Question

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 available. The requirements for ingredients ( in tons per batch) and the profit for each product (in thousand dollars per batch) are given in the chart. The company wants to determine how many batches of each product should be produced to maximize profit.

Answer 1

We have the given conditions,

Maximize $Z = 6x_1+2x_2+12x_3$ subject to

$2x_1+ 6x_2 + 3x_3\le 30$ (zinc, tons)

$4x_1+ x_2+3x_3 \le24$ (steel, tons)

$x_1, x_2, x_3\ge 0$

We can solve this inequality using the Simplex Method,

  

The last row In this table is negative. Hence, applying the row transformations.

$R_2\rightarrow \dfrac{1}{3}R_2$

$R_1\rightarrow R_1-3R_2$

$R_3\rightarrow R_3+ 12R_2$

Hence the new table formed after applying the transformation will be

From this table we get the values

We observe from the first column we get value of $x_1$ as 0, from the second column we get the value of $x_2$ as 0 from the third column we get the value of $x_3$ as 8 because it contains only elements with values 0 and 1.

Hence,

$x_1 =0 , x_2 = 0 , x_3 = 8$

and for maximization we get Z = 96