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A quick and practical introduction to linear programming

Updated: Apr 6

Making Better Decisions with Linear Programming

Every day, businesses and organizations face tough choices:

  • How many products should we make?

  • How do we minimize costs?

  • What’s the best way to allocate resources?

Many of these decisions involve multiple constraints—limited resources, budgets, time, or workforce availability. In this quick and practical introduction to Linear programming (LP), we will explore how it is a powerful tool that helps find the best solution while considering all these limitations.

In this post, we'll introduce linear programming, explain how it works, and show how mixed integer programming (MIP) can handle more complex decision-making needs.

What Is Linear Programming?

Linear programming is a mathematical method used to determine the best possible outcome in a given situation—whether it's maximizing profit, minimizing costs, or optimizing resource use. It does this by using a set of equations to define objectives and constraints.

A Simple Example: Maximizing Profit

Imagine a company that makes two products, A and B.

  • Each unit of Product A earns $3 profit

  • Each unit of Product B earns $2 profit

  • The company wants to maximize profit, but can only produce up to 100 total units

We can express this as:




An LP solver can find the best values for A and B that maximize profit while staying within production limits.

Key Parts of a Linear Program

✅ Objective Function – The equation you want to maximize or minimize (e.g., maximize profit).

✅ Constraints – The rules that limit the solution (e.g., production limits, budget caps).

How Do You Solve a Linear Program?

You have two main options:

1. Install and Use an Open-Source Solver (Free but Requires Setup)

These tools let you solve LP/MIP problems yourself:

💡 Pro: Free and customizable

💡 Con: Requires installation, setup, and knowledge of how to format problems

2. Use NPCompleter to Solve It for You (Easiest Option)

If you don’t want to install and configure solvers, NPCompleter offers a simple alternative:

✅ Just post your problem—someone else solves it for you

✅ All solutions are verified to ensure correctness

✅ Free for small problems (like those solvable by open-source solvers)

✅ Will support paid solutions for complex problems in the future

What If Some Variables Must Be Whole Numbers?

In many real-world problems, some decisions cannot be fractions:

  • You can’t hire 2.7 employees

  • You can’t schedule 1.5 delivery trucks

  • You can’t build half a warehouse

This is where Mixed Integer Programming (MIP) comes in.

Example: Staffing a Store

Imagine you're managing a retail store and need to decide how many full-time and part-time employees to hire to cover customer demand.

  • Full-time employees (F) work 8 hours per day

  • Part-time employees (P) work 4 hours per day

  • You need at least 40 hours of total labor

  • You must hire at least 1 full-time employee

  • You cannot hire more than 5 full-time employees

We can express this as:





An MIP solver can find the optimal combination of full-time and part-time employees that meets labor needs at the lowest cost.

Where Is Linear and Mixed Integer Programming Used?

🏭 Manufacturing – Optimize production schedules to maximize profit while reducing waste

🚚 Transportation – Find the cheapest and fastest way to move goods

💰 Finance – Create investment portfolios that balance risk and reward

🛒 Retail & Supply Chains – Manage inventory levels efficiently to avoid stockouts and overstocking

🏥 Healthcare – Assign limited medical staff efficiently across departments

Conclusion

Linear programming and mixed integer programming are powerful tools for decision-making in business, logistics, finance, and more.

🔹 If you’re comfortable with software tools, try an open-source solver.

🔹 If you want a quick, hassle-free solution, post your problem on NPCompleter!

🚀 Start solving optimization problems today!

 
 
 

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