Linear programming

Sometimes you want the best result.

You might want more money or less cost. This math helps you plan. It uses what you have to find an answer.

It can help a farmer plant crops.

A farmer must use the right amount of land. They also need the right amount of food for plants.

Many big jobs use this math. It helps with moving things or making things. Even Google uses it to help YouTube videos stay smooth. It makes hard choices much easier.

Sometimes we want the best result from a set of choices.

You might want the most profit or the lowest cost. This math is called linear programming. It helps you find that best answer. It uses rules to show what you can and cannot do. These rules are called constraints. For example, a farmer has a set amount of land. They also have a set amount of fertilizer.

The farmer must use these tools to make the most money. Linear programming shows the best way to plant crops.

Many people helped build this math. Leonid Kantorovich and Wassily Leontief studied how it works for business.

Later, George Dantzig made a fast way to solve these problems. He made the simplex method. This tool makes finding the best answer very quick. It is used in many jobs today. Companies use it for making goods or moving things. Even Google uses it to keep YouTube videos smooth.

It helps people make smart plans with what they have.

Imagine you are a farmer with a patch of land. You want to make the most money possible from your crops. However, you only have a certain amount of land and a set amount of fertilizer. You cannot plant more than your supplies allow. This is a puzzle about making the best choice under rules. In math, we call these rules constraints. Linear programming is a way to solve these puzzles. It helps us find the best outcome, like the highest profit or the lowest cost.

To solve the puzzle, math uses a special model. It looks at an objective function, which is the main goal you want to reach. It also looks at the constraints, which are the limits you must follow. When you draw these limits on a graph, they create a shape. This shape is called a convex polytope. Every point inside this shape is a possible choice you could make.

The math works by searching this shape for the one best point. This point gives you the largest or smallest value for your goal.

Many smart people helped develop these ideas over many years. In 1827, a mathematician named Fourier published a method for solving inequalities. Later, in the late 1930s, Leonid Kantorovich and Wassily Leontief studied how to use these ideas for economics.

Their work was very important but was ignored for a long time. During World War II, the math became a vital tool for moving supplies and planning. It helped leaders manage resources when things were very difficult. This success finally brought the method into the spotlight.

In 1947, George Dantzig invented the simplex method. This was a very fast way to solve these math problems. Before this, finding the best answer could take a very long time. Dantzig once used his method to assign 70 people to 70 different jobs. There were so many ways to do that job that the number of choices was bigger than the number of particles in the universe!

The simplex method found the best answer in just a moment. Other math breakthroughs followed, like Leonid Khachiyan's work in 1979 and Narendra Karmarkar's method in 1984.

Today, linear programming is used in many parts of our world. Big industries use it for energy, making goods, and moving items through telecommunications. Even Google uses it to keep YouTube videos playing smoothly. It helps companies manage their work and stay organized. By using these mathematical rules, people can make much better plans. It turns a hard job into a clear path forward.

Linear programming, also known as linear optimization, is a mathematical method for finding the best possible outcome within a model. This outcome might be maximizing a profit or minimizing a specific cost. The method works by using linear relationships to represent both the requirements and the objectives of a problem. It is a specialized branch of a broader field called mathematical programming.

A linear programming problem is built from two main components. The first is the objective function, which is a real-valued affine function. This function represents the goal, such as total revenue or total cost. The second component consists of linear equality and inequality constraints. These constraints define the limits of the problem, such as available land, time, or materials. When these constraints are combined, they create a feasible region. This region is a convex polytope, which is a set formed by the intersection of many half spaces.

To solve the problem, a linear programming algorithm searches the convex polytope for a specific point. This point is where the objective function reaches its largest or smallest value. In a two-dimensional problem, the feasible region might look like a polygon. In three dimensions, the region forms a convex polyhedron.

Finding the optimum involves navigating these shapes to find the most efficient solution. If the problem is written in standard form, it uses variables, vectors, and a matrix to express these relationships clearly.

The history of this field spans many decades and involves several key figures. The ability to solve systems of linear inequalities dates back to 1827, when Fourier published a specific method. This method is now known as Fourier–Motzkin elimination. In the late 1930s, Leonid Kantorovich and Wassily Leontief independently explored practical applications. Kantorovich focused on manufacturing schedules, while Leontief looked at economic applications. Although their work was groundbreaking, it was largely overlooked for many years.

World War II served as a major turning point for the recognition of linear programming. The method became a vital tool for solving complex wartime challenges. It was used for resource allocation, scheduling, and transportation logistics. These successes propelled the field into the spotlight after the war. In 1941, Frank Lauren Hitchcock formulated transportation problems using these methods. Later, between 1946 and 1947, George B. Dantzig developed a general formulation for the US Air Force. In 1947, Dantzig also invented the simplex method, which allowed for efficient solutions in most cases.

Dantzig's simplex method demonstrated incredible power through scale. One of his original examples involved assigning 70 people to 70 different jobs. The number of possible configurations for this task exceeds the number of particles in the observable universe. However, the simplex algorithm can find the optimum solution in just a moment. This is because the theory drastically reduces the number of solutions that must be checked. Other breakthroughs include Leonid Khachiyan showing the problem is solvable in polynomial time in 1979, and Narendra Karmarkar's 1984 interior-point method.

Linear programming also features a concept called duality. Every linear programming problem, known as the primal problem, can be converted into a dual problem. The dual problem provides an upper bound for the optimal value of the primal. There are two fundamental ideas in duality theory. First, the dual of a dual linear program returns the original primal program. Second, the weak duality theorem states that the dual's objective value is always greater than or equal to the primal's value at any feasible solution. If the primal has an optimal solution, the strong duality theorem states the dual also has one.

Today, the applications of linear programming are vast and diverse. It is used heavily in mathematics, business, economics, and engineering. Industries such as telecommunications, energy, and manufacturing rely on it for planning and design. It is also used in routing, scheduling, and assignment tasks. Even modern technology uses it, such as Google using linear programming to stabilize YouTube videos. By modeling complex decisions, linear programming helps manage limited resources to achieve the best possible results.