# INTRODUCTION 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. Taylor (2003) investigated a new damped multiplicative trend approach. 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. # II. # MATHEMATICAL FORMULATION FOR MSE AND MAD Different forecasting techniques and time series analysis are described by Montgomery and Johnson (1976). From 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 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 (2) ?? 2 = ???? 1 + ??(1 ? ??)?? 0(3)?? 3 = ???? 2 + ??(1 ? ??)?? 1 + ??(1 ? ??) 2 ?? 0(4)?? 4 = ???? 3 + ??(1 ? ??)?? 2 + ??(1 ? ??) 2 ?? 1 + ??(1 ? ??) 3 ?? 0 (5)?? 5 = ???? 4 + ??(1 ? ??)?? 3 + ??(1 ? ??) 2 ?? 2 + ??(1 ? ??) 3 ?? 1 + ??(1 ? ??) 4 ?? 0(6) And so on?.. The most commonly used measures of forecast accuracy are discussed by 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 # /?? 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 ]/??(7) Again, Mean Absolute Deviation (MAD) is given by, ?????? = ? |(?? ?? ? ?? ?? )| ?? ??=1 # /?? 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 ? ?? ?? }|]/??(8) 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. III. # RESULT AND DISCUSSION 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 1. 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. Minimum values of MSE and MAD and corresponding value of exponential smoothing constant is given in Table 4. # Conclusions 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. ©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 5. From the five period actual values, From the optimal value of ??, forecast values for different period are determined using equation ( 1 1![Figure 1: Variation of Mean Square Error for different values of ??](image-2.png "Figure 1 :") McKenzieand Gardner (2010) provided a theoretical rationale forthe damped trend method based on Brown's originalthinking about the form of underlying models forexponential smoothing. Hyndman (2002) provided anew approach to automatic forecasting based on anextended range of exponential smoothing methods.Corberan-Vallet et al. (2011) presented the Bayesiananalysis of a general multivariate exponential smoothingmodel that allows us to forecast time series jointly,subject to correlated random disturbances. 1Determination of Exponential SmoothingConstant To Minimize Mean Square Error and Mean Absolute DeviationApril 201132Volume XI Issue III Version IGlobal Journal of Research in EngineeringFor different values of exponential smoothing constant, MSE and MAD are calculates. Mean squareerror for different values of exponential smoothingconstant are determined using equation (7). Table 2shows the MSE values for different ??.Period (t)Actual Value (?? ?? ) (In thousands)01018214313412512.5©2011 Global Journals Inc. (US) 3Exponential smoothingMean Absolute Deviationconstant (??)(MAD)010.70.18.0380.26.0280.34.5350.43.4430.52.6630.62.2410.72.0410.751.9610.781.9190.81.8940.821.8690.831.8580.841.8670.881.9210.91.9490.952.0220.982.06812.1 4CriteriaMinimum Value Optimal value of ??Mean Square Error8.0310.88Mean Absolute1.8580.83Deviation 2Exponential smoothing constant (??)Mean Square Error (MSE)0133.850.176.3740.245.2410.328.30.418.95 5PeriodActual valuesForecast valuesForecast(in thousands)(in thousands)values (infor minimumthousands)MSEfor minimumMAD010------1810102148.0968.05131313.29112.75041213.03412.765512.512.12411.7946---12.45512.068 ©2011 Global Journals Inc. (US)technique is given by, * Forecasting correlated time series with exponential smoothing models ACorberan-Vallet JDBermudez EVercher International Journal of Forecasting 27 2011 * A state space framework for automatic forecasting using exponential smoothing methods RJHyndman ABKoehler RDSnyder SGrose International Journal of Forecasting 18 2002 * Exponential smoothing: The state of the art-Part II ESGardner International Journal of Forecasting 22 2006 * Damped trend exponential smoothing: A modelling viewpoint EMckenzi ESGardner International Journal of Forecasting 26 2010 * Forecasting technique familiarity, satisfaction, usage, and application JTMentzer KBKahn Journal of Forecast 14 1995 * Forecasting and time series analysis DCMontgomery Johnson LA 1976 McGraw-Hill New York * Forecasting for inventory control with exponential smoothing RDSnyder ABKoehler JKOrd International Journal of Forecasting 18 2002 * Exponential smoothing with a damped multiplicative trend JWTaylor International Journal of Forecasting 19 2003