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• A new method for construction of a mathematical model for reliability analysis. • Multi-Valued Decision Diagram construction for Multi-State System. • Reliability analysis based on uncertain and incompletely specified data. • The system importance analysis based on the proposed method. The development of the system's model is an important step in reliability analysis. The system's model in reliability analysis similar to other knowledge areas is used as an explanatory and research tool. In reliability analysis, this model is mathematical and allows us to determine quantitative characteristics to evaluate the behavior of the original system. Typically such a mathematical model approximates the behavior of the initial system and has some uncertainty. However, this uncertainty can increase significantly if the initial data for building the model is uncertain and incompletely specified. The uncertainty of the mathematical model also causes incorrect estimates of the system behavior and its reliability. Therefore, it is important to develop methods that allow us to take into account the uncertain nature of the initial data, first of all, take into account epistemic uncertainty in initial data. A new method for the development of a mathematical model of a Multi-State System (MSS) in the form of a Multi-Valued Decision Diagram based on incompletely specified and uncertain data is proposed. In other words, the proposed method takes into account the epistemic uncertainty of the initial data. The specific of this method is the use of Data Mining based classification procedures, in particular, the Fuzzy Decision Trees. [ABSTRACT FROM AUTHOR]

System availability evaluation, sensitivity analysis, Importance Measures, and optimal design are important issues that have become research topics for reliability engineering. There are different mathematical approaches to the development of these topics. The structure function based approach is one of them. Structure function enables one to analyse a system of any complexity. But computational complexity of structure function based methods is time consuming for large-scale networks. We propose to use two mathematical approaches for decision to this problem for system importance analysis. The first of them is Direct Partial Boolean Derivative. New equations for calculating the Importance Measures are developed in terms of these derivatives. The second is Binary Decision Diagram (BDD), that supports efficient manipulation of Boolean algebra. Two algorithms for calculating Direct Partial Boolean Derivative based on BDD of structure function are proposed in this paper. The experimental results show the efficiency of new algorithms for calculating Direct Partial Boolean Derivative and Importance Measures. [ABSTRACT FROM AUTHOR]

The Birnbaum importance is one of the measures often used in importance analysis. Many algorithms for the calculation of this measure have been proposed for coherent systems. Several interpretations of the Birnbaum importance for non-coherent systems also exist. These interpretations are usually based on the prime implicants of the structure function of the system. However, identification of the prime implicants is a complex problem. In this paper, we develop a method for computation of Birnbaum importance using Boolean expression (algebraic representation) and truth table (matrix representation). Both of these representations are calculated based on Logical Differential Calculus, in particular, Boolean derivatives. The Boolean expression derived based on Boolean derivatives is equal to known definitions of the Birnbaum importance. The matrix approach for the calculation of the Birnbaum importance based on Boolean derivatives is novel. This approach can be used for calculation of the Birnbaum importance for the structure function represented by truth table and does not need transformation of the structure function into Boolean expression. An important advantage of the proposed approach (in algebraic or matrix form) is a possibility to define the Birnbaum importance for the evaluation of the non-coherent system failure (restore) with respect to simultaneous changes of states of several system components. • The Boolean expression of Birnbaum importance for non-coherent system is proposed. • The Logical Differential Calculus is developed for importance analysis of non-coherent system. • The Birnbaum importance is defined based on Boolean Derivatives. • The Birnbaum importance of several components is defined based on Boolean Derivatives. • The matrix approach for Birnbaum importance is proposed [ABSTRACT FROM AUTHOR]

• The modification of the algorithm of signal classification. • The addition of new procedure (fuzzification) into the step of the preliminary signal classification. • Application of FDT as the fuzzy classifier for the signal classification. • The application of proposed algorithm for non-destructive diagnostic of aircraft engine blades. A new algorithm for non-destructive diagnosis based on signal classification is proposed. This algorithm permits classifying objects with different properties depending on signals characterizing the objects. In this paper, the algorithm is applied to diagnose blades of aircraft engine gas turbine according to vibration signal after a non-destructive shock excitation and to classify them as defective and faultless. Just like other signal classification algorithms, this algorithm consists of two steps that are signal preliminary transformation and classification. However, these steps are modified in the proposed algorithm. Unlike other algorithms, the fuzzy classifier is used in the classification step. The change of the classifier type causes modification of the preliminary transformation step because the attributes for the classification must be fuzzy. Therefore, fuzzification procedure is added into the step of preliminary transformation. The fuzzy classifier for the problem of aircraft engine blades diagnosis is ordered Fuzzy Decision Tree (oFDT) that is inducted by estimation of Cumulative Mutual Information. This induction has good efficiency for a small set of initial data. The accuracy of the classification of defective and faultless blades for the proposed algorithm is 98.5%, and oFDT for this classification is inducted based on 32 signals only. The comparison with other classification algorithms shows that oFDT based algorithm considered in this paper gives the best result for this problem. [ABSTRACT FROM AUTHOR]

• A system is mathematically represented by a structure function and survival signature. • A new method is developed for the calculation of structural importance measures based on survival signature. • The components' type importance based on the survival signature is defined. • The proposed method is approved for the evaluation of aircraft motor. The analysis of the influence of system components on system failure is a typical problem in reliability engineering which is known as importance analysis of a system. There are different methods for the calculation of the importance measures of the system components. In this paper, a new method for the calculation of structural importance measures that are one of the types of importance measures is presented. The novelty of the proposed method is the use of the survival signature instead of the structure function for the computation of the importance measures. The proposed method is based on the tool of logical differential calculus, in particular, direct partial logical derivatives. The advantages and practical applicability of the proposed method were confirmed in the analysis of the reliability characteristics of the fuel system of the propulsion complex of a passenger aircraft. [ABSTRACT FROM AUTHOR]

Importance analysis deals with the investigation of influence of individual system components on system operation. This investigation can be qualitative or quantitative. The qualitative analysis focuses on finding scenarios in which a degradation/improvement of a specific component or a group of components results in a degradation/improvement of the whole system, while the quantitative one deals with numerical estimation of components importance. In this paper, we propose a new approach for importance analysis of Multi-State Systems (MSSs). The qualitative or quantitative importance analysis is based on the identification of critical states. The new approach can be used for calculation of all types of critical states. It is based on the application of direct partial logic derivatives. These derivatives are one of the methods of logical differential calculus. In this paper, the specifics of these derivatives for importance analysis of MSSs are considered in detail. [ABSTRACT FROM AUTHOR]

Non-coherent systems represent a specific group of systems in context of reliability analysis. These systems are characteristic by existence of situations in which degradation of a system component can result in improvement of the whole system. Non-coherent systems are mostly investigated in case of Binary-State Systems (BSSs) that admit two performance levels for modeling system as well as its components. Properties of these systems have been studied and algorithms for calculation of most of reliability measures have been developed. However, in case of Multi-State Systems (MSSs), the situation is completely different. Non-coherent MSSs have been studied only marginally with focus on homogeneous MSSs. In this paper, we deal with heterogeneous non-coherent MSSs, generalize existing definitions for them and develop a new approach for computation of their critical system states, which play a key role in importance analysis. The analysis of non-coherent MSSs considered in this paper is implemented based on its representation in a form of time independent structure function. • Non-coherent Multi-State System (MSS) analysis. • The analysis of different definitions of component relevancy for MSSs. • The definition of critical system states for non-coherent MSSs. • The development of logical derivatives for analysis of non-coherent MSSs. • The computation of critical system states for non-coherent MSSs. [ABSTRACT FROM AUTHOR]