Julia’s Food Booth

(A) Formulate and solve an LP model for this case.
The objective here is to maximize profit. Profit is calculated for each variable by subtracting costs from the selling price. The decision variables used are X1 for pizza slices, X2 for a hotdog, and X3 for the BBQ sandwich.

X1 (pizza)
X2 (hotdog)
X3 (sandwich)

Sales Price
1.50
1.50
2.25

Cost
0.75
0.45
0.90

Profit
0.75
1.50
1.35

For Pizza Slice: Cost/Slice = $6/8 = $0.75 cost per slice
Maximize Z = 0.75 X1 + 1.05 X2 + 1.35 X3
Constraints:
Budget: 0.75X1 + 0.45X2 + 0.90X3 ≤ 1500
Oven Space: 24X1 + 16X2 + 25X3 ≤ 55,296 in2
The calculation for the oven space is as follows:
Pizza slice total space required for a 14 * 14 pizza = 196 in2. Since there are eight slices, we divide 196 by eight, and this gives us approx. 24 in2 per slice. The total dimension of the oven is the dimension of the oven shelf, 36 in * 48 in = 1728 in2, multiplied by 16 shelves = 27,648 in2, which is multiplied by 2, before kickoff and during the halftime, giving a total space of 55,296 in 2.

(B) Evaluate the prospect of borrowing money before the first game. The shadow price or dual value is $1.50 for each additional dollar Julia would increase her profit if she borrows some money. However, the upper limit of the sensitivity range is $1,658.88, so she should only borrow $158.77 and her additional profit would be $238.32 or a total profit of $2488.32.
(C) Evaluate the prospect of paying a friend $100/game to assist. Yes, she should hire her friend for $100/game for it is almost impossible for her to prepare all the food in such a short time. In order for Julia to prepare the hotdogs and barbeque sandwiches, she would need the additional help. With Julia being able to borrow the extra $158.88 she would be able to pay her friend.
(D) Analyze the impact of uncertainties on the model. The impact of uncertainties such as weather (too sunny, rainy, or cold), competition, increase in food cost, and the attendance at each of the six games could reduce the demand for the items sold by Julia. If it is raining or cold then there may not be as many patrons at the games and if it is too hot people may not want to eat before or during the games. The higher the uncertainties the demand shifts, therefore the solution of the LP model will change and so does her profit. She will not be able to produce a $1000 profit under high uncertainty.