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

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.


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, R s , 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][2][3][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: Subject to g j R, N ð Þ≤ u j , where R s is the system reliability; R = ðr 1 , r 2 ⋯ , r nsu Þ and N = ðn 1 , n 2 , ⋯, n nsu Þ are the component reliability vector and the redundancy allocation vector of the system, respectively, where r i and n i are, respectively, the reliability of each component and the number of components in subsystem i for i = 1, 2, ⋯, n su ; f ð•Þ is the objective function for the system reliability; g j ð•Þ and u j are the j th 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.
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 seriesparallel 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][20][21][22][23] and in SSO [12,13,15,[24][25][26][27][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  Journal of Sensors 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.

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 (n i > 1), then there is one initially operating component and n i − 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].
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 ρ i ðuÞ ≥ ρ i ðtÞ for all u ≤ t.
The limit of R i ðtÞ −R i ðtÞ is zero as ρ i ðtÞ approaches one. Hence, ρ i ðtÞ is usually close to 1.0 [5].
If the probability distribution of a component's time-tofailure is exponential, then Equation (5) can be expressed by treating the probability of subsystem failure as a homogeneous Poisson process prior to the n i th failure. In this case, the reliability of the subsystem is the probability that there are strictly less than n i failures, which is a Poisson distribution with parameter λ i . Hence [5,11], Air fuel mixture Gas turbine Mechanical and electrical over speed detection Figure 4: The overspeed protection system for a gas turbine.

Journal of Sensors
A convenient lower bound on subsystem reliability is determined as follows: In the mathematical formulation of cold-standby redundancy for the RRAP, λ i and n i are the two decision variables and r i is obtained on the basis of λ i from Equation (6).

The Four
Systems. This paper applies RRAP with coldstandby 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.

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 Journal of Sensors best, which is the best solution of all existing solutions among all pBests; there is only one gBest at a time. In the l th generation, each solution Y l i moves towards pBest P l−1 i and gBest P l−1 gBest [19,20] for l = 0,1,2, ⋯, N g and i = 1, 2, ⋯, N so . The velocities and positions are updated according to the following equation after both P l−1 i and P l−1 gBest are found.
where D l i and Y l i are the velocity and position of the i th solution at the l th generation, c 0 usually is equal 0.9999, c g ρ 1 and c p ρ 2 are the weights of the search directions, and 4 is the upper bound of c g + c p [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 c w , c p , c g , and c r be the probabilities of the new variable value updated from the variable in the same position of the current solution; the sum of c w , c p , c g , and c r 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 c w , c p , and c g are given: where i = 1, 2, ⋯, N so , j = 1, 2, ⋯, N su , l = 0,1,2, ⋯, N g , and x is a random number between the lower and upper bounds of the j th variable.
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 R penalty is the system reliability confirmed by the penalty function.

The PSSO
In this section, the PSSO is used together with an allvariable-UM (here termed n-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][18][19][20][32][33][34][35][36][37][38][39]. This study considers how the simultaneous application of those merits is conducive to computing the RRAP with cold-standby strategy, which is a mixedinteger 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 λ i 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 λ i 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 λ i of all components.

The n-UM.
Applying the SSO, the proposed n-UM updates the number of components, i.e., n su for su = 1, 2, ⋯, N su , in each subsystem. Let so = 1, 2, ⋯, N so , su = 1, 2, ⋯, N su , g = 1, 2, ⋯, N g , and n be a random number between the lower and upper bounds of the su th variable, and the mathematical model is as follows:

The λ-UM.
Applying the velocity function of PSO, the proposed λ-UM updates the parameters λ i 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 3. Update N g so and Λ g so based on Equations (18) and (19).
Step 4. If FðP so Þ < FðX g so Þ, let P so = X g so and FðP so Þ = FðX g so Þ and go to Step 5. Otherwise, go to Step 6.
Step 6. If so < N so , let so = so + 1 and go to Step 3.
Step 7. If g < N g , let g = g + 1 and go to Step 2. Otherwise, halt.
Þis a feasible solution, Journal of Sensors

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 (N g = 1000), the number of solutions was N so = 100, the mission time t = 1000, and the convenient lower bound on subsystem reliability ρ i ðtÞ = 0:99.
Four systems are provided to evaluate the performance of PSSO for cold-standby RRAP, which is the mixedinteger nonlinear reliability design. The corresponding input        [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 perfor-mance 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 N, λ i , and r i 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 ðR s PSSO − R s other Þ/ð1 − R s other Þ, where R s PSSO indicates the system reliability obtained by PSSO and R s 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  (1) The system reliabilities R s 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 R s of 0.99679154 obtained by PSSO for the fourth system is better than the PSO and the SSO (3) However, the system reliabilities R s 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 R s 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].

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 coldstandby 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 n-UM: Proposed update mechanism for the number variables of all components λ-UM: Proposed update mechanism for the λ variables of all components MPI: Maximum possible improvement.   The convenient lower-bound reliability of subsystem i r i ðtÞ: The reliability of each component in subsystem i at the mission time t f i ðkÞ : The pdf of subsystem i at the k th failure arrival for k = 0, 1, ⋯, n su − 1 ρ i ð•Þ: The failure detection/switching reliability λ i : The parameter of the Poisson distribution in subsystem i R s : The system reliability g j ðR, NÞ: The j th constraint function with respect to R and N α i , β i : The physical feature of each component in subsystem i for i = 1, 2, ⋯, N var u j : The resource limitation for the j th constraint function v i , c i , w i : The volume, cost, and weight, respectively, of each component in subsystem i, i = 1, 2, ⋯, N var V, C, W: The upper limits on the volume, cost, and weight of the system, respectively f ðR, NÞ: The fitness function with respect to R and N N g so , n g so,su : N g so = ðn g so,1 , n g so,2 , ⋯, n g so,Nsu Þ is the redundancy allocation vector of the so th solution at the g th generation, where n g so,su is the su th variable for su = 1, 2, ⋯, N su Λ g so , λ g so,su : Λ g so = ðλ g so,1 , λ g so,2 , ⋯, λ g so,Nsu Þ is the λ vector of the component parameters of the so th solution at the g th generation, where λ g so,su is the su th variable for su = 1, 2, ⋯, N su X g so : X g so = ðN g so , Λ g so Þ is the so th solution at the g th generation • ∧ , • ∧ gBest : 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.