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 present the results of the analysis of the experiments specified in the 9/18/2008 posting (Part 2).  

The following figures display graphically the relative significance of the six factors, i.e., A, B, C, D, AB and AC.   The figures show the average response at the factor low (-1) and high (+1) values.    Factors B and C are not nearly as significant as factors A and D since the average responses of B and C are nearly the same at their low and high values.  That is, a change in the factor levels for factors B and C has little effect on the response.  Also, the interaction factor AC is more significant than the interaction factor AB.  

We can test the significance of the factors using an Analysis of Variance (ANOVA).   Refer to Montgomery, Peck and Vining (2006).   Let SS_T be the total sum of squares.   That is:

where Y_i is the response on experiment i and ybar is the average response over the 8 experiments.   That is, SS_T is the sum of the 8 squared deviations between the experiment responses and the average response.   The value of ybar is 49.582, and the value of SS_T is 118.151.   Then we partition SS_T into a sum of squares due to the estimated effects (SS_R) and a sum of squared deviations from the estimated effects (SS_RES).  That is, SS_T = SS_R + SS_RES.  The value of SS_R is the same as a sum of squares due to an estimated regression function when we have a two-level experiment.   Consider the contribution of factor A to SS_R.    The posting on 9/18/2008 gives the estimated effect of factor A to be -6.067.  That is the difference between the average of the responses at the low values of factor A and the high values of factor A.    Thus the estimated average response at the high values of factor A is ybar - 6.067/2 = 46.5485.  Similarly, the estimated average response at the low values of factor A is  ybar + 6.067/2  = 52.6155.   The deviation between the mean response and the effect of A conditioned on whether A is high or low is 6.067/2.   Since we have 8 experiments, the contribution of factor A to SS_R is 8*(6.067/2)² = 73.60788.   For factor D and the interaction effect AC, the corresponding contributions to SS_R are 18.67308 and 11.38575.   Thus, SS_R is 103.6667.   The value of SS_RES is SS_T – SS_R = 14.48432.  We can test whether these three factors are statistically significant using the F statistic.    The F statistic assumes that the individual responses have a normal distribution.   The F statistic is:

where df_R = degrees of freedom for SS_R = 3 (the number of factors),
df_RES = degrees of freedom for SS_RES = 8-1-3 = 4 (we loose one degree of freedom due to estimating the mean and 3 due to estimating the 3 factor effects.
We can tell whether this value of F is statistically significant by calculating its PValue.    The PValue is the probability of obtaining this value of F, i.e., 9.543, or higher by chance when the factor effects have at true value of zero.   The PValue for this F is .027.    Usually, we regard a PValue as statistically significant when it is less than .05.   Thus the factors A, D and AC are statistically significant.   If we attempt to add a forth factor, i.e., AB,  the PValue becomes .0625; thus, we do not include AB. 

Higher values of the response S/N are desirable.   Thus, the low value of factor A (feed rate of .0008 mm/Revolution) and the low value of factor D (wheel grade of A54) are preferred.  Since the low value (-1) of the interaction effect AC is preferred, we select the high value of factor C which is a work speed of 360 RPM.   For the insignificant factor, the team chose its low value ( a wheel speed of 2200 RPM).

The posting on 2/28/2008 reports that the preferred factor levels specified above improved the process performance index (P_pk) from .49 to 1.25.   This is based on a sample of 40 parts.   The posting on 5/1/2008 defines the process capability index C_pk.   Process capability indices assume the process is stable.   When we have insufficient evidence the process is stable, we call the capability index a performance index and use the same equation.   

References

1. Montgomery, Douglas C., Elizabeth Peck, Geoffrey Vining (2006). Introduction to Linear Regression Analysis, John Wiley & Sons, p26.

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.

Experiment	Response (S/N)	Wheel Speed X Work Speed (BC)	Wheel Speed X Wheel Grade (BD)	Work Speed X Wheel Grade (CD)
1	53.4692	+1	+1	+1
2	50.9704	-1	-1	+1
3	49.0298	-1	+1	-1
4	56.991	+1	-1	-1
5	49.0298	+1	+1	+1
6	46.1079	-1	-1	+1
7	46.1079	-1	+1	-1
8	44.9483	+1	-1	-1
Effect		3.056	-0.345	0.625

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.

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.  The 9/15/2008 posting initiated the design of experiments portion of the case study.

The response variable was a measure of the variability of the outer diameter of the machined components.   One could use the estimated variance, i.e. s², for each set of experimental conditions.   That is, one would replicate the experiment for each set of experimental conditions and estimate s².  Gijo chose to use -10*ln(s²).   He lets the symbol S/N represent the -10*ln(s²).  Could S/N mean that the response is a Taguchi signal-to-noise ratio?   Montgomery (2005, p. 469) discourages the use of signal-to-noise ratios.   He states that a more effective approach is to model the mean and variance separately.   Hunter (1987) comes to the same conclusion.   Gijo does not justify the use of S/N other than a reference to the 3^rd edition of Montgomery’s book.

