Linear programming

Definition

Linear programming is a mathematical method used to find the best possible outcome in a given situation, usually maximizing or minimizing a linear objective function, subject to a set of linear constraints. This technique is essential for modeling real-world problems where resources are limited and decisions need to be optimized. By representing complex situations with linear inequalities, linear programming allows for efficient analysis and decision-making.

5 Must Know Facts For Your Next Test

1. Linear programming can be applied to various fields such as economics, engineering, military applications, and transportation to optimize resource allocation.
2. The graphical method is one approach to solve linear programming problems in two dimensions by plotting the constraints and identifying the feasible region.
3. The Simplex algorithm is a widely-used computational method for solving linear programming problems that may involve more than two variables.
4. Linear programming assumes that relationships between variables are linear, which means they can be represented with straight lines in graphs.
5. Sensitivity analysis in linear programming assesses how changes in the coefficients of the objective function or constraints affect the optimal solution.

Review Questions

• How does linear programming enable effective decision-making in real-world problems involving limited resources?
  • Linear programming provides a systematic approach to optimize decisions by modeling complex situations with mathematical equations. It allows decision-makers to represent their objectives and constraints clearly, leading to optimal solutions that maximize or minimize desired outcomes. By using linear inequalities, it helps visualize resource limitations and evaluate various scenarios, thus guiding effective resource allocation.
• Discuss the importance of the feasible region in linear programming and how it relates to finding optimal solutions.
  • The feasible region is crucial in linear programming as it represents all possible solutions that satisfy the given constraints. It is formed by plotting the constraints on a graph, with the area where they overlap indicating valid solutions. To find optimal solutions, one must evaluate points within this region, often at the vertices where objective functions are maximized or minimized. The feasible region ensures that any identified solutions adhere to limitations imposed by real-world conditions.
• Evaluate the impact of changing constraints on the optimal solution in a linear programming model and how this relates to sensitivity analysis.
  • Changing constraints in a linear programming model can significantly affect the optimal solution and may even alter the feasible region. Sensitivity analysis is used to examine how these changes impact both the objective function's value and overall decision-making. By understanding these effects, decision-makers can assess stability and robustness of their solutions, allowing them to adapt strategies effectively when facing uncertainty or shifts in resource availability.

Related terms

Objective Function: The function in linear programming that defines the goal of the optimization problem, typically expressed as either maximization or minimization.

Constraints:

Restrictions or limitations imposed on the variables in a linear programming problem, usually expressed as linear inequalities.

Feasible Region:

The set of all possible points that satisfy the constraints of a linear programming problem, where any point within this region represents a potential solution.