Graphical Method of Solving Linear Programming

Linear programming problems are a vital class of problem optimisation that helps in finding reliable solutions to acquire the lowest or the highest value of the function. There are mainly four types of linear programming problems. Over the years, several methods of solving linear programming have gained immense popularity and are used widely; however, they are not universal. The graphical method is one method that is universal. 

With the graphical method, any optimization linear programming issue which consists of two variables can be solved. Another significant advantage of using graphical methods over others is that it is visual and provide a clearer picture. Today, we will be discussing the graphical method to solve linear programming problems in detail. Here it begins! 

Explain Linear Programming 

Linear programming, also known as linear optimisation or LP, can be described as optimising everyday operations with certain constraints. The major objective of linear programming is to maximise or minimise the overall numerical value. It is considered one of the most important techniques for finding optimum resource utilisation. 

There are four major components of linear programming. These include decision variables, data, constraints, and objective functions. While working with linear programming, here are some assumptions that must be kept in mind for better outcomes-

• The relationship between the objective function and constraints should be linear. 
• The constraints can only be written in quantitative terms. 
• The linear functions need to be optimised.

Linear Programming Problems 

There are four different types of linear programming problems. These are as follows – 

• Optimal Assignment Problems
• Transportation Problems
• Manufacturing problems
• Diet Problems

Graphical Method – Linear Programming Problem 

We all have studied plotting points on the graph in the linear equation, which has two variables in lower classes. In linear programming, it is clear that there is more than a single simultaneous equation depending on the condition given. Here we will find solutions to the linear programming problems through the graphical method. 

Let’s understand this through an example – 

Z = 40x + 15y, 

We have, 

X+2y ≤ 100

X≥0,y

X+2y ≤ 70

Step 1: According to the given equations, x≥0 and y≥0; therefore, we will only focus on the first quadrant. 

Step 2: The linear equations will be plotted x + 2y = 100 on the graph by plotting (100,0) and (0,50) & x + y = 70 by plotting (0,70) and (70,0). 

After plotting the graph, it is important to detect the most feasible region. It is the region that is determined by the already given constraints. 

Every point on the feasible region is referred to as the choice and satisfies all the mentioned conditions. If there is any point outside the feasible region is called the infeasible solution and does not satisfy the conditions. 

According to the example mentioned above, we already know that x and y are most likely to satisfy conditions. This method can be considered for getting the optimal solution, which can be defined as any point that offers the minimum or maximum objective function’s value. Here, it is visible that several points satisfy the given constraints. 

Here is how you can get the minimum and maximum value of the objective function, which is Z = 40x + 15y?

We can follow the given theorem for finding the optimal solution – 

R here is the feasible region for a programming problem, whereas Z = Ax + By is the objective function. Here, the minimum and maximum optimal Z value will take place at the vertex, providing the optimal Z value exists. 

If the R is unbounded, there are higher chances that the maximum and minimum value of Z does not exist. However, if it does exist, it is most likely to occur at the corner point of the region R. 

Here are the corner points (70,0), (40,30), (0,0), and (0,50). If we substitute these values Z= 40x + 15y in the objective function, we get,

Z= 2050 for (40,30)

Z= 2800 for (70,0)

Z= 750 for (0,50)

Z= 0 for (0,0)

In this case, the minimum value occurring at Z is (0,0), whereas the maximum value that occurs is (70,0). 

Conclusion 

Linear programming, commonly known as linear optimisation, is often referred to as the problem to maximise or minimise the linear function which is subjected to the linear constraints. These can be either equalities or inequalities. Linear programming problems are a vital class of problem optimisation that helps in finding reliable solutions to acquire the lowest or the highest value of the function. There are mainly four types of linear programming problems. 

The best way to solve the linear programming problems is through the graphical representation method that we discussed here. The graphical method is one method that is universal. With the graphical method, any optimization linear programming issue which consists of two variables can be solved. Another significant advantage of using graphical methods over others is that it is visual and provide a clearer picture. At the end, we explained the graphical method to solve linear programming problems in step-wise for better understanding.