Transcript Document 7885097

CHAPTER 13
FORECASTING
Outline
• Forecasting and Choice of a Forecasting Methods
• Methods for Stationary Series:
– Simple and Weighted Moving Average
– Exponential smoothing
• Trend-Based Methods
– Regression
– Double Exponential Smoothing: Holt’s Method
• A Method for Seasonality and Trend
Forecasting
Decisions Based on Forecasts
• Production
– Aggregate planning,
inventory control,
scheduling
• Marketing
– New product
introduction, salesforce allocation,
promotions
• Finance
– Plant/equipment
investment, budgetary
planning
• Personnel
– Workforce planning,
hiring, layoff
Characteristics of Forecasts
• Forecasts are always
wrong; so consider
both expected value
and a measure of
forecast error
• Long-term forecasts
are less accurate than
short-term forecasts
• Aggregate forecasts
are more accurate than
disaggregate forecasts
Forecasting
•
•
•
•
Components of demand
Evaluation of forecasts
Time series: stationary series
Time series: trend
– Linear regression
– Double exponential smoothing
• Time series: seasonality
Components of Demand
• Average demand
• Trend
– Gradual shift in average demand
• Seasonal pattern
– Periodic oscillation in demand which repeats
• Cycle
– Similar to seasonal patterns, length and
magnitude of the cycle may vary
• Random movements
• Auto-correlation
Qantity
Components of Demand
Time
(a) Average: Data cluster about a horizontal line.
Quantity
Components of Demand
Time
(b) Linear trend: Data consistently increase or decrease.
Components of Demand
Quantity
Year 1
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J
F
M
A
M
J
J
A
S
O
N
D
Months
(c) Seasonal influence: Data consistently show
peaks and valleys.
Components of Demand
Quantity
Year 1
Year 2
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F
M
A
M
J
J
A
S
O
N
D
Months
(c) Seasonal influence: Data consistently show
peaks and valleys.
Components of Demand
Quantity
Components of Demand
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1
2
3
4
5
6
Years
(c) Cyclical movements: Gradual changes over
extended periods of time.
Components of Demand
Trend
Random
movement
Time
Demand
Demand
Components of Demand
Trend with
seasonal pattern
Time
Snow Skiing
Seasonal
Long term growth trend
Demand for skiing products increased
sharply after the Nagano Olympics
Choosing a Method
Forecast Error
Measures of Forecast Error
Et = At - Ft
RSFE = Et
MSE =
Et2
MAD =
n
MAPE =
 =
MSE
|Et |
n
[ |Et | (100) ] /At
n
Choosing a Method
Forecast Error
Month,
t
Demand,
At
Forecast,
Ft
1
2
3
4
5
6
7
8
200
240
300
270
230
260
210
275
225
220
285
290
250
240
250
240
-
Total
Error,
Et
Error
Squared,
Et2
Absolute
Absolute
Percent
Error,
Error,
|Et|
(|Et|/At)(100)
Choosing a Method
Forecast Error
Measures of Error
RSFE =
MSE =
MAD =
MAPE =
=
=
=
Choosing a Method
Forecast Error
Running Sum
of Forecast Errors
(RSFE - bias)
Mean Absolute
Deviation
(MAD)
23.1
69.8
17.1
15.5
14.0
18.4
65.6
41.0
14.8
15.3
Method
Simple moving average
Three-week (n = 3)
Six-week (n = 6)
Weighted moving average
0.70, 0.20, 0.10
Exponential smoothing
 = 0.1
 = 0.2
Choosing a Method
Tracking Signals
RSFE
Tracking signal =
MAD
+2.0 —
Control limit
Tracking signal
+1.5 —
+1.0 —
+0.5 —
0—
- 0.5 —
- 1.0 —
Control limit
- 1.5 —
0
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5
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10
15
20
Observation number
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25
Choosing a Method
Tracking Signals
Out of control
+2.0 —
Control limit
+1.5 —
Tracking signal
RSFE
Tracking signal =
MAD
+1.0 —
+0.5 —
0—
- 0.5 —
- 1.0 —
Control limit
- 1.5 —
0
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5
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10
15
20
Observation number
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25
Choosing a Method
Tracking Signals
Control Limit
Spread
(Number of
MAD)
1.0
1.5
.0
2.5
3.0
3.5
4.0
Equivalent Percentage of
Area within
Number of 
(=1.25 MAD) Control Limits
0.80
1.20
1.60
2.00
2.40
2.80
3.20
57.62
76.98
89.04
95.44
98.36
99.48
99.86
Problem 13-2: Historical demand for a product is:
Month
Jan Feb Mar Apr May Jun
Demand
12
11
15
12
16
15
a. Using a weighted moving average with weights of 0.60,
0.30, and 0.10, find the July forecast.
