In this Chapter, we imagine a decision maker (or a group of experts) trying to establish or examine fair procedures to combine opinions about alternatives related to different points of view.

Multiple criteria decision making

Recently, a new model based on Pythagorean fuzzy set PFS has been presented to manage the uncertainty in real-world decision-making problems. PFS has much stronger ability than intuitionistic fuzzy set to model such uncertainty. In this paper, we define some novel operational laws of PFSs and discuss their desirable properties. For the multicriteria decision-making problems with PFSs, we propose an extended technique for order preference by similarity to ideal solution method to deal effectively with them. In this approach, we first propose a score function based comparison method to identify the Pythagorean fuzzy positive ideal solution and the Pythagorean fuzzy negative ideal solution. Then, we define a distance measure to calculate the distances between each alternative and the Pythagorean fuzzy positive ideal solution as well as the Pythagorean fuzzy negative ideal solution, respectively. Afterward, a revised closeness is introduced to identify the optimal alternative. At length, a practical example is given to illustrate the developed method and to make a comparative analysis.

Extension of TOPSIS to Multiple Criteria Decision Making with Pythagorean Fuzzy Sets

We note that orthopair fuzzy subsets are such that that their membership grades are pairs of values, from the unit interval, one indicating the degree of support for membership in the fuzzy set and the other support against membership. We discuss two examples, Atanassov's classic intuitionistic sets and a second kind of intuitionistic set called Pythagorean. We note that for classic intuitionistic sets the sum of the support for and against is bounded by one, while for the second kind, Pythagorean, the sum of the squares of the support for and against is bounded by one. Here we introduce a general class of these sets called q-rung orthopair fuzzy sets in which the sum of the ${\rm{q}}$ th power of the support for and the ${\rm{q}}$ th power of the support against is bonded by one. We note that as q increases the space of acceptable orthopairs increases and thus gives the user more freedom in expressing their belief about membership grade. We investigate various set operations as well as aggregation operations involving these types of sets.

Generalized Orthopair Fuzzy Sets

Two decades was systematically reviewed on fuzzy MCDM techniques from 1994 to 2014.The database for review was 403 papers from more than 150 high-ranking journals.403 scholarly papers were grouped in four different main fields.Papers were classified based on utilizing, developing and proposing research papers. MCDM is considered as a complex decision-making tool involving both quantitative and qualitative factors. In recent years, several fuzzy FMCDM tools have been suggested to choosing the optimal probably options. The purpose of this paper is to review systematically the applications and methodologies of the fuzzy multi decision-making (FMCDM) techniques. This study reviewed a total of 403 papers published from 1994 to 2014 in more than 150 peer reviewed journals (extracted from online databases such as ScienceDirect, Springer, Emerald, Wiley, ProQuest, and Taylor & Francis). According to experts' opinions, these papers were grouped into four main fields: engineering, management and business, science, and technology. Furthermore, these papers were categorized based on authors, publication date, country of origin, methods, tools, and type of research (FMCDM utilizing research, FMCDM developing research, and FMCDM proposing research). The results of this study indicated that, in 2013, scholars have published papers more than other years. In addition, hybrid fuzzy MCDM in the integrated method and fuzzy AHP in the individual section were ranked as the first and second methods in use. Additionally, Taiwan was ranked as the first country that contributed to this survey, and engineering was ranked as the first field that has applied fuzzy DM tools and techniques.

Fuzzy multiple criteria decision-making techniques and applications - Two decades review from 1994 to 2014

Pythagorean fuzzy sets PFSs, originally proposed by Yager Yager, Abbasov. Int J Intell Syst 2013;28:436-452, are a new tool to deal with vagueness considering the membership grades are pairs µ,i¾? satisfying the condition µ2+i¾?2i¾?1. As a generalized set, PFSs have close relationship with intuitionistic fuzzy sets IFSs. PFSs can be reduced to IFSs satisfying the condition µ+i¾?i¾?1. However, the related operations of PFSs do not take different conditions into consideration. To better understand PFSs, we propose two operations: division and subtraction, and discuss their properties in detail. Then, based on Pythagorean fuzzy aggregation operators, their properties such as boundedness, idempotency, and monotonicity are investigated. Later, we develop a Pythagorean fuzzy superiority and inferiority ranking method to solve uncertainty multiple attribute group decision making problem. Finally, an illustrative example for evaluating the Internet stocks performance is given to verify the developed approach and to demonstrate its practicality and effectiveness.

