We consider the problem of designing an efficient system for allocating donated livers to patients waiting for transplantation. The trade-off between medical urgency and efficiency is at the heart of the liver allocation problem. We model the transplant waiting list as a multiclass fluid model of overloaded queues, which captures the disease evolution by allowing the patients to switch between classes, i.e., health levels. We consider the bicriteria objective of minimizing total number of patient deaths while waiting for transplantation (NPDWT) and maximizing total quality-adjusted life years (QALYs) through a weighted combination. On one hand, under the objective of minimizing NPDWT, the current policy of United Network for Organ Sharing (UNOS) emerges as the optimal policy, providing a theoretical justification for the current practice. On the other hand, under the metric of maximizing QALYs, the optimal policy is an intuitive dynamic index policy that ranks patients based on their marginal benefit from transplantation, i.e., the difference in benefit with versus without transplantation. Finally, we perform a detailed simulation study to compare the performances of our proposed policies and the current UNOS policy along the following metrics: total QALYs, NPDWT, number of patient deaths after transplantation, number of total patient deaths, and number of wasted livers. Numerical experiments show that our proposed policy for maximizing QALYs outperforms the current UNOS policy along all metrics except the NPDWT.