Redundancy strategies assessment and optimization of k-out-of-n systems based on Markov chains and genetic algorithms

Highlights

• We revisit the redundancy allocation problem under different strategies.

• We consider active, standby, mixed and K-mixed strategies for k-out-of-n configurations.

• We develop Markov chain models and compare the results to Monte Carlo simulation.

• A reliability optimization problem is formulated and solved using a genetic algorithm.

• The proposed approach is efficient for evaluating and optimization system reliability.

Abstract

The existing reliability formulation regarding mixed and K-mixed strategies considers only one component as the minimum number of required components for each subsystem. In this paper, we develop a Continuous-Time Markov Chain model for both mixed and K-mixed strategies while the minimum number of required components can be more than one and take any values. In addition, the drawbacks of the classical model, which are complicated formulation, approximate solution and time-consuming problem-solving process, have been addressed. The proposed model estimates the reliability under different redundancy strategies more efficiently and in a straightforward way. To validate the proposed approach, a sequential Monte Carlo model is also developed for the reliability analysis. Besides, the existing strategies are applied to a series-parallel system and an efficient genetic algorithm is developed to solve the resulting optimization problem. The numerical results confirm the accuracy of the Continuous-Time Markov Chain model in estimating k-out-of-n system reliability under standby, mixed and K-mixed strategies with a major reduction in the computation time.

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.