rcoleman@northropgrumman.com, (703)633-8300 x4536, 12/16/2015, 1 briefing, 35 th adodcas, scea, pmi,...

rcoleman@northropgrumman.com, (703)633-8300 x4536, 04/21/23, 1

Briefing, 35th ADoDCAS, SCEA, PMI, 2002

Schedule and Cost GrowthR. L. Coleman, J. R. Summerville, M. E. Dameron

35th ADoDCAS

PMI 2002 National Conference

November, 2002

Outline• Descriptive Statistics• Investigating the Hypothesis• Is There a Curve?• Normalizing for Dollar Size• Correction Factors and Their Use

– Correcting EACs and Risk Models

• Analysis Conclusions• Modeling Schedule Duration in Networks• How Networks Operate

– Some Toy Problems

Background

• At the MDA* Risk Working Group of 29/30 May 01, Schedule Risk was a major topic

• Action Item:– Investigate Schedule Risk

• Content variation • Cost risk* • PERT• Time and budget constraints

* The subject of this paper

This work was conducted for and funded by the IC CAIG and MDA

The Hypothesis

• Many people believe1 a graph of cost growth vs. schedule growth as illustrated below:

Schedule Growth Factor

Cost Growth Factor

1 E. g., Cost Risk Schedule – CEAC, Dr. M. Anvari, First BMDO Cost Symposium, 4 October 2001

The Data

• We analyzed data from the RAND Cost Growth Database with both the following characteristics:– Programs with E&MD only

• Because growth is different for those with and without PDRR

– Programs with schedule data in the requisite fields

• There were 59 points. The analysis follows.

Descriptive Statistics for Schedule Growth

• We will look at these descriptive statistics in the following slides– Distribution shape– Scatter plots– Dollar weighting

Schedule Growth Factor Histogram

0.68 0.94 1.19 1.44 1.69 1.95 2.20 2.45 2.71 More

Bin (SGF)

Schedule Growth Distribution

CDF for Schedule Growth

PDF for Schedule Growth

These two graphs look much like CGF graphs, but the PDF is tighter here, and the CDF is

steeper.

These two graphs look much like CGF graphs, but the PDF is tighter here, and the CDF is

steeper.

The distribution is highly skewed

Schedule Growth FactorCumulative Distribution Function

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

Note this region

Basic Statistics of Schedule ChangeAnalyzed data only

• Mean 1.29

• Standard Deviation 0.54

• CV 42%

• 75th %-ile 1.46

• 61st %-ile 1.29

• 50th %-ile 1.11

• 25th %-ile 1.00

• Shrinkers 9/59 15.3%

• Steady 12/59 20.3%

• Stretchers 38/59 64.4%

The distribution is highly skewed,as was seen in the histogram

But, many programs have little-to-no growth

There is some dispersion and tendency to extremes

Observations

Basic Scatterplots – SGF & Sked vs. Dollar Size

We see the usual size effect, analogous to that in CGF graphsBigger programs have less schedule growth

SGF vs Size

0 2000 4000 6000 8000 10000

Sked Grow th vs Size

0 2000 4000 6000 8000 10000

Size ($M)

RAND 93 - Procurement

0 10000 20000 30000 40000

DE Proc baseline (FY96$)

The “1/x Pattern”

Contract Data - RDT&E

$0 $200 $400 $600

Original Estimate

RAND 93 - RDT&E

0 2000 4000 6000 8000 10000

DE Baseline (FY 96$)

SW Growth

0 500000 1000000 1500000

Orginial Estimate

The 1/x pattern is virtually

universal.

CGF and SGF vs. Cost SizeCGF and SGF vs Size

0 2000 4000 6000 8000 10000

Size ($K)

1 Pt Removed for zoom

The pattern is similar, but CGF is generally more extreme:• Higher highs• Lower lows** See later plot

Basic Scatterplots – Dollar Size vs. Length

At Phase 2 start, there is a vague connection between length and sizeAt end, there is no connection

We would not say that longer programs are costlier

At Phase 2 start, there is a vague connection between length and sizeAt end, there is no connection

We would not say that longer programs are costlier

Program Size vs Schedule LengthInitial

y = 24.273x - 90.545R2 = 0.0659

0 50 100 150

Length (mos)

Program Size vs Schedule LengthFinal

y = 2.5013x + 1532.7R2 = 0.001

0 50 100 150 200

Length (mos)

Basic Scatterplots – Length vs. $ Size

At Phase 2 start, there is a vague connection between size and lengthAt end, there is no connection

