An important step in reliability evaluation of any system is selection of an appropriate mathematical representation. One of the possible mathematical representations is structure function that expresses dependency of system state on states of its components. This function must be completely specified for reliability evaluation of the analyzed system. The structure function is constructed based on complete information about the system structure and possible components states. However, there are a lot of practical problems when the complete information is not available because data from which it can be derived cannot be collected. As a rule, other mathematical representations and methods for evaluation of system reliability are used in these situations. In this paper, we propose a new method for construction of the structure function from uncertain or incomplete data. This method is developed based on application of Fuzzy Decision Tree.
As has been shown in paper [1], selecting a mathematical representation of an analyzed system is an important step in reliability analysis. Depending on the number of performance levels, two types of models can be recognized. These models are named as Binary-State Systems (BSSs) and Multi-State Systems (MSSs).
A BSS admits only two states in investigation of the system and its components: perfect functioning and complete failure. However, in practice, many systems can go through different performance levels between these two extreme states [1, 2]. A MSS is a mathematical model that is used to describe such systems since it allows defining more than two levels of performance [2, 3, 4].
There are different types of mathematical representations of a system. In reliability engineering, structure function, fault trees, reliability block diagrams, Markov models
- 362 and Petri nets are typically used for the mathematical representation of real systems under study. Historically, mathematical models based on the structure function have been proposed firstly. In this case, a system is modeled as a mapping that assigns system state to all possible combinations of component states. The system performance level is known based on the states of all its components. This interpretation of the structure function supposes the exact definition of all possible states of the system and its components. Therefore, any uncertainty cannot be considered and taken into account. However, this indicates that methods based on the structure function approach have some difficulties in application on real-world problems because, as a rule, data about behavior of such systems are uncertain. Two approaches can be used to solve this problem.
The first of them is development of a new model that takes uncertainties into account [5, 6, 7]. The application of a new mathematical model leads to a development of new mathematical methods for the analysis of this model. The second solution is to use one of the traditional models and develop new methods for construction of the structure function that will take uncertainties of the initial data into account.
Specifics of the uncertainty have to be analyzed before the development of the new
method for the structure function construction based on the uncertain data. There are
different factors of uncertain data. In our investigation, we will take into account two
of them. The first are ambiguity and vagueness of initial data. It means that initial data
about the system operation are collected based on (a) measurement that can be
inaccurate and with an error or (b) experts that can have different opinions on one situation.
Therefore, values of states of the components or system performance level cannot be
indicated as exact (integers). Ambiguity and vagueness in a real system have been
studied using the probability theory. However, it is worth pointing out that some
uncertainties that are not random in nature can play important roles in construction of
the structure function [
Secondly, situations in which it is impossible to indicate some values of the system components states or performance level can exist. For example, it can be very expensive, or it needs unacceptable long time. This implies that some information about the system behavior can be absent. Therefore, the data are incomplete.
Based on the previous text, we have a task of construction of exact and completely
specified structure function based on uncertain and incomplete data, what is a typical
problem of Data Mining [
In this paper, we propose a method based on the application of an FDT for
construction of the structure function. FDTs allow taking into account uncertainties of
two types. The first of them is ambiguity of initial data. This can occur when it is
expensive to obtain all data about real system behavior, or there are poorly
documented data. This type of uncertainty is covered by fuzzy values in an FDT. It means
that initial data can be defined and interpreted with some possibility and might not be
exact. The second type of uncertainty agrees with incompletely specified initial data.
As a rule, if the exact values of the actual data about the system behavior cannot be
determined, we need to rely on more data to get additional information necessary to
correct the used theoretical model [
This paper is structured as follows. Section 2 discusses the concept of the structure function. The principal steps of the proposed method are considered in sections 3 – 5. These steps are Collection of data into a repository (section 3), Representation of the system model in the form of an FDT (section 4), and Construction of the structure function based on the FDT (section 5). 2
The structure function as a mathematical model was introduced in reliability
engineering as one of the firsts [
Let us suppose that the system can be divided into n components (subsystems). A state of each component can be denoted by a random variable xi that can be in one of mi possible values. This variable takes value 0 if the component fails and one of values 1,…, mi -1 if the component works satisfactorily.
