Transcript 15.2 Single - Factor (One - Way) Analysis of Variance

Forecasting
To Accompany Russell and Taylor, Operations Management, 4th Edition,  2003 Prentice-Hall, Inc. All rights reserved.
Forecasting
 Predicting future events
 Usually demand behavior
over a time frame
Forecast:
• A statement about the future value of a
variable of interest such as demand.
• Forecasts affect decisions and activities
throughout an organization
– Accounting, finance
– Human resources
– Marketing
– MIS
– Operations
– Product / service design
Uses of Forecasts
Accounting
Cost/profit estimates
Finance
Cash flow and funding
Human Resources
Hiring/recruiting/training
Marketing
Pricing, promotion, strategy
MIS
IT/IS systems, services
Operations
Schedules, MRP, workloads
Product/service design New products and services
Uses of Forecasts
To help managers plan the system
To help managers plan the use of the
system.
Features Common to All Forecasts
• Assumes causal system
past ==> future
• Forecasts rarely perfect because of
randomness
• Forecasts more accurate for
groups vs. individuals
• Forecast accuracy decreases
as time horizon increases
I see that you will
get an A this semester.
Elements of a Good Forecast
Timely
Reliable
Accurate
Written
Time Frame in Forecasting
 Short-range to medium-range
 Daily, weekly monthly forecasts of
sales data
 Up to 2 years into the future
 Long-range
 Strategic planning of goals,
products, markets
 Planning beyond 2 years into the
future
Steps in the Forecasting Process
“The forecast”
Step 6 Monitor the forecast
Step 5 Prepare the forecast
Step 4 Gather and analyze data
Step 3 Select a forecasting technique
Step 2 Establish a time horizon
Step 1 Determine purpose of forecast
Forecasting Process
1. Identify the
purpose of forecast
6. Check forecast accuracy
with one or more measures
7.
Is accuracy
of forecast
acceptable?
8a. Forecast over
planning horizon
2. Collect historical
data
3. Plot data and
identify patterns
5. Develop/compute forecast
for period of historical data
4. Select a forecast model
that seems appropriate
for data
8b. Select new forecast
model or adjust parameters
of existing model
9. Adjust forecast based
on additional qualitative
information and insight
10. Monitor results and
measure forecast accuracy
Approaches to Forecasting
• Qualitative methods
– Based on subjective methods
• Quantitative methods
– Based on mathematical formulas
Approaches to Forecasting
• Judgmental (Qualitative)- uses
subjective inputs
• Time series - uses historical data
assuming the future will be like
the past
• Associative models - uses
explanatory variables to predict
the future
Judgmental Forecasts
• Executive opinions
• Sales force opinions
• Consumer surveys
• Outside opinion
•
Delphi method
– Opinions of managers and staff
– Achieves a consensus forecast
Time Series
A time series is a time-ordered
sequence of observations taken at
regular intervals (eg. Hourly, daily,
weekly, monthly, quarterly, annually)
Demand Behavior
 Trend
 gradual, long-term up or down movement
 Cycle
 up & down movement repeating over long
time frame; wavelike variations of more than
one year’s duration
 Seasonal pattern
 periodic oscillation in demand which repeats;
short-term regular variations in data
 Irregular variations caused by unusual
circumstances
 Random movements follow no pattern; caused by
chance
Demand
Demand
Forms of Forecast Movement
Random
movement
Time
(b) Cycle
Demand
Demand
Time
(a) Trend
Time
(c) Seasonal pattern
Time
(d) Trend with seasonal pattern
Forms of Forecast Movement
Irregular
variatio
n
Trend
Cycles
90
89
88
Seasonal variations
Time Series Methods
 Naive forecasts
Forecast = data from past period
Demand?
 Statistical methods using historical
data
 Moving average
 Exponential smoothing
 Linear trend line
 Assume patterns will
repeat
Naive Forecasts
Uh, give me a minute....
We sold 250 wheels last
week.... Now, next week
we should sell....
The forecast for any period equals
the previous period’s actual value.
