# Construction of Blocked Factorial Designs to Estimate Main Effects and Selected Two-Factor Interactions

## TL;DR

This paper develops a new method for constructing blocked two-level factorial designs that efficiently estimate main effects and selected interactions, using graph theory to improve upon traditional minimum aberration criteria.

## Contribution

It introduces a design construction approach based on graph theory, specifically vertex colorings, to optimize factorial designs for estimating effects and interactions.

## Key findings

- Designs with blocks of size four are effectively constructed using the proposed method.
- The new approach can outperform minimum aberration designs in certain estimation scenarios.
- Examples demonstrate the practical application of the construction strategy.

## Abstract

Two-level factorial designs are widely used in industrial experiments. For processes involving \(n\) factors, the construction of designs comprising \(2^n\) and \(2^{n-p}\) factorials, arranged in blocks of size \(2^q\) is investigated. The aim is to estimate all main effects and a selected subset of two-factor interactions. Designs constructed according to minimum aberration criteria are shown to not necessarily be the most appropriate designs in this situation. A design construction approach is proposed which exploits known results on proper vertex colourings in graph theory. Examples are provided to illustrate the results and construction strategies. Particular consideration is given to the special case of designs with blocks of size four and tables of designs are given for this block size.