A study of interval analysis for cold-standby system reliability optimization under parameter uncertainty

https://doi.org/10.1016/j.cie.2016.04.017Get rights and content

Highlights

  • We use interval analysis theory to deal with the uncertain parameters.

  • We incorporate interval analysis into cold-standby system optimization problem.

  • A new interval order relation reflecting decision maker’s preference is defined.

  • We propose the evaluation procedure of interval-valued system reliability and cost.

  • A genetic algorithm involving dual mutation and random keys technique is developed.

Abstract

This paper presents a study of interval analysis for solving cold-standby system reliability optimization problems with considering parameter uncertainty. Most works reported in existing literature have been based on the assumption that the probabilistic properties and statistical parameters have a known functional form, which is usually not the case. Very often the parameters are presented in form of an interval-valued number or bounds/tolerance from the engineering design. In this paper, interval analysis is used to incorporate this in the system optimization problems. A definition of interval order relation reflecting decision makers’ preference is proposed for comparing interval numbers. A computational algorithm is developed to evaluate the system reliability and expected mission cost, in which a discrete approximation approach and a technique of interval universal generating function are used. For illustration, an application to sequencing optimization for heterogeneous cold-standby system is given; a modified genetic algorithm is developed to solve the proposed optimization problem with interval-valued objective. The results indicate that the interval analysis exhibits a good performance for dealing with parameter uncertainty of cold-standby system optimization problems.

Introduction

The development of industrial technology involves an increasing amount of design of complex and interrelated systems. Reliability is an important performance measure of industrial systems, especially when it is of safety–critical concerns. Extensive research has been carried out on system reliability optimization; survey papers (Kuo and Prasad, 2000, Kuo and Wan, 2007, Tillman et al., 1977) have summarized many earlier studies on reliability optimal problems.

In the existing literature of system reliability optimization, most results are based on assumptions that the probabilistic properties or parameters of time-to-failure are deterministic. However, due to observation difficulties, resource limits and system complexity, uncertainties are usually unavoidable while modeling real industrial systems. For many engineering problems, it is overly difficult or costly to collect sufficient data about the uncertainties, especially at the very beginning of design processes. Stakeholders and decision makers have to deal with a variety of uncertainty issues when making decisions without sufficient information. In fact, many parameters are specified as intervals of some kind in engineering design.

Bayesian approach (Howson & Urbach, 2006) could be used to study the uncertainties associated with the estimation of parameters of a probability distribution (Pasanisi et al., 2012, Srivastava and Deb, 2013, Troffaes et al., 2014). The unknown parameters are assumed to be random variables. With the Bayesian approach, subjective judgments are required to estimate the Bayesian random variables. The estimation of the Bayesian random variables can be improved when more data become available. Before receiving more data, however, the Bayesian approach remains a subjective representation of uncertainty. Fuzzy theory is another commonly used method for analysing uncertainty issues (Dotoli et al., 2015, Hanss and Turrin, 2010, Wang and Watada, 2009). In the fuzzy approach, the imprecise parameters are represented as fuzzy numbers. However, the fuzzy sets and their membership functions are required to be known. It is a formidable task for decision makers to specify the appropriate membership functions in advance.

In order to overcome the drawbacks of probabilistic methods and fuzzy approaches, interval analysis first developed by Moore (1966) has recently received some attention. The interval analysis has been used to deal with problems of uncertainty in diverse fields, such as circuit analysis (Kolev, 1993), damage identification (Wang, Yang, Wang, & Qiu, 2012), structure safety analysis (Impollonia and Muscolino, 2011, Wang et al., 2014, Zhang et al., 2013), electric power system (Pereira & Da Costa, 2014), and so on. In these studies, interval variables were used to quantitatively describe the uncertain parameters in the face of limited information. Up to now, research on interval uncertainty problems has concentrated mainly in the aforementioned fields, while the application of interval analysis to system reliability optimization for complex industrial systems is relatively new.

In the existing literature of system reliability optimization, Feizollahi and Modarres (2012) suggested a robust deviation framework to deal with uncertain component reliabilities in constrained redundancy allocation problems; they addressed uncertainty by assuming that the component reliabilities belong to interval uncertainty sets. However, the interval numbers are not incorporated directly. Gupta et al., 2009, Bhunia et al., 2010 dealt with optimization problems for series systems; the reliability of each component was represented as an interval number. Sahoo, Bhunia, and Kapur (2012) studied the constrained multi-objective reliability optimization problem of systems with interval-valued component reliabilities. However, these studies dealt specifically with the series or series–parallel systems with active redundancy and given interval-valued component reliability, or placed greater attention on optimization algorithms.

In this paper, we present a study of interval analysis for cold-standby system optimization problems considering uncertain probabilistic parameters. Our study focuses on the evaluation and optimization of system reliabilities and expected mission costs. A discrete approximation approach based on Levitin et al., 2013a, Levitin et al., 2013b, Levitin et al., 2013c and the interval universal generating function (IUGF) technique (Li, Chen, Yi, & Tao, 2011) are used in the evaluation procedures for estimating the system reliability and the expected mission cost. IUGF is a technique which extends the universal generating function (UGF) (Levitin, 2005) for the situations with interval-valued parameters.

In solving the optimization problem with interval-valued objective, a set of interval values appear during the selection of the best alternative, which leads to a question related to the comparison of two arbitrary interval numbers. In this paper, we define a new order relation for two arbitrary interval numbers considering different levels of decision maker’s preference. The level of decision maker’s preference is measured by the ratio ρ0, where ρ0=1 stands for neutrality; ρ0>1 stands for optimistic preference; otherwise pessimistic.

