Trapezoidal Intuitionistic Fuzzy Numbers Research Articles

Intuitionistic Trapezoidal Fuzzy Group Decision-Making Based on Prospect Choquet Integral Operator and Grey Projection Pursuit Dynamic Cluster

In consideration of the interaction among attributes and the influence of decision makers’ risk attitude, this paper proposes an intuitionistic trapezoidal fuzzy aggregation operator based on Choquet integral and prospect theory. With respect to a multiattribute group decision-making problem, the prospect value functions of intuitionistic trapezoidal fuzzy numbers are aggregated by the proposed operator; then a grey relation-projection pursuit dynamic cluster method is developed to obtain the ranking of alternatives; the firefly algorithm is used to optimize the objective function of projection for obtaining the best projection direction of grey correlation projection values, and the grey correlation projection values are evaluated, which are applied to classify, rank, and prefer the alternatives. Finally, an illustrative example is taken in the present study to make the proposed method comprehensible.

An improved ranking method for comparing trapezoidal intuitionistic fuzzy numbers and its applications to multicriteria decision making

Problems with qualitative, quantitative and uncertain information can be modelled better using trapezoidal intutionistic fuzzy numbers (TrIFNs) than fuzzy numbers. Due to the partial ordering of TrIFNs, many ranking methods are available in the literature for comparing fuzzy and intuitionistic fuzzy numbers (Li in Comput Math Appl 60:1557–1570, 2010; Li and Yang in Math Comput Appl 20(1):25–38, 2015; De and Das in 12th international conference on intelligent systems design and applications (ISDA), Kochi, India, 27–29 Nov, IEEE, 2012; Dubey and Mehara in Proceedings of the seventh conference of the European society for fuzzy logic and technology, pp 563–569, 2011; Nehi in Int J Fuzzy Syst 12:80–86, 2010; Fangwei and Xu in Soft Comput, 2016. doi:10.1007/s00500-015-1932-x; Jun in Expert Syst Appl 38:11730–11734, 2011, Int J Gen Syst 41(7):729–739, 2009, Group Decis Negot 21:519–530, 2012; Nayagam and Geetha in Appl Soft Comput 11(4):3368–3372, 2011; Nayagam et al. in Expert Syst Appl 38(3):1464–1467, 2011, Soft Comput, 2016. doi:10.1007/s00500-016-2249-0; Sahin in Soft Comput 20(7):2557–2563, 2016; Shu-Ping and Dong in Appl Soft Comput 29:153–168, 2015; Zeng et al. in Sci World J 1–8, 2014; Ye in Expert Syst Appl 36:6899–6902, 2009; Zhang and Yu in Knowl Based Syst 30:115–120, 2012). In order to overcome the shortcomings and limitations of the existing methods, a new method for comparing TrIFNs based on the concept of improved value index and improved ambiguity index is introduced, and the significance of the proposed method is illustrated using numerical examples. Further to show the applicability of the proposed method, a new algorithm for solving multicriteria decision-making problem is established.

Cost-Sharing of Ecological Construction Based on Trapezoidal Intuitionistic Fuzzy Cooperative Games.

There exist some fuzziness and uncertainty in the process of ecological construction. The aim of this paper is to develop a direct and an effective simplified method for obtaining the cost-sharing scheme when some interested parties form a cooperative coalition to improve the ecological environment of Min River together. Firstly, we propose the solution concept of the least square prenucleolus of cooperative games with coalition values expressed by trapezoidal intuitionistic fuzzy numbers. Then, based on the square of the distance in the numerical value between two trapezoidal intuitionistic fuzzy numbers, we establish a corresponding quadratic programming model to obtain the least square prenucleolus, which can effectively avoid the information distortion and uncertainty enlargement brought about by the subtraction of trapezoidal intuitionistic fuzzy numbers. Finally, we give a numerical example about the cost-sharing of ecological construction in Fujian Province in China to show the validity, applicability, and advantages of the proposed model and method.

