Given $(A_{ij})_{1\leq i\leq 4,5\leq j\leq n}$. My question is how to compute the following Max-Min type formula, \begin{equation} \begin{aligned} \max_{B_1+B_2+B_3+B_4=1,B_i\in\{0,1\}}\min_{x_5,\cdots,x_n\in[0,\alpha]\cup[-\pi,\alpha-\pi]}&B_1\sum_{k=5}^n-A_{1k}\sin x_k+A_{2k}\sin(x_k-\alpha)\\ +&B_2\sum_{k=5}^nA_{3k}\sin x_k-A_{4k}\sin(x_k-\alpha)\\ +&B_3\sum_{k=5}^nA_{3k}\sin x_k+A_{2k}\sin(x_k-\alpha)\\ +&B_4\sum_{k=5}^n-A_{1k}\sin x_k-A_{4k}\sin(x_k-\alpha)\\ \end{aligned} \end{equation}
My thought is, first we evaluate the Min term. But I am not sure, whether I should take the feasible region of $B_i$ into account when I am computeting the Min? If yes, here is what I obtained. But I am very unsure about the correctiveness.
\begin{equation} \begin{aligned} =\max\Big\{&(N_{+}^{12}+ N_{-}^{12}) (-2\sin\frac{\alpha}{2})+(N_{+-}^{12}+N_{-+}^{12}) (-\sin\alpha),(N_{+}^{34}+ N_{-}^{34}) (-2\sin\frac{\alpha}{2})+(N_{+-}^{34}+N_{-+}^{34}) (-\sin\alpha),\\ +& (N_{+}^{32}+ N_{-}^{32}) (-\sin\alpha)+(N_{+-}^{32} +N_{-+}^{32}) (-2\sin\frac{\alpha}{2}),(N_{+}^{14} + N_{-}^{14}) (-\sin\alpha)+(N_{+-}^{14} +N_{-+}^{14}) (-2\sin\frac{\alpha}{2})\Big\}\\ \end{aligned} \end{equation}