A novel particle swarm optimization with fuzzy adaptive inertia weight for reliability redundancy allocation problems

Abstract

Particle swarm optimization (PSO) is a stochastic population based algorithm inspired by the social learning of fishes or birds. It is a swarm intelligence technique for optimization process. The parameters of PSO take an important role in the performance of PSO algorithm. Inertia weight is an important parameter of this algorithm which affects the convergence and exploration-exploitation trade-off in PSO. Since the introduction of this parameter, various developments for determining the value of inertia weight was proposed. In this paper a novel fuzzy adaptive particle swarm optimization (FAPSO) algorithm has been proposed considering the inertia weight as a triangular fuzzy number (TFN) which changes in every iteration. The algorithm outperforms the standard PSO as well as the previous adaptive approaches. Four reliability redundancy benchmarks are considered to display the performability of the proposed PSO that develops the strengths of PSO to enable optimizing the RRAP which belongs to mixed integer non-linear programming problem. Finally a statistical analysis has been done which indicates that the proposed FAPSO performs better than the algorithms existing in literature.

Keywords

Reliability Reliability Redundancy Allocation Problem (RRAP)Particle Swarm Optimization (PSO)

1. Introduction

Reliability is the probability of performing without failure during specific times under specific conditions. There are generally two methods to improve the reliability of a system. These are basically through increasing of component’s reliability or by adding redundant components. So, in the problem of reliability maximization, if the component reliabilities and redundancies both are considered as decision variables then the problem is called reliability redundancy allocation problem (RRAP).

Reliability redundancy allocation is one of the most popular research areas which has drawn attention of many researchers. Mishra and Ljubojevic [1] were the first who introduced the concept of RRAP. Also in [2], Mishra established a simple approach for constrained reliability redundancy allocation problem. Chern in [3] focused on the computational complexity of reliability redundancy allocation in a series system. Dhingra [4] constructed an optimal apportionment of reliability and redundancy in series system under multiple objectives. There are many real world applications of this problem in industrial systems like telecommunication system, electrical system, satellite system etc. In these applications either reliability or cost or both of them are considered as objective functions and volume, weight of the components are considered as constraints. These objectives and constraints are basically non-linear function. So RRAP is mainly a mixed integer non-linear programming problem [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14].

Researchers have not only focused on RRAP but also concentrated on developing meta-heuristic methods for optimizing the system reliability [15]. Different meta-heuristics such as genetic algorithm [5, 10, 12], artificial bee colony algorithm [11], immune algorithm [8], ant colony optimization [16] etc. have been proposed to address RRAP. Xu et al. [13] and Hikita et al. [7] described an iterative heuristic method to increase reliability. Yokota et al. [10] and Hsieh et al. [12] implemented genetic algorithms on RRAP. H. Kim and P. Kim [17] introduced parallel genetic algorithm for reliability redundancy allocation considering optimal redundancy.

The particle swarm optimization is one of the popular meta-heuristic algorithms for RRAP. Various developments of PSO have been done by several researchers [6, 18, 19, 20, 21, 22]. Yeh [21, 22] developed PSO as a simplified swarm optimization algorithm (SSO) considering the random movement of the particles. Based on both PSO and SSO, Huang [6] proposed a particle based simplified swarm optimization (PSSO) to optimize RRAP. Q. Wang and J. Wang [23] introduced bacterial foraging process into the adaptive particle swarm optimization algorithm for reliability redundancy allocation problem.

Inertia weight plays an important role in the balance between exploration and exploitation process of PSO algorithm. The inertia weight determines the contribution rate of a particle’s previous velocity to its current velocity. Generally a large inertia weight facilitates a global search while a small inertia weight helps a local search. Different types of inertia weight like random inertia weight, linearly decreasing inertia weight, sigmoidal increasing inertia weight, chaotic dynamic inertia weight etc have been introduced in literature [24, 25, 26, 27, 28] to increase the capabilities of PSO algorithm. But in iteration process sometimes local search and sometimes global search helps better convergence of PSO algorithm. So in this paper for the first time, a fuzzy adaptive inertia weight based novel particle swarm optimization algorithm is proposed to optimize RRAP as well as mixed integer non-linear problems. Here inertia weight is considered as a TFN which is adapted in every iteration. In this way, all the diversity of inertia weight can be covered which helps to attain the optimum result. The efficiency of the proposed approach is demonstrated by considering four benchmarks: a series system in benchmark 1, a network with series and parallel elements in benchmark 2, a complex (bridge) system in benchmark 3 and the over-speed protection of gas turbine system in benchmark 4.

