ATI's Quantitative Methods course: Bridging Project Management and System Engineering Technical Training Short Course

Video Sampler From ATI Professional Development Short Course

Quantitative Methods: Bridging Project Management and System Engineering

Instructor:
John C. Goodpasture, PMP

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Why number ideas are important for project management

Cardinal Ordinal Deterministic Random

• Metric • Rank choice • Numerical • Risk analysis
calculation & priority reporting to • Calculations
• Metric • Rank stakeholders and estimates
reporting complexity • Population of random or
• Budgets, sche • Give statistics probabilistic
dules, resourc numerical quantities
es visualization
to position
and rank

Copyright 2011 Square Peg Consulting, LLC, All Rights Reserved 2

Example: developer ranking of complexity

Histogram of developer opinion

Count [cardinal]
8 15
76th
30 Percentile
20
4 30 76% of rankings
15
are 4 or a 2

2 4 8
2 20
Rank [ordinal]
• Minimum 2
• Maximum 8
• Median 5
3
Copyright 2011 Square Peg Consulting, LLC, All Rights Reserved

Comparison of deterministic and random numbers

• Deterministic
– Single point, one value
– Certain knowledge
2
– Arithmetic on number values
• Probabilistic, aka random
– Range of possible values, with probabilities
– Different values occur from one trial or instance to the next
– Arithmetic on {value, value probability} pairs
– Most useful for project management if distribution is stationary
[invariant] with time and position
2.1
1 1.5
2 2.5
4


Arithmetic operations with random numbers

• Arithmetic operations require operations on distribution functions
– Functional operations are often quite complex
– Simulation methods substitute for direct calculations
• As a practical matter, distributions are not often known
– Only observations of distribution outcomes are known
– Arithmetic operations applied to outcomes
– Approximations are made using simpler functions as substitutes
– Simulation methods derive estimators for actual—but unknown—
functions


Logical operations with random numbers

• UNION and INTERSECTION
– Logical representation of addition and multiplication
• Logic operations provide practical and useful approximations of
outcomes

Union or Summation
A or B
A+ B

Intersection or Multiplication
A and B
A* B


The Project Balance Sheet Tool
Quantitative Methods in Project Management

7

Recall the “Project Balance Sheet”

Project Value from Project Estimate from
the Top Down the Bottom Up

Risk
Investor Value
Expectation &
Resource
Commitment Deliverables
Cost
Schedule

Management investment Project employment
of investment

8

Map from business to project
1. Disaggregate sponsor needs: break down expectations, judgments, and
commitments into component parts
2. Categorize component parts into capacity, capability, resource needs, and risk
2. Re-integrate component parts to identify gaps and missing parts

Resource
Sponsor Capacity Capability Risk
Needs

Expectations Resources, sk Schedule
ills, commitm Feature X
ent
All the
Value features and
Cost
judgments functions of
widget A
Environment, Dollars and
Resource tools People, proce schedule
Commitment ss, tools

9

Plot confidence in cost [or schedule]

Confidence that the
$_amount will not be
exceeded

Likely Risk

Very High
High
Medium
Low

Not to exceed cost $450K $475K $550K >$550K

10

Plot timeline of project expense and business value

Business value

Project Business value
expenses from sales

$450K

$550K

11

Sampling Metrics for Project Estimates


In the beginning, there is a population

• All the data values, events, or event
Population outcomes that share a common situation or
environment

• Space that holds all the values of the
Population space population

• May be deterministic or the outcome of a
Population values random process in/of the population

• Only those populations that bear upon
Population project results are important
importance • Because a population bears upon project
results, the population is important


Sampling risks

Accuracy Completeness

• Misunderstood • Excluded clusters or strata
exclusions, clusters, or strata • Unrepresentative data quality
• Unrepresentative sample data or deficiency
value outliers


Two risk assessments to be made

Margin of error Confidence interval

Estimated error around the
Interval that probably contains the
measurement, observation, or
true population parameter
calculation of statistics

