# PHPSimplex

## Optimizing resources with Linear Programming

### History of Operations Research

Along the history, is frequent to find collaboration among scientists and army officers with the same objective, ruling the optimal decision in battle. In fact that many experts consider the start of Operational Research in the III century B.C., during the II Punic War, with analysis and solution that Archimedes named for the defense of the city of Syracuse, besieged for Romans. Enter his inventions would find the catapult, and a system of mirrors that was setting to fire the enemy boats by focusing them with the Sun's rays.

Leonardo DaVinci took part, in 1503, like engineer in the war against Prisa due to he knew techniques to accomplish bombardments, to construct ships, armored vehicles, cannons, catapults, and another warlike machines.

Another antecedent of use of Operational Research obeys to F.W. Lanchester, who made a mathematical study about the ballistic potency of opponents and he developed, from a system of equations differential, Lanchester's Square Law, with that can be available to determine the outcome of a military battle.

Thomas Edison made use of Operational Research, contributing in the antisubmarine war, with his greats ideas, like shields against torpedo for the ships.

From the mathematical point of view mathematical, in centuries XVII and XVIII, Newton, Leibnitz, Bernoulli and Lagrange, worked in obtaining maximums and minimums conditioned of certain functions. Mathematical French Jean Baptiste Joseph Fourier sketched methods of present-day Linear Programming. And at late years of the century XVIII, Gaspar Monge laid down the precedents of the Graphical Method thanks to his development of Descriptive Geometry.

Janos Von Neumann published his work called "Theory of Games", that provided the basics Mathematicians to Linear Programming.At a later time, in 1947, he viewed the similitude among them Programming linear problems and the matrix theory that developed himself.

In 1939, mathematical Russian L. Kantorovich, in association with the mathematical Dutchman T. Koopmans, developed the mathematical theory called "Linear programming", thanks to that went rewarded with the Nobel.

In the late years 30, George Joseph Stigler presented a particular problem known as special diet optimal or more commonly known as problem of the diet, that came about by the worry of the USA army to guarantee some nutritional requests for his troops to the lowest price. It was solved with a heuristic method which solution only differ in some centimes against the solution contributed years later by the Simplex Method.

During the years 1941 and 1942, Kantorovich and Koopmans studied in independent ways the Transport Problem for first time, knowing this type of problems like problem of Koopmans-Kantorovich. For his solution, they used geometric methods that are related to Minkowski's theorem of convexity.

It is believed that Charles Babbage is the father of the Operational Research due to his research about the transportation's costs and sorting of mail realized for the Uniform Penny Post in in England in 1840.

But it does not considered than has been born a new science called Operations Research until the II World War, during battle of England, where Deutsche Air Force, that is the Luftwaffe, was submitting the Britishers to a hard Air raid, since these had an little aerial capability, although experimented in the Combat. The British government, looking for some method to defend his country, convoked several scientists of various disciplines for try to resolve the problem to get the peak of benefit of radars that they had. Thanks to his work determining the optimal localization of antennas and they got the best distribution of signals to double the effectiveness of the system of aerial defense.

To notice the range of this new discipline, England created another groups of the same nature in order to obtain optimal results in the dispute. Just like United States (USA), when joined the War in 1942, creating the project SCOOP (Scientific Computation Of Optimum Programs), where was working George Bernard Dantzig, who developed in 1947 the Simplex algorithm.

During the Cold War, the old Soviet Union (URRS), excluded of the Plan Marshall, wanted to control the terrestrial communications, including routes fluvial, from Berlin. In order to avoid the rendition of the city, and his submission to be a part of the deutsche communist zone, England and United States decided supplying the city, or else by means of escorted convoys (that would be able to give rise to new confrontations) or by means of airlift, breaking or avoiding in any event the blockage from Berlin. Second option was chosen, starting the Luftbrücke (airlift) at June 25, 1948. This went another from the problems in which worked the SCOOP group, in December of that same year, could carry 4500 daily tons, and after studies of Research Operations optimized the supplying to get to the 8000~9000 daily tons in March of 1949. This cipher was the same that would have been transported for terrestrial means, for that the Soviet decided to suspend the blockage at May 12, 1949.

After second World War, the order of United States' resources (USA) (energy, armaments, and all kind of supplies) took opportune to accomplish it by models of optimization, resolved intervening linear programming.

At the same time, that the doctrine of Operations Research is being developed, the techniques of computation and computers are also developing, thanks them the time of resolution of the problems decreased.

The first result of these techniques was given at the year 1952, when a SEAC computer from was used National Bureau of Standards in way to obtain the problem´s solution. The success at the resolution time was so encouraging that was immediately used for all kind of military problems, like determining the optimal height which should fly the planes to locate the enemy submarines, monetary founds management for logistics and armament, including to determine the depth which should send the charges to reach the enemy submarines in order to cause the biggest number of casualties, that was translated in a increase in five times in Air Force's efficacy.

During the 50's and 60's decade, grows the interest and developing of Operational Research, due to its application in the space of commerce and the industry. Take for example, the the problem of the calculation of the optimal transporting plan of sand of construction to the works of edification of the city of Moscow, which had 10 origins points and 230 destiny. To resolve it, was used and Strena computer, that took 10 days in the month of june of 1958, and such solution contributed a reduction of the 11 % of the expenses in relation to original costs.

Previously, this problems were presented in a discipline knew as Research Companies or Analysis Companies, that did not have so effective methods like the developed during Second World War (for example the Simplex method). No war applications of Operations Research there are so as you want imagine, with problems like nutrition of cattle raising, distribution of fields of cultivation in agriculture, goods transportation, location, personnel's distribution, and nets, queue problems, graphs, etc.

Real cases of using Operations Research and Contributed benefits.

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PHPSimplex
Version 0.81

Developed by:
Daniel Izquierdo Granja
Juan José Ruiz Ruiz

English translation by:
Luciano Miguel Tobaria

French translation by:
Ester Rute Ruiz

Portuguese translation by:
Rosane Bujes