Although it sounds like a modern-day process, linear programming is a type of mathematical problem-solving method that can be traced back to the 1930s. It has gained importance in recent years due to its application in coding and Artificial Intelligence (AI), and as a form of linear regression in data science. Though we have studied linear programming in its basic form in school and university, the answer to what is linear programming would still stump many of us. This blog will help define what is linear programming and how it can be used to solve complex real-life problems.
What is Linear Programming?
Linear programming is a way of achieving the best outcome, such as maximum profit or minimum cost, using a mathematical model represented by linear relationships. It is also known as ‘linear optimization’.
Example of Linear Programming
A farmer remains undecided on which crops to cultivate on a piece of land of size A. Given the type of soil and weather conditions, he has two options: wheat and millet. However, the farmer has constraints in terms of the amount he can invest in fertilizers (F) and pesticides (P). Let’s suppose cultivating a square meter of wheat crop requires F1 kilograms of fertilizer and P1 kilograms of pesticide, while millet crop requires F2 kilograms of fertilizer and P2 kilograms of pesticide. Let S1 be the selling price of wheat per square meter, and S2 be the selling price of millet. If we denote the area of land planted with wheat and millet by x1 and x2, respectively, then profit can be maximized by choosing optimal values for x1 and x2. It can be expressed in the following standard form:
Maximizing the objective function (in this case, it is revenue):
Given the following constraints:
Types of Linear Programming Problems
There are a variety of problems that can be solved using linear programming. However, the following types are the most common:
- Manufacturing problem: Mainly faced by production companies, this type of problem involves solving for making the maximum profit or minimum cost given various constraints like labor, output units, and machine runtime
- Diet problem: The main objective of this problem is to optimize for adequate nutrition considering the requirements of the body and the costs involved
- Transportation problem: This type of problem includes finding the right transportation solutions given the constraints of cost and time
- Resource allocation problem: This problem is concerned with managing the efficiency of the project. The primary objective is to complete the maximum number of tasks, given the constraints of man-hours and the types of resources available
Components of Linear Programming
1. Decision Variables
These are the unknown quantities in an optimization problem that need to be solved. For example, in the case of a company wanting to decide its production levels for the next twelve months, given various constraints, the production levels become the decision variables.
Constraints are the limitations one needs to consider while solving a given problem. For example, constraints can be regarding resources such as time, cost, and so on.
3. Objective Functions
Objective functions are the real-valued functions that need to be optimized for either minimum or maximum output given a set of constraints.
4. Non-Negativity Restriction
The decision variables should always take non-negative values, i.e., they should be greater than or equal to 0.
Importance of Linear Programming
Most business problems do not have straightforward solutions. Decision-making requires leaders to consider many variables and constraints, and this makes manual solutions difficult to achieve. Linear programming software helps leaders solve complex problems quickly and easily by providing an optimal solution.
The following are some of the key advantages of using linear programming:
- Attaining optimum use of resources
- A more objective way of arriving at decisions
- Ensuring due attention to bottlenecks before the problems occur
- Easy adaptation to changes in circumstances
Methods to Solve Linear Programming Problems
Common methods used to solve linear programming problems include:
- Graphical method
- Solving using R
- Using OpenSolver
- Simplex method
It should be noted that a linear program contains many variables making it near to impossible to solve using the graphical method. Hence businesses use Excel or solvers to remedy real-life problems involving linear programming.
We hope this article helped you gain insight into what is linear programming and its applications. With computer and information technology occupations projected to grow 15% from 2021 to 2031 in the U.S., there are plenty of opportunities for skilled professionals. If you are looking to advance your career in this field, check out the online courses by Emeritus to get expert guidance from top universities around the world.
By Priya S
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