Fuzzy Arithmetic Operations Research Articles

Reliability analysis of landing architecture of aircraft using Fermatean fuzzy arithmetic operation

Reliability analysis of landing architecture of aircraft using Fermatean fuzzy arithmetic operation

Examining multiplication and division on fuzzy numbers using composite tables

A fuzzy number is a concept that represents a real number abstractly. It doesn’t assign a single value, but instead a range of uncertain values, each with its own weight between 0 and 1. It is essential to comprehend the arithmetic operations of fuzzy numbers in the real number of fuzzy mathematics. One of the fundamental concepts that should not be ignored is the explanation of the interval operation. In this paper, a composition table is constructed to demonstrate the multiplication and division of fuzzy numbers. Fuzzy arithmetic operations are commonly utilized to solve mathematical equations that involve fuzzy numbers. In this paper, we explore the relationship between multiplication and division operations and fuzzy numbers. We also demonstrate that the shape of their membership functions heavily influences the outcome of our calculations when using fuzzy numbers.

An [formula omitted]-cut intervals based Fuzzy Best–Worst Method for Multi-Criteria Decision-Making

To address ambiguity arising from linguistic decision judgments, Guo and Zhao introduced a model of the well-known Multi-Criteria Decision-Making (MCDM) method, Best-Worst Method (BWM), using fuzzy sets, known as Fuzzy Best–Worst Method (FBWM). Despite its versatility, FBWM has limitations, such as criteria-weights being independent of fuzzy comparison values’ shape and reliance on approximate fuzzy arithmetic operations. In response to these limitations, we introduce a novel model of Fuzzy Best–Worst Method based on α-cut intervals. In this model, the calculation of optimal weights involves formulating an optimization problem that utilizes exact fuzzy arithmetic operations defined in terms of α-cut intervals. This approach aims to optimize the entire shape of fuzzy comparison values simultaneously. However, it turns out that, although we have proven the existence of optimal solution for this problem, solving it proves to be challenging. To resolve this issue, we approximate optimal weight set(s) using triangular fuzzy weight sets and assess the Error of Approximation (EA). The approximate weight set with the desired EA is then defuzzified to obtain a crisp weight set. In situations where there are multiple approximate weight sets with the same EA, interval-weights are calculated. Following that, we establish the necessary conditions for a Fuzzy Pairwise Comparison System (FPCS) to be consistent. Using these conditions, we calculate a lower bound of the Consistency Index (CI), providing a measure of the accuracy of weights. Finally, we discuss numerical examples and a real-world scenario, ranking of risk factors in supply chain 4.0, to validate the effectiveness of the proposed model.

Arithmetic Operations on Generalized Pentagonal Fuzzy Numbers

Fuzzy concepts have been widely used to treat imprecision in many fields of natural and social sciences. In most of the natural science fields such as applied mathematics, physics, chemistry, and engineering, triangular and trapezoidal fuzzy numbers are commonly used and arithmetic operations on those numbers are studied in detail. On the other hand, in engineering and social science fields such as sociology and psychology, while treating the uncertainties, these numbers are not applicable and fuzzy numbers with more parameters and clear definitions of their arithmetic operations are needed. In order to fill this gap in the literature, in this study we propose the generalized pentagonal fuzzy numbers, and we define fuzzy arithmetic operations based on both extension and the function principle.

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.

Fuzzy risk-based technique for the design of an ogee spillway in a diversion dam based on hydrological return period discharge and the resistance-load theory

This paper provides a Fuzzy methodology based on calculating the hydrological return period discharge and the resistance-load theory for designing an ogee spillway in a diversion dam. Hence, the uncertainties of rainfall intensity, catchment runoff coefficient, and spillway discharge coefficient are endowed with triangular Fuzzy numbers (TFNs), describing possible values the uncertain parameters may have. To do so, the rational formula and, later, the spillway discharge equation are analyzed here to obtain the Fuzzy products of the peak runoff and spillway discharge capacity. On the basis that both must match, the risk is calculated using the difference between the obtained Fuzzy product quantities with three Fuzzy arithmetic operations. It is observed that the standard approximation closely tracks the result of the alpha-cut (exact) method, providing a time-saving tool for Fuzzy calculations. Since the return period (T) and risk of failure are related quantities, the spillway dimensions are calculated for several recurrence intervals, assuming risk equal to 1/T. Then, the Fuzzy approach is compared with the conventional return-period approach for the design of an ogee crest in three experimental catchments in Iran with adequate historical data. The result indicates that the Fuzzy technique outperforms the conventional return-period approach for designing an ogee spillway, obtaining an average 29% larger spillway dimension in all case studies. Beyond that, the Fuzzy procedure is applied to another catchment with short rainfall-runoff data, which is impossible to implement the conventional return-period method for the hydrologic design of an ogee spillway.

