1. During the summer tourist season, Beach Treats, a local soft-serve ice cream parlor does a booming business. During the winter, Larry Beach, owner of Beach Treats, is contacted by his supplier of skim milk, cream, and chocolate syrup about signing a yearly contract for fixed weekly deliveries of these items at a very reasonable price. Larry signs the contracts in order to take advantage of the huge discounts. During the summer Larry has just enough supplies to meet his demands, but in the fall, the tourist season winds down. Larry finds that he has an excess supply of 500 gallons of cream, 1000 gallons of skim milk, and 240 lbs. of chocolate syrup per week. To utilize this surplus, Larry contracts with his brother, Jerry, a bottler, to deliver cases of whole milk and chocolate milk to the local schools. A case of whole milk uses I gallon of cream and 2 gallons of skim milk. The net profit on a case of whole milk is $2.50 A case of chocolate milk uses 0.5 gallons of cream, 2.5 gallons of skim milk, and 0.75 lbs. of chocolate syrup. The net profit on a case of chocolate milk is $4. A local dairy owner, Perry, who supplies milk to many of the local schools would like to increase his market share. He contacts Larry and offers to buy the weekly surplus of cream, milk, and syrup

(a) Formualte Larry's problem of maximizing his profit and Perry's problem of minimizing the price offered for the excess inventory. Verify that Perry's LP is the dual of Larry (the primal) Convert the primal problem to an equivalent LP with equality constraints by adding slack variables. Convert the dual to a form consistent with trying to find an initial feasible solution starting with all the original dual variables set to 0

(b)In the primal, the slack variables have the interpretation of representing the amount of inventory of each of the three goods remaining. In the primal, at the initial solution (0,0, 500,1000, 240), note that d1=(0, 1,-2,2,-4)is an improving direction. Convert di to a form representing the change in each variable upon changes to the variable s3. Call this direction di and find c dI. Interpret c di in terms of changes to the Inventory of chocolate syrup

(c) In the dual, find the(infeasible) solution where y3 becomes basic

(d) In the primal, move along direction dI as far as possible. At this location, note that direction d2=(1,0, -1,-2, 0) is a feasible direction. Convert d2 to a the change in each variable upon changes to the variable s2. Call tl orm representing find d2. Interpret cd2 in terms of changes to the inventory of skim milk

(e) In the dual find the solution where y2 and ys are basic

(f) In the primal, move along direction dz as much as possible. Verify the point that you reach is optimal

(g) At the optimal primal solution, find two nonimproving directions, ds and d4, that quantify the effects on the objective function of further decreasing the amount of $2 and s3. Perhaps this represents getting extra skim milk and chocolate syrup one week Compare these values to the optimal prices that you found in the dual and interpret these prices in terms of the primal and dual optimal solutions.