Logical-Trees Fault Tree, Decision Tree Analysis Computer Science Engineering (CSE) Notes | EduRev

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 1
Module 7 
Logical Trees 
Different sources of risk for an engineering system and / or activity can be analyzed with respect 
to their chronological and causal components using logical trees.  They are useful for the analysis 
of the overall risk as well as for the assessment of the risk contribution from the Individual 
components.  
Fault tree and event tree diagrams are the most well known and most widely applied type of  
logical trees in both qualitative and quantitative risk analysis. Even though more modern risk 
analysis techniques such as eg. Bayesian probabilistic nets have been developed over last years, 
fault tree and event tree are still main methods recommended (for US nuclear safety studies). 
Fault trees and event trees are in many ways similar and the choice of using one other or a 
combination of both in reality depends more on the traditions/preferences within a given industry 
than the specific characteristics of the logical tree.  
A significant difference between the two types of trees is though that the fault trees take basis in 
deductive (looking backwards) logic and the event trees are inductive (looking forward). In 
practical applications, a combination of fault trees and event trees is typically used. In this case, 
the fault tree part of the analysis is concerned about the representation of the sequences of 
failures, which may lead events with consequences and the event tree part of the analysis is 
concerned with the representation of the subsequent evolution of the consequence inducing 
events.  
Intersection between the fault tree and the event tree is in reality a matter of preference of the 
engineer performing the study. Small event tree / large fault tree and large event tree / small fault 
tree techniques may be applied to the same problem to supplement each other and provide 
additional insight with regard to the reliability of the considered system.  
Decision trees are often seen as a special type of event tree, but may be seen in much wider 
perspective and if applied consistently within the framework of decision theory, provide the 
theoretical basis for risk analysis.  The detailed analysis of the various types of logical trees 
Page 2


 1
Module 7 
Logical Trees 
Different sources of risk for an engineering system and / or activity can be analyzed with respect 
to their chronological and causal components using logical trees.  They are useful for the analysis 
of the overall risk as well as for the assessment of the risk contribution from the Individual 
components.  
Fault tree and event tree diagrams are the most well known and most widely applied type of  
logical trees in both qualitative and quantitative risk analysis. Even though more modern risk 
analysis techniques such as eg. Bayesian probabilistic nets have been developed over last years, 
fault tree and event tree are still main methods recommended (for US nuclear safety studies). 
Fault trees and event trees are in many ways similar and the choice of using one other or a 
combination of both in reality depends more on the traditions/preferences within a given industry 
than the specific characteristics of the logical tree.  
A significant difference between the two types of trees is though that the fault trees take basis in 
deductive (looking backwards) logic and the event trees are inductive (looking forward). In 
practical applications, a combination of fault trees and event trees is typically used. In this case, 
the fault tree part of the analysis is concerned about the representation of the sequences of 
failures, which may lead events with consequences and the event tree part of the analysis is 
concerned with the representation of the subsequent evolution of the consequence inducing 
events.  
Intersection between the fault tree and the event tree is in reality a matter of preference of the 
engineer performing the study. Small event tree / large fault tree and large event tree / small fault 
tree techniques may be applied to the same problem to supplement each other and provide 
additional insight with regard to the reliability of the considered system.  
Decision trees are often seen as a special type of event tree, but may be seen in much wider 
perspective and if applied consistently within the framework of decision theory, provide the 
theoretical basis for risk analysis.  The detailed analysis of the various types of logical trees 
 2
requires that the performance of the individual components of the trees already has been assessed 
in terms of failure rates and or failure probabilities. 
7.1. Fault Tree Analysis  
As mentioned earlier a fault tree is based on a deductive logic starting by considering an event of 
system failure and then aims to deduct which causal sequence of component failures could lead 
to the system failure. The system is thus often referred to as a top event. 
The logical interrelation of the sequences of component failures is represented through logical 
connections (logical gates) and the fault tree forms in effect a  
tree-like structure with the top event in the top and basic events at its extremities. The basic 
events are those events, for which failure rate data or failure probabilities are available and which 
cannot be dissected further. Sometimes the events are differentiated into initiating (or triggering) 
events and enabling events, where the initiating events are always the first event in a sequence of 
the enabling events are events, which may increase the severity of the initiated failure. 
The fault tree is a Boolean logical diagram comprised primarily of AND and OR gates. The 
output event of an AND gate occurs only if all of the input events occur simultaneously and the 
output event of an OR gate occur if any one of the input occurs. Figure 1 illustrates different 
commonly used symbols for AND and OR gates.  
 
