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Widya Eka Pranata, widyapranata983@gmail.com Department of Mathematics, Universitas
Negeri Gorontalo, Kota Gorontalo, Gorontalo 96128, Indonesia
The article can be accessed here. https://doi.org/10.15642/mantik.2021.7.1.31-40
Implementation of Dijkstra Algorithm and Welch-Powell
Algorithm for Optimal Solution of Campus Bus Transportation
Nurwan1, Widya Eka Pranata2* , Muhammad Rezky Friesta Payu3, Nisky
Imansyah Yahya4
1,2,3,4Department of Mathematics, Universitas Negeri Gorontalo, Gorontalo, Indonesia
Article history:
Received Sep 28, 2020
Revised Feb 19, 2021
Accepted May 30, 2021
Kata Kunci:
Rute terpendek,
Jadwal optimal,
Algoritma Dijkstra,
Algoritma Welch-Powell
Abstrak. Penelitian ini membahas tentang penerapan algoritma
Dijkstra dan algoritma Welch-Powell pada masalah transportasi bus
kampus. Tujuan peneitian ini adalah untuk menentukan rute
terpendek dan jadwal optimal untuk jalur trasportasi bus kampus
UNG. Dalam menentukan rute terpendek, setiap persimpangan
direpresentasikan sebagai simpul dan jalur yang dilalui
direpresentasikan sebagai sisi. Lintasan terpendek diperoleh
−
−
−
−
−
−
−
. Dalam menetukan jadwal optimal,
jumlah bus merepresentasikan simpul dan waktu merepresentasikan
sisi yang menghubungkan setiap simpul. Jadwal optimal bus dimulai
pukul 06.30 pagi sampai pukul 17.00 sore. Setiap bus mendapatkan 4
(empat) sesi keberangkatan dan 4 (empat) sesi kepulangan dengan
waktu tempuh masing-masing sesi 60 menit.
Keywords:
Shortest route,
Optimal schedule,
Dijkstra algorithm,
Welch-Powell algorithm
Abstract. This research deals with applying the Dijkstra algorithm
and Welch-Powell algorithm to on-campus bus transportation
problems. This research aims to determine the optimal solution of
campus bus transportation routes in the shortest routes and schedules.
In determining the shortest route, each intersection represented as a
node and the path described as the sides. The shortest path obtained
−
− −
−
−
−
−
. In determining the
optimal schedule, the number of buses represents the vertices, and the
time expresses the side that connects each node. The optimal program
of the bus starts from 06.30 am to 5.00 pm. Every bus gets four
sessions of departure and four sessions return with travel time each
session is 60 minutes.
How to cite:
Nurwan, W. E. Pranata, M. R. F. Payu, and N. I. Yahya, “Implementation of Dijkstra Algorithm
and Welch-Powell Algorithm for Optimal Solution of Campus Bus Transportation”, J. Mat.
Mantik, vol. 7, no. 1, pp. 31-40, May 2021.
Jurnal Matematika MANTIK
Vol. 7, No. 1, May 2021, pp. 31-40
ISSN: 2527-3159 (print) 2527-3167 (online)
Jurnal Matematika MANTIK
Vol. 7, No. 1, May 2021, pp. 31-40
32
1. Introduction
The development of a new campus of Gorontalo State University (UNG) at Bone
Bolango Regency is approximately 15 km from the center of Gorontalo City. Campus
transfer impacts academic activities or activities as many as 4 (four) faculties, with about
8643 students. The UNG campus transfer from the center of Gorontalo city to Bone
Bolango regency impacts transportation availability. The availability of transport is crucial
for students to support activities in the new UNG campus. Transportation equipment that
serves the route of Gorontalo City to the new campus Bone Bolango is a bus provided by
the Gorontalo Porvinsi Transportation Agency and the Bone Bolango Regency
Transportation Agency. Gorontalo Provincial Government prepares public transportation
services in the form of Bus Rapid Transit (BRT). BRT is a mass transportation facility for
the people, including UNG students, to the UNG Bone Bolango campus, even with a
limited number. BRT does not prioritize the needs of UNG students. Therefore, it has an
impact on the delay of students in eroding lectures. The issue of distance and the discovery
of optimal routes is the most crucial thing in transportation problems when students go to
Bone Bolango's new campus. This situation is to reduce student delays in participating in
academic activities.
The above conditions required a solution to get the optimal bus route that serves
students from the center of Gorontalo City to the New Campus of UNG Bone Bolango
Regency and the optimal schedule of companies or transportation service providers.
