forecasting ppt @ bec doms
TRANSCRIPT
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Forecasting
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Outline
GLOBAL COMPANY PROFILE: TUPPERWARE CORPORATION
WHAT IS FORECASTING? Forecasting Time Horizons The Influence of Product Life Cycle
TYPES OF FORECASTS THE STRATEGIC IMPORTANCE OF
FORECASTING Human Resources Capacity Supply-Chain Management
SEVEN STEPS IN THE FORECASTING SYSTEM
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Outline - Continued
FORECASTING APPROACHES Overview of Qualitative Methods Overview of Quantitative Methods
TIME-SERIES FORECASTING Decomposition of Time Series Naïve Approach Moving Averages Exponential Smoothing Exponential Smoothing with Trend
Adjustment Trend Projections Seasonal Variations in Data Cyclic Variations in Data
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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
MONITORING AND CONTROLLING FORECASTS Adaptive Smoothing Focus Forecasting
FORECASTING IN THE SERVICE SECTOR
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Learning Objectives
When you complete this chapter, you should be able to :Identify or Define:
Forecasting
Types of forecasts
Time horizons
Approaches to forecasts
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Learning Objectives - continued
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
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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
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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
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Tupperware - 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”
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What is Forecasting?
Process of predicting a future event
Underlying basis of all business decisions
Production
Inventory
Personnel
Facilities
Sales will be $200 Million!
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Short-range forecast Up to 1 year; usually less than 3 months Job scheduling, worker assignments
Medium-range forecast 3 months to 3 years Sales & production planning, budgeting
Long-range forecast 3+ years New product planning, facility location
Types of Forecasts by Time Horizon
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Short-term vs. Longer-term Forecasting
Medium/long range forecasts deal with more comprehensive issues and support management decisions regarding planning and products, plants and processes.
Short-term forecasting usually employs different methodologies than longer-term forecasting
Short-term forecasts tend to be more accurate than longer-term forecasts.
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Influence of Product Life Cycle
Stages of introduction and growth require longer forecasts than maturity and decline
Forecasts useful in projecting staffing levels,
inventory levels, and
factory capacity
as product passes through life cycle stages
Introduction, Growth, Maturity, Decline
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Strategy and Issues During a Product’s Life
Introduction Growth Maturity Decline
Standardization
Less rapid product changes - more minor changes
Optimum capacity
Increasing stability of process
Long production runs
Product improvement and cost cutting
Little product differentiation
Cost minimization
Over capacity in the industry
Prune line to eliminate items not returning good margin
Reduce capacity
Forecasting critical
Product and process reliability
Competitive product improvements and options
Increase capacity
Shift toward product focused
Enhance distribution
Product design and development critical
Frequent product and process design changes
Short production runs
High production costs
Limited models
Attention to quality
Best period to increase market share
R&D product engineering critical
Practical to change price or quality image
Strengthen niche
Cost control critical
Poor time to change image, price, or quality
Competitive costs become critical
Defend market position
OM
Str
ateg
y/Is
sues
Com
pany
Str
ateg
y/Is
sues
HDTV
CD-ROM
Color copiers
Drive-thru restaurants Fax machines
Station wagons
Sales
3 1/2” Floppy disks
Internet
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Types of Forecasts
Economic forecasts Address business cycle, e.g., inflation rate,
money supply etc.
Technological forecasts Predict rate of technological progress Predict acceptance of new product
Demand forecasts Predict sales of existing product
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Seven Steps in Forecasting
Determine the use of the forecast Select the items to be forecasted Determine the time horizon of the forecast Select the forecasting model(s) Gather the data Make the forecast Validate and implement results
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Product Demand Charted over 4 Years with Trend and Seasonality
Year1
Year2
Year3
Year4
Seasonal peaks Trend component
Actual demand line
Average demand over four years
Dem
and
for p
rodu
ct o
r ser
vice
Random variation
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Actual Demand, Moving Average, Weighted Moving Average
0
5
10
15
20
25
30
35
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Month
Sal
es D
eman
d
Actual sales
Moving average
Weighted moving average
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Realities of Forecasting
Forecasts are seldom perfect Most forecasting methods assume that there
is some underlying stability in the system Both product family and aggregated product
forecasts are more accurate than individual product forecasts
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Forecasting Approaches
♦ Used when situation is ‘stable’ & historical data exist
♦ Existing products♦ Current technology
♦ Involves mathematical techniques
♦ e.g., forecasting sales of color televisions
Quantitative Methods♦ Used when situation is
vague & little data exist♦ New products♦ New technology
♦ Involves intuition, experience
♦ e.g., forecasting sales on Internet
Qualitative Methods
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Overview of Qualitative Methods
Jury of executive opinion Pool opinions of high-level executives,
sometimes augment by statistical models
Delphi method Panel of experts, queried iteratively
Sales force composite Estimates from individual salespersons are
reviewed for reasonableness, then aggregated
Consumer Market Survey Ask the customer
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Involves small group of high-level managersGroup estimates demand by working together
Combines managerial experience with statistical models
Relatively quick
‘Group-think’disadvantage
© 1995 Corel Corp.
