An attraction based particle swarm optimization for solving multi-objective availability allocation problem under uncertain environment

Abstract

Reliability optimization and availability optimization are two classes of optimization problems in redundancy allocation problem (RAP). Contrary to reliability optimization, very few researchers have focused on availability optimization to find out the optimal redundancy. This paper proposes a multi-objective optimization problem of availability allocation in a series-parallel system with repairable components. The two objectives are maximizing the system availability and minimizing the total cost of the system. In real life situation, due to complexity of the systems and non-linearity of their behaviour, most of the data are usually uncertain and imprecise. Hence in order to make the model more reliable, fuzzy theory has been introduced in terms of triangular data for handling the uncertainties. Thus in fuzzy environment a fuzzy multi-objective availability allocation problem is formulated. In order to solve the problem a crisp optimization problem has been reformulated using fuzzy programming technique and finally an attraction based particle swarm optimization (APSO) has been proposed to solve this crisp optimization problem. The proposed APSO is compared with the traditional Particle swarm optimization(PSO) to show the efficiency and consistency of the proposed approach. Based on a numerical example,the statistical analysis of the experimental results establish that the proposed APSO has a better and consistent performance compared to traditional PSO.

Keywords

Series-Parallel System availability allocation problem fuzzy programming technique Attraction based particle swarm optimization (APSO)

1 Introduction

The complexity of a system increases gradually because of technological improvements of the system. Due to this complexity, any component’s failure can lead to major failure of the system. So reliability optimization is a vital issue that has attracted many researchers. The aim of this optimization is to obtain a reliable system structure considering minimum budget and other factors like number of components, mode of redundancy etc.

Generally, reliability is the probability of functioning without failure,a specific function under given conditions for a stated period of time. On the other hand, availability is defined as the probability that a system is in its intended functional condition and therefore capable of being utilized in a given environment [18]. The fundamental difference between reliability and availability is that availability is used for repairable components while reliability is for non-repairable components. Furthermore the term reliability can be utilized for the first failure of repairable components, while the term availability is utilized for whole life of repairable components[15].

In existing literature most of the researchers have focused on reliability redundancy allocation problem(RRAP) [6 , 16]. A lot of developments also have been done for RRAP under uncertain environment [17 , 21]. As compared to the reliability redundancy optimization, very few works have been done on availability optimization to obtain the optimal failure and repair rates of each component in a system. To increase the availability of a system there are different ways,such as: a) increasing the availability of every component, b) Using the redundant component, c) Considering both (a) and (b). In most cases availability optimization is considered as a multi-objective optimization problem to maximize the system availability and minimize the system cost.

Most importantly, if the reliability of a system diminishes towards 0 when its operating time tends to infinity (i.e, becomes very large) then its availability declines to inherent availability. Considering inherent availability different models have been developed for solving availability allocation problems. In 2015, Amiri [1] developed a bi-objective optimization model for solving the availability allocation problem in repairable series parallel systems and used NSGA-II to solve this problem. Zoulfaghai [15] developed a bi-objective redundancy allocation problem for a system with mixed repairable and non-repairable components.

Several researchers have focused this availability allocation problem in which the parameters of the components are assumed to be at fixed positive level. But It has been widely confessed that most of the informations of real life problems are collected under different operating and environmental conditions. Therefore it is not possible to determine a fixed specific numeric value of the parameters. Thus the available data are often imprecise, inaccurate and vague. The reasons might be human factors,internal factors and other factors related to environment which affect each component of the system differently. Hence to build a realistic and reliable mathematical model, the concept of fuzzy logic has been introduced here which can handle these practical situations. Thus multi-objective availability allocation problem in an uncertain environment has been considered for the first time.

The availability allocation problem is a non-linear integer programming problem belonging to the NP-hard class of optimization problems. Due to the complexity of such problems,numerous meta-heuristic algorithms such as genetic algorithm [5, 6, 12], artificial bee-colony algorithm [7], cuckoo search,particle swarm optimization [8, 9] and firefly algorithm [14] have been widely used in past decades. Several works have also been done on genetic algorithm to solve the multi-objective availability allocation problem for series-parallel system [2 , 15].

