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\title{Determination of Exponential Smoothing Constant to Minimize Mean Square Error and Mean Absolute Deviation}
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             \author[1]{Dr. Sanjoy Kumar  Paul}

             \affil[1]{  Bangladesh University of Engineering and Technology}

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\date{\small \em Received: 20 January 2011 Accepted: 17 February 2011 Published: 1 March 2011}

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\begin{abstract}
        


Exponential smoothing technique is one of the most important quantitative techniques in
forecasting.The accuracy of forecasting of this technique depends on exponential smoothing constant.Choosing an appropriate value of exponential smoothing constant is very crucial to minimize the error in forecasting. This paper addresses the selection of optimal value of exponential smoothing constant to minimize the mean square error (MSE) and mean absolute deviation (MAD). Trial and error method is used to determine the optimal value of exponential smoothing constant. An example is presented to discuss the method.

\end{abstract}


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\let\tabcellsep& 	 	 		 
\section[{INTRODUCTION}]{INTRODUCTION}\par
orecasting can be broadly considered as a method or a technique for estimating many future aspects of a business or other operations. All forecasting methods can be divided into two broad categories: qualitative and quantitative. Many forecasting techniques use past or historical data in the form of time series. A time series is simply a set of observations measured at successive points in time or over successive periods of time and forecasts essentially provide future values of the time series on a specific variable such as sales volume. Exponential smoothing is one the time series techniques which is widely used in forecasting. Exponential smoothing gives greater weight to more recent observations and takes into account all previous observations. In ordinary terms, an exponential weighting scheme assigns the maximum weight to the most recent observation and the weights decline in a systematic manner as older and older observations are included. Weight in the exponential smoothing technique is given by exponential smoothing constant ( ?? ). Forecast values are varied with the values of this constant. So, forecast errors are also depended on ?? . Many authors used exponential smoothing method in forecasting.  has shown that exponential smoothing remains appropriate under more general conditions, where the variance is allowed to grow or contract with corresponding movements in the underlying level. \hyperref[b7]{Taylor (2003)} investigated a new damped multiplicative trend approach. \hyperref[b2]{Gardner (2006)}  No research paper has found which determine the optimal value of exponential smoothing constant. In this paper, mean square error (MSE) and mean absolute deviation (MAD) are considered to determine the forecast error and to minimize MSE and MAD, optimal values of exponential smoothing constant are determined. 
\section[{II.}]{II.} 
\section[{MATHEMATICAL FORMULATION FOR MSE AND MAD}]{MATHEMATICAL FORMULATION FOR MSE AND MAD}\par
Different forecasting techniques and time series analysis are described by \hyperref[b5]{Montgomery and Johnson (1976)}. From \hyperref[b5]{Montgomery and Johnson (1976)}, forecast at period t is given for exponential smoothing?? 1 = ?? 0 ; When ?? = 1 And ?? ?? = ???? ???1 + ??(1 ? ??)?? ???2 + ??(1 ? ??) 2 ?? ???3 + ? ? ? ? ? + ??(1 ? ??) ???1 ?? 0 (1) When ?? > 1\par
About-Department of Industrial and Production Engineering Bangladesh University of Engineering and Technology Email: sanjoy.ipe@gmail.com F Where, ?? ?? = Forecast at period ?? ?? ???1 = Actual value of period ?? ? 1 ?? = Exponential smoothing constant From (1), for different period, it can be written as,?? 1 = ?? 0\par
(2)?? 2 = ???? 1 + ??(1 ? ??)?? 0\textbf{(3)}?? 3 = ???? 2 + ??(1 ? ??)?? 1 + ??(1 ? ??) 2 ?? 0\textbf{(4)}?? 4 = ???? 3 + ??(1 ? ??)?? 2 + ??(1 ? ??) 2 ?? 1 + ??(1 ? ??) 3 ?? 0 (\textbf{5})?? 5 = ???? 4 + ??(1 ? ??)?? 3 + ??(1 ? ??) 2 ?? 2 + ??(1 ? ??) 3 ?? 1 + ??(1 ? ??) 4 ?? 0\textbf{(6)}\par
