Weakest T-norm Research Articles

Intuitionistic fuzzy fault tree analysis of PCBA using novel arithmetic operations

PurposeThe main goal of this paper is to develop novel Tω(weakest t-norm)-based fuzzy arithmetic operations to analyze the intuitionistic fuzzy reliability of Printed Circuit Board Assembly (PCBA) using fault tree.Design/methodology/approachThe paper proposes a fuzzy fault tree analysis (FFTA) method for evaluating the intuitionistic fuzzy reliability of any nonrepairable system with uncertain information about failures of system components. This method uses a fault tree for modeling the failure phenomenon of the system, triangular intuitionistic fuzzy numbers (TIFNs) to determine data uncertainty, while novel arithmetic operations are adopted to determine the intuitionistic fuzzy reliability of a system under consideration. The proposed arithmetic operations employ Tω(weakest t-norm) to minimize the accumulating phenomenon of fuzziness, whereas the weighted arithmetic mean is employed to determine the membership as well as nonmembership degrees of the intuitionistic fuzzy failure possibility of the nonrepairable system. The usefulness of the proposed method has been illustrated by inspecting the intuitionistic fuzzy failure possibility of the PCBA and comparing the results with five other existing FFTA methods.FindingsThe results show that the proposed FFTA method effectively reduces the accumulating phenomenon of fuzziness and provides optimized degrees of membership and nonmembership for computed intuitionistic fuzzy reliability of a nonrepairable system.Originality/valueThe paper introduces Tω(weakest t-norm) and weighted arithmetic mean based operations for evaluating the intuitionistic fuzzy failure possibility of any nonrepairable system in an uncertain environment using a fault tree.

Novel approach to analyse vague reliability of repairable industrial systems

The purpose of the study is to develop some novel approaches to evaluate the vague reliability of repairable industrial systems by considering uncertainty and hesitation factors in data assessment. The paper introduces two novel vague reliability assessment techniques, named as generalized vague lambda-tau methodology (GVLTM), and generalized Tω (weakest t-norm) based vague lambda-tau methodology (GTBVLTM). These two techniques commonly apply fault-tree analysis, the lambda-tau method, and vague sets with different shapes for different components of a system. To compute system vague reliability, the GVLTM applies α-cut and interval arithmetic operations, whereas GTBVLTM employs α-cut and Tω coupled arithmetic operations defined on vague sets. To demonstrate the effectiveness of the developed approach, both proposed techniques and five other existing techniques are applied to analyse the vague reliability of a washing system in a paper plant. Long run reliability assessment, sensitivity analysis, and ranking of the system’s critical components have also been conducted to improve its performance. Results concluded that the proposed techniques are simple, easy to apply, and provide optimized vague reliability indices, when different vague sets for different components are applied to handle uncertainty and hesitation factors in data. Results can be used to design suitable action plans to enhance system performance.

Quantitative Risk Assessment of a Large-Scale Central Spherical Detector System by an Improved Fuzzy Fault Tree Analysis

Central spherical detector is one of the most essential and complicated support systems for the large-scale underground neutrino observatory. If a central spherical detector accident happens, the consequences may be disastrous with huge economic loss. However, there is very few published research works which focus on its quantitative risk analysis. In this paper, an improved fuzzy fault tree analysis (FFTA) method incorporated with the weakest t-norm (Tω) algorithm, the confidence level concept, and the analytic hierarchy process (AHP) approach is proposed to perform its risk assessment. By carrying out the identification of failure modes and failure reasons, fault tree (FT) model of central detector is constructed based on logical relationship among subcomponents. Fuzzy set theory is applied to obtain failure data, and Tω algorithm is exploited to eliminate fuzzy accumulation in the aggregation process. In addition, a confidence level coefficient and AHP approach are employed to enhance the reliability of the evaluation. Both importance and sensitivity analysis have been conducted to identify the critical basic events and provide improvement measures. Finally, the comparison of the occurrence possibility of detector failure is used to verify the applicability and the feasibility of proposed method. The calculated results indicate that the improved approach is more consistent with real situation and can provide a more effective engineering reference for the risk decision of central spherical detector.

