A column generation algorithm for the bus driver scheduling problem

Bus driver scheduling aims to find the minimum number of bus drivers to cover a published timetable of a bus company. When scheduling bus drivers, contractual working rules must be enforced, thus complicating the problem. In this research, we develop a column generation algorithm that decomposes this complicated problem into a master problem and a series of pricing subproblems. The master problem selects optimal duties from a set of known feasible duties, and the pricing subproblem augments the feasible duty set to improve the solution obtained in the master problem. The proposed algorithm is empirically applied to the realistic problems of several bus companies. The numerical results show that the proposed column generation algorithm can solve real‐world problems and obtain bus driver schedules that are better than those developed and used by the bus companies. Copyright © 2016 John Wiley & Sons, Ltd.     

Relaxation approaches to large scale bus driver scheduling problems

The bus driver scheduling (BDS) problem in a transit company consists of establishing, at minimum cost, a list of work-days in which a driver is assigned to each bus in the given time-table and all clauses of the union contract are respected. In this paper we present a general mathematical programming formulation for the BDS problem. Because, in general, the problem is too large to be solved directly, we introduce a relation of the problem and three different solution approaches. Computational results obtained on real life problems indicate that mathematical programming techniques can solve the BDS problem efficiently.     

Heuristic approaches for solving large-scale bus transit vehicle scheduling problem with route time constraints

This paper presents new models for multiple depot vehicle scheduling problem (MDVS) and multiple depot vehicle scheduling problem with route time constraints (MDVSRTC). The route time constraints are added to the MDVS problem to account for the real world operational restrictions such as fuel consumption. Compared to existing formulations, this formulation decreases the size of the problem by about 40% without eliminating any feasible solution. It also presents an exact and two heuristic solution procedures for solving the MDVSRTC problem. Although these methods can be used to solve medium size problems in reasonable time, real world applications in large cities require that the MDVSRTC problem size be reduced. Two techniques are proposed to decrease the size of the real world problems. For real-world application, the problem of bus transit vehicle scheduling at the mass transit administration (MTA) in Baltimore is studied. The final results of model implementation are compared to the MTA's schedules in January 1998. The comparison indicates that, the proposed model improves upon the MTA schedules in all respects. The improvements are 7.9% in the number of vehicles, 4.66% in the operational time and 5.77% in the total cost.     

Transit stop inspection and maintenance scheduling: A GPU accelerated metaheuristics approach

Bus stops are integral elements of a transit system and as such, their efficient inspection and maintenance is required, for proper and attractive transit operations. Nevertheless, spatial dispersion and the extensive number of bus stops, even for mid-size transit systems, complicates scheduling of inspection and maintenance tasks. In this context, the problem of scheduling transit stop inspection and maintenance activities (TSIMP) by a two-stage optimization approach, is formulated and discussed. In particular, the first stage involves districting of the bus stop locations into areas of responsibility for different inspection and maintenance crews (IMCs), while in the second stage, determination of the sequence of bus stops to be visited by an IMC is modelled as a vehicle routing problem. Given the complexity of proposed optimization models, advanced versions of different metaheuristic algorithms (Harmony Search and Ant Colony Optimization) are exploited and assessed as possible options for solving these models. Furthermore, two variants of ACO are implemented herein; one implemented into a CPU parallel computing environment along with an accelerated one by means of general-purpose graphics processing unit (GPGPU) computing. The model and algorithms are applied to the Athens (Greece) bus system, whose extensive number of transit stops (over 7500) offers a real-world test bed for assessing the potential of the proposed modelling approach and solution algorithms. As it was shown for the test example examined, both algorithms managed to achieve optimized solutions for the problem at hand while there were fund robust with respect to their algorithmic parameters. Furthermore, the use of graphics processing units (GPU) managed to reduce of computational time required.     

Bus maintenance systems and maintenance scheduling: model formulations and solutions

This paper deals with the problem of scheduling bus maintenance activities. The scheduling of maintenance activities is an important component in bus transit operations planning process. The other components include network route design, setting timetables, scheduling vehicles, and assignment of drivers. This paper presents a mathematical programming approach to the problem. This approach takes as input a given daily operating schedule for all buses assigned to a depot along with available maintenance resources. It, then, attempts to design daily inspection and maintenance schedules for the buses that are due for inspection so as to minimize the interruptions in the daily bus operating schedule, and maximize the utilization of the maintenance facilities. Three integer programming formulations are presented and different properties of the problem are discussed. Several heuristic methods are presented and tested. Some of these procedures produce very close to optimal solutions very efficiently. In some cases, the computational times required to obtain these solutions are less than 1% of the computational time required for the conventional branch and bound algorithm. Several small examples are offered and the computational results of solving the problem for an actual, 181-bus transit property are reported.     

