Systems of linear equations and inequalities can be solved using a collection of mathematical programming techniques known as linear programming, which can also be referred to as mathematical optimization. This methodology maximizes or minimizes a linear function. It is helpful in fields including technological sciences, energy, manufacturing, transportation, military, and scientific computers.
A variety of extensive and efficient linear programming tools are part of the Python environment. Simple to sophisticated tools are available, along with both free and paid options. Everything depends on your needs.
Why Linear Programming Works
You will learn the basics of linear programming in this section, as well as a related subject called mixed-integer linear programming. In the section that follows, you’ll see several actual world examples of linear programming. Python will be used later to address problems involving linear programming and mixed-integer linear programming.
What Is Linear Programming, Exactly?
Think about a set of linear equations with inequalities. Such systems usually offer a wide range of additional options. In order to discover a precise solution to this system that corresponds to the maximum or minimum of another linear function, you can use a collection of mathematical and computer techniques called linear programming.
What Does Mixed-Integer Linear Programming Mean?
Mixed-integer linear programming is a development of linear programming. It addresses issues where at least one variable lacks a continuous value in favor of an integer discrete value. Although at first appearance mixed-integer problems seem to be similar to continuous variable problems, they actually offer a number of advantages in terms of flexibility and accuracy.
When discussing values that are readily expressed as integers, such as the number of airplanes produced or customers served, integer variables are helpful.
An especially significant class of integer variable is the binary variable. It is helpful for making yes-or-no decisions, such as whether to build a plant or whether to turn on or off a machine, and it can only take the numbers zero or one. Additionally, they can emulate logical restrictions.
What Functions Does Linear Programming Serve?
A fundamental optimization method that has been used for many years in domains that require a lot of math and science is linear programming. It serves a range of helpful practical functions and is accurate and speedy.
Mixed-integer linear programming can resolve many of the limitations of linear programming. You can use semi-continuous variables to imitate logical constraints, piecewise linear functions to approximate non-linear functions, and more. Although it requires a lot of processing, advances in computer hardware and software are constantly increasing its usefulness.
The first consideration when creating and resolving an optimization problem is whether linear programming or mixed-integer linear programming can be applied.
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Linear Programming in Python
The basic approach for resolving linear programming issues is the simplex methodology, which has several variations. Another well-known technique is the interior-point approach.
More difficult and computationally expensive approaches, including the branch-and-bound method, which is based on linear programming, are used to solve mixed-integer linear programming problems. There are two variations of this procedure: the branch-and-cut method, which makes use of cutting planes, and the branch-and-price method.
There are several well-known Python tools for linear programming and mixed-integer linear programming. While some are open source and free, others are not. Whether you need a free or a paid tool depends on the size and complexity of your challenge, as well as your demand for speed and flexibility.
It is important to note that almost all commonly used libraries for linear programming and mixed-integer linear programming were created in Fortran, C, or C++. This is because linear programming demands computationally expensive work with (usually very large) matrices. Solvers are the term used to describe these libraries. The solvers are simply wrapped in the Python tools.
Linear programming examples
In this part, you’ll see two examples of linear programming problems:
- A straightforward problem that illustrates the idea of linear programming.
- A real-world application of linear programming principles to the problem of resource distribution.
Unsolved Problem in Linear Programming
A linear programming problem is infeasible if there is no solution to it. This typically happens when no solution can simultaneously satisfy all constraints.
Think about what would occur if the restriction x + y 1 was introduced. The decision variables (x or y) must then all have a negative value. The limitations x 0 and y 0 are violated by this. Such a system is referred to as infeasible because it is incapable of having a workable solution.
Another example is a second equality criterion parallel to the green line. There is no solution that satisfies both requirements because there is no shared point between these two lines.
Unbounded Linear Programming Problem
A linear programming problem is unbounded if the feasible region is not bounded and the solution is not finite. This means that the aim is boundless because at least one of your variables is unrestricted and can reach positive or negative infinity.
Take the first issue from above and remove the red and yellow restrictions. Removing restrictions from a problem is referred to as relaxing it. In this instance, x and y are not positively bound. They could be raised to a positive infinity, which would produce an infinitely large z value.
Resource Allocation Issues
In the previous sections, you explored a theoretical linear programming problem with no practical applicability. This section includes an optimization problem involving industrial resource allocation that is more concrete and applicable.