Interval Type-2 Fuzzy Numbers Research Articles

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.

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.

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.

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.

A chance-constrained programming method with credibility measure for solving the multi-skill multi-mode resource-constrained project scheduling problem

This paper addresses a multi-skilled extension of the multi-mode resource-constrained project scheduling problem (MRCPSP) with preemption of activities and generalized precedence relations under uncertainty. The problem is formulated mathematically as a single objective optimization model to minimize the project makespan. Concerning real-world circumstances, amount of non-renewable resources and availability are considered as interval type-2 fuzzy numbers. Therefore, a type-2 fuzzy chance-constrained programming (TFCP) method is developed to tackle the problem. Four unique characteristics are taken into account in this paper concurrently: (i) multi-skilled resources; (ii) renewable and non-renewable resources; (iii) preemptable activities; and (iv) interval type-2 fuzzy parameters. The existence of these practical contributions causes projects schedule, especially construction projects, to be closer to the real-world conditions. The proposed mathematical model is validated and tested on instances from PSPLIB (j12, j16 and j18) and MMLIB (MM50 and MM100) datasets. In this study, the computational results for the first time demonstrate that the number of preemption points depend heavily on fluctuations of non-renewable resources and number of skills. It is also shown that considering multi-skill renewable resources is an effective factor in project makespan. In J20, there were both a four-unit reduction and increase in the number of preemption points achieving by increasing the average amount of available non-renewable resources by 23 units and decreasing it by 22 units, respectively. Additionally, when seven skills of renewable resources were reduced, the number of preemption points increased by seven units. In this problem instance, project is completed within 28 days considering multi-skill renewable resources, whereas the project makespan is reported 33 days without considering them. Various sensitivity analyses are presented to evaluate the proposed mathematical model and the overall solution approach. Comparative analysis is conducted to compare the schedule in crisp and uncertain conditions. Eventually, the importance of considering multi-skill resources in project scheduling is illustrated compared to not considering them.

Interval type-2 trapezoidal fuzzy multi-attribute decision-making method and its application to the corporate investment selection

Interval type-2 trapezoidal fuzzy (IT2TrF) number is a powerful tool to depict fuzzy information. Information measures methods have received more and more attention in recent years as they play an important role in decision-making theory. A new multi-attribute decision-making (MADM) method supported by IT2TrF information measures is investigated in this paper under the IT2TrF information environment. Firstly, three axiomatic definitions of IT2TrF information measures are introduced, which include information entropy, similarity measure and cross-entropy. Secondly, with the help of the exponential function, we formulate some information measure formulas, which are followed by the proofs that the exponential entropy, exponential similarity measure and exponential cross-entropy fit the three axiomatic definitions. Subsequently, a novel IT2TrF MADM method is designed, in which the IT2TrF exponential entropy and cross-entropy are utilized to generate the attribute weights, the IT2TrF exponential similarity measure is employed to obtain the closeness degree of the ideal solution and derive the most satisfying solution. Lastly, we provide a numerical example of corporate investment to demonstrate the applicability and feasibility of the proposed MADM method. The robustness and merits of the developed MADM method are highlighted by the comparative analysis.

Uncertain project network analysis with fuzzy–PERT and Interval Type–2 fuzzy activity durations

Project network analysis is often subject to different uncertainties that can occur in real world scenarios. These uncertainties are not always measurable which makes its statistical estimation hard, so the use of third party information coming from experts is useful to provide an estimation of activity durations of the project. Projects usually have multiple experts who provide different estimates of their perceptions about activity durations which can be represented as Type-2 fuzzy sets in cases where experts do not agree on their estimates. Such disagreement/ambiguitites can be either expressed with Interval Type-2 fuzzy numbers or the proposed fuzzy-PERT set which is an extension of the classical PERT distribution to a fuzzy environment. To do so, a mathematical formulation of the fuzzy problem is presented, two solution methods are described and two illustrative examples are solved.

