Reliability Analysis, Safety Assessment and Optimization. Enrico Zio
rate from state i to state j at time t in Markov process
Notation: Part III
| ri | reliability of subsystem i |
| x=(x1,…,xn)T | decision variable vector |
| c=(c1,…,cn)T | coefficients of the objective function |
| b=(b1,…,bm)T | right-hand side values of the inequality constraints |
| z=(z1,z2,…,zM) | objective vector |
| xl*, l=1,2,…,L | set of optimal solutions |
| w=(w1,w2,…,wM) | weighting vector |
| x* | global optimal solution |
| R(⋅) | system reliability function |
| A(⋅) | system availability function |
| M(⋅) | system maintainability function |
| S(⋅) | system safety function |
| C(⋅) | cost function |
| Risk(⋅) | system risk function |
| RN | N-dimensional solution space |
| fi | i-th objective functions |
| gj | j-th equality constraints |
| hk | k-th inequality constraints |
| ω | random event |
| ξ=(q(ω)T, h(ω)T, T(ω)T) | second-stage problem parameters |
| W | recourse matrix |
| y(ω) | second-stage or corrective actions |
| Q(x) | expected recourse function |
| U | uncertainty set |
| u | uncertainty parameters |
| ζ | perturbation vector |
| Z | perturbation set |
| xu* | optimal solution under the uncertainty parameter u |
1 Reliability Assessment
Reliability is a critical attribute for the modern technological components and systems. Uncertainty exists on the failure occurrence of a component or system, and proper mathematical methods are developed and applied to quantify such uncertainty. The ultimate goal of reliability engineering is to quantitatively assess the probability of failure of the target component or system [1]. In general, reliability assessment can be carried out by both parametric or nonparametric techniques. This chapter offers a basic introduction to the related definitions, models and computation methods for reliability assessments.
1.1 Definitions of Reliability
According to the standard ISO 8402, reliability is the ability of an item to perform a required function, under given environmental and operational conditions and for a stated period of time without failure. The term “item” refers to either a component or a system. Under different circumstances, the definition of reliability can be interpreted in two different ways:
1.1.1 Probability of Survival
Reliability of an item can be defined as the complement to its probability of failure, which can be estimated statistically on the basis of the number of failed items in a sample. Suppose that the sample size of the item being tested or monitored is n0. All items in the sample are identical, and subjected to the same environmental and operational conditions. The number of failed items is nf and the number of the survived ones is ns, which satisfies
The percentage of the failed items in the tested sample is taken as an estimate of the unreliability,
Complementarily, the estimate of the reliability, R^(t), of the item is given by the percentage of survived components in the sample:
Example 1.1
A valve fabrication plant has an average output of 2,000 parts per day. Five hundred valves are tested during a reliability test. The reliability test is held monthly. During the past three years, 3,000 valves have failed during the reliability test. What is the reliability of the valve produced in this plant according to the test conducted?
Solution
The total number of valves tested in the past three years is
The number of failed components is
According to Equation 1.3, an estimate of the valve reliability is
1.1.2 Probability of Time to Failure
Let random variable T denote the time to failure. Then, the reliability function at time t can be expressed as the probability that the component does not fail at time t, that is,
Denote the cumulative distribution function (cdf) of T as F(t). The relationship between the cdf and the reliability is
Further, denote the probability density function (pdf) of failure time T as f(t). Then, equation (1.5) can be rewritten as