Linear Programming Case Study

You will be turning in two (2) deliverables, a short writeup of the project and the spreadsheet showing your work.

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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.

Below is the scenario to formulate the problem.

Bradleys Food Stand

John Bradley is a senior at Tech, and hes 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 Johnwas to borrow some more money from a friend before the first game to purchase more ingredients, could he increase hisprofit? 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)?

Our problem is a maximizing problem as we need to get the maximum profit at the set restrictions.

Let's define variables: x1, x2, x3 >=0 and integer

x1 - necessary amount of pizzasx2 - necessary quantity of hotdogs

x3 - necessary quantity of sandwichesLet's define our expenses:

800 $ pay for the stand for game

500 $ lease a warming oven for the six-game home season>>> 500/6 = 83,33$ for a match

4,40$ pizza (8 parts) = 0,55$ for a slice; 0,50$ hotdog; 0,85$ barbecue sandwich

1120$ - budget

0.55*x1+0.50*x2+0.85*x3 + (800+83.33) = < 1120

Let's count expenses of a necessary place in an warming oven for each product:

The oven has 16 shelves, and each shelf is 3 feet by 4 feet: 16*3*4=192 feet

14/8=1.75 feet pizzas slice

16 feet hotdog

25 feet sandwich

1.75* kh1 +16* kh2+25* kh3 =<192

Let's enter into restrictions of expectation of the seller:

1 pizza=2*(hotdog + sandwich) >>>>>> x1=2*(x2+x3)

1 hotdog=2*sandwiches >>>>>>>>>>>x2=2*x3

At last let's define our income (income=price-expenses):

pizza: 1.44-0.55=0.85$ hotdog: 1,45-0,5=0,95$ sandwich: 2-0,85=1,15$

0.85*x1+0.95*x2+1.15*x3 max

Let's write down everything together now (Model):

0.85*x1+0.95*x2+1.15*x3 max

0.55*x1+0.50*x2+0.85*x3 + (800+83.33) = < 1120

1.75* kh1 +16* kh2+25* kh3 =<192

x1=2*(x2+x3)

x2=2*x3

x1, x2, x3 >=0 and integer

Let's provide a solution from Excel

x1 x2 x3 12 4 2 F(x) 0,85 0,95 1,15 0 => 16,3

1,75 16 25 <= 192 135

0,55 0,5 0,85 883,33 <= 1120 893,63

1 -2 -2 = 0 0

0 1 -2 = 0 0

Optimum quantity of food at these restrictions the following:

12 slices of pizza, 4 hotdogs, 2 sandwiches and our income is 16.3$ (it is more less than we expected 1120$).

At restriction an integer we can't determine the shadow price and make the analysis of stability. In more detail in Excel. Having moved away him, we see that the shadow price for x1=0.12$, x2=0.638$, x3=0.295$.

Concerning a problem of a loan of money it is possible to tell that this action will be not favorable at these restrictions. We won't be able to gain desirable income even on condition of increase in expenses (Sheet 1 part 2). What sum we wouldn't enclose, we won't receive more than 93,25 dollars. Obtaining the bigger income requires change as external factors (rent price), and internal (production price).