Minimize $Z=3{X}_{1}+2{X}_{2}$ Subject to $5{X}_{1}+{X}_{2}\ge 10$ ${X}_{1}+{X}_{2}\ge 6{X}_{1}+4{X}_{2}\ge 12$ ${X}_{1}{X}_{2}\ge 0$ 0.1 $\to$ (90) 0.3 $\to$ (27) 0.5 $\to$ (15) 0.7 $\to$ (10) 0.9 $\to$ (7) 0.2 $\to$ (43) 0.4 $\to$ (20) 0.6 $\to$ (11) 0.8 $\to$ (9) 1.0 $\to$ (2) $Z=13$ ${X}_{1}=1$ ${X}_{2}=5$ Minimize $Z=20{X}_{1}+10{X}_{2}$ Subject to ${X}_{1}+2{X}_{2}\le 40$ $3{X}_{1}+{X}_{2}\ge 30$ $4{X}_{1}+3{X}_{2}\ge 60$ ${X}_{1}{X}_{2}\ge 0$ 0.1 $\to$ (90) 0.3 $\to$ (27) 0.5 $\to$ (15) 0.7 $\to$ (10) 0.9 $\to$ (7) 0.2 $\to$ (43) 0.4 $\to$ (20) 0.6 $\to$ (11) 0.8 $\to$ (9) 1.0 $\to$ (2) $Z=240$ ${X}_{1}=6$ ${X}_{2}=12$ Maximize $Z=16{X}_{1}+17{X}_{2}+10{X}_{3}$ Subject to ${X}_{1}+{X}_{2}+4{X}_{3}\le 2000$ $2{X}_{1}+{X}_{2}+{X}_{3}\le 3600$ ${X}_{1}+2{X}_{2}+2{X}_{3}\le 2400$ ${X}_{1}\le 30$ ${X}_{1},{X}_{2},{X}_{3}\ge 0$ 0.1 $\to$ (122) 0.3 $\to$ (37) 0.5 $\to$ (19) 0.7 $\to$ (12) 0.9 $\to$ (10) 0.2 $\to$ (58) 0.4 $\to$ (26) 0.6 $\to$ (15) 0.8 $\to$ (11) 1.0 $\to$ (3) $Z=20625$ ${X}_{1}=30$ ${X}_{2}=1185$ ${X}_{3}=0$