tackle a concrete example: designing a Mars exploration mission by optimizing the selection of scientific instruments, mission duration, and spacecraft configuration.

Problem Formulation

We’ll optimize three competing objectives:

minf1(x)=Cost
minf2(x)=Risk
maxf3(x)=Scientific Return

Or equivalently, minimizing −f3(x) for a consistent minimization framework.

Our decision variables include:

• Number and type of scientific instruments
• Mission duration
• Power system capacity
• Communication bandwidth

Complete Python Implementation

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from scipy.optimize import differential_evolution
import warnings
warnings.filterwarnings('ignore')

# Set random seed for reproducibility
np.random.seed(42)

class MarsExplorationMission:
    """
    Mars Exploration Mission Design Problem
    Decision variables:
    - x[0]: Number of cameras (0-5)
    - x[1]: Number of spectrometers (0-3)
    - x[2]: Number of drilling instruments (0-2)
    - x[3]: Mission duration in days (180-720)
    - x[4]: Solar panel area in m^2 (10-50)
    - x[5]: Communication bandwidth in Mbps (1-20)
    """
    
    def __init__(self):
        # Instrument specifications
        self.camera_cost = 15  # Million USD
        self.spectro_cost = 25  # Million USD
        self.drill_cost = 40   # Million USD
        
        self.camera_mass = 5   # kg
        self.spectro_mass = 12 # kg
        self.drill_mass = 20   # kg
        
        self.camera_power = 50  # Watts
        self.spectro_power = 80 # Watts
        self.drill_power = 150  # Watts
        
        # Base costs
        self.base_spacecraft_cost = 200  # Million USD
        self.daily_operation_cost = 0.5  # Million USD per day
        
    def calculate_cost(self, x):
        """Calculate total mission cost in Million USD"""
        n_cameras = int(x[0])
        n_spectros = int(x[1])
        n_drills = int(x[2])
        duration = x[3]
        solar_area = x[4]
        bandwidth = x[5]
        
        # Instrument costs
        instrument_cost = (n_cameras * self.camera_cost + 
                          n_spectros * self.spectro_cost + 
                          n_drills * self.drill_cost)
        
        # Spacecraft cost (increases with mass and power requirements)
        total_mass = (n_cameras * self.camera_mass + 
                     n_spectros * self.spectro_mass + 
                     n_drills * self.drill_mass)
        
        spacecraft_cost = self.base_spacecraft_cost + total_mass * 2
        
        # Power system cost
        power_cost = solar_area * 3  # $3M per m^2
        
        # Communication system cost
        comm_cost = bandwidth * 5  # $5M per Mbps
        
        # Operations cost
        ops_cost = duration * self.daily_operation_cost
        
        total_cost = (instrument_cost + spacecraft_cost + 
                     power_cost + comm_cost + ops_cost)
        
        return total_cost
    
    def calculate_risk(self, x):
        """Calculate mission risk score (0-100, higher is worse)"""
        n_cameras = int(x[0])
        n_spectros = int(x[1])
        n_drills = int(x[2])
        duration = x[3]
        solar_area = x[4]
        bandwidth = x[5]
        
        # Complexity risk (more instruments = more risk)
        n_instruments = n_cameras + n_spectros + n_drills
        complexity_risk = n_instruments * 5
        
        # Duration risk (longer mission = more exposure to failures)
        duration_risk = (duration / 720) * 30
        
        # Power margin risk
        required_power = (n_cameras * self.camera_power + 
                         n_spectros * self.spectro_power + 
                         n_drills * self.drill_power)
        available_power = solar_area * 100  # 100W per m^2
        power_margin = available_power - required_power
        
        if power_margin < 0:
            power_risk = 50  # Severe risk if insufficient power
        elif power_margin < required_power * 0.2:
            power_risk = 30  # High risk if < 20% margin
        elif power_margin < required_power * 0.5:
            power_risk = 15  # Medium risk if < 50% margin
        else:
            power_risk = 5   # Low risk with good margin
        
