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

Authors: Chia-Ling Huang; Yunzhi Jiang; Wei-Chang Yeh
Journal: Journal of Sensors
 (2021)

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## 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.


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## Body

## 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,Rs, 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:
(1)MaximizeRs=fR,N,SubjecttogjR,N≤uj,1≤j≤thenumberofconstraints,where Rs is the system reliability; R=r1,r2⋯,rnsu and N=n1,n2,⋯,nnsu are the component reliability vector and the redundancy allocation vector of the system, respectively, where ri and ni are, respectively, the reliability of each component and the number of components in subsystem i for i=1,2,⋯,nsu; f• is the objective function for the system reliability; gj• and uj are the jth 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.Figure 1
The series system.Figure 2
The series-parallel system.Figure 3
The complex (bridge) system.Figure 4
The overspeed protection system for a gas turbine.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. Section2 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 (ni>1), then there is one initially operating component and ni−1 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].
(2)Rit=rit+∑k=1ni−1∫0trit−ufikudu.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(3)Rit=rit+∑k=1ni−1∫0tρiurit−ufikudu.Fact 2: detection and switching mechanism only at time of failure(4)Rit=rit+∑k=1ni−1∫0tρikrit−ufikudu.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 ρiu≥ρit for all u≤t.
(5)Rit≥R~it=rit+ρit∑k=1ni−1∫0trit−ufikudu.The limit ofRit−R~it is zero as ρit approaches one. Hence, ρit 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 nith failure. In this case, the reliability of the subsystem is the probability that there are strictly less than ni failures, which is a Poisson distribution with parameter λi. Hence [5, 11],
(6)∫0trit−ufikudu=e−λitλitkk!.A convenient lower bound on subsystem reliability is determined as follows:(7)R~it=rit+ρit∑x=1ni−1e−λitλitxx!.In the mathematical formulation of cold-standby redundancy for the RRAP,λi and ni are the two decision variables and ri is obtained on the basis of λi from Equation (6).


### 2.2. The Four Systems

This paper applies RRAP with cold-standby strategy to four systems: a series system (Figure1), 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 Figure1 [11, 29]
(8)maxfR,N=∏i=1NsuRin,it,s.tg1R,N=∑i=1Nsuwivi2ni2≤V,g2R,N=∑i=1Nsuαi−1000lnritβini+expni4≤C,g3R,N=∑i=1Nsuwiniexpni4≤W,0≤rit≤1,rit∈realnumber,ni∈positiveinteger,i=1,⋯,Nsu.System 2.
The series-parallel system as in Figure2 [11, 29]
(9)maxfR,N=1−1−R1tR2t1−1−1−R3t1−R4tR5t,s.tg1R,N≤V,g2R,N≤C,g3R,N≤W,0≤rit≤1,rit∈realnumber,ni∈positiveinteger,i=1,⋯,Nsu.System 3.
The complex (bridge) system as in Figure3 [11, 29]:
(10)maxfR,N=R1tR2t+R3tR4t+R1tR4tR5t+R2tR3tR5t−R1tR2tR3tR4t−R1tR2tR3tR5t−R1tR2tR4tR5t−R1tR3tR4tR5t−R2tR3tR4tR5t+2R1tR2tR3tR4tR5t,s.tg1R,N≤V,g2R,N≤C,g3R,N≤W,0≤rit≤1,rit∈realnumber,ni∈positiveinteger,i=1,⋯,Nsu.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.
(11)maxfR,N=∏i=1Nsu1‐1‐ritni,s.th1R,N=∑i=1Nsuvini2≤V,h2R,N=∑i=1Nsuαi−1000lnritβini+expni4≤C,h3R,N=∑i=1Nsuwiniexpni4≤W,0.5≤rit≤1‐10‐6,rit∈realnumber,1≤ni≤10,ni∈positiveinteger,i=1,⋯,Nsu.


