A model for holding strategy in public transit systems with real-time information

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International Journal of Transport Management 2 (2004) 99–110

A model for holding strategy in public transit systems withreal-time information

Saeed Zolfaghari *, Nader Azizi, Mohamad Y. Jaber

Department of Mechanical and Industrial Engineering, Ryerson University, Toronto, Ont., Canada M5B 2K3

Abstract

Holding strategies are among the most commonly used operation-control strategies in public transit systems. These strategies are

most effective when used to control services characterized by high frequency. In this paper, a mathematical model for a holding con-

trol strategy is developed. Particularly, this model uses real-time information of locations of buses along a specified route. The objec-

tive of the developed model is to minimize the waiting time of passengers at all stops on that route. Furthermore, the model

developed in this paper is characterized by the flexibility of adopting situations where bus occupancy could be either high, or

low. A heuristic is developed to circumvent the complexity of the solution for the problem described. Numerical examples and com-

putational results are presented and discussed.

� 2005 Elsevier Ltd. All rights reserved.

Keywords: Public transit; Holding strategies; Real-time information; Simulated annealing; Heuristics; Passengers waiting time

1. Introduction

In most major cities, levels of traffic congestion arerising along with their associated problems such as tra-

vel delays and pollution. While any increase in public

transit rider-ship could reduce the level of traffic conge-

stion and related costs, most transit agencies are not

able to expand their existing services because of fiscal

and physical constraints. As a result, there is growing

interest in the use of new technologies to maximize tran-

sit system efficiency and productivity.In transit systems, public or private, providing a reli-

able service is the main objective of operation managers.

Dependable schedules, minimum passengers waiting

time, and short travel times are some instances of reli-

able services. Service reliability could be measured by

a number of indicators such as arrival delay, headway

1471-4051/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijtm.2005.02.001

* Corresponding author. Tel.: +1 416 979 5000x7735; fax: +1 416 979

5265.

E-mail address: [email protected] (S. Zolfaghari).

delay, and running time that can be categorized based

on frequency of the service. Service frequency is directly

related to the headway, which is defined as the timeinterval between successive vehicles of the same route

and the same direction as they pass a particular point

on the route. For high frequency services, passengers

are more concerned with regularity than with punctual-

ity, whereas, for low frequency services, passengers

expect vehicles to arrive on time (Abkowitz and Engel-

stein, 1984).

In public transit, reliability is an important measure ofservice quality that needs continuous attention and

improvement. This leads to the issue of control strate-

gies. These strategies are aimed at enhancing the reliabil-

ity of transit services (Turnquist, 1978). They are

categorized as either planning control, or real-time con-

trol strategies. Planning control strategies are long-

term and involve strategies such as re-structuring the

routes and schedules. Real-time control strategies areshort-term and involve strategies such as adding extra

buses and short-turning (Turnquist and Blume, 1980).

mailto:[email protected]

100 S. Zolfaghari et al. / International Journal of Transport Management 2 (2004) 99–110

Real-time control strategies are designed to enhance the

system�s ability to remedy specific problems as they

occur. These strategies could be divided into three cate-

gories: station controls; inter-station controls, and others

(Eberlein et al., 1999). The first category, station con-

trols, includes holding strategies, stop-skipping strate-gies, and short-turn strategies. These strategies are

among the most popular and frequently used by public

transit operators to reduce the passenger waiting time,

and/or to prevent the vehicle bunching along the route.

Holding control strategies are used to delay bus move-

ment deliberately when a vehicle is ahead of the schedule.

If adopted, holding control strategies could significantly

reduce the headway variance and the average waitingtime of passengers. Despite these advantages, holding

control strategies could also increase in-vehicle time of

passengers and vehicle travel time.

Stop skipping reduces the travel time of the vehicle of

interest. This would reduce waiting times for passengers

on board a vehicle, and those at downstream stops.

However, this might increase the waiting time for pas-

sengers at skipped stops, and those who are requestedby the driver to alight at a given stop to wait for the next

vehicle in service. For example, consider a case where

you have three buses, A, B, and C, where bus B is lo-

cated on the same route between buses A and C. Then,

an ideal scenario for stop skipping is to have a long pre-

ceding headway (between buses B and C), a short fol-

lowing headway (between buses A and B), and high

passenger demand beyond the segment where skippingis implemented (at stops between buses B and C).

Short-turning strategy involves turning a vehicle

around before it reaches the route terminus. This strat-

egy is usually adopted when the headway variance or

passenger waiting time in the opposite direction are to

be reduced. Referring to the earlier example, an ideal

scenario for short turning is to select a bus with a light

passenger load, a short preceding headway (betweenbuses B and C), and a short following headway (between

buses A and B). Short-turning strategy reduces passen-

gers waiting time in the opposite direction. This benefit

is at the expense of increasing waiting time for passen-

gers on board the vehicle who are requested by the dri-

ver to alight and transfer to the subsequent vehicle.

