Posted: August 4th, 2022

# Julia’s Food Booth Case Problem

Julia’s Food Booth Case Problem

Julia Robertson is a senior at Tech, and she’s 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 \$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 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.

Instructions

Address each of the issues A-D according to the instructions 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/game to assist.

(D)  Analyze the impact of uncertainties on the model.

Julia’s Food Booth Case Problem

Julia Robertson is a senior at Tech, and she’s 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 \$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 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.

Instructions

Address each of the issues A-D according to the instructions 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/game to assist.

(D) Analyze the impact of uncertainties on the model.

1. Read the grading rubric in the course guide and make sure that you have fulfilled all components in the rubric before you submit your assignment.

2. Linear Programming Model:

a. Decision Variables: The problem is to determine the number of slices of pizza, number of hot dogs, and the number of barbecue sandwiches that have to be sold in order to maximize profit. Let x1 = the number of slices of pizza, x2 = the number of hot dogs, and x3 = the number of BBQ sandwiches.

b. Objective function: The objective is to maximize profit. The price per unit and cost per unit of each item is given in the problem. For example the cost of pizza is given to be \$6 for 8 slices. Therefore the cost of one slice of pizza will be 6/8 = \$0.75. The price of one slice of pizza is given to be \$1.50. The net profit from selling x1 slices of pizza will then be (\$1.50 – \$0.75)*x1 = \$0.75*x1. Similarly you will find the net profit from selling x2 hot dogs and x3 BBQ sandwiches. The total will be the net profit and that will be the objective function.

c. Constraints: There are 4 constraints:

(i) There is a restriction on budget. Julia has only got \$1500. The total of costs of x1 slices of pizza, x2 hot dogs, and x3 BBQ sandwiches cannot exceed \$1500. The cost per unit of each item is given in the problem.

(ii) There is a restriction on the oven space. There are 16 shelves and each shelf is 3ft by 4ft (36 in by 48 in. note that 1ft = 12 in). The oven is going to be filled twice, one before the game (kickoff) and one during half time. The size of one big round (circular) pizza is given to be 14*14 = 196 square inches. Each big pizza includes 8 slices of pizza and therefore the space needed for a slice of pizza should be 196/8 = 24.5 square inches (you can take it to be approximately 24 square inches). Then the space needed for x1 slices of pizza will be 24*x1 square inches. The size of each hot dog and each BBQ sandwich is also given. Determine the space needed for x2 hot dogs and x3 BBQ sandwiches in a similar way. The total will then give you the total space needed for x1 slices of pizza, x2 hot dogs, and x3 BBQ sandwiches. The total space available is the dimension of the shelf and that is 16*36*48 = 27648 square inches. The oven space is used twice, one before kickoff and one at half time. Therefore the total space required will be 2*27648 = 55296 square inches. Thus the total space needed for x1 slices of pizza, x2 hot dogs, and x3 sandwiches should not exceed 55296 square inches.

(iii) She can expect to sell at least as many slices of pizza as hot dogs and BBQ sandwiches combined. This gives you the constraint x1 >= x2 + x3.

(iv) She anticipates selling at least twice as many hot dogs as BBQ sandwiches. This gives you the 4th constraint. You have to express this in terms of the variables.

3. Solve the model using EXCEL or QM for windows and find out the values of x1, x2, and x3 that will maximize the net profit. Also find the net profit.

4. Prospect of borrowing money: To answer this question, you need to look at the sensitivity analysis report from the computer solution output. Find the shadow price (also known as the dual value) for budget from the sensitivity report. The shadow price is a marginal value and is valid within the limits of the sensitivity range for budget. For example if the shadow price for budget is \$2, then for every dollar that she borrows she will make a profit of \$2. Since the shadow price is valid within the sensitivity range for budget, she can only borrow up to the maximum limit of the sensitivity range for budget. Her current budget is \$1500 and if the upper limit of the sensitivity range for budget is \$2000, then she can borrow up to a maximum of \$2000 – #1500 = \$500. Since the dual value is \$2, she will get an additional profit of \$2*\$500 = \$1000. The above is just an example I gave. You have to find the correct numbers from your computer solution output.

5. Prospect of paying a friend \$100 to assist her: Does she have enough money to pay her friend? The job cannot be done by one person.

6. Impact of uncertainties: The solution obtained in (3) is under the assumption that everything goes right. But that may not be the case. There could be many unforeseen situations that could affect the business. Discuss possible uncertainties that will affect the business. The solution obtained is in an ideal situation. She anticipates getting \$1000 profit. You have to consider her net profit margin and possible uncertainties to decide if it is worth for Julia to continue the business.

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