A Study on Intuitionistic Fuzzy Critical Path Problems Through Centroid Based Ranking Method

In this study the intuitionistic fuzzy version of the critical path method has been proposed to solve networking problems with uncertain activity durations. Intuitionistic fuzzy set [1] is an extension of fuzzy set theory [2] unlike fuzzy set, it focuses on degree of belonging, the degree of nonbelonging or non-membership function and the degree of hesitancy which helps the decision maker to adopt the best among the worst cases. Trapezoidal and the triangular intuitionistic fuzzy numbers are utilized to describe the uncertain activity or task durations of the project network. Here trapezoidal and triangular intuitionistic fuzzy numbers are converted into their corresponding parametric form and applying the proposed intuitionistic fuzzy arithmetic operations and a new method of ranking based on the parametric form of intuitionistic fuzzy numbers, the intuitionistic fuzzy critical path with vagueness reduced intuitionistic fuzzy completion duration of the project has been obtained. The authentication of the proposed method can be checked by comparing the obtained results with the results available in pieces of literature.

A user-configurable and explainable framework for ranking fuzzy numbers

Fuzzy number ranking is an important and fundamental subject matter within fuzzy set theory, just as the ranking of crisp values is important and fundamental to classical mathematics. For a user of fuzzy set theory, a fuzzy number ranking method that is user-configurable and user-explainable is highly desirable -- being user-configurable is important for ensuring the fitness of the fuzzy ranking method in relation with the assumptions of the application domain of interest, and being user-explainable is important for providing confidence to the user and the stakeholders of the appropriateness of the ranking outcomes in an expert system. In this article, a simple framework for ranking fuzzy numbers is proposed and developed. This simple framework consists of concrete constructs similar to the building blocks in a LEGO set, for the user to configure his/her own fuzzy ranking operator. The proposed framework is applied to examples taken from previous comparative studies from the fuzzy set theory literature, and to multiple-criteria-decision-making and other fuzzy ranking problems. First of all, these demonstrations will illustrate that fuzzy number ranking can indeed be done in a concrete, user-configurable, and user-explainable manner. Thus, the proposed framework represents a viable alternative for the fuzzy ranking literature, which currently consists of various ranking methods that are mathematically sophisticated yet oftentimes highly non-trivial for the user to interpret and to adapt/re-configure for his/her applications. Secondly, we note that the precursor of the proposed framework has been successfully employed to re-derive, instead of postulate, commonly used fuzzy set-theoretic operators, and to derive fuzzy arithmetic operators that do not produce anomalies plaguing other fuzzy arithmetic approaches. Thus, the applicability of such a framework to fuzzy ranking will further demonstrate its potential as a fundamental framework for tackling a wide range of issues in fuzzy set theory, rather than being an isolated approach limited to narrow aspects of fuzzy set theory.

A New Estimation of Failure Analysis in Fuzzy Environment, Case Study: The Electrical Model Failure for the Football Stadium

In this paper, we propose the new fuzzy arithmetic operations based on transmission average on pseudo-hexagonal fuzzy numbers from [F. Abbasi, T. Allahviranloo and S. Abbasbandy, A new attitude coupled with fuzzy thinking to fuzzy rings and fields, Journal of Intelligent and Fuzzy Systems 29 (2015) 851–861]. The properties of these propose operations and their fundamental qualities are discussed in detail. Several illustrative examples were given to show the accomplishment and ability of the proposed method. Finally, as a applied rudimentary case study an electric network model of a football stadium is considered in fuzzy environment. The reliability of components for the proposed model is considered as pseudo-hexagonal fuzzy number.

