MAT 540 Week 8 Assignment 1 Linear Programming Case Study Sam Allen
Your instructor will assign a linear programming project for this assignment according to the following specifications.
It will be a problem with at least three (3) constraints and at least two (2) decision variables. The problem will be bounded and feasible. It will also have a single optimum solution (in other words, it won’t have alternate optimal solutions). The problem will also include a component that involves sensitivity analysis and the use of the shadow price.
You will be turning in two (2) deliverables, a short writeup of the project and the spreadsheet showing your work.
Writeup.
Your writeup should introduce your solution to the project by describing the problem. Correctly identify what type of problem this is. For example, you should note if the problem is a maximization or minimization problem, as well as identify the resources that constrain the solution. Identify each variable and explain the criteria involved in setting up the model. This should be encapsulated in one (1) or two (2) succinct paragraphs.
After the introductory paragraph, write out the L.P. model for the problem. Include the objective function and all constraints, including any non-negativity constraints. Then, you should present the optimal solution, based on your work in Excel. Explain what the results mean.
Finally, write a paragraph addressing the part of the problem pertaining to sensitivity analysis and shadow price.
Excel.
As previously noted, please set up your problem in Excel and find the solution using Solver. Clearly label the cells in your spreadsheet. You will turn in the entire spreadsheet, showing the setup of the model, and the results.
Click here to view the grading rubric for this assignment.
Problem Introduction
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 daughter’s 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 fallFootball 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, that’s 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 (Sam’s Cheesy Pizza), hot dogs (Sam’s Red Hots), salted soft pretzels (Sam’s Salted Dough), and barbecue sandwiches (Sam’s 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 least as many slices of pizza (Sam’s Cheesy Pizza) as hot dogs and barbeque sandwiches combined. He expects to sell at least twice as many barbecue sandwiches (Sam’s Hot and Sassy Sandwiches) as he will sell hot dogs (Sam’s 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 Sam’s 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 her expenses (food, booth rental, oven rental – all on a per game basis), he believes that it will be worth leasing the booth.