Let us denote the structure function as (x). Then it agrees with the next map: (x) = (x1,…, xn): {0,…, m1 -1}×…×{0,…, mn -1}{0,…, M -1} , (1) where (x) defines system state from complete failure ((x) = 0) to perfect functioning ((x) = M -1); x = (x1,…, xn) is a state vector; xi is the i-th component state that changes from complete failure (xi = 0) to perfect functioning (xi = mi -1).
Next, let us suppose that the system is coherent. This means: (a) the system structure function is monotone: (xi, x) ≤ (xj, x) for any xi ≤ xj; and (b) there are no irrelevant components in the system.
Every system component is characterized by the probabilities of individual states: (2) (3) pi,s = Pr{xi = s}, s = 0,…, mi -1 .
Please note that the structure function of MSS (1) is transformed into the structure function of BSS if mi = M = 2.
Many reliability indices and measures can be calculated based on the system structure function. One of them is the probability of the system performance level that is calculated as follows [3]:
Aj = Pr{(x) = j}, j = 0,…, mi -1 .
The structure function also allows calculating the boundary system states [
As a rule, the structure function can be defined as a result of the system structure
analysis or based on expert data [
In terms of Data Mining, the structure function can be interpreted as a table of
decisions [
The method proposed in this paper includes the following steps: collection of data into the repository according to requests of FDT induction; representation of the system model in the form of an FDT that classifies components states according to the system performance levels; construction of the structure function as a decision table that is created by inducted FDT.
The structure function is constructed as a decision table that classifies the system performance level for each possible combination of components states. The decision table is formed based on the FDT that provides the mapping for all possible components states (input data) in M performance levels. The FDT is inducted using uncertain data that are presented in the form of a specified repository. 3
Collection of data in the form of a repository is provided by the monitoring of values of system component states and system performance level. This repository can be presented in the form of a table where the columns agree with the input and output attributes. The number of the input attributes is n and the i-th has mi possible values (the i-th column includes mi sub-columns). Every row contains a real sample of components states and the corresponding system performance level.
For example, let us consider the offshore electrical power generation system presented in [2]. The purpose of this system (Fig. 1) is to supply two nearby oilrigs with electric power. The system includes 3 generators: two main generators A1 and A3, and standby generator A2. Both main generators are at oilrigs. In addition, oilrig 1 has generator A2 that is switched into the network in case of outage of A1 or A3. The control unit U continuously supervises the supply from each of the generators with automatic control of the switches. If, for instance, the supply from A3 to oilrig 2 is not sufficient, whereas the supply from A1 to oilrig 1 is sufficient, U can activate A2 to supply oilrig 2 with electric power through the standby subsea cables L. This implies that the system consists of 5 relevant components (n = 5): generators A1, A2, and A3, control unit U, and the standby subsea cables L. Furthermore, according to the description of the system activity in [2], we assume that the system and all its components have 3 states/performance levels (M = 3 and mi = 3, for i = 1,…,5). Next, let us denote variables defining states of the system components in the following way: main generators A1 and A3 as x1 and x3 respectively, standby generator A2 as x2, and control unit U and standby subsea cables L as x4 and x5 respectively.
