Deluxe River Cruises operates a fleet of river vessels The f

Deluxe River Cruises operates a fleet of river vessels. The fleet has two types of vessels: A type -A vessels has 60 deluxe cabins and 160 standard cabins, whereas a type-B vessel has 80 deluxe cabins and 120 standard cabins. Under a charter agreement with Odyssey travel agency, Deluxe River cruises is to provide odyssey with a minimum of 360 deluxe and 680 standard cabins for their 15 - days cruise in may. It costs $44, 000 to operate a type-A vessel for that period. How many of each type of vessel (x type - A and type - B) should be used in order to keep the operating costs to a minimum?

Solution

Let the Deluxe and the standard cabins be denoted by D and S respectively. Let x number of type-A and y number of type-B vessels be provided to the odessey Travel Agency. Then the total number of D and S are xA + yB =

x (60D + 160S) + y (80D + 120S) = (60x + 80y)D + (160x + 120y)S . Since Odessey Travel Agency reuires 360D and 680S, we have 60x + 80y = 360 ...(1) and 160x + 120y = 680....(2). Ondividing bothe sides of 1st equation by 20 and the 2nd equation by 40, we have 3x + 4y = 18...(3) and 4x + 3y = 17...(4)

On multiplying the 3rd equation by 4 and the 2nd equation by 3 , we have 12x + 16y = 72...(5) and 12x + 9y = 51...(6) On subtracting the 6th equation from the 5th equation , we get 7y = 72 - 51 = 21.Therefore, y = 3. On substituting this value of y in the 3rd equation , we get 3x + 12 = 18, or, 3x = 6 or, x = 2. Thus Deluxe River Cruises should provide Odessey Travel Agency 2 type-A vessels and 3 Type B vessels.