On Pythagorean hesitant fuzzy sets and their application to group decision making problems

Abstract

The idea of multi-attribute decision making (MADM) has extensively been studied in the real life decision making and is of disquiet to researchers and proprietors. The leading purpose is to make accessible a complete solution by guessing and ranking alternatives based on differing attributes with respect to decision makers (DMs) choices and has broadly been used in engineering, economics, medical diagnoses and management. Acquiring sufficient and accurate data for practical decision making is difficult because of the high complexity of socioeconomics. In response to this issue, Zadeh [63] introduced the concept of fuzzy set, which has been widely used in many fields in our modern society. However, the fuzzy set theory is discreetly different with some limitations while decision makers suggest to deal with some ambiguous information convinced from numerous causes of vagueness, the attributes elaborate in decision making problems are not constantly specified in real numbers and roughly are enhanced, suitable to be represented by fuzzy values, for instance interval values (IVs) [8], linguistic variables (LVs) [19], intuitionistic fuzzy values (IFVs) [4, 5] and hesitant fuzzy elements (HFEs) [44, 45] Dual hesitant fuzzy elements (DHFEs) [6969], Pythagorean fuzzy values (PFVs) [ 59, 60], just to reference a few. This thesis consist of eight chapters In chapter one, we present some basic definitions such as aggregation operators, fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, Pythagorean hesitant fuzzy sets and intuitionistic hesitant fuzzy sets. The study of this chapter is the building block for our further study. In chapter two, we introduce the concept of Pythagorean hesitant fuzzy set (PHFS) as a generalization of intuitionistic hesitant fuzzy set (IHFS). The PHFS is characterized as a membership degree and a non-membership degree and full fill the condition that the square sum of the membership degrees is less than or equal to one. We define some basic operation for PHFS and discuss some properties of Pythagorean hesitant fuzzy numbers (PHFNs). We define score and deviation degree for the comparison between the PHFNs. We define a distance measure between PHFNs. On the bases of proposed operation we develop Pythagorean hesitant fuzzy aggregation operators namely, Pythagorean hesitant fuzzy weighted averaging (PHFWA) operator, Pythagorean hesitant fuzzy weighted geometric (PHFWG) operator, Pythagorean hesitant fuzzy ordered weighted averaging (PHFOWA) operator and Pythagorean hesitant fuzzy ordered weighted geometric (PHFOWG) operator. We study some properties such as idempotency, monotonicity and boundedness of the developed operators. We develop the maximizing deviation method for solving multi-attribute decision making (MADM) problems, in which the evaluation information provided by the decision makers (DMs) is expressed in Pythagorean hesitant fuzzy numbers and the information about attribute weights is incomplete. Moreover we present a MADM approach based on the developed operators. Furthermore a numerical is given to show the validity and practicality of the proposed method. Finally we compare the proposed approaches with existing methods. In chapter three, we develop generalized aggregation operators for Pythagorean hesitant fuzzy information namely generalized Pythagorean hesitant fuzzy weighted averaging (GPHFWAλ) operator, generalized Pythagorean hesitant fuzzy weighted geometric (GPHFWGλ) operator, generalized Pythagorean hesitant fuzzy ordered weighted averaging (GPHFOWAλ) operator, generalized Pythagorean hesitant fuzzy ordered weighted geometric (GPHFOWGλ) operator. We discuss some relationship between Pythagorean hesitant fuzzy numbers and the develop aggregation operators. Moreover we present a multi-attribute decision making (MADM) approach based on the developed operators. Finally a numerical is given to show the validity and practicality of the proposed method. In chapter four, we develop hybrid aggregation operators for Pythagorean hesitant fuzzy information namely, Pythagorean hesitant fuzzy hybrid weighted averaging (PHFHWA) operator, Pythagorean hesitant fuzzy hybrid weighted geometric (PHFHWG) operator generalized Pythagorean hesitant fuzzy hybrid weighted averaging (GPHFHWAλ) operator and generalized Pythagorean hesitant fuzzy hybrid weighted geometric (GPHFHWGλ) operator. These developed operators can weight both the argument and their ordered positions. Also some numerical examples are given to illustrate the developed operators. Moreover we develop a multiattribute group decision making (MAGDM) approach based on the proposed operators. Finally, we give a numerical example to show the effectiveness and flexibility of the proposed method. In chapter five, we develop Pythagorean hesitant fuzzy Choquet integral averaging (PHFCIA) operator, Pythagorean hesitant fuzzy Choquet integral geometric (PHFCIG) operator, generalized Pythagorean hesitant fuzzy Choquet integral averaging (GPHFCIAλ) operator and generalized Pythagorean hesitant fuzzy Choquet integral geometric (GPHFCIGλ) operator. We also discuss some properties such as idempotency, monotonicity and boundedness of the developed operators. Moreover we apply the developed operators to multi-attribute decision making (MADM) problem to show the validity and effectiveness of the developed operators. Finally a comparison analysis is given. In chapter six, we propose a novel approach based on Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) method and the maximizing deviation method for solving multi-attribute decision making (MADM) problems where the evaluation information provided by the decision maker is expressed in Pythagorean hesitant fuzzy numbers and the information about attribute weights is incomplete. To determine the attribute weight we develop an optimization model based on maximizing deviation method. Finally we provide a practical decision-making problem to demonstrate the implementation process of the proposed method. In chapter seven, we proposed a broad new extension of classical VIKOR method for multi-attribute decision making (MADM) problems with Pythagorean hesitant fuzzy information. Basically VIKOR method of compromise ranking determines a compromise solution, which provides a maximum "group utility" for the "majority" and a minimum of an "individual regret" for the "opponent" and is an effective tool to solve MADM problems. To do this first we give some basic definitions and analogous concepts, and the basic steps of classical VIKOR method are introduced. Different situations of attribute weight information are considered. If attribute weights are partly known a linear programming model is set up based on the idea that reasonable weights should make the relative closeness of each alternative evaluation value to the Pythagorean hesitant fuzzy positive ideal solution as large as possible. If attribute weights are unknown completely, an optimization model is set up based on the maximum deviation method. We describe a MADM problem and present the steps of VIKOR method under the Pythagorean hesitant fuzzy environment. Finally a numerical example is presented to illustrate feasibility and practical advantages of the proposed method. In chapter eight, we extend an acronym in Portuguese for Interactive Multi-Criteria Decision Making (TODIM) method to solve the MADM problems under Pythagorean hesitant fuzzy environment. First we introduce Pythagorean hesitant fuzzy Euclidean distance and then Pythagorean hesitant fuzzy TODIM approach is proposed for MADM problems. To show the effectiveness and applicability of the proposed method an energy policy selection problem has been given.