        # Communication risk
        data_rate = n_instruments * 10  # MB per day per instrument
        comm_capability = bandwidth * 60  # MB per day
        
        if comm_capability < data_rate:
            comm_risk = 40
        elif comm_capability < data_rate * 1.5:
            comm_risk = 20
        else:
            comm_risk = 5
        
        total_risk = complexity_risk + duration_risk + power_risk + comm_risk
        
        return min(total_risk, 100)
    
    def calculate_science_return(self, x):
        """Calculate scientific return score (0-100, higher is better)"""
        n_cameras = int(x[0])
        n_spectros = int(x[1])
        n_drills = int(x[2])
        duration = x[3]
        solar_area = x[4]
        bandwidth = x[5]
        
        # Base science from instruments
        camera_science = n_cameras * 10
        spectro_science = n_spectros * 20
        drill_science = n_drills * 30
        
        # Synergy bonus (instruments work better together)
        if n_cameras > 0 and n_spectros > 0:
            camera_science *= 1.3
            spectro_science *= 1.2
        
        if n_spectros > 0 and n_drills > 0:
            spectro_science *= 1.3
            drill_science *= 1.2
        
        # Duration factor (longer mission = more data, but diminishing returns)
        duration_factor = np.log(duration / 180 + 1) / np.log(5)
        
        # Communication limitation
        data_generated = (n_cameras + n_spectros + n_drills) * duration * 10
        data_transmitted = bandwidth * duration * 60
        transmission_efficiency = min(data_transmitted / (data_generated + 1), 1.0)
        
        total_science = (camera_science + spectro_science + drill_science) * \
                       duration_factor * transmission_efficiency
        
        return min(total_science, 100)
    
    def evaluate(self, x):
        """Evaluate all three objectives"""
        cost = self.calculate_cost(x)
        risk = self.calculate_risk(x)
        science = self.calculate_science_return(x)
        
        # Return as minimization problem (negate science)
        return cost, risk, -science


class NSGA2Optimizer:
    """NSGA-II Multi-objective Genetic Algorithm"""
    
    def __init__(self, problem, pop_size=100, n_gen=100):
        self.problem = problem
        self.pop_size = pop_size
        self.n_gen = n_gen
        
        # Bounds for decision variables
        self.bounds = [
            (0, 5),      # cameras
            (0, 3),      # spectrometers
            (0, 2),      # drills
            (180, 720),  # duration
            (10, 50),    # solar area
            (1, 20)      # bandwidth
        ]
    
    def initialize_population(self):
        """Create initial random population"""
        pop = np.random.rand(self.pop_size, 6)
        for i in range(6):
            pop[:, i] = pop[:, i] * (self.bounds[i][1] - self.bounds[i][0]) + self.bounds[i][0]
        return pop
    
    def fast_non_dominated_sort(self, objectives):
        """Fast non-dominated sorting"""
        n = len(objectives)
        domination_count = np.zeros(n)
        dominated_solutions = [[] for _ in range(n)]
        fronts = [[]]
        
        for i in range(n):
            for j in range(n):
                if i == j:
                    continue
                if self.dominates(objectives[i], objectives[j]):
                    dominated_solutions[i].append(j)
                elif self.dominates(objectives[j], objectives[i]):
                    domination_count[i] += 1
            
            if domination_count[i] == 0:
                fronts[0].append(i)
        
        k = 0
        while len(fronts[k]) > 0:
            next_front = []
            for i in fronts[k]:
                for j in dominated_solutions[i]:
                    domination_count[j] -= 1
                    if domination_count[j] == 0:
                        next_front.append(j)
            k += 1
            fronts.append(next_front)
        
        return fronts[:-1]
    
    def dominates(self, obj1, obj2):
        """Check if obj1 dominates obj2 (minimization)"""
        return all(obj1 <= obj2) and any(obj1 < obj2)
    
    def crowding_distance(self, objectives, front):
        """Calculate crowding distance for solutions in a front"""
        n = len(front)
        if n <= 2:
            return np.full(n, np.inf)
        