The second category of real-time control strategies,

inter-station controls, includes the use of traffic signals.Signal priority mechanisms reduce vehicle delays at sig-

nalled intersections. This mechanism either changes the

phase of a signal to green, or extends the duration of

the green phase when a vehicle approaches an intersec-

tion. In contrast to holding, which usually delays passen-

gers and increases running time; signal prioritization

reduces running times and decreases delay for passengers

on-board of buses. Although signal control can reducepassengers waiting time along the route, it might nega-

tively affect traffic flow at signalled intersections.

The third and last category of real-time control strat-

egies includes deadheading, expressing, and adding re-

serve vehicles. A deadheaded vehicle is a vehicle that

usually departs empty from a dispatching terminal

point, A, to a designated stop, E, skipping stops in be-

tween A and E. This strategy could reduce both thewaiting time of passengers at the stops beyond the

skipped ones, and the headway irregularity in the sys-

tem. However,deadheading comes at a cost (additional

waiting time) for those passengers who were not served

(skipped) by the deadheaded vehicle (Eberlein et al.,

1998). In a similar way to deadheading vehicles, express

vehicles might skip stops on a route. However, an ex-

press vehicle could be dispatched from any stop, andmight not necessarily run empty. Finally, reserve vehi-

cles are useful when there are unexpected interruptions

in the system (e.g., vehicle breakdown). In such circum-

stances, it may be desirable to add a reserve bus to fill

the existing gap in demand (i.e., passengers waiting at

stops). However, although adding a reserve vehicle to

a service can reduce passenger-waiting time and prevent

headway irregularity in the system, it might inflict addi-tional costs on transit agencies.

Among the strategies surveyed above, holding strate-

gies are the focus of this paper due to their popularity

among practitioners in public transit systems. Holding

control strategies could be classified into two types.

The first type uses threshold-base control models to hold

a bus at control stops to correct the headway between

consecutive buses. The second type use mathematicalprogramming models with holding times as decision

variables and passenger-waiting time as the cost func-

tion to be minimized.

Several studies have modeled threshold-base control.

Among the earliest studies was that of Osuna and

Newell (1972) who presented an analytic method that

determined the optimal holding strategy for a hypothet-

ical route consisting of one stop with either one or twovehicles. The objective function of the Osuna and New-

ell model was to minimize the average wait time of pas-

sengers. In their analysis, the passengers� arrival rate wasassumed to be uniformly distributed, and with no bus

bunching when considering a system of two buses. In

their study, Osuna and Newell (1972) provided an opti-

mal policy for the case of a single bus operating along

the route, while ignoring the case when there are twobuses or more. Barnett (1974) considered a route with

two terminals and one control stop in-between. Unlike,

the work of Osuna and Newell (1972), Barnett (1974)

developed a two-point, discrete, approximate distribu-

tion of vehicle delay, with the intention of reducing

the complexity of the problem discussed. In his model,

Barnett (1974) structured the cost function as the sum

of the weighted passengers� waiting time and on-boardpassengers delay time at the control stop using a thresh-

old-holding time as the decision variable. Koffman

S. Zolfaghari et al. / International Journal of Transport Management 2 (2004) 99–110 101

(1978) developed a simulation model to analyze a one-

way bus route. They tested several control strategies

for buses in real-time. For the holding strategy, two

threshold values, 75% and 65% of desirable headway,

were analyzed. Their results indicated that higher

threshold values correspond to lower waiting timesand higher bus travel time. Abkowitz and Engelstein

(1984) found that the optimal control point, defined as

a stop where a headway deviation is measured, is sensi-

tive to the ratio of passengers on board to those waiting

at downstream stops. Abkowitz et al. (1986) developed

an empirical headway deviation function using Monte

Carlo simulation to estimate the waiting time of passen-

gers. In their model, the cost function measures the wait-ing time of passengers along a route with threshold

value of holding and location of control point as deci-

sion variable. The results of Abkowitz et al. (1986) indi-

cate that using headway-based holding control could

reduce the wait times of passengers by 5–15%. They rec-

ommended that a control point be placed prior to high

demand stops.

The emergence of technologies such as automatic

vehicle location (AVL) and global positioning systems

(GPS) facilitated the design of computer-based real-

time decision support systems for public transit. Eber-

lein et al. (1999) presented the first research on

real-time routine control problems. Three types of con-

trol strategies were studied. These strategies were hold-

ing, deadheading, and expressing. They considered a

one-way loop transit network of two terminals andnumber of intermediate stations. On this route, vehicles

are assumed to operate with evenly scheduled headway.

