Transcript Lecture 18.pptx

Lecture 18
Forecasting (Continued)
Books
• Introduction to Materials Management, Sixth Edition, J. R. Tony Arnold, P.E., CFPIM, CIRM, Fleming
College, Emeritus, Stephen N. Chapman, Ph.D., CFPIM, North Carolina State University, Lloyd M.
Clive, P.E., CFPIM, Fleming College
• Operations Management for Competitive Advantage, 11th Edition, by Chase, Jacobs, and Aquilano, 2005,
N.Y.: McGraw-Hill/Irwin.
• Operations Management, 11/E, Jay Heizer, Texas Lutheran University, Barry Render, Graduate School of
Business, Rollins College, Prentice Hall
Objectives
When you complete this chapter you
should be able to :
 Compute three measures of forecast
accuracy
 Develop seasonal indexes
 Conduct a regression and correlation
analysis
 Use a tracking signal
Common Measures of Error
Mean Absolute Deviation (MAD)
∑ |Actual - Forecast|
MAD =
n
Mean Squared Error (MSE)
∑ (Forecast Errors)2
MSE =
n
Common Measures of Error
Mean Absolute Percent Error (MAPE)
n
MAPE =
∑100|Actuali - Forecasti|/Actuali
i=1
n
Comparison of Forecast Error
Quarter
Actual
Tonnage
Unloaded
Rounded
Forecast
with
a = .10
Absolute
Deviation
for
a = .10
1
2
3
4
5
6
7
8
180
168
159
175
190
205
180
182
175
175.5
174.75
173.18
173.36
175.02
178.02
178.22
5.00
7.50
15.75
1.82
16.64
29.98
1.98
3.78
82.45
Rounded
Forecast
with
a = .50
175
177.50
172.75
165.88
170.44
180.22
192.61
186.30
Absolute
Deviation
for
a = .50
5.00
9.50
13.75
9.12
19.56
24.78
12.61
4.30
98.62
Comparison of Forecast Error
MADActual
=
Quarter
∑ |deviations|
Rounded
Absolute
Tonnage
Unloaded
Forecast
n
with
a = .10
Deviation
for
a = .10
For a 180
= .10 175
5.00
168 = 82.45/8
175.5 = 10.31
7.50
1
2
3
4 For
5
6
7
8
159
a 175
= .50
190
205 =
180
182
174.75
173.18
173.36
98.62/8
175.02 =
178.02
178.22
15.75
1.82
16.64
12.33
29.98
1.98
3.78
82.45
Rounded
Forecast
with
a = .50
175
177.50
172.75
165.88
170.44
180.22
192.61
186.30
Absolute
Deviation
for
a = .50
5.00
9.50
13.75
9.12
19.56
24.78
12.61
4.30
98.62
Comparison of Forecast Error
∑ (forecast errors)2
MSE =
Quarter
Actual
Tonnage
Unloaded
Rounded
Forecast
n
with
a = .10
Absolute
Deviation
for
a = .10
For a 180
= .10 175
5.00
168
175.5 = 190.82
7.50
= 1,526.54/8
1
2
3
4 For
5
6
7
8
159
174.75
a 175
= .50 173.18
190
173.36
= 1,561.91/8
205
175.02=
180
178.02
182
178.22
MAD
15.75
1.82
16.64
195.24
29.98
1.98
3.78
82.45
10.31
Rounded
Forecast
with
a = .50
175
177.50
172.75
165.88
170.44
180.22
192.61
186.30
Absolute
Deviation
for
a = .50
5.00
9.50
13.75
9.12
19.56
24.78
12.61
4.30
98.62
12.33
Comparison of Forecast Error
n
∑100|deviationi|/actuali
i=1
MAPE =Actual
Quarter
1
2
3
4
5
6
7
8
Tonnage
Unloaded
Rounded
Forecast