We would not say that costlier programs are longer

At Phase 2 start, there is a vague connection between size and lengthAt end, there is no connection

We would not say that costlier programs are longer

Program Schedule Length vs SizeInitial

y = 0.0027x + 55.119R2 = 0.0659

0 2000 4000 6000 8000 10000

Program Schedule Length vs SizeFinal

y = 0.0004x + 72.394R2 = 0.001

0 2000 4000 6000 8000 10000

Basic Scatterplots – Cost Growth

There is no obvious connection between CGF and SGFThere is no obvious connection between CGF and SGF

Phase 2 SGF vs EMD only CGF

0.00 1.00 2.00 3.00 4.00

DE Only CGF

Tw o Points omitted for Zoom-in

Basic Scatterplots - Length

There is a slight tendency for longer programs to grow less

SGF vs Length

y = 3.4944x-0.2609

R2 = 0.0854

0 .0 0

2 .0 0

3 .0 0

4 .0 0

30 50 70 90 110 13 0

Length

Weighting by Length- and Dollar-Size

Schedule growth is less than cost growth

Weighting by Length- and Dollar-Size both reinforce size effects

Schedule growth is less than cost growth

Weighting by Length- and Dollar-Size both reinforce size effects

Raw vs Wtd GrowthAs GFs

0.000.250.500.751.001.251.501.75

RawAvg

L-w tdmean

$-WtdMn

L-WtShrinkFactor

$-WtShrinkFactor

Dollar Weighting shows a more severe effect

CGF & SGF both Sorted

Develop CGF

Phase 2 SGF

CGF & SGF both Sorted

1 6 11 16 21 26 31 36 41 46 51 56

Develop CGF

Phase 2 SGF

Sorted Graphs

This graph is a zoom-in

Sorted CGF shows more growth than Sorted SGF(To the left and right of the x-intercept, Pink y-values are more extreme)

Correlation and Other Joint Effects

Correlation and Other Joint Effects Between Schedule Growth and Cost Growth

• We will look for correlation– Parametric– Non-parametric– Trends in sorted data

• We will investigate the hypothesis for schedule growth vs. cost growth– We will normalize by dollar size to eliminate

any inadvertent distortion

Correlation - Parametric

There is no linear parametric correlationThere is no linear parametric correlation

Phase 2 SGF vs EMD only CGFAbridged

y = -0.0018x + 1.2284R2 = 5E-06

0.00 1.00 2.00 3.00 4.00

DE Only CGF

Tw o Points omitted for regression

Correlation – Non-Parametric

• Test – Cox Stewart Test for Trend test statistic of 18 is

within the critical values of 8.41 and 18.59• The non-parametric test cannot reject no correlation• Used CGF Sort because CGF had less ties, thus less ambiguity

– Previous parametric test cannot reject no correlation

– Moving averages of CGF do not show a rise

• Conclusion: Cannot reject “no correlation”• Visual presentations follow

Patterns in SGF and CGF

There is no strong rising pattern in either CGF or SGF after sorting on the other

CGF after SGF Sort

1 5 9 13

DevelopCGF

20 per.Mov. Avg.(DevelopCGF)

SGF after CGF Sort

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57

Phase 2SGF

20 per.Mov. Avg.(Phase 2SGF)

The gentle rise here conforms with

the near-critical test statistic

Investigating the Hypothesis

CGF by SGF Regimes

11.11.21.31.41.5

LT 1.0 EQ 1.0 LT 1.2 LT 1. 4 LT 1.6 LT 1.8

SGF Regime

Avg CGF

CGF by Regime

Programs divided into SGF Regimes show a marked pattern, like the hypothesis suggested

Largest CGF

Larger CGFs, but Some small n’s

Smallest CGF

CGF by SGF Regimes

11.11.21.31.41.5

LT 1.0 EQ 1.0 LT 1.2 LT 1. 4 LT 1.6 LT 1.8

SGF Regime

Avg CGF

CGF by Regime

Programs divided into SGF regimes look somewhat like the hypothesis suggested they would

There is a Patternbut

Is There a Curve?CGF

Is there a curve?

• There is no pattern on either side of the data

Stretch SGF vs CGF

0 1 2 3 4 5

Shrink SGF vs CGF

0 0.2 0.4 0.6 0.8 1 1.2

Is there a Curve?

CGF by SGF Regimes

11.11.21.31.41.5

LT 1.0 EQ 1.0 LT 1.2 LT 1. 4 LT 1.6 LT 1.8

SGF Regime

Avg CGF

There is no reasonable grouping of the

stretchers that will produce a curve.

Any grouping of points has the same average.

