Operations
Management
Forecasting
Chapter 4
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Outline




WHAT IS FORECASTING?
TYPES OF FORECASTS
THE STRATEGIC IMPORTANCE OF FORECASTING
FORECASTING APPROACHES


Overview of Qualitative Methods
Overview of Quantitative Methods







Naïve Approach
Moving Averages
Exponential Smoothing
Exponential Smoothing with Trend Adjustment
Trend Projections
Seasonal Variations in Data
Cyclic Variations in Data
 TIME-SERIES FORECASTING
 ASSOCIATIVE FORECASTING METHODS: REGRESSION AND
CORRELATION ANALYSIS
 MONITORING AND CONTROLLING FORECASTS
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What is Forecasting?
 Process of predicting a
future event
Sales will
be $200
Million!
 Underlying basis of
all business decisions




Production
Inventory
Personnel
Facilities
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Types of Forecasts by Time
Horizon
Short-range forecast
Up to 1 year; usually less than 3 months
 Job scheduling, worker assignments

Medium-range forecast
3 months to 3 years
 Sales & production planning, budgeting

Long-range forecast
 3+ years

New product planning, facility location
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Short-term vs. Longer-term Forecasting
Medium/long range forecasts deal with more
comprehensive issues and support
management decisions regarding planning and
products, plants and processes.
Short-term forecasting usually employs different
methodologies than longer-term forecasting
Short-term forecasts tend to be more accurate
than longer-term forecasts.
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Influence of Product Life Cycle
Introduction, Growth, Maturity, Decline
 Stages of introduction and growth require longer
forecasts than maturity and decline
 Forecasts useful in projecting



staffing levels,
inventory levels, and
factory capacity
as product passes through life cycle stages
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Strategy and Issues During a
Product’s Life
Growth
Maturity
Practical to change
price or quality image
Poor time to change
image, price, or quality
Competitive costs become
critical
Introduction
Company Strategy/Issues
Best period to
increase market
share
R&D product
engineering critical
Strengthen niche
Decline
Cost control
critical
Defend market position
Fax machines
Drive-thru restaurants
CD-ROM
Sales
3 1/2”
Floppy
disks
Station
wagons
Internet
Color copiers
HDTV
OM Strategy/Issues
Product design and
development critical
Frequent product and
process design changes
Short production runs
High production costs
Forecasting critical
Standardization
Product and process
reliability
Less rapid product
changes - more minor
changes
Competitive product
improvements and options
Increase capacity
Limited models
Shift toward product
focused
Attention to quality
Enhance distribution
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Optimum capacity
Increasing stability of
process
Long production runs
Product improvement and
cost cutting
4-7
Little product
differentiation
Cost minimization
Over capacity in the
industry
Prune line to eliminate
items not returning good
margin
Reduce capacity
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Types of Forecasts
Economic forecasts

Address business cycle, e.g., inflation rate, money
supply etc.
Technological forecasts
Predict rate of technological progress
 Predict acceptance of new product

Demand forecasts

Predict sales of existing product
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Seven Steps in Forecasting
Determine the use of the forecast
Select the items to be forecasted
Determine the time horizon of the forecast
Select the forecasting model(s)
Gather the data
Make the forecast
Validate and implement results
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Realities of Forecasting
Forecasts are seldom perfect
Most forecasting methods assume that there is
some underlying stability in the system
Both product family and aggregated product
forecasts are more accurate than individual
product forecasts
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Forecasting Approaches
Qualitative Methods
Quantitative Methods
 Used when situation is
vague & little data exist
 Used when situation is
‘stable’ & historical data
exist
 New products
 New technology
 Existing products
 Current technology
 Involves intuition,
experience
 Involves mathematical
techniques
 e.g., forecasting sales on
Internet
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 e.g., forecasting sales of
color televisions
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Overview of Qualitative Methods
Jury of executive opinion

Pool opinions of high-level executives, sometimes
augment by statistical models
Delphi method

Panel of experts, queried iteratively
Sales force composite

Estimates from individual salespersons are reviewed
for reasonableness, then aggregated
Consumer Market Survey

Ask the customer
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Jury of Executive Opinion
 Involves small group of high-level managers

Group estimates demand by working together
 Combines managerial experience with statistical
models
 Relatively quick
 ‘Group-think’
disadvantage
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© 1995 Corel Corp.
Delphi Method
Iterative group
process
3 types of people
Decision makers
 Staff
 Respondents