Cognitive component and Social component play an important role in the balance between exploration and exploitation process of PSO algorithm. In literature a large number of articles considered these components as constant. But in reality the value of these components are affected due to the attraction of particles towards the global best. So in this paper a novel attraction factor has been introduced in PSO algorithm.

This paper deals with the multi-objective availability allocation problem under uncertain environment. Here the parameters have been considered as triangular fuzzy numbers. To solve this problem a novel mixed integer non-linear optimization model under uncertain environment has been developed. An attraction based particle swarm optimization has been proposed to solve the problem. As there is no literature under uncertain environment, a comparison is made with ordinary PSO algorithm to show the efficiency of the proposed approach in handling such problems. The contribution in this paper is two fold. Firstly, it considers uncertainty in the parameters of the problem. Secondly, the PSO algorithm has been modified by introducing an attraction factor.

The rest of the paper is organised as follows: In Section 2 some basic preliminaries on availability and triangular fuzzy number are briefly discussed. The mathematical formulation of the problem under uncertain environment has been presented in Section 3, the solution methodology using fuzzy programming technique is provided in Section 4. Section 5 deals with the PSO and APSO that have been used for optimization. A numerical example and results are shown in Section 6, Section 7 respectively. Finally some conclusions are presented in Section 8.

2 Preliminaries

2.1 Definition 1

Availability: The availability function of a component at time t is given in [15] as: $A (t) = R (t) + \int_{0}^{t} Q (x) R (t - x) dx$

Q (x) is the inverse Laplace transform of $\frac{f_{Z}^{*} (s)}{1 - f_{Z}^{*} (s)}$ and $f_{Z}^{*} (s)$ is the Laplace transformation of the density function of Z = X + Y. Where X denotes the random variable of up time and Y denotes the random variable of down time of the component.

Based on the key renewal theory, the limiting value of availability function A (t) achieves a constant value known as inherent availability given by the following equation: $lim_{t \to \infty} A (t) = \frac{E (X)}{E (X) + E (Y)} = A$

For an example if the up time and down time of a component follow exponential distribution then the availability function is given by: $A (t) = \frac{λ}{λ + μ} + \frac{μ}{λ + μ} {exp}^{- t (\frac{1}{λ} + \frac{1}{μ})}$

And the inherent availability for this component is $A = \frac{λ}{λ + μ}$

In this paper inherent availability is considered as the availability function of Series-parallel system.

2.2 Definition 2

Triangular Fuzzy Number: A fuzzy number A = (c, d, e) is said to be a triangular fuzzy number if its membership function is given by $(x; c, d, e) = {\begin{matrix} 0 & if x \leq c; \\ \frac{x - c}{d - c} & if c \leq x \leq d; \\ \frac{e - x}{e - d} & if d \leq x \leq e; \\ 0 & if e \leq x . \end{matrix}$

3 Problem description

In this paper a series-parallel system is used as a well known structure for describing and demonstrating the proposed approaches. The general structure of a series-parallel system has been illustrated in Fig. 2. It is assumed that for every subsystem i, there are n_i functionally equivalent components and the sub-system needs at least n_min components to keep up its performance. This number of component will be treated as the minimum number of component for subsystem i. The maximum number of component of subsystem i is n_max. Every subsystem has different weight, cost and other characteristics. The aim is to select the number of components and the level of redundancy to maximize system availability. The notations and assumptions of the models are shown below.

Fig.1

Triangular Fuzzy Number.

Fig.2

General structure of series-parallel system.

3.1 Notation

Availability of the system.

Cost of the system.

