Type-2 Fuzzy Sets Research Articles

A Temporal Type-2 Fuzzy System for Time-Dependent Explainable Artificial Intelligence

Explainable Artificial Intelligence (XAI) is a paradigm that delivers transparent models and decisions, which are easy to understand, analyze, and augment by a non-technical audience. Fuzzy Logic Systems (FLS) based XAI can provide an explainable framework, while also modeling uncertainties present in real-world environments, which renders it suitable for applications where explainability is a requirement. However, most real-life processes are not characterized by high levels of uncertainties alone; they are inherently time-dependent as well, i.e., the processes change with time. In this work, we present novel Temporal Type-2 FLS Based Approach for time-dependent XAI (TXAI) systems, which can account for the likelihood of a measurement's occurrence in the time domain using (the measurement's) frequency of occurrence. In Temporal Type-2 Fuzzy Sets (TT2FSs), a four-dimensional (4D) time-dependent membership function is developed where relations are used to construct the inter-relations between the elements of the universe of discourse and its frequency of occurrence. The TXAI system manifested better classification prowess, with 10-fold test datasets, with a mean recall of 95.40\% than a standard XAI system (based on non-temporal general type-2 (GT2) fuzzy sets) that had a mean recall of 87.04\%. TXAI also performed significantly better than most non-explainable AI systems between 3.95\%, to 19.04\% improvement gain in mean recall. In addition, TXAI can also outline the most likely time-dependent trajectories using the frequency of occurrence values embedded in the TXAI model; viz. given a rule at a determined time interval, what will be the next most likely rule at a subsequent time interval. In this regard, the proposed TXAI system can have profound implications for delineating the evolution of real-life time-dependent processes, such as behavioural or biological processes.

A Lattice Structure on Hesitant Fuzzy Sets

In this paper we deal with the lattice-compatibility between several classes of extended fuzzy sets. Concretely, we treat the problem of finding a lattice structure on set-valued fuzzy sets (SVFSs) whose restriction to interval-valued fuzzy sets (IVFSs) and (type-1) fuzzy sets (FSs) match Zadeh's classical lattice operations. A prominent approach to this problem was given by Torra by means of the so-called hesitant fuzzy sets (HFSs). Nevertheless, despite their usefulness in group decision making problems, it is well-known that Torra's operations do not produce a lattice. Here, we mend partially this handicap by giving two lattice orders. Each of them preserves one of the Torra's operations and, additionally, reduces to Zadeh's orders on FSs and on IVFSs. As a counterpart, they cannot be defined on the whole class of HFSs, or SVFSs. Finally, we provide a full answer combining both orders. We define a partial order, that we call the symmetric order, on the whole class of non-empty subsets of [0,1]. This order extends the usual ones on [0,1] and on closed intervals of [0,1]. As a consequence, we find a lattice structure on HFSs whose restriction to FSs and IVFSs reduces to Zadeh's operations.

A barrier evaluation framework for forest carbon sink project implementation in China using an integrated BWM-IT2F-PROMETHEE II method

