**5. Results and Discussion**

The results of the analyzes of process variable capacity 12, the unit production process of the shafts with the grooves are shown in Figure 6.

**Figure 6.** Comparison of the results of the value of the unit production process of the shafts with grooves, for the 12 process variable, by the classic method (red), percentile method (yellow) and bootstrap method (green). Source: own study.

For the unit production process of an shafts with grooves, the lowest value is taken by the process coefficient of the lower limit of the confidence interval of the classic analysis. The process capability coefficient determined by the percentile method *CP <sup>P</sup>* falls within the process capability range determined by the classic method as well as in the process range determined by the bootstrap method.

The tests carried out for the unit production of shafts with grooves have shown that the analysis of the process capability for ten measurements is possible. In both examined cases, the average value of process capability determined by the classic method *CP* was greater than the lower bootstrap limit of the process capability. Whereas, the lower limit of the process capability determined by the classic method *CP MIN* was below the lower bootstrap limit *CP B MIN* (Figure 7).

**Figure 7.** Graphic presentation of the results of the production process capability assessment of the shafts with grooves, process variable 12. Source: own study.

Based on the analyzes and results obtained, new criterion standards for the assessment of process capability can be introduced, resulting from the lower limit of process capability assessment. Analyzing the lower limits of process capability confidence intervals, processes can be assessed without the need to analyze the sample size, because interval estimation of process capability using the bootstrap method does not require this. The proposed new criterion values for process capability assessment for the bootstrap method are presented in Table 1.

**Table 1.** Sample criterion values of process capability assessment for the lower limit of bootstrap process capability in relation to the process capability determined by the traditional method for normal distribution. Source: own study.


#### **6. Summary and Discussion**

The article reviews and critically evaluates the methods used to determine process capability. The capability of production processes was treated here as the basic property determining their quality, conditioning control and, as a consequence, process management. In the case of a process-based approach to organization management, process capability can and should be one of the key instruments of organization management, in particular, quality management in the organization.

Against the background of the methods used so far to determine the capability of production processes, traditional and percentile, the article presents the assumptions of the bootstrap method and discusses the effects of using this method in relation to specific cases of process flows [60]. An attempt was also made to use the bootstrap method to analyze production processes.

In addition, a procedure for determining the capability of production processes based on the bootstrap method was developed and this procedure was used to analyze the process of manufacturing shafts with grooves (production process). The above model for assessing the capability of production processes using the bootstrap method for low-volume production has been implemented in the defense industry.

On the basis of the literature studies, research and analysis of the results obtained, the following conclusions can be drawn:


4. New criterion values for process capability determined by the bootstrap method were developed. The introduced criteria make it possible to assess the capabilities of processes taking place in various branches of the economy (different classes of processes), determined by the bootstrap method, and are equivalent to the criteria used so far resulting from the use of the classic method of determining capacity. The developed criteria allow the assessment of the capability of the processes examined also on the basis of small sample sizes (*n* ≥ 10).

Based on the conducted research and literature study, the following possibilities of practical application of bootstrap analysis can be proposed:


**Author Contributions:** Conceptualization, P.S. and M.S.; methodology, P.S., M.S. and A.D.; formal analysis, M.S., A.D. and J.C.; resources, M.S., A.D. and J.C.; writing—original draft preparation, P.S., M.S., A.D. and J.C.; writing—review and editing, M.S., A.D. and J.C.; visualization, M.S., A.D. and J.C.; project administration, M.S.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.
