Redundancy strategies assessment and optimization of k-out-of-n systems based on Markov chains and genetic algorithms
Introduction
A system consisting of n components, of which only k components need to be functioning for system success, is called a "k-out-of-n" configuration. The redundancy strategy of this type of system can be planned to maximize its reliability. Several studies have been conducted to calculate system reliability of this type of system by considering different redundancy strategies (e.g., [1], [2], [3]). Generally, there are four types of redundancy strategies in the context of Redundancy Allocation Problem (RAP), namely active, standby, mixed, and K-mixed. In an active strategy, all redundant components start working when the system comes online, while in a standby strategy, the system functions with only a minimum number of required components. When one of these components fails, a switching system replaces the failed component with a redundant one. It should be noted that the switching system itself is also prone to breakdowns and component malfunctions and hence can fail. The overall system continues working until the number of active components drops down to k or the switching system cannot replace a failed component with a functional redundant one. In case when only one component is necessary for system operation, the failure of either the working component or the switching system leads to the breakdown of the entire system [4]. There are three cases for standby redundancy in the literature [3]: cold, warm, and hot standby which are named based on the component failure due to operational stress associated with system operation.
Many studies have been conducted on systems in which only one component is required for system operation. In some studies, active and standby strategies are considered as predefined strategies for each subsystem [5,6], while in other studies [7,8] redundancy strategies are considered as decision variables that need to be determined. Abouei et al. [9] proposed the mixed strategy that is a combination of active and standby strategies. According to this strategy, the numbers of active and standby components are considered as decision variables that must be optimally selected. When all active components fail, redundant components are replaced individually, and the system fails when all components break down. The authors applied this strategy on different cases and showed that it outperforms both active and standby strategies in most cases [9,10].
The K-mixed redundancy strategy, which is a generalization of the mixed strategy, was introduced by Peiravi et al. [11]. The authors showed that when the switching system is not highly reliable, the K-mixed strategy yields significant improvements over other strategies [12]. To evaluate the efficiency of their model, a series-parallel system was evaluated, and its results were compared with the existing methods in the literature. The results confirmed that the new approach outperforms previous strategies in several problem instances [12].
It is worth mentioning that the number of active and standby components under both mixed and K-mixed strategies are determined by solving an optimization model that aims to maximize system reliability. Nevertheless, it is computationally difficult to estimate system reliability by the aid of classical reliability models for systems with more than four redundant components. Furthermore, the existing approximation approaches provide a lower bound on the actual system reliability [11,12]. To measure the exact system reliability under a standby strategy, Kim and Kim [13] developed a Markov model and applied it in the context of a reliability-redundancy allocation problem. In another paper, the authors provided the same reliability model considering Phase-type time-to-failure distribution for components [14]. Recently, a similar reliability model was proposed in view of the mixed strategy [15].
However, no prior work has considered mixed and K-mixed strategies for k-out-of-n systems. The development of a Continuous-Time Markov Chain model (CTMC) for reliability calculation of k-out-of-n systems under mixed and K-mixed strategies is a novel contribution in this study. An example of a k-out-of-n system (2 out of 5) under K-mixed and mixed strategies is shown in Fig. 1. In this Figure, green, yellow, red, and non-colored boxes represent the required online components (k), extra online components, failed components, and standby components, respectively. In both strategies, the system starts functioning with more than k active components. Despite their similarities, there is a significant difference between the K-mixed and the mixed strategies with respect to the switching system. In the mixed strategy, the switching system is activated only when the number of active components drops below the minimum number of required components; therefore, switching failure results in the subsystem failure. However, in the K-mixed strategy, the switching system is used to substitute a redundant component when the first extra online component fails. In other words, the K-mixed strategy tries to keep the initial number of active components (K) functional during the whole operation time.
The field of k-out-n systems reliability has also been extensively studied. The redundancy allocation problem in k-out-of-n systems was studied for example in [3] under the assumption that the redundancy strategy (e.g., active or cold standby) for each subsystem was fixed beforehand. An algorithm proposed in [16] is used to determine the reliability of a weighted k-out-of-n system. In [17], a dynamic set of k-out-of-n component partnerships was described, where the number of components, k, changes dynamically in response to failures. A study by Li et al. [18] analyzed a sequential k-out-of-n system. Hamdan et al. [19] developed an optimal preventive maintenance model that takes average cost and availability into account for weighted k-out-of-n systems. An example of a load sharing k-out-of-n containing non-identical multi-state subsystems is given in [20].
The classical approach of reliability calculation has a complicated formula that involves several double integrals. Calculating the reliability of any configuration when the K-mixed or mixed strategy is implemented needs a few seconds on a personal computer. However, when it comes to solving combinatorial reliability optimization problems, there is a need to evaluate the objective function (i.e., the system reliability) for a large number of possibilities in a huge search space. Developing a fast accurate evaluation method is crucial to solve large combinatorial reliability optimization problems. To answer this need, our contribution is three-fold.