Let us suppose monitoring of the offshore power generation system that allowed
collecting 108 (from 243 possible) samples of the system behavior. Some of them are
shown in Table 1. The monitoring of this system permitted obtaining information
about some combinations of component states and the corresponding performance
levels of the system. However, this information is not complete because the data from
the real monitoring are uncertain. This uncertainty is caused by the ambiguity of
classification of component states and system performance levels into classes of exact
values [
The data obtained based on the monitoring and presented in Table 1 can be
interpreted as fuzzy data [
The data obtained based on the monitoring of the offshore electrical power generation system are incompletely specified because we have 108 of all 243 combinations of components states. Traditional mathematical approach for system reliability analysis based on the structure function cannot be used in this case. Therefore, construction of structure function (1) based on incomplete data requires a special transformation and development of new methods. In this paper, we suggest the new method for construction of the structure function based on an FDT. This method allows reducing indeterminate values and obtaining a completely specified structure function.
A decision tree is a formalism for expressing mappings of input attributes
(components states) to output attribute/attributes (system performance level), consisting of an
analysis of attribute nodes (input attributes) linked to two or more sub-trees and leafs
or decision nodes labeled with classes of the output attribute (in our case, a class
agrees with a system performance level) [
There are different methods for inducting an FDT [
A fuzzy set A with respect to a universe U is characterized by a membership function μA : U [0,1], which assign an A-membership degree, μA(u), to each element u in U. μA(u) gives us an estimation that u belongs to A. The cardinality measure of the fuzzy set A is defined by M(A) = uU μA(u), and it is measure of size of set A. For u U, μA(u) = 1 means that u is definitely a member of A and μA(u) = 0 means that u is definitely not a member of A, while 0 < μA(u) < 1 means that u is a partial member of A. If either μA(u) = 0 or μA(u) = 1 for all u U, A is a crisp set. The set of input attributes A is crisp if μA(u) = 0 or μA(u) = 1.
For example, let us consider input attributes A = {A1, A2, A3, A4, A5} and the
output attribute B for the offshore electrical power generation system in Fig. 1. This
system is represented by 5 input attributes. Each input attribute is defined as: Ai = {Ai,0,
Ai,1, Ai,2}, for i = 1,…, 5, and the output attribute is B = {B0, B1, B2}. The values of
the input attributes and the output attribute are defined in Table 3. These values are
obtained based on the data from Table 1 and are used for the FDT construction as a
training test. The principal difference of Table 1 and 3 is interpretation of initial date
as attributes. The induction of the FDT based on this training test can be implemented
using some of methods for FDT induction [
The FDT resulted from the training set presented in Table 3 has been inducted by
application of the cumulative information estimates using the method in [
According to [
Each non-leaf node is associated with an attribute Ai A, or in terms of reliability analysis: each non-leaf node is associated with a component. The non-leaf node agreeing with attribute Ai has mi outgoing branches. The s-th outgoing branch (s = 0,…, mi -1) from the non-leaf node corresponding to attribute Ai agrees with state s of the i-th component (xi = s). A path from the root to a leaf defines one or more state vectors (according to the values of the input attributes (component states) occurred in the path) for which the structure function takes value determined by the value of the output attribute. If any input attribute is absent in the path, all possible states have to be considered for the associated component.
Let us consider construction of the structure function of the offshore electrical power generation system from Fig. 1 using the FDT depicted in Fig. 2. All possible component states (all state vectors) have to be used for calculation of the system performance level by the FDT to form the decision table (structure function). Let us explain this idea for the first level of the FDT in more detail.
Preliminary analysis of the data obtained based on the monitoring (see Table 3) shows that possible values of the output attribute B are distributed as follows: value 0 – with confidence 0.493, value 1 – with confidence 0.209 and value 2 – with confidence 0.298. These values are implied by frequency of every output value in the training test (Table 3). Attribute A3 is associated with the FDT root. So, analysis of the data starts from this attribute. This attribute has the following possible values: A3,0, A3,1, and A3,2. Value A3,0 of this attribute makes the output attribute B to be B0 (the system is non-operational) with the confidence of 0.805. Other variants, B1 and B2, of output attribute B can be chosen with the confidence of 0.163 and 0.012 respectively. If the attribute A3 has other values, i.e. A3,1 or A3,2, then the analysis is done similarly.
