Assignment 3: Case problem “Julia’s Food Booth” Page 1 A. Julia Robertson is making an allowance for renting a food booth at her school. She is seeking ways to finance her last year and believed that a food booth outside her school’s stadium would be ideal. Her goal is to earn the most money possible thus increasing her earnings. In this case problem, she decided to sell pizza, hotdogs and BBQ sandwiches. The following LP model illustrates the maximum net profit and constraints that will determine whether or not to least the boot. Variables:

X1 – Pizza Slices
X2 – Hot Dogs
X3 – Barbeque Sandwiches
Subject to:
$0.75x1 + $0.45x2 + $0.90x3 ≤ $1,500
24x1 + 16x2 + 25x3 ≤ 55,296 in2 of oven space
X1 ≥ x2 + x3 (changed to –x1 + x2 + x3 ≤ 0 for constraint) X1, X2, X3 ≥ 0
Solution:
Variable | Status | Value |
X1 | Basic | 1250 |

Assignment 3 Case problem “Julia’s Food Booth” Page 2 X2 | Basic | 1250 |
X3 | NON Basic | 0 |
Slack 1 | NON Basic | 0 |
Slack 2 | Basic | 5296.0 |
Slack 3 | NON Basic | 0 |
Slack 4 | Basic | 1250 |
Optimal Value (Z) | | 2250 |
Built on the above LP model, Julia is estimated that she will earn a profit of $2,250.00. After paying for the rental lease, she has earned a net profit of $1,250.00. The model suggests that she rents the booth and sell only pizza and hotdog due to her spacing constraints. This is Julia best optimal results. B. Evaluate the prospect of borrowing money before the first game. In my opinion if Julia borrowed more money she could increase her profit. Any change in a coefficient in a parameter is carefully analyzed using sensitivity analysis. This analysis identifies any effect an independent variable might have on Julia’s given constraints, in this case her budget. The increase will generate an increase in product availability...

...(A) Formulate and solve an L.P. model for this case
Variable Food Cooking Area
x1 Pizza Slice 24in sq
x2 Hot Dogs 16in sq
x3 BBQ Sandwiches 25in sq
*The oven space required for a pizza slice is calculated by dividing the total area arequired for a whole pizza by the number of slices in a pizza 14 x 14 = 196 in2, by 8, or approximately 24 in2 per slice. The total space available is the dimension of a shelf, 36 in. x 48 in. = 1,728 in2, multiplied by 16 shelves, 27,648 in2, which is multiplied by 2, the times before kickoff and halftime the oven will be filled = 55,296 in2.
Maximize Z = $0.75x1 + 1.05x2 + 1.35x3
Subject to:
$.75x1 + $.45x2 + $.90x3≤1500
24x1 + 16x2 + 25x3 ≤ 55296in sq of oven space
x1 ≥ x2 + x3
x2/x3 ≥ 2
x1,x2,x3 ≥ 0
Solution:
X1 = 1250 slices of pizza
X2 = 1250 hotdogs
X3 = 0 BBQ sandwiches
Julia would profit $2250. Her lease per game for the tent is $1000.00 and $100.00 for the warming oven. This means she still clears $1150 which is more than her $1000 minimum profit to open the concession stand.
(B) Evaluate the prospect of borrowing money before the first game.
Julia should borrow money based on the given scenario. She could increase her profit if she borrows money from a friend. The shadow price, or dual value, is $1.50 for each extra dollar that she earns. The upper limit given in the model is $1,658.88. This means that she would max out her profit at $1,658.88 of spending. Since she already has $1,500...

...Complete the "Julia'sFoodBooth" case problem on page 109 of the text. Address each of the issues A - D according the instructions given.
(A) Formulate and solve an L.P. model for this case.
(B) Evaluate the prospect of borrowing money before the first game.
(C) Evaluate the prospect of paying a friend $100/game to assist.
(D) Analyze the impact of uncertainties on the model.
The assignment will be graded using the associated rubric.
Outcome Assessed:
Create sensitivity analysis on linear programming model parameters
Communicate issues in Management Science
Grading Rubric for Assignment - Assignment #4 Case Problem
There are 12 points in each of the five criteria for a total of 60 points possible
Criteria 0 Unacceptable (0 points) 1 Developing (6 points) 2 Competent (9 points) 3 Exemplary (12 points)
1. Formulate an LP model for this case. (Part A). Did not submit or LP model is not sufficiently attempted and does not demonstrate a. recognizable attempt to model this case. LP model is partially correct, but has errors in the objective function or constraints. Described with 70 - 79% accuracy, clarity, and completeness. LP model has objective function and most constraints correctly specified. Described with 80 - 89% accuracy, clarity, and completeness. LP model has objective function and all constraints fully and correctly specified. Described with 90 - 100% accuracy, clarity,...

