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.

This posting illustrates the use of model building to study cause and effect and reduce common-cause variation.  One approach to model building is to build a model such as a regression model based on either results from an experimental design or observed process data.  Another approach illustrated in this posting is to construct a simulation model based on the system flow chart or process map.    One application of a simulation model is to predict flow times or service times for complex systems.   In service or health system applications customer service or wait times could be useful quality measures.   One uses the simulation model by varying input variables such as the number of servers to predict their effect on customer service times.

Davies (2007) describes a case study involving the treatment of minor injuries and medical problems in an emergency department in England.   Receptionists route arriving patients with minor injuries or medical conditions are routed to the “Minors” department.   The standard processing procedure has receptionists in the Minors department assign patients to a queue for triage nurses who assess the patient condition and needs.   Then the triage nurse routes the patients to a doctor or nurse for treatment.   The nurses are qualified to assess and treat minor injuries but not to handle minor medical conditions which are handled by doctors.   These nurses are Emergency Nurse Practitioners (EPNs).  Call this procedure “See” and “Treat”.   The UK national health service recommended that emergency departments skip the triage nurse step.   The health service recommended that receptionists route patients to a doctor or ENP for diagnosis and treatment.  Call this procedure “See & Treat”.   The intent was to reduce patient system time by eliminating a step and its associated queuing time.   The following figure depicts the “See & Treat” patient flow.

Davies describes a simulation model for comparing the two procedures.   This model represents the processing of individual patients, their waiting times, and individual task processing times.   Inputs to the model would include distributions for task times, distributions for times between patient arrivals, and the numbers of doctors and EPNs.  The following figure presents some of the simulation results.   The new procedure “See & Treat” that eliminates the triage step gives the lowest system time.

References

1. Davies, R. (2007). "See and Treat" or "See" and "Treat" in an Emergency Department. 2007 Winter Simulation Conference. Washington, DC.