Availability Assessment of Multi-State Manufacturing Systems Using Universal Generating Function

The capability of a reconfigurable manufacturing system (RMS) configuration depends upon its availability to meet demands. This chapter presented a multi-objective and multi-state reconfigurable manufacturing systems model for optimal configuration selection. Two objectives were considered, namely, total cost and availability for optimal configuration selection. The availability of RMS was estimated using the modified universal generating function (UGF). An integrated objective function is formulated in this chapter considering total cost and availability of system. The decision makers considered various scenarios for configuration selection, such as no weight, 50–50% weight and 40–60% weight for total cost objective and availability objective. According to weights, Option-1 was found optimal for scenario with no weight and 50–50% weight for total cost objective and availability objective. Option-2 was found optimal for scenario with 44–60% weight for total cost objective and availability objective.

Large-scale adoption of gas-fired power plants (GPPs) significantly accelerates the integration of power systems and natural gas systems (NGSs). Because the operation of NGS has a significant impact on the reliability of power system, it is important to simultaneously evaluate the reliabilities of power system and NGS. Moreover, due to the stochastic failures and various operating characteristics of components, the integrated gas and power systems (IGPS) considering multiple states can be viewed as a multi-state system. Therefore, a multi-state model is proposed for reliability evaluation of IGPS based on universal generating function (UGF) techniques. First, a framework is proposed to evaluate the reliability of NGS: multi-state models for different components in the NGS are developed as the corresponding UGFs. These UGFs are aggregated utilizing the gas flow calculation operator to obtain the multi-state model of NGS. Moreover, a gas-to-power calculation operator is developed to convert the multi-state model of gas injection at each node to the power output models of GPPs in power system. In this manner, the impacts of NGS could be incorporated into the reliability evaluation of power system. Furthermore, nodal reliability indices for both the NGS and the power system are proposed to evaluate the reliability performances of IGPS. The proposed methods are validated using the integrated gas and power test system.

Universal generating function (UGF) technology is the basic approach to reliability evaluation of multi-state systems. For products with high reliability and long lifetime in practical engineering, their system structure is usually uncertain and the external working condition is extremely complex. Thus, accurate failure or degradation data of components are difficult to define. Data are often characterized by types of randomness, probability, grey, fuzzy and other uncertain numbers. Based on interval grey number, the concept of generalized uncertain number is proposed. In this paper, we first unify arbitrary type of uncertain numbers, propose the definition of generalized standard uncertain number and study algorithm such as addition, subtraction, multiplication and division between them. Based on generalized standard uncertain number, generalized universal generating function, which is an extension of fuzzy UGF, is proposed. Several commonly used generalized universal generating operators are discussed with system connection structure between components. According to the relationship between output performance and system demand level, reliability indexes such as availability, expected output performance and mean performance deficiency are evaluated. Comparisons in a real case show the effectiveness of the proposed approach. Therefore, relevant results can strengthen the theoretical research depth of uncertain reliability and definitely expand the application scope of fuzzy universal generating function.

In this paper, an algorithm based ant colony approach (ACA) is designed to find the optimal configuration for redundancy apportionment problem (RAP) of series-parallel multi-state system (MSS). In order to achieve maximum system reliability under cost and performance constraints, the algorithm takes advantage of ACA’s combinatorial optimization, uses heuristic information about reliability to arrange components from available choice. Different from traditional nominal performance and reliability system, universal generating function (UGF) is introduced and used to estimate the multi-state system reliability. Finally the experiment has shown that the presented algorithm can provide optimal and near optimal solutions, and have efficient and convenient calculation performance.

The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> -out-of- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> : G(F) majority voter consists of n components (or modules) and a number of the components are required to be operating correctly for the overall system to be correct. As per the state discretization of the components, such a system is usually classified as either a binary system or a multi-state system. In practice, the operating conditions of different components may contribute differently to the operation of the entire system. In this manuscript, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> -out-of- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> : G(F) majority voter is generalized as a consecutive-weighted- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> -out-of- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> : G(F) voter with either binary states or multiple states. To overcome the drawbacks of existing approaches, a stochastic analysis is proposed for assessing the system reliability. In the stochastic analysis, the input signal probabilities are encoded into non-Bernoulli sequences with fixed numbers of 0s and 1s for the Boolean case, or randomly permuted sequences for the multi-state scenario. By using stochastic logic, the reliability of a general system consisting of consecutive-weighted- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> -out-of- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> majority voters is efficiently and accurately predicted. The results are validated by an analysis of several case studies. Although the accuracy of the stochastic analysis is closely related with the employed sequence length, it is shown that a stochastic approach is more efficient than a universal generating function (UGF) method, while still retaining an acceptable accuracy.

Optimal power in wind farms has been a problem for constructors and managers during recent years, onshore wind farms are subject to more and more regulations and constraints. The goal is to define optimal installed capacity with maximum reliability and profits. A number of algorithms have thus been developed for the optimization of wind farm power plants. However, according to the literature, no methods have so far been tested on wind farms using real data and wind based conditions. In this Article, we continue the work that has recently been done on wind farm reliability optimization and apply an algorithm to find the best topology. The proposed resolution technique is based on finding the best topology of the system under the constraints of Performance (Availability), Costs and the Global Warming Potential of the system. A case study is done using data and constraints similar to those collected from wind farm constructors, managers and maintainers. Multi-state systems (MSS), Universal generating function (UGF), Wind and Load charge functions are applied. An economic study was conducted to assess the wind farm investment. Net present value (NPV) and levelized cost of energy (LCOE) were calculated for best topologies founded.

