Chapter 11, Problem 61RE

Minimizing Production Cost A factory produces gasoline engines and diesel engines. Each week the factory is obligated to deliver at least $20$ gasoline engines and at least $15$ diesel engines. Due to physical limitations, however, the factory cannot make more than $60$ gasoline engines or more than $40$ diesel engines in any given week. Finally, to prevent layoffs, a total of at least $50$ engines must be produced. If gasoline engines cost $\$450$ each to produce and diesel engines cost $\$550$ each to produce, how many of each should be produced per week to minimize the cost? What is the excess capacity of the factory? That is, how many of each kind of engine are being produced in excess of the number that the factory is obligated to deliver?