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 lets 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: either the probability distribution for the parameters ξ1, ξ2, In principle, it exists, but the corresponding statistical data cannot be obtained, or the probability distribution for the parameters ξ1, ξ2, does not exist at all.
Let us give an example related to the last, most «harmful» category of uncertainty. Lets assume that some commercial and industrial operation is planned, the success of which depends on the length of skirts ξ women will wear in the coming year. The probability distribution for the parameter ξ cannot, in principle, be obtained from any statistical data. One can only try to guess its plausible meanings in a purely speculative way.
Let us consider just such a case of «bad uncertainty»: the effectiveness of the operation depends on the unknown parameters ξ1, ξ2, , about which we have no information, but can only make suggestions. Lets try to solve the problem.
The first thing that comes to mind is to ask some (more or less plausible) values of the parameters ξ1, ξ2, and find a conditionally optimal solution for them. Lets assume that, having spent a lot of effort and time (our own and machine), we did it. So what? Will the conditionally optimal solution found be good for other conditions? As a rule, no. Therefore, its value is purely limited. In this case, it will be reasonable not to have a solution that is optimal for some conditions, but a compromise solution that, while not optimal for any conditions, will still be acceptable in their whole range. At present, a full-fledged scientific «theory of compromise» does not yet exist (although there are some attempts in this direction in decision theory). Usually, the final choice of a compromise solution is made by a person. Based on preliminary calculations, during which a large number of direct problems for different conditions and different solutions are solved, he can assess the strengths and weaknesses of each option and make a choice based on these estimates. To do this, it is not necessary (although sometimes curious) to know the exact conditional optimum for each set of conditions. Mathematical variational methods recede into the background in this case.
When considering the problems of operations research with «bad uncertainty», it is always useful to confront different approaches and different points of view in a dispute. Among the latter, it should be noted one, often used because of its mathematical certainty, which can be called the «position of extreme pessimism». It boils down to the fact that one must always count on the worst conditions and choose the solution that gives the maximum effect in these worst conditions for oneself. If, under these conditions, it gives the value of the efficiency indicator equal to W *, then this means that under no circumstances will the efficiency of the operation be less than W * («guaranteed winnings»). This approach is tempting because it gives a clear formulation of the optimization problem and the possibility of solving it by correct mathematical methods. But, using it, we must not forget that this point of view is extreme, that on its basis you can only get an extremely cautious, «reinsurance» decision, which is unlikely to be reasonable. Calculations based on the point of view of «extreme pessimism» should always be adjusted with a reasonable dose of optimism. It is hardly advisable to take the opposite point of view extreme or «dashing» optimism, always count on the most favorable conditions, but a certain amount of risk when making a decision should still be present.