forecasting passenger by using holt's exponential ...

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FORECASTING PASSENGER BY USING HOLT’S EXPONENTIAL SMOOTHING AND WINTER’S EXPONENTIAL SMOOTHING Asep Rusyana1), Nurhasanah2), Maulina Oktaviana3), Amiruddin4) 1)

Program Studi Statistika, Jurusan Matematika, FMIPA, Universitas Syiah Kuala email: [email protected] 2) Program Studi Statistika, Jurusan Matematika, FMIPA, Universitas Syiah Kuala email: nurhaasanah @yahoo.co.id 3) Program Studi Matematika, Jurusan Matematika, FMIPA, Universitas Syiah Kuala email: [email protected] 4) Program Studi Pendidikan Ekonomi, FKIP, Universitas Syiah Kuala email: [email protected]

Abstract Sultan Iskandar Muda International Airport is one of the airport under the management of PT. Angkasa Pura II, which serve domestic and international flights. A number of passengers at Sultan Iskandar Muda International Airport continues to increase every year. Therefore, forecasting is important to anticipate the possibility of bumping number of the passengers. This study aims to determine the best model to forecast the number of domestic passengers which is arriving and departing from Sultan Iskandar Muda International Airport. The best model is obtained by comparing the method of Holt's exponential smoothing and Winter's exponential smoothing, as data on the passenger numbers forms seasonal and trends patterns. The data which was used is the data of air traffic flow (DAU) in the year of 2008 to 2013. Based on the comparison of error measure, the best model to forecast the number of domestic arrivals is the method of Winter's exponential smoothing for multiplicative models with smoothing constant α = 0.1, γ = 0.1, and β = 0.1, the error measure MAE = 1230, MSE = 3215248, MPE = 0.18, and MAPE = 5.41. The best model to forecast the number of domestic departures is the method of Winter's exponential smoothing for multiplicative models with smoothing constant α = 0.3, γ = 0.1, and β = 0.1, the error measure MAE = 1220, MSE = 2671955, MPE = 0.02, and MAPE = 5.12. Keywords: forecasting, Holt's exponential smoothing, Winter's exponential smoothing, number of passengers

different level of accuracy depends on the data pattern which is formed. Single exponential smoothing method is used for the data with stationary pattern, double exponential smoothing method with the trend pattern, whereas triple exponential smoothing method is used for the data with trend and seasonal patterns. Based on observations, the number of airport passengers continues to increase. It shows trend and seasonal patterns at a given moment. So that the data will be analyzed and forecasted by using double exponential smoothing method (Holt’s exponential smoothing) and triple exponential smoothing (Winter’s exponential smoothing). One of the problem in using Holt’s exponential smoothing and Winter’s exponential smoothing methods is to determine the smoothing constant α, , and β which can minimize errors. Each of parameter α, , and β has the value between 0 and 1. The approach commonly uses to determine the parameter value is trial and error. Many combinations

1. INTRODUCTION Sultan Iskandar Muda International Airport is one of the airports under the management of PT. Angkasa Pura II, which served domestic and international flights. Nowadays, airplane isa kind of air transportation that provides various facilities. Therefore, aviation industry needs to forecast the number of passengers with the aim of estimating increment or decrement in demand of air travel so that the manager can anticipate all the possibilities in the future. Exponential smoothing is a kind of time series forecasting which is used to predict the future by using historical data and by way of looking at the data pattern that is formed. There are three methods that are used to forecast by using exponential smoothing namely single exponential smoothing, double exponential smoothing, and triple exponential smoothing. Each of the three methods has a 34

Prosiding SEMIRATA Bidang MIPA 2016; BKS-PTN Barat, Palembang 22-24 Mei 2016

that must be tested before the optimal value of α, , and β can be determined. Therefore, in this research the parameter of α, , and β is limited on the values 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9. Forecasting error is calculated with mean absolute error = MAE, mean square error = MSE, mean percentage error = MPE and mean absolute percentage error = MAPE. The aims of this study are to see the comparison of domestic passenger numbers that are arriving and departing from Sultan Iskandar Muda International Airport, to know the best forecasting model in predicting number of passengers, and to generate the forecasting values for the coming year.