The rest of the paper has been organized as follows: in Section 2, some preliminaries and notations are described on reliability. An overview on PSO algorithm has been given in Section 3. Section 4 proposes a novel fuzzy adaptive inertia weight based Particle swarm optimization algorithm and Section 5 provides the four benchmarks. The efficiency of the algorithm and results of the experiments are discussed in Section 6. Finally the conclusion has been presented in Section 7.

2. Preliminaries

2.1 Notations

$N_{\textit{var}}$ – number of subsystems in the system.

$N=(n_{1},n_{2},\ldots n_{\textit{var}})$ redundancy allocation of the system, where $n_{i}=$ number of component in $i^{\text{th}}$ subsystem for $i=1,2,\ldots N_{\textit{var}}$ .

$R=(r_{1},r_{2},\ldots r_{N_{\textit{var}}})$ is the system reliability, where $r_{i}=$ reliability of each component in $i^{\text{th}}$ subsystem for $i=1,2,\ldots N_{\textit{var}}$ .

$R_{i}(n_{i})=(1-q_{i}^{n_{i}})$ is the reliability of $i^{\text{th}}$ subsystem, where $q_{i}=(1-r_{i})$ is the failure probability of each component in $i^{\text{th}}$ subsystem for $i=1,2,\ldots N_{\textit{var}}$ .

$R_{s}=$ the system reliability.

$C,W,V=$ maximum limit of cost, weight and volume respectively.

$f(R,N)=$ fitness function.

$l_{j}=$ the resource limitation for the $j^{\text{th}}$ constraint function.

$\alpha_{i},\beta_{i}=$ the physical feature of each component in $i^{\text{th}}$ subsystem for $i=1,2,\ldots N_{\textit{var}}$ .

$g_{j}(R,N)=j^{\text{th}}$ constraint w.r.t $R$ and $N$ .

2.2 Basic Definitions

2.2.1 Reliability

Reliability is the probability of performing without failure, a specific function under given conditions for a specified period of time.

$\displaystyle R(t)=P(T>t)$

For series system, the total system reliability:

$\displaystyle R(t)=r_{1}(t)\cdot r_{2}(t)\ldots\cdot r_{n}(t)=\prod_{i=1}^{n}r% _{i}(t)$

For parallel system, the system reliability is:

$\displaystyle R(t)=1-[1-r_{1}(t)][1-r_{2}(t)]\ldots[1-r_{n}(t)]$

where $r_{i}(t)=$ reliability of the $i^{\text{th}}$ component.

If all the component have equal reliability, then total system reliability is:

$\displaystyle R(t)=1-[1-r(t)]^{n}$

2.2.2 Reliability redundancy allocation problem (RRAP)

Reliability redundancy allocation problem is a significant problem in the initial stages of reliability design. The main objective of RRAP is to maximize the overall system reliability by determining the number and reliability of the components in each subsystem considering multiple non-linear constraints. The problem is a mixed integer non-linear programming problem. The mathematical model is as follows:

$\displaystyle\text{Max}R_{s}=f(R,N)$

subject to, $g_{j}(R,N)\leqslant l_{j}$ , 1 $\leqslant j\leqslant$ number of constraints.

Figure 1.

Triangular fuzzy number.

2.2.3 Triangular fuzzy number (TFN)

A fuzzy number $A=(a,b,c)$ is said to be a triangular fuzzy number if its membership function is given by

$\displaystyle(x;a,b,c)=\left\{\begin{array}[]{ll}0&\text{if }x\leqslant a;\\ \frac{x-a}{b-a}&\text{if }a\leqslant x\leqslant b;\\ \frac{c-x}{c-b}&\text{if }b\leqslant x\leqslant c;\\ 0&\text{if }c\leqslant x.\end{array}\right.$

2.2.4 Expected value

The concept of expected value is used to convert triangular fuzzy number to a crisp number. The expected value is denoted by EV. If $w=(A,B,C)$ is a TFN then its expected value is:

$\displaystyle w=(A,B,C)\Rightarrow EV(w)=\frac{A+2B+C}{4}$

2.2.5 Uniform random number generator

The Uniform Random Number generator generates uniformly distributed random numbers over a specified interval. In matlab software, this is denoted by unifrnd().

