Ch 5-1© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Cost Analysis and Estimatingfor Engineering and Management

Chapter 5

Forecasting

Ch 5-2© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Overview Working with Data

Graphing, Statistics Regression / Curve FittingConfidence / Correlation

Time SeriesMoving Averages, Smoothing

Cost Indexes

Ch 5-3© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Business Forecasting Prediction

Price of MaterialCost/Amount of LaborMarket Demand/Price

TermShort 2 YearsMedium 2 - 5 YearsLong Range > 5 Years

Ch 5-4© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Graphical Analysis

Descriptive StatisticsCollect/Organize/Analyze DataSummarize/PresentDraw Conclusions/Make Decisions

Raw Data Communicates Little Information

Ch 5-5© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Simplifying DataPrice, ($/roll) Obs Rel Freq Cum Freq

12.35–12.75 1 0.003 0.003

12.75–13.15 6 0.019 0.022

13.15–13.55 33 0.102 0.124

13.55–13.95 51 0.157 0.281

13.95–14.35 121 0.373 0.654

14.35–14.75 50 0.154 0.808

14.75–15.15 44 0.136 0.944

15.15–15.65 13 0.040 0.984

15.65–16.05 5 0.016 1.000  324 1.000  

Ch 5-6© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Graphical Presentation

Ch 5-7© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Frequency Curves

Ch 5-8© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Mean

Average

121

n

x

n

x

n

xxxx

n

i

i

n

Eq 5.1

Ch 5-9© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Median & Mode

MedianAll Data from Lowest to HighestNumber in the Middle

ModeData Value(s) that Appear the Most Often

Ch 5-10© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Standard Deviation

Amount of Data Spread Around the Mean

Variance Is the Square of the Standard Deviation

11

2

n

xx

s

n

i Eq 5.2

Ch 5-11© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Graph the Data

Pure Statistics Can Be Misleading Any Set of Numbers

Will Have Mean, Std Dev, etc.May or May Not Be Relevant

Plot the DataVisual InterpretationApply “Judgment”

Ch 5-12© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Mathematical Model

Draw Line Through Data Half of Points Above Line, Half Below Straight Line

y = a + bx Determine a and b from Graph

Ch 5-13© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Example

Ch 5-14© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Why Graph?

Visual Analysis of What Is Happening Non-Linearity May Be Exposed Incorporates “Reasonableness” Mathematical Methods Can Assist in

Establishing the “Best Fit” Line Through the Data

Ch 5-15© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Regression Analysis

Finds Dependent y for Given x If x Is Time

Called Trend LineUsed for Forecasting

Ch 5-16© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Least Squares

Minimize Variation (Error) BetweenObserved (Real) ValuesFitted Curve (Predicted) Values

MinimizeSum of the Squares of the Errors

Ch 5-17© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Normal vs. Student-t Distribution

Ch 5-18© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Distribution Applied to Regression

Ch 5-19© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Mathematical Calculations

Error

Sum of the Squares

iii bxay )( Eq 5.4

2

11

2

n

i

ii

n

i

bxay Eq 5.5

Ch 5-20© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

The Least Squares Equation

y = a + bx

The Least Squares Line Goes Through (X, Y)

2

2

xxn

xyxyxa

22 xxn

yxxynb

Eq 5.8 Eq 5.9

Ch 5-21© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

ExampleYear

xIndex

y x2

 xy

 y

 

 2

0 87 0 0 84.875 2.125 4.516

1 89 1 89 87.264 1.736 3.013

2 90 4 180 89.654 0.346 0.120

3 92 9 276 92.043 -0.043 0.002

4 93 16 372 94.432 -1.432 2.051

… … … … … … …

105 1524 1015 11337     63.168

Ch 5-22© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Example Calculations

Find a and b

Y = 84.875 + 2.389X

ax y x xy

n x x

2

2 2

1015 1524 105 11 337

15 1015 10584 875

( )

( )( ) ( )( , )

( ) ( ).

389.2)105()1015(15

)1524)(105()337,11(1522)(2

xxn

yxxynb

Ch 5-23© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Confidence Limits

Ch 5-24© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Variance Around Regression Line

Degrees of Freedom

Equations

2

2

i

ys

2 n

Eq 5.19

Eq 5.11

Ch 5-25© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Confidence Limits

Based on Student-t Tables

Regression Line Passes Through yVariation of y Equals Constant Variation of regression line

y

tsy Eq 5.14

Ch 5-26© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

In General

Variance of y Due to Variance of y

Variance Applied to y

12

222

xx

xx

nss i

yyi

iyi

tsy

Eq 5.15

Eq 5.16

Ch 5-27© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Compounded Variance

Variance of Predicted Value

11

2

222

xx

xx

nss i

yyi

222yyy sss

ii

iyi tsy

Eq 5.17

Eq 5.18

Eq 5.19

Ch 5-28© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Confidence vs Prediction

Confidence Interval Variation Around Expected Y Value

Prediction IntervalVariation Around a Single Y ValueGreater In Magnitude

Ch 5-29© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Variance from Intercept

X = 0 at Intercept a

12

222

xx

x

nss ya

atsa

2

22

xx

ss

yb

btsb

Eq 5.20 Eq 5.22

Eq 5.21 Eq 5.23

Ch 5-30© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Confidence Intervals

Ch 5-31© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Non-Linear Relationships

Curvilinear Regression

y abx

y axb

..33

221

pp xbxbxbxbay

Exponential Eq 5.24

Power Eq 5.25

Polynomial

Eq 5.26

Ch 5-32© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Non-Linear Calculations

