Group Doubly Coupled Designs

Abstract

Doubly coupled designs (DCDs) have better space-filling properties between the qualitative and quantitative factors than marginally coupled designs (MCDs) which are suitable for computer experiments with both qualitative and quantitative factors. In this paper, we propose a new class of DCDs, called group doubly coupled designs (GDCDs), and provide methods for constructing two forms of GDCDs, within-group doubly coupled designs and between-group doubly coupled designs. The proposed GDCDs can accommodate more qualitative factors than DCDs, when the subdesigns for the qualitative factors are symmetric. The subdesigns of qualitative factors are not asymmetric in the existing results on DCDs, and in this paper, we construct GDCDs with symmetric and asymmetric designs for the qualitative factors, respectively. Moreover, detailed comparisons with existing MCDs show that GDCDs have better space-filling properties between qualitative and quantitative factors. Finally, the methods are particularly easy to implement.

1. Introduction

Computer experiments are an effective method for exploring complex systems and scientific problems [1,2].The space-filling properties, which measure the uniformity of the design points in the experimental space, are critical for effectively exploring the experimental region of computer experiments [2]. Latin hypercube designs (LHDs), proposed by [3], are widely used space-filling designs for computer experiments. Such designs are often used in computer experiments with quantitative factors because they achieve optimal univariate uniformity. Computer experiments involving only quantitative factors have received considerable attention [1,2]. However, researchers usually encounter computer experiments involving both qualitative and quantitative factors; see [1,4,5,6,7,8,9,10,11,12,13].

Sliced Latin hypercube designs (SLHDs) proposed by [14] are LHDs that can be partitioned into some LHD slices, which not only maintain the optimal univariate uniformity but for each slice as well. SLHDs are popular for computer experiments with both qualitative and quantitative factors; see [9,10,15] and the references therein. Each slice of an SLHD can be used at one level combination of the qualitative factors. However, its number of runs increases dramatically with the number of level combinations of the qualitative factors. This is thus suitable for situations where there are few level combinations of the qualitative factors or where the cost of runs is low.

Inspired by the notion of SLHD, [16] proposed the marginally coupled designs (MCDs). Their key feature is that for each level of any qualitative factor, the design points for the quantitative factors can form a small LHD, and they have fewer runs than SLHDs. In recent years, improvements for MCDs include, but are not limited to, its quantitative factors design with column orthogonality and multi-dimensional stratifications; for more details, refer to [17,18,19,20]. MCD, however, appears to be inapplicable when it is necessary to study the stratification between multiple qualitative factors and quantitative factors, whereas a design with such properties can be useful for studying the interaction between two qualitative factors and quantitative factors.

To this end, [21] proposed the doubly coupled designs (DCDs). It not only maintains the properties of MCDs, but also ensures that the design points for the quantitative factors can form an LHD corresponding to any level combination of any two qualitative factors. In a DCD, the subdesign for qualitative factors is an orthogonal array (OA). Equal-level and mixed-level orthogonal arrays are called symmetric and asymmetric orthogonal arrays, respectively. In the DCDs constructed by [21], the subdesign for qualitative factors is a symmetric orthogonal array. However, in real-world problems, there exist qualitative factors with mixed levels, and the design of the qualitative factors is usually an asymmetric OA. At present, there are no studies of DCDs with qualitative factors being asymmetric OAs. The latter construction cannot be a simple extension of the former. Moreover, the existing DCDs have an upper bound on the number of qualitative factors, namely, no more than the number of levels of qualitative factors. Therefore, existing DCDs are inapplicable when the qualitative factors are mixed-level or when the number of qualitative factors exceeds the number of their levels.

For a computer experiment with q s-level qualitative factors and p quantitative factors, an MCD is appropriate if there is no interaction effect between any two qualitative factors and all quantitative factors; if $q\le s$ and there is the interaction effect between any two qualitative factors and all quantitative factors, a DCD is applicable. However, neither an MCD nor a DCD is suitable, when $q>s$, some qualitative factors and all quantitative factors have such interaction effects, and some do not. Suppose that in an experiment there are four qualitative factors, the type of concentration of cell lysis reagent (A1, A2), the type of stain (Blue, Red, Pink), the shape of the cell slides (Thick, Moderate, Thin), and the cells’ activity (Dead, Alive) as well as other quantitative factors. We know that only the two qualitative factors, the type of concentration of cell lysis reagent and the shape of the cell slides, have the interaction effect with all quantitative factors. Obviously, both an MCD and a DCD are not suitable for such an experiment. Thus, we need to adopt a design that satisfies two properties: (i) the whole design is an MCD; and (ii) the columns of some qualitative factors and the columns of all quantitative factors form a DCD. In this paper, we focus on such designs and call them group DCD (GDCD).

In addition, not only can the GDCDs contain more qualitative factors, but the designs for the qualitative factors can be asymmetric OAs. Therefore, the level types of GDCDs are more flexible than those of DCDs. Our methods construct two forms of GDCDs, within-group DCDs and between-group DCDs. In a within-group DCD (WGDCD), the design of the qualitative factors can be divided into several groups, and the design of any two qualitative factors from the same group coupled with the design of the quantitative factors is a DCD. Thus, columns in the same group have excellent stratification properties between qualitative and quantitative factors. In a between-group DCD (BGDCD), the design of the qualitative factors can also be split into several groups, and the design of any two qualitative factors from different level groups combined with the design of quantitative factors is a DCD. The methods for constructing WGDCDs and BGDCDs are similar and easy to implement and are given in Section 3.1 and Section 3.2, respectively. Since the space-filling property of GDCDS is similar to that of DCDs, the space-filling property of GDCDs is better than that of MCDs.

The article is organized as follows: Section 2 introduces the basic notation and definitions. Methods for constructing GDCDs and the corresponding examples are given in Section 3. Comparison was made in Section 4. Section 5 provides conclusions and discussion. All proofs are deferred to Appendix A.

2. Definitions and Notation

Let $GF\left(s\right)=\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_{s-1}\}$ denote the Galois field of order s, where ${\alpha }_0=0$ and ${\alpha }_1=1$. An $r\times c$ matrix D is called a difference scheme over $GF\left(s\right)$, denoted by $D(r,c,s)$, if it has the property that every element of $GF\left(s\right)$ in the vector difference between any two distinct columns in D occurs $r/s$ times equally. For details of the difference schemes, refer to Section 6.1 of [22]. An $n\times m$ matrix is called an asymmetric orthogonal array of strength t, denoted by $OA(n,s_1^{m_1}{s}_2^{m_2}\cdots s_c^{m_c},t)$, $m_1+m_2+\cdots +m_c=m$, if any of its $n\times t$ submatrix satisfies all possible t-tuples occur equally often, where the level of the first $m_1$ columns is taken from $\{0,\dots ,s_1-1\}$, the level of the next $m_2$ columns is taken from $\{0,\dots ,s_2-1\}$, and so on. When all the $s_j$’s are equal to s, the orthogonal array is symmetric, denoted by $OA(n,s^m,t)$.

We now review the Rao-Hamming construction in Section 3.4 of [22]. For a prime power s, let $z_1$ and $z_2$ be two s-level columns of length $s^2$ with entries from $GF\left(s\right)$, $GF\left(s\right)=\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_{s-1}\}$, where ${\alpha }_0=0$ and ${\alpha }_1=1$. Suppose that $z_1$ and $z_2$ are independent. We apply the Rao-Hamming construction in [22] to obtain an $OA(s^2,s^{s+1},2)$Y, i.e., $\mathrm{Y}=(z_1,z_2)\Lambda$ over $GF\left(s\right)$, where $\Lambda =\left(\begin{array}{cccccc}1& 1& 1& \cdots & 1& 0\\ 0& 1& {\alpha }_2& \cdots & {\alpha }_s& 1\end{array}\right)$.

An $n\times p$ matrix that each column is a permutation of integers $\{0,1,\dots ,n-1\}$ is called an LHD, denoted by $LHD(n,p)$. ${\mathbf{0}}_n$ and ${\mathbf{1}}_n$ are two n-dimensional column vectors with all entries being zeros and ones, respectively. Let $A^T$ represent the transposition of matrix A. For an $n\times m$ matrix A and an $f\times g$ matrix B, $A\oplus B$ = $\left(a_{ij}+B\right)$ and $A\otimes B$ = $\left(a_{ij}B\right)$ represent the Kronecker sum and Kronecker product, respectively, where $a_{ij}$ is the $(i,j)$th entry of A.

Suppose there is an $OA(n,s_1^{q_1}{s}_2^{q_2},2)$, if its rows can be divided into $\frac{n}{{\alpha }_j{s}_j}$$OA({\alpha }_j{s}_j,s_1^{q_1}{s}_2^{q_2},1)$’s, and for j = 1, 2, ${\alpha }_j{s}_j$ remains the same, then called the array as $({\alpha }_1\times {\alpha }_2)$-resolvable orthogonal array, denoted by $({\alpha }_1\times {\alpha }_2)\mathit{-ROC}(n,s_1^{q_1}{s}_2^{q_2},2)$. Especially, when $s_1=s_2=s,{\alpha }_1={\alpha }_2=\alpha$, then the array reduces to $\alpha \mathit{-ROC}(n,s^{q_1+q_2},2)$. If $\alpha =1$, the array is called a completely resolvable orthogonal array (CROA).

