# Semi-Markov Model of the System of Repairs and Preventive Replacements by Age of City Buses
## 4 Conditions for the Existence of a Maximum of the Criterion Function
### Maximum of the Criterion Function--General Analysis
\(\text{z}_{2}<\text{z}_{5}\); means, that the unit cost generated in the state 2 (the state of repair carried out by the Technical Service units without the loss of the course) is lower than the unit cost generated in the state 5 (the state of preventive replacement). The above assumptions do not define the relationship between the state of repair performed by the Technical Service unit with loss of the course (state 3) and the state of preventive replacement (state 5). In the analysed system it is very difficult to unambiguously define the relation between the average values of \(\text{ET}3\) and \(\text{ET}5\) times and unit costs z3 and z5. However, based on the results of research on other systems of exploitation of this class of technical objects (means of transport), an additional assumption can be made regarding the unit costs generated in states 3 and 5:  
* Z12: \(\text{z}_{3}<\text{z}_{5}\); means that the unit cost generated in the state 3 (the state of repair performed by the Technical Service unit with the loss of the course) is lower than the unit cost generated in the state 5 (the state of preventive replacement). In the subsequent part of the paper, the following coefficients have been introduced to formulate the conditions for the existence of the extremum (maximum) of the criterion function (10):  
\[\alpha=\text{AB}_{1}-\text{A}_{1}\text{B}=\text{B}_{1}-\text{z}_{1}\text{B}\]  
\[\beta=\text{A}_{1}\text{C}-\text{AC}_{1}=\text{z}_{1}\text{C}-\text{C}_{1}\]  
\[\gamma=\text{B}_{1}\text{C}-\text{BC}_{1}\]  
where:  
\[\text{A}_{1}=\text{z}_{1}\]  
\[\text{B}_{1}=\text{p}_{12}\text{ET}_{2}\text{z}_{2}+\text{p}_{13}\text{ET}_{3 }\text{z}_{3}+\left(\text{p}_{13}\text{P}_{34}+\text{p}_{14}\right)\text{ET}_ {4}\text{z}_{4}-\left(\text{p}_{12}+\text{p}_{13}+\text{p}_{14}\right)\text{ET }_{5}\text{z}_{5}\]  
\[\text{C}_{1}=\text{ET}_{5}\text{z}_{5}\]  
\[\text{A}=1\]  
\[\text{B}=\text{p}_{12}\text{ET}_{2}+\text{p}_{13}\text{ET}_{3}+\left(\text{p}_{ 13}\text{P}_{34}+\text{p}_{14}\right)\text{ET}_{4}-\left(\text{p}_{12}+\text{ p}_{13}+\text{p}_{14}\right)\text{ET}_{5}\]  
\[\text{B}=\text{p}_{12}\text{ET}_{2}+\text{p}_{12}\text{p}_{23}\text{ET}_{3}- \text{p}_{12}\text{ET}_{4}\]  
\[\text{C}=\text{ET}_{5}\]The coefficients \(\alpha\), \(\beta\), and \(\gamma\) play an important role in formulating sufficient conditions for the existence of the extremes of a criterion function. For this purpose, the following formulates the sufficient conditions for the inequalities to be true \(\alpha<0\), \(\beta>0\), \(\gamma<0\). In regard to the above:  
* the coefficient \(\alpha\) is determined by the formula \[ \begin{array}{c}\alpha=\mathrm{p}_{12}\mathrm{ET}_{2}(z_{2}-z_{1})+\mathrm{ p}_{13}\mathrm{ET}_{3}(z_{3}-z_{1})+\big{(}\mathrm{p}_{13}\mathrm{P}_{34}+ \mathrm{p}_{14}\big{)}\mathrm{ET}_{4}(z_{4}-z_{1})\\ +\big{(}\mathrm{p}_{12}+\mathrm{p}_{13}+\mathrm{p}_{14}\big{)}\mathrm{ET}_{5}(z _{1}-z_{5})\end{array} \] (15) The inequality \(\alpha<0\) is equivalent to the inequality \[ \begin{array}{c}\big{(}\mathrm{p}_{13}+\mathrm{p}_{34}+\mathrm{p}_{14}\big{)} \quad\mathrm{ET}_{4}\\ >\mathrm{p}_{12}\mathrm{ET}_{2}(z_{2}-z_{1})/(z_{1}-z_{4})+\mathrm{p}_{13} \mathrm{ET}_{3}(z_{3}-z_{1})/(z_{1}-z_{4})\\ +(\mathrm{p}_{12}+\mathrm{p}_{13}+\mathrm{p}_{14})\mathrm{ET}_{5}(z_{1}-z_{5}) /(z_{1}-z_{4})\end{array} \] (16)
* the coefficient \(\beta\) is determined by the formula \[\beta=\mathrm{ET}_{5}(z_{1}-z_{5})\] (17) Based on the assumption Z1 made, it follows that \(\beta>0\).