Trapezoidal Interval Type-2 Fuzzy Sets Research Articles

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.

Canonical triangular interval type-2 fuzzy set linguistic distribution assessment TODIM approach: A case study of FMEA for electric vehicles DC charging piles

Linguistic distribution assessment (LDA) plays a prominent role in describing the subjective uncertainties and objective distribution of decision-maker (DM) opinions. Although there have been several studies on LDA, few of these studies have paid attention to the different understandings of language semantics provided by DMs. To address the issue, type-2 (T2) fuzzy sets (T2FSs) are a well-known approach because T2FSs can accurately model the understanding of individuals with linguistic uncertainty. A trapezium (or trapezoidal) interval type-2 fuzzy set (TrIT2FS) is a common application of T2 fuzzy logic. Accordingly, this paper discusses a novel LDA called a canonical triangular interval type-2 fuzzy set (CTriIT2FS)-LDA in a T2 fuzzy environment. The CTriIT2FS-LDA combines a CTriIT2FS simplified from a TrIT2FS with LDA, which can simplify the computation while retaining their advantages. To facilitate the calculation of the CTriIT2FS-LDA, the basic operations of the CTriIT2FSs and the CTriIT2F-LDA, such as the defuzzied (crisp) function, distance measurement, and entropy measurement, are defined. Based on the TODIM approach, a CTriIT2F-LDA-TODIM is developed for FMEA for interval type-2 fuzzy environments. Furthermore, the proposed approach is applied in a practical case for FMEA for the electric vehicle DC charging piles of the State Grid Corporation of China.

Knowledge-based prognostics and health management of a pumping system under the linguistic decision-making context

The process industry implements important maintenance strategies on the basis of the health status of system operation to reduce engineering maintenance cost and concurrently ensure reliability and safety. This study presents an architecture for knowledge-based prognostics and health management (K-PHM) by applying the trapezoidal interval type-2 fuzzy linguistic term sets. The proposed K-PHM methodology successfully provides operational information for diagnostics, prognostics, and subsequent actionable knowledge for health management in subjective decision making. It emphasizes knowledge-driven maintenance strategies to increase system reliability and safety within linguistic decision making. The main objective of this study is to focus on the uncertainty within the knowledge formation derived from data and subsequent information expressed with linguistic terms generated from domain experts in the K-PHM methodology. Fuzzy sets and fuzzy logic reasoning techniques are used in conjunction to enhance the capability of handling uncertainty throughout the PHM process. The effectiveness of the proposed K-PHM is verified with the authentic case study for the predictive maintenance of a chemical pumping system against abnormal vibration.

Site selection decision framework for photovoltaic hydrogen production project using BWM-CRITIC-MABAC: A case study in Zhangjiakou

Hydrogen production by photovoltaic power generation can not only effectively alleviate the problem of high cost of electrolysis water by making full use of light resources, but also is an efficient, clean and carbon-reducing way to produce hydrogen. The purpose of this paper is to provide a practical model for site selection decision of photovoltaic hydrogen production project (PVHPP) based on multi-criteria decision-making (MCDM) method, and take Zhangjiakou City, Hebei Province as an example for analysis. Firstly, a set of evaluation criteria system including natural factors, economic factors, technical factors, social factors and environmental factors is constructed. Secondly, probabilistic hesitation fuzzy linguistic term set (PHFLTS) and trapezoidal interval type-2 fuzzy set (TIT2FS) are combined to describe the information of qualitative criteria. Then, the subjective and objective weights are calculated by using best-worst method (BWM) and criteria importance through intercriteria correlation (CRITIC), and the comprehensive weights are obtained by using the idea of game theory. The multi-attribute boundary approximation regional comparison (MABAC) method is then used to sort the alternative plant sites. The ranking results show that A2 (Chongli District) performs the best, and A1 (Qiaodong District) is the second best. Finally, taking Zhangjiakou as an example, sensitivity analysis and comparative analysis are carried out to verify the validity and reliability of the model.

An Adoptable Multi-Criteria Decision-Making Analysis to Select a Best Hair Mask Product-Extended Weighted Aggregated Sum Product Assessment Method

Hair masks (HMs) act as one of the solutions for most of the hair problems like dandruff, frizziness, breakage, premature- greying and so on. Due to its various benefits, HM products are acquiring more popularity among the individuals. As there are different varieties of HM products available in the market, the confusion arises in choosing a HM which suits the individual’s hair profile and causes less side effects. Here, we have employed multi-criteria decision-making (MCDM) combined with fuzzy set theory to obtain better results. We used the extended Weighted Aggregated Sum Product Assessment (WASPAS) method based on trapezoidal interval type-2 fuzzy set (TIT2FS) in this research paper to handle vagueness and complexity in real-world problems. For determining the objective weights of the criteria, we used the entropy method of weight finding. An example of selecting a hair mask product (HMP) among four alternatives based on five criteria is provided to illustrate the applicability of the proposed method. In comparison to other MCDM methods, the approach yielded more practical results. By doing a sensitive study, the method’s stability is also assessed.

An interval type-2 fuzzy TOPSIS for MAGDM applied to solar power systems

This article proposes a new version of the technique of order preference by similarity to an ideal solution (TOPSIS) to solve fuzzy multi-attribute group decision making (MAGDM) problems using trapezoidal interval type-2 fuzzy sets (IT2FSs). The traditional TOPSIS ranks the alternatives according to their relative degree of closeness to the ideal solutions. On the other hand, TOPSIS based on similarity measure ranks the alternatives according to their total degree of similarity to the ideal solutions. This study extends TOPSIS using similarity measure using map distance to IT2FSs. First, the similarity measure based on map distance for interval-valued fuzzy sets (IVFSs) is extended to encompass IT2FSs due to the deficiency in IT2FSs similarity measures. Then, TOPSIS using similarity measure is applied. Hence, fuzzy MAGDM problems can be handled in a more flexible intelligent manner and avoiding defuzzification with its drawbacks. An illustrative example is given to explain the approach. Then, a practical problem in assessing thermal energy storage technologies in solar power systems is solved, where the weights of the attributes and the performance of the qualitative attributes are linguistic variables modeled by IT2FSs. The reliability of two normalization techniques is examined and the impact of the theoretical and empirical reference points on the solution is investigated.

An Area Similarity Measure for Trapezoidal Interval Type-2 Fuzzy Sets and Its Application to Service Quality Evaluation

An Area Similarity Measure for Trapezoidal Interval Type-2 Fuzzy Sets and Its Application to Service Quality Evaluation

An integrated interval type-2 fuzzy technique for democratic–autocratic​ multi-criteria decision making

This paper develops an integrated trapezoidal interval type-2 fuzzy (TrIT2F) technique for democratic–autocratic multi-criteria group decision making based on best–worstmethod (BWM) and VIKOR (VIsekriterijumska optimizacija i KOm-promisno Resenje), which is called TrIT2F-BW-VIKOR. In this technique, the pairwise comparisons and evaluations are represented by trapezoidal interval type-2 fuzzy sets (TrIT2FSs). The existing definition of TrIT2FS is perfected by adding two rational constraints proposed in this paper. A weight-normalizing theorem is initiated to normalize the TrIT2F weights. To determine the TrIT2F weights of junior decision makers (JDMs) and criteria, the classical BWM is extended into TrIT2F environment, which is called TrIT2F-BWM. In this TrIT2F-BWM, the weight-normalizing theorem is applied to normalize the TrIT2F weights, a consistency ratio is designed to check the reliability of the obtained TrIT2F weights. Based on the determined weights of JDMs and criteria, an extended VIKOR is developed to rank alternatives. The proposed technique can not only effectively retain the inherent fuzzy information of TrIT2FSs, but also flexibly handle different decision situations. The validity of the proposed technique is demonstrated with a makeshift (fangcang) hospital selection example on COVID-19. Some sensitivity and comparison analyses are provided to show the stability, flexibility, and superiorities of the proposed technique.

