Forecasting

Forecasting Provides a
Competitive Advantage for Disney

► Global portfolio includes parks in Shanghai,

Hong Kong, Paris, Tokyo, Orlando, and
Anaheim
► Revenues are derived from people – how
many visitors and how they spend their
money
► Daily management report contains only the
forecast and actual attendance at each park
 Forecasting Provides a
Competitive Advantage for Disney

► Disney generates daily, weekly, monthly,

annual, and 5-year forecasts
► Forecast used by labor management,
maintenance, operations, finance, and park
scheduling
► Forecast used to adjust opening times, rides,
shows, staffing levels, and guests admitted
 Forecasting Provides a
Competitive Advantage for Disney

► 20% of customers come from outside the

USA
► Economic model includes gross domestic
product, cross-exchange rates, arrivals into
the USA
► A staff of 35 analysts and 70 field people
survey 1 million park guests, employees, and
travel professionals each year
 Forecasting Provides a
Competitive Advantage for Disney

► Inputs to the forecasting model include airline

specials, Federal Reserve policies, Wall
Street trends, vacation/holiday schedules for
3,000 school districts around the world
► Average forecast error for the 5-year forecast
is 5%
► Average forecast error for annual forecasts is
between 0% and 3%
 What is Forecasting?
► Process of predicting a
future event
► Underlying basis
of all business
??
decisions
► Production
► Inventory
► Personnel
► Facilities
 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
 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
3. Short-term forecasts tend to be more
accurate than longer-term forecasts
 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
 Product Life Cycle
Introduction Growth Maturity Decline

Best period to Practical to change Poor time to Cost control

increase market price or quality change image, critical
share image price, or quality
Company Strategy/Issues

R&D engineering is Strengthen niche Competitive costs

critical become critical
Defend market
position
Hybrid engine vehicles Laptop computers

Boeing 787 Xbox One

DVDs
3D printers

Life Cycle Curve

Electric
vehicles
Apple Video
SmartWatch 3-D game physical
players rentals

Figure 2.5
 Product Life Cycle
Introduction Growth Maturity Decline
Product design and Forecasting critical Standardization Little product
development Product and Fewer rapid differentiation
critical process reliability product changes, Cost
Frequent product Competitive more minor minimization
and process changes
Strategy/Issues

product Overcapacity in
OMStrategy/Issues

design changes improvements and Optimum capacity the industry

Short production options Increasing stability Prune line to
runs Increase capacity of process eliminate items
High production Shift toward not returning
costs product focus good margin
Limited models Enhance Reduce
Long production capacity
Attention to quality distribution
OM

runs
Product
improvement and
cost cutting

Figure 2.5
 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
 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
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 the results
 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
 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
 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
Overview of Qualitative Methods

1. Jury of executive opinion

► Pool opinions of high-level experts,
sometimes augmented by statistical
models
2. Delphi method
► Panel of experts, queried iteratively
Overview of Qualitative Methods

3. Sales force composite

► Estimates from individual salespersons
are reviewed for reasonableness, then
aggregated
4. Market Survey
► Ask the customer
 Jury of Executive Opinion
► Involves small group of high-level experts
and managers
► Group estimates demand by working
together
► Combines managerial experience with
statistical models
► Relatively quick
► ‘Group-think’
disadvantage
 Delphi Method
► Iterative group
process, continues Decision Makers
(Evaluate responses
until consensus is and make decisions)
reached
► Three types of Staff
(Administering
participants survey)

► Decision makers
► Staff
► Respondents Respondents
(People who can make
valuable judgments)
 Sales Force Composite

► Each salesperson projects his or her

sales
► Combined at district and national
levels
► Sales reps know what customers want
► May be overly optimistic
 Market Survey
► Ask customers about purchasing
plans
► Useful for demand and product
design and planning
► What consumers say and what they
actually do may be different
► May be overly optimistic
 Overview of Quantitative
Approaches
1. Naive approach
2. Moving averages
3. Exponential Time-series
smoothing models
4. Trend projection
5. Linear regression Associative
model
 Time-Series Forecasting