First, by focusing on k-out-of-n configurations, a Continuous-Time Markov Chain (CTMC) model is developed to calculate the system reliability under standby strategy, where the switch is not continuously monitoring the system and is only triggered upon a component failure. It is noteworthy that this case is more realistic in industrial systems. For instance, to make sure of switch performance, it is triggered by the system operator twice or more during the mission. Second, the classical reliability models are more time-consuming to calculate the system reliability under mixed and K-mixed strategies, in comparison to active and standby strategies. Therefore, in this research, new CTMC modeling models are developed to calculate system reliability under mixed and K-mixed strategies for k-out-of-n configurations. Indeed, the proposed CTMC models calculate system reliability for any number of essential functioning components, k, including 1 out n systems. The system reliability is also evaluated using a Monte-Carlo simulation model to test the accuracy of the proposed approach. Lastly, we combine the suggested evaluation models with a genetic algorithm to efficiently solve a reliability optimization problem in the context of a series-parallel system and we compare the execution time with the one reported in literature.
The rest of the paper is structured as follows. In Section 2, we formulate the CTMC-based reliability model proposed for calculating k-out-of-n subsystem reliability under all the above-mentioned strategies, where the switching system can be either constantly monitored, or triggered in response to a failure. Section 3 presents a Monte Carlo simulation model to validate the obtained results by the Markov model. In Section 4, two numerical examples are presented to demonstrate the efficiency of the proposed methodology. Finally, conclusions are presented in Section 5.
Section snippets
Reliability model
Two cases for an imperfect switching/detection mechanism have been proposed in [4]: I) continuous monitoring and detection, and II) detection and switching in response to a failure. In Case I, the system performance is being observed for failure detection and, if one is detected, the right measures would be taken based on the imposed redundancy strategy. In Case II, the switching system is triggered only in direct response to a component failure. If the switch is failed upon triggering, it will
Monte-Carlo simulation model
To verify the accuracy of the transient matrix constructed in previous sections, a sequential Monte-Carlo Simulation Model (MCM) is developed in this section. The goal of this model is to estimate the reliability of a given subsystem under different redundancy allocation strategies considered in this study. The MCM mimics the components’ history of failure by using their state probability distributions. Statistics are then obtained and statistical computation is used to estimate different
Numerical results
To illustrate the efficiency of the proposed CTMC models, a single subsystem with four components is first considered; the minimum number of required components can be one or two. It is assumed that the components and switch reliability values are predefined. Afterward, the proposed reliability model is implemented for a series parallel system that has been widely studied in the literature.
Conclusion
Classical reliability models corresponding to the redundancy strategies, such as standby, mixed, and K-mixed suffer from a high degree of computational complexity and in some cases provide a lower bound on the actual system reliability. In addition, the mixed and K-mixed strategies have not been considered in k-out-of-n systems in the literature. In this paper, a CTMC modeling approach for the k-out-of-n system is proposed for calculating the exact system reliability under the aforementioned
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors would like to thank the Natural Sciences and Engineering Research Council of Canada (NSERC) for supporting this project. They would like to also thank the editor, the associate editor and the anonymous referees for their constructive comments and recommendations, which have significantly improved the presentation of this paper.
References (31)
Reliability and maintenance modeling for a load-sharing k-out-of-n system subject to hidden failures
Comput Ind Eng
(2020)- et al.
Variance of Reliability Estimate for k-out-of-n Systems with Cold Standby Units
- et al.
A study of interval analysis for cold-standby system reliability optimization under parameter uncertainty
Comput Ind Eng
(2016) - et al.
System reliability-redundancy optimization with cold-standby strategy by an enhanced nest cuckoo optimization algorithm
Reliab Eng Syst Saf
(2020) - et al.
Multi-objective optimization of reliability-redundancy allocation problem for multi-type production systems considering redundancy strategies
Reliab Eng Syst Saf
(2020) - et al.
Reliability–redundancy allocation problem considering optimal redundancy strategy using parallel genetic algorithm
Reliab Eng Syst Saf
(2017) - et al.
Reliability models for a nonrepairable system with heterogeneous components having a phase-type time-to-failure distribution
Reliab Eng Syst Saf
(2017) - et al.
A new Markov-based model for reliability optimization problems with mixed redundancy strategy
Reliab Eng Syst Saf
(2020) - et al.
The lost capacity by the weighted k-out-of-n system upon system failure
Reliab Eng Syst Saf
(2021) - et al.
Dynamic k-out-of-n system reliability with component partnership
Reliab Eng Syst Saf
(2015)
Reliability modeling for repairable circular consecutive-k-out-of-n: F systems with retrial feature
Reliab Eng Syst Saf
Optimal preventive maintenance for repairable weighted k-out-of-n systems
Reliab Eng Syst Saf
Optimal reliability design of a system with k-out-of-n subsystems considering redundancy strategies
Reliab Eng Syst Saf
Sequence optimization in reliability problems with a mixed strategy and heterogeneous backup scheme
Reliab Eng Syst Saf
Reliability allocation problem in a series–parallel system
Reliab Eng Syst Saf
Cited by (31)
Optimizing corrective maintenance for multistate systems with storage
2024, Reliability Engineering and System SafetyA novel evolutionary solution approach for many-objective reliability-redundancy allocation problem based on objective prioritization and constraint optimization
2024, Reliability Engineering and System SafetyMulti-task optimization in reliability redundancy allocation problem: A multifactorial evolutionary-based approach
2024, Reliability Engineering and System SafetyReliability analysis and redundancy optimization of k-out-of-n systems with random variable k using continuous time Markov chain and Monte Carlo simulation
2024, Reliability Engineering and System SafetyPerformance of load sharing repairable k-out-of-N: G systems
2024, Journal of Engineering Research (Kuwait)Allocation and activation of resource constrained shock-exposed components in heterogeneous 1-out-of-n standby system
2024, Reliability Engineering and System Safety