Type-2 Fuzzy Numbers Research Articles

Novel decision making approach for sustainable renewable energy resources with cloud fuzzy numbers

Decision making approaches depending on the assessments of individual decision makers produce inaccurate results due to the existence of multiple uncertainties. To model intrapersonal uncertainty, interpersonal uncertainty and randomness in decision making assessments, this research study proposes a novel approach by integrating linear and non-linear type of fuzzy numbers with cloud model theory using novel technique of computing entropy of these fuzzy numbers. A novel mathematical model known as cloud fuzzy numbers is introduced using the concepts of fuzzy numbers and cloud theory. The self-evaluated relative weights of experts are computed using a non-linear optimization method which is based on maximum deviation method and Lagrange multipliers of cloud fuzzy numbers. The new cloud fuzzy numbers are then combined with CODAS (combinative distance based assessment) approach that is based on the largest Euclidean and Taxicab distances for the selection of suitable criteria. Firstly, the linguistics evaluations are converted into the fuzzy numbers and then cloud fuzzy numbers using formulae of expectation and entropy ensuring that the obtained interval cloud values follows a normal distribution. Secondly, the cloud fuzzy weighted arithmetic averaging operator is used to aggregate cloud fuzzy numbers using the self-evaluated fuzzy weights Thirdly, the assessment score is determined to rank the alternatives by computing the distance between normalized weighted matrix and the negative ideal solution. Finally, a case study is discussed for the selection of best renewable energy resource in Turkey to elaborate the significance of the proposed research. The convergence and accuracy of the proposed model is proved with certain mathematical and theoretical results.

Causal Analysis of Roof Caving on Underground Mine: A New Theory and Optimized DEMATEL Approach

In the context of mines, roof-caving incidents constitute the most common and expensive accidents. To enhance the management and prevention of roof-caving accidents, it is imperative to investigate the factors that contribute to such incidents and comprehend the intricate causal relationships among them. This study aims to classify the causes of these accidents into three categories: basic factors, controllable factors, and sudden factors, based on the mechanism of roof caving. The categorization is primarily determined by two indicators: intervisibility and variability. Furthermore, the study delves into analyzing the mutual influence relationships among these factors and proposes the BCX theory (Basic-Controllable-Sudden causing theory) for roof caving. Subsequently, based on this theory, an index system called BCX is established for roof caving, and the DEMATEL method is employed to analyze the factors within this index system. To attain more accurate results, this study utilizes interval trapezoidal type-2 fuzzy number scale optimization and Tsallis relative entropy to address the limitations of the DEMATEL method. By comparing the outcomes of the traditional and optimal DEMATEL methods, it is observed that the optimal method exhibits superior applicability in the BCX index system of roof caving, with results that align closely with the actual scenario. Therefore, the optimal DEMATEL method’s analysis of centrality, importance, and chain relationships between the factors within the BCX index system will offer valuable guidance for preventing roof-caving accidents in mining operations.

Solving Interval Type-2 Fuzzy Pentagonal Polynomial Equation Using Horner-Muller's Method

The recent boom in interval type-2 fuzzy sets has attracted many researchers to the fields of fuzzy numbers, algebraic expressions, and iterative methods. Interval type-2 fuzzy pentagonal polynomial (IT2 FPP) is one of the algebraic expressions that combine the knowledge of interval type-2 pentagonal fuzzy numbers and polynomial equations. Solving IT2 FPP in the absence of an iterative method is becoming a common encounter as the solution is often difficult, expensive, and inexact. This paper defines IT2 FPP and the iterative method Horner-Muller’s used to find the approximate solution of the IT2 FPP. The IT2 FPP is introduced by combining the generalization of interval type-2 pentagonal fuzzy numbers and fuzzy polynomials, whereas the Horner-Muller’s method is a combination of Horner’s method and Muller’s method. A numerical example is presented to illustrate the computational procedures for finding the approximate solution. The approximate solution is obtained after two iterations with zero error. A Comparative analysis is also presented to validate the consistency and efficiency of the proposed method compared to benchmark iterative methods.

