Defining Linear Programming
Linear programming (LP) is a mathematical technique for solving real-world problems. It describes the situation, requirements, and constraints that define the problem in mathematical terms using variables and states the relationships of these variables in linear equations. The equations are solved through the application of algorithms, which are sets of instructions defining a process of calculations. The objective is to determine the best possible outcome, meaning the solution would show either the maximum or minimum values for the variables. The process of solving a problem expressed in linear programming fashion is called optimization.
Today, linear programming problems are more efficiently computed through the use of computers and software. However, the similarity of the name with the modern term 'programming', which refers to the creation of computer programs, is only accidental. Linear programming was developed during the Second World War when the military requested mathematicians to help plan their operations in such a way that the greatest possible loss to the enemy is achieved with the smallest amount of cost. The common military term for operational plans was 'programs', and this is where the name was derived. The name first arose in the 1940s when George B. Dantzig, one of the main developers of LP, wrote a paper entitled 'Programming in a Linear Structure'. This paper was an analysis of the United States Air Force's planning problems at that time of war, and it showed how the said problems could be formulated in a system of linear inequalities. Later, the title was shortened to linear programming. It was Dantzig who contributed the simplex method, or simplex algorithm. This is a numerical solution to LP problems that involves repeated computations, after which the optimal solution is extracted from the set of values generated.
Other methods and techniques for modeling and solving evolved as the field was developed further by other mathematicians. Integer programming, for example, is a type of linear programming where the variables are limited to taking on only integer values. Then there is quadratic programming, where the objective function—the mathematical statement of the problem—is a quadratic function (the variable or variables are squared), but the constraints are still expressed in linear equalities or inequalities. Standard linear programming problems are deterministic in nature, which means that the variables can be known. But of course, real-world problems are hardly ever certain. To compensate for uncertainty, stochastic programming was developed, and it takes a further step by considering random variables and using probability distributions. As the field grew and new methods for optimization were discovered, the term mathematical programming was eventually used, and this included all mathematical techniques that systematically solved for optimal solutions to problems, whether they were expressed as linear functions or otherwise.
Defining Linear Programming
Defining Linear Programming : Multi-thousand tips to make your life easier.
Linear programming (LP) is a mathematical technique for solving real-world problems. It describes the situation, requirements, and constraints that define the problem in mathematical terms using variables and states the relationships of these variables in linear equations. The equations are solved through the application of algorithms, which are sets of instructions defining a process of calculations. The objective is to determine the best possible outcome, meaning the solution would show either the maximum or minimum values for the variables. The process of solving a problem expressed in linear programming fashion is called optimization.
Today, linear programming problems are more efficiently computed through the use of computers and software. However, the similarity of the name with the modern term 'programming', which refers to the creation of computer programs, is only accidental. Linear programming was developed during the Second World War when the military requested mathematicians to help plan their operations in such a way that the greatest possible loss to the enemy is achieved with the smallest amount of cost. The common military term for operational plans was 'programs', and this is where the name was derived. The name first arose in the 1940s when George B. Dantzig, one of the main developers of LP, wrote a paper entitled 'Programming in a Linear Structure'. This paper was an analysis of the United States Air Force's planning problems at that time of war, and it showed how the said problems could be formulated in a system of linear inequalities. Later, the title was shortened to linear programming. It was Dantzig who contributed the simplex method, or simplex algorithm. This is a numerical solution to LP problems that involves repeated computations, after which the optimal solution is extracted from the set of values generated.
Other methods and techniques for modeling and solving evolved as the field was developed further by other mathematicians. Integer programming, for example, is a type of linear programming where the variables are limited to taking on only integer values. Then there is quadratic programming, where the objective function—the mathematical statement of the problem—is a quadratic function (the variable or variables are squared), but the constraints are still expressed in linear equalities or inequalities. Standard linear programming problems are deterministic in nature, which means that the variables can be known. But of course, real-world problems are hardly ever certain. To compensate for uncertainty, stochastic programming was developed, and it takes a further step by considering random variables and using probability distributions. As the field grew and new methods for optimization were discovered, the term mathematical programming was eventually used, and this included all mathematical techniques that systematically solved for optimal solutions to problems, whether they were expressed as linear functions or otherwise.
Defining Linear Programming
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Defining Linear Programming : Multi-thousand tips to make your life easier.