Total number of subsystems

minimum number of component in i^th subsystem.

maximum number of component in i^th subsystem.

number of component in i^th subsystem

Failure rate of a component in i^th subsystem.

repair rate of a component in i^th subsystem.

parameters used to calculate the weight and volume of components in i^th subsystem.

maximum volume of the system.

maximum weight of the system.

minimum and maximum bounds of failure rate in i^th subsystem.

minimum and maximum bounds of repair rate in i^th subsystem.

coefficients of the cost function.

3.2 Assumptions

The state of every component at any point of time is either good or bad states.

The components are independent of each other.

The overall system performs perfectly when each subsystem has at least one operable component.

The failure and repair rate of each component follows exponential distribution with failure rate λ_i and repair rate μ_i.

3.3 Mathematical model

For series-parallel system the system’s inherent availability is considered for maximization objective and cost is considered as minimization objective. The overall model is-

$Maximize A_{sys} = \prod_{i = 1}^{S} (1 - (\frac{λ_{i}}{λ_{i} + μ_{i}})^{n_{i}})$ (1) $Minimize C_{sys} = \sum_{i = 1}^{S} n_{i} (a_{i} (λ_{i})^{p_{i}} + b_{i} (μ_{i})^{q_{i}})$ (2)

Subject to;

$\sum w_{i} n_{i} \leq W$ (3) $\sum w_{i} {v_{i}}^{2} {n_{i}}^{2} \leq V$ (4) $n_{\min} \leq n_{i} \leq n_{\max} .$ (5) $L_{λ_{i}} \leq λ_{i} \leq U_{λ_{i}}$ (6) $L_{μ_{i}} \leq μ_{i} \leq U_{μ_{i}} for i = 1, 2, . . . . . ., S$ (7)

where a_i, b_i, p_i, q_i are real numbers such that: a_i > 0, b_i > 0, p_i < 0, q_i > 0

Here Equation (1) denotes the availability of the system as a maximizing objective and Equation (2) is related to the cost function which is a minimizing objective. Equations (3) and (4) indicate the constraints of weight and volume respectively. The boundaries of the decision variables are shown by Equations (5, 6) and (7) respectively.

In decision making process many factors are involved in reliability or availability optimization problems. Usually these factors and limitations of these problems are not exactly known. Hence in order to handle these conditions fuzzy concept has been introduced here. Therefore, the availability optimization model can be represented by non-linear fuzzy programming technique to make the model more reliable. Thus in fuzzy environment the corresponding problem becomes, $\begin{matrix} Maximize : {\tilde{A}}_{sys} (\tilde{λ_{i}}, \tilde{μ_{i}}, n_{i}), \\ Minimize : {\tilde{C}}_{sys} (\tilde{λ_{i}}, \tilde{μ_{i}}, n_{i}) \\ subject to : g_{j} (n_{i}) \leq b_{j} \\ n_{\min} \leq n_{i} \leq n_{\max} \\ L_{λ_{i}} \leq λ_{i} \leq U_{λ_{i}} \\ L_{μ_{i}} \leq μ_{i} \leq U_{μ_{i}}, for i = 1, 2, . . . . . ., S . \end{matrix}$ (8)

Here, λ_i and μ_i are considered as a triangular fuzzy number i.e. λ_i = [f_i - I, f_i, f_i + I] (say) is the failure rate of the component of i^th subsystem. Similarly, μ_i = [r_i - I, r_i, r_i + I] (say) is the repair rate of the component of i^th subsystem. Where I is the triangular fuzzy function parameters, as specified by the decision maker/system analyst corresponding to the failure rate and repair rate of each component of the system.

Considering these triangular fuzzy numbers each objective is presented in the form of [Z_L, Z_C, Z_R] and hence the optimization problem becomes:

$\begin{matrix} Minimize [Z_{tL} (f_{i} - I, r_{i} - I, n_{i}); \\ Z_{tC} (f_{i}, r_{i}, n_{i}); Z_{tR} (f_{i} + I, r_{i} + I, n_{i})] \\ subject to; g_{j} (n_{i}) \leq b_{j} \\ n_{\min} \leq n_{i} \leq n_{\max} . \\ L_{λ_{i}} \leq λ_{i} \leq U_{λ_{i}} \\ L_{μ_{i}} \leq μ_{i} \leq U_{μ_{i}} for i = 1, 2, . . . . . ., S . \\ t = 1, 2, . . . . . . . . ., number of objectives . \end{matrix}$ (9)

This type of formulation helps the decision maker to select any one of the three functions Z_L, Z_C, Z_R for minimization according to their needs.

