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Optimal loading and protection of multi-state systems considering performance sharing mechanism
... At present, in this era of rapid development of science and technology, the modernization level of industrial production is gradually improving. Under this development trend, various engineering systems have gradually become huge and complex, showing the characteristics of multi-state and performance sharing, especially in power, communication, computer, intelligent transportation and other systems abound with high techniques [1][2][3][4][5][6]. As these systems play a vital role in the stable operation and development of society and economy, once the system fails, it may cause major economic losses and disastrous consequences that affect society [7][8][9]. ...
... Then, in order to improve the reliability of such complex systems, scholars introduced a performance sharing mechanism in the study of multi-state systems. Performance sharing means that each unit in the system can transfer its surplus performance to other units with deficient performance on the premise of meeting its own requirements, so as to reduce the waste of resources and improve the system reliability [3,5,11]. Lisnianski and Ding [16] first introduced this important type of redundancy, performance sharing, in the study of multistate systems. ...
... 0,0 0,5 5,0 [1,2] 0,5 [1,3] 0,0 5,0 0,5 [1,4] 5,0 0,5 0,0 [1,5] ...
With the development of science and technology, the structure of engineering system has become increasingly large and complex. In order to ensure the safety and stability of the system in operation, the reliability evaluation of complex system has become an important research field. Based on the actual engineering system, this paper proposes a multi-state system with multi-level performance sharing mechanism. On this basis, we established a system reliability evaluation model using universal generating function technique. Through numerical examples, the application of the model and analyze the influence of different parameters on system reliability are demonstrated. In addition, we also use genetic algorithm to optimize the allocation of components in the system, so as to improve the reliability of the system. Different from the previous studies on system reliability with common bus performance sharing, the system proposed in this study is more general.
... The common bus is used to connect these components and transmit surplus supply to the component that is experiencing performance deficiency. Such systems are widely seen in power distribution systems, distributed computing systems, data transmission systems, communication systems, etc. (Lisnianski and Ding, 2009;Levitin, 2011;Xiao et al., 2016). Lisnianski and Ding (2009) were the first to conduct a reliability analysis of this type of performance sharing model. ...
... Qiu and Ming (2019a) studied a multi-state series performance sharing system under epistemic uncertainty. In addition, some related optimal problems of performance sharing systems have also been studied (Xiao and Peng, 2014;Xiao et al., 2016;Zhai et al., 2017;Peng, 2019;Yan et al., 2020). ...
In this article, a performance sharing system with transmission loss and a shock operation environment is studied. Such systems are widely found in power distribution systems, distributed computing systems, data transmission systems, communication systems, and so on. The system consists of n components and each of them works to satisfy its demand and shares its performance surplus with others through a common bus. When the system operates, it may suffer a variety of stresses from its operating environment, which can be regarded as random external shocks, and the transmission loss is also wildly seen in engineering systems. Therefore, the random shocks and transmission loss are considered in this article. The performance level of a component is affected by three types of random external shocks – invalid shocks, valid shocks and extreme shocks. The system fails if at least one component cannot satisfy its demand. A finite Markov chain imbedding approach and phase-type distributions are used to estimate the performance level for each component and the universal generating function technique is applied to analyze system reliability. Analysis of a power distribution system is given to show the application of the model under study and the effectiveness of the proposed method.
... Probability of m particular components working normally in a common cause group composed of n identical components at time t [25] considered performance sharing in a series MSS, where the performance can be shared among all units through the common bus. Xiao et al. studied the optimal allocation and maintenance [26], the optimal loading and protection [27], and optimal design [28] problems for systems with performance sharing. Yu et al. [29] evaluated the instantaneous availability of the repairable MSS with performance sharing. ...
... Calculate the UGF of each subsystem at each phase after performance sharing among subsystems and performance transmission between phases by Equations (22)(23)(24)(25)(26)(27)(28). 10: ...
Multi-state phased mission systems (MS-PMSs) with common bus performance sharing widely exist in practice. Their subsystems are often designed as redundant k-out-of-n(G) systems to ensure the reliable operation of the whole system, such as power generating systems. However, common cause failures (CCFs) of components can decrease the reliability of such redundant subsystems. Considering the above two factors, we propose a new reliability model for the MS-PMS with k-out-of-n(G) subsystems and performance sharing subject to CCFs. Each subsystem contains multiple multi-state components, whose behaviors are analyzed using the Markov chain. Components in each subsystem may belong to a CCF group, and each CCF group suffers multiple independent CCF modes. We propose a Binary Indicator Universal Generating Function (BIUGF) method to evaluate the system reliability by adding an indicator variable and a pointer vector in the traditional universal generating function method, and the impacts of CCFs are taken into account using an implicit analysis method. The method is illustrated and validated through applications to power supply systems, which justify the effectiveness of the proposed method.
... For example, Levitin and Lisnianski [29], Nourelfath et al. [30,31], and Liu et al. [31] put forth the joint formulations for redundancy and preventive maintenance optimization of MSSs with the aim of minimizing the lifecycle cost. The maintenance optimization of MSSs with common bus performance sharing was investigated in [32] and [33]. Nourelfath et al. [23] and Fitouhi and Nourelfath [34] considered the trade-off between production planning and maintenance scheduling of MSSs, and the lot-sizing and preventive maintenance strategies were optimized in a joint manner. ...
... If the time duration for maintenance is non-ignorable, it is oftentimes assumed to be either a constant or exponentially distributed to reduce computational complexity. For example, the constant maintenance time was adopted by Xiao et al. [32] for series-parallel MSSs. In de Smidt-Destombes et al. [42], Wu et al. [43], Zhang et al. [44], Lisnianski and Ding [40], Moghaddass et al. [41], and Mendes et al. [45], the time duration of maintenance was assumed to be exponentially distributed, the deteriorating and repairing processes of an MSS can be, therefore, characterized by the continuous-time Markov models. ...
In engineering scenarios, failures of some components in a system may not always lead to the failure of an entire system. In such cases, the system can continuously operate while some components are being repaired. On the other hand, due to limited maintenance capacity, such as manpower and/or repair facility, maintenance actions can only be executed serially rather than in parallel. In this study, a new maintenance optimization problem for multi-state systems with single maintenance capacity is studied. The homogeneous continuous-time Markov process is used to characterize the deterioration of multi-state components in a system. In contrast to the exponential assumption for the distribution of maintenance time in most reported works, the time for each maintenance task can be arbitrarily distributed in our study. The embedded Markov chain is constructed to model the state transition process of a system by introducing decision epochs. Two optimization problems are formulated by treating either the stationary availability or the expected performance capacity of a system as an objective under the constraint of the average maintenance cost per unit time. The genetic algorithm is customized to resolve the resulting optimization problems. An illustrative example is given to demonstrate the effectiveness of the proposed method.
... Moreover, the common bus performance sharing systems with arbitrary number of multi-state components were developed in [18] where traditional steady-state reliability evaluation was investigated. Availability of systems with performance sharing considering loading and protection was analyzed in [19]. Reliability of non-repairable phased-mission systems with performance sharing was evaluated where the state of a producer was binary in [20]. ...
... The method for evaluating steady-state reliability of systems with performance sharing in [17]- [19] was universal generating function technique. The combined stochastic processes 8 methods and universal generating function or Lz transform techniques were utilized for timevarying reliability evaluation in [16], [25]. ...
With the increasing interconnection of the power grids, the imbalanced distribution between power generation and demand in different areas has been effectively alleviated. In practical power systems, the subsystems in different areas need to meet the load requirements of each subsystem, and the performance sharing among different subsystems is one way to increase system reliability. Moreover, each subsystem can be configured with redundancy techniques especially warm standby, which consumes less energy than hot standby and has a shorter recovery time than cold standby. Furthermore, both the generating units and the performance sharing mechanism may have more than binary states in practice. Therefore, in this paper, the reliability evaluation of power systems with multi-state warm standby and multi-state performance sharing mechanism is proposed. Arbitrary state transition time distributions are allowed, and the successful activation probabilities for warm standby generating units are also embedded in the proposed model. The multi-state decision diagram (MSDD) technique is developed for system reliability evaluation. Time-dependent reliability is presented in illustrative examples to validate the proposed model and technique.
... To achieve maximum system reliability, the optimal element loading was studied for linear sliding window systems [14] as well as series systems that are exposed to external shocks [15]. On the other hand, to minimize expected total cost, the joint optimization of condition-based loading level and maintenance was investigated for mission-critical systems [16], and optimal component loading was studied for systems operating in dynamic environments [17] and multistate systems [18]. ...