3. Overview of PSO

Particle swarm optimization is one of the most successful stochastic global optimization techniques inspired by the social behaviour of fish schooling or bird flocking. Initially Kennedy and Eberhart [29] proposed this algorithm. This is a population based randomized search technique and widely used algorithm for solving non-linear optimization problems. Here every solution is called particle which flies around the multi-dimensional search space. During movement each particle modifies its position according to its own best position (pbest, $p^{j}$ ) as well as the best position of its neighbors (gbest, $g^{j}$ ). Depending upon this pbest and gbest information, the velocity and position of the particles are updated as follows:

$\displaystyle v_{k+1}^{i}=w\cdot v_{k}^{i}+c_{1}\cdot ud\cdot(p_{k}^{i}-x_{k}^% {i})+c_{2}\cdot Ud\cdot(p_{k}^{g}-x_{k}^{i})$ $\displaystyle x_{k+1}^{i}=x_{k}^{i}+v_{k+1}^{i}$

where $w$ is the inertia weight whose value is traditionally set to 0.9999; $i=1,2,\ldots,N$ denotes the number of particles in the population, $k=1,2,\ldots,k_{max}$ denotes the iterations, $c_{1}$ and $c_{2}$ are the cognitive and social components which are the acceleration coefficients responsible for varying the particle’s velocity towards pbest and gbest. $c_{1}+c_{2}$ is generally limited to 4.

Variables ud and Ud are two random functions in the range [0,1]. The basic steps of PSO are:

Basic steps of PSO algorithm:

1.
Set parameters of PSO.
2.
Initialize population of particles with position and velocity.
3.
Evaluate initial fitness of each particle and select pbest and gbest.
4.
Set iteration count $k=$ 1.
5.
Update velocity and position of each particle.
6.
Evaluate fitness of each particle and update pbest and gbest.
7.
Process continues until requirements are met.

4. Proposed fuzzy adaptive particle swarm optimization (FAPSO)

Our proposed FAPSO is a fuzzy adaptive inertia weight based swarm optimization technique where inertia weight is considered as a triangular fuzzy number. In this proposed approach inertia weight is adapted in every iteration by uniform random number generator and then defuzzified by expected value method. The algorithm is run through the following steps:

Step1: Initially swarm of particles is generated over the search space. Also different acceleration parameters $c_{1},c_{2}$ and velocity $v$ are initially considered.

Step2: In the first iteration, pbest of each particle is set as its initial position. Correspondingly after calculating the fitness of all particles, the index of the best particle is found. After-that in each iteration, pbest is computed based on the following rules:

1)
If the current solution is dominated by the pbest solution, then pbest solution remains unchanged.
2)
If the pbest solution is dominated by the current solution, then the current solution is stored as pbest.
3)
Otherwise one solution is selected as pbest at random.

Finally gbest solution is formed using the fitness values at pbest solutions.

Step3: In each iteration, the velocity of each particle is computed by the equation:

$\displaystyle v_{k+1}^{i}=w\cdot v_{k}^{i}+c_{1}\cdot ud\cdot(p_{k}^{i}-x_{k}^% {i})+c_{2}\cdot Ud\cdot(p_{k}^{g}-x_{k}^{i})$

and the new position of each particle is allotted by:

$\displaystyle x_{k+1}^{i}=x_{k}^{i}+v_{k+1}^{i}$

Here proposed inertia weight $w$ is:

$\displaystyle\!\!\!\!\!\!\!\!\!\!w=\left(0.5-\frac{\textit{unifrnd}(0,1)}{2},0% .5,0.5+\frac{\textit{unifrnd}(0,1)}{2}\right)$

Here in every iteration, inertia weight is adapted by uniform random number generator and then the expected value method is applied to calculate the crisp value of $w$ .

$\displaystyle EV(w)=$

Step4: Steps (1) to (4) are repeated until the number of iteration reaches to the number that is specified before.

Constraint handling strategy: There are four different strategies for constraint handling. The strategies are:

(a)
Modify: In this case, the structure of solution is designed in such a manner that always feasible solutions are found.
(b)
Reject: Every infeasible solution is rejected by this strategy.
(c)
Repair: In this strategy, every infeasible solution is changed to reach to a feasible solution.
(d)
Penalty function: This strategy accepts infeasible solutions by assigning a penalty function to the fitness function.

