Component orthogonal arrays (COAs), as subsets of all possible permutations on experimental factors, are suitable for designing order-of-addition experiments due to their pairwise balance property between any two positions of the orders. When the levels of components can be changed, the standard statistical design method has been to use the Cartesian product design, which often results in a large number of experimental runs. In this article, we consider combining COAs and 2(m-p) factional factorial designs by using a Subcartesian product. A systematic method is given for design construction. In the construction method, both the COA and 2(m-p) design are sliced properly, and then the two designs are combined by the Subcartesian product. We show that the number of runs obtained by the Subcartesian product method is usually much smaller than that obtained by the Cartesian product method. In addition, the constructed design satisfies the pairwise balance property and has the same D-efficiency as the full m! x 2(m) design under the main-effect model. Examples are given to illustrate the proposed method.

Sequential Latin hypercube designs (SLHDs) have recently received great attention for computer experiments, with much of the research restricted to invariant spaces. The related systematic construction methods are inflexible, and algorithmic methods are ineffective for large designs. For designs in contracting spaces, systematic construction methods have not been investigated yet. This paper proposes a new method for constructing SLHDs via good lattice point sets in various experimental spaces. These designs are called sequential good lattice point (SGLP) sets. Moreover, we provide efficient approaches for identifying the (nearly) optimal SGLP sets under a given criterion. Combining the linear level permutation technique, we obtain a class of asymptotically optimal SLHDs in invariant spaces, where the L1-distance in each stage is either optimal or asymptotically optimal. Numerical results demonstrate that the SGLP set has a better space-filling property than the existing SLHDs in invariant spaces. It is also shown that SGLP sets have less computational complexity and more adaptability.

A digital twin is a simulator of a physical system, which is built upon a series of models and computer programs with real-time data (from sensors or devices). Digital twins are used in various industries, such as manufacturing, healthcare, and transportation, to understand complex physical systems and make informed decisions. However, predictions and optimizations with digital twins can be time-consuming due to the high computational requirements and complexity of the underlying computer programs. This poses significant challenges in making well-informed and timely decisions using digital twins. This paper proposes a novel methodology, called the “digital triplet”, to facilitate real-time prediction and decision-making. A digital triplet is an efficient representation of a digital twin, constructed using statistical models and effective experimental designs. It offers two noteworthy advantages. Firstly, by leveraging modern statistical models, a digital triplet can effectively capture and represent the complexities of a digital twin, resulting in accurate predictions and reliable decision-making. Secondly, a digital triplet adopts a sequential design and modeling approach, allowing real-time updates in conjunction with its corresponding digital twin. We conduct comprehensive simulation studies to explore the application of various statistical models and designs in constructing a digital triplet. It is shown that Gaussian process regression coupled with sequential MaxPro designs exhibits superior performance compared to other modeling and design techniques in accurately constructing the digital triplet.

This paper studies circular designs for interference models, where a treatment assigned to a plot also affects its neighboring plots within a block. For the purpose of estimating total effects, the circular neighbor balanced design was shown to be universally optimal among designs which do not allow treatments to be neighbors of themselves. Our study shows that self-neighboring block sequences are actually the main ingredient for an optimal design. Here, we adopt the approximate design framework and study optimal designs in the whole design space. Our approach is flexible enough to accommodate all possible design parameters, that is the block size and the number of blocks and treatments. This approach can be broken down into two main steps: the identification of the minimal supporting set of block sequences and the optimality condition built on it. The former is critical for reducing the computational time from almost infinity to seconds. Meanwhile, the task of finding the minimal set is normally achieved through numerical methods, which can only handle small block sizes. Our approach is of a hybrid nature in order to deal with all design sizes. When block size is not large, we provide explicit expressions of the minimal set instead of relying on numerical methods. For larger block sizes when a typical numerical method would fail, we theoretically derived a reasonable size intermediate set of sequences, from which the minimal set can be quickly derived through a customized algorithm. Taking it further, the optimality conditions allow us to obtain both symmetric and asymmetric designs. Lastly, we also investigate the trade-off issue between circular and noncircular designs, and provide guidelines on the choices.

The objective of the order-of-addition (OofA) problem is to find the optimal (addition) order. Existing literature concentrated on the responses of different orders with homoscedasticity. Study was made here for the cases of heteroscedasticity, where the dispersion effects for replicated OofA experiments should be considered. This paper proposes some approaches to speculate optimal orders for the replicated OofA experiment. Based on the pair-wise-order (PWO) model, the obtained orders from the proposed methodologies not only achieve the goal of OofA experiment, but also minimize the standard deviation within the OofA framework. Theoretical support is given under the specific setups. Simulation studies are used to illustrate these methodologies. It is shown that the proposed methods perform well for replicated OofA experiments.

Composite designs are frequently utilized for fitting response surfaces in practice. This paper proposes a new type of composite designs, orthogonal uniform composite designs (OUCDs), which combine orthogonal arrays and uniform designs. Such designs not only inherit the advantages of orthogonal-array composite designs such as high estimation efficiencies and ability for multiple analysis for cross validation, but also have more flexible run sizes than central composite designs and orthogonal-array composite designs. Moreover, OUCDs are more robust than other types of composite designs under certain conditions. Some construction methods for OUCDs under the maximin distance criterion are provided and their properties are also studied. It is shown that many constructed OUCDs are maximin distance designs.

Good lattice point (GLP) sets are frequently used in quasi-Monte Carlo method and computer experiments. However, the space-filling property of GLP sets needs to be improved especially when the number of factors is large. This paper shows that the generalized GLP (GGLP) sets, constructed simply and fast by the linear level permutation of the GLP sets, have better space-filling property than the GLP sets and the orthogonal Latin hypercube designs (OLHD) in the sense of maximin distance criterion and uniformity criterion, especially for high dimensional cases. Unlike the OLHD, the number of runs of the GGLP sets can be chosen as any integer. It is also shown that the GLP sets are better than Latin hypercube designs as the starting design for linear and nonlinear level permutation. The GGLP sets are recommended for the designs with large number of factors and/or large number of runs.

Foldover and semifoldover techniques are two important ways to augment the number of runs of a design. However, these methods constrain the number of augmented runs to be equal to or half of that of the initial design. Such constraint may be relaxed in practical application. Moreover, some extra factors may be added in the follow-up designs, and one may augment the number of factors. In this paper, the wrap-around L2-discrepancy criterion is considered to augment the number of runs and the number of factors for mixed two- and three-level designs. The corresponding designs are called as the row augmented uniform designs and the column augmented designs, respectively. The lower bounds of the augmented designs are obtained, and a construction algorithm is also given. Some examples show that many augmented designs can reach the corresponding lower bounds. Compared with the foldover designs and semifoldover designs, the augmented uniform designs are more uniform and flexible.

In many industrial trials, the second-order models may not be enough to fit the non linearity of the underlying model, and the third-order models may be considered. In this article, the orthogonal-array composite design (OACD), combined with two-level OA and four-level OA and denoted by OACD(4), is proposed to estimate the second-order and third-order models. It is shown that OACD(4) has good properties and has higher efficiency than other types of designs for the third-order models, and OACD(4) can perform multiple analysis for cross-validation. The usefulness of OACD(4) is also shown by a case study for polymer synthesis experiment.