b. Using a simple three-month moving average, find the July
forecast.
c. Using single exponential smoothing with =0.20 and a June
forecast =13, find the July forecast.
d. Using simple regression analysis, calculate the regression
equation for the preceding demand data
e. Using regression equation in d, calculate the forecast in
July
Problem 13-15: In this problem, you are to test the validity of
your forecasting model. Here are the forecasts for a model
you have been using and the actual demands that
occurred:
Week
1
2
3
4
5
6
Forecast
800 850 950 950 1,000 975
Actual
900 1,000 1,050 900 900 1,100
Compute MAD and tracking signal. Then decide whether the
forecasting model you have been using is giving
reasonable results.
Methods for Stationary Series
Time Series Methods
Simple Moving Averages
450 —
Patient arrivals
430 —
410 —
390 —
370 —
0
Actual patient
arrivals
|
5
|
10
|
15
Week
|
20
|
25
|
30
Time Series Methods
Simple Moving Averages
450 —
Week
Patient
Arrivals
1
2
3
400
380
411
Patient arrivals
430 —
410 —
390 —
370 —
0
Actual patient
arrivals
|
5
|
10
|
15
Week
|
20
|
25
|
30
Time Series Methods
Simple Moving Averages
450 —
Week
Patient
Arrivals
1
2
3
400
380
411
Patient arrivals
430 —
410 —
390 —
F4 =
370 —
0
Actual patient
arrivals
|
5
|
10
|
15
Week
|
20
|
25
|
30
Time Series Methods
Simple Moving Averages
450 —
Week
Patient
Arrivals
2
3
4
380
411
415
Patient arrivals
430 —
410 —
390 —
F5 =
370 —
0
Actual patient
arrivals
|
5
|
10
|
15
Week
|
20
|
25
|
30
Time Series Methods
Simple Moving Averages
450 —
3-week MA
forecast
Patient arrivals
430 —
410 —
390 —
370 —
0
Actual patient
arrivals
|
5
|
10
|
15
Week
|
20
|
25
|
30
Time Series Methods
Simple Moving Averages
450 —
6-week MA
forecast
3-week MA
forecast
Patient arrivals
430 —
410 —
390 —
370 —
0
Actual patient
arrivals
|
5
|
10
|
15
Week
|
20
|
25
|
30
Taco Bell determined that
the demand for each 15minute interval
can be estimated from a 6week simple moving
average of sales.
The forecast was used to
determine the number of
employees needed.
