Question

Problem 8-12 (Algorithmic)

Many forecasting models use parameters that are estimated using nonlinear optimization. The basic exponential smoothing model for forecasting sales is

F_{t + 1} = αY_t + (1 – α)F_t

where

Ft + 1	=	forecast of sales for period t + 1
Yt	=	actual value of sales for period t
Ft	=	forecast of sales for period t
α	=	smoothing constant 0 ≤ α ≤ 1

This model is used recursively; the forecast for time period t + 1 is based on the forecast for period t, F_t; the observed value of sales in period t, Y_t and the smoothing parameter α. The use of this model to forecast sales for 12 months is illustrated in the table below with the smoothing constant α = 0.3. The forecast errors, Y_t - F_t, are calculated in the fourth column. The value of α is often chosen by minimizing the sum of squared forecast errors, commonly referred to as the mean squared error (MSE). The last column of Table shows the square of the forecast error and the sum of squared forecast errors.

EXPONENTAL SMOOTHING MODEL FOR α=0.3
Week ()	Observed Value ()	Forecast	Forecast Error ()			Squared Forecast Error
1	16	16.00		0.00			0.00
2	20	16.00		4.00			16.00
3	18	17.20		0.80			0.64
4	24	17.44		6.56			43.03
5	21	19.41		1.59			2.53
6	16	19.89		-3.89			15.13
7	19	18.72		0.28			0.08
8	21	18.80		2.20			4.84
9	24	19.46		4.54			20.61
10	22	20.82		1.18			1.39
11	12	21.17		-9.17			84.09
12	19	18.42		0.58		0.34
SUM=188.68

In using exponential smoothing models, we try to choose the value of α that provides the best forecasts. Build an Excel Solver or LINGO optimization model that will find the smoothing parameter, α, that minimizes the sum of squared forecast errors. You may find it easiest to put table into an Excel spreadsheet and then use Solver to find the optimal value of α. If required, round your answer for α to three decimal places and the answer for the resulting sum of squared errors to two decimal places.

The optimal value of α is  and the resulting sum of squared errors is .

Answer 1

week	observed	0.3	error	error^2
1	16	16	0	0
2	20	16	4	16
3	18	17.2	0.8	0.64
4	24	17.44	6.56	43.0336
5	21	19.408	1.592	2.534464
6	16	19.8856	3.8856	15.09789
7	19	18.71992	0.28008	0.078445
8	21	18.80394	2.196056	4.822662
9	24	19.46276	4.537239	20.58654
10	22	20.82393	1.176067	1.383135
11	12	21.17675	9.176753	84.21279
12	19	18.42373	0.576273	0.332091
			SS	188.7216

Formulas

week	observed	0.3	error	error^2
1	16	16	=ABS(C2-B2)	=D2*D2
2	20	=(1-$C$1)*C2+$C$1*B2	=ABS(C3-B3)	=D3*D3
3	18	=(1-$C$1)*C3+$C$1*B3	=ABS(C4-B4)	=D4*D4
4	24	=(1-$C$1)*C4+$C$1*B4	=ABS(C5-B5)	=D5*D5
5	21	=(1-$C$1)*C5+$C$1*B5	=ABS(C6-B6)	=D6*D6
6	16	=(1-$C$1)*C6+$C$1*B6	=ABS(C7-B7)	=D7*D7
7	19	=(1-$C$1)*C7+$C$1*B7	=ABS(C8-B8)	=D8*D8
8	21	=(1-$C$1)*C8+$C$1*B8	=ABS(C9-B9)	=D9*D9
9	24	=(1-$C$1)*C9+$C$1*B9	=ABS(C10-B10)	=D10*D10
10	22	=(1-$C$1)*C10+$C$1*B10	=ABS(C11-B11)	=D11*D11
11	12	=(1-$C$1)*C11+$C$1*B11	=ABS(C12-B12)	=D12*D12
12	19	=(1-$C$1)*C12+$C$1*B12	=ABS(C13-B13)	=D13*D13
			SS	=SUM(E2:E13)

our objective function is to minimize SS by varying alpha

data -> solver

we get alpha = 0.2305 = 0.231

minimum SS = 187.1287 = 187.13