Fuzzy probability and α‐cut based‐fault tree analysis approach to evaluate the reliability and safety of complex engineering systems

AbstractReliability and safety are very critical in complex engineering systems such as chemical and nuclear industries. Fault tree analysis (FTA) has been widely applied to evaluate the reliability and safety of complex engineering systems. This approach assumes that systems always have exact component failure probabilities. However, this assumption is not always true in real‐world applications. To deal with this, a number of scholars have integrated fuzzy approach into FTA of engineering systems whose component exact probabilities cannot be collected. Furthermore, simple fuzzy arithmetic operations on membership function have also been proposed to quantify Boolean gates for propagating uncertainties from basic event fuzzy probabilities into the top event fuzzy probabilities. In this study, fuzzy arithmetic operations on α‐cuts, as opposed to simple fuzzy arithmetic operations, are proposed. The significance of the α‐cut is that each α‐cut can uniquely represents each fuzzy probability. In order to verify the applicability of the proposed approach, the reliability of the long‐term cooling system of the Westinghouse AP1000 to mitigate a large break loss of coolant accident is evaluated and the results are compared to the reliability of the same system quantified by conventional FTA published in qualified scientific journals. The results of the benchmarking confirm that the proposed approach can be practicably implemented to estimate the reliability and safety of complex engineering systems whose basic event reliability are characterized by fuzzy probabilities. To be applied in other complex engineering systems, safety analysts have to concern for some steps, which are particularly developed for a specific application, such as failure possibility distribution and its corresponding fuzzy probability distribution and the selection of a defuzzification technique.

A study on multiplication operation on triangular fuzzy numbers

The arithmetic operations on fuzzy number are basic content in fuzzy mathematics. But still the operations of fuzzy arithmetic operations are not established. There are some arithmetic operations for computing fuzzy number. Certain are analytical methods and further are approximation methods. In this paper we, compare the multiplication operation on triangular fuzzy number between α-cut method and standard approximation method and give some examples.

Choquet integral-based intuitionistic fuzzy arithmetic aggregation operators in multi-criteria decision-making

Intuitionistic fuzzy set (IFS), a classical extension of fuzzy set, is a powerful tool to describe fuzzy information in multi-criteria decision-making (MCDM) because of its membership and non-membership functions. Intuitionistic fuzzy aggregation operators are the standard mathematical tools for the combination of several inputs with respect to criteria into one unique output. The traditional intuitionistic fuzzy aggregation operator-based methods have been criticized because of reasons that include disregard for comprehensive correlative relationships of criteria and ignorance for combination of fuzzy measures of criteria and weights of positions. In this paper, the effectiveness of traditional aggregation operator-based MCDM techniques is improved by a novel method to select the best alternative(s) under the intuitionistic fuzzy environment. We begin by defining the Choquet integral-based intuitionistic fuzzy arithmetic aggregation (CIIFAA) operator, the remarkable properties of which are proved in details. Further, the Choquet integral based intuitionistic fuzzy hybrid arithmetic aggregation (CIIFHAA) operator is defined as well as the proofs of its properties. The CIIFHAA operator can not only capture the comprehensive correlative relationships of criteria in a simpler manner, but also combine the weights of positions. Then, we put forward a MCDM method based on CIIFHAA operator with intuitionistic fuzzy evaluations. An illustrative example is solved, the best alternative and the ranking of alternatives are obtained, which can verify the feasibility of the proposed method. Finally, two comparisons are conducted to illustrate the stability and advantages of the proposed method.

Arithmetic Operations and Expected Values of Regular Interval Type-2 Fuzzy Variables

High computation complexity restricts the application prospects of the interval type-2 fuzzy variable (IT2-FV), despite its high degree of freedom in representing uncertainty. Thus, this paper studies the fuzzy operations for the regular symmetric triangular IT2-FVs (RSTIT2-FVs)—the simplest IT2-FVs having the greatest membership degrees of 1. Firstly, by defining the medium of an RSTIT2-FV, its membership function, credibility distribution, and inverse distribution are analytically and explicitly expressed. Secondly, an operational law for fuzzy arithmetic operations regarding mutually independent RSTIT2-FVs is proposed, which can simplify the calculations and directly output the inverse credibility of the functions. Afterwards, the operational law is applied to define the expected value operator of the IT2-FV and prove the linearity of the operator. Finally, some comparative examples are provided to verify the efficiency of the proposed operational law.

