Question

Given the following sales data:

Year 1, Spring: 108

Year 1, Summer: 46

Year 1, Fall: 68

Year 1, Winter: 26

Year 2, Spring: 96

Year 2, Summer: 44

Year 2, Fall: 76

Year 2, Winter: 22

Forecast sales for Spring, Year 3 using the seasonal model.

Your Answer:

Answer 1

The provided sales data is:

Year	Period	Sales
Year 1	Spring	108
Year 1	Summer	46
Year 1	Fall	68
Year 1	Winter	26
Year 2	Spring	96
Year 2	Summer	44
Year 2	Fall	76
Year 2	Winter	22

We will now find the seasonality for each period

We know that, seasonality for a period = average of that period/average of the whole data

Season	Year 1 Sales	Year 2 Sales	Seasonal Average	Seasonality = Seasonal average/total average
Spring	108	96	102	1.68
Summer	46	44	45	0.74
Fall	68	76	72	1.19
Winter	26	22	24	0.40

Here total average is = (108 + 46 + 68 + 26 + 96 + 44 + 76 + 22)/8 = 486/8 = 60.75

Hence deseasonalized forecast for all periods = sales/seasonality of period

Year	Period	Sales	Seasonality	Deseasonalized Forecast
Year 1	Spring	108	1.68	64.29
Year 1	Summer	46	0.74	62.16
Year 1	Fall	68	1.19	57.14
Year 1	Winter	26	0.40	65.00
Year 2	Spring	96	1.68	57.14
Year 2	Summer	44	0.74	59.46
Year 2	Fall	76	1.19	63.87
Year 2	Winter	22	0.40	55.00

Hence now the data is:

Year	Period	Period number (x)	Forecast (y)
Year 1	Spring	1	64.29
Year 1	Summer	2	62.16
Year 1	Fall	3	57.14
Year 1	Winter	4	65.00
Year 2	Spring	5	57.14
Year 2	Summer	6	59.46
Year 2	Fall	7	63.87
Year 2	Winter	8	55.00

Forecasting using Seasonal model requires us to find the deseasonalized data first. Then we go through the trend analysis in order to find the deseasonalized forecasting trend line. Using the trend line we will do the deseasonalized forecast for any period. Then the seasonality will be multiplied in order to get the seasonal forecast.

We know that, the trend line is

y = a + b*x

Where y is forecast, x is number of period

a = {(∑y)*(∑x2) – (∑x)*(∑xy)} / {(n)*(∑x2) – (∑x)*(∑x)}

b = {(n)*(∑xy) – (∑x)*(∑y)} / {(n)*(∑x2) – (∑x)*(∑x)}

Here,

Year	Period	Period number (x)	Forecast (y)	x*y	x^2
Year 1	Spring	1	64.29	64.29	1
Year 1	Summer	2	62.16	124.32	4
Year 1	Fall	3	57.14	171.42	9
Year 1	Winter	4	65	260	16
Year 2	Spring	5	57.14	285.7	25
Year 2	Summer	6	59.46	356.76	36
Year 2	Fall	7	63.87	447.09	49
Year 2	Winter	8	55	440	64
Total		36	484.06	2149.58	204

So,

∑x = 36

∑y = 484.06

∑x*y = 2149.58

∑x² = 204

n (number of observations) = 8

Putting the values in formula

a = {(∑y)*(∑x²) – (∑x)*(∑xy)} / {(n)*(∑x²) – (∑x)*(∑x)}

= {(484.06*204) – (36*2149.58)}/{(8*204) – (36*36)}

= (98748.24 – 77384.88)/(1632 – 1296)

= (21363.36/336)

= 63.58

a = 63.58

b = {(n)*(∑xy) – (∑x)*(∑y)} / {(n)*(∑x²) – (∑x)*(∑x)}

= {(8*2149.58) – (36*484.06)}/{(8*204) – (36*36)}

= (17196.64 – 17426.16)/(1632 – 1296)

= (-229.52/336)

= (-0.68)

b = -0.68

Hence the trend line is y = 63.58 – 0.68*x

For Year 3 – Spring, x = 9

Hence forecast = 63.58 – 0.68*9 = 63.58 – 6.12 = 57.46

Now the seasonality for spring = 1.68

Hence, Seasonal forecast = forecast*seasonality = 57.46*1.68 = 96.53

Hence, Answer is: 96.53

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