Naïve Forecasts
•
•
•
•
•
•
•
Simple to use
Virtually no cost
Quick and easy to prepare
Data analysis is nonexistent
Easily understandable
Cannot provide high accuracy
Can be a standard for accuracy
Uses for Naïve Forecasts
• Stable time series data
– F(t) = A(t-1)
• Seasonal variations
– F(t) = A(t-n)
• Data with trends
– F(t) = A(t-1) + (A(t-1) – A(t-2))
Techniques for Averaging
• Moving Average
• Weighted Moving Average
• Exponential Smoothing
Averaging techniques smooth
fluctuations in a time series.
Moving Average
 Average several
periods of data
 Dampen, smooth out
changes
 Use when demand is
stable with no trend
or seasonal pattern
n
At-i

i=1
MAn =
n
where
n = number of periods in
the moving average
At-i= actual demand in
period t-i
Moving Averages
• Moving average – A technique that
averages a number of recent actual values,
updated as new values become available.
Ft = MAn=
At-n + … At-2 + At-1
n
Ft = Forecast for time period t
MAn= n period moving average
Simple Moving Average
MONTH
Jan
Feb
Mar
Apr
May
June
July
Aug
Sept
Oct
ORDERS
PER MONTH
120
90
100
75
110
50
75
130
110
90
F11 =MA3
90 + 110 + 130
=
3
= 110 orders for Nov
Simple Moving Average
MONTH
Jan
Feb
Mar
Apr
May
June
July
Aug
Sept
Oct
Nov
ORDERS
PER MONTH
120
90
100
75
110
50
75
130
110
90
–
THREE-MONTH
MOVING AVERAGE
–
–
–
103.3
88.3
95.0
78.3
78.3
85.0
105.0
110.0
Simple Moving Average
MONTH
Jan
Feb
Mar
Apr
May
June
July
Aug
Sept
Oct
Nov
ORDERS
PER MONTH
120
90
100
75
110
50
75
130
110
90
–
THREE-MONTH
MOVING AVERAGE
–
–
–
103.3
88.3
95.0
78.3
78.3
85.0
105.0
110.0
F11 MA5 =
90 + 110 + 130 + 75 + 50
5
= 91 orders for Nov
Simple Moving Average
MONTH
Jan
Feb
Mar
Apr
May
June
July
Aug
Sept
Oct
Nov
ORDERS
PER MONTH
120
90
100
75
110
50
75
130
110
90
–
THREE-MONTH
MOVING AVERAGE
–
–
–
103.3
88.3
95.0
78.3
78.3
85.0
105.0
110.0
FIVE-MONTH
MOVING AVERAGE
–
–
–
–
–
99.0
85.0
82.0
88.0
95.0
91.0
Smoothing Effects
150 –
125 –
Orders
100 –
75 –
50 –
Actual
25 –
0–
|
Jan
|
Feb
|
Mar
|
|
Apr May
|
|
June July
Month
|
|
Aug Sept
|
Oct
|
Nov
Smoothing Effects
150 –
125 –
Orders
100 –
75 –
50 –
3-month
Actual
25 –
0–
|
Jan
|
Feb
|
Mar
|
|
Apr May
|
|
June July
Month
|
|
Aug Sept
|
Oct
|
Nov
Smoothing Effects
150 –
5-month
125 –
Orders
100 –
75 –
50 –
3-month
Actual
25 –
0–
|
Jan
|
Feb
|
Mar
|
|
Apr May
|
|
June July
Month
|
|
Aug Sept
|
Oct
|
Nov
Weighted Moving Average
n
WMAn =  Wi At-i
 Adjusts
i=1
moving
where
average
Wi = the weight for period i,
method to
between 0 and 100
more closely
percent
reflect data
fluctuations
 W = 1.00
i
Weighted Moving Averages
• Weighted moving average – More recent
values in a series are given more weight in
computing the forecast.
Ft = WMAn=
wnAt-n + … wn-1At-2 + w1At-1
n
Weighted Moving Average
Example
MONTH
August
September
October
WEIGHT
DATA
17%
33%
50%
130
110
90
3
November forecast WMA3 =
Wi Ai

i=1
= (0.50)(90) + (0.33)(110) + (0.17)(130)
= 103.4 orders
Exponential Smoothing
Ft = Ft-1 + (At-1 - Ft-1)
• Premise--The most recent observations
might have the highest predictive
value.
