Order-of-Addition Orthogonal Arrays with High Strength

Abstract

In order-of-addition experiments, the full order-of-addition designs are often unaffordable due to their large run sizes. The problem of finding efficient fractional OofA designs arises. The order-of-addition orthogonal arrays are a class of optimal fractional order-of-addition designs for the prevalent pair-wise ordering model, under a variety of widely used design criteria. In the literature, the studies on order-of-addition orthogonal arrays focused on strength 2 while the order-of-addition orthogonal arrays of higher strength have not been investigated yet. In this paper, we focus on order-of-addition orthogonal arrays of strength 3. First, the method of constructing order-of-addition orthogonal arrays of strength 3 is proposed. Second, a theoretical result that states that the order-of-addition orthogonal arrays of strength 3 have better balance properties than those of strength 2 is developed. Third, we provide thorough simulation studies which show that the constructed order-of-addition orthogonal arrays of strength 3 have desirable performance for estimating optimal orders of addition.

1. Introduction

In many experiments, the response is definitely affected by the order of processing of the materials or components. We call this class of experiments the order-of-addition (OofA) experiments. An example is the famous experiment of a lady testing tea in which two different orders, “tea preceding milk” and “milk preceding tea”, were tested [1]. To illustrate the characteristics of OofA experiments, we introduce one more example from [2]. In [2], three anti-tumor drugs (coded as $c_1$, $c_2$, and $c_3$, respectively) were added into tumor cells either sequentially (following the six orders $c_1\to c_2\to c_3,c_1\to c_3\to c_2,c_2\to c_1\to c_3,c_2\to c_3\to c_1,c_3\to c_1\to c_2$, and $c_3\to c_2\to c_1$) or simultaneously. The percentage of tumor inhibitions, a larger-the-better response, was measured at 12 h after the last drug was administrated. The largest response was yielded when the three anti-tumor drugs were administrated following the order $c_2\to c_3\to c_1$ rather than simultaneously. The OofA effect also matters in many other scientific disciplines including chemical science [3], bio-chemistry [4], food science [5], and manufacturing [6]. More applications of the order of addition can be found in [7,8] and the references therein.

Three prevalent models for the OofA problem have been proposed. Ref. [9] proposed the pair-wise ordering (PWO) model which will be detailed in Section 2. Ref. [2] proposed the component-position (CP) model which assumes that a component has different OofA effects when it is processed at different positions in an order. Ref. [8] proposed using the mapping-based universal Kriging model for OofA experiments with blocking. In this paper, we consider the OofA experiments without blocking effects which are not suitable for the the mapping-based universal Kriging model. Compared to the CP model, the PWO model has stronger interpretability and fewer parameters to be estimated, indicating less experimental cost. With this in mind, this paper carries out studies under the PWO model.

Suppose m components, denoted as $c_1,c_2,\dots ,c_m$, are considered in an OofA experiment, there are $m!$ different orders. We call a design which consists of these $m!$ different orders the full OofA design. It is often unaffordable to perform OofA experiments by using full OofA designs especially when m is large. For example, when $m=6$, the full OofA design contains 6!(=720) different orders. Thus, the study on efficient fractional OofA designs becomes important. Under the PWO model, [10] proposed a class of fractional OofA designs called the OofA orthogonal arrays (OofA-OAs) which will be defined in Section 2. Ref. [11] proved that OofA-OAs are optimal for the PWO model under some widely used design criteria including D-criterion, where the D-criterion is defined as follows. Suppose $\mathit{X}$ is a model matrix of a design under a certain model, N is the run size of $\mathit{X}$ and m is the number of columns in $\mathit{X}$, then D-efficiency is defined as ${(\mathrm{d}\mathrm{e}\mathrm{t}({\mathit{X}}^T\mathit{X}))}^{1/m}/N,$ where the superscript T denotes transpose. A design with a larger D-efficiency is better. Ref. [11] provided a closed-form construction method for OofA-OAs of strength 2 which have quite large run sizes. Ref. [12] provided smaller OofA-OAs of strength 2 compared to those in [11] via block designs. Ref. [13] proposed a systematical construction method of OofA-OAs of strength 2 which further reduced the run sizes compared to the work in [11,12]. Ref. [2] proposed the component orthogonal arrays which are D-optimal for the CP model. Refs. [2,14,15] respectively proposed different methods of constructing the component orthogonal arrays. Some other work under the PWO model can be found in [16] which extended the PWO model by entertaining interactions of PWO factors, and [17] which proposed a class of minimal-point OofA designs that have good D-efficiencies for the PWO model.

Throughout the literature on efficient fractional OofA designs for the PWO model, there is no study on the OofA-OAs of strength 3 which are D-optimal for the PWO model while saving a considerable amount of experimental costs compared to the full OofA designs, and have better balance properties than those of strength 2, as will be proved in this paper. The contributions of this paper are threefold: (1) we first propose a method of constructing OofA-OAs of strength 3 which is capable of finding non-isomorphic OofA-OAs of strength 3; (2) some balance properties of OofA-OAs of strength 3 are developed; (3) thorough simulation studies are conducted which show that the constructed OofA-OAs of strength 3 have desirable performance on estimating the optimal orders of addition.

The rest of the paper is organized as follows. In Section 2, we introduce the formulation of PWO model and the definition of OofA-OAs. The isomorphism of OofA designs is also defined in this section. Section 3 gives a construction method of OofA-OAs of strength 3. Section 4 explores the balance properties of OofA-OAs of strength 3. The thorough simulation studies, which show that the constructed OofA-OAs of strength 3 have desirable performance on estimating optimal orders of addition, are included in Section 5. Section 6 includes results and discussions. The conclusions are given in Section 7. Some proofs and useful design tables are deferred to Appendix A and Appendix B, respectively.

