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In diesem Kapitel wird ein stochastisches Optimierungsmodell vorgestellt, das darauf abzielt, die Pünktlichkeit des öffentlichen Busverkehrs in überlasteten städtischen Umgebungen zu verbessern. Das Modell erreicht dies durch präventive Steuerung von Ampeln, um Abweichungen von den geplanten Ankunftszeiten unter Berücksichtigung verschiedener Verkehrsszenarien zu minimieren. Der Ansatz stellt sicher, dass Änderungen der Ampeln für das gesamte Netzwerk optimal sind und nicht nur einzelne Kreuzungen. Das Kapitel veranschaulicht das Problem anhand eines vereinfachten Netzes mit zwei Buslinien und diskutiert vorläufige Experimente, die an einem Netz mit drei Buslinien durchgeführt wurden. Die Ergebnisse zeigen die Effektivität des Modells bei der Reduzierung von Fahrplanabweichungen unter unterschiedlichen Verkehrsbedingungen. Das Kapitel untersucht auch das Potenzial einer Erweiterung des Modells, um mehrere Zeithorizonte und unterschiedliche Risikoniveaus zu berücksichtigen, wodurch es zu einem vielseitigen Werkzeug zur Verbesserung der Zuverlässigkeit öffentlicher Verkehrsmittel in intelligenten Städten wird.
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Abstract
This paper describes the development of an optimization model that considers the travel times of buses and traffic patterns to minimize deviations from planned schedules for routes across a network in a European city. Traffic congestion is a significant challenge which negatively affects the on-time performance and reliability of bus services. To address this issue, smart traffic management systems such as SCATS (Sydney Coordinated Adaptive Traffic System) are implemented to enhance traffic flow. While SCATS allows buses to request priority at road junctions controlled by traffic signals, conflicts may arise when different buses have opposing requests. To address this, we use a stochastic optimization approach to determine an optimal time-specific traffic interference solution that network owners can impose, such as signal phasing, to minimize schedule deviations for buses throughout the entire network.
1 Introduction
In a highly congested European capital city, public bus transport service providers are faced with the constant challenge of ensuring the actual on-time arrival of buses at bus stops by preventing deviation from the planned schedule, whether buses arrive too early or too late. In this particular city’s operating environment, the majority of the deviations are delays, where buses arrive late at stops. There is also the phenomenon of “ghost buses,” where buses that are scheduled to arrive never actually do.
In order to better support bus transport service providers, we develop an optimization model where buses are given priority to minimise schedule deviations by preemptively controlling traffic light signals which then clears traffic congestion ahead of buses. The approach also incorporates an overall view of the network, ensuring that changes in traffic signals is optimal for the network as a whole. The model can be utilized as part of a ‘smart mobility’ solution, a core element of a ‘smart city’s’ centralized traffic solution [1], with applications in disruption management.
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We illustrate the problem using a simplified \(3\times 3\) grid network with two bus routes, as shown in Fig. 1. In this network, each node in the network corresponds to a traffic intersection. There are two bus routes: Route 1 (red) which traverses along 1–4–5–6–9, and Route 2 (blue) which traverses along nodes 3–2–5–8–7. Nodes 4 and 6 are bus stops on Route 1, and nodes 2 and 8 are bus stops on Route 2.
Fig. 1.
Example \(3\times 3\) grid network with two bus routes. (Color figure online)
Suppose that the wait time at a red signal is set at 0.5 min, and a fixed delay of 0.25 min occurs when the bus makes a left or right turn at an intersection. Consider three scenarios based on the traffic conditions, each of which occurs with a probability 0.2, 0.5, and 0.3, respectively. Table 1 provides the planned and actual arrival times for Route 1 and Route 2.
