Multi-state Series-parallel Systems Research Articles

Bounds for joint probabilities of multistate systems using preservation of log‐concavity

AbstractLog‐concavity of multivariate distributions is an important concept in general and has a very special place in the field of Reliability Theory. An attempt has been made in this paper to study preservation results for (i) the discrete version of multivariate log‐concavity for multistate series and multistate parallel systems consisting of independent components, states of both components and systems being represented by elements in a subset of and (ii) the continuous version of multivariate log‐concavity under multistate series and multistate parallel systems made up of independent components and states of both, systems and components, taking values in the set . These results for discrete and continuous versions of log‐concavity have also been extended to systems that are formed using both multistate series and multistate‐parallel systems. Further, the results in (ii) have been used to obtain important and useful bounds on joint probabilities related to times spent by multistate components, multistate series, multistate parallel systems, and the combinations thereof.

A DNN-based reliability evaluation method for multi-state series-parallel systems considering semi-Markov process

In order to evaluate the reliability of the multi-state series-parallel system considering semi-Markov process (MSSPS-SMP), the integral equations have been utilized to calculate the state probability distributions. Nevertheless, to solve the formulated integral equations, it is impossible to avoid time-consuming convolution operations for any numerical method, which can result in significant reliability evaluation complexity. To address the above problems, a deep neural network (DNN)-based method is proposed for the reliability evaluation of the MSSPS-SMP. For a multi-state component, the reliability parameters representing the arbitrary distributions of the SMP are firstly extracted as the input feature to DNN, while the corresponding state probability distributions serve as the output of DNN naturally. On this basis, the DNN is deployed to establish a direct mapping relationship between the reliability parameters and state probability distributions. Instead of repeating complicated calculations of SMP-related convolution operation, the well-trained DNN model can effectively determine the performance distributions of multi-state components given the varying reliability parameters. On this basis, the Lz-transform technique is utilized to develop the unified representations of dynamic reliability models of various multi-state components considering SMP. Combined with the Lz-transform, the time-varying performance distribution of the MSSPS-SMP with complicated structures of several components can be determined.

The methods for exactly solving redundancy allocation optimization for multi-state series–parallel systems

Redundancy allocation problems (RAPs) is a classical family of reliability optimization problems. RAP for multi-state systems (MSSs) is among the most difficult RAPs. Multi-state series–parallel system (MSSPS) is among the most commonly-applied structure of MSSs. To the knowledge of the authors, in literature there is no exact approach to solve MSSPS RAP. In this work, we propose an efficient exact solution method based on dynamic programming to solve it. In literature, there are two formulations of RAP, namely maximizing the system reliability and minimizing of the system cost under the resource constraints. In this work, we investigate the conversion between the two formulations such that the two different RAP formulations can be exactly solved via our proposed method. Experiments are conducted on the well-known benchmarks. The results are compared with the published ones achieved by meta-heuristics. Our methods confirm that majority of the published solutions are in fact global-optimal. In addition, our methods find three new global-optimal solutions. The experiments also illustrate that on systems with more subsystems and fewer component types, our proposed method significantly outperforms the heuristic method in MSSPS RAP with exact optimal solutions and less running time.

Robust optimization on redundancy allocation problems in multi-state and continuous-state series–parallel systems

In this paper, we consider the redundancy allocation problem (RAP) with uncertainties in component parameters for multi-state series–parallel system (MSSPS) and continuous-state series–parallel system (CSSPS). In real-world cases, the component parameters such as costs and reliabilities are often uncertain due to epistemic uncertainty. The existing research works mainly focused on binary-state RAP with data uncertainties. Few studies considered the epistemic uncertainty in MSSPS RAP. To the knowledge of the authors, nearly no research work addressed it in CSSPS. Therefore, in this paper we focus on MSSPS RAP and CSSPS RAP with uncertainties and propose a common model suitable for both of them. Moreover, the epistemic uncertainty of component state is handled by a state distribution distributed in an ambiguity set. The uncertain cost parameters are considered as the interval values. Given the partial information of the probability distribution of uncertain data, we establish a robust model to deal with different types of uncertain parameters. The robust model we proposed has a strong risk-averse capacity against the epistemic uncertainties and can help the ambiguity-averse managers design a system where all parameters are evaluated over the worst-case situation within the ambiguity set. Due to its intractability, we reformulate this proposed model as a mixed-integer linear programming problem via duality theory. The performance of the proposed model is illustrated by numerical experiments on the well-known benchmark problem for MSSPS RAP from three aspects: the robustness of the solutions under different conservative levels; the performance of robust solutions to hedge against the uncertainty of component state; the comparison of stochastic programming model and robust model to hedge against the uncertainties of component cost and state.

