Linear Programming Problem

Definition

A linear programming problem is a mathematical optimization problem in which the goal is to maximize or minimize a linear objective function, subject to a set of linear equality and inequality constraints. This type of problem involves decision variables that are continuous, and it can be visually represented in two dimensions, making it possible to use graphical methods for solutions, especially when there are two variables. Understanding the distinction between basic and non-basic variables is crucial in formulating and solving these problems effectively.

5 Must Know Facts For Your Next Test

1. In a linear programming problem, the feasible region is defined by the intersection of the constraints, where solutions must lie to be considered valid.
2. Graphical methods for solving two-variable linear programming problems involve plotting the constraints on a graph to find the vertices of the feasible region.
3. Basic variables in a linear programming problem correspond to the variables included in the solution at any given time, while non-basic variables are those not included in the current solution.
4. The optimal solution for a linear programming problem can often be found at one of the vertices of the feasible region.
5. Sensitivity analysis can be performed after solving a linear programming problem to understand how changes in the coefficients of the objective function or constraints affect the optimal solution.

Review Questions

• How does the graphical method help in identifying the optimal solution for a linear programming problem with two variables?
  • The graphical method allows you to visualize the feasible region created by plotting the constraints on a graph. By identifying where these constraints intersect, you can find the vertices of this region. The optimal solution will occur at one of these vertices since it maximizes or minimizes the objective function within those boundaries. This visual approach simplifies understanding how different constraint combinations impact potential solutions.
• Discuss the roles of basic and non-basic variables in solving a linear programming problem and their importance during optimization.
  • Basic variables are those that are actively part of the solution and correspond to points in the feasible region, while non-basic variables do not contribute to that particular solution. During optimization, understanding which variables are basic helps to define potential solutions and adjust them accordingly. When solving problems using techniques like the Simplex method, switching between basic and non-basic variables is essential for iterating towards an optimal solution.
• Evaluate how changes in constraints can affect the feasible region and optimal solutions in a linear programming problem.
  • Changes in constraints can shift or reshape the feasible region dramatically, potentially altering where optimal solutions lie. For example, tightening a constraint could shrink the feasible area, possibly eliminating previously valid solutions. Conversely, loosening a constraint might expand options and introduce new potential optimal points. By performing sensitivity analysis post-optimization, you can determine how robust your solution is to these changes and adapt your decision-making accordingly.

Related terms

Objective Function: The function that needs to be maximized or minimized in a linear programming problem, typically expressed as a linear combination of decision variables.

Constraints: The limitations or restrictions in a linear programming problem, which can be expressed as linear inequalities or equations that the solution must satisfy.

Feasible Region: The set of all possible points that satisfy the constraints of a linear programming problem, often represented graphically as an area on a coordinate plane.