64.0
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 value. Finding the best a is the key to success in using this method.
The method is simple and effective, since it does not require a lot of data other than for the variable involved. One disadvantage of the method, however, is that it does not include industrial or economic factor such as market conditions, prices, or the effects of competitors’ actions.
19. Regression Analysis
Introduction
Regression analysis is a statistical procedure for estimating mathematically the average relationship between the dependent variable and the independent variable(s). The least-squares method is widely used in regression analysis for estimating the parameter values in a regression equation. Simple regression involves one independent variable, price or advertising in a demand function, whereas multiple regression involves two or more variables, that is price and advertising together.
How is it computed?
We will assume a simple (linear) regression to illustrate the least-squares method, which means that we will assume the Y = a + bX relationship, where a = intercept and b = slope. The regression method includes all the observed data and attempts to find a line of best fit. To find this line, a technique called the least-squares method is used.
The Least-Squares Method
To explain the least-squares method, we define the error as the difference between the observed value and the estimated one and denote it with u. Symbolically,
u = Y - Y′
whereY = observed value of the dependent variable
Y′ = estimated value based on Y′ = a + bX
The least-squares criterion requires that the line of best fit be such that the sum of the squares of the errors (or the vertical distance in Figure 19.1 from the observed data points to the line) is a minimum, i.e.,
Minimum: Σu2 = ∑(y − y′)2= Σ(y-a-bX)2
Using differential calculus we obtain the following equations, called normal equations:
ΣY = na + bΣX
ΣXY = aΣX + bΣX2
Solving the equations for b and a yields
Figure 19.1: Y AND Y¢
Example
