Managing Set-up Managing Set-up Variation and Short Variation and Short

Production RunsProduction Runs

Brian Macpherson Ph.D, Professor of Statistics,

University of Manitoba

Tom BinghamStatistician, The Boeing Company

The Process Set-Up Problem

Processes must be repeatedly re-targeted for product specific requirements

- Set-up is affected by variation intrinsic to the process and

measurement system.

- Short runs require rapid establishment of control limits that

can be tailored to multiple product lines.

- Test runs can be expensive or defect prohibitive.

Four Possible Scenarioswith a Single Measurement Observed

1 Centered: Observation in the upper tail

2 Off target : Observation close to mean

3 Same as 1 with higher variability

4 True mean and observation on opposite sides of target.

Adjustments Without Knowledge of Variability

 

Table 1

 

Scenario Actual situation Consequence

1 The process is already centered but the first response was in the upper tail of its distribution

The process is moved off target by the amount of the adjustment

2 The process is off target by the amount the operator believes based on the actual deviation from the measured result.

The process will be adjusted correctly, although a second test is likely to be off target due to inherent random variation

3 This situation is much like the first, but the process has greater variability.

The consequence is greater than scenario one due to the increased variability. A second point may show degradation or falsely improvement since the region of likely results will lie on both sides of the target (see scenario 4)

4 The process is off target but opposite from what is indicated by the observation

The operator will adjust the process in the wrong direction.

The process is moved off target by the amount of the adjustment

The process will be adjusted correctly, although a second test is likely to be off targetdue to inherent random variation

The consequence is greater than scenario 1 due tothe increased variability. A second point may show degradation or falsely improvement since the region

of likely results will lie on both sides of the target (see scenario 4)

The operator will adjust the process in the wrong direction.

Focus on Two Objectives

1. Bring the process to the target with a minimum number of set-up trials.

2. Obtain sufficient data during set-up toestimate initial control limits

The Linear Setup Model assumptions

 

1.    A process parameter is available to the operator that can be usedto change the process mean.

2.   The output (Y) is affected linearly with small adjustments in theparameter (X).

3.   The parameter can be set arbitrarily within reasonable manufacturing limits.

4.    Changes in the parameter do not affect the process output’s variance.

5.    The measurement error () is approximately normally distributed as N ().

6.    Observations (y1, y2, y3, …) corresponding to process settings

(x1, x2, x3, …) have mutually independent errors (, …)

7.    The effect of changing the process parameter (x) of the initial two runs is substantially larger than the error variability ().

8.    Two starting conditions (x1, x2) are such that x1< xt < x2, or xt

does not lie too far outside the interval [x1, x2].

The Linear Setup Model Regression Model

  Y = + X + where :

Y = process output (key characteristic)y = a value of Y which was observed on some given run.T = target value of Y (usually the engineering nominal dimension)X = deterministic parameter that is used to adjust the process output.x = a specific value to which the variable X is set.xt = the value of X which results in the E(Y) = T.

(E(Y) = expected value of Y) = true (but unknown) process average when x = 0 = true (but unknown) effect that the parameter has on the output = process and measurement variation, assumed N(0,) = true (but unknown) standard deviation when no adjustments are being made to the process

Identifying the Centering Conditions

€

yp=̂ μ +̂ β xSubstituting the target T into yp

and xt for x we have:

T =

€

ˆ μ +̂ β xt

€

xt=T−̂ μ ˆ β

Estimating the Regression Parameters

€

ˆ β =(xi−x )(yi−y )

i=1

n∑(xi−x )2

i=1

n∑,

€

ˆ μ =y −̂ β x

and when n>2 we may estimatethe process variance as:

€

ˆ σ 2=(yi−̂ μ −̂ β xi)2i=1

n∑n−2

Confidence Interval for the Process MeanTo be used to assist in stopping rule for set-up runs

€

ˆ μ +̂ β x±t(1+γ)/2ˆ σ 1n+(x−x )2(xi−x )2

i=1

n∑

n = number of measurements used in the setupxi = process parameter setting corresponding to

setup run ix = arbitrary test setting of interest

€

t(1+γ)/2 = Student’s t value with n-2 degrees of

freedom

Confidence Interval for the Process Variability

To bound the variability applicable to on-going production€

(n − 2) ˆ σ 2

χ (1+γ ) /22 ≤ σ 2 ≤

(n − 2) ˆ σ 2

χ (1−γ ) /22

Rules for Stopping Set-up Process

2. Estimate process capability (Cp, Cpk) after each run and stop when it reachesacceptable level.

1. Estimate the Natural Tolerance Limitsand stop when these fall inside the Specification Limits.

Simulation of Simple Iterative TargetingOne observation per run

Each new parameter setting based on predicted value

Convergence is elusive however. Repeated simulations shows less stability than desired.