In engineering practice, mode switching, loading adjustment, and preventive maintenance are widely recognized as effective methods for enhancing system availability and reducing operational costs. Simultaneously considering these strategies can enhance the overall system performance, especially in scenarios with stochastic task arrivals. This article explores the integrated optimization of mode switching, loading levels, and maintenance for systems involved in processing dynamic tasks. Dynamic switching between working, idle, and offline modes is considered, leading to different cost items and failure risk. Therefore, mode switching decisions are optimized. An arrival task will be lost when the task queue is full. Then, the optimal loading level, i.e., processing rate, is investigated jointly with mode switching to balance different types of costs. Additionally, age-based preventive maintenance is optimized to strike a balance between system reliability and maintenance costs. The expected total cost under the optimization model is derived using a recursive formula. The numerical example demonstrates the superiority of the proposed joint optimization model.
... Yu et al. (2014) further discussed the repairable system model with common bus performance sharing in Levitin (2011). Xiao et al. (2016) considered a multi-state repairable loading system with performance sharing mechanism. Zhao et al. (2018) investigated the reliability of a repairable performance sharing k-out-ofn(G) structure MSS according to the Markov chain and universal generating function (UGF) technique. ...
Motivated by real-world engineering systems, this paper presents an in-depth exploration of a novel reliability model for a complex multi-state series system (MSSS) that incorporates a performance sharing mechanism. Specifically, the MSSS is composed of m distinct subsystems featuring a k-out-of-n: G structure. The ith subsystem consists of n i elements, such that a minimum of k i functioning elements is required for normal operation. Transmission devices (TDs) are present between adjacent subsystems to share surplus performance. Both element performance and subsystem demand are treated as random variables, and the performance of all elements in each subsystem is cumulated to meet its individual random demand. Subsystem failure can result from either an inability to meet performance demands or an insufficient number of functioning elements. The surplus performance of a subsystem can only be transferred to its adjacent subsystem via the TD. The entire MSSS will fail if at least one subsystem does not work properly by using performance sharing mechanism. To analyze the reliability indexes of the MSSS, a new algorithm based on the generalized universal generating function (GUGF) has been developed. Numerical examples are provided to illustrate the accuracy and effectiveness of the proposed model and method.
... For performance sharing policy optimization, Xiao et al. [25] and Xiao et al. [1] investigated the optimal performance sharing policy of a sliding window system(SWS) and a distributed computing system, respectively. Moreover, the loading and protection optimization [26] and defense and attack optimization [27] are also studied in the performance sharing systems It is worth noticing that most of the literature applied enumeration method or heuristic algorithm to solve the optimization problem. The solution space will become too large to find optimal solutions as the complexity of the systems increases. ...
Increasing research efforts have been devoted to the reliability analysis of multi-state systems (MSSs) with different performance sharing mechanisms. This paper investigates the reliability optimization problem of an MSS with two performance sharing groups. Such systems consist of multiple components and two common buses. The components connected by the same common bus make up a group and the performance can be shared in the group to meet the demand. Specially, components can be in one group or both. Taking advantage of this property, the components with larger performance and demand are preferentially placed in the overlapping part of two groups since more performance can be shared. The objective is to efficiently derive the optimal configuration policy that maximizes the system reliability. Universal generating function is utilized to develop the reliability evaluation algorithm. Then, stochastic orders are utilized to determine the components to be placed in the overlapping part, which reduces the solution space. Genetic algorithm is applied to find the optimization results in the reduced solution space. Numerical examples validate the effectiveness of the proposed reliability evaluation algorithm. The improved optimization method also outperforms the method without solution space reduction in improving the system reliability.
... For example, Yu et al. [4], Zhao et al. [5], and Peng et al. [6] studied the performance-sharing mechanisms for the series-parallel repairable binary-state system, multi-state k-out-of-n: G system, and system with two performance sharing groups, respectively. Furthermore, the research on performance-sharing mechanisms has been enriched by considering the multiple stages of operation process [7][8][9], transmission loss during performance sharing [9][10][11], related optimization problems [12][13][14], and so on. ...
Modern engineering systems are designed and utilized to realize complicated functions, and their operation mechanisms are becoming more complex. Nevertheless, prior related research mainly focused on the reliability evaluations of the systems with a single operation mechanism, which are not appropriate to depict the operation process of systems with multiple operation mechanisms. Faced with the research gaps and practical needs, this paper establishes a new reliability model for the multi-state k-out-of-n: F system composed of n subsystems, which runs under multiple interactive operation mechanisms, including performance sharing, balanced requirement, and protection strategy. The units in each subsystem can share the performance via a common bus, with the purpose of regulating the performance of all equal units. A new triggering criterion of the protection device in each subsystem is proposed based on the total performance of the units. Due to the protection from the device, the degradation rate of the units between two adjacent states decreases to a lower rate. Each subsystem breaks down when the total performance of the units reaches a critical value. According to the number of failed subsystems, the state of the entire system can be divided into multiple states. The Markov process imbedding method combined with the finite Markov chain imbedding approach is developed to obtain the probabilistic indexes of each subsystem and the entire system. The applicability of the proposed model and the effectiveness of the method can be sufficiently demonstrated by illustrative examples and sensitivity analyses.
... Failures of some components may lead to the degradation of the system performance. Thus the reliability analysis of the MSS has recently received substantial attention 2,3 . There are four main methods for reliability analysis of the MSS: stochastic process method 4 , Universal Generating Function (UGF) 5,6 , Bayesian Network (BN) 7 and Monte Carlo simulation 8 . ...
In the complex multi-state system (MSS), reliability analysis is a significant research content, both for equipment design, manufacturing, usage and maintenance. Universal Generating Function (UGF) is an important method in the reliability analysis, which efficiently obtains the system reliability by a fast algebraic procedure. However, when structural relationships between subsystems or components are not clear or without explicit expressions, the UGF method is difficult to use or not applicable at all. Bayesian Network (BN) has a natural advantage in terms of uncertainty inference for the relationship without explicit expressions. For the number of components is extremely large, though, it has the defects of low efficiency. To overcome the respective defects of UGF and BN, a novel reliability analysis method called UGF-BN is proposed for the complex MSS. In the UGF-BN framework, the UGF method is firstly used to analyze the bottom components with a large number. Then probability distributions obtained are taken as the input of BN. Finally, the reliability of the complex MSS is modeled by the BN method. This proposed method improves the computational efficiency, especially for the MSS with the large number of bottom components. Besides, the aircraft reliability-based design optimization based on the UGF-BN method is further studied with budget constraints on mass, power, and cost. Finally, two cases are used to demonstrate and verify the proposed method.
... In the reliability assessment of traditional MSS of which performance can be shared by the common bus system, reliability of the MSS is expressed as the probability of meeting the requirements of all components (Xiao and Peng et al., 2014;Yu et al., 2014;Xiao et al., 2016;Peng et al., 2016). For example, Peng et al. (2017) investigated the reliability of a system with a performance group of limited size. ...
In a multi-state system with common bus performance sharing, the residual performance of any sub-unit can be shared to other sub-units with insufficient performance by the common bus with stochastic sharing capacity. This study extends the reliability evaluation model by considering both the minimum and maximum requirements of elements, which should not only meet the random minimum requirements of each single element, but also require that there should be no excess performance. The reliability of the multi-state system (MSS) is estimated by using the Universal Generating Function (UGF) technique. Illustrative example is presented to illustrate the procedures. In addition, the genetic operator is used to obtain the optimal location of the components with maximum reliability.
... Furthermore, UGF is served as an efficient approach towards evaluating the reliability of multi-state systems [42]. For example, Su et al. [43], Cheng et al. [44], Levitin et al. [45] and Xiao et al. [46] used UGF to derive the reliability of various systems with common bus performance sharing. ...
In previous studies of shock models, the probability of shocks causing damage to the system during operation is always constant. However, it is more practical for the system to become more vulnerable to shocks under a severe condition and the probabilities of subsequent shocks causing damage to the system increase. Furthermore, few studies have focused on the shock environment of weighted k-out-of-n: F systems. To fill these research gaps, this paper proposes a novel mixed shock model for a multi-state weighted k-out-of-n: F system with a consideration of its resistance against shocks. The components in the system operate in a two-stage process and different weights are taken for their different states. The system has multiple states determined by the total system weight/performance. System failure results from the competing failure criteria of insufficient system weight/performance and from reaching the critical number of failed components. A combination of finite Markov chain imbedding approach, universal generating function technique and phase-type distribution is adopted to analyze the probabilistic indices of the components and entire system. Numerical illustrations based on a power-generating plant are presented to demonstrate the applicability of new shock model and the effectiveness of proposed method.
... A linear sliding window structure which took performance sharing into account was studied by Xiao et al. [26]. Moreover, some optimizations for the performance sharing systems were also done in recent years [27][28][29][30][31][32]. ...
