Vehicle and crew scheduling for urban bus lines

A solution to the urban transportation problem is given by vehicle and crew schedules. These schedules must meet the passenger demand and satisfy technical and contractual restrictions stemming from the daily operation of the lines, while optimizing some measure of operational cost. This work describes a computational tool developed to solve the urban transportation problem in the large metropolitan area of São Paulo, Brazil. The techniques used are based on integer programming models coupled with heuristics. The former produces good feasible solutions, and the latter improves the quality of the final solutions. While the operational and labor restrictions are specific to the city of São Paulo, the same ideas can inspire similar approaches for solving the urban transportation problem arising in other metropolitan areas.     

A model for the periodic optimization of bus dispatching times

We model the problem of dispatching time control in rolling horizons following a periodic optimization approach reactionary to travel time and passenger demand disturbances. This model provides more flexibility to transport planners allowing them to adjust the bus schedules during the daily operations. We prove that our periodic optimization model is a convex quadratic program, guaranteeing the global optimality of its solution. To reduce the computational burden, we introduce an iterative algorithm that uses gradient approximations to obtain an approximate dispatching solution. The proposed solution method is found to be significantly faster than exact optimization approaches for quadratic programming and maintains an (almost) negligible optimality gap in realistic bus operation scenarios. Finally, we show that our periodic optimization method outperforms myopic methods that adjust the dispatching time of each bus trip in isolation using operational data from bus line 302 in Singapore.     

A mathematical programming model for the bus deviation route problem

In small towns, or in those peripherical metropolitan areas in which the demand for public transportation is relatively low, the objectives of the bus route planner are different from those faced in highly congested networks. Some towns, also in Italy, are experimenting with urban public transportation systems where regular bus routes are designed which allow users located at specific points outside the main line to signal their presence to the bus driver, who then deviates from the main route to satisfy this demand. This way the bus line is a mixture between a regular line and a dial-a-ride system. The bus deviation route problem is concerned with the design problem which arises in planning the location of the demand points outside the line. A model is presented which takes into account both the advantage of passengers served by this deviation device and the disadvantage suffered by passengers on the bus, whose travel time increases during deviations, and by passengers downstream of the deviation whose waiting time also increases. Through some modeling assumption we are able to represent this problem as a mixed integer linear programming problem, whose relatively low dimension allows for exact solution through standard simplex-based branch and bound code. The proposed model has been applied to a real case and some results of this are presented and discussed.     

A non-linear congestion network model for planning internal movement in the Hajj

A non-linear congestion network model is developed for the Hajj case. The Hajj is one of the World's largest mass movements, according to religious rituals. Every year about two and a half million people gather to perform religious rituals. The objective of the model is to minimize traffic congestion and overcrowding of holy sites. The constraints of the model are religious, spatial and time constraints. The model developed and applied for this case assisted in easing overcrowding and planning for expansions in routes and holy sites. The model is effective in providing quantitative background for general policy decision on the Hajj transportation.     

Focusing attention on auxiliary lines when introduced into geometric problems

The introduction of auxiliary elements as a method of solving problems in high-school geometry is considered here from two perspectives: first, to elicit recalling some known result or concretizing a definition and, second, as a means of shifting the focus and structure of the students’ attention. We present and compare various examples of how auxiliary elements can be introduced in various problems and proofs and characterize their auxiliary quality. Some auxiliary elements unite previously unrelated components of the original diagram; others divide a given complex entity into manageable ones. Implications for further educational research and mathematics instruction are proposed.     

A simultaneous bus route design and frequency setting problem for Tin Shui Wai,Hong Kong

A bus network design problem for Tin Shui Wai, a suburban residential area in Hong Kong, is investigated, which considers the bus services from the origins inside this suburban area to the destinations in the urban areas. The problem aims to improve the existing bus services by reducing the number of transfers and the total travel time of the users. This has been achieved by the proposed integrated solution method which can solve the route design and frequency setting problems simultaneously. In the proposed solution method, a genetic algorithm, which tackles the route design problem, is hybridized with a neighborhood search heuristic, which tackles the frequency setting problem. A new solution representation scheme and specific genetic operators are developed so that the genetic algorithm can search all possible route structures, rather than selecting routes from the predefined set. To avoid premature convergence, a diversity control mechanism is incorporated in the solution method based on a new definition of hamming distance. To illustrate the robustness and quality of solutions obtained, computational experiments are performed based on 1000 perturbed demand matrices. The t-test results show that the design obtained by the proposed solution method is robust under demand uncertainty, and the design is better than both the current design and the design obtained by solving the route design problem and the frequency setting problem sequentially. Compared with the current bus network design, the proposed method can generate a design which can simultaneously reduce the number of transfers and total travel time at least by 20.9% and 22.7% respectively. Numerical studies are also performed to illustrate the effectiveness of the diversity control mechanism introduced and the effects of weights on the two objective values.     

