Question

The company “Le chocolat d ́elicieux” has you solve an LP problem, the ob- jective being measured in dollars of profit. You solve the LP problem, arriving at a final dictionary that is non-degenerate. Currently they are getting 3000 kilos of cocoa ingredients from the contractor “Cocoa-good” at the price $5 per kilo. The company wants to increase their production, but, the contractor “Cocoa-good” cannot provide additional amount. Then, another supplier “Cocoa-better” offers to provide a tiny bit of cocoa ingredients at the price $7 per kilo. Everything else in the LP problem will remain the same. How do you determine whether the company should buy a tiny bit of such ingredients to make more production? In other words, using what criteria would you advise the company buy or not a tiny bit of such ingredient? Explain your answer clearly using relevant theorems.

Answer 1

we need to know what additional revenue we generate.

The cost now becomes : 5*3000 + 7*t = 15000 + 7t

where 't' is the number of kilograms of cocoa bought from Cocoa-better.

we need to find the profit, which is (3000 + t)*SP - (15000 + 7t)

where SP is the selling price per kilogram of raw cocoa ingredients bought.

Two cases can arise:

i) SP > 7. In such a case, the additional cocoa doesnt hamper the profits. The profits still continue to increase. just that it is (SP - 7)$ per kg raw cocoa, whereas it was (SP - 5)$ per kg raw cocoa earlier.

ii) SP between 5 and 7. In this case, profits will decrease. We need to find the value of t upto which profits are more than zero. And it would be advisable to have the quantity of cocoa bought from Cocoa-better lesser than this quantity so as to sustain profits.