The paper refers to the evaluation of the unavailability of systems made by repairable binary independent components subjected to aging phenomena. Exponential, exponential-linear, and Weibull distributions are assumed for the components failure times. We assume that components failure rate increases only slightly during the maintenance period, but we recognize the effectiveness of preventive maintenance only in presence of aging phenomena. Importance measures allow the ranking of the input variables. We propose analytical equations that allow the estimation of the first-order Differential Importance Measure (DIM) on the basis of the Birnbaum measures of components, under the hypothesis of uniform percentage changes of parameters. Without further information than that used for the estimation of “DIM for components,” “DIM for parameters” allows considering separately the importance of random failures, aging phenomena, and preventive and corrective maintenance. A two-step process is proposed for the system improvement, by increasing the components reliability and maintainability performance as much as possible (within the applicable technological limits) and then by optimizing preventive maintenance on them. Some examples taken from the scientific literature are solved in order to verify the correctness of the analytical equations and to show their use.
Several studies have demonstrated that components do not contribute to system performance in the same way [
In this paper, we are interested in the unavailability of systems made by repairable binary components, under “perfect,” corrective, and preventive maintenance (i.e., components are “as good as new” after maintenance). We assume the exponential distribution for the components repair times.
Aging phenomena are introduced into the model through time-dependent failure rates, by assuming the Exponential-linear distribution [
Importance measures allow the ranking of the input variables of the model which can be the components unavailability and/or the parameters that define their failure and repair probability distributions, according to the contribution of their changes to the model output (system unavailability). We adopt the first-order Differential Importance Measure (DIM), which provides the fraction of total change in the model output that is due to “small” one at time changes in the input variables [
The estimation of the “DIM for components” only requires the knowledge of the first-order partial derivatives of the system unavailability with respect to the components unavailability. They can be estimated on the basis of the “system function” that defines the status of the system on the basis of the state of components.
The estimation of the “DIM for parameters” provides information about the importance of the random failure, aging phenomena, and corrective and preventive maintenance separately but requires the knowledge of the first-order partial derivatives of the system unavailability with respect to each parameter. The number of partial derivatives increases as well as their complexity because of nonlinear terms.
The paper proposes analytical equations for the estimation of the first-order Differential Importance Measure (DIM) for parameters, on the basis of the same information used for the estimation of DIM for components (i.e., Birnbaum measures). Specifically, proposed equations can be applied to systems made by “independent” components. Independence among components (and among parameters that define their failure and repair probability distributions) allows considering separately the dependence of system Unavailability on components Unavailability and the dependence of each component Unavailability on its parameters. If this assumption is not fulfilled (i.e., in presence of inter-component or functional dependences, e.g., cold spare, share load,…), and the system is described by a homogenous Markov process (this is not the case of components subjected to aging), DIM for Markov models can be used [
In the second paragraph, we provide some general information about systems made by repairable binary components under aging phenomena and preventive maintenance [
We assume an exponential distribution for the failure and repair times of a binary repairable component.
Components are assumed to age according to an exponential-linear distribution for the failure times [
Components aging is mitigated by preventive maintenance actions. We assume that the time interval between two consecutive maintenance actions, which is named maintenance period (
If the components failure rate increases only slightly during the maintenance period, we can refer to a constant “Effective value”; it is estimated by imposing that the failure probabilities for the exponential-linear and the exponential distributions coincide within the following maintenance period:
Components are assumed to age according to a Weibull distribution for the failure times [
The component unavailability comes from (
The unavailability of a system made by binary repairable components
Importance measures were originally introduced by Birnbaum [
“Traditional” importance measures (Birnbaum measure, Criticality measure, Risk Reduction Worth, Risk Achievement Worth measures, and Fussell-Vesely) require new evaluations of the model in order to estimate the importance of a combination or group of components/parameters [
DIM can be referred to the change of the components unavailability or to the change of the parameters that specify their failure and repair probability distributions. In the first case, DIM for the component “
Generally, DIM assumes different values and provides a different ranking of components under the two hypotheses; the uniform percentage changes are the more realistic ones (indeed, failure and repair rates of components differ significantly). Moreover, DIM is an additive measure, DIM for a group of components is the sum of DIM for components (
In the second case, DIM for the parameter “
The estimation of DIM for the component “
Generally, the measures estimated through (
It must be remarked that the opportunity to consider different changes of input variables (e.g., through the computation of DIM for Markov models by the Perturbation Analysis [
We refer to a system made by binary repairable components, and we assume the exponential distribution for their failure and repair times. Components are “as good as new” after corrective maintenance. Input variables are independent (i.e., there is not any relationship among the components unavailability
Because of the homogeneity of measure units of parameters, both hypotheses, uniform changes and uniform percentage changes of parameters, could be used [
By (
DIM for parameters
DIM for the component “
DIM for components and parameters can be estimated through (
This approach aims to consider separately the nonlinearity of the relationship between the unavailability of the system and its components, neglected in the first-order approximation of the
The comparison between (
Aging phenomena are introduced into the unavailability models of repairable components by assuming the exponential-linear or the Weibull distribution for their failure times. In the first case, we consider two different assumptions for the maintenance period. The first “basic” assumption is taken from paper [
Under the same general assumptions, aging phenomena are introduced into the model by assuming the exponential-linear distribution for the components failure times.