A response variable that has a constant variance over the set of experimental conditions facilitates regression analyses of the results.   Montgomery (2005, p. 83) recommends the use of the logarithmic transformation when the standard deviation of the response is proportional to its mean.   Let’s proceed by assuming the team used S/N since they wanted to estimate the contribution of the selected factors to the variance of the outer diameter and the standard deviation was roughly proportional to the mean.

The following table gives the experimental design and the observed response for each experiment.   The team replicated the experiment twice for each set of experimental conditions.   From the two observed outer diameters, they calculated a variance estimate, i.e., s², and from that computed the response value S/N.  The -1 and +1 symbols represent the lower and higher levels of the respective factors. 

Experiment	Feed Rate (A)	Wheel Speed (B)	Work Speed (C)	Wheel Grade (D)	Response (S/N)
1	-1	-1	-1	-1	53.4692
2	-1	-1	+1	+1	50.9704
3	-1	+1	-1	+1	49.0298
4	-1	+1	+1	-1	56.991
5	+1	-1	-1	-1	49.0298
6	+1	-1	+1	+1	46.1079
7	+1	+1	-1	+1	46.1079
8	+1	+1	+1	-1	44.9483
Effect	-6.067	-0.625	0.345	-3.056

Montgomery (2005, p208) shows how to calculate the average factor effects using the -1 and +1 coding.  For a single factor effect, we sum the products of the factor coding times the experiment response over all experiments.   Then we divide the sum by the number of -1, +1 pairs.   In this experiment, the number of pairs is 4.   The last row in the above table shows the estimated factor effects.   For an interaction effect, we multiply the experiment coding for each factor to get a coding for the interaction effect.

Experiment	Feed Rate X Wheel Speed AB	Feed Rate X Work Speed AC
1	+1	+1
2	+1	-1
3	-1	+1
4	-1	-1
5	-1	-1
6	-1	+1
7	+1	-1
8	+1	+1
Effect	-1.416	-2.386

Notice that the estimated AB and AC interaction effects are larger than the single factor B and C effects.

The next posting will examine the properties of 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. Hunter, J. S. (1987). "Signal-to-Noise Ratio Debated." Quality Progress 20(5): 7-9.
3. Montgomery, Douglas C. (2005). Design and Analysis of Experiments, 6^th Edition, John Wiley & Sons, Inc.

This posting describes a grinding process case study to illustrate the use of design and analysis of experiments to study cause and effect and reduce common-cause variation.  We continue the case study reported by Gijo (2005) in the 2/28/2005 posting.   That posting describes the construction of a cause-and-effect diagram by a team in an engineering organization identify potential causes of low grinding machine capability.  The team selected four factors for further analysis based on designed experiments.   These factors were Feed Rate, Wheel Speed, Work Speed, and Wheel Grade.  The team chose to perform experiments using two levels for each factor.   The following table shows the levels and factors selected for experimentation.  The levels with an * were existing operating levels.

Factor	Code	Low Level (-1)	High Level (+1)	Unit
Feed rate	A	.0008*	.0010	Mm/Rev
Wheel speed	B	2200	2450*	RPM
Work speed	C	250*	360	RPM
Wheel grade	D	A54	A60*	-

 

Experimental design terminology defines the effect of a factor as the change in the response produced by a change in the level of the factor.   Assume that the response in this experiment is the variance of the outer diameter measurements.   For example, if increasing the feed rate from .0008 to .0010 mm/revolution increases the variance of the outer diameter by .003 mm² then the feed-rate (factor A) effect is .003 mm².  When the difference in response to a factor level change is not the same at all levels of another factor, an interaction effect exists between the factors.   The factor A effect might be .003 mm² when the wheel speed is 2200 rpm and .005 mm² when the wheel speed (factor B) is 2400 rpm, then an interaction effect exists between factors A and B.   The magnitude of the interaction effect is the average difference between the two A effects.   Thus the AxB interaction effect is (.005-.003)/2 = .001 mm².

The team selected an experimental design the enables them to estimate the effects of the four factors in the above table.   They also wanted to estimate two interaction effects: 1. (AxB) between Feed Rate and Wheel Speed (AxB) and 2. (AxC) between Feed Rate and Work Speed.  The linear graph shown below depicts the effects the experimental design must be capable of estimating.  That is, the A, B, C and D effects, the AxB and AxC interaction effects and the error variance.

The next posting will describe 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.