4 Forecasting Demand

PowerPoint presentation to accompany

Heizer, Render, and Al-Zu’bi
Operations Management, Arab
World Edition

Original PowerPoint slides by Jeff Heyl

Adapted by Zu’bi Al-Zu’bi

4-1
 Outline

• What Is Forecasting?
• Forecasting Time Horizons
• The Influence of Product Life Cycle
• Types of Forecasts

4-2
 Outline – Continued

 The Strategic Importance of Forecasting

 Human Resources
 Capacity
 Supply Chain Management
 Seven Steps in the Forecasting System

4-3
 Outline – Continued

 Forecasting Approaches
 Overview of Qualitative Methods
 Overview of Quantitative Methods
 Time-Series Forecasting
 Decomposition of a Time Series
 Naive Approach

4-4
 Outline – Continued

• Time-Series Forecasting (cont.)

• Moving Averages
• Exponential Smoothing
• Measuring Forecast Error
• Exponential Smoothing with Trend Adjustment
• Trend Projections
• Seasonal Variations in Data
• Cyclical Variations in Data

4-5
 Outline – Continued
• Associative Forecasting Methods:
Regression and Correlation
Analysis
• Using Regression Analysis for
Forecasting
• Standard Error of the Estimate
• Correlation Coefficients for
Regression Lines
• Multiple-Regression Analysis
4-6
 Outline – Continued

• Monitoring and Controlling Forecasts

• Forecasting in the Service Sector

4-7
 What is Forecasting?

• Process of predicting a future event

• Underlying basis
of all business decisions
- Production
- Inventory
- Personnel
- Facilities

4-8
 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
4-9
 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

4 - 11
 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 products and
services

4 - 12
 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
4 - 13
 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

4 - 14
 Overview of Quantitative Approaches

1. Naive approach
2. Moving averages
3. Exponential time-series
models
smoothing
4. Trend projection
5. Linear regression associative
model

4 - 15
Time-Series Components

Trend Cyclical

Seasonal Random

4 - 16
 Components of Demand

Trend
component
Demand for product or service

Seasonal peaks

Actual demand
line

Average demand
over 4 years

Random variation
| | | |
1 2 3 4
Time (years)
Figure 4.1
4 - 17
Trend Component

 Persistent, overall upward or

downward pattern
 Changes due to population,
technology, age, culture, etc.
 Typically several years
duration

4 - 18
 Seasonal Component

 Regular pattern of up and down fluctuations

 Due to weather, customs, etc.
 Occurs within a single year

Number of
Period Length Seasons
Week Day 7
Month Week 4-4.5
Month Day 28-31
Year Quarter 4
Year Month 12
Year Week 52

4 - 19
 Cyclical Component

 Repeating up and down movements

 Affected by business cycle, political,
and economic factors
 Multiple years’ duration
 Often causal or
associative
relationships

0 5 10 15 20
4 - 20
 Random Component

 Erratic, unsystematic, ‘residual’

fluctuations
 Due to random variation or unforeseen
events
 Short duration
and nonrepeating

M T W T F
4 - 21
 1. Naive Approach

 Assumes demand in next

period is the same as
demand in most recent period
 e.g., If January sales were 68, then
February sales will be 68
 Sometimes cost effective and
efficient
 Can be good starting point

4 - 22
 2. 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

4 - 23
Moving Average Example

Actual 3-Month
Month Shed Sales Moving Average
January 10
February 12
March 13
April 16 (10 + 12 + 13)/3 = 11 2/3
May 19 (12 + 13 + 16)/3 = 13 2/3
June 23 (13 + 16 + 19)/3 = 16
July 26 (16 + 19 + 23)/3 = 19 1/3

4 - 24
Graph of Moving Average

Moving
Average
30 –
28 –
Forecast
26 – Actual
24 – Sales
Shed Sales

22 –
20 –
18 –
16 –
14 –
12 –
10 –
| | | | | | | | | | | |
J F M A M J J A S O N D

4 - 25
3. Weighted Moving Average

 Used when some trend might be

present
 Older data usually less important
 Weights based on experience and
intuition
∑ (weight for period n)
Weighted x (demand in period n)
moving average = ∑ weights

4 - 26
 Weights Applied Period
Weighted Moving Average
3 Last month
2 Two months ago
1 Three months ago
6 Sum of weights

