Type-2 Fuzzy Sets Research Articles

Container vessel selection for maritime shipping companies by using an extended version of the Grey Relation Analysis (GRA) with the help of Type-2 neutrosophic fuzzy sets (T2NFN)

Contrary to expectations, based on occurring changes in customer behaviours related to consuming and shopping triggered by the pandemic, the global container shipping market has continued to grow during COVID 19. Experts estimate that these increases will continue in the future due to changing consumption habits. However, container shipping companies (CSCs) may soon encounter many troubles and challenging situations. They have an extremely fragile structure and may be entirely unprotected when they encounter unexpected situations sourced from external factors. (i.e., due to grounding of a ship, complete blockage of the Suez channel for three weeks can be given as a clear example of that). Hence, selecting an appropriate container vessel type can help construct a healthier container shipping system less influenced by adverse conditions for decision-makers and practitioners. Besides, it can provide a more effective and productive maritime transportation environment for all stakeholders. However, selecting a proper container vessel type is a complicated decision-making problem since many conflicting criteria and complex ambiguities exist. The current paper proposes an extended version of the GRA technique with the help of type-2 neutrosophic fuzzy sets (T2NFN) for capturing and processing uncertainties better than the traditional MCDM frameworks. According to the obtained results, C6, container carrying capacity, is the most influential criterion and the type of post-Suezmax container vessel is the best option for the CSCs, as it provides advantages at a satisfactory level for almost all criteria than others. After the proposed model was applied, a comprehensive sensitivity analysis (SA) was performed to test the validity of the T2NFN GRA approach. The results of SA approve the applicability, effectivity, and robustness of the model.

On a Solution Set to a Mathematical Programming Problem With a Fuzzy Set of Constraint Indices

The present paper investigates a mathematical programming problem with a fuzzy set of constraint indices. To formalize the set of feasible alternatives, the concept of intersection of crisp sets with a fuzzy set of operands is used. The result of this operation is a type-2 fuzzy set (T2FS) (a fuzzy set whose membership function takes a fuzzy set of values). Furthermore, the corresponding membership function is given. It is shown that the T2FS of feasible alternatives can be represented in the form of the collection of T2FSs with constant secondary grades. The notion of the solution T2FS to the mathematical programming problem with a fuzzy set of constraint indices is proposed. The basic properties of the solution T2FS are investigated. The notion of a compromise maximizing solution is given. Several results are proved that make it possible to simplify its construction. Illustrative examples are given.

A Spatial Fuzzy Co-Location Pattern Mining Method Based on Interval Type-2 Fuzzy Sets

The goal of spatial co-location pattern mining is to find subsets of spatial features whose instances are often neighbors in a geographical space. In many practical cases, instances of spatial features contain not only spatial location information but also attribute information. Although there have been several studies that use type-1 fuzzy membership functions to mine spatial fuzzy co-location patterns, there is great uncertainty associated with such membership functions. To address this problem, we propose a spatial fuzzy co-location pattern mining method based on interval type-2 fuzzy sets. First, we collect the interval evaluation values of the interval data of attribute information from experts to form granular data. Next, the original type-1 fuzzy membership function is extended to a granular type-2 fuzzy membership function based on elliptic curves. We use a gradual method to adjust the parameters of the fuzzy membership function so that its footprint of uncertainty satisfies both the connectivity and the given confidence. Based on this granular type-2 fuzzy membership function, we fuzzify the attribute information of instances and define the concepts of fuzzy features and fuzzy co-location patterns. A fuzzy co-location pattern mining algorithm based on spatial cliques is then proposed, termed the FCPM-Clique algorithm. In order to improve the efficiency of the algorithm, we propose two pruning strategies. In addition, we extend two classical spatial pattern mining algorithms, the Join-based algorithm and the Joinless algorithm, to mine fuzzy co-location patterns based on interval type-2 fuzzy sets. Many experiments on synthetic and real-world datasets are conducted, the performance of the three algorithms is compared, and the effectiveness and efficiency of our proposed FCPM-Clique algorithm is demonstrated.

Dynamic collaborative optimization for disaster relief supply chains under information ambiguity.

