RiskReductionImportance

✖
`RiskReductionImportance`

✖

RiskReductionImportance[rdist,t]

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

✖

RiskReductionImportance[fdist,t]

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

Details

RiskReductionImportance is also known as risk reduction worth.
The risk reduction importance for component is the factor by which the system unreliability would be decreased if component were perfect. As such, it shows the potential for increase in system reliability by making component better.
The risk reduction importance at time t for component is given by , where is the probability that the system failed, given that the component never fails, and is the probability that the system has failed.
The results are returned in the component order given in the distribution list in rdist or fdist.

Examples

open allclose all

Basic Examples (3)Summary of the most common use cases

Two components connected in series, with different lifetime distributions:

In[1]:=1

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-e66rb

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

In[2]:=2

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-4wtin8

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-d43v3q

Out[3]=3

By improving a parallel component, the system reliability can be improved infinitely:

In[1]:=1

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-yn943z

In[2]:=2

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-8e7d7o

Out[2]=2

Use fault tree-based modeling to define the system:

In[1]:=1

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-28utav

In[2]:=2

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-fmkne5

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-0zv592

Out[3]=3

Scope (19)Survey of the scope of standard use cases

ReliabilityDistribution Models (10)

Two components connected in parallel, with identical lifetime distributions:

In[1]:=1

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-08flls

The importance is identical:

In[2]:=2

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-mkns50

Out[2]=2

Two components connected in series, with identical lifetime distributions:

In[1]:=1

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-rwabaw

The importance is identical:

In[2]:=2

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-f1wvq8

Out[2]=2

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

In[1]:=1

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-mji74e

The importance is identical:

In[2]:=2

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-xc5531

Out[2]=2

A simple mixed system with identical lifetime distributions:

In[1]:=1

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-bs7f7v

In[2]:=2

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-jy1k7y

Compute the importance:

In[3]:=3

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-6vuxir

Out[3]=3

Component is most important, and and are equally important because of symmetry:

In[4]:=4

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-fn2idk

Out[4]=4

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

In[1]:=1

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-8pfw08

In[2]:=2

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-mjkl5o

Compute the importance:

In[3]:=3

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-t1mcxq

Out[3]=3

Component is most important, but is not shown in the plot as it equals infinity:

In[4]:=4

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-xfjq50

Out[4]=4

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

In[1]:=1

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-3daofy

In[2]:=2

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-lqd3ci

Compute the importance:

In[3]:=3

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-srbs5o

Out[3]=3

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

In[4]:=4

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-xwt0s4

Out[4]=4

One component in parallel with two others, with different distributions:

In[1]:=1

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-pr0k1j

In[2]:=2

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-bkyutz

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

In[3]:=3

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https://wolfram.com/xid/0nx1wkv40hjeh4j2q-urftd7

Out[3]=3

As machine-precision results:

In[4]:=4

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https://wolfram.com/xid/0nx1wkv40hjeh4j2q-vf85i0

Out[4]=4

As symbolic expressions:

In[5]:=5

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https://wolfram.com/xid/0nx1wkv40hjeh4j2q-4n9ka6

Out[5]=5

Any valid ReliabilityDistribution can be used:

In[1]:=1

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-cjmrsj

In[2]:=2

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-6zfypk

In[3]:=3

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-6lxtrj

Out[3]=3

The less reliable component is more important:

In[4]:=4

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-iedm9b

Out[4]=4

ReliabilityDistribution can contain different distribution families:

In[1]:=1

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https://wolfram.com/xid/0nx1wkv40hjeh4j2q-kolqsy

In[2]:=2

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-kbcgxc

Out[2]=2

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

In[1]:=1

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-dcrxnl

In[2]:=2

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-2f3o3y

In[3]:=3

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-n8cikq

Out[3]=3

Plot the importance over time:

In[4]:=4

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-kf0c4n

Out[4]=4

FailureDistribution Models (9)

Either of two basic events leads to the top event:

In[1]:=1

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-l11ptk

The importance is identical:

In[2]:=2

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-6nc5zk

Out[2]=2

Only both basic events together lead to the top event:

In[1]:=1

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-xmgcyf

The importance is identical:

In[2]:=2

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-nlxcpv

Out[2]=2

A voting gate, with identical distributions on the basic events:

In[1]:=1

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-natpec

The importance is identical:

In[2]:=2

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-xz9lrc

Out[2]=2

A simple system with both And and Or gates:

