3. Forecasting
Homework problems: 2,4,5,6,7,11,12,14.
3.1. Providing Appropriate Forecast Information
The forecasting process involves much more than
just the estimation of future demand. The forecast
also needs to take into consideration the intended
use of the forecast, the methodology for aggregating
and disaggregating the forecast, and assumptions
about future conditions.
Selection of an appropriate forecast method is
determined by different levels of aggregation, cost
of data acquisition and processing, length of
forecast (timeframe), top management involvement,
forecast frequency, etc.
Figure 3.1
3.1. Forecast Information
• The forecast information and technique must
match the intended application:
For strategic decisions such as capacity or market
expansion highly aggregated estimates of general
trends are necessary.
Sales and operations planning (SOP) activities
require more detailed forecasts in terms of product
families and time periods.
Master production scheduling (MPS) and control
demand highly detailed forecasts, which only need
to cover a short period of time.
3.1.1 Forecasting for Strategic Business Planning
Forecast is presented in general terms (sales
dollars, tons, hours)
Aggregation level may be related to broad
indicators (gross national product (GNP), income)
Causal models and regression/correlation
analysis are typical tools
Managerial insight is critical and top
management involvement is intense
Forecast is generally prepared annually and
covers a period of years
3.1.2 Forecasting for Sales and Operations Planning
Forecast is presented in aggregate measures
(dollars, units)
Aggregation level is related to product families
(common family measurement)
Forecast is typically generated by summing
forecasts for individual products.
Managerial involvement is moderate, and limited
to adjustment of aggregate values
Forecast is generally prepared several times
each year and covers a period of several months
to a year.
3.1.3 Forecasting for MPS and Control
Forecast is presented in terms of individual
products (units, not dollars)
Forecast is typically generated by mathematical
procedures, often using software
Projection techniques are common
Assumption is that the past is a valid predictor of
the future
Managerial involvement is minimal.
Forecast is updated almost constantly and
covers a period of days or weeks.
3.2. Regression Analysis & Decomposition
Regression identifies a relationship between
two or more correlated variables.
Linear regression is a special case where the
relationship is defined by a straight line, used
for both time series and causal forecasting.
Data should be plotted to see if they appear
linear before using linear regression.
Y = a + bX
Y is value of dependent variable, a is the yintercept of the line, b is the slope, and X is
the value of the independent variable.
3.2 Least Squares Method
• Objective–find the Y – calculated dependent variable value
line that minimizes y – actual dependent variable point
a – y intercept
the sum of the
squares of the
b – slope of the line
vertical distance
x – time period
between each
data point and the
line
See Fig. 3.2
Y = a + bx
i
Sum of Squares  ( y1  Y1 ) 2  ( y2  Y2 ) 2    ( yi  Yi ) 2
Least Squares Regression Line (Fig.3.2)
Regression
errors are
the vertical
distance
from the
point to the
line
Least Squares Equation (3.3)
xy  n x y

b
 x  n(x )
2
2
Or
b
n  xy   x  y
n  x   x 
2
2
Least Squares Example (Fig. 3.3)
Sum
Quarter (x)
Sales (y)
xy
x2
y2
Y
1
600
600
1
360,000
801.3
2
1,550
3,100
4
2,402,500
1,160.9
3
1,500
4,500
9
2,250,000
1,520.5
4
1,500
6,000
16
2,250,000
1,880.1
5
2,400
12,000
25
5,760,000
2,239.7
6
3,100
18,600
36
9,610,000
2,599.4
7
2,600
18,200
49
6,760,000
2,959.0
8
2,900
23,200
64
8,410,000
3,318.6
9
3,800
34,200
81
14,440,000
3,678.2
10
4,500
45,000
100
20,250,000
4,037.8
11
4,000
44,000
121
16,000,000
4,397.4
12
4,900
58,800
144
24,010,000
4,757.1
78
33,350
268,200
650
112,502,500
Least Squares Example (Fig. 3.2)
xy  n x y

b
 x  n( x )
2
2
268,200  12 * 6.5 * 2,779.17

 359.6153
2
650  12 * 6.5
a  y  b x  2,779.17  6.5(359.6153)  441.6666
Y  a  bx  441.67  359.6x
Least Squares Example
Y  a  bx  441.67  359.6x
Quarter Calculation
Forecast
13
Y13=441.6+359.6(13)
5,119.4
14
Y14=441.6+359.6(14)
5,476.0
15
Y15=441.6+359.6(15)
5,835.6
16
Y16=441.6+359.6(16)
6,195.2
Standard Error of Estimate (Syx) – how well the line fits the data
n
S yx 
(y
i 1
i
 Yi ) 2
n2

