Construction of optimal fractional Order-of-Addition designs via block designs

doi:10.1016/j.spl.2020.108728

Statistics & Probability Letters

Volume 161, June 2020, 108728

https://doi.org/10.1016/j.spl.2020.108728 Get rights and content

Abstract

Order of addition (OofA) experiments have found wide applications. Each order (run) of an OofA experiment is a permutation of $m$ $(\geq 2)$ components. It is typically infeasible to compare all the $m!$ possible runs, especially when $m$ is large. This calls for experimentation with a subset or fraction of these $m!$ runs. However, only a few systematic results are available on the construction of such fractions ensuring optimality. We employ block designs to propose a systematic combinatorial construction method for optimal fractional OofA designs, and extend the method to construct highly efficient OofA designs, both in much smaller run sizes than the currently available optimal fractions.

Introduction

Many physical phenomena encountered in science and engineering are affected by the addition order of $m$ materials or components. In chemical experiments, the performance or the amount of reaction production is often determined by the order of adding reagents. For example, Song et al. (2014) conducted experiments with the $3!$ orderings of three reagents, namely, carbon dots, ${Fe}^{2 +}$ and $H_{2} O_{2}$ , and found that addition of ${Fe}^{2 +}$ and $H_{2} O_{2}$ before carbon dots leads to stronger photoluminescence intensities. Jiang and Ng (2014) reported chemical experiments where the output is influenced by the order of addition of four components, ${Hg}^{2 +}, {Cu}^{2 +}, S^{2 -}$ and light. Order of addition (OofA) experiments have found wide application also in bio-chemistry (Shinohara and Ogawa, 1998), nutritional science (Karim et al., 2000) and NP-hard ordering problem (Chen et al., 2019), just to name a few.

With $m$ components, there are $m!$ possible orders or runs, each is a permutation of ${1, 2, \dots, m}$ . It is infeasible to include all of these in an experiment, especially when $m$ is large. This warrants experimentation with a subset or fraction of these $m!$ runs. The problem of selecting such a fraction has received attention in the literature. For instance, Van Nostrand (1995) considered the design with “pseudo factors” or pairwise ordering (PWO) factors. Lin and Peng (2019) reviewed the latest work on the design and model of OofA experiments, and introduced some new thoughts. Voelkel (2019) provided some smaller OofA orthogonal arrays under some proposed design criteria. Peng et al. (2019b) considered different types of optimality criteria and proposed a systematic method to construct a class of robust optimal fractional OofA designs. Zhao et al. (2018) considered the minimal-point OofA designs. Yang et al. (2018) obtained a number of OofA designs called component orthogonal arrays that are optimal under their component-position model. Chen et al. (2019) introduced a statistical method to speculate solutions of NP-hard ordering problem by making use of design for OofA experiment.

The current literature on OofA designs focuses on computer search. A notable exception is Peng et al. (2019b) who present a systematic construction of optimal fractional OofA designs in $m! ∕ s!$ runs, where $m = 2 s$ for even $m$ and $m = 2 s + 1$ for odd $m$ . However, these are quite large for larger $m$ . Thus, with $m = 10$ , their construction requires 30240 runs which, despite being less than 1% of the run size ( $= 10!$ ) of the full design, is still rather large. With regard to smaller fractions, Peng et al. (2019b) hinted at the possibility of using block designs, but did not pursue the idea any further. We develop and formalize this approach here, and show in the subsequent sections how it can yield optimal or highly efficient fractional OofA designs in considerably smaller run sizes. The proofs are provided in the Appendix.

Section snippets

Optimality of fractional OofA designs

Denote a typical order by $π = (π_{1}, \dots, π_{m})$ , which is a permutation of ${1, \dots, m}$ . Based on a fraction $Π$ of size $N$ from the $m!$ possible orders, consider the pairwise order (PWO) model $y (π) = β_{0} + \sum_{i < j} β_{i j} I_{i j} (π) + ε, π \in Π,$ where $I_{i j} = + 1$ , if component $i$ precedes component $j$ ; and $I_{i j} = - 1$ otherwise; $y$ is the response of interest; and $ε$ is the observational error. As usual, such errors are assumed to be uncorrelated, all with mean zero and the same variance. There are $p = 1 + m (m - 1) ∕ 2$ parameters of interest as given by the

Optimal fractions

Theorem 1 and Proposition 1 lead to a systematic construction which is conveniently described in Algorithm 1.

Remark 1

Let $A_{3}$ denote the full OofA design for $m = 3$ , and $A_{4}, A_{5}, A_{6}$ denote OofA orthogonal arrays in 12, 12 and 24 runs for $m = 4, 5, 6$ , respectively (Voelkel, 2019). In Algorithm 1, $A$ or $\bar{A}$ may, in particular, be chosen as these arrays. An illustrative example follows.

Example 1

(a)

For $m = 8$ , obtain $d$ as in (II) above with $\tilde{d}$ constructed from a Hadamard matrix of order $8$ , and take $A = \bar{A} = A_{4}$ . Then $b = 7, R = 12$ , and $d^{*} = A_{8}$

Concluding remarks

Block designs were employed in this paper to obtain a systematic construction method leading to optimal and highly efficient fractional OofA designs in much smaller run sizes than the currently available optimal fractions. Future work on further reduction in run size in a systematic manner with theoretical assurance of optimality will be welcome.

While we worked in the framework of the PWO model (1), computations show that our designs remain highly efficient under variations of this model that

CRediT authorship contribution statement

Jianbin Chen: Writing - original draft, Methodology, Conceptualization. Rahul Mukerjee: Writing - review & editing, Methodology, Conceptualization. Dennis K.J. Lin: Writing - review & editing, Methodology, Conceptualization.

Acknowledgments

We thank the referees for their very constructive suggestions. The work of Jianbin Chen was supported by the National Natural Science Foundation of China (Grant No. 11771220). The work of Rahul Mukerjee was supported by the J.C. Bose National Fellowship of Government of India and a grant from Indian Institute of Management Calcutta. The work of Dennis Lin was supported by the National Science Foundation, USA via Grant DMS-18102925.

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Construction of optimal fractional Order-of-Addition designs via block designs

Abstract

Introduction

Section snippets

Optimality of fractional OofA designs

Optimal fractions

Concluding remarks

CRediT authorship contribution statement

Acknowledgments

J. Nutr.

Sequential logic operations with a molecular keypad lock with four inputs and dual fluorescence outputs

Angew. Chem. Int. Ed.

The order-of-addition experiments: a review and some new thoughts (with discussion)

Qual. Eng.

Order of addition modeling

Statist. Sinica

Swap-r algorithm for the construction of order-of-addition designs

Design of order-of-addition experiments

Biometrika