## 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 (ni>1), then there is one initially operating component and ni−1 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].
(2)Rit=rit+∑k=1ni−1∫0trit−ufikudu.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(3)Rit=rit+∑k=1ni−1∫0tρiurit−ufikudu.Fact 2: detection and switching mechanism only at time of failure(4)Rit=rit+∑k=1ni−1∫0tρikrit−ufikudu.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 ρiu≥ρit for all u≤t.
(5)Rit≥R~it=rit+ρit∑k=1ni−1∫0trit−ufikudu.The limit ofRit−R~it is zero as ρit approaches one. Hence, ρit 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 nith failure. In this case, the reliability of the subsystem is the probability that there are strictly less than ni failures, which is a Poisson distribution with parameter λi. Hence [5, 11],
(6)∫0trit−ufikudu=e−λitλitkk!.A convenient lower bound on subsystem reliability is determined as follows:(7)R~it=rit+ρit∑x=1ni−1e−λitλitxx!.In the mathematical formulation of cold-standby redundancy for the RRAP,λi and ni are the two decision variables and ri is obtained on the basis of λi from Equation (6).


## 2.2. The Four Systems

This paper applies RRAP with cold-standby strategy to four systems: a series system (Figure1), 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 Figure1 [11, 29]
(8)maxfR,N=∏i=1NsuRin,it,s.tg1R,N=∑i=1Nsuwivi2ni2≤V,g2R,N=∑i=1Nsuαi−1000lnritβini+expni4≤C,g3R,N=∑i=1Nsuwiniexpni4≤W,0≤rit≤1,rit∈realnumber,ni∈positiveinteger,i=1,⋯,Nsu.System 2.
The series-parallel system as in Figure2 [11, 29]
(9)maxfR,N=1−1−R1tR2t1−1−1−R3t1−R4tR5t,s.tg1R,N≤V,g2R,N≤C,g3R,N≤W,0≤rit≤1,rit∈realnumber,ni∈positiveinteger,i=1,⋯,Nsu.System 3.
The complex (bridge) system as in Figure3 [11, 29]:
(10)maxfR,N=R1tR2t+R3tR4t+R1tR4tR5t+R2tR3tR5t−R1tR2tR3tR4t−R1tR2tR3tR5t−R1tR2tR4tR5t−R1tR3tR4tR5t−R2tR3tR4tR5t+2R1tR2tR3tR4tR5t,s.tg1R,N≤V,g2R,N≤C,g3R,N≤W,0≤rit≤1,rit∈realnumber,ni∈positiveinteger,i=1,⋯,Nsu.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.
(11)maxfR,N=∏i=1Nsu1‐1‐ritni,s.th1R,N=∑i=1Nsuvini2≤V,h2R,N=∑i=1Nsuαi−1000lnritβini+expni4≤C,h3R,N=∑i=1Nsuwiniexpni4≤W,0.5≤rit≤1‐10‐6,rit∈realnumber,1≤ni≤10,ni∈positiveinteger,i=1,⋯,Nsu.


