Application of System of Linear Equations to Traffic Flow for a

International Journal of Contemporary Mathematical Sciences

Vol. 9, 2014, no. 14, 653 - 660

HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ijcms.2014.410109

Application of System of Linear Equations to

Traffic Flow for a Network of Four One-Way

Streets in Kumasi, Ghana

Isaac Kwasi Adu

Department of Mathematics

Valley View University, Techiman Campus

P. O. Box 183 B/A - Ghana

Douglas Kwasi Boah

Department of Mathematics

University for Development Studies

P. O. Box 24, Navrongo - Ghana

Vincent Tulasi

Department of Statistics and Mathematics

Ho Polytechnic

P. O. Box HP217 V/R - Ghana

Isaac Kwasi Adu, Douglas Kwasi Boah and Vincent Tulasi. This is an open

access article distributed under the Creative Commons Attribution License, which permits

unrestricted use, distribution, and reproduction in any medium, provided the original work is

properly cited.

Abstract

We used a system of linear equations to determine the number of vehicles that

should be allowed to route a four one-way streets in Kumasi, in order to keep traffic

flowing. The systems of equations used in the model were solved analytically using

the method of Gauss-Jordan elimination. Our work shows that if 155vph, 276vph,

and 240vph are allowed to route intersection A and B, A and D and D and C of the

654 Isaac Kwasi Adu et al.

model respectively, traffic congestion in the area of Kumasi discussed in the model

would be minimized. We recommend to the stakeholders of Kumasi to provide

exclusive lanes for public transport, ensure the use of traffic regulations and traffic

engineers to control the traffic, use innovative ideas to reduce traffic impacts on

public transport, provide traffic lights at the four intersections under discussion in

this research and adjust them in the direction of the results of this research.

Keywords: Mathematical model, traffic congestion, Traffic flow, traffic volume

1. Introduction

In mathematics and civil engineering, traffic flow is the study of interactions

between vehicles, drivers, and infrastructure (including highways, signage, and

traffic control devices), with the aim of understanding and developing an optimal

road network with efficient movement of traffic and minimal traffic congestion

problems [1].

Mathematical theory of traffic flow and traffic equilibrium analysis was first

introduced by Frank Knite in 1920’s, and was refined into Wardrop’s first and

second principles of equilibrium. Current traffic models use a mixture of

empirical and theoretical techniques. These models are then developed into traffic

forecast, to take account of proposed local or major changes, such as increased

vehicle use, changes in land use or changes in mode of transport and to identify

areas of congestion where the network needs to be adjusted [1].

Traffic congestion has a number of negative effects on humanity. These

include wasting time of motorists and passengers which therefore reduce regional

economic health; delays, which may result in late arrival for employment,

meetings and education, resulting in loss of businesses, disciplinary action or

other personal losses. Blocked traffic may interfere with the passage of emergency

vehicles travelling to their destinations where they are urgently needed; wasted

fuel, increasing air pollution and carbon dioxide emissions owing to increasing

idling, acceleration and breaking of vehicles; wear and tear on vehicles as a result

of idling in traffic and frequent accelerating and breaking, leading to more

frequent repairs and replacement of car parts; stressed and frustrated motorists,

encouraging road rage and reduced health of motorists[2].

In United States of America, the Texas Transportation Institute estimated

that, in 2000, the 75 largest metropolitan areas experienced 3.6 billion

vehicle-hours of delay, resulting in 5.7 billion U.S. gallons (21.6 billion liters) in

wasted fuel and $67.5 billion in lost productivity, or about 0.7% of the nation's

GDP [3].

In the United Kingdom, the inevitability of congestion in some urban road

networks has been officially recognized since the Department for Transport set

down policies based on the reported Traffic in Towns in 1963. The Department

for Transport sees growing congestion as one of the most serious transport

Application of system of linear equations 655

problems facing the UK [3]. Eddington (2006) published a UK

government-sponsored report into the future of Britain's transport infrastructure.

The Eddington Transport Study set out the case for action to improve road and rail

networks, as a "crucial enabler of sustained productivity and competitiveness".

