For each problem listed below, first write out an algebraic linear program as we did at the end of the first class.  Use the "standard" format, starting with clear definitions of the decision variables as  numbers. For example, write "x₁ = number of regular bags produced," not "x₁ = regular". Give this kind of definition for every variable. Once you have defined the variables, write the problem out in the form:

Customarily, the complicated constraints involving addition and multiplication are listed first, followed by the simpler constraints like "x₁, x₂  >  0".  (Note: these conventions are not critically important, but make grading easier.)

For each problem, you only have to write the formulation.  You will create and solve an accompanying spreadsheet model for a later assignment.
 

Q1. (30 points)   A bakery can sell chocolate cakes for $12 each, and vanilla cakes for $10 each.  A chocolate cake uses 20 minutes of baking time and 4 eggs, while a vanilla cake uses 40 minutes of baking time and 2 eggs.  If the bakery has 8 hours of baking time and 3 dozen eggs available, how many cakes of each kind should it bake to maximize its revenue?
 

Q2. (30 points)  The Honey Grove organic farm has up to 20 acres of land available to plant beans this season.  The farm markets two variety of beans: green beans and wax beans.  The farm manager estimates that the profit from an acre's worth of green beans is $120, and the profit from an acre's worth of wax beans is $90.  Harvesting an acre of green beans takes 8 person-hours of labor, and harvesting an acres of wax beans requires 6 person-hours of labor.  An acre of green beans also requires 5 cubic yards of compost, while an acre of wax beans needs 3 cubic yards of compost.  The manager anticipates having 140 person-hours of labor available to harvest beans, and the farm has 75 cubic yards of compost available.  To maximize its profit, how many acres of green beans should be planted, and how many acres of wax beans?
 

Q3. (40 points)  Your firm has a machine that takes large sheets of cardboard and makes them into boxes used by your shipping department.  The machine is set up to cut the sheets using two possible patterns.  When you use pattern 1, each cardboard sheet becomes 4 small boxes, 2 medium boxes, and 2 large boxes.  When you use pattern 2, each cardboard sheet becomes 2 small boxes, 3 medium boxes, and 4 large boxes.  This week, the shipping department needs at least 80 small boxes, at least 60 medium boxes, and at least 30 large boxes.  How can you supply their needs, and use the minimum possible number of cardboard sheets?