Problem:
Carson Metals Company produces metal alloys from six different ores it mines. The ores and the metal content of each are described in the table below. The company has an order from a customer to produce an alloy that contains four metals according to the following specifications: 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 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 refined, the impurities are removed.
a. Formulate a linear programming model that can be used to determine the amount of each ore to use per ton of the alloy while minimizing cost.
b. Determine the optimal solution including the total tonnage of each type of ore used and the total cost per ton of the alloy.