Large-scale disasters occur worldwide, with a continuing surge in the frequency and severity of disruptive events. Researchers have developed several optimization models to address the critical challenges of disaster relief supply chains (e.g., emergency material reserving and scheduling inefficiencies). However, most developed algorithms are proven to have low fault tolerance, which makes it difficult for disaster relief supply chain managers to obtain optimal solutions and meet the emergency distribution requirements within a limited time frame. Considering the uncertainty and ambiguity of disaster relief information and using Interval Type-2 Fuzzy Set (IT2TFS), this paper presents a collaborative optimization model based on an integrative emergency material supplier evaluation framework. The optimal emergency material suppliers are first selected using a multi-attribute group decision-making ranking method. Multi-objective fuzzy optimization is then run in three emergency phases: early -, mid-, and late-disaster relief stages. Focusing on a massive flash flood disaster event in Yunnan Province as a case study, a comprehensive numerical analysis tests and validates the developed model. The results revealed that the proposed optimization method can optimize emergency material planning while ensuring that reserve material safety inventory is always maintained at a reasonable level. The presented method suggests a fuzzy interval to prevent emergency materials’ safety inventory shortage and minimize continuous life/property losses in disaster-affected areas.

A new regret theory-based risk decision-making method for renewable energy investment under uncertain environment

Renewable energy is considered as a promising solution to alleviate the exploitation of fossil fuel energy and to keep sustainable development. Hence, renewable energy investments are attracting more and more attention. To solve the renewable energy investment problems under uncertain environment, this paper proposes a new multi-criteria risk decision-making method based on interval type-2 fuzzy sets (IT2FSs) and regret theory. First, to describe the uncertainty of real-world situation, the evaluation information is expressed by linguistic terms and encoded into IT2FSs. Second, a new interval type-2 fuzzy projection model is proposed to measure the relationships between alternatives under interval type-2 fuzzy environment. Then, to determine the criteria weights, a new interval type-2 fuzzy entropy weight determination model is developed. Moreover, some new rules are provided to explore the difference between alternatives under different spaces (i.e., positive space and negative space). Finally, the regret theory is extended to describe the psychological behavior of decision makers and select the optimal alternative under risk and uncertainty environment. A case study including comparative analysis is provided to manifest the feasibility and advantages of the proposed method.

A novel integrated MCDM framework based on Type-2 neutrosophic fuzzy sets (T2NN) for the selection of proper Second-Hand chemical tankers

Shipping companies may not present appropriate and reasonable shipping charges to their customers due to the high investment costs of chemical tanker vessels. Hence, the second-hand chemical tanker vessel selection may present many advantages and opportunities to the shipping companies. However, selecting a suitable chemical tanker represents a complicated process requiring overcoming many ambiguities and considering many conflicting criteria. For this purpose, the current paper proposes a new and integrated MCDM approach based on Type-2 Neutrosophic fuzzy sets. The proposed model combines the advantages of TOPSIS and WASPAS techniques, and it uses Type-2 Neutrosophic fuzzy sets to process the uncertainties. The proposed model was applied to solve decision-making problems in selecting appropriate second-hand chemical tankers among 15 alternatives to show effectivity, pertinence, and robustness. In the current paper, 14 different selection criteria were identified to evaluate the chemical tanker vessels. According to the results, C13″ Maintenance cost” is the most significant factor, and A3 is the best and most preferable chemical tanker. Also, a comprehensive sensitivity analysis was carried out to test the validity and applicability of the suggested model, and the sensitivity analysis approves the applicability, robustness, and validity of the model.

Sürdürülebilir Tedarikçi Seçimi için Aralık Tip-2 Bulanık Kural Tabanlı BWM Yaklaşımı

Fuzzy logic is a theory based on human-specific approximate reasoning. Therefore, fuzzy logic applications can bring simple and more effective solutions to situations that classical methods cannot overcome. The type-1 fuzzy set is a set, which has a continuous (crisp) membership degree to which a membership degree between 0 and 1 is assigned, and is characterised by membership functions. Type-2 fuzzy sets, which have the power to express uncertainty better, are expressed by membership functions, where the membership degrees of each element belonging to that set also specify a fuzzy set.Therefore, type-2 fuzzy sets allow us to include the membership functions uncertainty in fuzzy set theory. Using expert knowledge and using sensitivity of human to reflect the level of the decision maker influence is expressed as a fuzzy rule based system. Recently, it has been seen that fuzzy rules are frequently used together with multi-criteria decision making (MCDM) methods. Again, combining fuzzy rules with type-2 fuzzy numbers is also found. In this study, the Best Worst Method (BWM), one of the MCDM methods, has been integrated with fuzzy rules based interval type-2. The developed hybrid method was defined as Interval Type-2 Fuzzy Rule-Based BWM (IT2 FRB BWM). The proposed hybrid method has an important place when there are alternatives with similar ranking positions. Thus, even if there is a small difference in each alternative, it will show the difference better (more sensitively). This makes the proposed hybrid method forceful and unique.The proposed approach has been applied to a sustainable supplier selection problem comparatively with the BWM. The results show that the IT2 FRB BWM approach is more successful in ordering alternatives than the classical BWM method.

