Types of Linear Programming Problems

Linear programming can be described as a mathematical technique for optimising operations with a given set of constraints. The primary 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. Several problems related to minimising cost, maximising profits, and so on are parts of linear programming problems. We will briefly discuss them in this article.

This article talks about different types of linear programming problems. You will find brief information on the concept of linear programming in maths and types of linear programming problems, along with their solutions. 

What is linear programming? 

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

Here are some assumptions to keep in mind while working with linear programming: 

• The constraints can only be quantitatively expressed. 

• The linear functions should be optimised. 

• The objective function and constraints relationship should be linear. 

Components of linear programming 

There are four major components of linear programming. These include:

1. Data

2. Constraints

3. Decision variables

4. Objective functions

Different types of 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

 Let us discuss each of these in detail: 

• Optimal assignment problems

As the name suggests, the optimal assignment problems are mainly concerned with the completion of a task given to a group of employees within a given period. Optimal assignment problems are commonly seen in inefficiency in event management companies, large corporations, and other industries. Here, the primary objective function is the number of tasks completed, whereas constraints are the number of employees asked to complete those tasks.

• Transportation problems

Transportation problems are another type of linear programming problem, mainly concerned with studying effective transportation routes. Transportation problems mainly include studying how effectively one product is carried from one area to another so that the final transportation cost can be minimised. 

In large-scale organisations, it sometimes becomes nearly impossible to analyse transportation issues. Here, the objective function is the transportation cost, whereas constraints are the demand and supply patterns. 

• Manufacturing problems

The manufacturing problems involve maximising production and profits earned from manufactured products, which might be the function of workers engaged in the production, available workspace, packing materials utilised, machine hours, product’s market worth, raw materials required, etc. Here, the objective function is the production rate, whereas constraints are factors including the cost of packing materials, labour hours, and so on. 

• Diet Problems

The diet problems mainly comprise the intake of a specific food item, affecting the overall diet plan. The purpose of the diet problem is to find a set of food items that can meet the daily nutritional needs while spending the least amount of money. Here, the objective function is the cost of food consumed, whereas constraints are meeting nutritional needs, taking calories, etc. 

Methods to solve linear programming problems 

There are different methods to solve linear programming problems. These include simple graphical methods or using tools, such as open solver, R, etc. 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 optimisation linear programming issue that consists of two variables can be solved. Another significant advantage of using this method over others is that it is visual and provides a clearer picture. 

Conclusion 

As we learned about the different types of linear programming problems, we now know that these problems either increase or decrease the linear function, which can either be equalities or inequalities. The primary 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.

In this article, we covered the types of linear programming problems in detail, where we discussed four common types of linear programming problems faced in different sectors of life. These include optimal assignment problems, transportation problems, manufacturing problems, and diet problems that have been discussed in this study material. Here, we talked about these problems in detail and made a quick comparison for ease of understanding.