Solutions to Tutorial 3

By: Anuj Manuja

 

Q1. Unique Video Systems required a quarterly forecast for the upcoming year for its operations planning and budgeting. The following data are available about the past demand history:

 

 

Quarter	1995	1996	1997	1998
1	10	10	10
2	30	50	50
3	50	50	60
4	10	10	20

 

 

a.      Using a linear trend exponential smoothing model, compute the 1998 quarterly forecast. Assume a = b = 0.5, S_Q3,1997 = 50, T_Q3,1997 = 10.

 

Solution

 

For Linear trend exponential smoothing

S_t = a D_t + (1-a) (S_t-1 + T_t-1)  (1)

T_t = b (S_t – S_t-1) + (1-b) T_t-1   (2)

F_t+1 = S­_t + T_t                             (3)

 

Þ S_Q4 = a D_Q4 + (1-a) (S_Q3 + T_Q3) = 40

and T_Q4 = b (S_Q4 – S_Q3) + (1-b) T_Q3 = 0

 

Þ S₀ = 40 and T₀ = 0 (where subscript 0 represents Q4,1997, 1 represents Q1,1998 and so on)

 

Using (1), (2) and (3) we can now find

 

F_0,1 = 40

F_0,2 = 40

F_0,3 = 40

F_0,4 = 40

 

Where F_0,1 represents forecast for period 1 (Q1,1998) made in period 0 (Q2,1998). It can be seen that forecasts for all periods is same as T₀ = 0.

 

b.      Use a ratio seasonality and linear trend value model for forecasting demands. Estimate a good initial values using 1995 and 1996 data. Use hit and trial approach to come up with the smoothing constants (a, b, g) that best predict the 1997 demands.

 

Solution

 

For ratio seasonality and linear trend model the equations are a follows

 

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

T_t+1 = b (S_t+1 – S_t) + (1-b) T_t           (2)

I_t+1 = g D_t+1/S_t+1 + (1-g) I_t-L+1             (3)

F_t = (S_t + T_t)I_t-L+1                               (4)

 

For Initial values we use the method discussed in the class i.e.

 

	1995	1996
Q1	10	10
Q2	30	50
Q3	50	50
Q4	10	10
Total	100	120
Avg.	25	30

 

We assume the yearly averages to exist at the middle of the year and quarterly demands given are in the middle of a quarter, and hence

 

Increase per year = 30-25 = 5

Increase per quarter = 5/4 = 1.25 = T₀

 

Quarter	Trend Values (S)		Seasonal Indices (D/S)		Avg. I	Normalised i
	1995	1996	1995	1996
Q1	23.125	28.125	.432	1.467	0.394	0.394
Q2	24.375	29.375	1.231	1.702	1.467	1.468
Q3	25.625	30.625	1.951	1.633	1.792	1.794
Q4	26.875	31.875	0.372	0.314	0.343	0.343

 

Þ S₀ = 31.875, T₀ = 1.25, I_-3 = 0.394, I_-2 = 1.468, I_-1 = 1.794, I₀ = 0.343

 

F_0,1 = (S₀+T₀)I_-3 = 13.05

 

Now for determining the best possible values of a and b we use the availble actual demand for 1997 and minimize the error of forecasts by trying different combinations of a and b. This was done in excel (the file is available) and it was found that the MAD (Mean absolute deviation is minimum for a = 0.1 and b = 1.

 

NOTE – Best possible g cannot be determined unless the actual demands for 1998 is available as the g is used in forecast of 1997 and updated in 1997 for 1998 (for which actual demands are not available).

 

For a = 0.1 and b = 1

 

This is the fore cast for 1997 as done using data from 1995 and 1996

F_0,1 = 13.05

F_0,2 = 48.19

F_0,3 = 60.19

F_0,4 = 11.71  

 

Corresponding to the above forecast MAD = 3.43

 

c.      Develop forecast for 1998

 

Solution

 

Using the above calculations and assuming g = 0.2 we get

 

S₀ = 36.55, T₀ = 3 , I_-3 = 0.374, I_-2 = 1.466, I_-1 = 1.779, I₀ = 0.381

 

F_0,1 = (S₀+T₀)I_-3    » 15

F_0,2 = (S₀+2T₀)I_-2  » 62

F_0,3 = (S₀+3T₀)I_-1  » 81

F_0,4 = (S₀+4T₀)I₀   » 19

 

d.      Develop a 3-quarterly moving average forecast for 1997. Compare the error with that obtained by forecast model above

 

Solution

 

For 3-quarterly moving average Forecast = Avg. of previous 3 quarters

 

Using the data from part b. we get

 

Quarter	Actual Demand	Forecast from Moving Avg.	Error From Moving Avg. (abs)	Forecast from Part b (abs)	Error from Part b
Q1	10	36.67	26.67	13.05	3.05
Q2	50	23.33	26.67	48.19	1.81
Q3	60	23.33	36.67	60.19	0.19
Q4	20	40.00	20.00	11.71	8.29

 

S |e_t| = 110 for moving avg.

S |e_t| = 13.34 for ratio seasonality and linear trend

 

Hence the seasonality forecasting method is better.