Designing a Robust Approach to Resolve the Tehran Metro Trains Scheduling Problem in Uncertain Conditions
پژمان صالحی
1
(
PhD student of Islamic Azad University, Roodehen Branch
)
Mehran Khalaj
2
(
Department of Industrial Engineering, Parand Branch, Islamic Azad University, Parand, Iran
)
davood jafari
3
(
faculty of industry,islamic azad university of parand
)
الکلمات المفتاحية: Particle Swarm Method, Combinations Optimization, Robustness Approach, Scheduled Timetable and Tehran Metro,
ملخص المقالة :
Reliability and punctuality in the traffic supervision of the metro network are the key performance indicators for the evaluation of this industry's efficiency which can lead to the participants' satisfaction. A common technique to improve the level of the Metro traffic control system’s reliability and performance increasing the rate of train receives and dispatches from special position determined, particularly in the different periods of designing train timetables, which can be out-of-the-way or lead to a stable level. Therefore, to increase the permanency level of the metro timetables, the traffic control center’s operation in the metro network considers fixed time values as Buffer times between rail events with a high probability of flaws using normal procedures in the timetables. In this respect, the first delay in the metro network is the possibility of its spread in all of the lines and its effect that will be realized on all of the entities and the being of subsequent luckless consequences that will be prevented. Adding to this, Buffer time in the train schedule in some cases may cause the capacity of the metro network to be reduced, which could be the basis of traffic special effects for eventful lines such as Tehran Metro’s line one. However, the ability of time tables to add fixed times values as a Buffer time between two conflicting events in the metro network can be whispered subsequently in this case, it is necessary to allocate Buffer times with sufficient accuracy and according to the traffic priorities of dispatching and receiving trains. Some important measures in Metro passenger activities (such as unpredicted train dispatches) take place properly and prevent certain incidents. In current study, the objective of cultivating and improving the stability of train running schedule tables was accomplished by assigning optimum Buffer in the network which, times has been studied utilizing innovative methods. So, the resources for allocating optimum time have been modeled via combinatorial optimization algorithms and particle swarm optimization methods. In this algorithm, each Buffer time according to the priority of the events as well technical, economic, and operational criteria is shown to one of the prevailing assignments of the schedule tables which results and its effects in terms of time units (seconds). Ultimately, the validity of the proposed model that is presented applying the Tehran Metro line one is verified.
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Research Article
Designing a Robust Approach to Resolve the Tehran Metro Trains Scheduling Problem in Uncertain Conditions
Pejman Salehi1 | Mehran Khalaj2 | Davood Jafari3 |
1. Department of Industrial Engineering, Pa.C., Islamic Azad University, Parand, Iran, Email: Pezhmansalehi@piau.ac.ir |
2. Department of Industrial Engineering, Pa.C, Islamic Azad University, Parand, Iran , Email: khalaj@iau.ac.ir (Correspond Author) |
3. Department of Industrial Engineering, Pa.C, Islamic Azad University, Parand, Iran, Email: djafari@piau.ac.ir |
ABSTRACT
| DOR: https://doi.org/10.71584/MGT.2025.1207024 | |
Article Info: | Abstract: Reliability and punctuality in the traffic supervision of the metro network are the key performance indicators for the evaluation of this industry's efficiency which can lead to the participants' satisfaction. A common technique to improve the level of the Metro traffic control system’s reliability and performance increasing the rate of train receives and dispatches from special position determined, particularly in the different periods of designing train timetables, which can be out-of-the-way or lead to a stable level. Therefore, to increase the permanency level of the metro timetables, the traffic control center’s operation in the metro network considers fixed time values as Buffer times between rail events with a high probability of flaws using normal procedures in the timetables. In this respect, the first delay in the metro network is the possibility of its spread in all of the lines and its effect that will be realized on all of the entities and the being of subsequent luckless consequences that will be prevented. Adding to this, Buffer time in the train schedule in some cases may cause the capacity of the metro network to be reduced, which could be the basis of traffic special effects for eventful lines such as Tehran Metro’s line one. However, the ability of