Forest carbon sink project (FCSP) has received growing attention for its outstanding advantages in carbon emission reduction, environment protection, and achieving the carbon neutrality objective. Using a government-social capital cooperation model, FCSP helps to revitalize the ecological and economic benefits of the forest with standard and efficient management. However, the management and operation barriers of FCSP implementation are complex, multiple and indefinite in the long run, which requires a comprehensive barrier evaluation framework. Hence, this paper is devoted to constructing a barrier evaluation framework for FCSP implementation in China with an integrated multi-criteria decision-making (MCDM) method. To this end, critical barriers are identified and the criteria system is constructed with a two-stage systematic literature review and interaction with the experts. Later, an applicable and comprehensive barrier evaluation framework is built, wherein the interval type-2 fuzzy set (IT2FS) is used to process fuzziness and uncertain information, the best-worst method (BWM) is adopted to determine the criteria weight, and the Preference Ranking Organization Method for Enrichment Evaluations II (PROMETHEE II) is introduced for comprehensive barrier evaluation. The criteria weighting results indicate that FCSP implementation encounters challenges taking into account institutional and governance barriers (criteria weight: 0.3592), management barriers (0.2384), economic and market barriers (0.1637), knowledge barriers (0.0747), technical and infrastructure barriers (0.1639), C1 “Lack of clear leadership and policies” and C5 “Forest management performance” were determined the most significant criteria, A5 and A4 respectively performed the lowest and the highest barrier level. In sensitivity analysis, the ranking results were stable when the criteria changed, and Spearman’s rank correlations of B2, B4, and B5 are significant. The results were validated by a comparative analysis with other methods with the help of Spearman’s rank correlation coefficient, which reflects accepted high correlations (Spearman’s pro values of BWM-IT2F-PROMETHEE II (Gaussian rule), BWM-IT2F-TOPSIS, and BWM-TFN-VIKOR are 1.00, 1.00, 1.00). The findings of the paper help the decision makers to develop more rational strategies and successfully implement the FCSP.

How Will Autonomous Vehicles Decide in Case of an Accident? An Interval Type-2 Fuzzy Best–Worst Method for Weighting the Criteria from Moral Values Point of View

The number of studies on Autonomous Vehicle (AV) ethics discussing decision-making algorithms has increased rapidly, especially since 2017. Many of these studies handle AV ethics through the eye of the trolley problem regarding various moral values, regulations, and matters of law. However, the literature of this field lacks an approach to weighting and prioritizing necessary parameters that need to be considered while making a moral decision to provide insights about AVs’ decision-making algorithms and related legislations as far as we know. This paper bridges the gap in the literature and prioritizes some main criteria indicated by the literature by employing the best–worst method in interval type-2 fuzzy sets based on the evaluations of five experts from different disciplines of philosophy, philosophy of law, and transportation. The criteria included in the weighting were selected according to expert opinions and to the qualitative analysis carried out by coding past studies. The weighing process includes a comparison of four different approaches to the best–worst method. The paper’s findings reveal that social status is the most important criterion, while gender is the least important one. This paper is expected to provide valuable practical insights for Autonomous Vehicle (AV) software developers in addition to its theoretical contribution.

Comparative Study of Type-1 and Interval Type-2 Fuzzy Logic Systems in Parameter Adaptation for the Fuzzy Discrete Mycorrhiza Optimization Algorithm

The Fuzzy Discrete Mycorrhiza Optimization (FDMOA) Algorithm is a new hybrid optimization method using the Discrete Mycorrhiza Optimization Algorithm (DMOA) in combination with type-1 or interval type-2 fuzzy logic system. In this new research, when using T1FLS, membership functions are defined by type-1 fuzzy sets, which allows for a more flexible and natural representation of uncertain and imprecise data. This approach has been successfully applied to several optimization problems, such as in feature selection, image segmentation, and data clustering. On the other hand, when DMOA is using IT2FLS, membership functions are represented by interval type-2 fuzzy sets, which allows for a more robust and accurate representation of uncertainty. This approach has been shown to handle higher levels of uncertainty and noise in the input data and has been successfully applied to various optimization problems, including control systems, pattern recognition, and decision-making. Both DMOA using T1FLS and DMOA using IT2FLS have shown better performance than the original DMOA algorithm in many applications. The combination of DMOA with fuzzy logic systems provides a powerful and flexible optimization framework that can be adapted to various problem domains. In addition, these techniques have the potential to more efficiently and effectively solve real-world problems.

