4.3.1: Minimization By The Simplex Method (Exercises)

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SECTION 4.3 PROBLEM SET: MINIMIZATION BY THE SIMPLEX METHOD

In problems 1-2, convert each minimization problem into a maximization problem, the dual, and then solve by the simplex method.

1) \[\begin{aligned}
\text { Minimize } & \mathrm{z}=6 \mathrm{x}_{1}+8 \mathrm{x}_{2} \\
\text { subject to } & 2 \mathrm{x}_{1}+3 \mathrm{x}_{2} \geq 7 \\
& 4 \mathrm{x}_{1}+5 \mathrm{x}_{2} \geq 9 \\
& \mathrm{x}_{1}, \mathrm{x}_{2} \geq 0
\end{aligned} \nonumber \]

2) \[\begin{array}{ll}
\text { Minimize } & \mathrm{z}=5 \mathrm{x}_{1}+6 \mathrm{x}_{2}+7 \mathrm{x}_{3} \\
\text { subject to } & 3 \mathrm{x}_{1}+2 \mathrm{x}_{2}+3 \mathrm{x}_{3} \quad \geq 10 \\
& 4 \mathrm{x}_{1}+3 \mathrm{x}_{2}+5 \mathrm{x}_{3} \geq 12 \\
&\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3} \geq & 0
\end{array} \nonumber \]

SECTION 4.3 PROBLEM SET: MINIMIZATION BY THE SIMPLEX METHOD

In problems 3-4, convert each minimization problem into a maximization problem, the dual, and then solve by the simplex method.

3) \[\begin{array}{lr}
\text { Minimize } & \mathrm{z}=4 \mathrm{x}_1+3 \mathrm{x}_2 \\
\text { subject to } & \mathrm{x}_{1}+\mathrm{x}_{2} \geq 10 \\
& 3 \mathrm{x}_{1}+2 \mathrm{x}_{2} \geq 24 \\
& \mathrm{x}_{1}, \mathrm{x}_{2} \geq 0
\end{array} \nonumber \]

4) A diet is to contain at least 8 units of vitamins, 9 units of minerals, and 10 calories. Three foods, Food A, Food B, and Food C are to be purchased. Each unit of Food A provides 1 unit of vitamins, 1 unit of minerals, and 2 calories. Each unit of Food B provides 2 units of vitamins, 1 unit of minerals, and 1 calorie. Each unit of Food C provides 2 units of vitamins, 1 unit of minerals, and 2 calories. If Food A costs $3 per unit, Food B costs $2 per unit and Food C costs $3 per unit, how many units of each food should be purchased to keep costs at a minimum?