Figure 1 – Illustration of commonly used symbols for AND and OR gates 
 
Page 3


 1
Module 7 
Logical Trees 
Different sources of risk for an engineering system and / or activity can be analyzed with respect 
to their chronological and causal components using logical trees.  They are useful for the analysis 
of the overall risk as well as for the assessment of the risk contribution from the Individual 
components.  
Fault tree and event tree diagrams are the most well known and most widely applied type of  
logical trees in both qualitative and quantitative risk analysis. Even though more modern risk 
analysis techniques such as eg. Bayesian probabilistic nets have been developed over last years, 
fault tree and event tree are still main methods recommended (for US nuclear safety studies). 
Fault trees and event trees are in many ways similar and the choice of using one other or a 
combination of both in reality depends more on the traditions/preferences within a given industry 
than the specific characteristics of the logical tree.  
A significant difference between the two types of trees is though that the fault trees take basis in 
deductive (looking backwards) logic and the event trees are inductive (looking forward). In 
practical applications, a combination of fault trees and event trees is typically used. In this case, 
the fault tree part of the analysis is concerned about the representation of the sequences of 
failures, which may lead events with consequences and the event tree part of the analysis is 
concerned with the representation of the subsequent evolution of the consequence inducing 
events.  
Intersection between the fault tree and the event tree is in reality a matter of preference of the 
engineer performing the study. Small event tree / large fault tree and large event tree / small fault 
tree techniques may be applied to the same problem to supplement each other and provide 
additional insight with regard to the reliability of the considered system.  
Decision trees are often seen as a special type of event tree, but may be seen in much wider 
perspective and if applied consistently within the framework of decision theory, provide the 
theoretical basis for risk analysis.  The detailed analysis of the various types of logical trees 
 2
requires that the performance of the individual components of the trees already has been assessed 
in terms of failure rates and or failure probabilities. 
7.1. Fault Tree Analysis  
As mentioned earlier a fault tree is based on a deductive logic starting by considering an event of 
system failure and then aims to deduct which causal sequence of component failures could lead 
to the system failure. The system is thus often referred to as a top event. 
The logical interrelation of the sequences of component failures is represented through logical 
connections (logical gates) and the fault tree forms in effect a  
tree-like structure with the top event in the top and basic events at its extremities. The basic 
events are those events, for which failure rate data or failure probabilities are available and which 
cannot be dissected further. Sometimes the events are differentiated into initiating (or triggering) 
events and enabling events, where the initiating events are always the first event in a sequence of 
the enabling events are events, which may increase the severity of the initiated failure. 
The fault tree is a Boolean logical diagram comprised primarily of AND and OR gates. The 
output event of an AND gate occurs only if all of the input events occur simultaneously and the 
output event of an OR gate occur if any one of the input occurs. Figure 1 illustrates different 
commonly used symbols for AND and OR gates.  
 
Figure 1 – Illustration of commonly used symbols for AND and OR gates 
 
 3
Several other types of logical gates exists such as QUANTIFICATION and COMPARISON, 
however, these will not be elaborated in present text.  
Top events and basic events also have their specific symbols as shown in figure below 
 
Figure 2 – Symbols commonly used in fault tree representations 
 In the Figure 2 diamond shaped symbol represents an undeveloped scenario which has not been 
developed in to a system of sub events due to lack of information and data.  
 
