Scheduling heterogeneous repair channels in selective maintenance of multi-state systems with maintenance duration uncertainty

Highlights

• A selective maintenance with multiple heterogeneous repair channels is formulated.

• The stochastic time durations of maintenance tasks and breaks are considered.

• The system reliability is derived via the expected cumulative performance.

• A double loop algorithm embedded with the ACO is tailored to solve the model.

Abstract

Selective maintenance with multiple repair channels has received increasing attention in recent years. The problem aims to jointly identify a subset of units to be maintained and assign the selected maintenance tasks to multiple repair channels. Existing works, however, all assumed that the time durations of maintenance tasks and breaks are deterministic. Due to the randomness of the starting time of future missions and the variation of efficiency of repair channels, these two quantities inevitably possess uncertainty. In this article, by taking account of the uncertainties associated with the time durations of maintenance tasks and breaks, a new selective maintenance model with multiple heterogeneous repair channels is formulated. The new selective maintenance problem decides not only a subset of units to be maintained and the corresponding maintenance tasks, but also the number of repair channels with their specific skill levels and the sequences of maintenance tasks in each repair channel. The objective is to maximize the probability of a system successfully completing the next mission subject to a limited maintenance budget. The resulting optimization problem is resolved via a double-loop algorithm embedded with an ant colony optimization. A five-unit system and a multi-state coal transportation system are presented to demonstrate the effectiveness of the proposed method.

Introduction

In many industrial and military scenarios, engineered systems intend to execute a sequence of maintenance actions within a finite break between two adjacent missions. For example, commercial aircraft are maintained between flights, and military weapon systems are maintained between missions [1]. Due to limited maintenance resources, such as budget, time, and manpower, it is oftentimes impossible to perform all desirable maintenance tasks in a break. In such a circumstance, only a subset of the optional maintenance tasks can be performed during the break, and such a maintenance strategy is well-known as selective maintenance [2].

In the literature, selective maintenance problems (SMPs) have been extensively investigated. Rice et al. [3] provided a pioneering contribution to the area of SMP by developing a mathematical model to maximize the performance and reliability of a series-parallel system with limited maintenance time. Cassady et al. [2] presented a basic mathematical framework for SMPs. With the increasing complexity of engineered systems, research attention has shifted from binary-state systems to multi-state systems (MSSs), and a variety of selective maintenance models have been developed in the context of MSSs. Liu and Huang [4] introduced the imperfect maintenance to SMPs of MSSs. Diallo et al. [5] investigated an SMP for serial k-out-of-n: G system. Shahraki et al. [6] formulated an SMP for MSSs with stochastically dependent units and stochastic imperfect maintenance actions. Three general dependencies, viz., economic dependency, structural dependency, and stochastic dependency, were integrated into SMPs by Dao et al. [7], Dao et al. [8], and Dao et al. [9], respectively. To guarantee the robustness of selective maintenance strategy, Jiang et al. [10] proposed a bi-objective optimization model to maximize the expectation of system reliability while simultaneously minimizing its variance. Maillart et al. [11] and Jiang et al. [12] studied the SMPs of systems executing multiple missions. Xia et al. [13] introduced a branch and band algorithm with a rolling horizon strategy to optimize the multi-mission SMP. Using the sensor monitoring data, Hesabi et al. [14] investigated an SMP from a data-driven approach. With the increasing complexity of the maintenance strategy and system structure, the system state space becomes uncountable. As a consequence, approximate dynamic programming [15], two-stage programming [5,16] and deep reinforcement learning approaches [14,[17], [18], [19], [20]] were employed to overcome the “curse of dimensionality”.

In SMPs, a repair channel refers to a repair site with a repair crew as well as a set of repair facilities [21]. In most existing SMPs, only one repair channel is assumed to be available [22]. Consequently, the maintenance tasks should be performed in a sequential manner. In fact, multiple repair channels may be available in many industrial applications, such as the airline industry [23] and naval vessels [24]. For example, a navy vessel may be repaired by several repair crews without colliding and hindering their movements [24]. As a result, it is necessary to investigate the SMP under multiple repair channels. In the literature, several attempts have been made to investigate SMPs with multiple repair channels. For instance, Khatab et al. [23] incorporated two additional decision variables into SMPs, namely, the number of repairmen to hire and the assignment of specific units to each repairman. Diallo et al. [24] investigated a variation of the SMP for a k-out-of-n system with imperfect maintenance and multiple repair channels. Khatab et al. [25] proposed a two-phase decomposition method for reducing the complexity of the fleet-level SMP containing multiple repair channels with different skill levels. Chaabane et al. [26] studied the SMP that jointly optimizes the selective maintenance and multiple repair channels assignment in the case of multiple missions.

