Cost Analysis and Estimating for Engineering and Management

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1 Cost Analysis and Estimating for Engineering and Management Chapter 5 Forecasting 004 Pearson Education, Inc. Ch 5-1

2 Working with Data Overview Graphing, Statistics Regression / Curve Fitting Confidence / Correlation Time Series Moving Averages, Smoothing Cost Indexes 004 Pearson Education, Inc. Ch 5-

3 Prediction Business Forecasting Price of Material Cost/Amount of Labor Market Demand/Price Term Short Years Medium - 5 Years Long Range > 5 Years 004 Pearson Education, Inc. Ch 5-3

4 Graphical Analysis Descriptive Statistics Collect/Organize/Analyze Data Summarize/Present Draw Conclusions/Make Decisions Raw Data Communicates Little Information 004 Pearson Education, Inc. Ch 5-4

5 Simplifying Data Price, ($/roll) Obs Rel Freq Cum Freq Pearson Education, Inc. Ch 5-5

6 Graphical Presentation 004 Pearson Education, Inc. Ch 5-6

7 Frequency Curves 004 Pearson Education, Inc. Ch 5-7

8 Mean Average x x 1 + x n xi + L+ xn i 1 x Eq 5.1 n n n 004 Pearson Education, Inc. Ch 5-8

9 Median & Mode Median All Data from Lowest to Highest Number in the Middle Mode Data Value(s) that Appear the Most Often 004 Pearson Education, Inc. Ch 5-9

10 Standard Deviation Amount of Data Spread Around the Mean s n ( ) xi x 1 n 1 Eq 5. Variance Is the Square of the Standard Deviation 004 Pearson Education, Inc. Ch 5-10

11 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 004 Pearson Education, Inc. Ch 5-11

12 Mathematical Model Draw Line Through Data Half of Points Above Line, Half Below Straight Line y a + bx Determine a and b from Graph 004 Pearson Education, Inc. Ch 5-1

13 Example 004 Pearson Education, Inc. Ch 5-13

14 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 004 Pearson Education, Inc. Ch 5-14

15 Regression Analysis Finds Dependent y for Given x If x Is Time Called Trend Line Used for Forecasting 004 Pearson Education, Inc. Ch 5-15

16 Least Squares Minimize Variation (Error) Between Observed (Real) Values Fitted Curve (Predicted) Values Minimize Sum of the Squares of the Errors 004 Pearson Education, Inc. Ch 5-16

17 Normal vs. Student-t Distribution 004 Pearson Education, Inc. Ch 5-17

18 Distribution Applied to Regression 004 Pearson Education, Inc. Ch 5-18

19 Mathematical Calculations Error y ( a + bx ) ε i i i Eq 5.4 Sum of the Squares n i 1 n ε [ y ( + )] i a bx Eq 5.5 i i Pearson Education, Inc. Ch 5-19

20 The Least Squares Equation y a + bx a x y n x x xy ( x) b n xy x y n ( ) x x Eq 5.8 Eq 5.9 The Least Squares Line Goes Through (X, Y) 004 Pearson Education, Inc. Ch 5-0

21 Example Year x Index y x xy y ε ε Pearson Education, Inc. Ch 5-1

22 Example Calculations Find a and b a x y x xy n x ( x) ( 1015)( 154) ( 105)( 11, 337) 15( 1015) ( 105) b n xy x y n x ( x) 15(11,337) 15(1015) (105)(154) (105).389 Y X 004 Pearson Education, Inc. Ch 5-

23 Confidence Limits 004 Pearson Education, Inc. Ch 5-3

24 Equations Variance Around Regression Line s y ε i ν Eq 5.19 Degrees of Freedom ν n Eq Pearson Education, Inc. Ch 5-4

25 Confidence Limits Based on Student-t Tables y ± ts Eq 5.14 y Regression Line Passes Through y Variation of y Equals Constant Variation of regression line 004 Pearson Education, Inc. Ch 5-5

26 In General Variance of y Due to Variance of y s y i s y 1 + n ( ) xi x ( ) x x Variance Applied to y Eq 5.15 y ± ts i y i 004 Pearson Education, Inc. Eq 5.16 Ch 5-6

27 Compounded Variance Variance of Predicted Value s y i s y n + ( ) xi x ( ) x x Eq 5.17 s y i s y i + s y Eq 5.18 y i ± ts yi Eq Pearson Education, Inc. Ch 5-7

28 Confidence vs Prediction Confidence Interval Variation Around Expected Y Value Prediction Interval Variation Around a Single Y Value Greater In Magnitude 004 Pearson Education, Inc. Ch 5-8

29 Variance from Intercept X 0 at Intercept a s a s y 1 + n x ( ) x x s b s y ( ) x x Eq 5.0 Eq 5. a ± ts a b ± ts b Eq 5.1 Eq Pearson Education, Inc. Ch 5-9