4 Fuzzy programming technique

For solving the above formulated problem fuzzy programming technique has been used in this paper by considering membership function corresponding to each objective function. Linear as well as non-linear membership function has been discussed here. The steps involved in fuzzy programming technique are as follows:

Step 1: Solve the multi-objective availability optimization problem as a single objective problem using one objective at a time and ignoring the other objectives.

Step 2: From the solutions of above step, determine the corresponding values for each objective at every solution. Then find the minimum value(m_t) and maximum value (M_t) of each objective. $m_{t} = \min Z_{t} (x_{q}^{*}), M_{t} = \max Z_{t} (x_{q}^{*})$ q = 1, 2,…….., b.

Step 3: Both the linear and non-linear membership function for each objective function is considered here. The corresponding membership function of each objective are defined as follows:

Case I: Linear Membership Function:

$μ_{Z_{t}} (x) = {\begin{matrix} 1 & Z_{i} (x) \leq m_{i} \\ \frac{M_{i} - Z_{i} (x)}{M_{i} - m_{i}} & m_{i} \leq Z_{i} (x) \leq M_{i} \\ 0 & Z_{i} (x) \geq M_{i} \end{matrix}$ (10)

Case II: Non-linear membership function:

Herein quadratic function is considered for the non-linear membership function and is defined as follows:

$μ_{Z_{t}} (x) = {\begin{matrix} 1 & Z_{i} (x) \leq m_{i} \\ (\frac{M_{i} - Z_{i} (x)}{M_{i} - m_{i}})^{2} & m_{i} \leq Z_{i} (x) \leq M_{i} \\ 0 & Z_{i} (x) \geq M_{i} \end{matrix}$ (11)

Step 4: Equivalent Crisp Optimization Problem:

After formulating the membership functions for each of the objectives the original fuzzy problem can be converted as a equivalent crisp model in the form:

$\begin{matrix} Max ψ \\ Subject to; \\ μ_{z_{i}} (x) \geq ψ . \\ ψ \geq 0 \end{matrix}$ (12)

and the other constraints.

The obtained crisp optimization problem is solved by the proposed attraction based particle swarm optimization (APSO).

5 Overview of PSO [17]

Particle swarm optimization is a nature inspired meta-heuristic algorithm which was first proposed by Kennedy and Eberhart [9]. It is one of the most successful stochastic global optimization technique inspired by social behaviour of fish schooling or bird flocking. Compared to other optimization algorithms, the most important advantage of PSO is that it is easy to implement. In PSO the co-ordinates of each particle represent a probable solution called particles associated with position and velocity. Initially a population of particles randomly generated in the search space. The position and velocity of the i^th particle are denoted by n-dimensional vector x_i = (x_i1, x_i2,….., x_in) and v_i = (v_i1, v_i2,….., v_in) respectively. The particle moves towards the global optimum updating its current position and velocity. During movement,every particle modifies its present position according to its own experience of neighbouring particles, using the best position encountered by itself and its neighbours. The former one is personal best(Pbest,p^j) and the later one is global best (gbest,g^j). The velocity and position of the particle at k^th iteration are updated as follows:

$\begin{matrix} v_{k + 1}^{i} = w . v_{k}^{i} + c_{1} . ud . (p_{k}^{i} - x_{k}^{i}) + c_{2} . Ud . (p_{k}^{g} - x_{k}^{i}) \\ x_{k + 1}^{i} = x_{k}^{i} + v_{k + 1}^{i} \end{matrix}$ (13)

where w is the inertia weight,

i = 1, 2,…., N denotes the number of particles in the population,

k = 1, 2,….., k_max denotes the iterations,

c₁ and c₂, the positive constants are cognitive and social components respectively which are responsible for varying the particle velocity towards pbest and gbest respectively. c₁ + c₂ is generally limited to 4 and Variables ud and Ud are two random number in [0, 1].