Eberlein et al. (1999) adopted a rolling horizon ap-

proach to formulate control problems addressed. Each

time the optimization problem is solved considering

only a limited number of vehicles in the route, known

as impact set, where an impact set was defined as a

set of vehicles operating on a route. Then, the resultof optimal control policy is applied to the first vehicle

on the set. To illustrate their solution procedure, con-

sider a case where there are 10 vehicles operating on

a route. First, the optimization problem is solved con-

sidering only vehicles 1, 2, and 3 and the result of opti-

mal control policy is only applied to vehicle 1. In the

second step, vehicles 10, 1, and 2 are considered and

the optimization problem is solved for these vehicleset and the optimization control result is applied to

vehicle 10, the first vehicle in the impact set. This pro-

cess was repeated for all vehicles on the route. Eberlein

et al. (1999) probably used the concept of an impact set

to reduce the complexity of the problem. However, this

comes at the cost of eliminating the effect of vehicle

interaction on the one-way loop route. Eberlein et al.

(1999) attempted to minimize passengers waiting timeat stations beyond the impact set. The effectiveness of

these control models was tested using empirical data.

Eberlein et al. (1999) concluded that the holding con-

trol strategy was found to be more effective than the

other two strategies.

O�Dell and Wilson (1999) presented formulations for

disruption control problems in rail transit systems with

more than one rail branch. They investigated severalholding and short turning strategies. O�Dell and Wilson

(1999) objective was to minimize waiting time of passen-

gers within and beyond an impact set, with an impact set

being defined as a set of trains and stations preceding

and succeeding a disruption point. Using empirical

data, they concluded applying the holding control strat-

egies, passengers waiting time could be reduced by 15–

40%.Unlike earlier studies that investigated real-time hold-

ing control strategy for rail transit systems, this paper

presents a mathematical model for holding problem

applicable to urban bus transit operations. The model

presented in this paper could be viewed as an extension

to that of O�Dell and Wilson (1999), as their application

was limited to rail transit operations. This study pre-

sents a mathematical model that reflects some of the fea-tures inherent in bus operations, e.g., bus bunching. This

required altering the objective function and the addition

and/or alteration of constraint equations.

The study presented in this paper is part of a research

in the area of advanced public transportation systems

that include GPS/SO integrated AVL system, real-time

passenger information system, and real-time schedule

control as depicted in Fig. 1. Holding control presentedin this article is part of the third component in which

control strategies are devised to deal with possible ser-

vice irregularities. In the suggested framework, such

information as predicted arrival times of every bus to

a number of downstream stops and the actual departure

time of buses from recently visited stops are provided by

an enhanced AVL system. This information combined

with the historical data of the route is transmitted tothe real time schedule control component. Upon receiv-

ing updated information, the performance measurement

module computes the performance indicators. If these

indicators are below certain pre-specified critical values,

then appropriate control strategies are suggested. Upon

selection of the best strategy (e.g., holding control), the

predicted departure times at downstream stops are up-

dated. In an earlier study (Zolfaghari et al., 2002), amulti-attribute performance measurement model that

integrated headway and schedule adherences was pro-

posed. The study presented in this paper (shaded area

in Fig. 1) is the first step to extend the performance mea-

surement component in order to facilitate the develop-

ment of a decision support system for public transit

operations. The model has the flexibility of adapting

to both high-occupancy and low-occupancy of buses.The objective of this model is to minimize the passen-

gers� waiting time at any stops. Such waiting times

System Monitoringand Control

Process AVL System

HistoricalData

Data Collection Performance Measurement

Selection of Control Strategies

HoldingControl

ControlAction 1

ControlAction n

… …

Fig. 1. Framework of a computer-based real-time schedule control for public transit systems.


include the waiting time of new arrivals as well as that of

the left-behind passengers. In the model proposed here-

in, the optimization problem is solved simultaneously for

all buses on the route.

The remainder of this paper is organized as follows.Section 2 describes the problem of interest. Section 3

presents the mathematical model. A heuristic approach

used to solve this model is presented in Section 4. Sec-

tion 5 discusses a numerical example. Section 6, is for

the summary and conclusions.

