Solving Cold-Standby Reliability-Redundancy Allocation Problems with Particle-Based Simplified Swarm Optimization

Huang, Chia-Ling; Jiang, Yunzhi; Yeh, Wei-Chang

doi:https://doi.org/10.1155/2021/7833641

Journal of Sensors

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Abstract Introduction Preliminaries Experimental Results Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments Supplementary Materials References Copyright Related Articles

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Research Article | Open Access

Volume 2021 | Article ID 7833641 | https://doi.org/10.1155/2021/7833641

Solving Cold-Standby Reliability-Redundancy Allocation Problems with Particle-Based Simplified Swarm Optimization

Chia-Ling Huang,¹Yunzhi Jiang,²and Wei-Chang Yeh³

Academic Editor: Wei-Chiang Hong

Received22 Apr 2021

Revised15 Jul 2021

Accepted10 Aug 2021

Published21 Aug 2021

Abstract

Particle swarm optimization (PSO) and simplified swarm optimization (SSO) are two of the state-of-the-art swarm intelligence technique that is widely utilized for optimization purposes. This paper describes a particle-based simplified swarm optimization (PSSO) procedure which combines the update mechanisms (UMs) of PSO and SSO to determine optimal system reliability for reliability-redundancy allocation problems (RRAPs) with cold-standby strategy while aimed at maximizing the system reliability. With comprehensive experimental test on the typical and famous four benchmarks of RRAP, PSSO is compared with other recently introduced algorithms in four different widely used systems, i.e., a series system, a series-parallel system, a complex (bridge) system, and an overspeed protection system for a gas turbine. Finally, the results of the experiments demonstrate that the PSSO can effectively solve the system of RRAP with cold-standby strategy and has good performance in the system reliability obtained although the best system reliability is not obtained in all four benchmarks.

1. Introduction

The reliability-redundancy allocation problem (RRAP) is the best known reliability design problem and is a classical optimization problem that seeks to maximize system reliability. To optimize system reliability for RRAP, the development of the system designs involves the selection of the reliability and the redundancy levels of the components. Hence, RRAP belongs to the category of mixed-integer programming problems because the components’ reliabilities are denoted as continuous values that fall between zero and one, while the redundancy levels are integer values. RRAP formulations generally involve system constraints on allowable cost, weight, volume, etc.

Based on the system’s required functions, the entire system is made up of a specific number of subsystems. The goal of the RRAP is to select the best combination of components and their reliabilities in each subsystem to maximize system reliability, , given constraints such as cost, weight, and volume. In the RRAP literature, the objective is aimed at maximizing the system reliability subjected to several nonlinear constraints [1–4]. The mixed-integer nonlinear optimization programming model for RRAP is formulated as follows to maximize the system reliability by determining the number of components and the component reliabilities in each subsystem: where is the system reliability; and are the component reliability vector and the redundancy allocation vector of the system, respectively, where and are, respectively, the reliability of each component and the number of components in subsystem for ; is the objective function for the system reliability; and are the constraint function and its resource limitation, respectively.

The main goal of reliability engineering is to increase the system reliability. Two different strategies, i.e., active and cold-standby, are usually used to meet this goal. All components simultaneously start to operate from time zero, for the active strategy, although only one is required at any particular time. The cold-standby strategy first developed and studied on redundancy allocation problem (RAP) by Coit in 2001 [5]; the redundant components are protected from stresses associated with system operation so that no component fails before its start.

There has been much research on different solution algorithms for RRAP. For example, particle swarm optimization (PSO) [3, 6], nondominated sorting genetic algorithm II (NSGA-II) [7–9], artificial bee colony algorithm (ABC) [10], genetic algorithms (GA) [1, 2, 4, 11], simplified swarm optimization (SSO) [12, 13], nest cuckoo optimization algorithm [14], a hybrid of PSO and SSO (PSSO) [15], and stochastic fractal search (SFS) [16] have been employed to study for RRAP.

