12-01, 2016

公共交通规划是困扰城市规划的一个问题。新加坡拥有世界一流的公共交通系统。然而，随着城市人口的不断增长和新的组屋住宅区正在开发，容纳全国交通需求对城市规划者面临的挑战。由于日益增长的客流，公交线路体验高负荷高峰时段，乘客需要等待更长的时间，一辆公交车。在高峰时段的等待时间和行车时间，成为新加坡巴士系统的一个问题，在2006的等待时间的一项调查是两个属性，通勤者不太满意。

为了解决这一问题，新加坡当局正在加紧努力，以满足扩大运输的需要。事实上，从8月1日的2009，新的服务标准是由新加坡政府的巴士服务85%需要达到的频率不超过10分钟，高峰小时和15分钟的期间在非高峰小时，它可以转化成一个等待的时间要求不超过10分钟。作为新的服务标准的响应，据悉，新增加的公交车已由两总线服务提供商投入服务，SBS和SMRT。

这一举动当然减轻拥挤问题。然而，公交线路的其他设置可能与这种变化不兼容，必须调整适应了新的公共汽车。因此，这里来的问题：是目前的停止间距仍然适当，并应相应修改？

这个问题涉及到巴士站间距的优化。公交站距和公交车数量是影响总出行时间的两个重要因素。总出行时间是公共交通系统最重要的性能指标，从乘客的角度。因此，确定最佳的公交车站间距政策被选为我们的主要研究目标。在这种情况下，“最佳”的总线停止间隔被定义为停止间隔，最大限度地减少了一个单一的总行程时间的目标，受一个最小的等待时间约束。平均等待时间为10分钟，是服务提供商所规定的服务性能要求。

巴士站的数量和总行程时间之间的关系是不寻常的。常识告诉我们，过量的巴士站可能会导致巴士服务较慢。总行程时间花费在步行，旅行和等待时间的妥协。随着大量的停止，通勤者花费更少的时间在步行。但公交车会使更多的频繁的停止，导致更长的等待时间和更长的旅行时间。另一方面，太少的巴士站，使巴士行驶得更快，但乘客必须走很长的路要走的巴士站，以及到最终目的地。因此，巴士站间距的设置是至关重要的总行程时间的最小化，由于非普通的关系这需要进一步的调查。

Chapter 1 Introduction 简介

One of the problems that troubles urban planning today is public transportation planning. Singapore has a world class public transportation system, which has a total ridership of 1.48 billons in the year 2009. (SMRT, 2009 and SBS Transit, 2009). However, as the urban population continues to grow and new HDB residential areas are being developed, accommodating the nation's transportation needs poses a challenge to the city's planners. As a result of the growing ridership, bus lines experience a high load during peak hours and passengers need to wait longer for a bus. Longer waiting time and travelling time during peak hour becomes a issue for Singapore bus system, as in a survey in the year of 2006 waiting time is one of the two attributes with which commuters are less satisfied. (Land Transportation Authority, 2006)#p#分页标题#e#

To tackle this issue, Singapore's authority is stepping up its effort to cater for the enlarging transportation need. In fact, starting from Aug 1 2009, new service standards are implemented by Singapore government that 85% of the buses services are required to reach a frequency of no more than 10 minute during peak hour and 15 minute during off-peak hour (Lay, 2009),which can be translated into a waiting time requirement of no more than 10 minutes. As a response to the new service standards, it is reported that new additional buses have been put into service by the two bus service providers, SBS and SMRT.

This move certainly mitigates the crowding issue. However, other settings of the bus line may be incompatible with this change and have to be revamped to match up with the newly available buses. Hence here comes the question: Are the current stop spacing still appropriate and should them be modified accordingly?

This question involves optimization of bus stop spacing. Bus stop spacing and number of buses are two important factors that affect total trip time. And total trip time is the most important performance measure of a public transportation system, from the passenger's standpoint. Thus identifying optimal bus stop spacing policies is chosen as our primary research objective. In this case "optimal" bus stop spacing is defined as the stop spacing that minimizes a single objective of total trip time, subject to a minimum waiting time constraint. Average waiting time of 10 minutes is the service performance requirement imposed by the service providers.

