# Linear programming

Linear programming (LP) is a mathematical optimization technique used to achieve the best outcome, such as maximum profit or minimum cost, in a mathematical model whose requirements are represented by linear relationships.
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Updated: Jun 21, 2024

## 3 key takeaways

• Linear programming is an optimization method for achieving the best outcome in a model with linear relationships.
• The technique involves defining an objective function to maximize or minimize, subject to a set of linear constraints.
• Linear programming is used in diverse fields to optimize resource allocation, production planning, transportation, and more.

## What is linear programming?

Linear programming is a mathematical method used for optimizing a linear objective function, subject to a set of linear equality and inequality constraints. The objective function represents the goal of the optimization, such as maximizing profit or minimizing cost, while the constraints represent the limitations or requirements that must be satisfied.

### Example

Consider a company that produces two products, A and B. The profit from each unit of product A is $5, and from each unit of product B is$4. The company wants to maximize its profit but is limited by the availability of resources, such as labor and materials. The production constraints and the objective function can be formulated as a linear programming problem:

Objective function:
[ \text{Maximize } P = 5x_1 + 4x_2 ]

Subject to constraints:
[ 2x_1 + x_2 \leq 100 ] (labor constraint)
[ x_1 + 2x_2 \leq 80 ] (material constraint)
[ x_1, x_2 \geq 0 ]

where (x_1) and (x_2) represent the quantities of products A and B to produce, respectively.

## Key components of linear programming

### Objective function

The objective function is a linear equation that represents the goal of the optimization problem, such as maximizing profit or minimizing cost. It is expressed in terms of decision variables.

### Decision variables

Decision variables are the unknowns to be determined in the optimization problem. They represent the quantities to be optimized, such as the number of units of products to produce or the amount of resources to allocate.

### Constraints

Constraints are linear inequalities or equalities that represent the limitations or requirements of the problem. They define the feasible region within which the solution must lie. Constraints can represent resource limits, production capacities, or other restrictions.

### Feasible region

The feasible region is the set of all possible solutions that satisfy the constraints. It is a convex polytope in the decision variable space. The optimal solution to the linear programming problem lies within this region.

### Optimization

Optimization is the process of finding the best solution within the feasible region that maximizes or minimizes the objective function. This involves evaluating the objective function at the vertices of the feasible region and selecting the best value.

## Solving linear programming problems

### Graphical method

The graphical method is a simple way to solve linear programming problems with two decision variables. It involves plotting the constraints on a graph, identifying the feasible region, and evaluating the objective function at the vertices of the feasible region to find the optimal solution.

### Simplex method

The simplex method is an iterative algorithm used to solve linear programming problems with more than two decision variables. It systematically moves from one vertex of the feasible region to an adjacent vertex, improving the objective function value at each step, until the optimal solution is found.

### Interior-point methods

Interior-point methods are a class of algorithms used to solve large-scale linear programming problems. They work by exploring the interior of the feasible region rather than moving along the edges, providing an alternative to the simplex method for certain types of problems.

## Applications of linear programming

### Resource allocation

Linear programming is widely used in resource allocation problems, where the goal is to allocate limited resources to various activities to maximize or minimize an objective, such as profit, cost, or efficiency.

### Production planning

In manufacturing, linear programming helps optimize production schedules, inventory levels, and workforce assignments to meet demand while minimizing costs and adhering to constraints.

### Transportation and logistics

Linear programming is used in transportation and logistics to optimize routing, scheduling, and fleet management, reducing costs and improving efficiency.

### Financial portfolio management

Linear programming helps in financial portfolio optimization by selecting the best mix of assets to maximize returns or minimize risk, subject to investment constraints.

• Efficiency: Linear programming provides an efficient way to find the optimal solution to complex problems with multiple constraints and objectives.
• Flexibility: It can be applied to a wide range of problems in various fields, from economics to engineering.
• Simplicity: The linear nature of the model makes it relatively simple to formulate and solve using established algorithms.

• Linearity assumption: Linear programming assumes that relationships between variables are linear, which may not always be realistic in real-world problems.
• Sensitivity to data: The solutions can be highly sensitive to changes in input data, and small errors or changes can significantly impact the results.
• Scalability: While linear programming can handle large problems, extremely large or complex problems may require significant computational resources.
• Integer programming: Learn about optimization problems where some or all of the decision variables are restricted to integer values.
• Nonlinear programming: Explore optimization problems involving nonlinear relationships between variables and constraints.
• Operations research: Understand the broader field that encompasses various optimization techniques, including linear programming, for decision-making in complex systems.

Linear programming is a powerful optimization technique used to achieve the best outcome in problems with linear relationships. By understanding its components, methods, and applications, individuals and organizations can effectively utilize this tool to solve complex decision-making problems.

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