Planning-level optimisation of headway regularity

Planning-level optimisation of headway regularity | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Planning-level optimisation of headway regularity Pavle Pitka, Milan Simeunović, Milica Miličić, Tatjana Kovačević, and 1 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3993565/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Headway variability has a negative impact on the public transport passengers' perception of service quality. However, most of the existing methods aimed at improving the headway regularity operate in real time and require precise vehicle location data, making it difficult to implement them in practice. On the other hand, planning-level methods can be used to increase the resilience of public passenger transport (PPT) to the accumulation of headway disturbances. As this is typically done from the operator's perspective, the passengers' perspective tends to be overlooked, motivating the current work. In this article, an optimisation procedure for evaluating the viability of diametrical line splitting in terms of passenger travel time and headway regularity is proposed. The aim is to increase the robustness/resistance of the PPT system to the propagation of headway disturbances without reducing the service quality. The developed optimisation procedure was validated by applying it to real data pertaining to an urban PPT line. The results show that there is a positive correlation between the transport demand and the effects of the optimisation procedure, whereby an increase in the primary headway disturbance increases the sensitivity of the optimisation procedure to the transport demand. public passenger transport headway regularity disturbance propagation travel time Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 1 INTRODUCTION Public passenger transport (PPT) service operation reliability has a great influence on the passengers’ perception of service quality. Ample body of evidence indicates that headway variability has a negative impact on the PPT operational functioning, as it results in increased cycle time and delays in the next vehicle departure, as well as in vehicle loading above capacity and vehicle bunching (Newell and Potts 1964 ; Vuchic 1969 ; Turnquist and Bowman 1980 ). This operational variability also exerts a negative effect on the passengers’ journey experience, as it increases the expected waiting times (Osuna and Newell 1972 ; Delgado et al. 2009 ; Berrebi et al. 2015 ) and travel time uncertainty (Chang 2010 ; Duran-Hormazabal and Tirachini 2016) while reducing passenger comfort in the vehicle (Siemunović et al. 2012; Tirachini et al. 2013 ; Babaei et al. 2014 ; Cats et al. 2016 ). Accordingly, maintaining regular headways and consistent travel times are the key attributes of reliable PPT services (Chen et al. 2009 ; El-Geneidi et al. 2011; Berrebi et al. 2015 ; Munoz et al. 2020). In practice, however, effectively managing headway variability and vehicle bunching requires expert knowledge, data, technology and driver engagement (Tirachini et al. 2022 ). Consequently, many control methods have been developed to assist with this process and mitigate the negative effects of vehicle bunching with the aim of improving PPT service quality. Most of the methods developed at the operational level regulate the real-time movement of vehicles (vehicle holding, skipping stops, limited passenger access, earlier vehicle turning), while very few focus on increasing the robustness/resistance of the PPT system to the emergence and expansion of headway disturbances. In the latter case, this is typically achieved through prioritisation of PPT vehicles at signalised intersections (Diakaki et al. 2003 , Furth and Muller 2000 ; Kraus et al. 2010 ), line length optimisation (Levinson 2005 ; Chen et al 2009 ; van Oort and van Nes 2009 ), line splitting (Chen et al. 2009 ), optimising buffer time and cycle time (Newell 1977 ; Carey 1998 ; Zhao et al. 2006 ), introducing a PPT lane (Shalaby 1999 ; Nash 2003 ), and increasing passenger boarding intensity (Milkovits 2008 ; Sun et al. 2014 ). Lines with a greater number of stops create more opportunities for headway disturbances to accumulate (Chen et al. 2009 ; Pitka et al. 2017 ). On the other hand, owing to the technical − spatial conditions in the city centre (terminus), splitting diametrical lines into two radial lines can significantly improve the robustness of the PPT system to the propagation of headway disturbances, while also enhancing the headway regularity. However, line splitting also has some drawbacks, as it may result in journey disruption for a certain number of passengers who are required to make an additional transfer. In such cases, the splitting process can significantly increase the passenger travel time and reduce service quality. To examine these issues further, an optimisation procedure is proposed in this paper, allowing the practical utility of diametrical line splitting to be evaluated from the perspective of passenger travel time and headway regularity, with the aim of increasing the robustness/resistance of the PPT system to the propagation of headway disturbances. The effectiveness of this strategy is evaluated by applying it to real data from an urban PPT line. 2 LITERATURE REVIEW Welding ( 1957 ) was one of the first researchers to analyse headway variability. As a part of his work, Welding studied the operation of buses and trains in London, aiming to identify the causes and effects of headway variability as well as factors that lead to bus bunching on the line. This research motivated Newell and Potts ( 1964 ) to examine the movement of buses when a bus traveling on the same line arrives late at a stop. Their findings revealed that headway disturbances exert two basic effects on bus movement—delays resulting in late arrival at the stop, along with the tendency of vehicles following the late vehicle to move ahead—resulting in vehicle bunching. In contrast to previous research (Newell and Potts 1964 ; Vuchic 1969 ), Turnquist and Bovman (1980) emphasised the significant influence of vehicle movement variation between stops on vehicle bunching. According to these authors, the generated headway disturbance is propagated by stops along the line as well as by scheduled vehicles. Subsequent research confirms these findings, indicating that, if headway regularity is not actively controlled, vehicle bunching may occur (Chen et al. 2009 ; Feng and Figliozzi 2011 ; Byon et al. 2018 ; Iliopoulou et al. 2018 ). It has been empirically demonstrated in several studies that vehicle bunching is primarily caused by variability in vehicle movements at the beginning of the route, vehicle frequency, number of passengers boarding or alighting from the vehicle, boarding intensity, and number of stops/line length (Vuchic 1969 ; El-Geneidy et al. 2011 ; Diab et al. 2015 ; Pitka et al. 2017 ; Arriagada et al. 2019 ; Soza-Parra et al. 2021 ). Given the adverse impacts of vehicle bunching on the PPT service quality, many control methods have been developed to mitigate this issue, which can be broadly divided into two groups (Zolfaghari et al. 2004 ): Real-time strategies − operational-level methods aimed at increasing service quality by introducing additional vehicles to the line, stop skipping, and/or vehicle holding Strategies implemented at the planning level − long-term strategies that require route and timetable information, including any changes. Well-designed management methods can simultaneously reduce users' travel time and operational costs (Bueno-Cadena and Munoz 2017), which is typically achieved through vehicle holding based on either schedule or headway (Delgado et al. 2009 ). The schedule holding method involves holding vehicles at stops and/or additional control points that are ahead of schedule. While this method is effective in mitigating vehicle bunching, it also reduces the PPT operating speed. On the other hand, the aim of the headway-based holding method is maintaining the designed headway. Osuna and Newell ( 1972 ) were the first to apply this method to a circular route in an idealised environment. These authors analysed the movement of two buses using a single control point and concluded that bunching could be adequately mitigated using this approach. In the subsequent period, several attempts have been made to improve the vehicle holding method by applying it at only one control point (Eberlein et al. 2001 ; Fu and Yang 2002 ), or at several control points along the corridor (Sun and Hickman 2005 ; Bartholdi and Eisenstein 2012 ; Chen et al. 2013 ). In a number of the proposed models, the vehicle capacity is limited, i.e., passengers cannot board the vehicle once it has reached the maximum capacity (Zolfaghari 2004; Delgado et al. 2009 ; Cortés et al. 2010 ; Delgado et al. 2012 ). More recently, motivated by the advances in machine learning technologies, Wang and Sun ( 2020 ) adopted a reinforcement learning framework to determine the most optimal vehicle holding time. In addition to vehicle holding, other methods have been developed to manage headway variability in real time, whereby the most commonly studied methods are based on allowing buses to overtake each other (Wu et al. 2017 ; Sun and Schmöcker 2018 ), regulating speed (Chandrasekar et al. 2002 ; Daganzo and Pilachowski 2011 ; Munoz et al. 2013), or skipping stops (Nagatani 2001 ; Fu et al. 2003 ; Sun and Hickman 2005 ; Cortés et al. 2010 ; Wu et al. 2019 ; Larrain and Muñoz 2020). Considering that operations at stops have a large impact on vehicle bunching, a significant number of authors have opted to focus on reducing the vehicle dwell time by restricting boarding (Delgado et al. 2012 ; Bueno-Cadena and Munoz 2017), increasing the boarding intensity by allowing entry through all available gates (Vest and Cats 2017), or validating tickets before boarding (Ishak and Cats 2020). In a recent study, the authors did not directly restrict passenger boarding, but relied on passengers' desire for comfort and encouraged waiting for a less crowded vehicle. As a part of their work, Drabicki et al. ( 2023 ) examined the effect of providing passengers with real-time congestion information on vehicle bunching reduction. Similarly, Wu et al. ( 2017 ) demonstrated that, when passengers are sufficiently informed and can choose which vehicle to use, this not only improves the journey experience but also reduces vehicle bunching. In one of the recent studies dealing with the control strategy based on real-time information, Zhou et al. ( 2022 ) identified a significant gap between the recommendations based on recent theories and modern practice. According to these authors, closing this gap requires the adoption of appropriate technology and software for continuous communication between the control centre and the driver. On the other hand, Levinson ( 2005 ) examined the main contributors to headway reliability and reached the following conclusions: (I) bus lines with long routes, high occupancy, and mixed traffic have very low reliability; (II) a headway disturbance that occurs at the beginning of the line tends to propagate along the line; (III) accuracy is important for lines with long headways, whereas precedence should be given to uniformity for lines with short headways; (IV) service reliability and transport speed on bus lines can be improved by reducing the number of stops on the line, the dwell time, and the impact of mixed traffic on vehicle movement between stops. According to the available literature, the influence of line length and number of stops on headway uniformity and service reliability remains insufficiently studied (Levinson 2005 ; van Oort 2011 ; Soza-Parra et al. 2021 ). Using a case study in the Hague, van Oort ( 2011 ) investigated the effect of line length on service reliability. The author split the routes of longer lines and analysed the resulting service reliability and passenger travel time based on real data. The obtained results indicate that splitting the line increases service reliability, while reducing passenger waiting time at the bus stop as well as time spent in the vehicle. The author further noted that the time loss due to headway variability can be reduced by about 30% when the line is split at a stop with low passenger flow. Increasingly, however, technological advances are being harnessed to deal with headway disturbances in real time. In particular, methods based on the Automatic Vehicle Location (AVL) systems are being proposed despite their limited practical utility due to the lack of appropriate technology and software. Therefore, several authors advocate for the greater reliance on planning-level strategies as a means of increasing the PPT resistance to headway disturbances. While in most cases this approach has yielded positive results, the proposed methods are typically developed from the operator's perspective, offering limited insight into the effectiveness of these strategies from the passengers’ point of view. The work presented in this article attempts to fill these gaps. This is achieved by presenting a practical optimisation procedure that is implemented at the planning level, and its positive effects are evaluated on the basis of the average passenger travel time. 3 METHODS In this section, the optimisation procedure, along with the model for evaluating the impact of line splitting on average passenger travel time, is described in detail. Due to its complexity, the optimisation procedure is divided into four phases and is subject to the following limitations: The scheduled headway must be less than 15 minutes Terminal time can be used for delay recovery (Vuchic 2017 ), schedule adjustment (maintaining uniform headway) to prevent disruptions being passed on to the next departure of the same vehicle The existence of a terminus/turnpike at the line splitting point is mandatory Each line obtained by splitting, considered separately, must represent a functional unit that meets the passengers’ needs. By splitting the diametrical line into two radials, two components of the passenger travel time are affected. First, as a benefit of optimisation, the passenger waiting time at the stop is reduced due to the improved headway regularity. Second, passengers who cross the splitting point have to make an additional transfer, which increases their travel time. This is a notable drawback of the optimisation. Given that these time components have opposite effects on journey duration and that a large number of parameters influence their value, a model was developed to evaluate the influence of line splitting on average passenger travel time. 3.1 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 Fig. 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 (Fig. 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 (Fig. 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 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 Eq. 1 is adopted for this purpose, as it is based on the relationship between the passenger arrival rate and the passenger boarding rate (Newell and Potts 1964 ; Vuchic 1969 ; Daganzo 2009 ; Pitka et al. 2017 ). 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 that the disturbance from the first radial line would not be transmitted to the second line and vice versa. $${h}_{\text{n}}={{h}_{\text{p}}\left(1+\frac{\lambda }{\mu }\right)}^{n-1}$$ 1 where h n – headway disturbance for the n th stop (min); h p – primary headway disturbance (min); n – the number of sequential stops in relation to the primary disturbance location; λ – mean passenger boarding intensity (prs/min); and µ – 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 a large number of vehicle departures simultaneously. For this purpose, the Percentage Regularity Deviation Mean (PRDM) is commonly adopted (Eq. 2 ). PRDM is a popular index for describing bus service regularity and was first used by Hakkesteegt and Muller ( 1981 ) and was later adopted by other authors (Sorratini et al. 2008; van Oort and van Nes 2009 ; van Oort 2014 ; Simeunović et al. 2016 ; Zhang et al. 2018 ). A lower PRDM value indicates better bus service regularity. $${PRDM}_{\text{n}}=\frac{\sum _{\text{i}=1}^{\text{N}}\left|\frac{{H}_{\text{s}}-{H}_{\text{i}}}{{H}_{\text{s}}}\right|}{N}$$ 2 where PRDM n – percentage regularity deviation mean of the headway related to the n th stop; H i – headway value for the i th vehicle (min); H s – scheduled headway (min); N – 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. 