Novel arithmetic operations on type-2 intuitionistic fuzzy and its applications to transportation problem

Type-2 intuitionistic fuzzy sets possess many advantages over type-1 fuzzy sets because their membership functions are themselves fuzzy, making it possible to model and minimize the effects of uncertainty in type-1 intuitionistic fuzzy logic systems. This paper presents generalized type-2 intuitionistic fuzzy numbers and its different arithmetic operations with several graphical representations. Basic generalized trapezoidal intuitionistic fuzzy numbers considered for these arithmetic operations are formulated on the basis of (α,β)-cut methods. The ranking function of the generalized trapezoidal intuitionistic fuzzy number has been successively calculated. To validate the proposed arithmetic operations, we solved a type-2 intuitionistic fuzzy transportation problem by the ranking function for mean interval method. Transportation costs, supplies and demands of the homogeneous product are type-2 intuitionistic fuzzy in nature. A numerical example is presented to illustrate the proposed model.

Power Average of Trapezoidal Intuitionistic Fuzzy Numbers Using Strict t-Norms and t-Conorms

As an extension of intuitionistic fuzzy number (IFN), trapezoidal intuitionistic fuzzy number (TrIFN) is one of the useful tools to deal with ill-known quantities in decision data and decision making problems. In this paper, based on the strict t-norms and t-conorms, new operation laws for the normalized TrIFNs are defined. Specially, the operation laws of TrIFNs take the speculative and radical principle and have the property of closeness in the set of normalized TrIFNs. Then, the power average operator for real numbers is extended to four power average operators for TrIFNs, i.e., the triangular (co)norms-based (t-based) power average for TrIFNs, t-based weighted power average operator for TrIFNs, t-based power ordered weighted average operator for TrIFNs, and t-based power hybrid average operator for TrIFNs. To show the feasibility and reasonability in the applications of multiple attributes group decision making using the developed operations for TrIFNs, a numerical example is provided. It is shown that there is more flexibility in the choice of the parameters associated with the degree of the risk one can bear.

Contrasting Correlation Coefficient with Distance Measure in Interval Valued Intuitionistic Trapezoidal Fuzzy MAGDM Problems

In this paper, Multiple Attribute Group Decision Making (MAGDM) problems in which the data is of the form of Interval Valued Intuitionistic Trapezoidal Fuzzy Numbers (IVITzFNs) is presented. Some operational laws of IVITzFNs are introduced. Then some new aggregation operators including interval valued Intuitionistic Trapezoidal Fuzzy Weighted Averaging (IVITzFWA) operator, interval valued Intuitionistic Trapezoidal Fuzzy Ordered Weighted Averaging (IVITzFOWA) operator and Interval Intuitionistic Trapezoidal Fuzzy Hybrid Averaging (IVITzFHA) operator, are proposed and some desirable properties of these operators are studied, such as Commutativity, Idempotency and Monotonicity. A new distance function and correlation coefficient are proposed for IVITzFNs which will be utilized for ranking the alternatives in MAGDM problems. Finally, numerical illustrations are given to verify the developed approach and to demonstrate its practicality and effectiveness.

A new approach for ranking of intuitionistic fuzzy numbers using a centroid concept

Ranking of intuitionistic fuzzy numbers is a difficult task. Many methods have been proposed for ranking of intuitionistic fuzzy numbers. In this paper we have ranked both trapezoidal intuitionistic fuzzy numbers and triangular intuitionistic fuzzy numbers using the centroid concept. Some of the properties of the ranking function have been studied. Also, comparative examples are given to show the effectiveness of the proposed method.