Let $D=(A,L)$ be an n-run design with q qualitative factors and p quantitative factors, where the subdesigns A and L are qualitative factors and quantitative factors, respectively. The design D is called a marginally coupled design, denoted by $MCD(n,s_1^{q_1}{s}_2^{q_2}\cdots s_c^{q_c},p)$, $q_1+q_2+\cdots +q_c=q$, if it satisfies: (i) A is an $OA(n,s_1^{q_1}{s}_2^{q_2}\cdots s_c^{q_c},2)$; (ii) L is an $LHD(n,p)$; and (iii) the rows in L, corresponding to each level of any factor in A, form a small LHD. When $s_i=s$, $i=1,2,\dots ,c$, the MCD is denoted as $MCD(n,s^q,p)$.

Let $D^0=(D_1,D_2)$ be an $MCD(n,s_1^{q_1}{s}_2^{q_2}\cdots s_c^{q_c},p)$. The design $D^0$ is called a doubly coupled design, denoted by $DCD(n,s_1^{q_1}{s}_2^{q_2}\cdots s_c^{q_c},p)$, if it satisfies that the rows in $D_2$, corresponding to each level combination of any two factors in $D_1$, form a small LHD. When $s_i=s$, $i=1,2,\dots ,c$, the DCD is denoted as $DCD(n,s^q,p)$. Obviously, $D_1$ is an $OA(n,s_1^{q_1}{s}_2^{q_2}\cdots s_c^{q_c},2)$, $D_2$ is an $LHD(n,p)$, and the rows in $D_2$, corresponding to each level combination of any t factors in $D_1$, form a small LHD for $t=1,2$.

Definition 1. 

Let $D=(A,L)$ be an $MCD(n,s_1^{q_1}{s}_2^{q_2}\cdots s_c^{q_c},p)$, $q_1+q_2+\cdots +q_c=q$, where $A=(A_1,A_2,\dots ,A_c)$ is an $OA(n,s_1^{q_1}{s}_2^{q_2}\cdots s_c^{q_c},2)$, L is an $LHD(n,p)$, $A_i$ is an $OA(n,s_i^{q_i},2)$, $i=1,2,\dots ,c$.

(i) The D is called a within-group DCD (WGDCD), denoted by $WGDCD(n,s_1^{q_1}{s}_2^{q_2}\cdots s_c^{q_c},p)$, if $(A_i,L)$ is a DCD for $i=1,2,\dots ,c$. When $s_1=s_2=\dots =s_c=s$ and $q_1=q_2=\dots =q_c=u$, then we denote such D by $WGDCD(n,s^{u·c},p)$.

(ii) The D is called a between-group DCD (BGDCD), denoted by $BGDCD(n,s_1^{q_1}{s}_2^{q_2}\cdots s_c^{q_c},p)$, if $((A_i^j,A_m^n),L)$ is a DCD, where $A_i^j$ and $A_m^n$ are the jth column in $A_i$ and the nth column in $A_m$, respectively, for $1\le i\ne m\le c$, j = $1,2,\dots ,q_i$, n = $1,2,\dots ,q_m$. When $s_1=s_2=\dots =s_c=s$ and $q_1=q_2=\dots =q_c=u$, then we denote such D by $BGDCD(n,s^{u·c},p)$.

From Definition 1, it is easy to see that $(A_i,L)$ is a $DCD(n,s_i^{q_i},p)$ in a $WGDCD(n,s_1^{q_1}{s}_2^{q_2}$$\cdots s_c^{q_c},p)$, for $i=1,2,\dots ,c$. For any $DCD(n,s^q,p)$, Corollary 1 of [21] shows that $q\le s$. Similarly, we have the following Corollary 1.

Corollary 1. 

If a WGDCD $D=(A,L)$ with A being an $OA(n,s_1^{q_1}{s}_2^{q_2}\cdots s_c^{q_c},2)$ exists, then $q_j\le s_j$, $j=1,2,\dots ,c$.

Corollary 1 above tells us that the $WGDCD(n,s_1^{q_1}{s}_2^{q_2}\cdots s_c^{q_c},p)$ can accommodate up to $s_1+s_2+\cdots +s_c$ qualitative factors.

Here we provide some results on the existence of GDCDs. Recall the definition of GDCDs, a GDCD of n runs has two subdesigns, A and L, which are for q qualitative factors and p quantitative factors, respectively. Theorems 1 and 2 below establish the necessary and sufficient conditions of the existence of WGDCDs and BGDCDs, respectively. For ease of expression, for an $n\times 1$ column vector d, define $d^{\left(s_i\right)}$ and $d^{(s_i·s_m)}$ based on d. Let $d^{\left(v\right)}$ be the vth entry of d, $1\le v\le n$ and $d^{(v,s_i)}=⌊\frac{d^{\left(v\right)}}{s_i}⌋$, $d^{(v,s_i·s_m)}=⌊\frac{d^{\left(v\right)}}{s_i{s}_m}⌋$, where $d^{(v,s_i)}$ and $d^{(v,s_i·s_m)}$ are the vth entries in $d^{\left(s_i\right)}$ and $d^{(s_i·s_m)}$, respectively, and $⌊a⌋$ represents the largest integer not exceeding a.

Theorem 1. 

Suppose $A=(A_1,A_2,\dots ,A_c)$ is an $OA(n,s_1^{q_1}{s}_2^{q_2}\cdots s_c^{q_c},2)$, $L=(l_1,l_2,\dots ,l_p)$ is an $LHD(n,p)$, where $l_k$ is the kth column of L, $k=1,2,\dots ,p$. Let $A_i^j$ be the jth column of $A_i$ for $i=1,2,\dots ,c$, j = $1,2,\dots ,q_i$. Then design $D=(A,L)$ is a $WGDCD(n,s_1^{q_1}{s}_2^{q_2}\cdots s_c^{q_c},p)$ if and only if:

(i) $(A_i^j,l_k^{\left(s_i\right)})$ is an $OA(n,2,s_i(n/s_i),2)$, for any $i=1,2,\dots ,c$, $j=1,2,\dots ,q_i$, $k=1,2,\dots ,p$; and

(ii) $(A_i^j,A_i^f,l_k^{(s_i·s_i)})$ is an $OA(n,3,s_i^2(n/s_i^2),3)$, for any $i=1,2,\dots ,c$, $1\le j\ne f\le q_i$, $k=1,2,\dots ,p$.

Theorem 2. 

Suppose $A=(A_1,A_2,\dots ,A_c)$ is an $OA(n,s_1^{q_1}{s}_2^{q_2}\cdots s_c^{q_c},2)$, $L=(l_1,l_2,\dots ,l_p)$ is an $LHD(n,p)$, where $l_k$ is the kth column of L, $k=1,2,\dots ,p$. Let $A_i^j$ be the jth column of $A_i$ for $i=1,2,\dots ,c$, j = $1,2,\dots ,q_i$. Then design $D=(A,L)$ is a $BGDCD(n,s_1^{q_1}{s}_2^{q_2}\cdots s_c^{q_c},p)$ if and only if:

(i) $(A_i^j,l_k^{\left(s_i\right)})$ is an $OA(n,2,s_i(n/s_i),2)$, for any $i=1,2,\dots ,c$, $j=1,2,\dots ,q_i$, $k=1,2,\dots ,p$; and

(ii) $(A_i^j,A_m^n,l_k^{(s_i·s_m)})$ is an $OA(n,3,s_i{s}_m(n/s_i{s}_m),3)$, for any $1\le i\ne m\le c$, $j=1,2,\dots ,q_i$, $n=1,2,\dots ,q_m$, $k=1,2,\dots ,p$.

Theorems 1 and 2 establish the existence of GDCDs in terms of the relations between the individual columns in A and $l_k^{\left(s_i\right)}$, and between any pair of columns in A and $l_k^{(s_i·s_i)}$ or $l_k^{(s_i·s_m)}$.

3. Construction of GDCDs

Since the existing DCDs in [21] have an upper bound on the number of qualitative factors, that is not exceeding the number of levels. The subdesigns for qualitative factors are all symmetric OAs, in the DCDs constructed by [21]. This section, therefore, describes four main construction algorithms to produce different GDCDs that can contain more qualitative factors. Two forms of GDCDs, WGDCDs and BGDCDs, are provided for different needs. In Section 3.1, two algorithms were proposed for constructing WGDCDs with equal-level and mixed-level qualitative factors, respectively. In Section 3.2, two algorithms were proposed for generating different BGDCDs with mixed-level qualitative factors. These newly constructed designs with qualitative factors can be either symmetric or asymmetric. Different initial DCDs are used in the construction Algorithms 1, 3 and 5. The above constructions lead the resulting designs to entertain more qualitative factors and mixed-level qualitative factors, compared with the existing DCDs.

3.1. Construction of WGDCDs

Existing DCDs have few columns and do not work if the problem under study has $s+1$ or more factors (s is the number of levels of qualitative factors). To be able to study the relationship between more qualitative and quantitative factors, this section presents WGDCDs, which can accommodate more qualitative factors with equal or mixed levels than DCDs. The construction of WGDCDs is presented in the next two subsections.