Jury of Executive Opinion
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Sales Force Composite
Each salesperson projects his or her sales
Combined at district & national levels
Sales reps know customers’ wants
Tends to be overly optimistic
Sales
© 1995 Corel Corp.
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Delphi Method
Iterative group process 3 types of people
Decision makers
Staff
Respondents
Reduces ‘group-think’
Respondents
Staff
Decision Makers(Sales?)
(What will sales be? survey)
(Sales will be 45, 50, 55)
(Sales will be 50!)
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Consumer Market Survey
Ask customers about purchasing plans
What consumers say, and what they actually do are often different
Sometimes difficult to answer
How many hours will you use the
Internet next week?
© 1995 Corel Corp.
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Overview of Quantitative Approaches
Naïve approach Moving averages Exponential smoothing Trend projection
Linear regression
Time-series Models
Associative models
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Quantitative Forecasting Methods (Non-Naive)
QuantitativeForecasting
LinearRegression
AssociativeModels
ExponentialSmoothing
MovingAverage
Time SeriesModels
TrendProjection
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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
ExampleYear: 1998 1999 2000 2001 2002
Sales: 78.7 63.5 89.7 93.2 92.1
What is a Time Series?
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Trend
Seasonal
Cyclical
Random
Time Series Components
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Persistent, overall upward or downward pattern
Due to population, technology etc. Several years duration
Mo., Qtr., Yr.
Response
© 1984-1994 T/Maker Co.
Trend Component
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Regular pattern of up & down fluctuations Due to weather, customs etc. Occurs within 1 year
Mo., Qtr.
Response
Summer
© 1984-1994 T/Maker Co.
Seasonal Component
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Common Seasonal PatternsPeriod of Pattern
“Season” Length
Number of “Seasons” in
PatternWeek Day 7
Month Week 4 – 4 ½
Month Day 28 – 31
Year Quarter 4
Year Month 12
Year Week 52
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Repeating up & down movements Due to interactions of factors influencing
economy Usually 2-10 years duration
Mo., Qtr., Yr.
ResponseCycle
C
Cyclical Component
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Erratic, unsystematic, ‘residual’ fluctuations
Due to random variation or unforeseen
events
Union strike
Tornado
Short duration &
nonrepeating
© 1984-1994 T/Maker Co.
Random Component
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Any observed value in a time series is the product (or sum) of time series components
Multiplicative model Yi = Ti · Si · Ci · Ri (if quarterly or mo. data)
Additive model Yi = Ti + Si + Ci + Ri (if quarterly or mo.
data)
General Time Series Models
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Naive Approach
Assumes demand in next period is the same as demand in most recent period
e.g., If May sales were 48, then June sales will be 48
Sometimes cost effective & efficient
© 1995 Corel Corp.
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MA is a series of arithmetic means
Used if little or no trend
Used often for smoothing♦ Provides overall impression of data over
time
EquationMA
n
n= ∑ Demand in Previous Periods
Moving Average Method
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You’re manager of a museum store that sells historical replicas. You want to forecast sales (000) for 2003 using a 3-period moving average.
1998 41999 62000 52001 32002 7
© 1995 Corel Corp.
