## 2 Review of Literature
### Assessment of Process Reliability, Taking into Consideration the Timeliness of Operations  
In logistics and transportation systems, it is important whether the process has been executed while conforming to specific requirements or not. When analyzing processes, attention should also be paid to whether individual activities in the specific sub-processes were performed within the specified time, and if not, the cause of the delay should be determined. Thanks to this, constant process improvements are constantly possible.  
A reliability assessment method was developed that takes into consideration the three-state model, utilizing the event trees method. The assumption in this method is that an activity in each sub-process can be executed in three ways, that is, ending as a success, partial success, or failure. Success is when the activity is completed in a timely manner, partial success is when the action is delayed, and failure happens when the activity is not completed, which may interrupt the process execution.  
The steps to draw up an event tree for the process under study are the following [49]:  
Identifying the initiating operation;  
Development of the event tree for each activity assuming three states for the event: success, partial success and failure;  
Designating the probabilities or possibilities of the occurrence of three states: success, partial success and failure for each activity;  
Determination of reliability;  
Analysis and evaluation of the results.  
After identifying and analyzing the activities occurring, the probability of success, partial success, or failure should be assessed. For a three-state system, it is:  
\[PS(H_{m_{c}})+PSF(H_{m_{c}})+PF(H_{m_{c}})=1 \tag{9}\]  
where:  
\(PS(H_{m_{c}})\) is the probability of the success sequence for activity dm for the three-state system, for m = 1,..., M;  
\(PSF(H_{m_{c}})\) is the probability of the partial success sequence for the three-state system for activity dm, for m = 1,..., M;  
\(PF(H_{m_{c}})\) is the probability of failure sequence for the three-state system for activity dm, for m = 1,..., M;  
C is the consequent sequence branch for the three-state system.  
Determination of reliability consists in determining the probability of the implementation of the sequence of activities for the process under study in accordance with the event tree method, similar to the event tree for a two-state process [50]. The process reliability using the (three-state) event tree method is described by a set of subsequent sequences of the success of an activity \(SHD_{m_{c}}\) and partial success of an activity \(SFHD_{m_{c}}\), and each sequence is described by a set of subsequent activities dm.  
Thus, the process reliability for success is \(R_{pr(S)}\)[51; 52]:  
\[R_{pr(S)}=PS(H_{m_{c-1}})+\ldots+PS(H_{m_{c-c}}) \tag{10}\]  
where \(PS(H_{m_{c}})\) is the occurrence of the success sequence for the set of activities \(D\) for any \(c\) is determined on the basis of:  
\[PS(H_{m_{c}})=\prod_{c=1}^{C}PS(D_{m_{c}}) \tag{11}\]  
In contrast, the probability of the partial success sequence \(PSF(H_{m_{c}})\); for the set of activities, \(D\) is calculated as follows:  
\[PSF(H_{m_{c}})=\prod_{c=1}^{C}PSF(D_{m_{c}}) \tag{12}\]  
and the probability of partial success sequence \(PF(H_{m_{c}})\) for the set of activities, \(D\) is calculated from:  
\[PF\ (H_{m_{c}})=\prod_{c=1}^{C}PF(D_{m_{c}}) \tag{13}\]