– Therefore,
we should give more weight to
the more recent time periods when
forecasting.
Exponential Smoothing
Ft = Ft-1 + (At-1 - Ft-1)
• Weighted averaging method based on
previous forecast plus a percentage of
the forecast error
• A-F is the error term,  is the % feedback
or a percentage of forecast error
Ft =forecast for the next period
At -1 =actual demand for the present period
Ft -1 =previously determined forecast for the present period
α = weighting factor, smoothing constant
Exponential Smoothing
Ft = (1- α) Ft-1 + α At-1
Exponential Smoothing
 Averaging method
 Weights most recent data more strongly
 Reacts more to recent changes
 Widely used, accurate method
Effect of Smoothing Constant
0.0  1.0
If = 0.20, then Ft +1 = 0.20 At + 0.80 Ft
If = 0, then Ft +1 = 0 At + 1 Ft 0 = Ft
Forecast does not reflect recent data
If = 1, then Ft +1 = 1 At + 0 Ft = At
Forecast based only on most recent data
Exponential SmoothingExample 1
PERIOD
MONTH
DEMAND
1
2
3
4
5
6
7
8
9
10
11
12
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
37
40
41
37
45
50
43
47
56
52
55
54
Exponential Smoothing
PERIOD
MONTH
DEMAND
1
2
3
4
5
6
7
8
9
10
11
12
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
37
40
41
37
45
50
43
47
56
52
55
54
F2 = A1 + (1 - )F1
= (0.30)(37) + (0.70)(37)
= 37
F3 = A2 + (1 - )F2
= (0.30)(40) + (0.70)(37)
= 37.9
F13 = A12 + (1 - )F12
= (0.30)(54) + (0.70)(50.84)
= 51.79
Exponential Smoothing
PERIOD
MONTH
DEMAND
1
2
3
4
5
6
7
8
9
10
11
12
13
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Jan
37
40
41
37
45
50
43
47
56
52
55
54
–
FORECAST, Ft + 1
( = 0.3)
–
37.00
37.90
38.83
38.28
40.29
43.20
43.14
44.30
47.81
49.06
50.84
51.79
Exponential Smoothing
PERIOD
MONTH
DEMAND
1
2
3
4
5
6
7
8
9
10
11
12
13
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Jan
37
40
41
37
45
50
43
47
56
52
55
54
–
FORECAST, Ft + 1
( = 0.3)
( = 0.5)
–
37.00
37.90
38.83
38.28
40.29
43.20
43.14
44.30
47.81
49.06
50.84
51.79
–
37.00
38.50
39.75
38.37
41.68
45.84
44.42
45.71
50.85
51.42
53.21
53.61
Exponential Smoothing
Forecasts
70 –
60 –
Actual
Orders
50 –
40 –
30 –
20 –
10 –
0–
|
1
|
2
|
3
|
4
|
5
|
6
Month
|
7
|
8
|
9
|
10
|
11
|
12
|
13
Exponential Smoothing
Forecasts
70 –
60 –
Actual
Orders
50 –
40 –
 = 0.30
30 –
20 –
10 –
0–
|
1
|
2
|
3
|
4
|
5
|
6
Month
|
7
|
8
|
9
|
10
|
11
|
12
|
13
Exponential Smoothing
Forecasts
70 –
60 –
 = 0.50
Actual
Orders
50 –
40 –
 = 0.30
30 –
20 –
10 –
0–
|
1
|
2
|
3
|
4
|
5
|
6
Month
|
7
|
8
|
9
|
10
|
11
|
12
|
13
Exponential Smoothing-Example 2
Period
Actual
1
2
3
4
5
6
7
8
9
10
11
12
Alpha = 0.1 Error
42
40
43
40
41
39
46
44
45
38
40
42
41.8
41.92
41.73
41.66
41.39
41.85
42.07
42.36
41.92
41.73
Alpha = 0.4 Error
-2.00
1.20
-1.92
-0.73
-2.66
4.61
2.15
2.93
-4.36
-1.92
42
41.2
41.92
41.15
41.09
40.25
42.55
43.13
43.88
41.53
40.92
-2
1.8
-1.92
-0.15
-2.09
5.75
1.45
1.87
-5.88
-1.53
Picking a Smoothing Constant
Actual