Denote movements in North-South/South-North directions as ‘NS’ and movements in East-West/West-East directions as ‘EW.’ Assuming that the buses depart on time, an optimal arrangement of signal phases in this example network permits a green signal for NS traffic (red for EW) at node 1 and green signals for traffic flowing EW (red for NS) at nodes 3 and 5. The remaining signal phases in this simplified example network do not have an impact on the schedule adherence. Note that this is because Route 1 must move from node 1 to 4. At node 5, which is the only node where both routes cross paths, the EW movement of Route 1 is given priority to the NS movement of Route 2. This traffic signal configuration is optimized over the different traffic scenarios.
Table 1.
Actual versus planned time of bus arrivals at designated bus stops.
Table comparing planned and actual times for two routes. **Route 1:** - Node 4: Planned Time 0.5, Actual Times - S1: 0.35, S2: 1.16, S3: 2.64 - Node 6: Planned Time 2.5, Actual Times - S1: 1.29, S2: 5.16, S3: 8.17 **Route 2:** - Node 2: Planned Time 1, Actual Times - S1: 0.34, S2: 1.63, S3: 2.55 - Node 8: Planned Time 3, Actual Times - S1: 1.79, S2: 5.38, S3: 8.37
The remainder of this paper is organized as follows. Section 2 presents a review of the related literature. Section 3 introduces a stochastic optimization model to minimize deviations from bus planned schedules. We conduct computational studies of the optimization model on randomly generated grid networks and discuss the results in Sect. 4. Some concluding remarks and the future research directions are provided in Sect. 5.
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2 Literature Review
The European Commission defines smart city as “a place where traditional networks and services are made more efficient with the use of digital solutions for the benefit of its inhabitants and business” [2]. An intelligent transport network (or smart mobility) is one of the core elements in a smart city. In this section, we will provide an overview on current state of developments in intelligent transport systems that are most relevant to our work. For a more comprehensive review on intelligent transportation systems and signal control, see [3] and [4].
Concerning real-time traffic control, studies in the literature have explored dynamic programming [5], a multi-objective agent-based approach in which the traffic signals are modeled as intelligent agents [6], and a hybrid model that utilizes a vehicle location, arrival prediction, and priority selection algorithm [7]. Recent studies have also investigated machine learning approaches to manage traffic signals [8, 9].
With respect to managing traffic flows over the network, Sun et al. [10] developed a bi-level programming model to optimize the signal time cycle with bus priority, minimizing passenger and private vehicle delays at an intersection. Wong [11] focused on optimizing traffic signal timings at intersections using a group-based approach. Adacher [12] employed gradient-based methods and continuously updates system states to minimize the weighted sum of vehicle delays. Lee and Wang [13] proposed an adaptive traffic signal control method to minimize the total person-based delay instead of vehicle-based delays at intersections.
This article contributes a traffic signal control problem with bus priority considered. Existing work in the literature have focused on managing signal phases in real-time or minimizing the delays of one or multiple modes of transport. Our work seeks to improve the reliability of the public transport system by maximizing schedule adherence. That is, in addition to delays, we aim to minimize the penalties imposed when a vehicle is early. The proposed solution approach also allows the consideration of various traffic scenarios, allowing a network owner to control signal phases across the network preemptively and prevent schedule deviations from occurring.
3 Problem Formulation
We define a network \(G = (N, A)\), where N represents the set of nodes (intersections) and A represents the set of arcs (two-way streets). Let R be the set of all bus routes. For each route \(r\in R\), define \(N^r\) as the set of nodes and \(S^r\) as the set of bus stops on route r, where \(S^r \subseteq N^r\). In addition, \(V^r\) denotes the set of nodes in \(N^r\) such that a bus must move NS to arrive at a node in \(V^r\) (Recall that NS denotes both North-South and South-North movements). For instance, in Fig. 1, \(V^1\) contains node 4 as Route 1’s movement from node 1 to node 4 corresponds to a NS movement. Similarly, \(H^r\) denotes the set of nodes in \(N^r\) such that a bus must move EW on route r to arrive at a node in \(H^r\). Analogously, we use \(C^r\) to denote the set of intersections at which the bus must make a left or right turn on route r.
