Corresponding Author: Sapto
Priyanto
E-mail: [email protected]
INTRODUCTION
The population movement is
motivated by an imbalance in individual needs that cannot be obtained in the
area where they live (Kuciswara
et al., 2021). Fulfilment of needs is
inseparable from the existence of work sources to improve the quality of life
so that areas with better job opportunities will become targets for population
mobility (Fonna,
2019). The uneven distribution of
resources results in disparities in a region's economic growth rate (Effendy
& Djohan, 2022). In urban areas, mobility or commuting has become
commonplace (Tang, 2016). Changes in population, residence, employment,
economic growth and better transportation services have resulted in people
becoming commuters (Wang & Hu, 2017). Commuters are people who routinely commute to a
place of work that is outside their area of residence and return home the same
day. Based on BPS data for August 2021, commuters in Indonesia totalled 7.34
million, an increase of 4.49% from 2020.
The Yogyakarta and Solo regions are agglomeration
areas that the government continues to develop to improve the regional economy.
On November 10 2021, the Governments of Yogyakarta and Solo agreed to cooperate
in regional development in the fields of tourism, economy, transportation, and
culture. The development of transportation in the Jogja Solo area has been
carried out through the electrification of the railway line according to the
National Railway Master Plan (RIPNAS) (Darmaning Tyas et al.,
2015). This electrification will increase the capacity of
the number of passengers because it can cut travel time so that there will be
more and more trips. The electrification of the Jogja-Solo train line also
affected the termination of the operation of the Prambanan Ekspres (Prameks)
commuter train, which had previously served 27 years of Jogja-Solo commuter
travellers. The
Prameks train was replaced by an Electric Rail Train (KRL), which officially
operated on March 1, 2021. During its journey, the Jogja-Solo KRL serves 20
trips a day with a speed of 90 km/hour and a travel time of 68 minutes. During
its one year of operation since March 1 2021, Jogja Solo KRL users have reached
more than 2.2 million despite the Covid-19 pandemic. The government's plan to
extend the electrification of the KRL line to Kulonprogo and Madiun is an
opportunity to grow the number of Jogja Solo KRL passengers.
Based on the background above, the authors are
interested in predicting the number of KRL Jogja Solo passengers using the grey
(1,1), Moving average and exponential smoothing methods. This study aims to
find the best forecasting model for the number of KRL Jogja Solo passengers
among short-term forecasting models.
METHODS
The data used in this study is secondary data obtained
from PT. KAI (Persero) is the number of Jogja Solo KRL passengers from February
2021 to February 2023. Data on Jogja Solo
KRL passengers can be seen in the following table.
Table 1. Data on the Number of KRL Passengers
�Joogja - Solo
for the Period February 2021 to. February 2022
|
Period
|
Original
Data
|
|
Feb-21
|
94536
|
|
Mar
|
195018
|
|
Apr
|
183935
|
|
May
|
185092
|
|
Jun
|
190412
|
|
Jul
|
46340
|
|
Aug
|
49510
|
|
Sept
|
106384
|
|
Oct
|
186060
|
|
Nov
|
220388
|
|
dec
|
298190
|
|
Jan
|
291684
|
|
Feb
|
220712
|
The processing method used in this study is a number
series forecasting method that uses historical data as the basis for
forecasting (Pujiati et al., 2017). Forecasting methods used in this study include:
1. Gray Method (1,1)
Ju-Long develops a forecasting model that focuses on
conditions of uncertainty and insufficient information in analyzing and
understanding systems through research on the conditional analysis, prediction
and decision-making (Pratiwi & Achmad,
2019). This model is known as the Gray theory or GM (d,p),
where d indicates the number of differentiations and p indicates the many
research variables (Ahdika, 2018). This model is very suitable for forecasting with
limited data. This study uses the Gray Model (1,1), which in general has three
stages, namely AGO (Accumulated Generating Operation), IAGO and
the Gray Model (Marwati, nd).
