Comparing methods for design follow‐up: revisiting a metal‐cutting case study

Abstract

aAdding another fraction to an initial fractional factorial design is often required to resolve ambiguities with respect to aliasing of factorial effects from the initial experiment and/or to improve estimation precision. Multiple techniques for design follow-up exist; the choice of which is often made on the basis of the initial design and its analysis, resources available, experimental objectives, and so on. In this paper, we compare four design follow-up strategies: foldover, semifoldover, D-optimal, and Bayesian (MD-optimal) in the context of a metal-cutting case study previously utilized to compare fractional factorials of different run sizes. Follow-up designs are compared for each of a 2 6 3 III , 2 6 2 IV , and Plackett-Burman initial experiments. Our empirical results suggest that a single followup strategy does not outperform all others in every situation. This case study serves to illustrate design augmentation possibilities for practitioners and provides some basis for the selection of a follow-up experiment.

R Package OBsMD for Follow-Up Designs in an Objective Bayesian Framework

“…MetalCutting[c(62,15,6,17,41,28,36,55), 1:7] R> y <- MetalCutting[c(62,15,6,17,41,28,36,55) R> y <-c(y, y_TOP_DES) R> es8.aug.OBsProb <-OBsProb(X = X, y = y, abeta = 1, bbeta = 1, blk = 1, + mFac = 6, mInt = 2, nTop = 64) R> print ( The update posterior probability of factors is displayed in Figure 5 on the right side. The result is comparable with that reported in Edwards et al (2014).…”

“…The same whole procedure replicated for mInt = 3 (OBsProb() plus OMD() with the augmentation of runs 4 11 52 59; see the code in the supplementary material) generates the posterior probability of factors shown in Figure 2 (on the right side). We can observe that the results do not change and they are consistent to those reported in Figure 5 of Edwards et al (2014) who adopted different augmentation approaches for the same dataset. In conclusion this package allows, within the objective Bayesian approach, to reach two different goals.…”

“…Consider for instance the dataset MetalCutting included in the OBsMD package. This dataset was used in Mønness, Linsley, and Garzon (2007) and re-examined in Edwards, Weese, and Palmer (2014) with the aim to compare methods to design follow-up experiments. It is introduced here just for illustrative purposes and it is analyzed in Section 5 with the aim to identify the oracle model and active factors.…”

“…Table 3: Runs and response variable Ytransformed corresponding to the 2 6−2 IV fractional factorial design with generators E = ABC and F = ABD. E = ABC and F = ABD, where only n = 16, over the 64 runs, are performed; see Edwards et al (2014).…”

“…It corresponds to a full factorial design 2 6 that a researcher rarely has at his/her disposal, due to the cost per run and time limit constraints. In the following we consider and analyze this dataset only with the aim to have a benchmark to compare the performance of our approach with that of the other methods to specify follow-up designs (see Edwards et al 2014).…”