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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

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