Problem 2 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

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

Step-by-Step Solution

Answer

The minimization problem is solved using the graphical method. The feasible region is found by plotting the constraints: 1. \(3x + 4y \leq 24\) 2. \(7x - 4y \leq 16\) 3. \(x \geq 0, y \geq 0\) The feasible region is the convex polygon formed by the intersection of the shaded regions. The vertices of this polygon are found and the cost function C = -2x - 3y is evaluated at each vertex to find the minimum value. The optimal solution is the vertex with the minimum value of C.

Step 1: Plot the constraints on the graph

First, we'll plot the constraint inequalities on the Cartesian plane, which involve the following constraints: 1. \(3x + 4y \leq 24\) 2. \(7x - 4y \leq 16\) 3. \(x \geq 0, y \geq 0\) Note that the third constraint implies that the feasible region will lie in the first quadrant, since x and y are both non-negative. For the first two constraints, rewrite them as equalities to find the boundary lines. 1. \(3x + 4y = 24\) 2. \(7x - 4y = 16\) Plot these lines on the Cartesian plane, and since it's a linear programming problem, shade the regions that satisfy all the constraints.

Step 2: Identify the feasible region

The feasible region is the intersection of all the shaded regions from the constraints. From the plot, you should be able to see a convex polygon formed by the common shaded region of all constraints. Identify the vertices of the polygon. These vertices are the points at which the lines from the constraints intersect each other.

Step 3: Evaluate the objective function at each vertex

Now, we will evaluate the objective function C = -2x - 3y at each vertex of the feasible region. The minimum value of C will be the optimal solution.

Step 4: Determine the optimal solution

Find the minimal value of C among the results obtained from evaluating the objective function at each vertex. The vertex that gives the minimal value is the optimal solution to the minimization problem. Finally, write down the coordinates of the optimal solution and the minimum value of the objective function C.

Key Concepts

Optimization

In the world of linear programming, optimization is the process of finding the best, most efficient solution to a problem, given a set of constraints. In mathematical terms, optimization involves maximizing or minimizing an objective function. For instance, a business may want to minimize costs or maximize profits, and this goal is expressed through an optimization problem.

To solve an optimization problem, we first translate the real-world scenario into a mathematical model. This involves setting up an objective function that we aim to optimize, while considering certain constraints that restrict our solutions to a feasible set. In the given exercise, the task was to minimize the cost function, C, subject to constraints involving the variables x and y, which could represent quantities like resource allocation in a business setting.

Feasible Region

The feasible region is a cornerstone concept in linear programming. It refers to the set of all possible points that satisfy the problem's constraints, representing potential solutions to the optimization problem. In a geometric sense, these points form a shape on the graph, typically a polygon in two dimensions. Each corner or vertex of the feasible region represents a potential solution to the problem.

When dealing with a minimization or maximization problem, the best solutions often lie at the vertices of the feasible region. This is known as the corner-point principle. During the process of solving a linear programming problem, after plotting the constraints on a graph, the feasible region will emerge as the common overlap where all constraints are met. Only from within this region can the optimal solution be identified.

Constraint Inequalities

The constraints of a linear programming problem are formalized as constraint inequalities. These inequalities define the limits within which the variables of the problem can operate. They are essential in forming the feasible region, as each inequality represents a border of the region.

In the example problem, we have inequalities like \(3x + 4y \leq 24\) and \(7x - 4y \leq 16\), along with non-negativity constraints \(x \geq 0\) and \(y \geq 0\). These specify the allowable combinations of x and y. Constraint inequalities generally create straight lines on a graph when written in equality form. The solution space to which they confine the problem is pivotal in determining where the optimal solution lies. It is by testing the points of intersection — or vertices — within the feasible region outlined by these inequalities that we approach finding the optimal solution.

Objective Function

The objective function is a mathematical expression that defines the goal of an optimization problem, usually in terms of cost, profit, or some other quantity that needs to be minimized or maximized. It usually has the general form \(Z = ax + by\), where Z represents the objective function, x and y are the decision variables, and a and b are the coefficients reflecting their contribution to the outcome.

In the context of our exercise, the objective function to be minimized was \(C = -2x - 3y\). Evaluating this function at each vertex of the feasible region allowed us to find where C has its smallest value, thus determining the solution to the minimization problem. The optimal values of x and y yield the highest or lowest possible outcome that is still within the bounds set by the constraint inequalities.