Multiple attribute group decision making in interval type-2 fuzzy environment using a new distance formulation

Ranking of fuzzy numbers has become an important research direction for decision-making problems due to its role to find the best objects under uncertainty. In this paper, we propose a new approach to perform multiple attribute group decision-making (MAGDM) problems using the ranking of interval type-2 fuzzy sets. Initially, a new ranking method for interval type-2 fuzzy numbers based on centroid and rank index has been proposed. Next, we present a comparative study to analyze the ranking values of the proposed method with the existing approaches, where we explore the necessity of the proposed ranking method. After that, a new MAGDM approach has been developed using the proposed ranking procedure to solve uncertain MAGDM problems. Finally, the applicability of the proposed approach has been illustrated using two numerical examples and a case study related to car-sharing problems. The proposed study exhibits a useful way to solve fuzzy MAGDM problems with much efficient manner since it applies interval type-2 fuzzy sets compared to type-1 fuzzy sets to signify the evaluating values and weights of the attributes.

Fuzzy logic (FL) has been regarded as a useful methodology in modelling. The modeling performance of FL heavily relies on the modeling capability of a fuzzy set (FS). However, the MF of a FS cannot be arbitrarily accurate in practice and thus the centroid as a measure of a FS cannot be accurate to any degree. In face of inaccurate MF, the sensitivity of the centroid of a FS to the deviation of its MF from the ideal MF determines the accuracy of the centroid. To evaluate the modeling capability of a type-1 (T1) FS and an interval type-2 (IT2) FS, this paper will present a comparative study of the sensitivity of the centroid of a T1 FS and an IT2 FS to the deviation of their MFs from the ideal MFs. In this paper, equations relating the amount that the centroid of a T1 FS and an IT2 FS vary with the deviation of their MFs from the ideal MFs were established. Based on the derived equations, it was found that an IT2 FS is able to provide an additional design freedom to vary the sensitivity of its centroid compared to a T1 FS. This unique characteristic may enable an IT2 FS to minimize the sensitivity of the centroid and thus to achieve higher modeling accuracy. Furthermore, guidelines for designing an IT2 FS were provided.

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.

This paper investigates multiple attribute group decision making (MAGDM) with hesitant fuzzy preference information, which is a significantly import issue to be deeply studied in management and industrial engineering. Firstly, simultaneously considering optimistic and pessimistic attitudinal preference information, an improved distance measure for hesitant fuzzy set is defined. Then, utilizing the newly defined distance measure, a hesitant fuzzy multiple attribute group decision making approach based on TOPSIS method is constructed, which can effectively avoid high complexity of aggregating hesitant fuzzy information in traditional methods. Further, an application study on parts supplier selection has verified the practically and effectiveness of developed methods.

The Wang-Mendel Approach (WMA) focuses on combining the numerical as well as linguistic information for achieving greater explainability for inference models. The standard WMA models the linguistic information using type-1 (T1) fuzzy sets (FSs), which have a reduced capability to model the semantics of linguistic information. Therefore, we propose a novel Enhanced WMA, which models the linguistic information using the type-2 (T2) FSs. Further, our Enhanced T2 FS-based WMA can be modified to reflect the use of interval type-2 (IT2) FSs, for modelling linguistic uncertainty. IT2 FSs are suitable when better uncertainty handling capabilities are required compared to T1 FSs, however, at a computational cost lesser than the T2 FSs. Performance of Enhanced WMA is demonstrated through a real-world crop-yield prediction problem in smart agriculture and an additional exemplar application on users’ satisfaction ratings. Further, we have compared our approach with the performance obtained from the T1 FS-based WMA and the original estimations given in the original data. We found that our Enhanced WMA provides more precise estimates than the other two with 95% confidence level. To the best of our knowledge, this is the first proposal of a T2 FSs method for enhancing the modeling of linguistic uncertainty in the WMA.

