19† A company uses raw material to produce two products.

For c dollars, a unit of raw material can be purchased and processed into k1 units of product 1 and k2 units of product 2. If x1 units of product 1 are produced, they can be sold at p1(x1) dollars per unit. If x2 units of product 2 are produced, they can be sold at p2(x2) dollars per unit. Let z be the number of units of raw material that are purchased and processed. To maximize profits (ignoring non-negativity constraints), the following NLP should be solved:

a Write down the Kuhn–Tucker conditions for this problem. Let x1, x2, 1, 2 represent the optimal solution to this problem.
b Consider a modified version of the problem. The company can now purchase each unit of product 1 for 1 dollars and each unit of product 2 for 2 dollars.
Show that if the company tries to maximize profits in this situation, it will, as in part (a), produce x1 units of product 1 and x2 units of product 2. Also, show that profit and production costs will remain unchanged.
c Give an interpretation of 1 and 2 that might be useful to the company’s accountant.

Transcribed Image Text:

max w =x1p1(x1) + x2P2(x2) - cz s.t. x kz