Nevertheless, a majority of SMPs assumed that the time durations of maintenance tasks and breaks were deterministic. Due to the variability in repair efficiency across multiple repair channels and the randomness of the starting time of future missions, the uncertainties associated with the time durations of maintenance tasks and breaks inevitably exist. For instance, in military contexts, the starting time of the next mission/battle cannot be precisely predicted, resulting in the stochastic nature of the time duration of break between two adjacent missions or battles [27]. Ignoring these uncertainties may lead to inferior maintenance decision, which could result in huge economic losses and even threat to human life [28,29]. In the literature, only a few works were dedicated to addressing the uncertainty associated with the time durations of maintenance tasks and breaks [27,[30], [31], [32], [33]]. Khatab et al. [33] extended the basic SMP to deal with the case that both the time durations of breaks and missions are random quantities governed by known probability distributions. Liu et al. [27] presented a sequence planning model for the SMP of multi-state systems under stochastic maintenance time durations. The results indicated that different sequences of maintenance actions significantly affect the mission success when the uncertainty of the time durations of maintenance tasks and breaks were taken into account. Liu et al. [34] considered the stochasticity of the time durations of maintenance tasks, breaks, and missions simultaneously. Ghorbani et al. [16] proposed a two-stage stochastic programming model to address the SMP with the uncertainty of future operation conditions.

The aforementioned studies on selective maintenance are blocked in investigating the stochastic time durations and multiple repair channels individually. In practice, an effective selective maintenance strategy needs to jointly consider the above two aspects in a holistic manner. In aviation industry, for example, an aircraft is scheduled to complete multiple maintenance actions within a break between two consecutive flights. In the break, multiple repair crews are available when an aircraft undergoes overnight maintenance, and they are able to simultaneously overhaul multiple units, such as avionics, landing gears, structural units, engines, and wings [23]. Meanwhile, the time duration of the break between two scheduled flights of a civil aircraft may be abruptly increased or decreased due to a sudden change in weather conditions [35]. The maintenance time is also stochastic due to various skill levels of repair crews, the state of units, and the spare parts. Furthermore, due to limited maintenance resources, e.g., maintenance time and budget, the maintenance tasks at the front of the queue have a greater chance to be completed than these at back of the queue. Put another way, the maintenance sequence in a repair crew has an influence on the system state. Therefore, scheduling the maintenance tasks for each repair crew is also an important aspect of the studied selective maintenance strategy. Consequently, a natural motivation is to incorporate the uncertainties associated with heterogeneous repair channels into selective maintenance framework, so as to identify an optimal strategy.

In most literature on MSSs, reliability is defined as the probability of the system performance capacity to meet a specific user's demand [36], [37], [38]. Nevertheless, in many industrial scenarios, it is useful to concern about the amount of work produced by a system. To the best of our knowledge, the amount of work produced by repairable MSSs has been rarely investigated in the literature. Levitin et al. [39] studied a system with a single unit and defined the mission success as the system being able to deliver a specified amount of work. Levitin et al. [40] further extended their work in [39] to serial-parallel systems composed of binary-state repairable units. Jiang et al. [41] evaluated MSS reliability from the perspective of cumulative performance, which referred to the amount of work that an MSS produces throughout a finite period. Nevertheless, the system studied in [41] was only composed of a single multi-state unit. The cumulative performance can be viewed as a qualified measurement of the performance of an MSS. Our literature survey indicates that the works on cumulative performance are either for MSSs composed of binary-state units or for MSSs with a single unit. In practice, an MSS is typically a multi-unit system in which both the system and its units may exhibit multiple performance capacities [42], such as communication networks, computer systems, circuits, power systems, and transmission systems [43]. It is a challenging task to evaluate the cumulative performance of multi-unit MSSs due to the complex structure of the system and the multi-state nature of units.

To resolve the above research questions, we propose a new selective maintenance model for MSSs with multiple repair channels, and the uncertainties associated with the time durations of maintenance tasks and breaks are take into account. Compared with the existing works, the decision variables in our study are more comprehensive, as six factors are included: (1) the selection of the units to be maintained, (2) the state of each unit to be repaired, (3) the number of repair channels to be employed, (4) the assignment of repair channels, (5) the maintenance sequence of units within each repair channel, and (6) the skill level of each repair channel. As a consequence, it is challenging to evaluate the probability of a system successfully completing the next mission. In summary, compared with the existing works on SMPs, the unique contributions of this study are trifold:

(1) A new SMP with multiple repair channels is formulated for MSSs by taking account of the uncertainties associated with the time durations of maintenance tasks and breaks. To improve the flexibility of the maintenance strategy for the SMP, multiple heterogeneous repair channels are involved.

(2) The success probability of the next mission for an MSS is evaluated in terms of the expected cumulative performance rather than the instantaneous performance capacity of the system.

(3) A double-loop solution algorithm with an embedded ant colony optimization (ACO) is developed to efficiently solve the resulting combinatorial optimization problem.

The remainder of this study is rolled out as follows. Section 2 is the problem statement in which the basic assumptions of the studied MSS and the variables associated with the proposed SMP are explained in detail. In Section 3, we construct a new selective maintenance model with multiple heterogeneous repair channels. By taking the stochastic time durations of maintenance tasks and breaks into account, the mission success probability is derived in Section 4. The SMP is formulated in Section 5 and a double-loop solution algorithm is designed to resolve the SMP. Two illustrative examples, together with a set of comparative studies, are presented in Section 6, and it is followed by some conclusions in Section 7.