Cost analysis and estimating for engineering and management ch05

Cost Analysis and Estimatingfor Engineering and Management Chapter 5 Forecasting... Median & Mode Median  All Data from Lowest to Highest  Number in the Middle  Data Values that Appe

Cost Analysis and Estimating

for Engineering and Management

Chapter 5 Forecasting

Graphical Analysis

 Descriptive Statistics

 Collect/Organize/Analyze Data

 Summarize/Present

 Draw Conclusions/Make Decisions

 Raw Data Communicates Little

Information

Graphical Presentation

Frequency Curves

 Average

1 2

1

n

x n

x n

x x

x x

n i

Median & Mode

 Median

 All Data from Lowest to Highest

 Number in the Middle

 Data Value(s) that Appear the Most

Often

n

i

Eq 5.2

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 Data

 Visual Interpretation

 Apply “Judgment”

Mathematical Model

 Draw Line Through Data

 Half of Points Above Line, Half Below

 Straight Line

y = a + bx

 Determine a and b from Graph

Example

Why Graph?

 Visual Analysis of What Is Happening

 Non-Linearity May Be Exposed

Regression Analysis

 Finds Dependent y for Given x

 If x Is Time

 Called Trend Line

 Used for Forecasting

Least Squares

 Minimize Variation (Error) Between

 Observed (Real) Values

 Fitted Curve (Predicted) Values

 Minimize

 Sum of the Squares of the Errors

Normal vs Student-t Distribution

Distribution Applied to Regression

Mathematical Calculations

 Error

 Sum of the Squares

i i

i i

n i

bx a

y

The Least Squares Equation

n

xy x

y

x a

y x

xy

n b

( )( ) ( )( , )

389

2 )

105 (

) 1015 (

15

) 1524 )(

105 (

) 337 ,

11 (

15

2

2 ) (

xy n

b

Confidence Limits

 Variance Around Regression Line

 Degrees of Freedom

Equations

2 2



 i

y

Confidence Limits

 Based on Student-t Tables

 Regression Line Passes Through y

 Variation of y Equals Constant Variation

of regression line

  ts y

x n

x n

s

2 2

2

y y

Variance from Intercept

x n

s

  ts a

a 

  2   

2 2

x x

  ts b

b 

Confidence Intervals

3 3

2 2

1 x b x b x b p x p b

a

Exponential Eq 5.24 Power Eq 5.25 Polynomial

Eq 5.26

Non-Linear Calculations

 Convert to Log Representation

 For y = ab x (Exponential Function)

log

log log

log -

log log

2

2 2

2

y x

y x

n b

x x

n

y y

x y

x a

log log

   log

log

log log

log log

log log

2 2

2 2

2

x x

n

y x

y x

n b

x x

n

y x

x y

x a

Finding a and b

5152

0

38

y

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

2 2

2

2 2

n

y x

y x

n b

x x

n

y x

x y

x

Eq 5.30

p p



Correlation

2 2

n x

x n

y x

xy n

r

Multiple Linear Regression

 More than 1 Independent Variable

 Graphical Representation Difficult

 Mathematical Form

Eq 5.34

2 2

1

1 x b x b k x k

b a

Finding Constants

Eq 5.35

2 2 2

2 1 1

2 2

2 1 2

2 1 1

1 1

2 2

1 1

x b

x x b

x a

y x

x x b

x b

x a

y x

x b

x b

na y

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

Time Series Models

 Used for Forecasting

Time Series Data

 Collected at Successive Periods

 Usually Equally Spaced

 Is the Underlying Process

Typical Time Series Models

Moving Average

 Places More Reliance on Recent Data

 Recent Data Better Predicts Future

x

Smoothing Constant 

Cost Index

 Dimensionless Number

 Represents Change in Cost

 Over a Period of Time

 Relative to a “Benchmark” Period

 What is Costed Remains Constant

Using Cost Index

 Compares Known Cost at Period r

 Using Current I c and Reference I r

c

I

I C

C

Figuring Cost Indexes

 Benchmark Cost Used as Denominator

 Index for Benchmark Period = 1 or 100

 Costs for Other Periods Divided by

Benchmark Cost

Rate of Change

 Differences Between Periods

 Percent Change

Change Rate

 Figuring Average Rate of Change

 Using Rate of Change to Find Index

Eq 5.40

   100 1

/ 1

e

I I r

Composite Index

Material 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

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”

 Objective - Forecasting

 Methods for Working with Data

 Graphing, Statistics, Regression

 Data In Time Periods (Time Series)

 Cost Indexes

 Applications, Calculations, Caveats