Classic New Orleans desserts include pralines, pecan pie and Bananas Foster - all three are absolutely essential to eat on any trip to the city.Chefs may quibble about this ingredient list as there are dozens of variations on each recipe, but these ingredients are common to most formulations.A recipe for a dozen pralines takes a cup of pecans, two cups of brown sugar, a cup of granulated sugar, a half teaspoon of vanilla and a tablespoon of rum.A pecan pie takes two cups of pecans, a cup of brown sugar, a cup of cane syrup, a cup of granulated sugar, a teaspoon of vanilla, and a tablespoon of rum.A recipe for Bananas Foster requires a cup of pecans, a cup of brown sugar, a cup of granulated sugar, a banana, and a tablespoon of rum and makes two servings.Bananas Foster is considered a dessert for two and is priced accordingly at $15.Your pantry has on hand 24 cups of pecans, 40 cups of brown sugar, 12 cups of cane syrup, 20 cups of granulated sugar, 40 teaspoons of vanilla, twelve bananas, and 64 tablespoons of rum.Pralines sell for $22.68 per dozen and a pecan pie costs $22.40.Which of these is an appropriate objective function?
A)Max Z = $7.5BF + $11.34Pr + $11.20Pe
B)Max Z = $15BF + $22.68Pr 9 $22.40Pe
C)Max Z = $7.5BF + $22.68Pr + $22.40Pe
D)Max Z = $15BF + $11.34Pr + $11.20Pe

Question

Quizplus · Accepted Answer

The objective function should maximize the profit of selling pralines, pecan pies, and Bananas Foster. The total profit can be calculated as the sum of the profits earned from selling pralines, pecan pies, and Bananas Foster. 
The profit earned from selling one dozen pralines is $22.68, and the amount of brown sugar and pecans needed to make one dozen pralines is one cup each. Thus, the profit earned from selling 'Pr' dozens of pralines can be represented as $22.68Pr.
Similarly, the profit earned from selling one pecan pie is $22.4, and the amount of brown sugar and pecans needed to make one pecan pie is one cup each. Thus, the profit earned from selling 'Pe' pecan pies can be represented as $22.4Pe.
The profit earned from selling one Bananas Foster is $15, and the amount of brown sugar, pecans, and rum needed to make one Bananas Foster is one cup each and one tablespoon, respectively. Thus, to make 'BF' Bananas Foster, we need 'BF' cups of brown sugar, 'BF' cups of pecans, and '64BF' tablespoons of rum (since one Bananas Foster requires one tablespoon of rum). Since we have 12 bananas and each Bananas Foster requires one banana, we can make a maximum of 12 Bananas Foster.
Therefore, the objective function that maximizes the profit earned from selling pralines, pecan pies, and Bananas Foster is:
Max Z = $22.68Pr + $22.4Pe + $15BF
Subject to:
Pr ≤ 24 (since we have 24 cups of pecans)
Pr ≤ 40 (since we have 40 cups of brown sugar)
Pe ≤ 12 (since we have 12 cups of cane syrup)
Pe ≤ 20 (since we have 20 cups of granulated sugar)
BF ≤ 12 (since we have 12 bananas) 
BF*1 + Pr*1 + Pe*1 ≤ 40 (since we have 40 teaspoons of vanilla)
BF*64 + Pr*1 + Pe*1 ≤ 64 (since we have 64 tablespoons of rum)

Simplifying the objective function and constraints, we get:

Max Z = $22.68Pr + $22.4Pe + $15BF
Subject to:
Pr ≤ 24
Pr ≤ 40
Pe ≤ 12
Pe ≤ 20
BF ≤ 12 
BF + Pr + Pe ≤ 40
64BF + Pr ≤ 64

Therefore, the appropriate objective function is:
Max Z = $15BF + $22.68Pr + $22.40Pe.

Classic New Orleans Desserts Include Pralines, Pecan Pie and Bananas