In the Field of Data Science, What Exactly is Meant by the Term “Linear Programming”?

What Exactly is Meant by the Term "Linear Programming"?
Data Science – What Exactly is Meant by the Term “Linear Programming”?, Data science has evolved into a truly interdisciplinary field that draws inspiration from a variety of other disciplines, including computer science, mathematics, data analysis, and statistics, among others. Because of its advancements, businesses all over the world are now able to make decisions that are significantly better informed and supported by data.

Because of this, businesses of today understand the significance of the data they have accumulated over the course of their existence.

Data scientists make use of cutting-edge software to analyze ongoing business scenarios by analyzing existing data, deriving relationships, and locating patterns that are illuminating. Descriptive analytics is the name given to this methodology. In addition, data scientists investigate the effects as well as the factors that led to them. This line of inquiry, known as Predictive Analytics, requires them to think about a variety of dependent and independent variables.

Because it works by identifying cause and effect relationships, Predictive Analytics is beneficial for making insightful decisions for the future. [Cause and effect relationships] Nevertheless, there is more complexity to this than first meets the eye. Every company must navigate a complex landscape of shifting conditions, which may include shifting perspectives, bounded resources, and other factors.

In order to make accurate predictions, you need to take into account all of these different variables and determine the best possible solution. Linear programming enters the picture at this point to help solve the problem. The process of linear programming is an important technique that helps data scientists find the most optimal solution for a variety of problems. This technique operates in an algorithmic fashion. In order to arrive at the optimal solution, linear programming takes into account all of the critical variables, as well as equalities and inequalities. This guarantees that the prediction will always be accurate.

In this article, we will investigate what linear programming is, the various linear programming methods, and a sample linear programming problem!

In the field of Predictive Analytics, Linear Programming

Before diving into the specifics of the methodology, it is essential to clarify that the term “programming” in the context of linear programming does not refer to the process of writing code for a computer or software. On the other hand, Linear Programming is fundamentally an optimization method (known as Linear Optimization) that assists in determining the mathematical models that will produce the best possible results. In order to create a linear program, it is critical to have a solid grasp of the fundamentals of linear programming, which can be broken down into the following categories:

  • The terms “Decision Variables” and “Unknowns” refer to the variables that we would like to figure out, respectively.
  • The objective function is the linear function that represents the quantities that need to be minimized or maximized, depending on the situation.
  • The term “constraints” refers to a collection of inequalities or equality conditions that together represent all of the constraints placed on our decision variable.
  • The values of the decision variables must not be negative in order for this to be a valid restriction. This phrase is used to refer to an essential point of constraint.
  • Now that we have a firm grasp on the fundamentals, let’s examine the various strategies that can be utilized when attempting to solve a problem involving linear programming.

Obtaining Solutions for Linear Programming

To successfully solve a problem involving linear programming, we can proceed in the following four steps:

  • Identifying decision variables
  • Establishing the role of the objective function
  • Defining the parameters of the restrictions
  • Noting the restrictions on the absence of a negative

When we return to this topic later and examine a solved example of linear programming, we will go into greater depth regarding these steps. But before we get to that, let’s take a look at the different ways you can approach a problem involving linear programming.

There are basically four different approaches that can be taken:

Graphical Method The graphical method is the method that is used to solve a problem in linear programming with two variables. It is the most fundamental method. It is most commonly utilized in situations in which there are only two decision variables to take into consideration. The formation of a set of linear inequalities and their subsequent submission to the applicable conditions or constraints are the two main steps of the graphical method. After that, the equations are plotted on the X-Y plane, and the area that results when all linear equations are plotted is the feasible area. This region provides the best possible solution and reveals the values that a model predicts.

Simplex Method: This is a powerful method for solving Linear Programming problems, and it utilizes an iterative procedure to arrive at the best possible answer to the problem. In this method, the essential variables are changed one at a time until the maximum or minimum value (depending on the requirements) for the primary objective function is reached.

