Journal of Computer Science & Computational Mathematics, Volume 7, Issue 2, June 2017

DOI: 10.20967/jcscm.2017.02.003

The Application of the Shortest Path and Maximum
Flow with Bottleneck in Traffic Flow of
Kota Kinabalu
Noraini Abdullah1* , Ting Kien Hua2
1

Faculty of Science & Natural Resources, Universiti Malaysia Sabah, 88400 Kota Kinabalu, Sabah, Malaysia
2
Centre of Postgraduate Studies, Universiti Malaysia Sabah, , 88400 Kota Kinabalu, Sabah, Malaysia*
Corresponding author email: norainiabdullah.ums@gmail.com

Abstract: Urban mobility problems such as traffic flows that cause
traffic congestions have become a global transportation problem.
Traffic congestion happens as traffic volume exceeds the capacityof an existing road facility. The occurring traffic congestion is due
to freedom of owning private vehicles, poor traffic facilities andunrestricted urban population growth. In this study, theidentification of maximum flow, the bottleneck path and theshortest path were carried out. The scope of this study is a networkfrom Bandaraya Mosque to Kampung Air in Kota Kinabalu. All thepossible routes from source to sink were selected. Traffic volumedata were collected by manual traffic count with video recordings.
Distance data were collected from the Google Earth Software.
Directed network graphs were formed with all the obtained data.
The maximum flow problem was solved by using Ford-Fulkerson
Algorithm to find the maximum flow. The identification ofbottleneck path was done by using the max-flow and min-cut
theorem. Besides, the shortest path was determined by Dijkstra’s
Algorithm. Next, the maximum flow and the shortest path problem
was formulated using linear programming, and then was solved byusing excel solver in Microsoft Excel. The results obtained would
allow the traffic engineers to decide which roadway facilities
should be improved. Drivers can also refer to these results tochoose the desired alternative paths for their journeys.
Keywords: traffic congestion, capacity, maximum flow, bottleneck,shortest poath.

1. Introduction
Traffic congestion has become a major urban transportation
problem worldwide. This occurs when the number of traffic
items has exceeded the capacity of existing road facilities.
Besides affecting the economic productivity, traffic
congestion also affects the environment and the overall
quality of life for many people. Possible reasons ofcongestions may be due to slow improvement of traffic
facilities, unrestricted private car owning and behavior ofdrivers on road. Traffic congestion can be categorized into
two types, namely, recurring congestion and non-recurringcongestion. Recurring congestion occurs especially in peak
hours of weekdays, and they usually happen in the area ofCentral Business District (CBD). Non-recurring congestionis an unexpected congestion due to accidents, sudden roadclosures, and maintenance which will slow the traffic flow.
Moreover, non-recurring congestion is a troublesome traffic
problem because it is unpredictable. The happening of the
non-recurring congestion causes the temporary reduction of
roadway capacity [1].

The traffic volume and distances are collected to form the
network graph. After the capacitated network is formed, themaximal flow will be computed using the Ford-Fulkerson
algorithm, and will be followed by the max-flow and min-cut
theorem to find the bottleneck path of the network [2].
Weighted network graph is formed to find the shortest path,
while bottleneck path limits the maximum flow of a network.
The identification of the shortest path is carried out using the
Dijsktra’s algorithm. An alternative path with the shortest
distance and high maximum flow with bottlenecks can thus
be identified.
As reported by the Borneo Post, World Bank had
highlighted the increasing congestion and transportation
issues in Kota Kinabalu as a major economic hindrance [3].
The decrement of road capacity will result in the decrease of
maximal traffic flow and the speed of traffic vehicle will
drop dramatically. The scope of this study is a network from
Bandaraya Mosque in Kota Kinabalu (source node) toKampung Air (sink) where all the routes between source andsink node are established. The routes between Bandaraya
Mosque and Kampung Air are selected because this area ispart of a central business district for Kota Kinabalu where
the demand of traffic is higher than the other locations.
Hence, this study aspires to find the maximum flow of thedesired route as well as its bottleneck, and also to determine
the shortest path to reach a selected destination.