Redundancy optimization for multi-state series-parallel systems using ordinal optimization-based-genetic algorithm

Multi-state series-parallel systems (MSSPSs) are widely-used for representing engineering systems. In real-life cases, engineers need to design an optimal MSSPS structure by combining different versions and number of redundant components. The objective of the design is to ensure reliability requirements using the least costs, which could be formulated as a redundancy optimization problem under reliability constraints. The genetic algorithm is one of the most frequently used method for solving redundancy optimization problems. In traditional genetic algorithms, the population size needs to be determined based on the experience of the modeler. Often, this ends up creating a large number of unnecessary samples. As a result, the computational burden can be huge, especially for large-scale MSSPS structures. To solve these problems, this paper proposes an optimal structure designing method named as redundancy ordinal optimization. The universal generating function technique is applied to evaluate the reliabilities of the MSSPSs. Based on the reliabilities, an ordinal optimization algorithm is adapted to update the parent populations and the stopping criterion of genetic algorithm, so that the unnecessary structure designs can be eliminated. Numerical examples show that the proposed method improves the computational efficiency while remaining satisfactorily accurate.

Multi-Objective Electrical Power System Design Optimization Using a Modified Bat Algorithm

The design of energy systems is very important in order to reduce operating costs and guarantee the reliability of a system. This paper proposes a new algorithm to solve the design problem of optimal multi-objective redundancy of series-parallel power systems. The chosen algorithm is based on the hybridization of two metaheuristics, which are the bat algorithm (BA) and the generalized evolutionary walk algorithm (GEWA), also called BAG (bat algorithm with generalized flight). The approach is combined with the Ushakov method, the universal moment generating function (UMGF), to evaluate the reliability of the multi-state series-parallel system. The multi-objective design aims to minimize the design cost, and to maximize the reliability and the performance of the electric power generation system from solar and gas generators by taking into account the reliability indices. Power subsystem devices are labeled according to their reliabilities, costs and performances. Reliability hangs on an operational system, and implies likewise satisfying customer demand, so it depends on the amassed batch curve. Two different design allocation problems, commonly found in power systems planning, are solved to show the performance of the algorithm. The first is a bi-objective formulation that corresponds to the minimization of system investment cost and maximization of system availability. In the second, the multi-objective formulation seeks to maximize system availability, minimize system investment cost, and maximize the capacity of the system.

Redundancy Optimization for Multi-Performance Multi-State Series-Parallel Systems Considering Reliability Requirements

Various engineering systems operating with multiple performance measures are multi-state in nature. These systems are usually modelled as multi-performance multi-state series-parallel systems (MPS). In real-life cases, engineers need to design an optimal MPS structure by combining different versions and number of redundant multi-performance components. Moreover, to design a reliable structure, reliability evaluation is inevitable. However, the conventional redundancy optimization measured by single performance is difficult to model the optimal MPS structure. Furthermore, as the most frequently used method, randomly sizing of parent populations and stopping criterion of traditional genetic algorithm (GA) may lead to massive unnecessary structures. The computational burden may be huge for the redundancy optimization considering multi-performance measures. This paper proposes a redundancy optimization model considering multi-performance measuring systems. The optimization objective and the reliability constraints are described by multiple performance measures. The universal generating function technique is applied to evaluate the reliability. Based on the reliability, the efficient optimization method combining with ordinal optimization and genetic algorithms is proposed to design the optimal MPS structure. The ordinal optimization algorithm is employed to modify both the parent populations and stopping criterion of genetic algorithm. Finally, the proposed model and method are illustrated by numerical examples.