Time Series Methods
Weighted Moving Average
450 —
3-week MA
forecast
Weighted Moving Average
Patient arrivals
430 —
Assigned weights
t-1
t-2
t-3
410 —
390 —
F4 =
370 —
0
0.70
0.20
0.10
Actual patient
arrivals
|
5
|
10
|
15
Week
|
20
|
25
|
30
Time Series Methods
Weighted Moving Average
450 —
3-week MA
forecast
Weighted Moving Average
Patient arrivals
430 —
Assigned weights
t-1
t-2
t-3
410 —
390 —
F5 =
370 —
0
0.70
0.20
0.10
Actual patient
arrivals
|
5
|
10
|
15
Week
|
20
|
25
|
30
Time Series Methods
Exponential Smoothing
Patient arrivals
450 —
430 —
Exponential Smoothing
 = 0.10
410 —
Ft =  At-1 + (1 - )Ft - 1
390 —
370 —
0
|
5
|
10
|
15
Week
|
20
|
25
|
30
Time Series Methods
Exponential Smoothing
Patient arrivals
450 —
430 —
Exponential Smoothing
 = 0.10
410 —
Ft =  At-1 + (1 - )Ft - 1
390 —
F3 = (400 + 380)/2=390
A3 = 411
370 —
0
|
5
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10
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15
Week
|
20
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25
|
30
Time Series Methods
Exponential Smoothing
Patient arrivals
450 —
430 —
Exponential Smoothing
 = 0.10
410 —
Ft =  At-1 + (1 - )Ft - 1
390 —
F3 = (400 + 380)/2=390
A3 = 411
370 —
F4 =
0
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5
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10
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15
Week
|
20
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25
|
30
Time Series Methods
Exponential Smoothing
Patient arrivals
450 —
430 —
Exponential Smoothing
 = 0.10
410 —
Ft =  At + (1 - )Ft - 1
F4 =
A4 = 415
390 —
370 —
F5 =
0
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5
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10
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15
Week
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20
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25
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30
Time Series Methods
Exponential Smoothing
450 —
Patient arrivals
430 —
410 —
390 —
370 —
0
|
5
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10
|
15
Week
|
20
|
25
|
30
Comparison of Exponential
Smoothing and Simple Moving
Average
• Both Methods
–
–
–
–
Are designed for stationary demand
Require a single parameter
Lag behind a trend, if one exists
Have the same distribution of forecast error if
  2 /( N  1)
Comparison of Exponential
Smoothing and Simple Moving
Average
• Moving average uses only the last N periods
data, exponential smoothing uses all data
• Exponential smoothing uses less memory and
requires fewer steps of computation; store only
the most recent forecast!
Problem 13-20: Your manager is trying to determine what
forecasting method to use. Based upon the following
historical data, calculate the following forecast and specify
what procedure you would utilize:
Month
1 2 3 4 5 6 7 8 9 10 11 12
Actual demand
62 65 67 68 71 73 76 78 78 80 84 85
a. Calculate the three-month SMA forecast for periods 4-12
b. Calculate the weighted three-month MA using weights of
0.50, 0.30, and 0.20 for periods 4-12.
c. Calculate the single exponential smoothing forecast for
periods 2-12 using an initial forecast, F1=61 and =0.30
d. Calculate the exponential smoothing with trend component
forecast for periods 2-12 using T1=1.8,F1=60,=0.30,=0.30
e. Calculate MAD for the forecasts made by each technique in
periods 4-12. Which forecasting method do you prefer?
Trend-Based Methods
Turkeys have a long-term trend for increasing demand with a
seasonal pattern. Sales are highest during September to November
and sales are lowest during December and January.