3.1.1. Construction of WGDCDs with Symmetric Qualitative Factors

Suppose there exists a difference scheme $D(r,c,s)$ of strength 2, denoted by $D\left(1\right)$, where r is a multiple of s, $c\le r$, and an initial $DCD(n,s^q,p)$ $D^0=(D_1,D_2)$. Such difference scheme and $D^0$ are used in the following algorithm to construct a $WGDCD(rn,s^{q·c},pf)$ $D=(A,L)$, where A is an $OA(rn,s^{q·c},2)$, each group of the subdesign A is a symmetric $OA(rn,s^q,2)$. For clarity, let Kronecker sum ⊕ in Algorithm 1 be defined over the Galois field of order s ($GF\left(s\right)$).

Algorithm 1 Construction of WGDCDs with symmetric qualitative factors

Step 1.  Given that $D\left(1\right)$ is a difference scheme $D(r,c,s)$, $D_1$ is an $OA(n,s^q,2)$, $D_2$ is an $LHD(n,p)$ and $D^0=(D_1,D_2)$ is a $DCD(n,s^q,p)$, then construct a $\left(rn\right)\times \left(qc\right)$ matrix, as $A=(D\left(1\right)\oplus D_1)$ over $GF\left(s\right)$. Step 2.  Let C be an $r\times f$ matrix with all elements being ones, H be an $LHD(r,pf)$, construct an $\left(rn\right)\times \left(pf\right)$ matrix, as $L=C\otimes D_2+nH\otimes {\mathbf{1}}_n$. Step 3.  The resulting design is $D=(A,L)$.

Proposition 1. 

The design $D=(A,L)$ obtained by Algorithm 1 is a $WGDCD\left(rn,s^{q·c},pf\right)$.

The initial design $D^0$ in Step 1 is a DCD, which can be obtained from [21]. The subdesign A of the design D constructed by Algorithm 1 is an $OA(rn,s^{cq},2)$ and can be divided into c groups of q columns each, forming an $OA(rn,s^q,2)$. There are excellent stratification properties between the columns in the same group and L. DCDs with s-level qualitative factors can accommodate up to s qualitative factors according to Corollary 1 of [21]. However, the WGDCDs constructed by Algorithm 1 can accommodate up to $cs$ qualitative factors. The following example gives an illustration of Algorithm 1, where the initial DCD $D^0$ is taken from the first four columns of Table 1 in [21].

Example 1. 

Consider a design $WGDCD(32,2^{2·4},4)$ with the initial $DCD$ $D^0$ as in

Table 1. The $DCD(8,2^2,2)$ used in Example 1.

$D_1^T$	0	1	0	1	0	1	0	1
	0	1	1	0	0	1	1	0
$D_2^T$	1	0	6	7	4	5	3	2
	0	4	2	6	5	1	7	3

. By using the following difference scheme $D\left(1\right)$, $4\times 2$ matrix C and any $LHD(4,4)$ H, we can obtain A and L in Algorithm 1, respectively. The resulting design $D=(A,L)$ is listed in

Table 2. $WGDCD(32,2^{2·4},4)$ in Example 1.

Run	A								L				Run	A								L
1	0	0	0	0	0	0	0	0	1	16	25	8	17	0	0	0	0	1	1	1	1	25	24	17	0
2	1	1	1	1	1	1	1	1	0	20	24	12	18	1	1	1	1	0	0	0	0	24	28	16	4
3	0	1	0	1	0	1	0	1	6	18	30	10	19	0	1	0	1	1	0	1	0	30	26	22	2
4	1	0	1	0	1	0	1	0	7	22	31	14	20	1	0	1	0	0	1	0	1	31	30	23	6
5	0	0	0	0	0	0	0	0	4	21	28	13	21	0	0	0	0	1	1	1	1	28	29	20	5
6	1	1	1	1	1	1	1	1	5	17	29	9	22	1	1	1	1	0	0	0	0	29	25	21	1
7	0	1	0	1	0	1	0	1	3	23	27	15	23	0	1	0	1	1	0	1	0	27	31	19	7
8	1	0	1	0	1	0	1	0	2	19	26	11	24	1	0	1	0	0	1	0	1	26	27	28	3
9	0	0	1	1	0	0	1	1	9	8	9	16	25	0	0	1	1	1	1	0	0	17	0	1	24
10	1	1	0	0	1	1	0	0	8	12	8	20	26	1	1	0	0	0	0	1	1	16	4	0	28
11	0	1	1	0	0	1	1	0	14	10	14	18	27	0	1	1	0	1	0	0	1	22	2	6	26
12	1	0	0	1	1	0	0	1	15	14	15	22	28	1	0	0	1	0	1	1	0	23	6	7	30
13	0	0	1	1	0	0	1	1	12	13	12	21	29	0	0	1	1	1	1	0	0	20	5	4	29
14	1	1	0	0	1	1	0	0	13	9	13	17	30	1	1	0	0	0	0	1	1	21	1	5	25
15	0	1	1	0	0	1	1	0	11	15	11	23	31	0	1	1	0	1	0	0	1	19	7	3	31
16	1	0	0	1	1	0	0	1	10	11	10	19	32	1	0	0	1	0	1	1	0	18	3	2	27

. It is easy to check that the D is a $WGDCD(32,2^{2·4},4)$. Obviously, the subdesign A can be divided into 4 groups of 2 columns each, forming an OA. Let $A=(A_1,A_2,A_3,A_4)$, where $A_i$ is an $OA(16,2^2,2)$, then $(A_i,L)$ is a $DCD(16,2^2,4)$, $i=1,2,3,4$. The $WGDCD(32,2^{2·4},4)$ outperforms $DCD(32,2^2,4)$ obtained by [21] in terms of the number of qualitative factors. Figure 1 shows that the maximum one-dimensional projection uniformity of L with respect to each level combination of any one or two factors in $A_1$.

$D\left(1\right)=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 1& 0& 1\\ 0& 0& 1& 1\\ 0& 1& 1& 0\end{array}\right]$, $C=\left[\begin{array}{cc}1& 1\\ 1& 1\\ 1& 1\\ 1& 1\end{array}\right]$ and $H=\left[\begin{array}{cccc}0& 2& 3& 1\\ 1& 1& 1& 2\\ 3& 3& 2& 0\\ 2& 0& 0& 3\end{array}\right]$.

3.1.2. Construction of WGDCDs with Asymmetric Qualitative Factors

Next, we provide another method to construct a WGDCD $D=(A,L)$ with the subdesign A for qualitative factors being an asymmetric OA. Without loss of generality, we consider the case of $c=2$, i.e., $A=(A_1,A_2)$ is an $OA(n,s_1^{q_1}{s}_2^{q_2},2)$ in this paper. Here we assume that $s_2=s_1^2$ and n must be a multiple of $s_2^2$. The following algorithm is to construct a $WGDCD(\lambda s_2^2,s_1^{s_1}{s}_2^{s_2-1},p)$ $D=(A,L)$, with $A=(A_1,A_2)$ being an $OA(\lambda s_2^2,s_1^{s_1}{s}_2^{s_2-1},2)$, where $A_1$ is an $OA(\lambda s_2^2,s_1^{s_1},2)$ and $A_2$ is an $OA(\lambda s_2^2,s_2^{s_2-1},2)$, L is an $LHD(\lambda s_2^2,p)$.

We apply the Rao-Hamming construction in Section 3.4 of [22] to obtain the orthogonal arrays E and F in Algorithm 2. For example, if $s_1=2$, then an $OA(4,2^3,2)$ E can be obtain by the Rao-Hamming construction over $GF\left(2\right)$, as

$$E=\left(\begin{array}{cc}0& 0\\ 1& 0\\ 1& 1\\ 0& 1\end{array}\right)\left(\begin{array}{ccc}1& 1& 0\\ 0& 1& 1\end{array}\right)=\left(\begin{array}{ccc}0& 0& 0\\ 1& 1& 0\\ 1& 0& 1\\ 0& 1& 1\end{array}\right).$$

After row permutation, E can be transformed into

$$E=\left(\begin{array}{ccc}0& 0& 0\\ 0& 1& 1\\ 1& 0& 1\\ 1& 1& 0\end{array}\right).$$

It is easy to see that

$$E_0=(e_2,E^{*})={\left(\begin{array}{cccc}0& 1& 0& 1\\ 0& 1& 1& 0\end{array}\right)}^T.$$

Algorithm 2 Construction of WGDCDs with asymmetric qualitative factors

Step 1.  Given an $OA(s_1^2,s_1^{s_1+1},2)$ E and an $OA(s_2^2,s_2^{s_2+1},2)$ F, $s_2=s_1^2$. After row permutation, E and F can be transformed into $E=(e_1,e_2,E^{*})$ and $F=(f_1,f_2,F^{*})$, respectively, so that $e_1={\left({\mathbf{0}}_{s_1}^T,{\mathbf{1}}_{s_1}^T,\dots ,{({\mathbf{s}}_{\mathbf{1}}-\mathbf{1})}_{s_1}^T\right)}^T$, $e_2={\mathbf{1}}_{s_1}\otimes {(0,1,\dots ,s_1-1)}^T$, $f_1={\left({\mathbf{0}}_{s_2}^T,{\mathbf{1}}_{s_2}^T,\dots ,{({\mathbf{s}}_{\mathbf{2}}-\mathbf{1})}_{s_2}^T\right)}^T$, $f_2={\mathbf{1}}_{s_2}\otimes {(0,1,\dots ,s_2-1)}^T$. Delete the columns $e_1$ and $f_1$ of E and F, respectively. Denote the remaining columns as $E_0$ and $F_0$, respectively, i.e., $E_0=(e_2,E^{*})$ and $F_0=(f_2,F^{*})$. Step 2.  On base of $F_0$, the first to last rows of $E_0$ are used to replace the $s_2$ levels of $f_2$, respectively, and denote the replaced design as G, G is an asymmetric $OA(s_2^2,s_1^{s_1}{s}_2^{s_2-1},2)$. Let $A={\mathbf{1}}_{\lambda }\otimes G$. Step 3.  For a given $\lambda$ and p, let ${\pi }_k$ be a random permutation of ${(0,1,\dots ,\lambda -1)}^T$, $g_k$ be a random permutation of ${(0,1,\dots ,s_2-1)}^T$, and $v_k={(c_1^T,c_2^T,\dots ,c_{s_1}^T)}^T$, $k=1,2,\dots ,p$, where $c_i=(i-1)s_1\oplus z$, z is a random permutation of ${(0,1,\dots ,s_1-1)}^T$, $i=1,2,\dots ,s_1$. Construct an $LHD$$L=(l_1,l_2,\dots ,l_p)$, where $l_k=s_2^2({\pi }_k\otimes {\mathbf{1}}_{s_2^2})+s_2({\mathbf{1}}_{\lambda }\otimes (g_k\otimes {\mathbf{1}}_{s_2}))+{\mathbf{1}}_{\lambda s_2}\otimes v_k$, for $k=1,2,\dots ,p$. Step 4.  The resulting design is $D=(A,L)$.

