Logical Differential Calculus Research Articles

Importance analysis of a system based on survival signature by structural importance measures

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.

Logic Differential Calculus for Reliability Analysis Based on Survival Signature

The structure function is an often-used mathematical representation of the investigated system in reliability analysis. It is a binary function that models a system state according to the states of its components. The size of the structure function depends on the number of components and can be enormous for systems with many components. Therefore, the system reliability analysis based on the structure function needs special methods to decrease this dimension and to measure the system reliability. The concept of survival signature provides a useful transformation of the structure function to simplify reliability assessment for systems with many components of specified types. The survival signature is a complete probabilistic description of the system. The new methods and algorithms of system reliability analysis based on this mathematical representation should be developed. The Direct Partial Logic Derivative is one of the approaches that are effective in system reliability evaluation based on the structure function. This approach is used to determine different aspects of system failure depending on system components breakdowns. The development of this derivative for survival signature permits to obtain the new method for the reliability analysis of system failure caused by system component breakdown depending on components types.

Logical differential calculus for calculation of Birnbaum importance of non-coherent system

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.

NEW MATRIX BASED ALGORITHM FOR CALCULATION OF IMPORTANCE MEASURES

The system reliability/availability is complex term that is evaluated based on numerous indices and measures. There are different methods for the calculation of these indices and measures. Some of the most used are importance measures. These measures allow to evaluate the influence of fixed system components or set of components to the system reliability/availability. Importance measures are used to allow for various aspects of the impact of system elements on its failure or operability. Analysis of element importance is used in the system design, diagnosis, and optimization. In this paper new algorithm for the calculation some of importance measures is developed based on the matrix procedures. This paper goal is development of new algorithm to calculate importance measures of the system based on the matrix procedures that can be transformed in the parallel procedures/algorithm. This algorithm is developed based on the application of Logical Differential Calculus of Boolean logic for importance analysis of system. The application of parallel algorithm in importance analysis allows the evaluation of system of large dimension. Importance specific of the proposed matrix procedures for calculation of importance measures is the application of structure function for the mathematical representation of investigated system. This function defined the correlation of the system components states and system reliability/ availability. The structure function in this case is defined as truth vector to be used in the matrix transformation. The truth vector of Boolean function is column of the truth table of function if the values of the variables are lexicographically ordered

THE MATRIX PROCEDURES FOR CALCULATION OF IMPORTANCE MEASURES

System reliability/availability is a complex concept that is evaluated based on numerous indices and measures. There are different methods for the calculation of these indices and measures in reliability analysis. Some of the most used indices are important measures. These measures allow us to evaluate the influence of fixed system components or set of components to the system reliability/availability. Importance measures are used today to allow for various aspects of the impact of system elements on its failure or operability. Analysis of element importance is used in the system design, diagnosis, and optimization. In this paper new algorithms for the calculation, some of the important measures are developed based on the matrix procedures. This paper's goal is the development of a new algorithm to calculate importance measures of the system based on the matrix procedures that can be transformed in the parallel procedures/algorithms. These algorithms are developed based on the application of Logical Differential Calculus of Boolean logic for the important analysis of the system. The application of parallel algorithms in importance analysis allows the evaluation of the system of large dimensions. Importance specific of the proposed matrix procedures for calculation of importance measures is the application of structure-function for the mathematical representation of the investigated system. This function defined the correlation of the system components states and system reliability/ availability. The structure-function, in this case, is defined as a truth vector to be used in the matrix transformation. The truth vector of a Boolean function is a column of the truth table of function if the values of the variables are lexicographically ordered. Therefore, the structure-function of any system can be represented by the truth vector of 2n elements un-ambiguously.

Reliability analysis of multi-state system with application of multiple-valued logic

PurposeThe purpose of this paper is to develop a new mathematical method for the reliability analysis and evaluation of multi-state system (MSS) reliability that agrees with specifics of such system. It is possible based on the application of multiple-valued logic (MVL) that is a natural extension of Boolean algebra used in reliability analysis.Design/methodology/approachSimilar to Boolean algebra, MVL is used for the constriction of the structure function of the investigated system. The interpretation of the structure function of the MSS in terms of MVL allows using mathematical methods and approaches of this logic for the analysis of the structure function.FindingsThe logical differential calculus is one of mathematical approaches in MVL. The authors develop new method for MSS reliability analysis based on logical differential calculus, in particular direct partial logical derivatives, for the investigation of critical system states (CSSs). The proposed method allows providing the qualitative and quantitative analyses of MSS: the CSS can be defined for all possible changes of any system component or group of components, and probabilities of this state can also be calculated.Originality/valueThe proposed method permits representing the MSS in the form of a structure function that is interpreted as MVL function and provides the system analyses without special transformation into Boolean interpretation and with acceptable computational complexity.

