The Art of Mathematics in Business. Dr Jae K Shim

      How is it used and applied?

      Moving averages are used, for example, to project future sales. Once sales are projected, needed financing for production and inventory may be planned. Business owners can choose the number of periods to use on the basis of the relative importance attached to old versus current date.

      For example, one can compare two possibilities, a 5-month and a 3-month period. In terms of the relative importance of new versus old data, the old data receive a weight of ⅘ and current data ⅕. In the second possibility, the old data receive a weight of ⅔, while current observation receives a ⅓ weight. This is a special case of the exponential smoothing method, in which a smoothing constant is in effect the weight given to the most recent data. See Sec. 18, Exponential Smoothing.

      Sales forecast can be fairly accurate if the right number of observations to be averaged is picked. In order to pick the right number, the business manager may have to experiment with different moving-average periods. Measures of forecasting accuracy, such as the mean absolute deviation (MAD), can be used to pick the optimal number of periods. See Sec, 24, Measuring Accuracy of Forecasts.

18. Exponential Smoothing

      Introduction

      Exponential smoothing is a popular technique for short run business forecasting. It uses a weighted average of past data as the basis for a forecast. The procedure gives heaviest weight to recent information and smaller weights to observations in the more distant past. The reason for this is that the future is more dependent on the recent past than on the distant past.

      How is it computed?

      (The formula for exponential smoothing is

      ŷ_{t + 1} = αγ_t + (1 − γ)ŷ_t

      or

      ŷ_new = αy_old + (1 − γ)ŷ_old

      The higher the α, the greater is the weight given to the more recent information.

      Example

      The following data on sales are given for an appliance business:

      To initialize the exponential smoothing process, it is necessary to have the initial forecast. The first smoothed forecast to be used can be

      1.First actual observations.

      2.An average of the actual data for a few periods

      For illustrative purposes, let us use a six-period average as the initial forecast y, with a smoothing constant of α = 0.40. Then

      Note that y₇ = 70. Then ŷ₈ is computed as follows:

ŷ8	=	αy7 + (1 − α)ŷ7
	=	(0.40)(70) + (0.60)(63)
	=	28.0 + 37.80 = 65.80

      Similarly,

ŷ9	=	αy8 + (1 − α)ŷ8
	=	(0.40)(74) + (0.60)(65.80)
	=	29.60 + 39.48 = 69.08

      and

ŷ10	=	αy9 + (1 − α)ŷ9
	=	(0.40)(62) + (0.69)(69.08)
	=	24.0 + 41.45 = 66.25

      By using the same procedure, the values of ŷ₁₁, ŷ₁₂, ŷ₁₃, ŷ₁₄ and ŷ₁₅ can be calculated. Table 18.1 shows a comparison between the actual sales and predicted sales using the exponential smoothing method.

Table 18.1: Comparison of Actual Sales and Predicted Sales

      Because of the negative and positive differences between actual sales and predicted sales, the forecaster can use a higher or lower smoothing constant, α, in order to adjust the prediction as quickly as possible to large fluctuations in the data aeries. For example, if the forecast is slow in reacting to increased sales (that is, if the difference is negative), the forecaster may want to try a higher value of α. For practical purposes, the optimal a may be picked by minimizing the mean squared error (MSE), defined as:

      where i = the number of observations used to determine the initial forecast

      In our example, i = 6, so the mean squared error is

      The idea is to select the a that minimizes MSE, which is the average sum of the variations between the historical sales data and the forecast values for, the corresponding periods.

      Can a Computer Help?

      A manager will sometimes be confronted with complex problems requiring large sample data, necessitating trial of many different values of a for exponential smoothing. Excel has a routine for exponential smoothing.

      How is it used and applied?

      The exponential smoothing method is effective when there is randomness but no seasonal fluctuations in the data. The forecaster can use a higher or lower smoothing constant a in order to adjust the prediction as quickly as possible to large fluctuations it the data series. For example, if the forecast is slow in reacting to increases sales (that is, if the difference is negative), the forecaster may want to try a higher

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