Managing medical inventories, particularly for life-critical pharmaceuticals with high perishability, presents a paradoxical challenge: the necessity of high service levels against the backdrop of profound epistemic uncertainty. Traditional stochastic and basic fuzzy models often fail to capture the multi-layered hesitation inherent in human expert judgment during health crises. This paper proposes a novel mathematical framework for medical inventory management using Pythagorean Hesitant Fuzzy Sets (PHFS). By integrating the expanded membership space of Pythagorean logic with the flexibility of hesitant fuzzy elements, we model demand, deterioration rates, and lead times as complex uncertainty variables. We develop a non-linear programming model aimed at minimizing the total expected fuzzy cost while maximizing a "resilience index.” Theoretical proofs for the existence of an optimal policy in PHFS environments are provided. Numerical simulations based on emergency vaccine distribution scenarios demonstrate that our model significantly outperforms traditional intuitionistic fuzzy models in reducing stock-outs during demand surges.