Forecasting

4-1
 What is Forecasting?
► Art and science of
predicting a future
event
► Underlying basis
of all business
decisions
► Production
► Inventory
► Personnel
► Facilities 4-2
 Forecasting Time Horizons
1. Short-range forecast
► Up to 1 year, generally less than 3 months
► Purchasing, job scheduling, workforce levels,
job assignments, production levels
2. Medium-range forecast
► 3 months to 3 years
► Sales and production planning, budgeting
3. Long-range forecast
► 3+ years
► New product planning, facility location,
research and development
4-3
 Distinguishing Differences
1. Medium/long range forecasts deal with more comprehensive
issues and support management decisions regarding planning
and products, plants and processes
2. Short-term forecasting usually employs different methodologies
than longer-term forecasting. Mathematical techniques such as
moving averages, exponential smoothing and trend extrapolation.
Less quantitative methods are useful in predicting such issues as
whether a new product like the optical disk recorder should be
introduced into a company’s product line
3. Short-term forecasts tend to be more accurate than longer-term
forecasts
4-4
 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
4-5
 Product Life Cycle
Introduction Growth Maturity Decline
Product design and Forecasting critical Standardization Little product
development Product and Fewer product differentiation
critical process reliability changes, more Cost
Frequent product Competitive minor changes minimization
and process
OM Strategy/Issues

product Optimum capacity Overcapacity in

design changes improvements and the industry
Increasing stability
Short production options of process Prune line to
runs Increase capacity eliminate items
Long production
High production Shift toward runs not returning
costs product focus good margin
Product
Limited models Enhance improvement and Reduce
Attention to quality distribution cost cutting capacity

4-6
 Types of Forecasts
1. Economic forecasts
► Address business cycle – inflation rate, money
supply, housing starts, etc.
2. Technological forecasts
► Predict rate of technological progress
► Impacts development of new products
3. Demand forecasts
► Predict sales of existing products and services

4-7
 Strategic Importance of
Forecasting
► Supply-Chain Management – Good
supplier relations, advantages in product
innovation, cost and speed to market
► Human Resources – Hiring, training,
laying off workers
► Capacity – Capacity shortages can result
in undependable delivery, loss of
customers, loss of market share

4-8
Seven Steps in Forecasting
1. Determine the use of the forecast
2. Select the items to be forecasted
3. Determine the time horizon of the
forecast
4. Select the forecasting model(s)
5. Gather the data needed to make the
forecast
6. Make the forecast
7. Validate and implement results
4-9
4 - 10
 The Realities!
► Forecasts are seldom perfect,
unpredictable outside factors may
impact the forecast
► Most techniques assume an
underlying stability in the system
► Product family and aggregated
forecasts are more accurate than
individual product forecasts

4 - 11
 Forecasting Approaches
1. 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 - 12
 Overview of Qualitative Methods
1. Jury of executive opinion
► Opinions of a group of high-level experts or managers, often in combination
with statistical models, are pooled to arrive at a group estimate of demand.
► Grasp on future trends in the field
2. Delphi method
► Three different types of participants : decision makers, staff personnel and
respondents
► Decision makers usually group of 5-10 experts who make actual forecast.
► Staff personnel assist decision makers by preparing , distributing, collecting
and summarizing a series of questionnaires and survey results.
► Respondents are group of people, often located in different places whose
judgements are valued. This group provides inputs to the decision makers
before the forecast is made. 4 - 13
 Overview of Qualitative Methods
3. Sales force composite
► Estimates from individual salespersons are reviewed for
reasonableness, then aggregated at district and national levels to
reach an overall forecast.
4. Consumer Market Survey
► Solicits input from customer or potential customers regarding future
purchasing plans. It can help not only in preparing a forecast but
also in improving product design and planning for new products.
▶ The consumer market survey and sales force composite methods can
however suffer from overly optimistic forecasts that arise from
customer input. 4 - 14
 Forecasting Approaches
2. 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 - 15
 Time-Series Data
Trend
component
Demand for product or service

