Many studies have suggested that companies use control charts to perform process control. If the process is under statistical process control, the capability of the process will be evaluated32.33. It is assumed that each subsample contains not observations on the quality characteristics, and there are m sub-samples available.

In each subsample, we let (overline{X}_{i}) be the sample mean and (Whether}) be the sample variance of the Ie sub-sample, as shown below:

$$overline{X}_{i} = frac{1}{n}sumnolimits_{j = 1}^{n} {X_{ij} }$$

(seven)

and

$$S_{i} = sqrt {frac{1}{n – 1}sumnolimits_{j = 1}^{n} {left({X_{ij} – overline{X}_{ i} } right)^{2} } } .$$

(8)

We define the overall sample mean and pooled sample variance as follows:

$$overline{overline{X}} = frac{1}{m}sumlimits_{i = 1}^{m} {overline{X}_{i} }$$

(9)

and

$$overline{S} = frac{1}{m}sumlimits_{i = 1}^{m} {S_{i} } .$$

(ten)

The estimator (hat{Q}_{pi}) index (Q_{pi}) is displayed as follows:

$$hat{Q}_{pi} = frac{{USL – overline{overline{X}}}}{{overline{s}}},,,{text{or} },,,,frac{{overline{overline{X}} – LSL}}{{overline{s}}}$$

(11)

For a one-sided six sigma quality index (Q_{pi})the (100 times left( {1 – alpha } right)%) lower and upper confidence limits (LQ_{pi}) and (UQ_{pi}) for (Q_{pi}) satisfied

$$P(LQ_{pi} le Q_{pi} le UQ_{pi} ) = 1 – alpha ,,,,{text{where}},,i = u, ,,{text{or}},,,l.$$

(12)

Based on Choi and Owen34, (sqrt {mn} times hat{Q}_{pi}) follows a central step you broadcast with (mleft( {n – 1} right)) degrees of freedom, where not is the subsample size and m is the number of sample groups. The non-central parameter of (delta = sqrt {mn} times Q_{pi}) is expressed as (T^{prime}_{n – 1} left( {delta = sqrt {mn} times Q_{pi} } right)). Then the above two equations can be rewritten as

$$Pleft[ {T^{prime}_{n – 1} left( {delta = sqrt {mn} times Q_{Li} } right) le sqrt {mn} times hat{Q}_{pi} } right] = 1 – frac{{alpha^{prime}}}{2}$$

(13)

and

$$Pleft[ {T^{prime}_{n – 1} left( {delta = sqrt {mn} times Q_{Ui} } right) le sqrt {mn} times hat{Q}_{pi} } right] = frac{{alpha^{prime}}}{2},$$

(14)

where I (= u) Where (I) and (alpha^{prime} = {alpha mathord{left/ {vphantom {alpha q}} right. kern-nulldelimiterspace} q}); (q) is the total number of quality characteristics for a multi-process product, (m) is the number of sample groups, and (not) is the sample size for each subsample.

The online schedule S1 displays the lower limit ((LQ_{pi})) and upper bounds ((UQ_{pi})) 95% confidence intervals for different (Q_{pi}) values ​​with (n = 11), (m = 30)and (q = 6left( 1 right)10). For example, given the subsample size (n = 11), (m = 30), (q = 6), (alpha = 0.05)when (hat{Q}_{pi} = 3.60)based on the results of the SAS program, the (LQ_{pi}) and (UQ_{pi}) the values ​​are 3.222 and 3.981. The confidence intervals for different (Q_{pi}) the values ​​can be obtained in the online appendix S1. The online Appendix S1 only provides the numerical values ​​used in this study. Different sample size, different number of sample groups or different number of quality characteristics could possibly occur in the practical application. For a simpler explanation, we only perform the selected sample size and the selected number of sample groups and the results are listed in the online Appendix S1. The numbers of quality characteristics are calculated from 6 to 10 in the online appendix S1.

Suppose there is (q) quality characteristics of a product, the process capability index for the Ith characteristic will become

$$S_{pkj} = frac{1}{3}Phi^{ – 1} left{ {frac{1}{2}Phi (Q_{puj} ) + frac{1}{ 2}Phi (Q_{plj} )} right}$$

(15)

Let the confidence intervals of the indices (Q_{puj}) and (Q_{plj}) are designated by (left( {LQ_{puj} ,UQ_{puj} } right)) and (left( {LQ_{plj} ,UQ_{plj} } right)), respectively. And hint (S_{pkj}) is a function of indices (Q_{puj}) and (Q_{plj})the confidence intervals for the index (S_{pkj}) can be described as follows.

Lower confidence interval for the (j)th characteristic ((LS_{pkj})):

$$LS_{pkj} = frac{1}{3}Phi^{ – 1} left{ {frac{1}{2}Phi (LQ_{puj} ) + frac{1}{ 2}Phi (LQ_{plj} )} right}$$

(16)

Upper confidence interval for the (j)th characteristic ((US_{pkj})):

$$US_{pkj} = frac{1}{3}Phi^{ – 1} left{ {frac{1}{2}Phi (UQ_{puj} ) + frac{1}{ 2}Phi (UQ_{plj} )} right}$$

(17)

Thus, the integrated process capability index for the whole product is

$$S_{pk}^{T} = frac{1}{3}Phi^{ – 1} left{ {frac{1}{2}left( {prodlimits_{j = 1}^{q} {left[ {2Phi (3S_{pkj} ) – 1} right]} + 1} right)} right}$$

(18)

Let the confidence intervals of the indices (S_{pk}^{T}) are designated by (left( {LS_{pk}^{T} ,US_{pk}^{T} } right))and the confidence intervals for the index (S_{pk}^{T}) can be described as follows.

Lower confidence interval for the integrated product ((LS_{pk}^{T})):

$$LS_{pk}^{T} = frac{1}{3}Phi^{ – 1} left{ {frac{1}{2}left( {prodlimits_{j = 1}^{q} {left[ {2Phi (3LS_{pkj} ) – 1} right]} + 1} right)} right}$$

(19)

Upper confidence interval for the integrated product ((US_{pk}^{T})):

$$US_{pk}^{T} = frac{1}{3}Phi^{ – 1} left{ {frac{1}{2}left( {prodlimits_{j = 1}^{q} {left[ {2Phi (3US_{pkj} ) – 1} right]} + 1} right)} right}$$

(20)

As mentioned before and based on the equations. (16, 17, 19, 20), these confidence intervals allow statistical inferences to be made to assess whether the product’s processability and all quality characteristics are at the required level.