Two-stage simplified swarm optimization for the redundancy allocation problem in a multi-state bridge system
Introduction
The last three decades have seen a growing importance placed on research for the redundancy allocation problem (RAP) due to its wide and valuable application [1], [2]. RAP involves configuring an optimal system structure with high reliability and low cost either by alternating the elements with more reliable elements and/or forming them redundantly. RAP can be divided into two categories based on the system framework: binary-state systems and multi-state systems [3], [4], [5], [6]. Only the two states of operating or complete failure can be experienced in a binary-state system. Compared to a binary-state system, a multi-state system, which can experience more than two extreme states, is more practical in the real world [7], [8]. It is becoming an important performance measurement tool for real world engineering systems from the design phase to the control phase [1], [2].
RAP has been proved to be NP-hard because the burdens of computational complexity grow with the size of the system [9]. For reducing those burdens, existing approaches mainly focus on developing approximation methods, including the Monte-Carlo simulation [10], the linear programming with an approximate linear objective function [11], tabu search [12], genetic algorithm (GA) [13], [14], [15], [16], [17], ant colony optimization [18], particle swarm optimization (PSO) [19], [20], artificial immune algorithm [21] and artificial bee colony algorithm [22].
When RAP is applied to a multi-state system, availability is a common measurement to evaluate the reliability of a system [23], [24], [25], [26]. However, the availability of some systems cannot be evaluated by the abovementioned methods because those systems cannot be reduced to series and parallel combinations of elements [23], [26], [27], [28]. One of them is a bridge system. The simplest bridge system is depicted in Fig. 1 in which subsystems 1 and 2; and subsystems 3 and 4 have the same functionality. Those subsystems construct two series substructures connected in parallel. Subsystem 5 performs as an intermediator to balance the load between two parallel flow lines. This system structure is widely used in various engineering systems for load balancing and control [23], [24], [25], [26], [29], [30], [31], [32], [33].
A multi-state bridge system (MSBS) allows the mixture of element type in the subsystem which will greatly increase the computational complexity of the problem. Consider the simplest bridge system in Fig. 1. Suppose that m represents the number of element types can be chosen for each subsystem and n represents the largest number of elements for each type that can be simultaneously mounted in the subsystem. Thus, there are a total of (n+1)m ways to construct each subsystem. Then the search space for the simplest bridge system is (n+1)5m.
As the bridge system cannot be reduced to the series-parallel combinations and its computational complexity sharply increases with the size of the system, Levitin and Lisnianki first addressed MSBS with three different constraints and evaluated the availability of MSBS by GA [23], [24], [25], [31]. Recently, Wang and Li combined PSO and local refining strategies to address this problem [26]. The result shows that they have made important contributions to find nearer optimal solutions in a more efficient way than GA. However, there is still some room to improve the efficiency of the above two works. Hence, this paper proposes a novel algorithm called two-stage simplified swarm optimization (SSOTS) that offers an alternative method for solving MSBS.
The rest of this paper is organized as follows: Section 1 describes the formulation of the problem and structure representation. The applications of the universal generating function (UGF) for RAP in MSBS and a short description of simplified swarm optimization (SSO) are given in Section 3. The proposed SSOTS and its overall procedure are detailed in Section 4. The two experiments implemented for validating SSOTS are illustrated in Section 5. Finally, the conclusions are presented in Section 6.
Section snippets
Problem formulation and structure representation
Consider a MSBS, let Nsub be the number of subsystems in the system and let Mj be the number of element types available for the jth subsystem. Let xj,k represents the number of k type elements used in the jth subsystem, and Xj=(xj,1, xj,2, …, xj,k) be the jth subsystem vector (structure), where k=1, 2, …, Mj. Each element is characterized by its availability ajk, capacity wjk and cost cjk. The capacity of an element is the quantitative measure of its performance and measured as percentage of
UGF for the MSBS availability evaluation
The UGF introduced by Ushakov is commonly applied to find the entire multi-state system performance distribution based on performance states of its fundamental units by using algebraic procedures [31], [34]. In this paper, UGF is implemented to evaluate the availability of MSBS, and those related techniques are briefly introduced as follows by using the structure in Fig. 3 as an example.
Consider a discrete variable Y that take on a finite number S of possible values. The probabilistic
Proposed methods
Intuitively, it is more efficient to find a crude solution constructed by single-type subsystem, which only consists of at most one element type, versus the original problem. Therefore, the proposed algorithm is designed to be composed of two main stages. The first stage, called the vector-update stage, updates the structure of solutions by each subsystem vector instead of its each variable to efficiently obtain crude solutions. Then, the second stage, called the variable-update stage, updates
Experiment results and discussion
Two experiments, Ex-1 and Ex-2, using the MSBS benchmark reported in [23], [26], [31] as shown in Fig. 2 are implemented in this study to validate the performance of the proposed SSOTS. In each experiment, the performance level constraint is fixed at 100 (i.e., W0=100), and the availability constraint is 0.90, 0.95 and 0.99 (i.e., A0=0.90, 0.95 and 0.99). The purposes of Ex-1 are to verify the effect of and find the best setting for MTO and RIP using 20 designed treatments. Ex-2 compares SSOTS
Conclusions
In this paper, the two-stage simplified swarm optimization known as SSOTS was proposed to solve RAP in MSBS. The effectiveness and efficiency of SSOTS were validated using two experiments. To the best of our knowledge, PSO is currently the best-known approach in the literature for RAP in MSBS [26]. For all examined problems, SSOTS can yield equivalent or better solutions with a higher successful rate and computational efficiency than those obtained by PSO. Thus, the proposed SSOTS retains the
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