Intuitionistic fuzzy multi-attribute group decision-making with an application to plant location selection based on a new extended VIKOR method

This paper presents a new decision method for multi-attribute group decision-making (MAGDM) problems in general and plant location selection (PLS) problem in particular, with intuitionistic fuzzy information captured through trapezoidal intuitionistic fuzzy numbers (TrIFNs). We assume that the weights of the decision-makers and attributes are completely unknown. The ratings of alternatives with respect to each attribute are considered as linguistic terms, which are mapped to the appropriate TrIFNs. To reduce subjective randomness in the decision-process, we determine attribute weights using the Shannon entropy theory, and weights of the decision-makers by integrating the Evidence theory with Bayes approximation. Furthermore, we extend the classical VIKOR method to solve MAGDM problems under intuitionistic fuzzy environment based on the TrIFNs. Considering that the PLS problem is essentially a MAGDM problem that involves evaluation of the alternatives on several conflicting attributes based on the vague and imprecise assessments of the decision-makers, we demonstrate utility of the proposed decision method by applying it solve the PLS problem. A detailed comparison is presented to demonstrate the advantages of the proposed methodology over the existing methods used for both the intuitionistic fuzzy MAGDM problems and PLS problem.

Ranking of incomplete trapezoidal information

Any information system or decision model which consists of combinations of quantitative, qualitative, imprecise and incomplete informations can be modelled better using trapezoidal intuitionistic fuzzy numbers (TrIFNs) than interval valued intuitionistic fuzzy numbers. Ranking of TrIFNs plays an important role in intuitionistic fuzzy decision-making or intuitionistic fuzzy information system. In this paper, a new method for ranking of TrIFNs using membership, non-membership, vague and precise score functions which generalises the membership, non-membership, vague and precise score functions defined in Geetha et al. (Expert Syst Appl 41:1947–1954, 2014) is proposed and a new algorithm for solving information system problem with incomplete information is introduced. Further, the significance of our proposed method over the existing methods is studied by an illustrative example.

Robust decision making using intuitionistic fuzzy numbers

Robustness is defined as a system’s ability to withstand under disturbances. In real-life applications, where problem parameters are often uncertain, incorporating robustness in decision making is important. In this study, we propose a robust decision making (RDM) approach using intuitionistic trapezoidal fuzzy number (ITrFN). Fuzzy linguistic quantifier (FLQ) is used in the proposed approach to compute the uncertain optimism degree of the decision maker. Initially, decision maker expresses his/her opinion using linguistic terms, which are presented numerically using ITrFNs. The aggregated ITrFN for each of the alternatives is evaluated using intuitionistic trapezoidal fuzzy ordered weighted averaging operator (ITrFOWA). Then, we find out the expected value and variance of the aggregated ITrFN for each alternative, which are subsequently used for robust decision making. A collective measure of these two values of each alternative is considered to find an interval of the corresponding alternative, known as optimal interval. Alternative with maximum optimal interval is selected as the robust solution. Applicability of the proposed approach has been demonstrated on a site selection problem of nuclear power plant. Site selection for installing a nuclear power plant has become a crucial problem throughout the world, especially after the Fukushima (2011) and Chernobyl (1986) nuclear disasters.

Multiple Attribute Group Decision Analysis for Intuitionistic Triangular and Trapezoidal Fuzzy Numbers

Solving Multiple Attribute Group Decision Making (MAGDM) problems has become one of the most important researches in recent days. In situations where the information or the data is of the form of an Intuitionistic Triangular Fuzzy Number (ITrFN) or Intuitionistic Trapezoidal Fuzzy Number (ITzFN), a new distance function is defined for ranking the alternatives in the decision making process. After processing the decision information through a sequence of arithmetic aggregation operators, namely, the Intuitionistic Triangular Fuzzy Weighted Arithmetic Averaging (ITrFWAA), Intuitionistic Triangular Fuzzy Ordered Weighted Averaging (ITrFOWA) operator and the Intuitionistic Triangular Fuzzy Hybrid Aggregation (ITrFHA) operator, the proposed distance function is utilized to rank the best alternative. A model is proposed to solve MAGDM problems using the developed distance formula defined for ITrFNs. Numerical illustration is provided and comparisons are made with some of the existing MAGDM models and ranking procedures.