In this paper, we develop the parametric graded mean integration representation (PGMIR) for the interval type 2 fuzzy sets (IT2FS). The PGMIR value can be served as the type reduced or defuzzification of an IT2FS. A set of IT2FSs can then be ordered by the corresponding PGMIR values. Firstly, the parametric representation of the embedded fuzzy sets is defined for the general IT2FS. The IT2FS is characterized by the set of embedded fuzzy sets induced from the parametric function. Then, the graded mean integration representation (GMIR) of a type 1 fuzzy set (T1FS) is extended to the IT2FS through the parametric form of an IT2FS. The GMIR of the parametric embedded T1FS is defined and is called the PGMIR. In this manner, the IT2FSs are defuzzified and can be ranked by sorting the PGMIR values. The average value of the PGMIR is used to determine IT2FS ranking. A numerical example is presented to show that the proposed approach is superior to the ranking method from IT2FS centroid.

Picture fuzzy sets (PFSs) utilize the positive, neutral, negative and refusal membership degrees to describe the behaviours of decision-makers in more detail. In this article, we expound the application of extended TODIM based on cumulative prospect theory under picture fuzzy multiple attribute group decision making (MAGDM). In addition, we adopt Information Entropy, which is used to ascertain the weighting vector of attributes to improve the availability of the TODIM method. At last, we exercise the improved TODIM into a numerical case for super market location and testify the effectiveness of this new method by comparing its results with other methods’ results.

A new ranking methodology for general interval type 2 fuzzy sets (IT2FS) using extended alpha cuts representation is proposed. First, the extended alpha cut representation of the IT2FS is defined. Then, the parametric representation of the type 1 fuzzy sets (T1FS) is formulated by constructing the embedded pessimistic, modal and optimistic functionals. The corresponding parametric centroid functional of the IT2FS is determined. Finally, the average centroid of the IT2FS is calculated by taking the integral average of the parametric centroid functional. In this manner, the IT2FSs are defuzzified and can be ranked by sorting the average centroids. A numerical example is presented for verifying the proposed method.

Type-2 fuzzy sets (T2FSs), characterized by a fuzzy membership function, are much useful tool for representing the decision knowledge in the decision making process. Interval Type-2 fuzzy sets (IT2FSs) are the most commonly used T2FSs. In this study, a method based upon ranking values of IT2FSs is used to tackle multi-criteria decision making (MCDM) problems. First, some basic concepts and arithmetic operations for IT2FSs are presented. Then, three kinds of fuzzy ranking methods, proposed by [1], based on arithmetic average (AA), geometric average (GA) and harmonic average (HA) operators to compute the ranking values of IT2FSs are applied. Finally, the outcomes of MCDM methods based on the ranking values of IT2FSs are obtained and also compared with the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method based on Type-1 fuzzy sets (T1FSs) for a numerical example.

Gestures of a person symbolizing analogous pathological disorder are not always exclusive. Naturally, the gestural features of a subject suffering from the same pathological disorder vary widely over different instances. In presence of two or more gestural features, the variation of the attributes together makes the problem of pathological disorder recognition more convoluted. Wider variance in gestural features is the main source of uncertainty in the pathological disorder recognition, which has been addressed here using type-2 fuzzy sets (T2FSs). First, a type-2 fuzzy gesture space is created with the background knowledge of gestural features of different subjects for different pathological disorders. Second, the pathological disorder of an unknown gestural expression is recognized based on the consensus of the measured gestural features of the fuzzy gesture space. A fusion of interval and general type-2 Fuzzy Sets has been used to construct the fuzzy gestural space. An interval valued fuzzy set (IVFS) or interval type-2 fuzzy set (IT2FS) is constructed with primary membership functions for M gestural features obtained from N subjects, each having L instances of gestural expressions for a given pathological disorder. The corresponding general type-2 fuzzy set (GT2FS) includes the secondary memberships for individual primary upper and lower membership curves in the respective IVFS. The secondary membership of a given source, characterizing the reliability in its primary membership assignment, is determined based on the amalgamated knowledge of all the subjects’ primary membership functions. The representatives of each pathological disorder class are then evaluated by utilizing the attractive features of both IVFS and GT2FS. The pathological disorder of an unknown gestural expression is finally determined by identifying its minimal discrepancy with the existing class representatives. The adopted T2FS-based uncertainty management strategy, when applied to gender independent training and testing datasets, results in a classification accuracy of 93.41%.

