## 3 Materials and Methods
### Models and Indicators of Operational Effectiveness Assessment  
As indicated in the literature, efficiency measurement is of key importance for production management processes. Analysis of data from Formula (1) can be the basis for productivity analysis. One of the researchers emphasized that productivity is often treated as a relationship between the size of the \"output\" of the process and the \"entrance\", which means that the concepts of effectiveness and productivity are related [46]:  
\[E=\frac{e}{n},\% \tag{1}\]  
where:  
\(E\)--efficiency,  
\(e\)--effects,  
\(n\)--outlays. Based on Formula (1), there are several ways to improve efficiency, including:  
* reduction of expenditure while maintaining existing effects,
* reduction of expenditure when increasing the effects,
* maintaining expenditure when increasing the effects,
* increasing expenditure to drastically increase the effects. The \(OEE\) indicator is a tool to determine the degree of use of production resources and assess the effectiveness of work in production systems. It takes into account the following elements: performance, availability and quality. \[OEE=\text{({availability})}\times\text{({efficiency})}\times\text{({quality})},\% \tag{2}\]  
The availability indicator presents the relationship between the planned working time of the machine and its actual time of operation in a certain period. This indicator is mainly affected by failures, delays caused by a lack of materials and unforeseen events that can extend the duration of the production process. \[\textit{Availability}=(\textit{real working time})/(\textit{planned working time}),\% \tag{3}\]  
The performance indicator expresses the relationship between the planned time of making one product unit and the actual time needed to produce it. You can also describe performance as the ratio of the number of actually made products to the maximum possible number in a given period. The performance is influenced by, among others, errors made by operators and the need to carry out additional transport activities. These errors also affect the availability of machines, because their operation is directly dependent on the correct service by employees. \[\textit{Efficiency}=(\textit{real production time})/(\textit{planned production time}),\% \tag{4}\]  
The quality indicator determines the proportion of products that meet standards to the total number of products produced in a given period. The quality of production is influenced by many factors, including human errors, such as improper use of machines. \[\textit{Quality}=(\textit{number of good elements})/(\textit{number of all elements}),\% \tag{5}\]  
The level of the OEE indicator considered satisfactory is over 85%, but the key is continuous improvement of processes and elimination of the causes of losses. In practice, however, many companies achieve results around 60%. The \(\textit{MTTR}\) indicator (_Mean Time to Repair_) determines the average time needed to restore the system to efficiency after the failure. It may include the time needed to diagnose the fault, the waiting time for the service technician's arrival and the time to repair the system itself. \(\textit{MTTR}\) can be used to assess the efficiency of the movement maintenance team and the quality of their repairs [47]:  
\[\textit{MTTR}=\frac{\sum_{i=1}^{N}T_{i}}{N},h \tag{6}\]  
where:  
\(N\)--number of failures,  
\(T_{i}\)--failure time. \(\textit{MTTF}\) (_Mean Time to Failure_) is the average operation time of the device until the failure. It specifies the period from the start of use or the last repair until the next defect. Most often it is measured in hours. \[\textit{MTTF}=\frac{T_{D}-\sum_{i=1}^{N}T_{i}}{N+1},\ h \tag{7}\]  
where:  
\(T_{D}\)--the time when the machine should work (time available),  
\(N\)--number of failures,  
\(T_{i}\)--failure time.