A bi‐level programming approach — Optimal transit fare under line capacity constraints

The fare of a transit line is one of the important decision variables for transit network design. It has been advocated as an efficient means of coordinating the transit passenger flows and of alleviating congestion in the transit network. This paper shows how transit fare can be optimized so as to balance the passenger flow on the transit network and to reduce the overload delays of passengers at transit stops. A bi‐level programming method is developed to optimize the transit fare under line capacity constraints. The upper‐level problem seeks to minimize the total network travel time, while the lower‐level problem is a stochastic user equilibrium transit assignment model with line capacity constraints. A heuristic solution algorithm based on sensitivity analysis is proposed. Numerical example is used to illustrate the application of the proposed model and solution algorithm.     

Modeling and optimizing urban bus transit considering headway variation for cost and service reliability analysis

Due to the stochastic nature of traffic conditions and demand fluctuations, it is a challenging task for operators to maintain reliable services, and passengers often suffer from longer travel times. A failure to consider this issue while planning bus services may lead to undesirable results, such as higher costs and a deterioration in level of service. Considering headway variation at route stops, this paper develops a mathematical model to optimize bus stops and dispatching headways that minimize total cost, consisting of both user and operator costs. A Genetic Algorithm is applied to search for a cost-effective solution in a real-world case study of a bus transit system, which improves service reliability in terms of a reduced coefficient of variation of headway.     

Modeling a rail transit alignment considering different objectives

An optimization model for station locations for an on-ground rail transit line is developed using different objective functions of demand and cost as both influence the planning of a rail transit alignment. A microscopic analysis is performed to develop a rail transit alignment in a given corridor considering a many-to-one travel demand pattern. A variable demand case is considered as it replicates a realistic scenario for planning a rail transit line. A Genetic Algorithm (GA) based on a Geographical Information System (GIS) database is developed to optimize the station locations for a rail transit alignment. The first objective is to minimize the total system cost per person, which is a function of user cost, operator cost, and location cost. The second objective is to maximize the ridership or the service coverage of the rail transit alignment. The user cost per person is minimized separately as the third objective because the user cost is one of the most important decision-making factors for planning a transit system from the users’ perspective. A transit planner can make an informed decision between various alternatives based on the results obtained using different objective functions. The model is applied in a case study in the Washington, DC area. The optimal locations and sequence of stations obtained using the three objective functions are presented and a comparative study between the results obtained is shown in the paper. In future works we will develop a combinatorial optimization problem using the aforementioned objectives for the rail transit alignment planning and design problem.     

Dispatching control at transfer stations in multi‐hub transit networks

Among dispatching control approaches, the holding option has attracted the most attention in bus control. However, holding a vehicle at a transfer station may exacerbate the delays because more passengers might accumulate at downstream stations and may also affect other connecting routes at other transfer stations. Our problem is to minimize the total costs of dispatching ready vehicles at each transfer station along coordinated routes in a multi‐hub transit network. The total costs include the waiting cost for on‐board passengers, the missed connection costs for late arrival passengers at the subject transfer station and possible transfer costs at downstream transfer stations. We develop a heuristic algorithm to optimize the holding times based on real time information about late vehicles. The results show that ready vehicles should be held longer when the arrival variances of late vehicles are small or when many late connecting passengers are expected.     

A stochastic optimization model for transit network timetable design to mitigate the randomness of traveling time by adding slack time

Transit network timetabling aims at determining the departure time of each trip of all lines in order to facilitate passengers transferring either to or from a bus. In this paper, we consider a bus timetabling problem with stochastic travel times (BTP-STT). Slack time is added into timetable to mitigate the randomness in bus travel times. We then develop a stochastic integer programming model for the BTP-STT to minimize the total waiting time cost for three types of passengers (i.e., transferring passengers, boarding passengers and through passengers). The mathematical properties of the model are characterized. Due to its computational complexity, a genetic algorithm with local search (GALS) is designed to solve our proposed model (OPM). The numerical results based on a small bus network show that the timetable obtained from OPM reduces the total waiting time cost by an average of 9.5%, when it is tested in different scenarios. OPM is relatively effective if the ratio of the number of through passengers to the number of transferring passengers is not larger than a threshold (e.g., 10 in our case). In addition, we test different scale instances randomly generated in a practical setting to further verify the effectiveness of OPM and GALS. We also find that adding slack time into timetable greatly benefits transferring passengers by reducing the rate of transferring failure.     

