1. MANAGEMENT SCIENCE AN INTRODUCTION TO 4 7 11 5 6 8 9 7 3 2 2 7 8 4
QUANTITATIVE
APPROACHES TO
DECISION
MAKING 7 11 5 6 8 9 7 3 2 2 7 8 4
SLIDES PREPARED BY
JOHN LOUCKS
ANDERSON
SWEENEY
WILLIAMS
© 1997 West Publishing Company

2. Chapter 16 Forecasting 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

3. Quantitative Approaches to Forecasting
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.

4. 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 - b1t
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

5. Using Regression Analysis in Forecasting
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.

6. Using Regression as a Forecasting Method
Restaurant
Quarterly Sales
Population 1 58 2
105 6 3 88 8 4
118 5
117 12
137 16 7
157 20
169 9
149 22 10
202 26
Sum
1300
1400
Mean
130
140

7. Scatter Plot

8. Measures of Central Tendency
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”

9. Measure of variability (Variance & standard deviation)
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.

10. 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

11. Positive covariance

12. Negative Covariance

13. 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.

14. 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 = Cov (x,y)/var x
B1=S (xi - ¯x)(yi - ¯y)/S (xi - ¯X)2

15. y=Qtrly Sales
x=stu pop
yi-mean
xi-mean
f*g
d*d
e*e
58000
2000
-72000
-12000
105000
6000
-25000
-8000
88000
8000
-42000
-6000
118000
117000
12000
-13000
-2000
137000
16000
7000
157000
20000
27000
169000
39000
149000
22000
19000
202000
26000
72000
140000
1.573E+10
130000
14000
cov(x,y)
cor(x,y)
slope 5

16. b1 = 5
b0 = Y bar - b1 x bar = 130 – 5 *14 = 130 – 70 = 60
The estimated regression equation Y carrot = x
Y^ represents predicted value.
What is the expected qt sales for a new restaurant located near a campus with students?

17. The End of Chapter 16