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Chapter 4: Problem 21

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? What is the minimum cost?

Short Answer

Expert verified
Deluxe River Cruises should use 9 type-A vessels and no type-B vessels to minimize the operating cost for the 15-day cruise in May. The minimum cost is \(\$ 396,000\).

Step by step solution

01

Define the variables

Let x be the number of type-A vessels and y be the number of type-B vessels.
02

Write down the constraints

We have two constraints: 1. The total number of deluxe cabins should be at least 360: \(60x + 80y \geq 360\) 2. The total number of standard cabins should be at least 680: \(160x + 120y \geq 680\) Additionally, we have non-negativity constraints: \(x \geq 0\) \(y \geq 0\)
03

Write down the cost function

The cost function that we want to minimize is: C(x, y) = 44,000x + 54,000y
04

Solve the linear programming problem

To find the minimum cost, we can use graphical methods or various optimization techniques. In this case, we can solve graphically by plotting the constraints on a graph and finding the feasible region. The vertices of the feasible region will be the possible solutions, and we can evaluate the cost function at each vertex to find the minimum cost. 1. For constraint 1: \(60x + 80y \geq 360\), when x=0: \(y \geq 4.5\) 2. For constraint 2: \(160x + 120y \geq 680\), when y=0: \(x \geq 4.25\) By plotting these lines, we can find the feasible region. In the case of this problem, the vertices of the feasible region are (6, 4.5), (4.25, 5), (9, 0). Note that we cannot have fractional numbers of vessels, so we will take the nearest integer values of these vertices.
05

Evaluate the cost function at vertices

We will evaluate the cost function at the nearest integer values of the feasible region vertices: 1. \(C(6, 5) = 44,000 \cdot 6 + 54,000 \cdot 5 = 540,000\) 2. \(C(4, 5) = 44,000 \cdot 4 + 54,000 \cdot 5 = 476,000\) 3. \(C(9, 0) = 44,000 \cdot 9 + 54,000 \cdot 0 = 396,000\)
06

Select the minimum cost

The minimum cost is obtained for the solution \(C(9, 0) = 396,000\), which means that Deluxe River Cruises should use 9 type-A vessels and no type-B vessels to minimize the operating cost for the 15-day cruise in May. The minimum cost is \(\$ 396,000\).

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Most popular questions from this chapter

Determine whether the given simplex tableau is in final form. If so, find the solution to the associated regular linear programming problem. If not, find the pivot element to be used in the next iteration of the simplex method. $$ \begin{array}{rrrrrrrr|r} x & y & z & s & t & u & v & P & \text { Constant } \\ \hline \frac{5}{2} & 3 & 0 & 1 & 0 & 0 & -4 & 0 & 46 \\ 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 9 \\ 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 12 \\ 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 6 \\ \hline-180 & -200 & 0 & 0 & 0 & 0 & 300 & 1 & 1800 \end{array} $$

Use the technique developed in this section to solve the minimization problem. $$ \begin{aligned} \text { Minimize } & C=2 x-3 y-4 z \\ \text { subject to } &-x+2 y-z \leq 8 \\ & x-2 y+2 z \leq 10 \\ & 2 x+4 y-3 z \leq 12 \\ & x \geq 0, y \geq 0, z \geq 0 \end{aligned} $$

Use the method of this section to solve each linear programming problem. $$ \begin{aligned} \text { Maximize } & P=x-2 y+z \\ \text { subject to } & 2 x+3 y+2 z \leq 12 \\ & x+2 y-3 z \geq 6 \\ & x \geq 0, y \geq 0, z \geq 0 \end{aligned} $$

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If at least one of the coefficients \(a_{1}, a_{2}, \ldots, a_{n}\) of the objective function \(P=a_{1} x_{1}+a_{2} x_{2}+\cdots+a_{n} x_{n}\) is positive, then \((0,0, \ldots, 0)\) cannot be the optimal solution of the standard (maximization) linear programming problem.

Use the technique developed in this section to solve the minimization problem. $$ \begin{array}{ll} \text { Minimize } & C=-2 x-3 y \\ \text { subject to } & 3 x+4 y \leq 24 \\ & 7 x-4 y \leq 16 \\ & x \geq 0, y \geq 0 \end{array} $$
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