Hybrid Decision-Making Frameworks under Complex Spherical Fuzzy <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <mi>N</mi> </math>-Soft Sets

This paper presents the novel concept of complex spherical fuzzy N -soft set ( id="M3"> C S F f ) which is capable handling two-dimensional vague information with parameterized ranking systems. First, we propose basic notions for a theoretical development id="M4"> s , including functions, comparison rule, and fundamental operations (complement, union, intersection, sum, product). Furthermore, look into some properties id="M5"> . We then produce three algorithms multiattribute decision-making that take advantage these elements. demonstrate their applicability assistance numerical problem (selection best third-party app year). A performance Pythagorean id="M6"> sets speaks superiority our approach. Moreover, an aim to expand range techniques group problems, design id="M7"> -TOPSIS method. use id="M8"> weighted average operator in order aggregate decisions all experts according power attributes features alternatives. present normalized-Euclidean distances (from alternatives both id="M9"> positive negative ideal solutions, respectively) revised closeness index feasible alternative. As illustration, mathematical model selection physiotherapist doctor Mayo hospital, Lahore. conduct existing TOPSIS method confirms stability proposed reliability its results.