3.2 MODEL DEVELOPMENT According to Vuchic ( 2017 ), passenger travel time from the origin to the destination consists of t od = t a + t w + t ol + t f + t a' , where t od – origin–destination passenger travel time (min); t a – the time required for reaching the PPT stop (min); t w – waiting time at the stop (min); t ol – on-line travel time (min); t f – transfer time (min); t a' – the time required for covering the distance from a PPT stop to the final destination (min). The key parameters for this model are: waiting time, on-line travel time and transfer time. As noted by Bowman and Turnquist ( 1981 ), the expected passenger waiting time is related to both the distribution of passenger arrival times at a stop and the distribution of schedule deviations in bus arrival times at that stop. For short headways, several authors assume that passengers arrive at random times, independent of bus arrival schedules (Osuna and Newell 1972 ; Bowman and Turnquist 1981 ; Vuchic 2017 ). Under such conditions, headway regularity along the line is extremely important. When headway regularity is poor, the average passenger waiting time ( t w ) depends on the scheduled headway ( H s ) and headway disturbances ( h ), as indicated by Eq. 3 : $${t}_{\text{w}}=\frac{{H}_{\text{s}}+h}{2}$$ 3 The time a passenger spends in a PPT vehicle – on-line time ( t ol ) – is the sum of the vehicle dwell time at the stop ( t d ) and the vehicle running time between adjacent stops ( t r ), i.e., t ol = ∑ t d + ∑ t r . According to the extant studies in this domain, dwell time is determined by the required passenger boarding time (Daganzo 2009 ; Bellei and Gkoumas 2010 ; Vuchic 2017 ). Hence, if there is a headway disturbance, the dwell time can be defined by Eq. 4 : $${t}_{\text{d}}=\frac{\lambda }{\mu }\left({H}_{\text{s}}+h\right)$$ 4 In the model developed as a part of this work, transfer time is calculated in the same way as passenger waiting time (Eq. 3 ) because the walking distance between two radial lines is negligible and vehicle departures are uncoordinated. In order to evaluate the passenger travel time, a deterministic model was first created for one passenger's journey and was subsequently extended to all passengers on the line. The model calculates the difference in passenger travel time between the two previously described variants. On line AB, the passenger journey is observed from the time of their arrival at stop k to their departure at stop m (Fig. 2 ). The model does not consider the total passenger travel time, as it focuses solely on the time the passenger spends in the PPT (excluding the time spent walking to/from the stop). If the transport is realised with a single diametrical line, the time the passenger spends in the PPT ( T D ) is expressed by Eq. 5 : $${T}_{D}={t}_{\text{w}}+\sum {t}_{\text{d}}+\sum {t}_{\text{r}}$$ 5 In the case of a headway disturbance, determination of the time the passenger spends in the PPT requires consideration of several basic parameters that define the headway disturbance propagation, as shown in Eq. 6 . The expression in Eq. 6 is obtained by incorporating Eq. 3 and Eq. 4 into Eq. 5 . $${T}_{\text{D}}=\frac{{H}_{\text{s}}+{h}_{\text{k}}}{2}+\sum _{j=k+1}^{m-1}\left({H}_{\text{s}}+{h}_{\text{j}}\right)\left(\frac{{\lambda }_{\text{j}}}{\mu }\right)+\sum _{j=k}^{m-1}{t}_{\text{r}\text{j}}$$ 6 To simplify the expressions in Eq. 6 , we modify β j = λ j / µ and H j = H s + h j , where β j is the ratio of passenger boarding intensity and passenger accumulation intensity at the j th stop, and H j is the actual headway at the j th stop, resulting in Eq. 7 : $${T}_{\text{D}}=\frac{{H}_{\text{k}}}{2}+\sum _{j=k+1}^{m-1}{H}_{\text{j}}{\beta }_{j}+\sum _{j=k}^{m-1}{t}_{\text{r}\text{j}}$$ 7 If the transport is realised with two radial lines, Eq. 5 is expanded for the transfer time. As before, the time the passenger spends in the PPT ( T 2R ) comprises several components and is defined by Eq. 8 : $${T}_{2\text{R}}={t}_{\text{w}}+\sum _{k+1}^{l-1}{t}_{\text{d}}+\sum _{k}^{l-1}{t}_{\text{r}}+{t}_{\text{f}}+\sum _{l+1}^{m-1}{t}_{\text{d}}+\sum _{l}^{m-1}{t}_{\text{r}}$$ 8 Once again, in the case of a headway disturbance, the time the passenger spends in the PPT (two radial lines) is described by the basic parameters that define the headway disturbance propagation (Eq. 9 ). Accordingly, Eq. 9 is obtained by expressing Eq. 8 via Eq. 3 and Eq. 4 . $${T}_{2\text{R}}=\frac{{H}_{\text{k}}}{2}+\sum _{j=k+1}^{l-1}{H}_{\text{j}}{\beta }_{j}+\sum _{j=k}^{l-1}{t}_{\text{r}\text{j}}+\frac{{H}_{\text{l}}^{\prime }}{2}+\sum _{j=l+1}^{m-1}{H}_{\text{j}}^{\prime }{\beta }_{\text{j}}+\sum _{j=l}^{m-1}{t}_{\text{r}\text{j}}$$ 9 where H` j is the actual headway at the j th stop on the second radial line and is given by H` j = H s + h` j . The difference in passenger travel time between the two variants (Eq. 10 ) is obtained by subtracting Eq. 9 from Eq. 7 . $${T}_{2\text{R}-\text{D}}=\frac{{H}_{\text{l}}}{2}-{H}_{\text{l}}{\beta }_{\text{l}}-\sum _{j=l+1}^{m-1}\left({h}_{\text{j}}-{h}_{\text{j}}^{\prime }\right){\beta }_{\text{j}}$$ 10 From Eq. 7 , Eq. 9 and Eq. 10 , it can be concluded that the influence of line splitting on each passenger’s travel time depends on the position of the boarding and alighting stops on the line. By splitting the diametrical line into two radial ones, three characteristic areas have been created, which have different effects on passenger journey durations. Considering the direction of travel from terminus A to terminus B, the following sections can be distinguished (Fig. 3 ): Section from A to C (section AC) Line splitting point (section C) Section from C to B (section CB) Depending on the sections defined above and the locations of stops at which passengers enter and leave the line, three distinct passenger groups can be recognised: 1. Passengers travelling on the AC section only and not crossing the line split point (C). As this group is not affected by the line splitting, they experience no change in their travel time. 2. Passengers travelling on section CB only. This group will experience only the positive effects of line splitting, as their waiting time at the stop and the driving time will be reduced. 3. Passengers crossing the line splitting point (C) during their journey will note both positive and negative effects on their journey duration. While their travel time will be shortened, they will be required to transfer to another vehicle at the line splitting point (C), which will lead to an additional transfer time and greater travel time uncertainty. The number of passengers transported between stops represents the passenger flow ( q ), while the number of passengers entering ( U ) at a stop “j” is expressed by U j = ( H s + h j ) λ j = H j λ j . According to the previously presented arguments, when the transport is realised with the diametrical line, the total spent time of all passengers ( TT D ) in the PPT on line AB is defined by Eq. 11 : $${TT}_{\text{D}}=\sum _{j=1}^{n-1}\frac{{H}_{j}^{2}}{2}{\lambda }_{\text{j}}+\sum _{j=2}^{n-1}{\beta }_{\text{j}}{H}_{\text{j}}{q}_{\text{j}-1}+\sum _{j=1}^{n-1}{t}_{\text{r}\text{j}}{q}_{j}$$ 11 If the transport is realised with two radial lines, the total spent time of all passengers ( TT 2R ) in the PPT on line AB is defined by Eq. 12 : $$T{T}_{2\text{R}}=\sum _{j=1}^{l-1}\left(\frac{{H}_{j}^{2}}{2}{\lambda }_{\text{j}}+{t}_{\text{r}\text{j}}{q}_{j}\right)+\sum _{j=2}^{l-1}{\beta }_{\text{j}}{H}_{\text{j}}{q}_{\text{j}-1}+\frac{{H}_{s}}{2}\left({\lambda }_{\text{l}}{H}_{\text{l}}^{\prime }+{q}_{l}\right)+\sum _{j=l+1}^{n-1}\left(\frac{{H}_{j}^{` 2}}{2}{\lambda }_{\text{j}}+{\beta }_{\text{j}}{H}_{\text{j}}^{\prime }{q}_{\text{j}-1}\right)+\sum _{j=l}^{n-1}{t}_{\text{r}\text{j}}{q}_{j}$$ 12 The difference in the time passengers spend in the PPT ( TT 2R − D ) between the first and the second variant (Eq. 13 ) is obtained by subtracting Eq. 11 from Eq. 12 . $${TT}_{2\text{R}-\text{D}}=\sum _{j=l+1}^{n-1}\left(\frac{{H}_{j}^{`2}}{2}{\lambda }_{\text{j}}+{\beta }_{\text{j}}{H}_{j}^{\prime }{q}_{\text{j}-1}\right)+\frac{{H}_{s}}{2}\left({\lambda }_{\text{l}}{H}_{\text{l}}^{\prime }+{q}_{l}\right)-\sum _{j=l}^{n-1}\left(\frac{{H}_{j}^{2}}{2}{\lambda }_{\text{j}}+{\beta }_{\text{j}}{H}_{\text{j}}{q}_{\text{j}-1}\right)$$ 13 Based on the expression given above, the average passenger spent time represents the arithmetic mean of the total spent time of all passengers in the PPT. 4 CASE STUDY In accordance with the model limitations, the practical implementation was carried out in the PPT (Novi Sad, Serbia) on a diametrical line (Line 3: Petrovaradin − Centar − Detelinara) that connects the boroughs of Detelinara and Petrovaradin, and its route passes through the Novi Sad city centre. The practical application was carried out for two characteristic periods, whereby the first represents the line operation mode with maximum load (peak hour, 1:00 − 2:00 pm), while the second represents the line operation mode with the lowest load (off-peak hours, 9:00 − 10:00 am). 4.1 Diametrical line splitting (Phase I) On the basis of the static elements of Line 3, the Uspenska−Šafarikova stop (Stop No. 11 in direction A, and Stop No. 10 in direction B, show in Table 1 ) was determined as the intersection point of the diametrical line. This bus stop is located in the city centre and is designed as a terminus. By dividing the diametrical line at a defined point, two radial lines with a common terminus are created (Fig. 4 ): the Petrovaradin − Centar and Centar − Detelinara lines. The length of the new line Petrovaradin − Centar is 5.0 km in direction A and 4.9 km in direction B, with 11 stops in each direction. The second radial line (Centar − Detelinara) is 4.05 km and 3.8 km long in direction A and B, respectively. This line has 9 stops in direction A and 10 stops in direction B. 4.2 Simulation of the headway disturbance propagation (Phase II) In the optimisation procedure, a simulation of the headway disturbance propagation was performed for both variants, using the real collected data (Table 1 and Table 2 ), along with different primary headway disturbance values. Table 1 Passenger boarding and alighting at stops Stop No. Direction A Direction B 9:00 − 10:00 am 1:00 − 2:00 pm 9:00 − 10:00 am 1:00 − 2:00 pm Boarding Alighting Boarding Alighting Boarding Alighting Boarding Alighting 1 43 0 70 0 72 0 81 0 2 59 1 123 2 23 0 73 1 3 19 1 36 5 36 1 74 15 4 16 2 18 3 49 15 71 23 5 26 6 20 12 25 18 49 39 6 25 7 38 70 27 9 17 15 7 8 0 22 5 21 20 25 11 8 35 5 39 7 53 41 63 59 9 29 37 81 44 42 41 39 34 10 47 74 132 99 35 66 66 105 11 45 43 70 84 34 71 87 134 12 33 72 58 60 19 47 57 61 13 36 73 69 92 4 21 7 24 14 11 18 14 58 11 22 4 33 15 12 31 35 61 4 15 50 49 16 3 26 15 79 14 15 14 38 17 2 18 2 49 1 18 9 32 18 4 27 1 79 3 14 2 55 19 0 29 0 47 0 31 2 36 20 - - - - 0 18 0 27 Table 2 Input data for headway disturbance propagation simulation Parameter Values Time period 9:00 − 10:00 am 1:00 − 2:00 pm Projected headway (min) 10 8.3 Average passenger boarding intensity (prs/min) 14.3 Primary headway disturbance (min) 0 − 4 For both variants, the same values of the primary headway disturbance were used in the simulation procedure while the disturbance duration ranged from 0 to 4 minutes. In the case of the optimised variant, headway disturbance propagation is interrupted at the splitting point, after which the accumulation starts from the ideal state of headway regularity (Fig. 5 ). 4.3 Evaluation of headway regularity (Phase III) The evaluation of headway regularity was carried out for different combinations of primary headway disturbance, based on the simulations carried out in the second phase (Fig. 5 ). In Fig. 6 and Fig. 7 , the hypothetical primary headway disturbance is given on the x-axis and the realised PRDM values are presented on the y-axis. It is evident from the graphs that the primary headway disturbance has a linear effect on the PRDM, and that an improvement in the degree of regularity was achieved during peak as well as off-peak hours. The optimisation effects also depend on the line direction and the observation period. The greatest improvements in headway regularity are achieved during the peak period, when the following headway is the shortest and the number of boarding passengers per stop is the greatest (Fig. 6 and Fig. 7 ). 4.4 Evaluation of passenger travel time (Phase IV) The evaluation of passenger travel time in the PPT was performed for different primary headway disturbance combinations, based on the difference in the passenger travel time. Figure 8 shows the difference in average passenger travel times for the two analysed variants. The data are presented for two observation periods and both line directions. If the difference in passenger travel time is less than zero, the optimised (split) line design (the second variant) is more efficient than the current (diametrical) line (the first variant). The observed optimisation effects, in terms of average passenger travel times, vary significantly depending on the observation period and the primary headway disturbance duration. The reduction in average passenger travel times is only achieved during the peak period when the primary headway disturbance on the diametrical line is at least 2 minutes. If the primary headway disturbance in the peak period is 4 minutes, the optimisation achieves a 1.5-minute and 1.8-minute reduction in travel time in direction A and B, respectively. In the optimised variant for the off-peak period, the average travel time would be up to 4 minutes longer irrespective of the primary headway disturbance value. 