Enhancing fuzzy evidential reasoning approach using dynamic adjustment mechanism and new rule-based transformation for engineering emergency response evaluation

With the frequent occurrence of various emergency events in engineering field, engineering emergency response (EER) evaluation plays an increasingly significant role in handling such situations and provides great challenges to research since the uncertain information and the urgent response time. Aiming at achieving timely and effective emergency response, an enhanced evidential reasoning (ER) approach based on the dynamic adjustment mechanism and new rule-based transformation is proposed. First, the linguistic terms to represent various preference information provided by experts are encoded into the trapezoidal interval type-2 fuzzy sets (TrIT2FSs) with different granularities. Second, for ensuring the validity of the information, based on the definition of the expert decision risk preference coefficients, a dynamic adjustment mechanism is constructed to identify and adjust the preference information. Meanwhile, combined with social network, the experts’ weights can be calculated and revised several times to obtain group information. Then, a new rule-based transformation and related optimization models are proposed to convert the TrIT2FSs into interval belief structures. Furthermore, considering the importance of attributes, the relative weights and interval belief structures are combined. Finally, according to integrated interval belief structures obtained by the analytical ER algorithm, a new ranking approach with the optimism degree and decision tendency degree is constructed to rank the interval expected utility and score utility of each alternative. To further show the effectiveness, superiorities, and stability of the proposed method, a case study on the EER evaluation is preformed and some comparisons and discussions are provided.

Multi-criteria decision making based on vector similarity measures of picture type-2 hesitant fuzzy sets

Multi-criteria decision making based on vector similarity measures of picture type-2 hesitant fuzzy sets

Interval type-2 fuzzy set based block-SBU for image fusion technique

Image fusion is the process of examining one or more obscure images, combining only the critical information in those images to make them a good quality image. Due to the large gap between the high dynamic range of landscape scenes and the low quality of consumer quality cameras, a single-shot image cannot record the information of a single set. A part of image processing called image fusion is used to overcome the above problem. In this method, authors propose a new technology for image fusion with intuitionistic fuzzy sets. During the processing, the given images are converted into fuzzy images and then intuitionistic fuzzy images (IFIs). Thus, a significant change occurs between the given image and IFIs. The resulting IFIs are converted into interval type-2 fuzzy images (IT2FIs) to overcome this problem. The proposed technique is compared to other procedures such as discrete cosine harmonic wavelet transform (DCHWT), multi-resolution single value decay (MSVD), primary component analysis (PCA), standard wavelet transform (SWT), two scale image fusion (TSIF), and wavelet transform (WT). The proposed method gives the best results based on performance analysis such as Entropy, standard deviation (STD), average gradient (AG), mutual information (MIF), and blind referenceless image spatial quality evaluator (BRISQUE). Using the proposed technique, one can obtain the best results for entropy, STD, AG, MIF, and BRISQUE respectively as 7.6241, 49.6745, 69.2100, 5.7276, and 3.3091. Hence it reveals that the proposed method performs better than other methods regarding overall visual quality and performance measurements.

A data-driven fuzzy system for the automatic determination of fuzzy set type based on fuzziness

Existing research mainly uses prior knowledge to set all fuzzy sets in a fuzzy system to the same type. In data-driven fuzzy modeling, automatically determining the type of fuzzy set is still a challenging problem. In this paper, a novel data-driven design method is proposed to automatically determine the types of fuzzy sets (type-1 fuzzy sets (T1-FSs) or interval type-2 fuzzy sets (IT2-FSs)) based on fuzziness. In the hybrid-type fuzzy system (HTFS), the fuzzy set type is determined by fuzziness to improve the performance of the fuzzy system, and the interpretability of the fuzzy system is determined by the integrity, distinguishability, and redundancy of the fuzzy sets. First, fuzzy clustering based on fuzzy entropy is used to initialize the rule base and the types of fuzzy sets. Second, social learning particle swarm optimization (SLPSO) is used to optimize the parameters of the HTFS, and the types of the fuzzy sets are determined by fuzziness, which means that the types of the fuzzy sets will change dynamically in the optimization process. Because the fuzzy set types change, a parameter change strategy is also proposed for adaptive fuzzy set types. Finally, the HTFS is compared with the T1 fuzzy system on 10 real-world datasets, and it is shown that the HTFS has better performance than the T1 fuzzy system. Comparing the HTFS with other advanced methods, it is shown that the HTFS has better prediction performance while taking into account the interpretability of the fuzzy set distribution.