Demand
50
.4
45
 .1
40
35
1
2
3
4
5
6
7
Period
8
9 10 11 12
Linear Trend Line
y
= a + bx
where
a
b
x
y
=
=
=
=
intercept (at period 0)
slope of the line
the time period
forecast for demand for period x
Linear Trend Line
y = axy
+ bx
- nxy
b = x2 - nx2
where
a
b
x
y
= intercept
0)
a = (at
y - period
bx
= slope of the line
= where
the time period
n =for
number
of periods
= forecast
demand
for period x
x
x =
= mean of the x values
n
y
y = n = mean of the y values
Calculating a and b
n  (ty) -  t  y
b =
2
2
n t - (  t)
 y - b t
a =
n
Linear Trend Calculation Example
x(PERIOD)
y(DEMAND)
1
2
3
4
5
6
7
8
9
10
11
12
73
40
41
37
45
50
43
47
56
52
55
54
78
557
Linear Trend Calculation Example
x(PERIOD)
y(DEMAND)
xy
x2
1
2
3
4
5
6
7
8
9
10
11
12
73
40
41
37
45
50
43
47
56
52
55
54
37
80
123
148
225
300
301
376
504
520
605
648
1
4
9
16
25
36
49
64
81
100
121
144
78
557
3867
650
78
Linear Trend Calculation
Example
x =
= 6.5
x(PERIOD)
y(DEMAND)
1
2
3
4
5
6
7
8
9
10
11
12
73
40
41
37
45
50
43
47
56
52
55
54
78
557
12
557 x2
xyy =
= 46.42
12
37
xy 1- nxy
80
4
b =
x2 9- nx2
123
148
225
300
301
376
a
504
520
605
648
3867
386716- (12)(6.5)(46.42)
25
=
650 - 12(6.5)2
36
=
=
=
=
49
1.7264
y - bx81
100- (1.72)(6.5)
46.42
121
35.2
144
650
78
Linear Trend Calculation
Example
x =
= 6.5
x(PERIOD)
1
2
3
4
5
6
7
8
9
10
11
12
78
Linear trend line12
557 x2
y(DEMAND)
xy
y
=
= 46.42
y = 35.2
+
1.72x
12
73
37
xy 1- nxy
40
80
4
b =
x2 9- nx2
41
123
37
45
50
43
47
56
52
55
54
148
225
300
301
376
a
504
520
605
648
557
3867
386716- (12)(6.5)(46.42)
25
=
650 - 12(6.5)2
36
=
=
=
=
49
1.7264
y - bx81
100- (1.72)(6.5)
46.42
121
35.2
144
650
78
Least Squares
x =Example
= 6.5
12
x(PERIOD)
1
2
3
4
5
6
7
8
9
10
11
12
78
Linear trend line
557 x2
y(DEMAND)
xy
y
=
= 46.42
y = 35.2
+
1.72x
12
73
37
1
xy
- nxy
Forecast
for
period
13
40
80
4
b =
x2 9- nx2
41
123
y148
= 35.2 + 1.72(13)
37
386716- (12)(6.5)(46.42)
45
225
y = =57.56 25
units
650 - 12(6.5)2
50
300
36
43
47
56
52
55
54
301
376
a
504
520
605
648
557
3867
=
=
=
=
49
1.7264
y - bx81
100- (1.72)(6.5)
46.42
121
35.2
144
650
Linear Trend Line
70 –
60 –
Demand
50 –
40 –
30 –
20 –
10 –
0–
|
1
|
2
|
3
|
4
|
5
|
|
6
7
Period
|
8
|
9
|
10
|
11
|
12
|
13
Linear Trend Line
70 –
60 –
Actual
Demand
50 –
40 –
30 –
20 –
10 –
0–
|
1
|
2
|
3
|
4
|
5
|
|
6
7
Period
|
8
|
9
|
10
|
11
|
12
|
13
Linear Trend Line
70 –
60 –
Actual
Demand
50 –
40 –
Linear trend line
30 –
20 –
10 –
0–
|
1
|
2
|
3
|
4
|
5
|
|
6
7
Period
|
8
|
9
|
10
|
11
|
12
|
13
Trend-Adjusted Exponential
Smoothing
A variation of simple exponential
smoothing can be used when a time series
exhibits trend and it is called trend-adjusted
exponential smoothing or double smoothing
If a series exhibits trend, and simple
smoothing is used on it the forecasts will all
lag the trend: if the data are increasing, each
forecast will be too low; if the data are
decreasing, each forecast will be too high.