Interval Type-2 Fuzzy Disturbance Observer-Based T–S Fuzzy Control for a Pneumatic Flexible Joint

In this article, an interval type-2 fuzzy disturbance observer (IT2FDO) is proposed for the trajectory tracking control of a flexible joint actuated by a pneumatic artificial muscle and a torsion spring. Interval type-2 fuzzy sets are introduced into the fuzzy disturbance observer (FDO) to enhance the fuzzy approximation capability. A novel adaptive law that updates all the parameters of the FDO, including the control representative values, the center vectors, and the width vectors, is developed. A new control strategy incorporating the proposed IT2FDO and Takagi–Sugeno (T–S) fuzzy model-based control is designed for the trajectory tracking task. With the newly designed adaptive law, the stability of the IT2FDO and that of the closed-loop system are guaranteed. Simulation studies verify the effectiveness of the proposed IT2FDO, and real-world experiments demonstrate the robustness of the proposed control strategy to the external disturbance.

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.

Multiobjective Evolutionary Interpretable Type-2 Fuzzy Systems With Structure and Parameter Learning for Hexapod Robot Control

This article proposes a data-driven multiobjective evolutionary interval type-2 fuzzy-logic system (IT2FLS) learning approach that considers both system performance and rule interpretability. One characteristic of the evolutionary learning framework is that the IT2FLS structure is incrementally generated instead of searching from an initially huge grid-type rule base, which significantly reduces the parameter search space. Another one is that a new constrained objective function is proposed to improve the distinguishability of the generated interval type-2 fuzzy sets (FSs) without restricting them to be evenly distributed. Because of the tradeoff between system performance and interpretability, a new multiobjective ant colony optimization (ACO) algorithm is proposed to optimize the IT2FLS parameters and improve optimization performance. The evolutionary IT2FLS learning approach is applied to control a real wall-following hexapod robot. The approach shows the advantages of model-free design and interpretability and robustness to noise in the evolved type-2 fuzzy rules. Simulations with performance comparisons and experiments in controlling a real robot show the advantages of the learning approach.

A New Optimization Approach Based on Bipolar Type-2 Fuzzy Soft Sets

The main objective of this research study is to amplify the schematic representation of human reasoning by launching the most generalized fantastic theories of bipolar type-2 fuzzy set (BT2FS) and bipolar type-2 fuzzy soft set (BT2FS f S). These incredible models are exclusively developed for the simultaneous capturing of both polarity and abstruseness inherent in equivocal interpretations. The proposed BT2FS f S theory renders an outstanding parameterized framework that skilfully wipes out the high-order uncertainty of imprecise knowledge-based systems. First, we keenly provide the formal structure of both proposed models along with the deep exploration of their elementary properties. Moreover, we explore rudimentary set-theoretical operations of developed frameworks inclusive of equality, subset, complement, union, and intersection with their noticeable results. We brilliantly formulate a highly proficient algorithm using proposed theory to disentangle the real-world multiattribute decision-making conundrums with two-sided ambiguous information. Finally, we holistically scrutinize an empirical analysis for the selection best way to cultivate the dessert city to demonstrate the remarkable accountability of the proposed methodology.

The distributivity of extended uninorms over extended overlap functions on the membership functions of type-2 fuzzy sets

This paper mainly investigates the distributivity of an extended uninorm over an extended overlap function when the uninorm is continuous in (0,1)2. It first studies the distributivity of the extended representable uninorms over the extended overlap functions. Then, it presents the distributivity of the extended uninorms over the extended overlap functions when the uninorms belong to one of the usual classes Ucos,min, Ucos,max.

Multi-attribute decision making application using hybridly modelled Gaussian Interval Type-2 Fuzzy sets with uncertain mean

Multi-attribute decision making application using hybridly modelled Gaussian Interval Type-2 Fuzzy sets with uncertain mean

Interval type-2 outlier-robust picture fuzzy clustering and its application in medical image segmentation