## 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 Pil−1, is the local best solution among Yi0, Yi1,…,Yil−1; each solution has its own pBest. PgBestl−1 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 lth generation, each solution Yil moves towards pBest Pil−1 and gBest PgBestl−1 [19, 20] for l=0,1,2,⋯,Ng and i=1,2,⋯,Nso. The velocities and positions are updated according to the following equation after both Pil−1 and PgBestl−1 are found.
(12)Dil=c0⋅Dil−1+cg⋅ρ1PgBestl−1−Yil−1+cp⋅ρ2Pil−1−Yil−1,(13)Yil=Yil−1+Dil,where Dil and Yil are the velocity and position of the ith solution at the lth generation, c0 usually is equal 0.9999, cgρ1 and cpρ2 are the weights of the search directions, and 4 is the upper bound of cg+cp [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 cw, cp, cg, and cr be the probabilities of the new variable value updated from the variable in the same position of the current solution; the sum of cw, cp, cg, and cr 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 cw, cp, and cg are given:
(14)xijl=xijl−1ifρ0,1∈0,Cw=cw,pijl−1ifρ0,1∈Cw,Cp=Cw+cp,gjifρ0,1∈Cp,Cg=Cp+cg,xifρ0,1∈Cg,1,where i=1,2,⋯,Nso, j=1,2,⋯,Nsu, l=0,1,2,⋯,Ng, and x is a random number between the lower and upper bounds of the jth 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 cw, cp, and cg in Equation (14), and column 4 is the factor of the parameter c0 in Equation (12).Table 1
The orthogonal array.
TestsColumns111112122231333421235223162312731328321393321The 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 cw=0.1, 0.15, 0.2, cp=0.25, 0.3, 0.35, and cg=0.25, 0.35, 0.5, as presented in Table 3.Table 2
The parameter values of SSO.
cwcpcgReference0.10.30.5[39]0.150.250.35[32]0.10.30.50.10.30.30.10.30.5[34]0.150.250.35[35]0.150.250.35[30]0.150.350.25[33]0.150.350.25[36]0.10.30.5[37]0.20.10.19[38]Table 3
The 3 most frequent parameter values of SSO.
cwFrequencycpFrequencycgFrequency0.1050.2530.2520.1550.3050.3530.2010.3520.504The 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).
(15)c0=0.9999−g∗0.1Ng,(16)c0=c0⋅.9c0Ng.The nine combinations of the parameter values are presented in Table4 from the information above.Table 4
The combinations of the parameter values.
FactorsSSOPSOTestscgcpcwc010.250.250.100.999920.250.300.15Equation (15)30.250.350.20Equation (16)40.350.250.15Equation (16)50.350.300.200.999960.350.350.10Equation (15)70.500.250.20Equation (15)80.500.300.10Equation (16)90.500.350.150.9999


### 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.
(17)Rpenalty=RsifX=N,Λisafeasiblesolution,RsminVg1R,N,Cg2R,N,Wg3R,N,πotherwise,where Rpenalty is the system reliability confirmed by the penalty function.


## 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 Pil−1, is the local best solution among Yi0, Yi1,…,Yil−1; each solution has its own pBest. PgBestl−1 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 lth generation, each solution Yil moves towards pBest Pil−1 and gBest PgBestl−1 [19, 20] for l=0,1,2,⋯,Ng and i=1,2,⋯,Nso. The velocities and positions are updated according to the following equation after both Pil−1 and PgBestl−1 are found.
(12)Dil=c0⋅Dil−1+cg⋅ρ1PgBestl−1−Yil−1+cp⋅ρ2Pil−1−Yil−1,(13)Yil=Yil−1+Dil,where Dil and Yil are the velocity and position of the ith solution at the lth generation, c0 usually is equal 0.9999, cgρ1 and cpρ2 are the weights of the search directions, and 4 is the upper bound of cg+cp [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 cw, cp, cg, and cr be the probabilities of the new variable value updated from the variable in the same position of the current solution; the sum of cw, cp, cg, and cr 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 cw, cp, and cg are given:
(14)xijl=xijl−1ifρ0,1∈0,Cw=cw,pijl−1ifρ0,1∈Cw,Cp=Cw+cp,gjifρ0,1∈Cp,Cg=Cp+cg,xifρ0,1∈Cg,1,where i=1,2,⋯,Nso, j=1,2,⋯,Nsu, l=0,1,2,⋯,Ng, and x is a random number between the lower and upper bounds of the jth 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 cw, cp, and cg in Equation (14), and column 4 is the factor of the parameter c0 in Equation (12).Table 1
The orthogonal array.
TestsColumns111112122231333421235223162312731328321393321The 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 cw=0.1, 0.15, 0.2, cp=0.25, 0.3, 0.35, and cg=0.25, 0.35, 0.5, as presented in Table 3.Table 2
The parameter values of SSO.
cwcpcgReference0.10.30.5[39]0.150.250.35[32]0.10.30.50.10.30.30.10.30.5[34]0.150.250.35[35]0.150.250.35[30]0.150.350.25[33]0.150.350.25[36]0.10.30.5[37]0.20.10.19[38]Table 3
The 3 most frequent parameter values of SSO.
cwFrequencycpFrequencycgFrequency0.1050.2530.2520.1550.3050.3530.2010.3520.504The 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).
(15)c0=0.9999−g∗0.1Ng,(16)c0=c0⋅.9c0Ng.The nine combinations of the parameter values are presented in Table4 from the information above.Table 4
The combinations of the parameter values.
FactorsSSOPSOTestscgcpcwc010.250.250.100.999920.250.300.15Equation (15)30.250.350.20Equation (16)40.350.250.15Equation (16)50.350.300.200.999960.350.350.10Equation (15)70.500.250.20Equation (15)80.500.300.10Equation (16)90.500.350.150.9999