Shadowed AHP for Multi-Criteria Supplier Selection

Numerous techniques of multi-criteria decision-making (MCDM) have been proposed in a variety of business domains. One of the well-known methods is the Analytical Hierarchical Process (AHP). Various uncertain numbers are commonly used to represent preference values in AHP problems. In the case of multi-granularity linguistic information, several methods have been proposed to address this type of AHP problem. This paper introduces a novel method to solve this problem using shadowed fuzzy numbers (SFNs). These numbers are characterized by approximating different types of fuzzy numbers and preserving their uncertainty properties. The new Shadowed AHP method is proposed to handle preference values which are represented by multi-types of uncertain numbers. The new approach converts multi-granular preference values into unified model of shadowed fuzzy numbers and utilizes their properties. A new ranking approach is introduced to order the results of aggregation preferences. The new approach is applied to solve a supplier selection problem in which multi-granular information are used. The features of the new approach are significant for decision-making applications.

Analyzing risk factors of tuberculosis using type-2 interval-valued trapezoidal fuzzy numbers with Einstein aggregation operators extended to MCDM

The principal motive of this work is to evolve and initiate an extension from interval-valued fuzzy sets to type-2 interval-valued fuzzy sets (T2IVFS) related to weighted aggregation functions containing the Einstein operator. The chief reason for this extension is that the constancy of the terms can also be taken into data during the aggregation operation. The main goal of this article is to compose the aggregation operators and their characteristics such as the Type-2 interval-valued fuzzy Einstein weighted arithmetic aggregating operator (T2IVFEWA), Type-2 interval-valued fuzzy Einstein weighted geometric aggregating operator (T2IVFEWG), and the characteristics are expressed. At last, to intimate the effectiveness of the suggested approach and explicate the purpose of these operators, a hybrid multi-criteria decision-making problem (MCDM) to select the best risk factor for Tuberculosis (TB) is considered and the result is compared with the outcome of the existing operators and methods. Additionally, a sensitivity analysis was conducted to verify the robustness of the proposed decision-making process.

A new defuzzification method and ranking method for type-2 fuzzy numbers

The type-2 fuzzy number (T2 FN) is a special type-2 fuzzy set whose all secondary membership functions are fuzzy numbers. This paper proposes a new defuzzification method and ranking method for T2 FNs. The defuzzification method can be expressed in a closed-form formula or can be divided into two computational steps. The first step computes the centroid of each secondary membership function to obtain a type-reduced set using the method derived from generalizing the existing alpha-cut defuzzification methods by introducing a parameter that controls the importance of secondary membership degrees, and the second step calculates the centroid of the type-reduced set. The defuzzified values are used to provide position information of T2 FNs. Secondly, we define a function to calculate the area enclosed by the image of the type-reduced set and the coordinate axes, and prove that this function is continuous. The values of this function are used to provide shape information of T2 FNs. Finally, to rank T2 FNs, we define a family of quasi-orders based on their position and shape information, and illustrate the validity and rationality of these quasi-orders through examples.

A blockchain platform selection method with heterogeneous multi-criteria Decision-Making based on hybrid distance measures and an AHP-EWM weight method

The blockchain platforms (BPs) have emerged as important tools for improving enterprise data security. However, the selection of a suitable BP for is a challenging issue for enterprises. In this paper, a novel heterogenous multi-criteria decision-making (MCDM) model is proposed to facilitate the BP selection and evaluation process. First, the evaluation criteria for BPs are defined through an analysis of popular BPs. Second, based on the defined evaluation criteria, multiple types of fuzzy numbers are adopted to accurately represent the evaluation criteria. Then, to deal with multiple types of fuzzy numbers, a hybrid distance measure is proposed to determine the negative and positive ideal separation matrices of the evaluation criteria. In addition, using the axiomatic design’s information content metric the as basis, an improved method is proposed to unify the evaluation of multiple types into a single number. An improved weight method is then proposed based on the combination of objective and subjective weighting to obtain the comprehensive weights of the criteria. Finally, a closeness degree metric is introduced to obtain the ranking order of all BP alternatives. An example case is analyzed to illustrate how to evaluate and select a suitable BP. The superiority of the proposed heterogeneous model is verified by comparisons with state-of-the-art methods.

