Linear Programming

Definition

Linear programming is a mathematical optimization technique used to solve problems involving the maximization or minimization of a linear objective function subject to linear constraints. It is a powerful tool for decision-making in various fields, including business, economics, and engineering.

5 Must Know Facts For Your Next Test

1. The goal of linear programming is to find the values of the decision variables that maximize or minimize the objective function while satisfying the given constraints.
2. Linear programming problems can be solved using various methods, such as the simplex method or the interior-point method.
3. The feasible region in a linear programming problem is a convex polyhedron, and the optimal solution is always located at one of the vertices of this polyhedron.
4. Linear programming can be used to model and solve a wide range of optimization problems, including production planning, resource allocation, transportation, and portfolio optimization.
5. The graphical method is a useful technique for solving linear programming problems with two decision variables, as it allows for the visualization of the feasible region and the optimal solution.

Review Questions

• Explain how the concept of linear programming relates to the topic of graphing linear inequalities in two variables.
  • The graphical method for solving linear programming problems is closely related to the topic of graphing linear inequalities in two variables. In both cases, the feasible region is defined by a set of linear inequalities, and the goal is to find the optimal solution within this region. The process of graphing the linear inequalities and identifying the feasible region is a crucial step in solving linear programming problems using the graphical method. Understanding the properties of linear inequalities, such as their slopes and y-intercepts, is essential for constructing the feasible region and determining the optimal solution.
• Describe how the concept of the feasible region in linear programming relates to the graphing of linear inequalities in two variables.
  • The feasible region in a linear programming problem is the set of all possible solutions that satisfy the given constraints, which are typically expressed as a system of linear inequalities. When graphing linear inequalities in two variables, the feasible region is the shaded area on the graph that satisfies all the inequalities. This feasible region corresponds to the feasible region in a linear programming problem, where the optimal solution must lie. Understanding the properties of the feasible region, such as its shape and boundaries, is crucial for solving linear programming problems using the graphical method, as it allows you to identify the optimal solution within the feasible region.
• Analyze how the concept of the objective function in linear programming relates to the optimization of linear inequalities in two variables.
  • In linear programming, the objective function is the linear function that is to be maximized or minimized. This objective function represents the quantity that the decision-maker wants to optimize, such as profit, cost, or resource utilization. When graphing linear inequalities in two variables, the objective function can be represented as a line on the graph. The goal is to find the point on the feasible region that corresponds to the optimal value of the objective function. This is achieved by moving the objective function line in the direction of optimization (e.g., maximizing or minimizing) until it reaches the boundary of the feasible region, which represents the optimal solution. Understanding the relationship between the objective function and the graphing of linear inequalities is essential for solving linear programming problems using the graphical method.

Related terms

Objective Function: The linear function that is to be maximized or minimized in a linear programming problem.

Constraints: The set of linear inequalities or equations that define the feasible region for the decision variables in a linear programming problem.

Feasible Region:

The set of all possible solutions that satisfy the constraints in a linear programming problem.