Decision Makers
Staff
(What will
(Sales?)
(Sales will be 50!)
sales be?
survey)
Reduces ‘group-think’
Respondents
(Sales will be 45, 50, 55)
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Sales Force Composite
 Each salesperson projects
his or her sales
 Combined at district &
national levels
 Sales reps know
customers’ wants
 Tends to be overly
optimistic
Sales
© 1995 Corel Corp.
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Consumer Market Survey
 Ask customers about
purchasing plans
 What consumers say,
and what they
actually do are often
different
 Sometimes difficult to
answer
How many hours will
you use the Internet
next week?
© 1995 Corel
Corp.
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Overview of Quantitative Approaches
Naïve approach
Moving averages
Exponential smoothing
Trend projection
Time-series
Models
Linear regression
Associative
models
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What is a Time Series?

Set of evenly spaced numerical data


Forecast based only on past values


Obtained by observing response variable at regular time periods
Assumes that factors influencing past and present will continue
influence in future
Example
Year:
Sales:
1998
78.7
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1999
63.5
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2000
89.7
2001
93.2
2002
92.1
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Time Series Components
Trend
Cyclical
Seasonal
Random
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Product Demand Charted over 4
Years with Trend and Seasonality
Demand for product or service
Seasonal peaks
Trend component
Actual
demand line
Random
variation
Year
1
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Year
2
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Average demand
over four years
Year
3
Year
4
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Trend Component
Persistent, overall upward or downward pattern
Due to population, technology etc.
Several years duration
Response
Mo., Qtr., Yr.
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© 1984-1994 T/Maker Co.
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Seasonal Component
Regular pattern of up & down fluctuations
Due to weather, customs etc.
Occurs within 1 year
Summer
Response
© 1984-1994 T/Maker Co.
Mo., Qtr.
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Cyclical Component
Repeating up & down movements
Due to interactions of factors influencing economy
Usually 2-10 years duration
Cycle
Response

Mo., Qtr., Yr.
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Random Component
Erratic, unsystematic, ‘residual’ fluctuations
Due to random variation or unforeseen events
© 1984-1994 T/Maker Co.

Union strike

Tornado
Short duration &
nonrepeating
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General Time Series Models
Any observed value in a time series is the
product (or sum) of time series components
Multiplicative model
 Yi = Ti · Si · Ci · Ri
(if quarterly or mo. data)
Additive model
 Yi = Ti + Si + Ci + Ri
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Naive Approach
 Assumes demand in next
period is the same as demand
in most recent period

e.g., If May sales were 48, then
June sales will be 48
 Sometimes cost effective &
efficient
© 1995 Corel Corp.
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Moving Average Method
 MA is a series of arithmetic means
 Used if little or no trend
 Used often for smoothing

Provides overall impression of data over time
 Equation
Demand in Previous n Periods

MA 
n
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Moving Average Example
You’re manager of a museum store that sells
historical replicas. You want to forecast sales
(000) for 2003 using a 3-period moving average.
1998
4
1999
6
2000
5
2001
3
2002
7
© 1995 Corel Corp.
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Moving Average Solution
Time
1998
1999
2000
2001
2002
2003
Response
Yi
4
6
5
3
7
Moving
Total
(n=3)
NA
NA
NA
4+6+5=15
Moving
Average
(n=3)
NA
NA
NA
15/3 = 5
NA
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Moving Average Solution
Time
1998
1999
2000
2001
2002
2003
Response
Yi
4
6
5
3
7
Moving
Total
(n=3)
NA
NA
NA
4+6+5=15
6+5+3=14
Moving
Average
(n=3)
NA
NA
NA
15/3 = 5
14/3=4 2/3
NA
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Moving Average Solution
Time
1998
1999
2000
2001
2002
2003
Response
Yi
4
6
5
3
7
NA
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Moving
Total
(n=3)
NA
NA
NA
4+6+5=15
6+5+3=14
5+3+7=15
4-31
Moving
Average
(n=3)
NA
NA
NA
15/3=5.0
14/3=4.7
15/3=5.0
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Moving Average Graph
Sales
8
Actual
6
Forecast
4
2
95
96
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97 98
Year
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99
00
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Weighted Moving Average Method
Used when trend is present

