I'm trying to solve a linear programming task (using the graphing method) for profit maximization.
Say, there's a small business producing peanut and raisin mixtures called PR and RP.
The mixture PR contains 2/3 kg raisins and 1/3 kg peanuts, whereas mixture RP contains 1/2 kg raisins and 1/2 kg peanuts. The produce these mixtures, the businessman can buy 90 kg raisins and 60 kg of peanuts, with 1 kg of peanuts costing £0.6 and 1 kg of raisins costing £1.5. Packages labelled PR are sold at a price of 2.9 £/kg and packages labelled RP 2.55 £/kg.
The owner assumes that no more than 150 mixture packages will be sold.
How many different mixture packages should the small business produce to maximise profit?
I figure the variables should be: x1 - number of PR mixture packages produced, x2 - number of RP mixture packages produced.
The constraints, I think, would be: 1/3x1 + 1/2x2 <= 60 (maximum amount of peanuts)
2/3x1 + 1/2x2 <= 90 (maximum amount of raisins)
x1 + x2 <= 150 (maximum amount of mix packages)
The optimization equation is what I can't quite figure out, though. I figure I could just write it F = 2.9x1 + 2.55x2 --> max. But this formula would calculate revenue maximisation, not profit. So any way to make the formula take costs into account as well?
I did draw the graph (the feasibility region) and point of maximization does in fact show (90;60), which should be the correct answer. Point is, how to use the optimization equation to calculate the comparative values of all the corner points of the feasibility region?