Seasonal peaks

Trend Cyclical
Actual demand
line

Average demand
over 4 years
Seasonal Random

Random variation
| | | |
1 2 3 4
Time (years)

Not all time-series data have all these elements 4 - 16

 Trend Component
► The long-term general direction of data
► Persistent, overall upward or downward pattern
► Changes due to population, technology, age, culture, etc.
► Typically several years duration

4 - 17
 Cyclical Component
► Repeating up and down movements
► Affected by business cycle, political, and economic factors
► Often causal or associative relationships
► Time-series data that do not extend over a long period of
time may not have enough “history”
to show cyclical effects

0 5 10 15 20
4 - 18
 Seasonal Component
► Regular pattern of up and down fluctuations
► Due to weather, customs, etc.
► Occurs within a single year

PERIOD LENGTH “SEASON” LENGTH NUMBER OF “SEASONS” IN PATTERN

Week Day 7
Month Week 4 – 4.5
Month Day 28 – 31
Year Quarter 4
Year Month 12
Year Week 52

4 - 19
 Random Component
► Erratic, unsystematic, ‘residual’ fluctuations
► Due to random variation or unforeseen events
► Short duration and nonrepeating
► They are subject to momentary change and are
often unexplained.

4 - 20
M T W T F
 Time-Series Forecasting

► Models predict on the assumption that the future is a

function of the past.
► Look at what has happened over a period of time and use
a series of past data to make a forecast.
► Time-series data that contain no trend, cyclical, or
seasonal effects are said to be stationary. Techniques
used to forecast stationary data analyze only the irregular
fluctuation effects.

4 - 21
 Smoothing Techniques
► Several techniques are available to forecast time-series data that are
stationary or that include no significant trend, cyclical, or seasonal
effects. These techniques are often referred to as smoothing
techniques because they produce forecasts based on “smoothing out”
the irregular fluctuation effects in the time-series data.

1. Naïve Forecasting Models

2. Averaging Models

3. Exponential Smoothing 4 - 22
 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 - 23
 2. Averaging Models
► Many naïve model forecasts are based on the value of one
time period. Often such forecasts become a function of
irregular fluctuations of the data; as a result, the forecasts are
“oversteered.”
► Using averaging models, a forecaster enters information from
several time periods into the forecast and “smoothes” the
data.
► Averaging models are computed by averaging data from
several time periods and using the average as the forecast
for the next time period
4 - 24
 2.1 Moving Average Method
► MA is a series of arithmetic means
► Used if little or no trend
► Provides overall impression of data over time

Moving average =
å demand in previous n periods
n
► Limitations
► it is difficult to choose the optimal length of time for which to
compute the moving average, and
► moving averages do not usually adjust for such time-series
effects as trend, cycles, or seasonality 4 - 25
 Moving Average Example
MONTH ACTUAL SHED SALES 3-MONTH 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
August 30 (19 + 23 + 26)/3 = 22 2/3
September 28 (23 + 26 + 30)/3 = 26 1/3
October 18 (29 + 30 + 28)/3 = 28
November 16 (30 + 28 + 18)/3 = 25 1/3
December 14 (28 + 18 + 16)/3 = 20 2/3

4 - 26
2.2 Weighted Moving Average
► Used when some trend might be
present
► Older data usually less important
► Weights based on experience and
intuition

Weighted
moving =
å (( Weight for period n) (Demand in period n))
average å Weights

4 - 27
 Weighted Moving Average
MONTH ACTUAL SHED SALES 3-MONTH WEIGHTED MOVING AVERAGE
January 10
February 12
March 13
April 16 [(3 x 13) + (2 x 12) + (10)]/6 = 12 1/6
May 19
June WEIGHTS
23 APPLIED PERIOD