Trend-Adjusted Exponential
Smoothing
TAFt+1 = St + Tt
Where
St = Smoothed forecast
Tt = Current trend estimate and
St =TAFt + α (At – TAFt )
Tt= Tt-1 + β (TAFt – TAFt-1 – Tt-1 )
α and β are smoothing constants
Seasonal Adjustments
Repetitive increase/ decrease in demand
Models of seasonality:
Additive (seasonality is expressed as a
quantity that is added to or subtracted from
the series average)
Multiplicative (seasonality is expressed as a
percentage of the average (or trend)amount)
Seasonal Adjustments
 The seasonal percentages
in the multiplicative
model are referred to as
seasonal relatives or
seasonal indexes
Seasonal Adjustments
 Use seasonal factor
to adjust forecast
Di
Seasonal factor = Si =
D
Seasonal Adjustment
YEAR
1999
2000
2001
Total
DEMAND (1000’S PER QUARTER)
1
2
3
4
Total
12.6
14.1
15.3
42.0
8.6
10.3
10.6
29.5
6.3
7.5
8.1
21.9
17.5
18.2
19.6
55.3
45.0
50.1
53.6
148.7
Seasonal Adjustment
YEAR
1999
2000
2001
Total
DEMAND (1000’S PER QUARTER)
1
2
3
4
Total
12.6
14.1
15.3
42.0
8.6
10.3
10.6
29.5
6.3
7.5
8.1
21.9
17.5
18.2
19.6
55.3
45.0
50.1
53.6
148.7
D1
42.0
S1 =
=
= 0.28
D 148.7
D3
21.9
S3 =
=
= 0.15
D 148.7
D2
29.5
S2 =
=
= 0.20
D 148.7
D4
55.3
S4 =
=
= 0.37
D 148.7
Seasonal Adjustment
YEAR
DEMAND (1000’S PER QUARTER)
1
2
3
4
Total
1999
2000
2001
Total
12.6
14.1
15.3
42.0
8.6
10.3
10.6
29.5
6.3
7.5
8.1
21.9
17.5
18.2
19.6
55.3
Si
0.28
0.20
0.15
0.37
45.0
50.1
53.6
148.7
Seasonal Adjustment
YEAR
DEMAND (1000’S PER QUARTER)
1
2
3
4
Total
1999
2000
2001
Total
12.6
14.1
15.3
42.0
8.6
10.3
10.6
29.5
6.3
7.5
8.1
21.9
17.5
18.2
19.6
55.3
Si
0.28
0.20
0.15
0.37
45.0
50.1
53.6
148.7
For 2002
y = 40.97 + 4.30x
= 40.97 + 4.30(4)
= 58.17
Seasonal Adjustment
YEAR
DEMAND (1000’S PER QUARTER)
1
2
3
4
Total
1999
2000
2001
Total
12.6
14.1
15.3
42.0
8.6
10.3
10.6
29.5
6.3
7.5
8.1
21.9
17.5
18.2
19.6
55.3
Si
0.28
0.20
0.15
0.37
45.0
50.1
53.6
148.7
For 2002
y = 40.97 + 4.30x
= 40.97 + 4.30(4)
= 58.17
SF1 = (S1) (F5)
= (0.28)(58.17) = 16.28
SF3 = (S3) (F5)
= (0.15)(58.17) = 8.73
SF2 = (S2) (F5)
= (0.20)(58.17) = 11.63
SF4 = (S4) (F5)
= (0.37)(58.17) = 21.53
Centered Moving Average
 A commonly used method for representing the
trend portion of a time series involves a
centered moving average.
 By virtue of its centered position it looks
forward and looks backward, so it is able to
closely follow data movements whether they
involve trends, cycles, or random variability
alone.
