Availability modeling and optimization of dynamic multi-state series–parallel systems with random reconfiguration
Introduction
The multi-state series–parallel system (MSSPS) is among the most popular multi-state systems (MSS) being studied [21], [16], [33], [1], [38]. The typical architecture of a MSSPS consists of N subsystems connected in series, and in each subsystem si there are ni components connected in parallel (see Fig. 1). Based on this general structure, most existing studies on MSSPS optimization intend to optimize the types and the numbers of the components in each subsystem [10], [12]. One key assumption of these studies is that the system topology remains unchanged throughout the entire system life time.
Levitin et al. [22] proposed a recursive method for the exact reliability evaluation of phased-mission systems consisting of non-identical independent nonrepairable multistate elements. A structure optimization problem was also studied for a binary system working in multiple phases [3]. Though phased-mission systems can have different structures in different phase, the duration for each phase is constant. In practice, a number of complex industrial systems, such as oil transportation systems [35], shipping systems [14], and railway transportation systems [43] have several operation phases with random duration, at each only a fraction of the subsystems are operating to perform certain task. For instance, Soszynska [35] described a real-world oil transportation system with three pipeline subsystems connected in series to perform five tasks; each involves at most three subsystems at operation. Fig. 2 depicts a few ‘snapshots’ of the operational phases of a MSSPS, where an operation phase b is associated with a certain probability representing the likelihood that the system remains at phase b throughout its life time. It is seen that the stable structure in Fig. 1 can be regarded as a special case (i.e. with all subsystems functioning at all time) of the dynamic structure implied by Fig. 2.
At the component level, it is well known that the multi-state components also exhibit dynamic behaviors. For example, the multi-state components are often subject to aging process [12], [28] and maintenance activities [44]. These situations indicate that component state probability is not always a constant throughout time.
Moreover, the costs associated with the dynamics (both at system level and component level) should also be considered in the optimization problem. To the best of our knowledge, most existing studies in this field compute the total system cost by taking into account only the capital cost of the component, which is the one-time expense to construct or purchase such component. In practice, the operation cost is incurred by almost every type of equipment (e.g. railways [42], telecommunication devices [13], etc)—unless the equipment has no power/energy consumption, does not deteriorate and thus requires no maintenance.
To address the issues above, we propose an analytic approach that combines Markov process to model the dynamics at both system and component levels, the universal generating function (UGF) to derive [39] the system availability function, and the Markov reward model to compute the operation costs associated with system dynamics. Based on this approach, an optimization problem is formulated with the consideration of the availability of the system and its operation costs. Though there are different approaches, such as Pareto dominance [36], [34], [25] and weighted sum [7] to deal with multi-objective optimization problems, an usual way is to optimize one objective constrained by the other ones [5], [23], [30], [31], [32], [41]. In particular, this paper aims to minimize the total system cost with the constraint that system availability is greater than a predetermined level. In order to solve the optimization problem which is combinatorial in nature, a genetic algorithm (GA) technique is adopted. The rest of this paper is organized as follows: Section 2 develops the general model of the dynamic multi-state series–parallel system (DMSSPS), derives the system availability function and the system operation cost function. Section 3 presents the optimization problem and the solution technique. Section 4 presents a numeric example on oil transportation system design. Section 5 concludes this study and points out directions for future extensions.
Section snippets
General model of dynamic multi-state series–parallel system
The assumptions of the DMSSPS are presented as follows:
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All components are statistically independent from each other. This assumption appears in most previous multi-state reliability/availability studies [16], [17], [36].
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The system operation has several phases. Each requires a fraction of the subsystems being in operation to complete certain task.
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Each component has different states corresponding to different performance levels. The component state transition rates and the system phase transition
Optimization problem formulation
To achieve the optimal reliability/availability of the multi-state system, redundant components are often introduced. Usually, this is known as redundancy optimization problem (ROP) [11]. In the literature, it has been addressed in numerous research works. Kuo and Prasad [10] presented a comprehensive overview of the related works. In the majority of existing works, ROP aims to obtain an optimal system structure to minimize the system costs while maintain a desired reliability/availability
Numerical example of maritime oil transportation system
A numerical example from maritime oil transportation system (see Fig. 5) is used to illustrate the proposed model and the optimization approach. This example is modified from the real-world system studied in Soszynska [35]. The oil transportation system between Debogorze and the Port of Gdynia is designed to transport the oil products like petrol and fuel oil from ships to carriages or cars. This purpose is fulfilled by several terminal parts. In this example, six terminal parts are considered
Conclusions and limitations
The dynamic behaviors widely exist in many complex industrial systems, however there is limited literature focusing on this character. This paper aims to take account of all types of dynamics and the associated costs in the context of multi-state system. In this paper, Markov process is used to model the dynamics at both component level and system level; Markov reward model is used to calculate the costs associated with the dynamics; UGF technique is used to derive the system availability.
Acknowledgments
The research report here was partially supported by the NSFC under Grant numbers 71231001 and 71301009, China Postdoctoral Science Foundation funded project under Grant 2013M530531, the Fundamental Research Funds for the Central Universities of China under Grant numbers FRF-MP-13-009A and FRF-TP-13-026A, and by the MOE PhD supervisor fund, 20120006110025.
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