StructuralImportance
StructuralImportance[rdist]
gives the structural importances for all components in the ReliabilityDistribution rdist.
StructuralImportance[fdist]
gives the structural importances for all components in the FailureDistribution fdist.
StructuralImportance[bexpr,{x1,x2,…}]
gives the structural importance for the components x1, x2, … in the Boolean expression bexpr.
Details
- StructuralImportance is also known as Birnbaum's structural importance.
- The structural importance for component
is the fraction of the system states in which component 
is working, where a failure of component 
will result in a failure of the system. - For StructuralImportance[fdist] and StructuralImportance[rdist], the results are returned in the component order given in the distribution list in rdist or fdist.
- For StructuralImportance[bexpr,{x1,x2,…}], the results are returned in the order {x1,x2,…}.
- StructuralImportance[bexpr,…] is defined when UnateQ[bexpr] is True.
Examples
open all close allBasic Examples (4)
Two components connected in series:
The result is given in the same order as the distribution list in ReliabilityDistribution:
Two components connected in parallel:
Use the structure function directly, without lifetime distributions:
Scope (18)
Boolean Expression Models (5)
ReliabilityDistribution Models (7)
Two components connected in parallel:
Both components are equally important:
Two components connected in series:
A system where two out of three components need to work:
Components are equally important:
A system with a series connection in parallel with a component:
The single component 
is more important:
One component in parallel with two others, with different distributions:
Find the importance measures at one specific point in time as exact results:
Model the system in steps to get the importance measure for a subsystem:
FailureDistribution Models (6)
Any of two basic events lead to the top event:
Both events have the same importance:
Only both basic events together lead to the top event:
Serial events are equally important:
A voting gate, with identical distributions on the basic events:
The events have the same structural importance:
A simple system with both And and Or gates:
The basic event 
is most important:
A simple system with both And and Or gates:
Model the system in steps to get importance measure for a subsystem:
Applications (4)
Two points in a city are connected through a network of water pipes 
. Find the pipes most critical to maintain the supply of water:
Analyze what component is most likely to have caused a failure at 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:
Deicing and fuel storage have the highest structural importance:
StructuralImportance is also called the Banzhaf power index and measures the probability of changing the outcome of a voting procedure. Consider a county board with the following districts and votes:
Find all combinations of district votes that will give a majority:
The Boolean expression for a decision achieving a majority:
None of the Oyster Bay, Glen Cove, and Long Beach districts can change the outcome of a vote alone:
Compute the Banzhaf power index of the states in the 1804 US presidential election:
Compare with the population in the respective states:
Population on the inner pie, power index on the ring:
As a plot of population and power index:
Compare the power index with the dates they joined the union:
Properties & Relations (4)
Structural importance is independent of the lifetime distributions:
The result does not contain any of the distribution parameters:
It is equivalent to the specification without the distributions:
Structural importance is BirnbaumImportance where components have probability 
:
Irrelevant components do not influence the importance of other components:
Irrelevant components have importance 0:
Structural importance for bexpr in ReliabilityDistribution is equal to bexpr in FailureDistribution:
Text
Wolfram Research (2012), StructuralImportance, Wolfram Language function, https://reference.wolfram.com/language/ref/StructuralImportance.html.
CMS
Wolfram Language. 2012. "StructuralImportance." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/StructuralImportance.html.
APA
Wolfram Language. (2012). StructuralImportance. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/StructuralImportance.html