Large-scale application of MILATRAS: case study of the Toronto transit network

This paper documents the efforts to operationalize the conceptual framework of MIcrosimulation Learning-based Approach to TRansit Assignment (MILATRAS) and its component models of departure time and path choices. It presents a large-scale real-world application, namely the multi-modal transit network of Toronto which is operated by the Toronto Transit Commission (TTC). This large-scale network is represented by over 500 branches with more than 10,000 stops. About 332,000 passenger-agents are modelled to represent the demand for the TTC in the AM peak period. A learning-based departure time and path choice model was adopted using the concept of mental models for the modelling of the transit assignment problem. The choice model parameters were calibrated such that the entropy of the simulated route loads was optimized with reference to the observed route loads, and validated with individual choices. A Parallel Genetic Algorithm engine was used for the parameter calibration process. The modelled route loads, based on the calibrated parameters, greatly approximate the distribution underlying the observed loads. 75% of the exact sequence of transfer point choices were correctly predicted by the off-stop/on-stop choice mechanism. The model predictability of the exact sequence of route transfers was about 60%. In this application, transit passengers were assumed to plan their transit trip based on their experience with the transportation network; with no prior (or perfect) knowledge of service performance.     

Fixed‐route taxi system: route network design and fleet size minimization problems

This paper introduces the taxi route network design problem (TXRNDP) for a fixed‐route taxi service operating in Iran and, in similar form, in various other developing countries. The service operates fairly similar to regular transit services in that vehicles are only permitted to follow a certain predetermined route on the network. The service is provided with small size vehicles and main features are that vehicles only depart if full and that there are no intermediate boarding stops. In Iran the service attracts a high modal share but requires better coordination which is the main motivation for the present study. We develop a mathematical programming model to minimize the total travel time experienced by passengers while constraining the number of taxi lines, the trip transfer ratio and the length of taxi lines. A number of assumptions are introduced in order to allow finding an exact rather than heuristic solution. We further develop a linear programming solution to minimize the number of taxis required to serve the previously found fixed‐route taxi network. Results of a case study with the city of Zanjan, Iran, illustrate the resulting taxi flows and suggest the capability of the proposed model to reduce the total travel time, the total waiting time and the number of taxi lines compared to the current taxi operation. Copyright © 2016 John Wiley & Sons, Ltd.     

Transit network design based on travel time reliability

This paper presents a transit network optimization method, in which travel time reliability on road is considered. A robust optimization model, taking into account the stochastic travel time, is formulated to satisfy the demand of passengers and provide reliable transit service. The optimization model aims to maximize the efficiency of passenger trips in the optimized transit network. Tabu search algorithm is defined and implemented to solve the problem. Then, transit network optimization method proposed in this paper is tested with two numerical examples: a simple route and a medium-size network. The results show the proposed method can effectively improve the reliability of a transit network and reduce the travel time of passengers in general.     

A structured flexible transit system for low demand areas

Public transit structure is traditionally designed to contain fixed bus routes and predetermined bus stations. This paper presents an alternative flexible-route transit system, in which each bus is allowed to travel across a predetermined area to serve passengers, while these bus service areas collectively form a hybrid “grand” structure that resembles hub-and-spoke and grid networks. We analyze the agency and user cost components of this proposed system in idealized square cities and seek the optimum network layout, service area of each bus, and bus headway, to minimize the total system cost. We compare the performance of the proposed transit system with those of comparable systems (e.g., fixed-route transit network and taxi service), and show how each system is advantageous under certain passenger demand levels. It is found out that under low-to-moderate demand levels, the proposed flexible-route system tends to have the lowest system cost.     

Equity-oriented skip-stopping schedule optimization in an oversaturated urban rail transit network

In this study, we focus on improving system-wide equity performance in an oversaturated urban rail transit network based on multi-commodity flow formulation. From the system perspective, an urban rail transit network is a distributed system, where a set of resources (i.e., train capacity) is shared by a number of users (i.e., passengers), and equitable individuals and groups should receive equal shares of resources. However, when oversaturation occurs in an urban rail transit network during peak hours, passengers waiting at different stations may receive varying shares of train capacity leading to the inequity problem under train all-stopping pattern. Train skip-stopping pattern is an effective operational approach, which holds back some passengers at stations and re-routes their journeys in the time dimension based on the available capacity of each train. In this study, the inequity problem in an oversaturated urban rail transit network is analyzed using a multi-commodity flow modeling framework. In detail, first, discretized states, corresponding to the number of missed trains for passengers, are constructed in a space-time-state three-dimensional network, so that the system-wide equity performance can be viewed as a distribution of all passengers in different states. Different from existing flow-based optimization models, we formulate individual passenger and train stopping pattern as commodity and network structure in the multi-commodity flow-modeling framework, respectively. Then, we aim to find an optimal commodity flow and well-designed network structure through the proposed multi-commodity flow model and simultaneously achieve the equitable distribution of all passengers and the optimal train skip-stopping pattern. To quickly solve the proposed model and find an optimal train skip-stopping pattern with preferable system-wide equity performance, the proposed linear programming model can be effectively decomposed to a least-cost sub-problem with positive arc costs for each individual passenger and a least-cost sub-problem with negative arc costs for each individual train under a Lagrangian relaxation framework. For application and implementation, the proposed train skip-stopping optimization model is applied to a simple case and a real-world case based on Batong Line in the Beijing Subway Network. The simple case demonstrates that our proposed Lagrangian relaxation framework can obtain the approximate optimal solution with a small-gap lower bound and a lot of computing time saved compared with CPLEX solver. The real-world case based on Batong Line in the Beijing Subway Network compares the equity and efficiency indices under the operational approach of train skip-stopping pattern with those under the train all-stopping pattern to state the advantage of the train skip-stopping operational approach.     