Actual 3-Month Weighted

Month Shed Sales Moving Average
January 10
February 12
March 13
April 16 [(3 x 13) + (2 x 12) + (10)]/6 = 121/6
May 19 [(3 x 16) + (2 x 13) + (12)]/6 = 141/3
June 23 [(3 x 19) + (2 x 16) + (13)]/6 = 17
July 26 [(3 x 23) + (2 x 19) + (16)]/6 = 201/2

4 - 27
 Moving Average And
Weighted Moving Average

Weighted
moving
30 – average
25 –
Sales demand

20 – Actual
sales
15 –
Moving
10 – average

5 –
| | | | | | | | | | | |
Figure 4.2
J F M A M J J A S O N D
© 2013 Pearson Education 4 - 28
 Potential Problems With
Moving Average

 Increasing n smooths the forecast

but makes it less sensitive to
changes
 Does not forecast trends well
 Requires extensive historical data

4 - 29
4. Exponential Smoothing

 Form of weighted moving average

 Weights decline exponentially
 Most recent data give most weight
 Requires smoothing constant ()
 Ranges from 0 to 1
 Subjectively chosen
 Involves little record keeping of past
data

4 - 30
 Exponential Smoothing

Previous period’s forecast

+ a (Previous period’s actual demand
– Previous period’s forecast)

Ft = Ft – 1 + a(At – 1 - Ft – 1)

where Ft = new forecast

Ft – 1 = previous forecast
a = smoothing (or weighting)

constant (0 ≤ a ≤ 1)
4 - 31
 Exponential Smoothing Example

Predicted demand = 142 Ford Mustangs

Actual demand = 153
Smoothing constant a = .20

4 - 32
 Exponential Smoothing Example

Predicted demand = 142 Ford Mustangs

Actual demand = 153
Smoothing constant a = .20

New forecast = 142 + .2(153 – 142)

4 - 33
 Exponential Smoothing Example

Predicted demand = 142 Ford Mustangs

Actual demand = 153
Smoothing constant a = .20

New forecast = 142 + .2(153 – 142)

= 142 + 2.2
= 144.2 ≈ 144 cars

4 - 34
 Effect of
Smoothing Constants

Weight Assigned to
Most 2nd Most 3rd Most 4th Most 5th Most
Recent Recent Recent Recent Recent
Smoothing Period Period Period Period Period
Constant (a) a(1 - a) a(1 - a) 2
a(1 - a) 3
a(1 - a)4

a = .1 .1 .09 .081 .073 .066

a = .5 .5 .25 .125 .063 .031

4 - 35
 Impact of Different 

225 –

Actual a = .5
demand
200 –
Demand

175 –
a = .1
| | | | | | | | |
150 –
1 2 3 4 5 6 7 8 9
Quarter

4 - 36
 Impact of Different 
 Chose high values of  when
underlying average is likely to change
 Choose low values of  when
underlying average is stable

225 –
Actual a = .5
demand
Demand

200 –

a = .1
175 –

| | | | | | | | |
150 –
1 2 3 4 5 6 7 8 9
Quarter 4 - 37
Common Measures of Error

Mean Absolute Deviation (MAD)

∑ |Actual - Forecast|
MAD =
n

Mean Squared Error (MSE)

∑ (Forecast Errors)2
MSE =
n
4 - 38
Common Measures of Error

Mean Absolute Percent Error (MAPE)

n
∑100|Actuali - Forecasti|/Actuali
MAPE = i=1
n

4 - 39
 Comparison of Forecast Error

Rounded Absolute Rounded Absolute

Actual Forecast Deviation Forecast Deviation
Tonnage with for with for
Quarter Unloaded a = .10 a = .10 a = .50 a = .50
1 180 175 5.00 175 5.00
2 168 175.5 7.50 177.50 9.50
3 159 174.75 15.75 172.75 13.75
4 175 173.18 1.82 165.88 9.12
5 190 173.36 16.64 170.44 19.56
6 205 175.02 29.98 180.22 24.78
7 180 178.02 1.98 192.61 12.61
8 182 178.22 3.78 186.30 4.30
82.45 98.62