A “Basic” assumption is made for the Maintenance period: according to [
Equation (
According to [
In order to estimate the Differential Importance Measure for parameters and components, (
DIM for the component “
DIM for the component “
Then, an “alternative” assumption is made for the maintenance period. According to (
According to (
DIM for the component “ by considering all parameters as unknown (unfixed) ones,
by assuming that
Equivalently, DIM for components can be estimated as the sum of DIM for its (unfixed) parameters, which are computed through the following equations:
DIM for parameters provides information about the Importance of random failure
By (
Equation (
In order to estimate the Differential Importance Measure for parameters and components, (
DIM is evaluated under the hypothesis of uniform percentage changes of parameters (
DIM for the component “ by considering all parameters as unknown (unfixed) ones,
by assuming that by assuming that
Equivalently, DIM for components can be estimated as the sum of DIM for its (unfixed) parameters, which are computed through the following equations:
DIM for parameters provides information about importance of random failure
Having
DIM for parameters can be added over components in order to estimate the importance of each kind of parameter
The above equations are consistent with the ones previously provided for the exponential-linear distribution by assuming that
By introducing
The first-order Differential Importance Measure allows the ranking of the input variables, according to the fraction of total change in the model output (system unavailability), that is, due to their small, one at a time changes. DIM for parameters provides information about the importance of the random failure, aging phenomena, and corrective and preventive maintenance. DIM for components provides information about the importance of the components themselves.
Information about the effect of the parameters changes is provided by the sign of the first-order partial derivatives of the system unavailability, while the sign of DIM depends on the values of the input variables. By assuming the exponential-linear distribution for the components failure times, the increase of
The absolute values of DIM define a ranking of parameters that provides information about the effectiveness of their changes in the reduction of the system unavailability.
The improvement of the system performance (i.e., reduction of the system unavailability) can be supported by the information coming from a two-step process.
The first step aims to identify the “critical” parameters that define the failure and repair probability distributions of components, by assuming the same preventive maintenance “strategy” for all components (i.e., a fixed parameter
The second step aims to support the specification of “optimal” values for the Maintenance period of different components, through a heuristic approach. Specifically, we look for the values of
In order to verify the correctness of the analytical equations proposed in Section
System reliability block diagram.
The first case study refers to repairable components without aging. In the second and third case studies, the exponential-linear distribution is assumed for the components failure times. The second case study refers to the “basic” assumption for the maintenance period, while the third case study refers to the “alternative” one. In the fourth case study, the Weibull distribution is assumed for the components failure times; the “optimal” values are looked for the parameters
We consider a system made by repairable components without aging phenomena. Table
Case study 1, data and Differential Importance Measure.
Component |
|
|
|
|
|
|
|
---|---|---|---|---|---|---|---|
1 | 0.0050 | 0.0200 | 0.2000 | 0.197 | 0.158 | −0.039 | 0.118 |
2 | 0.0050 | 0.0200 | 0.2000 | 0.197 | 0.158 | −0.039 | 0.118 |
3 | 0.0005 | 0.0300 | 0.0164 | 0.960 | 0.776 | −0.013 | 0.763 |
Following the approach described in Section
Aging phenomena are introduced into the model through the exponential-linear distribution for the components failure times. The effective value of the failure rate
Table
Case study 2, data and basic characteristics of components.
Component |
|
|
|
|
|
|
|
---|---|---|---|---|---|---|---|
1 | 0.0050 |
|
1.30 | 131.400 |
|
0.226 | 0.190 |
2 | 0.0050 |
|
131.400 |
|
0.226 | 0.190 | |
3 | 0.0005 |
|
132.566 |
|
0.162 | 0.949 |
DIM is estimated by assuming that
Case study 2, Differential Importance Measure.
Component |
|
|
|
|
---|---|---|---|---|
1 | −0.060 | −0.354 | 0.475 | 0.060 |
2 | −0.060 | −0.354 | 0.475 | 0.060 |
3 | −0.879 | −0.083 | 1.841 | 0.879 |
| ||||
Total | −1.000 | −0.791 | 2.791 | 1.000 |
The ranking of components and parameters defined by DIM estimated through the equations in Section
In paper [
We assume the exponential-linear distribution for the components failure times, and we recognize that preventive maintenance is effective only in presence of aging phenomena; the maintenance period is provided by (
In order to evaluate the effects of the two different assumptions for the maintenance period, we consider the same data of the case study 2. The different values of the parameter
The only difference regarding the parameter
Case study 3, data and basic characteristic of the components.