2. Literature Review
A study [4], presented an applied minimum-cut maximumflow using cut set of a weighted graph on the traffic flow
network. A capacitated graph is a resulting graph with a real
number of capacity which serves as a structural model intransportation. The traffic control strategy of minimal cut
and maximum flow is to minimize number of edges in
network and maximum capacity of vehicles which can movethrough these edges. With a minimal cut in traffic network, it
allows to minimize the waiting time of traffic participants fora smooth and uncongested traffic flow.
As presented in one study [5], there are some empiricalmethods to estimate the capacity on Indian urban roads. Twomain types of capacity estimation were Direct Empirical
Methods and Indirect Empirical (Simulation) Methods. Dueto the complexity and high traffic volume on Indian urban
roads, it was appropriate to use direct empirical method for

38

The Application of the Shortest Path and Maximum Flow with Bottleneck in Traffic Flow of Kota Kinabalu

capacity estimation. By using direct empirical methods, theobserved traffic data like Headway, volume and speed wereneeded. In direct empirical method, three approaches weresuggested which were Headway method, Observed volume
method and Fundamental diagram method. From the resultsof this study [5], the headway method was able to achieve ahigh accuracy of capacity estimation by comparing all thethree approaches.
Based on another study [6], the maximum flow problem in
Ethiopian Airlines was investigated. This paper mainly
studied the maximum flow problem and solution algorithm
which was Ford and Fulkerson algorithm. By using FordFulkerson algorithm, different number of augmenting paths,
and flow of augmenting path might be different, but theobtained maximum flow value was the same. It meant thatthe solution of the maximum problem could have different
augmenting path and different number of augmenting path,but the maximum flow value was unique.
The maximum flow in road networks with speed
dependent capacities application to Bangkok traffic wasstudied in another investigation [7]. A traffic maximum flow
problem had arcs represented as capacity of road (maximum
vehicles per hour) that were functions of the traffic speed(kilometer per hour) and traffic density (vehicles per
kilometer). To estimate road capacities for a given speed,
empirical data on safe vehicle separations for a given speed
were used. A modified version of the Ford-Fulkerson
algorithm was developed to solve maximum flow problems
with speed-dependent capacities, with multiple source andtarget nodes. It was found that the maximum safe traffic flow
occurs at the speed of 30 km/hr.
A case study [8] was presented for method of pathselection in the graph. Dijkstra’s algorithm was used in thispaper to find additional paths among nodes in the maritime
sector. The shortest path was not always the best alternative
path because it involved single criterion. Hence, otherparameters were calculated such as the average time, numberof indirect vertices, and the safety factor. The method in
selecting one desirable path from several paths was multicriteria decision making. Dempster-Shafer theory was a
method that could be applicable to a fused data andcombination of evidences.

