# ImprovementImportance

ImprovementImportance[rdist,t]

gives the improvement importances for all components in the ReliabilityDistribution rdist at time t.

ImprovementImportance[fdist,t]

gives the improvement importances for all components in the FailureDistribution fdist at time t.

# Details

• ImprovementImportance is also known as improvement potential.
• The improvement importance of a component is the increase of the SurvivalFunction if is replaced with a perfect component.
• The improvement importance at time for component is given by , where is the probability that the system is working, given that the component never fails, and is the probability that the system is working.
• The importance results are returned in the component order given in the distribution list in rdist.

# Examples

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## Basic Examples(3)

Two components connected in series, with different lifetime distributions:

The result is given in the same order as the distribution list in ReliabilityDistribution:

Two components connected in parallel, with different lifetime distributions:

Use fault tree-based modeling to define the system:

## Scope(17)

### ReliabilityDistribution Models(9)

Two components connected in parallel with identical lifetime distributions:

Both components have the same potential for improvement:

Two components connected in series with identical lifetime distributions:

Both components have the same potential for improvement:

A system where two out of three components need to work, with identical lifetime distributions:

Both components have the same potential for improvement:

A simple mixed system with identical lifetime distributions:

Compute the importance:

Component would be best to replace by a perfect component:

A system with a series connection in parallel with a component:

Calculate the importance:

Component would be best to replace by a perfect component:

Study the effect of a change in parameter in a simple mixed system:

Compute the importance:

Show the changes in importance when worsening one of the parallel components, :

One component in parallel with two others with different distributions:

Find the importance measures at one specific point in time as exact results:

As machine-precision results:

As symbolic expressions:

Any valid ReliabilityDistribution can be used:

It is better for the reliability to replace the less reliable component with a perfect component:

Model the system in steps to get the importance measure for a subsystem:

The subsystem has less potential for improvement, as it is already more reliable:

### FailureDistribution Models(8)

Any of two basic events lead to the top event:

Both components have the same potential for improvement:

Only both basic events together lead to the top event:

Both components have the same potential for improvement:

A voting gate with identical distributions on the basic events:

Identical events in a voting gate have the same potential for improvement:

A simple system with both And and Or gates:

Calculate the improvement importance:

Event has the most potential for improvement:

A simple system with both And and Or gates:

Show the importance:

Event has the most potential for improvement:

Study the effect of a change in parameter in a simple mixed system:

Compute the importance:

Show the changes in importance when worsening one of the basic events, :

Any valid FailureDistribution can be used:

Plot the importance measures:

Model the system in steps to get the importance measure for a subsystem:

Plot the importance over time:

## Applications(3)

Find out which component is best to improve in a system that has to last for three hours:

Component is best to improve according to the improvement importance:

Study a system with one component in series and two components in parallel. Determine which component is the most important according to the improvement importance measure:

The components and are the most important components:

Analyze what components have the best potential for improving the reliability of the launch of an aircraft. The hangar door can be opened electronically or manually:

Two fuel pumps require power to run:

Two more pumps run on reliable batteries, giving the following fuel transfer structure:

Also needed is deicing of the aircraft and a fuel storage tank:

Compute the importance:

Improving the pumps has the largest possibility for an increase in reliability:

## Properties & Relations(3)

ImprovementImportance can be defined in terms of Probability:

System reliability when the component is replaced by a perfect component:

Compute the difference to the base system reliability:

Compare to the definition:

ImprovementImportance can be defined in terms of BirnbaumImportance:

Compute BirnbaumImportance for all components:

Multiply by the unreliability of the components:

Compare with the definition:

Irrelevant components have importance 0:

Wolfram Research (2012), ImprovementImportance, Wolfram Language function, https://reference.wolfram.com/language/ref/ImprovementImportance.html.

#### Text

Wolfram Research (2012), ImprovementImportance, Wolfram Language function, https://reference.wolfram.com/language/ref/ImprovementImportance.html.

#### CMS

Wolfram Language. 2012. "ImprovementImportance." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ImprovementImportance.html.

#### APA

Wolfram Language. (2012). ImprovementImportance. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ImprovementImportance.html

#### BibTeX

@misc{reference.wolfram_2023_improvementimportance, author="Wolfram Research", title="{ImprovementImportance}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/ImprovementImportance.html}", note=[Accessed: 13-April-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2023_improvementimportance, organization={Wolfram Research}, title={ImprovementImportance}, year={2012}, url={https://reference.wolfram.com/language/ref/ImprovementImportance.html}, note=[Accessed: 13-April-2024 ]}