$($a$)$ Solve the following goal programming model graphically.

Minimize $,P_1{d}_1^{+}+P_2{d}_2^{-}+P_3{d}_3^{+}+P_3{d}_4^{-}$

Subject to $4x_1+8x_2+d_1^{-}-d_1^{+}=45$

$8x_1+24x_2+d_2^{-}-d_2^{+}=100$

$x_1+2x_2+d_3^{-}-d_3^{+}=10$

$x_1+d_4^{-}-d_4^{+}=6$

$($b$)$ Repeat the above problem with the objective to minimize $P_1{d}_1^{-}+P_2{d}_2^{-}$