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 = M [W (a1, a2,..; х1, х2,..; o1, x2, …)]
and choose such a solution x1, x2,.., in which this average value turns into a maximum.
Note that this is exactly what we did, choosing in a number of examples of operations, the outcome of which depends on random factors, as an indicator of efficiency, the average value of the value that we wanted to turn into a maximum (minimum). This is the «average income» per unit of time, «average relative downtime», etc. In most cases, this approach to solving stochastic problems of operations research is fully justified. If we choose a solution based on the requirement that the average value of the performance indicator is maximized, then, of course, we will do better than if we chose a solution at random.
But what about the element of uncertainty? Of course, to some extent it remains. The success of each individual operation carried out with random values of the parameters ξ1, ξ2, …, can be very different from the expected average, both upwards and, unfortunately, downwards. We should be comforted by the following: by organizing the operation so that the average value of W is maximized and repeating the same (or similar) operations many times, we will ultimately gain more than if we did not use the calculation at all.
Thus, the choice of a solution that maximizes the average W value of the W efficiency indicator W is fully justified when it comes to operations with repeatability. A loss in one case is compensated by a gain in the other, and in the end our solution will be profitable.
But what if we are talking about an operation that is not repeatable, but unique, carried out only once? Here, a solution that simply maximizes the average value of W will be imprudent. It would be more cautious to guard yourself against unnecessary risk by demanding, for example, that the probability of obtaining an unacceptably small value of W, say, W˂w0, be sufficiently small:
P (W ˂w0) ≤ γ,
where γ is some small number, so small that an event with a probability of γ can be considered almost impossible. The condition-constraint can be taken into account when solving the problem of solution optimization along with others. Then we will look for a solution that maximizes the average value of W, but with an additional, «reinsurance» condition.
The case of stochastic uncertainty of conditions considered by us is relatively prosperous. The situation is much worse when the unknown factors ξ1, ξ2, … cannot be described by statistical methods. This happens in two cases: