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.
Lets 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 enemys 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, lets 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 of significant difficulties associated with the uncertainty of the conditions of the operation, we have still considered only the simplest problems of operations research, when the criterion by which the effectiveness is evaluated is clear, and it is necessary to turn into a maximum (or minimum) a single indicator of efficiency W. It is he who is the criterion by which one can judge the effectiveness of the operation and the decisions made.
Finally, lets 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.
Despite a number of significant difficulties associated with the uncertainty of the conditions of the operation, we have still considered only the simplest problems of operations research, when the criterion by which the effectiveness is evaluated is clear, and it is necessary to turn into a maximum (or minimum) a single indicator of efficiency W. It is he who is the criterion by which one can judge the effectiveness of the operation and the decisions made.
Unfortunately, in practice, such tasks, where the evaluation criterion is clearly dictated by the target orientation of the operation, are relatively rare, mainly when considering small-scale and modest-value activities. As a rule, the effectiveness of large-scale, complex operations affecting the diverse interests of participants cannot be exhaustively characterized using a single performance indicator W. To help him, he has to attract other, additional ones. Such operations research tasks are called «multi-criteria».
Such a multiplicity of criteria (indicators), of which it is desirable to turn some into a maximum, and others into a minimum, is characteristic of any somewhat complex operation. We suggest the reader to formulate in the form of an exercise a number of criteria (performance indicators) for the operation in which the work of urban transport is organized. Fleet of mobile vehicles (trams, buses, trolleybuses) it is considered set; the solution swings routes and stops. When choosing a system of indicators, think about which of them is the main one (most closely related to the target orientation of the operation), and arrange the rest (additional) in descending order of importance. Using this example, you will see that a) none of the indicators can be chosen as the only one and b) the formulation of the indicator system is not such an easy task as it may seem at first glance.
So, typical for a large-scale task of operations research is multi-criteria the presence of a number of quantitative indicators W1, W2,, one of which is desirable to turn into a maximum, and others into a minimum, and others into a minimum («so that the wolves are fed and the sheep are safe»).
The question is, is it possible to find a solution that satisfies all these requirements at the same time? With all frankness, we answer: no. A solution that turns any indicator to a maximum, as a rule, neither turns into a maximum, nor a minimum of others. Therefore, the phrase «achieving maximum effect at minimum cost», which is widely used in everyday life, is nothing more than a phrase and should be discarded in scientific analysis. Another wording will be legitimate: «achieving a given effect at minimal cost» or «achieving the maximum effect at a given cost» (unfortunately, these legal formulations seem to be somehow not «elegant» enough in oral speech).
What if you still have to evaluate the effectiveness of the operation by several indicators?
In practice, the following technique is often used for this: they try to compile one of several indicators and, when choosing a solution, use such a «generalized» indicator. Often it is composed in the form of a fraction, where in the numerator are those values, the increase of which is desirable, and in the denominator the increase of which is undesirable. For example, the enemys losses are in the numerator, own losses and the cost of funds spent are in the denominator, etc.
In practice, another, slightly more intricate method of compiling a «generalized» performance indicator is often used. They take individual private indicators, attribute some «weights» to them (a1, a2,), multiply each indicator by the corresponding weight and add them up; Those indicators that need to be maximized are with a minus sign.
With the arbitrary assignment of weights attributed to particular indicators, this method is no better than the first. Proponents of this technique refer to the fact that a person, making a compromise decision, also mentally weighs the pros and cons, attributing greater importance to factors that are more important to him. This may be true, but, apparently, the «weighting coefficients» with which different indicators are included in the mental calculation are not constant, but change depending on the situation.
Here we meet with an extremely typical technique for such situations «the transfer of arbitrariness from one instance to another.» Indeed, the simple choice of a compromise solution in a multi-criteria problem based on a mental comparison of the advantages and disadvantages of each solution seems too arbitrary, not «scientific» enough. But manipulating a formula that includes, albeit equally arbitrarily assigned coefficients, gives the solution the features of some kind of «scientificity». In fact, there is no science here one transfusion from empty to empty.
It turns out that the mathematical apparatus cannot help us in solving multi-criteria problems? Far from it, it can help, and very significantly. Firstly, with the help of this device, it is possible to successfully solve direct problems of operations research and establish what advantages and disadvantages each of the solutions has according to different criteria. The mathematical model gives us the opportunity to calculate not only the value of the main performance indicator, but also all additional ones, and the complexity of the calculation increases little. Comparison of the results of solving a set of such direct problems provides the decision maker with a certain «accumulated scientific experience». Knowing what he wins and what he sacrifices, a person can evaluate each of the decisions and choose the most acceptable for himself.
A perplexing question may arise: what about the mathematical methods of optimization, about which he heard a lot and which he hoped so much? The trouble is that each of these methods makes it possible to find only an optimal solution for a single, scalar criterion W. Evaluate by the vector criterion (W1, W2,) Modern mathematics does not yet know how. Indeed, not every «better» or «worse» is directly related to «more» or «less», and mathematical methods so far speak only the language of «more-less». Of all the devices known to us, so far only a person is able to make reasonable decisions not according to the scalar, but according to the vector criterion. How he does this is not clear. Maybe each time he reduces the vector to a scalar, forming some function (linear, nonlinear) from its components? Possibly, but not plausible. Most likely, when choosing a solution, he thinks not formally, but generally, instinctively assessing the situation as a whole, discarding insignificant details, subconsciously using all the experience he has, if not literally such, but similar situations. At the same time, the (informal) choice of a compromise solution can significantly help a person with a mathematical apparatus. In any case, it helps to discard in advance obviously unsuccessful solutions, which are not worth thinking about.
Lets demonstrate one of these methods of preliminary «rejection» of unsuccessful decisions. Let us have to make a choice between several solutions: X1, X2,, Xn (each option is a vector, the components of which are the elements of the solution). The effectiveness of the operation is evaluated by two indicators: the productivity of P and the cost of S. The first indicator is desirable to maximize, and the second to minimize.
Similarly, unsuitable options are discarded in the case when there are not two, but more. (With more than three of them, the geometric interpretation loses clarity, but the essence of the matter remains the same: the number of competitive solutions decreases sharply.) As for the final choice of the solution, it still remains the prerogative of man this unsurpassed «master of compromise».
However, the procedure for choosing the final solution, being repeated repeatedly, in different situations, can serve as the basis for which convenient formalization. We are talking about the construction of so-called «heuristic methods» of decision-making. Such methods are widely used in attempts to automate the solution of some informal tasks. For example, in order to force the automaton to solve difficult-to-formalize tasks (for example, reading handwritten text, recognizing images or sounds of live speech), so-called training automata are created. The program according to which such a machine works is not laid down in it in advance, but is formed gradually, in the process of familiarization with an increasingly wide range of situations. The initial model for the machine is an experienced person who knows how to perform an informal task, say, to make a decision according to a vector criterion. Subsequently, there may be further improvement of the program (already in the order of «self-study»).