Linear programming is a mathematical optimization technique used to maximize or minimize a linear objective function subject to a set of linear constraints. It is a widely used optimization method in various fields, including business, economics, and engineering.

One of the main applications of linear programming is in the field of production and operations management. It can be used to determine the optimal production levels of different products to maximize profits, while also taking into account factors such as available resources, production costs, and market demand. For example, a company may use linear programming to determine the optimal mix of products to manufacture in order to maximize profits, given the available resources and constraints on production.

Linear programming is also used in the field of transportation and logistics. It can be used to determine the most cost-effective routes for transporting goods and materials, taking into account factors such as distance, fuel costs, and vehicle capacity. It can also be used to optimize the allocation of resources, such as vehicles and drivers, to different routes to ensure that the transportation network is operating efficiently.

In the field of finance, linear programming can be used to optimize investment portfolios. It can be used to determine the optimal mix of assets to include in a portfolio in order to maximize returns while minimizing risk. It can also be used to model financial risks and determine the optimal strategies for managing them.

Linear programming has also found applications in a variety of other fields, including agriculture, military planning, and environmental management. It is a powerful tool for optimizing complex systems and making informed decisions under constraints.

In conclusion, the scope of linear programming is wide and varied, and it is a valuable tool for optimizing systems and making informed decisions in many different fields. It is a powerful and widely used optimization method that is essential for solving complex problems and making informed decisions in a variety of contexts.

## Linear programming

This problem is also classified as NP-hard, and in fact the decision version was one of If only some of the unknown variables are required to be integers, then the problem is called a mixed integer linear programming MIP or MILP problem. This technique is based on the assumption of linear relations between inputs and outputs. Equalities or inequalities can be used as restraints. Algorithm, in the forms of heuristics or exact methods, such as Branch-and-Cut or Column Generation, can also be implemented. As a result, we are interested in knowing the maximum Simplex pivot methods preserve primal or dual feasibility.