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Figure 4. Situation of the PR and the modified tolerance region respect to each other In the mentioned situations, the first component of the proposed index accounts for the process variation and the second com- ponent is the inverse of distance between the process mean and the target. Using this component, we can distinguish between two processes with equal first component but different distance between target and process mean.

Figure 4 Situation of the PR and the modified tolerance region respect to each other In the mentioned situations, the first component of the proposed index accounts for the process variation and the second com- ponent is the inverse of distance between the process mean and the target. Using this component, we can distinguish between two processes with equal first component but different distance between target and process mean.

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One of the problems with the NMPCV is that it is too sensitive to the angle between the large axis of the process region ellipsoic and the horizontal axis. This angle completely depends on the variance-covariance structure. As an example, for two processes with equa variation and different correlation between quality characteristics, NMC,,,, may compute two values with large difference (see Figures 2¢ and 2b). As shown in Figures 2a and 2b, two processes with equal variation but different correlation between quality characteristic:
One of the problems with the NMPCV is that it is too sensitive to the angle between the large axis of the process region ellipsoic and the horizontal axis. This angle completely depends on the variance-covariance structure. As an example, for two processes with equa variation and different correlation between quality characteristics, NMC,,,, may compute two values with large difference (see Figures 2¢ and 2b). As shown in Figures 2a and 2b, two processes with equal variation but different correlation between quality characteristic:
Figure 1. Example of the modified tolerance region (extracted from Shahriari and Abdollahzadeh"*)
Figure 1. Example of the modified tolerance region (extracted from Shahriari and Abdollahzadeh"*)
Figure 2. (a and b) Two processes with the same variation but different correlations between quality characteristics where PV is the P value of hypothesis test Ho: = flo against Hy : 4 jlo, which is computed by using Hotelling T* test statistic and
Figure 2. (a and b) Two processes with the same variation but different correlations between quality characteristics where PV is the P value of hypothesis test Ho: = flo against Hy : 4 jlo, which is computed by using Hotelling T* test statistic and
Figure 3. Two process with different variations, which are completely within the modified tolerance region
Figure 3. Two process with different variations, which are completely within the modified tolerance region
Figure 7. (a) A bivariate quality characteristic with bilateral specification limits. (b) A bivariate quality characteristic with both unilateral and bilateral specification limit: Figure 6. Unilateral quality characteristic without values greater than the process mean
Figure 7. (a) A bivariate quality characteristic with bilateral specification limits. (b) A bivariate quality characteristic with both unilateral and bilateral specification limit: Figure 6. Unilateral quality characteristic without values greater than the process mean
Figure 5. A unilateral quality characteristic (with LSL)
Figure 5. A unilateral quality characteristic (with LSL)
2.2.2. Proposed index “onsider the PR shown in Figure 8 by color black. The PR can be located anywhere, even out of the tolerance region. However, it is lesired that the PR lies in the modified tolerance region. To develop the PCI, we computed the CV and the NCV regions of modified tolerance and PR and considered as variation measures. By using computed regions value (see Appendix A), the proposed index is computed as follows: where 4 is a constant value and will be discussed in Section 2.3. The first component of the proposed index does not distinguis between processes A and B, as shown in Figure 9. However, the second component, which we call location part, can identify the mor capable process. It is obvious that process B is more capable because Y> is the smaller and better quality characteristic. The location part can be computed by Equation (15)as follows:
2.2.2. Proposed index “onsider the PR shown in Figure 8 by color black. The PR can be located anywhere, even out of the tolerance region. However, it is lesired that the PR lies in the modified tolerance region. To develop the PCI, we computed the CV and the NCV regions of modified tolerance and PR and considered as variation measures. By using computed regions value (see Appendix A), the proposed index is computed as follows: where 4 is a constant value and will be discussed in Section 2.3. The first component of the proposed index does not distinguis between processes A and B, as shown in Figure 9. However, the second component, which we call location part, can identify the mor capable process. It is obvious that process B is more capable because Y> is the smaller and better quality characteristic. The location part can be computed by Equation (15)as follows:
Figure 9. Effect of location part on the proposed index
Figure 9. Effect of location part on the proposed index
where D;; and Diz are the D;- and D>-type distance for the ith quality characteristic, respectively. 2.3. Discussion about the process capability sensitivity parameter (A)
where D;; and Diz are the D;- and D>-type distance for the ith quality characteristic, respectively. 2.3. Discussion about the process capability sensitivity parameter (A)
Equation (14) can be easily rewritten as a general equation for multivariate case,
Equation (14) can be easily rewritten as a general equation for multivariate case,
Figure 11. Tolerance region, which is not within the modified tolerance region (bilateral case)
Figure 11. Tolerance region, which is not within the modified tolerance region (bilateral case)
Figure 12. (a) Capable process when the PR is equal to the modified tolerance region. (b) Capable process with a correlation between quality characteristic In this section, similar to Pearn et al.7*, a confidence interval for the first component of the proposed index is developed, assuming the sensitivity parameter of the proposed PCI (A) is equal to zero. In Section 2.3, we discussed the sensitivity parameter and recommended the value of 0.27 for this parameter. However, as explained in the Section 2.3, when the correlation between quality characteristics
Figure 12. (a) Capable process when the PR is equal to the modified tolerance region. (b) Capable process with a correlation between quality characteristic In this section, similar to Pearn et al.7*, a confidence interval for the first component of the proposed index is developed, assuming the sensitivity parameter of the proposed PCI (A) is equal to zero. In Section 2.3, we discussed the sensitivity parameter and recommended the value of 0.27 for this parameter. However, as explained in the Section 2.3, when the correlation between quality characteristics
Figure 13. Tolerance region, which is not within the modified tolerance region (unilateral case)
Figure 13. Tolerance region, which is not within the modified tolerance region (unilateral case)
where Fy(z =f fy(y)dy; thus, a 100(1 — «)% confidence interval for the first component of MPCycy can be obtained as increases, it is desired to use the smaller value of the sensitivity parameter. Hence, assuming A=0, the first component of the proposed index is ee. Note that PR volume follows a chi-square distribution with v degrees of freedom. Hence, a 100(1 — «)% confidence interval for the first component of MPCycy can be established as follows:
where Fy(z =f fy(y)dy; thus, a 100(1 — «)% confidence interval for the first component of MPCycy can be obtained as increases, it is desired to use the smaller value of the sensitivity parameter. Hence, assuming A=0, the first component of the proposed index is ee. Note that PR volume follows a chi-square distribution with v degrees of freedom. Hence, a 100(1 — «)% confidence interval for the first component of MPCycy can be established as follows:
Table I. The characteristics of processes for example 1
Table I. The characteristics of processes for example 1
3.2. Example 2
3.2. Example 2
Figure 15. Values of NMPCV and MPCycy for the first example Figure 14. Graphical representation of the six processes for the first example
Figure 15. Values of NMPCV and MPCycy for the first example Figure 14. Graphical representation of the six processes for the first example
A biological reduction of the ethyl 4-chloro acetoacetate process for the production of optically pure compound is applied to demon- strate the effectiveness of the proposed index.*”° S-4-chloro-3-hydroxybutyric acid ethyl ester (or S-CHBE) can be used to synthesize various optically active compounds such as antihypertensive drugs and antibiotics. The compound can be produced by adding ethyl 4-chloro acetoacetate to baker's yeast. The reaction produces S-CHBE with a small amount of enzymes in the yeast cell.
A biological reduction of the ethyl 4-chloro acetoacetate process for the production of optically pure compound is applied to demon- strate the effectiveness of the proposed index.*”° S-4-chloro-3-hydroxybutyric acid ethyl ester (or S-CHBE) can be used to synthesize various optically active compounds such as antihypertensive drugs and antibiotics. The compound can be produced by adding ethyl 4-chloro acetoacetate to baker's yeast. The reaction produces S-CHBE with a small amount of enzymes in the yeast cell.
4. Numerical examples for unilateral situation In this section, we investigate the performance of the proposed index in the unilateral cases through two examples which are real cases. 4.1. Example 4
4. Numerical examples for unilateral situation In this section, we investigate the performance of the proposed index in the unilateral cases through two examples which are real cases. 4.1. Example 4
Figure 18. (a) Relationship between the tolerance region and the modified tolerance region for example 4. (b-c) PR and the modified tolerance region for processes o example 4 tae a, Dee The results of the index in Table V considering Figures , 18a-18c show the effectiveness of the proposed index in measuring the ‘apability of the processes. Note that because two processes are completely in the modified tolerance region, the first component of oroposed index only consider the variation of processes and distance from target is measured with the second component. To com- ute the proposed index, 2 =0.27/4 = 0.0675 is used because two quality characteristics are unilateral, and there is one corner in the nodified tolerance region.
Figure 18. (a) Relationship between the tolerance region and the modified tolerance region for example 4. (b-c) PR and the modified tolerance region for processes o example 4 tae a, Dee The results of the index in Table V considering Figures , 18a-18c show the effectiveness of the proposed index in measuring the ‘apability of the processes. Note that because two processes are completely in the modified tolerance region, the first component of oroposed index only consider the variation of processes and distance from target is measured with the second component. To com- ute the proposed index, 2 =0.27/4 = 0.0675 is used because two quality characteristics are unilateral, and there is one corner in the nodified tolerance region.
4.2. Example 5
4.2. Example 5
Figure A1. Tolerance and PRs for a unilateral process To compute CV and NCV regions for the proposed index, we used integral. Consider a unilateral process, which is depicted in Figure Al. Firct. we divided CV reaion in two narts from nrocess mean of Y. (Figure A171).
Figure A1. Tolerance and PRs for a unilateral process To compute CV and NCV regions for the proposed index, we used integral. Consider a unilateral process, which is depicted in Figure Al. Firct. we divided CV reaion in two narts from nrocess mean of Y. (Figure A171).
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Applied MathematicsStatistical Process ControlElectrical and Electronic EngineeringQuality and Reliability Engineering
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