In this paper, a penalization technique is used. Therefore the fitness function is defined as the summation of the objective function and a penalty function determined by the relative degree of infeasibility. To provide an efficient search through the infeasible region but to ensure that the final solution is feasible, the following fitness functions are proposed:

$\displaystyle\!\!\!\!\!\!\!\!\!\!f=\left\{\begin{array}[]{l}\textit{objective % function }(R_{s})\\ \quad\textit{if $x$ is a feasible solution};\\ R_{s}\!+\!\textit{number of constraints}\!\times\!\textit{penalty}\!\times\!% \text{sum}(c);\\ \quad\textit{otherwise}.\\ \end{array}\right.$

Here sum( $c$ ) is calculated in the following way:

At first all constraints are converted to less or equals to ( $\leqslant$ ) type then:

$\displaystyle i=1:\textit{number of constraints},$

If constraint function( $i$ ) $>$ 0, $c(i)=$ 1; else $c(i)=$ 0.
5. Numerical Illustration on four benchmarks

The mixed integer non-linear programming problem is used to optimize the system reliability of the four benchmarks. Mathematical model for each benchmark is given as follows:

Benchmark 1. The series system in Fig. 2[6, 8].

For this system the problem is given as:

$\displaystyle\text{Max}f(R,N)=\prod_{i=1}^{N_{\textit{var}}}R_{i}(n_{i})$

Subject to

$\displaystyle g_{1}(R,N)=\sum_{i=1}^{N_{\textit{var}}}w_{i}v_{i}^{2}n_{i}^{2}% \leqslant V.$ $\displaystyle g_{2}(R,N)=\sum_{i=1}^{N_{\textit{var}}}\alpha_{i}(-1000/\ln r_{% i})^{\beta_{i}}\cdot(n_{i}+\exp(n_{i}/4))\leqslant C.$ $\displaystyle g_{3}(R,N)=\sum_{i=1}^{N_{\textit{var}}}w_{i}n_{i}\exp(n_{i}/4)% \leqslant W.$

0 $\leqslant r_{i}\leqslant$ 1, $r_{i}\in$ real number, $n_{i}\in$ positive integer. $i=1,\ldots N_{\textit{var}}$ .

Benchmark 2. The network with series and parallel elements in Fig. 3 [6, 8].

For this system the problem is given as:

$\displaystyle\!\!\!\!\!\!\text{Max}f(R,N)=1-(1-R_{1}R_{2})\cdot(1-[1-(1-R_{3})% (1-R_{4})]R_{5})$

Subject to

$\displaystyle g_{1}(R,N)\leqslant V$ $\displaystyle g_{2}(R,N)\leqslant C$ $\displaystyle g_{3}(R,N)\leqslant W$

0 $\leqslant r_{i}\leqslant$ 1, $r_{i}\in$ real number, $n_{i}\in$ positive integer. $i=1,\ldots N_{\textit{var}}$ .

Benchmark 3. The complex (bridge) system in Fig. 4 [6, 8].

For this system the problem is given as:

$\displaystyle\text{Max}f(R,N)=R_{1}R_{2}+R_{3}R_{4}+R_{1}R_{4}R_{5}+$ $\displaystyle\quad R_{2}R_{3}R_{5}-R_{1}R_{2}R_{3}R_{4}-R_{1}R_{2}R_{3}R_{5}-$ $\displaystyle\quad R_{1}R_{2}R_{4}R_{5}-R_{1}R_{3}R_{4}R_{5}-R_{2}R_{3}R_{4}R_% {5}+$ $\displaystyle\quad 2R_{1}R_{2}R_{3}R_{4}R_{5}$

Subject to

$\displaystyle g_{1}(R,N)\leqslant V$ $\displaystyle g_{2}(R,N)\leqslant C$ $\displaystyle g_{3}(R,N)\leqslant W$

0 $\leqslant r_{i}\leqslant$ 1, $r_{i}\in$ real number, $n_{i}\in$ positive integer. $i=1,\ldots N_{\textit{var}}$ .

Benchmark 4. The overspeed protection system in Fig. 5 [4, 6, 8, 10, 11].

For this system the problem is given as:

$\displaystyle\text{Max}f(R,N)=\prod_{i=1}^{N_{\textit{var}}}[1-(1-r_{i})^{n_{i% }}]$

Figure 2.

The series system.

Figure 3.