The relationship between the number of bus stops and the total trip time is not trivial. Common sense tells us an excess of bus stops may lead to slow bus services. Total trip time compromises of time spent on walking, travelling and waiting. With a large number of stops, the commuters spend less time in walking. But buses make more frequent stops, resulting in longer waiting time and longer travelling time. On the other hand, too few bus stops make the bus traveling faster, but the passengers have to walk a long way to reach the bus stop, as well as to the final destination. Thus, the setting of bus stop spacing is crucial in the minimization of total trip time, which requires further investigation due to the nontrivial relationship.

In all, assisted by a real-time simulation C++ model, this project attempts to explore the optimal bus stops policies. Also in this project some measures that mitigate peak time waiting time are proposed and their effects are discussed.

The whole project is structured as follows: Chapter 2 identifies the problem and describes them in detail. In Chapter 2 definition of terms and performance measures are also introduced. Chapter 3 reviews the literature on optimal bus spacing and computation model of in-vehicle time and discusses their relevance to this project. In chapter 4 assumptions are introduced to simplify the problem. In the same chapter, the simulation model is built, providing detailed information such as underlying mechanism, model assumption and statistical background. In chapter 5 a "fixed-and-then vary" research methodology is proposed and implemented. Then in chapter 6 this study is implemented by a simulation optimization of stop spacing and the results of simulation are analyzed. Finally the project is concluded in chapter 7 and the limitations are also discussed.#p#分页标题#e#

Chapter 2 Problem description 问题描述

Bus stop spacing optimization problem

In the bus stop spacing optimization problem, our objective is to find out the best bus stop spacing that minimizes average total trip time. Number of bus stops, which is inversely proportional to bus stop spacing given the bus line is fixed, is the variable of the optimization problem. Service requirement is posed as a constraint which is that average waiting time of passengers should not be more than 10 minutes.

Objective: To find out the optimal level of bus stop spacing that minimizes the average total trip time during peak hour.

Variable: Number of bus stops

Constraint: Average waiting time must be no more than 10 minutes.

Subject of study

Bus 96 is chosen as our subject of study as it exhibits large waiting time during peak hour. The subject is also chosen because the writer's own experience travelling with Bus 96. Bus 96 mainly serves people going to National University of Singapore and Clementi MRT station. The simulation closely resembles that of bus 96 operation.

Definition of terms

Total trip time is the total time a passenger spends on this way from his/her place of origin to the destination.

Total trip time = access time + waiting time + in-vehicle time

Access time: The time spent by commuter to walk from his place to the nearest bus stop of origin and to walk from the bus stop of destination to his destination place.

Waiting time: the time spent by the commuter to wait for the next available bus to come

In-vehicle time: the time spent by a commuter between the time when he gets on the bus and the time when he gets off the bus.

Deceleration /Acceleration time: the time the bus takes to brake to a stop when approaching it and then regain cruising speed after leaving the stop. Simple dynamic formulas can be used to calculate this value, which can be found in Appendix 1.1.

Dwelling time: the time taken for the passengers to alight and board. Bus stop spacing will not have much impact on total dwelling time, assuming no significant change in arrival rate.

Headway: The duration in time that separates two vehicles traveling the same route.

Bus occupancy rate: the percentage of the bus capacity that is occupied. Average bus occupancy rate is used, which is measured by taking average over time.

Performance measures

Performance measures are parameters which are chosen to represent the performance of a bus line. There are three performance measures adopted in this project, namely average total trip time, average waiting time and average occupancy rate.#p#分页标题#e#

The performance measures are chosen primarily from the passenger's perspective. Average total trip time is the most important performance measure in the bus stop spacing optimization study because it is critical to passengers' travelling experience. Moreover, average waiting time is also an important performance measure because it is usually the most important service requirement imposed by the transportation authority. Therefore, average waiting time is used as a service criterion or a constraint in the optimization problem.Furthermore, average occupancy rate of all the buses is chosen to reflect the average utility of the bus line to assist the action of adding or remove buses from the fleet, as LTA (Land Transportation Authority of Singapore, 2007) guidelines that require bus occupancy rate cannot exceed 95%.