The α, β-Cut Intervals and Weakest t-Norm Based Importance Measure for Criticality Analysis in Intuitionisitic Fuzzy Fault Tree Analysis of LNG-ESD System

Different importance measures have been developed and widely utilized in system reliability analysis to measure the importance of system components. During operation period of a complex system, it becomes more difficult to collect exact failure data of system components i.e. the collected failure data may have some sort of uncertainties due to various reasons. Generally, triangular shaped fuzzy numbers or intuitionistic fuzzy numbers (IFNs) are used to quantify the various types of uncertainty in failure data of the system’s components and fuzzy arithmetic operations are utilized which computes each reliability index with wide range of prediction due to accumulating phenomenon of fuzziness. In this paper, different types of intuitionistic fuzzy number (IFN) are used to quantify the failure data uncertainty of the system’s components while α, β-cut intervals and weakest t-norm (Tw)-based arithmetic operations are employed to reduce the range of prediction for each reliability index. The objective of the present study is to develop Tw-based importance measure for criticality analysis in intuitionistic fuzzy fault tree analysis (IF-FTA) of a system, when different types of uncertainties in crisp failure data of basic events are quantified by different types of membership and non-membership functions. The developed importance measure applies system fault tree, α-cut and β-cut intervals, different types of intuitionistic fuzzy number (IFN) and weakest t-norm (Tw) based intuitionistic fuzzy arithmetic operations. To demonstrate the applicability of the proposed measure, a case study of liquefied natural gas terminal emergency shutdown system (LNG-ESD system) is performed. The IF-FTA of LNG-ESD system is presented to evaluate the fault intervals of the system at different levels using Tw-based arithmetic operations on different types of IFNs. The developed importance measure is applied to identify and rank the critical basic events in IF-FTA of LNG-ESD system. The proposed measure is also compared with some existing importance measures.

Evaluating the environmental protection strategy of a printed circuit board manufacturer using a Tw fuzzy importance performance analysis with Google Trends

Printed circuit boards (PCBs) are very important materials in consumer electronic products. The process of making PCBs usually uses chemicals, and large quantities of water are needed. PCB manufacturers are devoted to improving the process in order to conform to environmental protection rules and regulations. However, evaluating the performance of environmental protection strategies by quantitative methods is very difficult. Therefore, this study attempts to evaluate the environmental protection strategy of a PCB manufacturer using the novel weakest t-norm (Tw) fuzzy importance-performance analysis with Google Trends (TFIPA-Google). The TFIPA-Google methodology obtain advantages of Tw operations, IPA, and Google Trends, which can handle uncertainty based on a fuzzy matrix of IPA, reduce fuzzy accumulation using the Tw operators, and analyze social media viewpoints using Google Trends. This empirical example based on the TFIPA-Google method shows that the recovered waste material management system should be a priority for PCB manufacturers. Moreover, the TFIPA-Google method can provide more creditable information, based on Tw operations and volume of Google Trends for decision-makers, than a conventional importance-performance analysis (IPA) model.

Measuring Pearson's correlation coefficient of fuzzy numbers with different membership functions under weakest t-norm

In statistical theory, the correlation coefficient has been widely used to assess a possible linear association between two variables and often calculated in crisp environment. In this study, a simplified and effective method is presented to compute the Pearson's correlation coefficient of fuzzy numbers with different membership functions using weakest triangular norm (t-norm)-based approximate fuzzy arithmetic operations. Different from previous research studies, the correlation coefficient computed in this paper is a fuzzy number rather than a crisp number. The proposed method has been illustrated by computing the correlation coefficient between the technology level and management achievement from a sample of 15 machinery firms in Taiwan. The correlation coefficient computed by proposed method has less uncertainty and obtained results are more exact. The computed results have also been compared with existing approaches.

