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Problem 12

# TraNSPoRTATION Deluxe River Cruises operates a fleet of river vessels. The fleet has two types of vessels: A type-A vessel 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 -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?

Expert verified
The graph of the constraints shows a feasible region with three corner points: (4, 3), (6, 1.5), and (0, 8.5). Evaluate the objective function at each corner point: 1. Corner point 1: C(4, 3) = 44,000(4) + 54,000(3) = \$302,000 2. Corner point 2: C(6, 1.5) = 44,000(6) + 54,000(1.5) = \$312,000 3. Corner point 3: C(0, 8.5) = 44,000(0) + 54,000(8.5) = \$459,000 The minimum cost is \$302,000, which occurs when 4 type-A vessels and 3 type-B vessels are used.
See the step by step solution

## Step 1: Graph the Constraints

First, we need to graph the constraints to find the feasible region. The constraints are: 1. $$60x + 80y \geq 360$$ OR $$y \geq (\frac{9}{4} - \frac{3}{4}x)$$ 2. $$160x + 120y \geq 680$$ OR $$y \geq (\frac{17}{3} - \frac{4}{3}x)$$ 3. $$x \geq 0$$ (non-negative constraint) 4. $$y \geq 0$$ (non-negative constraint) Plot these lines on a graph and shade the area that represents the feasible region.

## Step 2: Find the Corner Points

Next, we need to find the corner points of the feasible region, where our minimum cost might occur. There are three corner points: 1. The intersection of lines 1 and 2: $$60x + 80y = 360$$ and $$160x + 120y = 680$$ 2. The intersection of lines 1 and 4: $$60x + 80y = 360$$ and $$y = 0$$ 3. The intersection of lines 2 and 3: $$160x + 120y = 680$$ and $$x = 0$$ Solve these systems of equations to find the coordinates of the corner points.

## Step 3: Evaluate the Objective Function

Now evaluate the objective function $$C(x,y) = 44000x + 54000y$$ at each corner point: 1. Corner point 1: $$(x_1, y_1)$$ 2. Corner point 2: $$(x_2, y_2)$$ 3. Corner point 3: $$(x_3, y_3)$$

## Step 4: Determine the Minimum Cost

Now, we will find the minimum cost by comparing the values of the objective function at each corner point. The lowest value of the function will be the optimal number of each type of vessel which minimizes the operating costs. Find the minimum value of C(x,y) and the corresponding values of x and y that give the minimum cost. The optimal solution will provide the number of each type of vessel to be used to keep the operating costs to a minimum.

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