Figure 3 – Principal shape of fault tree 
Page 4


 1
Module 7 
Logical Trees 
Different sources of risk for an engineering system and / or activity can be analyzed with respect 
to their chronological and causal components using logical trees.  They are useful for the analysis 
of the overall risk as well as for the assessment of the risk contribution from the Individual 
components.  
Fault tree and event tree diagrams are the most well known and most widely applied type of  
logical trees in both qualitative and quantitative risk analysis. Even though more modern risk 
analysis techniques such as eg. Bayesian probabilistic nets have been developed over last years, 
fault tree and event tree are still main methods recommended (for US nuclear safety studies). 
Fault trees and event trees are in many ways similar and the choice of using one other or a 
combination of both in reality depends more on the traditions/preferences within a given industry 
than the specific characteristics of the logical tree.  
A significant difference between the two types of trees is though that the fault trees take basis in 
deductive (looking backwards) logic and the event trees are inductive (looking forward). In 
practical applications, a combination of fault trees and event trees is typically used. In this case, 
the fault tree part of the analysis is concerned about the representation of the sequences of 
failures, which may lead events with consequences and the event tree part of the analysis is 
concerned with the representation of the subsequent evolution of the consequence inducing 
events.  
Intersection between the fault tree and the event tree is in reality a matter of preference of the 
engineer performing the study. Small event tree / large fault tree and large event tree / small fault 
tree techniques may be applied to the same problem to supplement each other and provide 
additional insight with regard to the reliability of the considered system.  
Decision trees are often seen as a special type of event tree, but may be seen in much wider 
perspective and if applied consistently within the framework of decision theory, provide the 
theoretical basis for risk analysis.  The detailed analysis of the various types of logical trees 
 2
requires that the performance of the individual components of the trees already has been assessed 
in terms of failure rates and or failure probabilities. 
7.1. Fault Tree Analysis  
As mentioned earlier a fault tree is based on a deductive logic starting by considering an event of 
system failure and then aims to deduct which causal sequence of component failures could lead 
to the system failure. The system is thus often referred to as a top event. 
The logical interrelation of the sequences of component failures is represented through logical 
connections (logical gates) and the fault tree forms in effect a  
tree-like structure with the top event in the top and basic events at its extremities. The basic 
events are those events, for which failure rate data or failure probabilities are available and which 
cannot be dissected further. Sometimes the events are differentiated into initiating (or triggering) 
events and enabling events, where the initiating events are always the first event in a sequence of 
the enabling events are events, which may increase the severity of the initiated failure. 
The fault tree is a Boolean logical diagram comprised primarily of AND and OR gates. The 
output event of an AND gate occurs only if all of the input events occur simultaneously and the 
output event of an OR gate occur if any one of the input occurs. Figure 1 illustrates different 
commonly used symbols for AND and OR gates.  
 
Figure 1 – Illustration of commonly used symbols for AND and OR gates 
 
 3
Several other types of logical gates exists such as QUANTIFICATION and COMPARISON, 
however, these will not be elaborated in present text.  
Top events and basic events also have their specific symbols as shown in figure below 
 
Figure 2 – Symbols commonly used in fault tree representations 
 In the Figure 2 diamond shaped symbol represents an undeveloped scenario which has not been 
developed in to a system of sub events due to lack of information and data.  
 
Figure 3 – Principal shape of fault tree 
 4
It is noted that a fault tree comprising an AND gate represents a parallel system, i.e. all 
components must fail for the system to fail. Such a system thus represents some degree of 
redundancy because the system will still function after one component has failed. Fault trees 
comprising an OR gate on the other hand represents a series system, i.e. a system without any 
redundancy in the sense that it fails as soon as any one of its components has failed. Such as 
system is often denoted a weakest component system. Systems may be represented alternatively 
by reliability block diagrams, see Figure 4. 
 
Figure 4 – Reliability block diagrams for OR and AND gates 
 In accordance with the rules of probability theory the probability of the event for an AND gate is 
evaluated by  
?
=
=
n
i
i
p P
1
  
And for an OR gate by 
?
=
- - =
n
i
i
p P
1
) 1 ( 1
  
Where n is the number of ingoing event to the gate .P
i 
are the probabilities of  the failure  of  
ingoing events and  it is assumed that the ingoing are independent. 
System failure modes are defined by so-called cut-sets, i.e. combinations of basic events, which 
with certainty will lead to the top event. The number of such combinations can be rather large - 
several hundreds for a logical tree with about 50 basic events. It is important to note that the top 
event may still occur event though not all basic events in a cut set occur. A minimal cut set is the 
Page 5