An integrated interval type-2 fuzzy rough technique for emergency decision making

This paper aims to develop a hybrid emergency decision-making (EDM) method by combining best–worst method (BWM), multi-attributive border approximation area comparison (MABAC) and prospect theory (PT) in trapezoidal interval type-2 fuzzy rough (TrIT2FR) environment. In this hybrid method, the decision information is represented by trapezoidal interval type-2 fuzzy rough numbers (TrIT2FRNs). Firstly, this paper defines the TrIT2FRN and analyzes its desirable properties. Then, the TrIT2FR-BWM is developed to determine criteria weights. To develop the TrIT2FR-BWM, this paper completes the following three core issues: (i) propose an effective theorem to normalize the TrIT2FR weights; (ii) build a crisp programming model to transform the minmax objective of weight-determining model for the TrIT2FR-BWM; (iii) design a consistency ratio for the TrIT2FR-BWM to check the reliability of the determined criteria weights. Afterwards, this paper extends the classical MABAC into TrIT2FR environment to calculate the border approximation area (BAA). Subsequently, the PT is used to rank the alternatives, in which the calculated BAA is selected as the reference point. Lastly, the validity of the proposed method is certificated with a real site selection case of makeshift hospitals on COVID-19. Sensitivity analysis and comparative analyses are conducted to illustrate the robustness and superiorities of the proposed method. Some valuable results are summarized as follows: (i) the best alternative determined by the proposed method conforms with the actual selection result, (ii) the proposed models in the TrIT2FR-BWM have strong robustness, (iii) PT is helpful to improve the decision quality of EDM.

Dynamic portfolio allocation for financial markets: A perspective of competitive-cum-compensatory strategy

Portfolio allocation is an important research branch in the realm of financial management and financial engineering. In this paper, a dynamic portfolio allocation problem considering the competitive-cum- compensatory relationship among decision objectives is discussed. Interval type-2 fuzzy numbers that provide more flexibility for processing uncertainty are innovatively utilized to characterize asset returns. To capture the behavioral characteristics of investors’ bounded rationality, prospect theory with dynamic updating of loss aversion rate and reference wealth is introduced. The expected semi-absolute deviation and the entropy function based on the Minkowski measure are adopted to describe the risk and diversification degree of portfolio allocation, respectively. With this description, a dynamic multi-objective portfolio allocation model is formulated. Regarding the multi-dimensional characteristics of the problem, competitive-cum-compensatory strategy-based fuzzy goal programming is embedded in the whole optimization process; thus, the model is transformed into a single-objective form for the solution. Several interesting conclusions are drawn from empirical studies in two financial markets. The robustness and superiority of the proposed model are verified by multi-angle comparison and sensitivity analysis. This research not only enriches and extends the field of dynamic portfolio allocation in the fuzzy context, but also offers an effective means for the optimization of multi-objective portfolio models.

An Extended Interval Type-2 Fuzzy VIKOR Technique with Equitable Linguistic Scales and Z-Numbers for Solving Water Security Problems in Malaysia

Interval Type-2 Fuzzy VIseKriterijumska Optimizacija I Kompromisno Resenje (IT2FVIKOR) technique is one of the techniques of Interval Type-2 Fuzzy Multi-Criteria Decision Making (IT2FMCDM), which was developed to solve problems involving conflicting and multiple objectives. Most of the IT2FVIKOR methods are created from linguistic variables based on Interval Type-2 Fuzzy Set (IT2FS) and its generalization, such as Interval Type-2 Fuzzy Numbers (IT2FNs). Recent literature suggests that equitable linguistic scales can offer a better alternative, particularly when IT2FSs have some limitations in handling uncertainty and imbalance. This paper proposes the extended IT2FVIKOR with an equitable linguistic scale and Z-Numbers, where its linguistic scale introduces an equitable balance of positive and negative scales added to the restriction and reliability approach. Different from the typical IT2FVIKOR, which directly utilizes IT2FNs with a positive membership, the proposed method introduces positive and negative membership where each side considers a restriction and reliability approach. Besides, this paper also offers objective weights using fuzzy entropy-based IT2FS to calculate the weights of the extended IT2FVIKOR. The obtained solutions would help decision makers (DMs) identify the best solution to enhance water security projects in terms of finding the best strategies for water supply security in Malaysia.

A Generalized Multiobjective Reliability Redundancy Allocation With Uncertainties

