Operations
Management
Chapter 4 Forecasting
PowerPoint presentation to accompany
Heizer/Render
Principles of Operations Management, 6e
Operations Management, 8e
© 2006
Prentice
Hall, Inc. Hall, Inc.
©
2006
Prentice
4–1
Outline
 Global Company Profile:
Tupperware Corporation
 What Is Forecasting?
 Forecasting Time Horizons
 The Influence of Product Life Cycle
 Types Of Forecasts
© 2006 Prentice Hall, Inc.
4–2
Outline – Continued
 The Strategic Importance Of
Forecasting
 Human Resources
 Capacity
 Supply-Chain Management
 Seven Steps In The Forecasting
System
© 2006 Prentice Hall, Inc.
4–3
Outline – Continued
 Forecasting Approaches
 Overview of Qualitative Methods
 Overview of Quantitative Methods
© 2006 Prentice Hall, Inc.
4–4
Outline – Continued
 Time-series Forecasting





Decomposition of a Time Series
Naïve Approach
Moving Averages
Exponential Smoothing
Exponential Smoothing with Trend
Adjustment
 Trend Projections
 Seasonal Variations in Data
 Cyclical Variations in Data
© 2006 Prentice Hall, Inc.
4–5
Outline – Continued
 Associative Forecasting Methods:
Regression And Correlation
Analysis
 Using Regression Analysis to
Forecast
 Standard Error of the Estimate
 Correlation Coefficients for
Regression Lines
 Multiple-Regression Analysis
© 2006 Prentice Hall, Inc.
4–6
Outline – Continued
 Monitoring And Controlling
Forecasts
 Adaptive Smoothing
 Focus Forecasting
 Forecasting In The Service Sector
© 2006 Prentice Hall, Inc.
4–7
Learning Objectives
When you complete this chapter, you
should be able to :
Identify or Define:
 Forecasting
 Types of forecasts
 Time horizons
 Approaches to forecasts
© 2006 Prentice Hall, Inc.
4–8
Learning Objectives
When you complete this chapter, you
should be able to :
Describe or Explain:
 Moving averages
 Exponential smoothing
 Trend projections
 Regression and correlation analysis
 Measures of forecast accuracy
© 2006 Prentice Hall, Inc.
4–9
Forecasting at Tupperware
 Each of 50 profit centers around the
world is responsible for
computerized monthly, quarterly,
and 12-month sales projections
 These projections are aggregated by
region, then globally, at
Tupperware’s World Headquarters
 Tupperware uses all techniques
discussed in text
© 2006 Prentice Hall, Inc.
4 – 10
Tupperware’s
Process
© 2006 Prentice Hall, Inc.
4 – 11
Three Key Factors for
Tupperware
 The number of registered
“consultants” or sales
representatives
 The percentage of currently “active”
dealers (this number changes each
week and month)
 Sales per active dealer, on a weekly
basis
© 2006 Prentice Hall, Inc.
4 – 12
Forecast by Consensus
 Although inputs come from sales,
marketing, finance, and production,
final forecasts are the consensus of
all participating managers
 The final step is Tupperware’s
version of the “jury of executive
opinion”
© 2006 Prentice Hall, Inc.
4 – 13
What is Forecasting?
 Process of
predicting a future
event
 Underlying basis of
all business
decisions
??
 Production
 Inventory
 Personnel
 Facilities
© 2006 Prentice Hall, Inc.
4 – 14
Forecasting Time Horizons
 Short-range forecast
 Up to 1 year, generally less than 3 months
 Purchasing, job scheduling, workforce
levels, job assignments, production levels
 Medium-range forecast
 3 months to 3 years
 Sales and production planning, budgeting
 Long-range forecast
 3+ years
 New product planning, facility location,
research and development
© 2006 Prentice Hall, Inc.
4 – 15
Distinguishing Differences
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
© 2006 Prentice Hall, Inc.