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.     

An activity-based approach for scheduling multimodal transit services

This paper proposes a new activity-based transit assignment model for investigating the scheduling (or timetabling) problem of transit services in multi-modal transit networks. The proposed model can be used to generate the short-term and long-term timetables of multimodal transit lines for transit operations and service planning purposes. The interaction between transit timetables and passenger activity-travel scheduling behaviors is captured by the proposed model, as the activity and travel choices of transit passengers are considered explicitly in terms of departure time choice, activity/trip chain choices, activity duration choice, transit line and mode choices. A heuristic solution algorithm which combines the Hooke–Jeeves method and an iterative supply–demand equilibrium approach is developed to solve the proposed model. Two numerical examples are presented to illustrate the differences between the activity-based approach and the traditional trip-based method, together with comparison on the effects of optimal timetables with even and uneven headways. It is shown that the passenger travel scheduling pattern derived from the activity-based approach is significantly different from that obtained by the trip-based method, and that a demand-sensitive (with uneven headway) timetable is more efficient than an even-headway timetable.     

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.     

Flexible feeder transit route design to enhance service accessibility in urban area

To improve the accessibility of transit system in urban areas, this paper presents a flexible feeder transit routing model that can serve irregular‐shaped networks. By integrating the cost efficiency of fixed‐route transit system and the flexibility of demand responsive transit system, the proposed model is capable of letting operating feeder busses temporarily deviate from their current route so as to serve the reported demand locations. With an objective of minimizing total bus travel time, a new operational mode is then proposed to allow busses to serve passengers on both street sides. In addition, when multiple feeder busses are operating in the target service area, the proposed model can provide an optimal plan to locate the nearest one to response to the demands. A three‐stage solution algorithm is also developed to yield meta‐optimal solutions to the problem in a reasonable amount of time by transforming the problem into a traveling salesman problem. Numerical studies have demonstrated the effectiveness of the proposed model as well as the heuristic solution approach. Copyright © 2015 John Wiley & Sons, Ltd.     

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).     

Designing large-scale bus network with seasonal variations of demand

Creating a bus network that covers passenger demand conveniently is an important ingredient of the transit operations planning process. Certainly determination of optimal bus network is highly sensitive to any change of demand, thus it is desirable not to consider average or estimated figures, but to take into account prudently the variations of the demand. Many cities worldwide experience seasonal demand variations which naturally have impact on the convenience and optimality of the transit service. That is, the bus network should provide convenient service across all seasons. This issue, addressed in this work, has not been thoroughly dealt with neither in practice nor in the literature. Analyzing seasonal transit demand variations increases further the computational complexity of the bus-network design problem which is known as a NP-hard problem. A solution procedure using genetic algorithm efficiently, with a defined objective-function to attain the optimization, is proposed to solve this cumbersome problem. The method developed is applied to two benchmarked networks and to a case study, to the city of Mashhad in Iran with over 3.2 million residents and 20 million visitors annually. The case study, characterized by a significant seasonal demand variation, demonstrates how to find the best single network of bus routes to suit the fluctuations of the annual passenger demand. The results of comparing the proposed algorithm to previously developed algorithms show that the new development outperforms the other methods between 1% and 9% in terms of the objective function values.     

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.     

Transportation service procurement problem with transit time

In this work, we investigate transit time in transportation service procurement, which is conducted by shippers using auctions to purchase transportation service from carriers in the planning stage. Besides cost, we find that many shippers are most concerned with transit time in practice; shorter transit time indicates better transportation service. To minimize both the total cost and transit time, the problem faced by shippers is the biobjective transportation service procurement problem with transit time. To solve the problem, we introduce a biobjective integer programming model that can also accommodate some important business constraints. A biobjective branch-and-bound algorithm that finds all extreme supported nondominated solutions is developed. To speed up the algorithm, two fast feasibility checks, a network flow model for particular subproblems, and lower bounds from relaxation are proposed. In addition, a sophisticated heuristic is introduced to meet shipper’s requirements in some situations. Computational experiments on evaluating the performance of the algorithms are conducted on a set of test instances that are generated from practical data.     