A new intuitionistic fuzzy scheme of data envelopment analysis for evaluating rural comprehensive health service centers

In this study, the classical CCR-IO DEA model (an input-oriented form of the DEA model of Charnes, Cooper, and Rhodes [24] is extended to an intuitionistic fuzzy environment. For this aim, the trapezoidal intuitionistic fuzzy numbers are used to show the values of inputs and outputs. An intuitionistic fuzzy number is characterized by membership, non-membership, and hesitancy degrees. Therefore, such type of fuzzy numbers is more complete and contains more information to reflect a fuzzy type of uncertainty comparing to the conventional fuzzy values. In order to solve the proposed intuitionistic fuzzy CCR-IO model, for the first time a possibility-based approach is developed for the intuitionistic fuzzy DEA models. In this approach for the first time the ME measure of fuzzy events is extended to intuitionistic fuzzy DEA models, therefore, some new definitions and propositions are presented and clarified. This measure is more flexible and effective comparing to the other measures such as credibility, necessity, and possibility measures. A case study from the health care sector of Iran is considered to evaluate the proposed solution approach numerically. The case study is solved under different scenarios, a full sensitivity analysis is performed, and finally some useful managerial insights are presented. In addition, the results of the proposed intuitionistic CCR-IO model are compared to the classical models of the literature.

Fuzzy Uncertainty Analysis of A Fractional Order HIV Dynamic Model with Type-1 and Interval Type-2 Parameters

Abstract In recent years, the study of mathematical models for the Human Immunodeficiency Virus (HIV) has attracted considerable interest due to its importance in comprehending and combating the propagation of the virus. Typically, the model's governing equations are a system of ordinary differential equations. In order to explain the inheritance behaviour, fractional order HIV models may be more helpful than integer order models. In addition, the presence of uncertainty in real world phenomena can not be avoided, and fuzzy numbers are of great use in these scenarios. In view of the above, the numerical solution of the fuzzy fractional order HIV model is analysed in this paper. The model takes into account the interactions between susceptible, asymptomatic, and symptomatic populations, as well as the effects of fractional order derivatives and fuzzy uncertainty. Here, the differentiation of the fuzzy parameters is considered in granular sense. The uncertain model parameters are addressed with triangular fuzzy numbers (TFNs) and interval type-2 triangular fuzzy numbers (IT2TFNs). The use of interval type-2 fuzzy numbers rather than type-1 fuzzy numbers to express the imprecise parameters may be helpful in some instances where the membership grade is unclear. The generalised modified Euler method (GMEM) is used to derive the corresponding solutions. Lastly, the behaviour of various populations in crisp as well as uncertain environments is also studied using graphical results.

An optimization of the fully fuzzy transportation problem with triangular type-2 fuzzy parameters using novel ranking technique

This paper focuses on solving the fully fuzzy transportation problem in which the parameters are triangular Type-2 fuzzy numbers due to the instinctive of human imprecision. To deal with uncertainty more precisely, a triangular Type-1 fuzzy transportation problem is reformed as a transportation problem with triangular Type-2 fuzzy parameters in this paper. In order to compare triangular Type-2 fuzzy numbers, a new ranking(ordering) technique is proposed by extending the Yager’s function. However, two efficient algorithmic approaches namely, triangular Type-2 fuzzy zero suffix method (TT2FZSM) and triangular Type-2 fuzzy zero average method (TT2FZAM) are proposed to generate the initial transportation cost of the fully triangular Type-2 fuzzy transportation problem. Both TT2FZSM and TT2FZAM are converging towards an optimal solution. In addition to TT2FZSM and TT2FZAM, the modified distribution method is applied to ensure optimality. Subsequently, we carry out a comprehensive discussion of the obtained results to establish the validation of the proposed approach.