(600  801.3) 2  (1,550  1,160.9) 2  (1,500  1,520.5) 2    (4,900  4,757.1) 2
10
=363.9
Regression Using Excel
Time Series Decomposition
A time series can consist of one
or more components of demand
Trend–the long
term growth (or
decrease) of
demand
Seasonal–
Changes in
demand
associated with
portions of the
year (may be
additive or
multiplicative)
Cyclical–
repetitive
patterns not
associated with
seasonal
demand
Autocorrelation–
Random–
changes in
changes in
demand
demand that
associated with
can’t be linked to
previous
a specific cause
demand levels
Seasonality
Seasonality may or may
not be relative to the
general demand trend
Additive seasonal variation
is constant regardless of
changes in average
demand
Multiplicative seasonal
variation maintains a
consistent relationship to
the average demand (this
is the more common case)
Seasonality
Additive seasonal variation is
constant regardless of
changes in average demand
Forecast=Trend + Seasonal
Multiplicative seasonal
variation maintains a
consistent relationship to the
average demand (this is the
more common case)
Forecast= Trend x Seasonal
factors
Seasonal Factor/Index
To account for seasonality within the forecast,
the seasonal factor/index is calculated.
The amount of correction needed in a time
series to adjust for the season of the year
Season
Past
Sales
Average Sales
Seasonal Factor
for Each Season
Spring
200
1000/4=250
Actual/Average=200/250=0.8
Summer
350
1000/4=250
350/250=1.4
Fall
300
1000/4=250
300/250=1.2
Winter
150
1000/4=250
150/250=0.6
Total
1000
Seasonal Factor/Index
• If we expect (forecast) next year’s sales
to be 1,100 units, the seasonal forecast
is calculated using the seasonal factors:
Season Expected Average Sales for
Sales
Each Season
Seasonal
Factor
Forecast
Spring
1100/4=275
X 0.8
= 220
Summer
1100/4=275
X 1.4
= 385
Fall
1100/4=275
X 1.2
= 330
Winter
1100/4=275
X 0.6
= 165
Total
1,100
Seasonality–Trend and Seasonal Factor
Quarter
Amount
I – 2008
300
II – 2008
200
III – 2008
220
IV – 2008
530
I – 2009
520
II – 2009
420
III – 2009
400
IV - 2009
700
Estimate of trend, use
linear regression software
to obtain more precise
results
Trend = 170 +55t
Seasonality–Trend and Seasonal Factor
Seasonal factors are calculated for each
season, then averaged for similar seasons
Seasonal Factor = Actual/Trend
Seasonality–Trend and Seasonal Factor
Forecasts for 2010 are calculated by extending
the linear regression and then adjusting by the
appropriate seasonal factor
FITS–Forecast Including Trend and Seasonal
Factors
Decomposition Using Least Squares Regression
1. Decompose the time series into its components
a. Find seasonal component
b. Deseasonalize the demand
c. Find trend component
2. Forecast future values for each component
a. Project trend component into future
b. Multiply trend component by seasonal
component
Decomposition Using Least Squares Regression
Period
Quarter
Actual
Demand
Average of Same Quarter
of Each Year
1
I
600
(600+2400+3800)/3=2266.7
2
II
1,550
3
III
1,500
4
IV
1,500
5
I
2,400
6
II
3,100
7
III
2,600
8
IV
2,900
9
I
3,800
10
II
4,500
11
III
4,000
12
IV
4,900
Total
33,350
Seasonal
Factor
Calculate average of
same period values
Decomposition Using Least Squares Regression
Period
Quarter
Actual
Average of Same Quarter
Demand of Each Year
Seasonal Factor
1
I
600
(600+2400+3800)/3=2266.7
2
II
1,550
(1550+3100+4500)/3=3050
3
III
1,500
(1500+2600+4000)/3=2700
4
IV
1,500
(1500+2900+4900)/3=3100
5
I
2,400
6
II
3,100
7
III
2,600
8
IV
2,900
9
I
3,800
10
II
4,500
11
III
4,000
12
IV
4,900
Total
33,350
2266.7/(33350/12)=0.82
Calculate seasonal factor
for each period
Decomposition Using Least Squares Regression
Perio
d
Deseasonalized
Demand
1
735.7
2
1412.4
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.929653282
R Square
0.864255225
Adjusted R Square
0.850680748
512.8180268
3
1544.0
Standard Error
4
1344.8
Observations
5
2942.6
ANOVA
6
2824.7
7
2676.2
8
2599.9
9
4659.2
10
4100.4
11
4117.3
12
4392.9
Use linear regression to
fit trend line to
deseasonalized data
12
df
SS
Regression
MS
1
16743469.64
16743469.64
Residual
10
2629823.286
262982.3286
Total
11
19373292.92
Coefficients
Standard Error
t Stat
F
63.66766059
P-value
Intercept
555.0045455
315.6176776
1.758471039
0.109173704
Period
342.1800699
42.88399775
7.979201751
1.20464E-05
Y= 555.0 + 342.2x
Significance F
1.20464E-05
Create Forecast by
Projecting Trend and Reseasonalizing
Period Quarter
Y from Regression
13
I
555+342.2*13=5003.5
X 0.82
= 4102.87
14
II
555+342.2*14=5345.7
X 1.10
= 5880.27
15
III
555+342.2*15=5687.9
X 0.97
= 5517.26
16
IV
555+342.2*16=6030.1
X 1.12
= 6753.71
Project Linear Trend
Y= 555.0 + 342.2x
Seasonal
Factor
Forecast
Project Seasonality
3.3. Short-term Forecasting Technique
Some basic concepts:



dependent/independent demand
aggregate/disaggregate demand
long-term/short-term forecast (regression and correlation vs.
smoothing out the random fluctuations)
The underlying assumption of time series models is that the
future values of the time series can be predicted based upon
previous time series values (i.e., past conditions that produced
the historical data won’t change !)
The need for some forecasting techniques (Fig. 3.11)
What’s wrong with drawing a line (i.e., use the regular averaging
process)?
3.3. Short-term Forecasting Techniques
Moving Average:
Q: what n to use? large or small (longer or shorter)?
Q: Drawback of (simple) moving average?
Weighted Moving Average:
Example. Use weighted moving average with weights
of 0.1, 0.2, and 0.3 to forecast demand for period 33.
Sol:
3.3. Short-term Forecasting Techniques
Exponential Smoothing Forecasting (ESF):
ESFt = ESFt-1 + α(actual demand t – ESFt-1)
=α(actual demand t ) + (1- α) ESFt-1
…….. (3.6)
…….. (3.7)
where:
α= the proportion of the forecast error, under or over estimate, that will be
incorporated into (next) forecast (i.e., smoothing constant).
ESFt-1 = Exp. smoothing forecast made at the end of period t-1
= Exp. smoothing forecast for period t
3.3. Short-term Forecasting Techniques
Exponential Smoothing Forecasting (ESF):
Q: Why is it called “exponential” smoothing? Proof
Q: What happens when α=0 or α=1?
Q: what αvalue to use? Small or large and the effects.
3.3. Short-term Forecasting Techniques
Bias = Σ(actual demand i – forecast demand i) /n

Bias (mean error) measures consistently high or low forecast
MAD = Σ|actual demand i – forecast demand i| /n
MSE=




MAD (mean absolute deviation) measures the magnitude of
forecast error
What is a good (ideal) forecast?
Which (Bias or MAD) is more critical?
When the forecast errors are normally distributed, the standard
deviation of forecast errors = 1.25 MAD
3.3. Some Insights




Focus forecasting: uses the one forecasting model that
would have performed the best in the recent past to make
the next forecast.
Simple models usually outperform more complex methods,
especially for short-term forecasting.
There is no one model that would consistently outperform
all the others.
It might be a good idea to average the forecasts from
several models used in each period (combination
technique).
3.4. Using the Forecasts
Aggregating Forecasts:
 Long-term or product-line forecasts are more
accurate than short-term or detailed forecasts.
 Theorem: Suppose that X and Y are
independent random variables with normal
distribution N(μ1 , σ12 ) and N(μ2 , σ22 ),
respectively. Let Z=X + Y, then Z is a normal
distribution N(μ1+ μ2 ,σ12 +σ22 ).
 Application: Figure 3.17
3.4. Using the Forecasts
Pyramid Forecasting:

To coordinate, integrate, and assure (force)
consistency between forecasts prepared in different
parts/levels of the organization and company goals or
constraints. Figs. 18~20.
Incorporating External Information:


Change the forecast directly, if we know the activities
that will influence demand for sure. e.g., promotions,
product changes, competitor’s action, etc.
Change the forecast model, if we are not sure of the
impact of the activities. e.g., use larger α to be more
responsive to demand change.