What Is Linear Programming? Meaning, Methods, and Examples

Linear programming is defined as a technique in algebra that uses linear equations to figure out how to arrive at the optimal situation (maximum or minimum) as an answer to a mathematical problem, assuming the finiteness of resources and the quantifiable nature of the end optimization goal. This article explains how linear programming works with examples.

What Is Linear Programming?

Linear programming is a technique in algebra that uses linear equations to determine how to arrive at the optimal situation (maximum or minimum) as an answer to a mathematical problem, assuming the finiteness of resources and the quantifiable nature of the end optimization goal. 

Linear programming (LP) uses many linear inequalities pertaining to a given scenario to determine the “optimal” value one can obtain under those constraints. A classic example would be calculating the “optimal” production levels to maximize profits, given the restrictions of supplies and personnel.

In the “real world,” linear programming is an essential subfield of mathematics known as optimization methods. This area of research (or at least its applicable findings) is used in resource allocation and management. These “real-world” systems may have dozens or even hundreds of variables. In algebra, however, you will only see the basic (and graphable) linear example with two variables.

Graphing the inequalities (called “constraints”) to construct a walled-off zone on the x,y plane is the typical method for solving linear programming problems (known as “feasibility region”). Then, you determine the dimensions of the extremities of this feasible zone (i.e., the intersection locations of the different pairs of lines) and evaluate these vertices in the equation (termed “optimization equation”) in which you’re attempting to get the maximum or minimum value.

Linear programming (LP) is among the most straightforward optimization techniques. It simplifies specific, complicated linear programming and optimization issues to help you reach a solution. Data analysts will always encounter applications and challenges requiring linear programming solutions.

Linear programming formula

A linear programming issue includes choice parameters, a nonlinear objective, constraints, and nonnegative limitations. The outcome of the LP model is determined by the choice variables x and y, which also reflect the ultimate answer.

Z is the function that must be optimized (maximized or minimized) to arrive at an answer. The constrictions are the limits placed on the variables to limit their values. According to the non-negative limitations, the variables must always possess a non-negative value.

The linear programming formula may be regarded as follows:

• The function of the formula: ax + by = Z
• The formula’s operating limitations: cx + dy ≤ e and fx + gy ≤ h
• Other, non-negative restrictions: x ≥ 0, y ≥ 0

You need to know a few terms to understand the meaning of linear programming. First come the decision variables. These elements fight for limited resources, including products, services, etc. They are referred to as decision variables if they are connected with a linear connection capable of determining the most optimal option.

The next component would be the objective function. The challenge must have had a quantitatively measurable aim, such as maximizing profit, reducing costs, etc. These constraints also apply to the available resources, such as limited equipment, people, materials, etc. Some constraints are observably existent but do not hamper the process of the studied issue; they are referred to as redundant constraints.

A viable solution is the collection of all potential solutions that fulfill the constants in the format of variables. In addition, an optimum value is the best possible solution that effectively supports the problem’s aim.

The most crucial step in addressing a linear programming issue is formulating it using the provided data. The following are the steps for solving linear programming problems:

• Determine the choice factors
• Develop the objective function 
• Determine whether the function should be decreased or maximized
• Record the limitations
• Verify that decision variables are either larger than or equal to 0. (Non-negative inhibition)
• Utilize either the simplex or graphical method to resolve the linear programming issue

See More: Top Five Free Cloud Platforms to Learn Kubernetes Online

Why is linear programming necessary?

We all encounter several target-based scenarios daily. Suppose a student has 15 days to finish an assignment, or a salesperson has one month to reach their sales quota, while another individual gets $600 to spend on an electronic device.

Let’s imagine that the student’s purpose for this endeavor is to get the highest possible grade. The salesman would strive for the highest possible monthly sales total. The purchaser of a device would want to decrease the price as much as feasible. They would attempt to purchase a device within their budget. The purpose of each situation described above is to maximize the benefits or reduce costs. Such issues are optimization challenges that may be resolved by linear programming.

An optimization issue in mathematics may include maximizing profit, minimizing costs, or minimizing resource use. We have previously described the aims of the three presented circumstances; we can now examine their respective limiting considerations. What does it imply?

In every instance, resources are scarce. In the first scenario, the deadline for completing the assignment is tight. Similarly, in example two, the individual must sell the greatest amount of things within a given time frame. In the third scenario, the individual must purchase the device within a defined budget; hence, the quantity of money is the restricting element. The lack of available resources hinders the search for optimal solutions to the presented challenges.