Chapter 3 Literature Review 文献综述

Current literature reveals that bus stop spacing in some countries is not optimized with respect to total trip time. Ammons (2001) reveals that in his multi-city study on bus stop spacing standards that in most US cities, stop spacing varies from 200 to 600 meters. On the contrary, bus stop spacing recommended from the research of Transportation and Research Board of United States is 750 feet (229 meters) for urban areas and 600 feet (183 meters) for central core business areas in the city (Transportation Board, 2009). This contradiction between stop spacing in practice and theoretic recommendation has given the motivation for the project

Furthermore, Kehoe's study (2004) shows that bus stop spacing is not a result of scientific study but rather formed based on convention and user request. In many routes along the USA, the bus stops were defined through time, as a result of user's request to the bus company. It is also true in the case of Singapore, where for buses which going to residential areas, a lot of stops have been placed under HDBs blocks. However, it remains unclear if these bus stops resulted from user request are in the best benefit of the commuters.

As in the case of Singapore, the bus stops are placed in a lower density generally than that of United States in urban areas. Public Transport Council of Singapore dictates 350 meters to 400 meters spacing between bus stops (Wikipedia, 2010). With no other studies on Singapore's public transportation, the following question emerges: in Singapore, how can the bus stops be placed in order to optimize the total trip time for the commuters?

Some of the studies in literature have attempted the problem of optimal bus stop spacing. Levinson (1983) concludes that capacity of the bus system can be improved best by keeping the number of stops to a minimum. However, he mainly focuses on system performance but he does not consider the time of the traveler. Homero et al. (2008) study a bus line in the city of San Paolo using a model based in Voronoi Diagrams to find the best bus-stop spacing that minimizes total trip time of travelers. Their model captures a tradeoff between time to access the bus stops and travelling time. By varying the bus stop spacing, their study shows the traveler's average travel time will decrease from larger bus spacing to a lowest point of 250 meters, after which it will slowly picks up again. Saka (2001) uses lost time in acceleration and deceleration to represent the effect of bus stops on travelling time and finds out "proper spacing of stops significantly improves the quality of transit service and decreases travel time". Van Nes and Bovy (2000) study bus stop spacing in Netherlands which ranges from 300 to 450 meters and recommends the bus stop spacing should be increased to 500 to 800 meters. Their study is done through a linear programming approach with a constrained objective function.#p#分页标题#e#

However, there are several limitations of the aforementioned researches which render the research result not applicable to the bus stop spacing problem of bus 96. Firstly, all of these literatures on bus stop spacing employ a mathematical approach. However, passenger arrivals at the bus stop and the bus arrivals at the bus stops are dynamic, which cannot be modeled simply as a statistical random variable. To imitate the probabilistic distribution of inter-arrival times, simulation is a better tool. Secondly, capacity of the bus is not taken into account in previous researches, which assume that buses have unlimited capacity. But it is important to note limited space of the bus is the main contributing factor that leads to large total trip time for some of the cases. Thus capacity of the bus has to be considered and it can be addressed easily by a simulation model. Thirdly, interactions exist among the passengers and buses, for example the number of passengers that board the bus will affect how fast the bus actually runs. In addition, this interaction contributes to the bus-bunching problem, where one bus delays and picks up more passengers than it should have in the non-delay situation. The subsequent bus thus takes in fewer passengers than in the non-delay situation and speeds up to crunch into the slower front-running bus. Again these problems can only be addressed using a real-time simulation model.

Following the above discussion, this project employs a real time simulation model in C++. The model is designed in such a way that it can be extended to a real time forecast system that provides real time bus arrival time to users in the future, which will benefit both the passengers and the bus company.

Relevant literature offers good insights on establishing the upper threshold value for bus stop spacing. It is easy to see that extreme small stop spacing slows the bus down and gives rise to prohibitively large waiting time. On the other hand, most people are unwilling to walk more than 800 meters to bus stops, as shown by past research (Pushkarev and Zupan, 1975). In this project, the upper threshold for stop spacing is 1000 meters, which offers a large safety margin for the optimization problem. Lower bound is arbitrarily set to be 200 meters as none of the optimal values (in literature and also in this project ) occur at value smaller than that.