Moving Average Example
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Moving Average SolutionTime Response
Yi Moving Total (n=3)
Moving Average
(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 15/3 = 5 2002 7 2003 NA
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Moving Average SolutionTime Response
Yi Moving Total (n=3)
Moving Average
(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 15/3 = 5 2002 7 6+5+3=14 14/3=4 2/3 2003 NA
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Moving Average SolutionTime Response
Yi Moving Total (n=3)
Moving Average
(n=3) 1998 4 NA NA 1999 6 NA NA 2000 5 NA NA 2001 3 4+6+5=15 15/3=5.0 2002 7 6+5+3=14 14/3=4.7 2003 NA 5+3+7=15 15/3=5.0
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95 96 97 98 99 00Year
Sales
2
4
6
8 Actual
Forecast
Moving Average Graph
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Used when trend is present Older data usually less important
Weights based on intuition Often lay between 0 & 1, & sum to 1.0
Equation
WMA =Σ(Weight for period n) (Demand in period n)
ΣWeights
Weighted Moving Average Method
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Actual Demand, Moving Average, Weighted Moving Average
0
5
10
15
20
25
30
35
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Month
Sal
es D
eman
d
Actual sales
Moving average
Weighted moving average
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Increasing n makes forecast less sensitive to changes
Do not forecast trend well Require much historical
data© 1984-1994 T/Maker Co.
Disadvantages of Moving Average Methods
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Form of weighted moving average Weights decline exponentially
Most recent data weighted most
Requires smoothing constant (α) Ranges from 0 to 1
Subjectively chosen
Involves little record keeping of past data
Exponential Smoothing Method
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Ft = αAt - 1 + α(1-α)At - 2 + α(1- α)2·At - 3
+ α(1- α)3At - 4 + ... + α(1- α)t-1·A0
Ft = Forecast value
At = Actual value
α = Smoothing constant
Ft = Ft-1 + α(At-1 - Ft-1) Use for computing forecast
Exponential Smoothing Equations
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During the past 8 quarters, the Port of Baltimore has unloaded large quantities of grain. (α = .10). The first quarter forecast was 175.. Quarter Actual
1 180 2 168
3 1594 1755 190
6 2057 1808 1829 ?
Exponential Smoothing Example
Find the forecast for the 9th quarter.
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Ft = Ft-1 + 0.1(At-1 - Ft-1)
Quarter ActualForecast, Ft
( α = .10)
1 180 175.00 (Given)
2 168
3 159
4 175
5 190
6 205
175.00 +
Exponential Smoothing Solution
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Quarter ActualForecast, Ft
( α = .10)
1 180 175.00 (Given)
2 168 175.00 + .10(
3 159
4 175
5 190
6 205
Exponential Smoothing SolutionFt = Ft-1 + 0.1(At-1 - Ft-1)
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Quarter ActualForecast, Ft
( α = .10)
1 180 175.00 (Given)
2 168 175.00 + .10(180 -
3 159
4 175
5 190
6 205
Exponential Smoothing SolutionFt = Ft-1 + 0.1(At-1 - Ft-1)
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Quarter ActualForecast, Ft
( α = .10)
1 180 175.00 (Given)
2 168 175.00 + .10(180 - 175.00)
3 159
4 175
5 190
6 205
Exponential Smoothing SolutionFt = Ft-1 + 0.1(At-1 - Ft-1)
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Quarter ActualForecast, Ft
( α = .10)
1 180 175.00 (Given)
2 168 175.00 + .10(180 - 175.00) = 175.50