Some Results for Pythagorean Fuzzy Sets

Hesitant fuzzy set (HFS), which allows the membership degree of an element to a set represented by several possible values, is considered as a powerful tool to express uncertain information in the process of multi-attribute decision making (MADM) problems. In this paper, we develop a novel approach based on TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) and the maximizing deviation method for solving MADM problems, in which the evaluation information provided by the decision maker is expressed in hesitant fuzzy elements and the information about attribute weights is incomplete. There are two key issues being addressed in this approach. The first one is to establish an optimization model based on the maximizing deviation method, which can be used to determine the attribute weights. According to the idea of the TOPSIS of Hwang and Yoon [1], the second one is to calculate the relative closeness coefficient of each alternative to the hesitant positive-ideal solution, based on which the considered alternatives are ranked and then the most desirable one is selected. An energy policy selection problem is used to illustrate the detailed implementation process of the proposed approach, and demonstrate its validity and applicability. Finally, the extended results in interval-valued hesitant fuzzy situations are also pointed out.

Hesitant fuzzy multi-attribute decision making based on TOPSIS with incomplete weight information

The Pythagorean fuzzy set, as a new extension of intuitionistic fuzzy set, has recently been developed to manage the complex uncertainty in practical group decision problems. The purpose of this article is to develop a new decision method based on similarity measure to address multiple criteria group decision making problems within Pythagorean fuzzy environment based on Pythagorean fuzzy numbers PFNs. The contribution of this article is fivefold: 1 An accuracy function of PFNs is defined and a new ranking method for PFNs is proposed; 2 new Pythagorean fuzzy aggregating operators are developed; 3 a novel similarity measure for PFNs is presented, and some desirable properties are discussed; 4 a simple and effective Pythagorean fuzzy group decision method is introduced; and 5 The proposed method is applied to address the selection problem of photovoltaic cells.

A Novel Approach Based on Similarity Measure for Pythagorean Fuzzy Multiple Criteria Group Decision Making

Hesitant fuzzy set (HFS) is used to deal with the situations in which the decision makers hesitate among several values to assess an indicator, alternative, variable, etc. Recently, the multi-criteria decision making (MCDM) problems with hesitant fuzzy information have received increasing attentions and many corresponding MCDM methods have also been developed, but none of them can be used to solve the MCDM problems in case of considering the decision maker's psychological behavior. In this study, we extend the TODIM (an acronym in Portuguese of interactive and multi-criteria decision making) method, which is based on prospect theory and can effectively capture the decision maker's psychological behavior, to solve this type of problems under hesitant fuzzy environment. Firstly, we develop two novel measured functions for comparing the magnitude of hesitant fuzzy elements and interval-valued hesitant fuzzy elements, which are more reasonable and effective compared with the existing measured functions. Then, we calculate the dominance degree of each alternative related to the others based on novel measured functions and distance measures. By aggregating these dominance degrees, we can further obtain the overall value of each alternative and whereby rank the alternatives. Finally, a decision making problem that concerns the evaluation and ranking of the service quality among domestic airlines is used to illustrate the validity and applicability of the proposed method.

The TODIM analysis approach based on novel measured functions under hesitant fuzzy environment

We define the consensus index from the perspective of the ranking of decision information and derive the experts' weights on the basis of the idea of the maximizing consensus.We extend the TOPSIS approach to rank all the alternatives from the perspective of the magnitude of decision information under interval-valued intuitionistic fuzzy environment.Combining with the multi-choice goal programming models, our approach can not only find the optimal alternative(s) but also determine their corresponding optimum quantities. Multi-attribute group decision making (MAGDM) is an important research topic in decision theory. In recent decades, many useful methods have been proposed to solve various MAGDM problems, but very few methods simultaneously take them into account from the perspectives of both the ranking and the magnitude of decision data, especially for the interval-valued intuitionistic fuzzy decision data. The purpose of this paper is to develop a soft computing technique based on maximizing consensus and fuzzy TOPSIS in order to solve interval-valued intuitionistic fuzzy MAGDM problems from such two aspects of decision data. To this end, we first define a consensus index from the perspective of the ranking of decision data, for measuring the degree of consensus between the individual and the group. Then, we establish an optimal model based on maximizing consensus to determine the weights of experts. Following the idea of TOPSIS, we calculate the closeness indices of the alternatives from the perspective of the magnitude of decision data. To identify the optimal alternatives and determine their optimum quantities, we further construct a multi-choice goal programming model based on the derived closeness indices. Finally, an example is given to verify the developed method and to make a comparative analysis.

Xiaolu Zhang

Papers

Extension of TOPSIS to Multiple Criteria Decision Making with Pythagorean Fuzzy Sets

Hesitant fuzzy multi-attribute decision making based on TOPSIS with incomplete weight information

A Novel Approach Based on Similarity Measure for Pythagorean Fuzzy Multiple Criteria Group Decision Making

The TODIM analysis approach based on novel measured functions under hesitant fuzzy environment

Soft computing based on maximizing consensus and fuzzy TOPSIS approach to interval-valued intuitionistic fuzzy group decision making