# Formulate and solve a linear programming model for Sam that will help you to advise him if he should lease the booth.

MAT540 Week 8 Assignment

Friday Night Lights

Case Study Background:

Sam Allen is a parent of several students at Mattson Area High School in a small community southwest of Dallas, TX.   He is investigating different ways to finance his daughters first year at Southwest Texas State University and he is considering leasing a food booth outside the Mattson Area HS stadium at home football games this fall.

Football is a big family activity on Friday nights throughout most high schools in Texas.  In fact, at least one TV series and several movies have been made depicting the strong interest in high school football in Texas.  Mattson Area is one of the larger high schools near Dallas and has a football stadium with a seating capacity of over 20,000 (Yup, thats right!!!).

Like most high schools, Mattson Area typically sells out every home game, and Sam knows from attending the games himself that everyone comes to the stadium to do two things: go crazy cheering during the game and stuff themselves with food.     He has to pay \$1,000 per game for a booth, and the booths are actually not very large.  Vendors can sell either food or drinks on Mattson Area but not both.  This seems to be a strange rule to Sam but he has to go with the flow on this.   Only the Mattson Area athletic department concession stands can sell both inside the stadium.   He thinks that he should bet on several key food items that are typically the most popular with the fans and has even given them special names:   cheese pizza (Sams Cheesy Pizza), hot dogs (Sams Red Hots), salted soft pretzels (Sams Salted Dough), and barbecue sandwiches (Sams Hot and Sassy Sandwiches).   He is planning just to sell these four items.

Most food items are sold during the hour before the game starts and during half time; thus it will not be possible for Sam to prepare the food while he is selling it.   He must prepare the food ahead of time and then store it in a warming over.  For \$1,200, he can lease a warming oven for the six-game home season.  The oven has 8 shelves, and each shelf is 3 feet by 6 feet.  He plans to fill the oven with the four food items before the game and then again before half time.

Sam has negotiated with a local pizza delivery company to deliver 16-inch square cheese pizzas twice each game 2 hours before the game and right after the opening kickoff.  Each pizza will cost her \$10 and will include 8 slices.   He estimates that it will cost him \$0.90 for each hot dog (hot dog, bun, and condiments), and \$2.00 for each barbecue sandwich if he makes the barbecue himself the night before the game.  The soft pretzels are likely to cost about \$0.75 per pretzel.

He measured a hot dog and found that it takes up about 16 square inches of space, whereas a barbecue sandwich takes up about 25 square inches of space.   He would only put whole pizzas into the warming over; not a fraction of a pizza.   He also finds that the soft salted pretzel take up about 20 square inches of space.  He plans to sell a slice of pizza for \$2.50, a hot dog for \$3.00, a soft pretzel for \$2.00, and a barbeque sandwich for \$5.00.

Sam had \$2,500 in case available to purchase and prepare the food items for the first home game; for the remaining five games he will purchase his ingredients with money he has made from the previous game.  He has decided that each game should be self-sufficient for food costs and that his food costs should not exceed \$2,500 per game.

Sam has talked to some students and vendors who have sold food at previous football games at Mattson Area as well as other high schools.   From this he has discovered that he can expect to sell at most as many slices of pizza (Sams Cheesy Pizza) as hot dogs and barbeque sandwiches combined.   He expects to sell at least twice as many barbecue sandwiches (Sams Hot and Sassy Sandwiches) as he will sell hot dogs (Sams Red Hots) as he has a reputation for fantastic barbecue sandwiches.   He expects to sell almost as many soft pretzels as he does barbeque sandwiches but he is not certain of the market for Sams Salted Dough.   Being the rather optimistic salesman that he is, Sam believes that he will sell everything he can stock and develop a customer base for the season if he follows these general guidelines for demand.

If Sam clears at least \$1.500 in profit for each game after paying all his expenses (food, booth rental, oven rental all on a per game basis), he believes that it will be worth leasing the booth.

Case Study Specific Requirements:

1. Formulate and solve a linear programming model for Sam that will help you to advise him if he should lease the booth.
2. If Sam were to borrow some money from a friend before the first games to purchase more ingredients, could he increase his profit? If so, how much should he borrow and how much additional profit would he make?   What factor constrains him from borrowing even more money than this amount (indicated in your answer to the previous question)?
3. If Sam were to rent a larger oven to preheat his food, could he increase his profit? If so, by how much would he increase his profit per sq. inch of oven?   How much additional profit would he make if he increased the oven size by 10%?   Explain your answers using the appropriate information from QM for Windows.
4. When Sam looked at the solution in (A), he realized that it would be physically difficult for him to prepare all the hot dogs and barbecue sandwiches indicated in this solution. He believes that he can hire of friend of his to help him for \$150 per game.  Based on the results of (A) and (B), is this something you think he could reasonably do and should do?
5. Sam seems to be basing his analysis on the assumption that everything will go as he plans. What are some of the uncertain factors in the model that could go wrong and adversely affect Sams analysis?  Given these uncertainties and the results in (A), (B), and (C), what do you recommend that Sam do?

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Approximately 250 words
\$12