Mathematical programming involves the implementation of methods for finding the optimal solution. The solution to these types of problems is connected with the study of functions on extremality. Methods of mathematical programming are quite common in the applied direction of cybernetics.
A large number of tasks that appear in society are often associated with phenomena that are based on a conscious basis of decisions. It is with the necessary choice of the possible course of action used in different areas of human life that the problems of mathematical programming find their application.
The history of the development of society shows that a limited amount of information has always prevented the adoption of the right decision, and the optimal decision was mainly based on intuition and experience. Subsequently, with an increase in the amount of information , direct calculations began to be used to make a decision.
The picture at a modern enterprise looks completely different, where, thanks to the wide range of goods produced there, the flow of input information is simply huge. Its processing is possible only using modern electronic technologies. And if you need to choose the best one from the proposed solutions, then you certainly can not do without electronics here.
Therefore, mathematical programming goes through the following main stages.
The first stage involves ranking all the factors by importance and establishing between them the patterns to which they are able to obey.
The second stage is the construction of a model of the problem in mathematical expression. In other words, it is an abstraction of reality represented using mathematical symbols. The mathematical model is able to establish a relationship between the control parameters and the selected phenomenon. This stage should include the construction of such a characteristic in which each greater or lesser value corresponds to the optimal situation from the position of the decision being made.
Based on the results of the implementation of the above steps, a mathematical model is formed that uses certain mathematical knowledge.
The third stage involves the study of variables that have a significant impact on the objective function. This period should include the possession of certain mathematical knowledge that will help in solving problems arising in the second stage of decision making.
The fourth stage consists in comparing the results of the calculations obtained in the third stage with a simulated object. In other words, at this stage, the adequacy of the model with the simulated object is established within the limits of achieving the necessary accuracy of the source data. Making a decision at this stage depends on the result of the study. So, upon receipt of unsatisfactory results of comparison, the input data about the simulated object are specified. If the need arises, then the statement of the problem is refined, followed by the construction of a new mathematical model, the solution of the mathematical problem and a new comparison of the results.
Mathematical programming allows you to use two main directions of calculations:
- solving deterministic problems that imply the certainty of all the source information;
- stochastic programming, which allows solving problems containing elements of uncertainty or when the parameters of these tasks are random. For example, production planning is often carried out in conditions of incomplete display of real information.
Basically, mathematical programming has the following programming sections in its structure : linear, nonlinear, convex and quadratic.