The network with Series and Parallel elements.

Figure 4.

The complex (bridge) system.

Figure 5.

The overspeed protection System.

Subject to

$\displaystyle g_{1}(R,N)=\sum_{i=1}^{N_{\textit{var}}}v_{i}n_{i}^{2}\leqslant V$ $\displaystyle g_{2}(R,N)=\sum_{i=1}^{N_{\textit{var}}}\alpha_{i}(-1000/\ln r_{% i})^{\beta_{i}}\quad(n_{i}+\exp(n_{i}/4))\leqslant C$ $\displaystyle g_{3}(R,N)=\sum_{i=1}^{N_{\textit{var}}}w_{i}n_{i}\exp(n_{i}/4)% \leqslant W.$

0.5 $\leqslant r_{i}\leqslant 1-10^{-6}$ , $r_{i}\in$ real number, 1 $\leqslant n_{i}\leqslant$ 10, $n_{i}\in$ positive integer. $i=1,\ldots N_{\textit{var}}$ .

6. Results and discussion

To establish the supremacy of our proposed algorithm, the algorithm is coded in matlab programming language and has been applied on all the problems described in Section 5. During the evolution process, the integer variables $n_{i}$ are taken as real variables and finally the real values are converted to nearest integer values for calculating the value of objective function. The acceleration coefficients $c_{1}$ , $c_{2}$ of PSO are taken as $c_{1}=$ 2, $c_{2}=$ 2. Maximum number of iterations and maximum number of particles both are considered as 100. Particles initial position and velocity are initiated by the linear expressions [initial position $=$ lower bound of the variable $+$ uniform random number (0,1) $\times$ (upper bound of the variable $-$ lower bound of the variable)] and [initial velocity $=$ 0.1 $\times$ initial position] respectively. The termination criteria has been set either to maximum number of iterations or to the order of relative error ( $=10^{-12}$ ) whichever is achieved first. Here the penalty value is determined as 10,000.

Table 1
Data used in benchmark 1 and benchmark 3

Subsystem	$10^{5}\alpha_{i}$	$\beta_{i}$	$w_{i}v_{i}^{2}$	$w_{i}$	$V$	$C$	$w$
1	2.330	1.5	1	7	110	175	200
2	1.450	1.5	2	8
3	0.541	1.5	3	8
4	8.050	1.5	4	6
5	1.950	1.5	2	9

Table 2

Data used in benchnark 2

Subsystem	$10^{5}\alpha_{i}$	$\beta_{i}$	$w_{i}v_{i}^{2}$	$w_{i}$	$V$	$C$	$w$
1	2.500	1.5	2	3.5	180	175	100
2	1.450	1.5	4	4.0
3	0.541	1.5	5	4.0
4	0.541	1.5	8	3.5
5	2.100	1.5	4	4.5

Table 3

Data used in benchmark 4

Subsystem	$10^{5}\alpha_{i}$	$\beta_{i}$	$w_{i}v_{i}^{2}$	$w_{i}$	$V$	$C$	$w$
1	1	1.5	1	6	250	400	500
2	2.3	1.5	2	6
3	0.3	1.5	3	8
4	2.3	1.5	2	7

Figure 6.

Iteration vs Fitness function value for benchmark 1.

Figure 7.

Iteration vs Fitness function value for benchmark 2.

Figure 8.

Iteration vs Fitness function value for benchmark 3.

Figure 9.

Iteration vs Fitness fuction value for benchmark 4.

Four RRAP based benchmarks are considered to check the performance of the proposed PSO. The related structures of the benchmarks and the corresponding parameters which are provided in Section 5, are the same values used by Dhingra [4], Huang [6], Hikita et al. [7], Chen [8], Yokota et al. [10], Hsieh et al. [12], Xu et al. [13], Kuo et al. [15] and Garg [30] and these are given in Tables 1–3 respectively.

