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# Formulate a linear programming model for Julia

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Julia is a senior at Tech, and she is investigating different ways to finance her final year at school. She is considering leasing a food booth outside the Tech stadium at home football games. Tech sells out every home game, and Julia knows, from attending the games herself, that everyone eats a lot of food. She has to pay \$1,000 per game for a booth, and the booths are not very large. Vendors can sell either food or drinks on Tech property, but not both. Only the Tech athletic department concession stands can sell both inside the stadium. She thinks slices of cheese pizza, hot dogs, and barbecue sandwiches are the most popular food items among fans and so these are the items she would sell.

Most food items are sold during the hour before the game starts and during half time; thus it will not be possible for Julia to prepare the food while she is selling it. She must prepare the food ahead of time and then store it in a warming oven. For \$600 she can lease a warming oven for the six-game home season. The oven has 16 shelves, and each shelf is 3 feet by 4 feet. She plans to fill the oven with the three food items before the game and then again before half time.

Julia has negotiated with a local pizza delivery company to deliver 14-inch cheese pizzas twice each game - 2 hours before the game and right after the opening kickoff. Each pizza will cost her \$4.50 and will include 6 slices. She estimates it will cost her \$0.50 for each hot dog and \$1.00 for each barbecue sandwich if she makes the barbecue herself the night before. She measured a hot dog and found it takes up about 16 square inch of space, whereas a barbecue sandwich takes up about 25 square inch. She plans to sell a piece of pizza for \$1.50 and a hot dog for \$1.60 each and a barbecue sandwich for \$2.25. She has \$1,500 in cash available to purchase and prepare the food items for the first home game; for the remaining five games she will purchase her ingredients with money she has made from the previous game.

Julia has talked to some students and vendors who have sold food at previous football games at Tech as well as at other universities. From this she has discovered that she can expect to sell at least as many slices of pizza as hot dogs and barbecue sandwiches combined. She also anticipates that she will probably sell at least twice as many hot dogs as barbecue sandwiches. She believes that she will sell everything she can stock and develop a customer base for the season if she follows these general guidelines for demand.

If Julia clears at least \$1,000 in profit for each game after paying all her expenses, she believes it will be worth leasing the booth.

A. Formulate a linear programming model for Julia that will help you to advise her if she should lease the booth. Formulate the model for the first home game. Explain how you derived the profit function and constraints and show any calculations that allow you to arrive at those equations.

B. Solve the linear programming model using a computer for Julia that will help you advise her if she should lease the booth. In this solution, determine the number of pizza slices, hot dogs and barbecue sandwiches she should sell at each game. Also determine the revenues, cost and profit; and do an analysis of how much money she actually will make each game given the expenses of each game.

Do an analysis of the profit solution and what impact it has on Julia's ability to have sufficient funds for the next home game to purchase and prepare the food. What would you recommend to Julia?

C. If Julia were to borrow some money from a friend before the first game to purchase more ingredients, she feels she can increase her profits. What amount, if any, would you recommend to Julia to borrow?

D. Food prices have been rising lately. Assume purchase costs for the food is now \$6.00 for each pizza, \$0.75 for each hot dog, and \$1.25 for each barbecue sandwich. Repeat the analysis of Part B. What would you recommend to Julia to do at this point?

E. Julia seems to be basing her analysis on the assumptions that everything will go as she plans. What are some of the uncertain factors in the model that could go wrong and adversely affect Julia's analysis? Given these uncertainties and the results in (B), (C), and (D), what do you recommend that Julia do? Take into consideration her profit margin for each game.

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## Formulate and solve a linear programming model for Julia.

Please read the Case Problem below and address each of the issues A-D according to the instruction given.

(A) Formulate and solve a Linear Programming model for this case.
(B) Evaluate the prospect of borrowing money before the first game.
(C) Evaluate the prospect of paying a friend \$100 to help her per game.
(D) Analyze the impact of uncertainties on the model.

CASE PROBLEM: JULIA'S FOOD BOOTH

Julia Robertson is a senior at Tech, and she is investigating different ways to finance her final year at school. She is considering leasing a food booth outside the Tech stadium at home football games. Tech sells out every home game, and Julia knows, from attending the games herself, that everyone eats a lot of food. She has to pay a \$1000 per game for a booth, and the booths are not very large. Vendors can sell either food or drinks on Tech property, but not both. Only the Tech athletic department concession stands can sell inside the stadium. She thinks slices of cheese pizza, hot dogs, and barbecue sandwiches are the most popular food items among fans and so these are the items she would sell.

Most food items are sold during the hour before the game starts and during half time; thus it will not be possible for Julia to prepare the food while she is selling it. She must prepare the food ahead of time and then store it in a warming oven. For \$600 she can lease a warming oven for the six-game home season. The oven has 16 shelves, and each shelf is 3 feet by 4 feet. She plans to fill the oven with the three food items before the game and then again before half time.

Julia has negotiated with a local pizza delivery company to deliver 14 -inch cheese pizzas twice each game- 2 hours before the game and right after the opening kickoff. Each pizza will cost her \$6 and will include 8 slices. She estimates it will cost her \$0.45 for each hot dog and \$0.90 for each barbecue sandwich if she makes the barbecue herself the night before. She measured a hot dog and found it takes up about 16 square inches of space, whereas a barbecue sandwich takes up about 25 square inches. She plans to sell a slice of pizza and a hot dog for \$1.50 apiece and a barbecue sandwich for \$2.25, she has \$1,500 in cash available to purchase and prepare the food items for the first home game; for the remaining five games she will purchase her ingredients with money she has made from the previous game.

Julia has talked to some students and vendors who have sold sites. From this she has discovered that she can expect to sell at least as many slices of pizza as hot dogs and barbecue sandwiches combined. She also anticipates that she will probably sell at least twice as many hot dogs as barbecue sandwiches. She believes that she will sell everything she can stock and develop a customer base for the season if she follows these general guidelines for demand.

If Julia clears at least \$1,000 in profit for each game after paying all her expenses, she believes it will be worth leasing the booth.

A. Formulate and solve a linear programming model for Julia that will help you advise her if she should lease the booth.

B. If Julia were to borrow some more money from a friend before the first game to purchase more ingredients, could she increase her profit? If so, how much should she borrow and how much additional profit would she make? What factor constrains her from borrowing even more money than this amount (indicated in your answer to the previous question)?

C. When Julia looked at the solution in (A), she realized that it would be physically difficult for her to prepare all the hot dogs and barbecue sandwiches indicated in this solution. She believes she can hire a friend of hers to help her for \$100 per game. Based on the results in (A) and (B), is this something you think she could reasonably do and should do?

D. Julia seems to be basing her analysis on the assumption that everything will go as she plans. What are some of the uncertain factors in the model that could go wrong and adversely affect Julia's analysis? Given these uncertainties and the results in (A), (B), and (C), what do you recommend that Julia do?

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