5 SENSITIVITY ANALYSIS As the optimisation procedure yields different results for peak and off-peak periods, which can be distinguished by the headway and the transport demand, these parameters were considered in the sensitivity analysis. First, the sensitivity of the optimisation procedure was examined by simulating a change in transport demand by decreasing/increasing the number of passengers boarding and alighting at each stop by 10% and 20%, resulting in the five cases shown in Table 3 . For each case, the impact of transport demand on travel time and headway regularity was analysed. Table 3 Passenger boarding and alighting at stops per case Case Percentage decrease/increase in the number of passengers 1 -20% 2 -10% 3 0% 4 + 10% 5 + 20% Sensitivity of headway regularity to changes in transport demand is obtained by the difference between PRDM of radial lines and diametrical line (Table 4 ). As can be seen from Fig. 9 , changes in transport demand have a significant impact on the headway regularity. In the example depicted on the graph, for a primary headway disturbance of 3 minutes after optimisation, a -0.184 reduction in PRDM is achieved in CASE 1, while − 0.219, -0.262, -0.306, and − 0.358 reduction is obtained for the remaining four cases. For the same values of primary headway disturbance, by increasing the transport demand, a better headway regularity is achieved, i.e., the optimisation procedure results in a greater PRDM reduction. Table 4 Sensitivity of headway regularity to changes in transport demand Variant Case Primary headway disturbance (min) 0 1 2 3 4 Diametrical line 1 0 0.282 0.556 0.821 1.094 2 0 0.256 0.600 0.891 1.208 3 0 0.352 0.652 0.996 1.269 4 0 0.370 0.722 1.057 1.400 5 0 0.413 0.783 1.153 1.531 Radial lines 1 0 0.212 0.434 0.637 0.850 2 0 0.204 0.451 0.672 0.911 3 0 0.265 0.486 0.733 0.946 4 0 0.265 0.521 0.751 0.998 5 0 0.291 0.547 0.795 1.068 Figure 10 and Table 5 show that changes in transport demand also have a significant impact on the passenger travel time. For example, if a primary disruption of 3 minutes occurs after optimisation, passengers will experience a -0.59 reduction in travel time in CASE 1, while − 0.80, -1.08, -1.31, and − 1.58 reduction is achieved in CASE 2 − 5. Correspondingly, with an increase in transport demand on the same line and with the same values of primary headway disruption, greater savings in passenger travel time are achieved. Table 5 Sensitivity of passenger travel time to changes in transport demand Case Primary headway disturbance (min) 0 1 2 3 4 1 1.55 0.61 0.03 -0.59 -1.23 2 1.52 0.84 -0.14 -0.80 -1.51 3 1.44 0.54 -0.26 -1.08 -1.69 4 1.48 0.46 -0.46 -1.31 -2.11 5 1.45 0.31 -0.65 -1.58 -2.40 Based on the aforementioned analysis, it can be concluded that the transport demand and the effects of the optimisation procedure (travel time and headway regularity) are positively correlated. Specifically, increasing the primary headway disturbance increases the sensitivity of the optimisation procedure to the transport demand. As a part of the sensitivity analysis, a simulation of headway was also carried out. Five cases were created for this purpose (Table 6 ) and the impact of the headway on travel time and headway regularity was analysed for each case. Table 6 Headway duration per case Case Headway (min) 6 6 7 7 8 8 9 9 10 10 Sensitivity of headway regularity to changes in headway is obtained by the difference between PRDM of radial lines and diametrical line (Table 7 ). It is evident from Fig. 11 that changes in headway duration have a significant impact on the headway regularity. For instance, if a primary disruption of 3 minutes occurs after optimisation, a -0.326 reduction in PRDM is achieved in CASE 6, while smaller reductions are attained in the remaining four cases (-0.280 in CASE 7, -0.262 in CASE 8, -0.225 in CASE 9, and − 0.182 in CASE 10). After applying the optimisation procedure, increasing the headway for the same values of the primary headway disturbance results in a worse headway regularity. That is, on lines with longer headway, the effects of optimisation procedure, as measured by PRDM, are worse. Table 7 Sensitivity of headway regularity to changes in headway Variant Case Primary headway disturbance (min) 0 1 2 3 4 Diametrical line 6 0 0.400 0.869 1.281 1.716 7 0 0.403 0.775 1.108 1.491 8 0 0.352 0.652 0.996 1.269 9 0 0.305 0.556 0.862 1.120 10 0 0.219 0.466 0.727 0.987 Radial lines 6 0 0.307 0.648 0.955 1.273 7 0 0.303 0.585 0.828 1.111 8 0 0.265 0.486 0.733 0.946 9 0 0.220 0.416 0.636 0.833 10 0 0.170 0.361 0.545 0.750 As shown in Table 8 and Fig. 12 , changes in headway also exert a significant influence on the passenger travel time. Using the earlier example, if a primary disruption of 3 minutes occurs after optimisation, passengers will experience a -1.31, -1.16, -1.08, -0.85, and − 0.56 reduction in travel time in CASE 6 − 10. Therefore, after applying the optimisation procedure, increasing the headway for the same values of the primary headway disturbance will result in a shorter passenger travel time. However, on lines with longer headways, the optimisation procedure will lead to smaller travel time savings. Table 8 Sensitivity of passenger travel time to changes in headway Case Primary headway disturbance (min) 0 1 2 3 4 6 1.12 0.28 -0.62 -1.31 -2.03 7 1.25 0.32 -0.47 -1.16 -1.90 8 1.44 0.54 -0.26 -1.08 -1.69 9 1.67 0.68 -0.01 -0.85 -1.49 10 1.78 1.08 0.32 -0.56 -1.24 Based on the results presented above, it can be surmised that there is a negative correlation between the headway values and the optimisation procedure effects. With respect to headway regularity, it is evident that increasing the primary headway disturbance increases the sensitivity of the optimisation procedure to the headway value, but not on passenger travel time. 6 CONCLUSION This research was carried out with the aim of creating a tool that could be applied in practice to evaluate the benefits/drawbacks of splitting a diametrical line from the point of view of passenger travel time and headway regularity. The application of the optimisation procedure presented here is foreseen in the system design phase, as it can contribute to the robustness/resistance of the PPT to the propagation of headway disturbances. Line splitting interrupts the accumulation of headway disturbances and improves headway regularity. However, evaluating the positive and negative effects of line splitting is a complex problem, considering that these effects are a function of the primary headway disturbance values, transport demand, actual headways, number of stops and other parameters. As a part of this work, an optimisation procedure was developed to determine the change in average passenger journey due to line splitting. Most of the currently available methods are based on mitigating the headway disturbances in real time, whereas the optimisation procedure developed as a part of the present study increases the PPT resistance to the propagation of headway disturbances in the design stage. This is also the first attempt to develop a model that can determine the change in average passenger travel time owing to line splitting. The validity of the optimisation procedure was confirmed on a real example, which indicated that line splitting is beneficial during peak periods, as it results in significantly less primary headway disturbances. Moreover, the sensitivity analysis focusing on headway and transport demand demonstrated that the transport demand is positively correlated to the effects of the optimisation procedure, whereby increasing the primary headway disturbance increases the sensitivity of the optimisation procedure to the transport demand. On the other hand, there is a negative correlation between the headway duration and the optimisation procedure effects. Specifically, the sensitivity of the optimisation procedure to headway increases as the primary disturbance value increases. However, this effect is only observed for headway regularity, while the sensitivity does not increase when passenger travel time is observed. These analyses confirm the validity of the proposed model, but also indicate that the optimisation procedure (line splitting) is most beneficial for lines with short headway and high transport demand where the primary headway disturbances exceed 2 minutes. While these findings are certainly beneficial, the proposed optimisation procedure can be further enhanced by introducing additional criteria such as the load factor, allowing for multi-criteria evaluation. In addition, as the value and location of the headway disturbances is in practice rarely known with certainty, the existing model could be extended by incorporating a fuzzy logic approach. Finally, authors of future studies in this domain could develop a hybrid method that would combine the proposed optimisation procedure at the planning level with some of the models for increasing headway regularity in real time. DECLARATIONS Funding: This research has been supported by the Ministry of Science, Technological Development and Innovation (Contract No. 451-03-65/2024-03/200156) and the Faculty of Technical Sciences, University of Novi Sad through project “Scientific and Artistic Research Work of Researchers in Teaching and Associate Positions at the Faculty of Technical Sciences, University of Novi Sad” (No. 01-3394/1). Data Availability: The manuscript contains data supporting the results of this study. Author Contribution: Conceptualization, P.P. and T.K.; Methodology, P.P. and T.K.; Validation, P.P. and M.M.; Investigation, P.P., M.S. and M.S.; Data Curation, P.P., M.S. and M.S.; Writing-Original Draft Preparation, P.P., M.S., M.M., T.K. and M.S.; Writing-Review & Editing, P.P., M.S., M.M., T.K. and M.S.; Visualization, P.P. and T.K. REFERENCES Ap. Sorratini, J., Liu, R., Sinha, S.: Assessing bus transport reliability using micro-simulation. Transport. Plan. Techn . 31 (3), 303-324 (2008). https://doi.org/10.1080/03081060802086512 Arriagada, J., Gschwender, A., Munizaga, M.A., Trépanier, M.: Modeling bus bunching using massive location and fare collection data. J. Intell. Transport. S. 23 (4), 332-344 (2019). https://doi.org/10.1080/15472450.2018.1494596 Babaei, M., Schmöcker, J.D., Shariat-Mohaymany, A.: The impact of irregular headways on seat availability. 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Kovačević","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA1ElEQVRIiWNgGAWjYHACZiCWANEHSNXCxpZAkhYgYOMxIE69fPsZY4OPbRbyDPI936QL/jDI8zfwPv6AT4vBmRzjxJltEoYNbLzbpGe2MRjOOMBuJoFXC0Na8mGeMxIJDCAtvA0MjBuAvsLvsP5nMC08z6R5/jDYA7Uw43UYw43kw8k8FWAtbNI8bAyJQC0M+B124/FhwxkVEoZtbGnG1rxtEskzDrOx4dUi35/YLPHBoE6en/nww9s8f2xs+9vbCDgMBqBeloDH0ygYBaNgFIwCCgAAdfI3LzRMYmkAAAAASUVORK5CYII=","orcid":"","institution":"Faculty of Technical Sciences, University of Novi Sad","correspondingAuthor":true,"prefix":"","firstName":"Tatjana","middleName":"","lastName":"Kovačević","suffix":""},{"id":275744249,"identity":"3d965d02-501a-41f0-a9a1-7af2219f69d5","order_by":4,"name":"Milja Simeunović","email":"","orcid":"","institution":"Faculty of Technical Sciences, University of Novi Sad","correspondingAuthor":false,"prefix":"","firstName":"Milja","middleName":"","lastName":"Simeunović","suffix":""}],"badges":[],"createdAt":"2024-02-27 10:15:02","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3993565/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3993565/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":52023814,"identity":"ed232f38-7453-4e06-a483-1664ce6ebce9","added_by":"auto","created_at":"2024-03-05 15:33:45","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":767673,"visible":true,"origin":"","legend":"\u003cp\u003eThe optimisation procedure framework in its entirety\u003c/p\u003e","description":"","filename":"Fig1.png","url":"https://assets-eu.researchsquare.com/files/rs-3993565/v1/90b900e7b44a208b8690b0f9.png"},{"id":52023811,"identity":"58a94308-2850-4c51-8763-4057905f497d","added_by":"auto","created_at":"2024-03-05 15:33:44","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":70794,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic representation of passenger travel along the route\u003c/p\u003e","description":"","filename":"Fig2.png","url":"https://assets-eu.researchsquare.com/files/rs-3993565/v1/e780bbfb9a17baed7171cc1c.png"},{"id":52024627,"identity":"3d9502fb-06b2-4fe1-8bbb-e020fa23e80b","added_by":"auto","created_at":"2024-03-05 15:41:45","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":114241,"visible":true,"origin":"","legend":"\u003cp\u003eLine characteristic areas\u003c/p\u003e","description":"","filename":"Fig3.png","url":"https://assets-eu.researchsquare.com/files/rs-3993565/v1/cc178d8f36b76e4413b73b87.png"},{"id":52023813,"identity":"583f53f9-886d-4fe1-b52d-812a848ae467","added_by":"auto","created_at":"2024-03-05 15:33:45","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":617507,"visible":true,"origin":"","legend":"\u003cp\u003eMap of the routes of two new lines created by splitting Line 3\u003c/p\u003e","description":"","filename":"Fig4.png","url":"https://assets-eu.researchsquare.com/files/rs-3993565/v1/090371c27e55030724472991.png"},{"id":52023815,"identity":"39a78bd2-8e86-4749-8348-89ec4d708201","added_by":"auto","created_at":"2024-03-05 15:33:45","extension":"jpg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":501199,"visible":true,"origin":"","legend":"\u003cp\u003eSimulated headway\u003c/p\u003e","description":"","filename":"5.jpg","url":"https://assets-eu.researchsquare.com/files/rs-3993565/v1/a38e750be1e6e1ce6cbf0fd2.jpg"},{"id":52023818,"identity":"5cdc3665-3321-4844-b62f-a4acb1b0cb4a","added_by":"auto","created_at":"2024-03-05 15:33:45","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":109582,"visible":true,"origin":"","legend":"\u003cp\u003eEvaluation of headway regularity for the off-peak period (9:00−10:00 am)\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-3993565/v1/fb97f1eee2d4ff607378737a.png"},{"id":52023817,"identity":"d2c261fe-3aaf-4e38-bb69-5ea696602d76","added_by":"auto","created_at":"2024-03-05 15:33:45","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":129336,"visible":true,"origin":"","legend":"\u003cp\u003eEvaluation of headway regularity for the peak period (1:00−2:00 pm)\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-3993565/v1/3d0f94e3add8da1ef21b782e.png"},{"id":52024628,"identity":"a1c6898a-c274-4f54-80e6-e32d8557bed3","added_by":"auto","created_at":"2024-03-05 15:41:45","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":117052,"visible":true,"origin":"","legend":"\u003cp\u003eDifference in the average passenger travel time\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-3993565/v1/6550c6bcc86cdd729eadbf75.png"},{"id":52023821,"identity":"c16e7d12-4977-4e59-87d1-8f25c3e05639","added_by":"auto","created_at":"2024-03-05 15:33:45","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":138365,"visible":true,"origin":"","legend":"\u003cp\u003eSensitivity of headway regularity to changes in transport demand\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-3993565/v1/0c326e827b5788192960261e.png"},{"id":52024629,"identity":"999d5d86-e6fe-487e-bd19-b47a61ba0981","added_by":"auto","created_at":"2024-03-05 15:41:45","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":149026,"visible":true,"origin":"","legend":"\u003cp\u003eSensitivity of passenger travel time to changes in transport demand\u003c/p\u003e","description":"","filename":"10.png","url":"https://assets-eu.researchsquare.com/files/rs-3993565/v1/d216876823a78d00453d4202.png"},{"id":52023819,"identity":"efaa6c9d-474d-46f2-93bb-56041f04b11f","added_by":"auto","created_at":"2024-03-05 15:33:45","extension":"png","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":165067,"visible":true,"origin":"","legend":"\u003cp\u003eSensitivity of headway regularity to changes in headway\u003c/p\u003e","description":"","filename":"11.png","url":"https://assets-eu.researchsquare.com/files/rs-3993565/v1/c82ef8cb95cf6bdf7b4fb640.png"},{"id":52023822,"identity":"74947bbe-f677-47d0-81f4-163d2dcbd2bd","added_by":"auto","created_at":"2024-03-05 15:33:45","extension":"png","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":194188,"visible":true,"origin":"","legend":"\u003cp\u003eSensitivity of passenger travel time to changes in headway\u003c/p\u003e","description":"","filename":"12.png","url":"https://assets-eu.researchsquare.com/files/rs-3993565/v1/c23c1af67d3c895823b154e1.png"},{"id":53810528,"identity":"6ba87fdd-e701-4c23-8fe9-7810d085c225","added_by":"auto","created_at":"2024-03-31 14:07:31","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2434155,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3993565/v1/831522a6-b29d-461b-be86-0c971e7c55a2.pdf"},{"id":52024626,"identity":"c84305a5-b75b-47a8-900c-bdc70b2a4bc1","added_by":"auto","created_at":"2024-03-05 15:41:44","extension":"docx","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":13164,"visible":true,"origin":"","legend":"","description":"","filename":"BIOGRAPHY.docx","url":"https://assets-eu.researchsquare.com/files/rs-3993565/v1/28ee453388bb1a548122cf3c.docx"}],"financialInterests":"No competing interests reported.","formattedTitle":"Planning-level optimisation of headway regularity","fulltext":[{"header":"1 INTRODUCTION","content":"\u003cp\u003ePublic passenger transport (PPT) service operation reliability has a great influence on the passengers\u0026rsquo; perception of service quality. Ample body of evidence indicates that headway variability has a negative impact on the PPT operational functioning, as it results in increased cycle time and delays in the next vehicle departure, as well as in vehicle loading above capacity and vehicle bunching (Newell and Potts \u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e1964\u003c/span\u003e; Vuchic \u003cspan citationid=\"CR59\" class=\"CitationRef\"\u003e1969\u003c/span\u003e; Turnquist and Bowman \u003cspan citationid=\"CR55\" class=\"CitationRef\"\u003e1980\u003c/span\u003e). This operational variability also exerts a negative effect on the passengers\u0026rsquo; journey experience, as it increases the expected waiting times (Osuna and Newell \u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e1972\u003c/span\u003e; Delgado et al. \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2009\u003c/span\u003e; Berrebi et al. \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2015\u003c/span\u003e) and travel time uncertainty (Chang \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2010\u003c/span\u003e; Duran-Hormazabal and Tirachini 2016) while reducing passenger comfort in the vehicle (Siemunović et al. 2012; Tirachini et al. \u003cspan citationid=\"CR53\" class=\"CitationRef\"\u003e2013\u003c/span\u003e; Babaei et al. \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2014\u003c/span\u003e; Cats et al. \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2016\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eAccordingly, maintaining regular headways and consistent travel times are the key attributes of reliable PPT services (Chen et al. \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2009\u003c/span\u003e; El-Geneidi et al. 2011; Berrebi et al. \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2015\u003c/span\u003e; Munoz et al. 2020). In practice, however, effectively managing headway variability and vehicle bunching requires expert knowledge, data, technology and driver engagement (Tirachini et al. \u003cspan citationid=\"CR54\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). Consequently, many control methods have been developed to assist with this process and mitigate the negative effects of vehicle bunching with the aim of improving PPT service quality.