4 – 16
Influence of Product Life
Cycle
Introduction – Growth – Maturity – Decline
 Introduction and growth require longer
forecasts than maturity and decline
 As product passes through life cycle,
forecasts are useful in projecting
 Staffing levels
 Inventory levels
 Factory capacity
© 2006 Prentice Hall, Inc.
4 – 17
Product Life Cycle
Company Strategy/Issues
Introduction
Growth
Maturity
Best period to
increase market
share
Practical to change
price or quality
image
Poor time to
change image,
price, or quality
R&D engineering is
critical
Strengthen niche
Competitive costs
become critical
Defend market
position
CD-ROM
Internet
Sales
Decline
Cost control
critical
Fax machines
Drive-through
restaurants
Color printers
Flat-screen
monitors
DVD
3 1/2”
Floppy
disks
Figure 2.5
© 2006 Prentice Hall, Inc.
4 – 18
Product Life Cycle
OM Strategy/Issues
Introduction
Product design
and
development
critical
Frequent
product and
process design
changes
Growth
Forecasting
critical
Product and
process
reliability
Maturity
Standardization
Less rapid
product changes
– more minor
changes
Competitive
product
improvements
and options
Optimum
capacity
High production
costs
Shift toward
product focus
Long production
runs
Limited models
Enhance
distribution
Product
improvement
and cost cutting
Short production
runs
Attention to
quality
Increasing
stability of
Increase capacity process
Decline
Little product
differentiation
Cost
minimization
Overcapacity
in the
industry
Prune line to
eliminate
items not
returning
good margin
Reduce
capacity
Figure 2.5
© 2006 Prentice Hall, Inc.
4 – 19
Types of Forecasts
 Economic forecasts
 Address business cycle – inflation rate,
money supply, housing starts, etc.
 Technological forecasts
 Predict rate of technological progress
 Impacts development of new products
 Demand forecasts
 Predict sales of existing product
© 2006 Prentice Hall, Inc.
4 – 20
Strategic Importance of
Forecasting
 Human Resources – Hiring, training,
laying off workers
 Capacity – Capacity shortages can
result in undependable delivery, loss
of customers, loss of market share
 Supply-Chain Management – Good
supplier relations and price advance
© 2006 Prentice Hall, Inc.
4 – 21
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
© 2006 Prentice Hall, Inc.
4 – 22
The Realities!
 Forecasts are seldom perfect
 Most techniques assume an
underlying stability in the system
 Product family and aggregated
forecasts are more accurate than
individual product forecasts
© 2006 Prentice Hall, Inc.
4 – 23
Forecasting Approaches
Qualitative Methods
 Used when situation is vague
and little data exist
 New products
 New technology
 Involves intuition, experience
 e.g., forecasting sales on Internet
© 2006 Prentice Hall, Inc.
4 – 24
Forecasting Approaches
Quantitative Methods
 Used when situation is ‘stable’ and
historical data exist
 Existing products
 Current technology
 Involves mathematical techniques
 e.g., forecasting sales of color
televisions
© 2006 Prentice Hall, Inc.
4 – 25
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
© 2006 Prentice Hall, Inc.
4 – 26
Overview of Qualitative
Methods
 Sales force composite
 Estimates from individual
salespersons are reviewed for
reasonableness, then aggregated
 Consumer Market Survey
 Ask the customer
© 2006 Prentice Hall, Inc.
4 – 27
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
© 2006 Prentice Hall, Inc.
4 – 28
Sales Force Composite
 Each salesperson projects his or
her sales
 Combined at district and national
levels
 Sales reps know customers’ wants
 Tends to be overly optimistic
© 2006 Prentice Hall, Inc.
4 – 29
Delphi Method
 Iterative group
process,
continues until
consensus is
reached
Staff
(Administering
 3 types of
survey)
participants
 Decision makers
 Staff
 Respondents
© 2006 Prentice Hall, Inc.
Decision Makers
(Evaluate
responses and
make decisions)
Respondents
(People who can
make valuable
judgments)
4 – 30
Consumer Market Survey
 Ask customers about purchasing
plans
 What consumers say, and what
they actually do are often different
 Sometimes difficult to answer
© 2006 Prentice Hall, Inc.