The selection of reform models for provincial rural credit cooperatives unions in China using an extended CPT-TODIM method based on novel type-2 fuzzy numbers

The selection of a suitable reform model for provincial rural credit cooperatives unions (PRCCUs) is a pivotal aspect of China’s ongoing financial reform endeavors. This involves optimizing internal processes, enhancing efficiency, minimizing operational costs, elevating service quality, strengthening financial innovation capabilities, and promoting the healthy growth of rural financial markets. However, current research on this matter primarily concentrates on qualitative analysis. As of yet, a scientifically grounded decision-making framework from a quantitative standpoint has not been developed. This absence of quantitative analysis results in inaccuracies in research findings, thereby hindering the practical applicability of outcomes. To tackle this formidable challenge, we propose an innovative quantitative analysis framework employing the multi-criteria group decision-making method. This framework, which builds upon the cumulative prospect theory (CPT), the TODIM method, and the most recent advances in type-2 fuzzy numbers (T2FNs), is referred to as the T2FNs-CPT-TODIM. Initially, we define a novel score function, followed by the introduction of a fresh method for computing the distance between the T2FNs. Subsequently, our model pioneers an exclusive weight allocation strategy for decision-makers, grounded in the principles of Dice Similarity. Furthermore, the objective weights of criteria can be computed utilizing the Distance Correlation-based CRITIC approach. We then optimize the rankings by applying the advantage of gains and losses preference in the CPT-TODIM. Ultimately, we apply our proposed model to the practical task of selecting the optimal reform model for a specific PRCCU. Thorough comprehensive sensitivity and comparative analyses, the results validate the theoretical robustness and practical efficacy of our proposed method.

An ordered multi-objective fuzzy stochastic approach to sustainable water resources management: a case study from Taiyuan City, China

Abstract The uncertainty of socioeconomic development and climate change poses challenges to the sustainable management of water resources in Taiyuan, China. The study proposes a type-2 fuzzy chance-constrained ordered multi-objective fractional programming (T2FCC-ORMOFP) model to address the allocation of water under uncertainties. The model incorporates type-2 fuzzy programming and chance-constrained programming into an overall framework to handle multiple uncertainties under multi-level and multi-objective conflicts, while using fractional programming to reflect the marginal benefits of the system. It prioritizes the principles of optimal, fair, and stable distribution by the upper-level governing authorities, while considering the social and economic environmental benefits of lower-level decision-making. The optimal water allocation scenarios were obtained under different hydrological guarantee rates, violation levels of water supply constraints, and net economic benefits of type-2 fuzzy numbers for 2030. Additionally, groundwater is replaced by reclaimed water in varying proportions for flexible water supply. The results show that the water demand in Taiyuan cannot be ignored, and the risk of water shortage is more sensitive for the agricultural and industrial sectors. The recycled water substitution strategy can optimize the water supply structure and improve social and economic benefits (9–12%). Also T2FCC-ORMOFP can improve overall efficiency (20%) compared with a single-level model.

Legendre wavelet method based solution of fractional order prey–predator model in type-2 fuzzy environment

Researchers have recently examined several fractional-order prey–predator population models in a crisp or type-1 fuzzy environment. But incompleteness in the membership may also be present. In this regard, type-2 fuzzy numbers are employed in this inquiry to address the system’s imprecision since they are more realistic than type-1 fuzzy numbers in light of the uncertain membership grades. To numerically estimate the solution, the Legendre wavelet method (LWM) is used in conjunction with the r2-cut of the r1-plane form of the type-2 fuzzy parameters. To test the applicability of the present method, three cases are examined. The present solutions are quantitatively compared with the RK4 solutions in the crisp case. Approximations of the type-2 fuzzy prey and predator populations are also included. In addition, two- and three-dimensional plots are depicted to examine the behaviour of prey and predator populations in crisp and type-2 fuzzy cases. To assess the LWM’s efficacy, comparison graphs of the acquired solutions and the results of the RK4 approach are also provided. Numerical validation against the RK4 method in crisp cases demonstrates strong agreement, while revealing contrasting population behaviours in first two scenarios, whereas in the third case, both populations reduce with time. Further, the behaviour of the fuzzy solution has also been shown graphically using the footprint of uncertainty (FOU).