There is also other contribution in the literature about actual dwelling time model. Levinson (1983) used the survey data across US cities to model the dwelling time as:

T = 2.75n+5 seconds

Where T is the "total stopped time per bus" and n is the number of interchanging passengers per bus. It can be seen that the T is linearly related to n. Later in chapter 5.2, a different model is adopted for the sake of simulation but the linear relation still holds.#p#分页标题#e#

Chapter 4 Model Formulation 模型表述

After reviewing against existing literature, the stop spacing optimization problem is simplified by making some assumptions.

Problem simplifications

There are three exogenous factors that need to be considered in planning stop spacing, namely fleet size, arrival pattern and operation rule of the bus line. Firstly, the number of buses in the fleet needs to be known. Secondly, the arrival pattern and the departure pattern should be identified. Thirdly, any operation rule of the bus should be considered. The writer has made requests for the above information to SBS transit; unfortunately they have dismissed the requests. Thus simplifications are made to reduce complexity of the problem while keeping it practical and realistic.

Firstly, the fleet size is treated as a constant initially where the fleet size is chosen by requiring a waiting average of no more than 10 minutes. Secondly, the arrival rate is assumed to be the peak arrival rate for the whole simulation period because we are interested in the peak-time performance of bus line, where bus bunching and large waiting time occur the most. Thirdly, it is assumed that there is only one interchange and during peak hour and the buses keep running without additional rest at the interchange.

Further Assumptions

General assumption about the problem

Traffic Condition

Traffic condition is not considered in the problem for the sake of simplicity.

Cost

Bus operating cost, which represents cost undertaken by the bus company, and the monetary cost bear by passengers are both excluded from the study. Therefore, the optimization of stop spacing is based on time instead of monetary value.

Assumptions about the simulation model

bus line

The bus line is a unidirectional loop line with bus stops on one side.

The bus stops are uniformly distributed over the bus line.

The first bus stop on the route is the only interchange station, which holds all the buses before the start of the simulation.

Buses do not stop or rest at the bus interchange once they depart from the interchange initially.

Buses

Buses have a limited capacity. Once a bus is full, it does not admit more passengers and it will pass by a bus stop unless there is someone to alight.

Passengers

Passengers always walk to the nearest bus stop and take the bus from there.

Passengers have a uniform arrival pattern and uniform alighting pattern, which means that each passenger has an equal probability of originating from any point and ending the journey at any point.#p#分页标题#e#

Simulation model

Model description

As it is very difficult to model uncertainty analytically and models usually become intractable when capacity limits of the buses are imposed, a real-time simulation model in C++ is used. This model is set up to imitate the behavior of bus 96. However, the model allows for changes in parameters like bus acceleration and speed. It also supports O-D (origin-destination) patterns by separating the whole bus line into different zones with different arrival rates. In other words, this model supports all bus lines that are circular with a single interchange station with assumptions described in chapter 4.

The detailed documentation of the model and the process can be found in Appendix 2.

Underlying mechanism

The following is an illustration of how the simulation works. The passenger arrival events are pre-generated before the simulation starts to run. During the simulation, which is discrete-time event- driven, each passenger originates from one point on the bus line and reaches the nearest bus stop. From there the passenger takes the next available bus to go to the bus stop near his or her destination, where he or she will walk again to reach the destination.

The in-vehicle time between two selective bus stops is the aggregate sum of interval travel time of all the bus stops intervals (Tinterval-travel Â). Tinterval-travel stands for the travel time from starting at one bus stop to reaching the subsequent one, which is calculated as below:

Tinterval-travel Â= Tad + Ted + T0

In which:

Tad = acceleration/deceleration time.

Ted = dwelling time due to passenger boarding and alighting.

Ted = maximum (1Tb , 2Ta ) + 3. 12 represent the number of people that board and alight respectively. Ta is the time per passenger spent on alighting and Tb is the time per passenger spent on boarding and. T3 is the dwelling time on the bus stop not caused by passenger boarding or alighting, which contains the open/close door time and re-enter traffic time. According to the study by Pline (1992), Ta = 2 seconds, Tb =3 seconds and T3 = 30 seconds. Note that alighting and boarding is taking place simultaneously, thus the dwelling time is taken as the maximum of the two.

T0 = cruising time of the bus. For details of calculation of Tad and T0, please refer to the Appendix 1.