: forecast value : smoothing constant : actual value The method includes the exponential smoothing method, among others: a. Single exponential smoothing  Single exponential smoothing with a one-parameter  Single exponential smoothing with adaptive approach b. Double exponential smoothing is used to handle the trend pattern in the data  One-parameter linear method by Brown  Two-parameter method by Holt c. Triple exponential smoothing is used to handle trend and seasonal patterns in the data  One parameter quadratic method by Brown  Three parameter trend and seasonal method by Winter d. Pegels classification exponential smoothing refers to the exponential smoothing trend and seasonal multiplicative

2. METHODOLOGY Forecasting in this research is quantitative with time series method. Data used is secondary data or data that have been collected monthly by Sultan Iskandar Muda International Airport fellow in form of time series data with the number of domestic passengers that are arriving and departing from 2008 to 2013, about 72 months. The tool used is Zaitun Time Series 0.1.3.0 software. The stages of data analysis are as follows. 1. Creating time series data plot for the number of domestic passengers that are arriving and departing from 2008 to 2013 in order to see the patterns formed. Data patterns can be divided into four types. (1). Horizontal pattern occurs when the data fluctuates around the constant average value. (2). Seasonal pattern occurs when a series is influenced by seasonal factors. (3). Cyclical pattern occurs when data are affected by longterm economic fluctuations such as something related to the business cycle. (4). Trends pattern occurs when there is an increment or decrement in the longterm secular data. 2. Describing the data. 3. Choosing appropriate forecasting method. Exponential smoothing forecasting method is a method that shows a group of weighted exponential decrease toward the value of the older observations. The general form of forecasting with exponential smoothing

4. Creating forecasting model. a. Holt’s exponential smoothing method Holt’s exponential smoothing equation as follows. The exponentially smoothed series The trend estimate Forecast m period into the future for n = 1, 2, …, N The stages to generate forecasting with Holt method is as follows.  Specifying the initialization parameters ofα and  Estimating the initial value of and  Calculating the value of exponentially smoothed series, trend estimate and forecast  Calculating forecasting error b. Winter’s exponential smoothing method (additive and multiplicative models) This method is divided into two models, (1) namely additive and multiplicative models. Calculations performed with the additive

where 35




model if the plot of the original data shows that seasonal fluctuation is relatively stable. Winter’s exponential smoothing equation for additive model. Level component



Calculating the value of the level component, trend component, seasonality component, and forecast Calculating forecasting error

5. Choosing the best forecasting model with the minimum error. (5) The accuracyof a forecasting method is the suitability of a method that ultimately show (6) how far the forecasting model is able to reproduce the data already known. Forecasting accuracy is used as follows. (7) a. Mean Absolute Error (MAE)

Trend component Seasonality component The m-step-ahead forecast is calculated as

b. Mean Squared Error (MSE) Winter’s exponential smoothing equation for a multiplicative model. Level component

(8)

c. Mean Percentage Error (MPE) d. Mean Absolute (MAPE)

Percentage

Error (9)

Trend component (10)

where Seasonality component

(11)

e. Percentage Error The m-step-ahead forecast is calculated as

(12) for n= 1, 2, …, l-1.

If the forecasting approach is not bias, then the resulting value of MPE will approach zero. Here are the criteria of MAPE value.

where: : the level component of the series : the smoothing constant for level : actual value at n time : the estimate of the trend component : the smoothing constant for the trend estimate : the estimate of the seasonality component : the smoothing constant for seasonality estimate : forecast for m-step-ahead l : seasonal length(eg.the number of months or quarters in a year)

Table 1. MAPE Criteria MAPE 50%

6. Forecasting the number of arriving and departing domestic passengers from Sultan Iskandar Muda International Airport for the coming year based on the best model.

The stages to generate forecasting with Winter method is as follows.  Specifying the initialization 

parameters of α, β, and Estimating the initial value of

,

Definition Excellent Good Adequate Poor

3. RESULT AND DISCUSSION Each of time seriesdata plot for the number of arriving and departing domestic passengersfrom Sultan Iskandar Muda

,

and 36


International Airport from 2008 to 2013 is shown in figure1 and figure 2 below.

in 2008 recorded 255 952 passengers as shown in the figure 3 below.