Older data usually less important
Weights based on intuition

Often lay between 0 & 1, & sum to 1.0
Equation
WMA =
Σ(Weight for period n) (Demand in period n)
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ΣWeights
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Actual Demand, Moving Average,
Weighted Moving Average
35
Sales Demand
30
25
Weighted moving average
Actual sales
20
15
10
Moving average
5
0
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Month
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Disadvantages of
Moving Average Methods
Increasing n makes forecast less
sensitive to changes
Do not forecast trend well
Require much historical
data
© 1984-1994 T/Maker Co.
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Exponential Smoothing Method
Form of weighted moving average
Weights decline exponentially
 Most recent data weighted most

Requires smoothing constant ()
Ranges from 0 to 1
 Subjectively chosen

Involves little record keeping of past data
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Exponential Smoothing Equations
Ft : Forecast for period t, t=1,2,...,n.
At : Actual value in period t, t=1,2,...,n
 = Smoothing constant

Ft
= Ft-1 + (At-1 - Ft-1) OR

Ft
= At-1 + (1-  )Ft-1

n
  (1   )i 1 At i
i 1
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Exponential Smoothing Example
During the past 8 quarters, the Port of Baltimore has unloaded large
quantities of grain. ( = .10). The first quarter forecast was 175..
Quarter
Actual
1
2
3
4
5
6
7
8
9
180
168
159
175
190
205
180
182
?
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Find the forecast
for the 9th quarter.
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Exponential Smoothing Solution
Ft = Ft-1 + 0.1(At-1 - Ft-1)
Quarter
Actual
1
180
2
168
3
159
4
175
5
190
6
205
Forecast, F t
(α = .10)
175.00 (Given)
175.00 +
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Exponential Smoothing Solution
Ft = Ft-1 + 0.1(At-1 - Ft-1)
Forecast, F t
(α = .10)
Quarter Actual
1
180
2
168
3
159
4
175
5
190
6
205
175.00 (Given)
175.00 + .10(
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Exponential Smoothing Solution
Ft = Ft-1 + 0.1(At-1 - Ft-1)
Quarter
Actual
1
180
2
168
3
159
4
175
5
190
6
205
Forecast, Ft
(α = .10)
175.00 (Given)
175.00 + .10(180 -
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Exponential Smoothing Solution
Ft = Ft-1 + 0.1(At-1 - Ft-1)
Forecast, Ft
(α = .10)
Quarter Actual
1
180
2
168
3
159
4
175
5
190
6
205
175.00 (Given)
175.00 + .10(180 - 175.00)
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Exponential Smoothing Solution
Ft = Ft-1 + 0.1(At-1 - Ft-1)
Forecast, Ft
(α = .10)
Quarter Actual
1
180
2
168
3
159
4
175
5
190
6
205
175.00 (Given)
175.00 + .10(180 - 175.00) = 175.50
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Exponential Smoothing Solution
Ft = Ft-1 + 0.1(At-1 - Ft-1)
Forecast, F t
(α = .10)
Quarter
Actual
1
180
2
168
175.00 + .10(180 - 175.00) = 175.50
3
159
175.50 + .10(168 - 175.50) = 174.75
4
175
5
190
6
205
175.00 (Given)
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Exponential Smoothing Solution
Ft = Ft-1 + 0.1(At-1 - Ft-1)
Forecast, F t
(α = .10)
Quarter Actual
1995
180
175.00 (Given)
1996
168
175.00 + .10(180 - 175.00) = 175.50
1997
159
175.50 + .10(168 - 175.50) = 174.75
1998
175
174.75 + .10(159 - 174.75)= 173.18
1999
190
2000
205
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Exponential Smoothing Solution
Ft = Ft-1 + 0.1(At-1 - Ft-1)
Forecast, F t
(α = .10)
Quarter Actual
1
180
175.00 (Given)
2
168
175.00 + .10(180 - 175.00) = 175.50
3
4
159
175.50 + .10(168 - 175.50) = 174.75
175
174.75 + .10(159 - 174.75) = 173.18
5
190
173.18 + .10(175 - 173.18) = 173.36
6
205
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Exponential Smoothing Solution
Ft = Ft-1 + 0.1(At-1 - Ft-1)
Forecast, F t
(α = .10)
Quarter Actual
1
180
175.00 (Given)
2
168
175.00 + .10(180 - 175.00) = 175.50
3
159
175.50 + .10(168 - 175.50) = 174.75
4
175
174.75 + .10(159 - 174.75) = 173.18
5
190
173.18 + .10(175 - 173.18) = 173.36
6
205
173.36 + .10(190 - 173.36) = 175.02
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Exponential Smoothing Solution
Ft = Ft-1 + 0.1(At-1 - Ft-1)
Actual
Forecast, F t
(α = .10)
4
175
174.75 + .10(159 - 174.75) = 173.18
5
190
173.18 + .10(175 - 173.18) = 173.36
6
7
205
180
173.36 + .10(190 - 173.36) = 175.02
175.02 + .10(205 - 175.02) = 178.02
Time
8
9
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Exponential Smoothing Solution
Ft = Ft-1 + 0.1(At-1 - Ft-1)
Time
Forecast, F t
(α = .10)
Actual
4
175
174.75 + .10(159 - 174.75) = 173.18
5
190
173.18 + .10(175 - 173.18) = 173.36
6
7
205
180
8
9
182
?
173.36 + .10(190 - 173.36) = 175.02
175.02 + .10(205 - 175.02) = 178.02
178.02 + .10(180 - 178.02) = 178.22
178.22 + .10(182 - 178.22) = 178.58
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Forecast Effects of
Smoothing Constant 
Ft =  At - 1 + (1- )At - 2 + (1- )2At - 3 + ...
Weights
=
Prior Period
2 periods ago 3 periods ago

