Rayleigh Curves A Tutorial

Transcription

1 Rayleigh Curves A Tutorial Hear F. Chelson Richard L. Coleman Jessica R. Summerville Steven L. Van Drew SCEA 2004 Manhattan Beach, CA June 2004

2 Outline Background Description Application The N-R Curve Generation Tool Risk Analysis considerations Refining Rayleigh after Program Start Fitting N-R Curve in Mature Programs Conclusions

3 Background Studies done by Norden, Lee ors have shown that cumulative costs of R&D projects, derived from earned value systems, typically follow Rayleigh distribution 1 quite closely V(t) = d(1-e -at2 ) The Rayleigh distribution models buildup, peak taper of a development program s effort over time Using Rayleigh curve, forecasting EACs, given sufficient earned value data, is a matter of predicting d a variables in above equation to yield a value for V(t final ). $250,000.0 Cum Expenditures V(t) = d(1 e^(-a*t^2) ) Dollars (in $k) $200,000.0 $150,000.0 $100,000.0 $50,000.0 $ Time Rayleigh Cumulative Distribution V(t) V(t) = d(1-e d(1-e -at2 ) -at2 ) 1. Norden-Raleigh Analysis: A Useful Tool for EVM in Development Projects, David Lee, Logistics Management Institute, The Measurable News, March 2002

4 Detailed Description

5 Norden-Rayleigh Model Cumulative distribution function for Rayleigh: V(t) = d(1-e -at2 ) V(t) = Total effort expended t = Time d = Scale factor of distribution a = Shape parameter Probability density function for Rayleigh: v(t) = 2adte -at2

6 Rayleigh Curve Use in Modeling Funding Profiles Expenditures v(t) = 2adte^(-a*t^2) Dollars (in $M) Time Funding Profile Over Over Time Dollars (in $M) Cum Expenditures V(t) = d(1 e^(-a*t^2) ) Time Cumulative Funding Over Over Time

7 The Norden-Rayleigh Funding Model Models time-phasing of expenditures for Development programs Given expenditures vs. time data, useful for forecasting Cost-to-go Time-to-go Models typical programs that rapidly ramp-up labor efforts n taper off Ideally reflected in manufacturing programs as well as incremental software development efforts

8 Application

9 Application of Rayleigh Curve Valid tool for assessing funding cost of Development programs Assessing funding profiles: Rayleigh Model offers a stard of comparison for reasonableness of a project s planned funding phasing Assessing cost: An assumed scale (d) shape factor (a) can be used to build a profile But uncertainties attached to project end time, or t f means that Rayleigh Curve methodology cannot reasonably predict cost until re is sufficient earned value data to estimate d a Valid tool for generating an EAC Must have following information Computed d a from ACWP data already completed

10 When Rayleigh Model Does Not Apply When schedule contains a great deal of uncertainty When programs* are comprised of distinct sub programs with starts stops, e. g.: When a contract funds more than one development program within same funding profile Software programs that release periodic versions or upgrades within same funding profile * * If If a a program program is is an an aggregation aggregation of of subprograms, subprograms, cannot cannot be be predicted predicted in in toto, toto, it it must must be be broken broken into into independent independent component component sub-programs, sub-programs, Rayleigh Rayleigh applied applied to to each each sub-program sub-program

