Intuitionistic Fuzzy Set

In this paper, we define the notion of intuitionistic fuzzy characteristic ideal (IFCI) of a Γ-ring which is analogue of a characteristic ideal in the ordinary ring theory and derive various new results. The correlation between the set of all automorphisms of Γ-ring and the corresponding automorphisms of its operator rings have been innovated. Then a one to one correlation between the set of all intuitionistic fuzzy characteristic ideals of Γ-ring and that of its operator ring has been constituted. This is used to obtain a similar bijection for characteristic ideals.

In this article, the crisp, fuzzy and intuitionistic fuzzy optimization problem is formulated. The basic definitions and notations related to optimization problems are given in the preliminaries section. Algorithms for solving the optimization problems using fuzzy and intuitionistic fuzzy set is presented in this article. Then, with the help of the proposed algorithm the optimal solution of the crisp, fuzzy and intuitionistic fuzzy optimization problems are determined. A new theorem related to type-2 fuzzy/type-2 intuitionistic fuzzy optimization problems is proposed and proved. Some new and concrete results related to type-2 fuzzy/type-2 intuitionistic fuzzy optimization problems are presented. To illustrate the proposed method, some real-life numerical examples are presented. The proposed article provides seven fully worked examples with screenshots of output summaries from the software used in the computations for better understanding. The advantages of the proposed approach as compared to other existing work are also specified. Detail analyses of the comparative study as well the discussion are given. To show the advantages of the proposed approach, superiority analysis is discussed. Comparison analysis and the advantages of the proposed operators are also discussed. Some managerial applications and the advantages of the proposed approach are given. Finally, conclusion and future research directions are also given.

When people solve real-life SAP they tend to face the uncertainty state as well as hesitation due to many uncontrollable factors. To deal with uncertainty and hesitation many authors have suggested the intuitionistic fuzzy representation for the data. In this article, the author tried to categorise the SAP under uncertain environment. He formulates the IFSAP and utilizes the TIFN to deal with uncertainty and hesitation. The SAP has uncertainty and hesitation in cost/time/profit/production is known as FIFSAP. The PSK (P. Senthil Kumar) method for finding an optimal solution for FIFAP is extended to solve the FIFSAP and the optimal objective value of FIFSAP is obtained in terms of TIFN. The main advantage of this method is that the optimal solution/assignment of FIFSAP is obtained without using the Hungarian method and intuitionistic fuzzy reduction method. Moreover, the proposed method is computationally very simple and easy to understand. The numerical example is presented to demonstrate computing procedure. The results affirm efficiency of the proposed method.

Analytic hierarchy process (AHP) is a utility theory based decision-making technique, which works on a premise that the decision-making of complex problems can be handled by structuring them into simple and comprehensible hierarchical structures. However, AHP involves human subjective evaluation, which introduces vagueness that necessitates the use of decision-making under uncertainty. The vagueness is commonly handled through fuzzy sets theory, by assigning degree of membership. But, the environmental decision-making problem becomes more involved if there is an uncertainty in assigning the membership function (or degree of belief) to fuzzy pairwise comparisons, which is referred to as ambiguity (non-specificity). In this paper, the concept of intuitionistic fuzzy set is applied to AHP, called IF-AHP to handle both vagueness and ambiguity related uncertainties in the environmental decision-making process. The proposed IF-AHP methodology is demonstrated with an illustrative example to select best drilling fluid (mud) for drilling operations under multiple environmental criteria.

This paper sustains a sound mathematical and computing background. In this paper, the software-based approach for solving intuitionistic fuzzy solid assignment problem (IFSAP) is presented. The IFSAP is formulated and it is solved by using Lingo 17.0 software tool. Theorems related to IFSAP is proved. The IFSAP and its crisp solid assignment problem both are solved at a time and their optimal solution is obtained. In addition, the optimal objective values of both the IFSAP and its crisp solid assignment problem (SAP) are estimated with the help of substituting the optimal solution(s) to their respective decision variables in the objective functions. Some new and important results are proposed. To illustrate the efficiency of the proposed method the numerical example is presented. The reliability of the proposed results are verified by using the numerical example. Strengths and weakness of the paper is mentioned. The novelty of the analysis is given into a coherent, concise, and meaningful manner of analysis. Social issue (real-life problem) is converted into a mathematical model and it is solved by the proposed method. At the end, the advantages of the proposed algorithm is explained.

In real-life decisions, usually we happen to suffer through different states of uncertainties. In order to counter these uncertainties, in this paper, the author formulated a transportation problem in which costs are triangular intuitionistic fuzzy numbers, supplies and demands are crisp numbers. In this paper, a simple method for solving type-2 intuitionistic fuzzy transportation problem (type-2 IFTP) is proposed and optimal solution is obtained without using intuitionistic fuzzy modified distribution method and intuitionistic fuzzy zero point method. So, the proposed method gives the optimal solution directly. The solution procedure is illustrated with the help of three real life numerical examples. Defect of existing results proposed by Singh and Yadav (2016a) is discussed. Validity of Pandian's (2014) method is reviewed. Finally, the comparative study, results and discussion are given.

This paper addresses the fuzzy system reliability analysis using different types of intuitionistic fuzzy numbers. Till now, in the literature, to analyze the fuzzy system reliability, it is assumed that the failure rates of all components of a system follow the same type of fuzzy set or intuitionistic fuzzy set. However, in practical problems, such type of situation rarely occurs. Therefore, in the present paper, a new algorithm has been introduced to construct the membership function and non-membership function of fuzzy reliability of a system having components following different types of intuitionistic fuzzy failure rates. Functions of intuitionistic fuzzy numbers are calculated to construct the membership function and non-membership function of fuzzy reliability via non-linear programming techniques. Using the proposed algorithm, membership functions and non-membership functions of fuzzy reliability of a series system and a parallel systems are constructed. Our study generalizes the various works of the literature. Numerical examples are given to illustrate the proposed algorithm.► We analyze fuzzy system reliability using different types of intuitionistic fuzzy numbers. ► The membership and non-membership functions of fuzzy reliability of a series and a parallel system are constructed. ► Components of the system follow different types of intuitionistic fuzzy failure rate functions. ► Functions of intuitionistic fuzzy numbers are calculated.

Medical decision making can be regarded as a process, combining both analytical cognition and intuition. It involves reasoning within complex causal models of multiple concepts, usually described by uncertain, imprecise, and/or incomplete information. Aiming to model medical decision making, we propose a novel approach based on cognitive maps and intuitionistic fuzzy logic. The new model, called intuitionistic fuzzy cognitive map (iFCM), extends the existing fuzzy cognitive map (FCM) by considering the expert's hesitancy in the determination of the causal relations between the concepts of a domain. Furthermore, a modification in the formulation of the new model makes it even less sensitive than the original model to missing input data. To validate its effectiveness, an iFCM with 34 concepts representing fuzzy, linguistically expressed patient-specific data, symptoms, and multimodal measurements was constructed for pneumonia severity assessment. The results obtained reveal its comparative advantage over the respective FCM model by providing decisions that match better with the ones made by the experts. The generality of the proposed approach suggests its suitability for a variety of medical decision-making tasks.

The notion of intuitionistic fuzzy sets was introduced by Atanassov as a generalization of the notion of fuzzy sets. Y.B. Jun and S.Z. Song introduced the notion of intuitionistic fuzzy points. In this paper we find some relations between the intuitionistic fuzzy ideals of a semigroup S and the set of all intuitionistic fuzzy points of S: