In this paper, we define the new notion of Interval-valued Pythagorean fuzzy ideals in semigroups and established the properties of its with suitable examples. Also, we introduce the concept of Interval valued Pythagorean fuzzy sub-semigroup, Interval valued Pythagorean fuzzy left (resp. right) ideal, Interval valued Pythagorean fuzzy bi-ideal, Interval valued Pythagorean fuzzy interior ideal and homomorphism of an Interval valued Pythagorean fuzzy ideal in semigroups with suitable illustration. We show that every Interval valued Pythagorean fuzzy left (resp. right) ideal is an Interval valued Pythagorean fuzzy bi-ideal.

Interval-valued Pythagorean fuzzy set (IVPFS), originally proposed by Peng and Yang, is a novel tool to deal with vagueness and incertitude. As a generalized set, IVPFS has close relationship with Interval-valued intuitionistic fuzzy set (IVIFS). IVPFS can be reduced to IVIFS satisfying the condition $\mu^++\nu^+ \leq 1$. However, the related operations of IVPFS do not take different conditions into consideration. In this paper, we initiate some new Interval-valued Pythagorean fuzzy operators ($\diamondsuit, \Box, \spadesuit, \clubsuit, \maltese, \rightarrow, \$ $) and discuss their properties in relation with some existing operators $(\cup, \cap, \oplus, \otimes)$ in detail. It will promote the development of Interval-valued Pythagorean fuzzy operators. Later, we propose an algorithm to deal with multi-attribute decision making (MADM) problem based on proposed $\spadesuit$ operator. Finally, the effectiveness and feasibility of approach is demonstrated by mine emergency decision making example.

In this paper, an Interval-valued Pythagorean Triangular fuzzy number (IVPTFN) as a useful tool to handle decision-making problems with vague quantities is defined. Then, their operational laws are developed. By introducing a novel method of making a decision on the concept of possibility theory, a multi-attribute group decision-making (MAGDM) problem is considered, in which the attribute values are expressed with the IVPTFN and the information on the decision makers’ (DM) weights is completely unknown. Two novel forms of a multi-attributive border approximation area comparison (MABAC) technique are proposed to solve the problem. One of them is applied to compute the weights of the decision makers, and the other is used to rank the preference order of alternatives, that is based on the possibility expected value and standard deviation and has no aggregation of information. Finally, to illustrate the practicality and effectiveness of proposed method in real-world problems, the proposed method is applied in a real case study of an Iranian transport complex to sustainability assessment of its transport projects.

In this paper, we first use a fuzzy preference relation with a membership function representing preference degree for comparing two Interval-valued fuzzy numbers and then utilize a relative preference relation improved from the fuzzy preference relation to rank a set of Interval-valued fuzzy numbers. Since the fuzzy preference relation is a total ordering relation that satisfies reciprocal and transitive laws on Interval-valued fuzzy numbers, the relative preference relation is also a total ordering relation. Practically, the fuzzy preference relation is more reasonable on ranking Interval-valued fuzzy numbers than defuzzification because defuzzification does not present preference degree between fuzzy numbers and loses messages. However, fuzzy pair-wise comparison for the fuzzy preference relation is more complex and difficult than defuzzification. To resolve fuzzy pair-wise comparison tie, the relative preference relation takes the strengths of defuzzification and the fuzzy preference relation into consideration. The relative preference relation expresses preference degrees of Interval-valued fuzzy numbers over average as the fuzzy preference relation does, and ranks fuzzy numbers by relative crisp values as defuzzification does. In fact, the application of relative preference relation was shown in traditional fuzzy numbers, such as Triangular and trapezoidal fuzzy numbers, for previous approaches. In this paper, we extend and utilize the relative preference relation on Interval-valued fuzzy numbers, especially for Triangular Interval-valued fuzzy numbers. Obviously, Interval-valued fuzzy numbers based on the relative preference relation are easily and quickly ranked, and able to reserve fuzzy information.

In the present paper, a new accuracy function is provided to overcome the limitations of the existing score/accuracy functions for Interval-valued Pythagorean Fuzzy Sets (PFSs). The proposed accuracy function is validated and discussed in detail through illustrative examples. Furthermore, a new distance measure for Interval-valued Pythagorean fuzzy numbers is proposed and used in terms of the existing weighted averaging operators. Finally, with regard to the proposed accuracy function, distance measure, and weighted averaging operators, a numerical example of Multi-Criteria Decision Making (MCDM) process is presented to validate the methodology.

Interval-valued Pythagorean fuzzy sets (IVPFSs) as enhanced type of Pythagorean fuzzy sets (PFSs) improve the expression of the degrees of membership, non-membership and hesitancy in comparison with intuitionistic fuzzy sets (IFSs). In this paper, to use the advantages of IVPFSs a new group decision-making method is introduced based on linear assignment method (LAM). In this approach, subjective and objective weights of criteria are taken into account. Moreover, the method applies a new ranking method for IVPFSs. To avoid the shortcomings of first aggregation methods, the introduced decision-making approach involves a last aggregation approach. Finally, the method is used in a case study of sustainable project evaluation in order to depict the applicability of this method.

In this paper, we expose cosine, jaccard and dice similarity measures and rough Interval Pythagorean mean operator. Some of the important properties of the defined similarity measures have been established. Then the proposed methods are applied for solving multi attribute decision making problems. Finally, a numerical example is solved to show the feasibility, applicability and effectiveness of the proposed strategies.