with
a = .10
n
Absolute
Deviation
for
a = .10
For a180= .10 175
5.00
168
175.5
7.50
= 44.75/8
= 5.59%
For
159
a175=
190
205
180
182
.50
=
174.75
173.18
173.36
54.05/8
175.02
178.02
178.22
MAD
MSE
15.75
1.82
16.64
= 29.98
6.76%
1.98
3.78
82.45
10.31
190.82
Rounded
Forecast
with
a = .50
175
177.50
172.75
165.88
170.44
180.22
192.61
186.30
Absolute
Deviation
for
a = .50
5.00
9.50
13.75
9.12
19.56
24.78
12.61
4.30
98.62
12.33
195.24
Comparison of Forecast Error
Quarter
Actual
Tonnage
Unloaded
Rounded
Forecast
with
a = .10
1
2
3
4
5
6
7
8
180
168
159
175
190
205
180
182
175
175.5
174.75
173.18
173.36
175.02
178.02
178.22
MAD
MSE
MAPE
Absolute
Deviation
for
a = .10
5.00
7.50
15.75
1.82
16.64
29.98
1.98
3.78
82.45
10.31
190.82
5.59%
Rounded
Forecast
with
a = .50
175
177.50
172.75
165.88
170.44
180.22
192.61
186.30
Absolute
Deviation
for
a = .50
5.00
9.50
13.75
9.12
19.56
24.78
12.61
4.30
98.62
12.33
195.24
6.76%
Exponential Smoothing with Trend Adjustment
When a trend is present, exponential smoothing
must be modified
Forecast
including (FITt) =
trend
Exponentially
smoothed (Ft) +
forecast
Exponentially
(Tt) smoothed
trend
Exponential Smoothing with Trend Adjustment
Ft = a(At - 1) + (1 - a)(Ft - 1 + Tt - 1)
Tt = b(Ft - Ft - 1) + (1 - b)Tt - 1
Step 1: Compute Ft
Step 2: Compute Tt
Step 3: Calculate the forecast FITt = Ft + Tt
Exponential Smoothing with Trend
Adjustment Example
Month(t)
1
2
3
4
5
6
7
8
9
10
Actual
Demand (At)
12
17
20
19
24
21
31
28
36
Smoothed
Forecast, Ft
11
Smoothed
Trend, Tt
2
Forecast
Including
Trend, FITt
13.00
Exponential Smoothing with Trend
Adjustment Example
Month(t)
1
2
3
4
5
6
7
8
9
10
Actual
Demand (At)
12
17
20
19
24
21
31
28
36
Smoothed
Forecast, Ft
11
Smoothed
Trend, Tt
2
Forecast
Including
Trend, FITt
13.00
Step 1: Forecast for Month 2
F2 = aA1 + (1 - a)(F1 + T1)
F2 = (.2)(12) + (1 - .2)(11 + 2)
= 2.4 + 10.4 = 12.8 units
Exponential Smoothing with Trend
Adjustment Example
Month(t)
1
2
3
4
5
6
7
8
9
10
Actual
Demand (At)
12
17
20
19
24
21
31
28
36
Smoothed
Forecast, Ft
11
12.80
Smoothed
Trend, Tt
2
Forecast
Including
Trend, FITt
13.00
Step 2: Trend for Month 2
T2 = b(F2 - F1) + (1 - b)T1
T2 = (.4)(12.8 - 11) + (1 - .4)(2)
= .72 + 1.2 = 1.92 units
Exponential Smoothing with Trend
Adjustment Example
Month(t)
1
2
3
4
5
6
7
8
9
10
Actual
Demand (At)
12
17
20
19
24
21
31
28
36
Smoothed
Forecast, Ft
11
12.80
Smoothed
Trend, Tt
2
1.92
Forecast
Including
Trend, FITt
13.00
Step 3: Calculate FIT for Month 2
FIT2 = F2 + T1
FIT2 = 12.8 + 1.92
= 14.72 units
Exponential Smoothing with Trend
Adjustment Example
Month(t)
1
2
3
4
5
6
7
8
9
10
Actual
Demand (At)
12
17
20
19
24
21
31
28
36
Smoothed
Forecast, Ft