\u003c/p\u003e \u003cp\u003eMost of the methods developed at the operational level regulate the real-time movement of vehicles (vehicle holding, skipping stops, limited passenger access, earlier vehicle turning), while very few focus on increasing the robustness/resistance of the PPT system to the emergence and expansion of headway disturbances. In the latter case, this is typically achieved through prioritisation of PPT vehicles at signalised intersections (Diakaki et al. \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e2003\u003c/span\u003e, Furth and Muller \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e2000\u003c/span\u003e; Kraus et al. \u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e2010\u003c/span\u003e), line length optimisation (Levinson \u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e2005\u003c/span\u003e; Chen et al \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2009\u003c/span\u003e; van Oort and van Nes \u003cspan citationid=\"CR58\" class=\"CitationRef\"\u003e2009\u003c/span\u003e), line splitting (Chen et al. \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2009\u003c/span\u003e), optimising buffer time and cycle time (Newell \u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e1977\u003c/span\u003e; Carey \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e1998\u003c/span\u003e; Zhao et al. \u003cspan citationid=\"CR67\" class=\"CitationRef\"\u003e2006\u003c/span\u003e), introducing a PPT lane (Shalaby \u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e1999\u003c/span\u003e; Nash \u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e2003\u003c/span\u003e), and increasing passenger boarding intensity (Milkovits \u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e2008\u003c/span\u003e; Sun et al. \u003cspan citationid=\"CR52\" class=\"CitationRef\"\u003e2014\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eLines with a greater number of stops create more opportunities for headway disturbances to accumulate (Chen et al. \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2009\u003c/span\u003e; Pitka et al. \u003cspan citationid=\"CR45\" class=\"CitationRef\"\u003e2017\u003c/span\u003e). On the other hand, owing to the technical\u0026thinsp;\u0026minus;\u0026thinsp;spatial conditions in the city centre (terminus), splitting diametrical lines into two radial lines can significantly improve the robustness of the PPT system to the propagation of headway disturbances, while also enhancing the headway regularity. However, line splitting also has some drawbacks, as it may result in journey disruption for a certain number of passengers who are required to make an additional transfer. In such cases, the splitting process can significantly increase the passenger travel time and reduce service quality.\u003c/p\u003e \u003cp\u003eTo examine these issues further, an optimisation procedure is proposed in this paper, allowing the practical utility of diametrical line splitting to be evaluated from the perspective of passenger travel time and headway regularity, with the aim of increasing the robustness/resistance of the PPT system to the propagation of headway disturbances. The effectiveness of this strategy is evaluated by applying it to real data from an urban PPT line.\u003c/p\u003e"},{"header":"2 LITERATURE REVIEW","content":"\u003cp\u003eWelding (\u003cspan citationid=\"CR62\" class=\"CitationRef\"\u003e1957\u003c/span\u003e) was one of the first researchers to analyse headway variability. As a part of his work, Welding studied the operation of buses and trains in London, aiming to identify the causes and effects of headway variability as well as factors that lead to bus bunching on the line. This research motivated Newell and Potts (\u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e1964\u003c/span\u003e) to examine the movement of buses when a bus traveling on the same line arrives late at a stop. Their findings revealed that headway disturbances exert two basic effects on bus movement\u0026mdash;delays resulting in late arrival at the stop, along with the tendency of vehicles following the late vehicle to move ahead\u0026mdash;resulting in vehicle bunching. In contrast to previous research (Newell and Potts \u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e1964\u003c/span\u003e; Vuchic \u003cspan citationid=\"CR59\" class=\"CitationRef\"\u003e1969\u003c/span\u003e), Turnquist and Bovman (1980) emphasised the significant influence of vehicle movement variation between stops on vehicle bunching. According to these authors, the generated headway disturbance is propagated by stops along the line as well as by scheduled vehicles. Subsequent research confirms these findings, indicating that, if headway regularity is not actively controlled, vehicle bunching may occur (Chen et al. \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e2009\u003c/span\u003e; Feng and Figliozzi \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e2011\u003c/span\u003e; Byon et al. \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e2018\u003c/span\u003e; Iliopoulou et al. \u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e2018\u003c/span\u003e). It has been empirically demonstrated in several studies that vehicle bunching is primarily caused by variability in vehicle movements at the beginning of the route, vehicle frequency, number of passengers boarding or alighting from the vehicle, boarding intensity, and number of stops/line length (Vuchic \u003cspan citationid=\"CR59\" class=\"CitationRef\"\u003e1969\u003c/span\u003e; El-Geneidy et al. \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e2011\u003c/span\u003e; Diab et al. \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e2015\u003c/span\u003e; Pitka et al. \u003cspan citationid=\"CR45\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Arriagada et al. \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Soza-Parra et al. \u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e2021\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eGiven the adverse impacts of vehicle bunching on the PPT service quality, many control methods have been developed to mitigate this issue, which can be broadly divided into two groups (Zolfaghari et al. \u003cspan citationid=\"CR69\" class=\"CitationRef\"\u003e2004\u003c/span\u003e):\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003eReal-time strategies\u0026thinsp;\u0026minus;\u0026thinsp;operational-level methods aimed at increasing service quality by introducing additional vehicles to the line, stop skipping, and/or vehicle holding\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eStrategies implemented at the planning level\u0026thinsp;\u0026minus;\u0026thinsp;long-term strategies that require route and timetable information, including any changes.\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e \u003cp\u003eWell-designed management methods can simultaneously reduce users' travel time and operational costs (Bueno-Cadena and Munoz 2017), which is typically achieved through vehicle holding based on either schedule or headway (Delgado et al. \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2009\u003c/span\u003e). The schedule holding method involves holding vehicles at stops and/or additional control points that are ahead of schedule. While this method is effective in mitigating vehicle bunching, it also reduces the PPT operating speed. On the other hand, the aim of the headway-based holding method is maintaining the designed headway. Osuna and Newell (\u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e1972\u003c/span\u003e) were the first to apply this method to a circular route in an idealised environment. These authors analysed the movement of two buses using a single control point and concluded that bunching could be adequately mitigated using this approach.\u003c/p\u003e \u003cp\u003eIn the subsequent period, several attempts have been made to improve the vehicle holding method by applying it at only one control point (Eberlein et al. \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e2001\u003c/span\u003e; Fu and Yang \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e2002\u003c/span\u003e), or at several control points along the corridor (Sun and Hickman \u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e2005\u003c/span\u003e; Bartholdi and Eisenstein \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2012\u003c/span\u003e; Chen et al. \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2013\u003c/span\u003e). In a number of the proposed models, the vehicle capacity is limited, i.e., passengers cannot board the vehicle once it has reached the maximum capacity (Zolfaghari 2004; Delgado et al. \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e2009\u003c/span\u003e; Cort\u0026eacute;s et al. \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2010\u003c/span\u003e; Delgado et al. \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2012\u003c/span\u003e). More recently, motivated by the advances in machine learning technologies, Wang and Sun (\u003cspan citationid=\"CR61\" class=\"CitationRef\"\u003e2020\u003c/span\u003e) adopted a reinforcement learning framework to determine the most optimal vehicle holding time.\u003c/p\u003e \u003cp\u003eIn addition to vehicle holding, other methods have been developed to manage headway variability in real time, whereby the most commonly studied methods are based on allowing buses to overtake each other (Wu et al. \u003cspan citationid=\"CR64\" class=\"CitationRef\"\u003e2017\u003c/span\u003e; Sun and Schm\u0026ouml;cker \u003cspan citationid=\"CR51\" class=\"CitationRef\"\u003e2018\u003c/span\u003e), regulating speed (Chandrasekar et al. \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2002\u003c/span\u003e; Daganzo and Pilachowski \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e2011\u003c/span\u003e; Munoz et al. 2013), or skipping stops (Nagatani \u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e2001\u003c/span\u003e; Fu et al. \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e2003\u003c/span\u003e; Sun and Hickman \u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e2005\u003c/span\u003e; Cort\u0026eacute;s et al. \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2010\u003c/span\u003e; Wu et al. \u003cspan citationid=\"CR65\" class=\"CitationRef\"\u003e2019\u003c/span\u003e; Larrain and Mu\u0026ntilde;oz 2020).\u003c/p\u003e \u003cp\u003eConsidering that operations at stops have a large impact on vehicle bunching, a significant number of authors have opted to focus on reducing the vehicle dwell time by restricting boarding (Delgado et al. \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2012\u003c/span\u003e; Bueno-Cadena and Munoz 2017), increasing the boarding intensity by allowing entry through all available gates (Vest and Cats 2017), or validating tickets before boarding (Ishak and Cats 2020). In a recent study, the authors did not directly restrict passenger boarding, but relied on passengers' desire for comfort and encouraged waiting for a less crowded vehicle. As a part of their work, Drabicki et al. (\u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e2023\u003c/span\u003e) examined the effect of providing passengers with real-time congestion information on vehicle bunching reduction. Similarly, Wu et al. (\u003cspan citationid=\"CR64\" class=\"CitationRef\"\u003e2017\u003c/span\u003e) demonstrated that, when passengers are sufficiently informed and can choose which vehicle to use, this not only improves the journey experience but also reduces vehicle bunching.\u003c/p\u003e \u003cp\u003eIn one of the recent studies dealing with the control strategy based on real-time information, Zhou et al. (\u003cspan citationid=\"CR68\" class=\"CitationRef\"\u003e2022\u003c/span\u003e) identified a significant gap between the recommendations based on recent theories and modern practice. According to these authors, closing this gap requires the adoption of appropriate technology and software for continuous communication between the control centre and the driver.\u003c/p\u003e \u003cp\u003eOn the other hand, Levinson (\u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e2005\u003c/span\u003e) examined the main contributors to headway reliability and reached the following conclusions: (I) bus lines with long routes, high occupancy, and mixed traffic have very low reliability; (II) a headway disturbance that occurs at the beginning of the line tends to propagate along the line; (III) accuracy is important for lines with long headways, whereas precedence should be given to uniformity for lines with short headways; (IV) service reliability and transport speed on bus lines can be improved by reducing the number of stops on the line, the dwell time, and the impact of mixed traffic on vehicle movement between stops.\u003c/p\u003e \u003cp\u003eAccording to the available literature, the influence of line length and number of stops on headway uniformity and service reliability remains insufficiently studied (Levinson \u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e2005\u003c/span\u003e; van Oort \u003cspan citationid=\"CR56\" class=\"CitationRef\"\u003e2011\u003c/span\u003e; Soza-Parra et al. \u003cspan citationid=\"CR49\" class=\"CitationRef\"\u003e2021\u003c/span\u003e). Using a case study in the Hague, van Oort (\u003cspan citationid=\"CR56\" class=\"CitationRef\"\u003e2011\u003c/span\u003e) investigated the effect of line length on service reliability. The author split the routes of longer lines and analysed the resulting service reliability and passenger travel time based on real data. The obtained results indicate that splitting the line increases service reliability, while reducing passenger waiting time at the bus stop as well as time spent in the vehicle. The author further noted that the time loss due to headway variability can be reduced by about 30% when the line is split at a stop with low passenger flow.\u003c/p\u003e \u003cp\u003eIncreasingly, however, technological advances are being harnessed to deal with headway disturbances in real time. In particular, methods based on the Automatic Vehicle Location (AVL) systems are being proposed despite their limited practical utility due to the lack of appropriate technology and software. Therefore, several authors advocate for the greater reliance on planning-level strategies as a means of increasing the PPT resistance to headway disturbances. While in most cases this approach has yielded positive results, the proposed methods are typically developed from the operator's perspective, offering limited insight into the effectiveness of these strategies from the passengers\u0026rsquo; point of view. The work presented in this article attempts to fill these gaps. This is achieved by presenting a practical optimisation procedure that is implemented at the planning level, and its positive effects are evaluated on the basis of the average passenger travel time.