The individual fairness k-median problem is a frequently encountered problem in applications involving center location, which generalizes the standard k-median problem by assigning each point a neighborhood radius, allowing connections only to centers within a constant factor of this radius. In this paper, we present a randomized polynomial-time approximation scheme (PTAS) framework with \((2+O(\epsilon ))\) -fairness violation for the individual fairness k-median problem, improving upon the previous best approximation ratio of \((7.081+\epsilon )\) and fairness violation of 3. We propose a new dynamic programming approach to deal with the challenges caused by the individual fairness requirements, which is the crucial step in getting the improved ratio.