A Simplified Approach for Correlation Coefficient Similarity Measure.

In 1991, Gerstenkorn and Manko introduced a correlation coefficient similarity measure for intuitionistic fuzzy sets, aiming to investigate statistical relationships between objectives. However, their proof of the upper bound for the correlation coefficient similarity measure is overly complex and can be simplified. In 1995, Bustince and Burillo developed a similar correlation coefficient similarity measure for interval-valued intuitionistic fuzzy sets, but their proof of the upper bound closely resembles that of Gerstenkorn and Manko. Both proofs utilized the Schwarz inequality, yet failed to fully harness its power as a tool. In this paper, we begin by simplifying the intricate computations presented in the works of Gerstenkorn and Manko (1991) and Bustince and Burillo (1995). Subsequently, we demonstrate that their proofs can be directly derived by applying the Schwarz inequality. For a related transit bus model, we derive a new approximated solution that is very closed to the approximated solution of Wang et al. (2023) with the relative error less than 5%. We also consider an inventory model to show a possible new direction to locate the formulated closed-form optimal solution. [ABSTRACT FROM AUTHOR]