...Julia’sFoodBooth
A) Formulate and solve an L.P. model:
Variables:
x1 – Pizza Slices
x2 – Hot Dogs
x3 – Barbeque Sandwiches
Subject to:
$0.75x1 + $0.45x2 + $0.90x3 ≤ $1,500
24x1 + 16x2 + 25x3 ≤ 55,296 in2 of oven space
x1 ≥ x2 + x3 (changed to –x1 + x2 + x3 ≤ 0 for constraint)
x2/x3 ≥ 2 (changed to –x2 +2x3 ≤ 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 |
B) Evaluate the prospect of borrowing money before the first game.
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.
C) Prospect of paying a friend $100/game to assist
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...

...Julia’sFoodBooth
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 foodbooth 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 thebooths 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...

...A. Julia Robertson is considering renting a foodbooth at her school. She is seeking ways to finance her last year and thought that a foodbooth outside her school’s stadium would be ideal. Her goal is to earn the most money possible thereby increasing her earnings. In this case problem, she decided to sell pizza, hotdogs and BBQ sandwiches. The following LP model illustrates the maximum net profit and constraints that will determine whether or not to least the booth.
Z = $ .75(X1) + $1.05(X2) + $1.35(X3)
Given the following remains true:
$ .75(X1) + $1.05(X2) + $1.35(X3) =0
and,X2/X3 >=2; Solve for 0 for Excel: X2 >= 2(X3); X2 – 2(X3)>=0
Where X1, X2, and X3 are Pizza, Hotdog and BBQ Sandwiches respectively and are greater than 0.
Based on the above LP model, Julie is expected to earn a profit of $2,250.00. After paying for rental lease, she has earned a net profit of $1250.00. The model suggests that she rents the booth and sell only pizza and Hotdog due to her spacing constraints. This will be the best option to achieve optimal results.
B. If Julia were to borrow more money to purchase more ingredients this would change her above profit. Any change in a coefficient in a parameter is carefully analyzed using a sensitivity analysis. This analysis identifies any effect an independent variable might have on Julia’s given constraints, in this case, her...

...A. Formulate a linear programming model for Julia that will help you to advise her if she should lease the booth.
Let, X1 =No. of pizza slices,
X2 =No. of hot dogs,
X3 = No. of barbeque sandwiches
* Objective function co-efficient:
The objective is to maximize total profit. Profit is calculated for each variable by subtracting cost from the selling price.
For Pizza slice, Cost/slice=$4.5/6=$0.75
| X1 | X2 | X3 |
SP | $1.50 | $1.60 | $2.25 |
-Cost | 0.75 | $0.50 | $1.00 |
Profit | $0.75 | $1.10 | $1.25 |
Maximize Total profit Z = $0.75X1 + 1.10X2 +1.25X3
* Constraints:
1. Budget constraint:
0.75X1+0.50X2+1.00X3<=$1500
2. Space constraint:
* Total space available=3*4*16=192 sq feet =192*12*12=27,648 in- square
The oven will be refilled during half time.
Thus, the total space available=2*27,648= 55,296 in-square
* Space required for a pizza=14*14=196 in-square
Space required for a slice of pizza=196/6=32.667in-square approximately.
Thus, space constraint can be written as:
33X1 + 16X2 +25X3 <= 55,296 (In-square Of Oven Space)
3. at least as many slices of pizza as hot dogs and barbeque sandwiches combined
X1>=X2 + X3 (at least as many slices of pizza as hot dogs and barbeque sandwiches combined)
4. at least twice as many hot dogs as barbeque sandwiches
X2/X3>= 2.0 (at least twice as many hot dogs as barbeque sandwiches)
This constraint can be rewritten as:
X2-2X3>=0
X1,...

...00 |$B$10>=0 |Not Binding |1250.00 |
| |$C$10 |X2 |1250.00 |$C$10>=0 |Not Binding |1250.00 |
| |$D$10 |X3 |0.00 |$D$10>=0 |Binding |0.00 |
From the table, we get the optimum solution as follows:
X1 = 1250; X2 = 1250 and X3 = 0 and Maximum value of Z = $2250
Julia should stock 1250 slices of pizza and 1250 numbers of Hot dogs. She need not stock sandwiches.
Maximum Profit that can be expected is $2250.
Lease cost for the booth per game = $1000
Lease cost for the oven per game = $100
Net profit after all the expenses = 2250 – 1100 = $1150
Now it is clear that as per the strategy it is worth leasing the booth.
B) The sensitivity report of the solution is given below
| | | | | | | |
| | |Final |Reduced |Objective |Allowable |Allowable |
|Cell |Name |Value |Cost |Coefficient |Increase |Decrease |
|$B$10 |X1 |1250.00 |0.00 |0.75 |1...