Problem 21 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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?

Step-by-Step Solution

Answer

The optimal solution is to use 3 type-A vessels and 4 type-B vessels. The minimum operating cost for the 15-day cruise in May is \(\$216,000\).

Step 1: Define variables

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

Step 2: Construct objective function

Since we want to minimize the operating costs, we will construct the objective function as follows: $$ C = 44000x + 54000y $$

Step 3: Construct constraints

We need to satisfy the minimum cabin requirements, which gives us two constraints: 1. For deluxe cabins: $$60x + 80y \ge 360$$ 2. For standard cabins: $$160x + 120y \ge 680$$ We also need to ensure that the number of vessels used is non-negative: $$x \ge 0$$ and $$y \ge 0$$.

Step 4: Simplify constraints

To simplify our calculations, we can divide each constraint by 20: 1. For deluxe cabins: $$3x + 4y \ge 18$$ 2. For standard cabins: $$8x + 6y \ge 34$$

Step 5: Graph the constraints

Plot the constraint lines on a coordinate plane. The feasible region will be the area where all constraints are satisfied simultaneously.

Step 6: Identify corner points

There are 4 corner points of the feasible region. Determine the coordinates of each corner point: 1. Intersection of both constraints: Solve for $$x$$ and $$y$$ using substitution or elimination method. 2. Intersection of constraint 1 with $$x$$-axis: Set $$y=0$$ and solve for $$x$$. 3. Intersection of constraint 2 with $$x$$-axis: Set $$y=0$$ and solve for $$x$$. 4. Intersection of constraint 1 with $$y$$-axis: Set $$x=0$$ and solve for $$y$$.

Step 7: Evaluate the objective function at corner points

Find the value of the objective function $$C$$ at each corner point. The minimum cost will correspond to the lowest value of $$C$$.

Step 8: Determine the optimal solution and minimum cost

Compare the values of $$C$$ obtained in step 7. The optimal solution is the one with the lowest cost. Identify the number of type-A and type-B vessels corresponding to this solution and calculate the minimum cost.

Key Concepts

Objective Function

In the context of linear programming, an objective function is a mathematical expression that we aim to maximize or minimize. For example, in the Deluxe River Cruises problem, the objective function represents the total operating cost, which is given by the formula
\[ C = 44000x + 54000y \]
Here, our goal is to minimize this cost. The variables \(x\) and \(y\) correspond to the number of type-A and type-B vessels, respectively. In real-world problems, the objective function could represent various goals such as profit, distance, volume, or any other quantity subject to optimization.

Constraints

Constraints are the conditions or limits that an optimization problem must respect. In our case, they come from the minimum cabin requirements set by Odyssey Travel Agency:
\[ 60x + 80y \ge 360 \]
\[ 160x + 120y \ge 680 \]
These two inequalities ensure that enough cabins of both types are available. Additionally, we have non-negativity constraints \(x \ge 0\) and \(y \ge 0\), meaning that negative numbers of vessels are not possible. Constraints define the feasible set of solutions from which we can choose the optimal one.

Feasible Region

The feasible region is the set of all possible solutions that satisfy the constraints of the problem. This area is visually represented on a graph where feasible solutions lie within a bounded space. For Deluxe River Cruises, after plotting the constraints on a graph, the feasible region would be at or above the lines represented by the constraints, confined within the first quadrant formed by the \(x\) and \(y\) axes since the number of vessels cannot be negative.

Graphical Method of Linear Programming

The graphical method is a solution technique for linear programming problems with two variables. It involves plotting the constraint equations on a graph and identifying the feasible region. Once the feasible region is established, the vertices or corner points of the region (where constraints intersect) are evaluated using the objective function. The optimal solution is located at one of these vertices, which yields the maximum or minimum value of the objective function. In the provided cruise ship optimization, the graphical method helps illustrate the relationship between costs and the number of vessels used.

Optimization Problem

An optimization problem involves finding the best solution from a set of feasible solutions. These problems can be incredibly complex, with many variables and constraints, but linear programming offers an organized method to systematically find the optimal solution. In the context of Deluxe River Cruises, the optimization problem is to determine the number of each type of vessel to operate while minimizing the total cost within the constraints of cabin availability and non-negativity.