## 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.
(17)Rpenalty=RsifX=N,Λisafeasiblesolution,RsminVg1R,N,Cg2R,N,Wg3R,N,πotherwise,where Rpenalty 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 termedn-UM and λ-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 parametersλi 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 λi of a component is a real number. Hence, two different UMs, termed n-UM and λ-UM, are proposed to update the numbers of all components and the parameters λi of all components.


### 4.1. Then-UM

Applying the SSO, the proposedn-UM updates the number of components, i.e., nsu for su=1,2,⋯,Nsu, in each subsystem. Let so=1,2,⋯,Nso, su=1,2,⋯,Nsu, g=1,2,⋯,Ng, and n be a random number between the lower and upper bounds of the suth variable, and the mathematical model is as follows:
(18)Nso,sug=n∧gBest,suifρ0,1∈0,Cg=cg,n∧so,suifρ0,1∈Cg,Cp=Cg+cp,nso,sug‐1ifρ0,1∈Cp,Cw=Cp+cw,nifρ0,1∈Cw,1.


### 4.2. Theλ-UM

Applying the velocity function of PSO, the proposedλ-UM updates the parameters λi of the Poisson distributions (Equation (7)) of the components in each subsystem. The mathematical model is as follows:
(19)Λsog=c0⋅Λsog‐1+cg⋅ρ1Λ∧gBestg‐1−Λsog‐1+cp⋅ρ1Λ∧sog‐1−Λsog‐1.


### 4.3. Pseudo-Code for PSSO

The pseudo-code of PSSO is as follows.Step 0.
GenerateXso0=Nso0,Λso0 randomly, calculate FXso0, and let Pso=Xso0 and FPso=FXso0 for so=1,2,⋯,Nso.Step 1.
Find gBest such thatFXso0≤FXgBest0 for so=1,2,⋯,Nso.Step 2.
Letg=1.Step 3.
UpdateNsog and Λsog based on Equations (18) and (19).Step 4.
IfFPso<FXsog, let Pso=Xsog and FPso=FXsog and go to Step 5. Otherwise, go to Step 6.Step 5.
IfFPgBest<FPso, let gBest=so.Step 6.
Ifso<Nso, let so=so+1 and go to Step 3.Step 7.
Ifg<Ng, let g=g+1 and go to Step 2. Otherwise, halt.


## 4.1. Then-UM

Applying the SSO, the proposedn-UM updates the number of components, i.e., nsu for su=1,2,⋯,Nsu, in each subsystem. Let so=1,2,⋯,Nso, su=1,2,⋯,Nsu, g=1,2,⋯,Ng, and n be a random number between the lower and upper bounds of the suth variable, and the mathematical model is as follows:
(18)Nso,sug=n∧gBest,suifρ0,1∈0,Cg=cg,n∧so,suifρ0,1∈Cg,Cp=Cg+cp,nso,sug‐1ifρ0,1∈Cp,Cw=Cp+cw,nifρ0,1∈Cw,1.


## 4.2. Theλ-UM

Applying the velocity function of PSO, the proposedλ-UM updates the parameters λi of the Poisson distributions (Equation (7)) of the components in each subsystem. The mathematical model is as follows:
(19)Λsog=c0⋅Λsog‐1+cg⋅ρ1Λ∧gBestg‐1−Λsog‐1+cp⋅ρ1Λ∧sog‐1−Λsog‐1.


