Deluxe River Cruises operates a fleet of river vessels. The fleet has two types of vessels: A type-A vessel has 50 deluxe cabins and 190 standard cabins, whereas a type-B vessel has 80 deluxe cabins and 140 standard cabins. Under a charter agreement with Odyssey Travel Agency, Deluxe River Cruises is to provide Odyssey with a minimum of 340 deluxe and 800 standard cabins for their 15-day cruise in May. It costs $44,000 to operate a type-A vessel and $54,000 to operate a type-B vessel for that period. How many of each type vessel should be used in order to keep the operating costs to a minimum? What is the minimum cost?
A) 3 type-A vessels; 2 type-B vessels; $240,000
B) 0 type-A vessels; 6 type-B vessels; $324,000
C) 2 type-A vessels; 3 type-B vessels; $250,000
D) 7 type-A vessels; 0 type-B vessels; $308,000

Question

Quizplus · Accepted Answer

Let x be the number of type-A vessels and y be the number of type-B vessels needed to meet the minimum requirement.

The objective function is to minimize the operating cost, given by:

C = 44000x + 54000y

The constraints are:

50x + 80y ≥ 340 (minimum number of deluxe cabins)
190x + 140y ≥ 800 (minimum number of standard cabins)

Simplifying these constraints, we get:

5x + 8y ≥ 34
19x + 14y ≥ 40

Solving these constraints, we get:

x ≥ 2
y ≥ 3

So, the minimum integers that satisfy these constraints are x = 2 and y = 3.

Substituting these values in the objective function, we get:

C = 44000(2) + 54000(3) = $250,000

Therefore, the best choice is C, where 2 type-A vessels and 3 type-B vessels should be used to keep the operating costs to a minimum, and the minimum cost is $250,000.

Deluxe River Cruises Operates a Fleet of River Vessels