Standby Redundancy Optimization with Uncertain Lifetimes Xiang Li Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
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Zhongfeng Qin Department of Risk Management and Insurance, Beihang University, Beijing 100191, China
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Meilin Wen Department of System Engineering of Engineering Technology, Beihang University, Beijing, 100191, China
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Abstract Uncertain variable is defined as a measurable function from an uncertainty space to the set of real numbers for describing subjective uncertain phenomena, which is employed to characterize the element, component and system lifetimes in this paper. Within the framework of uncertainty theory, reliability measure and percentile lifetime are defined to measure the system reliability in this paper. Based on these definitions, two chance-constrained programming standby redundancy optimization models are proposed by maximizing the system reliability under certain system constraints such as cost, weight, volume and so on. Finally, a numerical example is illustrated to show the effectiveness of the proposed methods. Keywords: Standby redundancy optimization; Uncertain variable; Uncertain measure; Chance-constrained programming.
1
Introduction
The primary goal of reliability engineering is to improve the system reliability, and component redundancy is an effective method to achieve it. Generally speaking, two underlying component redundancy techniques are considered. The first one is called parallel redundancy because all redundant elements work simultaneously, which is usually employed when the system is required to operate for a long time without interruption. The second one is called standby redundancy because one of the redundant elements begins to work only when the active one failed, which is usually employed when the replacement between elements takes a negligible amount of time and it does not cause system failure. In order to determine the number of redundant elements, several redundancy optimization models are designed by maximizing the system reliability subject to certain system constraints on element attributes such as cost, weight, volume and so on. There are three main kinds of the system reliability measures including the mean time to failure, percentile life and reliability (see [15]). According to the different definitions of reliability measures, the redundancy optimization models are mainly divided into expected value model [25], chance-constrained programming model [7, 8] and dependent-chance programming model [33, 34]. In order to solve these models, a large number of algorithms have been proposed including both exact algorithm [2, 5, 9, 26] and evolutionary algorithm [1, 6, 12, 13, 27]. Comprehensive overviews about the redundancy optimization models and algorithms may be found in [11, 14, 15]. All above researches are based on the assumption that the element lifetimes are random variables. Although this assumption has been adopted and accorded with the facts in widespread cases, it is not reasonable in a vast range of situations such as space shuttle system, where the estimations of probability distribution functions or density functions of element lifetimes are impossible due to the imprecision of data. In this case, fuzzy theory [16, 18, 19, 30, 32], which is proposed as a branch of nonadditive measure theory for dealing with 1
subjective uncertain phenomena, is widely employed in term of characterizing the element lifetime as fuzzy variables, and fuzzy redundancy optimization models are then widely studied [3, 28, 29, 34]. Beyond fuzzy theory, uncertainty theory initialized by Liu [20] is another branch of nonadditive measure theory for describing subjective uncertain phenomena, which has been well developed [10, 22, 31] and widely applied to uncertain logic [17], uncertain process [21], uncertain differential equation [4] and so on. The purpose of this paper is to study the standby redundancy optimization problem within the framework of uncertainty theory which was first proposed by Liu [23]. For this purpose, the rest of the paper is organized as follows. Section 2 recalls some basic concepts and properties about uncertain measure and uncertain variable. In Section 3, two system reliability measures are defined as the uncertainty that the system lifetime is larger than or equal to a given level and the optimistic value of the system lifetime, respectively. By maximizing these reliability measures, two chance-constrained programming standby redundancy optimization models are proposed in Section 4, both of which are proved to be crisp nonlinear integer programming models. In order to show the effectiveness of the proposed methods, a numerical example is given on bridge system in Section 5. At the end of this paper, a brief summary about the paper is given.
2
Preliminaries
Let Γ be a nonempty set, and let A be a σ-algebra over Γ. Each element A of A is called an event. In order to present an axiomatic definition of uncertain measure, it is necessary to assign to each event a number which indicates the degree that the event will occur. In order to ensure that the number has certain mathematical properties, Liu [20] proposed the following four axioms: Axiom 1. (Normality) {Γ} = 1; Axiom 2. (Monotonicity) {A} ≤ {B} whenever A ⊂ B; Axiom 3. (Self-Duality) is self-dual, i.e., {A} + {Ac } = 1 for any event A; Axiom 4. (Countable Subadditivity) for any countable sequence of events {Ai }, we have
M M M
M
M
M
M {∪ A } ≤ X M{A }. ∞
i
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i
i=1
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Definition 2.1 (Liu [20]) Let Γ be a nonempty set, and let A be a σ-algebra over Γ. The set function on A is called an uncertain measure if it satisfies the normality, monotonicity, self-duality and countable subadditivity axioms. If is an uncertain measure defined on A, then the triplet (Γ, A, ) is called an uncertainty space.
M
M
Definition 2.2 (Liu [20]) An uncertain variable is defined as a measurable function from an uncertainty space (Γ, A, ) to the set of real numbers.
M
Definition 2.3 (Liu [22]) Uncertain variables ξ1 , ξ2 , · · · , ξm are said to be independent if and only if for any Borel sets B1 , B2 , · · · , Bm of