Most previous research of RRAP in the literature has been devoted to the active strategy [1–4, 8, 10, 13, 15]. Several of these researches of RRAP using the cold-standby strategy can be distinguished which are aimed at solving the multiobjective [9] and focusing on the single objective of maximizing the system reliability [11, 12, 14, 16]. In addition, some researches of RRAP adopt the mixed strategy of active and cold-standby [6, 7]. In this work, the research of RRAP using the cold-standby strategy with single objective of system reliability that experimented on the four typical and famous benchmarks of RRAP including a series system (Figure 1), a series-parallel system (Figure 2), a complex (bridge) system (Figure 3), and an overspeed protection system for a gas turbine (Figure 4) as shown in Section 2.2 is studied. The research in [11, 12, 14, 16] studied RRAP using the cold-standby strategy with single objective of system reliability but only experimented on the first three famous benchmarks of RRAP including a series system, a series-parallel system, and a complex (bridge) system. Therefore, in this paper, the cold-standby strategy is used to increase system reliability in the RRAP formulation while focusing on the single objective of maximizing system reliability, and a solution methodology is presented to optimize system reliability for comprehensive experiments on all four famous benchmarks of RRAP.

Since the early 1990s, soft computing (SC) has been utilized to obtain optimal or good-quality solutions to difficult optimization problems. Swarm intelligence (SI) is a newly developed branch of SC that belongs in the category of population-based stochastic optimization. Particle swarm optimization (PSO) that was first developed by Kennedy and Eberhard in 1995 [17] and simplified swarm optimization (SSO) that was originally exploited by Yeh in 2009 [18] are two of the most well-known algorithms in SI. In recent years, we have seen an increasing interest both in PSO [3, 6, 15, 19–23] and in SSO [12, 13, 15, 24–28] for solving larger problems in science and technology.

The goal of this paper is to optimize the system reliability using RRAP with cold-standby strategy that belongs to the mixed-integer optimization programming model. Therefore, the merits of PSO and SSO, which are used to search for optima in real and discrete numbers, respectively, are adopted in this work. That is, a hybrid algorithm of PSO and SSO (PSSO) [15], which has only been used in RRAP with active strategy, is the first time used to optimize the system reliability using RRAP with cold-standby strategy. To demonstrate the efficiency of PSSO, a comprehensive comparative performance study with another recently introduced algorithm is presented for four different widely used systems. In summary, the novelty and contributions of this work are the RRAP using the cold-standby strategy with single objective of system reliability that has comprehensive experiments on all the four famous benchmarks of RRAP.

This paper is organized as follows. Section 2 presents the mathematical formulation of the cold-standby redundancy strategy for RRAP and four systems. Section 3 provides respective descriptions of PSO and SSO and the orthogonal array test. The PSSO and related UM are discussed in Section 4. A comprehensive comparative study of the performances of PSSO optimizing the four systems is given in Section 5. Finally, the discussion and conclusion are given in Section 6.

2. The Cold-Standby Redundancy RRAP and Four Systems

2.1. The Cold-Standby Redundancy RRAP

Cold-standby redundancy is more difficult to implement than active redundancy because of the necessity to detect failures as they occur and activate the redundant component. If more than one component is used (), then there is one initially operating component and components in cold standby waiting to be activated. The subsystem reliability for any distribution of component times-to-failure can be modeled as follows [5, 11].

A detection and switching mechanism is required to sense the occurrence of the component failure and to activate (switch to) a redundant component for cold-standby redundancy. However, the switch itself may fail. For the two imperfect operations, detection and switching, the subsystem reliability for any component time-to-failure distribution with imperfect failure detection and switching can be modeled as follows [5, 11].

Fact 1: continual detection and switching mechanism

Fact 2: detection and switching mechanism only at time of failure

In this study, we investigate the continual detection and switching mechanism. It is difficult to determine a closed form of Equation (3). A convenient lower bound on subsystem reliability can be determined as follows because for all .

The limit of is zero as approaches one. Hence, is usually close to 1.0 [5].