Theorem 3. 

The design $D=(A,L)$ obtained by Algorithm 2 is a $WGDCD(\lambda s_2^2,s_1^{s_1}{s}_2^{s_2-1},p)$ with $s_2=s_1^2$.

Clearly, in Theorem 3, A can be divided into two groups, denoted as $A=(A_1,A_2)$, where $A_1$ and $A_2$ are $OA(\lambda s_2^2,s_1^{s_1},2)$ and $OA(\lambda s_2^2,s_2^{s_2-1},2)$, respectively. Obviously, $(A_1,L)$ and $(A_2,L)$ are a $DCD(\lambda s_2^2,s_1^{s_1},p)$ and a $DCD(\lambda s_2^2,s_2^{s_2-1},p)$, respectively. It is not difficult to find that the number of columns in each group of the subdesign A in the design D constructed by Algorithm 2 almost reaches its number of levels. The total number of qualitative factors is $s_1+(s_2-1)$. The designs constructed by Algorithm 2 cannot be constructed in [21], because the subdesigns for qualitative factors, constructed by [21], are all symmetric OAs. An illustration of Algorithm 2 is given in the following example.

Example 2. 

Construct a $WGDCD(32,2^2{4}^3,2)$, when $\lambda =2$. In Step 1, E and F (after rows permuting) are obtained using the Rao-Hamming construction as follows,

$E^T=\left[\begin{array}{cccc}0& 0& 1& 1\\ 0& 1& 1& 0\\ 0& 1& 0& 1\end{array}\right]$, $F^T=\left[\begin{array}{cccccccccccccccc}0& 0& 0& 0& 1& 1& 1& 1& 2& 2& 2& 2& 3& 3& 3& 3\\ 0& 1& 2& 3& 0& 1& 2& 3& 0& 1& 2& 3& 0& 1& 2& 3\\ 0& 2& 3& 1& 1& 3& 2& 0& 2& 0& 1& 3& 3& 1& 0& 2\\ 0& 3& 1& 2& 1& 2& 0& 3& 2& 1& 3& 0& 3& 0& 2& 1\\ 0& 1& 2& 3& 1& 0& 3& 2& 2& 3& 0& 1& 3& 2& 1& 0\end{array}\right]$. In Step 2, obtain $G^T=\left[\begin{array}{cccccccccccccccc}0& 1& 1& 0& 0& 1& 1& 0& 0& 1& 1& 0& 0& 1& 1& 0\\ 0& 1& 0& 1& 0& 1& 0& 1& 0& 1& 0& 1& 0& 1& 0& 1\\ 0& 2& 3& 1& 1& 3& 2& 0& 2& 0& 1& 3& 3& 1& 0& 2\\ 0& 3& 1& 2& 1& 2& 0& 3& 2& 1& 3& 0& 3& 0& 2& 1\\ 0& 1& 2& 3& 1& 0& 3& 2& 2& 3& 0& 1& 3& 2& 1& 0\end{array}\right]$. Let $A={\mathbf{1}}_2\otimes G$. In Step 3, let ${\pi }_1={(1,0)}^T,{\pi }_2={(0,1)}^T,g_1={(0,3,2,1)}^T,g_2={(1,3,0,2)}^T,v_1={(1,0,3,2)}^T,v_2={(0,1,3,2)}^T$. We can get L is an $LHD(32,2)$. In Step 5, the resulting design $D=(A,L)$ is a $WGDCD(32,2^2{4}^3,2)$ and shown in

Table 3. $WGDCD(32,2^2{4}^3,2)$ in Example 2.

Run	A					L		Run	A					L
1	0	1	0	0	0	17	4	17	0	1	0	0	0	1	20
2	1	0	2	3	1	16	5	18	1	0	2	3	1	0	21
3	1	1	3	1	2	19	7	19	1	1	3	1	2	3	23
4	0	0	1	2	3	18	6	20	0	0	1	2	3	2	22
5	0	1	1	1	1	29	12	21	0	1	1	1	1	13	28
6	1	0	3	2	0	28	13	22	1	0	3	2	0	12	29
7	1	1	2	0	3	31	15	23	1	1	2	0	3	15	31
8	0	0	0	3	2	30	14	24	0	0	0	3	2	14	30
9	0	1	2	2	2	25	0	25	0	1	2	2	2	9	16
10	1	0	0	1	3	24	1	26	1	0	0	1	3	8	17
11	1	1	1	3	0	27	3	27	1	1	1	3	0	11	19
12	0	0	3	0	1	26	2	28	0	0	3	0	1	10	18
13	0	1	3	3	3	21	8	29	0	1	3	3	3	5	24
14	1	0	1	0	2	20	9	30	1	0	1	0	2	4	25
15	1	1	0	2	1	23	11	31	1	1	0	2	1	7	27
16	0	0	2	1	0	22	10	32	0	0	2	1	0	6	26

. A visualization of this example is shown in Figure 2. Clearly, Figure 2 shows the design points of quantitative factors enjoy the maximum one-dimensional stratification corresponding to each level combination of any two qualitative factors in $A_1$ and $A_2$, respectively.

If a small MCD can be constructed, then a large MCD with more columns can be constructed following Construction 3 of [16]. Similar to Construction 3 of [16], based on the $WGDCDs$ obtained by Algorithm 2, a series of new WGDCDs with more columns can be constructed by Corollary 2 as follows.

Corollary 2. 

Let $A=(A_1,A_2)$ be an $OA(\lambda s_2^2,s_1^{s_1}{s}_2^{s_2-1},2)$, where $A_j$ is the orthogonal array with $s_j$ levels, $j=1,2$. If for some u, there are difference schemes $D(u,c_j,s_j)$ of strength two, denoted by $D\left(j\right)$, for j = 1,2, then the design $A_{new}$ = $(D\left(1\right)\oplus A_1,D\left(2\right)\oplus A_2)$ over $GF\left(s_1\right)$ and $GF\left(s_2\right)$ is an $OA(\lambda u{s}_2^2,s_1^{s_1·c_1}{s}_2^{(s_2-1)·c_2},2)$. Let C be an $u\times f$ matrix with all elements being ones, H be an any $LHD(u,pf)$, obtain an LHD $L_{new}$ = $C\oplus L+\lambda s_2^{2}H\oplus {\mathbf{1}}_{\lambda s_2^2}$, then ($A_{new}$,$L_{new}$) is a $WGDCD(\lambda u{s}_2^2,s_1^{s_1·c_1}{s}_2^{(s_2-1)·c_2},pf)$.

3.2. Construction of BGDCDs

Section 3.1 is devoted to constructing WGDCDs with either equal-level or mixed-level qualitative factors, where each group of A in a WGDCD achieves excellent stratification between any two qualitative factors and quantitative factors. We now construct the second type of GDCDs, the BGDCDs with mixed-level qualitative factors. In a $WGDCD(n,s_1^{q_1}{s}_2^{q_2},p)$, a multiple relation between $s_1$ and $s_2$ is required, i.e., $s_2=s_1^2$. But, in a $BGDCD(n,s_1^{q_1}{s}_2^{q_2},p)$, this relation may or may not be present. Compared to WGDCDs, the BGDCDs have better stratification properties between any two qualitative factors from different groups of A and quantitative factors. Since the qualitative factor designs in DCDs constructed by [21] are symmetric OAs, we focus on the case that the qualitative factor designs are asymmetric OAs in Section 3.2.

3.2.1. Construction of BGDCDs Based on $OA(s_1{s}_2,s_1^1{s}_2^1,2)$

In this section, we also construct BGDCDs using an initial DCD $D^0$. Here we only discuss the case of BGDCDs with asymmetric qualitative factors. Since $D_1$ in an initial DCD $D^0$ is a symmetric OA, then WGDCDs like the one in Section 3.1.1 can be obtained by Algorithm 3 below. In light of Corollary 2, we propose Algorithm 3 below. In Step 2 of Algorithm 3, the ⊕ operator is based on $GF\left(s_1\right)$ and $GF\left(s_2\right)$.