11 Benefits Endorsements Benefits Good cross check to EAC Fast The methodology is in use elsewhere AFCAA OSD ASC

12 The N-R Curve Generation Tool

13 N-R Curve Generation Tool This N-R Curve Generation Tool is a basic tool that can be used early in program to generate a program s total funding profile Useable at outset to develop or check planned funding profile Usable throughout a program as a cross check or early indicator Early in program (before ~20% complete) plot will provide a good cross check when plotted against immature ACWP profile, is an early indicator of trends According to Christensen, et al, it is at 20% that a program stabilizes to a degree that claim can be made that Cum CPI will not change by more than 10% from its value at 20% point. 1 The 20% point is a forward looking point actual percent complete is unclear until later, but thumb rule is still valid This tool is also useful at any point in program to provide a cross-check on EVM data that may appear suspect INPUTS Norden-Rayleigh Tool Expenditure Curves Program Estimate $ 250,000.0 Program Start Date 1-May-00 Program Completion Date 15-Jun-04 t f 50 a (Shape Parameter) d (Scale Parameter computed as a cross check of total funding) 257,732 yellow cells are for input (in $K) $300,000.0 $250,000.0 $200,000.0 $150,000.0 $100,000.0 $50,000.0 $- OUTPUT N-R Expenditure Curve Double-click on picture to launch N-R tool. Months 1. Is CPI-Based EAC a Lower Bound to Final Cost of Post A-12 Contracts?, David S. Christensen, Ph.D., David A. Rees, Ph.D., The Journal of Cost Analysis Management, Winter 2002.

14 Determining a d (Early in Program) Early in program (because ACWP is immature), pdf parameters a d can only be found from schedule variables. Below are equations for calculating a d. V(t) = d(1 e -at2 ), at t f, V(t f ) = d(1 e -at f 2 ) Given V(t f ) =.97d, solve for a Because Because V(t) V(t) does does not not reach reach v v 0 in 0 in finite finite time, time, project s project s end end time time is is usually usually 1 1 defined defined as as time time at at which: which: V(t V(t f ) f ) = = 97% 97% of of v v 0, 0, or, or, V(t V(t f ) f ) = =.97d.97d 1. Analyzing Development Programs Expenditure with 1. Norden-Rayleigh Analyzing Development Model, David Programs Lee, Expenditure 32nd ADoDCAS, with February Norden-Rayleigh 1999, p21. Model, David Lee, 32nd ADoDCAS, February 1999, p21. V(t f ) = d(1 e -at f 2 ).97d = d(1 e -at f 2 ).97 = (1 e -at f 2 ) e -at f 2 =.03 -at f2 = ln(.03) a = -ln(.03) / t f 2 V(t) = d(1 e -(-ln.03/t f 2 )t 2 ) d = V(t) / (1 e -(-ln.03/t f 2 )t 2 ), where t f is known The The authors authors recommend recommend using using this this computation computation only only as as a a rough rough cross cross check check to to program program plan, plan, particularly particularly for for curve curve generation. generation. A mismatch mismatch between between this this derivation derivation of of d d program program funding funding should should be be viewed viewed as as an an indicator indicator of of schedule schedule funding funding misalignment misalignment Warning: SDD Completion Date is difficult to estimate, refore t f is almost always unknown as is evidenced by existence (in fact commonness) of schedule growth. This limits reliability of Norden-Rayleigh method until sufficient data are available.

15 Use of Curve Generator for Risk The previous tool will produce a Norden-Rayleigh curve when program planning data are input Start date End date Total budget A cross check of total funding is available, computed from t f, or t final, but it is not considered reliable The same tool can produce useful outputs for risk estimates If a risk estimate is done, in eir cost or schedule or both, different values for end date total funding will yield an alternative profile Even if a formal risk analysis is not done, nominal (average) growth factors can be applied to yield a profile with typical growth

16 Refining Rayleigh after Program Start

17 Refining Raleigh Curve As program begins to gar stable ACWP data, Rayleigh curve should be updated to reflect improved availability of information a d can be furr refined by finding peak of funding profile Finding a d in terms of peak of pdf (t peak ) firms up value of a d Due to previously noted volatility in schedules, t final is a poor basis a d dependent on t final should only be used when t peak cannot be determined (derivation on following slide )

