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AN INTRODUCTION TOMANAGEMENTSCIENCE
QUANTITATIVE
APPROACHES TO
DECISION
MAKING
ANDERSON SWEENEY WILLIAMS
SLIDES PREPARED BY JOHN LOUCKS
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© 1997 West Publishing Company
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Chapter 16Chapter 16ForecastingForecasting
Quantitative Approaches to ForecastingQuantitative Approaches to Forecasting The Components of a Time SeriesThe Components of a Time Series Measures of Forecast AccuracyMeasures of Forecast Accuracy Forecasting Using Smoothing MethodsForecasting Using Smoothing Methods Forecasting Using Trend ProjectionForecasting Using Trend Projection Forecasting with Trend and Seasonal Forecasting with Trend and Seasonal
ComponentsComponents Forecasting Using Regression ModelsForecasting Using Regression Models Qualitative Approaches to ForecastingQualitative Approaches to Forecasting
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Quantitative Approaches to ForecastingQuantitative Approaches to Forecasting Quantitative methodsQuantitative methods are based on an analysis of are based on an analysis of
historical data concerning one or more time series.historical data concerning one or more time series. A A time seriestime series is a set of observations measured at is a set of observations measured at
successive points in time or over successive periods of successive points in time or over successive periods of time.time.
If the historical data used are restricted to past values If the historical data used are restricted to past values of the series that we are trying to forecast, the of the series that we are trying to forecast, the procedure is called a procedure is called a time series methodtime series method..
Three time series methods are: Three time series methods are: smoothingsmoothing, , trend trend projectionprojection, and , and trend projection adjusted for seasonal trend projection adjusted for seasonal influenceinfluence..
If the historical data used involve other time series that If the historical data used involve other time series that are believed to be related to the time series that we are believed to be related to the time series that we are trying to forecast, the procedure is called a are trying to forecast, the procedure is called a causal causal methodmethod. .
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Trend ProjectionTrend Projection
Using the method of least squares, the formula for the Using the method of least squares, the formula for the trend projection is: trend projection is: TTtt = = bb00 + + bb11tt. .
where where TTtt = trend forecast for time period = trend forecast for time period tt
bb11= slope of the trend line= slope of the trend line
bb00 = trend line projection for time 0 = trend line projection for time 0
bb11 = = nntYtYtt - - ttYYtt bb00 = = YY - - bb11tt
nntt22 - ( - (tt))22
where where YYtt = observed value of the time series at time = observed value of the time series at time
period period tt
YY = average of the observed values for = average of the observed values for YYtt
tt = average time period for the = average time period for the nn observationsobservations
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Using Regression Analysis in ForecastingUsing Regression Analysis in Forecasting
Regression analysis is to develop a mathematical Regression analysis is to develop a mathematical equation showing how variables are related. equation showing how variables are related.
Types of variables are:Types of variables are:independent variablesindependent variablesdependent variablesdependent variables
Simple linear regressionSimple linear regressionRegression analysis involving one independent Regression analysis involving one independent variable and one dependent variable.variable and one dependent variable.The relationship between the variables is The relationship between the variables is approximated by a straight line.approximated by a straight line.
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Using Regression as a Forecasting MethodUsing Regression as a Forecasting MethodRestaurantRestaurant Quarterly SalesQuarterly Sales PopulationPopulation
11 5858 22
22 105105 66
33 8888 88
44 118118 88
55 117117 1212
66 137137 1616
77 157157 2020
88 169169 2020
99 149149 2222
1010 202202 2626
SumSum 13001300 14001400
MeanMean 130130 140140
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Scatter PlotScatter Plot
0
50000
100000
150000
200000
250000
0 10000 20000 30000
Series1
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Measures of Central Tendency Measures of Central Tendency
A Statistic is a descriptive measure computed A Statistic is a descriptive measure computed from a sample of data from a sample of data
The sample mean ¯XThe sample mean ¯X• The sum of the data values divided by the The sum of the data values divided by the
number of observationsnumber of observations ¯X=(¯X=(xi)/n = (x1+ x2 ….. + xn)/nxi)/n = (x1+ x2 ….. + xn)/n
means “to add”means “to add”
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Measure of variability (Variance & standard Measure of variability (Variance & standard deviation)deviation)
1.1. VarianceVariance Sample variance, sSample variance, s22, is the sum of , is the sum of
the squared differences between the squared differences between each observation and the sample each observation and the sample mean divided by the sample size mean divided by the sample size minus 1. minus 1.
SS2 =2 =xxi i - - ¯X)¯X)22 / n-1 / n-1
Standard deviation, s.Standard deviation, s.
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Summarizing Descriptive RelationshipsSummarizing Descriptive Relationships
Scatter plotScatter plot Covariance and correlation coefficientCovariance and correlation coefficient
• Covariance: Covariance: a measure of joint variability for two variablesa measure of joint variability for two variables A measure of the linear relationship between A measure of the linear relationship between
two variables.two variables.• a positive (negative) covariance value indicates a positive (negative) covariance value indicates
a increasing (decreasing) linear relation ship.a increasing (decreasing) linear relation ship.
Cov(x,y) = S Cov(x,y) = S xy = xy = xxi i - - ¯x)(y¯x)(yi i - - ¯y)/ n-1¯y)/ n-1
– Where n is the sample size
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Positive covariancePositive covariance
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Series1
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Negative CovarianceNegative Covariance
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Correlation CoefficientCorrelation Coefficient
Correlation Coefficient is a standardized measure Correlation Coefficient is a standardized measure of the linear relationship between two variablesof the linear relationship between two variables
Correlation Coefficient is computed by dividing Correlation Coefficient is computed by dividing the covariance by the product of the standard the covariance by the product of the standard deviation of the two variables, Sdeviation of the two variables, Sxx,, SSy.y.
RRxy xy = Cov (x,y)/S= Cov (x,y)/Sx x SSy.y.
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Finding the slope of the regression lineFinding the slope of the regression line
RRxy xy = Cov (x,y)/S= Cov (x,y)/Sx x SSyy..
BB1 1 = R= Rxyxy * * SSy.y.// SSx x
oror
BB1 = 1 = Cov (x,y)/SCov (x,y)/Sx x SSyy * * SSy.y.// SSx x
= = Cov (x,y)/var xCov (x,y)/var x
BB11==xxi i - ¯x)(y- ¯x)(yi i - ¯y)/- ¯y)/xxi i - ¯X)- ¯X)22
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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
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bb1 = 51 = 5
bb00 = = Y barY bar - - bb1 1 x barx bar = 130 – 5 *14 = 130 – 70 = 60 = 130 – 5 *14 = 130 – 70 = 60
The estimated regression equationThe estimated regression equation
Y carrot = 60 + 5 xY carrot = 60 + 5 x Y^ represents predicted value.Y^ represents predicted value.
What is the expected qt sales for a new What is the expected qt sales for a new restaurant located near a campus with restaurant located near a campus with 18000 students? 18000 students?
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The End of Chapter 16The End of Chapter 16
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