# Planning-Level Optimisation of Headway Regularity
## 3 Methods
### Optimisation Procedure Phases  
In order to apply the proposed procedure, it is necessary to collect the following data: headway, number of stops, passenger flow, passenger accumulation intensity along the line, and average passenger boarding intensity. As previously noted, the optimisation procedure is performed in four phases, as shown in Figure 1:Phase I: Diametrical line splitting  
Phase II: Simulation of headway disturbance propagation  
Phase III: Evaluation of headway regularity  
Phase IV: Evaluation of passenger travel time  
The outputs of the first phase are the basic operating elements for the two variants. The first variant represents the existing operation based on one diametrical line (AB), while the second variant consists of two radial lines (AC and CB), the optimisation variant. The output data of Phase III and Phase IV (Figure 1) represent the optimisation procedure results, namely the degree of line regularity and the difference in the average passenger travel time between the first and the second variant.  
Phase I: In the first phase, the diametrical line AB (Figure 2) is split into two radial lines (AC and CB). The splitting point is a terminus/turnpike in the city centre (C) or a stop that allows for the creation of a terminus/turnpike according to the spatial and technical conditions. The newly created radial lines have the same headway and capacity as the diametrical line (AB).  
Phase II: In the second phase, a simulation of the headway disturbance propagation is carried out for both variants. The deterministic model shown in Equation (1) is adopted for this purpose, as it is based on the relationship between the passenger arrival rate and the passenger boarding rate [1; 2; 39; 76]. In the simulation, the same conditions for the propagation of headway disturbances are used for both variants. After line splitting, the vehicle departures on the radial lines from the common terminus are not coordinated, so the disturbance from the first radial line would not be transmitted to the second line and vice versa.  
\[h_{\rm n}=h_{\rm p}\bigg{(}1+\frac{\lambda}{\mu}\bigg{)}^{n-1} \tag{1}\]  
where \(h_{\rm n}\) is the headway disturbance for the \(n^{th}\) stop (min); \(h_{\rm p}\) is the primary headway disturbance (min); \(n\) is the number of sequential stops in relation to the primary disturbance location; \(\lambda\) is the mean passenger boarding intensity (prs/min); and \(\mu\) is the passenger accumulation intensity along the line (prs/min).  
Phase III: In the third phase, the headway regularity is evaluated for both variants. Quantifying headway regularity is a complex process that requires spatio-temporal analysis of many vehicle departures simultaneously. For this purpose, the Percentage Regularity Deviation Mean (PRDM) is commonly adopted (Equation (2)). PRDM is a popular index for describing bus service regularity and was first used by [PERSON] and [PERSON] [77] and was later adopted by other authors [30; 78; 79; 80; 81]. A lower PRDM value indicates better bus service regularity.  
\[PRDM_{\rm n}=\frac{\sum_{i=1}^{\rm N}\left|\frac{H_{\rm n}-H_{\rm i}}{H_{\rm s }}\right|}{N} \tag{2}\]  
where \(PRDM_{\rm n}\) is the percentage regularity deviation mean of the headway related to the \(n^{th}\) stop; \(H_{\rm i}\) is the headway value for the \(i^{th}\) vehicle (min); \(H_{\rm s}\) is the scheduled headway (min); \(N\) is the number of vehicles crossing at the \(n^{th}\) stop during the studied time interval.  
Phase IV: In the fourth phase, the passenger travel time is evaluated based on the newly developed model that outputs the difference in passenger travel time between the two variants.  
Figure 1: The optimisation procedure framework in its entirety.