One cannot use the typical calculus and marginal analysis techniques in these circumstances. Calculus approaches can only handle precisely equal constraints, a limitation that linear programming does not have.

Numerous real-world applications make use of linear programming. It serves as the foundation for mathematical models that represent real-world connections. To organize and schedule production, manufacturing businesses employ linear programming extensively. To decrease travel time and fuel consumption, delivery services employ linear programming to determine the shortest route. Financial institutions establish the spectrum of investment instruments that may be supplied to customers using linear programming.

Linear programming delivers vital insights into business challenges by facilitating the identification of the ideal solution in each given circumstance.

Traits of a linear programming task

A problem being solved through linear programming will have the following traits or characteristics:

• Subject to constraints: Regarding the resource, one should represent the restrictions in mathematical form.
• Geared towards an objective function: The objective function of an issue should be described quantitatively.
• A linear relationship: The function’s connection across two or more independent variables must be linear. It indicates that the variable’s degree is one.
• Includes only finite numbers: There should be output and input numbers that are both finite and infinite. The optimum solution is not implementable if the function contains an unlimited number of elements.
• Does not include negative values: The variable’s value must be zero or positive. The value should not be negative.
• Hinges on decision variables: The result is determined by the decision variable. It provides the final solution to the issue. The first step in solving any issue is to determine the decision factors.

See More: What Is COBOL Programming Language? Definition, Examples, Uses, and Challenges

Linear Programming Methods

There are several approaches to solving linear programming problems. The four most important approaches are:

1. The simplex method 

The simplex method is a typical methodology for tackling optimization problems in linear programming. Typically, it consists of a function and some restrictions written as inequalities. The inequality defines a polygonal area, with the solution often located at a vertex. This approach is a method for systematically examining the vertices as potential solutions.

The technique approaches and finally achieves the maximum or lowest value of the goal function via an iterative procedure. In addition, the strategy aids the decision-maker in identifying duplicate restrictions, a complete solution, several alternatives, and an infeasible issue, thereby providing a thorough grasp of the business situation.

A twin problem accompanies every linear programming issue. One may easily derive the answer to this issue using the simplex approach from resolving the initial problem.

George Dantzig created the simplex technique for linear programming. Dantzig developed planning systems for the United States Air Defense during World War II using a desk calculator. In 1946, a coworker challenged him to automate the planning procedure to prevent him from accepting another position. Dantzig defined the issue as linear inequalities, although he did not include an aim in his formulation at the time. Without a goal, a wide variety of plausible options exist. Therefore, military-specific “ground rules” should be applied to identify the best possible alternative.

Dantzig’s key realization was that most such ground rules might be expressed as a linear function of objectives that must be maximized.

2. Solving linear programming problems using R

Linear programming is an excellent tool for decision-making optimization. Several R programs, such as the lpSolve R package, enable the solution of linear programming difficulties. lpSolve is an R extension that allows links to a C-based framework for linear programming problem-solving. With only a few bits of open-source code, you may get statistically significant information (sensitivity analysis).

Whereas other free optimization solutions are available, having a linear programming R code in one’s individual code repository may save a considerable amount of time by eliminating the need to start writing the formula from scratch and requiring only the modification of the coefficients and signs of the respective matrices. This is helpful since R is regularly used for data science and statistical analysis. 

3. Graphical linear programming

The technique of resolving a linear equation system by generating a graph is often referred to as the graphical method. The same holds true for linear programming issues.

Using graphical approaches, it is simple to solve any optimization programming issue with just two variables. These variables may be referred to as x1 and x2, and most of the analysis can be performed on a two-dimensional graph using these variables. The graphical approach for solving linear programming employs the extreme or corner points method and the iso-profit (cost) efficiency line method.

The iso-cost or iso-profit approach identifies the point combination that yields the same costs/profits as any other point combinations on the same line. This is accomplished by drawing parallel lines to the gradient of the equation.

4. Linear programming using OpenSolver

OpenSolver is a tool designed to solve models of linear and integer programming. OpenSolver is an Excel VBA add-on that expands the capabilities of Excel’s built-in Solver. Andrew Mason developed and updated it with students at the University of Auckland’s Engineering Science department. In addition, it permits you to resolve linear and mixed-integer optimization methods in Google Sheets without arbitrary size restrictions.

