For each problem listed below, first write out an algebraic formulation.  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 (relatively) 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.)

Second, for each problem, you set up a spreadsheet model and solve it with Solver.  Hand in the standard printouts, carefully following the instructions.  Failure to follow instructions can lead to assignments that are very time-consuming and difficult to grade, resulting in deduction of points.
 

Q1. (30 points)  You operate a small brewery that makes beer and ale.  Making a barrel of beer requires 3 pounds of corn, 5 pounds of wheat, and 2 pounds of hops, while making a barrel of ale requires 4 pounds of corn, 3 pounds of wheat, and 3 pounds hops.  Your profit from making beer is $3/barrel, and your profit from making ale is $2/barrel.  If you have 60 pounds of corn, 45 pounds of wheat, and 36 pounds of hops available, how much beer and ale should you make to obtain the highest possible profit?
 

Q2. (30 points)  Problem 3 on page 20 of the course pack (the farmer with 45 acres).  Remember that the farmer may plant at most 45 total acres (assume that she do not have to plant the entire 45 acres, but can leave some land unplanted if she chooses).  For this assignment, you should disregard part (b) in the textbook; just follow the instructions above.
 

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 1 large box.  When you use pattern 2, each cardboard sheet becomes 2 small boxes, 3 medium boxes, and 2 large boxes.  This week, the shipping department needs at least 72 small boxes, at least 52 medium boxes, and at least 25 large boxes.  Each time you use pattern 1, the net cost is $4.00, and each time you use pattern 2, the net cost is $3.00.  How can you supply the shipping department's needs at the smallest possible total net cost?