Component |
|
|
|
|
|
---|---|---|---|---|---|
1 | 1.75 | 296.50 |
|
0.257 | 0.2188 |
2 | 296.50 |
|
0.257 | 0.2188 | |
3 | 119.52 |
|
0.150 | 0.9338 |
The comparison between Table
DIM is estimated by assuming that
Case study 3, Differential Importance Measure.
Component |
|
|
|
|
---|---|---|---|---|
1 | −0.461 | −0.089 | 0.639 | 0.089 |
2 | −0.461 | −0.089 | 0.639 | 0.089 |
3 | −0.172 | −0.822 | 1.817 | 0.822 |
| ||||
Total | −1.094 | −1.000 | 3.094 | 1.000 |
By comparing the values provided in Table
The absolute values of DIM produce the following ranking of parameters:
DIM for parameters can be added over components, in order to estimate the “overall” importance of random failure, aging phenomena, and corrective and preventive maintenance. With reference to Table
In order to verify the correctness of the equations proposed in Section
We introduce aging phenomena through the Weibull distribution for the components failure times. The maintenance period is provided by (
Case study 4, data and basic characteristics of components.
Component |
|
|
|
|
|
---|---|---|---|---|---|
1 | 0.00255 | 2.00 |
|
0.257 | 0.2188 |
2 | 0.00255 | 2.00 |
|
0.257 | 0.2188 |
3 | 0.00635 | 2.00 |
|
0.150 | 0.9338 |
DIM is estimated by assuming that
Case study 4, Differential Importance Measures.
Component |
|
|
|
|
|
---|---|---|---|---|---|
1 | −0.822 | −0.317 | 0.089 | 1.138 | 0.089 |
2 | −0.822 | −0.317 | 0.089 | 1.138 | 0.089 |
3 | −0.306 | −2.941 | 0.823 | 3.247 | 0.823 |
| |||||
Total | −1.950 | −3.574 | 1.000 | 5.523 | 1.000 |
In order to verify the correctness of the equations proposed in Section
If technological constraints prevent the improvement of components unavailability by changing the parameters that define their failure
By assuming
If we assume that
Case study 4, Differential Importance Measures for
Component |
|
|
|
|
|
---|---|---|---|---|---|
1 | −0.231 | −0.044 | 0.319 | 0.044 | 0.089 |
2 | −0.231 | −0.044 | 0.319 | 0.044 | 0.089 |
3 | −0.086 | −0.411 | 0.908 | 0.411 | 0.822 |
| |||||
Total | −0.547 | −0.500 | 1.547 | 0.500 | 1.000 |
This paper refers to systems made by repairable binary components and subjected to aging phenomena and “perfect” maintenance, by assuming the exponential, the exponential-linear, or the Weibull distribution for their failure times. According to paper [
Generally, the estimation of DIM for components requires the computation of the first-order partial derivatives of the system unavailability with respect to the components unavailability (Birnbaum measure), which only depends on the system function. The estimation of DIM for parameters requires the computation of more and more complex partial derivatives of the system unavailability (for each parameter of each component).
In this paper, we proposed to consider separately the dependence of the system unavailability on the components unavailability, from the dependence of the component unavailability on the parameters that specify its failure and repair probability distributions. Under the assumptions of independent components and “perfect” maintenance, this approach leads to the analytical equations provided in Section
Through the proposed equations, the estimation of DIM for parameters does not require further information than that used for the estimation of DIM for components and allows considering separately the importance of the random failure, aging phenomena, and preventive and corrective maintenance. This information supports the improvement of the system performance.
We solved some case studies in order to verify the correctness of the analytical equations and to show the advantages of their use.
Results coming from the analytical equations (numeric values of the measure and ranking of parameters and components) were compared with values taken from paper [
Results coming from the case studies (2 and 3) show that preventive maintenance can be optimized (i.e., the maintenance period can be increased) by recognizing that it is effective only in presence of aging phenomena.
Results coming from the last case study (4) show the opportunity to apply a two-step process for the improvement of the system performance. The first step regards the improvement of the reliability (reduction of random failure and/or aging phenomena) and maintainability (increase of the repair rate) performance of components, within the applicable technological limits. The second step regards the specification of the “optimal” strategy for preventive maintenance (i.e., values of the maintenance periods that reduce differences among values of DIM).
The implementation of the proposed analytical equations into a software tool will allow their application to more complex and realistic systems.
Growth parameter for an exponential-linear distribution
Birnbaum measure for the component “
Cumulative density function
Differential Importance Measure for the component “
Differential Importance Measure for the parameter
(Constant) repair rate
Failure rate
Effective value of the failure rate
Initial value of the failure rate
Shape parameter for a Weibull distribution
Scale parameter for a Weibull distribution
System function
Probability density function
Probabilistic risk assessment
Reliability, availability, maintainability, and safety
Maintenance period
Unavailability.
This study has been performed within the Project Support System for the Availability Analysis of Products and Processes (SSAAPP), partially financed by the Emilia Romagna region (Italy) and developed by NIER Ingegneria with the support of the Department of Energy, Nuclear and Environmental Control Engineering (DIENCA) of the University of Bologna.