Spare Parts Forecasting Based on Reliability

Stochastic models for spare parts forecasting have not been widely researched in scientific literature from the aspect of their reliability. In this chapter, the authors present models which analyze standard reliability parameters of technical systems’ parts/components. By analyzing system reliability and failure rate, we estimate the required number of spare parts in the moment of expected failure or when reliability falls below the predefined level. Two different approaches based on data availability are presented herewith

Implementation of gradient methods for optimization of underage costs in aviation industry

Underage costs are not easily quantifiable in spare parts management. These costs occur when a spare part is required and none are available in inventory. This paper provides another approach to underage cost optimization for subassemblies and assemblies in aviation industry. The quantity of spare parts is determined by using a method for airplane spare parts forecasting based on Rayleigh's model. Based on that, the underage cost per unit is determined by using the Newsvendor model. Then, by implementing a transformed accelerated double-step size gradient method, the underage costs for spare sub-assemblies and assemblies in airline industry are optimized

Information and communications technologies in function of teaching process

This paper, based on the extensive research of relevant professional literature, will present and elaborate possible aspects of influence which information-communications technologies (ICT) have on educational process. Contemporary information-communications technologies and their use in teaching process resulted in numerous enhancements of teaching process altering traditional methods of learning and teaching. With the aim of confirming theoretical postulates researches were conducted on candidates with specific educational degrees in various forms of traditional teaching and teaching with educational software. The survey was conducted on a control and experimental group, on a sample of 1542 candidates. The candidates were observed in a period of approximately five months. The results of this research indicate that the candidates taught by use of educational software have shown far greater level of achieved knowledge and significantly higher motivation for learning. The participants of the experimental group actively participated in the acquisition of their competences. The contribution of this paper is in the new methodological approach to the organization of education, which significantly improves the teaching process, its goals and outcomes

A Stochastic Model for Achieving Required Level of Availability Based on the Repair Rate Analysis

The aim of this paper is to set framework for the repair rate in order to achieve required availability of a repairable system with alternating renewal process where the failure rate is Weibull distributed. In order to accomplish this goal, this paper proposes a novel model for a complete stochastic analysis of the repair rate. The resulting equation for probability density function of the repair rate enables precise modelling of the repair rate process for related values of availability and failure rate. The obtained results can be used for dynamic forecasting of system’s characteristics, planning of maintenance activities, inventory and servicing capacities

New Results on F-Contractions in Complete Metric Spaces

The main purpose of this paper is to improve, generalize, unify, extend and enrich the recent results established by Dung and Hang (2015), Piri and Kumam (2014, 2016), and Singh et al. (2018). In our proofs, we only use the property (F1) of Wardowski’s F-contraction, while the many authors in their papers still use all tree properties of F-contraction as well as two new properties introduced by Piri and Kumam. Our approach in this paper indicates that for most results with F-contraction, property (F1) is sufficient. It is interesting to investigate whether (F1) is sufficient in the case of multi-valued mappings

A Transformation of Accelerated Double Step Size Method for Unconstrained Optimization

A reduction of the originally double step size iteration into the single step length scheme is derived under the proposed condition that relates two step lengths in the accelerated double step size gradient descent scheme. The proposed transformation is numerically tested. Obtained results confirm the substantial progress in comparison with the single step size accelerated gradient descent method defined in a classical way regarding all analyzed characteristics: number of iterations, CPU time, and number of function evaluations. Linear convergence of derived method has been proved

Determination of accelerated factors in gradient descent iterations based on Taylor's series

In this paper the efficiency of accelerated gradient descent methods regarding the way of determination of accelerated factor is considered. Due to the previous researches we assert that the use of Taylor's series of posed gradient descent iteration in calculation of accelerated parameter gives better final results than some other choices. We give a comparative analysis of efficiency of several methods with different approaches in obtaining accelerated parameter. According to the achieved results of numerical experiments we make a conclusion about the one of the most optimal way in defining accelerated parameter in accelerated gradient descent schemes

A Novel Approach for Determination of Reliability of Covering a Node from K Nodes

In this paper, a stochastic problem of multicenter location on a graph was formulated through the modification of the existing p-center problem to determine the location of a given number of facilities, to maximize the reliability of supplying the system. The system is represented by a graph whose nodes are the locations of demand and the potential facilities, while the weights of the arcs represent the reliability, i.e., the probability that an appropriate branch is available. First, k locations of facilities are randomly determined. Using a modified Dijkstra’s algorithm, the elementary path of maximal reliability for every demand node is determined. Then, a graph of all of elementary paths for demand node is formed. Finally, a new algorithm for calculating the reliability of covering a node from k nodes (k—covering reliability) was formulated

The Nullity, Rank, and Invertibility of Linear Combinations of k-Potent Matrices

Baksalary et al. (Linear Algebra Appl., doi:10.1016/j.laa.2004.02.025, 2004) investigated the invertibility of a linear combination of idempotent matrices. This result was improved by Koliha et al. (Linear Algebra Appl., doi:10.1016/j.laa.2006.01.011, 2006) by showing that the rank of a linear combination of two idempotents is constant. In this paper, we consider similar problems for k-potent matrices. We study the rank and the nullity of a linear combination of two commuting k-potent matrices. Furthermore, the problem of the nonsingularity of linear combinations of two or three k-potent matrices is considered under some conditions. In these situations, we derive explicit formulae of their inverses

Mathematical Modeling of Integral Characteristics of Repair Process under Maintenance Contracts

The repair rate is a very important parameter for system maintainability and can be defined as a frequency of successfully performed repair actions on a failed component per unit of time. This paper analyzes the integral characteristics of a stochastic repair rate for corresponding values of availability in a system operating under maintenance contracts. The probability density function (PDF) of the repair rate has been studied extensively and it was concluded that it is not a symmetric function so its mean value does not correspond to its maximum. Based on that, the equation for the envelope line of the PDF maximums of the repair rate has been provided. Namely, instead of repair rate PDF equations, we can use envelope line parameters for certain calculations, which are expressed in a simpler mathematical form. That will reduce time required for calculations and prediction and enhance reactions in failure events. Further, for the analytical and numerical evaluation of a system performance, the annual repair rate PDFs are analyzed, such as particular solutions of corresponding differential equation, while the existence of a singular solution is considered and analyzed under different conditions. Moreover, we derived optimal values of availability for which the PDF maximums have been obtained. Finally, in order to generalize the behavior of the repair process, a partial differential equation, as a function of the repair rate process and availability parameter, has been formed