4 – 31
Overview of Quantitative
Approaches
1. Naive approach
2. Moving averages
3. Exponential
smoothing
Time-Series
Models
4. Trend projection
5. Linear regression
© 2006 Prentice Hall, Inc.
Associative
Model
4 – 32
Time Series Forecasting
 Set of evenly spaced numerical
data
 Obtained by observing response
variable at regular time periods
 Forecast based only on past
values
 Assumes that factors influencing
past and present will continue
influence in future
© 2006 Prentice Hall, Inc.
4 – 33
Time Series Components
Trend
Cyclical
Seasonal
Random
© 2006 Prentice Hall, Inc.
4 – 34
Demand for product or service
Components of Demand
Trend
component
Seasonal peaks
Actual
demand
Random
variation
|
1
|
2
|
3
Year
© 2006 Prentice Hall, Inc.
Average
demand over
four years
|
4
Figure 4.1
4 – 35
Trend Component
 Persistent, overall upward or
downward pattern
 Changes due to population,
technology, age, culture, etc.
 Typically several years
duration
© 2006 Prentice Hall, Inc.
4 – 36
Seasonal Component
 Regular pattern of up and
down fluctuations
 Due to weather, customs, etc.
 Occurs within a single year
© 2006 Prentice Hall, Inc.
Period
Length
Number of
Seasons
Week
Month
Month
Year
Year
Year
Day
Week
Day
Quarter
Month
Week
7
4-4.5
28-31
4
12
52
4 – 37
Cyclical Component
 Repeating up and down movements
 Affected by business cycle, political,
and economic factors
 Multiple years duration
 Often causal or
associative
relationships
0
© 2006 Prentice Hall, Inc.
5
10
15
20
4 – 38
Random Component
 Erratic, unsystematic, ‘residual’
fluctuations
 Due to random variation or
unforeseen events
 Short duration and
nonrepeating
M
© 2006 Prentice Hall, Inc.
T
W
T
F
4 – 39
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 and
efficient
© 2006 Prentice Hall, Inc.
4 – 40
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
∑ demand in previous n periods
Moving average =
n
© 2006 Prentice Hall, Inc.
4 – 41
Moving Average Example
Month
Actual
Shed Sales
3-Month
Moving Average
January
February
March
April
May
June
July
10
12
13
16
19
23
26
(10 + 12 + 13)/3 = 11 2/3
(12 + 13 + 16)/3 = 13 2/3
(13 + 16 + 19)/3 = 16
(16 + 19 + 23)/3 = 19 1/3
© 2006 Prentice Hall, Inc.
4 – 42
Shed Sales
Graph of Moving Average
30
28
26
24
22
20
18
16
14
12
10
Moving
Average
Forecast
–
–
–
–
–
–
–
–
–
–
–
Actual
Sales
|
J
© 2006 Prentice Hall, Inc.
|
F
|
M
|
A
|
M
|
J
|
J
|
A
|
S
|
O
|
N
|
D
4 – 43
Weighted Moving Average
 Used when trend is present
 Older data usually less important
 Weights based on experience and
intuition
Weighted
moving average =
© 2006 Prentice Hall, Inc.
∑ (weight for period n)
x (demand in period n)
∑ weights
4 – 44
Weights Applied
3
2
1
6
Period
Last month
Two months ago
Three months ago
Sum of weights
Weighted Moving Average
Month
Actual
Shed Sales
January
February
March
April
May
June
July
10
12
13
16
19
23
26
© 2006 Prentice Hall, Inc.
3-Month Weighted
Moving Average
[(3 x 13) + (2 x 12) + (10)]/6 = 121/6
[(3 x 16) + (2 x 13) + (12)]/6 = 141/3
[(3 x 19) + (2 x 16) + (13)]/6 = 17
[(3 x 23) + (2 x 19) + (16)]/6 = 201/2
4 – 45
Potential Problems With
Moving Average
 Increasing n smooths the forecast
but makes it less sensitive to
changes
 Do not forecast trends well
 Require extensive historical data
© 2006 Prentice Hall, Inc.