5.1 Proposed attraction based PSO (APSO)

In PSO algorithm, during the movement of particles in search space, velocity is updated by the Equation (13). Here c₁ and c₂ are the important components called cognitive and social component respectively. In literature most of the researchers consider these components as a constant value. But in our point of view,these factors are affected by the attractiveness towards the leading personality. So in this paper an attraction factor is proposed which affects c₁ and c₂. The attraction factor for each particle depend on the distance between the position of the particle and the position of the global best(leading personality). This proposed attraction factor is denoted by α = α₀ exp ^-γr². Where α is the attraction towards the global best particle, α₀ is the attraction of the individual particle at r=0 where r is the distance between the position of an individual particle and position of an global best particle in each iteration, and γ is a fixed confidence declination coefficient based on confidence on global best. There exist two limiting cases when γ is small and large. When γ → 0, the attractiveness become α₀. That means all particles have equal attraction towards the global best. When γ tends to very large value then the attractiveness decreases remarkably. This means all particles move nearly casually.

In each iteration the distance between the point of individual particle and the global best is given by the equation: $r = | | p_{k}^{g} - x_{k}^{i} | | = \sqrt{\sum_{k = 1}^{maxit} (p_{k}^{g} - x_{k}^{i})^{2}},$

where $p_{k}^{g}$ is the position of global best in iteration k and $x_{k}^{i}$ is the position of the i^th individual in iteration k. Here α₀ is considered as 1 which is the attraction at r=0 and the confidence declination coefficient γ is also taken as 1. We consider that this proposed attractiveness factor has a social impact. So we subtract this factor from c₁ (cognitive component) and add this with c₂ (socialcomponent.)

c₁= Initial consideration of c₁ (cognitive component) - attractiveness factor (α)

c₂= Initial consideration of c₂ (social component) + attractiveness factor (α)

In the early stage, a large c₁ and small c₂ allow the particles to move around the whole search space instead of moving towards the population best. In later stage, a small c₁ and a large c₂ allow the particles to converge into the global best. So the proposed velocity equation becomes:

$v_{k + 1}^{i} = w . v_{k}^{i}$ +(Initial consideration of c₁-α) $. ud . (p_{k}^{i} - x_{k}^{i})$ +(Initial consideration of c₂+α). Ud. $(p_{k}^{g} - x_{k}^{i})$ .

Initially c₁ and c₂ is considered as c₁ = c₂ = 2 which changes in each iteration based on attractiveness factor.

Basic steps of proposed APSO algorithm:

Set parameters of APSO except cognitive component and social component.

Initialize population of particles with position and velocity.

Evaluate initial fitness of each particle and select pbest and gbest.

set iteration count k = 1.

Update velocity and position of each particle considering attraction factor.

Evaluate fitness of each particle and update pbest and gbest.

Process continues Until requirements are met.

Constraint handling strategy:

There are four different strategies in meta-heuristic for constraint handling. The strategies are modify, reject, repair and penalty function. Between these strategies, penalty function technique is the most useful strategy. This strategy accepts infeasible solutions by assigning them a penalty function.

In this work a penalization technique is considered. Here 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 considered:

$f = {\begin{matrix} objective function (R_{s}) & if x is a feasible solution; \\ R_{s} + penalty \times sum (c) & ; otherwise . \end{matrix}$

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

At first make all the constraints as ≤ type. Then:

for i = 1 : number of constraints,

If constraint function (i) > 0; $\begin{matrix} c (i) = 1; \\ else c (i) = 0 \end{matrix}$

6 Numerical Example

This portion of the paper includes a numerical illustration whose information is taken from [15]. Here, the system considers five subsystems where component of the system are repairable. Maximum permissible volume and weight for the system are 190 (unit of volume) and 150 (unit of weight) respectively. Other details about the system are presented in Table 1.