Network equilibrium for congested multi‐mode networks with elastic demand

This paper proposes an elastic demand network equilibrium model for networks with transit and walking modes. In Hong Kong, the multi‐mode transit system services over 90% of the total journeys and the demand on it is continuously increasing. Transit and walking modes are related to each other as transit passengers have to walk to and from transit stops. In this paper, the multi‐mode elastic‐demand network equilibrium problem is formulated as a variational inequality problem where the combined mode and route choices are modeled in a hierarchical logit structures and the total travel demand for each origin‐destination pair is explicitly given by an elastic demand function. In addition, the capacity constraint for transit vehicles and the effects of bi‐directional flows on walkways are considered in the proposed model. All these congestion effects are taken into account for modeling the travel choices. A solution algorithm is developed to solve the multi‐mode elastic‐demand network equilibrium model. It is based on a Block Gauss‐Seidel decomposition approach coupled with the method of successive averages. A numerical example is used to illustrate the application of the proposed model and solution algorithm.     

Evaluating critical lines and stations considering the impact of the consequence using transit assignment model ‐case study of London's underground network

As the problem of full transit vehicles is encountered daily by passengers in most of the big cities, previous research evaluated the consequence of overcrowding in terms of on‐board crowding and passengers not being able to board with full vehicles. The impact of overcrowding in the real world is, however, not necessarily proportional to these numbers. This paper attempts to specify the critical lines and stations of a network by considering the number of passengers failing to board and attempting to evaluate its impact on service quality and safety risks. The hypothesis is that larger stations with wider platforms can often cope better with overcrowding than smaller stations. Therefore a station size dependent satisfaction function is proposed, which takes values from 0 to 1. The method is applied to London's underground network with a number of scenarios which show critical stations in the network if delays occur.     

Evaluation of an existing bus network using a transit network optimisation model: a case study of the Hiroshima City Bus network

This study evaluates an existing bus network from the perspectives of passengers, operators, and overall system efficiency using the output of a previously developed transportation network optimisation model. This model is formulated as a bi-level optimisation problem with a transit assignment model as the lower problem. The upper problem is also formulated as bi-level optimisation problem to minimise costs for both passengers and operators, making it possible to evaluate the effects of reducing operator cost against passenger cost. A case study based on demand data for Hiroshima City confirms that the current bus network is close to the Pareto front, if the total costs to both passengers and operators are adopted as objective functions. However, the sensitivity analysis with regard to the OD pattern fluctuation indicates that passenger and operator costs in the current network are not always close to the Pareto front. Finally, the results suggests that, regardless of OD pattern fluctuation, reducing operator costs will increase passenger cost and increase inequity in service levels among passengers.     

User perspectives in public transport timetable optimisation

The present paper deals with timetable optimisation from the perspective of minimising the waiting time experienced by passengers when transferring either to or from a bus. Due to its inherent complexity, this bi-level minimisation problem is extremely difficult to solve mathematically, since timetable optimisation is a non-linear non-convex mixed integer problem, with passenger flows defined by the route choice model, whereas the route choice model is a non-linear non-continuous mapping of the timetable. Therefore, a heuristic solution approach is developed in this paper, based on the idea of varying and optimising the offset of the bus lines. Varying the offset for a bus line impacts the waiting time passengers experience at any transfer stop on the bus line.In the bi-level timetable optimisation problem, the lower level is a transit assignment calculation yielding passengers’ route choice. This is used as weight when minimising waiting time by applying a Tabu Search algorithm to adapt the offset values for bus lines. The updated timetable then serves as input in the following transit assignment calculation. The process continues until convergence.The heuristic solution approach was applied on the large-scale public transport network in Denmark. The timetable optimisation approach yielded a yearly reduction in weighted waiting time equivalent to approximately 45 million Danish kroner (9 million USD).     