Ranking products through online reviews: A novel data-driven method based on interval type-2 fuzzy sets and sentiment analysis

As an essential information resource, online reviews play an important role in consumers’ decision-making processes. To solve the product ranking problem through online reviews, two important issues are involved: sentiment analysis (SA) for online reviews and product ranking based on multi-criteria decision-making (MCDM) methods. However, merely a few studies have considered the impact of SA accuracy, which can significantly affect the final decision-making process. This paper proposes a novel data-driven method for ranking products through online reviews based on interval type-2 fuzzy sets (IT2FSs) and SA. In this method, after acquiring online reviews, the explicit and implicit attributes are extracted from the website itself and the latent Dirichlet allocation (LDA) model, respectively. Thereafter, a deep learning model is adopted to identify the five sentiment intensities of online reviews, based on which the SA results are represented as IT2FSs by considering the classification effect. After type-reduction for IT2FSs, the ranking order is obtained based on the exponential TODIM (ExpTODIM) method. Furthermore, a case study on ranking travel products from Trip.com Group through online reviews is provided to illustrate the effectiveness and applicability of the proposed method.

Industrial and Management Applications of Type-2 Multi-Attribute Decision-Making Techniques Extended with Type-2 Fuzzy Sets from 2013 to 2022

The ongoing research in the field of decision-making can be analyzed from different perspectives. Research trends indicate that multi-attribute decision-making (MADM) methods have a significant impact on engineering and management scientific areas. Since many of the problems existing in the mentioned areas are associated with a certain level of uncertainty, type 2 fuzzy sets represent a common solution for the enhancement of conventional MADM methods. In this way, the decision-makers are encouraged to use linguistic expressions for the assessment of attributes’ relative importance and their values. The purpose of this paper is to review a determination of attributes’ relative importance, and their values, as well as the extension of ranking methods with type 2 fuzzy sets. The papers are systematically adjoined to groups consisting of hybrid models with the following characteristics: (1) indicating the procedure for modeling attribute relative importance and their values, (2) determining the extension of MADM methods with type 2 fuzzy sets to determine attributes’ vector weights, and (3) the extension of MADM for attributes ranking with type 2 fuzzy sets. This study reviewed a total of 42 papers in the domain of engineering and management published from 2013 to 2023 in different journals indexed by the Springer, Science Direct, Emerald, Wiley, ProQuest, Taylor, and Francis research platforms.

Multiobjective Optimization Design of Interpretable Evolutionary Fuzzy Systems With Type Self-Organizing Learning of Fuzzy Sets

This paper proposes a new multiobjective optimization approach for designing a self-generated interpretable fuzzy logic system (FLS). The types of fuzzy sets can be constructed automatically by self-organizing method, so as to form a hybrid fuzzy system. Different from the existing evolutionary type-1 fuzzy system, which is full of type-1 fuzzy sets, and the evolutionary interval type-2 fuzzy system, which is full of interval type-2 fuzzy sets, there are both type-1 fuzzy sets and interval type-2 fuzzy sets in the hybrid fuzzy system. A new transparency-oriented objective function is defined, and the constraint of the footprint of uncertainty (FOU) of the interval type-2 (IT2) fuzzy set (FS) is considered for the first time. A new FS merging criterion focusing on the proximity of the cores of fuzzy sets is proposed, which is easy to calculate and maintains the characteristics of classical similarity measures. Combined with the new merging criterion, the online cluster and fuzzy set updating (OCFU) algorithm is employed to initialize the reference rule base and the type of fuzzy sets, as it is assumed that no training data are collected in advance. Based on the reference rule base, the advanced multiobjective front-guided continuous ant colony optimization (AMO-FCACO) algorithm is introduced to optimize all the free parameters of the FLS. With the operation mentioned above, the self-generated FLSs achieve a good balance between interpretability and performance. The effectiveness of the proposed method is verified by three nonlinear system tracking problems.