► Set of evenly spaced numerical data

► Obtained by observing response
variable at regular time periods
► Forecast based only on past values, no
other variables important
► Assumes that factors influencing past
and present will continue influence in
future
Time-Series Components

Trend Cyclical

Seasonal Random
 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
 Trend Component
► Persistent, overall upward or
downward pattern
► Changes due to population,
technology, age, culture, etc.
► Typically several years duration
 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
 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
 Random Component
► Erratic, unsystematic, ‘residual’
fluctuations
► Due to random variation or unforeseen
events
► Short duration
and nonrepeating

M T W T
F
 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
 Moving Averages

► MA is a series of arithmetic means

► Used if little or no trend
► Used often for smoothing
► Provides overall impression of data
over time

Moving average =
å demand in previous n periods
n
 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 (26 + 30 + 28)/3 = 28
November 16 (30 + 28 + 18)/3 = 25 1/3
December 14 (28 + 18 + 16)/3 = 20 2/3
 Weighted Moving Average
► Used when some trend might be
present
► Older data usually less important
► Weights based on experience and
intuition

(( )(
Weighted å Weight for period n Demand in period n
moving =
))
average å Weights
 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
 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
 Potential Problems With
Moving Average
1. Increasing n smooths the forecast but
makes it less sensitive to changes
2. Does not forecast trends well
3. Requires extensive historical data
 Graph of Moving Averages
Weighted moving average (from Example 2)

30 –

25 –
Sales demand

20 –

15 – Actual sales

10 – Moving average
(from Example 1)
5–
| | | | | | | | | | | |

J F M A M J J A S O N D
Figure 4.2 Month
 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
 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 Ft = new forecast

Ft – 1 = previous period’s forecast
 = smoothing (or weighting) constant (0 ≤  ≤ 1)
At – 1 = previous period’s actual demand
 Exponential Smoothing
Example
Predicted demand = 142 Ford Mustangs
Actual demand = 153
Smoothing constant  = .20
 Exponential Smoothing
Example
Predicted demand = 142 Ford Mustangs
Actual demand = 153
Smoothing constant  = .20

New forecast = 142 + .2(153 – 142)

 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
 Effect of
Smoothing Constants
▶ Smoothing constant generally .05 ≤  ≤ .50
▶ As  increases, older values become less
significant

WEIGHT ASSIGNED TO
MOST 2ND MOST 3RD MOST 4th MOST 5th MOST
RECENT RECENT RECENT RECENT RECENT
SMOOTHING PERIOD PERIOD PERIOD PERIOD PERIOD
CONSTANT ( ) (1 – ) (1 – )2 (1 – )3 (1 – )4
 = .1 .1 .09 .081 .073 .066

 = .5 .5 .25 .125 .063 .031

 Impact of Different 
225 –

Actual  = .5
demand
200 –
Demand

175 –

 = .1
150 – | | | | | | | | |
1 2 3 4 5 6 7 8 9
Quarter
 Impact of Different 
225 –

Actual  = .5
Choose high
► 200 – of 
values
demand
when underlying average
Demand

is likely to change
175 –
► Choose low values of 
when underlying average  = .1
is–stable
150 | | | | | | | | |
1 2 3 4 5 6 7 8 9
Quarter
 Selecting the Smoothing
Constant
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 according to one of three preferred
measures:
► Mean Absolute Deviation (MAD)
► Mean Squared Error (MSE)
► Mean Absolute Percent Error (MAPE)
Common Measures of Error

Mean Absolute Deviation (MAD)

MAD =
å Actual - Forecast
n
 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

 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
Common Measures of Error

Mean Squared Error (MSE)

å (Forecast errors)
2

MSE =
n
 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 (Forecast errors)
2

MSE = = 1,526.52 / 8 = 190.8

n
Common Measures of Error

Mean Absolute Percent Error (MAPE)

n

å100 Actual -Forecast

i i
/ Actuali
MAPE = i=1
n
 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
 Comparison of Measures
TABLE 4.1 Comparison of Measures of Forecast Error