July 26 3 Last month

August 30 2 Two months ago

September 28 1 Three months ago

October 18 6 Sum of the weights

November Forecast for

16this month =
December 3 x14
Sales last mo. + 2 x Sales 2 mos. ago + 1 x Sales 3 mos. ago
Sum of the weights

4 - 28
 Weighted Moving Average
MONTH ACTUAL SHED SALES 3-MONTH WEIGHTED MOVING AVERAGE
January 10
February 12
March 13
April 16 [(3 x 13) + (2 x 12) + (10)]/6 = 12 1/6
May 19 [(3 x 16) + (2 x 13) + (12)]/6 = 14 1/3
June 23 [(3 x 19) + (2 x 16) + (13)]/6 = 17
July 26 [(3 x 23) + (2 x 19) + (16)]/6 = 20 1/2
August 30 [(3 x 26) + (2 x 23) + (19)]/6 = 23 5/6
September 28 [(3 x 30) + (2 x 26) + (23)]/6 = 27 1/2
October 18 [(3 x 28) + (2 x 30) + (26)]/6 = 28 1/3
November 16 [(3 x 18) + (2 x 28) + (30)]/6 = 23 1/3
December 14 [(3 x 16) + (2 x 18) + (28)]/6 = 18 2/3

4 - 29
 Graph of Moving Averages
Weighted moving average
30 –

Sales demand 25 –

20 –

15 – Actual sales

10 – Moving average

5–
| | | | | | | | | | | |

J F M A M J J A S O N D
Figure 4.2 Month

4 - 30
Potential Problems With Moving Average

► Increasing n smooths the forecast but makes it less

sensitive to changes
► Does not forecast trends well
► Need to continually carry a large amount of historical data
(also with regression)

4 - 31
 3 Exponential Smoothing
► Form of weighted moving average
► Weights decline exponentially, most recent data weighted most
► Oldest observation is dropped and the new forecast is calculated.
The most recent occurrences are more indicative of the future than
those in the more distant past
► Requires smoothing constant ()
► Ranges from 0 to 1 (subjectively chosen)
► Involves little record keeping of past data
► It is the most used forecasting techniques. It is integral part of virtually all
computerized forecasting programs, and is widely used in ordering
inventory in retail firms, wholesale companies and service agencies. 4 - 32
 Exponential Smoothing
► Exponential smoothing techniques have become well accepted for six
major reasons :
1. Exponential models are surprisingly accurate
2. Formulating an exponential model is relatively easy
3. The user can understand how the model works
4. Little computation is required to use the model
5. Computer storage requirements are small because of the limited use
of historical data
6. Tests for accuracy as to how well the model is performing are easy to
compute
4 - 33
 Exponential Smoothing
New forecast = Last period’s forecast
+  (Last period’s actual demand
– Last period’s forecast)
Ft = Ft – 1 + (At – 1 - Ft – 1)
= At – 1 + (1- Ft – 1
where Ft = new forecast
Ft – 1 = previous period’s forecast
 = smoothing (or weighting) constant (0 ≤  ≤ 1)
At – 1 = previous period’s actual demand

Smoothing constant () : The parameter in the exponential smoothing equation that
controls the speed of reaction to differences between forecasts and actual demand
4 - 34
 Effect of Smoothing Constants
► Smoothing constant generally .05 ≤  ≤ .50 and is
determined by the forecaster
► If  is chosen to be less than .5, less weight is
placed on the actual value than on the forecast of
that value
► If  is chosen to be greater than .5, more weight is
being put on the actual value than on the forecast
value

4 - 35
 Effect of Smoothing Constants

Substituting this forecast value, Ft, into the preceding equation for Ft +1 produces

but

Substituting this value of Ft -1 into the preceding equation for Ft +1 produces

As  increases, older
values become less
significant
4 - 36
Exponential Smoothing Example

Predicted demand = 142 Ford Mustangs

Actual demand = 153
Smoothing constant  = .20

4 - 37
Exponential Smoothing Example

Predicted demand = 142 Ford Mustangs

Actual demand = 153
Smoothing constant  = .20

New forecast = 142 + .2(153 – 142)

4 - 38
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

4 - 39
 Exponential Smoothing

MONTH (t) ACTUAL DEMAND (At) MONTH (t) ACTUAL DEMAND (At)

1 12 6 21

2 17 7 31

3 20 8 28

4 19 9 36

5 24 10 ?

