1
A Simple Moving Average Example
Moving Average Forecast =
 demand in previous n periods
n
Where n is the number of periods in the moving average
Example: The demand for Gingerbread Men is shown in the table. Forecast the demand for
month 7.
Month
1
2
3
4
Demand
1500
2200
2700
4200
5
7800
6
5400
Calculation
Forecast
1500  2200  2700
 2133
3
2133
7
Bakery Gingerbread Man Sales
9000
8000
7000
Sales
6000
5000
Demand
4000
Forecast
3000
2000
1000
0
1
2
3
4
Period
5
6
7
2
A Weighted Moving Average Example
Weighted Moving Average Forecast =
 (weight for period n)  (demand in period n)
 weights
Example: The demand for Gingerbread Men is shown in the table. Forecast the demand for
month 7, weighting the past three months as follows: last month 3, two months ago 2, three
months ago 3
Period
Last Month
Two Months Ago
Three Months Ago
Sum of Weights
Month
1
2
3
4
Demand
1500
2200
2700
4200
5
7800
6
5400
Weight
3
2
1
Calculation
Forecast
(1500  1)  (2200  2)  (2700  3)
 2333
6
2333
7
Bakery Gingrbread Man Sales
9000
8000
7000
6000
Sales
5000
Series1
Series2
4000
3000
2000
1000
0
1
2
3
4
Period
5
6
7
3
An Exponential Smoothing Example
Exponential Smoothing Forecast Ft = Ft-1 + (At-1 – Ft-1)
Where
Ft
Ft-1

At-1
= New Forecast
= Previous Forecast
= Smoothing Constant: (0    1)
= Previous Period’s Actual Demand
Example: The demand for Gingerbread Men is shown in the table. Forecast the demand for
month 7, using a smoothing constant of 0.4. The forecast for month 1 is 1500 units.
Month
1
2
Demand
1500
2200
Calculation
1500 + 0.4(1500 – 1500)
Forecast
1500
1500
3
2700
1500 + 0.4(2200 – 1500)
1780
4
4200
5
7800
6
5400
7
Bakery Gingerbread Man Sales
9000
8000
7000
Sales
6000
5000
Demand
Forecast
4000
3000
2000
1000
0
1
2
3
4
Period
5
6
7
4
A Mean Absolute Deviation Example
MAD =
 forecast errors
n
Example: Calculate the MAD for the value of  used in the Exponential Smoothing Example.
Month
1
2
3
Demand
1500
2200
2700
Forecast
1500
1500
1780
4
4200
2148
5
7800
2969
6
5400
4901
Error
1500 – 1500 = 0
2200 – 1500 = 700
| Error |
0
700
Total:
MAD:
A Mean Squared Error Example
 forecast errors 
2
MSE =
n
Example: Calculate the MAD for the value of  used in the Exponential Smoothing Example.
Month
1
2
3
Demand
1500
2200
2700
Forecast
1500
1500
1780
4
4200
2148
5
7800
2969
6
5400
4901
Error2
0
490000
Error
1500 – 1500 = 0
2200 – 1500 = 700
Total:
MSE:
5
An Exponential Smoothing With Trend Adjustment Example
Forecast Including Trend Ft = (At-1) + (1 - )(Ft-1 + Tt-1)
Trend Tt = (Ft – Ft-1) + (1 - )T t-1
Where
Ft
Tt
At


= Exponentially smoothed forecast for period t
= Exponentially smoothed trend for period t
= Actual demand for period t
= Smoothing constant for the average (0    1)
= Smoothing constant for the trend (0    1)
Example: The demand for Gingerbread Men is shown in the table. Forecast the demand for
month 7, using a smoothing constant for the average of 0.4, and a smoothing constant for the
trend of 0.2. The forecast for month 1 is 1500 units and the trend for month 1 is 200 units.
Month
1
2
3
Demand Forecast
1500
1500
2200
0.4(1500) + 0.6(1700)
= 1620
2700
4
4200
5
7800
6
5400
Trend
200
0.2(1620 – 1500) + 0.8(200)
= 184
7
Bakery Gingerbread Man Sales
9000
8000
7000
Sales
6000
Demand
5000
Forecast
Trend
4000
FIT
3000
2000
1000
0
1
2
3
4
Period
5
6
7
FIT
1700
1804
6
Least Squares Trend Projection Example
ŷ  a  bx
Where
= computed value of variable to be predicted (ie dependant variable)
= y-axis intercept
= slope of regression line
= independent variable
ŷ
a
b
x
We can determine a and b with the equations:
b
 xy - nxy
 x  nx
2
a  y - bx
2
Example: The demand for Gingerbread Men is shown in the table. Forecast demand for
period 7 by fitting a single line trend to the data.
Month (x)
1
2
3
Demand (y)
1500
2200
2700
4
4200
5
7800
6
5400
x =
y =
x
b
x2
1
4
xy
(1)(1500) = 1500
(2)(2200) = 4400
x2 =
xy =
x 
y
n
 xy - nxy
 x  nx
2
2
y 
n
=
a  y - bx =
Thus, our trend equation is: ŷ =
+
x
To calculate the forecast for month x = 7, we have: ŷ =
7
Seasonal Forecast Example
Average Monthly Demand =
Seasonal Index =
 Average Annual Demand
12 months
Average Annual Demand
Average Monthly Demand
Example: The demand for gingerbread men over the past three years is shown in the table. If
we expect the total yearly demand in 2002 to be 45,000 units, what will be our forecasted
monthly demands in 2002?
Month 1999
1
1100
2
1800
2000
1300
2000
2001
1500
2200
3
2300
2500
2700
4
3800
4000
4200
5
4500
4700
4900
6
5000
5200
5400
7
5500
5700
5900
8
4800
5000
5200
9
3000
3200
3400
10
2200
2400
2600
11
1500
1700
1900
12
1200
1400
1600
Average Annual Demand
(1100 + 1300 + 1500) / 3 = 1300
 Average Monthly Demand =
Seasonal Index for January = 1300 /
Forecast for January 2002 =
45000

12
=
=
8
Regression Analysis Example
ŷ  a  bx
b
 xy - nxy
 x  nx
2
2
a  y - bx
Example: We think that there may be a relationship between park attendance and number of
gingerbread men sold. Data for the first six months are shown in the table. Forecast the
number of gingerbread men that will be sold in month 7 if monthly park attendance is forecast
as 25000 people.
Sales (y)
x2
xy
1
2
Attendance (x)
(,000)
8
12
1500
2200
64
(8)(1500) = 12000
3
14
2700
4
18
4200
5
19
7800
6
22
5400
x =
y =
x2 =
xy =
Month
x
b
x 
y
n
 xy - nxy
 x  nx
2
2
y 
n
=
a  y - bx =
Thus, our regression equation is: ŷ =
+
To calculate the forecast for month 7, we have: ŷ =
x
9
Standard Error of Estimate Example
S y ,x 
y
2
 a y - b xy
n2
Example: Compute the Standard Error of the Estimate for our regression analysis example
S y ,x 
y
2
 a y - b xy
n2
=
Correlation Coefficient Example
r
n x
n xy   x  y
2

  x  n  y 2   y 
2
2

Example: Compute the Correlation Coefficient of the data in our regression analysis example