Reliability stochastic optimization for a series system with interval component reliability via genetic algorithm

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Abstract

This paper deals with chance constraints based reliability stochastic optimization problem in the series system. This problem can be formulated as a nonlinear integer programming problem of maximizing the overall system reliability under chance constraints due to resources. The assumption of traditional reliability optimization problem is that the reliability of a component is known as a fixed quantity which lies in the open interval (0, 1). However, in real life situations, the reliability of an individual component may vary due to some realistic factors and it is sensible to treat this as a positive imprecise number and this imprecise number is represented by an interval valued number. In this work, we have formulated the reliability optimization problem as a chance constraints based reliability stochastic optimization problem with interval valued reliabilities of components. Then, the chance constraints of the problem are converted into the equivalent deterministic form. The transformed problem has been formulated as an unconstrained integer programming problem with interval coefficients by Big-M penalty technique. Then to solve this problem, we have developed a real coded genetic algorithm (GA) for integer variables with tournament selection, uniform crossover and one-neighborhood mutation. To illustrate the model two numerical examples have been solved by our developed GA. Finally to study the stability of our developed GA with respect to the different GA parameters, sensitivity analyses have been done graphically.

Introduction

In most of the probabilistic methods of the reliability engineering system, it is assumed that all the probabilities are precise. This means that every probability involved is perfectly determinable. In this case it is usually assumed that there exist some complete probabilistic information about system and component behavior. For the completeness of probabilistic information the following two conditions must be satisfied.

  • 1.

    All the probabilities or probability distributions are known or perfectly determinable.

  • 2.

    The system components are independent, i.e., all the random variables, described the component reliability behavior, are independent.

During the past, the assumption of uncertainty in most of the methods in reliability is based on precise probabilities and the reliabilities of the system components are to be known at a fixed positive number which lies in the open interval (0, 1). The precise system reliability can be computed theoretically if both the above two conditions are satisfied (it is assumed that the system structure is defined precisely and there exists a function linking the system time to failure as well as the times to failure of the components). If at least one condition is violated, then only the interval measure of reliability (Barlow and Proschan [1], [2], Coolen and Newby [3], Lindqvist and Langseth [4]) can be obtained. However, in real life situations, there are not sufficient statistical data in most of the cases where the system is new or exists only as a project. It is not always possible to observe the stability from the statistical point of view if such data exists. This means that only some partial information about the system components are known. So the reliability of a component of a system will be an imprecise number which can be represented by an interval number [5] and is calculated by imprecise probabilities (Gurov and Utkin [6], Utkin and Gurov [7], [8]) define reliability as the probability of survival that a system will perform satisfactorily at least up to a given period of time under stated conditions. Designing a highly reliable system, there arises a question as to how to get a balance between the reliability of a system and resources such as cost, volume and weight. As a result, inclusion of redundant components or the increase of the components’ reliability leads to increase in the system reliability.

In the last two decades, a number of techniques have been proposed for solving reliability optimization problems [9], [10], [11]. These techniques can be classified as dynamic programming method, branch and bound method, Lagrange multiplier method, etc (Kuo et al. [12], [13], [14]). Recently, Zhao et al. [15] developed stochastic programming technique for redundancy allocation problems. Stochastic reliability optimization problem is either an extension or reformulation of reliability optimization problem with random variations of parameters. Moreover, the resource elements vary and it is reasonable to regard them as stochastic variables. It is also proved that a stochastic programming problem is harder than all other combinatorial optimization problems.

In this paper, we have solved chance constrained [16] reliability optimization problem with interval valued values of component reliabilities. Here, various types of randomness have been discussed with known probability distributions, viz. uniform, normal and log normal distributions, when the resource variables are random. The corresponding chance constrained redundancy allocation problem for the series system has been solved with the help of GA. As the objective function of the redundancy allocation problem is interval valued, to solve this type of problem by GA, order relations between interval numbers are essential. In this paper, we have used the definition of Mahato and Bhunia [17] for order relations between interval numbers.