Algorithm 3 Construction of BGDCDs based on $OA(s_1{s}_2,s_1^1{s}_2^1,2)$

Step 1.  Let $(M,B)$ be an $MCD(s_1{s}_2,s_1^1{s}_2^1,p)$, where M is an $OA(s_1{s}_2,s_1^1{s}_2^1,2)$ and B is an $LHD(s_1{s}_2,p)$. Let $D_1={\mathbf{1}}_{\lambda }\otimes M$ and $D_2={(B^T,{(s_1{s}_2\oplus B)}^T,{(2s_1{s}_2\oplus B)}^T,\dots ,{((\lambda -1)s_1{s}_2\oplus B)}^T)}^T$, then construct an initial $DCD(\lambda s_1{s}_2,s_1^1{s}_2^1,p)$$D^0$, as $D^0=(D_1,D_2)$, where $D_1=(D_{11},D_{12})$, $D_{1j}$ is an $OA(\lambda s_1{s}_2,s_j^1,1)$, j = 1,2. Step 2.  Obtain two designs $A=(A_1,A_2)=(D\left(1\right)\oplus D_{11},D\left(2\right)\oplus D_{12})$ over $GF\left(s_1\right)$ and $GF\left(s_2\right)$, and $L=C\otimes D_2+\lambda s_1{s}_{2}H\otimes {\mathbf{1}}_{\lambda s_1{s}_2}$, where $D\left(j\right)$ is a difference scheme $D(u,c_j,s_j)$ for j = 1, 2, C is an $u\times f$ matrix with all elements being ones, and H is any $LHD(u,pf)$. Step 3.  The resulting design $D=(A,L)$.

Remark 1. 

The design $(M,B)$ is an MCD which can be obtained from [23]. The $D^0$ generated by Algorithm 3 has two cases: (i) When $s_1=s_2$ in M, the design $D^0$ is a DCD with equal-level qualitative factors; (ii) Otherwise the obtained design $D^0$ is a DCD with mixed-level qualitative factors. In addition, design $A=(A_1,A_2)$ obtained from Step 2 is an $OA(u\lambda s_1{s}_2,s_1^{c_1}{s}_2^{c_2},2)$, where $A_j$ is an $OA(u\lambda s_1{s}_2,s_j^{c_j},2)$, j = 1,2.

Proposition 2. 

According to Algorithm 3, we have the following results:

(i) The design $D^0=(D_1,D_2)$ obtained from Step 1 is a $DCD(\lambda s_1{s}_2,s_1^1{s}_2^1,p)$, where $D_1$ is an $OA(\lambda s_1{s}_2,2,s_1{s}_2,2)$, $D_2$ is an $LHD(\lambda s_1{s}_2,p)$;

(ii) The design $D=(A,L)$ obtained from Algorithm 3 is a $BGDCD(u\lambda s_1{s}_2,s_1^{c_1}{s}_2^{c_2},pf)$.

In a $WGDCD(n,s_1^{q_1}{s}_2^{q_2},p)$ requires $s_2=s_1^2$. But, for the BGDCDs constructed by Algorithm 3, this multiple relation is unnecessary. Next, we give an illustrative example of Algorithm 3.

Example 3. 

First, the design $(M,B)$ in

Table 4. $MCD(6,2^1{3}^1,2)$ in Example 3.

Run	M		B
1	0	0	0	3
2	0	1	2	5
3	0	2	4	1
4	1	0	3	0
5	1	1	5	2
6	1	2	1	4

is an $MCD(6,2^1{3}^1,2)$ of 6 runs for two qualitative factors $m_1,m_2$ and two quantitative factors $b_1,b_2$. Here M is an asymmetric $OA(6,2^1{3}^1,2)$ and B is an $LHD(6,2)$. In Step 1, let $\lambda =2$ and juxtapose the two M’s row by row to obtain $D_1$, which is an $OA(12,2^1{3}^1,2)$ and can be partitioned into two full factorial designs, while using B to obtain $D_2$. Then we can obtain design $D^0=(D_1,D_2)$ in

Table 5. The $DCD(12,2^1{3}^1,2)$ used in Example 3.

$D_1^T$	$D_{11}^T$	0	0	0	1	1	1	0	0	0	1	1	1
	$D_{12}^T$	0	1	2	0	1	2	0	1	2	0	1	2
$D_2^T$		0	2	4	3	5	1	6	8	10	9	11	7
		3	5	1	0	2	4	9	11	7	6	8	10

is a $DCD(12,2^1{3}^1,2)$. It can be easily checked that the rows in $D_2$ corresponding to any level combination of any one or two factors in $D_1$ form an LHD.

Second, take $D{\left(1\right)}^T=\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 0& 1& 0& 1& 0& 1\end{array}\right]$, $D{\left(2\right)}^T=\left[\begin{array}{cccccc}0& 0& 0& 0& 0& 0\\ 0& 1& 2& 0& 1& 2\\ 0& 2& 1& 0& 2& 1\end{array}\right]$, $C^T=\left[\begin{array}{cccccc}1& 1& 1& 1& 1& 1\\ 1& 1& 1& 1& 1& 1\end{array}\right]$ and $H^T=\left[\begin{array}{cccccc}0& 1& 2& 3& 4& 5\\ 2& 1& 5& 3& 0& 4\\ 3& 4& 2& 5& 1& 0\\ 4& 3& 0& 1& 5& 2\end{array}\right]$. Then by Step 2 and Step 3, we obtain that the design D is a $BGDCD(72,2^2{3}^3,4)$. The final design $D=(A,L)$ is shown in

Table 6. $BGDCD(72,2^2{3}^3,4)$ in Example 3.

Run	A					L				Run	A					L
1	0	0	0	0	0	0	27	36	51	37	0	1	0	0	0	36	39	60	15
2	0	0	1	1	1	2	29	38	53	38	0	1	1	1	1	38	41	62	17
3	0	0	2	2	2	4	25	40	49	39	0	1	2	2	2	40	37	64	13
4	1	1	0	0	0	3	24	39	48	40	1	0	0	0	0	39	36	63	12
5	1	1	1	1	1	5	26	41	50	41	1	0	1	1	1	41	38	65	14
6	1	1	2	2	2	1	28	37	52	42	1	0	2	2	2	37	40	61	16
7	0	0	0	0	0	6	33	42	57	43	0	1	0	0	0	42	45	66	21
8	0	0	1	1	1	8	35	44	59	44	0	1	1	1	1	44	47	68	23
9	0	0	2	2	2	10	31	46	55	45	0	1	2	2	2	46	43	70	19
10	1	1	0	0	0	9	30	45	54	46	1	0	0	0	0	45	42	69	18
11	1	1	1	1	1	11	32	47	56	47	1	0	1	1	1	47	44	71	20
12	1	1	2	2	2	7	34	43	58	48	1	0	2	2	2	43	46	67	22
13	0	1	0	1	2	12	15	48	39	49	0	0	0	1	2	48	3	12	63
14	0	1	1	2	0	14	17	50	41	50	0	0	1	2	0	50	5	14	65
15	0	1	2	0	1	16	13	52	37	51	0	0	2	0	1	52	1	16	61
16	1	0	0	1	2	15	12	51	36	52	1	1	0	1	2	51	0	15	60
17	1	0	1	2	0	17	14	53	38	53	1	1	1	2	0	53	2	17	62
18	1	0	2	0	1	13	16	49	40	54	1	1	2	0	1	49	4	13	64
19	0	1	0	1	2	18	21	54	45	55	0	0	0	1	2	54	9	18	69
20	0	1	1	2	0	20	23	56	47	56	0	0	1	2	0	56	11	20	71
21	0	1	2	0	1	22	19	58	43	57	0	0	2	0	1	58	7	22	67
22	1	0	0	1	2	21	18	57	42	58	1	1	0	1	2	57	6	21	66
23	1	0	1	2	0	23	20	59	44	59	1	1	1	2	0	59	8	23	68
24	1	0	2	0	1	19	22	55	46	60	1	1	2	0	1	55	10	19	70
25	0	0	0	2	1	24	63	24	3	61	0	1	0	2	1	60	51	0	27
26	0	0	1	0	2	26	65	26	5	62	0	1	1	0	2	62	53	2	29
27	0	0	2	1	0	28	61	28	1	63	0	1	2	1	0	64	49	4	25
28	1	1	0	2	1	27	60	27	0	64	1	0	0	2	1	63	48	3	24
29	1	1	1	0	2	29	62	29	2	65	1	0	1	0	2	65	50	5	26
30	1	1	2	1	0	25	64	25	4	66	1	0	2	1	0	61	52	1	28
31	0	0	0	2	1	30	69	30	9	67	0	1	0	2	1	66	57	6	33
32	0	0	1	0	2	32	71	32	11	68	0	1	1	0	2	68	59	8	35
33	0	0	2	1	0	34	67	34	7	69	0	1	2	1	0	70	55	10	31
34	1	1	0	2	1	33	66	33	6	70	1	0	0	2	1	69	54	9	30
35	1	1	1	0	2	35	68	35	8	71	1	0	1	0	2	71	56	11	32
36	1	1	2	1	0	31	70	31	10	72	1	0	2	1	0	67	58	7	34

. Figure 3 reflects the projection property of L, with respect to level combinations of ($A_1^1,A_2^1$) and ($A_1^2,A_2^2$), respectively.

