Formulation of the LPP
Suppose Julia stock X1 numbers of pizza slices, X2 numbers of hot dogs and X2 numbers of sandwiches. Constraints:
1. On the 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 during half time.
Thus total space available = 27648 x 2 = 55296
Space required for pizza = 14 x 14 = 196 sq. inches
Space required for pizza slice = 196/ 8 = 24.5 sq. inches Total Space required: 24.5 X1 + 16 X2 + 25 X3
Constraint: 24.5 X1 + 16 X2 + 25 X3 ≤ 55296

2. On the cost: Maximum fund available for the purchase = $1500 Cost per pizza slice = 6/8 = $0.75
Funds required: 0.75 X1 + 0.45 X2 + 0.90 X3
Constraint: 0.75 X1 + 0.45 X2 + 0.90 X3 ≤ 1500

3. On demand:
She can sell at least as many slices of pizza as hot dogs and barbecue sandwiches combined Constraint: X1 ≥ X2 + X3 [pic]X1 - X2 - X3 ≥ 0

She can sell at least twice as many hot dogs as barbecue sandwiches X2/X3 ≥ 2 (at least twice as many hot dogs as barbeque sandwiches) X2 ≥2 X3 [pic] X2 - 2 X3 ≥ 0

Objective Function (Profit):
Profit on pizza slice = $1.50 - $0.75 = $ 0.75
Profit on hot dog = $1.50 – 0.45 = $1.05
Profit on sandwich = $2.25 - $0.90 = $1.35
Profit function: Z = 0.75 X1 + 1.05 X2 + 1.35 X3
A) 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
Solution
Solution of the LPP by using Excel Solver gives the following Report

...Julia’sFoodBoothCaseProblem
Assignment 3
Max Z =Profit1x1+ Profit2x2+ Profit3x3
A - Formulation of the LP model
x1 - number of pizza slice
x2 - number of hot dogs
x3 - number of barbecue sandwiches
Constraints
Cost
Maximum fund available for food = $1500
Cost per pizza $6 ÷08 (slices) = $0.75
Cost for a hot dog = $0.45
Cost for a barbecue sandwich = $0.90
Constraint: 0.75x1+0.45x2+0.90x3 ≤1500
Oven space
Space available 16.3.4.2 = 384ft^2
384.144=55296 in ^2
Space required for pizza: 14.14 = 196 ^2 inches
Space for slice of pizza; 196 ÷8 = 24.50 in ^2
Space for hot dog: 16 in ^2
Space for barbecue = 25 in ^2
Constraint 24.50x1+16x2+25x3 ≤55296
Julia can sell at least as many slice of pizza (x1) as hot dogs (x2) and Barbecue sandwiches (x3) combined.
x1-x2-x3≥0
Julia can sell at least twice as many hot dogs as Barbecue sandwiches
+x2-2x3≥0
Non negative constraint
x1,x2,x3≥0
Objective Function
| SELL | COST | PROFIT |
Pizza slice (x1) | $1.50 | $0.75 | $0.75 |
Hot dog (x2) | $1.50 | $0.45 | $1.05 |
Barbecue Sandwich (x3) | $2.25 | $0.90 | $1.35 |
Profit = Sell - Cost
Max Z=0.75x1+1.05x2+1.35x3
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...

...MAT 540
Case Assignment #3
Julia’sFoodBooth
A. Formulate and solve a linear programming model for Julia that will help you advice her if she should lease the booth.
JULIA’S PRODUCT STORAGE
VARIABLES COST SELLING PRICE NEEDED
Pizza X1 $1.33 $1.50 14 inches
Hot Dogs X2 $0.45 $1.50 16 square inches
Barbecue X3 $0.90 $2.25 25 square inches
Maximize Z= $0.75x1, 1.05x2, 1.35x3
$0.75x1 + .0.45x2 + 0.90x3 = 1,500
24x1 + 16x2 + 25x3 < 55,296 in2 of oven space.
x1 > x2 + x3
>2.0
x1, x2, x3 > 0
(A) Solution: x1 = 1,250 pizza slices
x2= 1,250 hot dogs
x3= 0 barbecue sandwiches
Z= $2,250
B. 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?)
| |Final |Reduced |Objective |Allowable |Allowable |
|Name |Value |Cost |Coefficient |Increase |Decrease |
|X1 |1250.00 |0.00 |0.75 |1 |1.00000001 |...

...(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...

...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. 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,...

...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...