 1
Module 7 
Logical Trees 
Different sources of risk for an engineering system and / or activity can be analyzed with respect 
to their chronological and causal components using logical trees.  They are useful for the analysis 
of the overall risk as well as for the assessment of the risk contribution from the Individual 
components.  
Fault tree and event tree diagrams are the most well known and most widely applied type of  
logical trees in both qualitative and quantitative risk analysis. Even though more modern risk 
analysis techniques such as eg. Bayesian probabilistic nets have been developed over last years, 
fault tree and event tree are still main methods recommended (for US nuclear safety studies). 
Fault trees and event trees are in many ways similar and the choice of using one other or a 
combination of both in reality depends more on the traditions/preferences within a given industry 
than the specific characteristics of the logical tree.  
A significant difference between the two types of trees is though that the fault trees take basis in 
deductive (looking backwards) logic and the event trees are inductive (looking forward). In 
practical applications, a combination of fault trees and event trees is typically used. In this case, 
the fault tree part of the analysis is concerned about the representation of the sequences of 
failures, which may lead events with consequences and the event tree part of the analysis is 
concerned with the representation of the subsequent evolution of the consequence inducing 
events.  
Intersection between the fault tree and the event tree is in reality a matter of preference of the 
engineer performing the study. Small event tree / large fault tree and large event tree / small fault 
tree techniques may be applied to the same problem to supplement each other and provide 
additional insight with regard to the reliability of the considered system.  
Decision trees are often seen as a special type of event tree, but may be seen in much wider 
perspective and if applied consistently within the framework of decision theory, provide the 
theoretical basis for risk analysis.  The detailed analysis of the various types of logical trees 
 2
requires that the performance of the individual components of the trees already has been assessed 
in terms of failure rates and or failure probabilities. 
7.1. Fault Tree Analysis  
As mentioned earlier a fault tree is based on a deductive logic starting by considering an event of 
system failure and then aims to deduct which causal sequence of component failures could lead 
to the system failure. The system is thus often referred to as a top event. 
The logical interrelation of the sequences of component failures is represented through logical 
connections (logical gates) and the fault tree forms in effect a  
tree-like structure with the top event in the top and basic events at its extremities. The basic 
events are those events, for which failure rate data or failure probabilities are available and which 
cannot be dissected further. Sometimes the events are differentiated into initiating (or triggering) 
events and enabling events, where the initiating events are always the first event in a sequence of 
the enabling events are events, which may increase the severity of the initiated failure. 
The fault tree is a Boolean logical diagram comprised primarily of AND and OR gates. The 
output event of an AND gate occurs only if all of the input events occur simultaneously and the 
output event of an OR gate occur if any one of the input occurs. Figure 1 illustrates different 
commonly used symbols for AND and OR gates.  
 
Figure 1 – Illustration of commonly used symbols for AND and OR gates 
 
 3
Several other types of logical gates exists such as QUANTIFICATION and COMPARISON, 
however, these will not be elaborated in present text.  
Top events and basic events also have their specific symbols as shown in figure below 
 
Figure 2 – Symbols commonly used in fault tree representations 
 In the Figure 2 diamond shaped symbol represents an undeveloped scenario which has not been 
developed in to a system of sub events due to lack of information and data.  
 
Figure 3 – Principal shape of fault tree 
 4
It is noted that a fault tree comprising an AND gate represents a parallel system, i.e. all 
components must fail for the system to fail. Such a system thus represents some degree of 
redundancy because the system will still function after one component has failed. Fault trees 
comprising an OR gate on the other hand represents a series system, i.e. a system without any 
redundancy in the sense that it fails as soon as any one of its components has failed. Such as 
system is often denoted a weakest component system. Systems may be represented alternatively 
by reliability block diagrams, see Figure 4. 
 
Figure 4 – Reliability block diagrams for OR and AND gates 
 In accordance with the rules of probability theory the probability of the event for an AND gate is 
evaluated by  
?
=
=
n
i
i
p P
1
  
And for an OR gate by 
?
=
- - =
n
i
i
p P
1
) 1 ( 1
  
Where n is the number of ingoing event to the gate .P
i 
are the probabilities of  the failure  of  
ingoing events and  it is assumed that the ingoing are independent. 
System failure modes are defined by so-called cut-sets, i.e. combinations of basic events, which 
with certainty will lead to the top event. The number of such combinations can be rather large - 
several hundreds for a logical tree with about 50 basic events. It is important to note that the top 
event may still occur event though not all basic events in a cut set occur. A minimal cut set is the 
 5
cut set that represents the smallest combination of basic events leading to the top event, 
sometimes denoted the critical path. The top event will only occur if all events in die minimal cut 
set occur. An important aspect of fault tree analysis is the identification of the minimal cut sets as 
this greatly facilitates the numerical evaluations involved. 
7.2. Event trees  
An event tree is a representation of the logical order of events leading to some (normally 
adverse) condition of interest for a considered system. It should be noted that several different 
states for the considered system could be associated the important consequences.  
In contrast to the fault tree it starts from a basic initiating event and develops from there in time 
until all possible states with adverse consequences have been reached. The initiating events may 
typically arise as top events from fault tree analysis. The event tree is constructed from event 
definitions and logical vertices (outcomes of events), which may have a discrete sample space as 
well as a continuous sample space. Typical graphical representations of event trees are shown in 
Figure 5.   
 
Figure 5 – Illustration of the principal appearance of an event tree 
Event trees can become rather complex to analyze, This is easily realized by noting that for a 
system with n two-state components the total number of paths is 2
n
. If each component has m 
states the total number of branches is m
n
.