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# The Simplex Solution Method

25. The Eastern Iron and Steel Company makes nails, bolts, and washers from leftover steel and coats them with zinc. The company has 24 tons of steel and 30 tons of zinc. The following linear pro- gramming model has been developed for determining the number of batches of nails (x1), bolts (x2), and washers (x3) to produce to maximize profit:

maximize Z = 6x1 + 2x2 + 12x3 (profit, \$1,000s) subject to

( s t e e l , t o n s ) ( z i n c , t o n s ) x 1 , x 2 , x 3 Ú 0 2 x 1 + 6 x 2 + 3 x 3 3 0 4 x 1 + x 2 + 3 x 3 2 4

Solve this model using the simplex method. 26. Solve the following linear programming model using the simplex method:

maximize Z = 100x1 + 75x2 + 90x3 + 95x

4

subject to

3x1 + 2x

2

4x3 + x

4

1 200x + 250x

3

• 40

• 25

• 2,000

• 2,200

Ú0

x 1 , x 2 , x 3 , x 1 0 0 x 1 + 2 0 0 x 4 4

27. Solve the following linear programming model using the simplex method:

minimize Z = 20x1 + 16x2

1 3x + x

2

1 x +x

2

2x1 + 6x

2

x1, x

2

subject to

Ú6 Ú4 Ú 12 Ú0

• 28.

Solve the linear programming model in Problem 27 graphically, and identify the points on the graph that correspond to each simplex tableau.

• 29.

Transform the following linear programming model into proper form for solution by the simplex method:

minimize Z = 8x1 + 2x2 + 7x3 subject to

x 1 , x 2 , x 3 Ú 0 x 1 + 2 x 2 Ú 2 0 + x 2 + 2 x 3 5 0 3 x 2 + 4 x 3 Ú 6 0 2 x 1 + 6 x 2 + x 3 = 3 0 4x1

30. Transform the following linear programming model into proper form for solution by the simplex method:

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