MULTI-CRITERIA DECISION-MAKING METHOD BASED ON INTUITIONISTIC TRAPEZOIDAL FUZZY PRIORITISED OWA OPERATOR

In the real decision-making, there are many multiple attribute decision-making (MADM) problems, in which there exists the prioritised relationship among decision-making attributes. In this paper, with respect to the prioritised multi-criteria decision-making problems under intuitionistic trapezoidal fuzzy information, a new decision-making method on the basis of the intuitionistic trapezoidal fuzzy prioritised ordered weighted aggregation operator has been proposed. Firstly, the definitions, operational rules and characteristics of intuitionistic trapezoidal fuzzy numbers and POWA operator have been introduced. Then, intuitionistic trapezoidal fuzzy prioritised ordered weighted aggregation (ITFPOWA) operator has been defined as well as the computational method of associated weight, and some properties have been studied and proved. Furthermore, based on the ITFPOWA operator, an approach to the multi-criteria decision-making with intuitionistic trapezoidal fuzzy numbers has been established. Finally, an illustrative example has been given to prove the evaluation procedures of the developed approach and to demonstrate its practicality and validity.

A linear ordering on the class of Trapezoidal intuitionistic fuzzy numbers

Fuzzy numbers and intuitionistic fuzzy numbers are introduced in the literature to model problems involving incomplete and imprecise information in expert and intelligent systems. Ranking of TrIFNs plays an important role in an information system (Decision Making) with imprecise and inadequate information and the complete ranking on the class of trapezoidal intuitionistic fuzzy number is an open problem worldwide. Researchers from all over the world have been working in ranking of intuitionistic fuzzy numbers since 1985, but till date there is no common methodology that ranks any two arbitrary intuitionistic fuzzy numbers due to the partial ordering of TraIFNs. Different algorithms are available in the literature for solving intuitionistic fuzzy decision (or information system) problem, but each and every algorithm failed to give better result in some places due to the ranking procedure of TrIFNs. Intuitionistic fuzzy decision algorithm works better when it have a complete ranking procedure that ranks arbitrary intuitionistic fuzzy numbers. In this paper a linear (total) ordering on the class of trapezoidal intuitionistic fuzzy numbers using axiomatic set of eight different scores is introduced. The main idea of this paper is to classify and study the properties of eight different sub classes of the set of TrIFNs. Further new total order relations are defined on each of the subclasses of TrIFNs and they are extended to a complete ranking procedure on the set of TrIFNs. Finally the significance of the proposed method over existing methods is studied by illustrative examples.

A complete ranking of incomplete trapezoidal information

The intuitionistic fuzzy sets (IFSs) introduced by Atanassov are widely applied in all areas such as data analysis, artificial intelligence, decision support systems in modelling problems with incomplete and imprecise information due to their better accuracy. More precisely, trapezoidal intuitionis tic fuzzy numbers (TraIFNs) are able to model incomplete and imprecise information based on qualitative in nature. Many researchers have proposed different ranking methods on TraIFNs, but none of them has covered the entire class of TraIFNs and also almost all the methods have disadvantage that they ranked two different IFNs as the same at some point of time. In this paper, a complete ranking on the class of TraIFNs using axiomatic set of total ordering on some special subclasses of TraIFN based on different score functions and an algorithm for making decisions from information system with incomplete trapezoidal information using proposed ordering, have been studied and also the significance of our proposed method over existing methods is shown by comparing the proposed method with existing methods.

Intuitionistic fuzzy transportation problem with various kinds of uncertainties in parameters and variables

In real-life decisions usually we have to suffer through different states of uncertainties. In this article, we formulate a transportation problem in which costs, supplies and demands all are different types of real, fuzzy or intuitionistic fuzzy numbers that is the data has different types of uncertainties. We propose a ranking procedure for intuitionistic fuzzy numbers. Using the proposed ranking function intuitionistic fuzzy methods are proposed to find starting basic feasible solution in terms of trapezoidal intuitionistic fuzzy numbers. Intuitionistic fuzzy modified distribution method is proposed to find optimal solution. We illustrate the methodology by numerical examples.