2. Moving Averages
The moving average is averaging a group of observational
values whose average is sought to predict the next period (Siagian & Sugiarto,
nd). The longer the timeframe, the more visible the
smoothing effect on forecasting results using the moving average method (Hudaningsih et al., 2020). The moving average equation can be written as follows:
Were
Ft = Forecasting value for the 1st, 2nd, 3rd period... etc
Xt = Actual value of the 1st, 2nd, 3rd period... etc
N � = number of moving average periods
3. Exponential Smoothing
The exponential smoothing method is a data smoothing
method by repeating calculations continuously using the latest data
(Ginantra & Anandita, 2019). The data is weighted with a trial number until a small
forecasting error value is obtained. This study uses weights based on optimal
ARIMA. The exponential smoothing equation can be written as follows :
Ft =
F t−1 + α (A t−1 − F t−1)
Were
Ft� ������������� = forecasting
value for the − t period
F t−1 �������������������� = Forecast value of
the past period
Α � ������������� =
Smoothing constant/weight
A t−1�������������������� = Actual value of
the t period
RESULTS AND DISCUSSION
1.
Total
Passenger Data
Since its inauguration, the number of KRL Jogja-Solo passengers
for (February 2021 to February 2022) has fluctuated. The rise and fall of
passengers are due to the ongoing Covid 19 pandemic. They have an impact on
community social restrictions (Orinaldi, 2021). The sharp decline occurred in July-August 2021 due to the
implementation of Java-Bali Community Activity Restrictions (PPKM). After
August 2021, the number of KRL passengers will gradually increase each month
due to concessions for travellers. Data on the number of KRL Jogja-Solo
passengers for one year (February 2021-February 2022) can be seen in the
following figure.
Figure 1. Number of Jogja Solo KRL Passengers from February 2021 to February 2022
The Jogja-Solo KRL passenger data for
February 2021 to February 2022 will then be used to determine the best
forecasting model.
2.
Gray Method
(1,1)
In conducting an analysis using the gray method (1,1), several
steps must be followed to obtain the Rey (1,1) forecasting model, namely:
a.
Determines
the original (actual) data series, which is symbolized by
The original data line is data on
the number of passengers from February 2021 to February 2022, as presented in
the following table.
Table 2. Original
data series
|
Period
|
1
|
2
|
3
|
4
|
5
|
6
|
|
|
94,536
|
195,018
|
183,935
|
185,092
|
190,412
|
46,340
|
|
Period
|
7
|
8
|
9
|
10
|
11
|
12
|
|
|
49,510
|
106,384
|
186,060
|
220,388
|
298,190
|
291,684
|
|
Period
|
13
|
|
|
|
|
|
|
|
220,712
|
|
|
|
|
|
b.
Defines a
data line
�with 1-AGO
The data line
is obtained by adding up the value 
with the previous period or denoted by 
. The 1-AGO calculation is data smoothing, and
its value increases. The results of 1-AGO calculations can be presented in the
following table.
Table 3. Rows of data
|
Period
|
1
|
2
|
3
|
4
|
5
|
6
|
|
|
94,536
|
289,554
|
473,489
|
658,581
|
848,993
|
895,333
|
|
Period
|
7
|
8
|
9
|
10
|
11
|
12
|
|
|
944,843
|
1,051,227
|
1,237,287
|
1,457,675
|
1,755,865
|
2,047,549
|
|
Period
|
13
|
|
|
|
|
|
|
|
2,268,261
|
|
|
|
|
|
c.
Calculating
MGO (Mean Generating Operations)
�
The data line 
is obtained by calculating the average
value of two 
adjacent data. Calculation
results can be presented in the following table.
Table 4. Data
Rows
|
Period
|
1
|
2
|
3
|
4
|
5
|
6
|
|
|
-
|
192,045
|
381,522
|
566,035
|
753,787
|
872163
|
|
Period
|
7
|
8
|
9
|
10
|
11
|
12
|
|
|
920,088
|
998035
|
1,144,257
|
1,347,481
|
1,606,770
|
1,901,707
|
|
Period
|
13
|
|
|
|
|
|
|
|
2,157,905
|
|
|
|
|
|
d.
Determine the
parameters α and b
Parameters α and b are obtained from the equation 
�were where
�
e.
Calculates
AGO predictions
Values 
�are
obtained by entering the values of parameters a and b in the formula 
. The following results are obtained.
