Complexity

Volume 2018, Article ID 3154360, 10 pages

https://doi.org/10.1155/2018/3154360

## Redundancy Optimization of an Uncertain Parallel-Series System with Warm Standby Elements

School of Science, Yanshan University, Qinhuangdao 066004, China

Correspondence should be addressed to Linmin Hu; nc.ude.usy@uhnimnil

Received 31 January 2018; Revised 24 June 2018; Accepted 5 July 2018; Published 6 September 2018

Academic Editor: Roberto Natella

Copyright © 2018 Linmin Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The redundancy optimization problem is formulated for an uncertain parallel-series system with warm standby elements. The lifetimes and costs of elements are considered uncertain variables, and the weights and volumes of elements are random variables. The uncertain measure optimization model (UMOM), the uncertain optimistic value optimization model (UOVOM), and the uncertain cost optimization model (UCOM) are developed through reliability maximization, lifetime maximization, and cost minimization, respectively. An efficient simulation optimization algorithm is provided to calculate the objective values and optimal solutions of the UMOM, UOVOM, and UCOM. A numerical example is presented to illustrate the rationality of the models and the feasibility of the optimization algorithm.

#### 1. Introduction

The primary goal of reliability design is to improve the reliability of a system. To maintain the reliability to a higher level, the redundancy allocation is an effective method in the system design phase. While improving system reliability by a redundancy method, the cost, weight, and volume also increase. Thus, it is an important topic for system decision-makers to determine the optimal number of redundant elements under certain system constraints.

In the traditional redundancy optimization problem, various kinds of optimization models have been proposed under the assumption that the lifetimes of the elements are random variables. Due to imprecision of data for element lifetimes in certain situations, fuzzy redundancy optimization models [1–4] are then developed based on fuzzy set theory [5, 6]. Furthermore, Zhao and Liu [7] proposed three redundancy optimization models under the assumption that the lifetimes of the elements are presented as fuzzy variables. Wang and Watada [8] developed two fuzzy random redundancy allocation models for a parallel-series system when the lifetimes of the elements are treated as fuzzy random variables. Recently, some researchers have addressed reliability optimization designs of some systems by considering interval-valued component reliability in an uncertain environment. Roy et al. [9] applied the symmetrical form of interval numbers by interval-valued parametric functional form to evaluate the optimum system reliability and system cost of the redundancy allocation problem. Zhang and Chen [10] investigated an interval multiobjective optimization problem for reliability redundancy allocation of a series-parallel system. Moreover, some researchers concentrated on some hybrid uncertainty optimization problems for system reliability [11, 12].

The probability, interval, and fuzzy theories have been widely used to handle the high level of uncertainty in various real-world applications. With the development of the research on the uncertainty phenomena, the mathematical model based on the probability, interval, and fuzzy theories is not enough to solve all problems, especially when we have no available samples but belief degree from the experts. Belief degree function is a type of distribution function for indeterminate quantity. Since it usually deviates far from the frequency, using probability theory may lead to counterintuitive results. In this case, we should use uncertainty theory. The uncertainty theory provides a useful tool to study reliability modeling and optimization problem of systems with human uncertainty phenomena. The basic uncertainty theory was founded by Liu [13] in 2007. It was refined by Liu [14] in 2010 based on normality axiom, duality axiom, subadditivity axiom, and product axiom. Nowadays, uncertainty theory has become a branch of mathematics for modeling human uncertainty. Many theories and applications have been done based on uncertainty theory, for example, uncertain statistics [15, 16], uncertain programming [17, 18], uncertain logic [19–21], uncertain inference [22–24], uncertain process [25–27], uncertain differential equations [28, 29], and uncertain graph [30].

The system reliability via uncertainty measure was first studied by Liu [31]. Afterwards, Liu [32] investigated reliability of redundant systems with cold and warm redundant elements based on uncertainty theory. Liu et al. [33] studied the reliability and MTTF of unrepairable systems with uncertain lifetimes. Wen and Kang [34] analyzed an uncertain random system based on chance theory which is a generalization of both probability theory and uncertainty theory. Gao et al. [35] studied the reliability of -out-of- systems with uncertain random lifetimes. Gao et al. [36] studied the reliability of the -out-of- system with uncertain weights. Zeng et al. [37] defined belief reliability as an uncertain measure due to the explicit representation of epistemic uncertainty and investigated the belief reliability for coherent systems based on minimal cut sets.

Making use of an uncertain variable as a tool to characterize the lifetimes and costs of elements, we will discuss the redundancy optimization problems for a parallel-series system with warm standby elements in this paper. In this work, three uncertain optimization models are developed, and an efficient simulation optimization algorithm is given to solve these models. In Section 2, some basic concepts and theorems concerning uncertainty theory are presented. The problem formulation of an uncertain parallel-series system is considered in Section 3. Section 4 shows the three uncertain optimization models and gives a solution approach to these models. A numerical example is provided in Section 5, and Section 6 presents a general conclusion.

#### 2. Preliminaries

*Definition 1 (see [13, 38]). *Let be a -algebra on a nonempty set . A set function is called an uncertain measure if it satisfies the following axioms:

*Axiom 1 (normality axiom). * for the universal set .

*Axiom 2 (duality axiom). * for any event .

*Axiom 3 (subadditivity axiom). *For any countable sequence of events , we have
the triple is called an uncertainty space.

*Axiom 4 (product axiom). *Let be the uncertainty space for . Then, the product uncertain measure is an uncertain measure satisfying
where are arbitrarily chosen events from for , respectively.

*Definition 2 (see [13]). *An uncertain variable is a measure function from an uncertainty space to the set of real numbers; that is, for any Borel set of real numbers, the set
is an event.

*Definition 3 (see [13]). *The uncertainty distribution of an uncertain variable is defined as
for any real number .

*Definition 4 (see [13]). *An uncertainty distribution is said to be regular if it is a continuous and strictly increasing function with respect to at which , and
In addition, the inverse function is called the inverse uncertainty distribution of .

*Definition 5 (see [17]). *A variable is said to be an -optimistic value if is an uncertain variable and
where .

*Definition 6 (see [13]). *An uncertain variable is said to be linear if it has a linear uncertainty distribution.
which is denoted by . Apparently, the linear uncertain variable is regular and has an inverse uncertainty distribution .

*Definition 7 (see [13]). *An uncertain variable is said to be lognormal if has a normal uncertainty distribution.
denoted by , where and are real numbers with . The uncertain variable is regular, and its inverse uncertainty distribution is

Theorem 1 (see [13]). *Assume that are independent uncertain variables with regular uncertainty distributions , respectively. If the function is strictly increasing with respect to , then has an inverse uncertainty distribution .*

Theorem 2 (see [13]). *Let be an uncertain variable with a regular uncertainty distribution . Then, .*

Theorem 3 (see [13]). *Let and be independent uncertain variables with finite expected values. Then, for any real numbers and , we have
*

Theorem 4 (see [13]). *Assume that are independent uncertain variables with uncertainty distributions , respectively. Then, , , and have uncertainty distributions , , and , respectively.*

#### 3. Problem Formulation of an Uncertain Parallel-Series System

Consider a warm standby redundant parallel-series system composed of subsystems , and subsystem consists of components connected in series, as shown in Figure 1. The component in subsystem contains one original element and warm standby redundant elements, and .