MEASURE MEANING APPLICATION TO CHAPTER EXAMPLE

Mean absolute How much the forecast For  = .10 in Example 4, the forecast for grain
deviation (MAD) missed the target unloaded was off by an average of 10.31 tons.
Mean squared The square of how much For  = .10 in Example 5, the square of the forecast
error (MSE) the forecast missed the error was 190.8. This number does not have a
target physical meaning, but is useful when compared to the
MSE of another forecast.
Mean absolute The average percent For  = .10 in Example 6, the forecast is off by 5.59%
percent error error on average. As in Examples 4 and 5, some forecasts
(MAPE) were too high, and some were low.
 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
 Comparison of Forecast Error
Rounded Absolute Rounded Absolute
Actual ∑ |deviations|
Forecast Deviation Forecast Deviation
MAD
Quarter
=
Tonnage
Unloaded
with
 =n.10
for
 = .10
with
 = .50
for
 = .50
1 180 175 5.00 175 5.00
2 For  168
= .10 175.5 7.50 177.50 9.50
3 159 = 82.45/8
174.75 = 15.75
10.31 172.75 13.75
4 175 173.18 1.82 165.88 9.12
5 For  190
= .50 173.36 16.64 170.44 19.56
6 205 175.02 29.98 180.22 24.78
7 180 = 98.62/8
178.02 = 12.33
1.98 192.61 12.61
8 182 178.22 3.78 186.30 4.30
82.45 98.62
 Comparison of Forecast Error
Rounded Absolute Rounded Absolute
∑ (forecast
Actual errors)
Forecast
2
Deviation Forecast Deviation
MSE =Tonnage with for with for
Quarter Unloaded  =n.10  = .10  = .50  = .50
1 180 175 5.00 175 5.00
2 For  168
= .10 175.5 7.50 177.50 9.50
3 159 174.75
= 1,526.52/8 = 15.75
190.8 172.75 13.75
4 175 173.18 1.82 165.88 9.12
5 For  190
= .50 173.36 16.64 170.44 19.56
6 205 175.02 29.98 180.22 24.78
7 = 1,561.91/8
180 178.02 = 195.24
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
 Comparison of Forecast Error
n
∑
Actual
Rounded
Forecast
Absolute
100|deviation i |/actual
Deviation
Rounded
iForecast
Absolute
Deviation
MAPE
Quarter
= i=1
Tonnage
Unloaded
with
 = .10
for
 = .10
with
 = .50
for
 = .50
n
1 180 175 5.00 175 5.00
2 For 
168= .10 175.5 7.50 177.50 9.50
3 159 174.75
= 44.75%/8 15.75
= 5.59% 172.75 13.75
4 175 173.18 1.82 165.88 9.12
5 For 
190= .50 173.36 16.64 170.44 19.56
6 205 175.02 29.98 180.22 24.78
7 180 = 54.00%/8
178.02 =1.98
6.75% 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
 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% 6.75%
Exponential Smoothing with
Trend Adjustment
When a trend is present, exponential
smoothing must be modified
MONTH ACTUAL DEMAND FORECAST (Ft) FOR MONTHS 1 – 5

1 100 F1 = 100 (given)

2 200 F2 = F1 + (A1 – F1) = 100 + .4(100 – 100) = 100

3 300 F3 = F2 + (A2 – F2) = 100 + .4(200 – 100) = 140

4 400 F4 = F3 + (A3 – F3) = 140 + .4(300 – 140) = 204

5 500 F5 = F4 + (A4 – F4) = 204 + .4(400 – 204) = 282

Exponential Smoothing with
Trend Adjustment
Forecast Exponentially Exponentially
including (FITt) = smoothed (Ft) + smoothed (Tt)
trend forecast trend

