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Various
studies have been carried out to evaluate forecasting methods most especially
with a focus on moving average and exponential smoothing. The common approach
for evaluating the accuracy of moving average and exponential smoothing were
Mean Absolute Deviation (MAD), Mean Square Error (MSE) and Root Mean Square
Error (RMSE). This paper, therefore, examines single moving average and
exponential smoothing and adopts the coefficient of description (correlation
coefficient), coefficient of explanation (regression), Mean Absolute Deviation
(MAD), Mean Square Error (MSE) and Root Mean Square Error (RMSE) to evaluate
the accuracy of single moving averages and exponential smoothing with different
smoothing constants. For single moving average: n=2, n=3, n=4, n=5, n=6, n=7,
n=8, n=9, n=10; and for simple exponential smoothing: α=0.1, α=0.2, α=0.3,
α=0.4, α=0.5, α=0.6, α=0.7, α=0.8, α=0.9. The study relies on secondary data of
revenue passenger demand in Murtala Muhammed International Airport (MMIA) and
Mallam Aminu Kano International Airport (MAKIA) from the period of 1995 to
2017. The behaviors of data obtained on different airports were observed, as
there seems to be consistency in the MMIA demand than that of MAKIA demand. The
implication of the consistency is that the result that emanates for forecast
evaluation will be reliable. The study reveals that simple exponential smoothing
generates a reliable forecast than the single moving average.
INTRODUCTION
Forecasts serve a crucial need in making rational decisions and planning activities more precisely by handling uncertainty about the future. Efficient prediction is considered an important prerequisite for efficient administration and organization in different areas of application related application areas.
During planning, taking a decision on the most accurate forecasting technique to employ is quite challenging, it that requires a comprehensive analysis of empirical results. Recent findings reveal that the performance evaluation of forecasting models depend on the accuracy measures adopted (Nijat et al., 2016).
Evaluating the performance of the forecasting method is very crucial. In the last three decades, various accuracy measures have been adopted by many scholars as an evaluation criterion. A number of different forecast accuracy measures for both regression and classification problems have been proposed by earlier researchers together with the comments and recommendations on the use of the relevant measures (Mahmou, 1984; Makridakis, 1991; Hyndman & Koehler, 2006; Sokolova & Lapalme, 2009; Power, 2011; Nijat et al., 2016; Adeniran & Ben, 2017; Adeniran & Kanyio, 2018). Such accuracy measures provide necessary and decisive feedback to decision makers for calibrating and refining the model in an effort to improve the preciseness of outcomes (Armstrong & Collopy, 1992). However, research findings suggest that there is no best overall accuracy measure which can be used as a universally accepted single metric for choosing the appropriate forecasting method (Mahmou, 1984). Forecasting approaches can realize extremely different performances depending on the chosen metric. Empirical evaluations reveal that some approaches are superior when error based measures are adopted, while others perform better for the same dataset when different metrics are utilized (Armstrong & Collopy, 1992).
It is pertinent to note that decision makers may be unwilling to generalize forecast from prior research, believing that their situation is different. Also, previous research may have revealed a number of relevant forecasting methods and one would like to narrow the field, which is systematic. Most principles for testing forecasting methods are based on commonly accepted methodological procedures, such as to pre-specify criteria or to obtain a large sample of forecast errors. However, forecasters often violate such principles, even in academic studies. Some principles might be surprising, such as R-square, Mean Square Error, and other models to select the most accurate forecasting model (Armstrong, 2001).
Ryu & Sanchez (2003) evaluated the forecasting method for institutional food service facility. They identified the most appropriate forecasting method of forecasting meal count for an institutional food service facility. The forecasting method analyzed included: naϊve model 1, 2 and 3; moving average method, double moving method, exponential smoothing method, double exponential method, Holt’s method, Winter method, linear regression and multiple regression method. The accuracy of forecasting methods was measured using mean absolute deviation, mean squared error, mean percentage error, mean absolute percentage error method, root mean squared error and Theil’s U-statistic. Their result showed that multiple regressions were the most accurate forecasting method, but naϊve method 2 was selected as the most appropriate forecasting method because of its simplicity and high level of accuracy.
Pradeep & Rajesh (2014) studied the evaluation of forecasting methods and their application for sales forecasting of sterilized flavored milk in Chhattisgarh. They applied weekly data spreading from October 2011 to October 2012, on the sales of sterilized flavored milk in a liter. The forecasting method analyzed included: naϊve model, moving average, double moving average, simple exponential smoothing; and semi-average method. The accuracy of the forecasting method was measured using mean Forecast Error (MFE), Mean Absolute Deviation (MAD), Mean Square Error (MSE), root mean square Error (RMSE).
