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 ^(th) 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

open allclose all

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 x 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 x has the most potential for improvement:

A simple system with both And and Or gates:

Show the importance:

Event x 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, z:

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 v 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 x and y 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:

Define the lifetime distributions:

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_2024_improvementimportance, author="Wolfram Research", title="{ImprovementImportance}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/ImprovementImportance.html}", note=[Accessed: 30-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_improvementimportance, organization={Wolfram Research}, title={ImprovementImportance}, year={2012}, url={https://reference.wolfram.com/language/ref/ImprovementImportance.html}, note=[Accessed: 30-December-2024 ]}