Table 1
Parameters of the problem

Subsystem w _i $w_{i} v_{i}^{2}$ a _i b _i p _i q _i L _{λ
_i} U _{λ
_i} L _{μ
_i} U _{μ
_i} n _min n _max V W

1 5 4 0.040 0.04 -0.32 0.34 0.0004 0.002 0.20 0.340 1 5 190 150

2 5 1 0.20 0.02 -0.16 0.17 0.0005 0.002 0.35 0.595 1 5

3 6 2 0.50 0.10 -0.80 0.85 0.0004 0.002 0.40 0.680 1 5

4 7 2 0.80 0.08 -0.64 0.68 0.0005 0.002 0.45 0.765 1 5

5 7 3 0.12 0.12 -0.96 1.02 0.0003 0.002 0.35 0.595 1 5

Subsystem	w _i	$w_{i} v_{i}^{2}$	a _i	b _i	p _i	q _i	L _{λ _i}	U _{λ _i}	L _{μ _i}	U _{μ _i}	n _min	n _max	V	W
1	5	4	0.040	0.04	-0.32	0.34	0.0004	0.002	0.20	0.340	1	5	190	150
2	5	1	0.20	0.02	-0.16	0.17	0.0005	0.002	0.35	0.595	1	5
3	6	2	0.50	0.10	-0.80	0.85	0.0004	0.002	0.40	0.680	1	5
4	7	2	0.80	0.08	-0.64	0.68	0.0005	0.002	0.45	0.765	1	5
5	7	3	0.12	0.12	-0.96	1.02	0.0003	0.002	0.35	0.595	1	5

7 Computational results

7.1 Parameter settings

To solve this problem,here an attraction based particle swarm optimization is proposed. The optimization has been performed by MATLAB software and the program run on a computer with 2 GB of RAM. During the overall evaluation process the integer variable n_i are treated as real variable and finally in evaluating the objective functions, the real variable are converted to the nearest integer values. To show the efficiency of the proposed APSO, the problem has also been solved by ordinary PSO. For both the algorithms the selected population size,maximum number of iterations and inertia weight are considered as 100,100 and 0.6 respectively. The particles position and velocity are initiated by the linear expressions [initial position=lower bound of the variable+uniform random number (0,1)×(upper bound of the variable – lower bound of the variable)] and [initial velocity=0.1 × initial position] respectively for both the algorithm. The parameters c₁ and c₂, the acceleration coefficient are taken as c₁ = c₂ = 2 for ordinary PSO and for APSO as c₁ = 2 - α(attraction factor), c₂ = 2 + α (attraction factor). The termination criteria has been set either maximum number of generations or to be order of relative error equal to 10^-12 whichever is achieved first. The program has been run 10 times and the best values are chosen. For handling the constraints,in this article 100 is considered as a penalize value.

7.2 Results

In order to solve the numerical example,initially the availability allocation problem is formulated as a fuzzy multi-objective availability allocation problem as given in Equation (8). In order to take the failure rate and repair rate of the problem as triangular fuzzy number, the problem is converted to problem 9 for different values of I=0,0.00001,0.00003. Basically I= 0 corresponds to the parameter in which component’s failure rate and repair rate is in the form of crisp number rather than the triangular number. For solving the above formulated problem, different ideal values are obtained considering one objective at a time and ignoring all the others. Based on these ideal values,linear and non-linear membership functions are constructed corresponding to each of objective through Equations (10, 11). Finally a crisp optimization model is reformulated using these membership functions as given in Equation (12) and ordinary PSO, APSO have been used with the parameter settings given in Section 7 to solve this problem. The results of the numerical example for different values of I=0,0.00001,0.00003 are shown in Tables 2 and 3 corresponding to linear and non-linear membership.