A model for the configuration of incoming wats lines

WearGuard is a direct marketer and retailer of uniforms and work cloths, which relies primarily on phone orders for sales. For this purpose it maintains a series of toll-free 800-number lines, known as WATS lines, to receive its incoming calls. These lines are of several types, where each type serves a different portion of the country and has a different usage fee. In this paper we determine how many of each type of WATS lines should be employed. After defining the problem more completely, we develop a queueing model to describe the system and a dynamic program to solve the configuration problem to optimality. The model has been applied to the problem by WearGuard since 1984. We present an example and examine the sensitivity of the solution to variations in various parameters. We validate the model by comparing the results of this model to other approximate models.Formerly: WearGuard, Director of Decision Support Systems, Longwater Drive, Norwell, MAmis 02061, USA     

Nonlinear-map model for bus schedule in capacity-controlled transportation

We study the schedule of shuttle buses in the transportation system controlled by capacity. The bus schedule is closely related to the dynamic motion of buses. We present the nonlinear-map model for the dynamics of shuttle buses. The motion of shuttle buses depends on the inflow rate. The dependence of the fixed points on the inflow is derived. The dynamic transitions occur with increasing the value of inflow rate. At the dynamic transition point, the motion of buses changes from a stable state to an unstable state and vice versa. The shuttle buses display periodic, quasi-periodic, and chaotic motions in the unstable state. In the unstable state, the number of riding passengers fluctuates complexly with varying trips. The bus schedule is governed by the complex motion of buses.     

A time–space network based exact optimization model for multi-depot bus scheduling

The vehicle scheduling problem, arising in public transport bus companies, addresses the task of assigning buses to cover a given set of timetabled trips with consideration of practical requirements, such as multiple depots and vehicle types as well as depot capacities. An optimal schedule is characterized by minimal fleet size and minimal operational costs including costs for unloaded trips and waiting time. This paper discusses the multi-depot, multi-vehicle-type bus scheduling problem (MDVSP), involving multiple depots for vehicles and different vehicle types for timetabled trips. We use time–space-based instead of connection-based networks for MDVSP modeling. This leads to a crucial size reduction of the corresponding mathematical models compared to well-known connection-based network flow or set partitioning models. The proposed modeling approach enables us to solve real-world problem instances with thousands of scheduled trips by direct application of standard optimization software. To our knowledge, the largest problems that we solved to optimality could not be solved by any existing exact approach. The presented research results have been developed in co-operation with the provider of transportation planning software PTV AG. A software component to support planners in public transport was designed and implemented in context of this co-operation as well.     

A simulation-based assessment of route guidance benefits under variable network congestion conditions

The purpose of this study is to establish a quantitative relationship between network congestion and travel-time reduction benefits of a real-time route guidance user service. The approach of the study is to employ the INTEGRATION traffic simulation model and a 2,000 link network based on the Detroit, Michigan roadway system in a series of experiments. While holding the capacity of the roadway fixed, the value of route guidance is evaluated over a range of increasing demand levels. Network demand patterns and trip characteristics are comparable to current national averages. Measures of congestion such as average system commute speed either match or exceed current national averages. Congestion metrics measured for the lightest demand scenario match most current empirical national average data, while the heaviest demand scenario appears roughly comparable with 1994 Tokyo conditions.Results from this study indicate that route-guided vehicles benefit regardless of level of congestion, however, the amount of trip time savings achieved is highly dependent on network congestion conditions. Average benefits for route guided vehicles over unguided vehicles in the A.M. peak period range between 8–26% depending on overall traffic volume.The results indicate a two-part linear relationship between route guidance benefit and network congestion. As congestion increases in the network, benefits of route guidance increase until average network speed drops below 20mph. Beyond that point, benefits decline (but remain positive). This 20mph threshold in our network is the point where the dynamically growing and shifting mass of queued vehicles around bottlenecks begins to impede access to alternative routes for guided vehicles network-wide.In a related experiment, route guided vehicles that receive reliable data on network conditions (including incidents or demand variation) gain 3–9% travel time savings over unguided vehicles that follow optimal routes based on average time-variant network congestion conditions. Route guided vehicles may exploit information about unexpected delays in the network related to incidents as well as variability in daily traffic patterns. This experiment was conducted to isolate the value of route guidance with respect to experienced commuter traffic, rather than an aggregated model of driver behavior including both familiar and unfamiliar drivers.The preliminary results of this study have implications for ITS benefit assessment. First, the benefits of route guidance are directly related to the level of recurrent congestion in a network. Thus, a near-term poor market for route guidance may evolve over time into a good market for these services. Likewise, a good market for a route guidance user service may deteriorate if overall network congestion reaches very high levels. Second, a route guidance user service provides benefits compared to both a model of aggregated unguided traveler behavior and a model of experienced commuter behavior, regardless of congestion levels. Third, route guided vehicles are demonstrated to gain benefit by avoiding the worst congestion in the network. This minimization in day-to-day variability in commute time may be the most significant benefit of the route guidance system for the familiar driver.     