\u003c/p\u003e"},{"header":"3 METHODS","content":"\u003cp\u003eIn this section, the optimisation procedure, along with the model for evaluating the impact of line splitting on average passenger travel time, is described in detail. Due to its complexity, the optimisation procedure is divided into four phases and is subject to the following limitations:\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\n\u003cp\u003eThe scheduled headway must be less than 15 minutes\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eTerminal time can be used for delay recovery (Vuchic \u003cspan class=\"CitationRef\"\u003e2017\u003c/span\u003e), schedule adjustment (maintaining uniform headway) to prevent disruptions being passed on to the next departure of the same vehicle\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eThe existence of a terminus/turnpike at the line splitting point is mandatory\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eEach line obtained by splitting, considered separately, must represent a functional unit that meets the passengers\u0026rsquo; needs.\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eBy splitting the diametrical line into two radials, two components of the passenger travel time are affected. First, as a benefit of optimisation, the passenger waiting time at the stop is reduced due to the improved headway regularity. Second, passengers who cross the splitting point have to make an additional transfer, which increases their travel time. This is a notable drawback of the optimisation. Given that these time components have opposite effects on journey duration and that a large number of parameters influence their value, a model was developed to evaluate the influence of line splitting on average passenger travel time.\u003c/p\u003e\n\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e\n\u003ch2\u003e3.1 Optimisation procedure phases\u003c/h2\u003e\n\u003cp\u003eIn 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 Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e:\u003c/p\u003e\n\u003cp\u003ePhase I: Diametrical line splitting\u003c/p\u003e\n\u003cp\u003ePhase II: Simulation of headway disturbance propagation\u003c/p\u003e\n\u003cp\u003ePhase III: Evaluation of headway regularity\u003c/p\u003e\n\u003cp\u003ePhase IV: Evaluation of passenger travel time\u003c/p\u003e\n\u003cp\u003eThe 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)\u0026thinsp;\u0026minus;\u0026thinsp;the optimisation variant. The output data of Phase III and Phase IV (Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e) 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.\u003c/p\u003e\n\u003cp\u003ePhase I: In the first phase, the diametrical line AB (Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e) 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 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).\u003c/p\u003e\n\u003cp\u003ePhase II: In the second phase, a simulation of the headway disturbance propagation is carried out for both variants. The deterministic model shown in Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e is adopted for this purpose, as it is based on the relationship between the passenger arrival rate and the passenger boarding rate (Newell and Potts \u003cspan class=\"CitationRef\"\u003e1964\u003c/span\u003e; Vuchic \u003cspan class=\"CitationRef\"\u003e1969\u003c/span\u003e; Daganzo \u003cspan class=\"CitationRef\"\u003e2009\u003c/span\u003e; Pitka et al. \u003cspan class=\"CitationRef\"\u003e2017\u003c/span\u003e). 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 that the disturbance from the first radial line would not be transmitted to the second line and vice versa.\u003c/p\u003e\n\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ1\" class=\"mathdisplay\"\u003e$${h}_{\\text{n}}={{h}_{\\text{p}}\\left(1+\\frac{\\lambda }{\\mu }\\right)}^{n-1}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere \u003cem\u003eh\u003c/em\u003e\u003csub\u003en\u003c/sub\u003e \u0026ndash; headway disturbance for the \u003cem\u003en\u003c/em\u003e\u003csup\u003eth\u003c/sup\u003e stop (min); \u003cem\u003eh\u003c/em\u003e\u003csub\u003ep\u003c/sub\u003e \u0026ndash; primary headway disturbance (min); \u003cem\u003en\u003c/em\u003e \u0026ndash; the number of sequential stops in relation to the primary disturbance location; \u003cem\u003e\u0026lambda;\u003c/em\u003e \u0026ndash; mean passenger boarding intensity (prs/min); and \u003cem\u003e\u0026micro;\u003c/em\u003e \u0026ndash; passenger accumulation intensity along the line (prs/min).\u003c/p\u003e\n\u003cp\u003ePhase 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 a large number of vehicle departures simultaneously. For this purpose, the Percentage Regularity Deviation Mean (PRDM) is commonly adopted (Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e). PRDM is a popular index for describing bus service regularity and was first used by Hakkesteegt and Muller (\u003cspan class=\"CitationRef\"\u003e1981\u003c/span\u003e) and was later adopted by other authors (Sorratini et al. 2008; van Oort and van Nes \u003cspan class=\"CitationRef\"\u003e2009\u003c/span\u003e; van Oort \u003cspan class=\"CitationRef\"\u003e2014\u003c/span\u003e; Simeunović et al. \u003cspan class=\"CitationRef\"\u003e2016\u003c/span\u003e; Zhang et al. \u003cspan class=\"CitationRef\"\u003e2018\u003c/span\u003e). A lower PRDM value indicates better bus service regularity.\u003c/p\u003e\n\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ2\" class=\"mathdisplay\"\u003e$${PRDM}_{\\text{n}}=\\frac{\\sum _{\\text{i}=1}^{\\text{N}}\\left|\\frac{{H}_{\\text{s}}-{H}_{\\text{i}}}{{H}_{\\text{s}}}\\right|}{N}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere \u003cem\u003ePRDM\u003c/em\u003e\u003csub\u003en\u003c/sub\u003e \u0026ndash; percentage regularity deviation mean of the headway related to the \u003cem\u003en\u003c/em\u003e\u003csup\u003eth\u003c/sup\u003e stop; \u003cem\u003eH\u003c/em\u003e\u003csub\u003ei\u003c/sub\u003e \u0026ndash; headway value for the \u003cem\u003ei\u003c/em\u003e\u003csup\u003eth\u003c/sup\u003e vehicle (min); \u003cem\u003eH\u003c/em\u003e\u003csub\u003es\u003c/sub\u003e \u0026ndash; scheduled headway (min); \u003cem\u003eN\u003c/em\u003e \u0026ndash; number of vehicles crossing at the \u003cem\u003en\u003c/em\u003e\u003csup\u003eth\u003c/sup\u003e stop during the studied time interval.\u003c/p\u003e\n\u003cp\u003ePhase 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.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec5\" class=\"Section2\"\u003e\n\u003ch2\u003e3.2 MODEL DEVELOPMENT\u003c/h2\u003e\n\u003cp\u003eAccording to Vuchic (\u003cspan class=\"CitationRef\"\u003e2017\u003c/span\u003e), passenger travel time from the origin to the destination consists of\u003c/p\u003e\n\u003cp\u003e\u003cem\u003et\u003c/em\u003e \u003csub\u003eod\u003c/sub\u003e = \u003cem\u003et\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e + \u003cem\u003et\u003c/em\u003e\u003csub\u003ew\u003c/sub\u003e + \u003cem\u003et\u003c/em\u003e\u003csub\u003eol\u003c/sub\u003e + \u003cem\u003et\u003c/em\u003e\u003csub\u003ef\u003c/sub\u003e + \u003cem\u003et\u003c/em\u003e\u003csub\u003ea'\u003c/sub\u003e, where \u003cem\u003et\u003c/em\u003e\u003csub\u003eod\u003c/sub\u003e \u0026ndash; origin\u0026ndash;destination passenger travel time (min); \u003cem\u003et\u003c/em\u003e\u003csub\u003ea\u003c/sub\u003e \u0026ndash; the time required for reaching the PPT stop (min); \u003cem\u003et\u003c/em\u003e\u003csub\u003ew\u003c/sub\u003e \u0026ndash; waiting time at the stop (min); \u003cem\u003et\u003c/em\u003e\u003csub\u003eol\u003c/sub\u003e \u0026ndash; on-line travel time (min); \u003cem\u003et\u003c/em\u003e\u003csub\u003ef\u003c/sub\u003e \u0026ndash; transfer time (min); \u003cem\u003et\u003c/em\u003e\u003csub\u003ea'\u003c/sub\u003e \u0026ndash; the time required for covering the distance from a PPT stop to the final destination (min). The key parameters for this model are: waiting time, on-line travel time and transfer time.\u003c/p\u003e\n\u003cp\u003eAs noted by Bowman and Turnquist (\u003cspan class=\"CitationRef\"\u003e1981\u003c/span\u003e), the expected passenger waiting time is related to both the distribution of passenger arrival times at a stop and the distribution of schedule deviations in bus arrival times at that stop. For short headways, several authors assume that passengers arrive at random times, independent of bus arrival schedules (Osuna and Newell \u003cspan class=\"CitationRef\"\u003e1972\u003c/span\u003e; Bowman and Turnquist \u003cspan class=\"CitationRef\"\u003e1981\u003c/span\u003e; Vuchic \u003cspan class=\"CitationRef\"\u003e2017\u003c/span\u003e). Under such conditions, headway regularity along the line is extremely important.\u003c/p\u003e\n\u003cp\u003eWhen headway regularity is poor, the average passenger waiting time (\u003cem\u003et\u003c/em\u003e\u003csub\u003ew\u003c/sub\u003e) depends on the scheduled headway (\u003cem\u003eH\u003c/em\u003e\u003csub\u003es\u003c/sub\u003e) and headway disturbances (\u003cem\u003eh\u003c/em\u003e), as indicated by Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e:\u003c/p\u003e\n\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ3\" class=\"mathdisplay\"\u003e$${t}_{\\text{w}}=\\frac{{H}_{\\text{s}}+h}{2}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe time a passenger spends in a PPT vehicle \u0026ndash; on-line time (\u003cem\u003et\u003c/em\u003e\u003csub\u003eol\u003c/sub\u003e) \u0026ndash; is the sum of the vehicle dwell time at the stop (\u003cem\u003et\u003c/em\u003e\u003csub\u003ed\u003c/sub\u003e) and the vehicle running time between adjacent stops (\u003cem\u003et\u003c/em\u003e\u003csub\u003er\u003c/sub\u003e), i.e., \u003cem\u003et\u003c/em\u003e\u003csub\u003eol\u003c/sub\u003e = \u0026sum;\u003cem\u003et\u003c/em\u003e\u003csub\u003ed\u003c/sub\u003e + \u0026sum;\u003cem\u003et\u003c/em\u003e\u003csub\u003er\u003c/sub\u003e.\u003c/p\u003e\n\u003cp\u003eAccording to the extant studies in this domain, dwell time is determined by the required passenger boarding time (Daganzo \u003cspan class=\"CitationRef\"\u003e2009\u003c/span\u003e; Bellei and Gkoumas \u003cspan class=\"CitationRef\"\u003e2010\u003c/span\u003e; Vuchic \u003cspan class=\"CitationRef\"\u003e2017\u003c/span\u003e). Hence, if there is a headway disturbance, the dwell time can be defined by Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e:\u003c/p\u003e\n\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ4\" class=\"mathdisplay\"\u003e$${t}_{\\text{d}}=\\frac{\\lambda }{\\mu }\\left({H}_{\\text{s}}+h\\right)$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eIn the model developed as a part of this work, transfer time is calculated in the same way as passenger waiting time (Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e) because the walking distance between two radial lines is negligible and vehicle departures are uncoordinated.\u003c/p\u003e\n\u003cp\u003eIn order to evaluate the passenger travel time, a deterministic model was first created for one passenger's journey and was subsequently extended to all passengers on the line. The model calculates the difference in passenger travel time between the two previously described variants. On line AB, the passenger journey is observed from the time of their arrival at stop \u003cem\u003ek\u003c/em\u003e to their departure at stop \u003cem\u003em\u003c/em\u003e (Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e). The model does not consider the total passenger travel time, as it focuses solely on the time the passenger spends in the PPT (excluding the time spent walking to/from the stop).\u003c/p\u003e\n\u003cp\u003eIf the transport is realised with a single diametrical line, the time the passenger spends in the PPT (\u003cem\u003eT\u003c/em\u003e\u003csub\u003eD\u003c/sub\u003e) is expressed by Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e:\u003c/p\u003e\n\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ5\" class=\"mathdisplay\"\u003e$${T}_{D}={t}_{\\text{w}}+\\sum {t}_{\\text{d}}+\\sum {t}_{\\text{r}}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eIn the case of a headway disturbance, determination of the time the passenger spends in the PPT requires consideration of several basic parameters that define the headway disturbance propagation, as shown in Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e. The expression in Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e is obtained by incorporating Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e and Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e into Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e.\u003c/p\u003e\n\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ6\" class=\"mathdisplay\"\u003e$${T}_{\\text{D}}=\\frac{{H}_{\\text{s}}+{h}_{\\text{k}}}{2}+\\sum _{j=k+1}^{m-1}\\left({H}_{\\text{s}}+{h}_{\\text{j}}\\right)\\left(\\frac{{\\lambda }_{\\text{j}}}{\\mu }\\right)+\\sum _{j=k}^{m-1}{t}_{\\text{r}\\text{j}}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eTo simplify the expressions in Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e, we modify \u003cem\u003e\u0026beta;\u003c/em\u003e\u003csub\u003ej\u003c/sub\u003e\u0026thinsp;\u003cem\u003e=\u0026thinsp;\u0026lambda;\u003c/em\u003e\u003csub\u003ej\u003c/sub\u003e \u003cem\u003e/ \u0026micro;\u003c/em\u003e and \u003cem\u003eH\u003c/em\u003e\u003csub\u003ej\u003c/sub\u003e = \u003cem\u003eH\u003c/em\u003e\u003csub\u003es\u003c/sub\u003e + \u003cem\u003eh\u003c/em\u003e\u003csub\u003ej\u003c/sub\u003e, where \u003cem\u003e\u0026beta;\u003c/em\u003e\u003csub\u003ej\u003c/sub\u003e is the ratio of passenger boarding intensity and passenger accumulation intensity at the \u003cem\u003ej\u003c/em\u003e\u003csup\u003eth\u003c/sup\u003e stop, and \u003cem\u003eH\u003c/em\u003e\u003csub\u003ej\u003c/sub\u003e is the actual headway at the \u003cem\u003ej\u003c/em\u003e\u003csup\u003eth\u003c/sup\u003e stop, resulting in Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e:\u003c/p\u003e\n\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ7\" class=\"mathdisplay\"\u003e$${T}_{\\text{D}}=\\frac{{H}_{\\text{k}}}{2}+\\sum _{j=k+1}^{m-1}{H}_{\\text{j}}{\\beta }_{j}+\\sum _{j=k}^{m-1}{t}_{\\text{r}\\text{j}}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eIf the transport is realised with two radial lines, Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e is expanded for the transfer time. As before, the time the passenger spends in the PPT (\u003cem\u003eT\u003c/em\u003e\u003csub\u003e2R\u003c/sub\u003e) comprises several components and is defined by Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e8\u003c/span\u003e:\u003c/p\u003e\n\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ8\" class=\"mathdisplay\"\u003e$${T}_{2\\text{R}}={t}_{\\text{w}}+\\sum _{k+1}^{l-1}{t}_{\\text{d}}+\\sum _{k}^{l-1}{t}_{\\text{r}}+{t}_{\\text{f}}+\\sum _{l+1}^{m-1}{t}_{\\text{d}}+\\sum _{l}^{m-1}{t}_{\\text{r}}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eOnce again, in the case of a headway disturbance, the time the passenger spends in the PPT (two radial lines) is described by the basic parameters that define the headway disturbance propagation (Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e9\u003c/span\u003e). Accordingly, Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e9\u003c/span\u003e is obtained by expressing Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e8\u003c/span\u003e via Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e and Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e.\u003c/p\u003e\n\u003cdiv id=\"Equ9\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ9\" class=\"mathdisplay\"\u003e$${T}_{2\\text{R}}=\\frac{{H}_{\\text{k}}}{2}+\\sum _{j=k+1}^{l-1}{H}_{\\text{j}}{\\beta }_{j}+\\sum _{j=k}^{l-1}{t}_{\\text{r}\\text{j}}+\\frac{{H}_{\\text{l}}^{\\prime }}{2}+\\sum _{j=l+1}^{m-1}{H}_{\\text{j}}^{\\prime }{\\beta }_{\\text{j}}+\\sum _{j=l}^{m-1}{t}_{\\text{r}\\text{j}}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere \u003cem\u003eH`\u003c/em\u003e\u003csub\u003ej\u003c/sub\u003e is the actual headway at the \u003cem\u003ej\u003c/em\u003e\u003csup\u003eth\u003c/sup\u003e stop on the second radial line and is given by \u003cem\u003eH`\u003c/em\u003e\u003csub\u003ej\u003c/sub\u003e = \u003cem\u003eH\u003c/em\u003e\u003csub\u003es\u003c/sub\u003e + \u003cem\u003eh`\u003c/em\u003e\u003csub\u003ej\u003c/sub\u003e.