4 - 40
 Comparison of Forecast Error

Rounded Absolute Rounded Absolute

Actual Forecast Deviation Forecast Deviation
Tonnage with for with for
Quarter Unloaded a = .10 a = .10 a = .50 a = .50
1 180 175 5.00 175 5.00
2 168 175.5 7.50 177.50 9.50
3 159 174.75 15.75 172.75 13.75
4 175 173.18 1.82 165.88 9.12
5 190 173.36 16.64 170.44 19.56
6 205 175.02 29.98 180.22 24.78
7 180 178.02 1.98 192.61 12.61
8 182 178.22 3.78 186.30 4.30
82.45 98.62
MAD 10.31 12.33

4 - 41
 Comparison of Forecast Error

Rounded Absolute Rounded Absolute

Actual Forecast Deviation Forecast Deviation
Tonnage with for with for
Quarter Unloaded a = .10 a = .10 a = .50 a = .50
1 180 175 5.00 175 5.00
2 168 175.5 7.50 177.50 9.50
3 159 174.75 15.75 172.75 13.75
4 175 173.18 1.82 165.88 9.12
5 190 173.36 16.64 170.44 19.56
6 205 175.02 29.98 180.22 24.78
7 180 178.02 1.98 192.61 12.61
8 182 178.22 3.78 186.30 4.30
82.45 98.62
MAD 10.31 12.33
MSE 190.82 195.24
4 - 42
 Comparison of Forecast Error

Rounded Absolute Rounded Absolute

Actual Forecast Deviation Forecast Deviation
Tonnage with for with for
Quarter Unloaded a = .10 a = .10 a = .50 a = .50
1 180 175 5.00 175 5.00
2 168 175.5 7.50 177.50 9.50
3 159 174.75 15.75 172.75 13.75
4 175 173.18 1.82 165.88 9.12
5 190 173.36 16.64 170.44 19.56
6 205 175.02 29.98 180.22 24.78
7 180 178.02 1.98 192.61 12.61
8 182 178.22 3.78 186.30 4.30
82.45 98.62
MAD 10.31 12.33
MSE 190.82 195.24
MAPE 5.59% 6.76%
4 - 43
 5. Exponential Smoothing with Trend
Adjustment

When a trend is present, exponential

smoothing must be modified

Forecast Exponentially Exponentially

including (FITt) = smoothed (Ft) + smoothed (Tt)
trend forecast trend

4 - 44
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
4 - 45
Exponential Smoothing with Trend Adjustment
Example

Forecast
Actual Smoothed Smoothed Including
Month(t) Demand (At) Forecast, Ft Trend, Tt Trend, FITt
1 12 11 2 13.00
2 17
3 20
4 19
5 24
6 21
7 31
8 28
9 36
10 -
Table 4.1

4 - 46
 Exponential Smoothing with Trend Adjustment
Example

Forecast
Actual Smoothed Smoothed Including
Month(t) Demand (At) Forecast, Ft Trend, Tt Trend, FITt
1 12 11 2 13.00
2 17
3 20
4 19
5 24 Step 1: Forecast for Month 2
6 21
7 31 F2 = aA1 + (1 - a)(F1 + T1)
8 28
9 36 F2 = (.2)(12) + (1 - .2)(11 + 2)
10 - = 2.4 + 10.4 = 12.8 units
Table 4.1
4 - 47
 Exponential Smoothing with Trend Adjustment
Example

Forecast
Actual Smoothed Smoothed Including
Month(t) Demand (At) Forecast, Ft Trend, Tt Trend, FITt
1 12 11 2 13.00
2 17 12.80
3 20
4 19
5 24 Step 2: Trend for Month 2
6 21
7 31 T2 = b(F2 - F1) + (1 - b)T1
8 28
9 36 T2 = (.4)(12.8 - 11) + (1 - .4)(2)
10 = .72 + 1.2 = 1.92 units
Table 4.1
4 - 48
 Exponential Smoothing with Trend Adjustment
Example

Forecast
Actual Smoothed Smoothed Including
Month(t) Demand (At) Forecast, Ft Trend, Tt Trend, FITt
1 12 11 2 13.00
2 17 12.80 1.92
3 20
4 19
5 24 Step 3: Calculate FIT for Month 2
6 21
7 31 FIT2 = F2 + T2
8 28
9 36
FIT2 = 12.8 + 1.92
10 = 14.72 units
Table 4.1
4 - 49
 Exponential Smoothing with Trend Adjustment
Example