Multiple-valued Decision Diagrams for Multi-state System Performability Analysis

Background: In a multi-state system, either the entire system or its components will have more than two states. Different kind of multi-state systems have been investigated, such as multistate series-parallel systems, multi-state k-out-of-n systems and multi-state linear consecutive-k-outof- n systems. Methods: Performability analysis is concerned with the calculation of probability distribution of different system performance levels. In this paper, Multi-valued Decision Diagram (MDD) based approaches are discussed to conduct the performability analysis of multi-state systems. Results: Examples are provided to illustrate the use of MDD approaches for three types of multistate systems, e.g. multi-state series-parallel systems, multi-state k-out-of-n systems and multi-state linear consecutive-k-out-of-n systems. Conclusion: The empirical results show that the MDD approaches have lower computational complexity than the traditional methods in model generation and evaluation.

Optimal structure screening for large-scale multi-state series-parallel systems based on structure ordinal optimization

Multi-state series-parallel systems are widely-used for representing engineering systems. In real-life cases, engineers need to select an optimal system structure among many different multi-state series-parallel system structures. Screening of system structures is meaningful and critical. Moreover, to design a reliable structure, reliability evaluation is an indispensable part of the process. Due to the large number of available system structures, the computational burden can be huge when selecting the optimal one. Also, the number of components and possible states of each system can be enormous when the system scale is large, which causes significant complexity in exact reliability evaluation. To effectively select the optimal structure among numerous multi-state series-parallel systems under a reliability constraint, this article proposes an optimal structure screening method called the structure ordinal optimization. The proposed method combines the fuzzy universal generating function technique with an ordinal optimization algorithm. The fuzzy universal generating function technique is applied to reduce the computational time by approximately evaluating the reliability. Based on the approximate reliabilities, ordinal optimization helps to reduce the number of structure options and thus accelerate the screening process. Numerical examples show that the structure ordinal optimization method has advantages in computational efficiency with satisfactory accuracy.

A novel discrete bat algorithm for heterogeneous redundancy allocation of multi-state systems subject to probabilistic common-cause failure

This paper focuses on a heterogeneous redundancy allocation problem (RAP) for multi-state series-parallel systems subject to probabilistic common-cause failure and proposes a novel discrete bat algorithm to solve it. Although abundant research studies have been published for solving multi-state RAPs, few of them have studied probabilistic common cause failure, which motivates this paper. Due to the insufficient data of components, an interval-valued universal generating function is utilized to evaluate the availability of components and the whole system. The challenge of solving this kind of RAPs lies in not only the reliability estimation, but also the solution method. This paper presents a novel discrete bat algorithm (BA) for effectively dealing with the proposed RAP and alleviating the premature convergence of BA. Two main features of the adaptation are Hamming distance-based bat movement (HDBM) and Q learning-based local search (QLLS). HDBM transfers the Hamming distance between the current bat and the best bat in the swarm to the movement rate. Then, QLLS utilizes Q-learning to adjust the local search strategies dynamically during the iterations. The computational results from extensive experiments demonstrate that the proposed algorithm is powerful, which is more efficient than other state-of-the-arts on this sort of problems.

An improved methodology for evaluating availability of multistate series–parallel systems using the polynomial distribution function

PurposeThis paper employs new methods to evaluate the availability of multistate series–parallel systems, in which a number of similar components are available in each subsystem.Design/methodology/approachIn this paper, polynomial distribution function (PDF) is combined with universal generating function (UGF) and recursive algorithm (RA) methods to evaluate the availability of multistate series–parallel systems. To achieve this goal, the PDF is initially used to determine the performance rates and the probabilities corresponding to the performance states of the similar components in a subsystem. The obtained results are used to evaluate the system availability via the UGF and RA methods.FindingsIt is shown that the combined UGF and PDF (UGF-PDF) and also the combined RA and PDF (RA-PDF) methods require less computational time than did the UGF and RA methods, respectively.Originality/valueIn the UGF and RA methods, there is no difference in system availability evaluation time when considering similar or different components in each subsystem. But the proposed methods in this article do not have this restrictions; therefore, these methods can be used to evaluate system availability in optimal redundancy allocation problems. As a result, using these methods reduces the optimization time of those problems.