3. Methodology
3.1 Network Graph
Network is formed with paths that are connected with points.
Capacitated network graph and weighted network graph areneeded in this study to get the shortest path and maximal
flow. First, a capacitated network graph was formulated withall the edges. Each of the edges has a non-negative
capacity, 𝑐(𝑢, 𝑣) ≥ 0 and flows 𝑓(𝑢, 𝑣) that cannot be morethan capacity of the edge. The source node, s and sink node, tof a network are starting point, and ending point respectively.
A capacitated network must fulfill the conditions below:
First, the capacity constraint, ∀(𝑢, 𝑣) ∈ 𝐸 𝑓(𝑢, 𝑣) ≤ 𝑐(𝑢, 𝑣)
which flow of the edges must not exceed its own capacity.
Then, the next condition is skew symmetry, ∀𝑢, 𝑣 ∈𝑉, 𝑓(𝑢, 𝑣) = −𝑓(𝑣, 𝑢) which net flow from u to v and fromv to u must be opposite to each other. Lastly, flow
conservation constraints, ∀𝑢 ∈ 𝑉: 𝑢 ≠ 𝑠 𝑎𝑛𝑑 𝑢 ≠ 𝑡 =>∑(𝑠,𝑢)∈𝐸 𝑓(𝑠, 𝑢) = ∑(𝑣,𝑡)∈𝐸 𝑓(𝑣, 𝑡) is the net flow to a node iszero except source node and sink node and the flow from thesource node must be equal to the flow at the sink node.
Weighted network graph is a network graph that isformulated by the edges with the non-negative distances. It isalmost the same as the capacitated network graph [9].
3.2 Ford-Fulkerson Algorithm
Maximal flow in a capacitated flow network is the total flowfrom a source node to a sink node. First, find an augmenting
path from the source node to the sink node. After theformation of augmentation path, compute the bottleneck
capacity. Lastly, augment each edge and the total flow until
the capacity of sink node reaches maximum [10].
3.3 Dijkstra’s Algorithm
First, assign to every node a tentative distance value. Then,label starting node with zero, and to infinity for all othernodes. Set starting node as temporary, and mark other nodes
as unvisited nodes. For the temporary node, choose theunvisited arcs that are connected to the starting node with theleast value. Mark the temporary node as visited as all theneighbors of the temporary node are considered.
Temporarily label all the nodes that are connected to thepermanent labelled nodes with the distances from starting
node. Choose the temporary label of the least value. Repeat
the steps until the destination node has a permanent label[11].
3.4 Maximum Flow and Minimum Cut Theorem
The minimum capacity of an (s,t)-cut is equal to the
maximum value of a flow.
max{𝑣𝑎𝑙(𝑓)│𝑓 𝑖𝑠 𝑎 𝑓𝑙𝑜𝑤} =min{𝑐𝑎𝑝(𝑆, 𝑇)│(𝑆, 𝑇)𝑖𝑠 𝑎𝑛 {𝑠, 𝑡} − 𝑐𝑢𝑡}
3.5 Capacity Estimation: Direct Empirical Method
There are different methods to estimate the roadway capacity.
Basically, it can be categorized into two types which are
direct empirical method and indirect empirical method. The
needed data for the method just mentioned are roadway
width, traffic volume, headway, speed and density. The
direct empirical method makes use of observed data toestimate the capacity directly. For indirect empirical method,
it is a method that makes use of simulation software to get
the estimated capacity. Therefore, the direct empiricalmethod was selected to apply in this study because it was aneasy performing method that leads to high accuracy results.
Direct empirical method can be volume approach,fundamental diagram approach and headway approach [12].
Volume approach was selected in this study instead because
it would be a direct and easy method to be done regarding
the scope of this study.

Noraini Abdullah , Ting Kien Hua

39

3.5.1 Capacity Estimation: Direct Empirical Method(Volume Approach)
Volume approach takes peak hour data and uses the Selected
Maxima Model to estimate the capacity. The flows in each 5minute interval for different composition of vehicles wereextracted. The capacities were calculated in equivalent
Passenger Car Unit (PCU) based on the recommended valueby [13]. Among 5-minute intervals in one peak hour, one of
the 5-minute intervals with maximum flow was identified.
The obtained peak hour rate of 5-minute intervals was
multiplied by a factor 12 to get the estimate capacity flow
per hour.

4. Data Collection
In this study, traffic volume and distances as two types ofdata were collected. Traffic volume can be collected by two
main methods. It could be either manual traffic count or
automatic traffic count. Manual traffic count would take theleast cost, but required more manpower. On the other hand,automatic traffic count could achieve a high accuracy result,
but would cost higher than the manual count method.
Manual traffic count was selected in this study because it
was considered as one of the user friendly methods [14].
Traffic volume was collected through manual traffic countmethod by using a camera. The camera was placed on a
tripod stand, at a higher place to get a clearer view. The
paths between Bandaraya Mosque and Kampong Air wereselected for video shots. Videos were recorded in theevenings at peak hours from 5 pm to 6 pm. The video clips
were reviewed to do manual counting in order to get theactual traffic volume. The collected traffic volume wascategorized into the different compositions of traffic vehicles
which were motorcycle, car, bus or truck and medium goods
vehicle or heavy goods vehicle respectively. Besides,
distances were needed in this study to form the weighted
network graph. The distances data were collected from theGoogle Map. The distances can be obtained by entering theaddress of a starting point and a destination using the link on
Google Map.