Normalizing for Dollar SizeTo Remove Inadvertent Dollar Size Distortion

Size Normalization• We know there is a size effect in CGF

• We think there is a size effect in SGF

• We must investigate schedule effects free from size effects– First we will look at a scatter plot

– Then we will normalize1 all programs for dollar size, and compare to actuals

• If there is a pattern in any regime, we will worry

• If there is no regime pattern, we can conclude there is no dollar size distortion

• We chose to correct out dollar-size because it is stronger, and because we were worried about a length and SGF correlation causing mischief if we tried to correct it out

1 See backup for norming algorithm

CGF by SGF Regime and Size

0 2000 4000 6000 8000 10000

Size (Dollars)

Sked Grow

Sked Steady

Sked Shrink

One outlier left out for zoom in

Is there a Dollar-Size Bias?

Programs in the 3 regimes show no clear size bias, but a clear growth bias

“Steady” programs

are probably

attenuated vertically (growth

“Shrink” programs may be attenuated

horizontally (size bias)

“Growth” programs span the full range horizontally and

vertically

Normed vs Actual CGFs by Regime

Averages for size-normed programs show the same patterns, so there is no size distortion

Note: Corrected 20 Apr 02. Minor differences

CGFs vs SGF Regime

LT 1.0 EQ 1.0 GT 1.0

SGF Regime

F Avg Act

Avg Norm'd

CGFs vs SGF Regime

LT 1.0 EQ 1.0 GT 1.0

SGF Regime

F Avg Act

Avg Norm'd

Normed vs Actual CGFs by Regime

Both sets of bars look like the hypothesis suggested they wouldBoth sets of bars look like the hypothesis suggested they would

Hypothesis – The Answer• The Hypothesis was about right

– The below is all we can say for sure– Some liberties have been taken with

the graph

Schedule Growth Factor

Cost Growth Factor

1.121.24

NB 1: Nominal has growth

NB 2: The curve is not validated, just the 3 regimes

Correction Factors and Their Use

CGFs vs SGF Regime as Percent of Average

LT 1.0 EQ 1.0 GT 1.0

SGF Regime

CGFs vs SGF Regime as Percent of SGF=1.0

LT 1.0 EQ 1.0 GT 1.0

SGF Regime

Correction Factors and Their Use

• We must correct for schedule growth, if we can predict it. The form of the correction is unclear:

These factors describe what happens if schedules change.

We might use these factors to adjust an EAC if a schedule changed.

We might use these factors to correct a risk model’s nominal growth

factors

Conclusions• Schedule growth is less extreme than cost growth

– But patterns are the same

• There is a cost-size and length effect, just as for cost growth– Dollar-larger programs lengthen less – Longer programs lengthen less

• Neither cost nor length predict the other

• There is a difference in cost growth by schedule-growth regimeRelative to

Relative to

Regime CGF No ChangeAverage

– Programs that shorten 1.42 1.25 1.14– Programs that stay the same 1.13 1.00 0.91– Programs that lengthen 1.24 1.09 1.00

• We now have tools to correct EACs and risk analysesThe hypothesis was essentially true

But there is no curve in evidence

The hypothesis was essentially trueBut there is no curve in evidence

Modeling Schedule Duration of Networks

Schedule Growth Distributions• For schedule network models, a distribution is useful to model

durations

• We will provide a distribution for program-level network schedule growth– Useable for confidence intervals and predictions for single programs

– Useable for systems of systems, to simulate component systems as single entities

• This section will provide a detailed analysis for fitting the schedule growth data to a distribution– Lognormal and Extreme Value distributions show the most promise

– Extreme Value is the most theoretically compelling• Extreme value distributions are used to model the largest of a set of random variables,

and networks complete when the last event is finished

Schedule Growth Factor CDFsPhase 2 DE only (n=59)

0.50 1.00 1.50 2.00 2.50

Extreme Value

Lognormal

Empirical CDF

Best Fits vs. Empirical Data

• Extreme Value Distribution is what we expect theoretically• Extreme Value more peaked, appears to represent data better than Lognormal • But we will see the number of 1.0’s in the data base (schedules finishing “on

time”) creates problems in the fit statistics

Note disproportionate amount of 1.0’s

Schedule Growth Factor PDFs w/HistogramPhase 2 DE only (n=59)

0.00 1.00 2.00 3.00 4.00

Extreme Value

Lognormal

Histogram

Note disproportionate number of 1.0’s

Why are Values of 1 more Common?And who cares?

