Technical Manual. S-Curve Tool

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1 Tchnical Manual for S-Curv Tool Vrsion 1.0 (as of 09/1/1) Sonsord by: Naval Cntr for Cost Analysis (NCCA) Dvlod by: Tchnomics, Inc th Strt South, Suit 61 Arlington, VA 0 Points of Contact: Bruc Parkr, Bruc.M.Parkr@navy.mil, (703) Brian Flynn, Brian.Flynn.ctr@navy.mil, (703) Richard L, RL@tchnomics.nt, (571) Ptr Braxton, PBraxton@tchnomics.nt, (571)

2 Acknowldgmnts W would lik to thank and acknowldg Dick Colman, Paul Garvy, Bob Jons, Jack Smuck, Kvin Cincotta, Lucas Still, and Travis Manning for thir fforts in rviwing and rfining th S-Curv Tool and all rlatd documntation. Tabl of Contnts Acknowldgmnts... List of Tabls... 3 List of Figurs... 3 Introduction... 4 Emirical Estimat Ty... 6 Paramtric and Point Estimat Tys Historical Adjustmnts... 1 Chart Otions... 14

3 List of Tabls Tabl 1: All Possibl Scnarios for Dfining an S-Curv in th S-Curv Tool v Tabl : Convrsion Factors for Emirical (Raw) Data... 6 Tabl 3: Paramtrs Drivd From Emirical (Raw) Data... 7 Tabl 4: Dscritions for Calculating th 01 Points in th Smoothing Tchniqu... 7 Tabl 5: Drivations for th Paramtric and Point Estimat Tys Tabl 6: Alying s and/or CGFs to S-Curvs Using S-Curv Tool Tabl 7: Calculations for All Dislayd Data Points on Charts List of Figurs Figur 1: Flowchart Diagram of S-Curv Tool v Figur : Comarison btwn PDF of Emirical Data and Smoothing Tchniqu for trials gratr than 01 oints... 8 Figur 3: Comarison btwn CDF of Emirical Data and Smoothing Tchniqu for trials gratr than 01 oints... 8 Figur 4: Comarison btwn Emirical Data and Smoothing Tchniqu for trials lss than 01 oints

Introduction Th uros of this documnt is to dislay/xlain th mathmatics bhind th S-Curv Tool v1.0. Rfr to th Usr Guid for additional dtails on th diffrnt tabs in th tool.

4 Introduction Th uros of this documnt is to dislay/xlain th mathmatics bhind th S-Curv Tool v1.0. Rfr to th Usr Guid for additional dtails on th diffrnt tabs in th tool. Figur 1 shows a flowchart diagram of th S-Curv Tool v1.0. For th stimat(s), th usr chooss Emirical (i.., a st of outcoms from a Mont Carlo risk run), Paramtric (.g., nhancd Scnario-Basd Mthod (SBM) or aramtrs from an xtrnal risk analysis), or a Point Estimat (i.., risk analysis not yt don). Historical adjustmnts ar basd on th Naval Cntr for Cost Analysis s (NCCA s) analysis of Slctd Acquisition Rorts (SARs) and ar dndnt on fiv diffrnt inuts: (1) commodity, () lif cycl has, (3) milston, (4) inflation, and (5) quantity. If usrs dcid not to aly historical adjustmnts to th stimat, thy can rocd with th bas s-curv that was gnratd. Figur 1: Flowchart Diagram of S-Curv Tool v1.0 4

5 Tabl 1 dislays all th ossibl scnarios for dfining an s-curv in th S-Curv Tool v1.0. With th xction of th Point Estimat, Lognormal, Mdian cas (Scnario 11), all scnarios rtain th valu of th man throughout th s-curv; scnario 11 rtains th valu of th mdian for th s-curv. # of Scnarios Estimat Ty Distribution Ty of Inut 1 Emirical - - Man and 3 Normal Man and Scifid Cost (w/ %til) 4 and Scifid Cost (w/ %til) Paramtric 5 Man and 6 Lognormal Man and Scifid Cost (w/ %til) 7 and Scifid Cost (w/ %til) 8 Man Normal 9 Mdian Point Estimat 10 Man Lognormal 11 Mdian Tabl 1: All Possibl Scnarios for Dfining an S-Curv in th S-Curv Tool v1.0 5