No Src T X Y

€

X

€

Y ΣXY ΣX ΣY

€

ˆ

€

ˆ

€

ˆ+ˆX Xt=

€

(T−̂)/ˆ

€

ˆLCL UCL

1 A 10 0.00 12.53 0.00 12.53 0.00 0.00 0.00 N.A. N.A. N.A. N.A. N.A.

2 A 10 1.00 16.51 0.50 14.52 1.99 0.50 7.93 3.98 12.53 16.51 -0.64 N.A.

3 A 10 -0.64 10.66 0.12 13.24 4.91 1.36 17.86 3.61 12.80 10.50 -0.77 0.33 6.88 14.12

4 A 10 -0.77 8.84 -0.10 12.14 7.87 1.96 32.38 4.01 12.55 9.44 -0.63 0.64 7.54 11.34

5 A 10 -0.63 13.50 -0.21 12.41 7.29 2.19 33.86 3.33 13.10 10.99 -0.93 1.79 7.96 14.01

6 A 10 -0.93 7.03 -0.33 11.51 10.53 2.62 57.95 4.01 12.83 9.09 -0.71 1.98 6.06 12.13

7 A 10 -0.71 9.82 -0.38 11.27 11.07 2.74 60.40 4.03 12.82 9.97 -0.70 1.77 8.03 11.91

8 A 10 -0.70 7.66 -0.42 10.82 12.06 2.83 71.78 4.26 12.62 9.65 -0.61 1.84 7.89 11.40

9 A 10 -0.61 9.19 -0.44 10.64 12.34 2.86 74.14 4.31 12.55 9.90 -0.59 1.73 8.48 11.33

10 A 10 -0.59 10.60 -0.46 10.63 12.35 2.88 74.14 4.285 12.60 10.06 -0.61 1.63 8.84 11.29

11 A 10 -0.61 11.86 -0.47 10.75 12.18 2.90 75.50 4.20 12.73 10.18 -0.65 1.65 9.02 11.34

12 A 10 -0.65 11.47 -0.49 10.81 12.07 2.93 75.98 4.11 12.81 10.14 -0.687 1.62 9.04 11.23

13 A 10 -0.68 7.41 -0.50 10.54 12.68 2.97 86.63 4.27 12.69 9.77 -0.63 1.72 8.65 10.89

14 A 10 -0.63 8.11 -0.51 10.37 12.97 2.98 92.14 4.35 12.59 9.86 -0.60 1.73 8.82 10.89

15 A 10 -0.60 10.19 -0.52 10.36 12.98 2.99 92.17 4.34 12.60 10.01 -0.60 1.66 9.07 10.95

Convergence of Targeting Algorithm

Repeated trials of this algorithm reveals inconsistency of the convergence. Each table entry is the 25th iteration from a unique trial.

No Src T X Y

€

X

€

Y ΣXY ΣX ΣY

€

ˆ

€

ˆ

€

ˆ+ˆX Xt=

€

(T−̂)/ˆ

€

ˆ LCL UCL

25 A 10 -0.45 12.37 -0.418 10.31 14.58 2.51 176.84 5.80 12.73 10.10 -0.47 2.00 9.262 10.931

25 A 10 -0.71 7.46 -0.829 9.25 96.22 17.02 664.40 5.65 13.93 9.90 -0.70 2.29 8.940 10.853

25 A 10 -0.59 11.35 -0.169 11.72 9.54 2.57 120.63 3.71 12.35 10.15 -0.63 1.93 8.827 11.469

25 A 10 -0.58 8.10 -0.425 10.80 14.06 2.52 219.70 5.57 13.16 9.90 -0.57 2.48 8.756 11.054

25 A 10 -0.68 8.89 -0.724 9.77 41.69 10.21 214.51 4.08 12.73 9.96 -0.67 1.39 9.379 10.531

25 A 10 -0.63 7.47 -0.451 10.75 15.63 3.16 183.87 4.94 12.98 9.87 -0.60 2.15 8.879 10.869

25 A 10 -0.66 10.85 -0.512 10.62 12.22 3.16 137.69 3.86 12.60 10.04 -0.67 1.98 9.149 10.931

25 A 10 -0.60 8.78 -0.516 10.46 18.00 2.82 160.58 6.38 13.75 9.95 -0.59 1.41 9.349 10.548

25 A 10 -0.53 7.99 -0.456 10.27 12.73 2.62 162.77 4.86 12.48 9.92 -0.51 2.09 9.027 10.804

25 A 10 -0.50 9.75 -0.383 10.87 17.95 2.36 209.53 7.60 13.79 9.99 -0.50 1.78 9.199 10.778

Minimizing the Confidence IntervalRevise the algorithm’s choice of the next sampling point

–Previous approach uses Xt (value that sets the regression result to the target) as the next sampling location

–As an alternative, choose Xn to be the point at which the width of the confidence interval is minimized for X=Xt.