2. Preliminaries

Denote ${\mathit{O}}_m$ as the full OofA design of m components, where the orders in ${\mathit{O}}_m$ are arranged in reversed lexicographical order. For example, the orders of ${\mathit{O}}_3$ are displayed in Table 1.

Table 1. Full OofA design ${\mathit{O}}_3$ and full PWO design ${\mathit{P}}_3$.

	PWO Factors (${\mathit{P}}_3$)
${\mathit{O}}_{\mathbf{3}}$	${\mathit{z}}_{12}$	${\mathit{z}}_{\mathbf{13}}$	${\mathit{z}}_{\mathbf{23}}$
$c_3{c}_2{c}_1$	$-1$	$-1$	$-1$
$c_3{c}_1{c}_2$	1	$-1$	$-1$
$c_2{c}_3{c}_1$	$-1$	$-1$	1
$c_2{c}_1{c}_3$	$-1$	1	1
$c_1{c}_3{c}_2$	1	1	$-1$
$c_1{c}_2{c}_3$	1	1	1

Suppose ${\mathit{o}}_k$, $k=1,2,\dots ,m!$, is the k-th order in ${\mathit{O}}_m$. Denote $\tau ({\mathit{o}}_k)$ as the observation arising from ${\mathit{o}}_k$. The PWO model is established as

$$\tau ({\mathit{o}}_k)={\beta }_0+\sum _{i=1}^{m-1}\sum _{j=i+1}^m{\beta }_{ij}{\lambda }_{ij}({\mathit{o}}_k)+\epsilon ({\mathit{o}}_k),$$

(1)

where, for $i<j$, ${\lambda }_{ij}({\mathit{o}}_k)=1$ if component $c_i$ precedes $c_j$ in ${\mathit{o}}_k$, otherwise ${\lambda }_{ij}({\mathit{o}}_k)=-1$, $\epsilon ({\mathit{o}}_k)\sim N(0,{\sigma }^2)$ for any ${\mathit{o}}_k$, $\epsilon ({\mathit{o}}_k)$ is independent of $\epsilon ({\mathit{o}}_l)$ for $k\ne l$, and ${\beta }_0$, ${\beta }_{ij}$s are unknown parameters to be estimated. For example when $m=3$, ${\lambda }_{12}(c_1{c}_2{c}_3)=1$ as $c_1$ precedes $c_2$ and ${\lambda }_{12}(c_2{c}_3{c}_1)=-1$ as $c_2$ precedes $c_1$. Let ${\mathit{z}}_{ij}={({\lambda }_{ij}({\mathit{o}}_1),{\lambda }_{ij}({\mathit{o}}_2),\dots ,{\lambda }_{ij}({\mathit{o}}_{m!}))}^T$. We call ${\mathit{z}}_{ij}$ the PWO factor related to components $c_i$ and $c_j$. Column juxtaposing ${\mathit{z}}_{ij}$s, we call ${\mathit{P}}_m=({\mathit{z}}_{12},{\mathit{z}}_{13},\dots ,{\mathit{z}}_{(m-1)m})$ the full PWO design, where ${\mathit{z}}_{ij}$ is ahead of ${\mathit{z}}_{kl}$ if $i<k$; or if $i=k$ and $j<l$. For example, the PWO factors for $m=3$ and full PWO design ${\mathit{P}}_3$ are displayed in Table 1. Denoting $\mathit{D}$ as a fractional OofA design and ${\mathit{P}}_D$ as the fractional PWO design determined by the orders in $\mathit{D}$, we give the definition of the OofA-OA.

Definition 1.

An N-run fractional OofA design$\mathit{D}$is called an OofA-OA of strength t, denoted as OofA-OA$\mathit{(}\mathit{N}\mathit{,}\mathit{m}\mathit{,}\mathit{t}\mathit{)}$, if the ratios among the frequencies of all t-tuples in any t-column subarray of${\mathit{P}}_{\mathit{D}}$equal to the ratios among the frequencies of all t-tuples in the corresponding t-column subarray of${\mathit{P}}_{\mathit{m}}$.

Definition 2 defines isomorphic OofA designs.

Definition 2.

Two OofA designs are said to be isomorphic if one can be obtained from the other by relabeling components or permuting rows.

In [13], the authors showed that non-isomorphic OofA-OAs may have different performances in some situations. For example, under the CP model, the non-isomorphic OofA-OAs may have different D-efficiencies. This is not true for the isomorphic OofA-OAs. In this paper, the construction method we propose is capable of finding the non-isomorphic OofA-OAs of strength 3. For a detailed definition of the CP model, one is referred to [2].

3. Constructions of OofA-OAs of Strength 3

From Definition 1, in order to construct OofA-OAs of strength 3, we need to investigate the frequencies of the three-tuple $(a,b,c)$s, with $a=\pm 1,b=\pm 1$ and $c=\pm 1$, in each of the three-column subarrays $({\mathit{z}}_{ij},{\mathit{z}}_{kl},{\mathit{z}}_{vw})$s of ${\mathit{P}}_m$, where ${\mathit{z}}_{ij}$ is ahead of ${\mathit{z}}_{kl}$ in ${\mathit{P}}_m$, and ${\mathit{z}}_{kl}$ is ahead of ${\mathit{z}}_{vw}$ in ${\mathit{P}}_m$. Note that $m=6$ is the smallest m such that ${\mathit{P}}_m$ has the three-column subarrays $({\mathit{z}}_{ij},{\mathit{z}}_{kl},{\mathit{z}}_{vw})$s with $i,j,k,l,v$, and w being mutually different. As shown in Table 2, for $m\ge 6$, there are 20 different types of ratios among the frequencies of the eight three-tuple $(a,b,c)$s, and the run size of an OofA-OA of strength 3 should be a multiple of 24. Lemma 1 formally summarizes these findings.

Table 2. Classifications of three-column subarrays of ${\mathit{P}}_m$.