Table 2
Optimal results corresponding to linear membership

I = 0 I = 0.00001 I = 0.00003

PSO APSO PSO APSO PSO APSO

N (3,2,1,1,1) (5,2,1,1,1) (3,2,1,1,1) (3,2,1,1,1) (3,2,1,1,1) (5,2,1,1,1)

λ ₁ 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020

λ ₂ 0.0005 0.0005 0.0005 0.0005 0.0005 0.0005

λ ₃ 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020

λ ₄ 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020

λ ₅ 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020

μ ₁ 0.2 0.2779 0.34 0.3399 0.2 0.34

μ ₂ 0.5950 0.5950 0.595 0.3500 0.35 0.595

μ ₃ 0.68 0.68 0.68 0.6797 0.68 0.68

μ ₄ 0.7650 0.7572 0.765 0.7650 0.765 0.765

μ ₅ 0.5919 0.5950 0.5903 0.5950 0.595 0.595

A _L 0.9991 0.9992 0.9991 0.9992 0.9992 0.9992

A _C 0.9991 0.9992 0.9991 0.9992 0.9992 0.9992

A _R 0.9991 0.9992 0.9991 0.9992 0.9991 0.9991

C _L 230.2176 231.0115 229.5810 230.9236 232.1877 232.1708

C _C 230.2176 231.0115 228.6632 229.9958 229.3942 229.3757

C _R 230.2176 231.0115 227.7536 229.0765 226.6767 226.6566

	I = 0	I = 0.00001	I = 0.00003
N	(3,2,1,1,1)	(5,2,1,1,1)	(3,2,1,1,1)	(3,2,1,1,1)	(3,2,1,1,1)	(5,2,1,1,1)
λ ₁	0.0020	0.0020	0.0020	0.0020	0.0020	0.0020
λ ₂	0.0005	0.0005	0.0005	0.0005	0.0005	0.0005
λ ₃	0.0020	0.0020	0.0020	0.0020	0.0020	0.0020
λ ₄	0.0020	0.0020	0.0020	0.0020	0.0020	0.0020
λ ₅	0.0020	0.0020	0.0020	0.0020	0.0020	0.0020
μ ₁	0.2	0.2779	0.34	0.3399	0.2	0.34
μ ₂	0.5950	0.5950	0.595	0.3500	0.35	0.595
μ ₃	0.68	0.68	0.68	0.6797	0.68	0.68
μ ₄	0.7650	0.7572	0.765	0.7650	0.765	0.765
μ ₅	0.5919	0.5950	0.5903	0.5950	0.595	0.595
A _L	0.9991	0.9992	0.9991	0.9992	0.9992	0.9992
A _C	0.9991	0.9992	0.9991	0.9992	0.9992	0.9992
A _R	0.9991	0.9992	0.9991	0.9992	0.9991	0.9991
C _L	230.2176	231.0115	229.5810	230.9236	232.1877	232.1708
C _C	230.2176	231.0115	228.6632	229.9958	229.3942	229.3757
C _R	230.2176	231.0115	227.7536	229.0765	226.6767	226.6566

Table 3

Optimal results corresponding to Non-linear membership

	I = 0		I = 0.00001		I = 0.00003
	PSO	APSO	PSO	APSO	PSO	APSO
N	(3,2,1,1,1)	(5,2,1,1,1)	(5,2,1,1,1)	(3,2,1,1,1)	(4,2,1,1,1)	(3,2,1,1,1)
λ ₁	0.0020	0.0020	0.0020	0.0020	0.0020	0.0020
λ ₂	0.0005	0.0005	0.0005	0.0005	0.0005	0.0005
λ ₃	0.0020	0.0020	0.0020	0.0020	0.0020	0.0020
λ ₄	0.0020	0.0020	0.0020	0.0020	0.0020	0.0020
λ ₅	0.0020	0.0020	0.0020	0.0020	0.0020	0.0020
μ ₁	0.2	0.2779	0.34	0.3399	0.2	0.34
μ ₂	0.5950	0.5950	0.595	0.3500	0.35	0.595
μ ₃	0.68	0.68	0.68	0.6797	0.68	0.68
μ ₄	0.7650	0.7572	0.765	0.7650	0.765	0.765
μ ₅	0.5919	0.5950	0.5903	0.5950	0.595	0.595
A _L	0.9991	0.9992	0.9991	0.9992	0.9992	0.9992
A _C	0.9991	0.9992	0.9991	0.9992	0.9992	0.9992
A _R	0.9991	0.9992	0.9991	0.9992	0.9991	0.9991
C _L	228.9758	230.8995	229.6756	231.0259	232.3374	231.6679
C _C	228.9758	230.8995	228.7569	230.1011	229.5398	228.8775
C _R	228.9758	230.8995	227.8467	229.1848	226.8182	226.1629