11
12.80
15.18
17.82
19.91
22.51
24.11
27.14
29.28
32.48
Smoothed
Trend, Tt
2
1.92
2.10
2.32
2.23
2.38
2.07
2.45
2.32
2.68
Forecast
Including
Trend, FITt
13.00
14.72
17.28
20.14
22.14
24.89
26.18
29.59
31.60
35.16
Exponential Smoothing with Trend
Adjustment Example
35 –
Product demand
30 –
Actual demand (At)
25 –
20 –
15 –
Forecast including trend (FITt)
with a = .2 and b = .4
10 –
5 –
0 – |
1
|
2
|
3
|
4
|
5
|
6
Time (month)
|
7
|
8
|
9
Trend Projections
Fitting a trend line to historical data points to
project into the medium to long-range
Linear trends can be found using the least squares
technique
y^ = a + bx
where y^ = computed value of the variable to be
predicted (dependent variable)
a = y-axis intercept
b = slope of the regression line
x = the independent variable
Values of Dependent Variable
Least Squares Method
Actual observation
(y value)
Deviation7
Deviation5
Deviation6
Deviation3
Deviation4
Deviation1
(error)
Deviation2
^ a + bx
Trend line, y =
Time period
Values of Dependent Variable
Least Squares Method
Actual observation
(y value)
Deviation7
Deviation5
Deviation3
Deviation6
Least squares method minimizes
the sum of the squared errors
Deviation
(deviations)
4
Deviation1
Deviation2
^ a + bx
Trend line, y =
Time period
Least Squares Method
Equations to calculate the regression variables
y^ = a + bx
b=
Sxy - nxy
Sx2 - nx2
a = y - bx
Least Squares Example
Year
2001
2002
2003
2004
2005
2005
2007
Time
Period (x)
1
2
3
4
5
6
7
∑x = 28
x=4
Electrical Power
Demand
74
79
80
90
105
142
122
∑y = 692
y = 98.86
∑xy - nxy
b=
=
2
2
∑x - nx
x2
xy
1
4
9
16
25
36
49
∑x2 = 140
74
158
240
360
525
852
854
∑xy = 3,063
3,063 - (7)(4)(98.86)
= 10.54
140 - (7)(42)
a = y - bx = 98.86 - 10.54(4) = 56.70
Least Squares Example
Year
Time
Period (x)
Electrical Power
Demand
1999
1
74
2000
2
79
2001The trend
3 line is
80
2002
4
90
2003
105
y^ =5 56.70 + 10.54x
2004
6
142
2005
7
122
Sx = 28
Sy = 692
x=4
y = 98.86
b=
Sxy - nxy
=
2
2
Sx - nx
x2
xy
1
4
9
16
25
36
49
Sx2 = 140
74
158
240
360
525
852
854
Sxy = 3,063
3,063 - (7)(4)(98.86)
= 10.54
140 - (7)(42)
a = y - bx = 98.86 - 10.54(4) = 56.70
Power demand
Least Squares Example
160
150
140
130
120
110
100
90
80
70
60
50
Trend line,
y^ = 56.70 + 10.54x
–
–
–
–
–
–
–
–
–
–
–
–
|
2001
|
2002
|
2003
|
2004
|
2005
Year
|
2006
|
2007
|
2008
|
2009
Seasonal Variations In Data
The multiplicative
seasonal model can
adjust trend data for
seasonal variations in
demand
Seasonal Variations In Data
Steps in the process:
1. Find average historical demand for each season
2. Compute the average demand over all seasons
3. Compute a seasonal index for each season
4. Estimate next year’s total demand