6 Emirical Estimat Ty This chatr rovids furthr dtails for th Emirical stimat ty. Th S-Curv Tool follows th list of rocdurs to dislay th PDFs and CDFs of th mirical data. 1. Insrt th numbr of trials and all Mont Carlo simulation oututs into th tool. Th tool thn automatically convrts th cost units of th mirical (raw) data to th cost units slctd in th Inuts tab using th convrsion factors listd in Tabl. Units for Emirical (Raw) Data Convrtd to Convrsion Factor $ $K $K $K 1 $M $K 1,000 $B $K 1,000,000 $ $M $K $M $M $M 1 $B $M 1,000 $ $B $K $B $M $B $B $B 1 Tabl : Convrsion Factors for Emirical (Raw) Data. Th tool automatically ranks and sorts th ntrd valus. 3. Using Equation 1 th tool automatically calculats th CDF and alis it to th sortd valus. Calculations ar dndnt on th total # of trials that th usr scifid on th Inuts tab. 4. Tabl 3 dfins aramtrs that ar drivd from th Emirical (raw) data, that is from th Mont Carlo (or othr statistical) outut. (1) Drivd Paramtrs From Emirical (raw) Data Equation Man Standard Dviation Cofficint of Variation () Varianc Mdian Middl valu that sarats th ur half from th lowr half of th data st 6

7 Mod Th most frqunt valu in th data st Undrlying Man (Calculatd for Lognormal distributions only) Undrlying Standard Dviation (StDv) (Calculatd for Lognormal distributions only) ln1 Min Max Minimum valu mirical raw data st Maximum valu of mirical raw data st Z min Z max Intrval ( = 01 oints for currnt vrsion of th tool, rfr to Tabl 4) Tabl 3: Paramtrs Drivd From Emirical (Raw) Data 5. Th tool incororats a smoothing tchniqu for th mirical data. This tchniqu is only usd to facilitat th dislay of data, NOT to r-calculat th drivd aramtrs from th mirical (raw) data (shown in Tabl 3). Th conct bhind th smoothing tchniqu is simly to constrain th dislay of th CDF and PDF to 01 oints (00 intrvals). By slcting 01 oints, th PDF of th raw data can b asily rcognizd and th tool can run fastr, sinc it dosn t hav to stor and rform calculations for 10,000 data oints (maximum numbr of trials allowd for th tool). Tabl 4 shows th quations/dscritions for calculating th 01 oints in th smoothing tchniqu. Calculatd Valus Z i Equation/Dscrition From drivd calculations, srad Z min and Z max into qual intrvals of 01 oints X i X (smoothd) i Cumulativ Distribution Function (CDF) Probability Dnsity Function (PDF ) From th qually sacd intrvals, look u (aroximat) th closst valu to th raw data From th X(smoothd) column, look u th xact valu of th CDF in th raw data column Tabl 4: Dscritions for Calculating th 01 Points in th Smoothing Tchniqu Shown blow ar two xamls of th smoothing tchniqu. As shown in Examl 1 and Examl, th smoothing tchniqu is mainly usd to smooth th PDF. Thr is no drastic chang to th CDF. 7

8 Examl 1: Numbr of trials for mirical (raw) data is GREATER TN numbr of oints for smoothing tchniqu. In this cas, 01 oints from smoothing tchniqu vs 10,000 Mont Carlo trials from Emirical (raw) data. a. Auto scal y-axis b. Constraind y-axis Figur : Comarison btwn PDF of Emirical Data and Smoothing Tchniqu for trials gratr than 01 oints Figur 3: Comarison btwn CDF of Emirical Data and Smoothing Tchniqu for trials gratr than 01 oints Examl : Numbr of trials for mirical (raw) data is LESS TN numbr of oints for smoothing tchniqu. In this cas, 01 oints from smoothing tchniqu vs 100 Mont Carlo trials from Emirical (raw) data. 4a. Comarison of CDF 4b. Comarison of PDF Figur 4: Comarison btwn Emirical Data and Smoothing Tchniqu for trials lss than 01 oints 8

9 6. As dscribd in th Usr Manual, th tool allows th usr to mak adjustmnts basd on historical data. Using th aramtrs drivd from th mirical (raw) data in Tabl 3 and th quations/dscritions for th calculations of th 01 oints in Tabl 4, Equation can b usd to mak ths adjustmnts. This again facilitats a quickr rocss. whr is th historically adjustd x valu. () 9