time tables to add fixed times values as a Buffer time between two conflicting events in the metro network can be whispered subsequently in this case, it is necessary to allocate Buffer times with sufficient accuracy and according to the traffic priorities of dispatching and receiving trains. Some important measures in Metro passenger activities (such as unpredicted train dispatches) take place properly and prevent certain incidents. In current study, the objective of cultivating and improving the stability of train running schedule tables was accomplished by assigning optimum Buffer in the network which, times has been studied utilizing innovative methods. So, the resources for allocating optimum time have been modeled via combinatorial optimization algorithms and particle swarm optimization methods. In this algorithm, each Buffer time according to the priority of the events as well technical, economic, and operational criteria is shown to one of the prevailing assignments of the schedule tables which results and its effects in terms of time units (seconds). Ultimately, the validity of the proposed model that is presented applying the Tehran Metro line one is verified. Keywords: Particle Swarm Method,Combinations Optimization, Robustness Approach, Scheduled Timetable and Tehran Metro | |
Received Date: 2025/05/16 Accepted Date: 2025/07/11
Published Online: 2025/08/30
| ||
How to Cite: Salehi, P., Jafari, D., & Khalaj, M. (2025). Designing a robust approach to resolve the Tehran Metro trains scheduling problem in uncertain conditions. Journal of Industrial Strategic Management, 9(1): 41-79. |
1. Introduction
Metro is one of the urban transportation systems modes playing an effective role in the citizen's mobility, especially in great cities, due to the value of travel time (Headway) and the increasing ratio in passenger demands (Jamili & Aghaee, 2018). Planning and evolving train timetables are some of the most important and also routine issues in the Metro industry, which has been of illimitable interest to researchers in the field of transportation planning and design in recent years (Kang et al. al., 2018). Overall, there are three steps in the train movement planning and scheduling in the Metro: first, the process of train scheduling. second, the process of assigning train fleets to rail lines, and third, assigning crews to rail fleets (Larsen et al., 2017). The lifetime of the metro industry in Iran is a little more than two decades, therefore, from this time forth, the technologies related to the train running scheduling are considered a new phenomenon.
Figure 1. Comprehensive map of Tehran Metro lines
Source: Tehran Metro technical documents and central archive
In recent years, new approaches have been developed for planning Metro timetables known as the robust programming approach. Robust design is one of the divisions of systematic and structured scheduling, the purpose of which is to respond to uncertainties in metro Network scheduling (Mao et al., 2017). Metro is one of the industries that face ambiguity and uncertainty in its state space for various aims (Parbo et al., 2019). For all intents and purposes, in real conditions, the subway operation and train fleet movement in metro lines are different from the pre-planned nominal timetable. It causes the complete implementation of the nominal schedule. In addition to the mentioned case, there are many other reasons for this issue, some of which could be caused by the delay in the train’s dispatching from the origin stations or the point of the rail route, as well as the disaster of fixed and mobile rail equipment in the process of running trains. A delay in the initial scheduling plan causes the dispatching of trains not to be carried out according to the expectations of the Metro operating companies and passengers, and as a result, its implementation is different from the initial plan (Schmöcker et al., 2015). occurrence and spreading delays in the Metro network is a problem that imposes relatively great and sometimes irremediable costs on the Metro operating companies and its benefactors. One of the solutions to reduce these costs is routine dynamic programs that are resilient to uncertainties, which is called robust planning (Nachtigall & Voget, 2006). In recent years, through the increase in demand caused by the tension in the metro industry to improve the service level, dedication has been paid to the issues of Metro traffic robustness and distortions control. the advances accomplished in the computer microprocessors’ speed and operations research algorithms have made it possible to achieve steady programs and develop table retrieval tools based on optimization and integration (Puong & Wilson, 2011). There are three general approaches to solve the running and movement of Metro trains planning problem:
- Integrated Approach
- Robustness Approach
- Delay Management Approach (Tamannaei et al., 2019).