\u003c/p\u003e\n\u003cp\u003eThe difference in passenger travel time between the two variants (Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e10\u003c/span\u003e) is obtained by subtracting Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e9\u003c/span\u003e from Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e.\u003c/p\u003e\n\u003cdiv id=\"Equ10\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ10\" class=\"mathdisplay\"\u003e$${T}_{2\\text{R}-\\text{D}}=\\frac{{H}_{\\text{l}}}{2}-{H}_{\\text{l}}{\\beta }_{\\text{l}}-\\sum _{j=l+1}^{m-1}\\left({h}_{\\text{j}}-{h}_{\\text{j}}^{\\prime }\\right){\\beta }_{\\text{j}}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e10\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eFrom Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e, Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e9\u003c/span\u003e and Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e10\u003c/span\u003e, it can be concluded that the influence of line splitting on each passenger\u0026rsquo;s travel time depends on the position of the boarding and alighting stops on the line. By splitting the diametrical line into two radial ones, three characteristic areas have been created, which have different effects on passenger journey durations. Considering the direction of travel from terminus A to terminus B, the following sections can be distinguished (Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e):\u003c/p\u003e\n\u003cul\u003e\n\u003cli\u003e\n\u003cp\u003eSection from A to C (section AC)\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eLine splitting point (section C)\u003c/p\u003e\n\u003c/li\u003e\n\u003cli\u003e\n\u003cp\u003eSection from C to B (section CB)\u003c/p\u003e\n\u003c/li\u003e\n\u003c/ul\u003e\n\u003cp\u003eDepending on the sections defined above and the locations of stops at which passengers enter and leave the line, three distinct passenger groups can be recognised:\u003c/p\u003e\n\u003cp\u003e1. Passengers travelling on the AC section only and not crossing the line split point (C). As this group is not affected by the line splitting, they experience no change in their travel time.\u003c/p\u003e\n\u003cp\u003e2. Passengers travelling on section CB only. This group will experience only the positive effects of line splitting, as their waiting time at the stop and the driving time will be reduced.\u003c/p\u003e\n\u003cp\u003e3. Passengers crossing the line splitting point (C) during their journey will note both positive and negative effects on their journey duration. While their travel time will be shortened, they will be required to transfer to another vehicle at the line splitting point (C), which will lead to an additional transfer time and greater travel time uncertainty.\u003c/p\u003e\n\u003cp\u003eThe number of passengers transported between stops represents the passenger flow (\u003cem\u003eq\u003c/em\u003e), while the number of passengers entering (\u003cem\u003eU\u003c/em\u003e) at a stop \u0026ldquo;j\u0026rdquo; is expressed by \u003cem\u003eU\u003c/em\u003e\u003csub\u003ej\u003c/sub\u003e = (\u003cem\u003eH\u003c/em\u003e\u003csub\u003es\u003c/sub\u003e + \u003cem\u003eh\u003c/em\u003e\u003csub\u003ej\u003c/sub\u003e) \u003cem\u003e\u0026lambda;\u003c/em\u003e\u003csub\u003ej\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eH\u003c/em\u003e\u003csub\u003ej\u003c/sub\u003e \u003cem\u003e\u0026lambda;\u003c/em\u003e\u003csub\u003ej\u003c/sub\u003e. According to the previously presented arguments, when the transport is realised with the diametrical line, the total spent time of all passengers (\u003cem\u003eTT\u003c/em\u003e\u003csub\u003eD\u003c/sub\u003e) in the PPT on line AB is defined by Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e11\u003c/span\u003e:\u003c/p\u003e\n\u003cdiv id=\"Equ11\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ11\" class=\"mathdisplay\"\u003e$${TT}_{\\text{D}}=\\sum _{j=1}^{n-1}\\frac{{H}_{j}^{2}}{2}{\\lambda }_{\\text{j}}+\\sum _{j=2}^{n-1}{\\beta }_{\\text{j}}{H}_{\\text{j}}{q}_{\\text{j}-1}+\\sum _{j=1}^{n-1}{t}_{\\text{r}\\text{j}}{q}_{j}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e11\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eIf the transport is realised with two radial lines, the total spent time of all passengers (\u003cem\u003eTT\u003c/em\u003e\u003csub\u003e2R\u003c/sub\u003e) in the PPT on line AB is defined by Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e12\u003c/span\u003e:\u003c/p\u003e\n\u003cdiv id=\"Equ12\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ12\" class=\"mathdisplay\"\u003e$$T{T}_{2\\text{R}}=\\sum _{j=1}^{l-1}\\left(\\frac{{H}_{j}^{2}}{2}{\\lambda }_{\\text{j}}+{t}_{\\text{r}\\text{j}}{q}_{j}\\right)+\\sum _{j=2}^{l-1}{\\beta }_{\\text{j}}{H}_{\\text{j}}{q}_{\\text{j}-1}+\\frac{{H}_{s}}{2}\\left({\\lambda }_{\\text{l}}{H}_{\\text{l}}^{\\prime }+{q}_{l}\\right)+\\sum _{j=l+1}^{n-1}\\left(\\frac{{H}_{j}^{` 2}}{2}{\\lambda }_{\\text{j}}+{\\beta }_{\\text{j}}{H}_{\\text{j}}^{\\prime }{q}_{\\text{j}-1}\\right)+\\sum _{j=l}^{n-1}{t}_{\\text{r}\\text{j}}{q}_{j}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e12\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eThe difference in the time passengers spend in the PPT (\u003cem\u003eTT\u003c/em\u003e\u003csub\u003e2R\u0026thinsp;\u0026minus;\u0026thinsp;D\u003c/sub\u003e) between the first and the second variant (Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e13\u003c/span\u003e) is obtained by subtracting Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e11\u003c/span\u003e from Eq.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e12\u003c/span\u003e.\u003c/p\u003e\n\u003cdiv id=\"Equ13\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ13\" class=\"mathdisplay\"\u003e$${TT}_{2\\text{R}-\\text{D}}=\\sum _{j=l+1}^{n-1}\\left(\\frac{{H}_{j}^{`2}}{2}{\\lambda }_{\\text{j}}+{\\beta }_{\\text{j}}{H}_{j}^{\\prime }{q}_{\\text{j}-1}\\right)+\\frac{{H}_{s}}{2}\\left({\\lambda }_{\\text{l}}{H}_{\\text{l}}^{\\prime }+{q}_{l}\\right)-\\sum _{j=l}^{n-1}\\left(\\frac{{H}_{j}^{2}}{2}{\\lambda }_{\\text{j}}+{\\beta }_{\\text{j}}{H}_{\\text{j}}{q}_{\\text{j}-1}\\right)$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e13\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eBased on the expression given above, the average passenger spent time represents the arithmetic mean of the total spent time of all passengers in the PPT.\u003c/p\u003e\n\u003c/div\u003e"},{"header":"4 CASE STUDY","content":"\u003cp\u003eIn accordance with the model limitations, the practical implementation was carried out in the PPT (Novi Sad, Serbia) on a diametrical line (Line 3: Petrovaradin\u0026thinsp;\u0026minus;\u0026thinsp;Centar\u0026thinsp;\u0026minus;\u0026thinsp;Detelinara) that connects the boroughs of Detelinara and Petrovaradin, and its route passes through the Novi Sad city centre. The practical application was carried out for two characteristic periods, whereby the first represents the line operation mode with maximum load (peak hour, 1:00\u0026thinsp;\u0026minus;\u0026thinsp;2:00 pm), while the second represents the line operation mode with the lowest load (off-peak hours, 9:00\u0026thinsp;\u0026minus;\u0026thinsp;10:00 am).\u003c/p\u003e\n\u003cdiv id=\"Sec7\" class=\"Section2\"\u003e\n\u003ch2\u003e4.1 Diametrical line splitting (Phase I)\u003c/h2\u003e\n\u003cp\u003eOn the basis of the static elements of Line 3, the Uspenska\u0026minus;\u0026Scaron;afarikova stop (Stop No. 11 in direction A, and Stop No. 10 in direction B, show in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e) was determined as the intersection point of the diametrical line. This bus stop is located in the city centre and is designed as a terminus. By dividing the diametrical line at a defined point, two radial lines with a common terminus are created (Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e): the Petrovaradin\u0026thinsp;\u0026minus;\u0026thinsp;Centar and Centar\u0026thinsp;\u0026minus;\u0026thinsp;Detelinara lines.\u003c/p\u003e\n\u003cp\u003eThe length of the new line Petrovaradin\u0026thinsp;\u0026minus;\u0026thinsp;Centar is 5.0 km in direction A and 4.9 km in direction B, with 11 stops in each direction. The second radial line (Centar\u0026thinsp;\u0026minus;\u0026thinsp;Detelinara) is 4.05 km and 3.8 km long in direction A and B, respectively. This line has 9 stops in direction A and 10 stops in direction B.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e\n\u003ch2\u003e4.2 Simulation of the headway disturbance propagation (Phase II)\u003c/h2\u003e\n\u003cp\u003eIn the optimisation procedure, a simulation of the headway disturbance propagation was performed for both variants, using the real collected data (Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e and Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e), along with different primary headway disturbance values.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003ctable id=\"Tab1\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003ePassenger boarding and alighting at stops\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth rowspan=\"3\" align=\"left\"\u003e\n\u003cp\u003eStop\u003c/p\u003e\n\u003cp\u003eNo.\u003c/p\u003e\n\u003c/th\u003e\n\u003cth colspan=\"4\" align=\"left\"\u003e\n\u003cp\u003eDirection A\u003c/p\u003e\n\u003c/th\u003e\n\u003cth colspan=\"4\" align=\"left\"\u003e\n\u003cp\u003eDirection B\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003cth colspan=\"2\" align=\"left\"\u003e\n\u003cp\u003e9:00\u0026thinsp;\u0026minus;\u0026thinsp;10:00 am\u003c/p\u003e\n\u003c/th\u003e\n\u003cth colspan=\"2\" align=\"left\"\u003e\n\u003cp\u003e1:00\u0026thinsp;\u0026minus;\u0026thinsp;2:00 pm\u003c/p\u003e\n\u003c/th\u003e\n\u003cth colspan=\"2\" align=\"left\"\u003e\n\u003cp\u003e9:00\u0026thinsp;\u0026minus;\u0026thinsp;10:00 am\u003c/p\u003e\n\u003c/th\u003e\n\u003cth colspan=\"2\" align=\"left\"\u003e\n\u003cp\u003e1:00\u0026thinsp;\u0026minus;\u0026thinsp;2:00 pm\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eBoarding\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eAlighting\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eBoarding\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eAlighting\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eBoarding\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eAlighting\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eBoarding\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eAlighting\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e43\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e70\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e72\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e81\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e59\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e123\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e23\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e73\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e19\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e36\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e5\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e36\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e74\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e15\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e4\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e16\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e18\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e49\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e15\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e71\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e23\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e5\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e26\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e6\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e20\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e12\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e18\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e49\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e39\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e6\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e7\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e38\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e70\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e27\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e9\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e17\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e15\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e7\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e8\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e22\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e5\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e21\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e20\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e11\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e8\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e35\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e5\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e39\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e7\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e53\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e41\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e63\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e59\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e9\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e29\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e37\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e81\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e44\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e42\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e41\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e39\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e34\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e10\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e47\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e74\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e132\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e99\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e35\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e66\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e66\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e105\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e11\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e45\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e43\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e70\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e84\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e34\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e71\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e87\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e134\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e12\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e33\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e72\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e58\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e60\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e19\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e47\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e57\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e61\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e13\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e36\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e73\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e69\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e92\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e4\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e21\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e7\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