python linear programming

It’s important in fields like scientific computing, economics, technical sciences, manufacturing, transportation, military, management, energy, and so on. Decision variables are restricted to satisfy a set of linear equality and/or inequality constraints. Notice that the shape of this vector is (3,) and not (3, 1) or (1, 3).

Discrete optimization is a branch of optimization methodology which deals with discrete quantities i.e. non-continuous functions. It is quite ubiquitous in as diverse applications such as financial investment, diet planning, manufacturing processes, and player or schedule selection for professional sports. In Python there are many libraries (CVXOPT, PULP, CVXPY, ECOS, etc.) that basically come with their own LP solvers or act as wrappers around other LP solvers. Thus one can choose a wrapper-solver combination depending on the problem at hand.

Such a system doesn’t have a feasible solution, so it’s called infeasible. In this section, you’ll learn the basics of linear programming and a related discipline, mixed-integer linear programming. python linear programming In the next section, you’ll see some practical linear programming examples. Later, you’ll solve linear programming and mixed-integer linear programming problems with Python.

The output of the above code is Optimal which tells us that our model has been able to find an optimal solution to the problem. We can initialize the model by calling LpProblem() function. The first argument in the function represents the name we want to give to our model. The second argument tells our model whether we want to minimize or maximize our objective function. I’d recommend the package cvxopt for solving convex optimization problems in Python. A short example with Python code for a linear program is in cvxopt’s documentation here.

As you can see, this list contains the exact objects that are created with the constructor of LpVariable. You use the sense parameter to choose whether to perform minimization (LpMinimize or 1, which is the default) or maximization (LpMaximize or -1). This system is equivalent to the original and will have the same solution. The only reason to apply these changes is to overcome the limitations of SciPy related to the problem formulation. You need to find x and y such that the red, blue, and yellow inequalities, as well as the inequalities x ≥ 0 and y ≥ 0, are satisfied.

python linear programming

If a single tuple (min, max) is provided, then min and
max will serve as bounds for all decision variables. As discussed earlier, the optimal solutions to linear programming problems lie at the vertices of the feasible regions. In this case, the feasible region is just the portion of the green line between the blue and red lines. The optimal solution is the green square that represents the point of intersection between the green and red lines. In this tutorial, you’ll use SciPy and PuLP to define and solve linear programming problems. Mixed-integer linear programming problems are solved with more complex and computationally intensive methods like the branch-and-bound method, which uses linear programming under the hood.

Python Tutorial

However, we would need to install them on top of OR-Tools and get the appropriate licenses (which can be quite costly). Linear programming is a technique to optimize any problem with multiple variables and constraints. It’s a simple but powerful tool every data scientist should master. If you found this article useful, feel welcome to download my personal codes on GitHub. You can also email me directly at and find me on LinkedIn.

But this way we avoid the direct multiplication and keep the problem structure linear. Then we write a special code to link the usual food_vars and the binary food_chosen and add this constraint to the problem. The popular machine learning technique Support Vector Machine essentially solves a quadratic programming problem. The code is almost identical as before, so it is not repeated here. The only difference is that the variables are defined as belonging to Integer category as opposed to Continuous. As we can see that the optimal result came back with a set of fractional numbers of servings for the food items.

The second element is a human-readable name for that constraint. The optional parameter cat defines the category of a decision variable. If you’re working with continuous variables, then you can use the default value “Continuous”. Once you install it, you’ll have everything you need to start. Its subpackage scipy.optimize can be used for both linear and nonlinear optimization.


These two lines wouldn’t have a point in common, so there wouldn’t be a solution that satisfies both constraints. The next step is using PuLP to create our solver and find the maximum coordinates values for the end of the diagonal line. Books to learn programming languages typically follow a grammar book format. Each chapter covers a different aspect of coding grammar (such as data structures, loops, etc.) with brief examples that frequently use mathematics. Finding course books that gradually teach to think in a coding language is rare.

In fact, integer programming is a harder computational problem than linear programming. Integer variables make an optimization problem non-convex, and therefore far more difficult to solve. Memory and solution time may rise exponentially as you add more integer variables.

Modeling Steps

I’ll receive a portion of your membership fee if you use the following link, with no extra cost to you. The full solution contains all the variables including the ones with zero weights. But to us, only those variables are interesting which have non-zero coefficients i.e. which should be included in the optimal diet plan. So, we can scan through the problem variables and print out only if the variable quantity is positive. In any optimization scenario, the hard part is the formulation of the problem in a structured manner which is presentable to a solver. But before going to the Python library, let us get a sense of the kind of problem we can solve with it.

You’ll see how to use GLPK (in addition to CBC) with PuLP later in this tutorial. To follow this tutorial, you’ll need to install SciPy and PuLP. The examples below use version 1.4.1 of SciPy and version 2.1 of PuLP. Integer variables are important for properly representing quantities naturally expressed with integers, like the number of airplanes produced or the number of customers served. Imagine we have a square field of 100 meters by 100 meters where we must draw the longest 45 degrees diagonal line starting at point 0 up to the upper right corner. However, let’s imagine that there is a fence 5 meters away from the upper margin and another fence 5 meters away from the right margin.