Computing Seasonal Relatives
by Using Centered Moving
Averages
The ratio of demand at period i to the
centered average at period i is an
estimate of the seasonal relative at
that point.
Associative Forecasting
Predictor variables - used to predict
values of variable interest
Regression - technique for fitting a line to
a set of points
Least squares line - minimizes sum of
squared deviations around the line
Causal Modeling with Linear
Regression
 Study relationship between two
or more variables
 Dependent variable y depends
on independent variable x
y = a + bx
Linear Model Seems Reasonable
X
7
2
6
4
14
15
16
12
14
20
15
7
Y
15
10
13
15
25
27
24
20
27
44
34
17
Computed
relationship
50
40
30
20
10
0
0
5
10
15
20
25
A straight line is fitted to a set of sample points.
Linear Regression Formulas
a = y-bx
xy - nxy
b =
x2 - nx2
where
a = intercept (at period 0)
b = slope of the line
x
x =
= mean of the x data
n
y
y = n = mean of the y data
Linear Regression Example
x
(WINS)
y
(ATTENDANCE)
xy
x2
4
6
6
8
6
7
5
7
36.3
40.1
41.2
53.0
44.0
45.6
39.0
47.5
145.2
240.6
247.2
424.0
264.0
319.2
195.0
332.5
16
36
36
64
36
49
25
49
49
346.7
2167.7
311
Linear Regression Example
x
(WINS)
4
6
6
8
6
7
5
7
49
49
= 6.125
8y
(ATTENDANCE)
xy
346.9
y=
= 43.36
8
36.3
145.2
40.1 - nxy2
240.6
xy
b = 41.2
247.2
x2 - nx2
53.0
424.0
(2,167.7)
- (8)(6.125)(43.36)
44.0
264.0
=
2
45.6(311) - (8)(6.125)
319.2
39.0
195.0
= 4.06
47.5
332.5
x=
a = y346.7
- bx
2167.7
= 43.36 - (4.06)(6.125)
= 18.46
x2
16
36
36
64
36
49
25
49
311
Linear Regression Example
x
(WINS)
4
6
6
8
6
7
5
7
49
49
= 6.125
8y
(ATTENDANCE)
xy
x2
346.9
Regression
equation
y=
= 43.36
8
36.3
145.2+ 4.06x
16
y = 18.46
40.1 - nxy2
240.6
36
xy
Attendance forecast
for367 wins
b = 41.2
247.2
x2 - nx2
y = 18.46
53.0
424.0+ 4.06(7)
64
(2,167.7)
- (8)(6.125)(43.36)
44.0
264.0 or 46,880
36
= 46.88,
=
2
45.6(311) - (8)(6.125)
319.2
49
39.0
195.0
25
= 4.06
47.5
332.5
49
x=
a = y346.7
- bx
2167.7
= 43.36 - (4.06)(6.125)
= 18.46
311
Linear Regression Line
60,000 –
Attendance, y
50,000 –
40,000 –
30,000 –
20,000 –
10,000 –
|
0
|
1
|
2
|
3
|
4
|
5
Wins, x
|
6
|
7
|
8
|
9
|
10
Linear Regression Line
60,000 –
Attendance, y
50,000 –
40,000 –
30,000 –
Linear regression line,
y = 18.46 + 4.06x
20,000 –
10,000 –
|
0
|
1
|
2
|
3
|
4
|
5
Wins, x
|
6
|
7
|
8
|
9
|
10
Correlation and Coefficient of
Determination
 Correlation, r
 Measure of strength and direction of
relationship between two variables
 Varies between -1.00 and +1.00
 Coefficient of determination, r2
 Percentage of variation in dependent variable
resulting from changes in the independent
variable. Percentage of variability in the values
of the dependent variable that is explained by
the independent variable.