Model assumptions

Buses travels at constant speed except near a bus stop, it may decelerate, stop and accelerate to the constant speed again. Shown by formula of Tinterval-travel Â in chapter 5.2, the time spent at a bus stop Ted is positively related to the number of people alighting and boarding at the bus stop.

#p#分页标题#e#
Passengers walk at constant speed to the bus stop the bus stop and join queues to wait for the bus. Their places of alighting are determined immediately by a uniform random variable after getting on the bus.

Simulation starts with all of the buses scheduled to run from the first stop. Initially the buses are scheduled to depart at constant intervals in order to achieve constant bus headway. Simulation continues until the last passenger comes within the pre-generation time reaching his or her destination.

Parameters in the model

Bus capacity : 60 seats (estimated based on personal experience)

The bus speed is 40km/h under a clear traffic condition( assumed for analysis)

Bus line length: 8km(estimated from online map data)

The simulation time is 24 hours.

Peak time arrival rate : 25 persons per 8km per minute( assumed for analysis)

Commuters walking speed : 45 meters/minute (assumed for analysis)

Rate of acceleration is 0.5m/s2and deceleration 2.0m/s2(Pline, 1992)

Boarding time each passenger is 3 seconds and alighting time each passenger is 2 seconds (Pline, 1992)

As the varying range of stop spacing is set from 200 meters to 1000 meters, the number of bus stops is varied from 8 to 40, given a bus line length of 8km.

Statistical background

In this study, we are interested in finding population parameters of total trip time and waiting time. As these population parameters cannot be obtained, they are estimated by the corresponding sample statistics. In order to get a precise estimation, there are usually two ways to follow: one is to run multiple replications and to compute average of the replication mean, which is suitable for parameters evolve over time. The other is to run for a prolonged period, to divide the data into multiple batches and to compute the average of the batch mean, which is used for parameters that stabilize over the long run.

The batch average approach is adopted in the stop spacing study. Because the moving average of total trip time during peak time stabilizes over the long run, the batch average also stabilizes over the long run. The average of the batch average is therefore qualified as a good estimate of the population mean.

Chapter 5 Research Implementation 研究执行

Research strategy

In the stop spacing optimization study, the primary variable of interest is the bus stop spacing or the number of stops[[1]] However, two other exogenous factors affect the total trip time as well as the optimal number of bus stops.

Number of buses

Arrival rate

Firstly, the number of buses in the fleet has a great impact on the average waiting time. With too few buses, the bus line would fail to meet the demands of the passengers and the average waiting time will keep on escalating. Increase the number of buses will generally decrease average waiting time, keeping all other factors constant. Secondly, the arrival rate, together with the number of buses, affects average total trip time. Under different arrival rates, the numbers of buses required to reach a certain service criterion are different. Therefore, without knowledge of the actual arrival rate, the arrival rate has to be assumed to be constant in order for the study to be meaningful.#p#分页标题#e#

Thus, as partly mentioned in chapter 4, to isolate the effects of these two factors from bus stop spacing and also to assess their separate effect on the optimal stop spacing policy, the following "fixed-and-then-vary" strategy is taken, similarly to what is customarily done in a control experiment.

Implementation of the "fixed-and-then-vary" strategy

First part of the analysis is to seek for a fixed policy under different number of buses with arrival rate is kept constant. The minimum number of buses is determined by imposing the constraint in the optimization problem that average waiting time should not be more than 10 minutes [[2]]. Then using the number of buses no smaller than this lower threshold, we proceed to analyze how stop spacing affects average total trip time. Within the pre-defined range of number of buses, at each number of buses we seek for the optimal bus stop spacing that minimizes average total trip time.

Second part of the analysis focuses on how to separate the effect of arrival rate on optimal stop spacing at each number of buses. As aforementioned, arrival rate is firstly assumed to be constant in part 1 of the analysis. Nevertheless, in the second part, sensitivity analysis is performed by varying the arrival rate. Notably the bus stop spacing is optimized under different combinations of arrival rate and number of buses instead of for different arrival rates or different number of buses separately, as it is impossible to isolate the effect of arrival rate or number of buses on optimal stop spacing.

Warm-up analysis

As we are not interested in all of the data generated in the simulation, warm-up analysis helps us to recognize and select the data that is needed. It can be imagined that to study the peak-time performance of the bus line, those information that corresponds to early hours of operations should be excluded from the analysis.