 = .2 F1 = 11

Ft = Ft – 1 + (At – 1 - Ft – 1)
4 - 40
Exponential Smoothing Example
Ft = Ft – 1 + (At – 1 - Ft – 1)
Forecast with  - .2
SMOOTHED
FORECAST
MONTH ACTUAL DEMAND AVERAGE, Ft
1 12 11
2 17 11.2
3 20 12.36
4 19 13.89
5 24 14.91
6 21 16.73
7 31 17.58
8 28 20.27
9 36 21.81
10 — 24.65

4 - 41
 Exponential Smoothing
ACTUAL TONNAGE
QUARTER UNLOADED FORECAST WITH  = .10
1 180 175

2 168

3 159

4 175

5 190

6 205

7 180

8 182

9 ?

4 - 42
 Exponential Smoothing
ACTUAL TONNAGE
QUARTER UNLOADED FORECAST WITH  = .10
1 180 175

2 168 175.50 = 175.00 + .10(180 – 175)

3 159 174.75 = 175.50 + .10(168 – 175.50)

4 175 173.18 = 174.75 + .10(159 – 174.75)

5 190 173.36 = 173.18 + .10(175 – 173.18)

6 205 175.02 = 173.36 + .10(190 – 173.36)

7 180 178.02 = 175.02 + .10(205 – 175.02)

8 182 178.22 = 178.02 + .10(180 – 178.02)

9 ? 178.59 = 178.22 + .10(182 – 178.22)

4 - 43
 Exponential Smoothing
ACTUAL
TONNAGE FORECAST WITH
QUARTER UNLOADED FORECAST WITH  = .10  = .50
1 180 175 175

2 168 175.50 = 175.00 + .10(180 – 175)

3 159 174.75 = 175.50 + .10(168 – 175.50)

4 175 173.18 = 174.75 + .10(159 – 174.75)

5 190 173.36 = 173.18 + .10(175 – 173.18)

6 205 175.02 = 173.36 + .10(190 – 173.36)

7 180 178.02 = 175.02 + .10(205 – 175.02)

8 182 178.22 = 178.02 + .10(180 – 178.02)

9 ? 178.59 = 178.22 + .10(182 – 178.22)

4 - 44
 Exponential Smoothing
ACTUAL
TONNAGE FORECAST WITH
QUARTER UNLOADED FORECAST WITH  = .10  = .50
1 180 175 175

2 168 175.50 = 175.00 + .10(180 – 175) 177.50

3 159 174.75 = 175.50 + .10(168 – 175.50) 172.75

4 175 173.18 = 174.75 + .10(159 – 174.75) 165.88

5 190 173.36 = 173.18 + .10(175 – 173.18) 170.44

6 205 175.02 = 173.36 + .10(190 – 173.36) 180.22

7 180 178.02 = 175.02 + .10(205 – 175.02) 192.61

8 182 178.22 = 178.02 + .10(180 – 178.02) 186.30

9 ? 178.59 = 178.22 + .10(182 – 178.22) 184.15

4 - 45
 Impact of Different 
225 –

Actual  = .5
demand
200 –
Demand

175 –

 = .1
150 – | | | | | | | | |
1 2 3 4 5 6 7 8 9
Quarter
4 - 46
 Impact of Different 
225 –
► Chose high values of  when
underlying average is likely
Actualto  = .5
demand
change 200 –
Demand

► If the firm is experiencing growth

► It gives greater
175 – importance to
recent growth experience  = .1
► Choose low
150 values
–| | of |when| | | | | |
underlying average
1 2 is stable
3 4 5 6 7 8 9

► If a firm produced a standardQuarter

item
with relatively stable demand 4 - 47
 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

4 - 48
 Measuring Forecast Error

The overall accuracy of any forecasting model can

be determined by computing the forecasted values
with the actual or observed values.