Some of the Metro operators have solved the running and movement of trains' planning problems in their rail network through integrated approaches or delay management. Therefore, the operation of Metro lines is subject to some important uncertainties as the Metro network disturbances are often due to unscrupulous climate conditions, special unpredicted cases related to the corrective maintenance and emergency repairs of fixed and mobile equipment, safety controls that are applied via traffic centers, system considerations through Train and station safety, security cases or other similar circumstances occur (Kang et al., 2018). The objective of robust planning in the metro network is to produce a comprehensive and barrier-free timetable that is sensitive to metro network conflicts. The type of regulating train timetables is known as demand-based strategic functions in the rail network, which also includes the dimensions of events and network uncertainty (Lidén & Joborn 2020). Studying the research literature shows that two major approaches have been defined for the program, stable planning in the rail network and that, the stochastic programming model is used to reflect and explain the uncertainties. In this method, disturbance modeling is used to define and verify the conditions that consider the stochastic nature of the problem. For example, the research by Tirachini et al (2014) showed that the conventional models of assigning trains and crews to lines and fleets by constructing less stretch in the hubs and circulation loops of the rail track show better performance in punctuality. Also, the fact that robust planning models try to prevent the happening and spreading of delays and disturbances caused by this despite it in a preventive manner. Robust recovery models are based on the fact that in the event of a disturbance in the metro network according to the conditions, the best response indicates the possibility that one from the set of answers so that the effects of distortion on the network are reduced to the minimum potential and the spread of delay in the entire network is prevented (Ulusoy et al., 2013). The second approach is to use robust planning to set up Metro train timetables. In this method, train running and scheduling are done in such a way that the whole system is flexible and resilient to probable disruptions and delays and that is less affected by negative consequences (Van Aken et al., 2020). For that reason, the use of time Buffers (mediators) as a solution that could be used in the recovery phase to reschedule train timetables and train running in the metro network will appeal in the direction of reports and prevent the occurrence of cumulative and dispersion effects as well as the spread of probable delays. (Wang et al., 2017). Instead, since the use of these reserve times in case of no delay event means the waste of metro resources and the relative decline of the network capacity, therefore, one of the main challenges of stable planning is the being of unused Buffer time values during the scheduling of the journey and the mobility of Metro lines. Too, the objective function of train planning models is used to determine the initial scheduling method and minimize costs through linear programming, not the direct reduction of operational costs of train traffic control, as a result of these models which certainly leads to the fact, that such programs will not logically have the lowermost amount of Buffer time between dispatch rows in the timetable of sequential trains, which causes major problems such as described below in the metro operation lines:
- Increasing the prospect of disaster in the initial schedule and the uncertainty of the entire network in the face of possible long-term disruptions
- Reducing the coefficient of the table retrieval options utility
- Increasing the cumulative effects probability and delays spread (Xiong et al., 2018).
According to what mentioned above, reliability criteria and punctuality index in metro traffic management are midst the key performance tools for measuring the effectiveness of the train dispatch and reception systems affecting significantly the passenger experience attraction in the metro network (Yue et al., 2019). In the metro network, the train movement is based on an accurate and initial schedule including several components such as the scheduled time for dispatching and receiving trains at the origin, route length, and destination stations (ADENEY, 2004). An unambiguous rail route for tracking the trains' movement is founded on a written timetable as a sequence of departure and stop times at stations and railway sections. Important limitation of metro traffic planning is the necessity of trains' spatial separation in a similar infrastructure, such as rail blocks between two signals to prevent accidents among rail vehicles by keeping safety margins (Zhao et al., 2019). In metro operations management, this obligation is made automatically through control and signaling systems, such as intelligent monitoring, which can prevent possible accidents, unnecessary braking, and frequent train accelerations and as a result, unsafe activities.Therefore, in a dynamic and appropriate schedule, it is necessary to take into account the physical separation of trains and the minimum Headway in the network, microscopic inferences, and through the theory of time sequences, the journey length is calculated and achieved values would be added to the train travel times on the rail route (De-Los et al., 