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This paper used the integer linear programming algorithm to solve the optimal phasor measurement unit (PMU) placement problem for full system observability. As the integer linear programming method leads to various optimal solutions, the multiple solution sets are generated, and a method is suggested to find the best set of PMU placement among the multiple solution sets for assessing the dynamic stability. In this method, two indices are used to assess dynamic stability. One is based on the real part of dominant eigenvalues of the system and the other one is based on MSE (Mean Squared Error) of system response. Different PMU placements with full observability property are ranked based on these indices. The suggested method is tested on the IEEE 39-bus test system and the results are shown in this paper.