Experts’ Judgment-Based Mamdani-Type Decision System for Risk Assessment

Mamdani fuzzy inference system has been widely used for potential risk modelling and management. The decision-making is usually provided by multiple experts in the field. The conflicting information in sources from different experts become an open issue and has attracted some researchers to investigate further. Various risk factors in a project caused difficulties for decision makers to make reliable decisions on the whole project since it involves ambiguities, vagueness, and fuzziness. The introduction of the fuzzy inference system to the evaluation of construction risk is capable in explaining its reasoning process and, hence, overcoming such problems. Risk factors under the project management risk were identified through literature sources and from the opinion of experts. It is found that the likelihood and severity of risk is somehow interlinked with the concept of fuzzy theory. For model input and output linguistics variables, the triangular membership function was selected. The methodology employs a fuzzy aggregation system in which an appropriate control action can be determined by the acquisition of expert judgment. A total of 23 rules with logical OR operator, truncation implication, and Mean of Maxima (MoM) method for defuzzification were used to create an effective fuzzy model intended for making decisions. The framework determines the relationship between input and output parameters in if-then rules or mathematical functions using an effective fuzzy arithmetic operator. The study addresses the principle issues of multiexpert opinions based on Mamdani-type decision system and the illustrative example taken from one of medium-sized project held in Malaysia’s construction industry. By comparing with other experimental results, we verify the rationality and reliability of the proposed method.

On the properties of the fuzzy weighted average of fuzzy numbers with normalized fuzzy weights

Weighted average with normalized weights is a widely used aggregation operator that takes into account the varying degrees of importance of the numbers in a data set. It possesses some important properties, like monotonicity, continuity, additivity, etc., that play an important role in practical applications. The inputs of the aggregation as well as the normalized weights are usually not known precisely. In such a case, their values can be expressed by fuzzy numbers, and the fuzzy weighted average of fuzzy numbers with normalized fuzzy weights needs to be employed in the model. The aim of the paper is to reveal whether and in which way the properties of the weighted average operator can be observed also for its fuzzy extension. It is shown that it possesses three conditions characteristic for aggregation operators -- identity, monotonicity and boundary conditions, and besides that, also compensation, idempotency, stability for linear transformation, 1-lipschitzianity, and continuity. Furthermore, it is proved that it preserves strict monotonicity in case of positive fuzzy weights, and symmetry in case of equal fuzzy weights, although it does not coincide with the fuzzy arithmetic mean operator in such a case. One of the most valuable result of the study is the fact that in contrast to the crisp weighted average operator, it is not additive. The importance of the obtained results is discussed and illustrated by several illustrative examples.

A modification of fuzzy arithmetic operators for solving near-zero fully fuzzy matrix equation

Matrix equations have its own important in the field of control system engineering particularly in the stability analysis of linear control systems and the reduction of nonlinear control system models. There are certain conditions where the classical matrix equation are not well equipped to handle the uncertainty problems such as during the process of stability analysis and reduction in control system engineering. In this study, an algorithm is developed for solving fully fuzzy matrix equation particularly for ~ A ~X ~B 􀀀 ~X = ~ C, where the coefficients of the equation are in near-zero fuzzy numbers. By modifying the existing fuzzy multiplication arithmetic operators, the proposed algorithm exceeds the positive restriction to allow the near-zero fuzzy numbers as the coefficients. Besides that, a new fuzzy subtraction arithmetic operator has also been proposed as the existing operator failed to satisfy the both sides of the nearzero fully fuzzy matrix equation. Subsequently, Kronecker product and V ec-operator are adapted with the modified fuzzy arithmetic operator in order to transform the fully fuzzy matrix equation to a fully fuzzy linear system. On top of that, a new associated linear system is developed to obtain the final solution. A numerical example and the verification of the solution are presented to demonstrate the proposed algorithm.