Both the Northwest Corner Method and the Least Cost Method are specific types of methods that are primarily utilized for the purpose of resolving transportation-related issues and determining the most efficient way to transport products or goods. As a consequence of this, this optimization method is useful for addressing supply-demand issues. The use of this method is predicated on the idea that there is just a single product. On the other hand, there are a variety of origins for the demand for this product, and these origins contribute individually and collectively to the total supply. As a result, the objective of this method is to reduce the cost of transportation as much as possible.

Using R to solve problems: R is one of the tools that is utilized most frequently in the fields of data science and data analysis. Utilizing the IpSolve package in R makes it very simple to carry out optimization using only a few lines of code at the most.

Using open-source software to solve problems Open-source software is used in the final method, which solves optimisation problems using one of the many open-source tools available. One example of an open-source tool is OpenSolve, which is a linear optimiser for Excel that can work without a hitch for as many as one hundred different variables. In addition to that, CPLEX, MATLAB, Gurobi, and other similar programs are examples of other helpful open-source tools.

A Simplified Example of Graphical Linear Programming Solving

When it comes time to put together a user pack for the holiday season, a certain company takes into consideration two factors, X and Y. The total weight of the package must be at least 5 kilograms, and there can be no more than 4 kilograms of Y and at least 2 kilograms of X. The following is how much X and Y each contribute to the overall profit: Rs. 5 / kg for X, and Rs. 6 / kg for Y.

Let’s put our linear programming skills to use and see if we can find the optimal combination of products and services that will bring the most revenue to the business.

  1. Continuation of our work with our primary purpose

The maximization of profits is the target of our optimization effort for this problem. The problem statement provides us with information regarding X and Y’s respective contributions to the profit. Now,
Take one kilogram of X.
Allow b kilograms of Y
Our objective function is then rewritten as -> c = 5a + 6b, and our goal is to achieve the highest possible value for c.
As decision variables, we have a and b, and the required function for this situation is c.

  1. Deriving the limitations from the aforementioned problem

In order to help us solve the problem, we have been given the following constraints:
It is required that the gift pack weigh 5 kilograms, so a + b = 5
Therefore, if X weighs at least 2 kilograms and Y weighs less than 4 kilograms, we can deduce that X is greater than 2 and Y is less than 4.

  1. Constraints that are not negative

Both X and Y should have positive quantities, which means that a and b should be greater than 0.
Now, in a few words, let us summarize the entire problem as we have described it up until this point:

It is necessary for us to perform optimization on the formula c = 5a+6b under the following two conditions:

In order to solve this issue, we will be employing the graphical method, so let us first consider a two-dimensional graph with an X-Y axis and then make an attempt to plot the equations and inequations. The following items will be available to us at all times:
The straight line a + b = 5 cuts the x-axis at the point (5,0), and it cuts the y-axis at the point (0,5). Since there is an equality sign in the expression that we are working with, we know for a fact that the area where these lines intersect is where our feasible region will be found.
If a and b are equal, then the x-axis is cut by a straight line (2,0). As a result of the fact that our expression contains a constraint that reads “greater than,” the feasible region lies to the right of our line.

The line b = 4 is a straight line that cuts the y axis at the point (0,4). Because this is a less severe constraint, the region below the line represents the feasible space for us.

Because both a and b are considered to have a positive value, our focus should be on the first quadrant of the graph.

After plotting these lines and constraints on a graph sheet, you will have the final region that satisfies all of the necessary conditions. This region will be the one that you choose. The two points that are located on the end of this line at its most extreme are potential factors to take into consideration when trying to maximize profits. These are points number two and three (5,0). To determine which of these two options results in higher profits, all we need to do is plug the relevant data into our objective function and look to see which option provides the best output:
c = 5a + 6b ⬄ c = (52) +(63) = 28
c = 5a + 6b ⬄ z = (55) +(60) = 25
As can be seen, the option A yields a higher profit value for us to work with. Our solution, which results in the greatest amount of profit, is as follows: 2 kilograms of factor X and 3 kilograms of factor Y!

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