Transportation problems can be solved using graph theory to describe the pain to make it
easier to solve. One of the ideas developed in graph theory is coloring. There are three
kinds of color in graph theory: vertex coloring, face coloring, and region color [1].
Dijkstra's algorithm can also be called a greedy algorithm. It is one of the algorithms
used to solve the shortest path. It does not have a negative cost [2]. The optimal route is
completed using the Dijkstra algorithm to get the optimal bus schedule used welch-Powell
algorithm. The working principle of algorithm Dijkstra searches for the two smallest
trajectories, so this algorithm is advantageous in determining the shortest course from one
point to another [3]. Dijkstra algorithms often search for the shortest routes, using nodes
on a simple road network [3]. The Dijkstra algorithm's use to determine the shortest route
of a graph will result in the best route, namely selecting and analyzing the unselected node's
weight, then selecting the node with the most negligible weight [4].
The Dijkstra algorithm's application in determining the shortest route, among others,
finds an effective route to avoid traffic jams during rush hour [5]. Dijkstra algorithm is used
to calculate the closest distance from one point to the museum chosen to be the destination
[6]. Implementation of Dijkstra algorithms on urban rail transit networks [7].
Determination of the Shortest Route with Using Dijkstra's Algorithm on the Path School
bus [8]. One of the concepts of graphs to solve transportation scheduling problems is the
concept of graph coloring. Graph coloring is the coloration represented by the sorted
number [9] [10]. Use by coloring vertices based on the highest degree of all vertices [11].
The welch-Powell algorithm invented by Welch and Powell is very useful in scheduling.
The application of graph coloring uses the Welch-Powell algorithm in determining student
guidance schedules [12].
Another study about applying the Dijkstra algorithm was carried for selecting the route
to reduce traffic congestion in Purwokerto. This study aims to solve congestion by
determining alternative routes that are more effective and efficient. Application of
Dijkstra's Algorithm utilizing determine the most negligible weight of each road segment.
From this research, the rider can choose alternative routes to avoid congestion [13]. They
are searching for the shortest route with Dijkstra's Algorithm. This research aims to
simulate shortest path search using The Dijkstra algorithm to help find the shortest route
[14]. Application of Dijkstra's Algorithm in the Bus Route Search Application Trans
Nurwan, Widya E. Pranata, Muhammad R. F. Payu, Nisky I. Yahya
Implementation of Dijkstra Algorithm and Welch-Powell Algorithm for Optimal Solution of Campus Bus
Transportation
33
Semarang. This researcher proposes a digital application solution to search for Trans
Semarang Bus routes using the Dijkstra Algorithm [15].
Several studies related to Dijkstra's algorithm in transportation problems only focus
on determining the shortest route without optimal scheduling. To optimize students'
transportation routes to the UNG Bone Bolango campus and the opposite, the researchers
solved two problems transportation: shortest route and the optimal schedule. Therefore, the
researcher uses two different algorithms, namely the Dijkstra algorithm and the Welch
Powell algorithm.
The background presented above is needed for optimal bus transportation that serves
students from campus 1 in Gorontalo city to Bone Bolango campus. Researchers applied
Dijkstra algorithms to determine the shortest routes and Welch-Powell algorithms to design
schedules. This study will find the shortest route and planned bus departure schedule from
campus 1 to Bone Bolango campus and scheduled return from Bone Bolango campus to
Campus 1 Gorontalo City.
2. Methods
This study aims to find the optimal route solution of the bus by using the Dijkstra
algorithm and set the bus optimal schedule solution by using the Welch-Powell algorithm
with the following stages:
a. Take a screenshot on google maps in the form of an image.
b. Determines several routes from campus 1 to the Bone Bolango campus.
c. Specify a starting point.
d. Specify a destination point.
e. Specify multiple intersections as nodes in the graph.
f. Create a straight line from node to node as a side on a graph.
g. We create a weighted graph by connecting the vertices using the contents and giving
weight according to the distance.
h. Analysis of data using Dijkstra and Welch-Powell algorithms.
i. Make conclusions.
3. Results and Discussion
3.1 Bus Shortest Route to UNG Bone Bolango Campus
Researchers take screenshots on google maps based on previous research methods and
represented them in the figure's form. The resulting image is then graphed, as shown in
Figure 1.
