Question

National Scan, Inc., sells radio frequency inventory tags. Monthly sales for a seven-month period were as follows:

Month	Sales (000)Units
Feb.	17
Mar.	20
Apr.	12
May.	22
Jun.	19
Jul.	24
Aug.	26

b. Forecast September sales volume using each of the following:

(1) A linear trend equation.(Round your intermediate calculations and final answer to 2 decimal places.)
  
Yt ______ thousands

(4) The naive approach.

Naive approach _______ thousands

Answer 1

We have been given the sales data of National Scan Inc. The data for a 7 month period has been provided to us, and we are required to forecast the sales for the next month (September).

This is time series data. The sales volume can be estimated ONLY using the sales volume in the previous months, as we have not been provided any other information. This means that the sales in the next month is a function of the sales in the previous months.

This is a graphical representation of the data:

We must forecast the sales in September using a linear trend method, and a naive method.

Let us first look at the linear trend method:

(1) A linear trend equation

A linear trend equation is similar to a linear regression method, where our dependent variable here is the sales value, and the independent variable is time. We have been given values of t1,t2,t3...t7, and we must forecast the value of t8 by fitting a line.

y(predicted)= a+bx

Thus, we must find the values of a and b, and given an x value (time period here) we would be able to estimate y.

The formula for finding b which minimizes the difference between predicted value of the line and the actual value (i.e. the error) is:

Calculating the values of x-mean(x), y-mean(y), (x-mean(x))*(y-mean(y)) and (x-mean(x))^2 for each period (1,2,....7) and summing those values up, we can fit them in the above equation to get the slope of the trend equation. The calculated value is 42/28 which is 1.5. Thus, the slope of the quation is 1.5

The value of 'a' can be found out by substituting it in the eqaution:

The value of mean(y) is 20, mean(x) is 4 and b is 1.5

20=a+(1.5*4)

20=a+6

Thus, a=14.

Thus, our trend equation is:

y=14+1.5x.

Required x is 8. Thus y=14+12= 26.

Thus, the estimated value using linear trend is 26 thousands.

(4) The naive approach:

This approach is usually used when we believe that the data is not following a trend (or a visible change) and that the level of the data is constant. In this case, we might use a naive approach to forecasting the value of y.

We may use:

i) Simple average: This is a method to take the simple average of all values and forecast the value of the next term as this value. Using this method, our forecast for September would be 20 thousands. (This is a simple average of all sales values). Thus, using this approach, the forecast for the October period would also be 20, as the mean does not change.

ii) Last term value: This is another naive method to forecasting time series data where we would say that the value of the next period will simply be the value in the last period and will not change henceforth. Thus, using this approach, our prediction will be 26 thousands. (This is the value of August)

iii) Moving average: In this approach, we take a moving average of the data, that is, the average of only the last few values, as tvery old values might not be important to forecast the next period value. In this method, we must choose the number of periods whose average we are willing to take. Usually, this is chosen according to the problem. Let us forecast the moving average of the last 4 periods only. This would give us a value of 26+24+19+22 divided by 4 which is 91/4 which is 22.75 thousands.

iv) Simple exponential smoothing: This is the most widely used 'naive' approach when the data is not following a trend. In this approach we do not leave out any points (like we do in the moving average approach) and give at least some weight to every point. The weights decrease exponentially for older periods, thus the name exponential moving average.

'Alpha' is the weight or importance we give to every period. Higher the value of alpha. more is the importance given to most recent values. Let us choose a value of 0.5 for alpha.

Thus, our final answer will be 0.5*26+0.25*24+0.125*19+0.0625*22+0.03125*12+0.015625*20+0.0078125*17

=13+6+2.375+1.375+0.375+0.3125+0.1328125

=23.57

Thus, our forecast using the exponential smoothing method method would be 23.57 thousands.

I hope this answer helped you understand the basic methods of forecasting time series data. There are more elegant methods like the Holt-Winter's method for data which exhibits trend and seasonality, and the ARIMA and SARIMA methods.
If you have a doubt regarding any of the answers or spot a calculation error, please get let me know so I may correct it. Happy learning!