A modified coupled map car following model and its traffic congestion analysis

Based on the coupled map car following model which was presented by Konishi et al. [Konishi KJ, Kokame H, Hirate K. Phys Rev E 1999;60:4000-7.] (for short, KKH), a modified coupled map car following model is proposed. In this model two successive vehicles’ headway distances in front of the considered one are incorporated in the optimal velocity (for short, OV) function. The stability condition is given for the change of the speed of the preceding vehicle on the base of the control theory. The control scheme in KKH model is applied to the modified model and the feedback gains are determined. Comparison between the modified model and KKH model is carried out. And the corresponding numerical simulation results show that the temporal behavior obtained by our model is better than that by KKH model. The simulation results are in good agreement with the theoretical analysis.     

A hybrid Outer-Approximation/Benders Decomposition algorithm for the single allocation hub location problem under congestion

An efficient procedure that concurrently generates Outer-Approximation and Benders cuts is devised to tackle the single allocation hub location problem under congestion, an MINLP. The proposed method is able to optimally solve large instances (up to 200 nodes) in reasonable time. The combination of both cuts is an algorithmic novelty.     

A model for strategic planning under uncertainty

Summary In this paper the multi-period strategic planning problem for a consumer sumer product manufacturing chain is considered. Our discussion is focused on investment decisions which, are economically optimal over the whole planning horizonT, while meeting customer demands and conforming to technological requirements. In strategic planning, time and uncertainty play important roles. The uncertainties in the model are due to different levels of forecast demands, cost estimates and equipment behaviour. The main aim of this paper is to develop and analyse a multiperiod stochastic model representing the entire manufacturing chain, from the acquisitions of raw material to the delivering of final products. The resulting optimization problem is computationally intractable because of the enormous, and sometimes unrealistic, number of scenarios that must be considered in order to identify the optimal planning strategy. We propose two different solution approaches; firstly, we apply a scenario risk analysis giving the related results of experiments on a particular real data set. We then describe and investigate an Integer Stochastic Programming formulation of the problem and propose, as a solution technique, a variation of Benders decomposition method, namely theL-shaped method.     

A metaheuristic for the school bus routing problem with bus stop selection

Existing literature on routing of school buses has focused mainly on building intricate models that attempt to capture as many real-life constraints and objectives as possible. In contrast, the focus of this paper is on understanding the joint problem of bus route generation and bus stop selection – two important sub-problems – in its most basic form. To this end, this paper defines the school bus routing problem (SBRP) as a variant of the vehicle routing problem in which three simultaneous decisions have to be made: (1) determine the set of stops to visit, (2) determine for each student which stop (s)he should walk to, and (3) determine routes that lie along the chosen stops, so that the total traveled distance is minimized. An MIP model of this basic problem is developed.     

Revenue management model for on-demand IT services

This paper presents a model for applying revenue management to on-demand IT services. The multinomial logit model is used to describe customer choice over multiple classes with different service-level agreements (SLAs). A nonlinear programming model is provided to determine the optimal price or service level for each class. Through a numerical analysis, we examine the impacts of system capacity and customer waiting incentives on the service provider’s profit and pricing strategies.     

A cardinality-constrained robust model for the assignment problem in Home Care services

Home Care includes medical, paramedical and social services which are delivered to patients at their domicile rather than in hospital. Managing human and material resources in Home Care services is a difficult task, as the provider has to deal with peculiar constraints (e.g., the continuity of care, which imposes that a patient is always cared for by the same nurse) and to manage the high variability of patients’ demands. One of the main issues encountered in planning Home Care services under continuity of care requirement is the nurse-to-patient assignment. Despite the importance of this topic, the problem is only marginally addressed in the literature, where continuity of care is usually treated as a soft-constraint rather than as a hard one. Uncertainty is another relevant feature of nurse-to-patient assignment problem, and it is usually managed adopting stochastic programming or analytical policies. However, both these approaches proved to be limited, even if they improve the quality of the assignments upon those actually provided in practice. In this paper, we develop a cardinality-constrained robust assignment model, which allows exploiting the potentialities of a mathematical programming model without the necessity of generating scenarios. The developed model is tested on real-life instances related to a relevant Home Care provider operating in Italy, in order to evaluate its capability of reducing the costs related to nurses’ overtimes.     

A decision model for berth allocation under uncertainty

This paper studies the berth allocation problem (BAP) under uncertain arrival time or operation time of vessels. It does not only concern the proactive strategy to develop an initial schedule that incorporates a degree of anticipation of uncertainty during the schedule’s execution, but also studies the reactive recovery strategy which adjusts the initial schedule to handle realistic scenarios with minimum penalty cost of deviating from the initial schedule. A two-stage decision model is developed for the BAP under uncertainties. Moreover, a meta-heuristic approach is proposed for solving the above problem in large-scale realistic environments. Numerical experiments are conducted to validate the effectiveness and efficiency of the proposed method.