Effective Methods and Transportation Processes Management Models at the Railway Transport. Textbook. Vadim Shmal

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СКАЧАТЬ 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 extremum. Secondly, the extremum is often reached not at all at the point where the derivatives turn to zero (such a point may not exist at all), but somewhere at the boundary of the area of change of arguments. All the specific difficulties of the so-called «multidimensional variational problem in the presence of limitations» arise, sometimes unbearable in its complexity even for modern computers. In addition, we must not forget that the function W may not have derivatives at all, for example, be integer, or be given only with integer values of arguments. All this makes the task of finding an extremum far from being as easy as it seems at first glance. The optimization method should always be chosen based on the features of the W function and the type of constraints imposed on the elements of the solution. For example, if the function W linearly depends on the elements of the solution x1, x2,.., and the constraints imposed on x1, x2,.., have the form of linear equalities or inequalities, the problem of linear programming arises, which is solved by relatively simple methods (we will get acquainted with some of them later). If the W function is convex, special methods of «convex programming» are used, with their kind of «quadratic programming». To optimize the management of multi-stage operations, the method of «dynamic programming» can be applied. Finally, there is a whole set of numerical methods for finding the extremes of the functions of many arguments, specially adapted for implementation on computers. Thus, the problem of optimizing the solution in the considered deterministic case is reduced to the mathematical problem of finding the extremum of a function that can present computational, but not fundamental difficulties.

      Let’s not forget, however, that we have considered so far the simplest case, when only two groups of factors appear in the problem: the given conditions α1, α2,.. and solution elements x1, x2,… The real tasks of operations research are often reduced to a scheme where, in addition to two groups of factors α1, α2,.., x1, x2,.., there is a third – unknown factors ξ1, ξ2, …, the values of which cannot be predicted in advance.

      In this case, the W performance indicator depends on all three groups of factors:

      W = W (a1, a2,..; х1, х2,..; o1, x2, …)

      And the problem of solution optimization can be formulated as follows:

      Under given conditions, α1, α2,.. Taking into account the presence of unknown factors ξ1, ξ2, … find such elements of the solution x1, x2,…, which, if possible, provide the maximum value of the efficiency indicator W.

      This is another, not purely mathematical problem (it is not for nothing that the reservation «if possible» is made in its formulation). The presence of unknown factors translates the problem into a new quality: it turns into a problem of choosing a solution under conditions of uncertainty.

      However, uncertainty is uncertainty. If the conditions for the operation are unknown, we cannot optimize the solution as successfully as we would if we had more information. Therefore, any decision made under conditions of uncertainty is worse than a decision made under predetermined conditions. It is our business to communicate to our decision as much as possible the features of reasonableness. It is not for nothing that one of the prominent foreign experts in operations research, T.L. Saati, defining his subject, writes that «operations research is the art of giving bad answers to those practical questions to which even worse answers are given by other methods.»

      The task of making a decision in conditions of uncertainty is found at every step in life. Suppose, for example, that we are going to travel and put some things in our suitcase. The size of the suitcase is limited (conditions α1, α2,..), the weather in the travel areas is not known in advance (ξ1, ξ2,…). What items of clothing (x1, x2,..) should I take with me? This problem of operations research, of course, is solved by us without any mathematical apparatus, although based on some statistical data, say, about the weather in different areas, as well as our own tendency to colds; Something like optimizing the decision, consciously or unconsciously, we produce. Curiously, different people seem to use different performance indicators. If a young person is likely to seek to maximize the number of pleasant impressions from the trip, then an elderly traveler, perhaps, wants to minimize the likelihood of illness.

      And now let’s take a more serious task. A system of protective structures is being designed to protect the area from floods. Neither the moments of the onset of floods, nor their size are known in advance. And you still need to design.

      In order to make such decisions not at random, by inspiration, but soberly, with open eyes, modern science has a number of methodological techniques. The use of one or the other of them depends on the nature of the unknown factors, where they come from and by whom they are controlled.

      The simplest case of uncertainty is the case when the unknown factors ξ1, ξ2,… are random variables (or random functions) whose statistical characteristics (say, distribution laws) are known to us or, in principle, can be obtained. We will call such problems of operations research stochastic problems, and the inherent uncertainty – stochastic uncertainty.

      Here is an example of a stochastic operations research problem. Let the work of the catering enterprise be organized. We do not know exactly how many visitors will come to it the day before work, how long the service of each of them will continue, etc. However, the characteristics of these random variables, if we are not already at our disposal, can be obtained statistically.

      Let us now assume that we have before us a stochastic problem of operations research, and the unknown factors ξ1, ξ2,… – ordinary random variables with some (in principle known) probabilistic characteristics. Then the efficiency indicator W, depending on these factors, will also be a random value.

      The first thing that comes to mind is to take as an indicator of efficiency not the random variable W itself, but its average value (mathematical expectation)

      W = СКАЧАТЬ