Confidence expresses probability
Interval of possible values for the
that the true parameter is in the
statistic relative to the statistic
interval


Margin of error example

Margin of error % = 3 / 18 (x100) = 16.7%

18 Statistic

17 Sample Interval of statistic values: 3 20

½ Interval Margin of error % = +/- 1.5 / 18 (x100) = +/- 8.3%


Confidence interval

• For some probability—for example, 95%--the true population
statistic is within the interval
– 5% of the trials may not have intervals that contain the true population
– For a single trial, there is a 95% confidence that the true population
statistic is within the sample interval

For 95/100 trials
Sample interval contains the true population statistics
For 1 trial
5% chance the interval does not contain the true population statistics
18 Statistic

Sample Interval of statistic: 3
17 20


Confidence interval for proportional data

Interval = p +/- Z * [p * (1 - p) / N]
Where
Z is range value of standard Normal distribution
Z is normalized to the standard deviation
Z = 1 means 1 σ from the mean

Z range

Margin of error, proportional data

+/- Margin of Error = ½ Interval width / p
Where ½ Interval width = +/- Z * [p * (1 - p) / N]

Z = 1.96


Hypothesis Testing


What if …. ?

Design parameter change
– You change a system design parameter with an expectation that there
will be a difference in performance.
– Comparing the ‘before’ to the ‘after’, is the difference a matter of
chance, or has there been a systemic change in performance?


Distributions of X and Y

Sample X

Sample Y


• We don’t know the distributions of sample X and sample Y (usually)
– Not needed for hypothesis test
– Distributions of sample average are known approximately
Sample average distribution
Sample X

Sample Y


• H0 likely TRUE for difference values < 0.219
• Otherwise, likely FALSE
• With confidence of 95%
H0 distribution & confidence curve


Risk mitigation in time and resource schedules


Any issues?

Should you be equally confident of making the milestone?

Tandem path primitive

Parallel path primitive


Interpreting the Confidence “S” Curve

A. 68% confidence: value between -1 to +1
B. 16% confidence: value > 1
C. 84% confidence: value < 1
B
1
0.84
0.75

0.5 A
0.25
0.16
C
0
-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5

C A B


Schedule example for tandem tasks

Schedule network primitive Task duration distribution, D

Task Probability distribution
0.45
0.4
Task A Task B 0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
1 2 3 4 5 6

Duration range 1 - 6


Milestone accumulates task performance

Milestone
Expected Value = 3.5 + 3.5

0.3

0.25

0.2

0.15

0.1

0.05

0
1 2 3 4 5 6 7 8 9 10 11 12

Milestone range 1 - 12


Confidence for “schedule-at-mode”

Date 1/1 1/21
2/12
3/15
3/25

1.2 Low confidence in 3/25
1
p/v
0.8
Calculate Confidence 0.6

0.4

0.2
0.0 0.5 0

1-Apr
2-Apr

4-Apr
23-Mar
24-Mar
25-Mar
26-Mar
27-Mar
28-Mar
29-Mar
30-Mar

3-Apr

5-Apr
31-Mar

Parallel path primitive

What is the schedule
confidence at the milestone?

Distribution of tasks
0.45
0.4
0.35
0.3
0.25
0.2
Confidence: 80% at 4
1.2
0.15
0.1 1
0.05 0.8
0
1 2 3 4 5 6 0.6

0.4

0.2

0
1 2 3 4 5 6


“Critical Chain” buffers uncertainty
1 2

10 days 11 days 12 days Buffer Project Buffer

Path buffer mitigates
15 days 10 days
“shift right” at the
milestone of joining
path
Task on the critical path

Task with risky duration, not on critical path

Critical chain is a concept developed in the book
Critical Chain (Goldratt, 1997)


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Quantitative Methods: Bridging Project Management and System Engineering

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ATI's Quantitative Methods course: Bridging Project Management and System Engineering Technical Training Short Course

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