        distances = np.zeros(n)
        
        for m in range(3):  # 3 objectives
            sorted_indices = np.argsort([objectives[i][m] for i in front])
            distances[sorted_indices[0]] = np.inf
            distances[sorted_indices[-1]] = np.inf
            
            obj_range = objectives[front[sorted_indices[-1]]][m] - \
                       objectives[front[sorted_indices[0]]][m]
            
            if obj_range == 0:
                continue
            
            for i in range(1, n - 1):
                distances[sorted_indices[i]] += \
                    (objectives[front[sorted_indices[i + 1]]][m] - \
                     objectives[front[sorted_indices[i - 1]]][m]) / obj_range
        
        return distances
    
    def tournament_selection(self, population, objectives, fronts):
        """Binary tournament selection"""
        idx1, idx2 = np.random.choice(len(population), 2, replace=False)
        
        # Find fronts for both
        front1, front2 = None, None
        for i, front in enumerate(fronts):
            if idx1 in front:
                front1 = i
            if idx2 in front:
                front2 = i
        
        if front1 < front2:
            return population[idx1].copy()
        elif front2 < front1:
            return population[idx2].copy()
        else:
            # Same front, use crowding distance
            front = fronts[front1]
            distances = self.crowding_distance(objectives, front)
            pos1 = front.index(idx1)
            pos2 = front.index(idx2)
            
            if distances[pos1] > distances[pos2]:
                return population[idx1].copy()
            else:
                return population[idx2].copy()
    
    def crossover(self, parent1, parent2):
        """Simulated Binary Crossover (SBX)"""
        eta = 20
        child1 = parent1.copy()
        child2 = parent2.copy()
        
        for i in range(len(parent1)):
            if np.random.rand() < 0.5:
                if abs(parent1[i] - parent2[i]) > 1e-6:
                    u = np.random.rand()
                    if u <= 0.5:
                        beta = (2 * u) ** (1 / (eta + 1))
                    else:
                        beta = (1 / (2 * (1 - u))) ** (1 / (eta + 1))
                    
                    child1[i] = 0.5 * ((1 + beta) * parent1[i] + (1 - beta) * parent2[i])
                    child2[i] = 0.5 * ((1 - beta) * parent1[i] + (1 + beta) * parent2[i])
        
        return child1, child2
    
    def mutate(self, individual):
        """Polynomial mutation"""
        eta = 20
        mutated = individual.copy()
        
        for i in range(len(individual)):
            if np.random.rand() < 1.0 / len(individual):
                u = np.random.rand()
                lb, ub = self.bounds[i]
                
                delta1 = (individual[i] - lb) / (ub - lb)
                delta2 = (ub - individual[i]) / (ub - lb)
                
                if u < 0.5:
                    deltaq = (2 * u + (1 - 2 * u) * (1 - delta1) ** (eta + 1)) ** (1 / (eta + 1)) - 1
                else:
                    deltaq = 1 - (2 * (1 - u) + 2 * (u - 0.5) * (1 - delta2) ** (eta + 1)) ** (1 / (eta + 1))
                
                mutated[i] = individual[i] + deltaq * (ub - lb)
                mutated[i] = np.clip(mutated[i], lb, ub)
        
        return mutated
    
    def optimize(self):
        """Run NSGA-II optimization"""
        population = self.initialize_population()
        
        for gen in range(self.n_gen):
            # Evaluate population
            objectives = np.array([self.problem.evaluate(ind) for ind in population])
            
            # Non-dominated sorting
            fronts = self.fast_non_dominated_sort(objectives)
            
            # Generate offspring
            offspring = []
            while len(offspring) < self.pop_size:
                parent1 = self.tournament_selection(population, objectives, fronts)
                parent2 = self.tournament_selection(population, objectives, fronts)
                
                child1, child2 = self.crossover(parent1, parent2)
                child1 = self.mutate(child1)
                child2 = self.mutate(child2)
                
                offspring.append(child1)
                if len(offspring) < self.pop_size:
                    offspring.append(child2)
            
            offspring = np.array(offspring)
            
