Answer to Question #175711 in Operations Research for Nomi

Question

The Copperfield Mining Company owns two mines, both of which produce three grades of ore— high, medium, and low. The company has a contract to supply a smelting company with at least 12 tons of high-grade ore, 8 tons of medium-grade ore, and 24 tons of low-grade ore. Each mine produces a certain amount of each type of ore each hour it is in operation. Mine 1 produces 6 tons of high-grade, 2 tons of medium-grade, and 4 tons of low-grade ore per hour. Mine 2 produces 2 tons of high-grade, 2 tons of medium-grade, and 12 tons of low-grade ore per hour. It costs $200 per hour to mine each ton of ore from mine 1, and it costs $160 per hour to mine a ton of ore from mine 2. The company wants to determine the number of hours it needs to operate each mine so that contractual obligations can be met at the lowest cost. Formulate a linear programming model for this problem and solve using the simplex method.

Answer 1

Solution:

Let x = the number of hours that mine 1 operates,

And y = the number of hours that mine 2 operates

Constraint functions are:

$x_1 \ge0 \\x_2\ge 0 \\6x_1 + 2x_2 \ge 12\Rightarrow 3x_1+x_2\ge6 \\2x _1+ 2x_2 \ge8\Rightarrow x_1+x_2\ge4 \\4x_1 + 12x_2 \ge 24\Rightarrow x_1+3x_2\ge6$

Minimise cost, $C=200x+160y$

After introducing slack,surplus,artificial variables:

Similarly, doing this process to 6-iteration, we get, optimal solution is arrived with value of variables as :

$x_1=0,x_2=6$

Min Z=$960