Forecast
Actual Smoothed Smoothed Including
Month(t) Demand (At) Forecast, Ft Trend, Tt Trend, FITt
1 12 11 2 13.00
2 17 12.80 1.92 14.72
3 20 15.18 2.10 17.28
4 19 17.82 2.32 20.14
5 24 19.91 2.23 22.14
6 21 22.51 2.38 24.89
7 31 24.11 2.07 26.18
8 28 27.14 2.45 29.59
9 36 29.28 2.32 31.60
10 32.48 2.68 35.16
Table 4.1
4 - 50
Exponential Smoothing with Trend Adjustment
Example

35 –
Actual demand (At)
30 –
Product demand

25 –

20 –

15 –

10 –
Forecast including trend (FITt)
with  = .2 and  = .4
5 –

0 – | | | | | | | | |
1 2 3 4 5 6 7 8 9
Figure 4.3
Time (month)
4 - 51
6. 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
4 - 52
Least Squares Method
Values of Dependent Variable

Actual observation Deviation7

(y-value)

Deviation5 Deviation6

Deviation3

Deviation4

Deviation1
(error) Deviation2
Trend line, y^ = a + bx

Time period Figure 4.4

4 - 53
Least Squares Method
Values of Dependent Variable

Actual observation Deviation7

(y-value)

Deviation5 Deviation6

Deviation3 Least squares method

minimizes the sum of the
squared errors (deviations)
Deviation 4

Deviation1
(error) Deviation2
Trend line, y^ = a + bx

Time period Figure 4.4

4 - 54
Least Squares Method

Equations to calculate the regression variables

y^ = a + bx

Sxy - nxy
b=
Sx2 - nx2

a = y - bx

4 - 55
Least Squares Example

Time Electrical Power

Year Period (x) Demand x2 xy
2005 1 74 1 74
2006 2 79 4 158
2007 3 80 9 240
2008 4 90 16 360
2009 5 105 25 525
2010 6 142 36 852
2011 7 122 49 854
∑x = 28 ∑y = 692 ∑x2 = 140 ∑xy = 3,063
x=4 y = 98.86

∑xy - nxy 3,063 - (7)(4)(98.86)

b= = = 10.54
∑x - nx
2 2 140 - (7)(4 )
2

a = y - bx = 98.86 - 10.54(4) = 56.70

4 - 56
 The trend line is
Least Squares Example y^ = 56.70 + 10.54x

Time Electrical Power

Year Period (x) Demand x2 xy
2005 1 74 1 74
2006 2 79 4 158
2007 3 80 9 240
2008 4 90 16 360
2009 5 105 25 525
2010 6 142 36 852
2011 7 122 49 854
∑x = 28 ∑y = 692 ∑x2 = 140 ∑xy = 3,063
x=4 y = 98.86

∑xy - nxy 3,063 - (7)(4)(98.86)

b= = = 10.54
∑x - nx
2 2 140 - (7)(4 )
2

a = y - bx = 98.86 - 10.54(4) = 56.70

4 - 57
Least Squares Example

Trend line,
160 –
150 –
y^ = 56.70 + 10.54x
140 –
Power demand

130 –
120 –
110 –
100 –
90 –
80 –
70 –
60 –
50 –
| | | | | | | | |
2005 2006 2007 2008 2009 2010 2011 2012 2013
Year
4 - 58
 Seasonal Variations In Data

The multiplicative
seasonal model
can adjust trend
data for seasonal
variations in
demand

4 - 60
 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
4 - 61
Seasonal Index Example

Demand Average Average Seasonal

Month 2009 2010 2011 2009-2011 Monthly Index
Jan 80 85 105 90 94
Feb 70 85 85 80 94
Mar 80 93 82 85 94
Apr 90 95 115 100 94
May 113 125 131 123 94
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

4 - 62
 Average 2009-2011 monthly demand
Seasonal index =
Average monthly demand
= 90/94 = .957

Demand Average Average Seasonal

Month 2009 2010 2011 2009-2011 Monthly Index
Jan 80 85 105 90 94 0.957
Feb 70 85 85 80 94
Mar 80 93 82 85 94
Apr 90 95 115 100 94
May 113 125 131 123 94
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