            # Combine parent and offspring populations
            combined = np.vstack([population, offspring])
            combined_obj = np.array([self.problem.evaluate(ind) for ind in combined])
            
            # Select next generation
            fronts = self.fast_non_dominated_sort(combined_obj)
            new_population = []
            
            for front in fronts:
                if len(new_population) + len(front) <= self.pop_size:
                    new_population.extend([combined[i] for i in front])
                else:
                    # Sort by crowding distance
                    distances = self.crowding_distance(combined_obj, front)
                    sorted_indices = np.argsort(distances)[::-1]
                    remaining = self.pop_size - len(new_population)
                    new_population.extend([combined[front[i]] for i in sorted_indices[:remaining]])
                    break
            
            population = np.array(new_population)
            
            if (gen + 1) % 20 == 0:
                print(f"Generation {gen + 1}/{self.n_gen} completed")
        
        # Final evaluation
        objectives = np.array([self.problem.evaluate(ind) for ind in population])
        fronts = self.fast_non_dominated_sort(objectives)
        pareto_front_indices = fronts[0]
        
        pareto_solutions = population[pareto_front_indices]
        pareto_objectives = objectives[pareto_front_indices]
        
        return pareto_solutions, pareto_objectives


# Initialize problem
print("=" * 60)
print("Mars Exploration Mission Multi-Objective Optimization")
print("=" * 60)
print("\nObjectives:")
print("  1. Minimize Cost (Million USD)")
print("  2. Minimize Risk (0-100)")
print("  3. Maximize Scientific Return (0-100)")
print("\nRunning NSGA-II optimization...")
print("=" * 60)

mission = MarsExplorationMission()
optimizer = NSGA2Optimizer(mission, pop_size=100, n_gen=100)

# Run optimization
pareto_solutions, pareto_objectives = optimizer.optimize()

print("\n" + "=" * 60)
print(f"Optimization complete! Found {len(pareto_solutions)} Pareto-optimal solutions")
print("=" * 60)

# Convert objectives for display (negate science back to positive)
pareto_objectives_display = pareto_objectives.copy()
pareto_objectives_display[:, 2] = -pareto_objectives_display[:, 2]

# Create visualizations
fig = plt.figure(figsize=(20, 12))

# 3D Pareto Front
ax1 = fig.add_subplot(2, 3, 1, projection='3d')
scatter = ax1.scatter(pareto_objectives_display[:, 0], 
                     pareto_objectives_display[:, 1], 
                     pareto_objectives_display[:, 2],
                     c=pareto_objectives_display[:, 2], 
                     cmap='viridis', 
                     s=100, 
                     alpha=0.6,
                     edgecolors='black')
ax1.set_xlabel('Cost (Million USD)', fontsize=10, fontweight='bold')
ax1.set_ylabel('Risk Score', fontsize=10, fontweight='bold')
ax1.set_zlabel('Scientific Return', fontsize=10, fontweight='bold')
ax1.set_title('3D Pareto Front', fontsize=12, fontweight='bold')
plt.colorbar(scatter, ax=ax1, label='Scientific Return', shrink=0.5)

# Cost vs Risk
ax2 = fig.add_subplot(2, 3, 2)
scatter2 = ax2.scatter(pareto_objectives_display[:, 0], 
                       pareto_objectives_display[:, 1],
                       c=pareto_objectives_display[:, 2], 
                       cmap='viridis', 
                       s=100, 
                       alpha=0.6,
                       edgecolors='black')
ax2.set_xlabel('Cost (Million USD)', fontsize=10, fontweight='bold')
ax2.set_ylabel('Risk Score', fontsize=10, fontweight='bold')
ax2.set_title('Cost vs Risk Trade-off', fontsize=12, fontweight='bold')
ax2.grid(True, alpha=0.3)
plt.colorbar(scatter2, ax=ax2, label='Scientific Return')