3.2.2. Construction of BGDCDs Based on $(s_1\times 1)$-ROAs

For the $BGDCD$ $D=(A,L)$ constructed by Algorithm 3, the number of columns in A is determined by the number of columns in $D_1$, which is taken from the initial design $D^0=(D_1,D_2)$. Since the $D_1$ has only two columns, the number of columns in A is very small. To solve this problem, we propose Algorithms 4 and 5. The initial design is constructed using Algorithm 4, and based on this initial design, a BGDCD with a large number of qualitative factor columns can be constructed by Algorithm 5. Before presenting Algorithm 4, we give Theorem 4 and Proposition 3, which are extremely useful for Algorithm 4.

According to Lemma 1 derived by [17] and Lemma 2 derived by [21], we present the necessary and sufficient condition for the existence of DCDs when $D_1$ is an asymmetric $OA(n,s_1^{q_1}{s}_2,2)$ with $s_2=s_1^2$. To drive this result, we define matrices $D_2^{\prime }$, $D_2^{″}$ and $D_2^{‴}$ based on $D_2$. For $D_2=(d_1,d_2,\dots ,d_p)$ in a DCD $D^0=(D_1,D_2)$, $d_k$ is the kth column of $D_2$, $d_{ik}$ is the $(i,k)$th entry of $D_2$, $1\le i\le n,1\le k\le p$. Let $d_{ik}^{\prime }=⌊\frac{d_{ik}}{.}⌋$, $d_{ik}^{″}=⌊\frac{d_{ik}}{s^2}⌋$ = $⌊\frac{d_{ik}^{\prime }}{.}⌋$, and $d_{ik}^{‴}=⌊\frac{d_{ik}}{s^3}⌋$ = $⌊\frac{d_{ik}^{″}}{.}⌋$, where $d_{ik}^{\prime }$, $d_{ik}^{″}$ and $d_{ik}^{‴}$ are the $(i,k)$th entries in $D_2^{\prime }$, $D_2^{″}$ and $D_2^{‴}$, respectively, and s is the number of levels of qualitative factors in $D_1$, $⌊a⌋$ represents the largest integer not exceeding a. Let $d_k^{\prime }$, $d_k^{″}$ and $d_k^{‴}$ be the kth columns of $D_2^{\prime }$, $D_2^{″}$ and $D_2^{‴}$, respectively. Refs. [17,21] derived Lemmas 1 and 2 below, respectively.

Lemma 1 

([17]). Given $D_1$ is an $OA(n,q,s,2)$, $D_2$ is an $LHD(n,p)$ and $D_2^{\prime }$ is defined as above, then $(D_1,D_2)$ is an $MCD(n,s^q,p)$ if and only if for $k=1,2,\dots ,p$, $(D_1,d_k^{\prime })$ is an asymmetric $OA(n,s^m(n/s),2)$, where $d_k^{\prime }$ is the kth column of $D_2^{\prime }$.

Lemma 2 

([21]). Suppose that $D_1=(z_1,z_2,\dots ,z_q)$ is an $OA(n,q,s,2)$ and $D_2=(d_1,d_2,\dots ,d_p)$ is an $LHD(n,p)$. The design $D^0=(D_1,D_2)$ is a $DCD(n,s^q,p)$ if and only if

(i) $(z_i,d_k^{\prime })$ is an $OA(n,2,s(n/s),2)$, for any $1\le i\le q$, $1\le k\le p$; and

(ii) $(z_i,z_j,d_k^{″})$ is an $OA(n,3,s^2(n/s^2),3)$, for any $1\le i\ne j\le q$, $1\le k\le p$.

Theorem 4. 

Suppose that $D_1=(M,B)$ is an $OA(\lambda s_1{s}_2,s_1^{q_1}{s}_2,2)$, where $s_2=s_1^2$, $\lambda \ge 2$, M and B are the first $q_1$ columns and the last column of $D_1$, respectively, $D_2$ is an $LHD(\lambda s_1^3,p)=(d_1,d_2,\dots ,d_p)$, then the design $D^0=(D_1,D_2)$ is a $DCD(\lambda s_1^3,s_1^{q_1}{s}_2,p)$, if and only if, for $1\le k\le p$:

(i) $(m_i,d_k^{\prime })$ is an $OA(\lambda s_1^3,2,s_1\left(\lambda s_1^2\right),2)$, $m_i$ is the ith column of M, $1\le i\le q_1$;

(ii) $(B,d_k^{″})$ is an $OA(\lambda s_1^3,2,s_2\left(\lambda s_1\right),2)$;

(iii) $(m_i,m_j,d_k^{″})$ is an $OA(\lambda s_1^3,3,s_1^2\left(\lambda s_1\right),3)$, $1\le i\ne j\le q_1$;

(iv) $(m_i,B,d_k^{‴})$ is an $OA(\lambda s_1^3,3,s_1{s}_2\lambda ,3)$, $1\le i\le q_1$.

Proposition 3. 

When $D_1=(M,B)$ is an $OA(\lambda s_1^3,s_1^{q_1}{s}_2,2)$, $s_2=s_1^2$, $\lambda \ge 2$, $DCD(\lambda s_1^3,s_1^{q_1}{s}_2,p)$ exists, if and only if $D_1$ can be divided into λ$(s_1\times 1)\mathit{-ROC}{(s_1^3,s_1^{q_1}{s}_2,2)}^{\prime }s$, and M is a $CROA(\lambda s_1^3,s_1^{q_1},2)$.

Inspired by Proposition 3, we give Algorithm 4 to construct a DCD, which can be used as an initial design for Algorithm 5.

Algorithm 4 Construction of $DCD(\lambda s_1^3,s_1^{q_1}{s}_2,p)’s$

Step 1.  Given $F_i=(M_i,B_i)$ is an $(s_1\times 1)\mathit{-ROC}(s_1^3,s_1^{q_1}{s}_2,2)$, where $M_i$ is an $CROA(s_1^3,s_1^{q_1},2)$ and $B_i$ is an $OA(s_1^3,s_2,1)$, $1\le i\le \lambda$, then let $D_1={(F_1^T,F_2^T,\dots ,F_{\lambda }^T)}^T$. Note that all the $F_i$’s can be either the same or different. Step 2.  Let $e_k$ be a random permutation of ${\left(0,1,\dots ,\lambda -1\right)}^T$, $f_k$ be a random permutation of $({(f_{k1}\otimes {\mathbf{1}}_{s_1^2})}^T,{(f_{k2}\otimes {\mathbf{1}}_{s_1^2})}^T$, $\dots ,{(f_{k\lambda }\otimes {\mathbf{1}}_{s_1^2})}^T{)}^T$, where $f_{ki}$ is a random arrangement of $(0,1,\dots ,s_1-1),i=1,2,\dots ,\lambda .$ Step 3.  Let $d_k=\left(e_k\otimes {\mathbf{1}}_{s_1^3}\right)s_1^3+f_k{s}_1^2+{\left(h_{k1}^T,h_{k2}^T,\dots ,h_{k\lambda s_1}^T\right)}^T,1\le k\le p$. Let $D_2=(d_1,d_2,\dots ,$$d_p)$. Here $h_{ki}$ is a random arrangement of ${\left(h_{ki1}^T,h_{ki2}^T,\dots ,h_{ki{s}_1}^T\right)}^T$ and $h_{kij}$ is a random arrangement of ${\left((j-1)s_1,\dots ,j{s}_1-1\right)}^T$, $1\le k\le p$, $1\le i\le \lambda s_1$, $1\le j\le s_1$. Step 4.  Obtain $D^0=(D_1,D_2)$.

Theorem 5. 

The design $D^0=(D_1,D_2)$ obtained by Algorithm 4 is a $DCD(\lambda s_1^3,s_1^{q_1}{s}_2,p)$, where $D_1$ is an asymmetric $OA(\lambda s_1^3,s_1^{q_1}{s}_2,2)$, $D_2$ is an $LHD(\lambda s_1^3,p)$.

From Proposition 3, it is clear that Theorem 5 is true. According to Corollary 1 of [21], we have the following Corollary 3.

Corollary 3. 

For the design $D^0=(D_1,D_2)$ in Theorem 5, we have $q_1\le s_1$.

The proof of Theorem 4 is straightforward. In Algorithm 4, Step 1 is devoted to creating the $D_1$ satisfying the requirements in Proposition 3. The above steps produce $\lambda !·{(s_1!)}^{\lambda }·\lambda s_1{(s_1!)}^{s_1}$ different quantitative columns. In other words, Algorithm 4 provides the DCDs with many quantitative factors. The following example gives an illustration of Algorithm 4.

Example 4. 

Consider a (2×1)-resolvable orthogonal array $F_i=(M_i,B_i)$ is an $OA(8,2^{2}4,2)$, where $s_1=2,s_2=s_1^2=4$, $1\le i\le \lambda$. Let $\lambda =2,p=3$, $F_1={\left[\begin{array}{cccccccc}0& 1& 0& 1& 0& 1& 0& 1\\ 0& 1& 1& 0& 0& 1& 1& 0\\ 0& 2& 1& 3& 2& 0& 3& 1\end{array}\right]}^T$, $F_2={\left[\begin{array}{cccccccc}0& 1& 0& 1& 0& 1& 0& 1\\ 0& 1& 1& 0& 0& 1& 1& 0\\ 1& 3& 0& 2& 3& 1& 2& 0\end{array}\right]}^T$. According to Step 1, we obtain $D_1$. In Step 2 and Step 3, let $e_1=e_2=e_3={(0,1)}^T$, $f_1={(0,1)}^T,f_2={(1,0)}^T,f_3={(0,1)}^T$, $h_{1i}=h_{2i}={(0,1,2,3)}^T,h_{3i}={(1,0,3,2)}^T$, for $i=1,2\dots ,4$. Then we can obtain the corresponding $d_1,d_2,d_3$ in $D_2$. According to Step 4, a $DCD(16,2^{2}4,3)$$D^0$ is constructed as in

Table 7. $DCD(16,2^{2}4,3)$$D^0=(D_1,D_2)$ in Example 4, where $\lambda =2$.

