In some cases, it happens that the operation pursues a well-defined goal A, which alone can be achieved or not achieved (we are not interested in any intermediate results). Then the probability of achieving this goal is chosen as an indicator of effectiveness. For example, if you are shooting at an object with the sine qua non condition of destroying it, the probability of destroying the object will be an indicator of effectiveness.
Choosing the wrong KPI is very dangerous, as it can lead to incorrect recommendations. Operations organized from the point of view of an unsuccessfully chosen indicator can lead to large unjustified costs and losses (recall at least the notorious «shaft» as the main criterion for the economic activity of enterprises).
2.3 Different types of operations research problems and methods for solving them
The objectives of the study are divided into two categories: a) direct and b) reverse. Direct tasks answer the question: what will happen if, under the given conditions, we make such and such a decision? In particular, what will be equal to the selected performance indicator W in this decision?
Inverse problems answer the question: how should the elements of the solution be selected in order for the efficiency indicator W to turn to the maximum?
Naturally, direct problems are simpler than inverse ones. It is also obvious that in order to solve the inverse problem, first of all, one must be able to solve a straight line. This purpose is served by the mathematical model of the operation, which makes it possible to calculate the efficiency indicator W (and, if necessary, other characteristics) for any given conditions, with any solution.
If the number of possible solutions is small, then by calculating the W value for each of them and comparing the values obtained with each other, you can directly specify one or more optimal options for which the efficiency indicator reaches a maximum. However, when the number of possible solutions is large, the search for the optimal one among them «blindly», by simple search, is difficult, in some cases it is almost impossible. For this purpose, special methods of targeted search are used (we will get acquainted with some of them later). Now we will limit ourselves to the formulation of the problem of optimizing the solution in the most general form.
Let there be an operation «O», the success of which we can influence to some extent by choosing in one way or another the parameters (elements of the solution) that depend on us. The efficiency of the operation is characterized by the efficiency indicator W, which is required to be turned to the maximum.
Suppose that the direct problem is solved and the mathematical model allows you to calculate the value of W for any chosen solution, for any set of conditions.
Let us first consider 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.
Lets 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.»