Chapter 3
Forecasting
1
Forecast
• OM is mostly proactive not reactive
• It involves structured planning activities
• Planning requires data pertaining to the feature
• Forecast: A statement about the future
– Not necessarily numerical
• Weather forecasts
2
Uses of Forecasts
Accounting
Cost/profit estimates
Finance
Cash flow and funding
Human Resources
Hiring/recruiting/training
Marketing
Pricing, promotion, strategy
MIS
IT/IS systems, services
Operations
Schedules, MRP, workloads
Product/service design
New products and services
3
REMARKS
• Assume a causal system
– Future resembles the past
• Forecasts rarely perfect because of randomness
• Forecasts more accurate for groups vs. individuals.
– Forecasting errors among items in a group usually have a canceling
effect.
– Extremes in a group cancel each other
• Ex. I can forecast the class average from the midterm better than
Mrs. X’s individual grade.
– Sample variance of {-1,1,-1,1} is 1.
– Sample variance of {(-1+1)/2, {(-1+1)/2} is 0.
• Forecast accuracy decreases as time horizon for forecasts increases
• Ex. I can forecast this year’s class average better than next year’s
class average
4
Elements of a Good Forecast
Timely
Reliable
Accurate
Written
5
Steps in the Forecasting Process
“The forecast”
Step 6 Monitor the forecast
Step 5 Prepare the forecast
Step 4 Gather and analyze data
Step 3 Select a forecasting technique
Step 2 Establish a time horizon
Step 1 Determine purpose of forecast
6
Types of Forecasts
• Judgmental - Subjective analysis of subjective inputs
• Associative models – Analyzes historical data to reveal
relationships between (easily or in advance) observable
quantities and forecast quantities. Uses this relationship to
make predictions.
• Time series – Objective analysis historical data assuming the
future will be like the past
7
Judgmental Forecasts
• Executive opinions (long-range planning)
– There are factors hard to quantify
• Ex: Effects of November 2004 election on new houses built in 2005
• Sales force composite
– Retailer forecasts for the manufacturer
• Consumer surveys
– The guy at the mall who asks if you like cherry flavor in your shampoo
• Outside opinion
– Financial and consulting gurus and companies
• Opinions of managers and staff
– Delphi method: A series of questionnaires developed sequentially
8
Associative Forecasting
• Based on identification of related variables that can be used
to predict values of the variable of interest.
– Sales of mountain bikes in an area may be related to the percentage
of the young population living in that area.
– Sales of Harley-Davidson motorbikes is related to mid-aged men
population. Average age of H-D owners is 46.
– Ice cream sales can be related to temperature
– Home depot bases sales forecasts on mortgage refinancing rates,
smaller rates imply higher sales.
– Changes in Federal Reserve Board’s interest rate leads to certain
business activities
• House sales
• Industrial investments
– Increase in energy cost leads to price increases in products and
services
9
Associative Forecasting
• Find an association between the predictor and the
predicted
• Predictor variables - used to predict values of variable
interest, sometimes called independent variables
• Predicted variable = Dependent variable
• Regression - technique for fitting a line to a set of points
• Linear regression is the most widely used form of
regression
– The objective is to obtain an equation of a straight line that minimizes the
sum of squared vertical deviations of data points from the line.
10
Linear Regression (cont.)