Hybrid Framework for Safety Design of Human–Rail Vehicle Transportation System Using Stochastic Approach and Optimization

The human–rail vehicle transportation system safety design is complicated given that the complexity of the multilevel system with parameter uncertainties propagating from the vehicle structure (primary collision) to the interior human compartment (secondary collision). This study establishes a hybrid framework incorporating a stochastic approach and an integrated optimization strategy to improve train crashworthiness and reduce passenger crash injuries. The stochastic approach utilizes adaptive sparse polynomial chaos expansion models and variance-based sensitivity indices to evaluate the statistic characteristics of system responses and quantify the contribution ranking of uncertain parameters to response variations. The optimization strategy integrating the evolutionary algorithm and the multi-criteria decision making (MCDM) is proposed to solve the non-uniqueness of Pareto optimal solutions. In the optimization process, the modified DEMATEL–ANP method with interval type-2 fuzzy sets is developed to deal with vague linguistic judgments for the importance sequence of human injury responses. The q-rung orthopair trapezoidal fuzzy uncertain linguistic sets–TOPSIS method is established to address hesitant linguistic evaluations for the Pareto front and select the final optimal solution. Compared with the initial design, the driver Abbreviated Injury Scale (AIS) 3+ joint injury probability is reduced from 67.08% to 14.17% after optimization. Results prove that the proposed framework is a practical tool for improving the passive safety of railway industry.

Optimization the reliability of emergency safety barriers based on the subjective safety analysis and evidential reasoning theory. Case study

PurposeThis study aims to improve the reliability of emergency safety barriers by using the subjective safety analysis based on evidential reasoning theory in order to develop on a framework for optimizing the reliability of emergency safety barriers.Design/methodology/approachThe emergency event tree analysis is combined with an interval type-2 fuzzy-set and analytic hierarchy process (AHP) method. In order to the quantitative data is not available, this study based on interval type2 fuzzy set theory, trapezoidal fuzzy numbers describe the expert's imprecise uncertainty about the fuzzy failure probability of emergency safety barriers related to the liquefied petroleum gas storage prevent. Fuzzy fault tree analysis and fuzzy ordered weighted average aggregation are used to address uncertainties in emergency safety barrier reliability assessment. In addition, a critical analysis and some corrective actions are suggested to identify weak points in emergency safety barriers. Therefore, a framework decisions are proposed to optimize and improve safety barrier reliability. Decision-making in this framework uses evidential reasoning theory to identify corrective actions that can optimize reliability based on subjective safety analysis.FindingsA real case study of a liquefied petroleum gas storage in Algeria is presented to demonstrate the effectiveness of the proposed methodology. The results show that the proposed methodology provides the possibility to evaluate the values of the fuzzy failure probability of emergency safety barriers. In addition, the fuzzy failure probabilities using the fuzzy type-2 AHP method are the most reliable and accurate. As a result, the improved fault tree analysis can estimate uncertain expert opinion weights, identify and evaluate failure probability values for critical basic event. Therefore, suggestions for corrective measures to reduce the failure probability of the fire-fighting system are provided. The obtained results show that of the ten proposed corrective actions, the corrective action “use of periodic maintenance tests” prioritizes reliability, optimization and improvement of safety procedures.Research limitations/implicationsThis study helps to determine the safest and most reliable corrective measures to improve the reliability of safety barriers. In addition, it also helps to protect people inside and outside the company from all kinds of major industrial accidents. Among the limitations of this study is that the cost of corrective actions is not taken into account.Originality/valueOur contribution is to propose an integrated approach that uses interval type-2 fuzzy sets and AHP method and emergency event tree analysis to handle uncertainty in the failure probability assessment of emergency safety barriers. In addition, the integration of fault tree analysis and fuzzy ordered averaging aggregation helps to improve the reliability of the fire-fighting system and optimize the corrective actions that can improve the safety practices in liquefied petroleum gas storage tanks.