5. Divide this estimate of total demand by the
number of seasons, then multiply it by the
seasonal index for that season
Seasonal Index Example
Month
Demand
2005 2006 2007
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sept
Oct
Nov
Dec
80
70
80
90
113
110
100
88
85
77
75
82
85
85
93
95
125
115
102
102
90
78
72
78
105
85
82
115
131
120
113
110
95
85
83
80
Average
2005-2007
Average
Monthly
90
80
85
100
123
115
105
100
90
80
80
80
94
94
94
94
94
94
94
94
94
94
94
94
Seasonal
Index
Seasonal Index Example
Month
Demand
2005 2006 2007
Average
2005-2007
Average
Monthly
Jan
80
85 105
90
94
Feb
70
85
85
80
94
Mar
80
93
82
85 monthly demand
94
average
2005-2007
Seasonal index
Apr
90 = 95 115 average 100
monthly demand94
May
113 125 131
123
94
= 90/94 = .957
Jun
110 115 120
115
94
Jul
100 102 113
105
94
Aug
88 102 110
100
94
Sept
85
90
95
90
94
Oct
77
78
85
80
94
Nov
75
72
83
80
94
Dec
82
78
80
80
94
Seasonal
Index
0.957
Seasonal Index Example
Month
Demand
2005 2006 2007
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sept
Oct
Nov
Dec
80
70
80
90
113
110
100
88
85
77
75
82
85
85
93
95
125
115
102
102
90
78
72
78
105
85
82
115
131
120
113
110
95
85
83
80
Average
2005-2007
Average
Monthly
Seasonal
Index
90
80
85
100
123
115
105
100
90
80
80
80
94
94
94
94
94
94
94
94
94
94
94
94
0.957
0.851
0.904
1.064
1.309
1.223
1.117
1.064
0.957
0.851
0.851
0.851
Seasonal Index Example
Month
Demand
2005 2006 2007
Average
2005-2007
Average
Monthly
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sept
Oct
Nov
Dec
80
85 105
90
for 2008
70
85 Forecast
85
80
80
93
82
85
annual demand
90Expected
95 115
100 = 1,200
113 125 131
123
110 115 120 1,200 115
Jan 113
x105
.957 = 96
100 102
12
88 102 110
100
1,200
85
90
95
Feb
x90
.851 = 85
12
77
78
85
80
75
72
83
80
82
78
80
80
94
94
94
94
94
94
94
94
94
94
94
94
Seasonal
Index
0.957
0.851
0.904
1.064
1.309
1.223
1.117
1.064
0.957
0.851
0.851
0.851
Seasonal Index Example
2008 Forecast
2007 Demand
2006 Demand
2005 Demand
140 –
130 –
Demand
120 –
110 –
100 –
90 –
80 –
70 –
|
J
|
F
|
M
|
A
|
M
|
J
|
J
Time
|
A
|
S
|
O
|
N
|
D
San Diego Hospital
Trend Data
10,200 –
Inpatient Days
10,000 –
9,800 –
9573
9,600 – 9530
9,400 –
9551
9659
9616
9594
9637
9745
9702
9680
9724
9766
9,200 –
9,000 –
|
|
|
|
|
|
|
|
|
|
|
|
Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec
67 68 69 70 71 72 73 74 75 76 77 78
Month
San Diego Hospital
Seasonal Indices
Index for Inpatient Days
1.06 –
1.04 –
1.04
1.03
1.02
1.02 –
1.01
1.00 –
1.00
0.99
0.98
0.98 –
0.96 –
0.99
0.97
0.97
0.96
0.94 –
0.92 –
1.04
|
|
|
|
|
|
|
|
|
|
|
|
Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec
67 68 69 70 71 72 73 74 75 76 77 78