4 – 46
Moving Average And
Weighted Moving Average
Weighted
moving
average
Sales demand
30 –
25 –
20 –
Actual
sales
15 –
Moving
average
10 –
5 –
|
Figure 4.2
© 2006 Prentice Hall, Inc.
J
|
F
|
M
|
A
|
M
|
J
|
J
|
A
|
S
|
O
|
N
|
D
4 – 47
Exponential Smoothing
 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
© 2006 Prentice Hall, Inc.
4 – 48
Exponential Smoothing
New forecast = last period’s forecast
+  (last period’s actual demand
– last period’s forecast)
Ft = Ft – 1 + (At – 1 - Ft – 1)
where
© 2006 Prentice Hall, Inc.
Ft = new forecast
Ft – 1 = previous forecast
 = smoothing (or weighting)
constant (0    1)
4 – 49
Exponential Smoothing
Example
Predicted demand = 142 Ford Mustangs
Actual demand = 153
Smoothing constant  = .20
© 2006 Prentice Hall, Inc.
4 – 50
Exponential Smoothing
Example
Predicted demand = 142 Ford Mustangs
Actual demand = 153
Smoothing constant  = .20
New forecast = 142 + .2(153 – 142)
© 2006 Prentice Hall, Inc.
4 – 51
Exponential Smoothing
Example
Predicted demand = 142 Ford Mustangs
Actual demand = 153
Smoothing constant  = .20
New forecast = 142 + .2(153 – 142)
= 142 + 2.2
= 144.2 ≈ 144 cars
© 2006 Prentice Hall, Inc.
4 – 52
Effect of
Smoothing Constants
Weight Assigned to
Smoothing
Constant
Most
Recent
Period
()
 = .1
.1
.09
.081
.073
.066
 = .5
.5
.25
.125
.063
.031
© 2006 Prentice Hall, Inc.
2nd Most 3rd Most 4th Most 5th Most
Recent
Recent
Recent
Recent
Period
Period
Period
Period
2
3
(1 - ) (1 - )
(1 - )
(1 - )4
4 – 53
Impact of Different 
Demand
225 –
 = .5
Actual
demand
200 –
175 –
 = .1
150 – |
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
Quarter
© 2006 Prentice Hall, Inc.
4 – 54
Choosing 
The objective is to obtain the most
accurate forecast no matter the
technique
We generally do this by selecting the
model that gives us the lowest forecast
error
Forecast error = Actual demand - Forecast value
= At - Ft
© 2006 Prentice Hall, Inc.
4 – 55
Common Measures of Error
Mean Absolute Deviation (MAD)
MAD =
∑ |actual - forecast|
n
Mean Squared Error (MSE)
MSE =
© 2006 Prentice Hall, Inc.
∑ (forecast errors)2
n
4 – 56
Common Measures of Error
Mean Absolute Percent Error (MAPE)
n
100 ∑ |actuali - forecasti|/actuali
MAPE =
© 2006 Prentice Hall, Inc.
i=1
n
4 – 57
Comparison of Forecast
Error
Quarter
Actual
Tonnage
Unloaded
Rounded
Forecast
with
 = .10
Absolute
Deviation
for
 = .10
Rounded
Forecast
with
 = .50
1
2
3
4
5
6
7
8
180
168
159
175
190
205
180
182
175
176
175
173
173
175
178
178
5
8
16
2
17
30
2
4
84
175
178
173
166
170
180
193
186
© 2006 Prentice Hall, Inc.
Absolute
Deviation
for
 = .50
5
10
14
9
20
25
13
4
100
4 – 58
Comparison of Forecast
Error
∑ |deviations|
Rounded
Absolute
MADActual
=
Quarter
Tonage
Unloaded
Forecast
n
with
 = .10
Deviation
for
 = .10
For 180
= .10 175
168 = 84/8
176
= 10.50
1
2
3
4 For
5
6
7
8
© 2006 Prentice Hall, Inc.