Type	${\mathit{T}}_1$	${\mathit{T}}_2$	${\mathit{T}}_3$	${\mathit{T}}_4$	${\mathit{T}}_5$	${\mathit{T}}_6$	${\mathit{T}}_7$	${\mathit{T}}_8$	$({\mathit{z}}_{\mathit{i}\mathit{j}},{\mathit{z}}_{\mathit{k}\mathit{l}},{\mathit{z}}_{\mathit{v}\mathit{w}})$	Examples
$t_1$	$\frac{m!}{6}$	$\frac{m!}{6}$	0	$\frac{m!}{6}$	$\frac{m!}{6}$	0	$\frac{m!}{6}$	$\frac{m!}{6}$	$i=k,j=v,l=w$	12 13 23
$t_2$	$\frac{5m!}{24}$	$\frac{m!}{24}$	$\frac{m!}{8}$	$\frac{m!}{8}$	$\frac{m!}{8}$	$\frac{m!}{8}$	$\frac{m!}{24}$	$\frac{5m!}{24}$	$j=w,k=v$	14 23 24
$t_3$	$\frac{5m!}{24}$	$\frac{m!}{8}$	$\frac{m!}{8}$	$\frac{m!}{24}$	$\frac{m!}{24}$	$\frac{m!}{8}$	$\frac{m!}{8}$	$\frac{5m!}{24}$	$i=k,j=w$	13 14 23
$t_4$	$\frac{m!}{24}$	$\frac{m!}{8}$	$\frac{5m!}{24}$	$\frac{m!}{8}$	$\frac{m!}{8}$	$\frac{5m!}{24}$	$\frac{m!}{8}$	$\frac{m!}{24}$	$j=k,l=v$	12 23 34
$t_5$	$\frac{m!}{12}$	$\frac{m!}{4}$	$\frac{m!}{12}$	$\frac{m!}{12}$	$\frac{m!}{12}$	$\frac{m!}{12}$	$\frac{m!}{4}$	$\frac{m!}{12}$	$j=l=v$	13 23 34
$t_6$	$\frac{m!}{12}$	$\frac{m!}{12}$	$\frac{m!}{12}$	$\frac{m!}{4}$	$\frac{m!}{4}$	$\frac{m!}{12}$	$\frac{m!}{12}$	$\frac{m!}{12}$	$j=k=v$	12 23 24
$t_7$	$\frac{m!}{8}$	$\frac{5m!}{24}$	$\frac{m!}{24}$	$\frac{m!}{8}$	$\frac{m!}{8}$	$\frac{m!}{24}$	$\frac{5m!}{24}$	$\frac{m!}{8}$	$i=k,j=v$	12 13 24
$t_8$	$\frac{m!}{8}$	$\frac{5m!}{24}$	$\frac{m!}{8}$	$\frac{m!}{24}$	$\frac{m!}{24}$	$\frac{m!}{8}$	$\frac{5m!}{24}$	$\frac{m!}{8}$	$i=k,l=v$	12 13 34
$t_9$	$\frac{m!}{8}$	$\frac{m!}{24}$	$\frac{m!}{8}$	$\frac{5m!}{24}$	$\frac{5m!}{24}$	$\frac{m!}{8}$	$\frac{m!}{24}$	$\frac{m!}{8}$	$j=k,l=w$	12 24 34
$t_{10}$	$\frac{m!}{8}$	$\frac{m!}{8}$	$\frac{5m!}{24}$	$\frac{m!}{24}$	$\frac{m!}{24}$	$\frac{5m!}{24}$	$\frac{m!}{8}$	$\frac{m!}{8}$	$j=w,l=v$	14 23 34
$t_{11}$	$\frac{m!}{8}$	$\frac{m!}{8}$	$\frac{m!}{24}$	$\frac{5m!}{24}$	$\frac{5m!}{24}$	$\frac{m!}{24}$	$\frac{m!}{8}$	$\frac{m!}{8}$	$j=v,l=w$	13 24 34
$t_{12}$	$\frac{5m!}{24}$	$\frac{m!}{8}$	$\frac{m!}{24}$	$\frac{m!}{8}$	$\frac{m!}{8}$	$\frac{m!}{24}$	$\frac{m!}{8}$	$\frac{5m!}{24}$	$i=k,l=w$ or $j=l,k=v$	12 14 34 13 23 25
$t_{13}$	$\frac{m!}{4}$	$\frac{m!}{12}$	$\frac{m!}{12}$	$\frac{m!}{12}$	$\frac{m!}{12}$	$\frac{m!}{12}$	$\frac{m!}{12}$	$\frac{m!}{4}$	$i=k=v$ or $j=l=w$	12 13 14 14 24 34
$t_{14}$	$\frac{m!}{6}$	$\frac{m!}{6}$	$\frac{m!}{12}$	$\frac{m!}{12}$	$\frac{m!}{12}$	$\frac{m!}{12}$	$\frac{m!}{6}$	$\frac{m!}{6}$	$i=k$ or $j=l$	12 13 45 13 23 45
$t_{15}$	$\frac{m!}{6}$	$\frac{m!}{12}$	$\frac{m!}{12}$	$\frac{m!}{6}$	$\frac{m!}{6}$	$\frac{m!}{12}$	$\frac{m!}{12}$	$\frac{m!}{6}$	$k=v$ or $l=w$	12 34 35 12 35 45
$t_{16}$	$\frac{m!}{6}$	$\frac{m!}{12}$	$\frac{m!}{6}$	$\frac{m!}{12}$	$\frac{m!}{12}$	$\frac{m!}{6}$	$\frac{m!}{12}$	$\frac{m!}{6}$	$j=w$	14 25 34
$t_{17}$	$\frac{m!}{12}$	$\frac{m!}{6}$	$\frac{m!}{6}$	$\frac{m!}{12}$	$\frac{m!}{12}$	$\frac{m!}{6}$	$\frac{m!}{6}$	$\frac{m!}{12}$	$l=v$	12 34 45
$t_{18}$	$\frac{m!}{12}$	$\frac{m!}{6}$	$\frac{m!}{12}$	$\frac{m!}{6}$	$\frac{m!}{6}$	$\frac{m!}{12}$	$\frac{m!}{6}$	$\frac{m!}{12}$	$j=v$	13 24 35
$t_{19}$	$\frac{m!}{12}$	$\frac{m!}{12}$	$\frac{m!}{6}$	$\frac{m!}{6}$	$\frac{m!}{6}$	$\frac{m!}{6}$	$\frac{m!}{12}$	$\frac{m!}{12}$	$j=k$	12 23 45
$t_{20}$	$\frac{m!}{8}$	$\frac{m!}{8}$	$\frac{m!}{8}$	$\frac{m!}{8}$	$\frac{m!}{8}$	$\frac{m!}{8}$	$\frac{m!}{8}$	$\frac{m!}{8}$	*	12 34 56