In this paper, a performance-based balanced system with common bus performance sharing is established. The system consists of n components connected by a common bus. Each component has its individual performance which can be shared among all components via the common bus. The system is balanced if the performance of each component is the same and it can be rebalanced by redistributing performances among all components via the common bus if the system is out of balance. The system fails when the system is out of balance after performance sharing or the total performance of the system is less than a predetermined value, whichever occurs first. A Markov process is employed to describe the performance variation of each component and the universal generating function technique is applied to obtain analytical solution of the system reliability. A specific analysis of battery balancing of lithium battery pack is given to show the application of the proposed model and effectiveness of the proposed method.
... Then, many researches were carried out based on common bus performance sharing system. Xiao et al. discussed the optimal allocation and maintenance [22], the optimal loading and protection [23] of multi-state systems with common bus performance sharing. Peng et al. studied the model of series systems with two performance sharing groups [24], and the model of MSSs with a performance sharing group of limited size [25]. ...
In many real situations, because of the lack or inaccuracy of data, it is difficult to evaluate the performance levels and state probabilities of multistate components with precise values. Thus, the reliability evaluation of systems is always affected by the epistemic uncertainty. Existing research on epistemic uncertainty just focuses on simple multistate systems, without considering the phased mission and performance sharing characteristics of systems. Besides, the impact of transmission loss and performance storage on reliability needs to be considered in the modeling of performance sharing systems. In this article, considering the epistemic uncertainty, transmission loss, and performance storage simultaneously, an efficient reliability evaluation method for multi-state phased mission systems with common bus performance sharing is proposed. The transmission loss during the processes thatperformance sharing between subsystems and transferring between phases is considered in the system model. A modified Markov model combined with the mass function is adopted to measure the precise belief degree of component states at any given moment. Then, the belief universal generating function (UGF) method based on the Dempster—Shafer evidence theory is utilized to evaluate the uncertainty of the system instantaneous availability. Finally, two case studies are carried out to demonstrate the effectiveness of the proposed method and explore the dynamic system availability under epistemic uncertainty.
... They employed a universal generating function-based algorithm to evaluate the system availability and the genetic algorithm to solve the cost minimization problem. Xiao et al. (2016) considered the performance sharing problem in multi-state loading systems. Specifically, they integrated the effect of loading and external impact protection to maximize system availability. ...
The reliability of a system that shares a single type of resource with others has been studied in the literature. In reality, a system may require different types of resources. For example, a building service system may be powered by electricity (for lighting) and gas (for heating). However, in the literature, research in this area is scarce. This paper therefore investigates the reliability of a system with multiple nodes. Each node has demand requirement for two types of resources, which can be shared among the system nodes subject to their bandwidths, respectively. In addition, the resources may be able to substitute each other. This paper considers both unidirectional and bidirectional substitutions. The system is said failed if either resource supply in a node is smaller than its demand even after performance substitution and sharing. A universal generating function technique is proposed to evaluate the system reliability. Numerical examples are presented to illustrate the applicability of the model. The influences of bandwidths and substitution rates on system reliability are also discussed.
... Recently, there are amounts of studies concentrating on the reliability modelling of industrial systems with performance sharing mechanism (Xiao et al.,2015). Previous research widely assumed that the system will only function if all subsystems satisfy their demand individually (Peng et al.,2016). ...
Performance sharing mechanism is widely applied in energy systems, computing systems and other types of industrial systems to increase system reliability. In reality, some systems may allow certain extent of performance deficiency instead of requiring every subsystem in the system satisfy its demand. Moreover, the subsystems in systems may combine with amounts of components, say that, a series of generators should be simultaneously used to provide the necessary energy for a given area. In this paper, we consider a system with performance sharing mechanism where the system remains functional as long as the summed weighted deficiency for subsystems after performance sharing is smaller than a predesigned reliable threshold. The amount of shared performance is further assumed to restricted by the bandwidth capacity of the system. We further consider the case where the surplus can be redistributed to maximize system reliability. Universal generating function technique is employed to model the reliability of the system and numerical examples are provided to illustrate the applications.
... Levitin et al. (2019) studied the dynamic availability and performance deficiency of performance sharing systems with imperfectly repairable units. Based on these extended models, some related optimization problems were also studied (Levitin et al., 2016;Peng, 2019;Xiao & Peng, 2014;Xiao et al., 2016;Yan et al., 2020;Yi et al., 2019;Zhai et al., 2017). ...
From previous studies on performance sharing systems, the system structure is considered to be either series, or series-parallel, or k-out-of-n structure. To enrich the research and extend application on performance sharing systems, the consecutive-k-out-of-n: F system with two performance sharing groups is investigated in this paper. Such systems consist of n units and two common buses, and each unit has a random performance to satisfy its random demand. Units connected by the same common bus make up one performance sharing group. If a unit has surplus performance, the surplus performance can be transmitted to the units which are experiencing performance deficiency and belonging to the same performance sharing group via the common bus. If the demand of one unit is not satisfied, the unit has to be shut down. The system fails due to at least k consecutive shut-down units. The universal generating function technique is used to analyze the system reliability. Analytical and numerical examples of the heating system are presented to demonstrate the new model under study.
... The genetic algorithm was used to optimize elements allocation and replacement intervals such that the total maintenance cost is minimized. Xiao et al. [17] evaluated the steady availability of systems by integrating the effect of loading on the unit failure rate and the protection against external impacts on the system with performance sharing mechanism. Zhai et al. [18] studied defense and attack strategies for a performance-sharing system which is subject to intentional attacks. ...
This paper addresses the reliability analysis problem for a k-out-of-(n+1): G multi-state system (MSS) with performance sharing mechanism. This MSS consists of two categories of units connected into a star topology configuration. The first category of unit is a main unit of the system which occupies the central location of the star configuration, and the second category of units are reserve units distributed at the terminal locations. All reserve units are connected to the main unit in a point-to-point manner through the intermediate transmitters with limited performance transmission capacities. Both the performance and the demand of each unit are random variables. The surplus performance of any reserve unit can be transmitted to the main unit through the transmitter between them, and then the excess performance of the main unit can be further transmitted to other reserve units which are experiencing performance deficiency. The system is reliable if and only if the demands of the main unit and at least k-1 reserve units are satisfied after performance redistribution. An algorithm based on the universal generating function (UGF) technique is developed to calculate the reliability of the proposed MSS with performance sharing. An analytical and a numerical example are presented to illustrate the advantages and applications of the proposed method.
... The authors provided three metrics related to the phase transition to measure the robustness of the grid. Xiao et al. [17] and Xiao and Gao [18] focused on the characteristics of multi-state systems and utilized different performance-based measures to quantify and improve system reliability. Several stochastic failure models were developed to reflect failures of networks with different maintenance policies or other patterns [19][20][21]. ...
A cascading failure is a phenomenon in which the failure of one or several nodes triggers the failure of other nodes. In cyber-physical systems, the failure propagation of edge information adversely affects the network performance. Past research on cascade failure has primarily focused on the node distribution but has ignored the influence of the edges on the cascading failure and network stability. In addition, research is lacking on the internal connections of nodes and edges and the role of the constituent topology. In this paper, a cascading failure model that considers different types of nodes and edges and their contribution to the network is developed and analyzed based on cluster aggregation in cyber-physical systems. The topological changes in the networks after multiple aggregations of clusters during a cascading failure are described. The relative capacity of the flow of the edges is proposed to determine the importance rank of the network elements, and the integrated transfer efficiency of the network is used to assess the performance. The model can detect the most important nodes, edges, and critical failure paths. These factors are of great significance for adjusting the topology of the network or designing new networks. Examples of networks are used to demonstrate the proposed method.
... Then, many researches were carried out based on common bus performance sharing system. Xiao et al. discussed the optimal allocation and maintenance [22], the optimal loading and protection [23] of multi-state systems with common bus performance sharing. Peng et al. studied the model of series systems with two performance sharing groups [24], and the model of MSSs with a performance sharing group of limited size [25]. ...
Multi-state phased mission systems (MS-PMSs) with common bus performance sharing widely exist in practical engineering, such as intelligent transmission systems and power supply systems. The challenges in assessing the reliability of these systems come from the dependence among phases and the complexity of multi-state behaviors of components. Besides, the performance transfer between different phases and the performance transfer with transmission loss have never been studied in such systems. To effectively address these problems, this paper proposes a reliability model for MS-PMSs with common bus performance sharing considering transmission loss and performance storage. First, a novel Markov model for multi-state components is established to solve the dependence problem among phases. Then, considering the performance sharing among subsystems within each phase through the common bus and the performance transferring between phases by an energy storage device, an iteration method combined with transmission loss function is adopted to establish the system reliability model, where the performance transmission loss is proportional to transmission distance and transmission performance. Furthermore, a universal generating function (UGF) based method is used for evaluating the system reliability. Finally, an analytical example and a case study of the IEEE-24 bus system are utilized to verify the availability of the proposed method.
... Performance sharing has also been extended to a series-parallel system (Xiao and Peng, 2014;Xiao et al., 2016c), a phased mission system (Yu et al., 2018), a multi-state k-out-of-n system , and a capacitated system (Wu et al., 2019). ...