If the probability distribution of a component’s time-to-failure is exponential, then Equation (5) can be expressed by treating the probability of subsystem failure as a homogeneous Poisson process prior to the failure. In this case, the reliability of the subsystem is the probability that there are strictly less than failures, which is a Poisson distribution with parameter . Hence [5, 11],

A convenient lower bound on subsystem reliability is determined as follows:

In the mathematical formulation of cold-standby redundancy for the RRAP, and are the two decision variables and is obtained on the basis of from Equation (6).

2.2. The Four Systems

This paper applies RRAP with cold-standby strategy to four systems: a series system (Figure 1), a series-parallel system (Figure 2), a complex (bridge) system (Figure 3), and an overspeed protection system for a gas turbine (Figure 4).

Due to their structures, the four systems have different objective functions to maximize their reliabilities but are subject to similar multiple nonlinear constraints. The respective RRAPs with cold-standby redundancy are formulated as follows.

System 1. The series system as in Figure 1 [11, 29]

System 2. The series-parallel system as in Figure 2 [11, 29]

System 3. The complex (bridge) system as in Figure 3 [11, 29]:

System 4. The overspeed protection system [29]
An RRAP with a cold-standby redundancy formulation of an overspeed protection system with a time-related cost function [29] for a gas turbine is introduced for the first time. The model is formulated as follows.

3. Preliminaries

The PSO and SSO ought to be expounded at first because the PSSO is the hybrid of PSO and SSO. In addition, an orthogonal array test (OA) is introduced in this study to help improve solution quality and a penalty function is used to deal with constraints.

3.1. The PSO

PSO belongs to the family of swarm intelligence algorithms that was originally developed by Kennedy and Eberhard [17]. A population of random particles is initialized with random positions and velocities in the solution space; these are to be optimized by the fitness function to guide the direction of the solution. In each generation, pBest, denoted as , is the local best solution among , ,…,; each solution has its own pBest. is the global best, which is the best solution of all existing solutions among all pBests; there is only one gBest at a time. In the generation, each solution moves towards pBest and gBest [19, 20] for and . The velocities and positions are updated according to the following equation after both and are found. where and are the velocity and position of the solution at the generation, usually is equal 0.9999, and are the weights of the search directions, and 4 is the upper bound of [17].

3.2. The SSO

The SSO belongs to the swarm intelligence family and is a population-based dynamic optimization algorithm that was originally developed by Yeh [18]. It is also initialized with a population of random solutions inside the problem space and then searches for optimal solutions by updating subsequent generations. Let , , , and be the probabilities of the new variable value updated from the variable in the same position of the current solution; the sum of , , , and equals one. The fundamental concept of SSO is that to maintain population diversity and enhance the capacity to escape from a local optimum [18], each variable of any solution needs to be updated to a value related to its current value, its current pBest, the gBest, or a random feasible value. A random movement of SSO is based on the following model after , , and are given: where , , , and is a random number between the lower and upper bounds of the variable.

3.3. The Orthogonal Array Test (OA)

This paper adopts the orthogonal array test (OA) to improve solutions because the OA is helpful to systematically and efficiently produce a potentially good approximation [30, 31]. Table 1 illustrates the class of the three-level OA where the numbers 1, 2, and 3 in each column indicate the levels of factors, and an equal number of 1s, 2s, and 3s is contained in each column. Columns 1-3 are the factors of the parameters , , and in Equation (14), and column 4 is the factor of the parameter in Equation (12).

The numbers of parameter values that were set according to the published papers of the SSO algorithm [30, 32–39] are arranged in Table 2. The top three parameter values that are most frequently set are , 0.15, 0.2, , 0.3, 0.35, and , 0.35, 0.5, as presented in Table 3.

The three parameter values that are most frequently set in PSO are 0.9999, linearly decreasing as in Equation (15) and exponentially decreasing as in Equation (16).

The nine combinations of the parameter values are presented in Table 4 from the information above.

3.4. The Penalty Function

A penalty function as shown in Equation (17) is used to deal with constraints. That is, the penalty function in Equation (17) is a penalty mechanism for system reliability if any constraint exceeds the upper limit of cost, weight, or volume. where is the system reliability confirmed by the penalty function.