This paper introduces a new proposal to involve uncertainties in Fuzzy Markov Chains by using Interval Type-2 Fuzzy Sets (IT2 FS). A Type-1 Fuzzy Markov chain is an approach which uses Type-1 Fuzzy Sets (T1 FS) to describe the distributional behavior of a Discrete-Time Markov process, while the IT2 FS approach is an extension of its scope that allows to embed several T1 FS inside its Footprint of Uncertainty. In this paper, a finite state Fuzzy Markov Chain process is defined by an Interval Type-2 Fuzzy environment, finding their limiting properties and its Type-reduced behavior. To do so, two examples are provided.

Weight aggregation is the key process to solve a multiple-attribute group decision-making (MAGDM) problem. This paper is trying to propose a possible approach to objectivize subjective information and to aggregate information from attribute values themselves and decision-makers’ judgment. An MAGDM problem without information about decision-makers’ and attributes’ weight is considered. In order to define decision-makers’ subjective preference, their utility function is introduced. The attributes value matrix is converted into a subjective attributes value matrix based on their subjective judgment on attribute values. By utilizing the entropy weighting technique, decision-maker’s subjective weight on attributes and objective weight on attributes are determined individually based on the subjective attributes value matrix and attributes value matrix. Based on the principle of minimum cross-entropy, all decision-makers’ subjective weights are integrated into a single weight vector that is closest to all decision-makers’ judgment without any extra information added. Then, by applying the principle of minimum cross-entropy again, a weight aggregation method is proposed to combine the subjective and objective weight of attributes. Finally, an MAGDM example of project choosing is presented to illustrate the procedure of the proposed method.

Fuzzy multiple attribute group decision making (MAGDM) is an important topic both in fuzzy sets (FSs) theory and its applications (Zadeh 1965; Chen 2000; Kuo et al. 2006; Roghanian et al. 2010; Hatami-Marbini and Tavana 2011).

In this paper, the concept of multiple-attribute group decision-making (MAGDM) problems with interval-valued Pythagorean fuzzy information is developed, in which the attribute values are interval-valued Pythagorean fuzzy numbers and the information about the attribute weight is incomplete. Since the concept of interval-valued Pythagorean fuzzy sets is the generalization of interval-valued intuitionistic fuzzy set. Thus, due the this motivation in this paper, the concept of interval-valued Pythagorean fuzzy Choquet integral average (IVPFCIA) operator is introduced by generalizing the concept of interval-valued intuitionistic fuzzy Choquet integral average operator. To illustrate the developed operator, a numerical example is also investigated. Extended the concept of traditional GRA method, a new extension of GRA method based on interval-valued Pythagorean fuzzy information is introduced. First, utilize IVPFCIA operator to aggregate all the interval-valued Pythagorean fuzzy decision matrices. Then, an optimization model based on the basic ideal of traditional grey relational analysis (GRA) method is established, to get the weight vector of the attributes. Based on the traditional GRA method, calculation steps for solving interval-valued Pythagorean fuzzy MAGDM problems with incompletely known weight information are given. The degree of grey relation between every alternative and positive-ideal solution and negative-ideal solution is calculated. To determine the ranking order of all alternatives, a relative relational degree is defined by calculating the degree of grey relation to both the positive-ideal solution and negative ideal solution simultaneously. Finally, to illustrate the developed approach a numerical example is to demonstrate its practicality and effectiveness.

Modeling, design and control of a 4-arm delta parallel manipulator employing type-1 and interval type-2 fuzzy logic-based techniques for precision applications