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In this case, there’s an infinite number of feasible solutions. Mixed-integer linear programming is an extension of linear programming. It handles problems in which at least one variable takes a discrete integer rather than a continuous value. Although mixed-integer problems look similar to continuous variable problems at first sight, they offer significant advantages in terms of flexibility and precision.

Here, solution contains the value of the objective function, and x.value is the column matrix, containing the values of xi. The problem is defined by the objective function and the constraints. Leonard Kantrovich was awarded the 1975 Nobel Prize in Economics for the optimal allocation of resources using linear programming. A vector \(x\) that satisfies all of the constraints is called a feasible solution.

This is perfectly acceptable because the linear program is trying to simultaneously satisfy all the conditions we encoded, including our newly imposed bounds for each of the ingredients. The total is still 700g, butter is still half of sugar, butter is still less than 100g (a bound we imposed), flour +butter is still more than or equal to eggs + sugar and so on. It aims at satisfying every single condition it was passed, therefore returning the most optimum solution. SciPy in Python offers basic linear programming capabilities. To implement the above program using SciPy, we need to define all matrices accordingly.

  • It handles problems in which at least one variable takes a discrete integer rather than a continuous value.
  • Then we write a special code to link the usual food_vars and the binary food_chosen and add this constraint to the problem.
  • The meaning of the name “ Sequential search ” is definitely justified by the process followed by this search algorithm.
  • It is quite ubiquitous in as diverse applications such as financial investment, diet planning, manufacturing processes, and player or schedule selection for professional sports.

For a maximization problem, we can first transform it to an equivalent minimization problem and then follow the above steps above to construct the dual minimization problem. The optimal plan tells the factory to produce 2.5 units of Product 1 and 5 units of Product 2; that generates a maximizing value of revenue of 27.5. The following graph illustrates the firm’s constraints and iso-revenue lines.

The only time a graph is used to solve a linear program is for a homework problem. In all other cases, linear programming problems are solved through matrix linear algebra. Also, because the residual on the first inequality constraint is 39, we
can decrease the right hand side of the first constraint by 39 without
affecting the optimal solution. As you can see, the optimal solution is the rightmost green point on the gray background.

Compute the dot product of two or more arrays in a single function call, while automatically selecting the fastest evaluation order. In order to have a linear system, the values K₁, …, K₉ and b₁, b₂, b₃ must be constants. Now that you’ve gone through creating arrays, you’ll see how to perform operations with them. Here, the -1 that you provide as an argument to .reshape() represents the number of rows necessary for the new array to have just one column, as specified by the second argument.

Now that we have our variables and constraints, we want to define our goal (or objective function). Since the linprog function from Python’s SciPy library is programmed to solve minimization problems, it is necessary to perform a transformation to the original objective function. Every minimization problem can be transformed into a maximization problem my multiplying the coefficients of the objective function by -1 (i.e. by changing their signs). As for Python, while there are some pure-Python libraries, most people use a native library with Python bindings.

There is a wide variety of free and commercial libraries for linear programming. For a detailed list, see Linear Programming in Wikipedia or the Linear Programming Software Survey in OR/MS Today. Fortunately, the Python ecosystem offers several alternative solutions for linear programming that are very useful for larger problems. One of them is PuLP, which you’ll see in action in the next section. The order of the coefficients from the objective function and left sides of the constraints must match. The objective function (profit) is defined in condition 1.

There you will see how it reads the problem logically as a math problem. Maximize function x+y subject to x and Y being less than 95. Like previously mentioned, the objective function is our goal, what we want to accomplish, what we’re trying to solve. There are a couple of LP solvers packages available in Python. Or you could simply write your problem in Python and generate an MPS file (which most standard LP/MILP (CPLEX, Gurobi, GLPK) solvers will accept).

There are many libraries in the Python ecosystem for this kind of optimization problems. PuLP is an open-source linear programming (LP) package which largely uses Python syntax and comes packaged with many industry-standard solvers. It also integrates nicely with a range of open source and commercial LP solvers. More generally, the values of the standard objective function (compatible with its constraints) are always ≤ than the values of the dual objective function (compatible with its constraints). In other words, the dual problem provides an upper bound to the optimal value of the original problem.

In hierarchy, linear programming could be considered as the easiest operations research technique. Mixed-integer linear programming allows you to overcome many of the limitations of linear programming. You can approximate non-linear functions with piecewise linear functions, use semi-continuous variables, model logical constraints, and more. It’s a computationally intensive tool, but the advances in computer hardware and software make it more applicable every day. Besides offering the basic operations to work with matrices and vectors, NumPy also provides some specific functions to work with linear algebra in numpy.linalg. However, for those applications, scipy.linalg presents some advantages, as you’ll see next.