Convert to Log Representation For y = abx (Exponential Function)

logloglog

log-log

log

22

22

2

xxn

yxyxnb

xxn

yyxyxa

Eq 5.27

Eq 5.28

Ch 5-33© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Another Version

For y = axb (Power Function)

loglog

loglogloglog

loglog

loglogloglogloglog

22

22

2

xxn

yxyxnb

xxn

yxxyxa

Eq 5.29

Eq 5.30

Ch 5-34© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Example

x y log x log y (log x)2 log x log y

10 510 1.0000 2.7076 1.0000 2.7076

30 210 1.4771 2.3222 2.1819 3.4302

100 190 2.0000 2.2788 4.0000 4.5575

150 125 2.1761 2.0969 4.7354 4.5631

300 71 2.4771 1.8513 6.1361 4.5858    9.1303 11.2567 18.0534 19.8441

Ch 5-35© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Finding a and b

5152.038.1556 xy

5152.0)1303.9()0534.18(5

)2567.11)(1303.9()8441.19(5

) log() log(

log log) log log(

1921.3)1303.9()0534.18(5

)8441.19)(1303.9()2567.11)(0534.18(

) log() log(

) log (log log log) log(log

2

22

2

22

2

xxn

yxyxnb

xxn

yxxyxa Eq 5.29

Eq 5.30

Ch 5-36© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Polynomial Regression

Linear Relationship Unknown

Eq 5.31

2

1

221

2

n

i

ppi xbxbxbay

Ch 5-37© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Correlation

Ch 5-38© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Correlation Sometimes There Isn’t Any Quantitative Measure

-1r 1 Farther from 0, Stronger Correlation

Eq 5.33

2/12222

yynxxn

yxxynr

Ch 5-39© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Multiple Linear Regression

More than 1 Independent Variable Graphical Representation Difficult Mathematical Form

Eq 5.34 2211 kk xbxbxbay

Ch 5-40© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Finding Constants

Solve Simultaneously

Eq 5.35

22221122

21221111

2211

xbxxbxayx

xxbxbxayx

xbxbnay

Ch 5-41© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Regression Assumptions

The Values of x Are Controlled Regression is Linear Deviations are Mutually Independent Deviations Are Not a Function of x Deviations Are Normally Distributed Model Contains ALL Relevant Variables

Ch 5-42© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Time Series Models Used for Forecasting Fundamentals

Consistent Data CollectionTypes of BehaviorMoving AverageSmoothing

Data Added on Revolving Basis

Ch 5-43© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Time Series Data Collected at Successive Periods Usually Equally Spaced Is the Underlying Process

ConstantVariableTrend-CycleSeasonalRegular

Ch 5-44© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Typical Time Series Models

Ch 5-45© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Moving Average

Places More Reliance on Recent Data Recent Data Better Predicts Future

N Determines Rate of Response

Eq 5.36 11

N

xxxM Nttt

a

Ch 5-46© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Smoothing

“Weighted” Moving Average Exponential Smoothing

Eq 5.38 1 1 xSxxS ttt

Ch 5-47© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Smoothing Constant

  Variations in values

Drift in data Small, = 0 Little, = 0.5 Large, = 1

None None None None

Moderate Very small Small Moderate

Large Small Moderate Large

Ch 5-48© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Cost Index

Dimensionless Number Represents Change in Cost Over a Period of Time Relative to a “Benchmark” Period What is Costed Remains Constant Used to Forecast

Ch 5-49© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Using Cost Index

Compares Known Cost at Period r Using Current Ic and Reference Ir

Indexes

Eq 5.39

r

crc

I

ICC

Ch 5-50© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Figuring Cost Indexes

Benchmark Cost Used as Denominator Index for Benchmark Period = 1 or 100 Costs for Other Periods Divided by

Benchmark Cost

Ch 5-51© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Rate of Change

Differences Between PeriodsPercent Change

Current index, period 5 106.8

Less previous index, period 4 102.3

Index point change +4.5

Divide by previous index 102.3

Equals +0.044 = + 4.4%

Ch 5-52© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Change Rate

Figuring Average Rate of Change

Using Rate of Change to Find Index

Eq 5.40

Eq 5.41

1001

/1

n

b

e

I

Ir

100

1n

ber

II

Ch 5-53© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Composite IndexMaterial 0* 1 2 3

Laser glass $26,117 $24,027 $22,345 $21,228

Steel tubing 1913 2008 2129 2278

Al extrusion 418 426 439 456

Printed circuits 637 643 656 657

Harness cable 2103 2124 2134 2305

Glass tubing 4317 4187 4103 4185

Total $35,505 $33,415 $31,806 $31,119

Index, % 100.0 94.1 89.6 87.6

* Benchmark Period

Ch 5-54© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Indexes

Basis Changes (Qty, Qual, Mix, etc) Technology Creep Kinds of Indexes

MaterialLabor Geographical

Index Specific for Time and Location

Ch 5-55© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Caveats Results Depend on Good Data Cause and Effect Relationship Eliminating Spurious Data Backcast Period Forecast Period Limit Number of Variables Use Judgment Test for “Reasonableness”

Ch 5-56© 2004 Pearson Education, Inc.Pearson Prentice Hall, Pearson Education, Upper Saddle River, NJ 07458Ostwald and McLaren / Cost Analysis and Estimating for Engineering and Management

Summary

Objective - Forecasting Methods for Working with Data

Graphing, Statistics, Regression Data In Time Periods (Time Series) Cost Indexes

Applications, Calculations, Caveats