Note: T₁, T₂, T₃, T₄, T₅, T₆, T₇, T₈ represent the three-tuples (−, −, −), (−, −, +), (−, +, −), (−, +, +), (+, −, −), (+, −, +), (+, +, −), and (+, +, +), respectively; for each t_i, all the equal numbers among i, j, k, l, v, w are displayed in column “z_ij, z_kl, z_vw”, and * means that i, j, k, l, v, w are mutually different.

Lemma 1.

For $m\ge 6$, ${\mathit{P}}_m$ has 20 different types of ratios among the frequencies of the eight three-tuple $(a,b,c)$s, with $a=\pm 1,b=\pm 1$ and $c=\pm 1$, as shown in Table 2, and the run size of an OofA-OA of strength 3 should be a multiple of 24.

Remark 1 below shows the frequencies of the eight three-tuple $(a,b,c)$s for ${\mathit{P}}_4$ and ${\mathit{P}}_5$.

Remark 1.

In Table 2, the types $t_1$-$t_{13}$ apply to ${\mathit{P}}_4$ and the types $t_1$-$t_{19}$ apply to ${\mathit{P}}_5$. The run sizes of OofA-OAs of strength 3 for $m=4$ and $m=5$ should also be multiples of 24.

In the following, we introduce the method of constructing OofA-OAs of strength 3. Denote ${\mathbf{\varphi }}_{ij,kl,vw}(-,-,-)$ as an $m!$-dimensional vector in which the uth entry is 1 if the three-tuple $(-,-,-)$ appears in the uth row of the three-column subarray $({\mathit{z}}_{ij},{\mathit{z}}_{kl},{\mathit{z}}_{vw})$ of ${\mathit{P}}_m$, and is 0 otherwise. The $m!$-dimensional vectors ${\mathbf{\varphi }}_{ij,kl,vw}(-,-,+),$ ${\mathbf{\varphi }}_{ij,kl,vw}(-,+,-),$${\mathbf{\varphi }}_{ij,kl,vw}(-,+,+),$ ${\mathbf{\varphi }}_{ij,kl,vw}(+,-,-),$${\mathbf{\varphi }}_{ij,kl,vw}(+,-,+),$ ${\mathbf{\varphi }}_{ij,kl,vw}(+,+,-),$${\mathbf{\varphi }}_{ij,kl,vw}(+,+,+)$ are similarly defined. Let

$$\begin{array}{ccc}\hfill {\mathbf{\Phi }}_{ij,kl,vw}^T& =& ({\mathbf{\varphi }}_{ij,kl,vw}(-,-,-),{\mathbf{\varphi }}_{ij,kl,vw}(-,-,+),{\mathbf{\varphi }}_{ij,kl,vw}(-,+,-),{\mathbf{\varphi }}_{ij,kl,vw}(-,+,+),\hfill \\ \hfill & =& {\mathbf{\varphi }}_{ij,kl,vw}(+,-,-),{\mathbf{\varphi }}_{ij,kl,vw}(+,-,+),{\mathbf{\varphi }}_{ij,kl,vw}(+,+,-),{\mathbf{\varphi }}_{ij,kl,vw}(+,+,+)).\hfill \end{array}$$

(2)

For m, there are $\left(\genfrac{}{}{0pt}{}{\left(\genfrac{}{}{0pt}{}{m}{2}\right)}{3}\right)$ such ${\mathbf{\Phi }}_{ij,kl,vw}$s. Let $\mathbf{\Phi }$ be the matrix generated by row-juxtaposing all the $\left(\genfrac{}{}{0pt}{}{\left(\genfrac{}{}{0pt}{}{m}{2}\right)}{3}\right)$${\mathbf{\Phi }}_{ij,kl,vw}$s. For a fractional OofA design $\mathit{D}$ of m components, let $y_k(\mathit{D})=1$ if the order ${\mathit{o}}_k$ is in $\mathit{D}$ and $y_k(\mathit{D})=0$ otherwise, where $k=1,2,\dots ,m!$. Let ${\mathit{Y}}_D={(y_1(\mathit{D}),y_2(\mathit{D}),\dots ,y_{m!}(\mathit{D}))}^T$. We establish a sufficient and necessary condition for $\mathit{D}$ to be an OofA-OA of strength 3.

Theorem 1.

A fractional OofA design $\mathit{D}$ is an OofA-OA $(N,m,3)$ if and only if ${\mathit{Y}}_D$ is a solution of

$$\mathbf{\Phi }{\mathit{Y}}_D=(N/m!)\mathrm{vdiag}(\mathbf{\Phi }{\mathbf{\Phi }}^T),$$

(3)

where $\mathrm{vdiag}(·)$ is a column vector consisting of the diagonal elements of a matrix.