The batch mean method is used to estimate the population mean of total trip time. As we are interested in the long-run average waiting time, there may exist some undesirable initialization bias at the starting period. In this case, the data belongs to the bias region should be excluded from the final statistics.

Warm analysis is performed for a duration of 24 hours, which is the simulation duration set. Possible initialization bias is identified by plotting batch averages with the batch size of 20 minutes [[3]] against simulation time.

Three scenarios are plotted: 6 buses and 6 bus stops, 6 buses and 10 bus stops and 8 buses and 10 bus stops, shown by the three corresponding curves in Figure 5.2.

For a point on either of the curves, its function value represents the waiting time batch mean of passengers that arrives within 20 minutes, counting from the time value of the point. Each batch includes passengers that arrive within a 20 minute period. Batch size of 20 minutes is used because the average full-cycle travelling time of the bus ranges from 20 to 30 minutes.#p#分页标题#e#

Chapter 6 Results and Discussion 结果与讨论

In the phase I, to get the lower threshold of the number of buses, the performance data are collected for scenarios with possible combinations of number of buses and number of bus stops, with the constraint to be imposed that average waiting time should be no more than 10 minutes.

Subsequently in Phase II, setting the number of buses to this threshold, the minimum number of buses is obtained by solving a minimization problem with respect to numbers of buses given the average waiting time must be no more than 10 minutes. Using the obtained minimum number of buses, the optimal number of stops is then the one which minimizes average total trip time and the corresponding optimal bus stop spacing is determined.

In phase III, the optimal bus stop spacing problem is generalized to a problem with different numbers of buses by performing a sensitivity analysis. The optimal number of bus stops is solved for a given number of buses while the arrival rate is kept fixed, after which in phase IV the constant arrival rate assumption is relaxed and a sensitivity analysis is performed on arrival rate.

Phase I: finding the lower threshold value for number of buses

To determine the minimum number of buses that is required to produce a waiting time of no more than 10 minutes, configurations with different numbers of buses and stops are run and the average waiting times are collected.

The number of buses is varied from 5to 20[[4]]. And the number of bus stops is varied from 4 to 16, as previously justified. Among all the possible configurations, those configurations with average waiting time smaller than 10 or equal to 10 are selected and a scatter plot of these points is presented in Figure 6.1, which outlines the feasible region.

As the optimization is bounded by the feasible region, it would not be desirable to have a smaller feasible set so that the optimal value may not occurs within the set but on the boundary, which is not real optimal. As a result, the guiding principle of choosing the lower threshold is that we would like to choose a number of bus at which configurations with most number of bus stops are within the feasible region.

Thus we can start to choose from a candidate where most of the bus stops are feasible but they may not constitute the full feasible set. To have a closer look, the statistics of the circled part of the scatter plot are presented below in Table 6.1.1 and 6.1.2. The points with average waiting time slightly larger than the threshold of 10 minutes are also included to show the transition.

Finally eight is selected as the lower threshold for the number of buses .At 8, the feasible range of bus stops is from 4 to 14. To reiterate, number 8 is chosen as to facilitate the optimization with a reasonably large feasible set. For the same reason, 5 and 6 should not be chosen as the candidates. Nevertheless, it is worth noting that 8 may not be the fittest candidate; while it is preferred based on the author's understanding of the bus 96 line.#p#分页标题#e#

Phase II: optimal bus stop spacing policy at number of buses = 8

6.2.1 Individual effect

Temporarily fixing the number of buses at 8, we are to find out the optimal number of bus stops that leads to the minimum total trip time.

Evidence from the simulation data

Temporarily fixing number of bus equal at 8, we are to find out the optimal number of bus stops that gives the minimum total trip time. All three components of total trip time are affected by the number of bus stops, and the effects of increasing the number of bus stops on the three components of the total trip time are summarized in Table 6.2.1. The relationship between number of bus stops and each of the three components are shown respectively in Figure 6.2.1.1 to 6.2.1.3.