A Centroid-based Ranking Method of Trapezoidal Intuitionistic Fuzzy Numbers and Its Application to MCDM Problems

The objective of this paper is to introduce a novel method to compare trapezoidal intuitionistic fuzzy numbers (TrIFNs). Till now little research has been done regarding the ranking of TrIFNs. This paper first reviews the existing ranking methods and shows their drawbacks by using several examples. In order to overcome the drawbacks of the existing methods, a new ranking method of TrIFNs is developed by utilizing the concept of centroid point. For this purpose, centroid point for TrIFN is also defined. The rationality validation of the proposed centroid formulae is proved. Further, the ranking method is applied to a multi-criteria decision making (MCDM) problem in which the ratings of the alternatives on criteria are expressed with TrIFNs. Finally, the effectiveness and applicability of the proposed ranking method are illustrated with an aerospace research organization center selection example. This article has also justified the proposed approach by analyzing a comparative study.

Supplier risk assessment based on trapezoidal intuitionistic fuzzy numbers and ELECTRE TRI-C: a case illustration involving service suppliers

Many companies today have embraced the concept of risk management, usually in the form of enterprise risk management or supply chain risk management. Both are based on a holistic view of risks. Hence, risks related to specific functions within a company must be considered more broadly than previously. Risks, however, involve uncertainty, and the less specific the context in which risks are viewed, the more uncertainty will be involved. One particular way to express uncertainty is through trapezoidal intuitionistic fuzzy numbers (TrIFNs). In this paper, risks that are relevant for supplier risk assessments are first collected from the literature. Then it is illustrated how the multi-criteria decision analysis method ELECTRE TRI-C can be used for sorting suppliers into risk categories, when the risks as well as some of the method’s parameters are expressed with TrIFNs. In order to do this, we make use of a small modification of an existing method for converting TrIFNs into crisp values. The approach is illustrated in a case problem based on a company that is looking for service providers (suppliers) of electrical maintenance. The problem involves 20 suppliers that are sorted into three risk categories based on evaluations from 27 criteria. Results from the case study point to two low risk suppliers. A further ad-hoc analysis suggests one of these to be less risky than the other.

English

In this paper the Generalized trapezoidal intuitionistic fuzzy numbers and ranking method has been adopted and for finding an optimal solutionfor travelling salesman problem (TSP) Hungarian method is used. This method requires least iteration to obtain the optimality.The method is illustrated by a numerical example.

Iterative Goal Programming Approach for Solving Multi-objective Intuitionistic Fuzzy Linear Programming Problem

This paper presents an Iterative goal programming approach for solving Multi-objective Intuitionistic Fuzzy Linear Programming Problem (MOIFLPP). The MOIFLP problem involving aspiration level and the right hand side constraints are trapezoidal intuitionistic fuzzy number (TIFN). The concept of ∝-cut of IFN is used to convert the TIFN into crisp one. A solution algorithm is proposed to solve the MOIFLPP and a numerical example is given in detail to explain the solution procedure. Furthermore, the proposed method is easy to apply in real-life situations which give more accurate solution in the sense that the objective values are sufficiently closer to the aspiration levels. Finally, a real life example is used and a compromise solution is found out to satisfy the decision maker.

An ELECTRE Approach for Multicriteria Interval-Valued Intuitionistic Trapezoidal Fuzzy Group Decision Making Problems

The Multiple Criteria Decision Making (MCDM) is acknowledged as the most useful branch of decision making. It provides an effective framework for comparison based on the evaluation of multiple conflicting criteria. In this paper, a method is proposed to work out multiple attribute group decision making (MAGDM) problems with interval-valued intuitionistic trapezoidal fuzzy numbers (IVITFNs) using Elimination and Choice Translation Reality (ELECTRE) method. A new ranking function based on value and ambiguity is introduced to compare the IVITFNs, which overcomes the limitations of existing methods. An illustrative numerical example is solved to verify the efficiency of the proposed method to select the better alternative.