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).
2.3 Different types of operations research problems and methods for solving them
The objectives of the study are divided into two categories: a) direct and b) reverse. Direct tasks answer the question: what will happen if, under the given conditions, we make such and such a decision? In particular, what will be equal to the selected performance indicator W in this decision?
Inverse problems answer the question: how should the elements of the solution be selected in order for the efficiency indicator W to turn to the maximum?
Naturally, direct problems are simpler than inverse ones. It is also obvious that in order to solve the inverse problem, first of all, one must be able to solve a straight line. This purpose is served by the mathematical model of the operation, which makes it possible to calculate the efficiency indicator W (and, if necessary, other characteristics) for any given conditions, with any solution.
If the number of possible solutions is small, then by calculating the W value for each of them and comparing the values obtained with each other, you can directly specify one or more optimal options for which the efficiency indicator reaches a maximum. However, when the number of possible solutions is large, the search for the optimal one among them «blindly», by simple search, is difficult, in some cases it is almost impossible. For this purpose, special methods of targeted search are used (we will get acquainted with some of them later). Now we will limit ourselves to the formulation of the problem of optimizing the solution in the most general form.
Let there be an operation «O», the success of which we can influence to some extent by choosing in one way or another the parameters (elements of the solution) that depend on us. The efficiency of the operation is characterized by the efficiency indicator W, which is required to be turned to the maximum.
Suppose that the direct problem is solved and the mathematical model allows you to calculate the value of W for any chosen solution, for any set of conditions.
Let us first consider the simplest (so-called «deterministic») case, when the conditions for performing the operation are fully known, i.e. do not contain an element of uncertainty. Then all the factors on which the success of the operation depends are divided into two groups:
1) Predetermined, predetermined factors (conditions) α1, α2,… over which we have no influence (in particular, restrictions imposed on the decision);
2) Factors depending on us (elements of the solution) x1, x2,… which we, within certain limits, can choose at our discretion.
The W performance indicator depends on both groups of factors. We will write this in the form of a formula:
W = W (a1, a2,..; х1, х2,..).
It is believed that the type of dependence (1) is known to us and with the help of a mathematical model we can calculate for any given α1, α2,.., x1, x2,.. value of W (i.e., the direct problem is solved). Then the inverse problem can be formulated as follows:
Under given conditions, α1, α2,.. find such elements of the solution x1, x2,.., which turn the W indicator to the maximum.
Before us is a typically mathematical problem belonging to the class of so-called variational problems. Methods for solving such problems are analyzed in detail in mathematics. The simplest of these methods (the well-known «maximum and minimum problems») are familiar to every engineer. These methods prescribe to find the maximum or minimum (in short, the «extremum») of the function to differentiate it by arguments, equate the derivatives to zero and solve the resulting system of equations. However, this classical method has only limited application in the study of operations. First, in the case when there are many arguments, the task of solving a system of equations is often not easier, but more difficult than the direct search for an