Measuring Pearson's correlation coefficient of fuzzy numbers with different membership functions under weakest t-norm

Measuring Pearson's correlation coefficient of fuzzy numbers with different membership functions under weakest t-norm

Investigation of the minimum risk portfolio under conditions of hybrid uncertainty based on the weakest t-norm

Investigation of the minimum risk portfolio under conditions of hybrid uncertainty based on the weakest t-norm

Fuzzy reliability analysis of DFSMC system in LNG carriers for components with different membership function

The purpose of the present study is to analyse the fuzzy reliability of dual-fuel steam turbine mechanical propulsion conventional (DFSMC) system of LNG carriers for components with different membership function by applying Tω(weakest t-norm) based generalized fuzzy lambda-tau (TBGFLT) technique. TBGFLT technique applies fault-tree, Lambda-Tau method, different membership functions, α-cut and the weakest t-norm (Tω) based approximate fuzzy arithmetic operations for evaluating fuzzy reliability of the system. The computed results have been compared with results obtained from four other fuzzy reliability techniques. This comparative study concludes that TBGFLT reduces growing phenomenon of fuzziness at any confidence level when system components' follow different membership function. System long run reliability and availability, sensitivity analysis and its critical components’ ranking have also been conducted for analysing system behavior to improve its performance. Based on the results, some influential suggestions have been drawn which may be helpful for maintenance personnel to design and execute some action plans for improving system performance under vague environment.

Fuzzy reliability analysis of the washing system in a paper plant using the TBFLT technique

PurposeThe purpose of this paper is to analyze the reliability of the washing system in a paper plant in a more promising way under vague environment by reducing the accumulating phenomenon of fuzziness and accelerating the computation process using the Tω (weakest t-norm) based fuzzy lambda-tau (TBFLT) technique.Design/methodology/approachThis paper presents a unified approach for analyzing the fuzzy reliability of the washing system under vague environment. This approach applies the TBFLT technique which uses triangular fuzzy numbers for incorporating data uncertainty, fault tree and lambda-tau method for finding system failure rate and repair time mathematical expressions while simplified Tω-based arithmetic operations are applied for computing various reliability parameters of the system. The effectiveness of the TBFLT technique has been demonstrated by analyzing fuzzy reliability of the system using five different techniques including TBFLT. Moreover, this paper applies extended Tanaka’s (1983) approach to rank the critical components of the system.FindingsThe TBFLT technique has the advantage of low computation complexity in comparison to other techniques and effectively reduces the accumulating phenomenon of fuzziness. This main finding verifies the conclusion made by Chen (1994).Originality/valueThe author has suggested a simple and more applicable technique for analyzing the fuzzy reliability of any complex process industrial system under vague environment. The effectiveness of the technique has been demonstrated by computing various reliability parameters of the washing system of a paper plant.

The problem of possibilistic-probabilistic optimization

This paper studies the models and methods for solving optimization problems with hybrid possibilistic-probabilistic uncertainty. The models under consideration have a peculiarity that the interaction of fuzzy parameters is described by the weakest t-norm. We propose solution methods that are based on the integration of indirect optimization methods (the design of equivalent problems) and direct (stochastic quasi-gradient) optimization methods. We establish the results for the models that were not considered in the previous publications on the subject. The resulting models and methods allow us to construct the generalized portfolio analysis models that are intended for managing combined (hybrid) uncertainty.

