Design of Experiments: Grinding Process Example (Part 3)

This posting continues the grinding process case study (Gigo, 2008) that illustrates the use of design and analysis of experiments to reduce common-cause variation.  We examine the properties of the experimental design reported by Gijo.   The examination illustrates the potential for aliasing in an experimental design and shows how it can bias the results.   The experimental design described by Gijo uses an orthogonal array which Taguchi recommended.   We contrast the properties of that design with a standard fractional factorial.

The 9/15/2008 posting initiated the design of experiments portion of the case study.  The primary purpose of the experimental design was to reduce the variation in the outer diameter produced by a grinding operation.   That posting reports that the team was primarily interested in estimating the following effects:

A – Feed Rate
B – Wheel Speed
C – Work Speed
D – Wheel Grade
AB – Interaction between A and B
AC - Interaction between A and C

Gijo states that the experimental design was developed using an L₈ orthogonal array.  He references Phadke (1989) for use of orthogonal arrays to construct designs.   Taguchi made extensive use of orthogonal arrays in constructing robust designs.  Hicks and Turner (1999, p381) give a table for using an L₈ orthogonal array to construct a design with the desired properties.   That is, we do not want the A, B, C, D, AB, and AC effects aliased with each other.   Two effects that have the same estimator are aliased.  The previous posting on September 15 gives the design and estimates of the factor effects.   Clearly the design meets the desired criterion since the factor effect estimates are all different.

However, consider the estimates of the of the BC, BD and CD interaction effects shown in the following table.

Note that the BC interaction effect is exactly equal to the negative of the D effect, the BD interaction effect is equal to the negative of the C effect and the CD interaction effect equals the negative of the B effect.  That is true because the sequences of +1 and -1s in the BC, BD and CD columns are precisely the negatives of those in the D, C and B columns.    With this design, the BC and D effects are aliased.   That is, if the BC effect is not zero, then our estimate of the D effect is affected by the BC effect.   Similarly, the BD effect estimate is aliased with the C effect, and the CD effect is aliased with the B effect.  Then this design provides no information on whether the BC, BD and CD interaction effects are negligible.   Also, this design can give a biased estimate of the D effect if the BC interaction defect is significant.

Montgomery (2005, p. 288) gives a standard one-half fraction of the 2⁴ factorial design.   Call it the 2^4-1 design.  This design uses 8 experiments and has four factors.   The properties of this design are:
·        Estimates of the main effects are not aliased with any two-factor interactions.
·        Estimates of the main effects are aliased with three factor interactions.
·        Every two factor interaction is aliased with another two factor interaction.   That is AB=CD, AC=BD and BC=AD.

The 2^4-1 design might be superior to the one described by Gijo.   Estimates of the A, B, C and D effects are not aliased with any two factor interaction.  Also, estimates of the AB and AC effects are not aliased with a main effect.

The next posting will present results from the experimental design.

References

1. Gijo, E. V. (2005). "Improving Process Capability of Manufacturing Process by Application of Statistical Techniques." Quality Engineering 17(2): 309-315.
2. Hicks, Charles R. and Kenneth V. Turner Jr. (1999).  Fundamental Concepts in the Design of Experiments, Oxford University Press.
3. Montgomery, Douglas C. (2005). Design and Analysis of Experiments, 6^th Edition, John Wiley & Sons, Inc.
4. Phadke, Madhav S. (1989).  Quality Engineering Using Robust Design, Prentice Hall.