= 0.10
(1 - )
(1 - )2
10%
= 0.90
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Forecast Effects of
Smoothing Constant 
Ft =  At - 1 + (1- ) At - 2 + (1- )2At - 3 + ...
Weights
=
Prior Period
= 0.10
2 periods ago 3 periods ago

(1 - )
10%
9%
(1 - )2
= 0.90
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Forecast Effects of
Smoothing Constant 
Ft =  At - 1 + (1- )At - 2 + (1- )2At - 3 + ...
Weights
=
Prior Period
= 0.10
2 periods ago 3 periods ago

(1 - )
(1 - )2
10%
9%
8.1%
= 0.90
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Forecast Effects of
Smoothing Constant 
Ft =  At - 1 + (1- )At - 2 + (1- )2At - 3 + ...
Weights
=
Prior Period
2 periods ago 3 periods ago

(1 - )
(1 - )2
= 0.10
10%
9%
8.1%
= 0.90
90%
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Forecast Effects of
Smoothing Constant 
Ft =  At - 1 + (1- ) At - 2 + (1- )2At - 3 + ...
Weights
=
Prior Period
2 periods ago 3 periods ago

(1 - )
(1 - )2
= 0.10
10%
9%
8.1%
= 0.90
90%
9%
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Forecast Effects of
Smoothing Constant 
Ft =  At - 1 + (1- ) At - 2 + (1- )2At - 3 + ...
Weights
=
Prior Period
= 0.10
= 0.90
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2 periods ago 3 periods ago

(1 - )
(1 - )2
10%
9%
8.1%
90%
9%
0.9%
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Impact of 
250
Forecast (0.5)
Actual Tonage
200
150
Forecast (0.1)
Actual
100
50
0
1
2
3
4
5
6
7
8
9
Quarter
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Choosing 
Seek to minimize the Mean Absolute Deviation (MAD)
If:
Then:
Forecast error = demand - forecast
MAD 
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 forecast errors
n
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Exponential Smoothing with
Trend Adjustment
“Double Exponential Smoothing”
Ft = exponentially smoothed forecast in period t
Tt = exponentially smoothed trend in period t
At = actual demand in period t
 = smoothing constant for the average
 = smoothing constant for the trend
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Exponential Smoothing with
Trend Adjustment
FITt : Forecast including trend in period t, t=1,2,...,n
FITt = Ft +Tt
where Ft =  At-1 +(1- ) FITt-1
Tt = (Ft - Ft-1) + (1- )Tt-1
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Chapter 4: Example 7
Exponential Smoothing with Trend