2. Problem description

The problem considered herein investigates a holding

strategy that could be applied to control an operation of

i buses on a specified route with N stops. A route is de-

fined as a one-way loop that connects a sequence of geo-

graphical points labelled as stops and terminals. The

difference between a stop and a terminal is that at a stop,

passengers may board, alight or continue their trip ifthey are already on board, but at a terminal, all passen-

(a)

Stop k

Bus i

Headway

Stop k

Bus i

Headway

(b)

…

N

1

N/2+k N-1

k

…

Terminal

Fig. 2. The schematic presentation of

gers on board must alight, and then new passengers may

board to start a new trip. In this paper, we divide the

route into two directions. In the first direction (includes

stops 1,2, . . . ,N/2�1), buses are running from the

dispatching terminal (N) to the intermediate terminal(N/2), whereas in the second direction (includes stops

N/2 + 1,N/2 + 2, . . . ,N � 1), they are running from the

intermediate terminal (N/2) to the dispatching terminal

(N) following a given headway (see Fig. 2 for illustra-

tion). It is assumed that at any given instant, and using

a real-time passenger information system and automatic

vehicle location system (AVL system), the necessary

information is transmitted to controllers. This informa-tion includes: (1) the departure times of buses from the

last visited stops, (2) indices of last visited stops, (3)

the arrival times of buses to their downstream stops,

and (4) loads on buses departing their last visited stops.

However, there is information that will not be transmit-

ted, such as: (1) passenger arrival rate, (2) fraction of

passengers on board of buses alighting at stops, (3) dwell

time parameters, and (4) running time for each bus be-tween stops including acceleration and deceleration

Bus i-1

Stop k+

Bus i-1

Stop k+∆ k

N/2-1

N/2+1

N/2

…

…

Intermediate stop

the one-way loop and headway.


times. We assume that these parameters could be deter-

mined from available historical data.

2.1. Assumptions and limitations

The model is subject to the following assumptionsand limitations:

Assumptions:

• When the bus is not loaded to capacity, it is assumedthat all boarding takes place at the front door and

alighting takes place at the rear door.

• When the bus is loaded to capacity, the front door

can be used for both boarding and alighting. In such

cases, boarding can only start when front-door

alighting is completed.

Limitations:

• Running times are approximated by their expected

values.

• Dwell time function is approximated by a linear

function.

• During controlling the vehicles we consider a limited

number of downstream stops for each bus, which is

called ‘‘rolling horizon’’.

2.2. Data requirements

In this paper, it is assumed that the following infor-

mation is either available or could be estimated: Pro-

vided by an AVL system:

• Actual departure times of buses from the most

recently visited stops.

• Predicted arrival times of buses from downstream

stops (maximum eN istops for each bus i).

Estimated from historical data:

• Expected value of bus running times between stops.

• Expected value of bus dwell times at stops.

• Passenger arrival rates and alighting fractions at each

stop.

• Dwell time function parameters.

2.3. Notation list

The following variables and parameters are used in

the proposed formulations:

i index of vehicles, i = 1, . . . , Ik index of stops, k = 1, . . . ,Ndi,k departure time of bus i from stop k

Fi,k predicted arrival time of bus i at stop k

li,k load in bus i departing stop k

pi,k passengers left behind by bus i at stop k

Di,k demand for bus i at stop k

rk passenger arrival rate at stop k

vi,k 1, if bus i is loaded to capacity when it departstop k; 0, otherwise

Sui;k dwell time of bus i at stop k when the passenger

load is below the bus capacity

Sci;k dwell time of bus i at stop k when bus i is loaded

to capacity

Cj dwell time function parameter

Qk passenger alighting fraction at stop k

Afk passenger alighting fraction at stop k that use

the front door for alighting

Ark passenger alighting fraction at stop k that use

the rear door for alighting

Rk running time from stop k-1 to stop k, including

acceleration and deceleration

Lmax passenger capacity of bus

Mj sufficiently large numbers

K set of stations on the route K = {1,2,3, . . . ,N/2, . . . ,N}

I set of buses operating on the route

I = {1,2,3, . . . , i, . . . , I}eN itotal number of stops in the horizon of bus i

Th rolling horizon

g expected value of running times on a route

u expected value of bus dwell time at stops

tnow present time (time now)li index of the last visited stop by bus i

kfst the first stop in the horizon of bus i

2.4. Rolling horizon

As discussed above, the transit system considered in

this study is a one-way loop in which buses are operating

on the route following a planned headway. When con-

trolling the route, and for each bus, only a limited num-

ber of immediate downstream stops are considered. Thisway, the problem could be reduced to a reasonable size

if there are a large number of stops and buses on the

route. The number of downstream stops, eN i, depends

on the length of the rolling horizon T h as described in

Fig. 3. The rolling horizon is a common term in many

fields that could be defined differently depending on

the application (e.g., Newell, 1998; Choong et al.,

2001). In this paper, the rolling horizon is defined as areasonable length of time in which bus movement on

the route could be projected fairly. It is a ‘‘horizon’’ as

it is a finite future time period, and it is ‘‘rolling’’ be-

cause it moves forward as buses progress on their routes.