## 4.3. Pseudo-Code for PSSO

The pseudo-code of PSSO is as follows.Step 0.
GenerateXso0=Nso0,Λso0 randomly, calculate FXso0, and let Pso=Xso0 and FPso=FXso0 for so=1,2,⋯,Nso.Step 1.
Find gBest such thatFXso0≤FXgBest0 for so=1,2,⋯,Nso.Step 2.
Letg=1.Step 3.
UpdateNsog and Λsog based on Equations (18) and (19).Step 4.
IfFPso<FXsog, let Pso=Xsog and FPso=FXsog and go to Step 5. Otherwise, go to Step 6.Step 5.
IfFPgBest<FPso, let gBest=so.Step 6.
Ifso<Nso, let so=so+1 and go to Step 3.Step 7.
Ifg<Ng, let g=g+1 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 Section2.2: 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 (Ng=1000), the number of solutions was Nso=100, the mission time t=1000, and the convenient lower bound on subsystem reliability ρit=0.99.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.Table 5
The parameter values used in Systems1 and 3.
Subsystemi105αiβiwivi2wiVCW12.3301.51711017520021.4501.52830.5411.53848.0501.54651.9501.529Table 6
The parameter values used in System2.
Subsystemi105αiβiwivi2wiVCW12.5001.523.518017510021.4501.544.030.5411.554.040.5411.583.552.1001.544.5Table 7
The parameter values used in System4.
Subsystemi105αiβiviwiVCW111.516250.0400.0500.022.31.52630.31.53842.31.527The 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 Section3.3, 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.Table 8
The reliability of System1 for the nine parameter combinations.
OAMaximumMeanMinimumStandard deviation10.969577320.963856170.912637280.0081595520.969564820.965000150.951402250.0043413130.997004040.922722000.820283280.0490466740.996561290.918778560.768427760.0559460550.969579240.958949400.875816320.0162288660.969576120.964233950.934411240.0053850770.969572330.946844950.816609300.0290125580.996510260.927888710.815503240.0510623690.969579260.887616900.683550490.07417057Average0.978613850.939543420.842071240.03259477Table 9
The reliability of System2 for the nine parameter combinations.
OAMaximumMeanMinimumStandard deviation10.999988280.999985670.999970260.0000045120.999988270.999986170.999973580.0000036030.999988260.999986770.999973580.0000033240.999988260.999985170.999967400.0000053450.999988280.999976170.999570380.0000433360.999988270.999982450.999949130.0000072470.999988270.999958770.999717340.0000460180.999988260.999972950.999815480.0000290890.999988280.999440280.993140110.00138911Average0.999988270.9999193780.9991196960.000170171Table 10
The reliability of System3 for the nine parameter combinations.
OAMaximumMeanMinimumStandard deviation10.999975380.999971420.999902260.0000096320.999975350.999972880.999937340.0000060330.999975320.999973050.999909980.0000070340.999975320.999971760.999923350.0000065650.999975380.999953090.999304990.0000695960.999975360.999969810.999910110.0000112370.999975380.999866390.998509740.0002056380.999975350.999956310.999790330.0000309090.999975380.999368350.989818970.00141941Average0.9999753580.9998892290.9985563410.000196223Table 11
The reliability of System4 for the nine parameter combinations.
OAMaximumMeanMinimumStandard deviation10.996791540.996784690.996746920.0000158920.996791220.996788770.996766630.0000038230.996791350.996787790.996781780.0000018940.996791400.996787000.996744190.0000050750.996791540.996778630.996747080.0000202460.996791440.996788310.996746260.0000077270.996791490.996778510.996698260.0000205080.996791210.996787360.996743220.0000098590.996791540.993206810.908233570.01352766Average0.996791410.996387540.986911990.00151251Finally, Tables12–14 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.Table 12
Comparison of PSSO with previous work for System1.