Month
San Diego Hospital
Combined Trend and Seasonal Forecast
10,200 –
10068
9949
Inpatient Days
10,000 – 9911
9,800 –
9764
9724
9691
9572
9,600 –
9520 9542
9,400 –
9,200 –
9,000 –
9411
9265
9355
|
|
|
|
|
|
|
|
|
|
|
|
Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec
67 68 69 70 71 72 73 74 75 76 77 78
Month
Associative Forecasting
Used when changes in one or more independent
variables can be used to predict the changes in the
dependent variable
Most common technique is linear
regression analysis
We apply this technique just as we did in the
time series example
Associative Forecasting
Forecasting an outcome based on predictor
variables using the least squares technique
y^ = a + bx
where y^ = computed value of the variable to be
predicted (dependent variable)
a = y-axis intercept
b = slope of the regression line
x = the independent variable though to
predict the value of the dependent
variable
Associative Forecasting Example
Local Payroll
($ billions), x
1
3
4
4.0 –
2
1
3.0 –
7
Sales
Sales
($ millions), y
2.0
3.0
2.5
2.0
2.0
3.5
2.0 –
1.0 –
0
|
1
|
2
|
|
|
3 4
5
Area payroll
|
6
|
7
Associative Forecasting Example
Sales, y
2.0
3.0
2.5
2.0
2.0
3.5
∑y = 15.0
Payroll, x
1
3
4
2
1
7
∑x = 18
x = ∑x/6 = 18/6 = 3
y = ∑y/6 = 15/6 = 2.5
x2
1
9
16
4
1
49
∑x2 = 80
∑xy - nxy
b=
=
∑x2 - nx2
xy
2.0
9.0
10.0
4.0
2.0
24.5
∑xy = 51.5
51.5 - (6)(3)(2.5)
= .25
80 - (6)(32)
a = y - bx = 2.5 - (.25)(3) = 1.75
Associative Forecasting Example
y^ = 1.75 + .25x
4.0 –
3.25
3.0 –
Sales
If payroll next year is
estimated to be $6
billion, then:
Sales = 1.75 + .25(payroll)
Sales = 1.75 + .25(6)
Sales = $3,250,000
2.0 –
1.0 –
0
|
1
|
2
|
|
|
3 4
5
Area payroll
|
6
|
7
Standard Error of the Estimate
 A forecast is just a point estimate of a
future value
4.0 –
3.25
3.0 –
Sales
 This point is
actually the
mean of a
probability
distribution
2.0 –
1.0 –
0
|
1
|
2
|
|
|
3 4
5
Area payroll
|
6
|
7
Standard Error of the Estimate
Sy,x =
where
∑(y - yc)2
n-2
y = y-value of each data point
yc = computed value of the dependent
variable, from the regression
equation
n = number of data points
Standard Error of the Estimate
Computationally, this equation is
considerably easier to use
Sy,x =
∑y2 - a∑y - b∑xy
n-2
We use the standard error to set up
prediction intervals around the point
estimate
Standard Error of the Estimate
Sy,x =
∑y2 - a∑y - b∑xy
=
n-2
39.5 - 1.75(15) - .25(51.5)
6-2
Sy,x = .306
4.0 –
The standard error of
the estimate is
$306,000 in sales
Sales
3.25
3.0 –
2.0 –
1.0 –
0
|
1
|
2
|
|
|
3 4
5
Area payroll
|
6
|
7
Correlation
 How strong is the linear relationship between the
variables?
 Correlation does not necessarily imply causality!
