Problem 40 - 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

To minimize operating costs, Deluxe River Cruises should use 4 type-A vessels and 3 type-B vessels. The minimum cost is $$\$300,000$$.

Step 1: Define the variables

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

Step 2: Write the constraints involving deluxe and standard cabins

Deluxe cabins constraint: \(60x + 80y \ge 360\) Standard cabins constraint: \(160x + 120y \ge 680\)

Step 3: Create the objective function (total cost)

The objective function represents the total cost, C, to operate the vessels, and we want to minimize it: C = 44,000x + 54,000y

Step 4: Solve the system considering constraints and the objective function

We'll start by dividing the constraint equations to get simpler coefficients: Deluxe cabins constraint: \(3x + 4y \ge 18\) Standard cabins constraint: \(4x + 3y \ge 17\) Now, graph these constraint inequalities and find the feasible region where both conditions are met. The coordinates of the vertices of the feasible region are the points where the lines intersect. The vertices are (0,9), (4,3), and (17,0). Next, we substitute these vertices into the objective function to find the minimum cost: C(0,9) = 44,000(0) + 54,000(9) = 486,000 C(4,3) = 44,000(4) + 54,000(3) = 300,000 C(17,0) = 44,000(17) + 54,000(0) = 748,000 The minimum cost is $$\$300,000$$ at the point (4,3), meaning that Deluxe River Cruises should use 4 type-A vessels and 3 type-B vessels to minimize operating costs in order to meet the stated requirements of the Odyssey Travel Agency.

Key Concepts

Mathematical Optimization

Mathematical optimization is the selection of the best element from some set of available alternatives. In the context of linear programming, it refers to finding the maximum or minimum value of a linear objective function, a mathematical expression that may relate to costs, profits, or other quantities that need to be maximized or minimized. In the Deluxe River Cruises example, the objective function is the total cost of operating the vessels, which the company is aiming to minimize.

This process involves deciding how best to use limited resources, which could include anything from raw materials and money to time and manpower, in order to achieve a desired outcome while adhering to a set of predefined constraints.

Inequality Constraints

Inequality constraints define the conditions that solutions to an optimization problem must satisfy. These take the form of inequalities that represent limits or requirements that dictate feasible solutions to the problem. In the scenario given, we have two constraints: one for the number of deluxe cabins and one for the number of standard cabins. Each constraint consists of a linear inequality that must be met or exceeded, ensuring that the number of cabins provided meets the minimum demanded by the charter agreement with Odyssey Travel Agency.

Constraints are crucial in defining the 'feasible region', which is the set of all possible points that satisfy all inequality constraints. When graphed, these constraints help in visualizing and identifying the feasible region.

Objective Function

An objective function, in linear programming, is a formula used to assess the quality of any potential solution. Specifically, it defines what is being optimized and sets the criteria for evaluating the decisions made within the constraints of the problem. For Deluxe River Cruises, the objective function is the total cost of operating the fleet, which they seek to minimize. This function is expressed as a linear equation: the cost of operating each vessel (type-A and type-B) is multiplied by the number of vessels of that type in operation, and the sum of these products gives the total cost.

The goal is to find the values for the variables representing the number of vessels that will result in the lowest possible total cost within the feasible region.

Feasible Region

The feasible region in a linear programming problem is the set of all possible solutions that satisfy all the constraints of a problem. It is the graphical representation of the area where all inequality constraints overlap. Solutions outside this region are called 'infeasible' and do not meet the criteria defined by the problem.

In the Deluxe River Cruises problem, the feasible region is depicted on a graph where each axis represents the number of type-A or type-B vessels. Coordinates within this region satisfy both the deluxe and standard cabin constraints. The vertices of this polygonal region are especially important as they represent potential solutions, and for linear programming problems, the optimal solution will always reside at one of the vertices of the feasible region.

Operations Research

Operations research is a discipline that deals with the application of advanced analytical methods to help make better decisions. It involves techniques like mathematical modeling, statistical analysis, and mathematical optimization, all of which help in optimizing complex scenarios such as in resource allocation, inventory control, scheduling, and logistics.

Operations research strives to provide a framework to model the complexity of operational systems and synthesize decisions that will enhance performance. Linear programming is an essential part of operations research and is often used to solve problems involving maximization of profit or minimization of cost with a given set of resources and constraints, just like the problem faced by Deluxe River Cruises.