Generalized picture fuzzy Frank aggregation operators and their applications

Picture fuzzy sets with four-dimensional features are widely used in decision-making as a mathematical tool because they can capture the uncertainty of data. However, the methods and techniques based on matrix theory are difficult to solve the decision problem involving high-dimensional data in a picture fuzzy setting. Therefore, operators that can identify high-dimensional data in a picture fuzzy environment are proposed to address this challenge. In this paper, firstly, by integrating the Frank operators into the picture fuzzy tensor, the generalized picture fuzzy Frank weighted arithmetic (GPFFWA) and generalized picture fuzzy Frank weighted geometric (GPFFWG) operators are defined. Their specific expressions are discussed, and the idempotency, order-preservation, boundedness, and commutativity of the proposed operators are also given. Then, combining the GPFFWA and GPFFWG operators, an algorithm is designed to solve the multi-criteria decision-making problem with high-dimensional data features in the picture fuzzy environment. Finally, a numerical example and related analysis demonstrate the effectiveness, superiority, and flexibility of the suggested technique. This work provides new theoretical and methodological support for developing and practicing the decision-making discipline.