The maritime sector operates in an environment characterized by multidimensional uncertainty—technical, operational, and financial. Mechanical failures, volatile fuel prices, and fluctuating freight rates simultaneously influence vessel performance and profitability, making effective decision-making increasingly complex. This study develops an integrated reliability–financial modeling framework that combines Multi-State Systems (MSS) Theory, Markov Chains, Markov Switching Models, and the Universal Generating Function (UGF) methodology to capture both physical system reliability and market-driven volatility. Using the Panamax Index as a case study, the model evaluates vessel availability, fuel price dynamics, and freight rate fluctuations to forecast profitability under uncertain operating conditions. The findings indicate that transient probabilities of vessel availability converge faster than financial variables, suggesting higher predictability of technical systems compared with market indices. The proposed framework provides maritime stakeholders with a strategic decision-support tool capable of forecasting profitability and identifying optimal operating routes within a 26-week horizon. According to our Panamax data, 3/5 routes have positive expected profit, availability stabilizes by ~ week 3, and freight regimes by ~ week 15. By integrating reliability engineering with financial modeling, the study enhances operational planning and risk management in the maritime industry.

A multi-state system (MSS) has more than two discrete states corresponding to different performance rates. Usually, MSS is viewed as in a failure state once its performance rate falls below user demand, and maintenance is carried out immediately. Generally, the repaired system cannot be regarded as good as new, and oftentimes the system restoration is stochastic. We introduce an optimal replacement policy for MSSs, called policy <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> . Under this policy, a MSS is replaced whenever its failure number reaches <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> . We assess the dynamic element state probabilities of each aging multi-state element (MSE) using a stochastic process model which is identified as a non-homogeneous continuous time Markov model (NHCTMM), and we evaluate the state distribution of the entire MSS via the combination of the stochastic process, and the universal generating function (UGF). To quantify the quality of imperfect maintenance, a quasi-renewal process is used to describe the stochastic behavior of each individual MSE after repair. Moreover, we derive an explicit expression of the long-run expected profit per unit time, and determine the optimal failure number <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N*</i> to replace the entire system. The proposed models are demonstrated via an illustrative case, followed by some comparative studies.

Multi-state Systems in Nature and Engineering Modern Stochastic Process Methods for MSS Reliability Assessment Statistical Analysis of Reliability Data for Real-world MSS's Universal Generating Function (UGF) Models Combined UGF and Stochastic Process Technique Aging Multi-state Systems Reliability Associated Costs for MSS and Optimal Management Decisions Fuzzy Multi-state System

A stochastic approach for the reliability evaluation of multi-state systems with dependent components

Reliability assessment of multi-state phased mission systems with common bus performance sharing considering transmission loss and performance storage

Algorithm for estimating reliability confidence bounds of multi-state systems

Many systems are required to perform a series of missions with finite breaks between any two consecutive missions. In such a case, one of the most widely used maintenance policies is a selective maintenance in which a subset of feasible maintenance actions is chosen to be performed with the aim at achieving the subsequent mission success under limited maintenance resources. Traditional selective maintenance optimization reported in the literature only focuses on binary state systems. Most systems in industrial applications, however, have more than two states in the deterioration process. In this work, a selective maintenance policy for multi-state systems (MSS) consisting of binary state elements is investigated. Taking the imperfect maintenance quality into consideration, the Kijima model is reviewed, and a cost-maintenance quality relationship which considers the age reduction factor as a function in terms of maintenance cost is established. Moreover, with the assistance of the universal generating function (UGF) method, the probability of the repaired MSS successfully completing the subsequent mission is formulated. In place of enumerative methods, a genetic algorithm (GA) is employed to solve the complicated optimization problem where both multi-state systems, and imperfect maintenance models are taken into account. The effectiveness of the proposed method is demonstrated via a case study of a power station coal transportation system. Finally, a comparative analysis between the strategies with and without considering imperfect maintenance is conducted, and it is concluded that incorporating imperfect maintenance quality into selective maintenance achieves better outcomes.

Importance and joint importance measures in reliability engineering are used to identify the weak areas of a system and signify the roles of components in either causing or contributing to proper functioning of the system. This paper introduces, for two multistate components, Joint Birnbaum-Reliability Achievement Worth (JBRAW), Joint Birnbaum-Reliability Reduction Worth (JBRRW), Joint Birnbaum-Reliability Fussell-Vesely measure (JBRFV) and analogues joint risk importance measures for the multistate system. A steady state performance level distribution with restriction to the component’s states is used to obtain the introduced measures. Universal generating function (UGF) technique is applied for the evaluation of proposed joint importance measures. An illustrative example is provided.

Many practical methods and different approaches have been proposed to assess Multi-State Systems (MSS) reliability measures. The universal generating function (UGF) method, introduced in 1986, is known to be a very efficient way of evaluating the availability of different types of MSSs. In this paper, we propose an extension of the UGF method considering epistemic uncertainties. This extended method allows one to model ill-known probabilities and transition rates, or to model both aleatory and epistemic uncertainty in a single model. It is based on the use of belief functions which are general models of uncertainty. We also compare this extension with UGF methods based on interval arithmetic operations performed on probabilistic bounds.