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AN INTRODUCTION TO
MANAGEMENT
SCIENCE
QUANTITATIVE
APPROACHES TO
DECISION
8
5
6
MAKING
SLIDES PREPARED BY
JOHN LOUCKS
ANDERSON
SWEENEY
WILLIAMS
© 1997 West Publishing Company
Slide 1
Chapter 16
Forecasting
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Quantitative Approaches to Forecasting
The Components of a Time Series
Measures of Forecast Accuracy
Forecasting Using Smoothing Methods
Forecasting Using Trend Projection
Forecasting with Trend and Seasonal Components
Forecasting Using Regression Models
Qualitative Approaches to Forecasting
Slide 2
Quantitative Approaches to Forecasting

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Quantitative methods are based on an analysis of
historical data concerning one or more time series.
A time series is a set of observations measured at
successive points in time or over successive periods
of time.
If the historical data used are restricted to past values
of the series that we are trying to forecast, the
procedure is called a time series method.
Three time series methods are: smoothing, trend
projection, and trend projection adjusted for seasonal
influence.
If the historical data used involve other time series
that are believed to be related to the time series that
we are trying to forecast, the procedure is called a
causal method.
Slide 3
Trend Projection

Using the method of least squares, the formula for the
trend projection is: Tt = b0 + b1t.
where Tt = trend forecast for time period t
b1= slope of the trend line
b0 = trend line projection for time 0
b1 = nStYt - StSYt
b0 = Y - b1 t
nSt2 - (St)2
where Yt = observed value of the time series at time
period t
Y = average of the observed values for Yt
t = average time period for the n observations
Slide 4
Using Regression Analysis in Forecasting
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Regression analysis is to develop a mathematical
equation showing how variables are related.
Types of variables are:
independent variables
dependent variables
Simple linear regression
Regression analysis involving one independent variable
and one dependent variable.
The relationship between the variables is approximated
by a straight line.
Slide 5
Using Regression as a Forecasting Method
Restaurant
Quarterly Sales
Population
1
58
2
2
105
6
3
88
8
4
118
8
5
117
12
6
137
16
7
157
20
8
169
20
9
149
22
10
202
26
Sum
1300
1400
Mean
130
140
Slide 6
Scatter Plot
250000
200000
150000
Series1
100000
50000
0
0
10000
20000
30000
Slide 7
Measures of Central Tendency

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
A Statistic is a descriptive measure computed from a
sample of data
The sample mean ¯X
• The sum of the data values divided by the number of
observations
¯X=(S xi)/n = (x1+ x2 ….. + xn)/n
 S means “to add”

Slide 8
Measure of variability (Variance & standard deviation)
1.
Variance
 Sample variance, s2, is the sum of the
squared differences between each
observation and the sample mean
divided by the sample size minus 1.
S2 =S (xi - ¯X)2 / n-1
 Standard deviation, s.
Slide 9
Summarizing Descriptive Relationships


Scatter plot
Covariance and correlation coefficient
• Covariance:
a measure of joint variability for two variables
A measure of the linear relationship between two
variables.
• a positive (negative) covariance value indicates a
increasing (decreasing) linear relation ship.



Cov(x,y) = S xy = S (xi - ¯x)(yi - ¯y)/ n-1
– Where n is the sample size
Slide 10
Positive covariance
250000
200000
150000
Series1
100000
50000
0
0
10000
20000
30000
Slide 11
Negative Covariance
250000
200000
150000
Series1
100000
50000
0
0
10000
20000
30000
Slide 12
Correlation Coefficient


Correlation Coefficient is a standardized measure of the
linear relationship between two variables
Correlation Coefficient is computed by dividing the
covariance by the product of the standard deviation of
the two variables, Sx, Sy.
 Rxy =
Cov (x,y)/Sx Sy.
Slide 13
Finding the slope of the regression line

Rxy = Cov (x,y)/Sx Sy.

B1 = Rxy * Sy./ Sx
or
 B1
Cov (x,y)/Sx Sy * Sy./ Sx
=
=
 B1=S
Cov (x,y)/var x
(xi - ¯x)(yi - ¯y)/S (xi - ¯X)2
Slide 14
y=Qtrly Sales
x=stu pop
yi-mean
xi-mean
f*g
d*d
e*e
58000
2000
-72000
-12000
864000000
5184000000
144000000
105000
6000
-25000
-8000
200000000
625000000
64000000
88000
8000
-42000
-6000
252000000
1764000000
36000000
118000
8000
-12000
-6000
72000000
144000000
36000000
117000
12000
-13000
-2000
26000000
169000000
4000000
137000
16000
7000
2000
14000000
49000000
4000000
157000
20000
27000
6000
162000000
729000000
36000000
169000
20000
39000
6000
234000000
1521000000
36000000
149000
22000
19000
8000
152000000
361000000
64000000
202000
26000
72000
12000
864000000
5184000000
144000000
1300000
140000
0
0
2840000000
1.573E+10
568000000
130000
14000
0
0
315555555.6
1747777778
63111111
41806.4323
7944.2502
cov(x,y)
315555555.6
cor(x,y)
0.950122955
slope
5
Slide 15
b1 = 5
b0 = Y bar - b1 x bar
= 130 – 5 *14 = 130 – 70 = 60
The estimated regression equation
Y carrot = 60 + 5 x
Y^ represents predicted value.
What is the expected qt sales for a new restaurant
located near a campus with 18000 students?
Slide 16
The End of Chapter 16
Slide 17