Since stops are not equidistant, then the total number of

stops in a given rolling horizon could differ for each bus

Th

k = li+1 k = li+2 k = li+ Ñi

i…

Fig. 3. Rolling horizon for bus i.


operating on the same route. In order to ensure that the

waiting time of passengers at all stops on the route are

considered, the length of the rolling horizon, Th, must

be selected such that:

T h > max di;li�1 � tnow� �

8 i 2 I ð1Þ

Furthermore, the relationship between the rolling hori-

zon, T h, and the number of downstream stops in the

horizon of bus i, eN i, could be defined as follows:

eN i ¼ T h

gþ uð2Þ

where g is the expected value of the route running times,

and u is the expected value of the bus dwell time at

stops.

3. Mathematical model

3.1. Objective function

In this paper, the cost (objective) function sums the

passengers waiting times at stops along a route. If bus

i is crowded when departing from stop k, then there is

an additional waiting time for passengers who could

not board that bus. These passengers are assumed towait for the next bus; i.e., bus i + 1. The total waiting

time of passengers is then calculated as

XI

i¼1

XliþeN i

k¼liþ1

rkðdi;k � di�1;kÞ2

2þ pi;k diþ1;k � di;kð Þ

" #ð3Þ

The first term in the cost function computes the waitingtime for passengers who arrive randomly at stop k since

the last bus departed that stop. The second term com-

putes the additional waiting time for those who are left

behind at stop k because of an overloaded bus.

3.2. Dwell time function

Guenthner and Sinha (1983) presented a mathemati-cal model for estimating bus dwelling time at bus stop

using data from Milwaukee and Lafayette. They as-

sumed that the number of passenger boarding and

alighting at each bus stop follows a Poisson distribution,

and that the passenger demand is uniformly distributed

along a bus route. Dwell time is an important element in

determining the departure time of buses from stops. It is

directly related to the number of passengers boardingand alighting, whether a bus has reached capacity or

not. Although the true dwell time may be non-linear,

several studies indicate that for short headway services,

dwell time could be approximated by a linear function

(Wilson and Lin, 1992; Eberlein et al., 1998). Following

are two linear functions used to calculate bus dwell timeat stop in this paper. When the busload is below capac-

ity, the dwell time is computed as

Sui;k ¼ C0 þmax C1A

rkli;k�1;C2 rkðdi;k � di�1;kÞ þ pi�1;k

� �� ð4Þ

where the first term C0 is a constant, C1 and C2 are,

respectively, the alighting and boarding times per pas-

senger when load in a bus is less than the bus capacity.

The total number of passengers boarding bus i at stop k

is the summation of those passengers who arrive at stop

since last bus (i � 1) departed and the number of peopleleft behind by bus i � 1; i.e., [rk (di,k � di�1,k) + pi�1,k].

The number of passengers alighting at stop k is calcu-

lated using the load on bus when i departing previous

stop k � 1 multiplied by alighting fraction at stop

k;Arkli;k�1.

If bus i is loaded to capacity when departing stop k,

then the dwell time is computed as

Sfi;k ¼ C3 þmax C4A

rkli;k�1;C4A

fk li;k�1

�þ C5 minf Lmax � ð1� QkÞli;k�1½ �;rkðdi;k � di�1;kÞ þ pi�1;k

� �ggwhere C3 is a constant, and C4 and C5 are, respectively,the alighting and boarding times per passenger in

crowded condition. In this case, the number of passen-

gers boarding is either the number of passengers at stop

k who desire to board bus i, or the number of available

seats on bus i, which ever is minimum.

3.3. Formulations

Three types of holding strategy have been addressed

for rail transit systems in literature (O�Dell and Wilson,

1999). The first strategy, known as holding-all, permits

holding each train at any of the stations in the impact

set. An impact set represents the number of trains before

and after a disruption point. The second strategy is to

hold each train at the first station it reaches. The third

strategy is to hold each train at only one station in theimpact set that reduces the waiting time of passengers.

The holding station for this strategy could be any station

in the impact set.


In this section, and using the second strategies of

O�Dell and Wilson (1999), a mathematical model is

developed for holding strategy. The objective of this

model is to alleviate the effect of service irregularities

on passengers� waiting times in bus transit systems.

The question this paper attempts to answer is whichbus (buses) has (have) to be held at the first downstream

stop in order to minimize total passengers� waiting time.