A new schedule-based transit assignment model with travel strategies and supply uncertainties

This paper proposes a new scheduled-based transit assignment model. Unlike other schedule-based models in the literature, we consider supply uncertainties and assume that users adopt strategies to travel from their origins to their destinations. We present an analytical formulation to ensure that on-board passengers continuing to the next stop have priority and waiting passengers are loaded on a first-come-first-serve basis. We propose an analytical model that captures the stochastic nature of the transit schedules and in-vehicle travel times due to road conditions, incidents, or adverse weather. We adopt a mean variance approach that can consider the covariance of travel time between links in a space–time graph but still lead to a robust transit network loading procedure when optimal strategies are adopted. The proposed model is formulated as a user equilibrium problem and solved by an MSA-type algorithm. Numerical results are reported to show the effects of supply uncertainties on the travel strategies and departure times of passengers.     

The allocation of buses in heavily utilized networks with overlapping routes

Many transit systems outside North America are characterized by networks with extensively overlapping routes and buses frequently operating at, or close to, capacity. This paper addresses the problem of allocating a fleet of buses between routes in this type of system; a problem that must be solved recurrently by transit planners. A formulation of the problem is developed which recognizes passenger route choice behavior, and seeks to minimize a function of passenger wait time and bus crowding subject to constraints on the number of buses available and the provision of enough capacity on each route to carry all passengers who would select it. An algorithm is developed based on the decomposition of the problem into base allocation and surplus allocation components. The base allocation identifies a feasible solution using an (approx.) minimum number of buses. The surplus allocation is illustrated for the simple objective of minimizing the maximum crowding level on any route. The bus allocation procedure developed in this paper has been applied to part of the Cairo bus system in a completely manual procedure, and is proposed to be the central element of a short-range bus service planning process for that city.     

A bi-level programming for bus lane network design

This paper proposes a bi-level programming model to solve the design problem for bus lane distribution in multi-modal transport networks. The upper level model aims at minimizing the average travel time of travelers, as well as minimizing the difference of passengers’ comfort among all the bus lines by optimizing bus frequencies. The lower level model is a multi-modal transport network equilibrium model for the joint modal split/traffic assignment problem. The column generation algorithm, the branch-and-bound algorithm and the method of successive averages are comprehensively applied in this paper for the solution of the bi-level model. A simple numerical test and an empirical test based on Dalian economic zone are employed to validate the proposed model. The results show that the bi-level model performs well with regard to the objective of reducing travel time costs for all travelers and balancing transit service level among all bus lines.     

A stochastic multimodal reliable network design problem under adverse weather conditions

This paper formulates a network design problem (NDP) for finding the optimal public transport service frequencies and link capacity expansions in a multimodal network with consideration of impacts from adverse weather conditions. The proposed NDP aims to minimize the sum of expected total travel time, operational cost of transit services, and construction cost of link capacity expansions under an acceptable level of variance of total travel time. Auto, transit, bus, and walking modes are considered in the multimodal network model for finding the equilibrium flows and travel times. In the proposed network model, demands are assumed to follow Poisson distribution, and weather‐dependent link travel time functions are adopted. A probit‐based stochastic user equilibrium, which is based on the perceived expected travel disutility, is used to determine the multimodal route of the travelers. This model also considers the strategic behavior of the public transport travelers in choosing their routes, that is, common‐line network. Based on the stochastic multimodal model, the mean and variance of total travel time are analytical estimated for setting up the NDP. A sensitivity‐based solution algorithm is proposed for solving the NDP, and two numerical examples are adopted to demonstrate the characteristics of the proposed model. Copyright © 2014 John Wiley & Sons, Ltd.     

Network design for personal rapid transit under transit-oriented development

This study addresses guideway network design for personal rapid transit (PRT) favoring transit-oriented development. The guideway network design problem seeks to minimize both the guideway construction cost and users’ travel time. In particular, a set of optional points, known as Steiner points, are introduced in the graph to reduce the guideway length. The model is formulated as a combined Steiner and assignment problem, and a Lagrangian relaxation based solution algorithm is developed to solve the optimal solution. Numerical studies are carried on a real-sized network, and illustrate that the proposed model and solution algorithm can solve the PRT guideway network design problem effectively.     

Stochastic transit equilibrium

We present a transit equilibrium model in which boarding decisions are stochastic. The model incorporates congestion, reflected in higher waiting times at bus stops and increasing in-vehicle travel time. The stochastic behavior of passengers is introduced through a probability for passengers to choose boarding a specific bus of a certain service. The modeling approach generates a stochastic common-lines problem, in which every line has a chance to be chosen by each passenger. The formulation is a generalization of deterministic transit assignment models where passengers are assumed to travel according to shortest hyperpaths. We prove existence of equilibrium in the simplified case of parallel lines (stochastic common-lines problem) and provide a formulation for a more general network problem (stochastic transit equilibrium). The resulting waiting time and network load expressions are validated through simulation. An algorithm to solve the general stochastic transit equilibrium is proposed and applied to a sample network; the algorithm works well and generates consistent results when considering the stochastic nature of the decisions, which motivates the implementation of the methodology on a real-size network case as the next step of this research.