Eddington has estimated that congestion may cost the economy of England £22 bn

a year in lost time by 2025. He warned that roads were in serious danger of

becoming so congested that the economy would suffer [4].

In China, the August 2010 China National Highway 110 traffic jam in Hebei

province, is considered the world's worst traffic jam ever, as traffic congestion

stretched more than 100 kilometres (62 miles) from August 14 to 26, including at

least 11 days of total gridlock [5, 6, 7]. The New York Times has called this event

the "Great Chinese Gridlock of 2010 [8].

In Ghana, Transport experts have attributed the depressing vehicular

congestion to an increase in vehicular population on the city’s already inadequate

roads. There are an estimated 1.2 million vehicles in Ghana, 60 per cent in Accra

alone, and with a total road network of 1,632 kilometres of 1,310 kilometres are

tarred. Accra’s roads appear woefully inadequate as heavy traffic congestion

characterizes travelling on most roads within the national capital [9].

Traffic congestion on Kumasi roads is becoming an ever-present nightmare for

road users, negatively affecting productivity and the environment. The city of

Kumasi has spread out over the past years. Being the second largest city in Ghana,

it is experiencing rapid urbanisation and accelerated population growth and an

exploded traffic on its roads. Nowhere is this more evident than in the Central

Business District (CBD) and on other arterial roads. One result of this

phenomenon is the severe traffic congestion as witnessed on the Lake Road, 24th

February Road and the Sunyani Road which results in loss of working time,

affecting productivity, higher vehicle running cost and negative environmental

impact. Information at the Driver and Vehicular Licensing Authority (DVLA)

indicate that the number of vehicles imported into Kumasi keep increasing year

after year [10].

The current study therefore aims at applying a system of linear equations to

traffic flow for a network of four one-way streets in Kumasi, Ghana. It also aims

to determine the number of vehicles that should be allowed to route the four

one-way streets under study in the model in order to reduce traffic congestion in

Kumasi.

2. Mathematical Model

A system of linear equations was used to analyze the flow of traffic for a network

of four one-way streets in Kumasi, Ghana. The pioneering work done by Gareth

Williams on Traffic flow [11] has led to greater understanding of this research.

The variables and represent the flow of the traffic between the four

656 Isaac Kwasi Adu et al.

intersections in the network. The data was obtained by counting the number of

vehicles that travelled around the four one-way streets between the hours of 6am to

10pm, and 2pm to 6pm during the mid-week peak traffic hours. The arrows in the

diagram indicate the direction of flow of traffic in and out of the network that is

measured in terms of number of vehicles per hour (vph). The diagram in Figure 1

below describes the four one-way streets in Kumasi under study in the model:

Figure 1: Diagram of the four one-way streets, in Kumasi

Model Assumptions

The following assumptions were made in order to ensure the smooth flow of the

traffic;

i) Vehicles entering each intersection should always be equal to the number

of vehicles leaving the intersection.

ii) The streets must all be one-way with the arrows indicating the direction

of traffic flow.

The system of equations for the model was formulated as follows:

At intersection A: Traffic in , traffic out 241 190, thus,

431.

At intersection B: Traffic in 150 105, traffic out , thus

255

At intersection C: Traffic in , traffic out 230 +110, thus, = 340

150vph

105vph

230vph

280vph

241vph

190vph

110vph

236vph

Atonsu-Prempeh Rd

Asafo market- Kajetia Rd

Roman Hill Road

Labour-Asafo interchange

office Rd

Application of system of linear equations 657

Intersection D: Traffic in 280 236, Traffic out , thus, = 516

The constraints were written as a system of linear equations as follows:

431

+ 255

340

516

We then used the Gauss-Jordan elimination method to solve the system of

equations. The augmented matrix and reduced row-echelon form of the above

system are as follows:

The system of equations that corresponds to this reduced row-echelon form is;

+ 255

= 176

340

Expressing each leading variable in terms of the remaining variable, we had

255

176

+ 340

Row operations

658 Isaac Kwasi Adu et al.

If we take a construction limit on Labour-Asafo Interchange Rd ( ) to be 100vph,

then the values of and will be;

