Linear Programming “Possibility Restaurant” 1. The problem statement, all variables and given/known data Angela Fox and Zooey Caulfield were food and nutrition majors at State University, as well as close friends and roommates. Upon graduation Angela and Zooey decided to open a French restaurant in Draperton, the small town where the university was located. There were no other French restaurants in Draperton, and the possibility of doing something new and somewhat risky intrigued the two friends. They purchased an old Victorian home just off Main Street for their new restaurant, which they named “The Possibility”. Angela and Zooey knew in advance that at least initially they could not offer a full varied menu of dishes. They had no idea what their local customers’ taste in French cuisine would be, so they decided to serve only two full-course meals each night, one with beef and the other with fish. Their chief, Pierre, was confident he could make each dish so exciting and unique that two meals would be sufficient, at least until they could assess which menu items were most popular. Pierre indicated that with each meal he could experiment with different appetizers, soups, salads, vegetable dishes, and desserts until they were able to identify a full selection of menu items. The next problem for Angela and Zooey was to determine how many meals to prepare for each night so they could shop for ingredients and set up the work schedule. They could not afford too much waste. They estimated that they would sell a maximum of 60 meals each night, Each fish dinner, including all accompaniments, requires 15 minutes to prepare, and each beef dinner takes twice as long. There is a total of 20 hours of kitchen staff labor available each day. Angela and Zooey believe that because of the health consciousness of their potential clientele they will sell at least 3 fish dinners for every 2 beef dinners. However, they also believe that at least 10% of their customers will order beef dinners. The profit from each fish dinner will be approximately $12, and the profit from a beef dinner will be about $16. A) If Angela and Zooey increased the menu price on the fish dinners so that the profit for both dinners was the same, what effect would that have on their solution? Suppose Angela and Zooey reconsidered the demand for beef dinners and decided that at least 20% of their customers would purchase beef dinners. What effect would this have on their meal preparation plan? 2. Relevant equations Resource Availability 1200 minutes of labor per day 60 maximum meals each night Decision variables X1 = number of fish meal X2 = number of beef meal Objective function Maximize Z = $12X1 + $16X2 Where Z = total profit per day $12X1 = profit from fish meals $16X2 = profit from beef meals Resource Constraints X1 + X2 ? 60 (the maximum estimated meals each night) 15X1 + 30X2 ? 1200 (labor in minutes to prepare meals) 2X1 – 3X2 ? 0 (they’ll sell at least 3fish/2beef meal) X1 – 9X2 ? 0 (at least10% of customers will order beef meal) Non-Negativity Constrains X1 ? 0 ; X2 ? 0 3. The attempt at a solution Complete Linear Programming Model Maximize Z = $12X1 + $16X2 Subject to X1 + X2 ? 60 15X1 + 30X2 ? 1200 2X1 – 3X2 ? 0 X1 – 9X2 ? 0 X1 , X2 ? 0 Possibilities X1 X2 RHS Equation form Maximize 12 16 Max 12X1 + 16X2 Constraint 1 1 1 What will be the effect if the price is the same and also if the beef is atleast 20% now?