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I found the following linear programming question in a past exam paper while preparing for an upcoming exam:

"The daily production of a sweet factory consists of at most 100 kg chocolate covered nuts and at most 125 kg chocolate covered raisins which are then sold in two different mixtures.

  • Mixture A consists of equal amounts of nuts and raisins and is sold at a profit of 5.00 per kg

  • Mixture B consists of one third nuts and two thirds raisins and is sold at a profit of 4.00 per kg

Let there be x kg of mixture A and y kg of mixture B:

1.) Find values for x and y that will give a maximum profit.

2.) Calculate the maximum daily profit."

The question seems fairly simple and I arrived at an answer of x = 84.375 and y = 173.4375 with a maximum profit of 1115.63.

These values seem to fulfill the constraints but do not concur with the memorandum, which unfortunately only gives the final answers (x = 150; y = 75; max. profit 1050) and not the method by which to determine them.

In the face of an exam that's rumored to be insanely challenging I'm losing my mind over this stupid thing. Someone help please!

Yours desperately Zombie-eyed Student

That Fellow
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  • Did you graph your constraints so that you can determine the domain wihtin these lines? – imranfat Jun 06 '13 at 20:49
  • Thank you sincerely for your response. Yes, I determined using a search line of gradient m = -5/4 that the maximum profit will be found at the point where the constraints intersect but this is apparently not the case. Did you arrive at the answer in the memo? – That Fellow Jun 06 '13 at 20:54

1 Answers1

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Here are the two constraints:

$$\frac{x}{2}+\frac{y}{3}\le100\\\frac{x}{2}+\frac{2y}{3}\le125$$

Notice that your point doesn't satisfy the second constraint.

Here is the graph of the two constraints:

$\hspace{2in}$enter image description here

The critical region is below both lines, so we check the vertices of the critical region which are $(0,0),(200,0), (0,\frac{375}{2}),$ and $(150,75)$. You will find that $(150,75)$ maximizes the profit function $5x+4y$.

Jared
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  • I could not have presented it clearer! – imranfat Jun 06 '13 at 21:46
  • Ah ok thank you. I tried to get that point by solving the equations of the constraints simultaneously and I seem to have wrote down the wrong equation for the second constraint. – That Fellow Jun 09 '13 at 18:47