As more businesses realize the potential of mathematical optimization to help them make better decisions and allocate resources more efficiently, they may explore different techniques—such as optimization with linear programming (LP), a mathematical approach for solving problems within a set of linear constraints.

Whether organizations are solving challenges related to supply chain management, finance, logistics, or manufacturing, linear programming enables them to maximize efficiency, minimize costs, and make better, data-driven decisions.

Optimization with Linear Programming: Key Components

Linear programming is a mathematical technique for finding the best possible solution to a problem, subject to a set of linear constraints.

There are three key components in every optimization problem:

1. An Objective Function: This is what needs to be optimized (maximized or minimized). For example, a company may want to maximize profit or minimize costs.

2. Decision Variables: These variables influence the outcome, and can be adjusted to achieve optimal results. For example, in transportation and routing problems, this could be the number of vehicles assigned to a route.

3. Constraints: These are the limitations or conditions that must be satisfied (such as resource availability, time restrictions, or budget limitations).

The goal of optimization is to identify the decision variable values that optimize the objective function while satisfying all constraints.

How Is Linear Programming Used in Optimization?

Linear programming is used across industries to solve a wide range of optimization problems. Here are some common examples of real-world applications:

• Supply Chain Logistics: Many companies use linear programming to optimize inventory management, transportation scheduling, warehouse distribution, and other supply chain operations.

• Portfolio Optimization: Investment firms leverage LP for portfolio optimization to maximize returns while keeping risk at an acceptable level. By defining constraints such as capital availability and risk tolerance, LP helps create balanced investment strategies.

• Manufacturing and Production Planning: Linear programming is widely used in manufacturing to optimize resource utilization, production schedules, and workforce allocation. By minimizing waste and maximizing output efficiency, companies can improve profitability.

• Workforce Scheduling: Hospitals, airlines, call centers, and many others use LP to optimize employee schedules, balancing factors like labor availability, skill levels, and shift preferences to ensure efficient workforce deployment.

• Energy and Utilities Optimization: Energy companies apply LP to optimize electricity generation and distribution while meeting demand and regulatory constraints. This can help to reduce operational costs and enhance sustainability.

Linear Programming in Action

To illustrate a simple, real-world example of optimization with linear programming, consider a small bakery that produces croissants and muffins. The bakery wants to maximize profit, but it can only produce so much with its limited ingredients (flour and sugar) and labor hours.

Using linear programming, the bakery can determine the optimal number of croissants and muffins to bake to achieve the highest profit while staying within resource limits.

Here’s how the problem would be represented in a linear programming model:

• Decision Variables: These would be the number of croissants (x) and the number of muffins (y).

• Objective Function: Because the goal is to maximize profits, the objective function would be: Profit = 3x + 4y (where 3 and 4 are the profits per unit).

• Constraints:

  • Flour availability: 2x + 3y ≤ 100 (in kg)

  • Sugar availability: x + 2y ≤ 50 (in kg)

  • Labor hours: 3x + 2y ≤ 80

  • Non-negativity: x, y ≥ 0

This model can then be solved by an optimization solver like Gurobi to quickly identify the production levels that yield the highest profit without exceeding resource constraints.

Solving Linear Programming Problems

Optimization with linear programming is a fundamental technique that can help businesses make data-driven, efficient decisions.

By defining clear objectives, decision variables, and constraints, optimization with LP provides a structured approach to solving real-world problems across a wide range of industries.

Curious to learn more about how mathematical optimization works? Check out Gurobi’s educational resource hub to get started.