# Cost vs Science
ax3 = fig.add_subplot(2, 3, 3)
scatter3 = ax3.scatter(pareto_objectives_display[:, 0], 
                       pareto_objectives_display[:, 2],
                       c=pareto_objectives_display[:, 1], 
                       cmap='coolwarm_r', 
                       s=100, 
                       alpha=0.6,
                       edgecolors='black')
ax3.set_xlabel('Cost (Million USD)', fontsize=10, fontweight='bold')
ax3.set_ylabel('Scientific Return', fontsize=10, fontweight='bold')
ax3.set_title('Cost vs Scientific Return', fontsize=12, fontweight='bold')
ax3.grid(True, alpha=0.3)
plt.colorbar(scatter3, ax=ax3, label='Risk Score')

# Risk vs Science
ax4 = fig.add_subplot(2, 3, 4)
scatter4 = ax4.scatter(pareto_objectives_display[:, 1], 
                       pareto_objectives_display[:, 2],
                       c=pareto_objectives_display[:, 0], 
                       cmap='plasma', 
                       s=100, 
                       alpha=0.6,
                       edgecolors='black')
ax4.set_xlabel('Risk Score', fontsize=10, fontweight='bold')
ax4.set_ylabel('Scientific Return', fontsize=10, fontweight='bold')
ax4.set_title('Risk vs Scientific Return', fontsize=12, fontweight='bold')
ax4.grid(True, alpha=0.3)
plt.colorbar(scatter4, ax=ax4, label='Cost (M USD)')

# Decision variable distributions
ax5 = fig.add_subplot(2, 3, 5)
variables = ['Cameras', 'Spectros', 'Drills', 'Duration', 'Solar', 'Bandwidth']
for i in range(3):  # Plot only instrument counts
    ax5.hist(pareto_solutions[:, i], bins=10, alpha=0.5, label=variables[i])
ax5.set_xlabel('Count', fontsize=10, fontweight='bold')
ax5.set_ylabel('Frequency', fontsize=10, fontweight='bold')
ax5.set_title('Instrument Distribution in Pareto Solutions', fontsize=12, fontweight='bold')
ax5.legend()
ax5.grid(True, alpha=0.3)

# Mission configuration scatter
ax6 = fig.add_subplot(2, 3, 6)
total_instruments = (pareto_solutions[:, 0] + 
                    pareto_solutions[:, 1] + 
                    pareto_solutions[:, 2])
scatter6 = ax6.scatter(pareto_solutions[:, 3], 
                       total_instruments,
                       c=pareto_objectives_display[:, 2], 
                       cmap='viridis', 
                       s=100, 
                       alpha=0.6,
                       edgecolors='black')
ax6.set_xlabel('Mission Duration (days)', fontsize=10, fontweight='bold')
ax6.set_ylabel('Total Instruments', fontsize=10, fontweight='bold')
ax6.set_title('Mission Duration vs Instrument Count', fontsize=12, fontweight='bold')
ax6.grid(True, alpha=0.3)
plt.colorbar(scatter6, ax=ax6, label='Scientific Return')

plt.tight_layout()
plt.savefig('pareto_front_analysis.png', dpi=300, bbox_inches='tight')
plt.show()

# Statistical analysis
print("\n" + "=" * 60)
print("Statistical Analysis of Pareto-Optimal Solutions")
print("=" * 60)
print(f"\nCost (Million USD):")
print(f"  Min: ${pareto_objectives_display[:, 0].min():.2f}M")
print(f"  Max: ${pareto_objectives_display[:, 0].max():.2f}M")
print(f"  Mean: ${pareto_objectives_display[:, 0].mean():.2f}M")
print(f"  Std: ${pareto_objectives_display[:, 0].std():.2f}M")

print(f"\nRisk Score:")
print(f"  Min: {pareto_objectives_display[:, 1].min():.2f}")
print(f"  Max: {pareto_objectives_display[:, 1].max():.2f}")
print(f"  Mean: {pareto_objectives_display[:, 1].mean():.2f}")
print(f"  Std: {pareto_objectives_display[:, 1].std():.2f}")

print(f"\nScientific Return:")
print(f"  Min: {pareto_objectives_display[:, 2].min():.2f}")
print(f"  Max: {pareto_objectives_display[:, 2].max():.2f}")
print(f"  Mean: {pareto_objectives_display[:, 2].mean():.2f}")
print(f"  Std: {pareto_objectives_display[:, 2].std():.2f}")