Theorem 1 shows that once we have a solution to (3), we can construct an OofA-OA of strength 3 according to this solution. Example 1 shows this point.

Example 1.

Let ${\mathit{Y}}_D$ be a 5!(=120)-dimensional vector which has entries 1s in its 1, 6, 16, 22, 26, 28, 40, 46, 51, 53, 57, 59, 66, 71, 75, 77, 81, 83, 95, 99, 101, 105, 107, and 120-th rows and 0s in the remainder of its rows. It can be verified that ${\mathit{Y}}_D$ is a solution of (3) with $m=5$ and $N=24$. Then, an OofA-OA $(24,5,3)$ can be constructed according to ${\mathit{Y}}_D$. The resulting design is ${\mathit{A}}_{5.1}^{24}$ displayed in Table A1 of Appendix B.

It is an infeasible task to directly solve Equation (3), we employ 0–1 linear programming to find solutions. Corollary 1 below states this approach.

Corollary 1.

For a given $\mathit{c}\in R^{m!}$, an $m!$-dimensional vector, if ${\mathit{Y}}_D$ is a solution of the 0–1 linear optimization problem,

$$\begin{array}{c}min{\mathit{c}}^T{\mathit{Y}}_{D}subject to:\hfill \\ \mathbf{\Phi }{\mathit{Y}}_D=(N/m!)\mathrm{vdiag}(\mathbf{\Phi }{\mathbf{\Phi }}^T),and {\mathit{Y}}_D\in {\{0,1\}}^{m!},\hfill \end{array}$$

(4)

then the rows chosen according to ${\mathit{Y}}_D$ compose an OofA-OA $(N,m,3)$.

Remark 2.

For given m and N, if Equation (3) has solution(s), then the optimization problem (4) has solution(s) for any $\mathit{c}$. Note that our interest is to find the solution ${\mathit{Y}}_D$ instead of minimizing ${\mathit{c}}^T{\mathit{Y}}_D$. Any programming solver can be employed to find ${\mathit{Y}}_D$s. Here, we use intlinprog “from Matlab. Given m, N and any$\mathit{c}$, intlinprog” reports one ${\mathit{Y}}_D$ of (4) unless (3) has no solution for the given m and N.

As an illustration, we apply Theorem 1 and Corollary 1 to finding OofA-OA $(N,m,3)$s with $m=5,6$ and $N=24,48,72$. For given N and m, we use 2000 random $\mathit{c}$s to find different solutions of (3). With these different solutions, we display some non-isomorphic OofA-OAs of strength 3 which have larger relative D-efficiencies under the CP model, where the relative D-efficiency of a fractional OofA design is the ratio between the D-efficiency of this fractional OofA design and that of its corresponding full OofA design. By doing so, it shows that non-isomorphic OofA-OAs may have different D-efficiencies under the CP model as pointed out in [13].

With 2000 random $\mathit{c}$s, we found only one OofA-OA $(24,5,3)$, up to isomorphism, whose row numbers are displayed in Table A1 of Appendix B. The OofA-OA $(24,5,3)$${\mathit{A}}_{5,1}^{24}$ provides relative D-efficiency 0 under the CP model. In order to find more non-isomorphic OofA-OA $(24,5,3)$s, another 8000 random $\mathit{c}$s are used to find solutions of (3). All the resulting OofA-OA $(24,5,3)$s are isomorphic to ${\mathit{A}}_{5,1}^{24}$. We conjecture that there is only one OofA-OA $(24,5,3)$ up to isomorphism. With 2000 random $\mathit{c}$s, in Table A2 and Table A4, 10 non-isomorphic OofA-OA $(N,5,3)$s with $N=48,72$ alone with their relative D-efficiencies under the CP model are displayed, respectively. The OofA-OA $(24,6,3)$ does not exist. When using intlinprog” to find solutions to (3) with $m=6$ and $N=24$, it is reported that no solution can be found. With 2000 random $\mathit{c}$s, in Table A4 and Table A5, 10 non-isomorphic OofA-OA $(N,6,3)$s with $N=48,72$ alone with their relative D-efficiencies under the CP model are displayed, respectively.

4. Some Properties of OofA-OAs of Strength 3

In [13], it is pointed out that when projecting an OofA-OA $(N,m,2)$ onto any two components, the resulting design is an $N/2$-replication of ${\mathit{O}}_2$, and when projecting an OofA-OA $(N,m,2)$ onto its any three components, the resulting design is an $N/6$-replication of ${\mathit{O}}_3$. As will be seen in Theorem 2 below, an OofA-OA $(N,m,3)$ poses additional balance properties when it is projected onto its any four components.

Theorem 2.

For any OofA-OA$\mathit{(}\mathit{N}\mathit{,}\mathit{m}\mathit{,}\mathit{3}\mathit{)}$${D}$,

(i) 

when$\mathit{D}$is projected onto its any two components${\mathit{c}}_{\mathit{i}}$and${\mathit{c}}_{\mathit{j}}$, the resulting design is an$\mathit{N}/2$-replication of${\mathit{O}}_2$;

(ii) 

when$\mathit{D}$is projected onto its any three components${\mathit{c}}_{\mathit{i}}\mathit{,}{\mathit{c}}_{\mathit{j}}$, and${\mathit{c}}_{\mathit{k}}$, the resulting design is an$\mathit{N}/6$-replication of${\mathit{O}}_3$;

(iii) 

when$\mathit{D}$is projected onto its any four components${\mathit{c}}_{\mathit{i}},{\mathit{c}}_{\mathit{j}},{\mathit{c}}_{\mathit{k}}$, and${\mathit{c}}_l$, the resulting design is an$\mathit{N}/24$-replication of${\mathit{O}}_4$.

Example 2 below illustrates the balance properties stated in Theorem 2.

Example 2.