Fig 1. Time seriesplot for the number of domestic passenger arrival

Fig 3. Graph of arriving and departing domestic passenger total numbers

1.1. Forecasting The Number of Domestic Passengers Arrival from Sultan Iskandar Muda International Airport 1.1.1. Holt’s Exponential Smoothing Method In making equations on Holt's method, the first step to be done is to specify initialization parameters of α and which can minimize error. By trial and error, the parameter values obtained for the level is α = 0.1, and = 0.1for smoothing the trend pattern. The smoothing constant produces the minimum error, with the values of MAE = 1641, MSE = 4816930, MPE = -0.52, andMAPE = 7.36. According to the result of smoothing constant value, so that Holt’s exponential smoothing equation for predicting the numbers of domestic passengers arrival can be written below. The exponentially smoothed series

Fig 2. Time seriesplot for the number of domestic passenger departure

Figure1 and 2 show the data of domestic passenger numbers which is arriving and departing from Sultan Iskandar Muda International Airport. Data of domestic arriving and departing passenger numbers from Sultan Iskandar Muda International Airport were experiencing a trend increment from the year of 2008 to 2013. In addition, the number of passengers was experiencing seasonal fluctuations that showed pattern repeated in certain months. The number of arriving and departing domestic passengers has always decreased in February and April but has increased in March and May.The highest number of passenger arrivals occurred in November 2013 and the lowest in September 2008. The highest number of passenger departures occurred in June 2013 and the lowest in February 2010. In total, the number of arrivals and departures of domestic passengers has increased each year, except the number of domestic passenger departures in 2009, which dropped as much as 5315 passengers from the previous total passengers

The trend estimate Forecast m period into the future

1.1.2. Winter’s exponential smoothing method Same as the Holt method, the first step to be done for having Winter’s exponential smoothing equation is to specify initialization parameters of α, β, and which can minimize error. 1) Additive model Optimal smoothing constant for the levelα = 0.1, for trend pattern = 0.1, and for 37


seasonality estimateβ = 0.1. The smoothing constants producesthe minimum error, with the values of MAE = 1236, MSE = 3102796, MPE = 0.21, and MAPE = 5.46. According to the result of smoothing constant value, so that Winter’s exponential smoothing equation with additive model for predicting the numbers of domestic passengers can be written below. Level component

multiplicative model are presented in table b below. Thus, according to the result of forecasting Table 2. Forecasting accuracy comparison for the number of domestic passenger arrivals Error Measure Model MAE MSE MPE MAPE Holt Winter’sadditive Winter’s multiplicative

Trend component

1641 1236

4816930 3102796

-0.52 0.21

7.36 5.46

1230

3215248

0.18

5.41

(22)

accuracy value comparison from those three (23) models, it can be concluded that the best forecasting model for predicting the data domestic passanger numbers arrival from (24) Sultan Iskandar Muda International Airportis Winter’s exponential smoothing for multiplicative model with smoothing constant α = 0.1, = 0.1, and β = 0.1. This model (25) produces MPE = 0.18 and MAPE = 5.41. MPE value that is produced approcing zero, it means forecasting approach is not bias. MAPE value shows that forecasting result obtained are very good with forecasting error or deviation rate of the actual data is about 5.41%. Graph comparison of the actual value and the forecast value of the domestic passenger arrivals from Sultan Iskandar Muda International Airport in the year of 2008 to 2013 by using Winter’s exponential smoothing for multiplicative model with smoothing constant α = 0.1, = 0.1, and β = 0.1 is shown in figure 4 below.

Seasonality component

The m-step-ahead forecast is calculated as 2) Multiplicativemodel By trial and error, the optimal smoothing constant obtained for the level is α = 0.1, smoothing constant of trend pattern = 0.1 and seasonality β = 0.1.The smoothing constant value producesthe minimum error, with the values of MAE = 1230, MSE = 3215248, MPE = 0.18, dan MAPE = 5.41. According to the result of smoothing constant value, so that Winter’s exponential smoothing equation with multiplicative model for predicting the numbers of domestic passengers arrival can be written below. Level component

(26) Trend component (27) Seasonality component (28) The m-step-ahead forecast is calculated as (29) The next step is choosing the best forecasting model by comparing the size of forecasting error result. Forecasting error comparison of MAE, MSE, MPE, andMAPE which was produce by Holt’s exponential smoothing model, Winter’s exponential smoothing for additive model, Winter’s exponential smoothing for additive for

Fig 4.Graph of actual and forecast values comparison for the domestic passenger arrivals in 2008 to 2013 The number of domestic passengers arrival at Sultan Iskandar Muda International Airport for the coming one-year period, based on forecasting result, is increasing. It shows a seasonal pattern that is repeated as in previous 38


years. Forecasting result the number of domestic passengers arrival by Winter’s exponential smoothing for multiplicative model with smoothing constant α = 0.1, = 0.1, and β = 0.1 shows below.