Existing research on the linear sliding window system (SWS) has focused on modeling and extending to different system structures. Little research has devoted to analyzing the performance redistribution internally. This research considers an SWS with a common bus such that the performance can be redistributed among system elements. The reliability model of an SWS with a common bus is built with consideration of performance sharing. We extend the universal generating function technique to represent the states of the system, and formulate an optimization problem to determine the optimal performance sharing policy. We further suggest a reliability evaluation algorithm based on the representation function of the system states and the optimal performance sharing policy. Optimal allocation of the system elements is also considered in this work. Numerical studies show that the system reliability can be improved by considering a common bus performance sharing. The optimal allocation of system elements can also increase the system reliability without incurring extra costs.
Modern technological advances require unprecedented reliability improvements in engineering systems. This is especially true for the numerous complex structures used as infrastructure in modern industries such as communication and power systems. The structures of systems used in these industries are generally designed as centralized networks, mainly configured in a star structure, in which the end nodes are physically connected to a central one. The principal objective of the current study is to develop a novel approach for reliability analysis and design optimization of such networks, in which all nodes might consist of more than one multi-state heterogeneous component. The performance of each node will then be evaluated as affected by different activation strategies with switch failure also considered. Moreover, it is possible for both central and end nodes in star networks to share their surplus performance, in which case transmission losses must be considered. The universal generating function (UGF) method will be exploited to develop the relevant model. Another aspect of this study involves investigating the optimal component allocation, corresponding activation strategies, and resulting optimal system reliability. Finally, the model will be validated using a numerical example and a real case study.
Production systems with storages have recently attracted considerable attention from the reliability research community. The existing models assume a single type of resource consumed for product generation. This work extends the state of the art by considering a more general and practical model, where multiple resource supply subsystems (RSS) provide different kinds of necessary resources during the production cycle. Each RSS has a storage with limited capacity to save surplus resource, which may be supplied to the production unit when the RSS fails. The product unit may operate with different productivity levels, and RSS and storages can be chosen from multiple types. We formulate a new optimization problem, which chooses the productivity level of the production unit and the types of RSS and storage for each kind of resource required for the production to maximize the expected manufacturer's profit. A new numerical algorithm is proposed to evaluate the expected profit and the genetic algorithm is implemented to solve the proposed optimization problem. A detailed case study of a chemical reactor system with four RSSs supplying two reagents, catalyzer, and cooling water is provided to demonstrate the proposed model and influences of several model parameters on the optimization solutions.
In the complex multi-state system (MSS), reliability analysis is an important research content, both for equipment design, manufacturing, operation and maintenance. Universal Generating Function (UGF) is an essential method in reliability analysis, which efficiently obtains system reliability by a fast algebraic procedure. However, when structural relationships between subsystems or components are unclear or without explicit expressions, the UGF method is difficult to use or not applicable at all. Bayesian Network (BN) has a natural advantage in terms of reliability inference for the relationship without explicit expressions. When the number of components is extremely large, though, it has the defects of low efficiency. To overcome the respective shortcomings of UGF and BN, a novel reliability analysis method called UGF-BN is proposed for the complex MSS. In the UGF-BN framework, the UGF method is first used to analyze the bottom components with a large number. Then probability distributions obtained are taken as the input of BN. Finally, the reliability of the complex MSS is modeled by the BN method. This proposed method improves the computational efficiency, especially for the MSS with a large number of bottom components. Besides, the aircraft reliability-based design optimization based on the UGF-BN method is further studied with budget constraints on mass, power, and cost. Finally, two cases are used to demonstrate and verify the proposed method.
The load under which a system operates has a significant impact on failure behaviors of systems and their components. System reliability can be improved with not only an optimal maintenance strategy but also an optimal load distribution among components. This chapter proposes an approach to address the load distribution problem for multi-state systems using a selective maintenance strategy. A joint optimization model was formulated to optimize load distribution and allocation of limited maintenance budget to maximize the probability of the system successfully completing the next mission. A genetic algorithm was employed to solve the optimization problem. The results indicated that the proposed method achieves better results than traditional methods without considering load distribution.
This paper, for the first time, models and optimizes the uploading and downloading pace distribution in a production-dual storage system that must supply a certain demand during a specified mission time. The surplus product generated by the production unit is uploaded to available storage unit(s), which may be subsequently downloaded to supply the system demand in the event of the failure of the production unit. The uploading and downloading paces of the two storage units can greatly affect failure probabilities of the storage units and further the mission success probability (MSP). This paper makes advancement in the state of the art by suggesting a probabilistic approach for evaluating the MSP of the considered production-dual storage system. The optimal uploading and downloading pace distribution problem is then solved to maximize the MSP. Two case studies of water supply systems respectively with identical and dissimilar tanks are conducted to illustrate the proposed model and optimal uploading and downloading pace distribution solutions. Impacts of several system parameters (system demand, performance of the production unit, reliability and initial amount of product in storage units) and their interactions on the MSP and optimization solutions are also investigated through the case studies.
The loading level applied during a system's operation can greatly affect the system's productivity and time-to-failure distribution. The optimal loading problem aims to determine loading levels leading to the best system performance. While this problem has been solved for many types of technical systems, very little work was devoted to production systems (PSs) with storage and the existing model assumed the storage is fully reliable and failed to consider effects of the storage's loading levels. This paper makes contributions by solving the optimal loading problem for an imperfect PS having an unreliable storage with possibility to choose different load levels that determine the storage's uploading and downloading paces and load levels that determine the productivity of the PS. Moreover, the downloading load level is chosen dynamically depending on the beginning time of the storage downloading. We evaluate and minimize the expected cumulative unsupplied demand (EUD) during the mission, and perform a detailed case study of a water supply system to demonstrate the proposed model and optimal loading policy solutions. Influences of several model parameters (mission time, PS's reliability, storage's reliability, capacity and initial amount) and their interactions on the EUD and optimization solutions are also examined through examples.
A multi-state star network with bi-level performance sharing mechanism is studied. According to the proposed topology, the network consists of a main node, which occupies the central position in the network, and n terminal nodes, which are directly connected to the central part through primary transmission lines with limited capacities. Each terminal node contains groups of components connected to a secondary common bus with a specific transmission capacity. Also, each group has a demand to satisfy. The groups within the same terminal node are able to share their surplus performance through a secondary common bus while the different terminal nodes may also share surplus performance with the main node through their primary transmitters. An adverse feature of real performance sharing systems is transmission loss that has to be taken into account if accurate reliability estimates are sought. To address this problem, the present study considers transmission loss at two levels. Network reliability is, then, calculated using a universal generating function (UGF)-based method while the best component allocation to maximize network reliability is also determined. The proposed method is eventually validated through a numerical example and a case study.
This paper presents a new single-objective redundancy allocation problem (RAP) for a system with subsystems in a series-parallel configuration. Different types of binary-state components are available for each subsystem, without the choice of component mixing, but the allocated components must be identical. Moreover, the components have a continuous performance level, i.e., their performance level is between zero and their maximum performance level. The presented model aims to maximize the system's availability by determining the optimum number and type of the allocated components to each subsystem in some constraints, such as the system's cost and weight. Since the components have continuous performance levels, the performance levels of the subsystems and the system are also continuous. First, the real-time and instantaneous availability of components, subsystems, and systems is modeled and calculated by adopting a modified universal generating function. Then, the mathematical model for the above-mentioned RAP is presented and solved using a Genetic Algorithm (GA), a Particle Swarm Optimization (PSO), and a Teaching-Learning-Based Optimization (TLBO) meta-heuristics. Next, a full enumeration technique is used to validate the performance of the adopted meta-heuristics as well as to validate the presented mathematical model. The results show the superiority of the adopted GA to solve the presented RAP. Finally, sensitivity analyses of the model's input parameters are conducted using a GA, and the effects of changing the model's parameters on the system's availability are investigated.
In the operating process of multi-component systems, components may interact with each other in a way that the load of a failed component is taken up by its nearby components at varying proportions. As a result, the failure rates of the proximate components are accelerated while distant components are not affected. This paper develops a novel spatial model for estimating the reliability of a load-sharing system considering spatial dependence and proximity effects. The spatial model is applicable to systems with heterogeneous or homogeneous components and is suitable for systems with or without distance information. To investigate the importance and significance of the spatial effect, we compare our spatial model with an equal load-sharing through numerical examples. Our results demonstrate that our model is more accurate than standard load sharing models in evaluating system reliability when spatial effects exist.