3 159
4 175
5 190
6 205
Exponential Smoothing SolutionFt = Ft-1 + 0.1(At-1 - Ft-1)
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Ft = Ft-1 + 0.1(At-1 - Ft-1)
Quarter ActualForecast, Ft
( α = .10)
1 180 175.00 (Given)
2 168 175.00 + .10(180 - 175.00) = 175.50
3 159 175.50 + .10(168 - 175.50) = 174.75
4 175
5 190
6 205
Exponential Smoothing Solution
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Ft = Ft-1 + 0.1(At-1 - Ft-1)
Quarter ActualForecast, Ft
( α = .10)
1995 180 175.00 (Given)
1996 168 175.00 + .10(180 - 175.00) = 175.50
1997 159 175.50 + .10(168 - 175.50) = 174.75
1998 175
1999 190
2000 205
174.75 + .10(159 - 174.75)= 173.18
Exponential Smoothing Solution
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Ft = Ft-1 + 0.1(At-1 - Ft-1)
Quarter ActualForecast, Ft
( α = .10)
1 180 175.00 (Given)
2 168 175.00 + .10(180 - 175.00) = 175.50
3 159 175.50 + .10(168 - 175.50) = 174.75
4 175 174.75 + .10(159 - 174.75) = 173.18
5 190 173.18 + .10(175 - 173.18) = 173.36
6 205
Exponential Smoothing Solution
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Ft = Ft-1 + 0.1(At-1 - Ft-1)
Quarter ActualForecast, Ft
( α = .10)
1 180 175.00 (Given)
2 168 175.00 + .10(180 - 175.00) = 175.50
3 159 175.50 + .10(168 - 175.50) = 174.75
4 175 174.75 + .10(159 - 174.75) = 173.18
5 190 173.18 + .10(175 - 173.18) = 173.36
6 205 173.36 + .10(190 - 173.36) = 175.02
Exponential Smoothing Solution
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Ft = Ft-1 + 0.1(At-1 - Ft-1)
Time ActualForecast, Ft
( α = .10)
4 175 174.75 + .10(159 - 174.75) = 173.18
5 190 173.18 + .10(175 - 173.18) = 173.36
6 205 173.36 + .10(190 - 173.36) = 175.02
Exponential Smoothing Solution
7 180
8
175.02 + .10(205 - 175.02) = 178.02
9
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Ft = Ft-1 + 0.1(At-1 - Ft-1)
Time ActualForecast, Ft
( α = .10)
4 175 174.75 + .10(159 - 174.75) = 173.18
5 190 173.18 + .10(175 - 173.18) = 173.36
6 205 173.36 + .10(190 - 173.36) = 175.02
Exponential Smoothing Solution
7 180
8
175.02 + .10(205 - 175.02) = 178.02
9 178.22 + .10(182 - 178.22) = 178.58 182 178.02 + .10(180 - 178.02) = 178.22?
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Ft = α At - 1 + α(1- α)At - 2 + α(1- α)2At - 3 + ...
Forecast Effects of Smoothing Constant α
Weights
Prior Period
α
2 periods ago
α (1 - α )
3 periods ago
α (1 - α )2
α=
α= 0.10
α= 0.90
10%
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Ft = α At - 1 + α(1- α) At - 2 + α(1- α)2At - 3 + ...
Forecast Effects of Smoothing Constant α
Weights
Prior Period
α
2 periods ago
α (1 - α )
3 periods ago
α (1 - α )2
α=
α= 0.10
α= 0.90
10% 9%
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Ft = α At - 1 + α(1- α)At - 2 + α(1- α)2At - 3 + ...
Forecast Effects of Smoothing Constant α
Weights
Prior Period
α
2 periods ago
α (1 - α )
3 periods ago
α (1 - α )2
α=
α= 0.10
α= 0.90
10% 9% 8.1%
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Ft = α At - 1 + α(1- α)At - 2 + α(1- α)2At - 3 + ...
Forecast Effects of Smoothing Constant α
Weights
Prior Period
α
2 periods ago
α (1 - α )
3 periods ago
α (1 - α )2
α=
α= 0.10
α= 0.90
10% 9% 8.1%
90%
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Ft = α At - 1 + α(1- α) At - 2 + α(1- α)2At - 3 + ...
Forecast Effects of Smoothing Constant α
Weights
Prior Period
α
2 periods ago
α (1 - α )
3 periods ago
α (1 - α )2
α=
α= 0.10
α= 0.90
10% 9% 8.1%
90% 9%
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Ft = α At - 1 + α(1- α) At - 2 + α(1- α)2At - 3 + ...
