Problem 17 - Precalculus with Limits

Question

Solving a Linear Programming Problem, use a graphing utility to graph the region determined by the constraints. Then find the minimum and maximum values of the objective function and where they occur, subject to the constraints. $$ \begin{array}{c}{\text { Objective function: }} \\ {z=3 x+y} \\ {\text { Constraints: }} \\ {x \geq 0} \\ {y \geq 0} \\ {x+4 y \leq 60} \\ {3 x+2 y \geq 48}\end{array} $$

Step-by-Step Solution

Answer

The minimum value of the function is 0 at the point (0,0) and the maximum value is 48 at points (12,12) and (16,0).

Step 1: Identify the feasible region

Draw the graph for each inequality to identify the region that satisfies all the constraints. Here, the feasible region is the area that satisfies all the constraints simultaneously. Therefore, it will be the region where all constraints overlap. This is a quadrant of the xy-plane bounded by the lines x = 0, y = 0, x + 4y = 60 and 3x + 2y = 48.

Step 2: Identify possible maximum and minimum points

The maximum and minimum values of the objective function z = 3x + y will occur at the corners of the feasible region. So, you need to identify those corner points by finding the intersections of the boundary lines. Here those points are (0,0), (0,15), (12,12), and (16,0).

Step 3: Evaluate the objective function at the corner points

Substitute the corner points into the objective function to find the z values, which will give the minimum or maximum of the function. Must get: z(0,0) = 0, z(0,15) = 15, z(12,12) = 48, z(16,0) = 48.

Step 4: Interpret the results

Based on the obtained z values, the minimum value of the objective function is 0 that occurs at the point (0,0), and the maximum value is 48 which occurs at points (12,12) and (16,0).

Key Concepts

Objective Function Optimization

Understanding how to optimize an objective function is a foundational skill in linear programming. The objective function, typically represented as z, is the mathematical expression you're trying to maximize or minimize. In the context of our example, the objective function is z = 3x + y.

To optimize it, you would identify the feasible region, determine the vertices (corner points), and then compute the value of the objective function at each vertex. The maximum or minimum values of z will occur at these vertices due to the linearity of both the objective function and the constraints defining the region. This process is relatively straightforward for two-variable problems as it can be visualized on a graph and calculated pretty easily.

Graphical Method of Linear Programming

The graphical method of linear programming is an excellent tool for visualizing and solving problems with two decision variables. It involves plotting each inequality constraint on a graph to form a visual representation of the feasible region.

For example, in our exercise, we graphed the constraints x \(\geq 0\), y \(\geq 0\), x+4y \(\leq 60\), and 3x+2y \(\geq 48\) on a coordinate plane. The area of overlap, bounded by these lines and the axes, is our feasible region. By graphing, we convert an abstract concept into a concrete visual, creating a clear path to find the solution.

Feasible Region Determination

The feasible region is the set of all possible points that satisfy all the given constraints in a linear programming problem. It represents all permissible combinations of x and y that adhere to the constraints of the problem. In our example, after plotting the inequalities, the feasible region is the quadrant of the xy-plane bound by the x-axis, y-axis, and the lines x+4y=60 and 3x+2y=48.

The method to identify this region is critically important because only within this area can the optimum value of the objective function be found. By finding where the constraints overlap, we can identify the area where all conditions are met simultaneously.

Constraints in Linear Programming

Constraints are the conditions that must be satisfied for any solution to be viable in a linear programming problem. These constraints are usually equalities or inequalities involving the decision variables. In our exercise, the constraints include x \(\geq 0\), y \(\geq 0\), x+4y \(\leq 60\), and 3x+2y \(\geq 48\).

Understanding each constraint's role is critical to determining the feasible region, as the solution to the linear programming problem lies within the space where these constraints intersect. In a graphical method, these constraints shape the boundary of the feasible region, marking the limits of potential solutions.