It is a common practice to use product storage to enhance the system operation efficiency and mission success probability (MSP). However, very few studies in the reliability literature considered the storage component and none of the existing models addressed the optimal loading problem. This paper contributes by analyzing and maximizing the MSP for repairable systems with load-dependent performance and product storage of limited capacity. The storage is used to cumulate surplus product when the system performance exceeds the demand and compensate the deficiency when the system has insufficient performance or is failed and under repair. Both time-to-failure and time-to-repair are random, following arbitrary distributions. A numerical MSP evaluation algorithm is put forward for the considered repairable system with specified demand and mission time. As another contribution, the optimal loading problem is solved to determine the loading policy that maximizes the MSP. A case study on a repairable pump system in a chemical reactor is provided to examine the influences of storage capacity and initial storage amount on the MSP and the optimized loading policy.
In this article, we discuss an optimal loading for items with lifetimes described by failure models that are popular in reliability and statistics. The obtained results can be relevant, for example, for production/manufacturing systems. The expected productivity and the mission success probability are maximized with respect to the value of a load. It is shown that the optimal load for the considered settings is not necessarily the load that maximizes the production rate. The crucial function in our discussion is the production rate over the load. It is shown that, depending on the model, the optimal load can be equal, smaller or larger than the value of the load that achieves the maximum of this function. The accelerated life and the proportional hazards failure models are considered, as well as the additive hazards model. Illustrative examples confirm our theoretical findings.
This paper considers fuzzy multi-state systems (MSSs) with performance sharing between adjacent units, in which each fuzzy multi-state unit has a random performance and a random demand. If the performance of a unit exceeds its individual demand (performance > demand), its surplus performance can be transmitted to its adjacent unit that experiences performance deficiency (performance < demand). The performance/demand of a unit is represented by the performance/demand levels and the corresponding state probabilities. It is usually difficult to estimate the precise state probabilities due to the inaccuracy and insufficiency of data, therefore, fuzzy state probabilities are given instead of precise state probabilities. This paper proposes a fuzzy universal generating function (FUGF)-based method to evaluate the fuzzy reliability of series systems with performance sharing between adjacent units considering the parametric uncertainty related to the state probabilities of the performance/demand levels of units. The originality of this work lies in two aspects: i) the surplus performance of a unit can only be shared with its adjacent units which experience deficiency; ii) the parametric uncertainty related to the state probabilities is considered. An illustrative example is provided to validate the proposed method.
From the reliability modeling perspective, the warm standby component can fail both in the warm standby state and the normal working state, and its lifetime in the normal working state may depend on the duration of the warm standby state.
The redundant systems with warm standby components studied in the previous chapters are all simple parallel structures. In practice, the systems may have more complex structures, such as series-parallel system, consecutive k-n system, linear sliding window system, etc.
Redundancy techniques have been extensively utilized to enhance the reliability of engineering systems.
This book introduces the reliability modelling and optimization of warm standby systems. Warm standby is an attractive redundancy technique, as it consumes less energy than hot standby and switches into the active state faster than cold standby. Since a warm standby component experiences different failure rates in the standby state and active state, the reliability evaluation is challenging and the existing works are only restricted to very special cases. By adapting the decision diagrams, this book proposes the methodology to evaluate the reliability of different types of warm standby systems and studies the reliability optimization. Compared with existing works, the proposed methods allow the system to have an arbitrary number of components and allow the failure time distribution of components to observe arbitrary distributions. From this book, the readers can not only learn how to evaluate and optimize the reliability of warm standby systems but also use the methods to study the reliability of other complex systems.
The redundant system with warm standby components is still considered in this chapter.
Warm standby redundancy is a fault-tolerant technique balancing the low economical efficiency of hot standby and the long recovery time of cold standby.
Motivated by real-world complex engineering systems, this research proposes a multi-state system with two levels of performance sharing. In this system, there exists a unique performance transmitter between any two adjacent subsystems and any two adjacent elements within each subsystem. Surplus performance of a subsystem or an element can only be transmitted to its adjacent subsystems or elements. We build the reliability model of the proposed system and suggest a corresponding reliability evaluation algorithm by extending the existing universal generating function technique. Since the performance sharing is only allowed between adjacent subsystems and elements, the element allocation and sequencing will affect the system reliability. Due to the complexity of the combinatorial optimization problem, we use the genetic algorithm to find the optimal allocation and sequencing of the elements. An analytical example is provided to illustrate the reliability evaluation algorithm. Numerical experiments are carried out to demonstrate how the optimal allocation and sequencing as well as the capacities of the performance transmitters affect the system reliability.
In this research, we modeled the reliability of a power grid with consideration of the power sharing among different regions. By extending the universal generating function technique, a reliability evaluation algorithm for a general power grid was suggested. A joint optimization model was formulated to determine the optimal redundancy and maintenance strategy in order to minimize the economic cost while meeting the reliability requirement. Numerical examples were used to illustrate the effects of different grid configurations and maintenance strategies on the reliability and cost of the power grid. It was shown that the system operation cost could be minimized by determining the optimal allocation of generators and the optimal maintenance strategy.
Reconfigurable manufacturing systems (RMSs) must be accurately designed in enterprises as it is subjected to many types of disturbances. Modeling the direct impact of each type of disturbance on RMSs and analyzing their structural-level vulnerabilities is significant for operating and optimizing RMSs. In this study, the structural vulnerability of RMSs was evaluated based on the principle of entropy and a Markov model. The proposed approach includes (a) a multistate Markov transfer equation of the manufacturing unit, (b) a method for analyzing the state and calculating the brittleness entropy of manufacturing units, (c) a method for identifying the impact of the status and capacity of a buffer on the structural vulnerability, (d) an efficient state simplification technique for RMSs based on the universal generating function (UGF) method, and (e) a quantitative assessment of the structural vulnerability. Moreover, the proposed approach was used to evaluate the structural vulnerability based on pertinent-vulnerability analyses with a cartoon box production line as an example. The results show that (a) the units connected in series are more vulnerable and that the state of the unit affects the system vulnerability, (b) the buffers reduce the system vulnerability, and (c) the UGF method significantly improves the efficiency of the vulnerability evaluation.
The reliability of the transportation network is studied by simulating the attack on the important edges. The important edges are deliberately attacked, and the impedance of the attacking edges is set to infinity. Then the influence of the characteristics of single failure edge on network reliability change and the influence of failure edge number on network reliability change are studied. Two test schemes are designed to compare the changes of network reliability in three failure experiments. At last, for the transportation network ofZhengzhou-Europe International Block train, the simulation experiment of the transportation network reliability is curried out to demonstrate the proposed method. The reliability of the transportation network is studied through numerical simulation, which is helpful to improve the reliability of the transportation network and reduce the huge economic loss caused by the failure.
With the progress of technology, the model of linear multistate consecutively connected systems (LMCCS) is applied more and more widely, such as telecommunication systems, internet of things, etc. LMCCS contains a series of linearly arranged nodes and any disconnection between nodes will lead to the failure of the whole system. Thus, the reliability of a linear multistate system is affected by many factors such as the positioning of elements. In order to be able to incorporate different types of complicated practical factors, researchers have proposed different extensions for LMCCS model. Besides, models are proposed to study the optimal system configurations of such systems considering reliability and some other objectives or constraints. Facing amounts of works on LMCCS, this paper aims to review these literatures, classify them, and elicit some future research directions.
This paper models and evaluates the performability of a sliding window system (SWS) with multi-state components. Different components may have different numbers of states, characterized by state probability and performance rate distributions. Multiple consecutive components form groups with identical or different sizes. The accumulation (sum) of performance rates of components within the same group defines the group performance; the minimum of the group performance defines the system performance. The performability of an SWS is concerned with the probability that the system performs at a particular system performance. In this paper, a multi-valued decision diagram (MDD)-based analytical approach is proposed for the performability analysis of SWSs. The approach encompasses a compact system MDD generation based on the group MDD generation and combination, and evaluation of the resultant MDD model to obtain the system performability measures. Case studies are performed to demonstrate the proposed MDD approach as well as effects of component allocation on the system performability.
This paper considers optimal maintenance and allocation of elements in linear multi-state consecutively connected systems (LMCCS) which is important in signal transmission and other network systems. The system consists of N+1 linearly ordered positions (nodes) and fails if the first node (source) is not connected with the final node (sink). The reliability of LMCCS has been studied in the past restricted to the case when each system element has a constant reliability. In practice, system elements usually fail with increasing failure probability due to the aging effects. Furthermore, in order to increase the system availability, resources can be put into the maintenance of each element to increase the availability of the element. In this paper, a framework is proposed to solve the cost optimal maintenance and allocation strategy of this type of system subject to availability requirement. A universal generating function is used to estimate the availability of the system. A genetic algorithm is adopted for optimization. Illustrative examples are presented.
The current and future developments of electric power systems are pushing the boundaries of reliability assessment to consider distribution networks with renewable generators. Given the stochastic features of these elements, most modeling approaches rely on Monte Carlo simulation. The computational costs associated to the simulation approach force to treating mostly small-sized systems, i.e. with a limited number of lumped components of a given renewable technology (e.g. wind or solar, etc.) whose behavior is described by a binary state, working or failed. In this paper, we propose an analytical multi-state modeling approach for the reliability assessment of distributed generation (DG). The approach allows looking to a number of diverse energy generation technologies distributed on the system. Multiple states are used to describe the randomness in the generation units, due to the stochastic nature of the generation sources and of the mechanical degradation/failure behavior of the generation systems. The universal generating function (UGF) technique is used for the individual component multi-state modeling. A multiplication-type composition operator is introduced to combine the UGFs for the mechanical degradation and renewable generation source states into the UGF of the renewable generator power output. The overall multi-state DG system UGF is then constructed and classical reliability indices (e.g. loss of load expectation (LOLE), expected energy not supplied (EENS)) are computed from the DG system generation and load UGFs. An application of the model is shown on a DG system adapted from the IEEE 34 nodes distribution test feeder.
We apply new bilevel and trilevel optimization models to make critical infrastructure more resilient against terrorist attacks. Each model features an intelligent attacker (terrorists) and a defender (us), information transparency, and sequential actions by attacker and defender. We illustrate with examples of the US Strategic Petroleum Reserve, the US Border Patrol at Yuma, Arizona, and an electrical transmission system. We conclude by reporting insights gained from the modeling experience and many “red-team” exercises. Each exercise gathers open-source data on a real-world infrastructure system, develops an appropriate bilevel or trilevel model, and uses these to identify vulnerabilities in the system or to plan an optimal defense.
In this paper, we apply game theory to identify equilibrium strategies for both attacker and defender in a fully endogenous model of resource allocation for countering terrorism and natural disasters. The key features of our model include balancing protection from terrorism and natural disasters, and describing the attacker choice by a continuous level of effort rather than a discrete choice (i.e., attack or not). Interestingly, in a sequential game, increased defensive investment can lead an attacker to either increase his level of effort (to help compensate for the reduced probability of damage from an attack), or decrease his level of effort (because attacking has become less profitable). This can either reduce or increase the effectiveness of investments in protection from intentional attack, and can therefore affect the relative desirability of investing in protection from natural disasters. Subject classifications: decision analysis: risk; games/group decisions: noncooperative; utility/preference: applications. Area of review: Decision Analysis.
The acyclic multi-state information network (AMIN) is an extension of the multi-state network without having to satisfy the flow conservation law. A very straightforward convolution universal generating function method (CUGFM) is developed to find the exact symbolic one-to-all-target-subset reliability function of AMIN. The correctness and computational complexity of the proposed algorithm will be proven. Two illustrative examples demonstrate the power of the proposed CUGFM to solve the exact symbolic reliability functions of the one-to-all-target-subset AMIN problem more efficiently than the best-known UGFM.
We propose a novel class of game-theoretic models for the optimal assignment of defensive resources in a game between a defender and an attacker. Compared to the other game-theoretic models in the literature of defense allocation problems, the novelty of our model is that we allow the defender to assign her continuous-level defensive resources to any subset (or arbitrary layers) of targets due to functional similarity or geographical proximity. We develop methods to solve for equilibrium, and illustrate our model using numerical examples. Compared to traditional models that only allow for individual target hardening, our results show that our model could significantly increase the defender's payoff, especially when the unit cost of defense is high.
A production and conflict (P&C) model and a rent-seeking (RS) model are compared for one group, two groups and K groups. Adding a new agent enlarges the pie in the P&C model, but causes the fixed size pie to be allocated on one more rent seeker in the RS model. The total production or rent is distributed within and between groups according to the within-group and between-group decisiveness. Productive and fighting efficiencies and group sizes play a role. The collective action problem is more severe for the RS model. As group size increases, the ratio of within-group to between-group fighting increases marginally toward a constant for the P&C model, while it increases convexly for the RS model. Adding an additional agent to each of two groups is more detrimental to the utilities in RS groups than in P&C groups, while adding a second group of agents when there is already one group of agents gives the reverse result. The severe between-group fighting in the P&C model for many groups causes the P&C model to be preferable for few groups, while the RS model is preferable for many groups. Applications are considered to intergroup migration, inside versus outside ownership, divestitures, mergers and acquisitions, multidivisional versus single-tier firms and U form versus M form of economic organization. Copyright Springer Science + Business Media, Inc. 2005
This paper considers single-component repairable systems supporting different levels of workloads and subject to random repair times. The mission is successful if the system can perform a specified amount of work within the maximum allowed mission time. The system can work with different load levels, each corresponding to different productivity, time-to-failure distribution, and per time unit operation cost. A numerical algorithm is first suggested to evaluate mission success probability and conditional expected cost of a successful mission for the considered repairable system. The load optimization problem is then formulated and solved for finding the system load level that minimizes the expected mission cost subject to providing a desired level of the mission success probability. Examples with discrete and continuous load variation are provided to illustrate the proposed methodology. Effects of repair efficiency, repair time distribution, and maximum allowed time on the mission reliability and cost are also investigated through the examples.
Many engineering systems are designed to support load with different amounts. The load carried by the system has a significant effect on the deterioration of the system elements. This article presents an algorithm for determining the optimal load given to each multi-state element in a linear sliding window system such that the reliability of the system can be maximized. The model considers both the effect of load on the failure rate and the relationship between load and performance for each element. A reliability evaluation algorithm based on universal generating function is suggested for the proposed load-dependent linear sliding window system. The optimal load distribution problem is solved using genetic algorithms. Numerical experiments are presented to illustrate the proposed methods.
This paper considers the modelling and optimization of 1-out-of-N: G cold standby (CS) systems with non-repairable components functioning at different levels of productivity or load. The productivity heterogeneity leads to difference in component failure behaviour as well as in operational and replacement costs. Thus, the choice of load or productivity of components can greatly affect the system reliability and mission cost. To make the optimal choice of component loading, we first suggest a method for analysing the reliability and expected mission cost of 1-out-of-N: G non-repairable CS systems with heterogeneous components. The optimal dynamic load distribution problem is then formulated and solved, in which the component loading is chosen depending on the amount of work completed prior to the component activation. The optimal loading is aimed at minimizing the expected mission cost, while meeting a certain system reliability constraint. Examples are given to demonstrate the proposed methodology and the improvement in the optimal design solution through introducing the component productivity’s dependence on the completed work.
This paper presents a study on selective maintenance for multi-state series-parallel systems with economically dependent components. In the selective maintenance problem, the maintenance manager has to decide which components should receive maintenance activities within a finite break between missions. All the system reliabilities in the next operating mission, the available budget and the maintenance time for each component from its current state to a higher state are taken into account in the optimization models. In addition, the components in series-parallel systems are considered to be economically dependent. Time and cost savings will be achieved when several components are simultaneously repaired in a selective maintenance strategy. As the number of repaired components increases, the saved time and cost will also increase due to the share of setting up between components and another additional reduction amount resulting from the repair of multiple identical components. Different optimization models are derived to find the best maintenance strategy for multi-state series-parallel systems. A genetic algorithm is used to solve the optimization models. The decision makers may select different components to be repaired to different working states based on the maintenance objective, resource availabilities and how dependent the repair time and cost of each component are.
A parallel system consisting of components with different characteristics is studied. The
components can be unavailable due to internal failures or external impacts. Each component has an
increasing failure rate and is subjected to external impacts that can occur with fixed frequency. In order to
increase the system availability, three measures can be taken: 1. Increase of the components replacement
frequency; 2. Overarching protection of the system; 3. Individual component protection. The external
impact must penetrate both the overarching protection and the individual component protection in order
to destroy a component. The optimal resource allocation strategy which minimizes the total cost of
maintenance, protection and damage caused by unsupplied demand is studied. The proposed approach is
based on a universal generating function technique and a genetic algorithm.
As was described in Chapter 2, stochastic process methods are very effective tools for MSS reliability evaluation. According to these methods a state-space diagram of a MSS should be built and transitions between all the states defined. Then a system evolution should be represented by a continuous-time discrete-state stochastic process. Based on this process all MSS reliability measures can be evaluated. The main disadvantage of stochastic process models for MSS reliability evaluation is that they are very difficult for application to real-world MSSs consisting of many elements with different performance levels. This is so-called the “dimension curse”. First, state-space diagram building or model construction for such complex MSSs is not a simple job. It is a difficult nonformalized process that may cause numerous mistakes even for relatively small MSSs. The problem of identifying all the states and transitions correctly is a very difficult task. Second, solving models with hundreds of states can challenge the available computer resources. For MSSs consisting of n different repairable elements where every element j has k j different performance levels one will have a model with \( K = \prod^{n}_{j=1} k_{j} \) states. This number can be very large even for relatively small MSSs.