Ft = (At - 1) + (1 - )(Ft - 1 + Tt - 1)
Tt = (Ft - Ft - 1) + (1 - )Tt - 1
where Ft = exponentially smoothed forecast average
Tt = exponentially smoothed trend
At = actual demand
 = smoothing constant for average (0 ≤  ≤ 1)
 = smoothing constant for trend (0 ≤  ≤ 1)
Exponential Smoothing with
Trend Adjustment
Step 1: Compute Ft
Step 2: Compute Tt
Step 3: Calculate the forecast FITt = Ft + Tt
Exponential Smoothing with
Trend Adjustment Example
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  = .4
 Exponential Smoothing with
Trend Adjustment Example
TABLE 4.2 Forecast with  = .2 and  = .4
SMOOTHED FORECAST
FORECAST SMOOTHED INCLUDING TREND,
MONTH ACTUAL DEMAND AVERAGE, Ft TREND, Tt FITt
1 12 11 2 13.00
2 17 12.80
3 20
4 19
Step 1: Average for Month 2
5 24
6 21 F2 = A1 + (1 – )(F1 + T1)
7 31
8 28 F2 = (.2)(12) + (1 – .2)(11 + 2)
9 36 = 2.4 + (.8)(13) = 2.4 + 10.4
10 —
= 12.8 units
 Exponential Smoothing with
Trend Adjustment Example
TABLE 4.2 Forecast with  = .2 and  = .4
SMOOTHED FORECAST
FORECAST SMOOTHED INCLUDING TREND,
MONTH ACTUAL DEMAND AVERAGE, Ft TREND, Tt FITt
1 12 11 2 13.00
2 17 12.80 1.92
3 20
4 19
5 24 Step 2: Trend for Month 2
6 21
7 31 T2 = (F2 - F1) + (1 - b)T1
8 28
T2 = (.4)(12.8 - 11) + (1 - .4)(2)
9 36
10 — = .72 + 1.2 = 1.92 units
 Exponential Smoothing with
Trend Adjustment Example
TABLE 4.2 Forecast with  = .2 and  = .4
SMOOTHED FORECAST
FORECAST SMOOTHED INCLUDING TREND,
MONTH ACTUAL DEMAND AVERAGE, Ft TREND, Tt FITt
1 12 11 2 13.00
2 17 12.80 1.92 14.72
3 20
4 19
5 24 Step 3: Calculate FIT for Month 2
6 21
7 31 FIT2 = F2 + T2
8 28
FIT2 = 12.8 + 1.92
9 36
10 — = 14.72 units
 Exponential Smoothing with
Trend Adjustment Example
TABLE 4.2 Forecast with  = .2 and  = .4
SMOOTHED FORECAST
FORECAST SMOOTHED INCLUDING TREND,
MONTH ACTUAL DEMAND AVERAGE, Ft TREND, Tt 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
Exponential Smoothing with
Trend Adjustment Example
40 – Figure 4.3

35 – Actual demand (At)

30 –
Product demand

25 –
20 –
15 –
10 – Forecast including trend (FITt)
5 – with  = .2 and  = .4

0 –
| | | | | | | | |
1 2 3 4 5 6 7 8 9
Time (months)
 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
Values of Dependent Variable (y-values) Least Squares Method
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^ = a + bx

| | | | | | |
1 2 3 4 5 6 7
Figure 4.4
Time period
 Least Squares Method
Equations to calculate the regression variables

ŷ = a + bx

b=
å xy - nxy
å x - nx
2 2

a = y - bx
 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
 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
 Least Squares Example
å xy - nxy 3,063 - ( 7) ( 4) (98.86) 295
b= = POWER = = 10.54
å x - nxDEMAND (y)140 - (7) ( 4 ) x 28
ELECTRICAL
2 2 2 2
YEAR (x) 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
Demandx in
= =4 y=
å
28year 8 = 56.70 y+ 10.54(8)
=
692
= 98.86
n 7 = 141.02,
n or 141
7 megawatts
 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
Least Squares Requirements

1. We always plot the data to insure a

linear relationship
2. We do not predict time periods far
beyond the database
3. Deviations around the least squares
line are assumed to be random
Seasonal Variations In Data

The multiplicative
seasonal model can
adjust trend data for
seasonal variations
in demand
 Seasonal Variations In Data
Steps in the process for monthly seasons:

1. Find average historical demand for each month

2. Compute the average demand over all months
3. Compute a seasonal index for each month
4. Estimate next year’s total demand
5. Divide this estimate of total demand by the
number of months, then multiply it by the
seasonal index for that month
 Seasonal Index Example
DEMAND
AVERAGE AVERAGE
YEARLY MONTHLY SEASONAL
MONTH YEAR 1 YEAR 2 YEAR 3 DEMAND DEMAND INDEX
Jan 80 85 105 90
Feb 70 85 85 80
Mar 80 93 82 85
Apr 90 95 115 100
May 113 125 131 123
June 110 115 120 115
July 100 102 113 105
Aug 88 102 110 100
Sept 85 90 95 90
Oct 77 78 85 80
Nov 75 82 83 80
Dec 82 78 80 80
Total average annual demand = 1,128
 Seasonal Index Example
DEMAND
AVERAGE AVERAGE
YEARLY MONTHLY SEASONAL
MONTH YEAR 1 YEAR 2 YEAR 3 DEMAND DEMAND INDEX
Jan 80 85 105 90 94
Feb 70 85 85 80 94
Mar 80 93 82 85 94
Apr
Average
90 95 1,128
115 100 94
monthly = = 94
May 113 125 131
12 months 123 94
June
demand
110 115 120 115 94
July 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 82 83 80 94
Dec 82 78 80 80 94
Total average annual demand = 1,128
 Seasonal Index Example
DEMAND
AVERAGE AVERAGE
YEARLY MONTHLY SEASONAL
MONTH YEAR 1 YEAR 2 YEAR 3 DEMAND DEMAND INDEX
Jan 80 85 105 90 94 .957( = 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
Seasonal110
June Average
115 monthly
120 demand
115 for past 394
years
=
July index 100 102 Average
113 monthly
105 demand 94
Aug 88 102 110 100 94
Sept 85 90 95 90 94
Oct 77 78 85 80 94
Nov 75 82 83 80 94
Dec 82 78 80 80 94
Total average annual demand = 1,128
 Seasonal Index Example
DEMAND
AVERAGE AVERAGE
YEARLY MONTHLY SEASONAL
MONTH YEAR 1 YEAR 2 YEAR 3 DEMAND DEMAND INDEX
Jan 80 85 105 90 94 .957( = 90/94)
Feb 70 85 85 80 94 .851( = 80/94)
Mar 80 93 82 85 94 .904( = 85/94)
Apr 90 95 115 100 94 1.064( = 100/94)
May 113 125 131 123 94 1.309( = 123/94)
June 110 115 120 115 94 1.223( = 115/94)
July 100 102 113 105 94 1.117( = 105/94)
Aug 88 102 110 100 94 1.064( = 100/94)
Sept 85 90 95 90 94 .957( = 90/94)
Oct 77 78 85 80 94 .851( = 80/94)
Nov 75 82 83 80 94 .851( = 80/94)
Dec 82 78 80 80 94 .851( = 80/94)
Total average annual demand = 1,128
 Seasonal Index Example
Seasonal forecast for Year 4
MONTH DEMAND MONTH DEMAND

Jan 1,200 July 1,200

x .957 = 96 x 1.117 = 112
12 12
Feb 1,200 Aug 1,200
x .851 = 85 x 1.064 = 106
12 12
Mar 1,200 Sept 1,200
x .904 = 90 x .957 = 96
12 12
Apr 1,200 Oct 1,200
x 1.064 = 106 x .851 = 85
12 12
May 1,200 Nov 1,200
x 1.309 = 131 x .851 = 85
12 12
June 1,200 Dec 1,200
x 1.223 = 122 x .851 = 85
12 12
 Seasonal Index Example
Year 4 Forecast
140 – Year 3 Demand
130 – Year 2 Demand
Year 1 Demand
120 –
Demand

110 –
100 –
90 –
80 –
70 –
| | | | | | | | | | | |
J F M A M J J A S O N D
Time
 San Diego Hospital