If the value of attribute A3 is A3,0, than the next attribute in the analysis is A5, which have values A5,0, A5,1, and A5,2. Value A5,0 of this attribute agrees with a leaf representing the output attribute. Therefore, in this situation, the analysis is stopped and value of the output attribute is defined: value B0 of attribute B should be chosen with the confidence of 0.926, and values B1 and B2 with confidences 0.067 and 0.007 respectively. Similarly, the process of the analysis of the non-ordered FDT continues for the other input attributes and their values.
Next, let us consider state vector x = (0,0,0,0,0). The analysis based on the FDT starts with the attribute A3 (Fig. 2) that is associated with the 3-rd component. State of this component is 0 (x3 = 0) for the specified state vector. Therefore, the branch for value A3,0 of attribute A3 value is taken into account. According to this value, the identification of the output attribute value (system performance level) has to continue through attribute A5. According to the state vector, x5 = 0, therefore, attribute A5 has value A5,0. Now, value of the output attribute can be indicated because the branch has a leaf. So, value of the output attribute is defined as 0 with the confidence of 0.926 without analysis of other attributes. Analysis of other state vectors is similar and allows obtaining all possible values of the system performance level in the form of the structure function. The analysis of all possible state vectors from x = (0,0,0,0,0) to x = (2,2,2,2,2) allows us to construct the complete structure function of the offshore electrical power generation system depicted in Fig. 1.
It is important to note that this method of construction of the structure function based on FDTs permits to compute (restore) data missing from the monitoring.
A representation of the system using the structure function allows calculating
different indices and measures for estimation of system reliability. Probabilities of
system performance levels (3) are one of them. Suppose that probabilities of the
components states of the offshore electrical power generation system have values shown in
Table 4. In this case, the probabilities of system performance levels are: A2 = 0.73, A1
= 0.20 and A0 = 0.07. Other measures can be computed using the structure function
too. For example, importance measures for this system can be calculated using the
algorithms considered in [
Let us present a simple case study to verify the modelling approach described in
the previous sections. We use structure function of the offshore electrical power
generation system to examine efficiency and accuracy of the proposed method for
construction of the structure function based on uncertain data. Therefore, the structure
function must be transformed to ambiguous and incompletely specified form. In the
proposed methods, two types of uncertainty are included. The first is ambiguity of
data values. Therefore, all integer values of components states and performance level
have to be transformed to values with some possibilities. We can use algorithm from
[
This method can be considered for special cases and types of initial data with
application of other algorithms from Data Mining to improve the obtained results. For
example, initial data can be obtained for similar samples and, in this case, special
algorithms for pattern recognition and intelligent diagnosis can be used [
The first type of uncertainty deals with some state vectors missing from the initial data. In practical application, it can be caused by the impossibility to obtain or indicate all possible combinations of system component states.
These uncertainties are considered and taken into account in the interpretation of
the initial data as quasi-fuzzy data. This interpretation requires use of mathematical
methods of fuzzy logic for the analysis. In this paper, an FDT is used for system
behavior modeling and construction of the system structure function. This mathematical
method transforms incomplete and uncertain initial data to a correct decision [
The second type of uncertainty results from ambiguity of initial data. In this case, the system performance level and components states can be defined with some possibilities. According to the typical definition of the structure function (1), performance level can have only one value for every state vector from set {0, …, M -1}. However, the boundary between two neighboring values can be diffused in real applications. Both such values can be therefore indicated with some possibility. The proposed method takes such ambiguity into account and permits indicating performance level using some values ranging from 0 to M -1 with a possibility that is considered in the next steps of the method and is not disregarded. The component states are indicated in a similar manner and the state of the i-th component is considered as a value ranging from 0 to mi -1 with possibilities. For example, the data in Table 1 are presented with consideration of such ambiguity: every value is indicated with some possibility. 7
This work is supported by the grant grants VEGA 1/0498/14 and VEGA 1/0038/16.