Then the mathematical programming problem could be

presented as

MinimizeXI

i¼1

XliþeN i

k¼liþ1

rkðdi;k � di�1;kÞ2

2þ pi;kðdiþ1;k � di;kÞ

" #Subject to : li;k 6Di;k 8 i;k 2 ðI;KÞ ð6Þ

li;k 6 Lmax 8 i;k 2 ðI;KÞ ð7Þli;k P Lmaxvi;k 8 i;k 2 ðI;KÞ ð8Þli;k PDi;k �M1vi;k 8 i;k 2 ðI;KÞ ð9Þdi;k P F i;k þ Su

i;k �M2vi;k

8 i;k 2 ðI;KÞ ð10Þdi;k P F i;k þ Sc

i;k �M3ð1� vi;kÞ8 i;k 2 ðI;KÞ ð11Þ

di;k 6 F i;k þ Sui;k þM4vi;k

8 i;k 2 ðI;KÞjk 6¼ kfrst ð12Þdi;k 6 F i;k þ Sc

i;k þM5ð1� vi;kÞ8 i;k 2 ðI;KÞjk 6¼ kfrst ð13Þ

Di;k ¼ ð1�QkÞli;k�1

þ rkðdi;k � di�1;kÞþ pi�1;k

8 i;k 2 ðI;KÞ ð14Þpi;k ¼max 0;Di;k � Lmaxf g8 i;k 2 ðI;KÞ ð15Þ

Qk ¼ Afk þAr

k 8 k 2K ð16Þ06Qk 6 1 8 k 2K ð17Þvi;k 2 0;1f g 8 i;k ð18Þdi;k; li;k;pi;k;P 0 integer 8 i;k ð19Þ

Constraints (6)–(9) restrict the number of passengers

on board when the bus departs the stop. The number of

passengers on board is calculated as a function of the

load on the bus when arrives at station k and the num-

ber of boarding and alighting. When bus i at bus-stop k

is at full capacity, vik = 1, the constraints (7) and (8) are

binding (li,k = lmax). When bus i at bus-stop k is not at

full capacity, vik = 0, then constraints (6) and (9) are

binding. Constraints (10)–(13) restrict the earliest depar-

ture time of bus i at station k. To reduce headway irreg-

ularities in the service, the holding strategy assumed

herein, allows for each bus to be held at the first stop

it for a finite period (e.g., 5 min). When bus i is not

loaded to capacity as it departs stop k, vik = 0, and

depending on whether or not stop k is the first stop

reached by bus i, constraint (10) or constraints (10)

and (12) will apply. If stop k is the first stop reached

by bus i, constraint (10) is binding. Therefore, the earli-

est departure time of bus i could be equal to, or greaterthan, the sum of the predicted arrival time and the dwell

time at stop k. Otherwise, constraints (10) and (12) are

binding. When bus i is at full capacity, vik = 1, and

depending on whether or not stop k is the first stop

reached by bus i, constraint (11) or constraints (11)

and (13) will apply. If stop k is the first stop reached

by bus i, constraint (11) is binding. Otherwise, both con-

straints (11) and (13) are binding.The demand for bus i at stop k is calculated by con-

straint (14). Constraint (14) represents the load on the

bus i departing stop k � 1 less the number of passengers

alighting at stop k, added to the number of passengers

arrived at stop k since bus i � 1 departed, and to the

number of passengers who were left behind at stop k

by bus i � 1, pi,k, with the later given by constraint

(15). Note that the value pi,k depends on the demandfor bus i at the stop k, and on the number of available

spots on the bus after alighting. When the number of

available spots on bus i is more than, or equal to, the

demand for bus i at stop k, then the number of passen-

gers left behind at stop k is equal to zero. Otherwise, the

number of passengers left behind at stop k is the demand

at that stop less the capacity of bus i.

4. Simulated annealing algorithm

Simulated annealing (Kirkpatrick et al., 1983) is

among the most popular iterative search methods that

have been applied widely to solve many combinatorial

optimization problems. This method is a random local

search technique based on the principles of physics.The search starts from an initial feasible solution. Each

solution has a specific cost value. A small change in one

variable, or a combination of some variables can gener-

ate a neighbouring solution with a different cost value.

In each iteration, a neighbouring solution is randomly

generated according to a pre-specified neighbourhood

structure. If the cost value of the candidate solution is

lower than that of the current solution, the move tothe candidate solution is made. However, if the candi-

date does not improve the current solution, there is still

a chance of transition according to the following proba-

bility function.

PðtransitionÞ ¼ min 1; exp �DCi

hi

� �� ð20Þ

where DCi is the cost difference between the candidate

solution and the current solution in iteration i, and hiis the control parameter known as temperature. For


each iteration, the above transition probability value is

compared to a uniform random number generated. If

the transition probability value is greater than or equal

to the value of random number generated, then the tran-

sition to the worse solution is accepted, otherwise the

transition from the current solution to the candidatesolution is rejected. Then, another solution in the neigh-

bourhood of the current solution will be generated and

evaluated. The temperature hi has a significant impact

on the transition probability. The greater the tempera-

ture, the higher the probability will be. In a standard

simulated annealing, the initial temperature is usually

set at a high level to allow the search to move freely to

worse solutions. As the search continues, the tempera-ture declines according to a cooling schedule. Toward

the end of the search process, it is less likely to move

to a worse neighbouring solution. The change in temper-

ature values is controlled by a cooling schedule. To se-

lect a cooling schedule, it is necessary to decide on the

type of the algorithm, homogenous or inhomogeneous.