10 Paramtric and Point Estimat Tys This chatr rovids furthr dtails for th Paramtric and Point Estimat tys. As rviously statd, th valu of th man is rtaind for all stimat tys xct for th Point Estimat, Lognormal, Mdian cas, which rtains th valu of th mdian. Th man is qual to th mdian for Normal distributions, and thrfor, th Point Estimat, Normal, Man cas rtains th man. Tabl 5 dislays th quations for all tys of inuts for th Paramtric and Point Estimat tys (rfr to scnarios to 11 in Tabl 1). Th quations in Tabl 5 ar also catgorizd by Normal and Lognormal distributions. Clls that ar filld in grn contain quations rlatd to th usr inuts in th S-Curv Tool, whil all othr clls that ar not filld contain quations usd to calculat th drivd aramtrs from th usr inuts. Th Point Estimat ty can b viwd as a uniqu cas of th Paramtric Estimat ty. Th Paramtric Estimat with man and inuts (scnario - Normal distribution and scnario 5 - Lognormal distribution in Tabl 1) ar comarabl to th Point Estimat (Man) distributions (scnario 8 - Normal distribution and scnario 10 Lognormal distribution in Tabl 1). Ths comarabl scnarios ar highlightd in th hadr of Tabl 5 with blu filld clls. For th Point Estimat (Man) cas, th tool alis a of to th quations that ar listd for th Paramtric stimat with man and inuts. Th Point Estimat with and scifid cost (X, ) inuts (scnario 4 - Normal distribution and scnario 7 - Lognormal distribution in Tabl 1) ar comarabl to th Point Estimat (Mdian) distributions (scnario 9 - Normal distribution and scnario 11 Lognormal distribution in Tabl 1). Ths comarabl scnarios ar highlightd in th hadr of Tabl 5 with orang filld clls. For th Point Estimat (Mdian) cas, th tool alis a of and a rcnt () of 50% to th quations that ar listd for th Paramtric Estimat with and scifid cost inuts. 10

11 Lognormal Distribution Normal Distribution 1 Z Man and 11 Paramtric Man and Scifid Cost (X, ) Man (μ) (σ/μ) Std Dv (σ) and Scifid Cost (X, ) X 1 Z Man (assum =0.0001) X 1 Z X Z X 1 Z Point Estimat Mdian (assum =0.0001, =0.5)) X 1 Z X 1 Z Varianc (σ^) Mdian (= μ) Mod (= μ) Man (E[X]) E X X E X E (s. d.(x) / E[X]) 1 E X E X E X E X E Std Dv (s.d.[x]) X Varianc (Var[X]) Mdian (^μ) Mod (^(μ-σ^)) Man Undrlying Normal (μ) St. Dv. Undrlying ln EX EX 1 Normal (σ) ln1 ln 1 Z X ln EX Tabl 5: Drivations for th Paramtric and Point Estimat Tys Z Z ln X ln 1 EX ln 1 Z ln X ln 1 ln1 ln1 ln1

12 Historical Adjustmnts Th historical cofficints of variations (s) and cost growth factors (CGFs) ar basd on th Naval Cntr for Cost Analysis s (NCCA s) analysis of Slctd Acquisition Rorts (SARs) and ar dndnt on fiv diffrnt inuts: (1) commodity, () lif cycl has, (3) milston, (4) inflation, and (5) quantity. Th first st in NCCA s calculation of s and CGFs was to xtract th original stimat and th currnt stimat for a givn rogram from th SAR Summary Shts (both in bas yar and thn yar dollars). From th xtractd data, th CGF was calculatd by dividing th currnt stimat by th original stimat, shown in Equation 3. (3) NCCA s aroach in driving cofficints of variations (s) was to catgoriz th CGFs of all rograms by th fiv historical adjustmnt inuts. Th man and th standard dviation wr calculatd from th filtrd data st of CGFs. (4) Not that th calculatd is th of CGFs, NOT th of cost. Aftr th historical s and CGFs ar obtaind, usrs slct on of thr otions to aly th historical adjustmnt to th stimat: (1) aly only ( flattning th s-curv), () aly CGF only ( shifting th s-curv, or (3) aly & CGF ( flattning and shifting th s-curv). If usrs dcid not to aly historically adjustmnts to th stimat, thy can rocd with th bas s-curv that was gnratd from th tool. Tabl 6 shows how th historical s and/or historical CGFs ar alid to th bas s- curv. As statd in th rvious sctions, th Point Estimat (Mdian) cas (rfr to scnario 11 in Tabl 1) is th only scnario that rtains th mdian. Thrfor, historical adjustmnts that ar alid to this scnario ar tratd diffrntly from all othr cass (shown in th bottom of Tabl 6). For xaml, if th usr slcts Only for scnario 11 in Tabl 6, th historically adjustd s-curv ivots on th mdian, whras for all othr scnarios, th historically adjustd s-curv ivots on th man. 1

