So far we have studied that for Additive trend seasonality,

 

S­_t = a(D_t/I_t-L) + (1-a)(S_t-1 + T_t-1)

T_t = b(S_t-S_t-1) + (1-b)(T_t-1)

I_t = g(D_t/S_t) + (1-g)I_t-L

F_t+m = (S_t + mT_t)I_t-L+m

 

For a given data, it becomes a problem to determine initial points S_o,T_o. Here is an example to illustrate one method.

 

Example:

 

Following is sales data for 3 years for 4 quarters in each year.

 

	1992	1993	1994
Q1	10	12	16
Q2	20	23	33
Q3	26	30	34
Q4	17	22	26
Average ->	18.25	21.75

 

Approach: The whole problem is divided into 2 major parts

§       One to determine initial points

§       Other to extrapolate for forecast

 

Initial point determination

Average of sales for past 2 years is placed at center of year and line joining them is formed. First data is taken as S₀ and T₀ is determined by equation of line

          which gives

          T₀ = (21.75-18.25)/4 = 0.875 (per quarter)

 

Now, the estimation from the line is done. Values corresponding to each quarter that are lying on line are calculated. The results are

         

	92	93
Q1	16.94	20.4
Q2	17.8	21.3
Q3	18.7	22.2
Q4	19.6	23.06

 

There is a seasonality attached with each data. So, seasonality indices are determined (actual demand/demand forecast by line), which are:

         

	92	93	Average	Normalised	Symbol
Q­1­	0.5904	0.5872	=0.5872/2+0.5904/2 =0.5888	0.59	I-3
Q2	1.123	1.079	1.0100	1.11	I-2
Q3	1.391	1.352	1.3720	1.38	I-1
Q4	0.869	0.9539	0.9115	0.92	I0
			3.882	4.00

 

As we see, averages sum up to 3.882. To prevent things from going haywire, their summation is normalised to nearest integer=4. Normalisation is done by taking proportional values.

 

Also, I₀,I_-1...   have been denoted by considering end of 1993 as base.

 

Now, the forecasting part is as follows:

 

F_0,1 = (S₀+T₀)I_-3 = (23.06 + 0.875)*0.59 = 14.12

F_0,2 = (S₀+2T₀)I_-2 … and so on.

 

Now, D₁=16     and taking a=0.2, b=0.1, g=0.1, we have

 

S₁ = 0.2 (16/0.59) + (1-0.2)(S₀ + T₀) = 24.57

T₁ = 0.9385   

I₁ = 0.5961

 

Now, we have the forecasted data for Q_1,1994. Actual sales data is also available.

By using these two, renormalisation is done as done previously.

This gives better indices to use.

 

How predictable is underlying data?

This is a question that poses a question mark on degree of complexity involved in method used. One simple method may be as good as one of the typical onesthat are computationally very expensive.

So, one has to know where to stop. In most cases, keeping things simple gives as good results as otherwise.