2015). Sometimes, unexpected events and obvious changes during the Metro trains' headway may cause delays in dispatching trains from their origins or receiving points at destinations, and as a result, the precise implementation of a pre-planned timetable is impossible (Abbink et al. 2014). Additionally, in an eventful and high-traffic metro network, often a delayed load of the train fleet from one line could be moved to train passengers in other rail lines, especially at connection stations that have the similar infrastructure and their schedule for passengers moving continuously used for the crew and planned facilities are shifted. So, to improve the level of rail network stability and prevent the delays spread in the metro network and effect on the operational processes, a Buffer time is considered in the interval time of train movement in the form of numerical sequences (Barrena et al., 2017). In terms of Buffer time, the objectives are to absorb the possible disturbances of the metro network and as a result, the deviation of the train's running time from the initial timetable, which would prevent spreading delays to the next trains that run consecutively on the metro routes (Cacchiani et al., 2015). In this research, an approach based on "optimizing combination " has been used to add supplement time to high-traffic timetables in Tehran Metro. For any certain sequence of train fleets in the metro network, the time intervals formed in the schedule would be adjusted through the complements included in the planning process, which are distinguished as time residuals after integration calculations and employing Buffer times. It could be stored in the rail tracks for normal conditions and considered to manage and control distortion situations (Fischetti & Monaci, 2012). Decreasing the time reserves in Metro schedules is often done to increase the minimum Headway of running trains by setting specific amounts of supplement times by traffic controllers so that its effects are small for the metro network scale (Ghaemi et al., 2019). Therefore, the main question of the present study is how to allocate the Buffer times toward the interval times of the train fleet in the schedule tables so that the stability of the line schedule reaches its maximum value. This is followed by modeling the optimizing combinational algorithm. In this way, a specific Buffer time is assigned to each interval time in the train dispatching, which has a weight value equivalent to the timetable accumulative. The Buffer times values in the studied tables as well as the size of the deviations resulting from it to coup with the unexpected events of the metro network through parameters such as the delay effect, its spread in the metro network, and the analyzing sensitivity process in the effect delays logic on other trains and the stations are decorated. Therefore, in the present study, a challenge has been made to formulate the optimum allocation of supplement times utilizing the particle swarm algorithm. Therefore, the used tools which compared focusing on the alternative's quality and the computational solutions complexity. The case study of this research is the operational space in Tehran Metro line one, which mainly tries to study the appropriate allocation of Buffer times according to the traffic conditions of the Tehran Metro. So, in the current research, the scheduling framework of the train movements is explained as one of the important and effective subdivisions in the subway processes, which can verify the new approaches used in this part of the Metro industry for stable planning and delay management. Metro experts have mainly considered the delay as an inseparable part of the subway operation, which the operating companies in different subways aspect according to the type of passenger movements (in terms of fast or normal). Therefore, to resolve the delay problem in the metro network, various solutions have been proposed and developed, one of the most important and effective approaches is based planning, which can respond to ambiguous conditions and uncertain situations. This approach can identify the most important factors that cause delays in running trains and non-operationalization of the initial schedule and provide sustainable solutions to do away with limitations and optimize the schedule.
2. Literature Review
Xu et al. (2019) proposed three approaches to procurement of the amount of noise (distortions) in Metro routes, as: the first, a simple estimation method through comparing with a pre-planned timetable (for example, a schedule Delay in receiving the train at the destination up to three minutes caused by congestion and track conditions); the second, with mathematical expectation or average weighted of the delays that occurred for the trains fleet in the train schedule program, and,the third, using the data delay collection of the train fleet recorded in the traffic control centers database (Xu et al., 2019). The "distortion threshold" parameter is considered as the average non-negative delay acquired from the traffic data recorded in the train fleet database (Yan et al., 2013). The Buffer times for each train is the amount of time that is calculated schedule among that specific train and the other trains before it in the running metro track sequence. If this time is less than the distortion threshold time, the risk of conflict on the railway track increases. Therefore, the use of robust methods causes small improvements in terms of Buffer times in the rail route (Abkowitz & Tozzi, 2004).