Table 5. AGO Data Line
|
Period
|
1
|
2
|
3
|
4
|
5
|
6
|
|
|
94,536
|
219,176
|
351,765
|
492,811
|
642,851
|
802,460
|
|
Period
|
7
|
8
|
9
|
10
|
11
|
12
|
|
|
972,249
|
1,152,866
|
1,345,002
|
1,549,391
|
1,766,815
|
1,998,106
|
|
Period
|
13
|
|
|
|
|
|
|
|
2,244,147
|
|
|
|
|
|
f.
Calculating
Gray's Prediction Result (1,1)
The
prediction results of Gray (1,1) are obtained by the equation 
and the results are:
Table 6. Gray's Prediction Results (1,1)
|
Period
|
1
|
2
|
3
|
4
|
5
|
6
|
|
 �
|
124,640
|
132,589
|
141,045
|
150,040
|
159,609
|
169,788
|
|
Period
|
7
|
8
|
9
|
10
|
11
|
12
|
|
�
|
180,617
|
192,136
|
204,389
|
217,424
|
231,291
|
246,041
|
|
Period
|
13
|
|
|
|
|
|
|
�
|
261,733
|
|
|
|
|
|
The difference in the prediction results above, compared to the
actual data, can be seen in the following figure.
Table 7.
Comparison of Original Data with the Gray Method (1,1)
|
Period
|
Original
Data
|
Gray
(1,1)
|
|
1
|
94536
|
124640
|
|
2
|
195018
|
132589
|
|
3
|
183935
|
141045
|
|
4
|
185092
|
150040
|
|
5
|
190412
|
159609
|
|
6
|
46340
|
169788
|
|
7
|
49510
|
180617
|
|
8
|
106384
|
192136
|
|
9
|
186060
|
204389
|
|
10
|
220388
|
217424
|
|
11
|
298190
|
231291
|
|
12
|
291684
|
246041
|
|
13
|
220712
|
261733
|
Based on the prediction results using the gray method (1,1), a
change in numbers tends to increase in each period, resulting in a smoother
curve than the original data. The results of the model's accuracy analysis
against the original data can be shown in the following table.
Table 8. Accuracy Value of the Gray Model (1.1)
|
Accuracy Value
|
Gray (1,1)
|
|
MAPE
|
62
|
|
MAD
|
55,111
|
|
MSD
|
4,399,133,058
|
Figure 2. Comparison of Gray's Forecasting (1.1) with Actual Data
3.
Moving
Average Method
This study uses forecasting methods of moving averages of orders 2
and 3 to predict the number of KRL Jogja Solo passengers. The prediction
results using the moving average method can be seen as follows.
Table 9. Comparison of Original
Data with Moving
Average Method
|
Period
|
Original
Data
|
Moving
Averages (n=2)
|
Moving
Averages (n=3)
|
|
1
|
94536
|
-
|
-
|
|
2
|
195018
|
-
|
-
|
|
3
|
183935
|
144777
|
-
|
|
4
|
185092
|
189477
|
157830
|
|
5
|
190412
|
184514
|
188015
|
|
6
|
46340
|
187752
|
186480
|
|
7
|
49510
|
118376
|
140615
|
|
8
|
106384
|
47925
|
95421
|
|
9
|
186060
|
77947
|
67411
|
|
10
|
220388
|
146222
|
113985
|
|
11
|
298190
|
203224
|
170944
|
|
12
|
291684
|
259289
|
234879
|
|
13
|
220712
|
294937
|
270087
|
Table 10. Accuracy Value
of the Moving Average Model
|
Accuracy
Value
|
Moving Averages (n=2)
|
Moving Averages (n=3)
|
|
MAPE
|
63
|
71
|
|
MAD
|
63,822
|
73,035
|
|
MSD
|
5,682,822,147
|
7,606,377,523
|
Forecasting error values can be seen from the MAPE, MAD and MSD
values. Based on the data above, the higher the order value used in the moving
average method, the higher the error value will be. So, the second-order moving
average method is preferred over the third-order moving average method.
Figure 3. Comparison of Moving Average Forecasting with Actual Data
4.
Exponential
Smoothing Method
This study uses the forecasting method of single exponential and
double exponential smoothing with smoothing measurements using optimal ARIMA
(α) to predict the number of KRL Jogja Solo passengers (Susilawati & Sunendiari, 2022). The prediction results using the exponential smoothing method
can be seen as follows.
