The challenge in mixed-variable experimental design lies in the presence of both discrete and continuous variables. Discrete variables have a finite number of possible values, and their distribution may be either uniform or non-uniform. Existing methods for discrete variable sampling typically involve selecting values in the continuous domain and rounding them to the nearest discrete points. This approach inevitably results in some values being excluded or duplicated, thereby compromising the uniformity of the samples. Therefore, a new experimental design method for discrete variables, referred to as Discrete Variable Sampling Design (DVD), needs to be developed based on the nature of discrete variables.

The implementation process of the DVD method is as follows:

Form a set K containing all combinations of discrete variables, with the total number of combinations denoted as m, and the desired number of design samples as N.

1) Calculate the quotient a by rounding down N/m, and determine the remainder b.

2) Duplicate the set K times to obtain the design sample set X₁.

3) Randomly select b points from the set K to form the design sample set X₂.

4) Combine X₁ and X₂ to create the complete design matrix.

To illustrate the DVD method more intuitively, consider a simple example with discrete variables x₁ and x₂, each having five and six discrete values, respectively. The goal is to generate 64 sample points in a two-dimensional space. Following the outlined steps, the total number of combinations m for x₁ and x₂ is 30, with set K comprising these 30 combinations. When N/m is rounded down, a is determined to be 2, and the remainder b is 4. Consequently, X₁ is created by duplicating K two times, and X₂ is formed by randomly selecting 4 points from K. Finally, X₁ and X₂ are combined to produce a design matrix containing 64 sample points.

Latin Hypercube Design (LHD) is a representative method in the field of “space- filling” experimental design, renowned for its excellent uniformity and low-di- mensional projection performance. It has become one of the most popular sampling methods in computer experiments [22] . In this paper, we employ this method to address continuous variables. For a trial design problem of size N × d, where N is the number of samples, and d is the dimensionality of the samples, the process of the LHD method is outlined as follows:

Divide each dimension into N subsets, each subset having a length of 1/N. Select one point from each subset, resulting in N points for each dimension within the design space. Randomly permute the N values for each dimension, yielding an initial design matrix denoted as A.

A = [ x 11 x 12 ⋯ x 1 d x 21 x 22 ⋯ x 2 d ⋮ ⋮ ⋱ ⋮ x N 1 x N 2 ⋯ x N d ]

A mixed-variable set comprises d continuous variables, x₁, x₂, ..., x_d, and e discrete variables, y₁, y₂, ..., y_e, with a total of N generated samples. The LHD experimental design method is employed to handle the continuous variable x₁, resulting in a set {x₁₁, x₂₁, ..., x_N1}, which is incorporated as the first column of matrix X. Similarly, other continuous variables are processed until x_d, yielding a set {x_1d, x_2d, ..., x_Nd}, which forms the dth column of X.

The DVD experimental design method is then applied to handle the discrete variable y₁, obtaining a set {y₁₁, y₂₁, ..., y_N1}, which is included as the (d + 1)th column of X. This process is repeated for other discrete variables until y_e, resulting in a set (y_1e, y_2e, ..., y_Ne), which is added as the (d + e)th column of X. At this point, the complete design matrix X is obtained.

X = { x 11 ⋯ x 1 d y 11 ⋯ y 1 e x 21 ⋯ x 2 d y 21 ⋯ y 2 e ⋮ ⋱ ⋮ ⋮ ⋱ ⋮ x N 1 ⋯ x N d y N 1 ⋯ y N e }

Li proposed the Columnwise-Pairwise Algorithms (CP) with the objective of maximizing entropy, aiming to optimize the design matrix. CP algorithms are effective in handling permutation optimization problems. The method is particularly easier to keep the structure properties of a design in relation to columns, such as the balance and orthogonality properties. This article will utilize the Columnwise-Pairwise Algorithms to optimize the design matrix X. In contrast to Li’s proposed Columnwise-Pairwise Algorithms, this paper adopts a different set of convergence criteria. Specifically, the criteria employed here include the maximum and minimum distance criterion, the φp criterion, and the minimum potential energy criterion.

