Trend-robust and minimally aliased response surface designs by means of integer linear programming

Gaining competitive advantage requires today’s businesses to innovate at an increasing speed. New product development is at the heart of innovation and requires substantial experimentation. The field of “design of experiments” delivers systematic plans to determine the settings that lead to optimal products. In this Ph.D. thesis, we have focused on Response Surface Designs (RSDs), a core component of the Response Surface Methodology, which is widely used in industry. Our work addresses two gaps in the existing literature on RSDs.

The first RSD problem we tackled occurs when a multi-factor experiment is carried out over a period of time and the responses depend on a time trend. Unless the tests of the experiment are conducted in a proper order, the time trend has a negative impact on the precision of the estimates of the main effects, the interaction effects and the quadratic effects. Such a proper order of the tests is called a trend-robust or trend-resistant run order. In the literature, most of the methods used to construct such run orders are tailored to specific designs, and are not applicable to an arbitrary design. In contrast, we developed a sequential approach based on integer programming to find a trend-robust run order for any given design. Using that approach, we succeeded  in identifying trend-robust run orders for standard RSDs with two up to six factors.

The second research problem we addressed is concerned with the large gap between the numbers of runs of standard RSDs and definitive screening designs. Definitive screening designs tend to be too small and exhibit a shortage of information in the event many factors matter, while standard RSDs yield much information, but at a large experimental cost. For this reason, we define a new family of RSDs, for which there is no aliasing between the main effects and the second-order effects (two-factor interactions and quadratic effects). We name our designs MARS designs, where MARS stands for minimally aliased response surface designs. By means of a tailored integer programming framework with advanced symmetry reduction techniques, we implemented an efficient enumeration algorithm. By running the algorithm on the Open Science Grid, a vast computing grid infrastructure in USA, we were able to build an extensive database of novel cost-efficient MARS designs with attractive statistical properties. To explore these designs in a user-friendly fashion, we also created a user-friendly software prototype that guides a practitioner toward the most suitable design for a broad class of applications.