Optimal feeder bus routes on irregular street networks

The methodology presented here seeks to optimize bus routes feeding a major intermodal transit transfer station while considering intersection delays and realistic street networks. A model is developed for finding the optimal bus route location and its operating headway in a heterogeneous service area. The criterion for optimality is the minimum total cost, including supplier and user costs. Irregular and discrete demand distributions, which realistically represent geographic variations in demand, are considered in the proposed model. The optimal headway is derived analytically for an irregularly shaped service area without demand elasticity, with non‐uniformly distributed demand density, and with a many‐to‐one travel pattern. Computer programs are designed to analyze numerical examples, which show that the combinatory type routing problem can be globally optimized. The improved computational efficiency of the near‐optimal algorithm is demonstrated through numerical comparisons to an optimal solution obtained by the exhaustive search (ES) algorithm. The CPU time spent by each algorithm is also compared to demonstrate that the near‐optimal algorithm converges to an acceptable solution significantly faster than the ES algorithm.     

Optimization of bus routing strategies for evacuation

Efficient transportation of evacuees during an emergency has long been recognized as a challenging issue. This paper investigates emergency evacuation strategies that rely on public transit, where buses run continuously, rather than fixed route, based upon the spatial and temporal information of evacuee needs. We formulated an optimal bus operating strategy that minimizes the exposed casualty time rather than operational cost, as a deterministic mixed‐integer program, and investigated the solution algorithm. A Lagrangian‐relaxation‐based solution algorithm was developed for the proposed model. Numerical experiments with different problem sizes were conducted to evaluate the method. Copyright © 2013 John Wiley & Sons, Ltd.     

Multi-objective bilevel construction material transportation scheduling in large-scale construction projects under a fuzzy random environment

Abstract

This paper investigates a transportation scheduling problem in large-scale construction projects under a fuzzy random environment. The problem is formulated as a fuzzy, random multi-objective bilevel optimization model where the construction company decides the transportation quantities from every source to every destination according to the criterion of minimizing total transportation cost and transportation time on the upper level, while the transportation agencies choose their transportation routes such that the total travel cost is minimized on the lower level. Specifically, we model both travel time and travel cost as triangular fuzzy random variables. Then the multi-objective bilevel adaptive particle swarm optimization algorithm is proposed to solve the model. Finally, a case study of transportation scheduling for the Shuibuya Hydropower Project in China is used as a real world example to demonstrate the practicality and efficiency of the optimization model and algorithm.     

International experience with speed limits during and prior to the energy crisis of 1973–4

This paper presents an artificial neural network (ANN) based method for estimating route travel times between individual locations in an urban traffic network. Fast and accurate estimation of route travel times is required by the vehicle routing and scheduling process involved in many fleet vehicle operation systems such as dial‐a‐ride paratransit, school bus, and private delivery services. The methodology developed in this paper assumes that route travel times are time‐dependent and stochastic and their means and standard deviations need to be estimated. Three feed‐forward neural networks are developed to model the travel time behaviour during different time periods of the day‐the AM peak, the PM peak, and the off‐peak. These models are subsequently trained and tested using data simulated on the road network for the City of Edmonton, Alberta. A comparison of the ANN model with a traditional distance‐based model and a shortest path algorithm is then presented. The practical implication of the ANN method is subsequently demonstrated within a dial‐a‐ride paratransit vehicle routing and scheduling problem. The computational results show that the ANN‐based route travel time estimation model is appropriate, with respect to accuracy and speed, for use in real applications.     

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.     

Investigating the impact of stochastic vehicle arrivals to optimal stop spacing and headway for a feeder bus route

Stop spacing and service frequency (i.e., the inverse of headway) are key elements in transit service planning. The trade‐offs between increasing accessibility and reducing travel time, which affect transit system performance, need to be carefully evaluated. The objective of this study is to optimize stop spacing and headway for a feeder bus route, considering the relationship between the variance of inter‐arrival time (VIAT), which yields the minimum total cost (including user and operator costs). A solution algorithm, called successive substitution, is adapted to efficiently search for the optimal solutions. In a numerical example, the developed model is applied to planning a feeder bus route in Newark, New Jersey. The results indicate that the optimal stop spacing should be longer that those suggested by previous studies where the impact of VIAT was ignored. Reducing VIAT via certain operational control strategies (i.e., holding/stop‐skipping, transit signal priority) may shorten stop spacing and improve accessibility. Copyright © 2014 John Wiley & Sons, Ltd.     

Sustainability provisions in the bus-scheduling problem

The traditional vehicle scheduling problem attempts to minimize capital and operating costs. However, the carbon footprint and toxic air pollutants have become an increasingly important consideration. This paper studies the bus-scheduling problem and evaluates new types of buses that use alternative energy sources to reduce emissions, including some toxic air pollutants and carbon dioxide. A time-space network based approach is applied to formulate the problem to reduce the numbers of arcs in the underlying network; CPLEX is used to solve the problem. The results show that the bus-scheduling model can significantly reduce the bus emissions – hence reducing the carbon footprint of the transit operation – while only slightly increasing operating costs.