Traditionally in redundancy allocation problem (RAP), it is assumed that the redundant components are used based on a predefined active or standby strategies. Recently, some studies consider the situation that both active and standby strategies can be used in a specific system. However, these researches assume that the redundancy strategy for each subsystem can be either active or standby and determine the best strategy for these subsystems by using a proper mathematical model. As an extension to this assumption, a novel strategy, that is a combination of traditional active and standby strategies, is introduced. The new strategy is called mixed strategy which uses both active and cold-standby strategies in one subsystem simultaneously. Therefore, the problem is to determine the component type, redundancy level, number of active and cold-standby units for each subsystem in order to maximize the system reliability. To have a more practical model, the problem is formulated with imperfect switching of cold-standby redundant components and k-Erlang time-to-failure (TTF) distribution. As the optimization of RAP belongs to NP-hard class of problems, a genetic algorithm (GA) is developed. The new strategy and proposed GA are implemented on a well-known test problem in the literature which leads to interesting results.
This article considers the optimal allocation and maintenance of multi-state elements in series-parallel systems with common bus performance sharing. The surplus performance from a sub-system can be transmitted to any other sub-system which is experiencing performance deficiency. The amount that can be transmitted is subjected to a random transmission capacity. In order to increase the system availability, maintenance actions can be performed during the system lifetime and the system elements can be optimally allocated into the sub-systems. In this paper, we consider the element allocation and maintenance simultaneously in order to minimize the total maintenance cost subject to the pre-specified system availability requirement. An algorithm based on universal generating function is suggested to evaluate the system availability and the genetic algorithm is explored to solve the optimization problem. Numerical experiments are presented to illustrate the applications.
This paper proposes a new model that generalizes the linear multistate sliding window system to the case of multiple failures. In this model, the system consists of independent linearly ordered multistate elements. Each element can have different states: from complete failure up to perfect functioning. A performance rate is associated with each state. The system fails if, at least in groups of consecutive elements (windows), the sum of the performance rates of elements belonging to the group is less than a minimum allowable level. An algorithm for system reliability evaluation is suggested which is based on an extended universal moment generating function. Examples of evaluating system reliability and elements' reliability importance indices are presented.
This paper presents a maintenance optimisation method for a multi-state series–parallel system considering economic dependence and state-dependent inspection intervals. The objective function considered in the paper is the average revenue per unit time calculated based on the semi-regenerative theory and the universal generating function (UGF). A new algorithm using the stochastic ordering is also developed in this paper to reduce the search space of maintenance strategies and to enhance the efficiency of optimisation algorithms. A numerical simulation is presented in the study to evaluate the efficiency of the proposed maintenance strategy and optimisation algorithms. The simulation result reveals that maintenance strategies with opportunistic maintenance and state-dependent inspection intervals are more cost-effective when the influence of economic dependence and inspection cost is significant. The study further demonstrates that the optimisation algorithm proposed in this paper has higher computational efficiency than the commonly employed heuristic algorithms.
A production system containing a set of machines (also called components) arranged according to a series-parallel configuration is addressed. A set of products must be produced in lots on this production system during a specified finite planning horizon. This paper presents a method for integrating load distribution decisions, and tactical production planning considering the costs of capacity change and the costs of unused capacity. The objective is to minimize the sum of capacity change costs, unused capacity costs, setup costs, holding costs, backorder costs, and production costs. The main constraints consist in satisfying the demand for all products over the entire horizon, and in not exceeding available repair resource. The production series-parallel system is modeled as a multi-state system with binary-state components. The proposed model takes into account the dependence of machines' failure rates on their load. Universal generating function technique can be used in the optimization algorithm for evaluating the expected system production rate in each period. We show how the formulated problem can be solved by comparing the results of several multi-product lot-sizing problems with capacity associated costs. The importance of integrating load distribution decisions and production planning is illustrated through numerical examples.
To improve the design reliability of a mobile handset, this paper not only estimates the reliability of mobile handset of two leading makes but also makes a comparative study between the sets. The paper analyzes the causes of failure and the failure rate through Cox proportional hazard model. Baseline life distribution has been studied using the Weibull form and mean lives have been estimated under both absence of the causal variables and presence of the causal variables at their respective mean values.
The paper considers a system consisting of genuine elements and false targets that cannot be distinguished by the attacker's observation. The false targets can be destroyed with much less effort than the genuine elements. We show that even when an attacker cannot distinguish between the genuine elements and the false targets, in many cases it can enhance the attack efficiency using a double attack strategy in which it tries first to eliminate with minimal effort as many false targets as possible in the first attack and then distributes its entire remaining resource among all surviving targets in the second attack. The model for evaluating the system vulnerability in the double attack is suggested for a single genuine element, and multiple genuine elements configured in parallel or in series. This model assumes that in both attacks the attacking resource is distributed evenly among the attacked targets. The defender can optimize its limited resource distribution between deploying more false targets and protecting them better. The attacker can optimize its limited resource distribution between two attacks. The defense strategy is analyzed based on a two period minmax game. A numerical procedure is suggested that allows the defender to find the optimal resource distribution between deploying and protecting the false targets. The methodology of optimal attack and defense strategies analysis is demonstrated. It is shown that protecting the false targets may reduce the efficiency of the double attack strategy and make this strategy ineffective in situations with low contest intensity and few false targets.
This paper discusses a type of redundancy that is typical in a multi-state system. It considers two interconnected multi-state systems where one multi-state system can satisfy its own stochastic demand and also can provide abundant resource (performance) to another system in order to improve the assisted system reliability. Traditional methods are usually not effective enough for reliability analysis for such multi-state systems because of the “dimensional curse” problem. This paper presents a new method for reliability evaluation for the repairable multi-state system considering such kind of redundancy. The proposed method is based on the combination of the universal generating function technique and random processes methods. The numerical example is presented to illustrate the proposed method.
This paper presents a technique to determine the optimal reserve structure (reserve providers and the corresponding reserve capacity) for a restructured power generating system (GS). The reserve of a GS can be provided by its own generating units and can also be purchased from other GSs through the reserve agreements. The objective of reserve management for a GS is to minimize its total reserve cost while satisfying the reliability requirement. The reserve management is a complex optimization problem, which requires a large amount of calculations. In order to simplify the evaluation, a complex generating system (CGS) consisting of different GSs and the corresponding transmitting network is represented by its multi-state reliability equivalents. The universal generating functions (UGFs) of these equivalents are developed and the special operators for these UGFs are defined to evaluate the reliability of a particular GS, which has reserve agreements with other GSs in the CGS. The genetic algorithm (GA) has been used to solve the optimization problem. An improved power system-IEEE reliability test system is used to illustrate the technique.
The paper presents the way of solving problems concerning reliability with respect to concomitant variables. The Cox models are discussed and Weibull's proportional hazards model is defined. The paper shows how to estimate the model parameters and statistically verify the results.
Redundancy allocation problems (RAPs) have attracted much attention for the past thirty years due to its wide applications in improving the reliability of various engineering systems. Because RAP is an NP-hard problem, and exact methods are only applicable to small instances, various heuristic and meta-heuristic methods have been proposed to solve it. In the literature, most studies on RAPs have been conducted for single-level systems. However, real-world engineering systems usually contain multiple levels. In this paper, the RAP on multi-level systems is investigated. A novel memetic algorithm (MA) is proposed to solve this problem. Two genetic operators, namely breadth-first crossover and breadth-first mutation, and a local search method are designed for the MA. Comprehensive experimental studies have shown that the proposed MA outperformed the state-of-the-art approach significantly on two representative examples.
Continuous stress–strength interference (SSI) model regards stress and strength as continuous random variables with known probability density function. This, to some extent, results in a limitation of its application. In this paper, stress and strength are treated as discrete random variables, and a discrete SSI model is presented by using the universal generating function (UGF) method. Finally, case studies demonstrate the validity of the discrete model in a variety of circumstances, in which stress and strength can be represented by continuous random variables, discrete random variables, or two groups of experimental data.
The analysis of censored failure times is considered. It is assumed that on each individual are available values of one or more explanatory variables. The hazard function (age‐specific failure rate) is taken to be a function of the explanatory variables and unknown regression coefficients multiplied by an arbitrary and unknown function of time. A conditional likelihood is obtained, leading to inferences about the unknown regression coefficients. Some generalizations are outlined.
Reliability measures are examined, taking into account the reliability
function, the expected life, the failure rate and the hazard function,
the reliability and hazard function for well-known distributions, hazard
models and product life, the estimation of the hazard function and the
reliability function from empirical data, and comments on distribution
selection. Static reliability models are considered along with aspects
of probabilistic engineering design, the combination of random variables
in design, interference theory and reliability computations, reliability
design examples, time-dependent stress-strength models, dynamic
reliability models, the exponential distribution, the Weibull
distribution, sequential life testing, and questions of reliability
optimization. Aspects of Bayesian reliability in design and testing are
discussed, giving attention to a Bayesian approach to statistical
inference, a binomial distribution testing situation, and the
application of Bayes theorem in design for reliability.
