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. Let’s 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. Let’s 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.
Let us mention one, rather original method used when choosing a solution in conditions of «bad uncertainty» – the so-called method of expert assessments. It is often used in other fields, such as futurology. Roughly speaking, it consists in the fact that a team of competent people («experts») gathers, each of them is asked to answer a question (for example, name the date when this or that discovery will be made); then the answers obtained are processed like statistical material, making it possible (to paraphrase T. L. Saati) «to give a bad answer to a question that cannot be answered in any other way.» Such expert assessments for unknown conditions can also be applied to solving problems of operations research under conditions of «bad uncertainty». Each of the experts evaluates the degree of plausibility of various variants of conditions, attributing to them some subjective probabilities. Despite the subjective nature of the estimates of probabilities by each expert, by averaging the estimates of the whole team, you can get something more objective and useful. By the way, the subjective assessments of different experts do not differ as much as one might expect. In this way, the solution of the problem of studying operations with «bad uncertainty» seems to be reduced to the solution of a relatively benign stochastic problem. Of course, the result obtained cannot be treated too trustingly, forgetting about its dubious origin, but along with others arising from other points of view, it can still help in choosing a solution.
Let’s name another approach to choosing a solution in conditions of uncertainty – the so-called «adaptive algorithms» of control. Let the operation O in question belong to the category of repeating repeatedly, and some of its conditions are ξ1, ξ2,… Unknown in advance, random. However, we do not have statistics on the probability distribution for these conditions and there is no time to collect such data (for example, it takes a considerable amount of time to collect statistics, and the operation needs to be performed now). Then it is possible to build and apply an adapting (adapting) control algorithm, which gradually takes place in the course of its application. At first, some (probably not the best) algorithm is taken, but as it is applied, it improves from time to time, since the experience of application indicates how it should be changed. It turns out something like the activity of a person who, as you know, «learns from mistakes.» Such adaptable control algorithms seem to have a great future.
Finally, we will consider a special case of uncertainty, not just «bad» but «hostile.» This kind of uncertainty arises in so-called «conflict situations» in which the interests of two (or more) parties with different goals collide. Conflict situations are characteristic of military operations, partly for sports competitions; in capitalist society – for competition. Such situations are dealt with by a special branch of mathematics – game theory. (It is often presented as part of the discipline «operations research.») The most pronounced case of a conflict situation is direct antagonism, when two sides A and B clash in a conflict, pursuing directly opposite goals («us» and «adversary»).
The theory of antagonistic games is based on the proposition that we are dealing with a reasonable and far-sighted adversary, always choosing his behavior in such a way as to prevent us from achieving our goal. In the accepted proposals, game theory makes it possible to choose the optimal solution in some sense, i.e. the least risky in the fight against a cunning and malicious opponent.
However, such a point of view on the conflict situation cannot be absolutized either. Life experience suggests that in conflict situations (for example, in hostilities), it is not the most cautious, but the most inventive who wins, who knows how to take advantage of the enemy’s weakness, deceive him, go beyond the conditions and methods of behavior known to him. So in conflict situations, game theory provides an extreme solution arising from a pessimistic, «reinsurance» position. Yet, if treated with due criticism, it, along with other considerations, can help in the final choice.
Closely related to game theory is the so-called «statistical decision theory». It is engaged in the preliminary mathematical justification of rational decisions in conditions of uncertainty, the development of reasonable «strategies of behavior» in these conditions. One possible approach to solving such problems is to consider an uncertain situation as a kind of «game», but not with a consciously opposing, reasonable adversary, but with «nature». By «nature» in the theory of statistical decisions is understood as a certain third-party authority, indifferent to the result of the game, but whose behavior is not known in advance.
Finally, let’s make one general remark. When justifying a decision under conditions of uncertainty, no matter what we do, the element of uncertainty remains. Therefore, it is impossible to impose too high demands on the accuracy of solving such problems. Instead of unambiguously indicating a single, exactly «optimal» (from some point of view) solution, it is better to single out a whole area of acceptable solutions that turn out to be insignificantly worse than others, no matter what point of view we use. Within this area, the persons responsible for this should make their final choice.
2.4 Multi-criteria Operations Research Tasks
Despite a number