Multi-objective reliability-redundancy allocation problem (MORRAP) needs to maximize system reliability and minimize cost, weight, and volume with underlining constraints. In the systems’ design and analysis phase, uncertainties can occur from various sources, such as manufacturing variability, environmental conditions, user behavior, etc. To deal with this, we present a generalization of the traditional MORRAP under multiple empirical and ambiguous circumstances, named interval type-2 fuzzy multiobjective reliability redundancy allocation problem (IT2FMORRAP). The newly formulated IT2FMORRAP considers optimizing goals as reliability, cost, and weight for a series-parallel system with interval type-2 fuzzy number. The mathematical formulation is established under which the proposed IT2FMORRAP model reduces to T1FMORRAP (type-1 fuzzy MORRAP), IVMORRAP (interval-valued MORRAP), and classical MORRAP. An Enhanced Karnik-Mendel and NSGA-II algorithm-based solving strategy is developed for the proposed IT2FMORRAP. The real-world dataset is considered to demonstrate the efficacy of the solution method for the proposed problem. A K-mean clustering technique identifies the best solution sets from the knee region of the generated Pareto fronts. An experimental study on commonly used performance metrics reveals that IT2FMORRAP performs significantly better than T1FMORRAP and crisp MORRAP. Further, the statistical analysis also confirms the hypothesis established in the empirical research. Finally, a comparative performance study has been conducted with notable state-of-the-art papers from the literature to encounter an appropriate establishment for the proposed work in the domain.

A method for solving interval type-2 triangular fuzzy bilevel linear programming problem

In this paper, we consider the bilevel linear programming problem (BLPP) where all the coefficients in the problem are interval type-2 triangular fuzzy numbers (IT2TFNs). First, we convert a BLPP with IT2TFN parameters to an interval BLPP. In the next step, we solve BLPPs and obtain optimal solution as an IT2TFN.

Sustainable Urban Conveyance Selection through MCGDM Using a New Ranking on Generalized Interval Type-2 Trapezoidal Fuzzy Number

This article proposes a modified ranking technique for generalized interval type-2 trapezoidal fuzzy numbers. For demonstrating uncertainty and managing imprecision in decision-making information, interval type-2 fuzzy sets are beneficial. The proposed ranking methodology resolves the difficulty of multi-criteria group decision-making on sustainable urban conveyance. Additionally, the proposed ranking approach considers all crucial aspects of transportation sustainability, including the effectiveness of durable transportation systems from economic, social, and ecological perspectives in multi-criteria group decision-making scenarios. The new ranking methodology yields superior outcomes for choosing sustainable urban transportation options. In the numerical part, studies compared the proposed ranking approach to other methods currently used for various MCDM techniques.

Sustainable Supplier Selection and Evaluation for the Effective Supply Chain Management System

With increased environmental protection awareness, sustainability has been incorporated into supply chain management. Sustainable supplier selection and evaluation have become an acritical part of supply chain management. They can significantly improve the supply chain’s operational performance and enhance enterprises’ competitiveness. Based on trapezoidal interval type-2 fuzzy numbers (TIT2FNs) and cloud probability dominance relations (PDR), manufacturers can make more tangible and environmentally friendly decisions in the SSSE process. In this paper, a SSSE indicator system is first established using the necessary economic, environmental, and social factors. The importance of the indicators described in linguistic terms is transformed into TIT2FNs, and the weight of each indicator is calculated. In order to prevent candidate suppliers from promoting performance maliciously, different weights are given according to the impact of the enterprise’s historical performance on the present. Finally, the cloud PDR method is used to determine the optimal sustainable supplier. A case study and analysis are provided to show the feasibility and superiority of the proposed method.

COVID-19 medical waste transportation risk evaluation integrating type-2 fuzzy total interpretive structural modeling and Bayesian network

With the amount of medical waste rapidly increasing since the corona virus disease 2019 (COVID-19) pandemic, medical waste treatment risk evaluation has become an important task. The transportation of medical waste is an essential process of medical waste treatment. This paper aims to develop an integrated model to evaluate COVID-19 medical waste transportation risk by integrating an extended type-2 fuzzy total interpretive structural model (TISM) with a Bayesian network (BN). First, an interval type-2 fuzzy based transportation risk rating scale is introduced to help experts express uncertain evaluation information, in which a new double alpha-cut method is developed for the defuzzification of the interval type-2 fuzzy numbers (IT2FNs). Second, TISM is combined with IT2FNs to construct a hierarchical structural model of COVID-19 medical waste transportation risk factors under a high uncertain environment; a new bidirectional extraction method is proposed to describe the hierarchy of risk factors more reasonably and accurately. Third, the BN is integrated with IT2FNs to make a comprehensive medical waste transportation risk evaluation, including identifying the sensitive factors and diagnosing the event's causation. Then, a case study of COVID-19 medical waste transportation is displayed to demonstrate the effectiveness of the proposed model. Further, a comparison of the proposed model with the traditional TISM and BN model is conducted to stress the advantages of the proposed model.