Sustainable supply chain management using approximate fuzzy DEMATEL method

This study develops the approximate fuzzy Decision Making Trial and Evaluation Laboratory (AFDEMATEL) to analyze uncertain influential factors. The approximate fuzzy arithmetic operations under the weakest t-norm (Tω) arithmetic operations to evaluate sustainable supply chain management based on AFDEMATEL. The fuzzy DEMATEL is one of important decision-making method under uncertain environment. The fuzzy DEMATEL had to be developed for clearly display expert’s options with linguistic variables. The fuzzy operations usually adopt α-cut arithmetic in fuzzy DEMATEL. In this research, the fuzzy DEMATEL technology is substituted with the AFDEMATEL technology. In the sustainable supply chain management example, the AFDEMATEL is employed to find fuzzy cause and effect relationships among criteria. Particular note should be made of the following: [1] the fuzzy DEMATEL with α-cut arithmetic model cannot exactly handle fuzzy cause and effect relationships under uncertain environment, and the fuzziness accumulation phenomenon of the α-cut arithmetic may influence final fuzzy cause and effect relationships; and [2] the approximate fuzzy arithmetic operations gives a justifiable fuzziness spread to analyze fuzzy cause and effect relationships. In the case of selection of cans suppliers, the AFDEMATEL examines the influential factors. Proposed method provides justifiable fuzzy cause and effect relationships with approximate fuzzy arithmetic and can analyze across quadrants phenomenon under uncertain environment, since decision-makers usually want to accurately estimate uncertain influential factors.

Developing a Tω (the weakest t-norm) fuzzy GERT for evaluating uncertain process reliability in semiconductor manufacturing

This paper develops a novel weakest t-norm ( T ω ) fuzzy Graphical Evaluation and Review Technique (GERT) simulation technology. This proposal is designed to be useful in a realistic environment and improve upon the traditional fuzzy GERT insofar as it has been developed for analyzing complex systems in uncertain environments; the traditional system usually adopts α-cut arithmetic operations for its calculations. In this research, the fuzzy support system develops the T ω fuzzy GERT as a substitute for traditional fuzzy GERT technology. In the examples, the fuzzy support system constructs a model of 300 mm manufacturing processes in the context of a lithography area. Moreover, the manufacturing processes model is examined with regard to the fuzzy support system using two types of fuzzy arithmetic: α-cut arithmetic and the T ω operator. Notably: (1) both types of fuzzy arithmetic provide a reliable analysis of the fuzzy GERT model with regard to a lithography area; (2) under the traditional fuzzy GERT model, the α-cut arithmetic provides results such that the fuzziness of the model calculation was fuzzier than that of the T ω fuzzy arithmetic due to the accumulation of fuzziness of the α-cut arithmetic; (3) the α-cut arithmetic cannot effectively preserve the original shape of a membership function; and (4) the T ω arithmetic gives a justifiable fuzziness/fuzzy spread because it takes only the maximal fuzziness encountered and calculates that into the operation. Our proposed T ω fuzzy GERT can successfully analyze a 300 mm manufacturing process; this has been evidenced in the research. Additionally, the proposed model uses simple arithmetic operations rather than traditional fuzzy GERT with applications in complex manufacturing systems. Moreover, the T ω arithmetic provides more credible (or conservative) information/results with regard to the amount of fuzziness in the 300 mm manufacturing processing model.

Applying an approximate Tω (the weakest t-norm) fuzzy GERT to evaluate two-unit standby redundant system reliability

Abstract Standby redundant system is a powerful tool to enhance the reliability, availability, quality, and safety of operational plants. This study proposed an approximate Tω (the weakest t-norm) fuzzy Graphical Evaluation and Review Technique (GERT) simulation technology to simulate two-unit standby redundant system reliability. The approximate fuzzy arithmetic operations employ principle of interval arithmetic under the Tω arithmetic operations. Therefore, the novel fuzzy arithmetic operations may obtain fitter decision values, which have smaller fuzziness accumulating, under vague environment. In numerical examples the approximate fuzzy arithmetic operations has evidenced that it can successfully calculate results of fuzzy operations as interval arithmetic, and can more effectively reduce fuzzy spreads. In the real fuzzy repairable reliability model the performance also shows that the approximate fuzzy arithmetic operations successfully simulate/analyze the two-unit standby redundant system reliability and obtain more confident fuzzy results under vague environment.