y = a + bx
Where
y = predicted (dependent) variable
x = predictor (independent) variable
b = slope of the line
a = value of y when x = 0 (the height of line
at the y intercept)
11
Computing a and b
Given n data points, find the intercept a and the slope b to
Minimize the sum of squared errors 
Minimize the sum of deviations from the line 
n
Minimize
2
(
y

a

bx
)
 t
t
t 1
n
b
n
n
n xt yt   xt  yt
t 1
t 1
t 1
n


2
n xt    xt 
t 1
 t 1 
n
2
n
a
y
t 1
n
n
t
b
x
t 1
t
n
12
Linear Model Seems Reasonable
X
7
2
6
4
14
15
16
12
14
20
15
7
Y
15
10
13
15
25
27
24
20
27
44
34
17
Computed
relationship
50
40
30
20
10
0
0
5
10
15
20
25
13
Another Linear Regression Example
Variables: Weeks and Sales
t
Week
1
2
3
4
5
t2
1
4
9
16
25
 t = 15
t2 = 55
(t)2 = 225
y
Sales
150
157
162
166
177
ty
150
314
486
664
885
 y = 812  ty = 2499
14
Linear Trend Calculation
b =
5 (2499) - 15(812)
5(55) - 225
=
12495-12180
275 -225
= 6.3
812 - 6.3(15)
a =
= 143.5
5
y = 143.5 + 6.3 t
Sales in week t = 143.5 + 6.3 t
15
Linear Trend Calculation
y = 143.5 + 6.3t
When t = 0, the value of y is 143.45 and the
slope of the line is 6.3. meaning that the
value of of y will increase by 6.3 units for
each time period. If t = 10, the forecast is
143.5 + 6.3(10) = 206.5
Excel example
regression.xls
16
Linear Regression
Remember from Statistics
• Correlation (r) between variables: The strength and
direction of relationships between two variables
– 1.00 means changes in one variable are always matched
by changes in the other, vice versa.
– A correlation close to zero means little linear relationship
– The square of the correlation coefficient provides a
measure of the percentage of variability in the values of y
that is explained by the independent variable.(80% or
more: the independent variable is a good predictor of the
values of dependent variable)
17
Time series
• Time-ordered sequence of observations taken at regular
intervals over a period of time
• Future values of the series can be estimated from past values.
Types of Variations in Time Series Data
•
•
•
•
•
Trend - long-term movement in data
Seasonality - short-term regular variations in data
Cycles – wavelike variations of long-term
Irregular variations - caused by unusual circumstances
Random variations - caused by chance
18
Forecast Variations
Irregular
variation
Figure 3-1
Trend
Cyclical
Cycles
Year 01
00
99
Seasonal variations
19
Naïve Forecasts
• Uses a single previous value of a time series as the basis
of a forecast.
• Virtually no cost
• Data analysis is nonexistent
• Easily understandable
• Cannot provide high accuracy
– If it were true, future will always be the same as the past
Some notation: Forecast at time t is F(t)
Actual observation at time t is A(t)
Today is temperature is 98 F, A(Today)=98
F(Tomorrow)=98
F(Day after)=98
20
Uses for Naïve Forecasts
• Stable time series data
– Forecast is the same as the last actual observation
– F(t) = A(t-1)
• Seasonal variations
– Forecast is the same as the last actual observation when
we were in the same point in the cycle, where a cycle
lasts n periods.
– F(t) = A(t-n)
• Data with trends
– There is constant trend, the change from (t-2) to (t-1)
will be exactly as the change from (t-1) to (t)
– F(t) = A(t-1) + (A(t-1) – A(t-2))
21
Naive Forecasts
Uh, give me a minute....
We sold 250 wheels last
week.... Now, next
week we should sell....
22
Naïve (Cont.)
• Check if the resulting accuracy is acceptable
• The higher the accuracy, often the higher the cost.
• Do we really need our forecast that accurate? Is it
worth the additional resources?
– Why do you need forecasts for? How critical they are
for operations?
23
Time Series Models: Variations
What is random and what is not?
• Historical data contain random variations or noise
• Random variations are caused by relatively
unimportant factors.
– What is random? Can we not study everything to negligible
detail? “God does not roll dices” –A.E.
• The objective is to remove all randomness and have
real variations.
• Minor variations are random and large ones are real.
24
Techniques for Averaging
• Moving averages (MA)
– Naïve methods just trace the actual data with a lag of
one period, F(t)=A(t-1)
– They don’t smooth
– MA uses a number of the most recent actual data to
smooth
• Weighted moving averages
• Exponential smoothing
25
Simple Moving Average
Note the sensitivity of forecasts
Averaging (over time) techniques are used to smooth variations in the data.