Linear Regression
Dependent variable
Y
X
Independent variable
Linear Regression
Regression
equation:
Y = a + bX
Dependent variable
Y
X
Independent variable
Linear Regression
Regression
equation:
Y = a + bX
Dependent variable
Y
Actual
value
of Y
Value of X used
to estimate Y
X
Independent variable
Linear Regression
Dependent variable
Y
Regression
equation:
Y = a + bX
Estimate of
Y from
regression
equation
Actual
value
of Y
Value of X used
to estimate Y
X
Independent variable
Linear Regression
Dependent variable
Y
Deviation,
Estimate of or error
Y from
regression
equation
{
Regression
equation:
Y = a + bX
Actual
value
of Y
Value of X used
to estimate Y
X
Independent variable
Linear Regression
Month
Sales
(000 units)
Advertising
(000 $)
1
2
3
4
5
264
116
165
101
209
2.5
1.3
1.4
1.0
2.0
Linear Regression
Month
Sales, y Advertising, x
(000 units)
(000 $)
1
2
3
4
5
a = y - bx
264
116
165
101
209
2.5
1.3
1.4
1.0
2.0
b=
xy - nxy
x2 - n(x )2
Linear Regression
Month
Sales, y Advertising, x
(000 units)
(000 $)
xy
1
2
3
4
5
264
116
165
101
209
2.5
1.3
1.4
1.0
2.0
Total
y=
a = y - bx
x=
b=
xy - nxy
x 2 - nx 2
x2
300 —
Sales (000s)
250 —
200 —
150 —
100 —
50
|
1.0
Y=
|
1.5
|
2.0
b = 109.229
|
2.5
Linear Regression
Month
Sales, y Advertising, x
(000 units)
(000 $)
xy
x2
1
2
3
4
5
264
116
165
101
209
2.5
1.3
1.4
1.0
2.0
660.0
150.8
231.0
101.0
418.0
6.25
1.69
1.96
1.00
4.00
Total
855
y = 171
8.2
x = 1.64
1560.8
14.90
r=
nxy - x y
[nx 2 -(x) 2][ny 2 - (y) 2]
y2
Linear Regression
Month
Sales, y Advertising, x
(000 units)
(000 $)
xy
x2
y2
1
2
3
4
5
264
116
165
101
209
2.5
1.3
1.4
1.0
2.0
660.0
150.8
231.0
101.0
418.0
6.25
1.69
1.96
1.00
4.00
69,696
13,456
27,225
10,201
43,681
Total
855
y= 171
8.2
x = 1.64
1560.8
14.90
164,259
r = 0.98
r 2 = 0.96
Linear Regression
Forecast for Month 6:
Advertising expenditure = $1750
Y =
Time Series Methods
Linear Regression Analysis
Patient arrivals
80 —
Yn = a + bXn
70 —
where
60 —
Xn = Weekn
50 —
40 —
30 —
0
|
1
|
2
|
3
|
4
|
5
|
6
|
|
7 8
Week
|
9
|
|
|
|
10 11 12 13
|
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14 15
Time Series Methods
Linear Regression Analysis
Patient arrivals
80 —
Yn = a + bXn
70 —
where
60 —
Xn = Weekn
50 —
40 —
30 —
0
|
1
|
2
|
3
|
4
|
5
|
6
|
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7 8
Week
|
9
|
|
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10 11 12 13
|
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14 15
Time Series Methods
Linear Regression Analysis
• Standard error of estimate is computed as
follows:
n
S yx 
2
(
y

Y
)
 i i
i 1
n2
Time Series Methods
Linear Regression Analysis
• An use of the standard error of estimate:
– Suppose that a manager forecasts that the demand
for a product is 500 units and Syx is 20. If the
manager wants to accept a stockout only 2% time,
how many additional units should be held in the
inventory?
