Linear programming (LP) is the central tool of mathematical programming. LP models are flexible enough to adequately describe many realistic problems arising in modern industrial settings, while at the same time take advantage of the considerable expertise in computational linear algebra that has been developed during the last 50 years. As a result, LP models are abundantly used in logistics, transportation, finance and many other practical applications.
LP has undergone profound changes during the past 20 years, resulting in codes that are thousands (and sometimes millions) of times faster than those available just 15 years ago. Yet difficult challenges persist in the form of large-scale linear programming problems arising in routing, network design, chip design and other settings. In fact, large problem instances render even the best of codes nearly unusable.
LP was a mathematical model developed during the Second World War to plan expenditure and returns in a manner so as to reduce costs to the army and increase losses to the enemy. It was kept secret till 1947. In the post-WWII period many industries found its use in their daily business planning. Many practical problems in operations research can be expressed as LP problems. The examples are network flow problem and multi-commodity flow problems. These are considered important enough to have generated much research on specialized algorithms for their solutions. LP is heavily used in microeconomics and business management, either to maximize income or minimize costs of production schemes. Some other problems that can be expressed as LP problems are in the areas of food blending, inventory management, portfolio management, resource allocations (human and machine), business planning and advertisement campaigns. LP will have the following structure:
A single and well-defi ned objective with a set of decision variables (i.e., maximum profi t or minimum cost)
A set of constraints including non-negative constraints (i.e., representations of a limited supply of resources)
More than one solution to the problem exists (there are an infi nite number of solutions)
The objective and constraints are in the form of linear equations or inequalities
A linear function to be maximized, for example,
Problem constraints of the following formalities, for example,
Non-negative variable, for example,
The problem usually expressed in matrix form then becomes
Other forms such as minimization problems, problems with constraints on alternative forms and problems involving variables can always be written into an equivalent problem in standard formats and solved through LP.
Thus, the LP model can be used for solving problems related to product mix, investment, scheduling, transportation and assignment of a firm.
Source: Sople V.V (2013), Logistics Management, Pearson Education India; Third edition.
Great post, you have pointed out some fantastic points, I besides think this s a very great website.