Figure 1. Weighted Graph
Jurnal Matematika MANTIK
Vol. 7, No. 1, May 2021, pp. 31-40
34
Figure 1 is a route that a bus can take from the starting point or departure point located at
campus 1 UNG Gorontalo city to the endpoint located at the campus UNG Bone Bolango.
The route in this study represents the side that connects each node. We can see the definition
of nodes in Figure 1 in Table 1.
Table 1. Image Copyright Getty Images Image Caption The 1st graph
No
Node
Name (intersection)
1
Campus 1 UNG
2
The intersection of three Sentra Media (Jendral Sudirman street–Pangeran
Hidayat street)
3
The intersection of four Junior high school 6 Gorontalo (Jendral Sudirman
street– Jaksa Agung Suprapto street– Arif Rahman Hakim street)
4
The intersection of four Darul Muhtadin mosque (Arif Rahman Hakim
street –Prof Jhon Aryo Katili street – Bj. Pola Isa street)
5
The intersection of four Public health center (Pangeran Hidayat street –
Rusli Datau street – Prof. Aryo Katili street – Bj. Pola Isa street)
6
The intersection of four Baiturahim mosques (Nani Wartabone street–
Raja Eyato street– Sultan Botutihe steer)
7
The intersection of four Moodu Market (Sultan Botutihe street– Aloei
Saboe steer– Matolodula street)
8
The intersection of three UBM (Bj. Pola Isa street–Aloei Saboe street-
Tinaloga street)
9
The intersection of three Tinaloga gas station (Tinaloga steer–Toto
Tengah street)
10
The intersection of 4 Bypass Kabila (Toto Tengah street–B.J Habibie
street– Sabes street– Noho Hudji street)
11
The intersection of three Police office of Kabila (Pasar Minggu street-
Tapa Kabila street)
12
The intersection of three Al Munawarah mosque (Pasar Minggu street–
Muh. Van Gobel street)
13
The intersection of four Darul Muhaimin mosque ( B.J Habibie street–
Muh. Van Gobel street – El Madinah Road street)
14
The intersection of three Indomaret (Pasar Minggu street–Jembatan Merah
street)
15
The intersection of three Adipura Monument of Bone Bolango (Jembatan
Merah street–B.J Habibie street)
16
UNG Bone Bolango campus
Table 1 data is used to determine the shortest trajectory using the Dijkstra algorithm
with steps: 1). Node label with = 0, and for each node in other than , node label
with = . 2). Suppose with a minimum . 3). If = , stop, then the
shortest trajectory from s to t is 4). For each side = , ; replace label with
= minimum + w(e). 5). , and return to iteration 2 [16].
The shortest trajectory problem in all knot pairs is to find the shortest trajectory
between knot pairs.
,
in such a way that ≠ [17]. Matrices of agility formed from
graph weights with steps: 1). The distance of
node with
if there is a connecting side,
then in writing with the weight, 2) 0 if the
node is connected to
And 3) if no side
connects the
node with
After looking at the steps above then fill the matrix of agility
with the weight on the graph. The results of graph representations weighted into the
matrices of the neighboring show in Table 2.
Nurwan, Widya E. Pranata, Muhammad R. F. Payu, Nisky I. Yahya
Implementation of Dijkstra Algorithm and Welch-Powell Algorithm for Optimal Solution of Campus Bus
Transportation
35
Table 2. Representation of Graphs in The Matrices of Neighboring
14
Weight change of each iteration based on algorithm Dijkstra's steps obtained the
shortest route from each node
to the
as in Table 3.
Table 3. Results of the shortest route in the matrices of neighbouring
Representation of the neighbouring matrix in Table 3 results in the shortest route of
the bus with the following details:
a. The
is connected to
,
,
with a weight of
−
= 1 ,
−
= 6,
−
=
18. Because
the smallest weight choose and colour, it then lowered the
.
b.
connected with
and
with a
−
= 17. Added with colored weight (17 + 1
= 18). Because
the smallest weight, then the
selected and coloured, then lowered
the
.
Jurnal Matematika MANTIK
Vol. 7, No. 1, May 2021, pp. 31-40
36
c.
connected with
and
nodes with a weight of
−
= 19 Added with a
coloured weight (19+6 = 25). Furthermore, two nodes have the same weight
(smallest), namely
and
, selected
then lowered the
.
d.
connected with
,
,
with a weight of
−
= 14, Added with a coloured
weight (14 +18 = 32). Because the weight is greater than the previous weight, the
previous weight is lowered. Next
−
= 23 then add with the coloured weight
(23+18 = 41). Because
the smallest weight, then select and colour, then lowered
the
.
e.
connected with
and
nodes with a
−
= 17, then add with the coloured
weight (17 + 18 = 35). Because
smallest weight choose and colour then lowered
the
.
f.
connected with the
and
because the
and
already coloured, then further
lose the smallest weight. Since
smallest weight choose and colour, it is also lowered
g.
connected with
and
with a weight of
−
= 29, then add with the coloured
weight (29 +35 = 64) because the result is 64, greater than the previous weight, then
lower the previous weight. Next
−
= 21 then add with the colored weight (21 +
35 = 56). Since the smallest weighted V8 select and colour, the next lowered
.
h.
is connected to
and
with a weight of
−= 3.5, then add with the coloured
weight (3.5 + 41 = 44.5). Because
smallest weight choose and colour then lowered
the
.
i.
is connected to
, and
with a weight of
−
= 11, then add with the
coloured weight (11 + 44.5 = 55.5). Since V_10 smallest weight choose and colour,
the knot is further lowered
.
j.
connected with
,
,
with a weight of
−
= 24 then add with the
weight already coloured (24 + 55.5 = 79.5) because the result is 79.5, greater than the
previous weight, then lower the previous weight. Next
−
= 24 then add with
the colored weight (24 + 55.5 = 79.5). Because
smallest weight choose and colour
then lowered the
.
k.
connected with
,
,
with a weight of
−
= 22, then added with a
coloured weight (22 + 56 = 78). Because
smallest weight choose and colour then
lowered the V_12.
l.
is connected with
,
,
with a weight of
−
= 22, then add with the
coloured weight (22 + 78 = 100) because the result is 100, greater than the previous
weight, then lower the previous weight. Next
−V_14 = 18 then add the colored
weight (18+78 = 96). Because
smallest weight choose and colour then lowered the
.
m.
is connected with
,
,
with a weight of
−
= 20 then add with the
colored weight (20 + 79.5 = 99.5). Because
smallest weight choose and colour
then lowered the
.
n.
is connected
,
with a weight of
−
= 22 then 27 add with colored
weight (22 + 96 = 118). Because of the smallest weighted V16 select and colour, the
next lowered
.
o.
is connected to the
and
with a weight of
−
= 2, then add with the
coloured weight (2+99.5 = 101.5) because the result is 101.5, smaller than the previous
weight, then choose and colour. Next, I lowered the
.
p.
connected to the
and
because the
and
has been coloured, then it is
finished.
Figure 2 The shortest route a bus can take from campus 1 UNG to Bone Bolango's
new campus is From V_1 to V_16, or vice versa obtained the shortest route begins
−
−
−
−
−
−
−
or Campus 1 UNG - The intersection of three Sentra
Nurwan, Widya E. Pranata, Muhammad R. F. Payu, Nisky I. Yahya
Implementation of Dijkstra Algorithm and Welch-Powell Algorithm for Optimal Solution of Campus Bus
Transportation
37
Media (Jendral Sudirman street–Pangeran Hidayat street) - The intersection of four Public
health center (Pangeran Hidayat street – Rusli Datau street – Prof. Aryo Katili street – Bj.
Pola Isa street) - The intersection of three UBM campus (Bj. Pola Isa street–Aloei Saboe
street-Tinaloga street) - The intersection of three Tinaloga gas station (Tinaloga steer–Toto
Tengah street) - The intersection of four Bypass Kabila (Toto Tengah street–B.J Habibie
street– Sabes street– Noho Hudji street) - The intersection of four Darul Muhaimin mosque
( B.J Habibie street– Muh. Van Gobel street – El Madinah Road street) – UNG Bone
Bolango Campus.
Figure 2. Bus Shortest Route to UNG Bone Bolango Campus
3.2 Bus Schedule to UNG Bone Bolango Campus
The Welch-Powell algorithm is required to color the graph's vertices based on the
highest degree of all its nodes. From the data available, there are four buses in operation
[18]. These four buses run back and there, represented in a graph. The data of the number
of buses be defined as a node on the graph. There are no specific labelling rules, labelled
with an index to distinguish which buses operate back and away. Suppose each node with
with the following description:
1. a: A bus, a [1, 4]
2. b: Return or Departure (1 for departure, 2 for return)
Figure 3. Representation of Bus Data in
Graph
Figure 4. Graph Representation of
Bus Schedules
Jurnal Matematika MANTIK
Vol. 7, No. 1, May 2021, pp. 31-40
38
Each node in Figure 3 is then connected and forms aside. The side represents the bus's
operational time, and each bus does not operate simultaneously. The arrangement of the
vertices' location establishes a circle to make it easier to draw a straight line for the
relationship of each node. The node in Figure 4 represents the bus, and the side is a
representation of time. Node
,
,
,
is the bus departure node from campus 1
to campus UNG Bone Bolango and node
,
,
,
is the node from campus UNG
Bone Bolango to campus 1. what can see degrees on each node in Table 4.