Adeniran, Kanyio & Owoeye (2018) study forecasting methods for domestic air passenger demand in Nigeria using two years single moving average and simple exponential smoothing with smoothing constant of 0.9 to forecast the 2018 demand. The two methods of forecasting earlier identified were evaluated and compared with their Mean Squared Deviations (MSD) to determine which method gives the lowest deviation as it will produce the best forecast for the year 2018 domestic air passenger demand in Nigeria using the domestic airport passenger demand from the period of the year 2010 to 2017. It was revealed that the MSD of two yearly single moving averages gave the best the year 2018 forecast as it has a lower MSD when compared to the MSD of simple exponential smoothing with the smoothing constant of 0.9. Similarly, Adeniran & Stephens (2018) study the dynamics for evaluating different forecasting methods for international air passenger demand in Nigeria. They used two single moving averages, four single moving averages and six single moving average, simple exponential smoothing, with smoothing constants of 0.7, 0.8 and 0.9, respectively with the data between the periods of the year 2001 to the year 2017. Single moving average and simple exponential smoothing were compared using Mean Squared Deviation (MSD). It was revealed that simple exponential smoothing with constant 0.8 will give a better forecast.
Evaluation of different forecasting methods for international air passenger demand in Murtala Muhammed International Airport (MMIA) and Mallam Aminu Kano International Airport (MAKIA), Nigeria was carried out in this study. The forecasting methods analyzed include: single moving average (n=2, n=3, n=4, n=5, n=6, n=7, n=8, n=9, n=10) and simple exponential smoothing method (α=0.1, α=0.2, α=0.3, α=0.4, α=0.5, α=0.6, α=0.7, α=0.8, α=0.9). The accuracy measures of forecasting method were the coefficient of description (correlation coefficient), the coefficient of explanation (regression), Mean Absolute Deviation (MAD), Mean Square Error (MSE) and Root Mean Square Error (RMSE).
METHODOLOGY
This study examines single moving average and exponential smoothing, and adopts the coefficient of description (correlation coefficient), coefficient of explanation (regression), Mean Absolute Deviation (MAD), Mean Square Error (MSE) and Root Mean Square Error (RMSE) to evaluate the accuracy of single moving averages and exponential smoothing with different smoothing constants. For single moving average: n=2, n=3, n=4, n=5, n=6, n=7, n=8, n=9, n=10; and for simple exponential smoothing: α=0.1, α=0.2, α=0.3, α=0.4, α=0.5, α=0.6, α=0.7, α=0.8, α=0.9. The study relies on secondary data of revenue passenger demand in MMIA and MAKIA from the period of 1995 to 2017.
RESULTS AND DISCUSSION
Forecast Evaluation Using the Single Moving Average
From Tables 2 and 3, forecasts were obtained from two, three, four, five, six, seven, eight, nine and ten yearly single moving averages with twenty-three years data of revenue passenger demand in MMIA and MAKIA. From Figures 2 and 3, it was revealed that the lines of forecast and demand have similar trend from 1995 to 2017 which might be easily predictable without any critical analysis, but there seems to be a situation of rising and falling which might not be easily predictable without critical analysis. Hence, there is a need to evaluate the accurate forecasting technique that will produce a reliable 2018 forecast.
Forecast Evaluation Using the Simple Exponential Smoothing
From Tables 4 and 5, forecasts were obtained from simple exponential smoothing with smoothing constants of α=0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8 and 0.9. From Figures 4 and 5, it was revealed that the lines of forecast and demand have a similar trend from 1995 to 2017. It was also revealed that all forecasts follow a similar pattern. The suitability of forecasts produced by simple exponential smoothing is quite better and easily understandable than the forecasts produced by a single moving average. Although a mere examination of the line graph does not mean that the forecast of simple exponential smoothing will be more reliable than single moving average. Hence, there is a need to evaluate the accurate forecasting technique that will enhance a robust and reliable 2018 forecast.
EVALUATION OF FORECASTS
The accuracy of the forecasting methods adopted in this study was coefficient of description (correlation coefficient), the coefficient of explanation (regression), Mean Absolute Deviation (MAD), Mean Square Error (MSE) and Root Mean Square Error (RMSE).
Furthermore, the analysis is shown in Table 8 and Figures 14-17, coefficient of description (correlation coefficient) and coefficient of explanation (regression) for simple exponential smoothing for MMIA reveals that a smoothing constant of 0.7 is most accurate, as it has the highest correlation value and regression value. The correlation value and regression value implies that the forecast of smoothing constant of 0.7 has a strong relationship with revenue passenger demand in MMIA. Also, Mean Absolute Deviation (MAD) and Relative Mean Square Error (RMSE) for simple exponential smoothing for MMIA show that the smoothing constant of 0.9 has the lowest value. However, Mean Square Error (MSE) for simple exponential smoothing shows that the smoothing constant of 0.7 has the lowest value. Since three methods of assessing the forecasting techniques explain that simple exponential smoothing with smoothing constant of 0.7 gives the lowest value, hence the results of Mean Square Error (MSE) will be retained. On this note, it can be deduced that for simple exponential smoothing, 0.7 smoothing constant is the most accurate for forecasting as it tends closer to 1. This corroborates the study of Hossein (2015); Lucey (2007); Montogomery & (1997); Kahn & Mentzer (1995); Brown (1963) that the higher the values of smoothing constant nearer to 1, the more sensitive the forecast becomes the current condition.
