2. A gold processor has two sources of gold ore, source A and source B. In order

Question

2. A gold processor has two sources of gold ore, source A and source B. In order to kep his plant running, at least three tons of ore must be processed each day. Ore from source A costs $20 per ton to process, and ore from source B costs $10 per ton to process. Costs must be kept to less than $80 per day. Moreover, Federal Regulations require that the amount of ore from source B cannot exceed twice the amount of ore from source A It ore trom source A yields 2 oz, of gold per ton, and ore from source B yields 3 oz. of gold per ton, how many tons of ore from both sources must be processed each day to maximize the amount of gold extracted subject to the above constraints?

Explanation / Answer

Solution:

Let   x = the number of tons from source A

and   y = the number of tons from source B

Now by Given condition:

The objective is to maximize the amount of the gold yield. Since each ton of ore from source A yields 2oz. of gold and each ton of ore from source B yields 3oz. of gold, the amount of gold recovered will be

2x + 3y

After getting the unknowns and the objective out of the way, everything else in the problem is a constraint. The constraints are the processing

x + y 3

cost

20x + 10y 80

federal regulations

y 2x

Of course there are also the implied constraints

x 0   y 0

Now using simplex method

Maximize p = 2x + 3y subject to
x + y >= 3
20x + 10y <= 80
y-2x <=0

Tableau #1
x y s1 s2 s3 p
1 1 -1 0 0 0 3   
20 10 0 1 0 0 80
-2 1 0 0 1 0 0   
-2 -3 0 0 0 1 0   

Tableau #2
x y s1 s2 s3 p
1 1 -1 0 0 0 3   
0 -10 20 1 0 0 20
0 3 -2 0 1 0 6   
0 -1 -2 0 0 1 6   

Tableau #3
x y s1 s2 s3 p
1 1/2 0 1/20 0 0 4   
0 -1/2 1 1/20 0 0 1   
0 2 0 1/10 1 0 8   
0 -2 0 1/10 0 1 8   

Tableau #4
x y s1 s2 s3 p
1 0 0 1/40 -1/4 0 2   
0 0 1 3/40 1/4 0 3   
0 1 0 1/20 1/2 0 4   
0 0 0 1/5 1 1 16   

from this table

Optimal Solution: p = 16; x = 2, y = 4

therefore

The maximum yield of gold is 16oz. by processing 2 tons of ore from source A and 4 tons from source B.

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