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Multi-Objective Optimization with Desirability and Morris-Mitchell Criterion

Thomas Bartz-Beielstein, Eva Bartz, Alexander Hinterleitner, Christoph Leitenmeier, Ihab Abd El Hussein

Abstract

Industrial experimental designs frequently lack optimal space-filling properties, rendering them unrepresentative. This study presents a comprehensive methodology to refine existing designs by enhancing coverage quality while optimizing experimental outcomes. We discuss and analyse variants of the Morris-Mitchell criterion to quantify and improve spatial distributions. Based on potential theory, we analyze monotonicity properties and limitations of the Morris-Mitchell criteria. Practically, we implement a multi-objective optimization framework utilizing the Python packages spotdesirability and spotoptim. This framework uses desirability functions to combine surrogate-model predictions with space-filling enhancements into a unified score. Demonstrated through data from a compressor development case study, this approach optimizes performance objectives alongside design coverage. To facilitate implementation, we introduce novel infill-point diagnostics that visually guide the sequential placement of design points. This integrated methodology successfully bridges spatial theory with engineering application, balancing the crucial exploration and exploitation trade-off.

Multi-Objective Optimization with Desirability and Morris-Mitchell Criterion

Abstract

Industrial experimental designs frequently lack optimal space-filling properties, rendering them unrepresentative. This study presents a comprehensive methodology to refine existing designs by enhancing coverage quality while optimizing experimental outcomes. We discuss and analyse variants of the Morris-Mitchell criterion to quantify and improve spatial distributions. Based on potential theory, we analyze monotonicity properties and limitations of the Morris-Mitchell criteria. Practically, we implement a multi-objective optimization framework utilizing the Python packages spotdesirability and spotoptim. This framework uses desirability functions to combine surrogate-model predictions with space-filling enhancements into a unified score. Demonstrated through data from a compressor development case study, this approach optimizes performance objectives alongside design coverage. To facilitate implementation, we introduce novel infill-point diagnostics that visually guide the sequential placement of design points. This integrated methodology successfully bridges spatial theory with engineering application, balancing the crucial exploration and exploitation trade-off.
Paper Structure (31 sections, 4 theorems, 33 equations, 31 figures)

This paper contains 31 sections, 4 theorems, 33 equations, 31 figures.

Table of Contents

  1. Introduction
  2. The Dataset from Compressor Development
  3. Data Loading and Preparation
  4. Surrogate Models
  5. Random Forest Regressor
  6. The Evaluation Function for Multiple Models
  7. Gaussian Process Regressor
  8. Cross-validation Comparison of the Models
  9. An Intensified Morris-Mitchell Criterion
  10. The Default Morris-Mitchell Criterion
  11. The Intensified Morris-Mitchell Criterion
  12. Using mmphi_intensive in spotoptim
  13. Efficient Updates with mmphi_intensive_update
  14. Sensitivity of the Morris-Mitchell Criterion with Respect to Added Points
  15. Analysis of Monotonicity
  16. ...and 16 more sections

Key Result

Theorem 4.1

The intensified Morris-Mitchell criterion $\Phi^{\ast}$ decreases if and only if the average $d^{-q}$ contribution of the new point to the existing points (weighted by the new distance multiplicities $J_k'$), i.e., is strictly below the current design's overall average pair contribution (weighted by the existing distance multiplicities $J_j$), i.e., $\blacktriangleleft$$\blacktriangleleft$

Figures (31)

  • Figure 1: Pareto front of the original data. Both functions should be maximized
  • Figure 2: Predictions for z8 of the multi-objective RF model compared to the original data
  • Figure 3: Predictions for z1 of the multi-objective RF model compared to the original data
  • Figure 4: Pareto front of the Random Forest surrogate model predictions and the original Pareto front (with possible outliers)
  • Figure 5: Predictions for z8 of the multi-objective GP model compared to the original data
  • ...and 26 more figures

Theorems & Definitions (21)

  • Definition 4.1: Morris-Mitchell Criterion
  • Definition 4.2: Maximin and Minimax Criteria
  • Definition 4.3: Intensified Morris-Mitchell Criterion
  • Example 4.1: Usage of mmphi_intensive
  • Example 4.2: Usage of mmphi_intensive_update
  • Remark 4.1: Hypothesis on the Sensitivity of the Morris-Mitchell Criterion to Added Points
  • Theorem 4.1: Monotonicity behavior of $\Phi^{\ast}$
  • proof
  • Example 4.3: Adding an optimal point increases $\Phi^{\ast}$
  • Theorem 4.2: Scaling behavior of $\Phi$ and $\Phi^{\ast}$ for optimal designs
  • ...and 11 more