Interval Type-2 Fuzzy Differential Equations and Stability

This article is devoted to present a framework for studying type-2 fuzzy calculus. Some new concepts pertaining to interval type-2 fuzzy functions (IT2FF) and interval type-2 fuzzy differential equations (IT2FDE) with their stability analysis are introduced. A new definition of type-2 fuzzy derivative called type-2 granular derivative (T2gr-derivative) is defined to overcome the disadvantages of the previous definitions of type-2 fuzzy derivatives. Furthermore, the definitions of elementary arithmetic on the set of interval type-2 fuzzy numbers (IT2FNs) and the set of interval type-2 fuzzy complex numbers are presented. The notion of granular metric on the set of IT2FNs is also defined. Besides, the concepts of continuity, integral, and limit of IT2FFs are introduced. Using the concept of type-2 granular fuzzy Laplace transform and the approximations of T2gr-derivative proposed in this article, the exact and approximate solutions of interval T2FDE are presented. Then, some new concepts such as fuzzy node, fuzzy saddle spiral point, fuzzy star, and so forth lay the groundwork for the analysis of interval T2FDE stability. Several examples are given to illustrate the presented notions.

An interval type-2 fuzzy enhanced best–worst method for the evaluation of ship diesel generator failures

The environmental destruction in our century necessitates the implementation of more safety-based rules and the strict implementation of failure analysis, especially in the maritime sector. The generator system, considered the ship's powerhouse, is expected to be ready and well-maintained for efficient and safe operations. This study proposes a new methodology to analyze ship generator failures. This method includes a version of the best-worst multi-criteria decision-making method enriched with interval type-2 fuzzy logic (IT2FE-BWM). Enrichment with interval type-2 fuzzy numbers shows that a parameter about how much the decision-maker trusts the pairwise comparisons are added to the model. Thus, a more sensitive methodology has been proposed for the confidence of the maritime sector evaluators. A solid decision mechanism has been developed to prioritize generator failures. In addition, a comparison study with some other BWM derivatives and a creative sensitivity analysis were added to this study and its robustness was tested.

The assessment and selection of suppliers using AHP and MABAC with type-2 fuzzy numbers in automotive industry

The selection of appropriate suppliers in an uncertain environment influences the sustainability and the competitive advantage of the automotive industry and hence presents one of the significant management problems. In the literature, it is suggested that an acceptable supplier may be determined concerning many criteria. In this manuscript, criteria selection is based on the relevant literature. Handling of different uncertainties is performed by using the type-2 fuzzy sets that have the capability of handling more considered impreciseness and uncertainties. In the presented research, the two-state model is proposed. In the first stage, the relative importance of criteria and sub-criteria are described by pre-defined linguistic expressions which are modeled by the interval type-2 triangular fuzzy numbers (IT2TFNs). The fuzzy weights vectors are given by using the fuzzy Analytic Hierarchical Process with IT2TFNs (IT2FAHP). After that, the ranking of suppliers is based on the modified Multi-Attributive Border Approximation area Comparison method (MABAC) with IT2TFNs (IT2FMABAC). The extension of the conventional MABAC includes: (a) modeling of sub-criteria values of IT2TFNs, (b) the fuzzy criteria values are calculated by using fuzzy algebra rules, (c) belonging of each supplier to border approximation areas (BAAs) is given by using procedure, and (d) the distance of suppliers to BAAs is determined by using the normalized Euclidean distance formulas. The proposed model is tested on the real-life data from the automotive supply chain. Through the presented research, it is shown that the proposed IT2FMABAC is a useful and reliable tool for the rational purchasing decision-making process.

The novel approach for ranking generalized interval type-2 trapezoidal fuzzy numbers based on integral value

Interval type 2 fuzzy sets (IT2FS) give us with additional value of membership function to represent the uncertainty and the fuzziness of the real word than type 1 fuzzy sets (T1FS). In real problem, the ranking of interval type 2 fuzzy numbers is an important component in the fuzzy decision making system. The existing ranking approaches i.e. centroid is cited by many researchers and has wide application, and then integral value also often developed by researchers both in the form of T1FS and newly in the form IT2FS. However, the existing ranking approaches still have shortcomings of the ranking function. In this paper, the concepts of left and right of interval type-2 trapezoidal fuzzy numbers (IT2TrFN) i.e the novel total integral value with index of optimistic is proposed to overcome the shortcomings of the existing ranking approaches. By using several examples, the new total integral value is capable to cover the shortcomings of the existing approaches and has well-consistency in ranking IT2TrFN with equal and different hight value.