## What is Linear Programming?

The resources such as land, material, machine, or labor are always limited. But on the other side, each resource has alternative use also. Hence, we need to choose only those alternatives which can maximize the profit and reduce the cost of production. In other words, we want to optimize the available resources. According to the dictionary, “optimization” means “the action of making the best or most effective use of a situation or resource. To select the best strategy from the number of alternatives, linear programming is used.

Linear programming is a quantitative tool used in operation management. Linear programming is a mathematical modeling technique. This technique is used to find the optimum resource utilization. In this technique, a linear function is maximized or minimized when subjected to various constraints.

## Applications of Linear Programming

Linear programming, also referred to as Linear Optimization, is used in the field of mathematics, and it is also used in business planning in industries when quantitative decisions are to be made. In other words, linear programming helps in making decisions related to maximizing profits.

Industry | Use of Linear Programming |

Manufacturing industry | It is used for analyzing supply chain operations. |

Retail industry | It is used in optimizing the shelf space. |

Transportation industry | It is used for optimizing delivery routes for cost and time efficiency. |

Engineering industry | It is used to solve design and manufacturing problems using shape optimization. |

Energy industry | It is used to maximize the energy power system. |

The components of linear programming include:

- Decision Variables
- Constraints
- Data
- Objective Functions

## Characteristics of Linear Programming Model

There are five characteristics of the linear programming model:

- Objective functions: The objective should be stated clearly and should be possible to state in a quantitative way.
- Constraints: Resource constraints or limitations should be expressed in mathematical form.
- Non-negativity: The value of the variables should be either positive or zero. It should not be negative.
- Finiteness: The number of inputs and outputs should be fixed. When there are infinite factors, you cannot compute the optimal solution using linear programming.
- Linearity: The relationship between the two or more variables must be linear. According to the definition, “proportional relationship between two ‘or more variable, i.e., the degree of variables should be maximum one.”

## Assumptions are taken for Linear Programming

- Number of constraints should be expressed in quantitative terms.
- Relationship between the constraints and objective function should be linear.
- The linear function is to be optimized.

## Different types of Linear Programming

The different types of linear programming are:

- Solving linear programming by Simplex method.
- Solving linear programming using R.
- Solving linear programming using the graphical method.
- Solving linear programming with the use of an open solver.

## Case Study - Linear Programming

Sean deals in robotic toy cars and robotic dolls only. He has $50,000 to invest and has storage space for 80 pieces. A robotic toy car costs $1000 and a robotic doll $500. Sean estimates that he can make a profit of $300 from the sale of a robotic car and a profit of $100 from the sale of the robotic doll. He wants to know how many toys he should buy using the available money so as to maximize his total profit, assuming that he can sell all the items which he buys.

In this case study, we observe

1. Sean can invest his money in buying robotic cars or robotic dolls or a combination thereof. Also, he would earn different profits by following different investment strategies.

2. The constraints that Sean has:

a) Investment is limited to a maximum of $50,000

b) Storage/display space is limited; he can display only 80 pieces.

According to LPP, we come to a conclusion that the dealer can invest his money in different ways, and he would earn different profits by following different investment strategies.

Let us try to formulate the problem mathematically so that Sean can invest his money and get maximum profit.

## Advantages of Linear Programming

- Linear programming helps to solve multi-dimensional problems.
- Linear programming provides better insight into the business problem.
- Linear programming helps to calculate the cost and profit using various alternatives, and hence the most optimal solution can be selected.
- Linear programming helps in making adjustments according to changing conditions.

## Limitations of Linear Programming

- Linear programming cannot solve the problems of nonlinear functions.
- Linear programming requires a lot of mathematical calculation.
- Linear programming is not able to solve the problems in which variables cannot be stated quantitatively.

When you need to solve mathematical optimization problems that involve quadratic functions, you can use Quadratic programming (QP). QP is a type of nonlinear programming.

## Context and Applications

This course is significant for students pursuing undergraduate and graduate courses in the business and finance sector.

- B.Com
- CIMA
- MBA

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