e24\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e14\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e11\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e18\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e14\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e58\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e11\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e22\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e4\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e33\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e15\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e12\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e31\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e35\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e61\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e4\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e15\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e50\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e49\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e16\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e26\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e15\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e79\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e14\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e15\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e14\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e38\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e17\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e18\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e49\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e18\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e9\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e32\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e18\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e4\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e27\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e79\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e3\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e14\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e55\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e19\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e29\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e47\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e31\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e36\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e20\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e18\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e27\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003ctable id=\"Tab2\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003eInput data for headway disturbance propagation simulation\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eParameter\u003c/p\u003e\n\u003c/th\u003e\n\u003cth colspan=\"2\" align=\"left\"\u003e\n\u003cp\u003eValues\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eTime period\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e9:00\u0026thinsp;\u0026minus;\u0026thinsp;10:00 am\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1:00\u0026thinsp;\u0026minus;\u0026thinsp;2:00 pm\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eProjected headway (min)\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e10\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e8.3\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eAverage passenger boarding intensity (prs/min)\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd colspan=\"2\" align=\"left\"\u003e\n\u003cp\u003e14.3\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003ePrimary headway disturbance (min)\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd colspan=\"2\" align=\"left\"\u003e\n\u003cp\u003e0\u0026thinsp;\u0026minus;\u0026thinsp;4\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eFor both variants, the same values of the primary headway disturbance were used in the simulation procedure while the disturbance duration ranged from 0 to 4 minutes.\u003c/p\u003e\n\u003cp\u003eIn the case of the optimised variant, headway disturbance propagation is interrupted at the splitting point, after which the accumulation starts from the ideal state of headway regularity (Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e).\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec9\" class=\"Section2\"\u003e\n\u003ch2\u003e4.3 Evaluation of headway regularity (Phase III)\u003c/h2\u003e\n\u003cp\u003eThe evaluation of headway regularity was carried out for different combinations of primary headway disturbance, based on the simulations carried out in the second phase (Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e). In Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e and Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e, the hypothetical primary headway disturbance is given on the x-axis and the realised PRDM values are presented on the y-axis. It is evident from the graphs that the primary headway disturbance has a linear effect on the PRDM, and that an improvement in the degree of regularity was achieved during peak as well as off-peak hours. The optimisation effects also depend on the line direction and the observation period. The greatest improvements in headway regularity are achieved during the peak period, when the following headway is the shortest and the number of boarding passengers per stop is the greatest (Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e and Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e).\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec10\" class=\"Section2\"\u003e\n\u003ch2\u003e4.4 Evaluation of passenger travel time (Phase IV)\u003c/h2\u003e\n\u003cp\u003eThe evaluation of passenger travel time in the PPT was performed for different primary headway disturbance combinations, based on the difference in the passenger travel time.\u003c/p\u003e\n\u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e8\u003c/span\u003e shows the difference in average passenger travel times for the two analysed variants. The data are presented for two observation periods and both line directions. If the difference in passenger travel time is less than zero, the optimised (split) line design (the second variant) is more efficient than the current (diametrical) line (the first variant).\u003c/p\u003e\n\u003cp\u003eThe observed optimisation effects, in terms of average passenger travel times, vary significantly depending on the observation period and the primary headway disturbance duration. The reduction in average passenger travel times is only achieved during the peak period when the primary headway disturbance on the diametrical line is at least 2 minutes. If the primary headway disturbance in the peak period is 4 minutes, the optimisation achieves a 1.5-minute and 1.8-minute reduction in travel time in direction A and B, respectively. In the optimised variant for the off-peak period, the average travel time would be up to 4 minutes longer irrespective of the primary headway disturbance value.\u003c/p\u003e\n\u003c/div\u003e"},{"header":"5 SENSITIVITY ANALYSIS","content":"\u003cp\u003eAs the optimisation procedure yields different results for peak and off-peak periods, which can be distinguished by the headway and the transport demand, these parameters were considered in the sensitivity analysis.\u003c/p\u003e\n\u003cp\u003eFirst, the sensitivity of the optimisation procedure was examined by simulating a change in transport demand by decreasing/increasing the number of passengers boarding and alighting at each stop by 10% and 20%, resulting in the five cases shown in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e. For each case, the impact of transport demand on travel time and headway regularity was analysed.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003ctable id=\"Tab3\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003ePassenger boarding and alighting at stops per case\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eCase\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003ePercentage decrease/increase in the number of passengers\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-20%\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-10%\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0%\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e4\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e+\u0026thinsp;10%\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e5\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e+\u0026thinsp;20%\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eSensitivity of headway regularity to changes in transport demand is obtained by the difference between PRDM of radial lines and diametrical line (Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e). As can be seen from Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e9\u003c/span\u003e, changes in transport demand have a significant impact on the headway regularity. In the example depicted on the graph, for a primary headway disturbance of 3 minutes after optimisation, a -0.184 reduction in PRDM is achieved in CASE 1, while \u0026minus;\u0026thinsp;0.219, -0.262, -0.306, and \u0026minus;\u0026thinsp;0.358 reduction is obtained for the remaining four cases. For the same values of primary headway disturbance, by increasing the transport demand, a better headway regularity is achieved, i.e., the optimisation procedure results in a greater PRDM reduction.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003ctable id=\"Tab4\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003eSensitivity of headway regularity to changes in transport demand\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd rowspan=\"2\" align=\"left\"\u003e\n\u003cp\u003eVariant\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd rowspan=\"2\" align=\"left\"\u003e\n\u003cp\u003eCase\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd colspan=\"5\" align=\"left\"\u003e\n\u003cp\u003ePrimary headway disturbance (min)\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e4\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd rowspan=\"5\" align=\"left\"\u003e\n\u003cp\u003eDiametrical line\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.282\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.556\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.821\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.094\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.256\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.600\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.891\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.208\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.352\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.652\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.996\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.269\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e4\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.370\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.722\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.057\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.400\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e5\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.413\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.783\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.153\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.531\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd rowspan=\"5\" align=\"left\"\u003e\n\u003cp\u003eRadial lines\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.212\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.434\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.637\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.850\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.204\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.451\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.672\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.911\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.265\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.486\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.733\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.946\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e4\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.265\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.521\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.751\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.998\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e5\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.291\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.547\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.795\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.068\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eFigure \u003cspan class=\"InternalRef\"\u003e10\u003c/span\u003e and Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e show that changes in transport demand also have a significant impact on the passenger travel time. For example, if a primary disruption of 3 minutes occurs after optimisation, passengers will experience a -0.59 reduction in travel time in CASE 1, while \u0026minus;\u0026thinsp;0.80, -1.08, -1.31, and \u0026minus;\u0026thinsp;1.58 reduction is achieved in CASE 2\u0026thinsp;\u0026minus;\u0026thinsp;5. Correspondingly, with an increase in transport demand on the same line and with the same values of primary headway disruption, greater savings in passenger travel time are achieved.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003ctable id=\"Tab5\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 5\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003eSensitivity of passenger travel time to changes in transport demand\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth rowspan=\"2\" align=\"left\"\u003e\n\u003cp\u003eCase\u003c/p\u003e\n\u003c/th\u003e\n\u003cth colspan=\"5\" align=\"left\"\u003e\n\u003cp\u003ePrimary headway disturbance (min)\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e2\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e3\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e4\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.55\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.61\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.03\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-0.59\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-1.23\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.52\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.84\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-0.14\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-0.80\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-1.51\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.44\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.54\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-0.26\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-1.08\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-1.69\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e4\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.48\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.46\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-0.46\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-1.31\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-2.11\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e5\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.45\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.31\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-0.65\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-1.58\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-2.40\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eBased on the aforementioned analysis, it can be concluded that the transport demand and the effects of the optimisation procedure (travel time and headway regularity) are positively correlated. Specifically, increasing the primary headway disturbance increases the sensitivity of the optimisation procedure to the transport demand.\u003c/p\u003e\n\u003cp\u003eAs a part of the sensitivity analysis, a simulation of headway was also carried out. Five cases were created for this purpose (Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e) and the impact of the headway on travel time and headway regularity was analysed for each case.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003ctable id=\"Tab6\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 6\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003eHeadway duration per case\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eCase\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eHeadway (min)\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e6\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e6\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e7\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e7\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e8\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e8\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e9\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e9\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e10\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e10\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eSensitivity of headway regularity to changes in headway is obtained by the difference between PRDM of radial lines and diametrical line (Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e). It is evident from Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e11\u003c/span\u003e that changes in headway duration have a significant impact on the headway regularity. For instance, if a primary disruption of 3 minutes occurs after optimisation, a -0.326 reduction in PRDM is achieved in CASE 6, while smaller reductions are attained in the remaining four cases (-0.280 in CASE 7, -0.262 in CASE 8, -0.225 in CASE 9, and \u0026minus;\u0026thinsp;0.182 in CASE 10). After applying the optimisation procedure, increasing the headway for the same values of the primary headway disturbance results in a worse headway regularity. That is, on lines with longer headway, the effects of optimisation procedure, as measured by PRDM, are worse.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003ctable id=\"Tab7\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 7\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003eSensitivity of headway regularity to changes in headway\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd rowspan=\"2\" align=\"left\"\u003e\n\u003cp\u003eVariant\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd rowspan=\"2\" align=\"left\"\u003e\n\u003cp\u003eCase\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd colspan=\"5\" align=\"left\"\u003e\n\u003cp\u003ePrimary headway disturbance (min)\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e3\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e4\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd rowspan=\"5\" align=\"left\"\u003e\n\u003cp\u003eDiametrical line\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e6\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.400\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.869\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.281\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.716\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e7\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.403\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.775\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.108\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.491\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e8\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.352\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.652\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.996\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.269\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e9\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.305\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.556\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.862\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.120\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e10\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.219\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.466\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.727\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.987\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd rowspan=\"5\" align=\"left\"\u003e\n\u003cp\u003eRadial lines\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e6\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.307\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.648\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.955\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.273\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e7\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.303\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.585\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.828\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.111\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e8\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.265\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.486\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.733\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.946\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e9\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.220\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.416\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.636\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.833\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e10\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.170\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.361\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.545\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.750\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eAs shown in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e8\u003c/span\u003e and Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e12\u003c/span\u003e, changes in headway also exert a significant influence on the passenger travel time. Using the earlier example, if a primary disruption of 3 minutes occurs after optimisation, passengers will experience a -1.31, -1.16, -1.08, -0.85, and \u0026minus;\u0026thinsp;0.56 reduction in travel time in CASE 6\u0026thinsp;\u0026minus;\u0026thinsp;10. Therefore, after applying the optimisation procedure, increasing the headway for the same values of the primary headway disturbance will result in a shorter passenger travel time. However, on lines with longer headways, the optimisation procedure will lead to smaller travel time savings.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003ctable id=\"Tab8\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 8\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003eSensitivity of passenger travel time to changes in headway\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth rowspan=\"2\" align=\"left\"\u003e\n\u003cp\u003eCase\u003c/p\u003e\n\u003c/th\u003e\n\u003cth colspan=\"5\" align=\"left\"\u003e\n\u003cp\u003ePrimary headway disturbance (min)\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e0\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e1\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e2\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e3\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003e4\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e6\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.12\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.28\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-0.62\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-1.31\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-2.03\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e7\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.25\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.32\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-0.47\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-1.16\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-1.90\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e8\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.44\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.54\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-0.26\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-1.08\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-1.69\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e9\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.67\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.68\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-0.01\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-0.85\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-1.49\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e10\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.78\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.08\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.32\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-0.56\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e-1.24\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eBased on the results presented above, it can be surmised that there is a negative correlation between the headway values and the optimisation procedure effects. With respect to headway regularity, it is evident that increasing the primary headway disturbance increases the sensitivity of the optimisation procedure to the headway value, but not on passenger travel time.\u003c/p\u003e"},{"header":"6 CONCLUSION","content":"\u003cp\u003eThis research was carried out with the aim of creating a tool that could be applied in practice to evaluate the benefits/drawbacks of splitting a diametrical line from the point of view of passenger travel time and headway regularity. The application of the optimisation procedure presented here is foreseen in the system design phase, as it can contribute to the robustness/resistance of the PPT to the propagation of headway disturbances.\u003c/p\u003e \u003cp\u003eLine splitting interrupts the accumulation of headway disturbances and improves headway regularity. However, evaluating the positive and negative effects of line splitting is a complex problem, considering that these effects are a function of the primary headway disturbance values, transport demand, actual headways, number of stops and other parameters. As a part of this work, an optimisation procedure was developed to determine the change in average passenger journey due to line splitting.\u003c/p\u003e \u003cp\u003eMost of the currently available methods are based on mitigating the headway disturbances in real time, whereas the optimisation procedure developed as a part of the present study increases the PPT resistance to the propagation of headway disturbances in the design stage. This is also the first attempt to develop a model that can determine the change in average passenger travel time owing to line splitting.\u003c/p\u003e \u003cp\u003eThe validity of the optimisation procedure was confirmed on a real example, which indicated that line splitting is beneficial during peak periods, as it results in significantly less primary headway disturbances.\u003c/p\u003e \u003cp\u003eMoreover, the sensitivity analysis focusing on headway and transport demand demonstrated that the transport demand is positively correlated to the effects of the optimisation procedure, whereby increasing the primary headway disturbance increases the sensitivity of the optimisation procedure to the transport demand.\u003c/p\u003e \u003cp\u003eOn the other hand, there is a negative correlation between the headway duration and the optimisation procedure effects. Specifically, the sensitivity of the optimisation procedure to headway increases as the primary disturbance value increases. However, this effect is only observed for headway regularity, while the sensitivity does not increase when passenger travel time is observed.\u003c/p\u003e \u003cp\u003eThese analyses confirm the validity of the proposed model, but also indicate that the optimisation procedure (line splitting) is most beneficial for lines with short headway and high transport demand where the primary headway disturbances exceed 2 minutes.\u003c/p\u003e \u003cp\u003eWhile these findings are certainly beneficial, the proposed optimisation procedure can be further enhanced by introducing additional criteria such as the load factor, allowing for multi-criteria evaluation. In addition, as the value and location of the headway disturbances is in practice rarely known with certainty, the existing model could be extended by incorporating a fuzzy logic approach. Finally, authors of future studies in this domain could develop a hybrid method that would combine the proposed optimisation procedure at the planning level with some of the models for increasing headway regularity in real time.\u003c/p\u003e"},{"header":"DECLARATIONS","content":"\u003ch2\u003eFunding:\u003c/h2\u003e\n\u003cp\u003eThis research has been supported by the Ministry of Science, Technological Development and Innovation (Contract No. 451-03-65/2024-03/200156) and the Faculty of Technical Sciences, University of Novi Sad through project \u0026ldquo;Scientific and Artistic Research Work of Researchers in Teaching and Associate Positions at the Faculty of Technical Sciences, University of Novi Sad\u0026rdquo; (No. 01-3394/1).\u003c/p\u003e\n\u003ch2\u003eData Availability:\u003c/h2\u003e\n\u003cp\u003eThe manuscript contains data supporting the results of this study.\u003c/p\u003e\n\u003ch2\u003eAuthor Contribution:\u003c/h2\u003e\n\u003cp\u003eConceptualization, P.P. and T.K.; Methodology, P.P. and T.K.; Validation, P.P. and M.M.; Investigation, P.P., M.S. and M.S.; Data Curation, P.P., M.S. and M.S.; Writing-Original Draft Preparation, P.P., M.S., M.M., T.K. and M.S.; Writing-Review \u0026amp; Editing, P.P., M.S., M.M., T.K. and M.S.; Visualization, P.P. and T.K.\u003c/p\u003e"},{"header":"REFERENCES","content":"\u003col\u003e\n\u003cli\u003eAp. 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However, most of the existing methods aimed at improving the headway regularity operate in real time and require precise vehicle location data, making it difficult to implement them in practice. On the other hand, planning-level methods can be used to increase the resilience of public passenger transport (PPT) to the accumulation of headway disturbances. As this is typically done from the operator's perspective, the passengers' perspective tends to be overlooked, motivating the current work. In this article, an optimisation procedure for evaluating the viability of diametrical line splitting in terms of passenger travel time and headway regularity is proposed. The aim is to increase the robustness/resistance of the PPT system to the propagation of headway disturbances without reducing the service quality. The developed optimisation procedure was validated by applying it to real data pertaining to an urban PPT line. The results show that there is a positive correlation between the transport demand and the effects of the optimisation procedure, whereby an increase in the primary headway disturbance increases the sensitivity of the optimisation procedure to the transport demand.\u003c/p\u003e","manuscriptTitle":"Planning-level optimisation of headway regularity","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-03-05 15:33:40","doi":"10.21203/rs.3.rs-3993565/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"b525d80d-50ce-4c3a-9338-49e4a59ce38a","owner":[],"postedDate":"March 5th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2024-03-31T13:59:20+00:00","versionOfRecord":[],"versionCreatedAt":"2024-03-05 15:33:40","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-3993565","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-3993565","identity":"rs-3993565","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}