30 Confidence Intervals 004 Pearson Education, Inc. Ch 5-30

31 Non-Linear Relationships Curvilinear Regression y ab x Exponential Eq 5.4 y ax b Power Eq 5.5 Polynomial y a + b x + b x + b x + L b p x p.. Eq Pearson Education, Inc. Ch 5-31

32 Non-Linear Calculations Convert to Log Representation For y ab x (Exponential Function) loga logb x log y n ( x) n xlog y x log y n x x - x ( x) ylog y Eq 5.7 Eq Pearson Education, Inc. Ch 5-3

33 Another Version For y ax b (Power Function) loga b ( logx) log y logx ( logxlog y) Eq 5.9 n n ( logx) ( logx) ( logxlog y) logx log n ( logx) ( logx) y Eq Pearson Education, Inc. Ch 5-33

34 Example x y log x log y (log x) log x log y Pearson Education, Inc. Ch 5-34

35 Finding a and b (log x) log y log x (log x log y) log a Eq 5.9 n (log x) ( log x) ( )(11.567) (9.1303)( ) ( ) (9.1303) n (log xlog y) log x log y b n (log x) ( log x) 5( ) (9.1303)(11.567) ( ) (9.1303) Eq 5.30 y x 004 Pearson Education, Inc Ch 5-35

36 Polynomial Regression Linear Relationship Unknown ε n [ ( )] p yi a + b1 x + b x + K + bpx i 1 Eq Pearson Education, Inc. Ch 5-36

37 Correlation 004 Pearson Education, Inc. Ch 5-37

38 Correlation Sometimes There Isn t Any Quantitative Measure r n x n xy ( ) ( ) x n y y x y 1/ Eq r 1 λ Farther from 0, Stronger Correlation 004 Pearson Education, Inc. Ch 5-38

39 Multiple Linear Regression More than 1 Independent Variable Graphical Representation Difficult Mathematical Form y a + b x + b x + L+ b k x 1 1 k Eq Pearson Education, Inc. Ch 5-39

40 Ch Pearson Education, Inc. Finding Constants Solve Simultaneously Eq x b x x b x a y x x x b x b x a y x x b x b na y

41 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 004 Pearson Education, Inc. Ch 5-41

42 Time Series Models Used for Forecasting Fundamentals Consistent Data Collection Types of Behavior Moving Average Smoothing Data Added on Revolving Basis 004 Pearson Education, Inc. Ch 5-4

43 Time Series Data Collected at Successive Periods Usually Equally Spaced Is the Underlying Process Constant Variable Trend-Cycle Seasonal Regular 004 Pearson Education, Inc. Ch 5-43

44 Typical Time Series Models 004 Pearson Education, Inc. Ch 5-44

45 Moving Average Places More Reliance on Recent Data Recent Data Better Predicts Future M a x t + x t 1 + L + N x t N + 1 Eq 5.36 N Determines Rate of Response 004 Pearson Education, Inc. Ch 5-45

46 Smoothing Weighted Moving Average Exponential Smoothing S t () x α x + ( 1 α) S () 1 x t t Eq Pearson Education, Inc. Ch 5-46

47 Smoothing Constant α Variations in α values Drift in data Small, α 0 Little, α 0.5 Large, α 1 None None None None Moderate Very small Small Moderate Large Small Moderate Large 004 Pearson Education, Inc. Ch 5-47

48 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 004 Pearson Education, Inc. Ch 5-48

49 Using Cost Index Compares Known Cost at Period r Using Current I c and Reference I r Indexes C c C r I I c r Eq Pearson Education, Inc. Ch 5-49

50 Figuring Cost Indexes Benchmark Cost Used as Denominator Index for Benchmark Period 1 or 100 Costs for Other Periods Divided by Benchmark Cost 004 Pearson Education, Inc. Ch 5-50

51 Rate of Change Differences Between Periods Percent Change Current index, period Less previous index, period Index point change +4.5 Divide by previous index 10.3 Equals % 004 Pearson Education, Inc. Ch 5-51

52 Change Rate Figuring Average Rate of Change r I I e b 1/ n Eq 5.40 Using Rate of Change to Find Index Pearson Education, Inc. I e I b r n Eq 5.41 Ch 5-5

53 Composite Index Material 0* 1 3 Laser glass $6,117 $4,07 $,345 $1,8 Steel tubing Al extrusion Printed circuits Harness cable Glass tubing Total $35,505 $33,415 $31,806 $31,119 Index, % * Benchmark Period 004 Pearson Education, Inc. Ch 5-53

54 Indexes Basis Changes (Qty, Qual, Mix, etc) Technology Creep Kinds of Indexes Material Labor Geographical Index Specific for Time and Location 004 Pearson Education, Inc. Ch 5-54

55 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 004 Pearson Education, Inc. Ch 5-55

56 Summary Objective - Forecasting Methods for Working with Data Graphing, Statistics, Regression Data In Time Periods (Time Series) Cost Indexes Applications, Calculations, Caveats 004 Pearson Education, Inc. Ch 5-56

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