A novel optimistic multi-granulation roughness of pythagorean fuzzy set for medical decision analysis via soft relations over dual universes

Practical decision making particularly in selecting a course of medical treatment is complex and uncertain and should have advanced mathematical tools that are easier and interpretable to describe imprecision. Heart attack as one of the major causes of death is a major challenge in terms of choosing the best treatment options as a result of medical decision making complexity and uncertainty. To establish the basis of an optimistic multi-granulation rough Pythagorean fuzzy soft set model, this paper presents a comprehensive and integrated architecture of combining optimistic multi-granulation rough sets with Pythagorean fuzzy sets using soft binary relations. Here, we applied afterset and foreset techniques to add new lower approximation and upper approximation operators that are formulated under soft relations on dual universes. These operators generate two distinct Pythagorean fuzzy soft sets to provide a higher accuracy of approximation and flexibility in context as they represent optimistic multi-source information on two universes. To test the proposed mechanism in a practical way, one of the decision making models is introduced in the given framework. There are two algorithms, the formulations of which have been presented systematically. The usefulness of the framework has been shown using a case study on the choice of treatments of heart attack where treatments in the sigma and delta groups are compared using the criteria on treatment efficacy, quality of the hospital, and patient satisfaction. The relevance and usefulness of the given technique is also proved by a critical comparison with the current methods.