Smooth t-subnorms on finite scales

In this work smooth t-subnorms and boundary weak t-norms (bwt-norms) defined on a finite chain are studied. In the smooth case, it is proved that smooth bwt-norms reduces to smooth t-norms, whereas a classification of smooth t-subnorms as ordinal sums of a smooth t-norm and an Archimedean smooth t-subnorm is proved, in a similar way as in the case of [0,1]. With respect to Archimedean smooth t-subnorms, some cases depending on their zero-region are considered, classifying all of them in some cases and some specific families in others.

CpkIndex Estimation under Tw(the weakest t-norm)-based Fuzzy Arithmetic Operations

The measurement of performance of a process considering both the location and the dispersion of information about the process is referred to as the process capacity indices (PCIs) of interest, <TEX>$C_{pk}$</TEX>. This information is presented by the mean and standard deviation of the producing process. Linguistic variables are used to express the evaluation of the quality of a product. Consequently, <TEX>$C_{pk}$</TEX> is defined with fuzzy numbers. Lee [Eur. J. Oper. Res. 129(2001) 683-688] constructed the definition of the <TEX>$C_{pk}$</TEX> index estimation presented by fuzzy numbers and approximated its membership function using the "min" - norm based Zadeh's extension principle of fuzzy sets. However, Lee's result was shown to be invalid by Hong [Eur. J. Oper. Res. 158(2004) 529-532]. It is well known that <TEX>$T_w$</TEX> (the weakest t-norm)-based addition and multiplication preserve the shape of L-R fuzzy numbers. In this paper, we allow that the fuzzy numbers are of L-R type. The object of the present study is to propose a new method to calculate the <TEX>$C_{pk}$</TEX> index under <TEX>$T_w-based$</TEX> fuzzy arithmetic operations.

Weighted average of fuzzy numbers under TW(the weakest t-norm)-based fuzzy arithmetic operations

Many authors considered the computational aspect of sup-min convolution when applied to weighted average operations. They used a computational algorithm based on a-cut representation of fuzzy sets, nonlinear programming implementation of the extension principle, and interval analysis. It is well known that <TEX>$T_W$</TEX>(the weakest t-norm)-based addition and multiplication preserve the shape of L-R type fuzzy numbers. In this paper, we consider the computational aspect of the extension principle by the use of <TEX>$T_W$</TEX> when the principle is applied to fuzzy weighted average operations. We give the exact solution for the case where variables and coefficients are L-L fuzzy numbers without programming or the aid of computer resources.

On the construction of boundary weak triangular norms through additive generators

Recently, M. Urbański and J. Wa̧sowski [Fuzzy arithmetic based on boundary weak t-norms, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 13 (2005) 27–37] extended the triangular norm operators to a new class of operators called the boundary weak triangular norms. This extension is based on the replacement of the condition T ( x , 1 ) = x by the weaker one T ( x , 1 ) ≤ x for x ∈ [ 0 , 1 ) . In this paper we discuss the construction of the boundary weak triangular norms based on their additive generators. Continuous decreasing functions f : [ 0 , 1 ] → [ 0 , ∞ ] which generate proper boundary weak triangular norms are completely characterized. Several examples are included.

Fuzzy measures for a correlation coefficient of fuzzy numbers under TW(the weakest t-norm)-based fuzzy arithmetic operations

Recently, the sup-min convolution based on Zadeh’s extension principle has been used by Liu and Kao [Fuzzy measures for correlation coefficient of fuzzy numbers, Fuzzy Sets and Systems 128 (2002) 267–275], to calculate a fuzzy correlation coefficient. They used a mathematical programming approach to derive fuzzy measures based on the classical definition of the correlation coefficient. It is well known that T W (the weakest t-norm)-based addition and multiplication preserve the shape of L– R fuzzy numbers. In this paper, we consider the computational aspect of the T W-based extension principle when the principle is applied to a correlation coefficient of L– R fuzzy numbers. We give the exact solution of a fuzzy correlation coefficient without programming or the aid of computer resources.

Shape preserving multiplications of fuzzy numbers

In this note, we show that T W , the weakest t-norm, is the only t-norm T which induces a shape preserving multiplications of LR-fuzzy numbers.