	$D_{11}^T$	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
$D_1^T$		0	1	1	0	0	1	1	0	0	1	1	0	0	1	1	0
	$D_{12}^T$	0	2	1	3	2	0	3	1	1	3	0	2	3	1	2	0
		0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
$D_2^T$		4	5	6	7	0	1	2	3	12	13	14	15	8	9	10	11
		1	0	3	2	5	4	7	6	9	8	10	11	13	12	15	14

. From Figure 4, we can verify that the rows in $D_2$ corresponding to each level combination of any one or two factors in $D_1$ form an LHD. Obviously, this satisfies the definition of DCD.

Next, we propose another algorithm to construct $BGDCD(u\lambda s_1^3,s_1^{q_1·c_1}{s}_2^{c_2},pf)$ based on the DCD $D^0=(D_1,D_2)$ constructed by Algorithm 4. Similar to Corollary 2, mixed-level difference schemes are used in Algorithm 5.

Algorithm 5 Construction of BGDCDs based on $(s_1\times 1)$-ROAs

Step 1.  Obtain a $DCD(\lambda s_1^3,s_1^{q_1}{s}_2,p$) $D^0=(D_1,D_2)$ from Algorithm 4, where $D_1=(D_{11},D_{12})$ is an $OA(\lambda s_1^3,s_1^{q_1}{s}_2,2)$, $D_2$ is an $LHD(\lambda s_1^3,p)$, $s_2$ = $s_1^2$, $D_{11}$ is an $OA(\lambda s_1^3,s_1^{q_1},2)$, and $D_{12}$ is an $OA(\lambda s_1^3,s_2,1)$. Step 2.  Obtain two designs $A=(D\left(1\right)\oplus D_{11},D\left(2\right)\oplus D_{12})$ over $GF\left(s_1\right)$ and $GF\left(s_2\right)$, and $L=C\otimes D_2+\lambda s_1^{3}H\otimes {\mathbf{1}}_{\lambda s_1^3}$. Step 3.  Obtain a design $D=(A,L)$.

Theorem 6. 

The design $D=(A,L)$ obtained by Algorithm 5 is a $BGDCD(u\lambda s_1^3,s_1^{q_1·c_1}{s}_2^{c_2},pf)$, where A is an asymmetric $OA(u\lambda s_1^3,s_1^{q_1·c_1}{s}_2^{c_2},2)$, L is an $LHD(u\lambda s_1^3,pf)$.

The result of Theorem 6 just follows from the proofs of Propositions 1 and 2.

Remark 2. 

In Algorithm 5, since $D^0=(D_1,D_2)$ is a $DCD(\lambda s_1^3,s_1^{q_1}{s}_2,p)$, $(D_{11},D_2)$ is also a $DCD(\lambda s_1^3,s_1^{q_1},p)$. Therefore $(A_1,L)=\left((D\left(1\right)\oplus D_{11}),L\right)$ is a $WGDCD(u\lambda s_1^3,s_1^{q_1·c_1},pf)$, i.e., subgroup $A_{1j}$ of $A_1$ satisfies $(A_{1j},L)$ is a $DCD(u\lambda s_1^3,s_1^{q_1},pf)$, for $1\le j\le c_1$.

Example 5. 

Consider a design $BGDCD(64,2^{2·4}{4}^2,4)$, using the initial DCD $D^0$ as in Table 7, and delete the first column in $D_2$ (for saving space). By using the following difference schemes $D\left(1\right)$ and $D\left(2\right)$, $4\times 2$ matrix C and any $LHD(4,4)$ H, we can obtain A and L in Algorithm 5, respectively. The resulting design $D=(A,L)$ is listed in

Table 8. $BGDCD(64,2^{2·4}{4}^2,4)$$D=(A,L)$ in Example 5.

Run	A										$\mathit{L}$				Run	A										$\mathit{L}$
	$A_1$								$A_2$							$A_1$								$A_2$
1	0	0	0	0	0	0	0	0	0	0	4	33	52	17	33	0	0	0	0	1	1	1	1	0	2	52	49	36	1
2	1	1	1	1	1	1	1	1	2	2	5	32	53	16	34	1	1	1	1	0	0	0	0	2	0	53	48	37	0
3	0	1	0	1	0	1	0	1	1	1	6	35	54	19	35	0	1	0	1	1	0	1	0	1	3	54	51	38	3
4	1	0	1	0	1	0	1	0	3	3	7	34	55	18	36	1	0	1	0	0	1	0	1	3	1	55	50	39	2
5	0	0	0	0	0	0	0	0	2	2	0	37	48	21	37	0	0	0	0	1	1	1	1	2	0	48	53	32	5
6	1	1	1	1	1	1	1	1	0	0	1	36	49	20	38	1	1	1	1	0	0	0	0	0	2	49	52	33	4
7	0	1	0	1	0	1	0	1	3	3	2	39	50	23	39	0	1	0	1	1	0	1	0	3	1	50	55	34	7
8	1	0	1	0	1	0	1	0	1	1	3	38	51	22	40	1	0	1	0	0	1	0	1	1	3	51	54	35	6
9	0	0	0	0	0	0	0	0	0	0	12	41	60	25	41	0	0	0	0	1	1	1	1	0	2	60	57	44	9
10	1	1	1	1	1	1	1	1	2	2	13	40	61	24	42	1	1	1	1	0	0	0	0	2	0	61	56	45	8
11	0	1	0	1	0	1	0	1	1	1	14	42	62	26	43	0	1	0	1	1	0	1	0	1	3	62	58	46	10
12	1	0	1	0	1	0	1	0	3	3	15	43	63	27	44	1	0	1	0	0	1	0	1	3	1	63	59	47	11
13	0	0	0	0	0	0	0	0	2	2	8	45	56	29	45	0	0	0	0	1	1	1	1	2	0	56	61	40	13
14	1	1	1	1	1	1	1	1	0	0	9	44	57	28	46	1	1	1	1	0	0	0	0	0	2	57	60	41	12
15	0	1	0	1	0	1	0	1	3	3	10	47	58	31	47	0	1	0	1	1	0	1	0	3	1	58	63	42	15
16	1	0	1	0	1	0	1	0	1	1	11	46	59	30	48	1	0	1	0	0	1	0	1	1	3	59	62	43	14
17	0	0	1	1	0	0	1	1	0	1	20	17	20	33	49	0	0	1	1	1	1	0	0	0	3	36	1	4	49
18	1	1	0	0	1	1	0	0	2	3	21	16	21	32	50	1	1	0	0	0	0	1	1	2	1	37	0	5	48
19	0	1	1	0	0	1	1	0	1	2	22	19	22	35	51	0	1	1	0	1	0	0	1	1	0	38	3	6	51
20	1	0	0	1	1	0	0	1	3	0	23	18	23	34	52	1	0	0	1	0	1	1	0	3	2	39	2	7	50
21	0	0	1	1	0	0	1	1	2	3	16	21	16	37	53	0	0	1	1	1	1	0	0	2	1	32	5	0	53
22	1	1	0	0	1	1	0	0	0	1	17	20	17	36	54	1	1	0	0	0	0	1	1	0	3	33	4	1	52
23	0	1	1	0	0	1	1	0	3	0	18	23	18	39	55	0	1	1	0	1	0	0	1	3	2	34	7	2	55
24	1	0	0	1	1	0	0	1	1	2	19	22	19	38	56	1	0	0	1	0	1	1	0	1	0	35	6	3	54
25	0	0	1	1	0	0	1	1	0	1	28	25	28	41	57	0	0	1	1	1	1	0	0	0	3	44	9	12	57
26	1	1	0	0	1	1	0	0	2	3	29	24	29	40	58	1	1	0	0	0	0	1	1	2	1	45	8	13	56
27	0	1	1	0	0	1	1	0	1	2	30	26	30	42	59	0	1	1	0	1	0	0	1	1	0	46	10	14	58
28	1	0	0	1	1	0	0	1	3	0	31	27	31	43	60	1	0	0	1	0	1	1	0	3	2	47	11	15	59
29	0	0	1	1	0	0	1	1	2	3	24	29	24	45	61	0	0	1	1	1	1	0	0	2	1	40	13	8	61
30	1	1	0	0	1	1	0	0	0	1	25	28	25	44	62	1	1	0	0	0	0	1	1	0	3	41	12	9	60
31	0	1	1	0	0	1	1	0	3	0	26	31	26	47	63	0	1	1	0	1	0	0	1	3	2	42	15	10	63
32	1	0	0	1	1	0	0	1	1	2	27	30	27	46	64	1	0	0	1	0	1	1	0	1	0	43	14	11	62

. As we can verify that $(A_1,L)$ is a $WGDCD(64,2^{2·4},4)$, where $A_1$ is an $OA(64,2^8,2)$.