Analytical counterpart

Shown in figure 6.2.1.1 and figure 6.2.1.2, the average in-vehicle time and the average waiting time increase as the number of bus stops increases. In contrast, average access time decreases as number of bus stops increases. The rate of increase for in-vehicle time and waiting time is approximately linear. On the contrary, the rate of decrease for access time decreases over time. These effects can be justified also in an analytical manner:

In-vehicle time: as the bus stops more frequently, it spends more time at the bus stops. Thus in-vehicle time will be longer. In Appendix 1.3 it is proven that in-vehicle time is positively proportional to the number of bus stops.

Waiting time: as the in-vehicle time becomes longer, the passengers who have not boarded the bus need to wait for longer period. From a continuous point of view, the increase in average in-vehicle time of a passenger is equal to the increase in the average waiting time. Therefore, the average waiting time assumes a linear positive relation with respect to the number of bus stops.

Access time: Access time will be shorter and the access time is inversely related to the number of bus stops. An Analytical relation can be obtained by assuming the bus line length as A and the number of bus stops as x. According to assumptions that passengers go to their nearest bus stop and passenger arrivals are uniformly distributed, average access time is calculated to be which represents a parabolic function. The rate of decrease actually decreases over time and it decreases slower than that of a linear relationship as in-vehicle time or waiting time with number of bus stops.

Trade-off and optimal point

In addition, as access time is inversely affected by the number of bus stops while in-vehicle time and waiting time are positively affected, we can hypothesize a trade-off between these three components. The optimal number of bus stops is where the two are balanced out. A graphical method can be used to get the optimal value of bus stops by finding the intersection point of the average waiting time and average access time. Simulation data from number of buses = 8 are drawn in figure 6.2.2#p#分页标题#e#

In figure 6.2.2 the lower graph directly supports the hypothesis on the trend of the three components. In addition, the upper graph clearly shows the average total trip time is minimized at number of bus stops = 10, which is also the intersection point between average waiting time and average access time. This corresponds to 8000/10 = 800 meters of bus stop spacing.

Sensitivity analysis and generalization

Due to the assumptions made to the model, the final result of 800 meters as the optimal bus stop spacing may not be applicable to other circumstances. Nonetheless, it has two significant implications for future work. Firstly, it provides a sounding framework and methodology to find out the optimal stop spacing. Secondly, by relaxing some of the constraints and performing sensitivity analysis, the conclusion can be modified to fit the new circumstances. In addition, by analyzing how varying some of the parameters may affect the decision making, we are able to understand the relationship between the optimal stop spacing policy and exogenous variables and then we can use it to predict the trend in the future work.

Sensitivity analysis is first performed on the number of buses and then on arrival rates.

Sensitivity analysis on the number of buses

Since the optimization of bus stop spacing has been done for the number of bus = 8, simulations are run for number of buses from 5 to 20 and the number of bus stops is swept from 5 to 16. Table 6.3.1 shows for each number of buses in the range, the optimal numbers of bus stops that minimize the total trip time. Arrival rate is kept fixed at 25 arrivals per minute per 8km.

There are two conclusions that can be drawn from the figure 7.3.1.

Firstly, the optimal number of bus stops increases initially as the number of buses increases but fluctuates between number of bus 16 and 20. This initial positive correlation can be explained by the following hypothesis: increasing the number of bus stops slows the buses down by adding more time in acceleration, deceleration and dwelling time, which is more evident during a scenario with few buses.

For fewer buses, this slow down effect is significant enough to lead to a substantial increase in waiting time. Thus the more bus stops there are, the larger the average waiting time becomes. The optimal number of stops needs to stay small so that the increase in average waiting time does not outweigh reduction in access time. For instance, the optimal number of bus stops for 5 buses is only 8. However, for scenario with more buses, as the headway between the buses is much nearer and the time difference between two adjacent bus arrival times can be relatively small. Therefore adding more buses stop only lead to smaller increase in average waiting and in-vehicle time. On the contrary, the average access time is reduced considerably, due to the linearity of the relationship between access time and number of bus stops.#p#分页标题#e#

To summarize, as the increase in waiting and in-vehicle time does not outweigh the decrease in access time, the optimal number of bus stops with more buses is larger than that of fewer buses in general. But we are unable to represent this relationship in a simple mathematical manner. In addition, the increasing trend discontinues at large number of buses, namely from 16 to 20. Despite the two limitations above, the positive correlation between the number of bus stops and the number of buses are still very instructional to the bus planning practice. Apart from the trend, more detailed information is needed and similar simulation studies must be run to get the actual number.