Comparative Ranking Preferences Decision Analysis through a Novel Fuzzy TOPSIS Technique for Vehicle Selection

Acquiring a vehicle or financing its purchase is often considered a luxury, particularly for middle-class households. In addition to their primary concern, consumers prioritize distinguishing elements such as vehicle type, size, capacity, engine power, fuel efficiency, safety features, and life-cycle costs. Furthermore, the choice of vehicle has multiple factors and is contingent upon the financial prosperity of the household, allowing for precise expression. The decision-making process is complex and entails selecting the most appropriate alternatives. This study proposes a novel fuzzy technique for order of preference by similarity to the ideal solution (FTOPSIS) method to solve vehicle selection multi-criteria decision-making (MCDM) problems. Decision-makers express their opinions on each alternative and criterion in linguistic terms, in terms of generalized interval type-2 trapezoidal fuzzy numbers (GIT2TrFNs). The FTOPSIS process involved utilizing the defuzzification of GIT2TrFNs to acquire the normalization matrix. Finally, the proposed approach prioritizes the options and chooses the best vehicle when faced with conflicting criteria, as illustrated through a numerical illustration.

A new standardization-based ranking method for generalized trapezoidal picture fuzzy numbers

A generalized fuzzy number (GFN), whose height is not necessarily 1, is used in situations when expert opinions are not completely reliable. This subnormality complicates operations based on the extension principle. Moreover, complexity is inherited in non-standard fuzzy numbers (FNs). This paper aims to present a unified approach for comparing generalized and trapezoidal types of FNs, intuitionistic FNs (IFNs), and picture FNs (PFNs). If some of the hesitation, neutrality, and refusal are assumed to be resolved, then the uncertainty is reduced while making a non-standard FN standardized. The method uses the weighted average membership function (WAMF) to standardize generalized IFNs (GIFNs) and generalized PFNs (GPFNs). WAMF employs parameters describing the behavioral patterns when decision-makers encounter situations involving risk. Then, the ranking process can be continued with the calculation of the centroid point of the resulting GFN. One of the main advantages of this approach is that the computations are straightforward due to the presence of piecewise linearity, enabling us to employ numerical integration. Furthermore, we adapt operations for generalized trapezoidal PFNs (GTPFNs) to mitigate the counter-intuitive consequences resulting from utilizing the minimum operator. The effectiveness of the method is discussed through benchmarks and its implementation in multi-attribute decision-making (MADM).

How to implemented Knowledge management in supply chain management Best-Worst with D-number

There is no doubt that in this hectic world, many companies seek to implement Knowledge Management (KM) as a part of their supply chain management (SCM). However, many factors affect them in many ways. As a result of this research, we will be able to find out which of these factors influenced the implementation of knowledge management in supply chain management in healthcare facilities. In order to identify the most prominent factors in this research, the Best-Worst Method (BWM) has been applied. Due to the fact that decision makers have to deal with a vague environment, DMs have to be integrated with proper methods in order to cope with this problem. The D-number is a kind of fuzzy number that is one of these methods, one of the types of fuzzy numbers. To find and prioritize the best criterion that will impact the implementation of Knowledge Management in Supply Chain Management with the Best-Worst Method, we conducted this research. This model of decision-making attempts to eliminate the pitfall of Analytical Hierarchy Process by using a Multi-Attribute Decision-Making strategy. Using multi-criteria decision-making in uncertain environments, this paper reviews the literature on knowledge management applied to Supply Chain Management (SCM). These methods have not been implemented in the healthcare industry in previous studies, both theoretically and methodologically. Based on the results of the study, it should be noted that, among the 12 criteria, two of them have been eliminated with the use of the Delphi method; culture capacity is the most important criterion and should be targeted by healthcare facilities companies.

Fractional programming for Stackelberg game under type-2 fuzzy environment

This study intends to present a Stackelberg game model for design of the fractional type-2 fuzzy programming. In achieving this aspiration, we develop a probabilistic fuzzy multi-objective fractional linear programming where the entire parameters are of type-2 fuzzy numbers apart from the right-hand side of the constraints are follow Weibull distribution. In the projected approach, the membership function allied with each objective function is generated by using the first-order Taylor series approximation and converted into a single objective function by assuming the weights of the objective functions are equal. Type conversion is made in two ways by existing methods, and using stochastic programming, the probabilistic constraints are transformed into a deterministic form. The accessible model incorporates the non-linear programming viewpoint of the decision-maker and is solved with the help of intuitionistic fuzzy programming (IFS). A comparison study on the optimum results by genetic algorithm (GA) and particle swarm optimization (PSO) with the LINGO 15.0 iterative scheme is offered to resolve the created bi-level programming problem (BLPP) in the course of the Stackelberg game. To make obvious the feasibility of the projected representation and solution methodology, realistic data are measured and results are presented through several discussions.