Table 4

Comparison of the proposed algorithm solutions with other solutions for benchmark 1

	Hikita et al. [7]	Kuo and	Xu et al. [13]	Hsieh et al. [12]	Chen [8]	Garg [30]	PSO	PSSO [6]	FAPSO
		Hwang [15]							(Proposed)
$N$	(3,2,2,3,3)	(3,2,2,3,3)	(3,2,2,3,3)	(3,2,2,3,3)	(3,2,2,3,3)	(3,2,2,3,3)	(2,3,2,4,2)	(3,2,2,3,3)	(3,3,2,3,3)
$r_{1}$	0.777143	0.77960	0.77939	0.779427	0.779266	0.779405727	0.80059281	0.77946645	0.7929
$r_{2}$	0.867514	0.80065	0.87183	0.869482	0.872513	0.8718339252	0.74049316	0.87173278	0.8255
$r_{3}$	0.896696	0.90227	0.90288	0.902674	0.902634	0.9028816332	0.82914384	0.90284951	0.8940
$r_{4}$	0.717739	0.71044	0.71139	0.714038	0.710648	0.7114011594	0.63686144	0.71148780	0.6976
$r_{5}$	0.793889	0.85947	0.78779	0.786896	0.788406	0.7878058784	0.88704276	0.78781644	0.8350
$R_{s}$	0.93163	0.92975	0.931677	0.931578	0.9316782	0.9316823874	0.8885037	0.93168230	0.94152347
MPI	14.4704841%	16.7593879%	14.4116476%	14.5354857%	14.4101443%	14.4048983%	47.5529412%	14.4050078%

Table 5

Comparison of the proposed algorithm solutions with other solutions for benchmark 2

	Hikita et al. [7]	Hsieh et al. [12]	Chen [8]	Garg [30]	PSO	PSSO [6]	FAPSO (Proposed)
$N$	(3,3,1,2,3)	(2,2,2,2,4)	(2,2,2,2,4)	(2,2,2,2,4)	(4,3,2,1,2)	(2,2,2,2,4)	(3,3,1,3,4)
$r_{1}$	0.838193	0.785452	0.812485	0.8196583448	0.84025282	0.81958939	0.8139
$r_{2}$	0.855065	0.842998	0.843155	0.8449101406	0.88865099	0.84458412	0.8496
$r_{3}$	0.878859	0.885333	0.897385	0.8954871713	0.62375055	0.89534134	0.8797
$r_{4}$	0.911402	0.917958	0.894516	0.8955148963	0.93984950	0.89581626	0.8990
$r_{5}$	0.850355	0.870318	0.870590	0.8684681613	0.75158691	0.86852902	0.8554
$R_{s}$	0.99996875	0.99997418	0.99997658	0.9999766491	0.99985845	0.99997665	0.99998766
MPI	60.512%	52.207591%	47.3099915%	47.1540711%	91.2822324%	47.1520343%

Table 6

Comparison of the proposed algorithm solutions with other solutions for benchmark 3

	Hikita et al. [7]	Hsieh et al. [12]	Chen [8]	Garg [30]	PSO	PSSO [6]	FAPSO (Proposed)
$N$	(3,3,2,3,2)	(3,3,3,3,1)	(3,3,3,3,1)	(3,3,2,4,1)	(3,3,2,2,3)	(3,3,2,4,1)	(3,3,2,3,1)
$r_{1}$	0.814483	0.814090	0.812485	0.8280606892	0.77061588	0.82783292	0.8298
$r_{2}$	0.821383	0.864614	0.867661	0.8580404545	0.90109253	0.85771241	0.8601
$r_{3}$	0.896151	0.890291	0.861221	0.9141487486	0.89278651	0.91437458	0.8890
$r_{4}$	0.713091	0.701190	0.713852	0.6479689012	0.60083008	0.64861002	0.6755
$r_{5}$	0.814091	0.734731	0.756699	0.7042048796	0.73451002	0.70287554	0.6783
$R_{s}$	0.99978937	0.99987916	0.99988921	0.9998896364	0.99967140	0.99988964	0.99991134
MPI	57.9072307%	26.6302549%	19.974727%	19.6655419%	73.0188679%	19.6629213%

As a result, Figs 6–9 show the convergence graph obtained by the proposed FAPSO for each benchmark respectively and Tables 4–7 represent the numerical solution for four benchmarks respectively. The best solution of each benchmark is compared with the solutions of the existing literature. The first row of these tables indicate the optimal number of components in each subsystem respectively. The second-six rows, containing $r_{i}$ , denote the optimal reliability of components in $i^{\text{th}}$ subsystem. A statistical analysis is given in Table 8 based on ten run of the proposed FAPSO for each benchmark. To assess the performance of the proposed FAPSO algorithm, the best, worst, mean, SD and $(\mu\pm 2\sigma)$ values of optimal system reliability for each benchmark are shown in this table.