Estimation of Failure Using Fault Tree Analysis Based on New Operations on LR-Type Flat Fuzzy Numbers

In this paper, we analyze the reliability of fuzzy system (particularly, series and parallel system) with independent and non-identically distributed components using the new operations of TA [F. Abbasi, T. Allahviranloo and S. Abbasbandy, A new attitude coupled with fuzzy thinking to fuzzy rings and fields, Journal of Intelligent and Fuzzy Systems 29 (2015) 851–861.] due to the smaller results support, easier calculations and special properties than operations based on the extension principle (in the domain of the membership function) and the interval arithmetic (in the domain of the [Formula: see text]-cuts). we propose the new fuzzy arithmetic operations based on transmission average(TA) on LR type flat fuzzy numbers. In the proposed formulae, LR type flat fuzzy numbers are not restricted to have the same [Formula: see text] and [Formula: see text] shape functions. This allows arithmetic operator for arithmetic involving LR type flat fuzzy numbers of different and general shape. Finally, an imprecise failure to start of an automobile is considered in fuzzy environment. The reliability of components of the proposed model is considered as LR type flat fuzzy numbers.

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.

The Fuzzy Arithmetic Operations of Transmission Average on Pseudo-Hexagonal Fuzzy Numbers and Its Application in Fuzzy System Reliability Analysis

In most natural science fields, triangular and trapezoidal fuzzy numbers are commonly used while in engineering and social science fields such as sociology and psychology while treating the uncertainties these numbers are not applicable and fuzzy numbers with more parameters and clear definitions of their arithmetic operations are needed. In order to fill this gap in the literature, we propose the new fuzzy arithmetic operations based on transmission average on pseudo-hexagonal fuzzy numbers, which was already implied in [1] in its rudimentary form and was finally presented in its fully fledged form in [2]. Several illustrative examples were given to show the accomplishment and ability of the proposed method. We present a new method for fuzzy system reliability analysis based on the fuzzy arithmetic operations of transmission average, where the reliabilities of the components of a system are represented by pseudo- hexagonal fuzzy numbers defined in the universe of discourse [0, 1]. The proposed method has the advantages of modeling and analyzing fuzzy system reliability in a more flexible and more intelligent manner. Finally, a marine power plant is considered in fuzzy environment. The reliability of components of the proposed model is considered as pseudo-hexagonal fuzzy numbers.

Fuzzy fault tree analysis for controlling robot-related accidents involving humans in industrial plants: a case study

PurposeIn recent years, the application of robots in different industrial sectors such as nuclear power generation, construction, automobile, firefighting and medicine, etc. is increasing day by day. In large industrial plants generally humans and robots work together to accomplish several tasks and lead to the problem of safety and reliability because any malfunction event of robots may cause human injury or even death. To access the reliability of a robot, sufficient amount of failure data is required which is sometimes very difficult to collect due to rare events of any robot failures. Also, different types of their failure pattern increase the difficulty which finally leads to the problem of uncertainty. To overcome these difficulties, this paper presents a case study by assessing fuzzy fault tree analysis (FFTA) to control robot-related accidents to provide safe working environment to human beings in any industrial plant.Design/methodology/approachPresented FFTA method uses different fuzzy membership functions to quantify different uncertainty factors and applies alpha-cut coupled weakest t-norm () based approximate fuzzy arithmetic operations to obtain fuzzy failure probability of robot-human interaction fault event which is the main contribution of the paper.FindingsThe result obtained from presented FFTA method is compared with other listing approaches. Critical basic events are also ranked using V-index for making suitable action plan to control robot-related accidents. Study indicates that the presented FFTA is a good alternative method to analyze fault in robot-human interaction for providing safe working environment in an industrial plant.Originality/valueExisting fuzzy reliability assessment techniques designed for robots mainly use triangular fuzzy numbers (TFNs), triangle vague sets (TVS) or triangle intuitionistic fuzzy sets (IFS) to quantify data uncertainty. Present study overcomes this shortcoming and generalizes the idea of fuzzy reliability assessment for robots by adopting different IFS to control robot-related accidents to provide safe working environment to human. This is the main contribution of the paper.