And so on?.. The most commonly used measures of forecast accuracy are discussed by \hyperref[b4]{Mentzer and Kahn (1995)}. In this paper, mean square error and mean absolute deviation are considered to measure forecast accuracy. Now Mean Square Error (MSE) can be determined as,?????? = ? (?? ?? ? ?? ?? ) 2 ?? ??=1 
\section[{/??}]{/??}\par
This can be written as,?????? = [(?? 1 ? ?? 1 ) 2 + (?? 2 ? ?? 2 ) 2 + (?? 3 ? ?? 3 ) 2 + ? ? ? ? ? + (?? ?? ? ?? ?? ) 2 ]/?? Putting the value of ?? 1 , ?? 2 , ?? 3 , ? ? ? , ?? ?? equation of MSE is found, ?????? = [\{?? 0 ? ?? 1 \} 2 + \{???? 1 + ??(1 ? ??)?? 0 ? ?? 2 \} 2 + \{???? 2 + ??(1 ? ??)?? 1 + ??(1 ? ??) 2 ?? 0 ? ?? 3 \} 2 + ? ? ? + \{???? ???1 + ??(1 ? ??)?? ???2 + ??(1 ? ??) 2 ?? ???3 + ? ? ? ? + ??(1 ? ??) ???1 ?? 0 ? ?? ?? \} 2 ]/??\textbf{(7)}\par
Again, Mean Absolute Deviation (MAD) is given by,?????? = ? |(?? ?? ? ?? ?? )| ?? ??=1 
\section[{/??}]{/??}\par
This can be written as,?????? = [|(?? 1 ? ?? 1 )| + |(?? 2 ? ?? 2 )| + |(?? 3 ? ?? 3 )| + ? ? ? ? ? + |(?? ?? ? ?? ?? )|]/?? Putting the value of ?? 1 , ?? 2 , ?? 3 , ? ? ? , ?? ?? equation of MAD is found ?????? = [|\{?? 0 ? ?? 1 \}| + |\{???? 1 + ??(1 ? ??)?? 0 ? ?? 2 \}| + |\{???? 2 + ??(1 ? ??)?? 1 + ??(1 ? ??) 2 ?? 0 ? ?? 3 \}| + ? ? ? + |\{???? ???1 + ??(1 ? ??)?? ???2 + ??(1 ? ??) 2 ?? ???3 + ? ? ? ? + ??(1 ? ??) ???1 ?? 0 ? ?? ?? \}|]/??\textbf{(8)}\par
Now Trial and Error method is used to find minimum value of exponential smoothing constant. Putting the different values within 0 to 1 for exponential smoothing constant, MSE and MAD are calculated.\par
III. 
\section[{RESULT AND DISCUSSION}]{RESULT AND DISCUSSION}\par
The method is described by considering a problem. A company's historical sales data are given for some previous periods. Actual values of sales are given in Table \hyperref[tab_1]{1}.   \hyperref[fig_0]{1}. MAD is decreased with increasing ?? up to 0.83 and after that MAD is increased. Variation of MAD with ?? is shown in Figure 2. The value of exponential smoothing constant is 0.88 and 0.83 for minimum MSE and MAD respectively.  To find the optimal value of exponential smoothing constant, minimum values of MSE and MAD are selected and corresponding value of exponential smoothing constant is the optimal value for this problem.\par
Minimum values of MSE and MAD and corresponding value of exponential smoothing constant is given in Table \hyperref[tab_3]{4}.     
\section[{Conclusions}]{Conclusions}\par
An exponential smoothing technique is widely used by many organizations to predict the future events. But problems are raised to assign the value of exponential smoothing constant. In this paper, this problem is solved by determining the optimal value of exponential smoothing constant. Mean square error and mean absolute deviation are minimized to get optimal value of the constant and optimal values are 0.88 and 0.83 for minimum mean square error and mean absolute deviation respectively. In this method, any organization can compute the optimal value of exponential smoothing constant to enhance the accuracy of forecasting. This work can be extended to minimize some other forecast errors such as mean absolute percent error (MAPE), cumulative forecast error (CFE) etc.\par
©2011 Global Journals Inc. (US) a sale at sixth period is also forecasted. Forecasts at sixth period are 12.455 thousand and 12.068 thousands for minimum MSE and MAD respectively. is shown is Table \hyperref[tab_5]{5}. From the five period actual values, From the optimal value of ??, forecast values for different period are determined using equation (  {\ref 1} \begin{figure}[htbp]
\noindent\textbf{1}\includegraphics[]{image-2.png}
\caption{\label{fig_0}Figure 1 :}\end{figure}
    \begin{figure}[htbp]
\noindent\textbf{} \par 
\begin{longtable}{P{0.85\textwidth}}
McKenzie\\
and Gardner (2010) provided a theoretical rationale for\\
the damped trend method based on Brown's original\\
thinking about the form of underlying models for\\
exponential smoothing. Hyndman (2002) provided a\\
new approach to automatic forecasting based on an\\
extended range of exponential smoothing methods.\\
Corberan-Vallet et al. (2011) presented the Bayesian\\
analysis of a general multivariate exponential smoothing\\
model that allows us to forecast time series jointly,\\
subject to correlated random disturbances.\end{longtable} \par
 