Forecast Effects of Smoothing Constant α
Weights
Prior Period
α
2 periods ago
α (1 - α )
3 periods ago
α (1 - α )2
α=
α= 0.10
α= 0.90
10% 9% 8.1%
90% 9% 0.9%
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Impact of α
0
50
100
150
200
250
1 2 3 4 5 6 7 8 9
Quarter
Actua
l Ton
age Actual
Forecast (0.1)
Forecast (0.5)
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Choosing α
Seek to minimize the Mean Absolute Deviation (MAD)
If: Forecast error = demand - forecast
Then:n
errorsforecast ∑=MAD
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Exponential Smoothing with Trend Adjustment
Forecast including trend (FITt)
= exponentially smoothed forecast (Ft)
+ exponentially smoothed trend (Tt)
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Ft = Last period’s forecast
+ α (Last period’s actual – Last period’s forecast)
Ft = Ft-1 + α (At-1 – Ft-1)or
Tt = β (Forecast this period - Forecast last period)
+ (1- β )(Trend estimate last period
Tt = β (Ft - Ft-1) + (1- β )Tt-1 or
Exponential Smoothing with Trend Adjustment - continued
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Ft = exponentially smoothed forecast of the data series in period t
Tt = exponentially smoothed trend in period t
At = actual demand in period t
α = smoothing constant for the average β = smoothing constant for the trend
Exponential Smoothing with Trend Adjustment - continued
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Comparing Actual and Forecasts
0
5
10
15
20
25
30
35
40
1 2 3 4 5 6 7 8 9 10
Month
Dem
and
Actual Demand
Smoothed Forecast
Smoothed Trend
Forecast including trend
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Regression
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Least Squares
Deviation
Deviation
Deviation
Deviation
Deviation
Deviation
Deviation
Time
Valu
es o
f Dep
ende
nt V
aria
ble
bxaY +=ˆ
Actual observation
Point on regression line
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Actual and the Least Squares Line
0
20
40
60
80
100
120
140
160
1996 1997 1998 1999 2000 2001 2002 2003 2004
Year
Regress io n Line
Actual D em and
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Used for forecasting linear trend line Assumes relationship between response
variable, Y, and time, X, is a linear function
Estimated by least squares method Minimizes sum of squared errors
iY a bX i= +
Linear Trend Projection
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Y a bXi i= +b > 0
b < 0
a
a
Y
Time, X
Linear Trend Projection Model
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Scatter DiagramSales versus Payroll
0
1
2
3
4
0 1 2 3 4 5 6 7 8
Area Payroll (in $ hundreds of millions)
Sales
(in $
hund
reds o
f tho
usan
ds)
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Least Squares Equations
Equation: ii bxaY +=
Slope:22
1=
1=
−∑
−∑=
xnx
yxnyxb
i
n
i
ii
n
i
Y-Intercept: xbya −=
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X i Y i X i2 Y i
2 X iY i
X 1 Y 1 X 12 Y 1
2 X 1Y 1
X 2 Y 2 X 22 Y 2
2 X 2Y 2
: : : : :
Xn Yn Xn2 Yn
2 XnYn
ΣX i ΣY i ΣX i2 ΣY i
2 ΣX iY i
Computation Table
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Using a Trend Line
Year Demand
1997 74
1998 79
1999 80
2000 90
2001 105
2002 142
2003 122
The demand for electrical power at N.Y.Edison over the years 1997 – 2003 is given at the left. Find the overall trend.
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Finding a Trend LineYear Time
PeriodPower
Demandx2 xy
1997 1 74 1 74
1998 2 79 4 158
1999 3 80 9 240
2000 4 90 16 360
2001 5 105 25 525
2002 6 142 36 852
2003 7 122 49 854
Σx=28 Σy=692 Σx2=140 Σxy=3,063
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The Trend Line Equation
megawatts 151.56 10.54(9) 56.70 2005in Demand
megawatts 141.02 10.54(8) 56.70 2004in Demand
56.70 10.54(4) - 98.86 xb - y a
10.5428
295
(7)(4)140
86)(7)(4)(98.3,063
xnΣx
yxn -Σxy b
98.867
692
n
Σyy 4
7
28
n
Σxx
222
=+=
=+=
===
==−
−=−
=
======
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Actual and Trend ForecastElectric Power Demand
60
70
80
90
100
110
120
130
140
150
160
1997 1998 1999 2000 2001 2002 2003 2004 2005
Year
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Monthly Sales of Laptop ComputersSales Demand Average Demand
Month 2000 2001 2002 2000-2002 Monthly Seasonal IndexJan 80 85 105 90 94 0.957Feb 70 85 85 80 94 0.851Mar 80 93 82 85 94 0.904Apr 90 95 115 100 94 1.064May 113 125 131 123 94 1.309Jun 110 115 120 115 94 1.223Jul 100 102 113 105 94 1.117Aug 88 102 110 100 94 1.064Sept 85 90 95 90 94 0.957Oct 77 78 85 80 94 0.851Nov 75 72 83 80 94 0.851Dec 82 78 80 80 94 0.851
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Demand for IBM Laptops
0
20
40
60
80
100
120
140
Jan Feb M ar Apr M ay Jun Jul Aug Sep Oct Nov D ec
Month
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
Trend
Seaso nal Index
Fo recast : trend + seasonal
index
M onthly
Average
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San Diego Hospital – Inpatient Days
8800
9000
9200
9400
9600
9800
10000
10200
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
0.92
0.94
0.96
0.98
1
1.02
1.04
1.06
Seasonal Index
Trend
Combined Forecast
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Multiplicative Seasonal Model Find average historical demand for each “season” by
summing the demand for that season in each year, and dividing by the number of years for which you have data.