• There is intense pressure to complete on time, and late finishes are easily discerned

• The consequence of an early finish is to “ship” a flawed system– Flaws can be fixed after testing

• There is a temptation to drag out work if you are done early

• Perhaps the implication is that the customer should put less emphasis on finish time and more on test results?

• In any event, it is altogether likely that there would be cosmetic 1.0 SGFs, and the data would seem to reflect that

• We will find a way to deal with this in the analysis, and recommend a modeling approach

Extreme Value Distribution Fit• The CDF of the data is oddly shaped due to a large number of 1.0’s and

fails a Kolmogorov-Smirnov test for the Extreme Value Distribution

• We believe the disproportionate amount of 1.0’s is politically motivated and not a natural occurrence– This causes a “gap” between the empirical and fitted distributions

• We will next examine a hypothetical distribution with the 1.0’s redistributed along the “gap” area (using the Ext Val fit)

0.50 1.00 1.50 2.00 2.50

K-S stat = 0.161

95% Critical Value (n=59)

= 0.1131

1. Lilliefors methodology applied to Extreme Value distribution to generate critical value with Monte Carlo simulation

Empirical Schedule

Growth CDF vs Fitted

Extreme Value

Note “gap” caused by

1.0’s

“gap”

The Hypothetical “Natural” CDF Original and Revised Empiricals

0.50 1.00 1.50 2.00 2.50

Extreme Value Fit

Revised Empirical

Empirical CDF

12 points respread

12 points at 1.0

0.50 1.00 1.50 2.00 2.50

K-S stat = 0.093

= 0.113

1.0’s redistributed along the “gap” area (in red) better

represents what we believe to be the “natural” distribution

The revised empirical produces an Extreme Value fit with K-S stat below

the critical value. This suggests Extreme Value is a good representation of the

natural SGF distribution

Revised Empirical and Extreme Value Fit

Extreme Value: = 1.12 = 0.28

What the test showsAnd what it doesn’t show

• The redistributed data pass a K-S test

• But, the test cannot take the redistribution of data into account– This is analogous to loss of degrees of freedom,

but the literature provides no remedy

• We fully realize that this is not a “valid statistical test”– But it strongly suggests that the underlying

distribution is the Extreme Value distribution

Hybrid Distribution Alternative• The hypothetical natural (re-distributed) distribution is reasonable for use

– But, if you wish to capture the effects of too many programs appearing to finish “on schedule” then a hybrid distribution should be examined

• To do this we must consider the probability of 1.0 vs. the rest of the outcomes as discrete cases– P(1.0) = 12/59 = 20.3%

– P(Extreme Value) = 79.7%

• The Extreme Value parameters would then be estimated from the data with the 1.0’s removed

0.50 1.00 1.50 2.00 2.50

20.3% (i.e. 12/59) probability of 1.0Hybrid Schedule

Growth PDF with Histogram

(original SGF data)

79.7% probability of Extreme Value Distribution

(fitted w/o 1.0’s)

Hybrid Distribution Alternative

Extreme Value fit to data without 1.0s:K-S stat is less than the critical value. The Extreme Value is a good representation of

this data.

0.50 1.00 1.50 2.00 2.50

K-S stat = 0.087

= 0.1261

Extreme Value: = 1.16 = 0.32

1. Lilliefors methodology applied to Extreme Value distribution to generate critical value with Monte Carlo simulation

Simulated Hybrid Schedule Growth CDF

0.50 1.00 1.50 2.00 2.50

Hybrid 1000 trials Empirical CDF Plot Data

Hybrid Model

0.25 0.81 1.38 1.94 2.50

Overlay Chart

Results of simulation combining this distribution with a discrete 20.3% probability of a 1.0

Distribution Conclusions

• We have shown that the Extreme Value distribution is well supported as the natural distribution

• We have shown that the pieces of the hybrid distribution fit the data– And, the hybrid reproduces the actuals well

• We recommend using the hybrid– But if “political” or “cosmetic” effects are absent,

we recommend using the hypothetical natural distribution

How Networks OperateSome “Toy Problems”

Independent Tasks• Tasks 1 and 2 begin at the same

time and are independent

• Both tasks must be complete before the system is ready

• Duration is modeled as a uniform distribution ranging from Estimated – Note that it is symmetric!

• What is the Expected Duration?

EndStart

Task 1Duration 9

Task 2Duration 10

Task 1 Task 2 Max Dur7.02 11.91 11.917.08 11.62 11.628.22 11.27 11.27

10.00 10.91 10.919.94 8.77 9.949.03 10.94 10.949.54 8.39 9.54

10.05 10.09 10.0910.33 11.22 11.2210.59 11.64 11.64

Average 9.18 10.68 10.91Criticality 20% 80%

Independent Tasks

The average system duration is 10.91 months … longer than the estimated duration of either component task

The “shorter” Task 1 is the critical path 20% of

the time!