159
175
175
= .50 173
190
173
205 = 100/8
175 =
180
178
182
178
5
8
16
2
17
12.5030
2
4
84
Rounded
Forecast
with
 = .50
175
178
173
166
170
180
193
186
Absolute
Deviation
for
 = .50
5
10
14
9
20
25
13
4
100
4 – 59
Comparison of Forecast
Error2
∑ (forecast errors)
MSE = Actual
Quarter
Tonage
Unloaded
Rounded
Forecast
n
with
 = .10
Absolute
Deviation
for
 = .10
For 180
= .10 175
5
168
176
= 1,558/8
= 194.758
1
2
3
4 For
5
6
7
8
© 2006 Prentice Hall, Inc.
159
175
175
= .50 173
190
173
= 1,612/8175=
205
180
178
182
178
16
2
17
201.50
30
2
4
84
MAD
10.50
Rounded
Forecast
with
 = .50
175
178
173
166
170
180
193
186
Absolute
Deviation
for
 = .50
5
10
14
9
20
25
13
4
100
12.50
4 – 60
Comparison of Forecast
n
Error
100 ∑ |deviationi|/actuali
i =Rounded
1
Forecast
Tonage
with
Unloaded
 = .10
MAPE =
Actual
Quarter
1
2
3
4
5
6
7
8
n
Absolute
Deviation
for
 = .10
For 180
 = .10 175
5
168
176
8
= 45.62/8
= 5.70%
159
For 175
=
190
205
180
182
175
.50 173
173
= 54.8/8
175
178
178
MAD
MSE
© 2006 Prentice Hall, Inc.
16
2
17
= 6.85%
30
2
4
84
10.50
194.75
Rounded
Forecast
with
 = .50
175
178
173
166
170
180
193
186
Absolute
Deviation
for
 = .50
5
10
14
9
20
25
13
4
100
12.50
201.50
4 – 61
Comparison of Forecast
Error
Quarter
Actual
Tonnage
Unloaded
Rounded
Forecast
with
 = .10
1
2
3
4
5
6
7
8
180
168
159
175
190
205
180
182
175
176
175
173
173
175
178
178
MAD
MSE
MAPE
© 2006 Prentice Hall, Inc.
Absolute
Deviation
for
 = .10
5
8
16
2
17
30
2
4
84
10.50
194.75
5.70%
Rounded
Forecast
with
 = .50
175
178
173
166
170
180
193
186
Absolute
Deviation
for
 = .50
5
10
14
9
20
25
13
4
100
12.50
201.50
6.85%
4 – 62
Exponential Smoothing with
Trend Adjustment
When a trend is present, exponential
smoothing must be modified
Forecast
exponentially
exponentially
including (FITt) = smoothed (Ft) + (Tt) smoothed
trend
forecast
trend
© 2006 Prentice Hall, Inc.
4 – 63
Exponential Smoothing with
Trend Adjustment
Ft = (At - 1) + (1 - )(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
© 2006 Prentice Hall, Inc.
4 – 64
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
Table 4.1
© 2006 Prentice Hall, Inc.
4 – 65
Exponential Smoothing with
Trend Adjustment Example
Month(t)
1
2
3
4
5
6
7
8
9
10
Forecast
Including
Trend, FITt
13.00
Actual
Smoothed
Smoothed
Demand (At) Forecast, Ft
Trend, Tt
12
11
2
17
20
19
Step 1: Forecast for Month 2
24
21
F2 = A1 + (1 - )(F1 + T1)
31
28
F2 = (.2)(12) + (1 - .2)(11 + 2)
36
= 2.4 + 10.4 = 12.8 units
Table 4.1
© 2006 Prentice Hall, Inc.