Above width of confidence interval is minimized for x = average(x), yielding :

€

2t(1+γ ) /21n

+(x − x )2

(xi − x )2

i=1

n∑

€

xn = nxt − (n −1)x [n−1]

Incremental Targeting for repeated measurement and revised selection approach

Process Adjustment Table After 15 Set-up Runs of size 3

No X Y1 Y2 Y3Avg(Y)

€

X

€

Y ΣXY ΣX ΣY

€

ˆ

€

ˆ

€

ˆ+ˆX Xt=

€

(T−̂)/ˆ

€

ˆ LCL UCL

1 0.00 15.13 10.10 12.82 12.68 0.00 12.68 0.00 0.00 0.00 N.A. N.A. N.A. N.A. N.A.

2 1.00 16.88 19.76 20.03 18.89 0.50 15.79 3.10 0.50 19.25 6.20 12.68 18.89 -0.43 N.A.

3 -2.30 0.72 1.51 0.69 0.98 -0.43 10.85 30.72 5.72 165.49 5.37 13.17 0.83 -0.59 0.61 -6.76 8.41

4 -1.06 7.07 6.89 9.19 7.72 -0.59 10.07 32.21 6.02 172.85 5.35 13.23 7.53 -0.60 0.46 6.46 8.60

5 -0.65 14.03 9.79 7.26 10.36 -0.60 10.12 32.19 6.02 172.92 5.35 13.35 9.86 -0.63 0.50 9.16 10.57

6 -0.74 9.56 12.36 10.07 10.66 -0.63 10.21 32.13 6.04 173.16 5.32 13.55 9.59 -0.67 0.73 8.76 10.42

7 -0.91 8.71 7.25 7.55 7.84 -0.67 9.87 32.70 6.10 178.01 5.36 13.44 8.58 -0.64 0.74 7.83 9.33

8 -0.48 9.77 6.82 13.60 10.06 -0.64 9.90 32.74 6.13 178.04 5.34 13.33 10.77 -0.62 0.75 10.11 11.43

9 -0.47 13.84 13.24 13.28 13.45 -0.62 10.29 33.28 6.16 189.27 5.40 13.66 11.12 -0.68 1.17 10.18 12.05

10 -1.17 5.15 9.43 6.85 7.14 -0.68 9.98 34.82 6.43 198.21 5.42 13.65 7.33 -0.67 1.09 6.40 8.26

11 -0.63 9.95 8.39 5.44 7.93 -0.67 9.79 34.74 6.43 202.02 5.40 13.44 10.01 -0.64 1.26 9.15 10.87

12 -0.21 14.74 11.84 12.88 13.15 -0.64 10.07 36.16 6.62 212.38 5.46 13.54 12.38 -0.65 1.22 11.48 13.29

13 -0.81 7.99 7.87 8.24 8.03 -0.65 9.92 36.48 6.65 216.22 5.49 13.47 9.05 -0.63 1.21 8.29 9.80

14 -0.43 11.03 12.00 10.64 11.22 -0.63 10.01 36.75 6.69 217.82 5.49 13.49 11.11 -0.64 1.16 10.41 11.81

15 -0.66 6.95 8.83 13.48 9.75 -0.64 9.99 36.75 6.69 217.88 5.49 13.48 9.87 -0.63 1.11 9.25 10.49

Applications to SPC1. Variance estimate can be applied

directly to x-bar chart control limits using subgroups size = 3 (for this example)

2. Range charts control limits can be developed setting the estimate of sigma to R/d2.

3. Part families can be accommodated by maintaining the regression model as the process is re-targeted to support varying product dimension requirements. Control limits can be maintained across re-targeting by subtracting the expected response from the measurements prior to plotting.

Conclusions

• Incremental Targeting allows a model of the process output in terms of a controllable parameter.

• Underlying regression can be used to estimate on going control limits

• Multiple part requirements can be managed through a single control system using the regression model.