# Describe the Excel Solver and/or QM for Windows tool input)

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 write up 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.v

Describe the Excel Solver and/or QM for Windows tool input)

Pizza Slices x1       Hot Dogs x2           BBQ x3              RHS

Maximize

Food Costs                                                                                                                                               <=

Oven Space                                                                                                                                             <=      55296

Hot Dog to BBQ ratio demand                                                                                                             >=       0

Pizza to Hot Dog and BBQ ratio demand                                                                                            >=       0

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.

Instructor assigned Linear Program Project:

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.

Questions:

# Parameters/Background

The case study involving Julia’s food booth …. (provide background and parameters very similar to an Executive Summary in a Business Report).

Julia is considering leasing a food booth outside Tech Stadium at home (6) football games.

If she clears \$1000 in profit for each game she believes it will be worth leasing the booth.

\$1000 per game to lease the booth

\$600 to lease a warming oven

She has \$1500 to purchase food for first game and will for remaining 5 games she will purchase her ingredients with money made from previous game.

Each pizza costs \$6 for 8 slices which is ?  per slice, and she will sell it for \$1.50

Each hot dog costs 0.45, and she will sell it for \$150

Each BBQ Sandwich costs 0.90, and she will sell it for \$2.25

There are Food Cost, Oven and Ratio Constraints that include:

QM assessment (Describe the Excel Solver and/or QM for Windows tool input)

Pizza Slices x1       Hot Dogs x2           BBQ x3              RHS

Maximize

Food Costs                                                                                                                                               <=

Oven Space                                                                                                                                             <=      55296

Hot Dog to BBQ ratio demand                                                                                                             >=       0

Pizza to Hot Dog and BBQ ratio demand                                                                                            >=       0

### Equation form (fill in coefficients, amounts, etc.)

Maximize Z =  0.75Pizza Slices x1 +  _Hot Dogs x2 +  _BBQ x3

Food Cost Constraint:   _Pizza Slices x1 +  _Hot Dogs x2 +  _BBQ x3 <= ___

Oven Space Constraint:  _Pizza Slices x1 +  _Hot Dogs x2 +  _BBQ x3 <= 55296

Hot Dog to BBZ ratio Constraint:   _Hot Dogs x2 +  _BBQ x3 >= 0

Pizza to Hot Dog and BBQ Constraint:  _Pizza Slices x1 – _Hot Dogs x2 – _BBQ x3 >= 0

## Linear Programming Results (from Excel Solver and/or QM for Windows):

Optimal Value (Z) =

# Case Study Questions

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

Conclusion:  If Julia were to open a food booth at her college’s home football games, her optimal value would be _______with Pizza xvalue _____ Hot dogs x2  value of ____ and BBQ  x3 value of ______

1. 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)?

After solving the linear program in QM and utilizing the ranging function (see ranging function in QM assessment) the upper bound for food costs is ________.

Since Julia already is starting with \$1500 for food cost, she could increase her profit and the most she should borrow from her friend is \$_________

If she borrowed money from her friend the additional amount of profit she could generate is _________.

This is determined because when looking in the ranging section of the solution, the dual value is ______.  This means it is worth _______ to Julia for each additional dollar that she receives.  So with this is mind, we can conclude that …….

The factor that constrains her from borrowing even more money is ….

Conclusion:

1. 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?

In solution A,

In solution B,

Conclusion:

1. 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?

# Conclusions/Final thoughts

What I have done so Far:

# Parameters/Background

Julia is considering leasing a food booth outside Tech Stadium at home (6) football games.  If she clears \$1000 in profit for each game she believes it will be worth leasing the booth.

\$1000 per game to lease the booth

\$600 to lease a warming oven

She has \$1500 to purchase food for first game and will for remaining 5 games she will purchase her ingredients with money made from previous game.

Each pizza costs \$6 for 8 slices which is 0.75  per slice, and she will sell it for \$1.50

Each hot dog costs 0.45, and she will sell it for \$1.50

Each BBQ Sandwich costs 0.90, and she will sell it for \$2.25

There are Food Cost, Oven and Ratio Constraints that include:

# QM assessment (Describe the Excel Solver and/or QM for Windows tool input)

Pizza Slices x1       Hot Dogs x2           BBQ x3              RHS

Maximize

Food Costs                                                                                                                                               <=

Oven Space                                                                                                                                             <=      55296

Hot Dog to BBQ ratio demand                                                                                                             >=       0

Pizza to Hot Dog and BBQ ratio demand                                                                                            >=       0

### Equation form

Maximize Z (profit) = Pizza Slices 0.75x1 + Hot Dogs0.45×2 + BBQ 0.90 x3

Food Cost Constraint:   0.75×1 + 0.45×2 + 0.90×3 <= \$1500.00

Oven Space Constraint:  24×1 + 16×2 + 25×3 <= 55296 (see Excel Spreadsheet for explanation)

Hot Dog to BBQ ratio Constraint:   -x2 + 2×3 >= 0

Pizza to Hot Dog and BBQ ratio Constraint:  Pizza Slice -x1 +x2 -x3 >= 0

Optimal Value (Z) =         \$2250

# Case Study Questions

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

Variables:

x1 – Pizza Slices

x2 – Hot Dogs

x3 – Barbeque Sandwiches

Subject to:

\$0.75×1 + \$0.45×2 + \$0.90×3 ≤ \$1,500

24×1 + 16×2 + 25×3 ≤ 55,296 in2 of oven space

x1 ≥ x2 + x3 (changed to –x1 + x2 + x3 ≤ 0 for constraint)

x2/x3 ≥ 2 (changed to –x2 +2×3 ≤ 0 for constraint)

x1, x2, x3 ≥ 0

Solution:

Variable | Status | Value |

X1 | Basic | 1250 |

X2 | Basic | 1250 |

X3 | NONBasic | 0 |

slack 1 | NONBasic | 0 |

slack 2 | Basic | 5296.0 |

slack 3 | NONBasic | 0 |

slack 4 | Basic | 1250 |

Optimal Value (Z) | | 2250 |

Conclusion:  If Julia were to open a food booth at her college’s home football games, her optimal value would be _______with Pizza xvalue _____ Hot dogs x2  value of ____ and BBQ  x3 value of ______

1. 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)?

Yes, Julia would increase her profit if she borrowed some more money from a friend. The shadow price, or dual value, is \$1.50 for each additional dollar that she earns. The upper limit given in the model is \$1,658.88, which means that Julia can only borrow \$158.88 from her friend, giving her an additional profit of \$238.32.

After solving the linear program in QM and utilizing the ranging function (see ranging function in QM assessment) the upper bound for food costs is ________.

Since Julia already is starting with \$1500 for food cost, she could increase her profit and the most she should borrow from her friend is \$_________

If she borrowed money from her friend the additional amount of profit she could generate is _________.

This is determined because when looking in the ranging section of the solution, the dual value is ______.  This means it is worth _______ to Julia for each additional dollar that she receives.  So with this is mind, we can conclude that …….

The factor that constrains her from borrowing even more money is ….

Conclusion:

1. 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?

Yes, I believe Julia should hire her friend for \$100 per game. In order for Julia to prepare the hot dogs and barbeque sandwiches needed in a short period of time to make her profit, she needs the additional help. Also, with her borrowing the extra \$158.88 from her friend, Julia would be able to pay her friend for the time spent per game helping with the food booth.

Conclusion:

1. 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?