It’s important to note that almost all linear programming and mixed-integer linear programming libraries that are widely used are authored in Fortran, C, or C++ and are native to those languages. This is because linear programming requires a lot of work with (often sizable) matrices, which is hard to do in a language like Python. This kind of library is called a solver.

5. Mixed-integer linear programming

Linear programming can be made even more robust with mixed-integer linear programming. It can solve problems in which at least one variable has a discrete integer value instead of a continuous value. At first glance, mixed-integer problems look like continuous variable problems, but they are much better in flexibility and accuracy.

Integer variables are essential for accurately representing numbers expressed with integers, like the number of airplanes made or the number of customers served. The binary variable is a type of integer variable that is very important. It can only have the values 0 or 1 and helps make yes-or-no decisions, like whether or not to build a plant or turn on a machine. You can also use them to imitate logical constraints.

See More: Pivoting From Coder to Solution Architect: Four Skills and Certifications to Thrive

Examples of Linear Programming

The Linear Programming Problem (LPP) involves finding the best value of a given linear function. The ideal value may either be the largest or smallest one. Here, the linear function provided is regarded as an objective function. The objective function may include several variables dependent on the situation and must fulfill the linear constraints, a collection of linear inequalities. One may utilize linear programming issues to determine the ideal solution for manufacturing, diet, transportation, and allocation problems, among others.

Listed below are a few illustrations of the kind of issues commonly addressed by linear programming techniques:

Example 1: Optimizing dietary needs and cost constraints

A doctor wants to combine two food kinds such that the mixture’s vitamin content includes a minimum of 8 elements of vitamin A and ten elements of vitamin C. Food ‘I’ includes 2 vitamin A units per kilogram and 1 vitamin C unit per kilogram. Food ‘II’ has 1 vitamin A unit per kilogram and 2 vitamin C units per kilogram. Food ‘I’ is priced at $5 per kilogram, whereas Food ‘II’ costs $7 per kilogram. To minimize the price of such a combination, this may be expressed as a problem of linear programming.

Example 2: Optimizing food ingredients and food volume

One type of cake calls for 200g of flour and 25g of fat, but another one calls for 100g of flour and 50g of fat. This issue may be expressed as a linear programming problem to determine the highest proportion of cake that can be baked using 5kg of wheat and 1kg of fat. It also implies that there are sufficient quantities of the other cake-making components.

Example 3: Optimizing goods transportation costs

Consider a manufacturing business with two plants in cities F1 & F2 and three retail outlets in cities C1, C2, or C3. Monthly demand at retail locations is 8, 5, and 2 units, whereas monthly supply at manufacturers is 6 and 9, accordingly. Notice that the entire supply and demand are equal. We are also provided with the cost of transporting one unit from manufacturing to retail outlets. This linear programming challenge aims to estimate the amount that must be shipped from each manufacturer to each retail area to satisfy demand at the lowest possible total shipping cost.

Example 4: Optimizing product sales to arrive at maximum profit

A bakery produces two types of cookies: chocolate chip and caramel. The bakery anticipates daily demand for a minimum of 80 caramelized & 120 chocolate chip cookies. Due to a lack of raw materials and labor, the bakery can produce 120 caramel cookies and 140 chocolate chip cookies daily. For the bakery to be viable, it must sell a minimum of 240 cookies each day. Every chocolate chip cookie served generates $0.75 in profit, whereas each caramel biscuit generates $0.88. The solution to the number of chocolate chip and caramel cookies that the bakery must produce each day to maximize profit may be determined using linear programming.

See More: Cobol Programmer: Job Description, Key Skills, and Salary in 2022

Takeaway

Linear programming, like decision trees and fuzzy logic, is essential for computing algorithms. It states that, given a set of fixed resource constraints, an optimal or best solution exists. This has myriad applications in cognitive technologies such as AI or machine learning, which try and apply mathematical formulae and statistical models to real-world problems. 

Did this article adequately explain how linear programming works? Tell us on FacebookOpens a new window , TwitterOpens a new window , and LinkedInOpens a new window . We’d love to hear from you!

MORE ON IT STRATEGY

• What Is Unified Communication? Definition, System, Cloud Service, Best Practices, and Examples
• 10 Best Practices for Disaster Recovery Planning (DRP)
• Top 10 SIEM Solutions in 2022
• What is Gamification? Definition, Software, Examples, and Best Practices 2022
• Scrum vs. DevOps: Understanding the Key Differences