1. (2 points) The bartender of your local pub has asked you to assist him in finding the combination of mixed drinks that will maximize his profit. He has the following bottles available: - 2 litre King Car Conductor Single Malt (whiskey; cost 80/1) - 1 litre Cocchi Riserva La Venaria Reale Vermouth (cost 80/1) - 2 litres Hayman Family Reserve Gin (cost 48/1) Since he is new to the business, his knowledge is limited to the following drinks: Drink Alcohol Price 8 16 Whiskey Sour 4cl whiskey Manhattan 4cl whiskey, 2cl vermouth Martini 4cl gin, 2cl vermouth Pub Special 4cl gin, 4cl whiskey 16 24 All other ingredients needed in the drinks are assumed to be free. (a) Formulate an integer linear programming (ILP) problem to deter- mine the optimal number of each drink to be sold. Are the costs of the bottles relevant here? If not, leave them out of the problem formulation. (b) Solve the problem by using Excel Solver. What happens if you relax the integrality constraints? (c) Formulate the dual of the LP relaxation and solve it by Excel Solver. (d) Verify that the optimal values of the primal and dual objective func- tions are the same, and that the complementary slackness conditions hold. (e) Optional: Can you come up with an interpretation for the dual and the optimal values of its decision variables