Do you keep hitting a wall with linear programming problems, finding it hard to turn real-world scenarios into mathematical models, or getting tangled in constraints and objective functions? It’s frustrating when you know the formulas but still can’t reach the right solution, especially when every step feels like a guess. That constant uncertainty can make even simple problems feel like dead ends. In this article, we will break down the 6 essential steps on how to solve linear programming problems with clarity and confidence!
Linear Programming (LP) is a mathematical technique used to determine the optimal outcome in situations where relationships between factors are linear. Simply put, it's a way to maximize or minimize something, such as profit or cost, while working within a set of rules or limits, like available resources.
Before we dive into how to solve linear programming problems, it's important to understand the key benefits of LP:
A linear programming problem has several key components that help define the structure and guide the process of finding the best possible solution:
Solving a linear programming problem involves a clear, step-by-step process, from formulating the problem to finding and interpreting the solution. This approach ensures the problem is modeled accurately and the appropriate methods are used to solve it effectively.
The first step is to clearly identify your decision variables. Ask yourself: “What are the unknowns I need to decide to achieve my objective?” and define descriptive symbols for each of these unknowns. For example, in a production mix problem, you might define:
Each variable should represent a specific, measurable decision and include a clear interpretation or unit (like “number of units” or “hours worked”). In more complex problems, there could be dozens or even hundreds of decision variables, such as x₁, x₂, ..., x₁₀₀, but the process is the same.
📌 Note
By clearly naming and describing your decision variables, you set the stage for the rest of your model. All other components (the objective function and constraints) will be built using these variables. It’s a good habit to include a short explanation for each one to keep your model understandable and well-organized.
Now that you've identified your decision variables, the next step is to express the goal of the problem using those variables. This goal (what you're trying to maximize or minimize) is called the objective function. The objective function is written as a linear expression involving your decision variables. It shows how each variable contributes to the overall outcome.
If you're trying to maximize something like profit, revenue, output, or efficiency, the function might look like this: Maximize Z = c₁x₁ + c₂x₂ + ... + cₙxₙ. In this function:
For example, if each table generates a $50 profit and each chair generates a $30 profit, and you let x represent the number of tables and y represent the number of chairs, then the objective function would be: Maximize Z = 50x + 30y.
If you're trying to minimize something like cost, time, or waste, you would write: Minimize Z = c₁x₁ + c₂x₂ + ... + cₙxₙ.
For instance, if each unit of activity x costs $8 and each unit of activity y costs $5, and your goal is to minimize total cost, then the objective function would be: Minimize Z = 8x + 5y.
📌 Note
To properly define your objective function, you should keep the following key points in mind:
After writing your objective, the next step is to identify and list all the constraints that limit the values your decision variables can take. These constraints usually come from factors such as limited resources, business rules, or specific requirements.
Each constraint is typically written as a linear inequality (or sometimes an equality) involving your decision variables. The general form looks like this: a₁x₁ + a₂x₂ + ... + aₙxₙ ≤ b.
This format means that the total amount (a combination of your variables) must be less than or equal to a specified limit, such as available resources. You may also use:
For example, if each table uses 5 units of wood and each chair uses 3, and you have at most 60 units of wood, your function will look like this: 5x + 3y ≤ 60.
In most linear programming problems, it’s essential to include non-negativity constraints to ensure that decision variables cannot take on negative values unless specifically allowed. This reflects real-world situations: you can't produce a negative number of products or allocate a negative quantity of resources.
These constraints are simple to express:
Including non-negativity constraints ensures that your solution is realistic and feasible, and it is a required part of the standard form used in many solution methods, such as the simplex algorithm.
Now that your LP model is defined, it’s time to explore linear optimization methods for finding the best solution. The method you use to solve the problem depends largely on how complex or large the problem is. You can choose from:
When a problem has only two decision variables, it can be solved using a graph. To do this, you:
The solution will always lie at one of the corners (also called vertices) of the feasible region. You can either test each corner point or slide the objective line across the graph until it just touches the last possible point in the region, showing your optimal solution. This method is great for visualizing and understanding how linear programming works, but it only applies to problems with two variables.
For larger problems with numerous variables or constraints, the Simplex method is a common and powerful approach to finding the optimal solution. Instead of graphing, Simplex uses an algebraic, step-by-step approach to move from one corner (or vertex) of the feasible region to another, always improving the result at each step.
Even though the feasible region might be in many dimensions, Simplex doesn’t need to check every point. It cleverly follows a path along the edges and quickly finds the best solution. The process is often time-consuming and complex, which is why the Simplex method is typically performed using computer software that can efficiently handle thousands of variables in real-world scenarios.
One of the most common methods for solving linear programming (LP) problems is the use of computer-based tools. There are many powerful solvers that can handle these problems efficiently using algorithms like the Simplex method or interior-point methods.
Popular tools include Google OR-Tools (with its GLOP solver), COIN-OR’s CBC, and commercial options like Gurobi and CPLEX. If you prefer working in programming languages like Python, there are user-friendly libraries such as PuLP, SciPy, and Pyomo.
Now it’s time to solve the model and obtain the optimal values of the decision variables. If you’re doing this manually for a small problem, you might substitute corner-point values or run the simplex algorithm by hand. Given the formulated model from the previous steps, feed it into your chosen solver or tool. The solver will apply the algorithm internally and return the optimal solution (if one exists).
To demonstrate, we will solve our example using Python. Below is a code snippet using PuLP (a Python linear programming library) to maximize profit for the tables and chairs example:
In this code, we defined the LP with PuLP: we set up the decision variables x and y (both ≥ 0), added our objective (50*x + 30*y), and added three constraints exactly as formulated earlier. After calling model.solve(), the solver finds the optimal solution. In this example, the output will indicate:
x = 8.0, y = 6.0, Max Profit = 580.0
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