Salman Izadkhah

One of the main goals of reliability theory is to design systems with higher reliability. Among the methods of improving the reliability of a system, redundancy allocation means assigning one or more additional components to the system component or components. The different types of redundancy allocation are: active (hot) redundancy, cold standby redundancy and minimal repair redundancy. Although these allocations always increase the reliability of the system, a fundamental question is which component or components to apply redundancy to achieve a system with optimal reliability. In systems with dependent components, using the copula function theory, the lifetime of the system components and the dependence between them are modeled, and then, in terms of this dependence, the allocation of redundancy on the components has been investigated. Allocation of redundancy causes the change of the lifetime of the component to which the redundancy is assigned, the existence of this change has caused that in recent researches, the distortion function that is used in risk theory to model future risks is used to display The life span of the component can be used with redundancy or system allocation, and new and more general results have been obtained based on the shape of the distortion function. The effect of all the components is not the same during the lifetime of the system, and as a result, the allocation of redundancy does not have the same effect on all components, therefore the most effective component should be selected for the allocation of redundancy. In reliability, various measures are defined to measure the importance of the component, such as: Barlow-Proshan component importance measure and Birnbaum measure. Our main goal in this thesis is to find the optimal redundancy allocation by using common component importance measures and in systems with dependent components. For this purpose, the general framework of this thesis is proposed as follows: In the first chapter, we will explain the necessary introductions and definitions such as: concept of coherent system, structure function, concepts of aging, types of coherent systems, etc. An important tool in comparing the lifetime of systems is the theory of stochastic orderings, which we will discuss in this chapter. Among the other required concepts, there is the concept of copula function and distorsion function, which is expressed in a concise form. In the second chapter, we will introduce the Birnbaum measure and Barlow-Proshan component importance measure. We first define these measures for a system with independent components and then generalize these definitions for the dependent state. In the third chapter, we discuss the issue of redundancy allocation in a system with dependent components. In this chapter, we will first explain the concept of redundancy allocation and its types. And then we express results in the state of independence of the components and generalize them to the dependent state. In the fourth chapter, we examine the relationship between redundancy allocation and component importance measures for systems with dependent components. At the end, in the fifth chapter, we present the general conclusions about the conducted studies, and several suggestions for future studies are also presented.