Based on picture fuzzy set theory, picture fuzzy clustering has achieved good results on some data as more information is involved in the clustering process. However, current picture fuzzy clustering methods still suffer from two common weaknesses, i.e., the sensitivity to outliers and the neglect of the uncertainty caused by different fuzzy degrees, which influence their performance in practical applications like medical image segmentation. To solve these issues, we present two new picture fuzzy clustering methods in this paper. First, to improve immunity to outliers, we propose an outlier-robust picture fuzzy clustering method named ORPFC by using a robust distance measurement, which treats the data objects far away from cluster prototypes as outliers and limits their effects on the prototype update. Second, to handle the uncertainty caused by fuzzy degrees, we further present an interval type-2 enhanced method called IT2ORPFC by incorporating the interval type-2 fuzzy set theory into ORPFC. In each iteration, IT2ORPFC estimates positive memberships, neutral memberships, and refusal memberships according to different fuzzification coefficients and then conducts type reduction for reliable type-1 clustering results. In the experiments, the proposed methods obtain robust and reliable results on eleven datasets. Specifically, ORPFC and IT2ORPFC achieve rewarding performance on segmenting medical images with noise.

Design of single and double acceptance sampling plans based on interval type-2 fuzzy sets

Defectiveness of items is generally considered as a certain value in acceptance sampling plans (ASPs). It is clear that, it may not be certainly known in some real-case problems. Uncertainties of the inspection process such as measurement errors, inspectors’ hesitancies or vagueness of the process etc. should be taken into account to obtain more reliable results. The fuzzy set theory (FST) is one of the best methods to overcome these problems. There are some studies in the literature formulating the ASPs with the help of FST. Deciding the right membership functions of the fuzzy sets (FSs) has a vital importance on the quality of the uncertainty modeling. Additionally, the fuzzy set extensions have been offered to model more complicated uncertainties to achieve better modeling. As one of these extensions, type-2 fuzzy sets (T2FSs) gives an ability to model uncertainty in situations where it is not possible to determine exact membership function parameters. In this study, single and double ASPs based on interval T2FSs (IT2FSs) have been designed for binomial and Poisson distributions. Thus, it becomes possible to make more flexible, sensitive and descriptive sensitivity analyzes. The main characteristic functions of ASPs have been derived and the suggested formulations have been illustrated on a comparative application from manufacturing process. Results allowing for more comprehensive analysis as against to the traditional and T1FSs based plans have been obtained.

Soft clustering and interval type-2 fuzzy set based inference strategy for I.T. personnel selection

Impreciseness and uncertainty are the fabrics that make life interesting. For decades, human beings have developed strategies to cope with uncertainties and automate them. In personnel selection for the I.T. field, selectors often find it very difficult to select candidates by going through a set of resumes containing similar kinds of skills. Hence the selection task becomes a fuzzy decision making with the uncertainty involved. A combination of fuzzy clustering and Interval Type-2 fuzzy sets (IT2FS) is proposed in such scenarios. An experiment is conducted over a resume dataset containing fifteen hundred resumes for a particular job description. Firstly, Fuzzy C-means clustering (FCM) is applied for selective clustering, while decision-making under uncertainty is carried through IT2FS. The candidates in the selected cluster are given a score for ranking as per the skillset criteria. The final decision for shortlisting the resumes is carried through IT2FS. The model shows an average accuracy of 88.2% with an F1-score of 0.76 compared to (K-means + IT2FS) model with an F1-score of 0.72. Thus, the proposed model performs better while decision-making under uncertainty.

An extended multi-criteria group decision-making method with psychological factors and bidirectional influence relation for emergency medical supplier selection

The COVID-19 pandemic outbreak spread rapidly worldwide, posing a severe threat to human life. Due to its unpredictability and destructiveness, the emergency has aroused great common in society. At the same time, the selection of emergency medical supplier is one of the critical links in emergency decision-making, so undertaking appropriate decision-making using scientific tools becomes the primary challenge when an emergency outbreak occurs. The multi criteria group decision-making (MCGDM) method is an applicable and common method for choosing supplier. Nevertheless, because emergency medical supplier selection should consider regarding many aspects, it is difficult for decision makers (DMs) to develop a comprehensive assessment method for emergency medical supplier. Therefore, few academics have focused on emergency situation research by the MCGDM method, and the existing MCGDM method has some areas for improvement. In view of this situation, in this study, we propose a new MCGDM method, which considers the bidirectional influence relation of the criteria, consensus and the psychological factors of DMs. It providers a good aid in emergency decision-making and it could apply to other types of MCGDM research. Firstly, DMs give their assessment in interval type-2 fuzzy sets (IT2FSs). Secondly, an extended IT2FSs assessment method and a novel ISM-BWM-Cosine Similarity-Max Deviation Method (IBCSMDM) are used for weighing all alternatives. The TODIM (an acronym for interactive and multi-criteria decision-making in Portuguese) can obtain the ranking results under different risk attenuation factors. Eventually, this extended IT2FSs-IBCSMDM-TODIM method is applied in a real case in Wuhan in the context of COVID-19 to illustrate the practicability and usefulness.

