## 3 Research Results and Discussion  
### Maximum of the Criterion Function--General Analysis  
The parameters of the developed semi-Markov model of the renewal system (repairs and preventive replacements) are necessary to define the conditions of existence of the extremum (maximum) of the criterion function (10). These conditions consist of the elements of the matrix of probabilities of state changes of the model \(P=[p_{ij}]\), i, j = 1, 2, 3, 4, the average times spent in the states of the model \(ET_{i}\), i = 1, 2, 3, 4 and profits (costs) generated in the model states \(z_{i}\), i = 1, 2, 3, 4. The input data for the model are considered parameters. Their values depend on the type of the analyzed technical objects, the adopted strategy of operation (exploitation) and the actual conditions in which the repair and preventive replacement processes are carried out. The assumptions regarding the values of the parameters of the tested system are specified below. The adopted assumptions must take into account the real relationships between the parameters characterizing the processes of repairing damaged technical objects and preventive replacement:  
* Z1: \(z_{1}>0\), \(z_{2}<0\), \(z_{3}<0\), \(z_{4}<0\); the technical object (transport set) in state 1 generates profit (+), but, in states 2, 3 and 4, it generates costs (-);
* Z2: \(ET_{3}>ET_{2}\); the value of the average repair time carried out at the service station (perfect repair) is greater than the value of the average repair time carried out by the Technical Emergency Service (minimal repair);
* Z3: \(z_{3}>z_{2}\); the unit cost generated in state 3 (state of repair carried out at the service station--perfect repair) is higher than the unit cost generated in state 2 (state of repair carried out by the Technical Emergency Service--minimal repair);
* Z4: \(ET_{3}>ET_{4}\); the average time of repair carried out at the service station (perfect repair) is longer than the average time of preventive replacement;
* Z5: \(z_{3}>z_{4}\); the unit cost generated in state 3 (state of repair at the service station--perfect repair) is higher than the unit cost generated in state 4 (state of preventive replacement). For these assumptions, there were no links between the state of repair by the Technical Emergency Service (state of minimal repair) and the state of preventive replacement. In the presented example, it is difficult to clearly define the relationship between the average values of the times \(ET_{2}\) and \(ET_{4}\) and the unit costs \(z_{2}\) and \(z_{4}\). Despite this, based on the results of research on other operational systems of this class of technical objects (transport set), we find an additional assumption for the unit costs generated in states 2 and 4:  
* Z6: the unit cost generated in state 2 (repair by the Technical Emergency Service--minimal repair) is higher than the unit cost generated in state 4 (state of preventive replacement). Below, in order to simplify the formulae for the conditions of existence of the criterion function extremum (10), the coefficients were introduced (10):  
\[\alpha=B_{1}-z_{1}B\]  
\[\beta=z_{1}C-C_{1}\]  
\[\gamma=B_{1}C-B\,C_{1}\]  
where:\[ \begin{split} B_{1}=P_{12}\ ET_{2}\ z_{2}+P_{12}\ p_{23}\ ET_{3}\ z_{3}-P_{12}\ ET_{4}\ z_{4}\\ C_{1}=ET_{4}\ z_{4}\\ B=P_{12}\ ET_{2}+P_{12}\ p_{23}\ ET_{3}-P_{12}\ ET_{4}\\ C=ET_{4}\end{split} \]  
The coefficients \(\alpha\), \(\beta\) and \(\gamma\) are essential for formulating sufficient conditions for the existence of extremes of the criterion function.