Julia's Food Booth
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, X2, X3 >= 0
Model:
Maximize Total profit Z = $0.75X1 + 1.10X2 +1.25X3
Subject to:
0.75X1+0.50X2+1.00X3<=$1500 (Budget)
32.67X1 + 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)
B. If Julia were to borrow some more money from a friend before the first game to purchase more
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, X2, X3 >= 0
Model:
Maximize Total profit Z = $0.75X1 + 1.10X2 +1.25X3
Subject to:
0.75X1+0.50X2+1.00X3<=$1500 (Budget)
32.67X1 + 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)
B. If Julia were to borrow some more money from a friend before the first game to purchase more