of domectic pessangers departure written below. The exponentially smoothed series The trend estimate

Table 3. Forecasting result for the number of domestic passengersarrival in 2014 Period January

Forecast 29101

February March April

24872 28539 26927

May June July

28752 30741 31428

August September October

27408 28680 31244

November December

30524 29990

Total

348206

can be

The m-step-ahead forecast is calculated as

1.2.2. Winter’s exponential smoothing method 1) Additive model Optimal smoothing constant for the level α = 0.3, smoothing constant of trend pattern = 0.1, and seasonality β = 0.1.The smoothing constants producesthe minimum error, with the values of MAE = 1246, MSE = 2678435, MPE = 0.03, and MAPE = 5.26. According to the result of smoothing constant value, so that Winter’s exponential smoothing equation with additive model for predicting the numbers of domectic passengers departure can be written below. Level component

The figure of the trend pattern which is constantly increasing and seasonal pattern is repeated in 2014 can be seen in figure5 below.

Trend component Seasonality component The m-step-ahead forecast is calculated as Fig 5. Graph of forecasting value for domestic passenger arrivals in 2014

2) Multiplicative model By trial and error, the optimal smoothing constant obtained for the level is α = 0.3, smoothing constant of trend pattern = 0.1 and seasonality β = 0.1.The smoothing constant value produces the minimum error, with the values of MAE = 1220, MSE = 2671955, MPE = 0.02, dan MAPE = 5.12. According to the result of smoothing constant value, so that Winter’s exponential smoothing equation with multiplicative model for predicting the numbers of domestic passengers arrival can be written below. Level component

1.2. Forecasting The Number of Domestic Passengers Departure from Sultan Iskandar Muda International Airport 1.2.1. Holt’s exponential smoothing method Through trial and error by combining every single value of smoothing constant, the parameter values obtained for the the level is α = 0.1, and = 0.1 for smoothing the trend pattern. The smoothing constant value produces the minimum error, with the values of MAE = 1591, MSE = 4345339, MPE = 0.17, and MAPE = 6.66. According to the result of smoothing constant value, so that Holt’s exponential smoothing equation for predicting the numbers

Trend component

39

Prosiding SEMIRATA Bidang MIPA 2016; BKS-PTN Barat, Palembang 22-24 Mei 2016 Fig 6. Graph comparison of the actual value and (38) the forecast value of the domestic passengers departure from 2008 to 2013

Seasonality component

(39) The number of domestic passengers departure at Sultan Iskandar Muda International Airport for the coming one-year (40) period also increase. It shows a seasonal pattern that is repeated as in previous years. Forecasting result the number of domestic passengers departure by Winter’s exponential smoothing for multiplicative model with smoothing constant α = 0.3, = 0.1, and β = 0.1 shows below.

The m-step-ahead forecast is calculated as

Forecasting error value comparison of MAE, MSE, MPE, and MAPE which were produced by Holt’s exponential smoothing method, Winter’s exponential smoothing for additive, Winter’s exponential smoothing for multiplicative are presented in table d below.

Table 5. Forecasting result for the number of domestic passengersdeparture in 2014 Period Forecast January 29687 February 26528 March 29830 April 28443 May 30339 June 32991 July 32136 August 29104 September 30590 October 31788 November 31565 December 31730 Total 364731

Table 4. Forecasting accuracy comparison for the number of domestic passenger departure Error Measure Model MAE MSE MPE MAPE Holt Winter’sadditive Winter’s multiplicative

1591

4345339

-0.17

6.66

1248

2678435

0.03

5.26

1220

2671955

0.02

5.12

According to the result of forecasting accuracy value comparison from those three models, it can be concluded that the best forecasting model for predicting the data domestic passanger numbers departure from Sultan Iskandar Muda International Airport is Winter’s exponential smoothing for multiplicative model with smoothing constant α = 0.3, = 0.1, and β = 0.1.This model produces MPE = 0.02 and MAPE = 5.12. MPE value that is produced approcing zero, it means forecasting approach is not bias. MAPE value shows that forecasting result obtained is very good with forecasting error or deviation rate of the actual data is about 5.12%. Graphcomparison of the actual value and the forecast value of the domestic passenger departure from 2008 to 2013 is shown in figure 6 below.

The figure of the trend pattern which is constantly increasing and seasonal pattern is repeated in 2014 can be seen in figure7.

