This paper presents a robust location-allocation planning problem for emergency relief in a disaster situation. It is a multi-objective, multi-commodity, multivehicle, and multi-level logistics model that considers injury variety through service prioritizing for injuries. Furthermore, it incorporates unmet demand for particular item types in various damaged areas. Public donation of different relief goods through capacitated medical centers and emergency centers is also addressed with regard to damage type, capacitated relief distribution centers, and disaster management centers. The model is a non-linear mixed-integer programming that simultaneously optimizes three objectives, namely maximizing service fairness to damaged areas, maximizing fair commodity disaster management, and minimizing the total logistics cost. To solve such a hard problem, a Non-dominated Sorting Genetic Algorithm-II (NSGA-II) was developed and the Taguchi method was employed to adjust its parameters. The "-constraint method was used for the evaluation of the performance of the proposed algorithm. For more accurate validation, three comparison metrics including diversification, spacing, and mean ideal distance were adopted. The results verified the effectiveness of the algorithm in a reasonable computational time. Eventually, to examine the applicability of the presented model and the proposed algorithm, a case study was analyzed in an area located in the north of Iran, known for historical earthquake records and aggregated active faults.