Uncertainty in linear programming problems or optimization systems is one of the important issues in the metro industry. As a result, various approaches such as stochastic or fuzzy planning have been developed to deal with uncertainty in the subway through mathematical modeling. The robust planning approach is one of the most up-to-date methods of dealing with uncertainty in the Metro, which has been widespread due to its special capabilities in identifying limitations and optimizing the objective function. Therefore, the current research objectives are to analyze different planning methods and their applications in metro management network disturbances. The robust decision-making approach is a method in which the focus is on stability promoting against environmental ambiguities, and the method that has the least possible variations against unstable conditions is a resolution to the optimization problem in the general state spaces formal. They could have stability and optimum robustness (ADENEY, 2004). Possible stability means that the acquired solution should remain feasible for all situations or at most for parameters with uncertainty. Also, optimum stability is a situation in which the value of the objective function remains stable for the proposed solution, and therefore, for all or most of the parameters with uncertainty, it is neighboring to its optimum value and has the minimum possible deviation from the optimum value (Birge & Louveaux, 2014). CADARSO et al. (2016) have provided a framework for robust optimization, which includes two main concepts: "stable solution" and "stable model". This optimization method is related to approaches in which the data type is scenario-oriented. The results of this research state that the use of operational research in mathematical planning models leads to uncertainty through uncertain and ambiguous data, so opposite this type of data using methods of sensitivity analysis stochastic planning is problematical. It deals with implementation, which includes the structural (fixed) and the control parts. In this case, the linear programming model for optimization is written as follows:
Subject to: Ax=b Bx+Cy=e
| (1) |
| (2) |
s.t Ax=b Bx+Cy=e
| (3) |
symbol | Descriptions | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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tR | The termination time of the disturbance in the Metro and the start time of the timetable retrieval period | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
k | Index related to the dispatch of trains from the origins | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
i,j | Index of stations | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| Passenger arrival rate at station i at time t | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| The percentage matrix of travel demand in the destinations received by the train for time t | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
mk | The set of stations along the rail route of dispatching and receiving trains from k origins after time tR | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| The set of dispatches after the end of the rail disruption for trains whose journey has not yet started after time tR. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| The set of dispatches after the termination disturbance must be received at the travel destinations and have not yet arrived at the destination after time. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| The set of dispatches that occurred before the termination disturbance, which specify the train travel time in the direction of the dispatch route from u stations of the metro route. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| The set of dispatches before the termination disturbance for the trains that are going back to destinations d at the mentioned time. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
TA | The total set of transmissions that are sent from the origin after the disruption time is calculated through the contradictory relationship:
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TB | The total set of trains that have been dispatched from the origin before the time of disruption and the amount is obtained from the following relationship:
| |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| The maximum number of stations that the dispatch train can pass without stopping at time k. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| The number of passengers present at station i going to destination j at the time of disturbance tR | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| The number of passengers who were present in the dispatch train k at the time to and intend to reach the destination j. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| The time required to increase or decrease the acceleration of the train is assumed to be the same for all the railway parts of the running route, rendering the average slope of the line. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
h | The index related to the number of active trains on the Metro route | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
