Question

20.

a.

Shipping - Product Mix  A produce dealer in Florida ships oranges, grapefruits, and avocados to New York by truck. Each truckload consists of 100 crates, of which at least 20 crates must contain oranges, at least 10 crates must contain grapefruits, at least 30 crates must contain avocados, and there must be at least as many crates of oranges as grapefruits. The profit per crate is $5 for oranges, $6 for grapefruits, and $4 for avocados. How many crates of each type should be shipped per truckload in order to maximize profit?

Let o = the number of crates of oranges,
Let g = the number of crates of grapefruits, and
Let a = the number of crates of avocados

Which option (a, b, c, or d) shows the correct objective function and constraints for this application?

		Objective Function: Maximize Profit, P = 5o + 6g + 4a Constraints: o + g + d = 100, o = 20, g = 10, a = 30, g >= o, o >= 0, g >= 0, a >= 0
		Objective Function: Maximize Profit, P = 5o + 6g + 4a Constraints: o + g + d >= 100, o <= 20, g <= 10, a <= 30, o >= g, o >= 0, g >= 0, a >= 0
		Objective Function: Maximize Profit, P = 5o + 6g + 4a Constraints: o + g + d = 100, o >= 20, g >= 10, a >= 30, o >= g, o >= 0, g >= 0, a >= 0
		Objective Function: Maximize Profit, P = 5o + 6g + 4a Constraints: o + g + d >= 100, o >= 20, g >= 10, a >= 30, o >= g, o >= 0, g >= 0, a >= 0

b.

Production A factory that manufactures knives sells sets of kitchen knives. The Basis Set consists of 2 utility knives and 1 chef's knife. The Regular Set consists of 2 utility knives, 1 chef's knife, and 1 slicer. The Deluxe Set consists of 3 utility knives, 1 chef's knife, and 1 slicer. Each Basic Set yields a $30 profit. Each Regular Set yields a $40 profit and each Deluxe Set yields a $60 profit.   The factory has 800 utility knives, 400 chef's knives, and 200 slicers on hand. Assuming that all set are available for sale, how many sets of each type should be made so that profit is maximized?

Let b = the number of Basic Sets,
Let r = the number of Regular Sets, and
Let d = the number of Deluxe Sets

Which option (a, b, c, or d) shows the correct objective function and constraints for this application?

		Objective Function: Maximize Profit, P = 30b + 40r + 60d Constraints: 2b + 2r + 3d = 800, b + r + d <= 200, 0b + r + d <= 400, b >= 0, r >= 0, d >= 0
		Objective Function: Maximize Profit, P = 30b + 40r + 60d Constraints: 2b + 2r + 3d >= 800, b + r + d <= 400, 0b + r + d <= 200, b >= 0, r >= 0, d >= 0
		Objective Function: Maximize Profit, P = 30b + 40r + 60d Constraints: 2b + 2r + 3d <= 400, b + r + d <= 800, 0b + r + d <= 200, b >= 0, r >= 0, d >= 0
		Objective Function: Maximize Profit, P = 30b + 40r + 60d Constraints: 2b + 2r + 3d <= 800, b + r + d <= 400, 0b + r + d <= 200, b >= 0, r >= 0, d >= 0

Answer 1

20) a)

option-1: incorrect, because, from 2nd ,3rd and 4th constrains are wrong as constraints are at least meaning that greater than or equal.

option 2: incorrect; from first constraint as truck load capacity is 100 crates, so it should be equal to 100. from 2nd ,3rd and 4th constrains are wrong as constraints are at least meaning that greater than or equal.

option-3 : Correct : as per the problem all constrains were modeled correctly.

option-4: incorrect, from the first constraint as truck load capacity is 100 crates. but not at least 100 crates.

b)

option 1: incorrect; first three constraints are wrong as first one should be less than or equal to 800 utility knives because one may not predict that all utility knives will sold out. the rest of two, i.e second and third constraints total availability  (i.e. RHS quantity ) of Chief's knives and slicer were inter changed.

option 2 : incorrect; first one should be less than or equal to 800 utility knives because one may not predict that all utility knives may sold out or may not.

option 3 : incorrect; first two constraints were interchanged their total availability (i.e  RHS quantity) of Utility knives and Chief's knives.

option 4: correct ; as it was modeled correctly according to the given problem.