# Pareto-Front Optimization of Variance-Added Expected Loss with Interrelated Qualities

## TL;DR

This paper introduces a new optimization method that balances accuracy and uncertainty in quality control by considering both bias and variance together.

## Contribution

The novel contribution is a Pareto-front optimization framework that integrates bias and variance into a single loss function for interrelated quality characteristics.

## Key findings

- The proposed framework enables a more balanced optimization by capturing both deviation from targets and system uncertainty.
- Pareto-front analysis reveals trade-offs between expected loss and variance, allowing for flexible decision-making.
- The method is validated through examples and a case study, showing its effectiveness in quality design.

## Abstract

In industries, particularly in quality optimization, the trade-off between model bias and variance is inevitable, reflecting the tension between accuracy and uncertainty. Traditional methods often address these aspects separately, potentially leading to suboptimal decisions. This study proposes a Pareto-front optimization framework for a variance-added expected loss function within the context of interrelated quality characteristics. By integrating multivariate quadratic loss with a variance term, our approach simultaneously captures deviation from targets (bias) and system uncertainty (variance). Unlike sequential approaches that first minimize bias and then variance—often increasing total risk—our weighted formulation flexibly adjusts for their trade-offs. This enables a more balanced and efficient optimization process that identifies solutions with lower overall risk. Through Pareto-front analysis, we reveal trade-offs between expected loss and variance, allowing users to select optimal quality designs based on their preferred bias–variance balance. Representative examples and a case study adopted from the literature validate the effectiveness and practical applicability of the proposed method.