computed by taking the Max Flow in 5 Minutes in PCU andmultiplied it to the factor of 12.
Table 2: Composition of Traffic (Max Flow in 5 minutes
interval)
No

Location name

From

To

Car

Motorcycle

1
2
3
4
5

Jalan Tun Fuad Stephen 1
Jalan Pasir
Jalan Tun Fuad Stephen 2
Jalan Tuaran 1
Jalan Kompleks sukan and
Jalan Bunga Nasar
Jalan Kompleks sukan and
Jalan Bunga Nasar
Jalan Istiadat
Jalan Istiadat
Jalan Tuaran 2
Jalan K.K Bypass
Jalan Tuaran 3
Jalan
Tunku
Abdul
Rahman 1
Jalan
Tunku
Abdul
Rahman 2
Jalan Kemajuan
Jalan Laiman Diki
Jalan Coastal

s
s
V1
V2
V3

V1
V2
V5
V3
V4

171
114
173
162
114

V4

V3

V4
V1
V3
V5
V6
V7

V1
V4
V6
V7
V9
V8

6
7
8
9
10
11
12
13
14
15
16

Bus &
Truck

39
18
32
28
5

MGV
&
HGV
7
9
10
11
2

56

11

3

0

58
50
130
90
169
57

15
8
19
18
13
20

2
3
3
2
19
2

3
3
2
3
5
3

6
10
1
5
3

V8

V9

62

12

3

6

V9
V7
V10

V10
t
t

153
91
81

32
21
9

8
4
4

6
2
0

Table 2 depicted the number of vehicles in one of the 5minute maximum flow intervals which were counted andcategorized into different compositions as in one of studies.
Table 3: Capacity estimation of selected locations (5 mintime slice) using Selected Maxima Model
Location No
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16

Max Flow in 5 Minutes in PCU
226.25175.5
220242.75130.75
70.25
82.25
74
156.25118.5
231.758795
211
120.75
95.75

Total Capacity in PCU
2715
2106
2640
2913
1569
843
987
888
1875
1422
2781
1044
1140
2532
1449
1149

Table 1: Passenger Car Unit (PCU) value for Urban Road
Class of vehicles
Car, Taxi, Light Goods Vehicle (LGV)
Motorcycle, Scooter
Medium Goods Vehicle (MGV) or Heavy
Goods Vehicle (HGV)
Bus, Truck

Urban Road PCU value
1.00
0.75
2.00
3.00

In Table 1, the PCU values for different categories ofvehicle were displayed by referring to the Urban Traffic
System [15]. The estimation of capacity by the volume
approach was making use of Passenger Car Unit (PCU)
value in Table 1. Table 2 showed the composition of traffic
so as to calculate the total capacity of the selected locationsin PCU values.
In Table 3, the maximum flows for 5-minute intervals in
PCU were obtained from the multiplication of number ofvehicle with PCU values. Total capacity in PCU was

Table 4: Distance of the branches from Bandaraya
Mosque to Kampung Air
Location name
Jalan Tun Fuad Stephen 1
Jalan Pasir
Jalan Tun Fuad Stephen 2
Jalan Tuaran 1
Jalan Kompleks sukan and Jalan Bunga Nasar
Jalan Kompleks sukan and Jalan Bunga Nasar
Jalan Istiadat
Jalan Istiadat
Jalan Tuaran 2
Jalan K.K Bypass
Jalan Tuaran 3
Jalan Tunku Abdul Rahman 1
Jalan Tunku Abdul Rahman 2
Jalan Kemajuan
Jalan Laiman Diki
Jalan Coastal

