On component failure in coherent systems with applications to maintenance strategies

Providing optimal strategies for maintaining technical systems in good working condition is an important goal in reliability engineering. The main aim of this paper is to propose some optimal maintenance policies for coherent systems based on some partial information about the status of components in the system. For this purpose, in the first part of the paper, we propose two criteria under which we compute the probability of the number of failed components in a coherent system with independent and identically distributed components. The first proposed criterion utilizes partial information about the status of the components with a single inspection of the system, and the second one uses partial information about the status of component failure under double monitoring of the system. In the computation of both criteria, we use the notion of the signature vector associated with the system. Some stochastic comparisons between two coherent systems have been made based on the proposed concepts. Then, by imposing some cost functions, we introduce new approaches to the optimal corrective and preventive maintenance of coherent systems. To illustrate the results, some examples are examined numerically and graphically.

On the Lifetime and Signature of Constrained (k, d)-out-of-n: F Reliability Systems

In the present paper we carry out a reliability study of the constrained (k, d)-out-of-n: F systems with exchangeable components. The signature vector is computed by the aid of the proposed algorithm. In addition, explicit signature-based expressions for the corresponding mean residual lifetime and the conditional mean residual lifetime of the aforementioned reliability system are also provided. For illustration purposes, a well-known multivariate distribution for modelling the lifetimes of the components of the constrained (k, d)-out-of-n: F structure is considered.

Reliability Study of Systems: A Generating Function Approach

In this paper we carry out a reliability study of the <n, f, 2> systems with independent and identically distributed components ordered in a line. More precisely, we obtain the generating function of structure’s reliability, while recurrence relations for determining its signature vector and reliability function are also provided. For illustration purposes, several numerical results are presented and some figures are constructed and appropriately commented.

On the Consecutive k1 and k2-out-of-n Reliability Systems

In this paper we carry out a reliability study of the consecutive-k1 and k2-out-of-n systems with independent and identically distributed components ordered in a line. More precisely, we obtain the generating function of the structure’s reliability, while recurrence relations for determining its signature vector and reliability function are also provided. For illustration purposes, some numerical results and figures are presented and several concluding remarks are deduced.

Some new results on stochastic comparisons of coherent systems using signatures

We consider coherent systems with independent and identically distributed components. While it is clear that the system’s life will be stochastically larger when the components are replaced with stochastically better components, we show that, in general, similar results may not hold for hazard rate, reverse hazard rate, and likelihood ratio orderings. We find sufficient conditions on the signature vector for these results to hold. These results are combined with other well-known results in the literature to get more general results for comparing two systems of the same size with different signature vectors and possibly with different independent and identically distributed component lifetimes. Some numerical examples are also provided to illustrate the theoretical results.

Comprehensive biological interpretation of gene signatures using semantic distributed representation

Recent rise of microarray and next-generation sequencing in genome-related fields has simplified obtaining gene expression data at whole gene level, and biological interpretation of gene signatures related to life phenomena and diseases has become very important. However, the conventional method is numerical comparison of gene signature, pathway, and gene ontology (GO) overlap and distribution bias, and it is not possible to compare the specificity and importance of genes contained in gene signatures as humans do.This study proposes the gene signature vector (GsVec), a unique method for interpreting gene signatures that clarifies the semantic relationship between gene signatures by incorporating a method of distributed document representation from natural language processing (NLP). In proposed algorithm, a gene-topic vector is created by multiplying the feature vector based on the gene’s distributed representation by the probability of the gene signature topic and the low frequency of occurrence of the corresponding gene in all gene signatures. These vectors are concatenated for genes included in each gene signature to create a signature vector. The degrees of similarity between signature vectors are obtained from the cosine distances, and the levels of relevance between gene signatures are quantified.Using the above algorithm, GsVec learned approximately 5,000 types of canonical pathway and GO biological process gene signatures published in the Molecular Signatures Database (MSigDB). Then, validation of the pathway database BioCarta with known biological significance and validation using actual gene expression data (differentially expressed genes) were performed, and both were able to obtain biologically valid results. In addition, the results compared with the pathway enrichment analysis in Fisher’s exact test used in the conventional method resulted in equivalent or more biologically valid signatures. Furthermore, although NLP is generally developed in Python, GsVec can execute the entire process in only the R language, the main language of bioinformatics.

On the equivalence of systems of different sizes, with applications to system comparisons

The signature of a coherent system is a useful tool in the study and comparison of lifetimes of engineered systems. In order to compare two systems of different sizes with respect to their signatures, the smaller system needs to be represented by an equivalent system of the same size as the larger system. In the paper we show how to construct equivalent systems by adding irrelevant components to the smaller system. This leads to simpler proofs of some current key results, and throws new light on the interpretation of mixed systems. We also present a sufficient condition for equivalence of systems of different sizes when restricting to coherent systems. In cases where for a given system there is no equivalent system of smaller size, we characterize the class of lower-sized systems with a signature vector which stochastically dominates the signature of the larger system. This setup is applied to an optimization problem in reliability economics.

The Kruskal-Katona Theorem and a Characterization of System Signatures

We show how to determine if a given vector can be the signature of a system on a finite number of components and, if so, exhibit such a system in terms of its structure function. The method employs combinatorial results from the theory of (finite) simplicial complexes, and provides a full characterization of signature vectors using a theorem of Kruskal (1963) and Katona (1968). We also show how the same approach can provide new combinatorial proofs of further results, e.g. that the signature vector of a system cannot have isolated zeroes. Finally, we prove that a signature with all nonzero entries must be a uniform distribution.

The Kruskal-Katona Theorem and a Characterization of System Signatures

We show how to determine if a given vector can be the signature of a system on a finite number of components and, if so, exhibit such a system in terms of its structure function. The method employs combinatorial results from the theory of (finite) simplicial complexes, and provides a full characterization of signature vectors using a theorem of Kruskal (1963) and Katona (1968). We also show how the same approach can provide new combinatorial proofs of further results, e.g. that the signature vector of a system cannot have isolated zeroes. Finally, we prove that a signature with all nonzero entries must be a uniform distribution.