0,2
0,4
 
Month
Demand
1
12
2
17
3
20
4
19
5
24
6
21
7
31
8
28
9
36
10
Smoothed Smoothed Fore. inc.
Forecast, Ft
Trend Trend, FIT t
11
2
13
12,80
1,92
14,72
15,18
2,10
17,28
17,82
2,32
20,14
19,91
2,23
22,14
22,51
2,38
24,89
24,11
2,07
26,18
27,14
2,45
29,59
29,28
2,32
31,60
32,48
2,68
35,16
Average=
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Error
-1,00
2,28
2,72
-1,14
1,86
-3,89
4,82
-1,59
4,40
Abs. Error
1,00
2,28
2,72
1,14
1,86
3,89
4,82
1,59
4,40
Squared
1,00
5,20
7,41
1,31
3,45
15,14
23,24
2,54
19,36
0,94
BIAS
2,63
MAD
8,74
MSE
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Comparing Actual and Forecasts
40
35
Actual
Demand
30
Demand
25
20
15
Smoothed
Forecast
Forecast including
trend
10
Smoothed Trend
5
0
1
2
3
4
5
6
7
8
9
10
Month
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Linear Trend Projection
using Regression
 Used for forecasting linear trend line
 Assumes relationship between response
variable, Y, and time, X, is a linear function
Yi  a + bX i
 Estimated by least squares method

Minimizes sum of squared errors
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Least Squares
Values of Dependent Variable
Actual
observation
Deviation
Deviation
Deviation
Deviation
Deviation
Deviation
Deviation
Point on
regression
line
Yˆ  a + bx
Time
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Least Squares Equations
Equation:
Ŷi  a + bx i
n
Slope:
 x i y i  nx y
b  i n
 x i  nx 
i 
Y-Intercept:
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a  y  bx
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Using a Trend Line
Year
1997
1998
1999
2000
2001
2002
2003
The demand for
electrical power at
N.Y.Edison over the
years 1997 – 2003 is
given at the left. Find
the overall trend.
Demand
74
79
80
90
105
142
122
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Finding a Trend Line
Year
1997
1998
1999
2000
2001
2002
2003
Time
Power
x2
xy
Period Demand
1
74
1
74
2
79
4
158
3
80
9
240
4
90
16
360
5
105
25
525
6
142
36
852
7
122
49
854
x=28 y=692 x2=140 xy=3,063
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The Trend Line Equation
x
Σx 28

4
n
7
b
Σxy - nxy 3,063  (7)(4)(98. 86) 295


 10.54
2
2
2
28
Σx  nx
140  (7)(4)
y
Σy 692

 98.86
n
7
a  y - bx  98.86 - 10.54(4)  56.70
Demand in 2004  56.70 + 10.54(8)  141.02 megawatts
Demand in 2005  56.70 + 10.54(9)  151.56 megawatts
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Actual and Trend Forecast
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Seasonality Analysis
Critical in capacity planning when high demand occurs in:
-electric power during winter
-banks during Friday afternoons
-buses subways during rush hours,
-restaurants during lunch and dinner, etc.
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Sales for IBM Labtop Computers 2000-2002
140
120
Demand
100
80
actual
60
40
20
34
31
28
25
22
19
16
13
10
7
4
1
0
Month
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Ex. 9 Monthly Sales of IBM Laptops
Month
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sept
Oct
Nov
Dec
2000
80
70
80
90
113
110
100
88
85
77
75
82
Sales Demand
2001
2002
85
105
85
85
93
82
95
115
125
131
115
120
102
113
102
110
90
95
78
85
72
83
78
80
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Average Demand
2000-2002 Monthly
90
94
80
94
85
94
100
94
123
94
115
94
105
94
100
94
90
94
80
94
80
94
80
94
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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
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Question:
1) Let the forecasted demand in 2003 be 1200 units.
Give forecasts for January 2003 IBM labtop sales?
F37=average monthly sales in 2003*January index
=1200/12 * 0.957 = 96! (so we assume no trend at all)
2) If no info is given on next year’s demand, how would you
give the forecast for January 2003 IBM labtop sales?
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Answer:
Step 1: Apply trend analysis by using average monthly
ˆ i  a + bxi
sales data only. (use 12 data points), y
Step 2:
Forecast for month i in 2003  yˆ12+i * indexi
Alternative approach:
Deseasonalize the time series data by Yt ' Yt / index
where Yt ' is the deseasonalized value of Yt
Apply trend analysis by using all deseasonalized data
points (use 36 data points).
Alternative approach:
Apply exponential smoothing with trend to
deseasonalized data points.
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Demand for IBM Laptops
140
1,40
Seasonal Index
120
1,20
Demand
100
1,00
Trend
80
0,80
Monthly
Average
60
Forecast : trend +
seasonal index
0,60
40
0,40
20
0,20
0
0,00
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Month
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Example 10: San Diego Hospital used 66 data points
to reach to the following trend equation:
yˆ '  8090 + 21.5 x
ŷ '
x
= Patient days (deseasonalized value)
= time in months
Forecast the patient days in the next month?
yˆ '67  8090 + 21.5 x67  8090 + 21.6 * 67  9530
yˆt  yˆ 't *index  9530 *1.04  9911
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San Diego Hospital – Inpatient Days
10200
1,06
Combined
Forecast
10000
9800
1,04
Trend
1,02
9600
1
9400
0,98
Seasonal
Index
9200
0,96
9000
0,94
8800
0,92
Jan
Feb
Mar
Apr
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Jun
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Jul
Aug
Sep
Oct
Nov
Dec
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Multiplicative Seasonal Model
 Find average historical demand for each “season” by summing
the demand for that season in each year, and dividing by the
number of years for which you have data.
 Compute the average demand over all seasons by dividing the
total average annual demand by the number of seasons.
 Compute a seasonal index by dividing that season’s historical
demand (from step 1) by the average demand over all seasons.
 Estimate next year’s total demand
 Divide this estimate of total demand by the number of seasons,
then multiply it by the seasonal index for that season. This
provides the seasonal forecast.
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Regression
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Linear Regression Model
Shows linear relationship between dependent &
explanatory variables