4 – 66
Exponential Smoothing with
Trend Adjustment Example
Month(t)
1
2
3
4
5
6
7
8
9
10
Forecast
Including
Trend, FITt
13.00
Actual
Smoothed
Smoothed
Demand (At) Forecast, Ft
Trend, Tt
12
11
2
17
12.80
20
19
Step 2: Trend for Month 2
24
21
T2 = b(F2 - F1) + (1 - b)T1
31
28
T2 = (.4)(12.8 - 11) + (1 - .4)(2)
36
= .72 + 1.2 = 1.92 units
Table 4.1
© 2006 Prentice Hall, Inc.
4 – 67
Exponential Smoothing with
Trend Adjustment Example
Month(t)
1
2
3
4
5
6
7
8
9
10
Forecast
Including
Trend, FITt
13.00
Actual
Smoothed
Smoothed
Demand (At) Forecast, Ft
Trend, Tt
12
11
2
17
12.80
1.92
20
19
Step 3: Calculate FIT for Month 2
24
21
FIT2 = F2 + T1
31
28
FIT2 = 12.8 + 1.92
36
= 14.72 units
Table 4.1
© 2006 Prentice Hall, Inc.
4 – 68
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
Table 4.1
© 2006 Prentice Hall, Inc.
4 – 69
Exponential Smoothing with
Trend Adjustment Example
35 –
Product demand
30 –
Actual demand (At)
25 –
20 –
15 –
Forecast including trend (FITt)
10 –
5 –
0 – |
1
|
2
|
3
|
4
|
5
|
6
Time (month)
© 2006 Prentice Hall, Inc.
|
7
|
8
|
9
Figure 4.3
4 – 70
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
^ = computed value of the variable to
where y
be predicted (dependent variable)
a = y-axis intercept
b = slope of the regression line
x = the independent variable
© 2006 Prentice Hall, Inc.
4 – 71
Values of Dependent Variable
Least Squares Method
Actual observation
(y value)
Deviation7
Deviation5
Deviation3
Deviation4
Deviation1
Deviation2
Trend line, y^ = a + bx
Time period
© 2006 Prentice Hall, Inc.
Deviation6
Figure 4.4
4 – 72
Values of Dependent Variable
Least Squares Method
Actual observation
(y value)
Deviation7
Deviation5
Deviation3
Least squares method
minimizes the sum of the
Deviation
squared
errors (deviations)
4
Deviation1
Deviation2
Trend line, y^ = a + bx
Time period
© 2006 Prentice Hall, Inc.
Deviation6
Figure 4.4
4 – 73
Least Squares Method
Equations to calculate the regression variables
y^ = a + bx
b=
Sxy - nxy
Sx2 - nx2
a = y - bx
© 2006 Prentice Hall, Inc.
4 – 74
Least Squares Example
Year
1999
2000
2001
2002
2003
2004
2005
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
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)
∑xy - nxy
b=
=
= 10.54
2)
2
2
140
(7)(4
∑x - nx
a = y - bx = 98.86 - 10.54(4) = 56.70
© 2006 Prentice Hall, Inc.
4 – 75
Least Squares Example
Time
Period (x)
Electrical Power
Demand
x2
xy
1999
1
74
1
2000
2
79
4
line is 80
2001The trend
3
9
2002
4
90
16
2003
105
25
y^ 5= 56.70 + 10.54x
2004
6
142
36
2005
7
122
49
Sx = 28
Sy = 692
Sx2 = 140
x=4
y = 98.86
74
158
240
360
525
852
854
Sxy = 3,063
Year
3,063 - (7)(4)(98.86)
Sxy - nxy
b=
=
= 10.54
2)
2
2
140
(7)(4
Sx - nx
a = y - bx = 98.86 - 10.54(4) = 56.70
© 2006 Prentice Hall, Inc.
4 – 76
Power demand
Least Squares Example
160
150
140
130
120
110
100
90
80
70
60
50
Trend line,
y^ = 56.70 + 10.54x
–
–
–
–
–
–
–
–
–
–
–
–
|
1999
© 2006 Prentice Hall, Inc.
|
2000
|
2001
|
2002
|
2003
Year
|
2004
|
2005
|
2006
|
2007
4 – 77
Seasonal Variations In Data
The multiplicative seasonal model can
modify trend data to accommodate
seasonal variations in demand
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
© 2006 Prentice Hall, Inc.