4 - 63
Seasonal Index Example

Demand Average Average Seasonal

Month 2009 2010 2011 2009-2011 Monthly Index
Jan 80 85 105 90 94 0.957
Feb 70 85 85 80 94 0.851
Mar 80 93 82 85 94 0.904
Apr 90 95 115 100 94 1.064
May 113 125 131 123 94 1.309
Jun 110 115 120 115 94 1.223
Jul 100 102 113 105 94 1.117
Aug 88 102 110 100 94 1.064
Sept 85 90 95 90 94 0.957
Oct 77 78 85 80 94 0.851
Nov 75 72 83 80 94 0.851
Dec 82 78 80 80 94 0.851

4 - 64
Seasonal Index Example

Demand Average Average Seasonal

Month 2009 2010 2011 2007-2009 Monthly Index
Jan 80 85 105 90 94 0.957
Feb 70 85 Forecast
85 for802012 94 0.851
Mar 80 93 82 85 94 0.904
Apr 90Expected
95 115annual demand
100 = 1,200
94 1.064
May 113 125 131 123 94 1.309
Jun 110 115 120 1,200 115 94 1.223
Jul Jan 113
100 102 x .957 = 96 94
105 1.117
12
Aug 88 102 110 100 94 1.064
Sept 85 90 95 1,200 90 94 0.957
Feb x .851 = 85
Oct 77 78 85 12 80 94 0.851
Nov 75 72 83 80 94 0.851
Dec 82 78 80 80 94 0.851

4 - 65
Seasonal Index Example

2012 Forecast
140 – 2011 Demand
130 – 2010 Demand
2009 Demand
120 –
Demand

110 –
100 –
90 –
80 –
70 –
| | | | | | | | | | | |
J F M A M J J A S O N D
Time
4 - 66
 San Diego Hospital
Trend line,
Trend Data y = 8,090 + 21.5x

10,200 –

10,000 –
Inpatient Days

9745
9,800 – 9702
9616 9659
9573 9724 9766
9,600 – 9530 9680
9594 9637
9551
9,400 –

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
4 - 67
San Diego Hospital

Seasonal Indices

1.06 –
1.04 1.04
Index for Inpatient Days

1.04 – 1.03
1.02
1.02 – 1.01
1.00
1.00 – 0.99
0.98
0.98 – 0.99
0.96 – 0.97 0.97
0.96
0.94 –
| | | | | | | | | | | |
0.92 –
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
4 - 68
 San Diego Hospital

Combined Trend and Seasonal Forecast

10,200 – 10068
9949
10,000 – 9911
Inpatient Days

9764 9724
9,800 – 9691
9572
9,600 –
9520 9542
9,400 –
9411
9265 9355
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 4.8
4 - 69
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

4 - 70
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 4 - 71
 Associative Forecasting Example

Sales Area Payroll

($ millions), y ($ billions), x
2.0 1
3.0 3
2.5 4 4.0 –
2.0 2
2.0 1 3.0 –
3.5 7 Sales
2.0 –

1.0 –
| | | | | | |
0 1 2 3 4 5 6 7
Area payroll

4 - 72
Associative Forecasting Example

Sales, y Payroll, x x2 xy
2.0 1 1 2.0
3.0 3 9 9.0
2.5 4 16 10.0
2.0 2 4 4.0
2.0 1 1 2.0
3.5 7 49 24.5
∑y = 15.0 ∑x = 18 ∑x2 = 80 ∑xy = 51.5

∑xy - nxy 51.5 - (6)(3)(2.5)

x = ∑x/6 = 18/6 = 3 b= = = .25
∑x - nx
2 2 80 - (6)(3 2
)

y = ∑y/6 = 15/6 = 2.5 a = y - bx = 2.5 - (.25)(3) = 1.75

4 - 73
 Associative Forecasting Example

y^ = 1.75 + .25x Sales = 1.75 + .25(payroll)

If payroll next year

is estimated to be 4.0 –
$6 billion, then: 3.25

Shaer’s sales
3.0 –

Sales = 1.75 + .25(6) 2.0 –

Sales = $3,250,000
1.0 –
| | | | | | |
0 1 2 3 4 5 6 7
Area payroll

4 - 74
Standard Error of the Estimate

 A forecast is just a point estimate of a

future value
 This point is
4.0 –
actually the 3.25
mean of a
Shaer’s sales
3.0 –
probability
2.0 –
distribution
1.0 –
| | | | | | |
0 1 2 3 4 5 6 7
Area payroll
Figure 4.9