7.2.1 Comparison and statistical analysis

Based on the numerical evidence, proposed APSO is compared with ordinary PSO approaches. The parameter settings are same for both the algorithms except for the cognitive and social components. Twenty simulations are performed by setting the parameters for each algorithm. Considering the obtain results, a statistical analysis namely correlation analysis is performed for each algorithm between availability and cost for I=0,0.00001,0.00003 corresponding to linear as well as non-linear membership.

From the Tables 4 and 5, it can be conclude that the obtain results are highly negatively correlated for APSO than the ordinary PSO. Hence, the proposed APSO outperforms the other approach and statistically more consistent than ordinary PSO algorithm.

Table 4
Correlation between Availability and Cost corresponding to linear membership

I = 0 I = 0.00001 I = 0.00003

PSO APSO PSO APSO PSO APSO

Correlation _Left -0.93922 -0.99693 -0.65845 -0.99267 -0.76755 -0.98509

Correlation _Center -0.93922 -0.99693 -0.71753 -0.99325 -0.60497 -0.96513

Correlation _Right -0.93922 -0.99693 -0.81227 -0.99354 -0.5877 -0.97157

	I = 0	I = 0.00001	I = 0.00003
Correlation _Left	-0.93922	-0.99693	-0.65845	-0.99267	-0.76755	-0.98509
Correlation _Center	-0.93922	-0.99693	-0.71753	-0.99325	-0.60497	-0.96513
Correlation _Right	-0.93922	-0.99693	-0.81227	-0.99354	-0.5877	-0.97157

Table 5

Correlation between Availability and Cost corresponding to Non-linear membership

	I = 0		I = 0.00001		I = 0.00003
	PSO	APSO	PSO	APSO	PSO	APSO
Correlation _Left	-0.95362	-0.99387	-0.47896	-0.97292	-0.95987	-0.99466
Correlation _Center	-0.95362	-0.99387	-0.4795	-0.97816	-0.944	-0.99689
Correlation _Right	-0.95362	-0.99387	-0.57652	-0.97888	-0.9727	-0.99724

8 Conclusion

Reliability and availability are two important attributes in many complex systems especially mechanical and electrical systems. So a special attention is required to these features to provide a secure and reliable system. In a large portion of the past work in this area, it is constantly viewed as that systems are in deterministic environment. But this is the first time when multi-objective availability allocation problem is considered in uncertain environment. On the base of this consideration, a new mathematical model is presented and solved by proposed attraction based particle swarm optimization (APSO). The results produced by APSO shows the acceptable and appropriate availability of the system. To show the supremacy of the proposed APSO, a comparison is also made with an ordinary PSO algorithm. Finally a statistical analysis displays that the propose APSO is more consistent than ordinary PSO reported in the literature. This type of approach assists a decision maker to decide his future plan to obtain optimum performance of the system.

For future studies, one can consider the model in such a way that each component’s failure and repair rate follows any other distribution instead of exponential distribution. In this study, confidence declination coefficient is considered as a fixed value. This value might be tuned with chaotic maps instead of constant value to consider the ergodic, irregular and non-periodic behaviour of the particles. The study on comparison of the results obtained using other meta-heuristic techniques is also an important future research assignment.

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