# Using a Linear Programming Model Manually and Graphically

Directions:

Pretend that you have been hired as a business consultant, or you are consulting for your place of work. Given the data in both case studies, write a single consolidated report that outlines your findings. The linear programming model contains only two decision variables. Therefore, produce a graph that illustrates the feasible region from case study 1. (i.e., graph the constraints, identify the corner points, determine the z values for those corner points, and identify the feasible region.) Please solve the first case study manually (through graphical methods).

Case Study Part 1

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) Formulate a Linear programming model for Angela and Zooey that will help then estimate the number of meals they should prepare each night and solve this model graphically.

B) 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?

Directions:

The second case math portions please do in Excel, and the question portion answer in MS Word.

Case Study Part 2

In the Possibility Restaurant case problem, 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 chief, 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 them determine the number of beef and fish meals they should prepare each night. Solve Zooey and Angela's linear programming model using the computer.

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 and explain why?

B) Zooey and Angela are also 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 five hours. How would this affect their profit?

C) The final question they would like to explore is raising the price of the fish dinner. Angela believes that 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?

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#### Solution Summary

The solution will give you very extensive details about the L.P. Model. It has attachments explaining two parts of the solution, and gives an interesting web link to help you understand the Model. The first attachment is 870 words long with 2 graphical images. The second attachment is 480 words long and is accompanied with an excel file with 5 worksheets. The cases in the excel sheet will help you understand the effect of changing constraints.