Secondly, adding more buses to the bus line always improves the average total trip time and average waiting time at the optimal level. Thus adding buses is one of the viable options for the bus service provider to improve the quality of service. If bus operating cost and passengers' cost of time are included, this study can be further utilized to determine the optimal level of bus stop spacing given a budge constraint, which is more practical and applicable.

Sensitivity analysis on arrival rates

As we made an assumption that the arrival rate is 25 persons per minute per 8km, the result obtained in previous chapters may not be applicable to the scenarios with different arrival rates. Consider the case of an earlier morning and a rush hour: Is it always optimal to have the same number of bus stops? This is a question in our interest, because there is a practice that some bus stops is skipped during rush hour in order to speed up. To investigate the relationship between optimal stops number (or optimal stop spacing) and arrival rates, we then repeat the optimization procedure for arrival rates = 15, 25, 35 per minute per 8km and the optimal number of bus stops are plotted out with respect to the number of buses at each of these arrival rates. [DATA]

Figure 7.3.2 indicates that as number of buses >=8, the optimal level of bus spacing does not vary much for different arrival rates. In fact, there exists a great overlapping among the three curves after the line where number of buses = 8. Especially for curves correspond to arrival rate = 15 and 25, the same optimal bus spacing occurs at different number of buses so that the two curves overlaps the most.

Recall that for scenarios having more than 8 buses, the average waiting time is smaller than 10 minutes. Thus it can be concluded that to achieve a minimum waiting time of no more than 10 minutes, the optimal stop spacing is quite independent of arrival rate. To further this argument, a fixed stop spacing policy can be suggested: the optimal bus stops number stays at 10 irrespective of the number of buses and arrival rate. And as the optimal level of bus stops is approximately the same for arrival rate = 15 and 25, by interpolation the optimal bus stop spacing should be same for any arrival rate within 15 and 25, within which the optimal stop spacing is determined mainly by number of bus stops, not by arrival rate.#p#分页标题#e#

Chapter 7 Conclusion and Limitation 结论与局限性

This project is based on an analysis of a real world bus line, in which bus stop spacing are varied and total trip times are measured accordingly. The optimal bus stop spacing obtained from the optimization problem leads to shorter total trip time among all possible settings. With currently assumed arrival rate of 25 arrivals per minute per 8km or equivalently 3.125 arrivals per minutes per km, the stop spacing of bus 96 is found to be best at 800 meters. Sensitivity analysis shows after varying the arrival rate, the optimal stop spacing varies within the small range from 780 to 830 meters. Therefore even the arrival rate changes, the optimal stop spacing policy is robust to some extent.

Moreover, the C++ model is proven as an effective tool for bus stop spacing optimization and consolidation. The model can be generalized to simulate feeder bus lines that have similar features with assumptions imposed in chapter 4. In addition, this model can be easily modified to accommodate the needs of collecting statistics about all time aspects of the bus line and passenger information as well as ridership.

The limitations of the project mainly lie in the assumptions and simplification made to facilitate the study. The limitations are therefore ranked based on the importance of their implications on the conclusion of the project. More important limitations are ranked first.

Firstly, the arrival rate is assumed to be uniformly distributed across the bus line. With this assumption lifted, uniformly distributed bus stops may not be optimal. Because to accommodate the transportation need in the region with a higher arrival rate, more bus stops need to be placed. Thus, after relaxing the assumption of uniform arrival rate, this project can be extended to analyze scenarios with arrival rates which are not uniformly distributed geographically.

Secondly, traffic condition is not considered in this project. Traffic disturbance usually cause delay of buses, affecting both the lower threshold for the number of buses and optimal bus stop spacing, assuming no significant change in arrival rate. A more rigorous study can yield more realistic result by introducing traffic variance.

Thirdly, this project focuses on optimizing the bus spacing of one single bus line. In an actual situation, bus lines connect with each other. Passengers transit among these bus lines, bringing additional load to the bus lines. However, this consideration about the interactions is only made possible by gaining access to the real data, which may be realized by a future project.

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