Actual
MA(t,5)
47
45
43
41
39
37
MA(t,3)
35
1
2
3
4
5
6
7
8
9
10 11 12
t 1
Ft  MAt ,n 
MAt ,n
A
i t  n
i
,
n
: MA forecast made in period t - 1 using n actual observatio ns
26
Ex: Three period moving average forecast
Month
1
2
3
4
5
6
Demand
42
MA(6,3) = (43 + 40 + 41) / 3
40
= 41.33.
43
If A(6) = 39, then
40
MA(7,3) = (40 + 41 + 39) / 3
41
= 40.00
39
27
Weighted average
Moving Average
• Advantage=Easy to compute and easy to
understand
• Disadvantage=All values in the average are
weighted equally
Weighted Moving Average
• Similar to moving average
• It assigns more weight to the most recent values in
a time series
– Idea: most recent observations must be better indicators
of the future than older observations
28
Weighted average
Month
1
2
3
4
5
6
Demand
42
40
43
40
41
39
Compute a weighted average forecast using a
weight of 0.4 for the most recent period, 0.3
for the next most recent, 0.2 for the next and
0.1 for the next.
Continuing with the data on the left
F(6) = .40(41)+.30(40)+.20(43)+.10(40)=41.0
If the actual demand for period 6 is 39,
F(7) = .40(39)+.30(41)+.20(40)+.10(43)=40.2
• The weighted average is more reflective of
the most recent occurrences.
29
Exponential Smoothing
Forecast error:=Actual – Forecast =A(t-1)-F(t-1)
Ft  Ft 1   ( At 1  Ft 1 )
Forecast today=Forecast yesterday+(alpha)*(Forecast error yesterday)
Each new forecast is equal to the previous forecast plus a percentage of
the previous error.
Today’s forecast
Depends on yesterday’s (time-wise dependence, strong memory)
But it has to be corrected by forecast error
Therefore, we should give more weight to the more recent time
periods when forecasting.
– Alpha = smoothing constant = percentage of the forecast error.
30
Exponential Smoothing
as an Weighted Average
Ft  At 1  (1   ) Ft 1
Idea--The most recent observations might have the
highest predictive value along with the most recent
forecast errors. Let us balance them:
Ft
At 1
(1   ) Ft 1
31
Example of Exponential Smoothing
Forecasts made in a period and the period has the same color
Period
Actual
1
2
3
4
5
6
7
8
9
10
11
12
42
40
43
40
41
39
46
44
45
38
40
Forecast withAlpha
Error with
= 0.1
Forecast withError with
Alpha=0.1 Alpha=0.1 Alpha=0.4 Alpha=0.4
42
-2.00
42
-2
41.8
1.20
41.2
1.8
41.92
-1.92
41.92
-1.92
41.73
-0.73
41.15
-0.15
41.66
-2.66
41.09
-2.09
41.39
4.61
40.25
5.75
41.85
2.15
42.55
1.45
42.07
2.93
43.13
1.87
42.36
-4.36
43.88
-5.88
41.92
-1.92
41.53
-1.53
41.73
40.92
Ft  At 1  (1   ) Ft 1
32
Picking a Smoothing Constant:
Responsiveness vs. Smoothing
• The quickness of forecast adjustment to error is determined by the
smoothing constant.
• The closer the alpha is to zero, the slower the forecast will be to
adjust to forecast errors.
• Conversely, the closer the value of alpha is to 1.00, the greater the
responsiveness to the actual observations and the less the
smoothing
• Select a smoothing constant that balances the benefits of
responding to real changes if and when they occur.
Ft  Ft 1   ( At 1  Ft 1 )  At 1  (1   ) Ft 1
33
Picking a Smoothing Constant
Sensitivity of Forecasts
Actual
Demand
50
 0.1
45
40
0.4
35
1
2
3
4
5
6
7
8
9 10 11 12
Period
Excel example
exponential-smoothing.xls
34
Techniques for trend
• Develop an equation that will describe trend
• The trend component may be linear or it
may not
• Linear trend:
Y
Yt = a + bt
0 1 2 3 4 5
t
b is similar to the slope.
However, since it is
calculated with the variability
of the data in mind, its
formulation is not as
straight-forward as our usual
notion of slope.
