Chapter 6, Problem 41RE

In problems 38-41, construct a mathematical model in the form of a linear programming problem (The answer in the back of the book for these application problems include the model.) Then solve the problem by the simplex, dual problem, or big $M$ methods.

Blending-food processing. A company blends long-grain rice and wild rice to produce two brands of rice mixes: brand $A$, which is marketed under the company’s name, and brand $B$, which is marketed as a generic brand. Brand $A$ must contain at least $10\text{\%}$ wild rice, and brand $B$ must contain at least $5\text{\%}$ wild rice. Long-grain rice costs $\text{\$}0.70$ per pound, and wild rice costs $\text{\$}3.40$ per pound. The company sells brand $A$ for $\text{\$}1.50$ a pound and brand $B$ for $\text{\$}1.20$ a pound. The company has $8,000$ pounds of long-grain rice and $500$ pounds of wild rice on hand. How should the company use the available rice to maximize its profit? What is the maximum profit?