From
s
s
V1
V2
V3
V4
V4
V1
V3
V5
V6
V7
V8
V9
V7
V10

To
V1
V2
V5
V3
V4
V3
V1
V4
V6
V7
V9
V8
V9
V10
t
t

Distance (km)
1.5
0.63
2.3
2.4
1.22
1.22
0.8
0.8
2.4
1.6
0.72
0.65
0.2
0.49
0.34
0.84

The Application of the Shortest Path and Maximum Flow with Bottleneck in Traffic Flow of Kota Kinabalu

40

Table 4 showed the distances for the selected routes thatwere obtained from the Google Earth Software. The data in
Table 4 were used to form a weighted and capacitateddirected network graph as shown in Figure 1 and Figure 2
respectively.

From Figure 3, the shortest path in this weighted network
wss 𝑠 → 𝑉1 → 𝑉5 → 𝑉7 → 𝑡. It was about 5.74 km from theBandaraya Mosque to Kampung Air.
5.2 Results of Shortest Path Problem Using Excel Solver(Simplex Linear Programming)

Figure 1: Weighted directed network graph from Bandaraya
Mosque to Kampung Air

Figure 4: Shortest path problem solved by using simplex
linear programming in Microsoft excel

Figure 2: Capacitated directed network graph from
Bandaraya Mosque to Kampung Air

5. Results
5.1 Results of Shortest Path Problem Using Dijkstra’s
Algorithm

In Figure 4, column F represented the distance of the edge.
The objective function was the cell D21 which contained theformula of ‘=SUMPRODUCT (D4:D19,F4:F19)’ as shownin Table 5. The cell from I4 to I15 represented the net flow
were the constraints cells. The ‘On Route’ from cell D4 tocell D19 were the variable cells. The number ‘1’ showed in
the column of ‘on route’ denoted for the route selected, andnumber ‘0’ denoted for route unselected. Hence, the selectedroutes for shortest path were from node “s” to V1, V1 to V5,
V5 to V7’ and ‘V7 to t’. In the supply or demand column,that source node, “s” was set as number ‘1’ and the sink node,t was set as number ‘-1’ because both of the nodes were thestarting and the ending nodes [16].
Table 5: Formula of the cells in Figure 4
Cell
D21
I4

Figure 3: Result of Dijkstra’s Algorithm

Key Cell Formulas
Formula=SUMPRODUCT(D4:D19,F4:F19)=SUMIF($B$4:$B$19,$H$4,$D$4:$D$19)SUMIF($C$4:$C$19,$H$4,$D$4:$D$19)

Copied to
I5:I15Figure 5 showed the solver parameters for the shortest
path problem in Microsoft Excel. First, the cell D21 was set
as the Objective. Since the goal was to find the shortest path,therefore the minimum button was chosen in order tominimize the shortest path problem. The changing variable
cells were the cells from D4 to D19. Next, the constraints of
the Net Flow (cells I4 to I5) must be equal toSupply/Demand (cell J4 to J15). Before clicking on the solve
button, Simplex Linear Programming was selected as theNoraini Abdullah , Ting Kien Huasolving method. The output of the shortest path by using theexcel solver would be the same as the output of the
Dijkstra’s algorithm in Figure 3.

41

5.4 Results of Maximum Flow Problem Using Excel
SolverFigure 7: Maximum flow problem solved by using simplex
linear programming in Microsoft Excel

Figure 5: Solver parameter for shortest path problem in
Excel
5.3 Results of Maximum Flow Problem Using FordFulkerson Algorithm

In Figure 7, column G represented the capacity for eachedge. The objective function was the cell E22 whichcontained the formula of ‘=E20’. Cell E20 was the flow fromnode “t” to node “s”. The cell J4 to cell J15 represented the
net flow which was the constraints cells, as shown in Table 6.
The units of flow, from cell E4 to cell E20, were the variable
cells. The unutilized paths were from node V1 to V4, node V4
to V1, node V4 to V3 and node V3 to V4 respectively.
Table 6: Formula of the cell in Figure 7
Cell
E22
J4