Computing Correlation
r=
n xy -  x y
[n x2 - ( x)2] [n y2 - ( y)2]
(8)(2,167.7) - (49)(346.9)
r=
[(8)(311) - (49)2] [(8)(15,224.7) - (346.9)2]
r = 0.947
Coefficient of determination
r2 = (0.947)2 = 0.897
Multiple Regression
Study the relationship
of demand to two or more
independent variables
y =  0 +  1x 1 +  2x 2 … +  kx k
where
0 = the intercept
1, … , k = parameters for the
independent variables
x1, … , xk = independent variables
Important Points in Using
Regression
 Always plot the data to verify that a
linear relationship is appropriate
Check whether the data is timedependent. If so use time series
instead of regression
A small correlation may imply that
other variables are important
Forecast Accuracy
 Error = Actual - Forecast
 Find a method which minimizes error
 Mean Absolute
Deviation (MAD)
 Mean Squared Error (MSE)
 Mean Absolute
Percent Deviation (MAPE)
Mean Absolute Deviation
(MAD)
 At - Ft 
MAD =
n
where
t = the period number
At = actual demand in period t
Ft = the forecast for period t
n = the total number of periods
 = the absolute value
MAD Example
PERIOD
1
2
3
4
5
6
7
8
9
10
11
12
DEMAND, At
Ft ( =0.3)
37
40
41
37
45
50
43
47
56
52
55
54
37.00
37.00
37.90
38.83
38.28
40.29
43.20
43.14
44.30
47.81
49.06
50.84
557
MAD Example
PERIOD
1
2
3
4
5
6
7
8
9
10
11
12
DEMAND, At
Ft ( =0.3)
(At - Ft)
|At - Ft|
37
40
41
37
45
50
43
47
56
52
55
54
37.00
37.00
37.90
38.83
38.28
40.29
43.20
43.14
44.30
47.81
49.06
50.84
–
3.00
3.10
-1.83
6.72
9.69
-0.20
3.86
11.70
4.19
5.94
3.15
–
3.00
3.10
1.83
6.72
9.69
0.20
3.86
11.70
4.19
5.94
3.15
49.31
53.39
557
MAD Example
PERIOD
1
2
3
4
5
6
7
8
9
10
11
12
DEMAND, Dt
37
40
41
37
MAD
45
50
43
47
56
52
55
54
557
=
=
=
Ft ( =0.3)
(Dt - Ft)
|Dt - Ft|
37.00
–
37.00
3.00
37.90
3.10
 A
F

t
t -1.83
38.83
n
38.28
6.72
40.29
9.69
53.39
43.20
-0.20
11
43.14
3.86
44.30
11.70
47.81
4.19
4.85
49.06
5.94
50.84
3.15
–
3.00
3.10
1.83
6.72
9.69
0.20
3.86
11.70
4.19
5.94
3.15
49.31
53.39
MAD, MSE, and MAPE
MSE
=
 ( Actual
 forecast)
2
n -1
MAPE =
( Actual
 forecas
t
n
/ Actual*100)
Example 10
Period
1
2
3
4
5
6
7
8
MAD=
MSE=
MAPE=
Actual
217
213
216
210
213
219
216
212
2.75
10.86
1.28
Forecast
215
216
215
214
211
214
217
216
(A-F)
2
-3
1
-4
2
5
-1
-4
-2
|A-F|
2
3
1
4
2
5
1
4
22
(A-F)^2
4
9
1
16
4
25
1
16
76
(|A-F|/Actual)*100
0.92
1.41
0.46
1.90
0.94
2.28
0.46
1.89
10.26
Forecast Control
 Reasons for out-of-control forecasts
(sources of forecast errors)
 Change in trend
 Appearance of cycle
 Inadequate forecasts
 Irregular variations
 Incorrect use of forecasting technique
Controlling the Forecast
A forecast is deemed to perform adequately
when the errors exhibit only random
variations
• Control chart
– A visual tool for monitoring forecast errors
– Used to detect non-randomness in errors
• Forecasting errors are in control if
– All errors are within the control limits