Seasonality Indices for Adult Inpatient Days at San Diego Hospital

MONTH SEASONALITY INDEX MONTH SEASONALITY INDEX

January 1.04 July 1.03

February 0.97 August 1.04

March 1.02 September 0.97

April 1.01 October 1.00

May 0.99 November 0.96

June 0.99 December 0.98

Demand Y=8090+21.5X

Forecast for months 67 to 78

 San Diego Hospital
Trend Data Figure 4.6

10,200 –

10,000 –
Inpatient Days

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

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
 San Diego Hospital
Seasonal Indices Figure 4.7

1.06 –
1.04
Index for Inpatient Days

1.04
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
 San Diego Hospital

Period 67 68 69 70 71 72

Month Jan Feb Mar Apr May June

Forecast with 9,911 9,265 9,764 9,691 9,520 9,542

Trend &
Seasonality
Period 73 74 75 76 77 78

Month July Aug Sept Oct Nov Dec

Forecast with 9,949 10,068 9,411 9,724 9,355 9,572

Trend &
Seasonality
 San Diego Hospital
Combined Trend and Seasonal Forecast Figure 4.8

10,200 –
10,068
10,000 – 9,911 9,949
Inpatient Days

9,800 – 9,764 9,724

9,691
9,600 – 9,572

9,400 – 9,520 9,542

9,411
9,265 9,355
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
 Adjusting Trend Data
Management at Jagoda Wholesalers, in Calgary, Canada, has used time-
series regression based on point-of-sale data to forecast sales for the next 4
quarters. Sales estimates are $100,000, $120,000, $140,000, and $160,000
for the respective quarters. Seasonal indices for the four quarters have been
found to be 1.30, .90, .70, and 1.10, respectively

ŷseasonal = Index ´ ŷtrend forecast

Quarter I: ŷI = (1.30)($100,000) = $130,000

Quarter II: ŷII = (.90)($120,000) = $108,000
Quarter III: ŷIII = (.70)($140,000) = $98,000
Quarter IV: ŷIV = (1.10)($160,000) = $176,000
 Cyclical Variations

▶ Cycles – patterns in the data that

occur every several years
▶ Forecasting is difficult
▶ Wide variety of factors
 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
 Associative Forecasting
Forecasting an outcome based on predictor
variables using the least squares technique

y^ = a + bx

where ^y = value of the dependent variable (in our example,

sales)
a = y-axis intercept
b = slope of the regression line
x = the independent variable
 Associative Forecasting
Example
NODEL’S SALES AREA PAYROLL NODEL’S SALES AREA PAYROLL
(IN $ MILLIONS), y (IN $ BILLIONS), x (IN $ MILLIONS), y (IN $ BILLIONS), x
2.0 1 2.0 2
3.0 3 2.0 1
2.5 4 3.5 7

4.0 –
Nodel’s sales
(in $ millions)

3.0 –

2.0 –

1.0 –

| | | | | | |

0 1 2 3 4 5 6 7
Area payroll (in $ billions)
 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
Σy = 15.0 Σx = 18 Σx2 = 80 Σxy = 51.5

x=
å x 18
= =3 y=
å y 15
= = 2.5
6 6 6 6

b=
å xy - nxy 51.5 - (6)(3)(2.5)
= = .25 a = y - bx = 2.5 - (.25)(3) = 1.75
å x - nx
2 2
80 - (6)(3 ) 2
 Associative Forecasting
Example
SALES, y PAYROLL, x x2 xy
2.0 1 1 2.0
3.0 3
ŷ = 1.75
9
+ .25x 9.0
2.5 4 16 10.0
2.0 2 Sales = 1.75
4 + .25(payroll)
4.0
2.0 1 1 2.0
3.5 7 49 24.5
Σy = 15.0 Σx = 18 Σx2 = 80 Σxy = 51.5

x=
å x 18
= =3 y=
å y 15
= = 2.5
6 6 6 6

b=
å xy - nxy 51.5 - (6)(3)(2.5)
= = .25 a = y - bx = 2.5 - (.25)(3) = 1.75
å x - nx
2 2
80 - (6)(3 ) 2
 Associative Forecasting
Example
SALES, y PAYROLL, x x2 xy
2.0 1 1 2.0
4.0 –
3.0 3
ŷ = 1.75
9
+ .25x 9.0
Nodel’s sales
(in $ millions)