The algorithm for searching optimal CP is summarized as follows:

1) Start with the X.

2) Each iteration has L steps. At the ith step, the best two simultaneous exchanges within column i are found. The design matrix is updated accordingly.

3) If the resulting design is better with respect to the criterion, repeat Step 2. Otherwise, it is considered to be an “optimal design”, and the search is terminated.

With a sample size of 20 and two dimensions, where in x₁ is a continuous variable and x₂ is a discrete variable encompassing 5 discrete values, both processed using the ODCD and MLHD methods. The resulting sample distributions are illustrated in Figure 1 and Figure 2. It is evident that the sample uniformity obtained by the ODCD method is superior to the MLHD method.

To quantitatively characterize the spatial uniformity of the samples, this study employed three criteria: the Maximum and minimum distance criterion (d_min),

φ_p criterion(φ_p), and minimum potential energy criterion (U). Larger values of d_min indicate stronger spatial uniformity, while smaller values of φ_p and U indicate greater spatial uniformity. Below is a brief description of the three criteria.

1) Maximum and minimum distance criterion [23] . The objective of the minimum distance criterion is to maximize the minimum distance between experimental sample points, satisfying the condition:

max { d min } = max { min 1 ≤ i , j ≤ m , i ≠ j d ( x i , x j ) } (1)

where m is the number of sample points, and d(x_i, x_j) represents the distance between any two sample points, x_i and x_j

d ( x i , x j ) = d i j = [ ∑ k = 1 n | x i k − x j k | t ] 1 t ,       t = 1     or     2 (2)

2) φ_p criterion [15] . Morris and Mitchell extended the application of the minimum distance criterion, introducing the φ_p criterion. If an experimental design is referred to as a φ_p-optimal design, it satisfies the following conditions:

min { ϕ p } = min { [ ∑ i = 1 s J i d i − p ] 1 p } (3)

For a given experimental design, calculate the distance d_ij between any two sample points (here, t = 1). Sort these distances to obtain a list of distance function values (d₁, d₂, ..., d_s) and their corresponding index list (J₁, J₂, ..., J_s). The distances d_i represent different distance values, and d 1 < d 2 < ⋯ < d s , where J_i is the number of pairs with a distance of d_i, and S is the number of distinct distance values. In this formula, p is a positive integer set to 50.

3) Minimum Potential Energy Criterion [20] . This criterion is inspired by physical principles: when the system’s potential energy reaches a minimum, it is in an equilibrium state. The expression for potential energy U is given by:

U = ∑ i = 1 m − 1 ∑ j = i + 1 m d i j − 2 (4)

Ensuring identical variables, generate experiment samples of the same size using the ODCD method and MLHD method. To reduce experimental errors, repeat the process 100 times. Evaluate the test samples using d_min, φ_p, and U criteria, recording the optimal, worst, and average values for d_min, φ_p, and U. Table 1 presents the results for four variables, including two continuous variables and two discrete variables.

From the table, it can be observed that the average values of the performance criteria φ_p and U for the experiment samples generated using the ODCD method are smaller than those using the MLHD method, while the average value for d_min is greater than that of the MLHD method. This indicates that the ODCD method yields experiment samples with superior performance compared to the MLHD method. The results for optimal and worst values align with the average values.

To further illustrate the sample performance of the ODCD method, generate experiment samples of different scales using ODCD and MLHD methods, particularly in scenarios with numerous design variables. Evaluate the samples using the three criteria, and average the results over 50 iterations. The outcomes are presented in Table 2, where N denotes the sample quantity, A represents the sample dimensionality, a denotes the dimensionality of continuous variables, and b represents the dimensionality of discrete variables. Similar to the low-dimensional cases, the ODCD method demonstrates superior sample performance compared

to the MLHD method.