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’ tastes 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 chef, 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 three fish dinners for every two 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. Formulate a linear programming model for Angela and Zooey that will help them estimate the number of meals they should prepare each night and solve this model graphically. If Angela and Zooey increased the menu price on the fish dinner 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?
In “The Possibility” Restaurant case Part 1, Angela Fox and Zooey Caulfield opened a French restaurant called “The Possibility.” Initially, Angela and Zooey could not offer a full, varied menu, so their chef, Pierre, prepared two full-course dinners with beef and fish each evening. In the case problem, Angela and Zooey wanted to develop a linear programming model to help determine the number of beef and fish meals they should prepare each night. Solve Zooey and Angela’s linear programming model by using Excel. A. Angela and Zooey are considering investing in some advertising to increase the maximum number of meals they serve. They estimate that if they spend $30 per day on a newspaper ad, it will increase the maximum number of meals they serve per day from 60 to 70. Should they make the investment? B. Zooey and Angela are concerned about the reliability of some of their kitchen staff. They estimate that on some evenings they could have a staff reduction of as much as 5 hours. How would this affect their profit level? C. The final question they would like to explore is raising the price of the fish dinner. Angela believes the price for a fish dinner is a little low and that it could be closer to the price of a beef dinner without affecting customer demand. However, Zooey has noted that Pierre has already made plans based on the number of dinners recommended by the linear programming solution. Angela has suggested a price increase that will increase profit for the fish dinner to $14. Would this be acceptable to Pierre, and how much additional profit would be realized?
Case #4: Summer Sports Camp at State University Mary Kelly is a scholarship soccer player at State University. During the summer, she works at a youth all-sports camp that several of the university’s coaches operate. The sports camp runs for 8 weeks during July and August. Campers come for a 1-week period, during which time they live in the State dormitories and use the State athletic fields and facilities. At the end of a week, a new group of kids comes in. Mary primarily serves as one of the camp soccer instructors. However, she has also been placed in charge of arranging for sheets for the beds the campers will sleep on in the dormitories. Mary has been instructed to develop a plan for purchasing and cleaning sheets each week of camp at the lowest possible cost. Clean sheets are needed at the beginning of each week, and the campers use the sheets all week. At the end of the week, the campers strip their beds and place the sheets in large bins. Mary must arrange either to purchase new sheets or to clean old sheets. A set of new sheets costs $10. A local laundry has indicated that it will clean a set of sheets for $4. Also, a couple of Mary’s friends have asked her to let them clean some of the sheets. They have told her they will charge only $2 for each set of sheets they clean. However, while the laundry will provide cleaned sheets in a week, Mary’s friends can deliver cleaned sheets only in 2 weeks. They are going to summer school and plan to launder the sheets at night at a neighborhood Laundromat. The accompanying table lists the number of campers who have registered during each of the 8 weeks the camp will operate. Based on discussions with camp administrators from previous summers and on some old camp records and receipts, Mary estimates that each week about 20% of the cleaned sheets that are returned will have to be discarded and replaced. The campers spill food and drinks on the sheets, and sometimes the stains do not come out during cleaning. Also, the campers occasionally tear the sheets, or the sheets get torn at the cleaners. In either case, when the sheets come back from the cleaners and are put on the beds, 20% are taken off and thrown away. At the beginning of the summer, the camp has no sheets available, so initially sheets must be purchased. Sheets are thrown away at the end of the summer. Week Registered Campers 1 115 2 210 3 250 4 230 5 260 6 300 7 250 8 190 Mary’s major at State is management science, and she wants to develop a plan for purchasing and cleaning sheets by using linear programming. Help Mary formulate a linear programming model for this problem and solve it by using Excel.