Fig 7. Graph of forecasting value for domestic passengers departure in 2014 Forecasting the number of arriving and departing domestic passengersat Sultan Iskandar Muda International Airport for the coming one-year period which is based on the best model conclude in the table fbelow. 40

Prosiding SEMIRATA Bidang MIPA 2016; BKS-PTN Barat, Palembang 22-24 Mei 2016 3. Aswi dan Sukarna. Analisis Deret Waktu:Teori dan Aplikasi. Andira Publisher, Makassar. 2006. 4. Box, G.E.P., Jenkins, G.M., & Reinsel, G.C. Time Series Analysis: Forecasting and Control Third Edition. Pearson Prentice Hall, New Jersey.1994. 5. Fathony, R.Z.A., Wiboowo, S.H., Anas, K., Amelia, L (Tim Pengembang Aplikasi Zaitun Time Series). Petunjuk Penggunaan Zaitun Time Series Bahasa Indonesia. http://zaitunsoftware.com/system/files/zaitunTS_IDm anual.pdf. 2009. 6. Kalekar, P.S., Bernard. Time Series Forecasting using Holt-Winters Exponential Smoothing. Kanwal Rekhi School of Information Technology. Mumbai, India.2004. 7. Lazim, M. A. Introductory Business Forecasting a Practical Approach 3rd Ed. UiTM Press Universiti Teknologi Mara, Shah Alam.2011. 8. Makridakis, S., Wheelwright, S.C., dan McGee, V.E. Metode Dan Aplikasi Peramalan. TerjemahandariForecasting: Methods and Applications, oleh Hari Suminto, Binarupa Aksara, Jakarta.1999. 9. Montgomery, D.C., Jennings, C.L., Kulahci, M. Introduction To Time Series Analysis and Forecasting. John Wiley & Sons Inc, Hoboken. 2008. 10. Munawaroh, A.N. Peramalan Jumlah Penumpang Pada PT. Angkasa Pura I (Persero) Kantor Cabang Bandar Udara Internasional Adisutjipto Yogyakarta Dengan Metode Winter’sExponential Smoothing Dan Seasonal Arima. Skripsi. Universitas Negeri Yogyakarta, Yogyakarta.2010. 11. Noeryanti, Oktafiani, E., Andriyani, F. Aplikasi Pemulusan Eksponensial dari Brown dan dari Holt untuk Data yang Memuat Trend. Prosiding Seminar Nasional Aplikasi Sains & Teknologi (SNAST) Periode III. Yogyakarta, Indonesia.2012. 12. Pramita, W., Tanuwijaya, H. Penerapan Metode Exponential Smoothing Winter dalam Sistem Informasi Pengendalian Persediaan Produk dan Bahan Baku Sebuah Cafe. Seminar Nasional Informatika 2010 (Semnasif 2010). Yogyakarta, Indonesia.2010. 13. Render, B., dan Heizer, J. Prinsip-Prinsip Manajemen Operasi. Salemba Empat, Jakarta.2001. 14. Santoso, S. Business Forecasting: Metode Peramalan Bisnis Masa Kini dengan MINITAB dan SPSS. PT. Elex Media Komputindo,Jakarta.2009. 15. Soepeno, B. Manajemen Produksi Berbantuan Komputer: modul Peramalan Penjualan. Politeknik Negeri Malang, Malang.2012. 16. Subagyo, Pangestu. Forecasting Konsep dan Aplikasi. BPFE, Jakarta.2002.

Table 6. Forecasting the number of arriving and departing domestic passengers at Sultan Iskandar Muda International Airport 2014 Passanger numbers Period Arrival Departure January 29101 29687 February 24872 26528 March 28539 29830 April 26927 28443 May 28752 30339 June 30741 32991 July 31428 32136 August 27408 29104 September 28680 30590 October 31244 31788 November 30524 31565 December 29990 31730 Total 348206 364731 4. CONCLUSSION 1. The number of domestic passenger arrival and departure at Sultan Iskandar Muda International Airport pattern shows trendincrement and seasosal repetition for each year. 2. The number of domestic passenger arrival and departure has a relatively similar pattern.. 3. The best model for forecasting the number of arrival and departure at Sultan Iskandar Muda International Airport is Winter’s exponential smoothing. 4. Forecasting for the number of domestic passenger arrival and departure at Sultan Iskandar Muda International Airport on 2014 shows increment trend and seasonal repetition. 5. According to forecasting result for 2014, the highest number of arrival occurred on July while the highest number of departure occurred on June.

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