t | The index related to the row number of the Headway in the timetable | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
d | The index related to the direction of the running train on the two-way Metro track, which has an appearance value d=1,2 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
m | The number of metro stations | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
k | Indexes related to the number of metro stations | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
n | The number of active trains on the rail sections | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
T | The number of interval times counted in in the Metro tables | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Vave | The average speed of trains on the railway track | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Vmax | The maximum speed of trains on the railway track | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Dis k, k+1 | The distance between two consecutive stations k and k+1 in an assumed Metro line | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
C maximum | Maximum capacity of a train on metro lines for peak passenger hours | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| The minimum duration of train stops at metro stations | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
rest minimum | The minimum duration of metro trains stopping at each metro route stations | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
CD | Duration of delay caused by passenger congestion at each station (in seconds) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| The arrival rate of passengers to station k in direction d for interval times 't' in the Metro network | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| The rate of passengers getting off at station k in direction d for the Headway t in the Metro network | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
TP | The total number of subway passengers on a particular Metro line for a typical day | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
TT | The total number of train passengers dispatched for a normal day in the metro network | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| The number of passengers waiting at station k in direction d for the assumed interval time t in the Metro network | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| The number of passengers on the train h for the time interval t. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| The departure time of the lth train from station k for interval times t in direction d in the metro network | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| The number of passengers of the lth departure from station k in direction d for the metro network | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| The number of passengers in the lth train movement who have entered station k in direction d. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| The number of passengers on the Metro train in the lth departure at station k and in direction d | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
FR | Passenger transportation rates in the metro network | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| The duration of measuring of rail segments between station k and k+1 in the interval times t of the metro network | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| The stopping time of the lth train from station K in direction d for the metro network | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| interval times between l and l+1th running train in direction d for metro network |
s.t
| (15) |
s.t
| (21) |
So, we will have: | (27) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| (28) |
Min C0 s.t
| (33) |
Min C0 s.t
| (34) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| (35) | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| (36) |
Min C0 S.t
| (37) |
Min C0 S.t
| (38) |
Row number | The time on a non-holiday working day | The Headway of the interval time in the relevant floor (minute) | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 5:00 – 7:00 |
| ||||||||||||
2 | 7:00 – 9:00 |
| ||||||||||||
3 | 9:00 – 11:00 |
| ||||||||||||
4 | 11:00 – 13:00 |
| ||||||||||||
5 | 13:00 – 15:00 |
| ||||||||||||
6 | 15:00 – 17:00 |
| ||||||||||||
7 | 17:00 – 19:00 |
| ||||||||||||
8 | 19:00 – 21:00 |
| ||||||||||||
9 | 21:00 – 23:00 |
| ||||||||||||
10 | 21:00 – 23:00 |
|
| FR |
|
| T |
|
| Number of trains | Number of stations | Problem parameters | |||||
1800 | 48% | 95 | 35 | 7 | 80 km/h | 45 km/h | 25 | 29 | Values |
Optimal Headway (in seconds) | The level of conservatism
| The mean of the reaction variables Z1(s), Z2(%) | The standard deviation Reaction variables | Minimum variables Reaction | Maximum variables Reaction | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
H2=(630, 180,270,195,490) |
| Z1= 225 Z 2= 71 | Z 1=6/2 Z 2=5 | Z1=201 Z 2=65 | Z 1=239 Z 2=66 | |||||||||
H1=(740, 243,290,220,560) |
| Z 1= 254 Z 2= 72 | Z 1=7/2 Z2=9 | Z 1=208 Z 2=68 | Z 1=248 Z 2=73 | |||||||||
H1=(702, 256,311,256,690) |
| Z 1= 267 Z 2= 79 | Z 1=8/7 Z 2=6 | Z 1=239 Z 2=72 | Z 1=256 Z 2=77 | |||||||||
H1=(702, 283,259,261,786) |
