*5.3. Limitations*

However, some study limitations should be acknowledged. In the paper we used for cluster analysis the VPi4 index [58], which has some limitations. First, the VPi4 index does not include some Industry 4.0 technologies (block chain, laser technology, RFID, drones, 3D printers, track-n-trace, nanotechnology, autonomous vehicles, etc.). Second, the VPi4 index is based on the results of a questionnaire survey. Some limitations of this kind of method should therefore be considered. Third, the VPi4 index was not verified by confirmatory factor analysis and structural model equation method. This problem will be resolved in the future, after the third wave of questionnaire survey research on Industry 4.0. Currently, the results and structure of the VPi4 index are confirmed by relation to subjective perception of the level of Industry 4.0 by the enterprises and re-test in second wave of the research [58]. Fourth, the VPi4 index is based on the research in the Czech Republic and can be limited to the specifics of this Country. Our study was conducted in the same country, so it should not be a problem. Another limit may be the use of only factors in the field of technology, without taking into account, for example, financial data, which also prove to be very important in terms of sustainable development [102].

The questionnaire research on small- and medium-sized enterprises had a lower rate of return than we expected. Therefore, the margin of error (5.86%) was higher than 5% at 95% confidence level. The error, therefore, slightly exceeded the planned level. However, we assume that this will not a ffect the results. Moreover, the sample size was not representative in the case of proportionality of the Czech Republic enterprise population. Rather, the research sample was created in regard to the size of the enterprises and technological demands of the industry.

Other limitations include problems related to k-cluster analysis methods. The main issues of this method are handling empty clusters (special situation) and outliers (centroids may not be representative for all cases). K-means is not suitable for all types of data. It cannot handle non-globular clusters or data without a specific center [71]. Another issue could be selecting the wrong number of initial clusters. This can be solved by Elbow, Silhouette, or other methods.

The results of the factory analysis validation based on the Tukey HSD test shows similarities between clusters especially at the third level of the VPi4 index. Similarly, this overlap is also confirmed by the average Silhouette coefficient and the lower Dunn index. This clustering issue is based on the fact that most small- and medium-sized enterprises do not reach the highest level of the VPi4 index. Similarity is therefore in low index values. However, in spite of this problem, the different categories of clusters are different from each other. With the increasing level of implementation of Industry 4.0 in SMEs, this issue will be eliminated in the future, and the differences between the clusters will become more noticeable. Hovewer, the relation between Industry 4.0 implementation and the size of an enterprise is still significant [103].