 Coefficient of correlation, r, measures degree of
association
 Values range from -1 to +1
Correlation Coefficient
r=
nSxy - SxSy
[nSx2 - (Sx)2][nSy2 - (Sy)2]
y
y
Correlation Coefficient
nSxy - SxSy
r=
(a) Perfect positive
correlation:
r = +1
2 - (Sx)2][nSy2 - (Sy)2]
[nSx
x
x
(b) Positive
correlation:
0<r<1
y
y
(c) No correlation:
r=0
x
(d) Perfect negative
correlation:
r = -1
x
Correlation
 Coefficient of Determination, r2, measures the
percent of change in y predicted by the change in x
 Values range from 0 to 1
 Easy to interpret
For the Nodel Construction example:
r = .901
r2 = .81
Multiple Regression Analysis
If more than one independent variable is to be used in
the model, linear regression can be extended to
multiple regression to accommodate several
independent variables
y^ = a + b1x1 + b2x2 …
Computationally, this is quite complex and
generally done on the computer
Multiple Regression Analysis
In the Nodel example, including interest rates in the model
gives the new equation:
y^ = 1.80 + .30x1 - 5.0x2
An improved correlation coefficient of r = .96 means this
model does a better job of predicting the change in
construction sales
Sales = 1.80 + .30(6) - 5.0(.12) = 3.00
Sales = $3,000,000
Monitoring and Controlling Forecasts
Tracking Signal
 Measures how well the forecast is predicting
actual values
 Ratio of running sum of forecast errors (RSFE) to
mean absolute deviation (MAD)
 Good tracking signal has low values
 If forecasts are continually high or low, the forecast
has a bias error
Monitoring and Controlling Forecasts
Tracking
signal
Tracking
signal
RSFE
=
MAD
=
∑(Actual demand in
period i Forecast demand
in period i)
(∑|Actual - Forecast|/n)
Tracking Signal
Signal exceeding limit
Tracking signal
+
Upper control limit
Acceptable
range
0 MADs
–
Lower control limit
Time
Tracking Signal Example
Qtr
Actual
Demand
Forecast
Demand
Error
RSFE
Absolute
Forecast
Error
1
2
3
4
5
6
90
95
115
100
125
140
100
100
100
110
110
110
-10
-5
+15
-10
+15
+30
-10
-15
0
-10
+5
+35
10
5
15
10
15
30
Cumulative
Absolute
Forecast
Error
MAD
10
15
30
40
55
85
10.0
7.5
10.0
10.0
11.0
14.2
Tracking Signal Example
Qtr
1
2
3
4
5
6
Tracking
Signal
Actual
Forecast
(RSFE/MAD)
Demand
Demand
Error
RSFE
Absolute
Forecast
Error
90-10/10
100= -1 -10
95-15/7.5
100= -2 -5
= 0 +15
115 0/10
100
100-10/10
110= -1 -10
+5/11110
= +0.5+15
125
+35/14.2
140
110= +2.5
+30
-10
-15
0
-10
+5
+35
10
5
15
10
15
30
Cumulative
Absolute
Forecast
Error
MAD
10
15
30
40
55
85
10.0
7.5
10.0
10.0
11.0
14.2
The variation of the tracking signal between -2.0
and +2.5 is within acceptable limits
Adaptive Forecasting
It’s possible to use the computer to
continually monitor forecast error and adjust
the values of the a and b coefficients used in
exponential smoothing to continually
minimize forecast error
This technique is called adaptive smoothing
Focus Forecasting
Developed at American Hardware Supply, focus
forecasting is based on two principles:
1. Sophisticated forecasting models are not
always better than simple ones
2. There is no single technique that should be
used for all products or services
This approach uses historical data to test multiple
forecasting models for individual items
The forecasting model with the lowest error is then used to
forecast the next demand
Forecasting in the Service Sector
 Presents unusual challenges
 Special need for short term records
 Needs differ greatly as function of industry
and product
 Holidays and other calendar events
 Unusual events
Fast Food Restaurant Forecast
Percentage of sales
20% –
15% –
10% –
5% –
11-12
1-2
12-1
(Lunchtime)
3-4
2-3
5-6
4-5
7-8
6-7
(Dinnertime)
Hour of day
9-10
8-9
10-11
FedEx Call Center Forecast
12% –
10% –
8% –
6% –
4% –
2% –
0% –
2
4
6
8
A.M.
10
12
2
Hour of day
4
6
8
P.M.
10
12
Measuring Forecast Errors
• Mean Absolute Deviation (MAD) The sum of the
absolute value of the individual forecast errors divided
by the number of periods
Bias
• Bias exists when the cumulative actual demand varies
from the cumulative forecast.
Seasonal Index
Seasonal Index =
period average demand
avg. demand for all periods
End of Lecture 18