Figure 6: Results of Ford-Fulkerson Algorithm
Figure 6 showed the first route from 𝑠 → 𝑉1 → 𝑉5 →𝑉7 → 𝑡 which carried a maximum flow of 1449 vehicles perhour. The second route was from 𝑠 → 𝑉2 → 𝑉3 → 𝑉6 →𝑉9 → 𝑡 . The total maximum flow value using the FordFulkerson algorithm was 2598 vehicles per hour.

Key Cell Formulas
Formula=E20=SUMIF($C$4:$C$20,$I$4,$E$4:$E$20)SUMIF($D$4:$D$20,$I$4,$E$4:$E$20)

Copied to
J5:J15Figure 8 below shows the solver parameters for the
maximum flow problem in Microsoft Excel. First, the cell
E22 was set as the Objective. The minimum button waschosen in order to minimize the shortest path problem. Thechanging variable cells were the cells from D4 to D19. Next,
constraints were the Unit of Flow (cell E4 to E20) less thanor equal to Capacity (cells G4 to G20) and Net Flow (cells J4
to J15) must equal to the Supply/Demand (cells K4 to K15).
Simplex Linear Programming was selected as the solving
method. The augmenting path of the maximum flow path byexcel solver might not be the same as the augmenting path of
the Ford-Fulkerson algorithm as shown in Figure 6. However,
the maximum flow value for Ford-Fulkerson algorithm andexcel outputs were expected to be the same.

42

The Application of the Shortest Path and Maximum Flow with Bottleneck in Traffic Flow of Kota Kinabalu

5.5 Bottleneck Path using Maximum Flow and Minimum
Cut Theorem

Figure 10: Minimum Cut

Figure 8: Solver parameters for maximal flow problem

The cut capacity was the same as the sum of the capacities
of its arcs. Bottleneck capacity was the minimum residual
capacity of any edge in the augmenting path. The cut withthe smallest capacity gave the maximum flow in thecapacitated network. From Figure 10, the blue line denoted
the minimum cut with the smallest capacity which was thesame as the maximum flow that was computed by the Fordfulkerson method. The bottleneck routes of the network in
Figure 10 were denoted by V7 and V10 , namely Jalan Laiman
Diki, and Jalan Coastal respectively.

6. Conclusion
The shortest path in this weighted network was → 𝑉1 → 𝑉5 →𝑉7 → 𝑡 , viz. Bandaraya Mosque=> Jln. Tun Fuad 1=> Jln.
Tun Fuad 2=> Jln. KK Bypass=> Jln. Laiman Diki=>
Kampung Air, and it took about 5.74 km from Bandaraya
Mosque to reach Kampung Air in Kota Kinabalu. The
maximum flow was 2598 vehicles per hour, while thebottleneck paths were found to be Jalan Laiman Diki and
Jalan Coastal. Thus, with these outputs, traffic planar canmake decisions on the bottlenecks, and traffic drivers can
avoid this respective traffic congestion routes, and reach
their respective selected destinations in a shorter time.

Acknowledgement
Figure 9: Results of the maximum flow problem using
Excel Solver
In Figure 9, the first route from 𝑠 → 𝑉1 → 𝑉5 → 𝑉7 → 𝑡
carried a maximum flow of 1449 vehicles per hour. The
second route from 𝑠 → 𝑉2 → 𝑉3 → 𝑉6 → 𝑉9 → 𝑡 carried amaximum flow of 678 vehicles per hour, while the thirdroute from 𝑠 → 𝑉2 → 𝑉3 → 𝑉6 → 𝑉9 → 𝑉10 → 𝑡 carried amaximal flow of 471 vehicles per hour. The total maximum
flow value was 2598 vehicles per hour.

The authors would like to thank Universiti Malaysia Sabah
for funding this research under the grant number of
GUG0139-1/2017.

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