Cost Analysis and Estimating for Engineering and Management

1. Cost Analysis and Estimatingfor Engineering and Management Chapter 5 Forecasting

2. Overview • Working with Data • Graphing, Statistics • Regression / Curve Fitting • Confidence / Correlation • Time Series • Moving Averages, Smoothing • Cost Indexes

3. Business Forecasting • Prediction • Price of Material • Cost/Amount of Labor • Market Demand/Price • Term • Short  2 Years • Medium 2 - 5 Years • Long Range > 5 Years

4. Graphical Analysis • Descriptive Statistics • Collect/Organize/Analyze Data • Summarize/Present • Draw Conclusions/Make Decisions • Raw Data Communicates Little Information

5. Price, ($/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 Simplifying Data

6. Graphical Presentation

7. Frequency Curves

8. Mean • Average Eq 5.1

9. Median & Mode • Median • All Data from Lowest to Highest • Number in the Middle • Mode • Data Value(s) that Appear the Most Often

10. Standard Deviation • Amount of Data Spread Around the Mean • Variance Is the Square of the Standard Deviation Eq 5.2

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”

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

13. Example

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

15. Regression Analysis • Finds Dependent y for Given x • If x Is Time • Called Trend Line • Used for Forecasting

16. Least Squares • Minimize Variation (Error) Between • Observed (Real) Values • Fitted Curve (Predicted) Values • Minimize • Sum of the Squares of the Errors

17. Normal vs. Student-t Distribution

18. Distribution Applied to Regression

19. Mathematical Calculations • Error • Sum of the Squares Eq 5.4 Eq 5.5

20. The Least Squares Equation • y = a + bx • The Least Squares Line Goes Through (X, Y) Eq 5.8 Eq 5.9

21. Year x Index 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 Example

22. Example Calculations • Find a and b • Y = 84.875 + 2.389X

23. Confidence Limits

24. Equations • Variance Around Regression Line • Degrees of Freedom Eq 5.19 Eq 5.11

25. Confidence Limits • Based on Student-t Tables • Regression Line Passes Through y • Variation of y Equals Constant Variation of regression line Eq 5.14

26. In General • Variance of y Due to Variance of y • Variance Applied to y Eq 5.15 Eq 5.16

27. Compounded Variance • Variance of Predicted Value Eq 5.17 Eq 5.18 Eq 5.19

28. Confidence vs Prediction • Confidence Interval • Variation Around Expected Y Value • Prediction Interval • Variation Around a Single Y Value • Greater In Magnitude

29. Variance from Intercept • X = 0 at Intercept a Eq 5.22 Eq 5.20 Eq 5.23 Eq 5.21

30. Confidence Intervals

31. Non-Linear Relationships • Curvilinear Regression Exponential Eq 5.24 Power Eq 5.25 Polynomial Eq 5.26

32. Non-Linear Calculations • Convert to Log Representation • For y = abx (Exponential Function) Eq 5.27 Eq 5.28

33. Another Version • For y = axb (Power Function) Eq 5.29 Eq 5.30

34. 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 Example

35. Finding a and b Eq 5.29 Eq 5.30

36. Polynomial Regression • Linear Relationship Unknown Eq 5.31

37. Correlation

38. Correlation • Sometimes There Isn’t Any • Quantitative Measure • -1r 1 • Farther from 0, Stronger Correlation Eq 5.33

39. Multiple Linear Regression • More than 1 Independent Variable • Graphical Representation Difficult • Mathematical Form Eq 5.34

40. Finding Constants • Solve Simultaneously Eq 5.35

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

42. Time Series Models • Used for Forecasting • Fundamentals • Consistent Data Collection • Types of Behavior • Moving Average • Smoothing • Data Added on Revolving Basis

43. Time Series Data • Collected at Successive Periods • Usually Equally Spaced • Is the Underlying Process • Constant • Variable • Trend-Cycle • Seasonal • Regular

44. Typical Time Series Models

45. Moving Average • Places More Reliance on Recent Data • Recent Data Better Predicts Future • N Determines Rate of Response Eq 5.36

46. Smoothing • “Weighted” Moving Average • Exponential Smoothing Eq 5.38

47. 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 Smoothing Constant 

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

49. Using Cost Index • Compares Known Cost at Period r • Using Current Ic and Reference Ir Indexes Eq 5.39

50. Figuring Cost Indexes • Benchmark Cost Used as Denominator • Index for Benchmark Period = 1 or 100 • Costs for Other Periods Divided by Benchmark Cost