Forecast error = Actual demand – Forecast value

= At - Ft

4 - 49
 TE

Actual Model-1 Error Model-2 Error

80 70 85
120 130 110

4 - 50
Common Measures of Error

1. Mean Absolute Deviation (MAD)

MAD =
å Actual - Forecast
n

4 - 51
 Determining the MAD
ACTUAL
TONNAGE FORECAST WITH
QUARTER UNLOADED FORECAST WITH  = .10  = .50
1 180 175 175

2 168 175.50 = 175.00 + .10(180 – 175) 177.50

3 159 174.75 = 175.50 + .10(168 – 175.50) 172.75

4 175 173.18 = 174.75 + .10(159 – 174.75) 165.88

5 190 173.36 = 173.18 + .10(175 – 173.18) 170.44

6 205 175.02 = 173.36 + .10(190 – 173.36) 180.22

7 180 178.02 = 175.02 + .10(205 – 175.02) 192.61

8 182 178.22 = 178.02 + .10(180 – 178.02) 186.30

9 ? 178.59 = 178.22 + .10(182 – 178.22) 184.15

4 - 52
 Determining the MAD
ACTUAL FORECAST ABSOLUTE FORECAST ABSOLUTE
TONNAGE WITH DEVIATION WITH DEVIATION
QUARTER UNLOADED  = .10 FOR a = .10  = .50 FOR a = .50
1 180 175 5.00 175 5.00

2 168 175.50 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

Sum of absolute deviations: 82.45 98.62

Σ|Deviations|
MAD = 10.31 12.33
n

4 - 53
 MAD

Actual Model-1 Error Model-2 Error

80 85 80
120 110 105

4 - 54
Common Measures of Error

2. Mean Squared Error (MSE)

å (Forecast errors)
2

MSE =
n

4 - 55
 Determining the MSE
ACTUAL
TONNAGE FORECAST FOR
QUARTER UNLOADED  = .10 (ERROR)2
1 180 175 52 = 25
2 168 175.50 (–7.5)2 = 56.25
3 159 174.75 (–15.75)2 = 248.06
4 175 173.18 (1.82)2 = 3.31
5 190 173.36 (16.64)2 = 276.89
6 205 175.02 (29.98)2 = 898.80
7 180 178.02 (1.98)2 = 3.92
8 182 178.22 (3.78)2 = 14.29
Sum of errors squared = 1,526.52

å( )
2
Forecast errors
MSE = = 1,526.52 / 8 = 190.8
n
4 - 56
 MSE

Actual Model-1 Error Model-2 Error

80 85 90
120 110 115

4 - 57
 Common Measures of Error

3. Mean Absolute Percent Error (MAPE)

n

å 100 Actual
i
- Forecast i / Actuali
MAPE = i=1
n

A problem with both MAD and MSE is that their values depend on the
magnitude of the item being forecast. If the forecast item is measured in
thousands, the MAD and MSE values can be very large. To avoid this
MAPE is used. 4 - 58
 Determining the MAPE
ACTUAL
TONNAGE FORECAST FOR ABSOLUTE PERCENT ERROR
QUARTER UNLOADED  = .10 100(ERROR/ACTUAL)
1 180 175.00 100(5/180) = 2.78%
2 168 175.50 100(7.5/168) = 4.46%
3 159 174.75 100(15.75/159) = 9.90%
4 175 173.18 100(1.82/175) = 1.05%
5 190 173.36 100(16.64/190) = 8.76%
6 205 175.02 100(29.98/205) = 14.62%
7 180 178.02 100(1.98/180) = 1.10%
8 182 178.22 100(3.78/182) = 2.08%
Sum of % errors = 44.75%

