Brian Macpherson Ph.D, Professor of Statistics, University of Manitoba Tom Bingham Statistician, 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 Scenarios with 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 ScenarioActual situationConsequence 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 target due to inherent random variation

The consequence is greater than scenario 1 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)

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 to estimate initial control limits

The Linear Setup Model assumptions 1. A process parameter is available to the operator that can be used to change the process mean. 2. The output (Y) is affected linearly with small adjustments in the parameter (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 (y 1, y 2, y 3, …) corresponding to process settings (x 1, x 2, x 3, …) 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 (x 1, x 2 ) are such that x 1 < x t < x 2, or x t does not lie too far outside the interval [x 1, x 2 ].

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. x t = 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

Estimating the Regression Parameters

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

Confidence Interval for the Process Variability To bound the variability applicable to on-going production

Rules for Stopping Set-up Process 2.Estimate process capability (Cp, Cpk) after each run and stop when it reaches acceptable level. 1.Estimate the Natural Tolerance Limits and stop when these fall inside the Specification Limits.

Simulation of Simple Iterative Targeting One observation per run Each new parameter setting based on predicted value Convergence is elusive however. Repeated simulations shows less stability than desired.

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

Minimizing the Confidence Interval Revise the algorithm’s choice of the next sampling point –Previous approach uses X t (value that sets the regression result to the target) as the next sampling location –As an alternative, choose X n to be the point at which the width of the confidence interval is minimized for X=X t. Above width of confidence interval is minimized for x = average(x), yielding :

Incremental Targeting for repeated measurement and revised selection approach Process Adjustment Table After 15 Set-up Runs of size 3

Applications to SPC 1.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/d 2. 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.