The weapon target assignment (WTA) problem is a fundamental problem arising in defense-related applications of operations research, and the multi-stage weapon target assignment (MWTA) problem is the basis of dynamic weapon target assignment (DWTA) problems which commonly exist in practice. The MWTA problem considered in this paper is formulated into a multi-objective constrained combinatorial optimization problem with two competing objectives. Apart from maximizing damage to hostile targets, this paper follows the principle of minimizing ammunition consumption under the consideration of resource constraints, feasibility constraints and fire transfer constraints. In order to tackle the two challenges, two types of multi-objective optimizers: NSGA-II (domination-based) and MOEA/D (decomposition-based) enhanced with an adaptive mechanism are adopted to achieve efficient problem solving. Then a comparison study between adaptive NSGA-II (ANSGA-II) and adaptive MOEA/D (AMOEA/D) on solving instances of three scales MWTA problems is done, and four performance metrics are used to evaluate each algorithm. Numerical results show that ANSGA-II outperforms AMOEA/D on solving multi-objective MWTA problems discussed in this paper, and the adaptive mechanism definitely enhances performances of both algorithms.