In this paper, newly de ned level operators and modal-like operators over extensional Generalized intuitionistic fuzzy sets (GIFSB) are proposed. Some of the basic properties of the new operators are discussed.

Based on the point of view of geometrical representation of an intuitionistic fuzzy set, we take into account all three parameters describing intuitionistic fuzzy set, propose a kind of new method to calculate correlation and correlation coefficient of intuitionistic fuzzy sets which is similar to the cosine of the intersectional angle in finite sets and probability space, respectively. Further, we discuss some of their properties and give three numerical examples to illustrate our proposed method reasonable.

In the present paper, the acceptance single sampling plan is developed by the Generalized intuitionistic fuzzy numbers. The $\alpha_1$-cut and $\alpha_2$-cut sets of the Generalized intuitionistic fuzzy numbers are applied to construct the acceptance single sampling plan. We also investigate the operating characteristic curve, where the parameter is considered the Generalized intuitionistic fuzzy number. The $(\alpha_1,\alpha_2)$-cut set of Generalized intuitionistic fuzzy operating characteristics is constructed. The bands are represented with upper and lower bounds instead of curves for operating characteristics and evaluated in detail. Finally, the numerical example is given to illustrate the proposed approach.

The intuitionistic fuzzy sets (IFSs) are generalizations of Zadeh’s fuzzy sets, in which the elements of the universe have apart from Zadeh’s membership and the degree of non-membership in [0, 1]. This paper studies applications of intuitionistic fuzzy sets (FS) to assessment and multi-criteria decision making, which are very useful when uncertainty characterizes the grades or parameters respectively assigned to the elements of the universal set. Applications to everyday life situations are also presented illustrating our results.

In the dynamic world of financial investment, crafting an optimal stock portfolio that judiciously balances risk, return, and efficiency emerges as a critical challenge. Despite the wealth of research on financial portfolio optimization, prevailing methodologies predominantly emphasize either risk minimization or return maximization, often overlooking the imperative for a holistic strategy that simultaneously boosts efficiency and effectiveness. Addressing this gap in the literature, this study introduces an innovative four-objective model that intricately blends risk, return, and efficiency considerations for the strategic selection of stock portfolios. This model ingeniously integrates the foundational principles of Markowitz's mean-variance analysis with the sophisticated network data envelopment analysis (NDEA) techniques, significantly refining the portfolio selection methodology. It further distinguishes itself by incorporating returns represented as trapezoidal intuitionistic fuzzy numbers, adeptly capturing the inherent uncertainties in financial returns. Additionally, the model employs the network data envelopment analysis's cross-efficiency principle, providing a nuanced measure of company performance. To effectively navigate the complexities of this model, we deploy the Non-dominated Sorting Genetic Algorithm II (NSGA-II) and a multi-objective genetic algorithm, demonstrating the model's capability to unearth optimal solutions efficiently. The comparative analysis highlights that the proposed model significantly outperforms the efficiency and effectiveness of existing models, marking a substantial advancement in portfolio optimization strategies.