18 Refining a d To determine when funding is at max, we must find point (t p, or t-peak) at which first derivative of pdf is zero (stard math technique): Start with pdf v(t) = 2adte -at2 Taking first derivative v (t) = 2ad * [e -at2 * t * (-2at) + e -at2 ] = 2ad * (e -at2 * -2at 2 + e -at2 ) = 2ade -at2 * (-2at 2 + 1) Set v (t) = 0 0 = 2ade -at p 2 * (-2at p2 + 1) Solving, we get t p =1 / 2a So, a = 1 / (2t p2 ) And, d = v(t) / 2t p te -(1/ 2t p 2 )t 2 or d = V(t) / (1 e -(1/ 2t p 2 )t 2 ) Computing Computing 2 nd 2 nd derivative derivative yields yields a a negative negative number number (given (given that that a a d d are are greater greater than than 0), 0), indicating indicating that that t p t is p is at at max max point point vs. vs. a a min min point point of of curve: curve: v (t) v (t) = = a 2 dte a 2 dte -at2 -at2 (8at (8at 2-12), 2-12), substitute substitute t p t = p = 1/(sqrt(2a)) 1/(sqrt(2a)) => => v (t) v (t) = = -8a -8a 2 d/(sqrt(2ae)) 2 d/(sqrt(2ae)) By definition, time is greater than 0, so a must be greater than 0. Solving for d in terms of t p, since time is greater than 0 as is also v(t) [funding], so d must be greater than 0.

19 Fitting N-R Curve in Mature Programs

20 Fitting N-R Curve in Mature Programs After a program is 20% complete, earned value data should be sufficient to fit a Rayleigh distribution to data The 20% point is not empirically demonstrated, but authors believe that EACs are sufficiently stable at this point to use method based on work by Christle, Abba, Christensen ors The parameters a d are found by fitting a curve to data using least squares. This is difficult given that equation has two unknowns. Solutions: to best fit a Rayleigh curve to earned value data, analyst needs additional tools that will make se computations $250,000.0 Cum Expenditures v(t) = d(1 e^(-a*t^2) ) COTS COTS software software solutions: solutions: Rayleigh Rayleigh Analyzer, Analyzer, Logistics Logistics Management Management Institute Institute Premium Premium Solver Solver Platform Platform Versions Versions or or 5.5, 5.5, Frontline Frontline Systems Systems Inc. Inc. - - Used Used with with Microsoft Microsoft Excel Excel Solver Solver DLL DLL Platform, Platform, Frontline Frontline Systems Systems Inc. Inc. - - Used Used with with Visual Visual Basic Basic C++ C++ Dollars (in $k) $200,000.0 $150,000.0 $100,000.0 $50,000.0 $- t ptp N-R Curve ACWP Time Warnings: 1) Excel Solver uses an algorithm that finds local optimal solutions based on inputted start points for decision variables (changing cells) in non-linear equations. The answers provided may not be global optimal solutions. 2) The 20% point is a forward looking calculation. It may prove inexact, but is sufficient for use of thumb rule

21 Conclusions

22 Conclusions The Norden-Rayleigh model can be a valid tool for assessing performance (cost schedules) of DoD Development programs offers tests for reasonableness of a project s planned earned value phasing Caveat: reliability of model is dependent on maturity of earned value data to estimate a d ( shape scale parameters) A Summary of Different Methodologies ACWP data availablitity Basis of a d Concerns Comments Beginning of program Stabilized Program Mature Program Not available or Mature, stable Inititial data available insufficient available a is based on an a d based on a assumed schedule a d found by fitting a known curve critical critical t-final d is curve to data using t-peak to compute based on program plans least squares method curve checked with t-final Actual t-final is unknown due to reality of schedule variability Good for early planning Actual t-peak is difficult to determine until ACWP profile is well beyond peak t-peak can be sketchy if determined too early Difficult because equation has two unknowns (a d ) Needs lots of data (program past 20%)

23 References (also see footnotes) Analyzing Development Programs Expenditure with Norden- Rayleigh Model, David Lee, 32 nd ADoDCAS, February 1999 The Rayleigh Analyzer, John Dukovich, Scott Houser, David Lee, LMI Report At902C1, October 1999 Familiar Metric Management Effort, Development Time, Defects Interact, Lawrence H. Putnam, Ware Myers, Quantitative Software Management, Inc. Norden-Raleigh Analysis: A Useful Tool for EVM in Development Projects, David Lee, Logistics Management Institute, The Measurable News, March 2002 ASC/FMC Rayleigh Curve Overview, Ross Jackson, 60 th ASC Industry Cost Schedule Workshop, April 2003 Is CPI-Based EAC a Lower Bound to Final Cost of Post A- 12 Contracts?, David S. Christensen, Ph.D., David A. Rees, Ph.D., The Journal of Cost Analysis Management, Winter 2002.