Table 7

Comparison of the proposed algorithm solutions with other solutions for benchmark 4

	Dhingra [4]	Yokota et al. [10]	Chen [8]	Garg [30]	PSO	PSSO [6]	FAPSO (Proposed)
$N$	(6,6,3,5)	(3,6,3,5)	(5,5,5,5)	(5,5,4,6)	(4,6,3,5)	(5,5,4,6)	(5,6,4,5)
$r_{1}$	0.81604	0.965593	0.903800	0.9015629232	0.92952331	0.90166461	0.8895
$r_{2}$	0.80309	0.760592	0.874992	0.8882249447	0.81370356	0.88817296	0.8755
$r_{3}$	0.98364	0.972646	0.919898	0.9481559581	0.88663747	0.94821033	0.9413
$r_{4}$	0.80373	0.804660	0.890609	0.8499528951	0.89987183	0.84987084	0.8756
$R_{s}$	0.99961	0.999468	0.999942	0.9999546746	0.99990474	0.99995467	0.99996419
MPI	90.8179487%	93.268797%	38.2586207%	20.9935268%	62.4081461%	21.0015442%

Table 8

Statistical analysis on performance of proposed FAPSO

	Best performance	Worst performance	Mean( $\mu$ )	Standard deviation ( $\sigma$ )	( $\mu-2\sigma,\mu+2\sigma$ )
Benchmark 1	0.94152347	0.93698989	0.93954	0.001537	(0.936465,0.942615)
Benchmark 2	0.99998766	0.99997090	0.999982361	5.75485 $\times$ 10 ${}^{-6}$	(0.999970851, 0.999993871)
Benchmark 3	0.99991134	0.99985768	0.99989452	1.77337 $\times$ 10 ${}^{-5}$	(0.999859053, 0.999929987)
Benchmark 4	0.99996419	0.99995908	0.999962435	1.50019 $\times$ $10^{-6}$	(0.999959435, 0.999965435)

The optimal system reliability ( $R_{s}=$ 0.94152347, 0.99998766, 0.99991134, 0.99996419) of the four benchmarks obtained by proposed FAPSO are better than the solutions obtained in the existing literature and the normal PSO algorithm. The amount of improvement of the solutions obtained by this proposed algorithm is calculated using the formula MPI $=(R_{s-\textit{FAPSO}}-R_{s-\textit{other}})/(1-R_{s-\textit{other}})$ , where $R_{s-\textit{FAPSO}}$ denotes the system reliability obtained by proposed FAPSO and $R_{s-\textit{other}}$ denotes the system reliability obtained by other algorithms in the existing literature. In all these cases, MPI values show that the proposed approach is better than the existing algorithms in literature. From the statistical analysis done in Table 8, it can be concluded using Tchebysheff’s theorem that at least 75% of the results lie within the interval $(\mu\pm 2\sigma)$ for each of the benchmarks 1–4. It also establishes the stability of the method.

Finally, it is observed that the solutions of the four benchmarks found by proposed FAPSO are comparatively much better than those found by other algorithms. Thus the proposed FAPSO is superior to other methods for mixed-integer non-linear reliability redundancy allocation problems.

7. Conclusion

In this present study, a penalty guided fuzzy adaptive particle swarm optimization (FAPSO) technique is proposed. This development considers inertia weight as a triangular fuzzy number which is adapted in every iteration. Also a constraint handling strategy is proposed to overcome the infeasibility of the solutions. In order to evaluate the efficiency of the proposed algorithm, four benchmark problems are considered which are non-linear integer programming problem. Numerical results of these problems show that the proposed algorithm leads to higher reliability compared to other algorithms. This algorithm offers a greater flexibility to system designers and reliability analysts and leads to an appreciable improvement in the reliability of complex systems. In future this work can be expanded not only for RRAP but also in different large scale benchmarks with mixed integer non-linear programming problems.

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A novel particle swarm optimization with fuzzy adaptive inertia weight for reliability redundancy allocation problems

Abstract

Keywords

1. Introduction

2. Preliminaries

2.1 Notations

2.2 Basic Definitions

2.2.1 Reliability

2.2.2 Reliability redundancy allocation problem (RRAP)

2.2.4 Expected value

2.2.5 Uniform random number generator

3. Overview of PSO

Table 1 Data used in benchmark 1 and benchmark 3

References

Table 1
Data used in benchmark 1 and benchmark 3