Remoteness index-based Pythagorean fuzzy VIKOR methods with a generalized distance measure for multiple criteria decision analysis

The aim of this paper is to develop novel VIKOR-based methods for multiple criteria decision analysis involving Pythagorean fuzzy information. The concept of Pythagorean fuzzy sets possesses definite advantages in handling vagueness and complex uncertainty over well-known nonstandard fuzzy sets, such as intuitionistic fuzzy sets and interval-valued fuzzy sets. Taking the powerfulness of Pythagorean fuzzy sets into account when tackling imprecise and ambiguous information in multiple criteria decision-making problems, this paper proposes remoteness index-based Pythagorean fuzzy VIKOR methods, which are significantly different from the existing VIKOR techniques. The uniqueness of the proposed VIKOR methodology is the consideration of uncertain information represented by Pythagorean fuzzy values and the construction of certain valuable concepts of a generalized distance measure and displaced and fixed remoteness indices. A flexible and multipurpose definition of a distance measure for Pythagorean fuzzy information is presented based on the Minkowski distance model. The generalized Pythagorean fuzzy distance measure presented in this paper is then applied to establish useful concepts of remoteness indices, consisting of displaced positive- and negative-ideal remoteness indices as well as fixed positive- and negative-ideal remoteness indices. Unlike the canonical VIKOR ranking procedure, this paper provides a new way to rank candidate alternatives and determine the compromise solution depending on distinct preference structures for adapting to the particularities within the Pythagorean fuzzy environment. Several useful multiple criteria ranking indices, consisting of remoteness-based group utility, individual regret, and compromise indices, are developed to facilitate compromise ranking among alternatives. Four algorithmic procedures of the presented remoteness index-based Pythagorean fuzzy VIKOR methods are also provided in detail. Furthermore, some real-world applications and comparative analyses concerning a criteria satisfaction problem, two service quality and Internet stock performance evaluation problems, and two Internet stock and R&D project investment problems are conducted to demonstrate the effectiveness and advantages of the proposed methods in practice.