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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

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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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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