$D\left(1\right)=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 1& 0& 1\\ 0& 0& 1& 1\\ 0& 1& 1& 0\end{array}\right]$, $D\left(2\right)=\left[\begin{array}{cc}0& 0\\ 0& 1\\ 0& 2\\ 0& 3\end{array}\right]$, $C=\left[\begin{array}{cc}1& 1\\ 1& 1\\ 1& 1\\ 1& 1\end{array}\right]$ and $H=\left[\begin{array}{cccc}0& 2& 3& 1\\ 1& 1& 1& 2\\ 3& 3& 2& 0\\ 2& 0& 0& 3\end{array}\right]$.

4. Comparison

In this section, the GDCDs presented in this paper, including WGDCDs and BGDCDs, are compared with the existing DCDs and MCDs in terms of qualitative factors.

Table 9. Comparisons between DCDs $(D_1,D_2)$ in [21] and WGDCDs $(A,L)$.

s	Run	${\mathit{D}}_1$	A
2	16	$OA(16,2^2,2)$ *	$OA(16,2^4,2)$
	32	$OA(32,2^2,2)$ *	$OA(32,2^8,2)$
3	54	$OA(54,3^3,2)$ *	$OA(54,3^9,2)$
	162	$OA(162,3^3,2)$ *	$OA(162,3^{15},2)$
4	128	$OA(128,4^4,2)$ *	$OA(128,4^{16},2)$
	512	$OA(512,4^4,2)$ *	$OA(512,4^{24},2)$
5	250	$OA(250,5^5,2)$ *	$OA(250,5^{25},2)$
	1250	$OA(1250,5^5,2)$ *	$OA(1250,5^{35},2)$
6	216	$OA(216,6^2,2)$ *	$OA(216,6^4,2)$
	1296	$OA(1296,6^2,2)$ *	$OA(1296,6^{16},2)$

1 $D_1$: The subdesigns for qualitative factors in DCDs generated from [21]. Symbol * represents that the number of columns in $D_1$ reaches the upper bound. 2 A: The subdesigns for qualitative factors in WGDCDs constructed by Algorithm 1.

compares the qualitative factor designs of the WGDCDs constructed by Algorithm 1 with that of the DCDs constructed by [21]. As we can see in Table 9, as the run size of the design increases, the number of qualitative factors in WGDCDs also increases, but the number of qualitative factors in DCDs remains constant. Therefore, the WGDCDs constructed by Algorithm 1 can accommodate more equal-level qualitative factors than DCDs constructed by [21] under the same runs; details see Table 9. Moreover, the subdesigns for qualitative factors in the DCDs from [21] are all symmetric OAs. To this end, from Algorithm 4 we construct the DCDs with the qualitative factor subdesigns being asymmetric OAs.

On the other hand, we compare GDCDs, including WGDCDs and BGDCDs, with MCDs. Firstly, because the GDCDs have similar space-filling properties to DCDs, the GDCDs have better stratification properties between two qualitative factors and quantitative factors than the MCDs in [16,17,18,19,20]. Secondly, when the subdesigns for qualitative factors are mixed-level,

Table 10. Comparisons of GDCDs (WGDCDs and BGDCDs) with the existing MCDs.

$s_1$	$s_2$	${\mathit{A}}^{\left(1\right)}$	${\mathit{A}}^{\left(2\right)}$	${\mathit{A}}^{\left(3\right)}$	${\mathit{A}}^{\left(4\right)}$	${\mathit{A}}^{\left(5\right)}$
2	3	-	-	$OA(72,2^2{3}^3,2)$	-	-
2	4	$OA(32,2^4{4}^4,2)$	$OA(32,2^2{4}^3,2)$	$OA(64,2^4{4}^4,2)$	$OA(16,2^{2}4,2)$	$OA(64,2^8{4}^2,2)$
2	5	-	-	$OA(200,2^2{5}^2,2)$	-	-
2	6	$OA(72,2^2{6}^2,2)$	-	$OA(144,2^2{6}^2,2)$	-	-
2	7	-	-	$OA(392,2^2{7}^2,2)$	-	-
2	8	$OA(128,2^8{8}^2,2)$	-	$OA(256,2^8{8}^2,2)$	-	-
3	4	-	-	$OA(288,3^2{4}^2,2)$	-	-
3	5	-	-	$OA(450,3^2{5}^5,2)$	-	-
3	6	$OA(108,3^3{6}^2,2)$	-	$OA(216,3^3{6}^2,2)$	-	-
3	7	-	-	$OA(882,3^2{7}^2,2)$	-	-
3	8	-	-	$OA(1152,3^2{8}^2,2)$	-	-
3	9	$OA(243,3^5{9}^2,2)$	$OA(162,3^3{9}^8,2)$	$OA(486,3^5{9}^2,2)$	$OA(81,3^{3}9,2)$	$OA(729,3^{20}{9}^9,2)$
4	5	-	-	$OA(800,4^2{5}^2,2)$	-	-
4	6	-	-	$OA(576,4^2{6}^2,2)$	-	-
4	7	-	-	$OA(1568,4^2{7}^2,2)$	-	-
4	8	$OA(256,4^2{8}^2,2)$	-	$OA(512,4^2{8}^2,2)$	-	-

1 $A^{\left(1\right)}$: The subdesigns for qualitative factors in MCDs generated from Construction 3 of [16]. 2 $A^{\left(2\right)}$: The subdesigns for qualitative factors in WGDCDs constructed by Algorithm 2. 3 $A^{\left(3\right)}$: The subdesigns for qualitative factors in BGDCDs constructed by Algorithm 3. 4 $A^{\left(4\right)}$: The subdesigns for qualitative factors in DCDs constructed by Algorithm 4. 5 $A^{\left(5\right)}$: The subdesigns for qualitative factors in BGDCDs constructed by Algorithm 5.

compares the qualitative factor designs of the GDCDs (WGDCDs and BGDCDs) with that of the MCDs. In [16] there is a relation in an MCD, i.e., $s_2=\alpha s_1$, when the subdesign for qualitative factors is an $OA(n,s_1^{q_1}{s}_2^{q_2},2)$. When the constraint conditions $(s_2=s_1^2)$ are the same, the designs constructed by Algorithms 2, 4 and 5 have better space-filling properties between qualitative factors and quantitative factors than the MCDs in [16]. Especially, the relation, $s_2=\alpha s_1$, is not required in the BGDCDs constructed by Algorithm 3. Therefore, the level types of the BGDCDs constructed by Algorithm 3 are more flexible than those of MCDs in [16].

5. Conclusions and Future Research Directions

The existence of the interaction effects between any two qualitative factors and all quantitative factors in a computer experiment involving both qualitative and quantitative factors is very important for design selection. If no such effects exist, then an MCD is chosen; if the effects exist and the number of qualitative factors is no greater than the number of their levels, then a DCD is the best choice. When the number of qualitative factors exceeds the number of levels, neither an MCD nor a DCD can be used if some qualitative factors have the effects with quantitative factors and some do not. Inspired by this, we propose a new class of DCDs, namely GDCDs. A GDCD is an MCD, and the columns of some qualitative factors and all quantitative factors form a DCD. DCDs in [21] can only accommodate equal-level qualitative factors and the number of qualitative factors is also limited. Unlike DCDs, GDCDs can not only entertain more qualitative factors, but the qualitative factors can be either symmetric or asymmetric. In addition, GDCDs are equipped with better stratification properties between the qualitative and quantitative factors than the existing MCDs, whether the qualitative factors are symmetric or asymmetric.

In this paper, we propose two classes of GDCDs, namely WGDCDs and BGDCDs. While the algorithms for constructing WGDCDs and BGDCDs are similar and easy to implement, they differ in the initial DCDs used to construct the subdesign A. Four algorithms for constructing different GDCDs are provided. Algorithm 1 constructs WGDCDs based on the initial DCD $D^0=(D_1,D_2)$, where the design of the qualitative factors $D_1$ is an $OA(n,s^q,2)$. In contrast to DCDs in [21], whose number of qualitative factors is at most s, the WGDCDs obtained from Algorithm 1 can accommodate $cs$ qualitative factors. The WGDCDs obtained from Algorithm 2 can entertain two different levels of qualitative factors, and the number of qualitative factors in each group can almost reach its bound. Similar to Algorithm 1, the designs obtained from Algorithm 2 with more qualitative factors can be further extended by using the difference schemes with mixed levels. Algorithm 3 not only constructs BGDCDs with two different levels of qualitative factors, but also constructs DCDs with mixed-level qualitative factors, and there is no multiple relation between $s_1$ and $s_2$. Algorithm 4 provides the initial DCDs that contain more mixed-level qualitative factors and are needed in Algorithm 5. The BGDCDs obtained by Algorithm 5 can also be realized as WGDCDs in the groups extended by $D_{1i}$ being an OA. Moreover, according to the comparisons in Section 4, the GDCDs in this paper, including WGDCDs and BGDCDs, outperform both MCDs in [16,17,18,19,20] and DCDs in [21].

An interesting but challenging direction for future research is to construct initial DCDs with more qualitative factors possessing different levels. Another possible direction is to construct GDCDs with L having high-dimensional space-filling properties, such as 2 to 3 dimensions, or to consider adding column-orthogonality within or between groups. The construction of such GDCDs is not trivial and cannot be easily extended. We hope to investigate this and report the results in the near future.