Assessment of national innovation capabilities of OECD countries using trapezoidal interval type-2 fuzzy ELECTRE III method

PurposeFor a nation to become a superpower, it's scientific and technological advancement is essential. Each country is exploring how to improve themselves in terms of science and technology. The authors analyzed the innovation capabilities of 35 OECD countries that have not recently joined Lithuania.Design/methodology/approachIn recent years, a lot of research work has been done on trapezoidal interval type-2 fuzzy sets (TIT-2 FS), and many research works have been published. The trapezoidal interval type-2 fuzzy set helps effectively to represent the uncertainty comparatively than the type-1 fuzzy set. Taking advantage of this effectiveness, the authors extend the best multi-criteria decision making method (MCDM) for trapezoidal interval type-2 fuzzy sets. Here, ELimination and Choice Expressing REality III (ELECTRE III) method in the trapezoidal interval type-2 fuzzy set environment is proposed.FindingsThis analysis helps to the OECD countries to develop their level of innovation in the criteria. The authors are making this evaluation for the year 2018 based on the 31 criteria. Application of the proposed method expressed by evaluation of the national innovation capability problem. Based on the obtained results, the top five countries are United States, Switzerland, Canada, Germany and Japan.Originality/valueThe authors collected required data from different available data sources like OECD, IMD, USPTO, ITU and surveyed data reported by KISTEP. After collecting all the data from different sources, the authors calculated the standard values as KISTEP. After converting the standard values into trapezoidal interval type-2 fuzzy values, the authors construct a decision matrix based on these values. Then, the authors determined the possibility mean values and preference. Then, they calculated the concordance and discordance credibility degree values. Finally, they ranked OECD countries by the net credibility degree. The results are computed by using the MATLAB software.

Trapezoidal Interval Type-2 Fuzzy TOPSIS Using Alpha-Cuts

Fuzzy multi-criteria decision-making (FMCDM) provides solutions to the problems involving multiple criteria for decision-makers under uncertain environments. The Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is one of the most popular methods to address FMCDM problems. In this paper, we propose an extension of the TOPSIS method with trapezoidal interval type-2 fuzzy sets (IT2 FSs) using the concept of α-cut. We first propose a new method to calculate the distance between two trapezoidal IT2 FSs with the tool of α-cut and ordered weighted averaging (OWA) operator. Next, based on the distance method, we extend the TOPSIS method to the context of trapezoidal IT2 FSs and utilize it to solve FMCDM problems. Finally, an illustrative example is used to demonstrate the feasibility of the proposed method, and some comparisons with other existing works are presented to show the highlights of the proposed method. The proposed method provides a flexible solution for the FMCDM problems by taking the decision-makers’ attitudes into consideration.

An Extended TOPSIS Method Based on Gaussian Interval Type-2 Fuzzy Set

Compared with trapezoidal interval type-2 fuzzy set (IT2 FS), Gaussian IT2 FS is more concise in representation and can meanwhile capture sufficient uncertainties. Therefore, in this paper the Gaussian IT2 FS is used to model words in multi-attribute decision-making (MADM) process and a comprehensive method is developed under the Gaussian IT2 FS environment. Firstly, a new type of Gaussian IT2 FS is proposed and the ranking rule is constructed to determine the ideal solutions. Secondly, the Euclidean distance is extended to Gaussian IT2 FS to measure the difference between alternatives and ideal solutions. Further, a complete MADM method is established under the framework of TOPSIS (technique for order preference by similarity to an ideal solution). In the end, the effectiveness of the proposed method is illustrated by an investment decision-making example.

Trapezoidal interval type-2 fuzzy soft stochastic set and its application in stochastic multi-criteria decision-making

Trapezoidal interval type-2 fuzzy soft set (TrIT2FSS) (Zhang and Zhang, Appl Math Model 37:4948–4971, 2013) is a delicious generalization of soft set theory where the performance of the alternatives over some parameters is assigned by means of trapezoidal interval type-2 fuzzy numbers (TrIT2FNs). Zhang and Zhang (2013) proposed an approach to multi-criteria group decision-making (MCGDM) problems under trapezoidal interval type-2 fuzzy soft set. However, due to the increasing complexity of decision-making problems, the performance of the alternatives may be determined randomly with respect to some possible states related to the decision-making problems which are known as stochastic multi-criteria decision-making (SMCDM) problems. To handle such type of SMCDM problems based on trapezoidal interval type-2 fuzzy soft set, no research exists till now. To fill up this lacuna, here we have formulated a new generalization of TrIT2FSS to solve the MCDM problems under stochastic environment. First, we have introduced the notion of trapezoidal interval type-2 fuzzy soft stochastic set (TrIT2FSSS). Second, we deal with the obligatory definition of expected trapezoidal interval type-2 fuzzy soft set (ETrIT2FSS). Finally, we have proposed a methodology for handling the trapezoidal interval type-2 fuzzy soft stochastic set (TrIT2FSSS) based stochastic MCDM problems. Additionally, in this model we have assumed that the weights of each of the parameters are partially known. We have proposed a process to obtain the weights of the parameters using signed distance measurement. An application of our proposed approach regarding the selection of the best company for investment of a bank has also been given to describe the feasibility and effectiveness of our proposed approach.