The analysis is shown in Table 9 and Figures 18-21 revealed that coefficient of description (correlation coefficient) and coefficient of explanation (regression) for simple exponential smoothing for MAKIA reveals that smoothing constant of 0.5 and 0.6 is most accurate, as they have the highest correlation and regression value. The correlation and regression value implies that the forecast of smoothing constants of 0.5 and 0.6 has a strong relationship with revenue passenger demand in MAKIA. Also, Mean Absolute Deviation (MAD) for simple exponential smoothing for MAKIA shows that the smoothing constant of 0.9 has the lowest value. However, Mean Square Error (MSE) for simple exponential smoothing shows that the smoothing constant of 0.1 has the lowest value, meanwhile in support of MAD but contrary to MSE, Relative Mean Square Error (RMSE) for simple exponential smoothing shows that the smoothing constant of 0.9 has the lowest value. Hence, the evaluation of accuracy for simple exponential smoothing, MAD and RMSE will be acceptable, therefore 0.9 smoothing constant is the most accurate for forecasting. This corroborates the study of Lucey (2007) that the higher the value of smoothing constant nearer to 1, the more sensitive the forecast becomes the current condition.
In addition, there is a need to find out which forecasting method is more accurate from single moving average and simple exponential smoothing. The evaluation of forecast with data in MAKIA will not be used as it might give an inconclusive result, but the data in MMIA will be useful and reliable.
Hence, from the data of revenue passenger in MMIA, correlation and regression reveal that simple exponential smoothing with a smoothing constant of 0.7 has a strong relationship than 2 yearly single moving averages. This corroborates the findings achieved by Mean Square Deviation (MSD) which shows that simple exponential smoothing with smoothing constant of 0.7 has low deviation than 2 yearly single moving averages. This implies that simple exponential smoothing with smoothing constant of 0.7 will give accurate forecast than 2 yearly single moving averages. Also, simple exponential smoothing with a smoothing constant of 0.9 was revealed by Mean Absolute Deviation (MAD) and Relative Mean Square Error (RMSE) to have low deviation than 2 yearly single moving averages. This implies that simple exponential smoothing with smoothing constant of 0.9 will give accurate forecast than 2 yearly single moving averages. It can, therefore, be affirmed that simple exponential smoothing is more reliable than single moving average.
The findings of this study corroborate the findings of Adeniran & Stephens (2018); Hossein (2015); Lucey (2007); Hsiao (2003) & Wooldridge (2001) Montogomery & Johnson (1997); Kahn & Mentzer (1995); Brown (1963), but it opposes the findings of Adeniran, Kanyio & Owoeye (2018) which chooses single moving average over simple exponential smoothing. The findings of Adeniran, Kanyio & Owoeye (2018) seems different because the sample size of their data is lesser (n=7). In order to achieve a more plausible result for time series analysis, a larger sample size of non-experimental is requested. From simple exponential smoothing with smoothing constant of 0.9, the 2018 forecast of international air passenger travel demand in Murtala Muhammed International Airport will be 2,844,230.
CONCLUSION
This study identified the most appropriate forecasting method based on accuracy and ease of use (simplicity) to forecast the future demand of international air passenger in Murtala Muhammed International Airport. Data involved in MMIA and MAKIA are both historical and long on twenty-three years, but their evaluated results are quite different. This difference can be because the data of MMIA satisfies the assumption of linearity and consistency, while the data of MAKIA does not. Hence, the evaluation of forecast with data in MAKIA will not be used as it might give an inconclusive result, but the data in MMIA will be useful and reliable. From simple exponential smoothing with smoothing constant of 0.7, the 2018 forecast of revenue passenger demand in MMIA will be 2,870,005. Also, from simple exponential smoothing with smoothing constant of 0.9, the 2018 forecast of revenue passenger demand in MMIA will be 2,844,230.
The following contributions emanate from this study: time series analysis requires large sample size of non-experimental or observational data with respect to time; the observational data with respect to time must satisfy the assumption of linearity and consistency; the higher the value of smoothing constant nearer to 1, the more sensitive the forecast become the current conditions; the lower the value of n for the single moving average, the more realistic or reliable the forecast; and simple exponential smoothing is more reliable than single moving average.
Adeniran, A.O., Kanyio, A.O. & Owoeye, A.S. (2018). Forecasting methods for domestic air passenger demand in Nigeria. Journal of Applied Research on Industrial Engineering, 5(2), 146-155.
Adeniran, A.O. & Stephens, M.S. (2018). The dynamics for evaluating forecasting methods for international air passenger demand in Nigeria. Journal of Tourism and Hospitality, 7(4), 366.
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Hossein, A. (2015). Forecasting by smoothing techniques. Retrieved from: https://home.ubalt.edu/ntsbarsh/Business-stat/otherapplets/ForecaSmo.htm
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Pradeep, K.S. & Rajesh, K. (2014). Evaluation of forecasting methods and their application for sales forecasting of sterilized flavored milk in Chhattisgarh. International Journal of Engineering Trends and Technology, 8, 98-104.
Ryu, K. & Sanchez, A. (2003). The evaluation of forecasting method at an institutional food service dining facility. Journal of Hospitality Financial Management, 11(1), 27-45.
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