– No patterns, such as trends or cycles, are
present
Tracking Signal
 Compute each period
 Compare to control limits
 Forecast is in control if within limits
(At - Ft)
E
Tracking signal =
=
MAD
MAD
Use control limits of +/- 2 to +/- 5 MAD
Bias: persistent tendency for forecasts to be greater or
less than actual values
Tracking Signal Values
PERIOD
DEMAND
Dt
FORECAST,
Ft
1
2
3
4
5
6
7
8
9
10
11
12
37
40
41
37
45
50
43
47
56
52
55
54
37.00
37.00
37.90
38.83
38.28
40.29
43.20
43.14
44.30
47.81
49.06
50.84
ERROR
At - Ft
–
3.00
3.10
-1.83
6.72
9.69
-0.20
3.86
11.70
4.19
5.94
3.15
E =
(At - Ft)
MAD
–
3.00
6.10
4.27
10.99
20.68
20.48
24.34
36.04
40.23
46.17
49.32
–
3.00
3.05
2.64
3.66
4.87
4.09
4.06
5.01
4.92
5.02
4.85
Tracking Signal Values
PERIOD
DEMAND
At
1
2
3
4
5
6
7
8
9
10
11
12
37
40
41
37
45
50
43
47
56
52
55
54
FORECAST,
Ft
ERROR
At - Ft
E =
(At - Ft)
37.00
–
–
37.00
3.00
3.00
37.90
3.10
6.10
38.83
-1.83
4.27
38.28
6.72 for period
10.99 3
Tracking
signal
40.29
9.69
20.68
43.20
-0.20
6.10 20.48
43.14
TS3 = 3.86 =24.34
2.00
3.05
44.30
11.70
36.04
47.81
4.19
40.23
49.06
5.94
46.17
50.84
3.15
49.32
MAD
–
3.00
3.05
2.64
3.66
4.87
4.09
4.06
5.01
4.92
5.02
4.85
Tracking Signal Values
PERIOD
DEMAND
At
FORECAST,
Ft
1
2
3
4
5
6
7
8
9
10
11
12
37
40
41
37
45
50
43
47
56
52
55
54
37.00
37.00
37.90
38.83
38.28
40.29
43.20
43.14
44.30
47.81
49.06
50.84
ERROR
At - Ft
–
3.00
3.10
-1.83
6.72
9.69
-0.20
3.86
11.70
4.19
5.94
3.15
E =
(At - Ft)
MAD
–
3.00
6.10
4.27
10.99
20.68
20.48
24.34
36.04
40.23
46.17
49.32
–
3.00
3.05
2.64
3.66
4.87
4.09
4.06
5.01
4.92
5.02
4.85
TRACKING
SIGNAL
–
1.00
2.00
1.62
3.00
4.25
5.01
6.00
7.19
8.18
9.20
10.17
Tracking Signal Plot
Tracking signal (MAD)
3 –
2 –
1 –
0 –
-1 –
-2 –
-3 –
|
0
|
1
|
2
|
3
|
4
|
5
|
6
Period
|
7
|
8
|
9
|
10
|
11
|
12
Tracking Signal Plot
Tracking signal (MAD)
3 –
2 –
Exponential smoothing ( = 0.30)
1 –
0 –
-1 –
-2 –
-3 –
|
0
|
1
|
2
|
3
|
4
|
5
|
6
Period
|
7
|
8
|
9
|
10
|
11
|
12
Tracking Signal Plot
Tracking signal (MAD)
3 –
2 –
Exponential smoothing ( = 0.30)
1 –
0 –
-1 –
-2 –
Linear trend line
-3 –
|
0
|
1
|
2
|
3
|
4
|
5
|
6
Period
|
7
|
8
|
9
|
10
|
11
|
12
Statistical Control Charts
=
(At - Ft)2
n-1
 Using  we can calculate statistical
control limits for the forecast error
 Control limits are typically set at  3
Statistical Control Charts
18.39 –
12.24 –
Errors
6.12 –
0–
-6.12 –
-12.24 –
-18.39 –
|
0
|
1
|
2
|
3
|
4
|
5
|
6
Period
|
7
|
8
|
9
|
10
|
11
|
12
Statistical Control Charts
18.39 –
UCL = +3
12.24 –
Errors
6.12 –
0–
-6.12 –
-12.24 –
-18.39 –
|
0
LCL = -3
|
1
|
2
|
3
|
4
|
5
|
6
Period
|
7
|
8
|
9
|
10
|
11
|
12
Choosing a Forecasting Technique
• No single technique works in every
situation
• Two most important factors
– Cost
– Accuracy
• Other factors include the availability of:
– Historical data
– Computers
– Time needed to gather and analyze the data
– Forecast horizon