Importance analysis of multi-state systems based on tools of logical differential calculus

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.

Evaluation of Algorithms for Identification of Minimal Cut Vectors and Minimal Path Vectors in Multi-State Systems

Minimal Cut Vectors (MCVs) and Minimal Path Vectors (MPVs) are one of the key concepts of reliability analysis. They allow us to estimate system availability or to analyze influence of individual system components on the entire system. However, the main problem of their use, especially in reliability analysis of complex systems, lies in their identification. Several algorithms have been proposed to solve this task. Some of the most universal ones are based on logical differential calculus. These algorithms use integrated direct partial logic derivatives to find situations that can correspond to the MCVs (MPVs) and a special type of logic conjunction to select only those situations that really agree with the MCVs (MPVs). In this paper, we summarize the ideas behind these algorithms in more formal way and present results of some experiments performed to study their time complexity.

English

English

Reliability Analysis of Logic Network with Multiple Outputs

Reliability analysis of a logic network with multiple outputs is considered in this paper. One of the principal tasks of reliability engineering is identification of those system components that have the most influence on the system activity. There exist several measures that are used for this purpose. One of them is Structural Importance Measure (SIM) which focuses on topological importance of individual system components. This measure is calculated from the system structure function, i.e. function that defines dependency between system activity and activity of its components, using logical differential calculus. In this paper, we present a method that can be used to identify the structure function of a logic network with multiple outputs and, based on logical differential calculus, we propose several definitions of the SIM for this type of systems. As a case study, the topological analysis of a one-bit full adder is considered in the last part of this paper. This study demonstrates usefulness of our approach in reliability analysis of logic networks and indicates that its further development and practical implementation could be beneficial.

Importance analysis based on logical differential calculus and Binary Decision Diagram

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.

Reliability Analysis of Noncoherent Systems by Logical Differential Calculus and Binary Decision Diagrams

Reliability is the important characteristic of systems. Most of the systems consist of more than one component. Moreover, modern systems are very complex and they include a lot of components. Therefore, effective methods have to be developed for analysis of these systems. The techniques that have been already considered for this purpose are the Binary Decision Diagrams and the Logical Differential Calculus. However, these methods have been considered only for the coherent systems. In this paper, their use in the reliability analysis of the noncoherent systems is considered.

Multiple-Valued Logic mathematical approaches for multi-state system reliability analysis

A mathematical description of an examined system such as Multi-State System (MSS) permits the system reliability to be analyzed in more detail, because the MSS defines some performance levels (more than only working and failure). The structure function is one of the basic definitions and representations of MSS. But the dimension of the structure function increases critically depending on the number of system components. Therefore, the development of methods for examination and quantification of such a function is an actual problem in MSS reliability analysis. In this paper, a method for the analysis of the MSS structure function of high dimension is proposed. The principal point of this method is the interpretation of the MSS structure function as a Multiple-Valued Logic function. It allows effective and approved mathematical methods of Multiple-Valued Logic to be used for analysis and quantification of the MSS structure function. We propose to use two mathematical approaches of Multiple-Valued Logic for the MSS. One of them is a representation of the MSS structure function by a Multiple-Valued Decision Diagram. It is an effective approach for analysis and estimation of the function of high dimension in Multiple-Valued Logic. The other approach is Logic Differential Calculus. Logic Differential Calculus is a useful approach for analysis of the MSS state changes.

Importance analysis by logical differential calculus

The physical objects or processes include many interconnected components representing a complex systems. Their reliability analysis usually considers two states interpreted as failure and operability. They are described in terms of the binary mathematical model. Importance analysis of the system elements is a traditional component of the reliability analysis. It enables one to estimate the impact of individual components on the system's operability or failure. The present paper proposed a new approach to analysis and estimation of the system importance on the basis of the logical differential calculus.