In a homogeneous algorithm, the value of temperature

remains constant for a certain number of iterationsand then is decreased in between subsequent chains.

However, in an inhomogeneous algorithm, the value

of the temperature is decreased between subsequent

transitions (Van Laarhoven and Aarts, 1987). The

asymptotic convergence of both types has been well

studied in the literature (Van Laarhoven and Aarts,

1987), (Gidas, 1985), and (Hajek, 1988).

In this paper we adopt an inhomogeneous approachand we use an adaptive cooling schedule that may occa-

sionally increase the temperature according to the char-

acteristics of the search trajectory as follows.

h ¼ h þ k lnð1þ r Þ ð21Þ
i min i
Fig. 4. Stop and bus positions f

where hmin is the minimum temperature value, k is a coef-ficient that controls the rate of temperature rise, and ri is

the number of consecutive upward moves at iteration i.

The initial value of ri is zero, thus the initial temperature

h0 = hmin. At any point during the search, if a move is

made to a solution with a cost higher than that of the pre-vious solution (i.e., upward move), then the counter riincremented by 1. If the new solution has an equal cost,

ri remains unchanged; and if the cost is improved, then riis reset to zero. The two other main elements of any sim-

ulated annealing base algorithm are cost function and

neighbourhood generating mechanism. For the problem

described in Section 3, the cost function is defined as the

total passenger waiting time and a neighbour is gener-ated by adding/deducting a prescribed holding value to/

from the departure time of a randomly selected bus.

The heuristic used in this paper is explained in the

Appendix A. The algorithm was coded in Visual Basic

and implemented on a Pentium III with 450 MHz CPU.

5. Numerical example

Consider a case where there are 18 buses operating on

a route consisting of twenty four intermediate stops and

two terminals as illustrated in Fig. 4(a). The planned

headway is 6 min, H = 6, and the rolling horizon is

20 min, Th = 20. Based on the length of the rolling hori-

zon, there are eN i ¼ 5 stations for all buses. Two cases

might arise. Either all buses are running on scheduleor there is one or more headway irregularity in the sys-

tem. These cases are discussed below.

Case 1: Regular service Assume that the positions of

buses at clock-time 5 are as described in Fig.

4(a). Also assume that the information trans-

or the numerical example.

Table 1

Buses data for the numerical example

Stops 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Bus 1 The last visited stop *

Actual departure time from the last visited stop 3

Expected arrival time at down stream stops 6 9 13 18 22

Load on bus when departing the last visited stop 21









































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Table 1 (continued)

Stops 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26


















Actual departure time from the last visited stop 1 5











Stops data for the numerical example

Passenger

arrival rate

1.5 1.5 1.5 2 1.5 2 1.5 1.5 2 2.5 2 2 4 2 2 2 2.5 2.5 2 2 1.5 2 1.5 1.5 1.5 4

Passenger

alighting

fraction

0.1 0.1 0.2 0.2 0.3 0.2 0.3 0.2 0.2 0.2 0.3 0.2 1 0.1 0.1 0.1 0.2 0.2 0.2 0.3 0.2 0.3 0.2 0.1 0.1 1

Running

time

3 3 4 3 2 3 4 3 4 3 3 3 3 3 3 3 3 4 3 4 3 2 3 4 3 3

New transmitted information for bus 16 in the numerical example (case 2)





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Table 2

Computational results for the numerical example (Case 2)

Regular service Irregular service

Do nothing Holding first Do nothing Holding first

Total waiting time 3289 – 3390 3342

Additional waiting time – – 101 53

Saving additional waiting time (%) – – 0 47.5


mitted at that time includes: (1) the index of the

last visited stop, k � 1, with respect to bus i, (2)bus i’s actual departure time from the last vis-

ited stop, (3) the expected arrival time of bus i

from downstream stops, and (4) load on bus

when departing the last visited stop, with values

summarized in Table 1(a). Note that, in order to

generate those values for actual departure times

and expected arrival times of buses, it is

assumed that bus 1 has departed from stop 3at time 3. Then, based on the running time

between stop 3 and stop 4 it has to reach stop

4 at time 6. Adding 1 unit of time to the arrival

time of bus 1 at stop 4 as the expected dwell

time, the expected arrival time of bus 1 at stop

5 is at time 9. Using the value of planned head-

way, H, and running times between stops, the

actual departure time and expected arrival timeof the remaining buses from the last visited stop

and downstream stops will be determined.