An investigation of new local search strategies in memetic algorithm for redundancy allocation in multi-state series-parallel systems

Redundancy allocation is widely used for optimizing system reliability in various fields. Non-homogeneous redundancy allocation in multi-state series-parallel systems has captured much attention in recent years. Due to the large and complex solution space, it is among the most difficult types of redundancy allocation problems. Due to the multimodal landscape of solution neighborhood, ordinary evolutionary algorithms (EAs) could not sufficiently exploit the solution neighborhood and thus usually trap into local optimums. Therefore, local search (LS) becomes a critical attachment to the main EA to achieve better performance. This work proposes two originally designed inter-subsystem LS methods for non-homogeneous redundancy allocation in multi-state series-parallel systems. They provide a well-balanced mechanism to reduce system cost via inter-subsystem operation and ensure system availability. The inter-subsystem local search methods are combined with quantum-inspired evolutionary algorithm (QEA), establishing an efficient memetic algorithm. The proposed memetic algorithm is validated on five benchmark problems with different availability requirements. The statistical tests results indicate that the inter-subsystem LS strategies are significantly better than the LS strategies published in QEA approach proposed by one of the authors of this paper, at small computational costs. The comparisons to various published results show that the proposed memetic algorithm achieves the best published solutions using much fewer fitness evaluations and higher robustness.

Approximate Reliability Evaluation of Large-Scale Multistate Series-Parallel Systems

Multistate series-parallel system (MSSPS) is a widely used model for representing engineering systems, whose reliability has been extensively analyzed. Universal generating function (UGF) is an efficient method for evaluating the reliability of MSSPS. However, when facing the large-scale MSSPS, where the number of system components and possible states are enormous, calclating the exact system reliability can be rather time-consuming. To evaluate the reliability of large-scale MSSPS more efficiently, this paper proposes an approximation method, named continuization discretization approximation (CDA) method. The CDA approach consists of continuization and discretization processes. The continuization process applies Gaussian approximation method based on the central limit theory and the UGF technique to evaluate parallel subsystems. While the discretization process discretizes the continuous distribution to a discrete one, and proposes an algorithm to evaluate the series subsystems efficiently. The efficiency and accuracy performance of the CDA method can be adjusted by parameters according to the computational resource and the system scale. The newly proposed method is compared to the existing methods in evaluating the large-scale MSSPS. Numerical examples show that the CDA method has evident advantage in computational efficiency with satisfactory accuracy performance.

Reliability evaluation of multi-state series-parallel systems with common bus performance sharing considering transmission loss

In a multi-state system with common bus performance sharing, each multi-state unit has a random performance and should satisfy its individual random demand. If the performance of a unit exceeds its demand, the surplus performance can be transmitted to units that experience performance deficiency. Performance transmission is an important feature of systems with performance sharing. Transmission loss exists widely in many real systems with performance sharing mechanism. However, transmission loss has never been studied in such systems. The reliability of systems with performance sharing will be overestimated if the transmission loss is ignored. In this paper, the transmission loss defined as a function of the transmission distance and the performance transmitted through the common bus is taken into account in systems with common bus performance sharing, and a universal generating function (UGF)-based method is proposed to evaluate the system reliability. Analytical and numerical examples are given to validate the proposed method.

Reliability Assessment in Distributed Multi-State Series-Parallel Systems

The distributed system plays a major part in power system, and numerous researches are devoted to evaluate the reliability of system with distributed generators. However, the traditional reliability evaluation methods are confronted with a great challenge of huge computational burden, especially with the high penetration of distributed generators. Hence, in this paper, a new distributed system named as ideal line distributed series-parallel system is presented. In this system, the interaction between the basic components and distributed components are concerned and the structure between these components is described as series-parallel component. This defined distributed series-parallel system can simplify the reliability assessment process. Two analytical reliability evaluation approaches based on probability theory are proposed to release the computational burden, where the number of distributed generators is varying from 1 to m. The relationships between series-parallel components are quantified and the distributed generators are equivalent as traditional generators in these reliability evaluation approaches. The comparisons of computation cost and complexity are given to demonstrate the accuracy and efficiency of proposed reliability assessment methods.

