MINIMIZATION BY THE SIMPLEX METHOD.

convert each minimization problem into a maximization problem, the dual, and then solve by the simplex method.

1)

Minimize    z=6x1+8x2
subject to    2x1+3x2≥7
         4x1+5x2≥9
         x1,x2≥0

2)
Minimize    z=5x1+6x2+7x3
subject to    3x1+2x2+3x3≥10
         4x1+3x2+5x3≥12
         x1,x2,x3≥0


convert each minimization problem into a maximization problem, the dual, and then solve by the simplex method.

3)
Minimize    z=4x1+3x2
subject to    z=4x1+3x2
         x1+x2≥10
         3x1+2x2≥24
         x1,x2≥0

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?