4. The PSSO

In this section, the PSSO is used together with an all-variable-UM (here termed -UM and -UM) that retains the merits of PSO and SSO that are beneficial in searching for the optima in real and discrete numbers, respectively [15, 17–20, 32–39]. This study considers how the simultaneous application of those merits is conducive to computing the RRAP with cold-standby strategy, which is a mixed-integer programming model. Therefore, the key characteristics and merits of PSO and SSO are applied in this paper to optimize RRAP with cold-standby strategy.

The parameters of the Poisson distribution in Equation (7) and the numbers of all components need to be determined in cold-standby redundancy RRAP. Each number of components is an integer, and each parameter of a component is a real number. Hence, two different UMs, termed -UM and -UM, are proposed to update the numbers of all components and the parameters of all components.

4.1. The -UM

Applying the SSO, the proposed -UM updates the number of components, i.e., for , in each subsystem. Let , , , and be a random number between the lower and upper bounds of the variable, and the mathematical model is as follows:

4.2. The -UM

Applying the velocity function of PSO, the proposed -UM updates the parameters of the Poisson distributions (Equation (7)) of the components in each subsystem. The mathematical model is as follows:

4.3. Pseudo-Code for PSSO

The pseudo-code of PSSO is as follows.

Step 0. Generate randomly, calculate , and let and for .

Step 1. Find gBest such that for .

Step 2. Let .

Step 3. Update and based on Equations (18) and (19).

Step 4. If , let and and go to Step 5. Otherwise, go to Step 6.

Step 5. If , let .

Step 6. If , let and go to Step 3.

Step 7. If , let and go to Step 2. Otherwise, halt.

5. Experimental Results

While this paper aims at optimizing the system reliability, the studied RRAP with cold-standby strategy comprehensively applied to the typical and well-known four systems described in Section 2.2: a series system, a series-parallel system, a complex (bridge) system, and an overspeed protection system for a gas turbine is solved by PSSO [11, 29].

The PSSO implemented for RRAP with cold-standby strategy including the four systems was coded in the C⁺⁺ programming language and run on an Intel Core i7 3.07 GHz PC with 6 GB memory. The experiments used 1000 generations (), the number of solutions was , the mission time , and the convenient lower bound on subsystem reliability .

Four systems are provided to evaluate the performance of PSSO for cold-standby RRAP, which is the mixed-integer nonlinear reliability design. The corresponding input data and parameters are the same as in [11, 29] and are presented in the supplementary file “Data.docx” (available here) and Tables 5–7, respectively.

The experimental results solved by PSSO in terms of the statistical analysis for the maximum (the best), mean, minimum (the worst), and standard deviation of the related nine combinations, using the OA introduced in Section 3.3, are presented in Tables 8–11. The best solutions for the system reliability are 0.99700404, 0.99998828, 0.99997538, and 0.99679154 for the four systems, respectively.

Finally, Tables 12–14 illustrate the best performances of PSSO in comparison with previous results [11, 12, 14, 16], the PSO, and the SSO algorithms for the first three systems, respectively. The cold-standby strategy for RRAP is applied to the fourth system, i.e., overspeed protection of a gas turbine, for the first time, considering the PSSO applies the combined merits of PSO and SSO. Hence, the best performance of PSSO in comparison with the PSO and the SSO algorithms for the fourth system is illustrated in Table 15.

In Tables 12–15, the second row illustrates the solution to the system reliability. Other rows containing , , and indicate the number of components, the parameters of the Poisson distribution in Equation (7), and the reliability of components in each subsystem, respectively. Finally, the MPI rows give the improvements of the solutions found by the proposed solution over those of the best known previous solutions; the calculation equation is , where indicates the system reliability obtained by PSSO and indicates the system reliability obtained by a previous algorithm.

The results demonstrate that the PSSO performs better than the PSO and the SSO in terms of system reliability for the fourth system. The results of the first three systems in terms of system reliability obtained by ENCOA [14] are the best but the results found by PSSO are the second best. The detailed comparisons are as follows. (1)The system reliabilities of 0.99999, 0.99999999, and 0.99999995 obtained by ENCOA [14] for the first three systems are better than PSSO, those of the previous work, the PSO, and the SSO(2)The system reliability of 0.99679154 obtained by PSSO for the fourth system is better than the PSO and the SSO(3)However, the system reliabilities of 0.99700404, and 0.99998828 obtained by PSSO for the first two systems are better than those of the previous work, the PSO, and the SSO except ENCOA [14], and the system reliability of 0.99997538 obtained by PSSO for the third system is better than those of the previous work, the PSO, and the SSO except ENCOA [14] and SFS [16].

6. Conclusion and Future Work

A particle-based version of SSO called PSSO with a new UM to enhance the ability of traditional SSO is used to solve RRAP with cold-standby strategy. The RRAP cold-standby effectively maximizes the system reliability with the PSSO. Moreover, the UM is an important part of soft computing. This paper presents significant and novel modifications to SSO to optimize the cold-standby redundancy RRAP.

A comprehensive comparative study of the performances of the PSSO and previous work has been made. The system reliability obtained by the PSSO is better than the PSO and SSO for the fourth system. The system reliability obtained by the PSSO is the second best; those are second to ENCOA [14] for the first three systems. Roughly speaking, the PSSO based on UM has the ability to optimize the mixed-integer programming model and can be used to solve cold-standby redundancy RRAP efficiently. In future research, we will focus on strengthening SSO performance and will apply it to different optimization problems and solve practical engineering problems with larger-scale systems.

Acronyms

PSO:	Particle swarm optimization
SSO:	Simplified swarm optimization
PSSO:	Particle-based simplified swarm optimization
RAP:	Redundancy allocation problem
RRAP:	Reliability-redundancy allocation problem
pBest:	Local best
gBest:	Global best
-UM:	Proposed update mechanism for the number variables of all components
-UM:	Proposed update mechanism for the variables of all components
MPI:	Maximum possible improvement.

Notations

, , :	The number of subsystems in the system, solutions, and generations, respectively
:	is the redundancy allocation vector of the system, where is the number of components in subsystem for
:	is the component reliability vector of the system, where is the reliability of each component in subsystem for
:	is the reliability of subsystem , where is the failure probability of each component in subsystem for
:	The convenient lower-bound reliability of subsystem
:	The reliability of each component in subsystem at the mission time
:	The pdf of subsystem at the failure arrival for
:	The failure detection/switching reliability
:	The parameter of the Poisson distribution in subsystem
:	The system reliability
:	The constraint function with respect to and
:	The physical feature of each component in subsystem for
:	The resource limitation for the constraint function
, , :	The volume, cost, and weight, respectively, of each component in subsystem ,
, , :	The upper limits on the volume, cost, and weight of the system, respectively
:	The fitness function with respect to and
, :	is the redundancy allocation vector of the solution at the generation, where is the variable for
, :	is the vector of the component parameters of the solution at the generation, where is the variable for
:	is the solution at the generation
, :	The related pBest and gBest.

Data Availability

Data are available in the supplementary information file.

Conflicts of Interest

The authors declare no conflict of interest.

Authors’ Contributions

Conceptualization, software, investigation, resources, and writing—original draft preparation were contributed by all authors; methodology was contributed by Chia-Ling Huang; validation was contributed by Yunzhi Jiang; formal analysis was contributed by Yunzhi Jiang and Chia-Ling Huang; data curation was contributed by Chia-Ling Huang; supervision was contributed by Yunzhi Jiang and Chia-Ling Huang; project administration was contributed by Chia-Ling Huang; funding acquisition was contributed by Yunzhi Jiang. All authors have read and agreed to the published version of the manuscript.

Acknowledgments

This work was supported in part by the Natural Science Foundation of China and Ministry of Science and Technology, Taiwan, R.O.C., under grant 61702118, MOST 105-2221-E-424-003, and MOST 109-2221-E-424-002.

Supplementary Materials

The corresponding input data and parameters are the same as in [11, 29] and are presented in the supplementary file “Data.docx” and Tables 5–7, respectively. (Supplementary Materials)

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Copyright © 2021 Chia-Ling Huang 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.

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