For purposes of illustration, we propose the application of interval analysis theory to the sequencing optimization problem for heterogeneous cold-standby systems (Levitin et al., 2013a). In this paper, we model the parameters of component time-to-failure distributions as interval-valued numbers. A genetic algorithm (GA) is developed to solve the proposed sequencing optimization problem. In order to avoid premature convergence and to increase computational efficiency, dual mutation (Wang, Ma, & Wang, 2008) and the random keys technique (Bean, 1994) are introduced in the GA.

The rest of this paper is organized as follows. Section 2 provides some basics of interval analysis, gives the definitions of interval order relation and presents a general formulation of cold-standby system optimization problem with interval-valued objective functions. Section 3 proposes the computation procedure of the system reliability and the expected mission cost. Section 4 shows a study of the interval analysis for solving sequencing optimization problem for heterogeneous cold-standby systems, and a GA-based searching approach is developed. A numerical example is given in Section 5. Finally, conclusions are presented in Section 6.

Section snippets

Interval arithmetic

Interval arithmetic was introduced by Moore in its modern form as an extension of real arithmetic (Moore, 1979, Moore et al., 2009). In interval arithmetic, an uncertain variable a is represented as an interval number [a]=[a̲,a¯] with a̲aa¯, where a̲ and a¯ are the lower and upper bounds of a, respectively. If a̲=a¯, then a is a real number. The basic arithmetical operations of interval variables [a]=[a̲,a¯] and [b]=b̲,b¯ (a̲0,b̲0) are defined as[a]+[b]=a̲+b̲,a¯+b¯,[a]-[b]=a̲-b¯,a¯-b̲,[a]·[b

Cold-standby redundancy

Consider a cold-standby system that consists of N statistically independent components in parallel. The components can be similar or dissimilar with equivalent functionality. Let Fjt;η1,j,η2,j, be the cumulative distribution function (cdf) of the time-to-failure Tj of component j, where the parameters η1,j,η2,j, are uncertain but known as interval-valued numbers. The same situation suits the probability density function (pdf) fjt;η1,j,η2,j,, if applicable.

In the cold-standby system, one of

Problem formulation

In order to illustrate the proposed interval analysis method, a cold-standby sequencing optimization problem is studied in this section. Since the components are activated one by one in a cold-standby system, the system reliability equals to the probability that the sum of all components’ working time reaches mission time. When the set of components is fixed, the system reliability does not depend on initiation sequence of components. However, if the system consists of dissimilar components,

Numerical example of cold-standby sequencing optimization

The example is adapted from Levitin et al., 2013a, Levitin et al., 2013b, Levitin et al., 2013c, the probabilistic parameters of the system components are known as interval numbers. Ten components with Weibull time-to-failure distribution (F(t)=1-exp-t/η1η2,t0) are given in this example, and the parameters of distribution and cost are listed in Table 1. The GA parameters are set as: pop_size=20,max_gen=30,pc=0.95,pm1=0.3 and pm2=0.45. To run the GA by MATLAB, a computer with Intel Core i5-4590

Conclusions

A study of interval analysis has been carried out in this paper to better deal with parameter uncertainties of cold-standby system optimization problems. An order relation considering the decision makers’ preference was defined for comparing interval numbers. A general formulation was presented for cold-standby system optimization problems with considering uncertain-but-bounded probabilistic parameters by using interval arithmetic. Based on a discrete approximation approach of time-to-failure

Acknowledgements

The authors are grateful for the valuable comments and suggestions provided by the editor and anonymous referees. This research was partially supported by the National Natural Science Foundation of China (No. 61374026, No. 71371163) and a grant from University Grants Council of Hong Kong (No. GRF 9042183).

References (43)

  • G. Levitin et al.

    Optimal sequencing of warm standby elements

    Computers & Industrial Engineering

    (2013)
  • G. Levitin et al.

    Cold vs. hot standby mission operation cost minimization for 1-out-of-N systems

    European Journal of Operational Research

    (2014)
  • A. Pasanisi et al.

    Estimation of a quantity of interest in uncertainty analysis: Some help from Bayesian decision theory

    Reliability Engineering & System Safety

    (2012)
  • L.E.S. Pereira et al.

    Interval analysis applied to the maximum loading point of electric power systems considering load data uncertainties

    International Journal of Electrical Power & Energy Systems

    (2014)
  • L. Sahoo et al.

    Genetic algorithm based multi-objective reliability optimization in interval environment

    Computers & Industrial Engineering

    (2012)
  • P. Sevastjanov et al.

    Two-objective method for crisp and fuzzy interval comparison in optimization

    Computers & Operations Research

    (2006)
  • M. Troffaes et al.

    A robust Bayesian approach to modeling epistemic uncertainty in common-cause failure models

    Reliability Engineering & System Safety

    (2014)
  • C. Wang et al.

    Stochastic interval analysis of natural frequency and mode shape of structures with uncertainties

    Journal of Sound and Vibration

    (2014)
  • S. Wang et al.

    Modelling redundancy allocation for a fuzzy random parallel–series system

    Journal of Computational and Applied Mathematics

    (2009)
  • H. Zhang et al.

    Structural reliability analysis on the basis of small samples: An interval quasi-Monte Carlo method

    Mechanical Systems and Signal Processing

    (2013)
  • M.A. Ardakan et al.

    Reliability optimization of series–parallel systems with mixed redundancy strategy in subsystems

    Reliability Engineering & System Safety

    (2014)
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