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Solve the following linear programming problems using the simplex method.
Maximize $z={x}_{1}+{\mathrm{2x}}_{2}+{\mathrm{3x}}_{3}$
subject to $\begin{array}{ccccccc}{x}_{1}& +& {x}_{2}& +& {x}_{3}& \le & \text{12}\\ {\mathrm{2x}}_{1}& +& {x}_{2}& +& {\mathrm{3x}}_{3}& \le & \text{18}\end{array}$
${x}_{1},{x}_{2},{x}_{3}\ge 0$
${x}_{1}=0$ , ${x}_{2}=9$ , ${x}_{3}=3$ , $z=\text{27}$
Maximize $z={x}_{1}+{\mathrm{2x}}_{2}+{x}_{3}$
subject to $\begin{array}{ccccc}{x}_{1}& +& {x}_{2}& \le & 3\\ {x}_{2}& +& {x}_{3}& \le & 4\\ {x}_{1}& +& {x}_{3}& \le & 5\end{array}$
A farmer has 100 acres of land on which she plans to grow wheat and corn. Each acre of wheat requires 4 hours of labor and $20 of capital, and each acre of corn requires 16 hours of labor and $40 of capital. The farmer has at most 800 hours of labor and $2400 of capital available. If the profit from an acre of wheat is $80 and from an acre of corn is $100, how many acres of each crop should she plant to maximize her profit?
Wheat 80 acres, corn 20 acres; Profit $8400
A factory manufactures chairs, tables and bookcases each requiring the use of three operations: Cutting, Assembly, and Finishing. The first operation can be used at most 600 hours; the second at most 500 hours; and the third at most 300 hours. A chair requires 1 hour of cutting, 1 hour of assembly, and 1 hour of finishing; a table needs 1 hour of cutting, 2 hours of assembly, and 1 hour of finishing; and a bookcase requires 3 hours of cutting, 1 hour of assembly, and 1 hour of finishing. If the profit is $20 per unit for a chair, $30 for a table, and $25 for a bookcase, how many units of each should be manufactured to maximize profit?
The Acme Apple company sells its Pippin, Macintosh, and Fuji apples in mixes. Box I contains 4 apples of each kind; Box II contains 6 Pippin, 3 Macintosh, and 3 Fuji; and Box III contains no Pippin, 8 Macintosh and 4 Fuji apples. At the end of the season, the company has altogether 2800 Pippin, 2200 Macintosh, and 2300 Fuji apples left. Determine the maximum number of boxes that the company can make.
600 boxes; 400 of Box I, 200 of Box II, and none of Box III
In problems 1-2, convert each minimization problem into a maximization problem, the dual, and then solve by the simplex method.
Minimize $z={\mathrm{6x}}_{1}+{\mathrm{8x}}_{2}$
subject to $\begin{array}{ccccc}{\mathrm{2x}}_{1}& +& {\mathrm{3x}}_{2}& \ge & 7\\ {\mathrm{4x}}_{1}& +& {\mathrm{5x}}_{2}& \ge & 9\end{array}$
${x}_{1},{x}_{2}\ge 0$
Initial Simplex Tableau
image needed!!!!
Minimize $z={\mathrm{5x}}_{1}+{\mathrm{6x}}_{2}+{\mathrm{7x}}_{3}$
subject to $\begin{array}{ccccccc}{\mathrm{3x}}_{1}& +& {\mathrm{2x}}_{2}& +& {\mathrm{3x}}_{3}& \ge & \text{10}\\ {\mathrm{4x}}_{1}& +& {\mathrm{3x}}_{2}& +& {\mathrm{5x}}_{3}& \ge & \text{12}\end{array}$
${x}_{1},{x}_{2},{x}_{3}\ge 0$
In the next two problems, convert each minimization problem into a maximization problem, the dual, and then solve by the simplex method.
Minimize $z={\mathrm{4x}}_{1}+{\mathrm{3x}}_{2}$
subject to $\begin{array}{ccccc}{x}_{1}& +& {x}_{2}& \ge & \text{10}\\ {\mathrm{3x}}_{1}& +& {\mathrm{2x}}_{2}& \ge & \text{24}\end{array}$
$x,{x}_{2}\ge 0$
${x}_{1}=4$ , ${x}_{2}=6$ , $z=\text{34}$
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?
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