Compute the average demand over all seasons by dividing the total average annual demand by the number of seasons.
Compute a seasonal index by dividing that season’s historical demand (from step 1) by the average demand over all seasons.
Estimate next year’s total demand Divide this estimate of total demand by the number of
seasons, then multiply it by the seasonal index for that season. This provides the seasonal forecast.
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Y Xi i= a b
Shows linear relationship between dependent & explanatory variables Example: Sales & advertising (not time)
Dependent (response) variable
Independent (explanatory) variable
SlopeY-intercept
^
Linear Regression Model
+
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Y
X
Y aii
^i i
b Xi = =++ Error
Error
Observed value
Y a b X= =+
Regression line
Linear Regression Model
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Linear Regression Equations
Equation: ii bxaY +=
Slope:22
i
n
1i
ii
n
1i
xnx
yxnyx b
−∑
−∑=
=
=
Y-Intercept: xby a −=
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X i Y i X i2 Y i
2 X iY i
X 1 Y 1 X 12 Y 1
2 X 1Y 1
X 2 Y 2 X 22 Y 2
2 X 2Y 2
: : : : :
Xn Yn Xn2 Yn
2 XnYn
ΣX i ΣY i ΣX i2 ΣY i
2 ΣX iY i
Computation Table
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Slope (b) Estimated Y changes by b for each 1 unit
increase in X If b = 2, then sales (Y) is expected to increase by 2
for each 1 unit increase in advertising (X)
Y-intercept (a) Average value of Y when X = 0
If a = 4, then average sales (Y) is expected to be 4 when advertising (X) is 0
Interpretation of Coefficients
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Variation of actual Y from predicted Y Measured by standard error of estimate
Sample standard deviation of errors
Denoted SY,X
Affects several factors Parameter significance
Prediction accuracy
Random Error Variation
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Least Squares Assumptions
Relationship is assumed to be linear. Plot the data first - if curve appears to be present, use curvilinear analysis.
Relationship is assumed to hold only within or slightly outside data range. Do not attempt to predict time periods far beyond the range of the data base.
Deviations around least squares line are assumed to be random.
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Standard Error of the Estimate
( )
2
2
1 11
2
1
2
,
−
−−=
−
−=
∑ ∑∑
∑
= ==
=
n
yxbyay
n
yyS
n
i
n
iiii
n
ii
n
ici
xy
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Answers: ‘how strong is the linear relationship between the variables?’