13 Hist. Adj. Man (Man ) Hist. Adj. ( ) Hist. Adj. StDv (StDv ) Undrlying Man (only calculatd if Lognormal is slctd) Undrlying StDv (only calculatd if Lognormal is slctd) Hist. Adj. Mdian (Mdian ) Hist. Adj. Man (Man ) Hist. Adj. ( ) Hist. Adj. StDv (StDv ) Undrlying Man (μ ha ) Undrlying StDv (σ ha ) Hist. Adj. Mdian (Mdian ) Only CGF Only & CGF Only For ALL cass xct *Scial Cas: Point Estimat, Lognormal, Mdian Bas Man Bas Man*Suggstd CGF Bas Man*Suggstd CGF Suggstd Bas Suggstd Bas Man*Bas Bas Man*Bas Bas Man*Bas ln Man 1 ln 1 ln Man 1 ln Man 1 ln1 ln1 Normal Lognormal Normal Lognormal Normal Lognormal Man Man Man Man Man Man *Scial Cas: Point Estimat, Lognormal, Mdian Sam as abov ( for all cass ) Sam as abov ( for all cass ) ln X X Bas Mdian ln ln X Sam as abov ( for all cass ) Bas Mdian*Suggstd CGF Tabl 6: Alying s and/or CGFs to S-Curvs Using S-Curv Tool Bas Mdian*Suggstd CGF 13

14 Chart Otions This chatr rovids furthr dtails on th drivd aramtrs and all calculatd oints for th chart otions in th tool. Ths calculations ar mainly usd in th Bnchmarking and Rconciliation tabs. Th aramtrs that can b alid to a chart ar (1) Man, () Mdian, (3) Custom Cost, (4) 0 th Prcntil, (5) 80 th Prcntil, and (6) Custom Prcntil. For th Custom Cost slction, usrs inut a cost of thir choic and th tool calculats th rcntil. Similarly, for th Custom Prcntil slction, usrs inut a rcntil of thir choic and th tool calculats th cost. Th quations/aroachs usd to calculat th chart aramtrs for Emirical data, data with a Normal distribution (for both th Paramtric and Point Estimat cass), and data with Lognormal distribution (for both th Paramtric and Point Estimat cass) ar listd blow. 1. Emirical data (no distribution) Calculat all valus from raw Emirical data.. Equation 5 shows th Normal Distribution PDF for both th Paramtric and Point Estimat cass. (5) 3. Equation 6 shows Lognormal Distribution PDF for both th Paramtric and Point Estimat cass. (6) Furthr dtails ar rovidd in Tabl 7, which dislays quations for Z, cost (X), PDF, and CDF. Ths quations aly to all 01 oints usd to crat th s-curv. 14

15 Estimat Ty Distribution Z i Cost (x) i PDF i CDF i Emirical - blank Bas Estimat Paramtric Point Estimat Paramtric Point Estimat Normal Lognormal -4 to 4 in qual intrvals -3 to 6 in qual intrvals Emirical - Historical Adjustmnt (if chckbox is chckd, thn calculations blow will show, othrwis, sit out #N/A ) Paramtric Point Estimat Paramtric Point Estimat Normal Lognormal Equal to Bas Estimat Tabl 7: Calculations for All Dislayd Data Points on Charts 15

Solution of Assignment #2

Solution of Assignment #2 olution of Assignmnt #2 Instructor: Alirza imchi Qustion #: For simplicity, assum that th distribution function of T is continuous. Th distribution function of R is: F R ( r = P( R r = P( log ( T r = P(log