Chapter 6.3, Problem 45E

In Problems 49-58, construct a mathematical model in the form of a linear programming problem. (the answers in the back of the book for these application problems indicate the model.) then solve the problem by applying the simplex method to the dual problem.

Ice cream. A food processing company produces regular and deluxe ice cream at three plants. Per hour of operation, the Cedarburg plant produces $20$ gallons of regular ice cream and $10$ gallons of deluxe ice cream. The Grafton plant produces $10$ gallons of regular and $20$ gallons of deluxe, and the West Bend plant produces $20$ gallons of regular and $20$ gallons of deluxe. It costs $\text{\$}70$ per hour to operate the Cedarburg plant, $\text{\$}75$ per hour to operate the Grafton plant, and $\text{\$}90$ per hour to operate the West bend plant. The company needs to produce at least $300$ gallons of regular ice cream and at least $200$ gallons of deluxe ice cream each day. How many hours per day should each plant operate in order to produce the required amounts of ice cream and minimize the cost of production? What is the minimum production cost?