Julia's Food Booth

(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 of her own money she should borrow $158.88 from her friend, giving her an additional profit of $238.32.
(C) Evaluate the prospect of paying a friend $100/game to assist
If I were Julia I would not hire a friend for $100.00. Julia thought that it would be difficult to prepare barbeque sandwiches and hot dogs but based on the solution in section A Julia only has to prepare hot dogs. So long as Julia can prepare all of the hot dogs required on her own it is not cost effective to hire a friend. If Julia is unable to prepare all of the hot dogs that she has scheduled to sell
(D) Analyze the impact of uncertainties on the model
There is very little room for error when it comes to the margin that Julia has to meet her $1000.00 minimum profit. If everything goes as planned Julia would make $388.32 over her desired profit. If there weather were bad and the turn out were low to the game Julia would have product prepared that she could not sell and would then lose that profit. Julia may still make a profit but the likelihood that she would make her minimum every game is slim. Another uncertainty is the profit per unit. If the prices were to increase for either pizza or hot dogs it could throw off her profit and probably cause her to not meet the $1000 profit. It seems very risky to undertake the project with such a slim margin and so many uncontrollable variables that could impact the outcome.