Table 4. Vertices by degree
No
Node
Neighbors
Degrees
1
,
,
,
4
2
,
,
,
4
3
,
,
,
4
4
,
,
,
4
5
,
,
,
4
6
,
,
,
4
7
,
,
,
4
8
,
,
,
4
Table 4 shows the same degree on each node, then coloured by not giving the same
colour to the neighbouring nodes as in Figure 5. By doing the steps of the welch-Powell
algorithm, knot colouring is obtained as in Figure 6.
Figure 5. The Colouration of Vertices In Graph Figure 6. Final Result of Knot Coloring
In Figure 6, four kinds of coloring are obtained. Next, group the buses by chromatic
numbers. A graph has a chromatic number denoted by [19] [20]. Zero graphs have a
chromatic number of , while to color, a complete graph is required n color fruits
because all points are interconnected [19] [21]. The chromatic number of the bus schedule
representation graph is which means each bus schedule for return or departure is
at least four sessions. We can see the results of graph coloring for the campus bus schedule
in Table 5.
Table 5. Bus Departure and Return Schedule to/from UNG Bone Bolango Campus
No
Departure Schedule
Schedule of Return
Time
Bus
Time
Bus
1
06.30
BUS 1
07.30
BUS 1
2
07.30
BUS 2
08.30
BUS 2
3
08.00
BUS 3
09.00
BUS 3
Nurwan, Widya E. Pranata, Muhammad R. F. Payu, Nisky I. Yahya
Implementation of Dijkstra Algorithm and Welch-Powell Algorithm for Optimal Solution of Campus Bus
Transportation
39
No
Departure Schedule
Schedule of Return
Time
Bus
Time
Bus
4
08.30
BUS 4
09.30
BUS 4
5
09.00
BUS 1
10.00
BUS 1
6
09.30
BUS 2
11.00
BUS 2
7
10.00
BUS 3
12.30
BUS 3
8
11.00
BUS 4
13.00
BUS 4
9
12.30
BUS 1
13.30
BUS 1
10
13.00
BUS 2
14.00
BUS 2
11
13.30
BUS 3
14.30
BUS 3
12
14.00
BUS 4
15.00
BUS 4
13
14.30
BUS 1
15.30
BUS 1
14
15.00
BUS 2
16.00
BUS 2
15
15.30
BUS 3
16.30
BUS 3
16
16.00
BUS 4
17.00
BUS 4
Table 5 the bus departure schedule from campus 1 to UNG Bone Bolango campus
consists of 16 departure time sessions with departure time starting from 06:30 and ending
at 16:00 with a delay of 60 minutes each departure time. The bus return schedule from
UNG Bone Bolango campus to campus 1 consists of 16 sessions by the same bus. The bus
departure schedule from UNG Bone Bolango campus to campus starts at 07.30 am and
ends at 05.00 pm. The overall departure schedule consists of 32 departure sessions. Each
bus gets eight departure sessions per day with a minimum session break of 60 minutes,
including travel time.
4. Conclusions
The Dijkstra algorithm's calculation obtained the shortest bus route from campus 1 to
campus UNG Bone Bolango is the trajectory
−−
−
−
−
−
−
with a
distance of 9.95 km. The bus passed by the bus is Campus 1 UNG → The intersection of
3 Sentra Media → The intersection of 4 Public health center → The intersection of 3 UBM
campus → The intersection of 3 Tinaloga gas station → The intersection of 4 Bypass
Kabila → Intersection of 4 Darul Muhaimin mosque → UNG Bone Bolango Campus. The
coloring results using the Welch-Powell algorithm obtained the number of colors on the
bus schedule's color to UNG Bone Bolango in four colors. The bus departure and return
schedule are 16 sessions each, and the bus operating time starts from 06.30 am to 05.00
pm. Each bus gets four departure sessions and four return sessions per day with a travel
time of 60 minutes.
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