**1. Introduction**

In the modern globalized market, and in connection with the growing requirements of customers, one of the key problems and at the same time challenges of suppliers is the quality assurance of manufactured products throughout the entire production cycle. One of the aspects of product quality assurance is the quality of production processes and in particular their capability.

Process capability is an important factor in the cooperation between the supplier and the customer [1]. Currently, the recipients of components for the aviation, automotive or machine industry among others, in addition to product specifications and acceptable manufacturing defects, impose on their suppliers the required process capability, as the lack of process capability control can generate losses. To meet these requirements, the supplier should therefore monitor and measure the course of manufacturing processes so as to be able to correct and improve the quality of the product based on reliable information. That is why the problem of assessing the capability of monitored processes becomes so important, and shaping, analyzing and controlling the capability of processes is an important aspect of organization management, especially when it uses a process approach to management.

The problems of assessing the capability of production processes and maintaining the capability at the required level are a key element of business cooperation. When entering into contracts between interested parties, it is often impossible to sign such agreement. In most cases, this is a level set too high for the production process capability, i.e., the quality of the production. Elderly, time-worn machinery park and old technology mean low process capability; in turn, modern machinery park, automated production, precise control of production processes and modern technology allow achieving high capability of the production process [2]. That quality of processes, represented by their capability, is expected by recipients from manufacturers, especially from European Union countries. In the case of suppliers operating in the automotive sector having a quality management system certificate, continuous process capability analysis is even an obligatory action [3].

The results of the production process measurements depend on the complexity of the research methods used. An appropriate, structured research method is also a prerequisite for obtaining reliable information about the capability of the production process.

In the general case, the analysis of the capability of the production process consists in comparing the width of the tolerance range required with the distribution of results obtained in a selected range of the duration of this process [4]. Currently, two methods of testing process capabilities are used: classic and percentile analysis. Classic analysis is used for distributions that can be considered normal, while percentile analysis is used for distributions that deviate from the normal distribution. In both cases, the number of samples taken should exceed *n* = 100 measurement results [5]. Therefore, if in reality the tested process cannot be characterized by the required number of measurement results (less than 100 measurements), and in addition, their distribution deviates from the normal, the analysis of the process capability, carried out by classic or percentile methods, is not possible (assuming the correctness and reliability of the obtained capability values). Therefore, it seems that in the case of monitoring the course of such processes, the bootstrap method can be used to test their capability [6].

The bootstrap method involves drawing with return lots of small-scale bootstrap samples from a small number of results. Due to the size of the bootstrap sample set, this method is cumbersome to measure process capability in industrial practice. However, after using the appropriate software, it becomes accessible to operators who do not have extensive knowledge of statistical process control.

The basic use of bootstrap analysis to measure the capability of production processes can find place in industrial practice, mainly in cases where:


The bootstrap method can be used for what-if studies. It is used in many different areas, including in simulation models analyzing medical data [7–12], in financial analyzes [13–15], in solving problems in the area of logistics and distribution [16–18], in environmental protection [19–22], safety sciences [23], automotive [24], risk management [25,26] and in classic queuing models [27].