4 – 79
Seasonal Index Example
Month
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sept
Oct
Nov
Dec
© 2006 Prentice Hall, Inc.
Demand
2003 2004 2005
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
2003-2005
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
4 – 80
Seasonal Index Example
Month
Demand
2003 2004 2005
Average
2003-2005
Average
Monthly
Jan
80
85 105
90
94
Feb
70
85
85
80
94
Mar
80
93 average
82
85 monthly demand
94
2003-2005
Seasonal90index95= 115
Apr
100
94
average monthly
demand
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
© 2006 Prentice Hall, Inc.
Seasonal
Index
0.957
4 – 81
Seasonal Index Example
Month
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sept
Oct
Nov
Dec
© 2006 Prentice Hall, Inc.
Demand
2003 2004 2005
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
2003-2005
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
4 – 82
Seasonal Index Example
Month
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sept
Oct
Nov
Dec
© 2006 Prentice Hall, Inc.
Demand
2003 2004 2005
Average
2003-2005
Average
Monthly
80
85 105
90
94
for802006
70
85 Forecast
85
94
80
93
82
85
94
annual demand
= 1,200
90Expected
95 115
100
94
113 125 131
123
94
110 115 120 1,200 115
94
Jan 113
x
.957 = 96 94
100 102
105
12
88 102 110
100
94
1,200
85
90
95
Feb
x90
.851 = 85 94
77
78
85 12
80
94
75
72
83
80
94
82
78
80
80
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
4 – 83
Seasonal Index Example
2006 Forecast
2005 Demand
2004 Demand
2003 Demand
140 –
130 –
Demand
120 –
110 –
100 –
90 –
80 –
70 –
|
J
|
F
|
M
|
A
|
M
|
J
|
J
|
A
|
S
|
O
|
N
|
D
Time
© 2006 Prentice Hall, Inc.
4 – 84
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
9723
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
Figure 4.6
© 2006 Prentice Hall, Inc.
4 – 85
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
0.99
1.00 –
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
Figure 4.7
© 2006 Prentice Hall, Inc.
4 – 86
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
Figure 4.8
© 2006 Prentice Hall, Inc.
4 – 87
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
© 2006 Prentice Hall, Inc.
4 – 88
Associative Forecasting
Forecasting an outcome based on
predictor variables using the least squares
technique
y^ = a + bx
^ = computed value of the variable to
where y
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
© 2006 Prentice Hall, Inc.
4 – 89
Associative Forecasting
Example
Local Payroll
($000,000,000), x
1
3
4
4.0 –
2
1
3.0 –
7
Sales
Sales
($000,000), y
2.0
3.0
2.5
2.0
2.0
3.5
2.0 –
1.0 –
0
© 2006 Prentice Hall, Inc.
|
1
|
2
|
|
|
|
3 4
5 6
Area payroll
|
7
4 – 90
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
© 2006 Prentice Hall, Inc.
x2
1
9
16
4
1
49
∑x2 = 80
xy
2.0
9.0
10.0
4.0
2.0
24.5
∑xy = 51.5
51.5 - (6)(3)(2.5)
∑xy - nxy
b=
= 80 - (6)(32) = .25
∑x2 - nx2
a = y - bx = 2.5 - (.25)(3) = 1.75
4 – 91
Associative Forecasting
Example
If payroll next year
is estimated to be
$600 million, then:
Sales = 1.75 + .25(6)
Sales = $325,000
Sales = 1.75 + .25(payroll)
4.0 –
3.25
3.0 –
Sales
y^ = 1.75 + .25x
2.0 –
1.0 –
0
© 2006 Prentice Hall, Inc.
|
1
|
2
|
|
|
|
3 4
5 6
Area payroll
|
7
4 – 92
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
Figure 4.9
© 2006 Prentice Hall, Inc.
|
1
|
2
|
|
|
|
3 4
5 6
Area payroll
|
7
4 – 93
Standard Error of the
Estimate
Sy,x =
∑(y - yc)2
n-2
where y = y-value of each data point
yc = computed value of the dependent
variable, from the regression
equation
n = number of data points
© 2006 Prentice Hall, Inc.