EndStart

Task 2Duration 10

Task 1Duration 9

Each task is uniformly distributed from –20% to +20%

of the expected duration

Comparisons with Constant Critical Path

These all have Critical Path = 10

Network Comparisons

9.00 9.50 10.00 10.50 11.00 11.50

(4-5)//(5-5)crosslink

(4-5)//(5-5)

Average DurationMean 80th%-ile

Comparisons with Constant CP

Parallel is bad

Serial is good

Cross links are bad

Durations were modeled as uniform distributions ranging from 20% of the estimate. 5000 iterations were run.

Serial is good

These all have CP = 10 … but their probabilistic durations are all different

Network Growth Effect vs Number of Parallel Tasks

0 20 40 60 80 100

Number of Parallel Identical Tasks

10% CV

20% CV

29% CV

39% CV

Network Schedule Growth As a Function of Network Complexity … Parallel-Task Toy Problem

• This is another toy problem, to see what happens to a network as identical parallel tasks are added

Increasing the

number of tasks

increases the

schedule stretch

Network Growth Effect vs Coefficient of Variation (no mean shift)

0 0.2 0.4 0.6 0.8 1

Mean Stretch (Triangular)Mean Stretch (Normal)80th %-ile Stretch (Triangular)80th %-ile stretch (Normal)

Network Schedule Growth As a Function of Task Variance … Changing-CV Toy Problem

• This is a real network, with changing variance, to see what happens as variance grows

Increasing the

variance of tasks

increases the

schedule stretch

“Toy Problem” Conclusions• The duration of a network will be longer than any of the

component legs• Parallel tasks lengthen the average duration

– Independent tasks that must finish at the same time should make you worry about schedule

– The more parallel tasks, the more you stretch

• Serial tasks decrease the average duration– Serial tasks should make you feel a bit better about schedule– However, breaking a single task into smaller pieces will not improve your

schedule

• Interdependencies (cross links) increase the average duration– Tasks that depend on two or more other tasks should make you worry about

schedule

• Greater variability of the tasks will make the schedule duration grow

Backup

Size Adjustments

Prediction Equation - RAND RDT&E

RDT&E DE only

0 5000 10000 15000

Baseline

Actual

Predicted

Note that data is sparse on the right (large programs)

SSE = 72.56

RDT&E Predicted CGF = 1.8 * (MSII Baseline FY96$M)-0.3 + 1.1RDT&E Predicted CGF = 1.8 * (MSII Baseline FY96$M)-0.3 + 1.1

Prediction Equation - RAND RDT&ERDT&E DE only

zoom-in

0 1000 2000 3000 4000 5000

Baseline

FActual

Predicted

RDT&E Predicted CGF = 1.8 * (MSII Baseline FY96$M)-0.3 + 1.1RDT&E Predicted CGF = 1.8 * (MSII Baseline FY96$M)-0.3 + 1.1

Dispersion – Bounds

This graph shows the actual data, the CGF prediction line, and the Bounds. The next slide will zoom-in.

R&D DE only

0 2000 4000 6000 8000 10000 12000 14000

Baseline (FY96$M)

FActual

Predicted

Dispersion – Bounds

R&D DE onlyzoom in

0 200 400 600 800 1000

Baseline

Actual

Predicted

Note that the Upper and

Lower bounds are not

symmetric. Also, dispersion

is higher for smaller projects … an effect that is captured by the bounds.

Basic Statistics of Schedule ChangeAll available schedule data compared to analyzed data

Statistic Analyzed All Observations• Mean 1.29 1.25• Standard Deviation 0.54 0.51• CV 42% 41%• n 59 98

• 75th %-ile 1.46 1.365• %-ile of the mean 61% 63%• 50th %-ile 1.11 1.03• 25th %-ile 1.00 1.00

• Shrinkers 15.3% 20.4%• Steady 20.3% 22.4%• Stretchers 64.4% 57.1%

The larger data set is somewhat

less skewed

The larger data set is somewhat

less skewed

The larger data set has slightly less dispersion

The two data sets are quite similar,

but, use the smaller

one as your basis

The two data sets are quite similar,

but, use the smaller

one as your basis

rcoleman@northropgrumman.com, (703)633-8300 x4536, 12/16/2015, 1 briefing, 35 th adodcas, scea, pmi,...

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