| Z 1= 279 Z 2= 82 | Z 1=8/5 Z 2=6 | Z 1=240 Z 2=78 | Z 1=266 Z 2=80 | |||||||||
H1=(725, 296,400,280,890) |
| Z 1= 290 Z 2= 84 | Z 1=7/4 Z 2=4 | Z 1=261 Z 2=80 | Z 1=268 Z 2=80 |
Row | The title of the parameter | a numerical value (second) |
1 | The minimum interval time for trains | 150 |
2 | The maximum interval time of trains | 600 |
3 | Minimum stop time of the train at the passenger platform | 30 |
4 | The time required to retrieve the table | 4200 |
5 | Maximum delay time | 2100 |
In this case, to solve the problems by using the mixed integer linear programming method, it is necessary to determine the passenger demand for all the events of the state space and finally check the limitations of the model. The data related to passenger demand is obtained based on the outputs of the Tehran Metro comprehensive traffic system. The peak hours of travel in Tehran metro line four are in the morning and evening hours, which for this study is a part of the service period between the hours of 9:00 and 12:00. Based on the priorities of the comprehensive traffic system, the traffic amount of some stations of the 4 line such as NirooyeHavaei and Ebneina is assumed to be of little importance. This is due to the volume of passengers and the waiting time of the line 4 for the ease of calculation of the congestion variable. However, the capacity of A passenger is assumed to have constant and fixed service delivery times. Some traffic parameters of Tehran Metro Line 4 for a normal month of the year are shown in Table 6:
Table 6. Some traffic parameters of Tehran Metro Line 4 for a normal month of the year
Total number of planned trips | Total number of planned trips | Total number of extra passenger trips | Total number of canceled trips | Total number of delayed trips | Total delay time (in seconds) | Average delay of each passenger trip (seconds) |
11151 | 11133 | 31 | 18 | 28 | 5370 | 1/03 |
source: Tehran Metro Comprehensive Traffic System
Table 7 shows the demands of the origin, the stations along the route and the destinations of line 4 based on the outputs of the traffic system:
Table 7. Requests for point of origin, stations along the route, and destinations of Line 4 of Tehran Metro between the hours of 9 am and 12 pm
hours Statins | 9-10 | 10-11 | 11-12 |
Shahid Kolahdooz | 21773 | 28348 | 35798 |
Nirooye Havaei | 35859 | 43816 | 53472 |
Nabard | 36936 | 37106 | 48567 |
Piroozi | 17166 | 20887 | 25451 |
Ebn-e Sina | 24215 | 30308 | 40947 |
Meydan-e Shohada | 36308 | 42282 | 52861 |
Darvazeh Shemiran | 8733 | 8141 | 7621 |
Darvazeh Dowlat | 16861 | 15358 | 13959 |
Ferdowsi | 24136 | 31271 | 17504 |
Teatr-e Shahr | 34389 | 25319 | 17577 |
Meydan-e Enghelab | 67741 | 44894 | 32153 |
Towhid | 31614 | 26683 | 21653 |
Shademan | 25814 | 25481 | 25229 |
Doctor Habib-o-llah | 26214 | 21210 | 20604 |
Ostad Moein | 46263 | 28227 | 21480 |
Meydan-e Azadi | 62577 | 71180 | 84321 |
Bimeh | 78604 | 60807 | 51638 |
source: Tehran Metro Comprehensive Traffic System
The optimization of waiting time and passenger capacity was implemented in a case study of Tehran Metro Line 4 as a small-scale 296 km rail network. The simulation results show that just by reducing the number of trains and as a consequence passenger dispatches, the computational complexity is reduced to a significant extent. As a result, relatively favorable effectiveness is achievable in the conditions of line congestion, especially in intersection stations.
Figure 14. Distribution of passengers during the operation hours of Tehran Metro line 4
(Tehran Metro Comprehensive Traffic System, 2024)
Optimization passenger-oriented models were evaluated based on two objective functions with different weights. In these functions, cancellation of passenger movements due to disturbance is 
and for other situations such as 
delay and 
.
Table 8 shows the process of canceling passenger dispatches and train delays in implementing schedules and train movements in Tehran Metro line 4.
Table 8. Train cancellation and delay process
| cancellation | delay | ||||||||||||
| 18 | 2100 | ||||||||||||
| 18 | 4208 |
The weight | Shahid Kolahdooz | Nirooye Havaei | Nabard | Piroozi | Ebn-e Sina | Meydan-e Shohada | Darvazeh Shemiran | Darvazeh Dowlat | Ferdowsi | Towhid | Shademan | Habib-o-llah | Azadi | Bimeh |
Ωsc=0 | 0/104 | 0/042 | 0/23 | 0/143 | 0/144 | 0/177 | 0/264 | 0/362 | 0/191 | 0/151 | 0/183 | 0/288 | 0/167 | 0/254 |
Ωsc=4200 | 92 | 0/28 | 0/193 | 0/145 | 0/147 | 0/161 | 0/254 | 0/105 | 0/105 | 0/202 | 0/247 | 0/288 | 0/137 | 0/244 |
In Figure 15, each continuous line in the diagram shows the level of congestion of a station on line four of the Tehran Metro during the duration of operation. The dashed lines in the graph indicate the beginning and end of the disruption, according to the levels of rush hour traffic in the waiting time. Because many passengers cannot reach their destination due to the effects of the disruption, they use another means of transportation to their destination. For example, passengers with congestion caused to happen while waiting for a train at “Midan Azadi” station to reach “Eram Sabz” station should not continue their journey to get from their destination by train until the disruption ends.