4 – 94
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
© 2006 Prentice Hall, Inc.
4 – 95
Standard Error of the
Estimate
Sy,x =
∑y2 - a∑y - b∑xy
=
n-2
Sy,x = .306
39.5 - 1.75(15) - .25(51.5)
6-2
4.0 –
The standard error
of the estimate is
$30,600 in sales
Sales
3.25
3.0 –
2.0 –
1.0 –
0
© 2006 Prentice Hall, Inc.
|
1
|
2
|
|
|
|
3 4
5 6
Area payroll
|
7
4 – 96
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
© 2006 Prentice Hall, Inc.
4 – 97
Correlation Coefficient
r=
nSxy - SxSy
[nSx2 - (Sx)2][nSy2 - (Sy)2]
© 2006 Prentice Hall, Inc.
4 – 98
y
Correlation Coefficient
y
n∑xy - ∑x∑y
r=
2 - (∑x)2][n∑y2 - (∑y)2]
[n∑x
(a) Perfect positive x
(b) Positive
correlation:
0<r<1
correlation:
r = +1
y
x
y
(c) No correlation:
r=0
© 2006 Prentice Hall, Inc.
x
(d) Perfect negative x
correlation:
r = -1
4 – 99
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
© 2006 Prentice Hall, Inc.
4 – 100
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
© 2006 Prentice Hall, Inc.
4 – 101
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 = $300,000
© 2006 Prentice Hall, Inc.
4 – 102
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
© 2006 Prentice Hall, Inc.
4 – 103
Monitoring and Controlling
Forecasts
Tracking
RSFE
=
signal
MAD
∑(actual demand in
period i forecast demand
in period i)
Tracking
signal = (∑|actual - forecast|/n)
© 2006 Prentice Hall, Inc.
4 – 104
Tracking Signal
Signal exceeding limit
Tracking signal
+
Upper control limit
Acceptable
range
0 MADs
–
Lower control limit
Time
© 2006 Prentice Hall, Inc.
4 – 105
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
© 2006 Prentice Hall, Inc.
Cumulative
Absolute
Forecast
Error
MAD
10
15
30
40
55
85
10.0
7.5
10.0
10.0
11.0
14.2
4 – 106
Tracking Signal Example
Qtr
1
2
3
4
5
6
Tracking
Actual Signal
Forecast
(RSFE/MAD)
Demand
Demand
Error
RSFE
Absolute
Forecast
Error
90-10/10
100= -1 -10
95
-15/7.5
100= -2 -5
115 0/10
100
= 0 +15
100-10/10
110= -1 -10
125
+5/11110
= +0.5+15
140
+35/14.2
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
© 2006 Prentice Hall, Inc.
4 – 107
Adaptive Forecasting
It’s possible to use the computer to
continually monitor forecast error and
adjust the values of the  and b
coefficients used in exponential
smoothing to continually minimize
forecast error
This technique is called adaptive
smoothing
© 2006 Prentice Hall, Inc.
4 – 108
Focus Forecasting
Developed at American Hardware Supply,
focus forecasting is based on two principles:
1. Sophisticated forecasting models are not
always better than simple models
2. There is no single techniques 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
© 2006 Prentice Hall, Inc.
4 – 109
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
© 2006 Prentice Hall, Inc.
4 – 110
Fast Food Restaurant
Forecast
Percentage of sales
20% –
15% –
10% –
5% –
11-12
1-2
12-1
(Lunchtime)
© 2006 Prentice Hall, Inc.
3-4
2-3
5-6
4-5
7-8
6-7
(Dinnertime)
Hour of day
9-10
8-9
10-11
Figure 4.12
4 – 111