Risk and Opportunity Assessment Under Fuzzy Constraints

Problem context. It is necessary to develop methods for analyzing risks and opportunities in economic systems under the assumption that normative criteria are arbitrarily blurry. Research objective. To propose models and methods for analyzing risks and opportunities when norms are defined by general type fuzzy numbers (GTFN). Research methodology. When the efficiency factor and its norm are defined by GTFN, risk and opportunity assessments can be obtained using alpha cuts of GTFN and subsequent weighting according to the rule of arithmetic means. In certain cases, there are analytical expressions for risk and chance assessments. Practical application of the results. Expressions have been derived for cases where efficiency factors and norms are represented by scalars, intervals, triangular numbers, and GTFN. Corresponding risk functions and chance functions have been constructed. The proposed models and methods have been tested in assessing risks and opportunities for an industry economic system, with evaluation of boundary levels of industry indexes based on risk criteria. Originality and significance. The research is original and applies methods from fuzzy set theory and soft computing. A technology for analyzing risks and opportunities in economic systems has been proposed.

Performance Analysis of Interval Type-2 Fuzzy X¯ and R Control Charts

Statistical process control (SPC) is one of the most powerful techniques for improving quality, as it is able to detect special causes of problems in processes, products and services with a remarkable degree of accuracy. Among SPC tools, X¯ and R control charts are widely employed in process monitoring. However, scenarios involving vague, imprecise and even subjective data require a type-2 fuzzy set approach. Thus, X¯ and R control charts should be coupled with interval type-2 triangular fuzzy numbers (IT2TFN) in order to add further information to traditional control charts. This paper proposes a performance analysis of IT2TFN and X¯ and R control charts by means of average run length (ARL), standard deviation of the run length (SDRL) and RL percentile. Computer simulations were carried out considering 10,000 runs to obtain ARL, SDRL and the 5th, 25th, 50th, 75th and 95th RL percentiles. Simulation results reveal that the proposed control charts increased fault detection capability (speed of response) and slightly reduced the number of false alarms in processes under control. Moreover, it was observed that, in addition to superior performance, IT2TFN X¯-R control charts proved to be more robust and flexible when compared to traditional control charts.

A Comprehensive Study of New Fuzzy Logic Metric Based on Interval Type-2 Fuzzy Sets Using Real Application

One of the crucial subjects used in operations research is decision-making processes. Multi-Criteria Decision-Making (MCDM) is one of the topics of this problem that is utilized rather often. In terms of ease and effectiveness, Technique for Order Preference by Similarity to Ideal Solutions (TOPSIS) is one of the most used MCDM techniques. To increase the applicability and simplicity of MCDM methods, Interval Type-2 Fuzzy Numbers (IT2FN), a specific type of Type-2 Fuzzy Sets (T2FS), are widely employed in combination with them. Defuzzification methods are used to find the best solution before the problem's ultimate step of resolution, however. Before the last phase of the process, switching to crisp numbers diminishes the use of the IT2FNs. A brief overview of fuzzy IT2FNs applications is given in this study. The review is based on 25 studies carried out between 2015 and 2023. Supply chains, the environment, energy sources, business, and healthcare are just a few of the application areas that were reviewed and segregated from fuzzy logic articles of various sorts that were the most pertinent and often discussed. The analysis and comparison of fuzzy logic implementations takes into account a wide range of alternatives and criteria, including the use of fuzzy logic mixed with additional techniques, such as the fuzzy Analytic Hierarchical Process (AHP), or improvements for group decision-making, fuzzy sets, hesitant fuzzy sets, intuitionistic fuzzy sets, or improvements for fuzzy logic. The merits and limitations of a variety of FLS types that have been employed for system identification are evaluated. Their effectiveness in various applications is also evaluated.