PSOSSOGA, 2014 [11]New SSO, 2019 [12]ENCOA, 2020 [14]SFS, 2019 [16]PSSORs0.962689030.959892270.969577580.969579240.999990.9695790.99700404N(2,3,2,3,3)(3,2,2,4,2)(3, 2, 2, 3, 3)(3, 2, 2, 3, 3)(1, 3, 3, 2, 2)(3, 2, 2, 3, 3)(3,4,4,3,3)λ10.000153690.000305180.000268410.000265870.0005960.0002660.00003271λ20.000369000.000152590.000119310.000119240.0006610.0001190.00008385λ30.000067050.000095720.000088400.000088560.0005460.0000890.00002787λ40.000360140.000732420.000366000.000367280.0006240.0003670.00006836λ50.000263030.000099420.000253560.000253870.0005590.0002540.00008259r10.857541770.736993810.764593350.766530.7423860.7664980.96782041r20.691425910.858483440.887528920.887590.8168680.8875760.91957190r30.935145510.908717120.915395270.915250.7139630.9152930.97251610r40.697577160.480743280.693505440.692610.5750150.6925670.93392478r50.768716760.905362230.776031450.775790.7387210.7758250.92072882MPI91.97%92.53%90.15%90.15%90.15%Table 13
Comparison of PSSO with previous work for System2.
PSOSSOGA, 2014 [11]New SSO, 2019 [12]ENCOA, 2020 [14]SFS, 2019 [16]PSSORs0.999984180.999969290.999988240.999988270.999999990.999988270.99998828N(3,3,1,2,3)(3,3,1,2,3)(3,3,2,1,3)(3,3,1,2,3)(3,2,1,3,3)(3,3,1,2,3)(3,3,1,2,3)λ10.000202860.000305180.000192550.000191560.000425640.000191860.00019217λ20.000185300.000274660.000171000.000164170.000171870.000164980.00016433λ30.000231610.000061040.000096320.000106910.000334260.000107050.00010766λ40.000077070.000052760.000106800.000096390.000338070.000095090.00009650λ50.000160420.000183110.000144490.000148390.000259400.000148470.00014768r10.816389040.736993810.824846720.825670.682458680.825424440.82516974r20.830858950.759831790.842816570.848600.864737940.847911460.84846299r30.793256260.940790160.908173080.898600.699728960.898483730.89793059r40.925822340.948608800.898699000.908110.732103570.909294190.90800596r50.851786440.832680330.865463010.862090.700980380.862028710.86270308MPI25.91%61.83%0.34%0.09%0.09%Table 14
Comparison of PSSO with previous work for System3.
PSOSSOGA, 2014 [11]New SSO, 2019 [12]ENCOA, 2020 [14]SFS, 2019 [16]PSSORs0.999966020.999940090.999974130.999975370.999999950.999975380.99997538N(3,3,3,3,1)(3,3,2,3,2)(3,3,3,3,1)(3,3,2,4,1)(2,3,2,3,3)(3,3,2,4,1)(3,3,2,4,1)λ10.000269830.000335690.000217440.000199130.000480530.000199540.00020096λ20.000189390.000097090.000154120.000155820.000353080.000156580.00015568λ30.000167660.000067830.000142320.000068080.000580800.000068390.00006777λ40.000301170.000549320.000318020.000490700.000523600.000487750.00048910λ50.000179400.000457760.000268970.000279340.000283470.000275850.00027874r10.763510990.714842270.804572340.819440.706475730.819110280.81794334r20.827463170.907472810.857173050.855710.683275130.855063380.85583287r30.845642860.934423690.867346830.934180.888121380.933898580.93447620r40.739948680.577344340.727591620.612190.614311380.614004150.61317957r50.835771610.632696980.764166660.756280.702421050.758929560.75673635MPI27.54%58.90%4.83%0.04%Table 15
Comparison of PSSO with PSO and SSO for System4.
PSOSSOPSSORs0.996703040.996779640.99679154N(3,3,3,3)(3,3,3,3)(3,3,3,3)λ10.000070280.000067790.00007003λ20.000091440.000095270.00009313λ30.000053190.000045200.00004502λ40.000094730.000093760.00009315r10.932131570.934460760.93236397r20.912614200.909130420.91107606r30.948204110.955809620.95597900r40.909620700.910504800.91105944MPI2.68%0.36%In Tables12–15, the second row illustrates the solution to the system reliability. Other rows containing N, λi, and ri 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 Rs_PSSO−Rs_other/1−Rs_other, where Rs_PSSO indicates the system reliability obtained by PSSO and Rs_other 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 reliabilitiesRs 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 reliabilityRs of 0.99679154 obtained by PSSO for the fourth system is better than the PSO and the SSO(3)
However, the system reliabilitiesRs 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 Rs 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.



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*Source: 7833641-2021-08-21.xml*