MAPE =
å absolute percent error = 44.75% = 5.59%
n 8
4 - 59
 Comparison of Forecast Error
Rounded Absolute Rounded Absolute
Actual Forecast Deviation Forecast Deviation
Tonnage with for with for
Quarter Unloaded  = .10  = .10  = .50  = .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 - 60
 Comparison of Forecast Error
Rounded Absolute Rounded Absolute
Actual Forecast Deviation Forecast Deviation
Tonnage with for with for
Quarter Unloaded a = .10 a = .10  = .50  = .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 - 61
 Comparison of Forecast Error
Rounded Absolute Rounded Absolute
Actual Forecast Deviation Forecast Deviation
Tonnage with for with for
Quarter Unloaded a = .10 a = .10  = .50  = .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 - 62
 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  = .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 - 63
 Comparison of Forecast Error
Rounded Absolute Rounded Absolute
Actual Forecast Deviation Forecast Deviation
Tonnage with for with for
Quarter Unloaded  = .10  = .10  = .50  = .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 - 64
 Trend Analysis
► Analysis of trend involves developing an equation that
will suitably describe trend (assuming that trend is
present in the data). The trend component may be linear,
or it may not.
► One of the important techniques that can be used to
develop forecasts when trend is present.
1. Linear Regression Trend Analysis

4 - 65
Some Non-linear Trends

4 - 66
 Linear Regression Trend Analysis
• Regression analysis is the process of constructing a
mathematical model or function that can be used to predict or
determine one variable by another variable or other variables.
• In simple regression, the variable to be predicted is called the
dependent variable and is designated as y. The predictor is
called the independent variable, or explanatory variable,
and is designated as x.
4 - 67
 Linear Regression

The following are the assumptions of simple regression analysis.

1. The model is linear.
2. The error terms have constant variances.
3. The error terms are independent.
4. The error terms are normally distributed.

4 - 68
 Least Squares Method

Values of Dependent Variable (y-values)

Actual observation Deviation7
(y-value)

Deviation5 Deviation6

Deviation3
Least squares method minimizes the
sum of Deviation
the squared
4
errors (deviations)

Deviation1
(error) Deviation2
Trend line, y^ = b0 + b1x

| | | | | | |
1 2 3 4 5 6 7
Time period
4 - 69
 Least Squares Method
Equations to calculate the regression variables

y^ = b0 + b1x

4 - 70
Least Squares Method

4 - 71
 Problem
A specialist in hospital administration stated that the number of
FTEs (full-time employees) in a hospital can be estimated by
counting the number of beds in the hospital (a common measure
of hospital size). A healthcare business researcher decided to
develop a regression model in an attempt to predict the number of
FTEs of a hospital by the number of beds. She surveyed 12
hospitals and obtained the following data. The data are presented
in sequence, according to the number of beds.

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Solution

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 Solution

The least squares equation of the regression line is

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 Problem
Suppose the data displayed in are the
costs and associated number of
passengers for twelve 500-mile
commercial airline flights using Boeing
737s during the same season of the year.
Develop a regression model to predict
cost by number of passengers.

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Solution

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Solution

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 Problem

With the Huntsville Chemicals Company

data, forecast sales for the year 2012
using these data

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Solution

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Solution

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 Least Squares Example

ELECTRICAL ELECTRICAL
YEAR POWER DEMAND YEAR POWER DEMAND
1 74 5 105
2 79 6 142
3 80 7 122
4 90 8 ?