Table 1(b), summarizes the further information

including: (1) passenger arrival rate, (2) passen-

ger alighting fraction, and (3) the running times

of buses between two consecutive stops, where

this information could be determined from

available historical data. Then, substitutingthese values into the mathematical model

described by equations (3), (6)–(19), the cost

function, which computes the total waiting time

for passengers, yields a value of 3289 units of

time. This value represents the minimum value

of passenger�s waiting time in the next 20 units

of time (Th) for the route described above.

Case 2: Irregular service Now, assume bus 16 has beenrunning ahead of schedule. The position of the

buses on the route is as depicted in Fig. 4(b).

Under such conditions, the new transmitted

information related to bus 16 is summarized in

Table 1(c). Using the same data for case 1

except for bus 16 and substituting these values

into the mathematical model described by equa-

tions (3), (6)–(19), the cost function, which com-putes the total waiting time for passengers,

yields a value of 3390 min. Indicating that if

the controller does nothing, the expected total

passengers waiting time will increase by

101 min (3390 � 3289 = 101).

Applying the proposed holding model to the problem,

the best solution found after 3 s computational time is to

hold bus 16 at stop 10 for 2 min. By doing so, the total

expected passengers� waiting time yields the value

3342 min. The additional passengers� waiting time

caused by service irregularity has been reduced by

47.5% (see Table 2).

6. Summary and conclusion

Motivated by recent studies on real-time holding con-

trol for rail transit system, this paper presented a mod-

ified mathematical model for holding strategy

applicable to urban bus transit operations. The devel-

oped model is flexible in adapting to two different bus-loading conditions (full and less than full capacity) when

a given bus departs a stop along a given route. The cost

function of the developed model measures the passen-

gers� total waiting time. This cost function is the sum

of waiting times for new arrivals at stops and for those

passengers that are left behind by a full-load bus. In the

developed model, we proposed the concept of dwell time

functions that calculate the delay times at stops based onthe number of passengers boarding and alighting from

the front and the rear door of the bus. A heuristic algo-

rithm based on simulated annealing is presented for

solving the model. The efficiency of the algorithm was

tested using a numerical example. The first case assumes

a regular service, and considered as a benchmark point,

while the other case assumes an irregular service. When

solving the numerical example for the irregular case, twodecision situations where considered. First, a ‘‘do noth-

ing’’ decision (the extreme case), that is, do not correct

the irregularity of the bus service, with the second is to

correct the irregularity of the service by using the hold-

ing strategy concept incorporated in the developed

model. The results indicate that applying the holding

strategy significantly improved the cost function by

bringing its value closer to that of the benchmark case.The computational time to reach an optimal solution

was found to be insignificant, which is what a manager

desires when operating in a real-time environment.

To attain the desired performance of the proposed

model, it may be necessary to incorporate this model

into a decision support system framework for transit

operation control, similar to that presented in Fig. 1.


The holding strategy model proposed in this paper is a

first step in extending previously developed performance

monitoring modules to include control actions. This

model is capable of solving the problem simultaneously

for all buses on the route. This is considered as an

important aspect of the model presented herein as com-pared to previously developed models. The implementa-

tion, however, would be more effective if it is in

conjunction with other control strategies. This is impor-

tant since different services are subject to different

behavioural interruptions, thus requiring the implemen-

tation of different control strategies. For example, there

are cases where buses might be early while other buses

late. An early bus requires a decelerating action (e.g.,holding), while a late bus requires an accelerating action

(e.g., stop-skipping). Further research is still needed in

this direction to develop other control strategies for

urban transit operations. Such strategies may include

short-turning, stop-skipping, dead-heading, expressing,

adding/removing a trip and signal priority.

Acknowledgments

The authors wish to thank anonymous reviewers for

their valuable suggestions. The financial support of the

Geomatics for Informed Decision (GEOIDE) Network

of Centres of Excellence (Grant TCO#51) is also greatly

appreciated.

Appendix A. SA algorithm

Step 1:

Set i = 1 and select hmin

Select an initial solution and set the current solu-

tion Bi and the best solution Oi equal

to the initial solution.Step 2:

Select a candidate solution Bc from the neighbour-

hood of Bi

IF Z(Bc) < Z(Bi) THEN

Set Bi+1 = Bc

IF Z(Bc) < Z(Oi) THEN set Oi = Bc.

Go to Step 3

END IFGenerate a random number u from a U(0,1)

distribution;

Compute the transition probability P(Bi,Bc) =

exp[(Z(Bi) � Z(Bc))/hi]IF u 6 P(Bi,B

c), THEN set Bi+1 = Bc.

Step 3:

Select hi+1Set i = i + 1.

IF termination condition is satisfied, THEN stop;

ELSE, go to Step 2.

END

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