Question

True or False If a linear programming problem has a solution, it is located at a corner point of the graph of the feasible points.

Short answer

True

Step-by-step solution

Understand the Concept

A linear programming problem involves finding the maximum or minimum value of a linear objective function, subject to a set of linear inequalities. The feasible region is the set of all possible solutions that satisfy these inequalities.

Define the Feasible Region

Plotting the linear inequalities on a graph results in a feasible region, which is typically a polygon or polyhedron in higher dimensions.

Recognize Corner Points

Corner points, also known as vertices, are points where the boundaries of the feasible region intersect. These points are crucial in examining the possible solutions.

Apply the Fundamental Theorem of Linear Programming

The Fundamental Theorem of Linear Programming states that if there is an optimal solution to a linear programming problem, it occurs at one of the corner points of the feasible region.

Key concepts

Feasible Region

A feasible region is a key concept in linear programming. It is the set of all possible solutions that satisfy the given linear inequalities. Imagine you have multiple inequalities describing different conditions. When you plot these conditions on a graph, the overlapping area where all these conditions are true is called the feasible region.
This region is essential because any potential solution to the linear programming problem must lie within it.

• The feasible region is typically a polygon (a shape with many sides) in two dimensions.
• In three or more dimensions, it is called a polyhedron.

Understanding the feasible region helps visualize the problem and find solutions that meet all the constraints.

Corner Points

Corner points, also known as vertices, are crucial in linear programming for identifying optimal solutions. These are the points where the boundaries of the feasible region intersect, forming the vertices of the polygon or polyhedron.
The significance of corner points lies in their role in maximizing or minimizing the objective function.

• To find the corner points, you need to solve the system of equations formed by the intersecting lines.
• Corner points provide potential solutions that must be evaluated to find the optimal one.

Typically, the solution to a linear programming problem, if it exists, is found at one of these corner points, according to the Fundamental Theorem of Linear Programming.

Linear Inequalities

Linear inequalities form the constraints in linear programming. They describe the conditions or limits within which the solution must lie. A linear inequality looks similar to a linear equation but instead uses inequality symbols such as <, ≤, >, or ≥ instead of an equal sign.

• Linear inequalities can be represented in the form of lines on a graph.
• The solutions to these inequalities are the points that lie on one side of the line.

By combining multiple linear inequalities, we define the feasible region where all conditions are satisfied simultaneously.

Objective Function

The objective function is the function you are trying to optimize in a linear programming problem. It is usually represented as a linear equation that you want to maximize or minimize.
The objective function gives us a way to measure how 'good' a particular solution is by assigning a numerical value to it. For example:

• Maximize profit, represented by the function P = 3x + 2y where x and y are decision variables.
• Minimize cost, represented by the function C = 5x + 4y.

The goal of linear programming is to find the values of the decision variables (x and y in these examples) that result in the optimal (maximum or minimum) value of the objective function, subject to the given constraints (the linear inequalities).