Optimizing Dynamic Performance of Multistate Systems With Heterogeneous 1-Out-of-<inline-formula> <tex-math notation="LaTeX">${N}$ </tex-math> </inline-formula> Warm Standby Components

This paper models and optimizes dynamic performance of multistate systems with a general series parallel structure. Each system component is a 1-out-of- ${N}$ warm standby configuration of heterogeneous functional elements, which can be characterized by different time-to-failure distributions, performances, and costs. The entire system must satisfy a random demand specified by a time-dependent distribution. An iterative algorithm is developed for determining performance stochastic processes of particular components. A universal generating function technique is used for evaluating expected system availability and unsupplied demand over a particular mission time for the considered system. Two types of optimization problems are then identified and solved, with the objective of finding component structures and element activation sequences to maximize system availability, or minimize unsupplied system demand, or minimize total cost. Optimization results can facilitate the optimal decision on design and operation of multistate series parallel systems. A practical example of a power station coal transportation system is provided to illustrate application of the proposed methodology and optimization problems.

Reliability Evaluation of Compressor Systems Based on Universal Generating Function Method

At present, universal generating function (UGF) is a reliability evaluation technique which holds the bare-looking and easily program-realized merits in multi-state system. Thus, it is meaningful to apply this method to an actual industry system. Compressor systems in natural gas pipelines are series-parallel multi-state systems, where the compressor units in each compressor station work in a parallel way and these pressure-boosting stations in the pipeline are series connected. Considering the characteristic of gas pipelines, this paper develops two different UGFs to evaluate the system reliability. One (Model 1) establishes a system model from every compressor unit while the other (Model 2) considers the whole system as a combination of multi-state components. Besides, all the parameters of “weight” in UGFs are obtained from thermal-hydraulic models based on the actual engineering and “probability” from Monte Carlo simulation. The results show that the system reliabilities calculated by different UGFs are approximately equal. In addition, the demand of gas and the gas pipeline transportation system show a reverse trend. Because the number of parameters needed in Model 2 is far less than that needed in Model 1, Model 2 is simpler programming and faster solved.

Availability optimisation of multi-state series-parallel systems with human failures

Availability optimisation of multi-state series-parallel systems with human failures

Availability optimization of multi-state series-parallel systems with human failures

Nowadays, the availability of systems and tendency to commit errors has become an important issue during the design of many engineering systems. Humans play a key role in the overall availability of engineering systems because many systems are interconnected by human links to a certain degree. Obviously, if the system is multi-state, methods to achieve the availability values of the system have an important role. In this paper, a redundancy optimisation problem for multi-state systems subject to human failures is formulated to minimise the system cost. Human failures are recognised with a fault tree analysis method, the universal generating function algorithm is used to compute the system availability and the genetic algorithm to optimise the system structure. To illustrate the credibility and applicability of the model, a case study is offered and analysed. The result of the solved model is corroborated and shows the efficiency of the used methods.

On the optimal redundancy allocation for multi-state series–parallel systems under epistemic uncertainty

In this paper, we study the redundancy allocation problem (RAP) for multi-state series–parallel systems (MSSPSs). For each multi-state component, the exact values of its state probabilities are assumed to be unknown, due to epistemic uncertainty (EU), and only conservative lower and upper bounds of them are given. The objective of the RAP is to simultaneously maximize the supremum and infimum of the system's uncertain availability, under a cost constraint. The problem is two-stage and multi-objective. In this work, we: 1. provide a linear-time algorithm to obtain the component state distribution, under which the uncertain system availability will be at its supremum or infimum; 2. show that the problem is reducible to one-stage; 3. analyze the landscape of MSSPS RAP under EU and propose a modified NSGA-II, with targeted designs of repair and local search operation. The proposed algorithm is compared with standard NSGA-II on multiple benchmarks. The results show that the proposed algorithm significantly outperforms the standard NSGA-II in both optimality and time efficiency.