The fractional OofA design

$${\mathit{A}}_{5.1}^{24}={\left(\begin{array}{cccccccccccccccccccccccc}5& 5& 5& 5& 4& 4& 4& 4& 3& 3& 3& 3& 3& 3& 2& 2& 2& 2& 2& 1& 1& 1& 1& 1\\ 4& 4& 2& 1& 5& 5& 2& 1& 5& 5& 4& 4& 2& 1& 5& 5& 4& 4& 1& 5& 5& 4& 4& 2\\ 3& 1& 3& 3& 3& 2& 3& 3& 2& 1& 2& 1& 1& 2& 3& 1& 3& 1& 3& 3& 2& 3& 2& 3\\ 2& 2& 1& 2& 1& 1& 1& 2& 4& 4& 5& 5& 4& 5& 4& 4& 5& 5& 5& 4& 4& 5& 5& 4\\ 1& 3& 4& 4& 2& 3& 5& 5& 1& 2& 1& 2& 5& 4& 1& 3& 1& 3& 4& 2& 3& 2& 3& 5\end{array}\right)}^T$$

in Table A1 is an OofA-OA $(24,5,3)$, where we use the Arabic numbers $1,2,3,4,5$ instead of $c_1,c_2,c_3,c_4,c_5$ to denote the components, respectively, to save space. Projecting the design ${\mathit{A}}_{5.1}^{24}$ onto the components 1 and 2, we obtain design

$${\mathit{H}}_1={\left(\begin{array}{cccccccccccccccccccccccc}2& 1& 2& 1& 1& 2& 2& 1& 2& 1& 2& 1& 2& 1& 2& 2& 2& 2& 2& 1& 1& 1& 1& 1\\ 1& 2& 1& 2& 2& 1& 1& 2& 1& 2& 1& 2& 1& 2& 1& 1& 1& 1& 1& 2& 2& 2& 2& 2\end{array}\right)}^T.$$

In ${\mathit{H}}_1$, each order of the components 1 and 2 appears 12 times. Similar balance properties can be obtained when projecting ${\mathit{A}}_{5.1}^{24}$ onto other two components. Projecting the design ${\mathit{A}}_{5.1}^{24}$ onto the components 1, 2, and 3, we obtain design

$${\mathit{H}}_2={\left(\begin{array}{cccccccccccccccccccccccc}3& 1& 2& 1& 3& 2& 2& 1& 3& 3& 3& 3& 3& 3& 2& 2& 2& 2& 2& 1& 1& 1& 1& 1\\ 2& 2& 3& 3& 1& 1& 3& 3& 2& 1& 2& 1& 2& 1& 3& 1& 3& 1& 1& 3& 2& 3& 2& 2\\ 1& 3& 1& 2& 2& 3& 1& 2& 1& 2& 1& 2& 1& 2& 1& 3& 1& 3& 3& 2& 3& 2& 3& 3\end{array}\right)}^T.$$

In ${\mathit{H}}_2$, each order of the components 1, 2, and 3 appears 4 times. Similar balance properties can be obtained when projecting ${\mathit{A}}_{5.1}^{24}$ onto other three components. Projecting the design ${\mathit{A}}_{5.1}^{24}$ onto the components 1, 2, 3, and 4, we obtain design

$${\mathit{H}}_3={\left(\begin{array}{cccccccccccccccccccccccc}4& 4& 2& 1& 4& 4& 4& 4& 3& 3& 3& 3& 3& 3& 2& 2& 2& 2& 2& 1& 1& 1& 1& 1\\ 3& 1& 3& 3& 3& 2& 2& 1& 2& 1& 4& 4& 2& 1& 3& 1& 4& 4& 1& 3& 2& 4& 4& 2\\ 2& 2& 1& 2& 1& 1& 3& 3& 4& 4& 2& 1& 1& 2& 4& 4& 3& 1& 3& 4& 4& 3& 2& 3\\ 1& 3& 4& 4& 2& 3& 1& 2& 1& 2& 1& 2& 4& 4& 1& 3& 1& 3& 4& 2& 3& 2& 3& 4\end{array}\right)}^T.$$

In ${\mathit{H}}_3$, each order of the components 1, 2, 3, and 4 appears once. Similar balance properties can be obtained when projecting ${\mathit{A}}_{5.1}^{24}$ onto other four components.

The design $\mathit{A}$ in Example 3 is an OofA-OA $(24,5,2)$ but not an OofA-OA $(24,5,3)$. As will be seen, when projecting $\mathit{A}$ onto the components 1, 2, 3, and 4, the resulting design does not have the balance properties stated in Theorem 2 (iii).

Example 3.

Projecting the design

$$\mathit{A}={\left(\begin{array}{cccccccccccccccccccccccc}5& 5& 5& 5& 5& 4& 4& 4& 4& 4& 3& 3& 3& 3& 3& 2& 2& 2& 2& 1& 1& 1& 1& 1\\ 4& 3& 3& 2& 1& 5& 3& 2& 2& 1& 5& 5& 4& 2& 1& 5& 4& 3& 1& 5& 4& 3& 2& 2\\ 2& 4& 2& 1& 2& 1& 2& 5& 1& 3& 2& 1& 1& 4& 4& 4& 3& 1& 5& 4& 5& 2& 4& 3\\ 3& 1& 4& 3& 4& 2& 5& 1& 3& 5& 1& 4& 5& 5& 2& 3& 1& 4& 3& 3& 3& 5& 5& 5\\ 1& 2& 1& 4& 3& 3& 1& 3& 5& 2& 4& 2& 2& 1& 5& 1& 5& 5& 4& 2& 2& 4& 3& 4\end{array}\right)}^T$$

onto the components 1, 2, 3, and 4 obtains

$${\mathit{H}}_4={\left(\begin{array}{cccccccccccccccccccccccc}4& 3& 3& 2& 1& 4& 4& 4& 4& 4& 3& 3& 3& 3& 3& 2& 2& 2& 2& 1& 1& 1& 1& 1\\ 2& 4& 2& 1& 2& 1& 3& 2& 2& 1& 2& 1& 4& 2& 1& 4& 4& 3& 1& 4& 4& 3& 2& 2\\ 3& 1& 4& 3& 4& 2& 2& 1& 1& 3& 1& 4& 1& 4& 4& 3& 3& 1& 3& 3& 3& 2& 4& 3\\ 1& 2& 1& 4& 3& 3& 1& 3& 3& 2& 4& 2& 2& 1& 2& 1& 1& 4& 4& 2& 2& 4& 3& 4\end{array}\right)}^T.$$

