## Present Research

My research activities are focused on orthogonal arrays that could be useful for statistical designs of experiments . Here is an example of such an array.

0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 |

0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |

0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |

0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 |

The rows correspond with experimental factors, which can be varied at will in experimentation. Examples are temperature of a liquid, coarseness of material, and variety of wheat. Each row-entry is a factor-setting. So the first factor has four settings and the remaining three factors have two settings each. For example, one may wish to include four formulations of detergents and two washing temperatures in an experiment on cleaning towels.

The 16 columns of the orthogonal array shown here correspond with different experiments, or runs. When carried out in practice, each experiment will result in the measurement of one or more output variables that characterize the product or process one is interested in. The purpose of the series of experiments is to establish a relation between the outputs and the factor settings.

The array in the example has strength *t*=3, because each triple of rows is a full factorial design in three factors (repeated if the triple consists of two-level factors only). This property is convenient when you look at observations averaged over the runs with a common level of one of the factors. For example, there are four such averages for the four-level factor. We can study the main effect of this factor by comparing the four averages. Because of the strength-3 property, the difference in averages of a factor is not affected by the joint effect of any pair of other factors.

Strength, run size, and level numbers of the factors are collectively called the *parameters* of an orthogonal array. With the help of prof. Andries Brouwer (Eindhoven University of Technology, Netherlands), and Man Nguyen (University of Technology, Ho Chi Minh City, Vietnam) I formulated a general algorithm to enumerate all non-isomorphic orthogonal arrays of strenth 3 and given parameters (2005). Under my supervision, Ruben Snepvangers wrote a C program for parallel computing of the arrays of any strength given the parameters (2006). The program was good enough for small cases, but got stuck for cases with amny factors or experiments. In 2008, I had the good luck to meet Pieter Eendebak, who is a TNO colleague. We started a collaboration which resulted in a tremendous increase in efficiency. We collected an extensive collection of arrays; please refer to the ‘Series of Arrays’ page of this site. Since then, my research focuses on exploring the many different series of orthogonal arrays and generation of attractive subclasses. Hot issues at this moment are

- Are there efficient ways to employ the basic algorithm for generating attractive subclasses of arrays? (Joint research with Pieter Eendebak and Nha Vo-Thanh.)
- If we have a complete series of arrays, how do we select an array for practical use? For arrays with the same factors but different run sizes, do the extra runs pay off? (Joint research with Robert W. Mee and David Edwards.)
- How can we block the arrays with two crossed blocking factors (for example, six week days and four machines) in groups orthogonal to the main effects? (Joint research with Nha Vo-Thanh and Peter Goos.)
- How can we best join two orthogonal arrays such that an array with a larger run size and the same number of factors results that is better than the individual arrays? (joint research with Alan Vazquez and Peter Goos.)