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This study aims for multistate systems (MSSs) with aging multistate components (MSCs) to construct a time-replacement policy and thereby determine the optimal time to replace the entire system. The nonhomogeneous continuous time Markov models (NHCTMMs) quantify the transition intensities among the degradation states of each component. The dynamic system state probabilities are therefore assessed using the established NHCTMMs. Solving NHCTMMs is rather complicated compared to homogeneous continuous time Markov models (HCTMMs) in determining reliability related performance indexes. Often, traditional mathematics cannot acquire accurate explicit expressions, in particular, for multiple components that are involved in designed system configuration. To overcome this difficulty, this study uses Markov reward models and the bound approximation approach to assess rewards of MSSs with MSCs, including such things as total maintenance costs and the benefits of the system staying in acceptable working states. Accordingly, we established a long-run expected benefit (LREB) per unit time, representing overall MSS performance through a lifetime, to determine the optimal time to replace the entire system, at which time the LREB values are maximized. Finally, a simulated case illustrates the practicability of the proposed approach.

Conventional binary-state systems assume that component constituted system configurations involve either only perfect functioning or complete failure. The optimal system reliability design, maintenance policy, and repairing management are thereby determined. However, the design of modern devices tends to be of large scale and complicated; these devices normally confront different kinds of faults and errors, including damage, impacts, and aging factors, throughout their lifetimes. Various systems, such as computer server, telecommunication, and electricity distribution systems, become tolerant to these faults and errors. Even if a fault occurs, these systems continue working at an acceptable or degraded performance level. Accordingly, from being perfectly functioning, systems normally experience multiple intermediate states during the degradation process, before complete failure occurs. Confining systems to binary states can ignore the intrinsic multistate property of these systems and result in a biased evaluation of actual system performance. Therefore, constructing a multistate reliability theory can provide further insight into the complicated failure theory, lifetime prediction, and improvement of system reliability, which is an essential issue in both practical and theoretical aspects.

Multistate systems (MSSs) are regarded as failures when they degrade into unacceptable performance levels and cannot meet operational requirements. Preventive maintenance (PM) implementation is beneficial to sustain or improve system performance during the planning horizon. Incorporating imperfect maintenance theories into the stochastic process [

This study extends the work of Wang and Huang [

In general, some indices were proposed in evaluation of dynamic system over time mainly related with system availability; these indices can fall into two categories with expense-oriented and benefit-oriented indices. These two categories commonly refer to index like “long-run average cost” [

Liu and Huang [

Much of the previously discussed literatures aimed at binary systems to address the problem of replacement policy given imperfect maintenance from the system implementation perspective. Although some works address the replacement policy problems for multistate systems, the PM from the perspective of a system with binary components is given. Furthermore, in order to deduce the explicit expression of the MSS performance measure, a rather simple system configuration is given to illustrate the theoretical results. Practically, for a safety system, the PM policy from the component perspective can prevent a sudden system failure due to component degradation or failure, avoiding possible catastrophic consequences. In this regard, this study aims at MSSs with aging MSCs to establish an optimal time-replacement policy, given the implementation of a PM policy from a component perspective. The PM policy involves maintenance actions in the degradation states for aging MSCs. A performance index regarding the long-run expected benefit (LREB) per unit time for MSSs with aging MSCs determined the optimal time to replace an MSS by maximizing the LREB values throughout the lifetime. The term “benefit,” a general designate, corresponds to the profit making with system staying at acceptable states capable of functioning properly with distinct performances in this paper. Alternatively, given other considerations, the term “benefit” could relate to other measures such as productivity and delivery rate in a manufacturing system.

The Lisnianski and Levitin [

State-transition diagram of a component.

For component

When the MSS consists of

The nonhomogeneous Markov reward model [

Solving the NHCTMM to obtain the aging MSC/MSS performance indicators requires a lot of time. Often, the use of calculators embedded in the common mathematical tools, such as MATLAB or MATHCAD, may induce the problem of inaccuracy [

This study aims for MSSs with aging MSCs to determine the time-replacement policy in which a PM from the component perspective is implemented. The calculation procedure integrates the multiple states of all components to obtain the distinctive states for MSSs and, therefore, to determine MSS performances that are established on the basis of nonhomogeneous Markov models. As mentioned previously, the bound approximation approach [

The aging MSCs of the system degrade from perfectly functioning to complete failure over multiple states of degradation.

The failure rate of an individual component is an increasing function of time.

Components at degradation states can be restored to previous better states by appropriate maintenance.

Real-time monitoring of the system can identify the performance of individual components within the system.

A specific PM policy is implemented on the aging MSCs.

There are five maintenance activities:

No service or repair.

Minor service: enables restoration to state

Major service: enables restoration to state

Minor repair: enables restoration to state

Major repair: enables restoration to state

The proposed approach is elucidated on the basis of a series-parallel system [

Failure-rate function of each component between states.

Failure rate | Component | ||
---|---|---|---|

1 | 2 | 3 | |

| 0.24+0.07 | 0.24+0.07 | 0.34+0.14 |

| |||

| 0.18+0.04 | 0.18+0.04 | 0.28+0.08 |

| |||

| 0.14+0.02 | 0.14+0.02 | 0.24+0.04 |

| |||

| 0.12+0.01 | 0.12+0.01 | 0.22+0.02 |

| |||

| 0.26+0.08 | 0.26+0.08 | 0.36+0.16 |

| |||

| 0.20+0.05 | 0.20+0.05 | 0.30+0.10 |

| |||

| 0.16+0.03 | 0.16+0.03 | 0.26+0.06 |

| |||

| 0.28+0.09 | 0.28+0.09 | 0.38+0.18 |

| |||

| 0.22+0.06 | 0.22+0.06 | 0.32+0.12 |

| |||

| 0.30+0.10 | 0.30+0.10 | 0.40+0.20 |

Note:

Repair rate of each component between states (per hour).

Component | Repair rate | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

| | | | | | | | | | |

1 | 0.125 | 0.320 | 0.335 | 0.410 | 0.425 | 0.440 | 0.455 | 0.470 | 0.485 | 0.500 |

| ||||||||||

2 | 0.080 | 0.245 | 0.260 | 0.275 | 0.290 | 0.305 | 0.350 | 0.365 | 0.380 | 0.395 |

| ||||||||||

3 | 0.065 | 0.095 | 0.110 | 0.140 | 0.155 | 0.170 | 0.185 | 0.200 | 0.215 | 0.230 |

Note:

Cost parameters of maintenance activities.

Maintenance | Cost | ||
---|---|---|---|

Component 1 | Component 2 | Component 3 | |

Minor service | 54 | 72 | 120 |

Major service | 72 | 96 | 160 |

Minor repair | 90 | 120 | 200 |

Major repair | 180 | 240 | 400 |

Replacement parameters.

| | |
---|---|---|

50 | 2 | 200 |

MSS performance and benefits.

State | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 |
---|---|---|---|---|---|---|---|---|---|---|

Performance | 140 | 120 | 110 | 100 | 90 | 80 | 50 | 40 | 30 | 0 |

| ||||||||||

Benefit ($) | 1000 | 800 | 700 | 600 | 500 | 400 | 300 | 200 | 100 | 0 |

MSS configuration with MSC state-space diagrams.

The calculation procedure involved in the proposed approach has five steps.

Generate the possible states of the MSSs.

The proposed approach initially combines the multistate components constituting the system to form an MSS. For the simulated case, in total, 125 possible states are determined for this system, configuring three components each with five distinctive states with different performances.

Obtain the reduced MSS.

Uniting the systems states with an identical performance from step 1 can obtain the reduced MSS. This process can significantly reduce the subsequent calculation complex without loss of system performance information. Accordingly, a total of ten distinctive performance states for this system are determined; the state-transition diagram of this reduced MSS is also constructed. Figure

State-transition diagram of reduced MSS.

Determine the total maintenance cost of the system.

According to the state-transition diagrams for components 1–3 shown in Figure

Determine the MSS benefit.

Initially, the NHCTMMs and NHCTMRMs related to the reduced MSS are established, respectively, on the basis of the state-transition diagrams shown in Figure

Determine the optimal time to replace the system.

First, using (

According to the previously mentioned parameter settings, this study mimics two PM policies to illustrate the ramifications of the proposed approach. The first PM policy is derived from Huang and Wang [

State-transition diagram of component 1 with PM policy 1.

State-transition diagram of component 2 with PM policy 1.

State-transition diagram of component 3 with PM policy 1.

State-transition diagram of component 1 with PM policy 2.

State-transition diagram of component 2 with PM policy 2.

State-transition diagram of component 3 with PM policy 2.

Figure

LREB trend diagrams through lifetime with two PM policies.

To verify the proposed approach further, a sensitivity analysis with multiple increases in the failure rates for component 3 was performed by a given PM policy 1. Figure

Optimal replacement time T and LREB versus demand

PM policy | Demand | T (days) | LREB ($/day) |
---|---|---|---|

1 | 100 | 818 | 36,046 |

110 | 818 | 35,913 | |

120 | 818 | 35,824 | |

140 | 818 | 35,795 | |

| |||

2 | 100 | 635 | 34,933 |

110 | 632 | 34,678 | |

120 | 632 | 34,490 | |

140 | 632 | 34,283 |

LREB trend diagrams with multiple increases in failure rates for component 3 given PM policy 1.

LREB trend diagrams with multiple increases in repair rates for component 3 given PM policy 1.

This study aims at the PM model from the component perspective for the MSS with aging MSCs, in which the maintenance alternatives are implemented when the MSCs fall into degradation states, to propose a time-replacement policy. A time-dependent LREB index of system performance was developed on the basis of the continuous time Markov theory. By maximizing the LREB values throughout the system’s lifetime, we determined the optimal time to replace this type of MSS economically. The proposed approach provides further insight into the relationship between PM policy setting and long-term system benefits; it also verifies a time-replacement policy. For future study, the ramifications of the current study can be extended to situations that consider the repair difficulties as MSCs age. Namely, the repair rate is a decreasing function of time. Using this method, the analyzed PM models more adequately fulfill the practical requirements. However, the mathematical calculation involved is a difficult challenge in striving to obtain precise results.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

This work was supported by the Ministry of Science and Technology of Taiwan (NSC 104-2221-E-606-004 and 107-2221-E-012-001-MY2).