# Find interesting solutions
print("\n" + "=" * 60)
print("Representative Mission Configurations")
print("=" * 60)

# Lowest cost
idx_low_cost = np.argmin(pareto_objectives_display[:, 0])
print("\n1. LOWEST COST MISSION:")
print(f"   Cost: ${pareto_objectives_display[idx_low_cost, 0]:.2f}M")
print(f"   Risk: {pareto_objectives_display[idx_low_cost, 1]:.2f}")
print(f"   Science: {pareto_objectives_display[idx_low_cost, 2]:.2f}")
print(f"   Configuration:")
print(f"     - Cameras: {int(pareto_solutions[idx_low_cost, 0])}")
print(f"     - Spectrometers: {int(pareto_solutions[idx_low_cost, 1])}")
print(f"     - Drills: {int(pareto_solutions[idx_low_cost, 2])}")
print(f"     - Duration: {int(pareto_solutions[idx_low_cost, 3])} days")
print(f"     - Solar panels: {pareto_solutions[idx_low_cost, 4]:.1f} m²")
print(f"     - Bandwidth: {pareto_solutions[idx_low_cost, 5]:.1f} Mbps")

# Highest science
idx_high_science = np.argmax(pareto_objectives_display[:, 2])
print("\n2. HIGHEST SCIENCE MISSION:")
print(f"   Cost: ${pareto_objectives_display[idx_high_science, 0]:.2f}M")
print(f"   Risk: {pareto_objectives_display[idx_high_science, 1]:.2f}")
print(f"   Science: {pareto_objectives_display[idx_high_science, 2]:.2f}")
print(f"   Configuration:")
print(f"     - Cameras: {int(pareto_solutions[idx_high_science, 0])}")
print(f"     - Spectrometers: {int(pareto_solutions[idx_high_science, 1])}")
print(f"     - Drills: {int(pareto_solutions[idx_high_science, 2])}")
print(f"     - Duration: {int(pareto_solutions[idx_high_science, 3])} days")
print(f"     - Solar panels: {pareto_solutions[idx_high_science, 4]:.1f} m²")
print(f"     - Bandwidth: {pareto_solutions[idx_high_science, 5]:.1f} Mbps")

# Lowest risk
idx_low_risk = np.argmin(pareto_objectives_display[:, 1])
print("\n3. LOWEST RISK MISSION:")
print(f"   Cost: ${pareto_objectives_display[idx_low_risk, 0]:.2f}M")
print(f"   Risk: {pareto_objectives_display[idx_low_risk, 1]:.2f}")
print(f"   Science: {pareto_objectives_display[idx_low_risk, 2]:.2f}")
print(f"   Configuration:")
print(f"     - Cameras: {int(pareto_solutions[idx_low_risk, 0])}")
print(f"     - Spectrometers: {int(pareto_solutions[idx_low_risk, 1])}")
print(f"     - Drills: {int(pareto_solutions[idx_low_risk, 2])}")
print(f"     - Duration: {int(pareto_solutions[idx_low_risk, 3])} days")
print(f"     - Solar panels: {pareto_solutions[idx_low_risk, 4]:.1f} m²")
print(f"     - Bandwidth: {pareto_solutions[idx_low_risk, 5]:.1f} Mbps")

