Linear Optimization, typically known as Linear Programming (LP) is a mathematical method to identify the maximum or minimum values of a linear function over a convex polyhedron under different conditions, specified by linear equality and linear inequality constraints.
Linear Programming can be dated back to the Second World War as a method to plan costs and returns, for the army to reduce expenditures and increase the enemy's losses. It was utilized in secret up until 1947, and many industries have found it useful for daily planning and operations after the war.
Following are the founders of Linear Programming:
- Leonid Kantorovich - a Russian mathematician who, in 1939, developed different linear programming problems
- George B. Dantzig - developed the Simplex Method in 1947
- John von Neumann - devised the Theory of Duality, also in 1947
- Leonid Khachiyan - solved a linear programming problem using polynomial time in 1979 with the Ellipsoid Method
- Narendra Karmarkar - introduced Interior-Point Method, or Barrier Method, in 1984, where she combined the two methods, Ellipsoid Method and Simplex Method. Instead of utilizing and passing from vertex to vertex, Karmarkar's method would be passing through the interior of the feasible region.
Another branch of Applied Mathematics use Linear Programming to solve lots of practical problems. This branch is what we call Operational or Operations Research. Along with other methods such as Statistics, Algorithms and Mathematical Modeling, Linear Programming helps to determine and arrive with the optimal solution to complex problems. It is to help Management reach its goals of maximizing its profit and minimize the risks.
It may be a bit hard to understand at first, but to simplify the explanation for Linear Programming, basically, would be the process of taking the best and optimal value of different linear inequalities provided on a specific problem or situation. The best example for this one would be George B. Dantzig's solution to the problem of finding the best possible assignment of 70 jobs to 70 people.
We normally would think that mathematical theories and formulas will not actually be put to use in our day-to-day living. As mentioned earlier though, different industries have been using this mathematical method and have been applying it on daily operations. Just imagine how useful it is for every business. When one encounters a problem and comes up with hundreds of solutions for it, the best way to identify the most advantageous resolution is with Linear Programming. Instead of trying to perform every possible solution to one specific problem, resolving it with Linear Programming narrows down the list of solutions. Just imagine how much time, effort and money your business can save with this friendly mathematical method.