Multi-criteria group decision-making using an extended edas method with interval type-2 fuzzy sets

Multi-criteria decision-making (MCDM) methods are very useful in the real-world decision-making problems. We are usually confronted with the decision-making process in an uncertain environment, and the fuzzy set theory is an effi cient tool to handle this uncertainty. Interval type-2 fuzzy sets are one of the extensions of the fuzzy sets which are very fl exible to model an uncertain environment. This study is related to MCDM problems within the context of interval type-2 fuzzy sets (IT2FSs). The evaluation based on distance from average solution (EDAS) method is a new and effi cient MCDM method, and assessment of alternatives in this method is based on the distance of them from average solution with respect to all criteria. In the EDAS method, each alternative has positive and negative distances which are used to determine the appraisal score of it. In this research, we present an extended EDAS method, which is named EDAS-IT2FSs, for dealing with multicriteria group decision-making problems with interval type-2 fuzzy sets. Basic concepts of interval type-2 fuzzy sets and the arithmetic operations of trapezoidal IT2FSs are used to develop the extended EDAS method. A numerical example of multi-criteria subcontractor evaluation problem is used to illustrate the process of using the extended EDAS method. The example involves eight subcontractors that need to be evaluated with respect to seven criteria. A comparison and a sensitivity analysis based on different sets of criteria weights are also performed to show the validity of the proposed method. The results of these analyses show the effi ciency and stability of the extended EDAS method.

Comments on “A note on “A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets””

In a recently published paper “A note on “A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets””, Khalil and Hassan pointed out that assertions (3) and (4) of Theorem 3.2 in our previous paper “A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets” are not true [2]. Furthermore, they introduced the notions of a generalized trapezoidal interval type-2 fuzzy soft subset and a generalized trapezoidal interval type-2 fuzzy soft equal and used these two notions to correct the flaw in assertions (3) and (4) of Theorem 3.2 in our previous paper. In this paper, we show by a counterexample that Khalil and Hassan's correction is incorrect and provide the modified versions of assertions (3) and (4) of Theorem 3.2, along with a strict proof. In addition, Khalil and Hassan pointed out by a counterexample that assertions (3)–(6) of Theorem 3.5 in our paper are not true and proposed the corrections of those assertions. In this paper, we show that Khalil and Hassan's counterexample and corrections are incorrect and provide a new example to verify the inaccuracies of assertions (3) and (5) of Theorem 3.5 in our paper. Moreover, we offer the modified versions of assertions (3) and (5) of Theorem 3.5 and prove them. Finally, Khalil and Hassan's statement that assertions (4) and (6) of Theorem 3.5 in our previous paper are not true is proven to be incorrect, i.e. assertions (4) and (6) of Theorem 3.5 in our previous paper are true.

Trapezoidal interval type-2 fuzzy aggregation operators and their application to multiple attribute group decision making

A type-2 fuzzy set, which is characterized by a fuzzy membership function, involves more uncertainties than the type-1 fuzzy set. As the most widely used type-2 fuzzy set, interval type-2 fuzzy set is a very useful tool to model the uncertainty in the process of decision making. As a special case of interval type-2 fuzzy set, trapezoidal interval type-2 fuzzy set can express linguistic assessments by transforming them into numerical variables objectively. The aim of this paper is to investigate the multiple attribute group decision-making problems in which the attribute values and the weights take the form of trapezoidal interval type-2 fuzzy sets. First, we introduce the concept of trapezoidal interval type-2 fuzzy sets and some arithmetic operations between them. Then, we develop several trapezoidal interval type-2 fuzzy aggregation operators for aggregating trapezoidal interval type-2 fuzzy sets and examine several useful properties of the developed operators. Furthermore, based on the proposed operators, we develop two approaches to multiple attribute group decision making with linguistic information. Finally, a practical example is given to illustrate the feasibility and effectiveness of the developed approach.