Section snippets

Interval arithmetic

An interval number A is a closed interval defined by A=[aL,aR]={x:aLxaR,xR}, where aL and aR are the left and right limits, respectively and R is the set of all real numbers. An interval A can also be expressed in terms of centre and radius as A=aC,aW={x:aC-aWxaC+aW,xR}, where aC and aW are respectively, the centre and radius of the interval A, i.e., aC=(aL+aR)/2 and aW=(aR-aL)/2. Actually, each real number can be regarded as an interval number, such as for all xR, x can be written as

Mean and variance of interval numbers

Let Xi=[xiL,xiR],i=1,2,,n be the ith interval numbers. Then mean, variance and standard deviation of Xi=[xiL,xiR],i=1,2,,n are given byX¯=1ni=1nxiL,1ni=1nxiR,Var(X)=1ni=1nxiL-1ni=1nxiR,xiR-1ni=1nxiL2,σX=Var(X)=1ni=1nxiL-1ni=1nxiR,xiR-1ni=1nxiL2.

Order relations of interval numbers

Let A and B be two intervals. These two intervals A and B may be one of the following three types.

  • Type I: A and B are disjoint.

  • Type II: A and B are partially overlapping.

  • Type III: either AB or BA.

Here, we shall consider the definitions of order relations developed by Mahato and Bhunia [17].

There are two types of decision-making, viz. optimistic and pessimistic decision-making. For optimistic decision-making, decision maker selects the best alternative ignoring the uncertainty whereas for

Assumptions

The following assumptions have been used in the entire paper.

  • 1.

    Reliability of each component is imprecise and interval valued.

  • 2.

    If a component of any subsystem fails to function, the entire system will not be damaged or fail.

  • 3.

    All redundancy is active redundancy with out repair.

  • 4.

    The state of components and system has only two states like operating state or failure state.

  • 5.

    The resource constraints are chance constraints with resource vector as stochastic in nature.

Problem formulation

Let us consider a system consisting of n subsystems in series in which the jth (1jn) subsystem consists of xj components in parallel. Such a system is called parallel-series system or n-stage series system (Ref. Fig. 1). Assuming all the components in the jth subsystem as identical, the system reliability RS is given byRS(x)=j=1n[RSL(x),RSR(x)],where RSL(x)=1-(1-rjL)xj and RSR(x)=1-(1-rjR)xj.

The chance constrained optimization problem for a parallel-series system with m chance constraints

Genetic algorithm based constraints handling approach

In the application of GA for solving the reliability optimization problems, there arises an important question: how the algorithm handles the constraints relating to the problem. During the last few decades, several techniques have been proposed to handle the constraints in evolutionary Algorithms ([20], [21], [22], [23]). These methods can be classified into several types, viz. penalty-based method, methods that preserve the feasibility of solutions, methods that clearly distinguish between

Numerical examples

To illustrate our proposed GA based on Big-M penalty technique for solving the reliability stochastic optimization problem with interval valued as well as fixed valued reliabilities of components, we have considered two numerical examples. Each example has been formulated using Case 1. In the first example, the reliability of components are interval valued whereas the second one (taken from Yadavalli et al. [24]), the reliabilities of components are fixed. The proposed GA is coded in C

Sensitivity analysis

To study the performance of our developed GA based on Big-M penalty technique, sensitivity analyses have been done graphically on the interval valued system reliability with respect to GA parameters separately keeping the other parameters at their original values. These are shown in Fig. 2, Fig. 3, Fig. 4, Fig. 5. The graphs (Fig. 2, Fig. 3, Fig. 4, Fig. 5) have been drawn for lower and upper bounds of the system reliability in the same graph. In Fig. 2, the effect of population size (p_size)

Conclusions

In this paper, for the first time chance constrained reliability stochastic optimization problem in the series system with some resource constraints have been solved considering the reliability of each component as an interval number. The interval number representation is more appropriate among other representations like, random variable representation with known probability distribution, fuzzy set with known fuzzy membership function or fuzzy number. For handling the resource constraints, the

Acknowledgements

The first author would like to acknowledge the support of CSIR (Council of Scientific and Industrial Research), India, for conducting this work. Also the authors are thankful to the Editor and anonymous referee for their helpful comments and suggestions to improve the paper.

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