ch 5-1 © 2004 pearson education, inc. pearson prentice hall, pearson education, upper saddle river,...
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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