2.5 3.0 – 4 16 10.0

2.0 2 Sales = 1.75
4 + .25(payroll)
4.0
2.0 2.0 – 1 1 2.0
3.5 7 49 24.5
1.0 –
Σy = 15.0 Σx = 18 Σx2 = 80 Σxy = 51.5
| | | | | | |

x=
0å x 1 18 2
= =3
3 å
4 y 5 15 6
y =(in $ billions)
= = 2.5
7
Area payroll
6 6 6 6

b=
å xy - nxy 51.5 - (6)(3)(2.5)
= = .25 a = y - bx = 2.5 - (.25)(3) = 1.75
å x - nx
2
80 - (6)(3 )
2 2
 Associative Forecasting
Example

If payroll next year is estimated to be $6 billion,

then:

Sales (in $ millions) = 1.75 + .25(6)

= 1.75 + 1.5 = 3.25

Sales = $3,250,000
 Associative Forecasting
Example

If payroll next
4.0 year
– is estimated to be $6 billion,
then: 3.25
Nodel’s sales
(in $ millions)

3.0 –

2.0 –
Sales (in$ millions) = 1.75 + .25(6)
1.0 –
= 1.75 + 1.5 = 3.25
| | | | | | |
0 1 2 3 4 5 6 7
Sales = $3,250,000
Area payroll (in $ billions)
Standard Error of the Estimate
► A forecast is just a point estimate of a
future value
► This point is
actually the
4.0 –
mean or 3.25
Nodel’s sales
(in $ millions)
3.0 –
expected Regression line,
value of a 2.0 – ŷ =1.75+.25x

probability 1.0 –
distribution | | | | | | |
0 1 2 3 4 5 6 7
Figure 4.9 Area payroll (in $ billions)
Standard Error of the Estimate

S y,x =
å ( y - y c
) 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
Standard Error of the Estimate

Computationally, this equation is

considerably easier to use

S y,x =
å - aå y - bå xy
y 2

n-2

We use the standard error to set up

prediction intervals around the point
estimate
 Standard Error of the Estimate

S y,x =
å - aå y - bå xy
y 2

=
39.5 -1.75(15.0) - .25(51.5)
n-2 6-2
= .09375
= .306 (in $ millions)
3.6 –
Nodel’s sales 3.5 – + .306
(in $ millions) 3.4 –
3.3 –
The standard error 3.2 –
3.1 –
of the estimate is 3.0 –
– .306
$306,000 in sales 2.9 –

5 6
Area payroll (in $ billions)
 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
 Correlation Coefficient
Figure 4.10
y y

x x
(a) Perfect negative (e) Perfect positive
correlation, r = –1 y y correlation, r = 1

y
x x
(b) Negative correlation (d) Positive correlation

x
(c) No correlation, r = 0

High Moderate Low Low Moderate High

| | | | | | | | |

–1.0 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1.0
Correlation coefficient values
 Correlation Coefficient

nå xy - å xå y
r=
é ùé ù
êënå x - ( )
å x úûêënå y - ( )
å y úû
2 2
2 2
 Correlation Coefficient
y x x2 xy y2
2.0 1 1 2.0 4.0
3.0 3 9 9.0 9.0
2.5 4 16 10.0 6.25
2.0 2 4 4.0 4.0
2.0 1 1 2.0 4.0
3.5 7 49 24.5 12.25
Σy = 15.0 Σx = 18 Σx2 = 80 Σxy = 51.5 Σy2 = 39.5

(6)(51.5) – (18)(15.0)
r=
é(6)(80) – (18)2 ùé(16)(39.5) – (15.0)2 ù
ë ûë û

309 - 270 39 39
= = = = .901
(156)(12) 1,872 43.3
 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
 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

ŷ = a + b1x1 + b2 x2

Computationally, this is quite

complex and generally done on the
computer