Elena Zaitseva

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

Abstract 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.

Usage of New Information Estimations for Induction of Fuzzy Decision Trees

We introduce a technique to compute new summary information estimations (information and entropy) for fuzzy sets. Special features for these estimations are investigated. We give an algorithm for determine various information measures for fuzzy sets and fuzzy decision trees. Finally, we are using our estimations for induction of fuzzy decision trees from a group of training examples.

Dynamic reliability indices for multi-state system

The Multi-State System reliability is investigated in this paper. The new class of Reliability Indices is obtained. We have called these indices as Dynamic Reliability Indices. These indices estimate influence upon the Multi-State System reliability by the capacity levels of a system component. The Multiple-Valued Logic mathematical tools are used for calculation of Dynamic Reliability Indices.

Dynamic reliability indices for parallel, series and k-out-of-n multi-state systems

A reliability of multi-state system (MSS) is investigated in this paper. This system and its component can have more than two reliability states. Dynamic reliability indices (DRI) are interpreted as probabilities of changes of the system reliability depending on change of a component state. The structure function of MSS is used for calculation of DRI. These indices for typical MSS (parallel, series and k-out-of-n system) are investigated in this paper and new equations of DRI for measure these systems are getting. These equations depend on number of system components, discrete levels of MSS reliability and components probabilities only and allow to reduce computational complexity of reliability analysis for typical MSS

Minimal Cut Vectors and Logical Differential Calculus

Reliability analysis of Multi-State System is one of fields, where the methods of the multiple-valued logic are used efficiently. Multi-State System is a mathematical model in reliability engineering, which allows considering some performance level in the system reliability behavior. The principal problem in the reliability engineering is identification of the system components that have the most influence on the system reliability. One of possible approaches, which can be used for this task, is to identify minimal cut sets (or minimal cut vectors (MCVs)). MCVs represent situations in which the repair of any damaged component causes the system improvement. In this paper, the new theoretical background for identification of MCVs is considered. The presented method is based on logical differential calculus.

Multi-state system importance analysis based on direct partial logic derivative

Multi-State System (MSS) is mathematical model of investigated object in reliability analysis. MSS allows present the investigated object in more detail than traditional Binary-State System. The importance analysis allows MSS reliability estimation depending on the system structure and its components states. Importance measures quantify the criticality of a particular component. They have been widely used as tools for identifying system weaknesses, and to priorities reliability improvement activities. New algorithms for calculation of the importance measures based on the methods of Multiple-Valued Logic are proposed. Direct Partial Logic Derivative and Multi-Valued Decision Diagram are used for these algorithms.

Fuzzy Classifier Based on Fuzzy Decision Tree

A popular method for making a fuzzy decision tree for classification is Fuzzy ID3 algorithm. We introduce a new approach that uses cumulative information estimations of initial data. Based on these estimations we propose a new greedy version of fuzzy ID3 algorithm to be used to generate understandable fuzzy classification rules. The goal is to find a sequence of rules that causes near minimal classification costs.

Construction of a Reliability Structure Function Based on Uncertain Data

Structure function is one of the possible mathematical models of the real systems under study in reliability engineering. The structure function represents the correlation the system performance level and components states. The system performance level is defined from the states of all its components. It means that all possible components states and performance levels must be indicated and reflected in the structure function. Initial data for the analysis of real system is uncertain. New methods to construct a structure function based on initial uncertain data is proposed. Fuzzy decision trees (FDTs) are used in this method to transform initial uncertain data about a real system into an exact-defined a system structure function. The proposed method includes three principal steps for the structure function construction: 1) collection of data in the repository; 2) representation of the system model in the form of an FDT; and 3) construction of the structure function based on the FDT.

Investigation Multi-State System Reliability by Structure Function

Multi-State System is a mathematical model that is used in reliability analysis to present a system with some level of working efficiency. A structure function allows to describe behavior of system reliability depending on the efficiency of its components unambiguously. There are a lot of estimates of a Multi-State System on the basic of structure function. Dynamic Reliability Indices belong to these estimates and characterize the changes of Multi-State System reliability caused by changes in components efficiency. These indices are computed based on structure function and Logical Differential Calculus. The method of Dynamic Reliability Indices calculation is considered in this paper and is compared with another methods to evaluate Multi-State System reliability. Path (cut) method, method of Dynamic Reliability Indices calculation and method based on the Lower (Upper) Boundary Points are examined in particular.