Example: Sales & advertising (not time)
Y-intercept
Slope
^
Yi = a + bX i
Dependent
(response) variable
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Independent (explanatory)
variable
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Linear Regression Equations
Equation:
Ŷi  a + bx i
n
Slope:
b 
 x i y i  nx y
i 1
n
 x i2  nx 2
i 1
Y-Intercept:
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a  y  bx
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Interpretation of Coefficients
Slope (b)

Estimated Y changes by b for each 1 unit increase in X
 If b = 2, then sales (Y) is expected to increase by 2 for each 1
unit increase in advertising (X)
Y-intercept (a)

Average value of Y when X = 0
 If a = 4, then average sales (Y) is expected to be
4 when
advertising (X) is 0
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Least Squares Assumptions
Relationship is assumed to be linear.
Relationship is assumed to hold only within or
slightly outside data range. Do not attempt to
predict time periods far beyond the range of the
data base.
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Random Error Variation
Variation of actual Y from predicted Y
Measured by standard error of estimate, SY,X
Affects several factors
Parameter significance
 Prediction accuracy

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Standard Error of the Estimate
  yi  yˆ i 
n
2
i 1
S y,x 
n2
n


i 1
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yi2
n
n
i 1
i 1
 a  yi  b  xi yi
n2
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Assumptions on Error Terms
•The mean of errors for each x is zero.
•Standard deviation of error terms , SY,X
is the same for each x.
•Errors are independent of each other.
•Errors are normally distributed with mean=0 and
variance= SY,X. for each x.
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Correlation
Answers: ‘how strong is the linear relationship
between the variables?’
Correlation coefficient, r
Values range from -1 to +1
 Measures degree of association

Used mainly for understanding
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Sample Coefficient of Correlation
n
r
cov( X , Y )

sx s y
 ( xi  x )( yi  y ) /( n  2)
i 1
n
[  ( xi  x ) /( n  1)][  ( yi  y ) 2 /( n  1)]
i 1
r
n
2
i 1
n
n
n
i 1
i 1
i 1
n  xi yi   xi  yi
 n 2  n 2   n 2  n 2 
n  xi    xi   n  yi    yi  
 i 1    i 1
 i 1  
 i 1
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Coefficient of Correlation and
Regression Model
Y
r=1
Y
Y^i = a + b X i
r = -1
Y^i = a + b X i
X
Y
X
r = .89
Y^i = a + b X i
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Y
r=0
Y^i = a + b X i
X
4-89
X
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Coefficient of Determination
If we do not use any regression model, total sum of square of
errors, SST
SST   ( yi  y ) 2
If we use a regression model, sum of squares of errors
SSE   ( yi  yˆ )2   ( yi  a  bxi )2
Then sum of squares of errors due to regression
SSR  SST  SSE
We define coef. of determination
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SSR
r 
SST
2
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•Coefficient of determination r2 is the variation in y
that is explained and hence recovered/eliminated
by the regression equation !
•Correlation coeficient r can also be found by using
r  ( sign of b) r 2
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Guidelines for Selecting
Forecasting Model
You want to achieve:

No pattern or direction in forecast error, remember
independency ^of errors assumption.
 Error = (Yi - Yi) = (Actual - Forecast)
 Seen in plots of errors over time

Smallest forecast error
 Mean square error (MSE)
 Mean absolute deviation (MAD)
 Mean absolute percent error
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Pattern of Forecast Error
Residual Analysis: Validation
Trend Not Fully
Accounted for
Desired Pattern
Error
Error
0
0
Time (Years)
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Time (Years)
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Forecast Error Equations
 Mean Square Error (MSE)
 (y  ŷ )  forecast
n
2
MSE 
i 1
i
i
n

errors
2
n
 Mean Absolute Deviation (MAD)
 | y  yˆ |  | forecast
n
MAD 
i
i 1
n
i

errors |
n
 Mean Absolute Percent Error (MAPE)
n
MAPE  100

i 1
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actual i  forecast i
actual i
n
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Selecting Forecasting Model
Example
You’re a marketing analyst for Hasbro Toys. You’ve forecast sales with a linear
model & exponential smoothing. Which model do you use?
Actual
Linear Model
Year
Sales
Forecast
Exponential
Smoothing
Forecast (.9)
1998
1999
2000
2001
2002
1
1
2
2
4
0.6
1.3
2.0
2.7
3.4
1.0
1.0
1.9
2.0
3.8
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Linear Model Evaluation
Year
Yi
Y^ i
1998
1999
2000
2001
2002
1
1
2
2
4
0.6
1.3
2.0
2.7
3.4
Total
Error Error2
|Error|
0.4
-0.3
0.0
-0.7
0.6
0.16
0.09
0.00
0.49
0.36
0.4
0.3
0.0
0.7
0.6
0.0
1.10
2.0
|Error|
Actual
0.40
0.30
0.00
0.35
0.15
1.20
MSE = Σ Error2 / n = 1.10 / 5 = 0.220
MAD = Σ |Error| / n = 2.0 / 5 = 0.400
MAPE = 100 Σ|absolute percent errors|/n= 1.20/5 = 0.240
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Exponential Smoothing Model
Evaluation
Year
Y
1998
1999
2000
2001
2002
1
1
2
2
4
i
Y^
i
1.0
1.0
1.9
2.0
3.8
Total
Error
Error2
|Error|
0.0
0.0
0.1
0.0
0.2
0.00
0.00
0.01
0.00
0.04
0.0
0.0
0.1
0.0
0.2
0.3
0.05
0.3
|Error|
Actual
0.00
0.00
0.05
0.00
0.05
0.10
MSE = Σ Error2 / n = 0.05 / 5 = 0.01
MAD = Σ |Error| / n = 0.3 / 5 = 0.06
MAPE = 100 Σ |Absolute percent errors|/n = 0.10/5 = 0.02
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Comparison of the Models
Linear Model:
MSE = Σ Error2 / n = 1.10 / 5 = .220
MAD = Σ |Error| / n = 2.0 / 5 = .400
MAPE = 100 Σ|absolute percent errors|/n= 1.20/5 = 0.240
Exponential Smoothing Model:
MSE = Σ Error2 / n = 0.05 / 5 = 0.01
MAD = Σ |Error| / n = 0.3 / 5 = 0.06
MAPE = 100 Σ |Absolute percent errors|/n = 0.10/5 = 0.02
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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
Should be within upper and lower control limits
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Tracking Signal Equation
RSFE
TS 
MAD
n
 y i
 ŷ i 
 i 
MAD