Coefficient of correlation Sample correlation coefficient denoted r Values range from -1 to +1
Measures degree of association
Used mainly for understanding
Correlation
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Sample Coefficient of Correlation
−
−
−=
∑ ∑∑ ∑
∑ ∑ ∑
1=
2
1=
2
1=
2
1=
2
1= 1= 1=
n
i
n
iii
n
i
n
iii
n
i
n
i
n
iiiii
yynxxn
yxyxnr
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-1.0 +1.00
Perfect Positive
Correlation
Increasing degree of negative correlation
-.5 +.5
Perfect Negative
CorrelationNo Correlation
Increasing degree of positive correlation
Coefficient of Correlation Values
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r = 1 r = -1
r = .89 r = 0
Y
XYi = a + b Xi^
Y
X
Y
X
Y
XYi = a + b Xi^ Yi = a + b Xi
^
Yi = a + b Xi^
Coefficient of Correlation and Regression Model
r2 = square of correlation coefficient (r), is the percent of the variation in y that is explained by the regression equation
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You want to achieve: No pattern or direction in forecast error
Error = (Yi - Yi) = (Actual - Forecast)
Seen in plots of errors over time
Smallest forecast error Mean square error (MSE)
Mean absolute deviation (MAD)
Guidelines for Selecting Forecasting Model
^
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Time (Years)
Error
0
Desired Pattern
Time (Years)
Error
0
Trend Not Fully Accounted for
Pattern of Forecast Error
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Mean Square Error (MSE)
Mean Absolute Deviation (MAD)
Mean Absolute Percent Error (MAPE)
Forecast Error Equations
2
n
1i
2ii
n
errorsforecast
n
)y(yMSE ∑∑
=−
= =
nn
yyMAD
n
iii ∑∑
=−
= = |errorsforecast ||ˆ|
1
n
actual
forecastactual
100MAPE
n
1i i
ii∑=
−
=
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You’re a marketing analyst for Hasbro Toys. You’ve forecast sales with a linear model & exponential smoothing. Which model do you use?
ActualLinear Model Exponential
SmoothingYear Sales
Forecast Forecast (.9)
1998 10.6 1.0
1999 11.3 1.0
2000 22.0 1.9
2001 22.7 2.0
2002 43.4 3.8
Selecting Forecasting Model Example
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MSE = Σ Error2 / n = 1.10 / 5 = 0.220
MAD = Σ |Error| / n = 2.0 / 5 = 0.400
MAPE = 100 Σ|absolute percent errors|/n= 1.20/5 = 0.240
Linear Model EvaluationY i
1
1
2
2
4
Y i^
0.6
1.3
2.0
2.7
3.4
Year
1998
1999
2000
2001
2002
Total
0.4
-0.3
0.0
-0.7
0.6
0.0
Error
0.16
0.09
0.00
0.49
0.36
1.10
Error2
0.4
0.3
0.0
0.7
0.6
2.0
|Error||Error|
Actual0.400.30
0.000.35
0.15
1.20
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MSE = Σ Error2 / n = 0.05 / 5 = 0.01MAD = Σ |Error| / n = 0.3 / 5 = 0.06MAPE = 100 Σ |Absolute percent errors|/n = 0.10/5 = 0.02
Exponential Smoothing Model Evaluation
Year
1998
1999
2000
2001
2002
Total
Y i1
1
2
2
4
Y i1.0 0.0
1.0 0.0
1.9 0.1
2.0 0.0
3.8 0.2
0.3
^ Error
0.00
0.00
0.01
0.00
0.04
0.05 0.3
Error2
0.0
0.0
0.1
0.0
0.2
|Error||Error|
Actual0.000.00
0.050.000.05
0.10
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Exponential Smoothing Model Evaluation
Linear Model:MSE = Σ Error2 / n = 1.10 / 5 = .220
MAD = Σ |Error| / n = 2.0 / 5 = .400
MAPE = 100 Σ|absolute percent errors|/n= 1.20/5 = 0.240
Exponential Smoothing Model:MSE = Σ Error2 / n = 0.05 / 5 = 0.01
MAD = Σ |Error| / n = 0.3 / 5 = 0.06
MAPE = 100 Σ |Absolute percent errors|/n = 0.10/5 = 0.02
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Measures how well the forecast is predicting actual values
Ratio of running sum of forecast errors (RSFE) to mean absolute deviation (MAD) Good tracking signal has low values
Should be within upper and lower control limits
Tracking Signal
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Tracking Signal Equation
( )
MAD
errorforecast
MAD
yy
MADRSFE
TS
n
iii
∑
∑
=
−=
=
1=
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Mo Fcst Act Error RSFE AbsError
Cum MAD TS
1 100 90
2 100 95
3 100 115