Development of type-2 Pythagorean fuzzy set with its application to sustainable transport system selection

In resolving multicriteria decision-making (MCDM) problems, it is observed that the solution process is frequently affected by the available vague information together with intra-personal uncertainties associated with expert’s opinion. In most of the time fuzzy set or its existing variants fail to capture such uncertainties for making reasonable balance in decision making processes. To reduce the effects of uncertainties, this paper introduces a new variant of fuzzy set, viz., type-2 Pythagorean fuzzy set (T2PFS). The main advantage of using T2PFS in MCDM is that the membership and non-membership values of each argument are presented by Pythagorean fuzzy numbers rather than using fuzzy number or intuitionistic fuzzy number. In model formulation process, at first, T2PFS is explicitly formulated and some basic operations are defined. Subsequently, four aggregation operators are constructed to aggregate the arguments, and their properties are verified. Afterwards, two information measures and a score function are defined. Moreover, two MCDM algorithms based on the developed aggregation operators and TOPSIS are framed. To show the application potentiality of the proposed method, a case study relating to the selection of sustainable transport system in India is considered and solved. In the case study five different fuel-based vehicles, mostly used in India in recent times, are evaluated by the experts as transport alternatives with the consideration of thirteen sustainability related criteria. From the achieved result it is found that in future electric cars would become the most sustainable vehicle in India. Sensitivity analysis of the developed models and comparative study with existing fuzzy and non-fuzzy MCDM methods establish effectiveness of the developed models. This study may facilitate transport authority to select appropriate transport system in connection with sustainable development of a region.

Sustainable Supplier Selection and Order Allocation Using an Integrated ROG-Based Type-2 Fuzzy Decision-Making Approach

The sustainable Supplier Evaluation and Selection and Order Allocation (SSOA) problem has received significant attention in supply chain management due to its potential to enhance a company’s performance, improve customer satisfaction, and reduce costs. In this study, an integrated methodology is proposed to address the SSOA problem. The methodology combines multiple techniques to handle the uncertainties associated with supplier evaluation, including a new ranking method based on the concept of Radius of Gyration (ROG) for interval type-2 fuzzy sets. The methodology also incorporates both subjective weights obtained using the Simple Multi-Attribute Rating Technique (SMART) and expert preferences, and objective weights calculated using the Method based on the Removal Effects of Criteria (MEREC) method to determine the weights of evaluation criteria. Some criteria for sustainable development are used to evaluate supplier performance, resulting in type-2 fuzzy sets, which are evaluated using the Weighted Aggregated Sum Product Assessment (WASPAS) method. The ROG-based ranking method is employed to calculate the relative scores of suppliers. Finally, a multi-objective decision-making (MODM) mathematical model is presented to identify suitable suppliers and allocate their order quantities. The methodology is demonstrated in a sustainable SSOA problem and is shown to be efficient and effective, as the ROG-based ranking method allows for more accurate supplier performance evaluation, and the use of the criteria highlights the importance of sustainability in supplier selection and order allocation. The methodology’s practicality is further supported by the analysis conducted in this study, which demonstrates the methodology’s ability to handle the uncertainties associated with supplier evaluation and selection. The proposed methodology offers a comprehensive approach to the SSOA problem that can effectively handle the uncertainties in supplier evaluation and selection and promote sustainable practices in supply chain management.

