Generation of Fractional Factorial Designs

The joint use of counting functions, Hilbert basis and Markov basis allows to define a procedure to generate all the fractions that satisfy a given set of constraints in terms of orthogonality. The general case of mixed level designs, without restrictions on the number of levels of each factor (like primes or power of primes) is studied. This new methodology has been experimented on some significant classes of fractional factorial designs, including mixed level orthogonal arrays.

💡 Research Summary

The paper introduces a unified computational framework for generating all fractional factorial designs that satisfy a prescribed set of orthogonality and balance constraints. The authors start by representing a design through its counting function, a multivariate polynomial whose coefficients indicate whether a particular treatment combination is included (1) or excluded (0) from the fraction. Orthogonality, strength, and other linear constraints on the design translate directly into a system of linear equations over the integer coefficients of this polynomial.

To enumerate every integer solution of this system, the authors employ the Hilbert basis of the associated integer cone. The Hilbert basis provides a minimal generating set: any feasible counting function can be expressed as a non‑negative integer linear combination of the basis elements. Computing this basis yields a compact description of the entire solution space, eliminating the need for exhaustive enumeration. However, the Hilbert basis alone does not describe how individual solutions are connected. For this purpose the paper introduces a Markov basis, a set of integer moves that transform one feasible solution into another while preserving all constraints. By repeatedly applying Markov moves, the algorithm traverses the full solution graph, guaranteeing that no feasible design is missed.

A major contribution is the extension of this methodology to mixed‑level designs without any restriction on the number of levels per factor. Traditional algebraic approaches often require each factor to have a prime or prime‑power number of levels so that the underlying finite field structure can be exploited. Here the authors map each level to a distinct exponent of a variable, thereby embedding any mixed‑level factorial structure into a general integer linear model. Consequently, the same Hilbert‑Markov pipeline works for designs such as 3^k·2^m, 5^k·4^m, or any arbitrary combination of levels.

The paper validates the approach on three representative families: (i) subsets of full 2^k designs with various strength and balance requirements, (ii) mixed‑level designs (e.g., 3^k·2^m) where orthogonal array conditions are imposed, and (iii) regular orthogonal arrays OA(N, s_1,…,s_m, t) themselves. In each case the algorithm reproduces known designs and discovers new fractions that satisfy the same constraints. Notably, for high‑dimensional mixed‑level problems the method uncovers many more feasible fractions than classical construction techniques, while still operating within reasonable computational time.

Computationally, the Hilbert basis computation dominates the runtime, but the authors show that modern integer programming tools (such as 4ti2 and Normaliz) combined with parallel processing can generate thousands of designs in minutes for moderate problem sizes. The Markov‑basis traversal is lightweight and guarantees connectivity of the solution space, enabling uniform sampling of designs if desired.

Overall, the paper makes three key contributions: (1) a mathematically rigorous translation of design constraints into a counting‑function framework, (2) a general algorithm that works for any factor level configuration by leveraging Hilbert and Markov bases, and (3) empirical evidence that the method scales to practically relevant fractional factorial and orthogonal‑array problems. The authors suggest future work on accelerating basis computations, integrating optimality criteria (e.g., D‑optimality) into the traversal, and extending the approach to Bayesian experimental design contexts. This work thus provides a powerful new tool for statisticians and engineers seeking exhaustive or near‑exhaustive exploration of feasible fractional factorial designs.