Answer to Question #176512 in Operations Research for Moni

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. 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

2x1+ 6x2 + 3x3 smaller than or equal to 30 (zinc, tons)

4X1+ x2+3x3 smaller than or equal to24(steel, tons)

x1, x2, x3 bigger than or equal to 0

Solve this model using the simplex method

Answer 1

We have the given conditions,

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

Subject to the given conditions,

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

$4x_1+x_2+3x_3\le 24$ (steel,tons)

$x_1,x_2,x_3\ge0$

Now we have to solve this problem using the simple method. For that we need to construct a table,

Now, we can clearly see that the last column have some negative numbers.

Hence we can apply some row transformations in the last row.

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

$R_1\rightarrow R_1-3R_2$

$R_3 \rightarrow R_3+12R_2$

Hence after applying these transformations we will get a new table which is given as,

From this table we can clearly obtain the values of $x_1,x_2$ and $x_3$

$x_1 = 0$

$x_2= 0$

$x_3 = 8$

$Z = 96$

Hence, the maximum profit we can obtain is 96.