A Novel Classification Method Using the Takagi–Sugeno Model and a Type-2 Fuzzy Rule Induction Approach

The main purpose of this research was to introduce a classification method, which combines a rule induction procedure with the Takagi–Sugeno inference model. This proposal is a continuation of our previous research, in which a classification process based on interval type-2 fuzzy rule induction was introduced. The research goal was to verify if the Mamdani fuzzy inference used in our previous research could be replaced with the first-order Takagi–Sugeno inference system. In the both cases to induce fuzzy rules, a new concept of a fuzzy information system was defined in order to deal with interval type-2 fuzzy sets. Additionally, the introduced rule induction assumes an optimization procedure concerning the footprint of uncertainty of the considered type-2 fuzzy sets. A key point in the concept proposed is the generalization of the fuzzy information systems’ attribute information to handle uncertainty, which occurs in real data. For experimental purposes, the classification method was tested on different classification benchmark data and very promising results were achieved. For the data sets: Breast Cancer Data, Breast Cancer Wisconsin, Data Banknote Authentication, HTRU 2 and Ionosphere, the following F-scores were achieved, respectively: 97.6%, 96%, 100%, 87.8%, and 89.4%. The results proved the possibility of applying the Takagi–Sugeno model in the classification concept. The model parameters were optimized using an evolutionary strategy.

Risk Assessment Model of Chemical Process Based on Interval Type-2 Fuzzy Petri Nets

An interval type-2 fuzzy set and fuzzy Petri net combined risk assessment model for chemical production was proposed to solve the problems of disorganized hierarchy and poorly targeted measures, as well as the requirement for complex equipment associated with chemical production risk assessment. First, four different types of risk databases were established according to the production process of cyclohexane. Considering the intrinsic relationship between the risk factors in the fault database, the interval type-2 fuzzy set was used to improve the semantic transformation accuracy and calculate the confidence in the risk factors. The fuzzy Petri net model was used to simulate the dynamic development of accidents, and the parallel relationship between risk factors was intuitively described. Thereafter, the external relationship between risk factors was analyzed, and the net structure of each layer was divided to build a multilevel model. Finally, the catalyst activation process during cyclohexane production was taken as an example for risk assessment calculation, and the accident risk probability was calculated by multilevel fuzzy reasoning. The results demonstrate that the model is an improvement over traditional methods and can be used for precise prevention and control. Moreover, it can accurately analyze risk probability during chemical production, determine the risk associated with the reaction process, effectively prevent accidents, and provide a reference for risk evaluation and risk classification.

Youth Homelessness and Type-2 Fuzzy Sets

The United Nation’s Sustainable Development Goals (SDGs) encourage countries to solve many social problems. One of these problems is homelessness. We rank states in the United States with respect to youth homelessness and the achievement of these goals. We use fuzzy similarity measures to determine the degree of similarity between these rankings. Given a similarity measure between two categories, we introduce a method using type-2 fuzzy sets that determines a similarity of a subcategory of one category with the other category. In particular, given a similarity measure between youth homelessness and the SDGs, we determine a similarity measure between youth vulnerable to toxic stress and the SDGs and also youth vulnerable to human trafficking and the SDGs. We also use a relatively new similarity measure, an [Formula: see text]-dimensional similarity measure.

Applications to biogas-plant implementation problem based on type-2 picture fuzzy matrix game under new minkowski type measures

To manage ambiguity and uncertainty information in real life problems, certain intellectuals have initiated numerous modifications of fuzzy sets. But, in certain situations, the decision-makers have been neglected to take a feasible decision. For instance, when an intellectual provides information in the form of a two-dimension or type-1 fuzzy set, then the prevailing theories have been neglected. For this, in this study, we elaborated the new idea of type-2 picture fuzzy sets (T2-PFS) and their algebraic laws have been given. Moreover, by using the T2-PFS, Hamacher aggregation operators are proposed and discussed their important properties. The Minkowski sorts of measures under the T2-PFS are also initiated. In the current study, we aim to utilize the matrix game in a type-2 picture fuzzy environment. By using the proposed operators and measures based on the Hausdorff metric. The elaborated idea has been defined with the biogas-plant implementation dilemma to verify the pertinence and authenticity. Lastly, to discuss the sensitivity analysis and geometrical shown of the initiated works, certain examples are illustrated to determine the supremacy and consistency of the proposed approaches.