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 Least Squares Example
ELECTRICAL POWER
YEAR (x) DEMAND (y) x2 xy
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
Σx = 28 Σy = 692 Σx2 = 140 Σxy = 3,063

x=
å x = 28 = 4 y=
å y = 692 = 98.86
n 7 n 7

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 Least Squares Example
- nxy 3,063 - ( 7) ( 4) (98.86) 295
å xyELECTRICAL
b= = POWER = = 10.54
YEAR (x) å x - nxDEMAND (y)140 - (7) ( 4 ) x 28
2 2 2 2 xy
1 74 1 74

()
2 79 4 158
3
a = y - bx = 98.8680
- 10.54 4 = 56.70 9 240
4 90 16 360
5 105 ŷ = 56.70 +10.54x25
Thus, 525
6 142 36 852
7 122 49 854
Σx = 28 Σy = 692 Σx2 = 140 Σxy = 3,063

x=
å x = in28year
Demand
=4 y=
å y+=10.54(8)
8 = 56.70 692
= 98.86
n 7 = 141.02,
n or 7141 megawatts

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 Least Squares Example
Trend line,
160 – y^ = 56.70 + 10.54x
150 –
Power demand (megawatts)
140 –
130 –
120 –
110 –
100 –
90 –
80 –
70 –
60 –
50 –
| | | | | | | | |
1 2 3 4 5 6 7 8 9
Year Figure 4.5
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 Problem
Cell phone sales for a California-based
firm over the last 10 weeks are shown in
the following table. Plot the data, and
visually check to see if a linear trend line
would be appropriate. Then determine
the equation of the trend line, and
predict sales for weeks 11 and 12

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Solution

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Solution

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Solution

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 Seasonal Variations In Data
• Seasonal variations in data are regular up-and-down movements in
a time series that relate to recurring events such as weather or
holidays. It may be applied to hourly, daily, weekly, monthly or other
recurring patterns, usually in periods of time of less than one year
• Rush hour traffic, customers in theaters and restaurants
• Seasonality in a time series is expressed in terms of the amount that
actual values deviate from the average value of series
• There are two models – additive and multiplicative 4 - 90
Seasonality
• Additive Model : Seasonality is
expressed as a quantity e.g. 20
units which is added or subtracted
from the series average

• Multiplicative Model : Seasonality

is expressed as a percentage of
the average (or trend) e.g. 1.10,
0.90 which is then used to
multiply the value of a series to
incorporate seasonality. The
seasonal percentages are
referred to as seasonal relatives
or seasonal indexes 4 - 91
 Using Seasonal Relatives
Seasonal relatives are used in two different ways,
1. De-seasonalize data
• Remove the seasonal component from the data in order to get a
clearer picture of the demand. De-seasonalizing data is
accomplished by dividing each data point by its seasonal index
2. Incorporate seasonality
• When demand has both trend and seasonal components

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 Forecasting
Techniques

Qualitative Quantitative
Models Models

Time Series
Delphi Method Casual Models
Models

Sales Force Moving Simple

Composite Averages Regression

Consumer Exponential Multiple

Panel Survey Smoothing Regression

Trend
Projection 4 - 94
 Forecasting Across The Organization
• Marketing relies heavily on forecasting tools to generate forecasts of demand and future
sales. However, the marketing department also needs to forecast sizes of markets, new
competition, future trends, and changes in consumer preferences.
• Finance uses the tools of forecasting to predict stock prices, financial performance, capital
investment needs, and investment portfolio returns. The accuracy of demand forecasts, in
turn, affects the ability of finance to plan future cash flow and financial needs.
• Information systems play an important role in the forecasting process. Today’s forecasting
requires sharing of information and databases not only within a business but also between
business entities. Often companies share their forecasts or demand information with their
suppliers. These capabilities would not be possible without an up-to-date information
system.
• Human resources relies on forecasting to determine future hiring requirements. In addition,
forecasts are made of the job market, labor skill availability, future wages and compensation,
hiring and layoff costs, and training costs. In order to recruit proper talent, it is necessary to
forecast labor needs and availability.
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