Multi-Objective Process Optimization

# Multi-Objective Process Optimization

## Introduction & Motivation

Multi-Objective Process Optimization (MOPO) solves complex trade-offs in semiconductor and materials manufacturing where competing operational goals must be optimized simultaneously. For instance, increasing etch rate often worsens profile verticality and increases surface roughness.

Motivation: Identify the Pareto-optimal frontier of non-dominated process parameters rather than relying on single-objective scalar approximations.

Applications: Wafer throughput vs defect rate optimization, chemical usage vs etch selectivity trade-offs, thermal budget vs dopant activation.

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## Core Concepts & Theoretical Foundations

### Pareto Optimality

A process parameter vector $$\mathbf{x}^* \in \mathcal{X}$$ is Pareto-optimal if there exists no other $$\mathbf{x} \in \mathcal{X}$$ such that:

$$f_i(\mathbf{x}) \le f_i(\mathbf{x}^*) \quad orall i \in \{1, \dots, m\}$$

with at least one strict inequality.

### Weighted Chebyshev Scalarization

To explore non-convex Pareto frontiers, Chebyshev scalarization minimizes:

$$\min_{\mathbf{x} \in \mathcal{X}} \max_{i=1, \dots, m} \left\{ w_i \cdot \left| f_i(\mathbf{x}) - z_i^* ight| ight\}$$

where $$z_i^*$$ is the ideal reference point for objective $$i$$, and $$w_i$$ are positive weighting coefficients.

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## Python Laboratories & Practical Implementations

### Lab 1: Multi-Objective Manufacturing Surrogate Evaluator

import numpy as np

def manufacturing_objectives(x):
 # x[0]: RF Power (W), x[1]: Pressure (mTorr)
 power, pressure = x[0], x[1]
 
 # Objective 1: Maximize Etch Rate (minimize negative rate)
 etch_rate = 5.0 * power + 2.0 * pressure - 0.01 * (power ** 2)
 f1 = - etch_rate 
 
 # Objective 2: Minimize Surface Roughness (nm)
 roughness = 0.5 + 0.02 * power + 0.05 * pressure + 0.0001 * (power * pressure)
 f2 = roughness
 return np.array([f1, f2])

x_test = np.array([300.0, 20.0])
objs = manufacturing_objectives(x_test)
print(f"Parameters: Power={x_test[0]}W, Pressure={x_test[1]}mTorr")
print(f"Etch Rate: {-objs[0]:.2f} nm/min | Surface Roughness: {objs[1]:.2f} nm")

### Lab 2: Pareto Non-Dominated Sorting Engine

import numpy as np

def identify_pareto_front(costs):
 is_efficient = np.ones(costs.shape[0], dtype=bool)
 for i, c in enumerate(costs):
 if is_efficient[i]:
 is_efficient[is_efficient] = np.any(costs[is_efficient] < c, axis=1) | np.all(costs[is_efficient] == c, axis=1)
 is_efficient[i] = True
 return is_efficient

np.random.seed(42)
candidate_params = np.column_stack([
 np.random.uniform(100, 500, 100),
 np.random.uniform(5, 50, 100)
])

cost_matrix = np.array([manufacturing_objectives(p) for p in candidate_params])
pareto_mask = identify_pareto_front(cost_matrix)
print("Total Candidates Evaluated:", len(candidate_params))
print("Number of Pareto-Optimal Solutions Found:", np.sum(pareto_mask))

### Lab 3: Weighted Chebyshev Scalarization Optimizer

import numpy as np

def chebyshev_scalarize(objectives, weights, ideal_point):
 diffs = weights * np.abs(objectives - ideal_point)
 return np.max(diffs, axis=1)

ideal = np.min(cost_matrix, axis=0) # Best achievable values
weights_equal = np.array([0.5, 0.5])
scalarized_scores = chebyshev_scalarize(cost_matrix, weights_equal, ideal)

best_idx = np.argmin(scalarized_scores)
best_param = candidate_params[best_idx]
best_obj = cost_matrix[best_idx]

print(f"Optimal Chebyshev Parameter: Power={best_param[0]:.1f}W, Pressure={best_param[1]:.1f}mTorr")
print(f"Resulting Etch Rate: {-best_obj[0]:.2f} nm/min | Roughness: {best_obj[1]:.2f} nm")

### Lab 4: Hypervolume Indicator Calculation

import numpy as np

def calculate_hypervolume_2d(pareto_costs, ref_point):
 # Sort by first objective
 sorted_idx = np.argsort(pareto_costs[:, 0])
 sorted_costs = pareto_costs[sorted_idx]
 
 hv = 0.0
 last_y = ref_point[1]
 for cost in sorted_costs:
 if cost[0] < ref_point[0] and cost[1] < ref_point[1]:
 width = ref_point[0] - cost[0]
 height = last_y - cost[1]
 if height > 0:
 hv += width * height
 last_y = cost[1]
 return hv

pareto_costs = cost_matrix[pareto_mask]
ref_point = np.array([0.0, 10.0]) # Worst-case bound
hv = calculate_hypervolume_2d(pareto_costs, ref_point)
print(f"Calculated Hypervolume Indicator: {hv:.2f}")

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## Summary & Best Known Methods

1. Non-Dominated Sorting: Filter parameter sweep data using strict dominance criteria.
2. Chebyshev Scalarization: Avoid missing non-convex Pareto regions inherent to linear weighted sums.
3. Hypervolume Validation: Track hypervolume expansion across optimization iterations to verify convergence.