Let \(d_1\) be the amount of time a bus must wait at a red signal, and \(d_2\) the amount of additional time taken when making a turn. Define \(a^r_i\) as the scheduled arrival time of a bus on route r at node \(i\in N^r\), for each \(r\in R\). We also define K as the set of all possible scenarios. Let \(p^k\) denote the probability associated with scenario \(k \in K\) such that \(\sum _{k\in K}p^k = 1\). For each scenario \(k\in K\), \(\delta ^k_{ij}\) represents the actual time taken to traverse arc \((i,j) \in A\).
We use a binary variable \(x_i\) to represent the traffic signal phases at each node \(i \in N\), where \(x_i=0\) corresponds to a green signal for EW traffic (red for NS traffic) and \(x_i=1\), otherwise. Define \(t^{r,k}_{i}\) as the bus on route r’s actual arrival time at node i in scenario k, \(\forall r\in R, i \in N^r, k\in K\). Model () minimizes the expected value of total schedule deviation across the network.
$$\begin{aligned} & x \in \{0,1\}^{|N|}. \end{aligned}$$
(1g)
Constraints (1b)–(1d) ensure that the actual arrival time at node j is the sum of the actual arrival time at node i, the actual time taken to traverse arc (i, j), and any delays incurred on a route for each traffic scenario. Note that constraints (1b) and (1c) correspond to NS and EW movements, respectively. Thus, traffic can only flow without any delay due to red singals in either NS or EW movements at a given intersection, which is enforced through the use of binary variables \(x_i\). Constraint (1c) corresponds to when a bus makes a turn. Also observe that the expected total schedule deviation is given by \(\sum _{k\in K}\sum _{r \in R}\sum _{j\in S^r}p^k\left( |a^{r,k}_{j} - t^{r,k}_{j}|\right) \), which is replaced with objective function (1a) through the introduction of u-variables and constraints (1e)–(1f) to obtain a mixed-integer linear program. The objective function (1a) minimizes the weighted difference between the actual versus planned arrival times at all bus stops across the network.
The model can be expanded to be multi-modal with minor modifications to the objective function and additional variables. For this purpose, we can define variables \(w^r\), for each \(r \in R\), as the relative importance (i.e., priority) of any route of a particular mode of transport (e.g., taxis, personal cars, and bicycles). The revised objective function is given by \(\sum _{k\in K}\sum _{r \in R}\sum _{j\in S^r}p^kw^r\left( u^{r,k}_{1j} + u^{r,k}_{2j}\right) \). As other modes of transportation like taxis, personal cars or bicycle do not contain fixed stops along their routes, we consider their origins and destinations as the stations and the expected arrival times at the origin and destination, where applicable, are used as scheduled arrival times. Note that with the introduction of variables w, we also permit the differentiation of traffic flows at route level, enabling system owners to prioritize traffic at known bottlenecks in the network.
4 Preliminary Experiment
The preliminary experiments were conducted on a Dell vPRO equipped with 10 physical cores, an Intel Core i7 Processor, and 32 GB of RAM. The model was implemented using Pyomo library and CBC solver [14].
The experiment was conducted on a \(50\times 50\) grid network with a set of three bus routes \(R=\{1,2,3\}\). We consider the set of scenarios \(K = \{1,2,3\}\). Each scenario occurs with probabilities 0.6, 0.3, and 0.1, respectively. The actual travel times \(\delta ^k_{ij}\) across arc \((i,j) \in A\) were randomly generated following uniform distributions in [1, 4], [4, 6], and [6, 10] for scenarios \(k=1\), 2, and 3, respectively. We set \(d_1 = 0.5\) and \(d_2 = 0.25\).