Time Series Methods
Double Exponential Smoothing
• The method uses two smoothing constants 
and 
Ft  At 1  (1   )FITt 1
Tt   ( Ft  Ft 1 )  (1   )Tt 1
FITt  Ft  Tt
A Comparison of Methods
Demand
90
85
Actual
80
3-Mo MA
3-Mo WMA
75
Exp Sm
70
Double Exp Sm
65
60
0
5
10
Months
15
Methods for Seasonal Series
Time Series Methods
Seasonal Influences
Quarter
1
2
3
4
Total
Average
Year 1
Year 2
Year 3
Year 4
45
335
520
100
70
370
590
170
100
585
830
285
100
725
1160
215
1000
250
1200
300
1800
450
2200
550
Time Series Methods
Seasonal Influences
Quarter
1
2
3
4
Total
Average
Year 1
Year 2
Year 3
Year 4
45
335
520
100
70
370
590
170
100
585
830
285
100
725
1160
215
1000
250
1200
300
1800
450
2200
550
Actual Demand
Seasonal Index =
Average Demand
Time Series Methods
Seasonal Influences
Quarter
1
2
3
4
Total
Average
Year 1
Year 2
Year 3
Year 4
45
335
520
100
70
370
590
170
100
585
830
285
100
725
1160
215
1000
250
1200
300
1800
450
2200
550
Seasonal Index =
=
Time Series Methods
Seasonal Influences
Quarter
1
2
3
4
Total
Average
Year 1
Year 2
Year 3
Year 4
45/250 =
335
520
100
70
370
590
170
100
585
830
285
100
725
1160
215
1000
250
1200
300
1800
450
2200
550
Seasonal Index =
=
Time Series Methods
Seasonal Influences
Quarter
1
2
3
4
Year 1
Year 2
45/250 = 0.18
335/250 = 1.34
520/250 = 2.08
100/250 = 0.40
70/300 = 0.23
370/300 = 1.23
590/300 = 1.97
170/300 = 0.57
Year 3
Year 4
100/450 = 0.22 100/550 = 0.18
585/450 = 1.30 725/550 = 1.32
830/450 = 1.84 1160/550 = 2.11
285/450 = 0.63 215/550 = 0.39
Time Series Methods
Seasonal Influences
Quarter
1
2
3
4
Year 1
Year 2
45/250 = 0.18
335/250 = 1.34
520/250 = 2.08
100/250 = 0.40
70/300 = 0.23
370/300 = 1.23
590/300 = 1.97
170/300 = 0.57
Year 3
Year 4
100/450 = 0.22 100/550 = 0.18
585/450 = 1.30 725/550 = 1.32
830/450 = 1.84 1160/550 = 2.11
285/450 = 0.63 215/550 = 0.39
Quarter
Average Seasonal Index
1
2
3
4
(0.18 + 0.23 + 0.22 + 0.18)/4 = 0.20
Time Series Methods
Seasonal Influences
Projected Annual Demand = 2600
Average Quarterly Demand = 2600/4 = 650
Quarter
Average Seasonal Index
1
2
3
4
(0.18 + 0.23 + 0.22 + 0.18)/4 = 0.20
Forecast
Seasonal Influences
Demand
(a) Multiplicative influence
|
0
| | | | | | | | | | | | | | |
2
4
5
8
10 12 14 16
Period
Seasonal Influences
Demand
(b) Additive influence
|
0
| | | | | | | | | | | | | | |
2
4
5
8
10 12 14 16
Period
Time Series Methods
Seasonal Influences with Trend
Step 1: Determine seasonal factors
– Example: if the demands are quarterly, divide the average demand in
Quarter 1 by the average quarterly demand
Step 2: Deseasonalize the original data
– Divide the original data by the seasonal factors
Step 3: Develop a regression line on deaseasonalized data
–
–
–
–
–
Find parameters a and b in Y=a+bX
Where
yi = deseasonalized data (not the original data)
xi = time; 1, 2, 3, …, n
n = Number of periods
Time Series Methods
Seasonal Influences with Trend
Step 4: Make projection using regression line
– For each i = n+1, n+2, …, compute yi by substituting a, b and xi in
the regression equation yi = a+bxi
Step 5: Reseasonalize projection using seasonal factors
– Multiply the projected values by the seasonal factors
Problem 13-21: Use regression analysis on deseasonalized
demand to forecast demand in summer 2006, given the
following historical demand data:
Year
Season
Actual Demand
2004
Spring
205
Summer
140
Fall
375
Winter
575
2005
Spring
475
Summer
275
Fall
685
Winter
965
Reading and Exercises
• Chapter 13 pp. 518-539
• Problems 1, 7, 13, 14,16