Analytic Method for Probabilistic Cost and Schedule Risk Analysis Final Report 5 April 2013 PREPARED FOR: NATIONAL AERONAUTICS AND SPACE ADMINISTRATION (NASA) OFFICE OF PROGRAM ANALYSIS AND EVALUATION (PA&E) COST ANALYSIS DIVISION (CAD) Felecia L. London Contracting Officer NASA GODDARD SPACE FLIGHT CENTER, PROCUREMENT OPERATIONS DIVISION OFFICE FOR HEADQUARTERS PROCUREMENT, 210.H Phone: 301-286-6693 Fax: 301-286-1746 e-mail: [email protected] Contract Number: NNH10PR24Z Order Number: NNH12PV48D PREPARED BY: RAYMOND P. COVERT, COVARUS, LLC UNDER SUBCONTRACT TO GALORATH INCORPORATED [This Page Intentionally Left Blank] ii TABLE OF CONTENTS 1 Executive Summary .................................................................................................. 11 2 Introduction .............................................................................................................. 12 2.1 Probabilistic Nature of Estimates .................................................................................... 12 2.2 Uncertainty and Risk ....................................................................................................... 12 2.2.1 Probability Density and Probability Mass ................................................................ 12 2.2.2 Cumulative Probability ............................................................................................. 13 2.2.3 Definition of Risk ..................................................................................................... 14 2.3 Joint Probability Distributions ......................................................................................... 15 2.3.1 Marginal Distributions .............................................................................................. 16 2.3.2 Conditional Distributions .......................................................................................... 16 2.4 Statistics of a Random Variable ...................................................................................... 17 2.4.1 Moments ................................................................................................................... 17 2.4.2 Quantile Statistics ..................................................................................................... 19 2.4.3 Expectation Operator ................................................................................................ 19 2.4.4 Order Statistics .......................................................................................................... 20 2.5 Section Summary ............................................................................................................. 20 3 Cost and Schedule Estimates ................................................................................... 21 3.1 Nomenclature ................................................................................................................... 21 3.2 The Cost Estimating Problem .......................................................................................... 22 3.2.1 WBS structure ........................................................................................................... 22 3.2.2 Estimating Methods .................................................................................................. 22 3.2.3 Discrete Risks ........................................................................................................... 26 3.3 The Schedule Estimating Problem .................................................................................. 27 3.3.1 Using Workdays in a Schedule ................................................................................. 27 3.3.2 Arrangement of Tasks in a Network ......................................................................... 29 3.3.3 The Critical Path ....................................................................................................... 32 3.4 Mathematics of Estimates ................................................................................................ 33 3.4.1 Correlation between Random Variables ................................................................... 34 iii 3.4.2 Calculating Correlation Coefficients ........................................................................ 35 3.4.3 Correlation, Dependence and Independence ............................................................. 36 4 Probability Tools ...................................................................................................... 38 4.1 Statistical Simulation ....................................................................................................... 38 4.1.1 Sampling Techniques ................................................................................................ 38 4.1.2 Correlating Random Numbers .................................................................................. 42 4.1.3 Timing of Discovery of Correlation Methods .......................................................... 42 4.1.4 Benefits and Drawbacks of Statistical Simulation Techniques ................................ 43 4.2 Statistical Analysis .......................................................................................................... 44 4.2.1 Moments ................................................................................................................... 44 4.2.2 Method of Moments .................................................................................................. 44 4.3 MOM Operations and Analytic Method Description ...................................................... 48 4.3.1 Addition and Subtraction of Random Variables ....................................................... 48 4.3.2 Covariance of Random Variables ............................................................................. 51 4.3.3 Transformation of Random Variables ...................................................................... 52 4.3.4 Multiplication and Division of Random Variables ................................................... 54 5 Product of Dependent Random Variables.............................................................. 55 5.1 Product of Two Normal Random Variables .................................................................... 55 5.2 Product of Two Lognormal PDFs ................................................................................... 56 5.3 Product of Exponentiated Lognormal PDFs .................................................................... 59 5.3.1 Correlation Between Exponentiated Lognormal PDFs............................................. 60 5.4 Product of Multiple Lognormal PDFs ............................................................................. 60 5.5 Limitations of Statistical Simulations.............................................................................. 61 6 Mellin Transforms .................................................................................................... 62 6.1 Mellin Transform Properties ........................................................................................... 62 6.2 Mellin Transform of the Uniform Distribution ............................................................... 63 6.3 Mellin Transform of the Triangular Distribution ............................................................ 63 6.4 Mellin Transform Example ............................................................................................. 63 7 Propagation of Errors .............................................................................................. 67 7.1 Propagation of Errors Example ....................................................................................... 68 8 Functional Correlation between WBS Elements ................................................... 70 iv 8.1 Type I-1 Functional Correlation ...................................................................................... 72 8.1.1 Type I-1 Functional Correlation Example ................................................................ 73 8.2 Type I-2 Functional Correlation ...................................................................................... 75 8.2.1 Type I-2 Functional Correlation Example ................................................................ 76 8.3 Type II-1 Functional Correlation ..................................................................................... 84 8.3.1 Common Predecessor Functional Correlation .......................................................... 87 8.3.2 Type II-1 Functional Correlation Example ............................................................... 88 8.3.3 Type II-1 Functional Correlation between Multivariate Functions .......................... 90 8.4 Type II-2 Functional Correlation ..................................................................................... 91 8.5 Type III-1 Functional Correlation ................................................................................... 93 8.6 Type III-2 Functional Correlation ................................................................................... 95 8.6.1 Type III-2 Functional Correlation Example ............................................................. 96 8.7 Section Summary ............................................................................................................. 97 9 Discrete Risks ............................................................................................................ 98 9.1.1 Single Discrete Risk Case ........................................................................................