Consider a partially balanced fractional $2^{m_1+m_2}$ factorial design derived from a simple partially balanced array such that the general mean, all the $m_1+m_2$ main effects, some linear combinations of the $\binom{m_1}{2}$ two-factor interactions and of the $\binom{m_2}{2}$ two-factor ones and all the $m_1m_2$ two-factor ones are estimable, where the three-factor and higher-order interactions are assumed to be negligible, and $2 \le m_k \ (k=1,2)$. Furthermore we consider the situation in which the number of assemblies is less than the number of non-negligible factorial effects. Under these situations, this paper presents optimal designs with respect to the generalized A-optimality criterion, where $2 \le m_1\le m_2 \le 4$.