## Linear Programming in Optimization

Linear programming is a mathematical technique used to optimize a business objective while adhering to constraints. The variables are linear and the objective function, as well as constraints, are represented as a linear combination of these variables. Linear programming is used to determine the most efficient allocation of resources given restrictions in the usage of these resources.

## Sensitivity Analysis Definition

Sensitivity analysis, also known as post-optimality analysis, is a process of determining the effect of changes in the constraints or coefficients of the objective function on the optimal solution. When using linear programming, sensitivity analysis is essential in the decision-making process.

## Purpose of Sensitivity Analysis in Linear Programming

Why is sensitivity analysis important? The answer is simple. In the real world, the data that is used as input to the model can change. As a result, the impact of these changes on the optimal solution must be analyzed. The purpose of sensitivity analysis is to gain insight into the stability of the optimal solution, the sensitivity of the solution to changes in the input data, and how a deviation from the original data could impact future decisions.

During sensitivity analysis, different scenarios are created in which one or more inputs are changed to analyze the impact on the optimal solution.

## Application of Sensitivity Analysis

Sensitivity analysis is used in various industries like finance, engineering, logistics, and healthcare. In finance, it is used to calculate and understand the impact of market fluctuations, whereas in healthcare, it is used to manage the allocation of resources to meet future needs.

Let’s take the example of a manufacturing company that produces two products, A and B. Product A requires 1 hour of work on machine X and 2 hours of work on machine Y. Product B requires 3 hours of work on machine X and 1 hour of work on machine Y.

The limited availability of machines requires that a maximum of 12 hours be used on machine X, and a maximum of 18 hours be used on machine Y. Additionally, the profit per unit of A and B is $20 and $30, respectively.

The objective is to maximize profit while adhering to machine constraints. Therefore, we need to find the optimal quantity of A and B to produce. The linear programming model for this problem is represented below:

Maximize 20A + 30B

Subject to:

A + 3B <= 12

2A + B <= 18

Non-negativity Constraint: A, B >= 0

The optimal solution is A=4 and B=4 with a profit of $160. But what happens if the availability of working hours on machine Y is reduced from 18 to 15? Here we execute sensitivity analysis:

Maximize 20A + 30B

Subject to:

A + 3B <= 12

2A + B <= 15

Non-negativity Constraint: A,B >= 0

The optimal solution is A=4 and B=4 with a profit of $160. By minimizing the hours on machine Y, the company can still produce the same output with a small change in profit. Sensitivity analysis shows the organization the degree of stability of the model and the variance that can arise when there are even slight adjustments to the data.

## Conclusion

Sensitivity analysis plays a crucial role in linear programming optimization. It shows the stability of the optimal solution and the impact of the variations in constraints or objective function coefficients. Thus, one can be agile and can prepare for the future decisions in advance. The sensitivity analysis enhances the quality and reliability of the optimization model solution. Find extra details about the topic in this external resource we’ve specially prepared for you. https://www.analyticsvidhya.com/blog/2017/02/lintroductory-guide-on-linear-programming-explained-in-simple-english/, obtain worthwhile and supplementary details to enhance your comprehension of the topic.

Dive deeper into the topic with the related posts we’ve suggested below: