Question

Grafton Metalworks Company produces metal alloys from six different ores it mines. The company has an order from a customer to produce an alloy that contains four metals according to the following specications: at least 21% of metal A, no more than 12% of metal B, no more than 7% of metal C, and between 30% and 65% of metal D. The proportion of the four metals in each of the six ores and the level of impurities in each ore are provided in the following table:
Metal (%)
Ore - A - B - C - D - Impurities (%) - Cost/Ton
1 - 19 - 15 - 12 - 14 - 40 - $27
2 - 43 - 10 - 25 -7 - 15 - 25
3 - 17 - 0 - 0 - 53- 30 - 32
4 - 20 - 12 - 0 - 18 - 50 - 22
5 - 0 - 24- 10- 31 - 35 - 20
6 - 12 - 18 - 16 - 25 - 29 - 24
When the metals are processed and rened, the impurities are removed. The company wants to know the amount of each ore to use per ton of the alloy that will minimize the cost per ton of the alloy.
a. Formulate a linear programming model for this problem.
b. Solve the model by using the computer.

Answer 1

Answer:

Explanation:

Part (a):

Let $x_1,\; x_2,\; x_3,\; x_4,\;x_5 \;and \;x_6$ be the 6 ores. The constraints will be as follows:

For at least 21% of Metal A:

$19x_1+43x_2+17x_3+20x_4+0x_5+12x_6 \geq 21$

For at most 12% of Metal B:

$15x_1+10x_2+0x_3+12x_4+24x_5+18x_6 \leq 12$

For at most 7% of Metal C:

$12x_1+25x_2+0x_3+0x_4+10x_5+16x_6 \leq 7$

For at least 30% of Metal D:

$14x_1+7x_2+53x_3+18x_4+31x_5+25x_6 \geq 30$

For at most 65% of Metal D:

$14x_1+7x_2+53x_3+18x_4+31x_5+25x_6 \leq 65$

Practical constraint:

$x_1,\; x_2,\; x_3,\; x_4,\;x_5 \;and \;x_6\;\geq 0$

This is a minimization problem and the Cost Function Z is:

$Z_{(min)\;=\;}27x_1+25x_2+32x_3+22x_4+20x_5+24x_6$

Part (b):

The problem is converted to canonical form by adding slack, surplus and artificial variables as appropriate.

As the constraint 1 is of type '≥' we should add the surplus variable $x_{7}$ and the artificial variable $x_{12}$.

As the constraint 2 is of type '≤' we should add the slack variable $x_{8}$.

As the constraint 3 is of type '≤' we should add the slack variable$x_{9}$.

As the constraint 4 is of type '≥' we should add the surplus variable $x_{10}$ and the artificial variable $x_{11}$.

As the constraint 5 is of type '≤' we should add the slack variable $x_{11}$.

MAXIMIZE:

$Z = -27x_1 -25x_2-32x_3 -22x_4 -20x_5 -24 x_6 + 0 x_7 + 0 x_8 + 0 x_9 + 0 x_{10} + 0 x_{11} + 0 x_{12} + 0 x_{13}$

Subject to

$19 x_1 + 43 x_2 + 17 x_3 + 20 x_4 + 12 x_6 -1 x_7 + 1 x_{12} = 21\\\\15 x_1 + 10 x_2 + 12 x_4 + 24 x_5 + 18 x_6 + 1 x_8 = 12\\\\12 x_1 + 25 x_2 + 10 x_5 + 16 x_6 + 1 x_9 = 7\\\\14 x_1 + 7 x_2 + 53 x_3 + 18 x_4 + 31 x_5 + 25 x_6 -1 x_{10} + 1 x_{13} = 30\\\\14 x_1 + 7 x_2 + 53 x_3 + 18 x_4 + 31 x_5 + 25 x_6 + 1 x_11 = 65\\\\x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9, x_{10}, x_{11}, x_{12}, x_{13} \geq 0$

Using a solver, the optimal solution value is

Z = $23.9125  per ton of ore

$x_1$ = 0

$x_2$ = 0.2792

$x_3$ = 0.5292

$x_4$ = 0

$x_5$ = 0

$x_6$ = 0