The paper considers an object exposed to external intentional attacks. The defender distributes its resource between deploying false targets and protecting the object. The false targets are not perfect and there is a nonzero probability that a false target can be detected by the attacker. Once the attacker has detected a certain number of false targets, it ignores them and chooses such number of undetected targets to attack that maximizes the probability of the object destruction. The defender decides how many false targets to deploy in order to minimize the probability of the object destruction assuming that the attacker uses the most harmful strategy to attack. The optimal number of false targets and the optimal number of attacked targets are obtained for the case of single and multiple types of the false targets. A methodology of finding the optimal defence strategy under uncertain contest intensity is suggested.
This article extends an existing model for performance sharing among the multi-state units. The extended model considers an arbitrary number of units that must satisfy individual random demands. If a unit has a performance that exceeds the demand it can transmit the surplus performance to other units. The amount of transmitted performance is limited by the random capacity of a transmission system. The entire system fails if at least one demand is not satisfied. An algorithm based on the universal generating function technique is suggested to evaluate the system reliability and expected performance deficiency. Analytical and numerical examples are presented.
Weighted voting classifiers considered in this paper consist of N units each providing individual classification decisions. The entire system output is based on tallying the weighted votes for each decision and choosing the one which has total support weight exceeding a certain threshold. Each individual unit may abstain from voting. The entire system may also abstain from voting if no decision support weight exceeds the threshold. Existing methods of evaluating the reliability of weighted voting systems can be applied to limited special cases of these systems and impose some restrictions on their parameters. In this paper a universal generating function method is suggested which allows the reliability of weighted voting classifiers to be exactly evaluated without imposing constraints on unit weights. Based on this method, the classifier reliability is determined as a function of a threshold factor, and a procedure is suggested for finding the threshold which minimizes the cost of damage caused by classifier failures (misclassification and abstention may have different price.) Dynamic and static threshold voting rules are considered and compared. A method of analyzing the influence of units' availability on the entire classifier reliability is suggested, and illustrative examples are presented. © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 322–344, 2003.
This paper proposes a new model that generalizes the linear consecutive k-out-of-r-from-n:F system to multistate case with multiple failure criteria. In this model (named linear multistate multiple sliding window system) the system consists of n linearly ordered multistate elements (MEs). Each ME can have different states: from complete failure up to perfect functioning. A performance rate is associated with each state. Several functions are defined for a set of integer numbers ρ in such a way that for each r ∈ ρ corresponding function fr produces negative values if the combination of performance rates of r consecutive MEs corresponds to the unacceptable state of the system. The system fails if at least one of functions fr for any r consecutive MEs for r ∈ ρ produces a negative value. An algorithm for system reliability evaluation is suggested which is based on an extended universal moment generating function. Examples of system reliability evaluation are presented. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005.
In this paper, we present a practical approach for the joint reliability–redundancy optimization of multi-state series–parallel systems. In addition to determining the optimal redundancy level for each parallel subsystem, this approach also aims at finding the optimal values for the variables that affect the component state distributions in each subsystem. The key point is that technical and organizational actions can affect the state transition rates of a multi-state component, and thus affect the state distribution of the component and the availability of the system. Taking this into consideration, we present an approach for determining the optimal versions and numbers of components and the optimal set of technical and organizational actions for each subsystem of a multi-state series–parallel system, so as to minimize the system cost while satisfying the system availability constraint. The approach might be considered to be the multi-state version of the joint system reliability–redundancy optimization methods.
Recent results have used game theory to explore the nature of optimal investments in the security of simple series and parallel systems. However, it is clearly important in practice to extend these simple security models to more complicated system structures with both parallel and series subsystems (and, eventually, to more general networked systems). The purpose of this paper is to begin to address this challenge. While achieving fully general results is likely to be difficult, and may require heuristic approaches, we are able to find closed-form results for systems with moderately general structures, under the assumption that the cost of an attack against any given component increases linearly in the amount of defensive investment in that component. These results have interesting and sometimes counterintuitive implications for the nature of optimal investments in security.
This paper presents an algorithm for determining an optimal loading of elements in series–parallel systems. The optimal loading is aimed at achieving the greatest possible expected system performance subject to repair resource constraint. The model takes into account the dependence of elements’ failure rates on their load. The optimization algorithm uses a universal generating function technique for evaluating the expected system performance, and a genetic algorithm for determining the optimal load distribution. An illustrative example of load distribution optimization is presented.
This paper addresses the heterogeneous redundancy allocation problem in multi-state series-parallel reliability structures with the objective to minimize the total cost of system design satisfying the given reliability constraint and the consumer load demand. The demand distribution is presented as a piecewise cumulative load curve and each subsystem is allowed to consist of parallel redundant components of not more than three types. The system uses binary capacitated components chosen from a list of available products to provide redundancy so as to increase system performance and reliability. The components are characterized by their feeding capacity, reliability and cost. A system that consists of elements with different reliability and productivity parameters has the capacity strongly dependent upon the selection of constituent components. A binomial probability based method to compute exact system reliability index is suggested. To analyze the problem and suggest an optimal/near-optimal system structure, an ant colony optimization algorithm has been presented. The solution approach consists of a series of simple steps as used in early ant colony optimization algorithms dealing with other optimization problems and offers straightforward analysis. Four multi-state system design problems have been solved for illustration. Two problems are taken from the literature and solved to compare the algorithm with the other existing methods. The other two problems are based upon randomly generated data. The results show that the method can be appealing to many researchers with regard to the time efficiency and yet without compromising over the solution quality.
The Proportional Hazards (PH) model is an important type of failure time regression model which relates the occurrence probability of critical failures to influential factors. However, little research work has been done on detecting changes in the PH models fitted based on different sets of reliability data. This paper develops the methods for change detection in the Cox PH models, also known as Semiparametric PH model, for reliability prediction and/or assessment of the time-to-failure data collected from different subjects. The effectiveness of the developed methods is illustrated through numerical studies and real-world data analysis. The developed technique possesses wide applicability to the systems and processes where the Cox PH model fits the reliability data well. Copyright © 2010 John Wiley & Sons, Ltd.
This paper proposes and validates a methodology to measure explicitly the increase in the risk of a processor error with increasing workload. By relating the occurrence of a CPU related error to the system activity just prior to the occurrence of an error, the approach measures the dynamic CPU workload/failure relationship. The measurements show that the probability of a CPU related error (the load hazard) increases nonlinearly with increasing workload; i.e., the CPU rapidly deteriorates as end points are reached. The load hazard is observed to be most sensitive to system CPU utilization, the I/O rate, and the interrupt rates. The results are significant because they indicate that it may not be useful to push a system close to its performance limits (the previously accepted operating goal) since what we gain in slightly improved performance is more than offset by the degradation in reliability. Importantly, they also indicate that conventional reliability models need to be reevaluated so as to take system workload explicitly into account.
This study proposes a genetic algorithm based method integrating the minimal paths and the recursive sum of disjoint products to find maximal network reliability with optimal transmission line assignment for a stochastic electric power network. In our problem, a set of transmission lines is ready to be assigned to branches of the electric power network. Because each transmission line combined with several physical lines has multiple states, the capacity of the electric power network associated with any transmission line assignment is stochastic. Network reliability is the probability that the network can transmit d units of electric power from an electric power generator (origin) to a specific area (destination). The discussed problem exhibits the features of network reliability and assignment problems, and thus it is non-deterministic polynomial-time hard. A simple electric power network and a real one are adopted to demonstrate the efficiency of the proposed algorithm while comparing with several approaches.
The multi-state k -out-of-n system model finds wide applications in industry, and has been extensively studied in recent years. This model has also been generalized to the multi-state weighted k -out-of-n system model. Recursive methods, and universal generating functions (UGF) are two primary algorithms for exact performance evaluation of multi-state k-out-of-n systems. However the computational burden becomes the crucial factor when there is a “dimension damnation” problem caused by the increase in the number of components in the system, and the number of possible states a component may be in. In situations wherein exact values of system reliability are not necessary, we may use more efficient algorithms to approximate system reliability. In this paper, we develop a comprehensive framework for reliability approximation of multi-state weighted k -out-of-n systems. Two fuzzy based multi-state weighted k-out-of- n system models are defined. Procedures for building these two models from the conventional models are also introduced. The fuzzy recursive methods, and fuzzy UGF techniques are developed to evaluate such systems. The clustering technique, and curve fitting method are used to determine the fuzzy weights, and probabilities of states in the models.
Tournaments, conflict, and rent-seeking have been modelled as contests in which participants exert effort to increase their probability of winning a prize. A Contest Success Function (CSF) provides each player's probability of winning as a function of all players' efforts. In this paper the additive CSF employed in most contests is axiomatized, with an independence from irrelevant alternatives property as the key axiom. Two frequently used functional forms are also axiomatized: one in which winning probabilities depend on the ratio of players' efforts and the other in which winning probabilities depend on the difference in efforts.