A metaheuristic approach for interval type-2 fuzzy fractional order fault-tolerant controller for a class of uncertain nonlinear system*

A new optimum interval type-2 fuzzy fractional-order controller for a class of nonlinear systems with incipient actuator and system component faults is introduced in this study. The faults of the actuator and system component (leak) are taken into account using an additive model. The Interval Type-2 Fuzzy Sets (IT2FS) is used to design an optimal fuzzy fractional order controller, and two different nature inspired metaheuristic algorithms, Follower Pollination Algorithm (FPA) and Genetic Algorithm (GA), are used to optimize the parameters of the fuzzy PID controller and Interval Type-2 Fuzzy Tilt-Integral-Derivative Controller (IT2FTID) for nonlinear system. The suggested control approach consists of two parts: an Interval Type-2 Fuzzy Logic Controller (IT2FLC) controller and a fractional order TID controller. Additionally, the two inputs of the IT2FLC are also calibrated using two fine tuning parameters and , respectively. The stability of the proposed controller is presented with some conditions. In addition to unknown dynamics, some unknown process disturbances, such as rapid changes in the control variable, are taken into account to check the efficacy of the proposed control scheme. Two nonlinear conical two-tank level systems are used in the simulation as a case study. The performance of the suggested approach is also compared to that of a widely recognized Interval Type-2 Fuzzy Proportional-Integral-Derivative (IT2FPID) Controller. Finally, the proposed control scheme's fault-tolerant behaviour is demonstrated using fault-recovery time results and statistical Z-tests for both controllers, and the proposed IT2FTID controller's effectiveness is demonstrated when compared to IT2FPID and existing passive fault tolerant controllers in recent literature.

Investigating risks in renewable energy in oil-producing countries through multi-criteria decision-making methods based on interval type-2 fuzzy sets: A case study of Iran

Despite the importance of relying on renewable energy sources, oil-producing countries, especially Iran, encounter numerous risks associated with the generation of such energy sources. The present study investigated the risks in renewable energy in Iran. As the first investigation addressing this problem in Iran, the study methodologically improved the probability-impact matrix by employing multi-criteria decision-making (MDCM) methods and using interval type-2 fuzzy sets. The results showed that most important macro-level risks facing Iran's renewable energy generation were “inflation and price fluctuations”, “corruption”, “complex licensing procedures”, “research and development and the capacity of domestic technology”, “sanctions”, “exchange rate fluctuations”, and “property rights and contractual risks.” Following these macro-level risks, there were important specific risks both in the general field of energy and renewable energy including “geographical risks”, “non-renewable energy prices”, “electricity prices”, “a lack of a coherent renewable energy policy”, “the impact of climate change on revenue rates”, and “financial risks.” Finally, based on the outcome of the probability-impact matrix combined with the hierarchy of risks, the two highest-priority risks are “complex licensing procedures” and “non-renewable energy prices.” To respond to such risks, it would be necessary to formulate coherent and integrated policies.

Development, experimental validation, and comparison of interval type-2 Mamdani fuzzy PID controllers with different footprints of uncertainty

Interval Type-2 Fuzzy PID (IT2FPID) controllers use Interval Type-2 Fuzzy Sets (IT2FSs). The difference between Type-1 Fuzzy PID (T1FPID) controllers and IT2FPID controllers is the Footprint of Uncertainty (FOU) which occurs due to the use of IT2FSs. FOU gives an extra degree of freedom to the controller structure, which is capable of handling uncertainty in the plant. However, the effect of FOU on the analytical structures of the IT2FPID controllers is not yet explored when uniformly/non-uniformly distributed multiple IT2FSs are used on the input side and uniformly/non-uniformly distributed multiple singleton fuzzy sets are considered on the output side of the controller. In this paper, controllers with three different types of FOUs and two different defuzzification methods are considered, and their analytical structures are derived. The suitability of the controllers is examined, and the properties of the suitable controllers are explored. Effects of FOU on the nonlinearity, gain, and computational aspects of the suitable controllers are studied. The applicability of proposed IT2FPID controllers is shown through simulation and real-time studies. The effectiveness of the proposed controllers is depicted by comparing their performances with that of the recently developed Integer Order Fuzzy PID (IOFPID) and Fractional Order Fuzzy PID (FOFPID) controllers.