Figure 15. A timetable for traffic disruptions, in a case study ()
A case study of Tehran Metro line 4 provided the possibility to evaluate the effectiveness and practicality of the rail transport fleet. Based on the direction of a comparison between the theoretical models for optimizing the travel and movement of trains in the conditions of disruption and practical models. The results show that passenger-orientated models for intersection stations lead to providing more favorable solutions. However, in terms of operation, timetables require a considerable amount of cancellation of passenger journeys due to insufficient exit points of trains and the obstruction of railway parts in the disturbed line. Therefore, providing models that can face the cancellation of passenger travel provides appropriate strategies to face the disruption. This model in which ready trains are located along the route and in appropriate buffer locations improves the overall performance of trains on subway lines during disruptions. It provides a basis for trains to be used for control measures at a later stage. By using this method, instead of dealing with a long delay for each station, especially switching between stations. It is possible to use the buffer values presented in the tables and intersecting stations, which seems sufficient to minimize the total delay. In this research, change levels of computational complexity, and optimization models have been implemented on a smaller scale. The simulation results show that the model presented in this research can significantly reduce the maximum level of traffic congestion in the timetable. Using the model presented in this research, a proportional balance can be generated between the deviation from the initial schedule and the level of passenger congestion at the stations. The achievements of the study show that using different integer programming approaches in passenger-orientated modes is effective enough for long computing times and small networks. Hence, it could occur that the integer programming may not have the time to retrieve the timetables and reorganize the efficiency. One of the alternative methods in this case can be the neighborhood search method. To optimize the efficiency of the many components included in the model. In this way, the neighborhood search algorithm could minimize the space for finding the most optimal solution. The assumption is that the recognition operator searches for the optimal solution in the neighborhood. The model needs an initial solution that, in addition to determining the sequence of events on the schedule, enables minimizing the deviation from the initial schedule. The acceptability of mathematical relationships in complex integer programming to cross-sectional robustness of events is one of the strengths of this method. The overall results of the case study show that the small-scale neighborhood search method is faster than the mixed integer programming method. The preliminary results of the case study show that the neighborhood search method could find an answer in 523 seconds in the short set. This has been done while the mixed integer programming method obtained a justified solution in the solution set in about 11 hours.
5. Conclusion
A look at the history of the research conducted in the field of the Metro shows that in the category of optimizing the characteristic uncertainty in the demand of Metro passengers, it has been underestimated. This issue may lead to overcrowding of the passengers and the occurrence of random disturbances and as a result cause delays in planning the train’s schedule. In this study, the Tehran Metro was studied, taking into account the uncertainty in demand and the duration time of the blocks. To determine a stable schedule that can absorb small disturbances, a simulation optimization tool has been used. Metro model considers the limitations caused by the type of rails (single rail or multiple lines), the speed of the trains, the stop time limit at the stations, the limit related to the head distances, the limit of the number of lines in each station, and the safety limits and considerations. It was simulated in CPLEX software. Using the particle swarm algorithm, some points in the acceptable solution space were sampled, to estimate the relationship between the input (distances) and the output of the simulation model (passenger waiting time and average train rate) from random simulation, one for the objective function and the constraint were used and their suitability was evaluated. Also, Bertsmias and Sim's stable optimization method was used to formulate the stable counterpart problem for functions. The obtained stable peer model is solved using the particle swarm algorithm. For a case study, Line 1 of Tehran Metro was simulated, and stable optimal solutions for working days (Saturday to Wednesday) in 9 time periods with different demand rates for passengers were obtained. For future studies, the presented development is suggested to solve the scheduling problem in the entire network and to consider the connections and correlation between the lines. It is also suggested to use other pseudo-models such as regression pseudo-models' neural networks and other artificial intelligence techniques and compare the results obtained from the particle swarm method.
Funding
There is no funding support.
Declaration of Competing Interest
The author has no conflicts of interest to declare that are relevant to the content of this article.
ACKNOWLEDGEMENTS
We would like to express our gratitude to the anonymous reviewers for their valuable comments, which greatly contributed to improving our work.
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