Answer to Question #237048 in Operations Research for opr

Question

Q4). A gold processor has two sources of gold ore, source A and source B. In order to keep 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. If ore from 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?

Answer 1

QUESTION

A gold processor has two sources of gold ore, source A and source B. In order to keep 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. If ore from 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?

SOLUTION

Given source A and B, let's make source $A=x$ and source $B=y$

It is obvious that: $x \ge 0$ and $y\ge0$

The cost, $C=20x+10y\leq80$

Given the condition y2x$y\le2x$

We need to maximize the amount of gold extracted, $S=2x+3y$

We graph:

The region of (x, y) is the region in triangle ABD where A,B,D are the critical points

A(0,0) ,B(2, 4) ,D(4, 0)

The maximum happens at specific vital point so we can substitute:

$S(A)=0$; $S(B)=16 ;S(C)=8$ (B is the intersection of $y=2x$ and $20x+10y=80$ )

S is maximum at B, when $x=2$ and $y=4$

ANSWER: So, per day, the gold processor should process 2 tons of source A and 4 tons of source B to get maximum of 16 oz. of gold