35
Common Nonlinear Trends
Figure 3-5
Parabolic
Exponential
Growth
36
Adjusting for Trend with Double
Exponential Smoothing
• Simple exponential smoothing with no trend
Ft 1   At  1   Ft  Tt
• Add forecasted trend Tt
Tt   ( Ft  Ft 1 )  1   Tt 1
• This time trend is also smoothed, note that
previous trend (of t-1) and current trend (of t)
appear in the smoothing formula: Tt 1 and Ft  Ft 1
• See Table 3-2 for an exercise
37
Techniques for seasonality
• Regularly repeating upward or downward
movements in time series values
• Seasonality: weather variations, vacations and
holidays
• Seasonality: Expressed in terms of the amount
that actual values deviate from the average
value of the series
• Seasonality is expressed as a percentage of the
average amount
seasonal percentages = seasonal relatives = seasonal indices
38
Different models of seasonality
– Seasonal relative = 1.45 for the quantity of television
sold in August at Circuit City, meaning that TV sales
for that month are 45% above the monthly average.
– Seasonal factor=0.60 for the number of notebooks
sold at the UTD bookstore in April, meaning that
notebook sales are 40% below the monthly average.
– Seasonal indices are your vehicle to travel between
the seasonal and deseasonal worlds.
39
Use Seasonality Indices
to Deseasonalize and Seasonalize
Inputs
Analyze
t
Output
t
t
• Deseasonalize historical observations
• Divide them by seasonal indices
• Make the analysis = Generate forecasts
• Seasonalize forecasts
• Multiply them by seasonal indices
Excel example
seasonalforecast.xls
40
Forecast Accuracy
• Measurement is the first step to improve an
activity
– What value of smoothing constant is good?
• Accuracy measurement is a vital aspect of
forecasting
• Impossible to correctly predict future values
• Important to include an indication of how big the
forecast deviate from the actual values
41
Forecast Accuracy
• Error - difference between actual value and
predicted value
• Mean absolute deviation (MAD)
– Average absolute error (weights all errors evenly)
• Mean squared error (MSE)
– Average of squared error (weights errors according
to their squared values)
• Tracking signal
– Ratio of cumulative error and MAD
42
MAD & MSE
Forecast error  Actual  Forecast
n
MAD 
| A  F |
t 1
t
t
n
n
MSE 
 ( At  Ft )
n
2

t 1
n 1
2
(
A

F
)
 t t
t 1
n
n
Tracking Signal 
A F
t 1
t
t
MAD
Estimate of (forecast error) standard deviation  s  MSE
Statistics says : MSE is the unbiased estimator for the variance of forecast error.
43
Use for MAD & MSE
• Compare the accuracy of alternative
– forecasting methods using MAD and MSE.
– parameter (such as alpha) values used in forecasting
by using MAD and MSE
• Determine which method yields the lowest MAD
or MSE for a given set of data.
44
Controlling the quality of forecast
• Necessary to monitor forecast to ensure that the
forecast is performing adequately
• This is accomplished by comparing forecast errors to
predetermined values
• Errors that fall within the limits are considered
acceptable
• Errors outside either limit indicates that corrective
action is needed.
• Tracking signal values are compared to predetermined limits (+4,-4) based
on judgment and experience
• Upper and lower limits for individual forecast errors are calculated using
control chart techniques. We will learn about control charts in quality
chapters.
45
Choosing a forecasting technique
No single technique works best in every situation
No single technique works best in every situation
• The forecast horizon
• Forecasting frequency
– Forecasting is not free
– Consider cost and accuracy
• Weigh cost-accuracy trade-offs carefully
• Forecast detail, part / product level?
• Availability of
– historical data
– computers
– able users / decision makers
46
Choosing a forecasting technique (cont.)
• Moving Averages and Exponential
Smoothing are short range techniques. They
produce forecast for the next period
• Trend equations are used for much longer
time horizons.
• More than one forecasting techniques might
be used to increase confidence.
47
Summary
• We studied the steps of forecasting
• We examined three forecasting techniques:
– Judgmental
– Associative
– Time Series
• We learned about seasonality, trend, cyclical data
• Discussed monitoring forecast accuracy
48