Table 11. Forecasting with the Single Exponential Smoothing Method
|
Period
|
Original Data
|
Single Smoothing
�α = 1.29
|
Double Smoothing
α = 1.29
|
|
1
|
94536
|
63914
|
248630
|
|
2
|
195018
|
103673
|
230857
|
|
3
|
183935
|
222274
|
260104
|
|
4
|
185092
|
172495
|
201532
|
|
5
|
190412
|
188851
|
217751
|
|
6
|
46340
|
190878
|
216055
|
|
7
|
49510
|
3212
|
6692
|
|
8
|
106384
|
63325
|
77709
|
|
9
|
186060
|
119232
|
134503
|
|
10
|
220388
|
206001
|
228088
|
|
11
|
298190
|
224681
|
244096
|
|
12
|
291684
|
320124
|
347483
|
|
13
|
220712
|
283198
|
301160
|
Table 12. Exponential
Smoothing Model Accuracy Value
|
Accuracy Value
|
Single Exp Smoothing
|
Double Exp Smoothing
|
|
MAPE
|
51
|
64
|
|
MAD
|
50,308
|
61,501
|
|
MSD
|
3,891,632,651
|
6,041,280,143
|
The exponential smoothing forecast error value can be seen from
the MAPE, MAD and MSD values. Based on the data above, the single exponential
smoothing method produces a lower error value than the double exponential
smoothing method.
Figure 4. Comparison of Exponential Smoothing Forecasting
with Actual Data
Based on the calculations that have been done and by looking at
the error value of each forecasting method, it is obtained that the forecasting
method is more conical in 3 methods, namely Gray (1, 1), second-order moving
average and single exponential smoothing. Of the three methods, the MAPE, MAD
and MSD values are compared again to obtain the most appropriate forecasting
method to be used in forecasting in the next period. The results of comparing the predictions
of the three selected forecasting methods can be seen in the following table.
Table 13. Comparison of
Forecasting Methods
|
time
|
actual
|
Gray (1,1)
|
Moving Averages (n=2)
|
Single Exp Smooth
α = 1.29
|
|
1
|
94536
|
124,640
|
-
|
63914
|
|
2
|
195018
|
132,589
|
-
|
103673
|
|
3
|
183935
|
141,045
|
144777
|
222274
|
|
4
|
185092
|
150,040
|
189477
|
172495
|
|
5
|
190412
|
159,609
|
184514
|
188851
|
|
6
|
46340
|
169,788
|
187752
|
190878
|
|
7
|
49510
|
180,617
|
118376
|
3212
|
|
8
|
106384
|
192,136
|
47925
|
63325
|
|
9
|
186060
|
204,389
|
77947
|
119232
|
|
10
|
220388
|
217,424
|
146222
|
206001
|
|
11
|
298190
|
231,291
|
203224
|
224681
|
|
12
|
291684
|
246,041
|
259289
|
320124
|
|
13
|
220712
|
261,733
|
294937
|
283198
|
Figure 5. Comparison of Forecasting Methods
The error prediction value among
the three selected methods, namely Gray (1.1), second-order moving average and
single exponential smoothing, was obtained from the results of the single
exponential smoothing method, which has a mirror value compared to the others. So,
this method is the best method chosen to predict the number of Jogja Solo KRL
passengers in the next period.
Table 14. Model Accuracy Value
|
Accuracy
Value
|
Gray
(1,1)
|
Moving
Averages (n=2)
|
Single
Exp Smooth
α = 1.29
|
|
MAPE
|
62
|
63
|
51
|
|
MAD
|
55,111
|
63,822
|
50,308
|
|
MSD
|
4,399,133,058
|
5,682,822,147
|
3,891,632,651
|
Forecasting
the next period in the single exponential smoothing method is calculated using
data on the number of passengers in the previous period with α = 1.29 so that the equation can be written as follows:
�
To calculate the number of
passengers, period 14 can be calculated as follows.
So that the
prediction of the number of passengers in the 14th period is 202,067 people.
CONCLUSION
Based on a
comparison of the forecasting methods that have been carried out, the single
exponential smoothing method is the best because it has the mirror value, namely
MAPE of 51, MAD of 50,308 and MSD of 3,891,632,651. The prediction of period 14
passengers obtained results of 202,067 people.
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