Q. 32 - Precalculus Enhanced with Graphing Utilities

Question

Explain in your own words what a linear programming problem is and how it can be solved.

Step-by-Step Solution

Answer

Linear Programming Problems is a system process of finding the maximum or minimum value of any objective function under many constraints.

It can be solved by finding a feasible region for the constraints and using the corner points of the feasible region to find where the objective function is minimum or maximum and thus getting the required optimization.

Step 1. Given

A linear programming problem in two variables x and y consists of maximizing (or minimizing) a linear objective function subject to certain constraints.

Step 2. Definition

Linear Programming Problems is a system process of finding the maximum or minimum value of any objective function under many constraints.

Every linear programming problem has two components: 

1. A linear objective function that is to be maximized or minimized. 

2. A collection of linear inequalities that must be satisfied simultaneously. 

A solution to a linear programming problem consists of a feasible point that maximizes (or minimizes) the objective function, together with the corresponding value of the objective function. 

Step 3. How to solve a linear programming problem

The solution of a liner programming problem always lies in the corner points of the feasible region.

To solve a linear programming problem the steps involved are:

• First, we form the inequalities and the objective function.
• Then we graph the inequalities and find the feasible region
• We locate each corner points
• Then we find the value of the objective function for each point.
• The largest (or smallest) of these is the solution. 
• If no corner point has been found in the graph then the problem has no solution.