In ${\mathit{H}}_4$, the order 1432 appears two times and the order 4321 appears once. Clearly, ${\mathit{H}}_4$ is not an ${\mathit{O}}_4$ and thus does not have the balance properties stated in Theorem 2 (iii).

Remark 3.

The balance properties in Theorem 2 make OofA-OAs of strength 3 useful in the situation where $m-4$ or more components are found inactive after experimentations. The OofA-OAs of strength 3 may have larger run sizes compared to OofA-OAs of strength 2. For example, when $m=5$, the smallest run size of the OofA-OAs of strength 2 is 12 while the smallest run size of the OofA-OAs of strength 3 is 24. Note that a larger design run size implies more observations. People may choose to use OofA-OAs of strength 2 or 3 according to their practical needs. When $m-4$ components are found inactive after experimentations, the OofA-OAs of strength 3 would be better choices.

5. Simulation Studies

We conducted thorough simulation studies to investigate the performances of the constructed OofA-OAs presented in Table A1, Table A2, Table A3, Table A4, Table A5. It was shown that the constructed OofA-OAs of strength 3 have the desirable capability of estimating the optimal orders of addition. For saving space, we only use the OofA-OA $(24,5,3)$$A_{5.1}^{24}$ (in Table A1) and OofA-OA $(48,6,3)$$A_{6.1}^{48}$ to illustrate the simulation studies we have conducted. The other OofA-OAs of strength 3 presented in Table A2, Table A3, Table A4, Table A5 have either close or better performance of estimating optimal orders of addition than $A_{5.1}^{24}$ and $A_{6.1}^{48}$.

Without loss of generality, suppose the underling true optimal orders for $m=5$ and $m=6$ are

$$1\to 2\to 3\to 4\to 5 \mathrm{a}\mathrm{n}\mathrm{d}$$

(5)

$$1\to 2\to 3\to 4\to 5\to 6,$$

(6)

respectively. Establish the PWO models for $m=5$ and $m=6$ as

$$\tau ({\mathit{o}}_k)={\beta }_0+\sum _{i=1}^4\sum _{j=i+1}^5{\beta }_{ij}{\lambda }_{ij}({\mathit{o}}_k)++\epsilon ({\mathit{o}}_k)\mathrm{a}\mathrm{n}\mathrm{d}$$

(7)

$$\tau ({\mathit{o}}_k)={\beta }_0+\sum _{i=1}^5\sum _{j=i+1}^6{\beta }_{ij}{\lambda }_{ij}({\mathit{o}}_k)+\epsilon ({\mathit{o}}_k),$$

(8)

respectively, where ${\mathit{o}}_k$ is an order in $A_{5.1}^{24}$ and $A_{6.1}^{48}$, $\epsilon ({\mathit{o}}_k)\sim N(0,{\sigma }^2)$, and $\epsilon ({\mathit{o}}_k)$ is independent of $\epsilon ({\mathit{o}}_l)$ for $k\ne l$. For both models (7) and (8), the values of ${\beta }_0$ are set to be 1 and the values of ${\beta }_{ij}$s are set following the four scenarios

(S1)

all of the ${\beta }_{ij}$s in models (7) and (8) equal to $0.5$;

(S2)

all of the ${\beta }_{ij}$s in models (7) and (8) equal to 1;

(S3)

all of the ${\beta }_{ij}$s in models (7) and (8) equal to 3;

(S4)

all of the ${\beta }_{ij}$s with $1\le i\le 2$ and $1\le j\le 4$ in models (7) and (8) equal to $0.5$, and the other ${\beta }_{ij}$s in models (7) and (8) equal to 5.

For each scenario $S1$, $S2$, $S3$, and $S4$, the simulation procedure is designed as follows.

• For each order ${\mathit{o}}_k$ of $A_{5.1}^{24}$ (or $A_{6.1}^{48}$ for $m=6$), randomly draw $\epsilon ({\mathit{o}}_k)$ from $N(0,\sigma )$ with a given $\sigma$ ($=1,3,5$).
• Compute $\tau ({\mathit{o}}_k)$ by model (7) (or model (8) for $m=6$), where $\epsilon ({\mathit{o}}_k)$ is the one obtained in Step 1, ${\beta }_0=1$ and ${\beta }_{ij}$s are referred to each scenario of $S1$, $S2$, $S3$, and $S4$.
• Fit model (7) (or model (8) for $m=6$) to obatin a ${\beta }_{ij}$s’ least squre estimations ${\widehat{\beta }}_{ij}$s by $\mathit{Y}=(\tau ({\mathit{o}}_1),\tau {({\mathit{o}}_2,\dots ,\tau ({\mathit{o}}_N))}^T$ and $\mathit{X}=({\mathbf{1}}_N,\mathit{P})$, where $N=24$ (or $N=48$ for $A_{6.1}^{48}$), ${\mathit{o}}_k$s are from $A_{5.1}^{24}$ (or $A_{5.1}^{24}$), and $\mathit{P}$ is the PWO design corresponding to $A_{5.1}^{24}$ (or $A_{6.1}^{48}$).
• Test the significance of ${\beta }_{ij}$s. For the two-sided alternative $H_1:{\beta }_{ij}\ne 0$, the p-value is evaluated by $2\times \mathrm{P}\mathrm{r}\mathrm{o}\mathrm{b}(t>|\frac{{\widehat{\beta }}_{ij}}{\sqrt{c_{ij}}\widehat{\sigma }}|)$, where t follows the t-distribution with 13 (or 32) degrees of freedom for $A_{5.1}^{24}$ (or $A_{6.1}^{48}$), $c_{ij}$ is the diagonal entry of ${({\mathit{X}}^T\mathit{X})}^{-1}$ corresponding to ${\widehat{\beta }}_{ij}$, and the significance level is set to be $0.05$.
• Let $\mathit{S}=(s_{12},s_{13},\dots ,s_{(m-1)m})$, where $s_{ij}$ is the sign of ${\widehat{\beta }}_{ij}$ if ${\beta }_{ij}$ is significant, and otherwise $s_{ij}=0$.
• The underlying order is correctly estimated if there is no element $-1$ in $\mathit{S}$. Repeat this simulation procedure 10,000 times, and summarize the frequency (out of 10,000) of correct estimations of the underlying optimal order.