\caption{\label{tab_0}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{1} \par 
\begin{longtable}{P{0.21208897485493228\textwidth}P{0.6379110251450677\textwidth}}
\tabcellsep Determination of Exponential Smoothing\\
\tabcellsep Constant To Minimize Mean Square Error and Mean Absolute Deviation\\
April 2011\tabcellsep \\
32\tabcellsep \\
Volume XI Issue III Version I\tabcellsep \\
Global Journal of Research in Engineering\tabcellsep For different values of exponential smoothing constant, MSE and MAD are calculates. Mean square\\
\tabcellsep error for different values of exponential smoothing\\
\tabcellsep constant are determined using equation (7). Table 2\\
\tabcellsep shows the MSE values for different ??.\\
Period (t)\tabcellsep Actual Value (?? ?? ) (In thousands)\\
0\tabcellsep 10\\
1\tabcellsep 8\\
2\tabcellsep 14\\
3\tabcellsep 13\\
4\tabcellsep 12\\
5\tabcellsep 12.5\\
\multicolumn{2}{l}{©2011 Global Journals Inc. (US)}\end{longtable} \par
 
\caption{\label{tab_1}Table 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{3} \par 
\begin{longtable}{P{0.3755813953488372\textwidth}P{0.47441860465116276\textwidth}}
Exponential smoothing\tabcellsep Mean Absolute Deviation\\
constant (??)\tabcellsep (MAD)\\
0\tabcellsep 10.7\\
0.1\tabcellsep 8.038\\
0.2\tabcellsep 6.028\\
0.3\tabcellsep 4.535\\
0.4\tabcellsep 3.443\\
0.5\tabcellsep 2.663\\
0.6\tabcellsep 2.241\\
0.7\tabcellsep 2.041\\
0.75\tabcellsep 1.961\\
0.78\tabcellsep 1.919\\
0.8\tabcellsep 1.894\\
0.82\tabcellsep 1.869\\
0.83\tabcellsep 1.858\\
0.84\tabcellsep 1.867\\
0.88\tabcellsep 1.921\\
0.9\tabcellsep 1.949\\
0.95\tabcellsep 2.022\\
0.98\tabcellsep 2.068\\
1\tabcellsep 2.1\end{longtable} \par
 
\caption{\label{tab_2}Table 3 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{4} \par 
\begin{longtable}{P{0.40765306122448974\textwidth}P{0.37295918367346936\textwidth}P{0.06938775510204082\textwidth}}
Criteria\tabcellsep \multicolumn{2}{l}{Minimum Value Optimal value of ??}\\
Mean Square Error\tabcellsep 8.031\tabcellsep 0.88\\
Mean Absolute\tabcellsep 1.858\tabcellsep 0.83\\
Deviation\tabcellsep \tabcellsep \end{longtable} \par
 
\caption{\label{tab_3}Table 4 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2} \par 
\begin{longtable}{P{0.4163265306122449\textwidth}P{0.4336734693877551\textwidth}}
Exponential smoothing constant (??)\tabcellsep Mean Square Error (MSE)\\
0\tabcellsep 133.85\\
0.1\tabcellsep 76.374\\
0.2\tabcellsep 45.241\\
0.3\tabcellsep 28.3\\
0.4\tabcellsep 18.95\end{longtable} \par
 
\caption{\label{tab_4}Table 2 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{5} \par 
\begin{longtable}{P{0.052870813397129186\textwidth}P{0.1748803827751196\textwidth}P{0.3131578947368421\textwidth}P{0.3090909090909091\textwidth}}
Period\tabcellsep Actual values\tabcellsep Forecast values\tabcellsep Forecast\\
\tabcellsep (in thousands)\tabcellsep (in thousands)\tabcellsep values (in\\
\tabcellsep \tabcellsep for minimum\tabcellsep thousands)\\
\tabcellsep \tabcellsep MSE\tabcellsep for minimum\\
\tabcellsep \tabcellsep \tabcellsep MAD\\
0\tabcellsep 10\tabcellsep ---\tabcellsep ---\\
1\tabcellsep 8\tabcellsep 10\tabcellsep 10\\
2\tabcellsep 14\tabcellsep 8.096\tabcellsep 8.051\\
3\tabcellsep 13\tabcellsep 13.291\tabcellsep 12.750\\
4\tabcellsep 12\tabcellsep 13.034\tabcellsep 12.765\\
5\tabcellsep 12.5\tabcellsep 12.124\tabcellsep 11.794\\
6\tabcellsep ---\tabcellsep 12.455\tabcellsep 12.068\end{longtable} \par
 
\caption{\label{tab_5}Table 5 :}\end{figure}
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\end{document}