The purpose of operations research is a preliminary quantitative justification of optimal solutions.
Sometimes (relatively rarely) as a result of the study, it is possible to indicate a single, strictly optimal solution. Much more often there are cases – to highlight the area of almost equivalent optimal solutions, within which the final choice can be made. The decision-making itself goes beyond the scope of the operations study and falls within the competence of the responsible person, more often a group of persons who are given the right of final choice.
In this choice, they can take into account, along with the recommendations arising from the mathematical calculation, also a number of considerations (quantitative and qualitative) that were not taken into account by this calculation.
The indispensable presence of a person (as the final instance of decision-making) is not canceled even in the presence of a fully automated control system, which, it would seem, makes the optimal decision depending on the situation without human intervention. We must not forget that the very creation of the control algorithm, the choice of one of its possible options, is also a decision, and a very responsible one. With the development of ACS and ITS, human functions are not canceled, but simply move from one elementary level to another, higher.
The parameters that combine to form a solution are called solution elements. For example, if you plan to transport goods, the elements of the solution will be numbers that indicate how much cargo will be sent from each point of origin to each destination, the routes of the goods and the time of delivery.
In the simplest problems of operations research, the number of solution elements can be relatively small. However, in most tasks of practical importance, the number of elements of the solution is very large, which, of course, makes it difficult to analyze the situation and make recommendations. As a rule, any task of operations research results in a whole scientific study performed collectively, which takes a lot of time and requires the mandatory use of computer technology.
In addition to the elements of the solution, which we, within some limits, can dispose of, in any problem of operations research there are also given, «disciplining» conditions that are fixed from the very beginning and cannot be violated. In particular, such conditions include the means (material, technical, technological, human) that we have the right to dispose of, and various kinds of restrictions relying on solutions.
2.2 Mathematical modeling of operations
For the application of quantitative research methods in any field, some kind of mathematical model is always required. When constructing a mathematical model, a real phenomenon (in our case, an operation) is always simplified, schematized, and the resulting scheme is described using one or another mathematical apparatus. The more successfully the mathematical model is chosen, the better it will reflect the characteristic features of the phenomenon, the more successful the study will be and the more useful the recommendations arising from it.
There are no general ways to construct mathematical models. In each case, the model is selected based on the target orientation of the operation and the research task, taking into account the required accuracy of the solution and the accuracy with which we can know the initial data. If the initial data is known inaccurately, then, obviously, there is no point in building a very detailed, subtle and accurate model of the phenomenon and wasting time (your own and machine) on subtle and accurate optimization of the solution. Unfortunately, this principle is often neglected in practice and excessively detailed models are chosen to describe phenomena.
The model should reflect the most important features of the phenomenon, i.e. it should take into account all the essential factors on which the success of the operation most depends. At the same time, the model should be as simple as possible, not «clogged» with a mass of small, secondary factors, since taking them into account complicates mathematical analysis and makes the results of the study difficult to see. In a word, the art of making mathematical models is precisely the art, and experience in this matter is acquired gradually. Two dangers always lie in wait for the compiler of the model: the first is to drown in detail («you can’t see the forest because of the trees»); The second is to coarsen the phenomenon too much («throw out the baby with the bathwater»). Therefore, when solving problems of operations research, it is always useful to compare the results obtained by different models, to arrange a kind of «model dispute». The same problem is solved not once, but several, using different systems of assumptions, different apparatus, different models.
If scientific conclusions change little from model to model, this is a serious argument in favor of the objectivity of the study. If they differ significantly, it is necessary to revise the concepts underlying the various models, to see which of them is most adequate to reality. It is also characteristic of the operations study to re-refer to the model (after the study in the first approximation has already been performed) to make the necessary adjustments to this model.
The construction of a mathematical model is the most important and responsible part of the study, which requires deep knowledge not only and not so much of mathematics, but of the essence of the phenomena being modeled. As a rule, «pure» mathematicians do not cope with this task well without the help of specialists in this field. They focus on the mathematical apparatus with its subtleties, and not the correspondence of the model to the real phenomenon. Experience shows that the most successful models are created by specialists in this field of practice, who have received deep mathematical training in addition to the main one, or by groups that unite specialists and mathematicians.
The mathematical training of a specialist wishing to engage in the study of operations in his field of practice should be quite wide. Along with classical methods of analysis, it should include a number of modern branches of mathematics, such as optimization methods, including linear, nonlinear, dynamic programming, methods of machine search for extremes, etc. Special requirements for probabilistic training are related to the fact that most operations are carried out in conditions of incomplete certainty, their course and outcome depend on random factors – such as meteorological conditions, fluctuations in supply and demand, failures of technical devices, etc. Therefore, creative work in the field of operations research requires a good command of probability theory, including its newest sections: the theory of stochastic processes, information theory, theory of games and static solutions, theory of queuing.
When constructing a mathematical model, a mathematical apparatus of varying complexity can be used (depending on the type of operation and research tasks). In the simplest cases, the model is described by simple algebraic equations. In more complex ones, when it is necessary to consider the phenomenon in dynamics, the apparatus of differential equations, both ordinary and partial derivatives, is used. In the most difficult cases, if the development of the operation in time depends on a large number of intricately intertwined random factors, the method of statistical modeling is used. As a first approximation, the idea of the method can be described as follows: the process of development of the operation, as it were, is «copied», reproduced on a machine (computer) with all the accompanying accidents. Thus, one instance (one implementation) of a random process (operation) is built, with a random course and outcome. By itself, one such implementation does not give grounds for choosing a solution, but, having received a set of such implementations, we process them as ordinary statistical material (hence the term «statistical modeling»), derive the average characteristics for a set of implementations and get an idea of how, on average, the conditions of the problem and the elements of the solution affect the course and outcome of the operation.
In the study of operations, the course of which is influenced by random factors, the so-called «stochastic problems of operations research», both analytical and statistical models are used. Each of these types of models has its advantages and disadvantages. Analytical models are coarser than statistical ones, take into account fewer factors, and inevitably require some assumptions and simplifications. These models can describe the phenomenon only approximately,