 forecast
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error
MAD
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Tracking Signal Computation
Mo Fcst
Act Error RSFE Abs Cum MAD
Error |Error|
1
100
90
2
100
95
3
100 115
4
100 100
5
100 125
6
100 140
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TS
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Tracking Signal Computation
Mo Forc
Act Error RSFE Abs Cum MAD
Error |Error|
1
100
90
2
100
95
3
100 115
4
100 100
5
100 125
6
100 140
TS
-10
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Error = Actual - Forecast
= 90 - 100 = -10
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Tracking Signal Computation
Mo Forc
Act Error RSFE Abs Cum MAD
Error |Error|
1
100
90
2
100
95
3
100 115
4
100 100
5
100 125
6
100 140
-10
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TS
-10
RSFE =  Errors
= NA + (-10) = -10
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Tracking Signal Computation
Mo Forc
Act Error RSFE Abs Cum MAD
Error |Error|
1
100
90
2
100
95
3
100 115
4
100 100
5
100 125
6
100 140
-10
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-10
TS
10
Abs Error = |Error|
= |-10| = 10
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Tracking Signal Computation
Mo Forc
Act Error RSFE Abs Cum MAD
Error |Error|
1
100
90
2
100
95
3
100 115
4
100 100
5
100 125
6
100 140
-10
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-10
10
TS
10
Cum |Error| =  |Errors|
= NA + 10 = 10
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Tracking Signal Computation
Mo Forc
Act Error RSFE Abs Cum MAD
Error |Error|
1
100
90
2
100
95
3
100 115
4
100 100
5
100 125
6
100 140
-10
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Management, 7e
-10
10
TS
10 10.0
MAD =  |Errors|/n
= 10/1 = 10
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Tracking Signal Computation
Mo Forc
Act Error RSFE Abs Cum MAD
Error |Error|
1
100
90
2
100
95
3
100 115
4
100 100
5
100 125
6
100 140
-10
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-10
10
10 10.0
TS
-1
TS = RSFE/MAD
= -10/10 = -1
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Tracking Signal Computation
Mo Forc
Act Error RSFE Abs Cum MAD
Error |Error|
1
100
90
-10
2
100
95
-5
3
100 115
4
100 100
5
100 125
6
100 140
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-10
10
10 10.0
TS
-1
Error = Actual - Forecast
= 95 - 100 = -5
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Tracking Signal Computation
Mo Forc
Act Error RSFE Abs Cum MAD
Error |Error|
1
100
90
-10
-10
2
100
95
-5
-15
3
100 115
4
100 100
5
100 125
6
100 140
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Principles of Operations Management, 5e, and Operations
Management, 7e
10
10 10.0
TS
-1
RSFE =  Errors
= (-10) + (-5) = -15
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Tracking Signal Computation
Mo Forc
Act Error RSFE Abs Cum MAD
Error |Error|
1
100
90
-10
-10
10
2
100
95
-5
-15
5
3
100 115
4
100 100
5
100 125
6
100 140
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Principles of Operations Management, 5e, and Operations
Management, 7e
10 10.0
TS
-1
Abs Error = |Error|
= |-5| = 5
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Tracking Signal Computation
Mo Forc
Act Error RSFE Abs Cum MAD
Error |Error|
1
100
90
-10
-10
10
2
100
95
-5
-15
5
3
100 115
4
100 100
5
100 125
6
100 140
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Principles of Operations Management, 5e, and Operations
Management, 7e
10 10.0
TS
-1
15
Cum Error =  |Errors|
= 10 + 5 = 15
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Tracking Signal Computation
Mo Forc
Act Error RSFE Abs Cum MAD
Error |Error|
1
100
90
-10
-10
10
2
100
95
-5
-15
5
3
100 115
4
100 100
5
100 125
6
100 140
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Management, 7e
10 10.0
15
TS
-1
7.5
MAD =  |Errors|/n
= 15/2 = 7.5
4-112
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Tracking Signal Computation
Mo Forc
Act Error RSFE Abs Cum MAD
Error |Error|
1
100
90
-10
-10
10
2
100
95
-5
-15
5
3
100 115
4
100 100
5
100 125
6
100 140
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TS
10 10.0
-1
15
-2
7.5
TS = RSFE/MAD
= -15/7.5 = -2
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Plot of a Tracking Signal
Signal exceeded limit
+
Upper control limit
0
-
Tracking signal
Acceptable range
Lower control limit
Time
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160
140
120
100
80
60
40
20
0
3
2
Forecast
1
Actual demand
0
Tracking Signal
-1
-2
Tracking Singal
Actual Demand
Tracking Signals
-3
0
1
2
3
4
5
6
7
Time
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Forecasting in the Service Sector
Presents unusual challenges
special need for short term records
 needs differ greatly as function of industry and product
 issues of holidays and calendar
 unusual events

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Forecast of Sales by Hour for Fast
Food Restaurant
20
15
10
5
0
+11-12
+1-2
11-12 12-1 1-2 2-3
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+3-4
+5-6
3-4 4-5 5-6
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+7-8
+9-10
6-7 7-8 8-9 9-10 10-11
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