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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, schematically, but the results of such modeling are more visual and more clearly reflect the patterns inherent in the phenomenon. And most importantly, analytical models are more suitable for finding optimal solutions, which can also be carried out by analytical methods, using all the means of modern mathematics.

Statistical models, in comparison with analytical ones, are more accurate and detailed, do not require such crude assumptions, and allow us to take into account a large (in theory, infinitely large) number of factors. It would seem that they are closer to reality and should be preferred. However, they also have their drawbacks: comparative bulkiness, high consumption of computer time; poor visibility of the results obtained and the difficulty of comprehending them. And most importantly, the extreme difficulty of finding the optimal solutions that have to be sought «by touch», by guesses and trials.

Young professionals, whose experience in operations research is limited, having at their disposal modern computers, often unnecessarily begin research with the construction of its statistical model, trying to take into account in this model a huge number of factors (the more, the better). As a result, many of these models remain «stillborn», since they have not developed a methodology for applying and comprehending the results, translating them into the rank of recommendations.

The best results are obtained when analytical and statistical models are used together. A simple analytical model allows you to understand the basic laws of the phenomenon, outline, as it were, its «contour», and indicate a reasonable solution in the first approximation. After that, any refinement can be obtained by statistical modeling. If the results of statistical modeling do not diverge too much from the results of analytical modeling, this gives us reason not only in this case, but also in many similar ones, to apply an analytical model. If the statistical model gives significantly different results compared to the analytical one, a system of corrections to the analytical solution can be developed such as «empirical formulas» that are widely used in technology.

When optimizing solutions, it can also be very useful to optimize them in advance on an analytical model. This will allow, when using a more accurate statistical model, to search for the optimal solution not quite at random, but in a limited area containing solutions that are close to the optimal ones in the analytical model. Given that in practice we are rarely interested in a single, exactly optimal solution, more often it is necessary to indicate the area in which it lies, analytical optimization methods, tested and supported by statistical modeling, can be a valuable tool for making recommendations.

The construction of a mathematical model of operations is not important in itself, but is aimed at identifying optimal solutions. It is advisable to choose a solution that ensures operations of maximum efficiency. Under the effectiveness of the operation, of course, the measure of its success is the degree of its adaptability to achieve the goal before it.

In order to compare various solutions in terms of effectiveness, it is necessary to have some kind of quantitative criterion, an indicator of effectiveness (it is often called the «target function»). This indicator is selected so that it best reflects the target orientation of operations. To choose a performance indicator, you must first ask yourself: what do we want, what do we strive for when undertaking an operation? When choosing a solution, we prefer one that turns the performance indicator into a maximum (or minimum).

Very often, the cost of performing operations appears as performance indicators, which, of course, need to be minimized. For example, if the operation aims to change the production technology so as to reduce the cost of production as much as possible, then it will be natural to take the average cost as an indicator of efficiency and prefer the solution that will turn this indicator into a minimum.

In some cases, it happens that the operation pursues a well-defined goal A, which alone can be achieved or not achieved (we are not interested in any intermediate results). Then the probability of achieving this goal is chosen as an indicator of effectiveness. For example, if you are shooting at an object with the sine qua non condition of destroying it, the probability of destroying the object will be an indicator of effectiveness.

Choosing the wrong KPI is very dangerous, as it can lead to incorrect recommendations. Operations organized from the point of view of an unsuccessfully chosen indicator can lead to large unjustified costs and losses (recall at least the notorious «shaft» as the main criterion for the economic activity of enterprises).