# Balanced solution (closest to ideal point)
normalized_obj = (pareto_objectives_display - pareto_objectives_display.min(axis=0)) / \
                 (pareto_objectives_display.max(axis=0) - pareto_objectives_display.min(axis=0))
normalized_obj[:, 2] = 1 - normalized_obj[:, 2]  # For science (higher is better)
distances = np.sqrt(np.sum(normalized_obj**2, axis=1))
idx_balanced = np.argmin(distances)

print("\n4. BALANCED MISSION (Best compromise):")
print(f"   Cost: ${pareto_objectives_display[idx_balanced, 0]:.2f}M")
print(f"   Risk: {pareto_objectives_display[idx_balanced, 1]:.2f}")
print(f"   Science: {pareto_objectives_display[idx_balanced, 2]:.2f}")
print(f"   Configuration:")
print(f"     - Cameras: {int(pareto_solutions[idx_balanced, 0])}")
print(f"     - Spectrometers: {int(pareto_solutions[idx_balanced, 1])}")
print(f"     - Drills: {int(pareto_solutions[idx_balanced, 2])}")
print(f"     - Duration: {int(pareto_solutions[idx_balanced, 3])} days")
print(f"     - Solar panels: {pareto_solutions[idx_balanced, 4]:.1f} m²")
print(f"     - Bandwidth: {pareto_solutions[idx_balanced, 5]:.1f} Mbps")

print("\n" + "=" * 60)

Code Explanation

Problem Definition Class

The MarsExplorationMission class encapsulates the mission design problem:

Decision Variables:

• Instruments: Number of cameras (0-5), spectrometers (0-3), and drilling instruments (0-2)
• Mission Parameters: Duration (180-720 days), solar panel area (10-50 m²), communication bandwidth (1-20 Mbps)

Objective Functions:

1. Cost Function (calculate_cost):

   • Combines instrument costs, spacecraft costs (mass-dependent), power system costs, communication system costs, and operational costs
   • Formula: Cost=Cinst+Cs/c+Cpower+Ccomm+Cops

2. Risk Function (calculate_risk):

   • Evaluates complexity risk (more instruments = higher failure probability)
   • Duration risk (longer missions have more exposure to failures)
   • Power margin risk (insufficient power is critical)
   • Communication capacity risk (data downlink bottlenecks)

3. Science Return (calculate_science_return):

   • Base scientific value from each instrument type
   • Synergy bonuses when complementary instruments are combined
   • Diminishing returns with mission duration: f(duration)=log(duration/180+1)log(5)
   • Data transmission efficiency factor

NSGA-II Algorithm Implementation

The NSGA2Optimizer class implements the Non-dominated Sorting Genetic Algorithm II, specifically designed for multi-objective optimization:

Key Components:

1. Fast Non-dominated Sorting: Classifies solutions into Pareto fronts based on dominance relationships. A solution dominates another if it’s better in at least one objective and not worse in any other.

2. Crowding Distance: Measures solution density in objective space. Solutions in less crowded regions are preferred to maintain diversity:
   di=M∑m=1fi+1m−fi−1mfmaxm−fminm

3. Simulated Binary Crossover (SBX): Creates offspring by mimicking single-point crossover behavior with a distribution parameter η=20.

4. Polynomial Mutation: Introduces variation while respecting variable bounds, also using η=20.

5. Tournament Selection: Selects parents based on Pareto front rank first, then crowding distance as a tiebreaker.

Optimization Process

The algorithm iterates for 100 generations with a population of 100 individuals:

1. Initialize random population within variable bounds
2. Evaluate all three objectives for each solution
3. Perform non-dominated sorting to identify Pareto fronts
4. Create offspring through selection, crossover, and mutation
5. Combine parent and offspring populations
6. Select the best solutions for the next generation based on Pareto dominance and crowding distance

Results and Visualization

============================================================
Mars Exploration Mission Multi-Objective Optimization
============================================================

Objectives:
  1. Minimize Cost (Million USD)
  2. Minimize Risk (0-100)
  3. Maximize Scientific Return (0-100)

