Please answer questions A,B,C & D Formulate the Linear Programming Model using t
ID: 387230 • Letter: P
Question
Please answer questions A,B,C & D
Formulate the Linear Programming Model using the 3 step process (Step 1: Define the decision variables, Step 2: Define the objective function, Step 3: Define the constraints)
ULIA's FoOD BOOTH Most food items are sold during the hour before th game starts and during half time; thus it will not be pos- sible for Julia to prepare the food while she is selling it. ulia Robertson is a senior at Tech, and she's investigat- She must prepare the food ahead of time and then store it g different ways to finance her final year at school. She in a warming oven. For $600 she can lease a warming oven is considering leasing a food booth outside the Tech sta- for the six-game home season. The oven has 16 shelves dium at home football games. Tech sells out every home and each shelf is 3 feet by 4 feet. She plans to fill the oven game, and Julia knows, from attending the games herself, with the three food items before the game and then again before half time 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 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 S0.45 for each hot dog and S0.90 for each bar becue sandwich if she makes the barbecue herself the night nds 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 before. She measured a hot dog and found it takes up about 112 CHAPTER 3 LINEAR PROGRAMMING: COMPUTER SOLUTION AND SENSITIVITY ANAL 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 sand- wich 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. B. If Julia w ere to borrow some more money from friend before the first game to purchase more ingre- ients, 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 (indi- ated in your answer to the previous question 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 bar- becue sandwiches. She believes that she will sell everything she can stock and develop a customer base for the season if C. When Julia looked at the solution in (A), she realized that it would be physically difficult for her to prepare Il the hot dogs and barbecue sandwiches indicated i this solution. She believes she can hire a friend of hers help her for $100 per game. Based on the resul in (A) and (B), is this something you think she could reasonably do and should do? D. Julia seems to be basing her analysis on the assump- tion 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? she follows these general guidelines for demand If Julia clears at least $1,000 in profit for each game after paying all her expenses, she believes it will be worth leasing the booth. A. Formulate and solve a linear programming model for Julia that will help you advise her if she should lease the boothExplanation / Answer
, the profit function and constraints and calculations is broken down to the equations.
Let, X1 =No of pizza slices,
X2 =No of hot dogs,
X3 = barbeque sandwiches
Formulation:
1. Calculating Objective function co-efficient:
The objective is to Maximize total profit. Profit is calculated for each variable by subtracting cost from the selling price.
X1
X2
X3
SP
$ 1.50
$ 1.50
$ 2.25
-Cost
$ 0.75
$ 0.45
$ 0.90
Profit
$ 0.75
$ 1.05
$ 1.35
The oven will be refilled during half time.
Thus, the total space available=2*27,648= 55,296 in-square
Space required for a slice of pizza=196/8=24 in-square approximately.
Therefore, Objective function for the model can be written as:
Maximize Total profit Z = $0.75X1 + 1.05X2 +1.35X3
Subject to constraints:
$0.75X1 + .0.45X2 + 0.90X3 <= 1,500 (Budget constraint)
24X1 + 16X2 +25X3 <= 55,296 (Inch square Of Oven Space)
X1>=X2 + X3 (at least as many slices of pizza as hot dogs and barbeque sandwiches combined)
X2/X3>= 2.0 (at least twice as many hot dogs as barbeque sandwiches)
This constraint can be rewritten as:
X2-2X3>=0
X1, X2, X3 >= 0
Final Model:
Maximize Total profit Z = $0.75X1 + 1.05X2 +1.35X3
Subject to:
$0.75X1 + .0.45X2 + 0.90X3 <= 1,500 (Budget)
24X1 + 16X2 +25X3 <= 55,296 (In-square Of Oven Space)
X1-X2 - X3>=0 (at least as many slices of pizza as hot dogs and barbeque sandwiches combined)
X2-2X3>=0 (at least twice as many hot dogs as barbeque sandwiches)
X1, X2, X3 >= 0 (Non negativity constraint)
X1
X2
X3
SP
$ 1.50
$ 1.50
$ 2.25
-Cost
$ 0.75
$ 0.45
$ 0.90
Profit
$ 0.75
$ 1.05
$ 1.35