Non-Uniqueness of E(s2)-Optimal Supersaturated Designs for N ≡ 2 (mod 4) Runs with Application to the Case N = 10 Runs

An algorithm for constructing mixed-level [formula omitted]-circulant supersaturated designs

Supersaturated designs (SSDs) have been widely used in factor screening experiments. The present paper aims to prove that the maximal balanced designs are a kind of special optimal SSDs under the E(fNOD) criterion. We also propose a new method, called the complementary design method, for constructing E(fNOD) optimal SSDs. The basic principle of this method is that for any existing E(fNOD) optimal SSD whose E(fNOD) value reaches its lower bound, its complementary design in the corresponding maximal balanced design is also E(fNOD) optimal. This method applies to both symmetrical and asymmetrical (mixed-level) cases. It provides a convenient and efficient way to construct many new designs with relatively large numbers of factors. Some newly constructed designs are given as examples.

Optimal mixed-level k-circulant supersaturated designs

Supersaturated designs (SSDs) have been highly valued in recent years for their ability of screening out important factors in the early stages of experiments. Recently, Liu and Lin (in Statist Sinica 19:197–211, 2009) proposed a method to construct optimal mixed-level SSDs from smaller multi-level SSDs and transposed orthogonal arrays (OAs). This paper extends their method to construct more equidistant optimal SSDs by replacing the multi-level SSDs and transposed OAs with mixed-level SSDs and general transposed difference matrices, respectively, and then proposes two practical methods for constructing weak equidistant SSDs based on this extended method. A large number of new optimal SSDs can be constructed from these three methods. Some examples are provided and more new designs are listed in “Appendix” for practical use.

Supersaturated designs (SSDs) are crucial in factor screening experiments, especially when factor sparsity is assumed, meaning only a few factors are expected to be significant. Building on the foundational work of Jones and Majumdar [1], who introduced the UE(\(S^2 )\) criterion as an improvement over the E(\(S^2 )\) criterion by Booth and Cox [2], this study simplifies the construction of UE(\(S^2 )\) -optimal designs. The UE(\(S^2 )\) criterion is similar to the E(\(S^2 )\) criterion but removes the requirement for factor level balance. Our contribution lies in further simplifying these methods, explaining them with practical examples, and providing proofs for lower bounds for UE(\(S^2 )\) designs. Through this study, we aim to make the concepts and applications of supersaturated designs more accessible and easier to understand for practitioners. These methods can significantly optimize resource use and reduce costs in industrial, biological, and agricultural experiments. The study's implications extend to any field requiring efficient factor screening, offering a robust framework for future research.

A supersaturated design (SSD) is a factorial design whose run size is not enough for estimating all the main effects. Such designs have received much recent interest because of their potential in factor screening experiments. This paper first shows that the design obtained by the Kronecker sum of a balanced design and a generalized Hadamard matrix (i.e., a matrix with both itself and its transpose being difference matrices) has some nice properties. Based on these findings, some new methods for constructing -optimal SSDs via generalized Hadamard matrices are developed. Meanwhile, the non-orthogonality of the proposed designs is well controlled by the source designs. In addition, some generalized Hadamard matrices with nice properties are constructed for obtaining -optimal SSDs. The proposed methods are easy to implement and many new SSDs can then be constructed.

Some theory and the construction of mixed-level supersaturated designs

Supersaturated designs are an important class of factorial designs in which the number of factors is larger than the number of runs. These designs supply an economical method to perform and analyze industrial experiments. In this paper, we consider generalized Legendre pairs and their corresponding matrices to construct E(s 2)-optimal two-level supersaturated designs suitable for screening experiments. Also, we provide some general theorems which supply several infinite families of E(s 2)-optimal two-level supersaturated designs of various sizes.

A supersaturated design is essentially a factorial design with the equal occurrence of levels property and no fully aliased factors in which the number of main effects is greater than the number of runs. It has received much recent interest because of its potential in factor screening experiments. A packing design is an important object in combinatorial design theory. In this paper, a strong link between the two apparently unrelated kinds of designs is shown. Several criteria for comparing supersaturated designs are proposed, their properties and connections with other existing criteria are discussed. A combinatorial approach, called the packing method, for constructing optimal supersaturated designs is presented, and properties of the resulting designs are also investigated. Comparisons between the new designs and other existing designs are given, which show that our construction method and the newly constructed designs have good properties.

Supersaturated design (SSD) has received much recent interest because of its potential in factor screening experiments. In this paper, we provide equivalent conditions for two columns to be fully aliased and consequently propose methods for constructing $E(f_{\mathrm{NOD}})$- and $\chi^2$-optimal mixed-level SSDs without fully aliased columns, via equidistant designs and difference matrices. The methods can be easily performed and many new optimal mixed-level SSDs have been obtained. Furthermore, it is proved that the nonorthogonality between columns of the resulting design is well controlled by the source designs. A rather complete list of newly generated optimal mixed-level SSDs are tabulated for practical use.

Optimal mixed-level supersaturated design with general number of runs

A cyclic construction of saturated and supersaturated designs

Supersaturated designs (SSDs) are often used to reduce the number of experimental runs in screening experiments with a large number of factors. As more factors are used in the study, the search for an optimal SSD becomes increasingly challenging because of the large number of feasible selection of factor level settings. This article tackles this discrete optimization problem via an algorithm based on swarm intelligence. Using the commonly used E(s2) criterion as an illustrative example, we propose an algorithm to find E(s2)-optimal SSDs by showing that they attain the theoretical lower bounds found in previous literature. We show that our algorithm consistently produces SSDs that are at least as efficient as those from the traditional CP exchange method in terms of computational effort, frequency of finding the E(s2)-optimal SSD, and also has good potential for finding D3-, D4-, and D5-optimal SSDs. Supplementary materials for this article are available online.

Supersaturated designs robust to two-factor interactions

Addition of runs to an s-level supersaturated design