Running NSGA-II optimization...
============================================================
Generation 20/100 completed
Generation 40/100 completed
Generation 60/100 completed
Generation 80/100 completed
Generation 100/100 completed

============================================================
Optimization complete! Found 100 Pareto-optimal solutions
============================================================

============================================================
Statistical Analysis of Pareto-Optimal Solutions
============================================================

Cost (Million USD):
  Min: $317.45M
  Max: $862.86M
  Mean: $589.44M
  Std: $137.68M

Risk Score:
  Min: 15.98
  Max: 100.00
  Mean: 47.45
  Std: 20.14

Scientific Return:
  Min: 0.00
  Max: 100.00
  Mean: 57.39
  Std: 31.19

============================================================
Representative Mission Configurations
============================================================

1. LOWEST COST MISSION:
   Cost: $317.45M
   Risk: 16.81
   Science: 0.00
   Configuration:
     - Cameras: 0
     - Spectrometers: 0
     - Drills: 0
     - Duration: 163 days
     - Solar panels: 10.2 m²
     - Bandwidth: 1.0 Mbps

2. HIGHEST SCIENCE MISSION:
   Cost: $679.28M
   Risk: 74.37
   Science: 100.00
   Configuration:
     - Cameras: 4
     - Spectrometers: 2
     - Drills: 1
     - Duration: 344 days
     - Solar panels: 7.7 m²
     - Bandwidth: 1.2 Mbps

3. LOWEST RISK MISSION:
   Cost: $372.65M
   Risk: 15.98
   Science: 0.00
   Configuration:
     - Cameras: 0
     - Spectrometers: 0
     - Drills: 0
     - Duration: 143 days
     - Solar panels: 29.5 m²
     - Bandwidth: 2.5 Mbps

4. BALANCED MISSION (Best compromise):
   Cost: $575.83M
   Risk: 44.20
   Science: 61.89
   Configuration:
     - Cameras: 2
     - Spectrometers: 2
     - Drills: 1
     - Duration: 220 days
     - Solar panels: 10.3 m²
     - Bandwidth: 1.3 Mbps

============================================================

Interpretation of Results

The visualization suite provides comprehensive insights:

3D Pareto Front

The three-dimensional plot reveals the complete trade-off surface. The color gradient (viridis colormap) shows scientific return values, making it easy to identify regions where high science can be achieved despite cost or risk constraints.

2D Projections

Three 2D scatter plots show pairwise relationships:

• Cost vs Risk: Generally, lower-cost missions tend to have higher risk (fewer redundancies)
• Cost vs Science: Higher investment typically enables better science, but with diminishing returns
• Risk vs Science: This often shows an interesting trade-off where maximizing science may require accepting moderate risk

Decision Variable Analysis

The instrument distribution histogram reveals which configurations appear most frequently in Pareto-optimal solutions. This indicates robust choices across different priority weightings.

The mission duration vs instrument count plot shows how longer missions can justify carrying more instruments due to increased data collection opportunities.

Representative Configurations

The code identifies four key mission archetypes:

1. Lowest Cost: Minimal viable mission with essential instruments
2. Highest Science: Flagship mission with comprehensive instrumentation
3. Lowest Risk: Conservative design with high reliability margins
4. Balanced: Compromise solution minimizing distance to an ideal point where all objectives are simultaneously optimized

Practical Applications

Mission planners can use this Pareto frontier to:

• Understand fundamental trade-offs between competing objectives
• Justify design decisions based on quantitative analysis
• Adapt configurations as budget constraints or risk tolerance changes
• Identify whether incremental cost increases provide proportional science gains
• Support stakeholder discussions with concrete alternatives

This multi-objective optimization framework is applicable beyond space exploration to any complex system design problem involving competing objectives: aerospace engineering, infrastructure planning, resource allocation, and portfolio optimization.

The NSGA-II algorithm’s strength lies in producing a diverse set of non-dominated solutions rather than forcing premature convergence to a single “optimal” design, acknowledging that different stakeholders may have different priority weightings among the objectives.