4 100 100
5 100 125
6 100 140
|Error|
Tracking Signal Computation
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Mo Forc Act Error RSFE AbsError
Cum MAD TS
1 100 90
2 100 95
3 100 115
4 100 100
5 100 125
6 100 140
-10
Error = Actual - Forecast = 90 - 100 = -10
|Error|
Tracking Signal Computation
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Mo Forc Act Error RSFE AbsError
Cum MAD TS
1 100 90
2 100 95
3 100 115
4 100 100
5 100 125
6 100 140
-10 -10
RSFE = Σ Errors = NA + (-10) = -10
|Error|
Tracking Signal Computation
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Mo Forc Act Error RSFE AbsError
Cum MAD TS
1 100 90
2 100 95
3 100 115
4 100 100
5 100 125
6 100 140
-10 -10 10
Abs Error = |Error| = |-10| = 10
|Error|
Tracking Signal Computation
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Mo Forc Act Error RSFE AbsError
Cum MAD TS
1 100 90
2 100 95
3 100 115
4 100 100
5 100 125
6 100 140
-10 -10 10 10
Cum |Error| = Σ |Errors| = NA + 10 = 10
|Error|
Tracking Signal Computation
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Mo Forc Act Error RSFE AbsError
Cum|Error|
MAD TS
1 100 90
2 100 95
3 100 115
4 100 100
5 100 125
6 100 140
-10 -10 10 10 10.0
MAD = Σ |Errors|/n = 10/1 = 10
Tracking Signal Computation
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Mo Forc Act Error RSFE AbsError
Cum MAD TS
1 100 90
2 100 95
3 100 115
4 100 100
5 100 125
6 100 140
-10 -10 10 10 10.0 -1
TS = RSFE/MAD = -10/10 = -1
|Error|
Tracking Signal Computation
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Mo Forc Act Error RSFE AbsError
Cum MAD TS
1 100 90
2 100 95
3 100 115
4 100 100
5 100 125
6 100 140
-10 -10 10 10 10.0 -1
-5
Error = Actual - Forecast = 95 - 100 = -5
|Error|
Tracking Signal Computation
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Mo Forc Act Error RSFE AbsError
Cum MAD TS
1 100 90
2 100 95
3 100 115
4 100 100
5 100 125
6 100 140
-10 -10 10 10 10.0 -1
-5 -15
RSFE = Σ Errors = (-10) + (-5) = -15
|Error|
Tracking Signal Computation
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Mo Forc Act Error RSFE AbsError
Cum MAD TS
1 100 90
2 100 95
3 100 115
4 100 100
5 100 125
6 100 140
-10 -10 10 10 10.0 -1
-5 -15 5
Abs Error = |Error| = |-5| = 5
|Error|
Tracking Signal Computation
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Mo Forc Act Error RSFE AbsError
Cum MAD TS
1 100 90
2 100 95
3 100 115
4 100 100
5 100 125
6 100 140
-10 -10 10 10 10.0 -1
-5 -15 5 15
Cum Error = Σ |Errors| = 10 + 5 = 15
|Error|
Tracking Signal Computation
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Mo Forc Act Error RSFE AbsError
Cum MAD TS
1 100 90
2 100 95
3 100 115
4 100 100
5 100 125
6 100 140
-10 -10 10 10 10.0 -1
-5 -15 5 15 7.5
MAD = Σ |Errors|/n = 15/2 = 7.5
|Error|
Tracking Signal Computation
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Mo Forc Act Error RSFE AbsError
Cum MAD TS
1 100 90
2 100 95
3 100 115
4 100 100
5 100 125
6 100 140
-10 -10 10 10 10.0 -1
-5 -15 5 15 7.5 -2
|Error|
TS = RSFE/MAD = -15/7.5 = -2
Tracking Signal Computation
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Plot of a Tracking Signal
Time
Lower control limit
Upper control limit
Signal exceeded limit
Tracking signal
Acceptable range
MAD
+
0
-
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Tracking Signals
020406080
100120140160
0 1 2 3 4 5 6 7
Time
Act
ual D
eman
d
-3
-2
-1
0
1
2
3
Tra
ckin
g Si
ngal
Tracking Signal
Forecast
Actual demand
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Forecasting in the Service Sector
Presents unusual challenges special need for short term records
needs differ greatly as function of industry and product
issues of holidays and calendar
unusual events
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Forecast of Sales by Hour for Fast Food Restaurant
0
5
10
15
20
+11-12+1-2 +3-4 +5-6 +7-8 +9-1011-12 12-1 1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11