During the Warring States Period in China, there was a horse race that was passed down from generation to generation. I believe everyone knows that this is the Tian Ji Horse Racing. The story of horse racing in Tianji shows that under the existing conditions, after planning, arranging and selecting the best scheme, the best effect will be achieved. It can be seen that the planning arrangement is very important.
It is generally believed that operational research is a branch of modern applied mathematics, which mainly refines some general operational research problems in production, management and other events, and then solves them by mathematical methods. The former provides models, while the latter provides theories and methods.
The idea of operational research has been produced in ancient times. When the enemy and I are at war, we must make the best plan to deal with the enemy on the basis of understanding the situation of both sides. This is the saying that "strategic planning wins the battle thousands of miles away".
However, as a mathematical discipline, it is too late to solve the selection and arrangement of the optimal method by pure mathematics. It can also be said that operational research is a branch that began to rise in the 1940s.
Operational research mainly studies the planning and management problems that can be expressed quantitatively in economic activities and military activities. Of course, with the development of objective reality, many contents of operational research not only study economic and military activities, but also go deep into daily life. Operations research can get various results through mathematical analysis and operation according to the requirements of the problem, and finally put forward a comprehensive and reasonable arrangement, which has achieved the best results.
As a discipline used to solve practical problems, operational research generally has the following steps when dealing with various problems: determining goals, making plans, establishing models and making solutions.
Although it is unlikely that operational research can deal with large-scale objects, some abstract models have been formed in the development of operational research, which can be applied to solve large-scale practical problems.
With the development of science and technology and production, operational research has penetrated into many fields and played an increasingly important role. Operational research itself is also developing constantly, and now it is a mathematics department including several branches. For example: mathematical programming (including linear programming; Nonlinear programming; Integer programming; Combinatorial planning, graph theory, network flow, decision analysis, queuing theory, reliability mathematics theory, inventory theory, game theory, search theory, simulation and so on.
Brief introduction of each branch
The research object of mathematical planning is the arrangement and evaluation in planning management. The main problem to be solved is to find the optimal scheme of arrangement according to a certain measurement index under given conditions. It can be expressed as the problem of finding the minimum value of a function under constraints.
Mathematical programming is essentially different from the classical problem of seeking extreme value. Classical methods can only deal with simple expressions and simple constraints. However, in modern mathematical programming, the objective function and constraints of the problem are very complex, and some accurate numerical solutions are needed, so the research on the algorithm is particularly valued.
The simplest problem here is linear programming. If the constraint and objective function are linear, it is called linear programming. To solve linear programming problems, linear equations should be solved theoretically, so the method of solving linear equations and the knowledge of determinant and matrix are very necessary tools in linear programming.
The emergence of linear programming and its solution-simplex method has greatly promoted the development of operational research. Many practical problems can be solved by linear programming, and simplex method is an effective algorithm, and the emergence of computers makes the solution of some large and complex practical problems become a reality.
Nonlinear programming is the further development and continuation of linear programming. Many practical problems, such as design problems and economic balance problems, belong to the category of nonlinear programming. Nonlinear programming not only expands the application scope of mathematical programming, but also raises many basic theoretical problems for mathematicians, which makes convex analysis and numerical analysis in mathematics develop. There is also a time-related planning problem called "dynamic planning". In recent years, it has become an important tool commonly used in optimal control problems in engineering control, technical physics and communication.
Queuing theory is another branch of operational research, which is called stochastic service system theory. The purpose of its research is to answer the question of how to improve the service objects of service institutions or organizations and make some indicators reach the optimal level. For example, how many docks should a port have and how many maintenance personnel should a factory have.
Queuing theory was first studied by Danish engineer Erlang in the early 20th century on the efficiency of telephone exchange. In order to estimate the capacity of airport runway in World War II, it has been further developed, and its corresponding discipline renewal theory and reliability theory have also been developed.
Because queuing phenomenon is a random phenomenon, probability theory is mainly used as the main tool to study queuing phenomenon. In addition, there are differential and differential equations. Queuing theory describes the image of the object it wants to study when customers come to the service desk to ask for reception. If the service desk is occupied by other customers, there will be a queue. On the other hand, the service desk is sometimes idle and sometimes busy. It is necessary to obtain the probability distribution of customer waiting time and queue length by mathematical method.
Queuing theory is widely used in daily life, such as the regulation of reservoir water volume, the arrangement of production line, the dispatching of railway approach, the design of power grid and so on.
Game theory is also called game theory. The aforementioned horse racing in Tian Ji is a typical game theory problem. As a branch of operational research, the development of game theory is only a few decades. The mathematician who systematically founded this subject is now recognized as the Hungarian-American mathematician and the father of computers-von Neumann.
At first, the study of game theory by mathematical methods began with chess-how to determine the winning method. Because this is a problem of studying the conflict between the two sides and winning countermeasures, this subject has very important applications in the military. In recent years, mathematicians have also studied the fighting and tracking between mines and ships, fighters and bombers, and put forward a mathematical theory that both sides can make decisions independently. In recent years, with the further development of artificial intelligence research, more new requirements have been put forward for game theory.
Search theory is a branch of operational research that emerged because of the need of war in the Second World War. This paper mainly studies the theory and method of how to design and find the optimal scheme of a certain target and implement it under the condition of limited resources and detection means. In World War II, the allied air force and navy were born in the process of studying how to identify the submarine activities, fleet transportation and force deployment of the Axis countries. Search theory has also made many achievements in practical application. For example, in the 1960s, the United States successfully searched for the nuclear submarines "oil tankers" and "scorpions" missing in the Atlantic Ocean and the hydrogen bombs missing in the Mediterranean Sea.
The application field of operational research is very wide, which has penetrated into service, inventory, search, population, confrontation, control, timetable, resource allocation, site selection, energy, design, production, reliability and so on.