Sensitivity analysis for process performance (P pk )

If the deviation of the controlled process is small enough, the measured value of the product characteristic never reaches the specification limit. The aim of this simu-lation is to answer the question where is the point regarding process performance, where the consideration of measurement uncertainty can still decrease the total de-cision cost. In other words, where is the limit, beyond that measurement uncertainty has no impact to the total decision cost. This analysis also determines the limitations of the proposed method.

In the simulation, the real value of the measurand is centered to the target value, and process performance (P_pk) is modified with the alteration of the standard devi-ation. Figure5.2presents the result of the simulation. Six scenarios were simulated with different cost structures:

• Case A: General cost structure

• Case B: Type I. error cost is twice as type II. error cost

• Case C: Cost of type II. error is increased significantly

• Case D: Cost of correct rejection is enlarged

• Case E: Cost of correct acceptance is high

• Case F: Cost of type II. error is extremely high

Ppk

A: Generic (initial) cost structure

Ppk

C: Cost of type II. error is increased significantly

D: Cost of correct rejection is high

Ppk

E: Cost of correct acceptance (production) is high

F: Cost of type II. error is extremely high c11=1

FIGURE5.2: Sensitivity analysis for process performance (Ppk)

For cases A, C, E and F, the same phenomena can be observed. When the P_pk is low,K^∗fluctuates around a given value based on the cost structure and measure-ment uncertainty.K^∗ >0 because the most significant cost is the cost of type II. error;

therefore, the acceptance interval must be narrowed. IfP_pk improves, the value of K^∗ decreases dynamically. Finally, whenP_pkapproaches approximately 1.3-1.5, the correction component tends to zero, meaning that this is the limit where the mea-surement uncertainty does not have any impact on the decisions.

because the dominant cost is the cost of type I. error (acceptance interval needs to be extended). K^∗ tends to zero whenP_pk approaches 1.3-1.5 as in cases A, C, E and F. The model is sensitive to the process performance when P_pk is low however, it can not find better solution forKin the case of a process with strong performance, because no significant amount of scrap arises.

This analysis showed that the benefit of measurement uncertainty consideration strongly depends on process performance. Nevertheless, it can be assumed that the standard deviation of measurement error can influence this statement. Therefore, this sensitivity analysis was extended with an additional variable. As a further step, the value ofK^∗ was investigated as the functions of P_pk and standard deviation of measurement error distribution (σε) compared to the standard deviation of the pro-cess (σ_x) (Figure5.3).

0.0025 0.005 0.0075

0.01 0.01250.015 0.0175 A: Generic (initial) cost structure

<₀/<_x C: Cost of type II. error is increased

significantly D: Cost of correct rejection is high

<0/< E: Cost of correct acceptance

(production) is high F: Cost of type II. error is extremely high

<0/<

FIGURE 5.3: Sensitivity analysis for process performance (Ppk) and standard deviation of measurement error

On Figure5.3, axis "x" shows the ratio of measurement error- and process stan-dard deviation (σε/σx), axis "y" denotes the process performance and each contour map area represents the value ofK^∗at each combination ofσε/σxratio andP_pkvalue.

The same cost structures were applied to each simulation (A-F) as it was introduced by Figure5.2. It is clearly observable that not only process performance but also the ratio of measurement error- and process standard deviation impacts the optimal acceptance strategy. The relationship between P_pk and K^∗ remains the same in all the cases however, this analysis highlights that standard deviation of measurement error (compared to process standard deviation) has significant impact onK^∗. That is to say, process performance is not enough to judge the limitation of the proposed method because it can find better solution even at strong process performance if σε/σx ratio is high. In order to decide if measurement uncertainty is beneficial to

Conclusion of the analysis:

In the case of processes with strong performance index, the consideration of mea-surement uncertainty cannot decrease the overall decision cost since practically no measured value can be observed near to the acceptance limit. Nevertheless, the benefit of measurement uncertainty consideration can be judged through the joint investigation ofP_pkandσε/σx.

5.2 Risk-based multivariate control chart

During the sensitivity analysis, the effect of the following parameters are analyzed:

• Decision cost of type II. error (c01)

• Sample size (n)

• Skewness of the probability density function of product characteristic 1 (γ₁)

• Standard deviation of product characteristic 1 (σ₁)

• Standard deviation of the measurement error (σε)

• Number of controlled product characteristics (p)

These parameters were chosen, because they can strongly influence the applica-bility of the proposed control chart. The cost of type II. error may determine the stringency of the control policy. Furthermore, the applicability of the RBT² chart can be analyzed under non-normality by modifying the skewness of the distribu-tion regarding the monitored product characteristic. The performance of the control chart is also analyzed under different sample sizes and different level of standard deviation of measurement error.