In[1]:=1

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-4jsp53

In[2]:=2

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-3a7nr5

The basic event is most important:

In[3]:=3

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-wh2s2m

Out[3]=3

Both and are equally important because of symmetry:

In[4]:=4

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-tynywc

Out[4]=4

A simple system with both And and Or gates:

In[1]:=1

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-fecls

In[2]:=2

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-sr2e4s

Show the importance:

In[3]:=3

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-gbr67k

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-88i8fa

Out[4]=4

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

In[1]:=1

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-ig97mn

In[2]:=2

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-xdjp3g

Compute the importance:

In[3]:=3

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-4smvi5

Out[3]=3

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

In[4]:=4

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-pcd9ai

Out[4]=4

Any valid FailureDistribution can be used:

In[1]:=1

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-34mp4o

In[2]:=2

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-i1m9g0

In[3]:=3

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-9qepae

Out[3]=3

The less reliable component is more important:

In[4]:=4

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-cmc5tk

Out[4]=4

FailureDistribution can contain different distribution families:

In[1]:=1

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-egoopq

In[2]:=2

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-cvkcqm

Out[2]=2

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

In[1]:=1

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-lxyr6j

In[2]:=2

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-mvuep3

In[3]:=3

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-okc513

Out[3]=3

Plot the importance over time:

In[4]:=4

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-z93seu

Out[4]=4

Applications (3)Sample problems that can be solved with this function

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

In[1]:=1

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-z9j4b1

Two fuel pumps require power to run:

In[2]:=2

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-nefezh

In[3]:=3

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-l66sqs

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

In[4]:=4

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-lbq8mc

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

In[5]:=5

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-qek6nb

Define the lifetime distributions:

In[6]:=6

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-nkxog6

In[7]:=7

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-2q97ci

In[8]:=8

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-o4f20d

In[9]:=9

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-ifr45q

Compute the importance:

In[10]:=10

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-yz5yqc

Improving the power would give a good reliability increase in the beginning, but for long-term reliability the various pumps should be improved:

In[11]:=11

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-lzpwgc

Out[11]=11

Find out which component is best to improve in a system that has a mission time of three hours:

In[1]:=1

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-bqrgiq

In[2]:=2

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-c1oyjo

In[3]:=3

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-wgnax0

In[4]:=4

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-ud0au2

Out[4]=4

Show the importance over time:

In[5]:=5

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-7l5tir

Out[5]=5

Component can improve system reliability the most, assuming a mission time of three hours:

In[6]:=6

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-k1mgn7

Out[6]=6

The importance is the factor with which the system unreliability can be reduced by improving :

In[7]:=7

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-fqy9xf

In[8]:=8

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-kc7jl3

Out[8]=8

In[9]:=9

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-sqvup7

Out[9]=9

Consider a water pumping system with one valve and two redundant pumps. The reliability of the components is given as probabilities:

In[1]:=1

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-dd2ary

Find out which components can improve system reliability most:

In[2]:=2

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-q9ocrf

Out[2]=2

Properties & Relations (5)Properties of the function, and connections to other functions

RiskReductionImportance can be defined in terms of Probability:

In[2]:=2

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-dmwz4w

Compute the base unreliability for the system:

In[3]:=3

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-gaprad

Out[3]=3

The unreliability for when the component is always working:

In[4]:=4

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-bf4xi

Out[4]=4

The ratio of the base unreliability and the unreliability for the always working components:

In[5]:=5

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-mktkuk

Out[5]=5

Compare with definition:

In[6]:=6

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-7lf1xg

Out[6]=6

RiskReductionImportance is related to CriticalityFailureImportance:

In[1]:=1

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-qczapv

Compute the risk-reduction importance:

In[2]:=2

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-r4rx21

Out[2]=2

Compare with the definition:

In[3]:=3

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-m0jfac

Out[3]=3

A system with all components in parallel will have RiskReductionImportance equal to :

In[1]:=1

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-m5wq70

In[2]:=2

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-2xkxww

Out[2]=2

For all importances that are not , the RiskReductionImportance approaches 1 as :

In[1]:=1

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-iuca3a

In[2]:=2

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-v2jra3

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-q1tobo

Out[3]=3

Irrelevant components have importance 1:

In[1]:=1

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-qx43x6

In[2]:=2

✖

https://wolfram.com/xid/0nx1wkv40hjeh4j2q-kvr29j

Out[2]=2

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