Ordering factorial designs with good stratification properties

Ordering factorial designs (OFDs) play a crucial role when both the level combinations of factors and the addition orders of components affect the responses. All the existing construction methods of OFDs are proposed for optimal designs under some specific models. When the true model is unknown or complex, space-filling designs can be considered. However, the current space-filling criteria cannot be used for OFDs directly. In this article, we define the stratification property of an OFD to measure the low-dimensional space-filling property of a design. Two construction methods are also provided for OFDs with good stratification properties. Some of the resulting designs have fewer run sizes, more flexible numbers of components, or can accommodate more factors with the same run sizes than the existing OFDs. Moreover, simulations on a modified traveling salesman problem demonstrate the superiority of the constructed designs.