4 - 75
Standard Error of the Estimate

∑(y - yc)2
Sy,x =
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
4 - 76
Standard Error of the Estimate

Computationally, this equation is

considerably easier to use

∑y2 - a∑y - b∑xy

Sy,x =
n-2

We use the standard error to set up

prediction intervals around the
point estimate

4 - 77
 Standard Error of the Estimate

∑y2 - a∑y - b∑xy = 39.5 - 1.75(15) - .25(51.5)

Sy,x =
n-2 6-2

Sy,x = .306 4.0 –

3.25
Shaer’s sales3.0 –
The standard error
of the estimate is 2.0 –
$306,000 in sales
1.0 –
| | | | | | |
0 1 2 3 4 5 6 7
Area payroll

© 2013 Pearson Education 4 - 78

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

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Correlation Coefficient

nSxy - SxSy
r=
[nSx2 - (Sx)2][nSy2 - (Sy)2]

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Correlation Coefficient
y

nSxy - SxSy
r=
[nSx2 - (Sx)2][nSy2 - (Sy)2]
(b) Positive x
correlation:
0<r<1

y y y

(c) No x (d) Perfect negative x

(a) Perfect positive x correlation:
correlation: correlation: r = -1
r=0 r = +1
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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 Shaer Construction example:

r = .901
r2 = .81
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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
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Multiple Regression Analysis

In the Shaer 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
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Monitoring and Controlling Forecasts

Tracking Signal
 Measures how well the forecast is predicting
actual values
 Ratio of cumulative forecast errors to mean
absolute deviation (MAD)
 Good tracking signal has low values
 If forecasts are continually high or low,
the forecast has a bias error

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Monitoring and Controlling Forecasts

Tracking Cumulative error

signal =
MAD

∑(Actual demand in
period i -
Forecast demand
Tracking in period i)
signal = (∑|Actual - Forecast|/n)

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Tracking Signal

Signal exceeding limit

Tracking signal
Upper control limit
+

Acceptable
0 MADs range

– Lower control limit

Time

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Tracking Signal Example

Cumulative
Absolute Absolute
Actual Forecast Cumm Forecast Forecast
Qtr Demand Demand Error Error Error Error MAD

1 90 100 -10 -10 10 10 10.0

2 95 100 -5 -15 5 15 7.5
3 115 100 +15 0 15 30 10.0
4 100 110 -10 -10 10 40 10.0
5 125 110 +15 +5 15 55 11.0
6 140 110 +30 +35 30 85 14.2

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Tracking Signal Example

Tracking
Signal Cumulative
(Cumm Absolute Absolute
Actual Forecast Cumm Forecast Forecast
Qtr Error/MAD)
Demand Demand Error Error Error Error MAD

1 90-10/10
100= -1 -10 -10 10 10 10.0
2 95
-15/7.5
100= -2 -5 -15 5 15 7.5
3 115 0/10
100= 0 +15 0 15 30 10.0
4 100-10/10
110= -1 -10 -10 10 40 10.0
5 125
+5/11110
= +0.5+15 +5 15 55 11.0
6 140
+35/14.2
110= +2.5
+30 +35 30 85 14.2

The variation of the tracking signal

between -2.0 and +2.5 is within acceptable
limits
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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
 Holidays and other calendar events
need to be factored in
 Unusual events can affect actual
demand

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 Fast Food Restaurant Forecast

20% –
Percentage of sales

15% –

10% –

5% –

11-12 1-2 3-4 5-6 7-8 9-10

12-1 2-3 4-5 6-7 8-9 10-11
(Lunchtime) (Dinnertime)
Figure 4.12
Hour of day
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Call Center Forecast

12% –

10% –

8% –

6% –

4% –

2% –

0% – 2 4 6 8 10 12 2 4 6 8 10 12
A.M. P.M.
Hour of day
Figure 4.12
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Lecture 2 Assignment: Forecasting

• Reading Chapter 4
• Reading solved problems p122-123
• Problems:
– 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.7, 4.9, 4.12, 4.15, 4.16, 4.17,
4.18, 4.19

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