Practice Problems: Chapter 4, Forecasting
Problem 1:
Auto sales at Carmen’s Chevrolet are shown below. Develop a 3-week moving average.
Week Auto
Sales
1 8
2 10
3 9
4 11
5 10
6 13
7 -
Problem 2:
Carmen’s decides to forecast auto sales by weighting the three weeks as follows:
Weights Period
Applied
3 Last week
2 Twoweeks
ago
1 Three weeks
ago
6 Total
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Problem 3:
A firm uses simple exponential smoothing with α = 0.1 to forecast demand. The forecast
for the week of January 1 was 500 units whereas the actual demand turned out to be 450
units. Calculate the demand forecast for the week of January 8.
Problem 4:
Exponential smoothing is used to forecast automobile battery sales. Two value of α are
examined, α = 0.8 and α = 0.5. Evaluate the accuracy of each smoothing constant. Which
is preferable? (Assume the forecast for January was 22 batteries.) Actual sales are given
below:
Month Actual Forecast
Battery
Sales
January 20 22
February 21
March 15
April 14
May 13
June 16
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Problem 5:
Use the sales data given below to determine: (a) the least squares trend line, and (b) the
predicted value for 2008 sales.
Year Sales
(Units)
2001 100
2002 110
2003 122
2004 130
2005 139
2006 152
2007 164
To minimize computations, transform the value of x (time) to simpler numbers. In this
case, designate year 2001 as year 1, 2002 as year 2, etc.
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Problem 6:
Given the forecast demand and actual demand for 10-foot fishing boats, compute the
tracking signal and MAD.
Year Forecast Actual
Demand Demand
1 78 71
2 75 80
3 83 101
4 84 84
5 88 60
6 85 73
Problem: 7
Over the past year Meredith and Smunt Manufacturing had annual sales of 10,000
portable water pumps. The average quarterly sales for the past 5 years have averaged:
spring 4,000, summer 3,000, fall 2,000 and winter 1,000. Compute the quarterly index.
Problem: 8
Using the data in Problem 7, Meredith and Smunt Manufacturing expects sales of pumps
to grow by 10% next year. Compute next year’s sales and the sales for each quarter.
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ANSWERS:
Problem 1:
Moving average =
∑ demand in previous n periods
n
Week Auto Three-Week Moving
Sales Average
1 8
2 10
3 9
4 11 (8 + 9 + 10) / 3 = 9
5 10 (10 + 9 + 11) / 3 = 10
6 13 (9 + 11 + 10) / 3 = 10
7 - (11 + 10 + 13) / 3 = 11
1/3
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Problem 2:
Weighted moving average =
∑ (weight for period n)(demand in period n)
∑ weights
Week Auto Three-Week Moving Average
Sales
1 8
2 10
3 9
4 11 [(3*9) + (2*10) + (1*8)] / 6 = 9 1/6
5 10 [(3*11) + (2*9) + (1*10)] / 6 = 10 1/6
6 13 [(3*10) + (2*11) + (1*9)] / 6 = 10 1/6
7 - [(3*13) + (2*10) + (1*11)] / 6 = 11 2/3
Problem 3:
Ft = Ft −1 + α ( A t −1 − Ft −1 ) = 500 + 0.1( 450 − 500) = 495 units
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Problem 4:
Month Actual Rounded Absolute Rounded Absolute
Battery Sales Forecast Deviation Forecast Deviation
with a =0.8 with a =0.8 with a =0.5 with a =0.5
January 20 22 2 22 2
February 21 20 1 21 0
March 15 21 6 21 6
April 14 16 2 18 4
May 13 14 1 16 3
June 16 13 3 15 1
Sum = 15 16
2.5 2.75
SE 3.7 4.1
On the basis of this analysis, a smoothing constant of a = 0.8 is preferred to that of a
= 0.5 because it has a smaller MAD.
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Problem 5:
Year Time Sales X2 XY
Period (Units)
(X) (Y)
2001 1 100 1 100
2002 2 110 4 220
2003 3 122 9 366
2004 4 130 16 520
2005 5 139 25 695
2006 6 152 36 912
2007 7 164 49 1148
S X = S Y S S XY
2
28 =917 X =140 = 3961
x=
∑ x = 28 = 4
n 7
y=
∑ y = 917 = 131
n 7
b=
∑ xy − nxy = 3961 − (7)(4)(131) = 293 = 10.46
∑ x − nx 2 2
140 − ( 7)( 4 ) 2
28
a = y − bx = 131 − (10.46 × 4) = 89.16
Therefore, the least squares trend equation is:
y$= a + bx = 89.16 + 10.46 x
To project demand in 2008, we denote the year 2008 as x = 8, and:
Sales in 2008 = 89.16 + 10.46 * 8 = 172.84
Problem 6:
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Year Forecast Actual Error RSFE
Demand Demand
1 78 71 -7 -7
2 75 80 5 -2
3 83 101 18 16
4 84 84 0 16
5 88 60 -28 -12
6 85 73 -12 -24
MAD =
∑ Forecast errors =
70
= 11.7
n 6
Year Forecast Actual |Forecast Cumulative MAD Tracking
Demand Demand Error| Error Signal
1 78 71 7 7 7.0 -1.0
2 75 80 5 12 6.0 -0.3
3 83 101 18 30 10.0 +1.6
4 84 84 0 30 7.5 +2.1
5 88 60 28 58 11.6 -1.0
6 85 73 12 70 11.7 -2.1
RFSE −24
Tracking Signal = = = 2.1 MADs
MAD 11.7
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Problem 7:
Sales of 10,000 units annually divided equally over the 4 seasons is 10,000 / 4 = 2,500
and the seasonal index for each quarter is: spring 4,000 / 2,500 = 1.6; summer
3,000 / 2,500 = 1.2; fall 2,000 / 2,500 =.8; winter 1,000 / 2,500 =.4.
Problem 8:
Next years sales should be 11,000 pumps (10,000 * 110. = 11,000). Sales for each quarter
should be 1/4 of the annual sales * the quarterly index.
Spring = (11,000 / 4) *1.6 = 4,400;
Summer = (11,000 / 4) *1.2 = 3,300;
Fall = (11,000 / 4) *.8 = 2,200;
Winter = (11,000 / 4) *.4.= 1,100.
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