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This 3-day course is de¬signed for the professional program manager, system engineer, or project manager engaged in technically challenging projects where close technical collaboration between engineering and management is a must. To that end, this course addresses major topics that bridge the disciplines of project management and system engineering. Each of the selected topics is presented from the perspective of quantitative methods. Students first learn a theory or narrative, and then related methods or practices. Ideas are demonstrated that are immediately applicable to programs and projects. Attendees receive a copy of the instructor’s text, Quantitative Methods in Project Management.
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Boost Your Skills with On-Site Courses Tailored to Your Needs The Applied Technology Institute specializes in training programs for technical professionals. Our courses keep you current in the state-of-the-art technology that is essential to keep your company on the cutting edge in today’s highly competitive marketplace. Since 1984, ATI has earned the trust of training departments nationwide, and has presented on-site training at the major Navy, Air Force and NASA centers, and for a large number of contractors. Our training increases effectiveness and productivity. Learn from the proven best. For a Free On-Site Quote Visit Us At: http://www.ATIcourses.com/free_onsite_quote.asp For Our Current Public Course Schedule Go To: http://www.ATIcourses.com/schedule.htm
Why number ideas are important for project management
Cardinal
• Metric calculation
• Metric reporting
• Budgets, schedules, resources
Ordinal
• Rank choice & priority
• Rank complexity
• Give numerical visualization to position and rank
Deterministic
• Numerical reporting to stakeholders
• Population statistics
Random
• Risk analysis• Calculations
and estimates of random or probabilistic quantities
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Example: developer ranking of complexity
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Histogram of developer opinion
2 4 8Rank [ordinal]• Minimum 2• Maximum 8• Median 5
Count [cardinal]
20
30
1530
20
1576th
Percentile
76% of rankings are 4 or a 2
2
4
8
Comparison of deterministic and random numbers
• Deterministic– Single point, one value– Certain knowledge– 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
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2
2 2.51 1.5
2.1
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
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Logical operations with random numbers
• UNION and INTERSECTION– Logical representation of addition and multiplication
• Logic operations provide practical and useful approximations of outcomes
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Union or SummationA or BA + B
Intersection or MultiplicationA and BA * B
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7
The Project Balance Sheet ToolQuantitative Methods in Project Management
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Project Value from the Top Down
Project Estimate from the Bottom Up
Investor Value Expectation &
ResourceCommitment
Management investment
Risk
DeliverablesCost
Schedule
Project employment of investment
Recall the “Project Balance Sheet”
Map from business to project
Sponsor
Expectations
Value judgments
Resource Commitment
Capacity
Resources, skills, commitm
ent
Environment, tools
Capability
All the features and functions of
widget A
Resource Needs
Schedule
Cost
People, process, tools
Risk
Feature X
Dollars and schedule
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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
Plot confidence in cost [or schedule]
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$450K $550K$475K
Likely Risk
Confidence that the $_amount will not be exceeded
HighMedium
Low
Very High
>$550KNot to exceed cost
Plot timeline of project expense and business value
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$450K
$550K
Business value
Project expenses
Business value from sales
Sampling Metrics for Project EstimatesQuantitative Methods in Project Management
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In the beginning, there is a population
• All the data values, events, or event outcomes that share a common situation or environment
Population
• Space that holds all the values of the populationPopulation space
• May be deterministic or the outcome of a random process in/of the populationPopulation values
• Only those populations that bear upon project results are important
• Because a population bears upon project results, the population is important
Population importance
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Sampling risks
Accuracy
• Misunderstood exclusions, clusters, or strata
• Unrepresentative sample data value outliers
Completeness
• Excluded clusters or strata• Unrepresentative data quality
or deficiency
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Two risk assessments to be made
Margin of error
Estimated error around the measurement, observation, or
calculation of statistics
Interval of possible values for the statistic relative to the statistic
Confidence interval
Interval that probably contains the true population parameter
Confidence expresses probability that the true parameter is in the
interval
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Margin of error example
Margin of error % = 3 / 18 (x100) = 16.7%
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Sample Interval of statistic values: 3
Statistic
17 20
18
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
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Sample Interval of statistic: 3
Statistic
17 20
18
For 95/100 trialsSample interval contains the true population statistics
For 1 trial5% chance the interval does not contain the true population statistics
Confidence interval for proportional data
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Interval = p +/- Z * [p * (1 - p) / N]WhereZ is range value of standard Normal distributionZ is normalized to the standard deviationZ = 1 means 1 σ from the mean
Z range
Margin of error, proportional data
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+/- Margin of Error = ½ Interval width / pWhere ½ Interval width = +/- Z * [p * (1 - p) / N]
Z = 1.96
Hypothesis TestingQuantitative Methods in Project Management
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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?
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Distributions of X and Y
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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
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Sample X
Sample Y
Sample average distribution
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H0 distribution & confidence curve
• H0 likely TRUE for difference values < 0.219• Otherwise, likely FALSE• With confidence of 95%
Risk mitigation in time and resource schedulesQuantitative Methods in Project Management
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Any issues?
Should you be equally confident of making the milestone?
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Tandem path primitive
Parallel path primitive
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Interpreting the Confidence “S” Curve
27
B
B
A
A
C
C
0.25
0.5
0.75
1
0-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5
0.84
0.16
A. 68% confidence: value between -1 to +1B. 16% confidence: value > 1C. 84% confidence: value < 1
Schedule example for tandem tasks
Schedule network primitive Task duration distribution, D
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00.05
0.10.15
0.20.25
0.30.35
0.40.45
1 2 3 4 5 6
Task A Task B
Duration range 1 - 6
Task Probability distribution
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Milestone accumulates task performance
0
0.05
0.1
0.15
0.2
0.25
0.3
1 2 3 4 5 6 7 8 9 10 11 12
29
MilestoneExpected Value = 3.5 + 3.5
Milestone range 1 - 12
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Confidence for “schedule-at-mode”
30
0
0.2
0.4
0.6
0.8
1
1.2
23-M
ar
24-M
ar
25-M
ar
26-M
ar
27-M
ar
28-M
ar
29-M
ar
30-M
ar
31-M
ar
1-Ap
r
2-Ap
r
3-Ap
r
4-Ap
r
5-Ap
r
Low confidence in 3/25
1/12/12
1/21
3/25
3/15
Date
0.0 0.5
p / v
Calculate Confidence
Parallel path primitive
00.05
0.10.15
0.20.25
0.30.35
0.40.45
1 2 3 4 5 6
What is the schedule confidence at the milestone?
0
0.2
0.4
0.6
0.8
1
1.2
1 2 3 4 5 6
Confidence: 80% at 4
Distribution of tasks
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“Critical Chain” buffers uncertainty
Critical chain is a concept developed in the book Critical Chain (Goldratt, 1997)
Buffer11 days10 days
15 days 10 days
Task on the critical path
Task with risky duration, not on critical path
1 2
Project Buffer12 days
Path buffer mitigates “shift right” at the milestone of joining path
32
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