A note on “A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets”

Zhang and Zhang (2013) proposed an extended soft set model called trapezoidal interval type-2 fuzzy soft sets by combining trapezoidal interval type-2 fuzzy sets and soft sets and showed some examples of its operations, along with an application in decision making. However, we found that some results are incorrect. Thus, the purpose of this note is to point out that assertions (3) and (4) of Theorem 3.2 in Zhang and Zhang (2013) are not true, as well, assertion (3) of Theorem 3.5 is generally not true, while assertions (4)–(6) of Theorem 3.5 can be similarly proven to be incorrect. We also introduce the notion of a generalized trapezoidal interval type-2 fuzzy soft subset and fuzzy soft equal, and use these notions to correct the flaw in assertions (3) and (4) of Theorem 3.2 of Zhang and Zhang (2013). Finally we introduce two new definitions and a theorem to improve the work of Zhang and Zhang (2013), along with two examples.

A note on “A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets”

Zhang and Zhang (2013) proposed the arithmetic operations of trapezoidal interval type-2 fuzzy numbers having different left and right heights and hence the arithmetic operations of trapezoidal interval type-2 fuzzy soft sets having different left and right heights. In this paper, it is pointed out that the complement operation of a trapezoidal interval type-2 fuzzy number, proposed by Zhang and Zhang, is not valid and hence, the complement operation of trapezoidal interval type-2 fuzzy soft set as well as all the results, proposed by Zhang and Zhang in which complement operation is used, are not valid. The results, proposed by Zhang and Zhang, are valid only for such trapezoidal interval type-2 fuzzy numbers and trapezoidal interval type-2 fuzzy soft sets in which left and right heights are equal.

An assessment approach for non-governmental organizations in humanitarian relief logistics and an application in Turkey

The ever-increasing natural disasters have been causing the loss of lives, properties and resources. By the preparedness and response ability of non-governmental organizations, it is aimed to minimize these losses. In this paper, first, the critical success factors of humanitarian relief logistics management operations are determined and categorized. Then, by considering these factors, a hybrid method that consists of trapezoidal interval type-2 fuzzy sets, AHP and TOPSIS, is proposed to evaluate emergency preparedness and response ability performance of non-governmental relief organizations. The proposed hybrid method is applied for non-governmental relief organizations in Turkey to evaluate their performance, and to the factors need to be improved for each determined organization.

Multi-attribute group decision making method based on geometric Bonferroni mean operator of trapezoidal interval type-2 fuzzy numbers

In this paper, we investigate the fuzzy multi-attribute group decision making (FMAGDM) problems in which all the information provided by the decision makers (DMs) is expressed as the trapezoidal interval type-2 fuzzy sets (IT2 FS). We introduce the concepts of interval possibility mean value and present a new method for calculating the possibility degree of two trapezoidal IT2 FS. Then, we develop two aggregation techniques called the trapezoidal interval type-2 fuzzy geometric Bonferroni mean (TIT2FGBM) operator and the trapezoidal interval type-2 fuzzy weighted geometric Bonferroni mean (TIT2FWGBM) operator. We study its properties and discuss its special cases. Based on the TIT2FWGBM operator and the possibility degree, the method of FMAGDM with trapezoidal interval type-2 fuzzy information is proposed. Finally, an illustrative example is given to verify the developed approaches and to demonstrate their practicality and effectiveness.

Fuzzy Multiple Attributes Group Decision-Making Based on Ranking Interval Type-2 Fuzzy Sets and the TOPSIS Method

In this paper, we present a new method for fuzzy multiple attributes group decision-making based on the proposed ranking method of trapezoidal interval type-2 fuzzy sets and the TOPSIS method. Firstly, we present a new method for ranking trapezoidal interval type-2 fuzzy sets. Secondly, we construct the decision matrix and the average decision matrix, respectively. Thirdly, we construct the weighting decision matrix and average weighting decision matrix, respectively. Fourthly, we construct the ranking decision matrix by calculating the ranking values of trapezoidal interval type-2 fuzzy sets in the average decision matrix. Then, we get the absolute positive ideal solution and the absolute negative ideal solution with respect to the attributes based on the ranking decision matrix. Then, we calculate the distance between each alternative and the absolute positive ideal solution and calculate the distance between each alternative and the negative ideal solution, respectively. Finally, we calculate the relative degree of closeness of each alternative. The larger the value of the relative degree of closeness of an alternative, the better the preference order of the alternative. The proposed fuzzy multiple attributes group decision-making method can overcome the drawbacks of the existing method.