Linear programming

programming method to achieve the best outcome in a mathematical model

Linear programming or linear optimization is a mathematical method for determining a way to achieve the best outcome. such as maximum profit or lowest cost, in a given mathematical model for some list of requirements represented as linear relationships. Linear programming is a specific case of mathematical programming or mathematical optimization.

A pictorial representation of a simple linear program with two variables and six inequalities.
CONTENT : A - F , G - L , M - R , S - Z , See also , External links

Quotes edit

Quotes are arranged alphabetically per author

A - F edit

  • The word model is used as a noun, adjective, and verb, and in each instance it has a slightly different connotation. As a noun "model" is a representation in the sense in which an architect constructs a small-scale model of a building or a physicist a large-scale model of an atom. As an adjective "model" implies a degree of perfection or idealization, as in reference to a model home, a model student, or a model husband. As an adjective "model" implies a degree or perfection or idealization, as in reference to a model home, a model student, or a model husband. As a verb "to model" means to demonstrate, to reveal, to show what a thing is like.
  • Scientific [mathematical] models have all these connotations. They are representations of states, objects, and events. They are idealized in the sense that they are less complicated than reality and hence easier to use for research purposes. These models are easier to manipulate and "carry" than the real thing. The simplicity of models, compared with reality, lies in the fact that only the relevant properties of reality are represented.
    • Russell L. Ackoff ( 1962) Scientific method: optimizing applied research decisions, p. 108 as cited in: Joe H. Ward, Earl Jennings (1973) Introduction to linear models. p. 4
  • There is no America. There is no democracy. There is only IBM and ITT and AT&T and DuPont, Dow, Union Carbide, and Exxon. Those are the nations of the world today. What do you think the Russians talk about in their councils of state — Karl Marx? They get out their linear programming charts, statistical decision theories, minimax solutions, and compute the price-cost probabilities of their transactions and investments, just like we do. We no longer live in a world of nations and ideologies, Mr. Beale. The world is a college of corporations, inexorably determined by the immutable bylaws of business. The world is a business, Mr. Beale. It has been since man crawled out of the slime. And our children will live, Mr. Beale, to see that perfect world in which there's no war or famine, oppression or brutality — one vast and ecumenical holding company, for whom all men will work to serve a common profit, in which all men will hold a share of stock, all necessities provided, all anxieties tranquilized, all boredom amused.
  • Linear programming is viewed as a revolutionary development giving man the ability to state general objectives and to find, by means of the simplex method, optimal policy decisions for a broad class of practical decision problems of great complexity. In the real world, planning tends to be ad hoc because of the many special-interest groups with their multiple objectives.
    • George Dantzig (1983) "Reminiscences about the origins of linear programming". In: Mathematical programming : the state of the art. New York, 1983, p. 78-86.

G - L edit

  • "Linear relationships can be captured with a straight line on a graph. Linear relationships are easy to think about....Linear equations are solvable... Linear systems have an important modular virtue: you can take them apart, and put them together again — the pieces add up."
    • James Gleick (1987) Chaos: Making a New Science, p. 23 as cited in: James R. Hansen (2004) Trees of Texas: An Easy Guide to Leaf Identification. p. 246
  • There are two types of systems engineering - basis and applied. There is no need to attempt to define the term systems engineering in a manner acceptable to everybody, as Chalmer Jones brings out in his article herein. Systems engineering is, obviously, the engineering of a system. It usually, but not always, includes dynamic analysis, mathematical models, simulation, linear programming, data logging, computing, optimating, etc., etc. It connotes an optimum method, realized by modern engineering techniques. Basic systems engineering includes not only the control system but also all equipments within the system, including all host equipments for the control system. Applications engineering is — and always has been — all the engineering required to apply the hardware of a hardware manufacturer to the needs of the customer. Such applications engineering may include, and always has included where needed, dynamic analysis, mathematical models, simulation, linear programming, data logging, computing, and any technique needed to meet the end purpose - the fitting of an existing line of production hardware to a customer's needs. This is applied systems engineering.
    • Instruments and Control Systems, (1958) Vol. 31. p. 72
  • Linear programming was a new branch of applied mathematics that – in the USA – came into being as a direct consequence of mathematicians’ war work. It was not done under contract with the AMP but by some of the mathematicians employed directly by the armed forces.15 The source was a concrete practical problem within the US Air Forces,16 a logistic problem that eventually led to the mathematical theory of linear programming, and from there to mathematical programming. The person normally associated with the origin of linear programming in the USA is George B. Dantzig. Dantzig was one of the mathematicians hired directly by the armed forces for the war effort. In 1941 he began working at the Combat Analysis Branch of the United States Air Force Headquarters Statistical Control under the leadership of Tex Thornstons. During the war Dantzig worked on what was called “programming planning” methods to calculate Air Force programs. An Air Force program was a kind of activity plan.
    • Tinne Hoff Kjeldsen (2003) "New Mathematical Disciplines and Research in the Wake of World War II"

M - R edit

  • Linear Programming is a generalization of Linear Algebra. It is capable of handling a variety of problems, ranging from finding schedules for airlines or movies in a theater to distributing oil from refineries to markets. The reason for this great versatility is the ease at which constraints can be incorporated into the model.
  • Linear programming was developed as a discipline in the 1940's, motivated initially by the need to solve complex planning problems in wartime operations. Its development accelerated rapidly in the postwar period as many industries found valuable uses for linear programming. The founders of the subject are generally regarded as George B. Dantzig, who devised the simplex method in 1947, and John von Neumann, who established the theory of duality that same year. The Nobel prize in economics was awarded in 1975 to the mathematician Leonid Kantorovich (USSR) and the economist Tjalling Koopmans (USA) for their contributions to the theory of optimal allocation of resources, in which linear programming played a key role. Many industries use linear programming as a standard tool, e.g. to allocate a finite set of resources in an optimal way.
  • In another part of the management forest, the mechanistic school was gathering its forces and preparing to outflank the forces of light. First came the numbers men — the linear programmers, the budget experts, and the financial analysts — with their PERT systems and cost-benefit analyses. From another world, unburdened by most of the scientific management ideology and untouched by the human relations school, they began to parcel things out and give some meaning to those truisms, "plan ahead" and "keep records." Armed with emerging systems concepts, they carried the "mechanistic" analogy to its fullest—and it was very productive. Their work still goes on, largely untroubled by organizational theory; the theory, it seems clear, will have to adjust to them, rather than the other way around.
    • Charles Perrow, "The short and glorious history of organizational theory." Organizational Dynamics 2.1 (1973): 3-15. p. 6

See also edit

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