The advancements in next-generation sequencing have made it possible to effectively detect somatic mutations, which has led to the development of personalized neoantigen cancer vaccines that are tailored to the unique variants found in a patient's cancer. These vaccines can provide significant clinical benefit by leveraging the patient's immune response to eliminate malignant cells. However, determining the optimal vaccine dose for each patient is a challenge due to the heterogeneity of tumors. To address this challenge, we formulate a mathematical dose optimization problem based on a previous mathematical model that encompasses the immune response cascade produced by the vaccine in a patient. We propose an optimization approach to identify the optimal personalized vaccine doses, considering a fixed vaccination schedule, while simultaneously minimizing the overall number of tumor and activated T cells. To validate our approach, we perform in silico experiments on six real-world clinical trial patients with advanced melanoma. We compare the results of applying an optimal vaccine dose to those of a suboptimal dose (the dose used in the clinical trial and its deviations). Our simulations reveal that an optimal vaccine regimen of higher initial doses and lower final doses may lead to a reduction in tumor size for certain patients. Our mathematical dose optimization offers a promising approach to determining an optimal vaccine dose for each patient and improving clinical outcomes.