Ten replications of this experiment were conducted and the results are reported in Table 2. For each route and scenario, Table 2 reports the planned arrival time and actual arrival times in minutes in columns Planned Time and Actual Time, respectively. Column Deviation reports route r’s deviation from schedule for each bus stop i in each scenario k, which is given by \((|t^{r,k}_{i} - a^{r,k}_{i}|/a^{r,k}_{i})\times 100\%\). Under Planned Time, Actual Time, and Deviation, columns St 1, St 2, and St 3 correspond to the 1st, 2nd, and 3rd bus stop, respectively, on each route. At optimality, the objective function value representing the total expected schedule deviations of the entire network is 174.7 min.
Table 2.
Network performance under an optimal signal phasing solution.
Route
Scenario k
Planned Time
Actual Time
Deviation
(\(p^k\))
St 1
St 2
St 3
St 1
St 2
St 3
St 1
St 2
St 3
1
1 (0.6)
30
60
70
30.9
57.8
69.2
3%
3%
1%
2 (0.3)
43.8
95.6
117.0
46%
59%
67%
3 (0.1)
72.7
157.8
190.0
142%
163%
171%
2
1 (0.6)
25
50
65
25.7
53.5
60.2
3%
7%
7%
2 (0.3)
44.3
91.2
114.1
77%
82%
75%
3 (0.1)
68.7
153.81
177.9
175%
208%
174%
3
1 (0.6)
35
60
75
38.3
60.4
64.3
10%
1%
14%
2 (0.3)
46.48
95.69
104.5
33%
59%
39%
3 (0.1)
64.0
152.6
167.5
83%
154%
123%
Scenario 1 represents the arc travel time planned arrival times for buses at their stop stations under normal traffic conditions. Scenarios 2 and 3 represent heavy traffic and ultra-heavy traffic conditions, which have longer travel times. The deviations between actual and expected times in the most likely scenario (Scenario 1) are minimal. As travel times of the arcs increase in the second scenario, these deviations also grow larger. In the least probable scenario where travel times are significantly longer, the deviations become notably significant. However, by considering the probability of all scenarios, the overall performance of the traffic signal model is optimized across all defined scenarios.
In addition, we also conducted experiments to understand how the objective function value changes when the probabilities associated with different scenarios change. The modified probability for scenario k is reported under column \(p^k\), for each \(k = 1, 2, 3\), in Table 3. In this set of experiments, five replications were conducted and the objective function values (i.e., weighted time deviations over the network) were obtained, whose average is reported under column Average Objective Function Value in Table 3.
Table 3.
Mean network time deviation under varying scenario probabilities
ID
\(p^1\)
\(p^2\)
\(p^3\)
Average Objective Function Value
1
0.90
0.05
0.05
69.7
2
0.60
0.30
0.10
177.5
3
0.33
0.33
0.33
368.8
4
0.10
0.40
0.50
499.0
As previously mentioned, the first scenario pertains to normal traffic conditions, characterized by lower values of travel time between arcs. In the second and third scenarios, travel times escalate due to heavy and ultra-heavy traffic conditions. Consequently, with the decrease in the probability of the first scenario, the total delay for all buses increases, in accordance with expectations for heavier traffic conditions.
5 Conclusion
We proposed a stochastic optimization model that specifically prioritizes buses in traffic signal phasing and optimizes the performance of all bus routes over the network. For future research, we plan to extend the current model to account for multiple time horizons to enhance its practicality and applicability. Different penalties for earliness and tardiness may be considered, as is usually the case in practice. As the proposed model in this article is risk-neutral, we may also consider different risk levels in future studies to adapt to different profiles of network owners. Finally, we plan to apply the model in a highly-congested urban European city to evaluate its practicality and needs for further developments, while benchmarking its performance against current best practices.
Acknowledgements
The authors are grateful to anonymous referees and the editorial team, whose feedback has helped improve this article.
This study was funded by the Science Foundation Ireland, grant number 22/NCF/FD/10957.
Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.
The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
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