-100 + 255 = 155vph

100 + 176 = 276vph

-100 + 340 = 240vph

Discussion of Results

The system of the modeling equations has many solutions, and therefore many

traffic flows are possible. A driver has a certain amount of choice at the intersection,

due to the nature of the model. Considering the stretch DC, it is desirable to have

small traffic flow as possible along this stretch of road. The flows can therefore

be controlled along the various branches by the use of traffic lights. According to

the model, the third equation in the system shows that will be a minimum when

is as large as possible, as long as it does not exceeds 340. The largest value

can be assumed without causing negative values of , or is 255. Thus the

smallest value of is -255+340 or 85. Any road work on Asafo interchange to

Roman hill down should allow for traffic volume of at least 85vph. Therefore, to

keep the traffic flowing 240vph must be routed between and , 155vph between

and B and 276vph between the intersections and

Conclusion

We have established that traffic congestion at the four one-way street linking

Labour-Prempeh Rd, Roman Hill Rd, Asafo-Roman Rd and Asafo interchange-

Labour Rd can be minimized if any road work on Asafo interchange to Roman hill

down should allow for traffic volume of at least 85vph. Therefore, to keep the

traffic flowing, 240 vehicles per hour must be routed between and , 155vph

between and B, and 276vph between the intersections and respectively

Application of system of linear equations 659

Recommendations

In order to reduce the impact of traffic congestion and ensure the free flow of traffic

in Kumasi, we made the following recommendations to the stakeholders of

Kumasi:

 provide exclusive lanes for public transport

 use regulations and traffic engineers to control the traffic

 ensure the use of innovative ideas to reduce traffic impacts on public

transport.

 provide traffic lights at the four intersections under discussion and adjusted in

the direction of the results of this research.

References

[1] Wikipedia encyclopedia, Traffic flow.

Available: http://en.m.wikipedia.org/wiki/Traffic-flow. (Assessed: 12-10-2014)

[2] Reduce Traffic Congestion.

Available:http://pages.greencitystreets.com/improve-public-transport/reduce-traffi

c-congestion/. (Assessed: 12-07-2014)

[3] Traffic Congestion Plummets Worldwide: INRIX Traffic Scorecard Reports

30 Percent Drop in Traffic Across the U.S.". INRIX. 2012-05-24,

Available:http://en.wikipedia.org/wiki/Traffic_congestion#cite_note-65. Accessed:

2014 - 07-14.

[4] Tackling congestion on our roads, Department for Transport, Available:

http://en.wikipedia.org/wiki/Traffic_congestion#cite_note-65.

(Accessed 2014-06-20).

[5] Delivering choice and reliability, Department for Transport. Available:

http://en.wikipedia.org/wiki/Traffic_congestion#cite_note-65.

(Accessed: 2014-07-11).

[6] L. Hickman, Welcome to the world's worst traffic jam, The Guardian, 2010.

Available:http://www.guardian.co.uk/technology/2010/aug/23/worlds-worst-traffi

c-jam. (Accessed: 15/06/2014).

[7] The great crawl of China, The Economist.

Available: http://www.economist.com/node/1690916?story_id =1690916.

(Accessed: 18/07/2014).

660 Isaac Kwasi Adu et al.

[8] M. Wines, China’s Growth Leads to Problems Down the Road, New York

Times, 2010. http://www.nytimees.com/2010/08/28/world/asia/28china.html

[9] J. Watts. Gridlock is a way of life for Chinese, The Guardian, 2010

Available:http://www.guardian.co.uk/world/2010/aug/24/china-60-mile-motorwa

y-tailback. (Accessed: 15/08/2014).

[9] Traffic congestion in Accra worsens. Available:

http://bentil.blogspot.com/2012/01/traffic-congestion-in-accra-worsens.html.

(Accessed on 15/09/2014)

[10] Bumper To Bumper Traffic Intensifies In Kumasi. Available:

http://enochdarfahfrimpong.blogspot.com/2007/08/bumper-to-bumper-traffic-inte

nsifies-in.html. (Accessed: 15/10/2014).

[11] G. William, Linear Algebra with Applications, 7

edition, Jones and Barnetlett

Publishers, LLC, UK (2011) , 28 - 30.

Received: October 19, 2014; Published: November 19, 2014