The simulation results are displayed in Table 3 with different values of $\sigma$ (=$1,3$, and 5).

Table 3. The frequencies of correct estimations for the true underling orders.

Designs	$\left(\left(\genfrac{}{}{0pt}{}{\mathit{m}}{2}\right)+1\right)/\mathit{m}!$	$\mathit{\sigma }$	$\mathit{S}1$	$\mathit{S}2$	$\mathit{S}3$	$\mathit{S}4$
		1	0.999	1	1	0.997
$A_{5.1}^{24}$	0.2 $(24/120)$	3	0.993	1	1	0.986
		5	0.997	0.996	0.994	0.990
		1	1	0.996	1	0.993
$A_{6.1}^{48}$	0.07 $(48/720)$	3	0.961	0.994	1	0.985
		5	0.955	0.990	0.995	0.973

From Table 3, the OofA-OAs(24,5,3) $A_{5.1}^{24}$ and OofA-OAs(48,6,3) $A_{6.1}^{48}$ have quite high frequecies (out 10,000) of correct estimations of the true underlying optimal orders while saving a significant experimental cost compared to the full OofA designs as shown in the second column of Table 3.

6. Results and Discussions

The OofA-OAs are a class of D-optimal fractional OofA designs under the prevalent PWO model. The OofA-OAs of strength 2 have been studied in a few studies, we defer to [11,12,13]. However, there is no study on the OofA-OAs of strength 3. This paper studies the OofA-OAs of strength 3 for the first time in the literature.

In Theorem 2, it is shown that the OofA-OAs of strength 3 have better balance properties than those of OofA-OAs of strength 2. These balance properties make OofA-OAs of strength 3 more useful when $m-4$ or more components are found inactive after experimentations. For such a motivation, we propose a systematical construction method for OofA-OAs of strength 3 in Theorem 1. The proposed construction method is capable of finding non-isomorphic OofA-OAs of strength 3, noting that non-isomorphic OofA-OAs may have different performances for other OofA models such as the CP model. In Table A1, Table A2, Table A3, Table A4, Table A5, non-isomorphic OofA-OAs of strength 3 are provided which provides quite high D-efficiencies under the CP model. When models are not prespecified, the OofA-OAs of strength 3 which can provide higher D-efficiencies for both PWO and CP models are desirable.

To further show the efficiencies of the constructed OofA-OAs of strength 3, thorough simulation studies are provided in Section 5. From Table 3, the constructed OofA-OAs of strength 3 can provide quite high frequencies (out of 10,000) of correct estimations of the true underlying orders. This indicates that the OofA-OAs of strength 3 are capable of estimating the optimal order of addition.

7. Conclusions

As a class of efficient fractional OofA designs, OofA-OAs are optimal for the PWO model under a variety of widely used design criteria [11]. In the literature, the studies on OofA-OAs were focused on strength 2 while OofA-OAs of strength 3 have not been studied yet. The high strength results in two major challenges of this work. The first one is the classification of three-column subarrays of ${\mathit{P}}_m$ (as shown in Table 2). As previously stated, ${\mathit{P}}_6$ is the smallest full PWO design to investigate the classification of the three-column subarrays of ${\mathit{P}}_m$ with respect to the ratios among the frequencies of the three-tuple $(a,b,c)$s with $a=\pm 1$, $b=\pm 1$ and $c=\pm 1$. The PWO design ${\mathit{P}}_6$ has $\left(\genfrac{}{}{0pt}{}{\left(\genfrac{}{}{0pt}{}{6}{2}\right)}{3}\right)=455$ three-column subarrays to be classified which is not so easy as the counterpart problem in the case of strength 2. One is referred to [10,11] for the classification of the two-column subarrays of ${\mathit{P}}_m$. The second one is the derivation of Theorem 2. The derivation of Theorem 2 concerns analyses of an equation system consisting of 3640 equations, i.e., the equation system (3) for $m=6$. These large number of equations make the derivation of Theorem 2 more challengeable as indicated by the proof of Theorem 2.

Despite the challenges stated above, this paper provides a threefold contribution. First, this paper provides a method of constructing OofA-OAs of strength 3. This method is capable of finding non-isomorphic OofA-OAs of strength 3. Second, some balance properties of this class of designs are developed. It is shown that OofA-OAs of strength 3 have better balance properties than OofA-OAs of strength 2. For example, when projecting an OofA-OA of strength 3 onto any four components, all of the 24(=4!) orders in the resulting design appear equally often. This balance property is useful when $m-4$ components are found inactive after experimentations. For practical usage, some non-isomorphic OofA-OAs of strength 3 are also provided. Third, the thorough simulation studies are conducted which show that the constructed OofA-OAs of 3 are quite capable of estimating optimal orders of addition.