Question

Assignment 1. Linear Programming Case Study 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. QM for Windows or Excel As previously noted, please set up your problem in QM for Windows or 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.

Answer 1

Sol:

Bradley’s Food Stand

John Bradley is a senior at Tech, and he’s investigating different ways to finance his final year at school. He is considering leasing a temporary food facility outside the Tech stadium at home football games. Tech sells out every home game, and he knows, from attending the games himself, that everyone eats a lot of food. He has to pay $800 per game for the stand, and the food stands 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. He thinks slices of cheese pizza, hot dogs, and barbecue sandwiches are the most popular food items among fans and so these are the items he 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 John to prepare the food while he is selling it. He must prepare the food ahead of time and then store it in a warming oven. For $500 he 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. He plans to fill the oven with the three food items before the game and then again before half time.

John 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 him $4.40 and will include 8 slices. He estimates it will cost him $0.50 for each hot dog and $0.85 for each barbecue sandwich if he makes the barbecue himself the night before. He 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. He plans to sell a slice of pizza for $1.40, a hot dog for $1.45, and a barbecue sandwich for $2.00. He has $1,120 in cash 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.

John 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 he has discovered that he can expect to sell at least as many slices of pizza as twice hot dogs and barbecue sandwiches combined. He also anticipates that he will probably sell at least twice as many hot dogs as barbecue sandwiches. He 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 John clears at least $1,200 in profit for each game after paying all his expenses, he believes it will be worth leasing the booth.

(1). Formulate and solve a linear programming model for John that will help you advise him if he should lease the food stand.

(2). If John was to borrow some more money from a friend before the first game 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)?

A: Formulation of the LP Model

X1(Pizza), X2(hotdogs), X3(barbecue sandwiches)

Constraints:

Cost:

Maximum fund available for the purchase = $1500

Cost per pizza slice = $6 (get 8 slices) =6/8 = $0.75

Cost for a hotdog = $.45

Cost for a barbecue sandwich = $.90

Constraint: 0.75X1 + 0.45X2+ 0.90(X3) ≤ 1500



Oven space:

Space available = 3 x 4 x 16 = 192 sq. feet = 192 x 12 x 12 =27648 sq. inches

The oven will be refilled before half time- 27648 x 2 = 55296

Space required for pizza = 14 x 14 = 196 sq. inches

Space required for pizza slice = 196/ 8 = 24.50 sq. inches

Space required for a hotdog=16

Space required for a barbecue sandwich = 25

Constraint: 24.50 (X1) + 16 (X2) + 25 (X3) ≤ 55296



Constraint:

Julia can sell at least as many slices of pizza(X1) as hot dogs(x2) and barbecue sandwiches (X3) combineds

Constraint: X1 ≥ X2 + X3 = X1 - X2 - X3 ≥ 0

Julia can sell at least twice as many hot dogs as barbecue sandwiches

X2/X3 ≥ 2 = X2 ≥2 X3 =X2 - 2 X3 ≥ 0

X1, X2, X3 >= 0 (Non negativity constraint)



Objective Function (Maximize Profit):

Profit =Sell- Cost

Profit function: Z = 0.75 X1 + 1.05 X2 + 1.35 X3



LPP Model:

Maximize Z = 0.75 X1 + 1.05 X2 + 1.35 X3

Subject to 24.5 X1 + 16 X2 + 25 X3 ≤ 55296

0.75 X1 + 0.45 X2 + 0.90 X3 ≤ 1500

X1 - X2 - X3 ≥ 0

X2 - 2 X3 ≥ 0

X1≥ 0, X2≥ 0 and X3 ≥0



Solve the LPM

Based on the excel solution the optimum solution:

Pizza (X1) = 1250; Hotdogs(X2) = 1250 and Barbecue sandwiches (X3) = 0

Maximum value of Z = $2250

Julia should stock 1250 slices of pizza, 1250 hot dogs and no barbecue sandwiches.

Maximum Profit = $2250.

|Maximum Profit |$ 2,250.00 |
|Booth Rent per game |$ (1,000.00) |
|Warming Oven 600 for total of 6 home |$ (100.00) |
|games 600/6 =100