Operations research has been gaining an increasingly broad field of applications in recent years. This science belongs to the comparatively young, recently formed disciplines; its boundaries and content are not clearly defined.
Operations Research is taught at many universities, but the content assigned to this term is far from uniform. Some authors understand "operations research" to mean primarily mathematical optimization methods, such as linear, nonlinear, and dynamic programming. Others, on the contrary, do not include these branches of mathematics in operations research, approaching the latter mainly from the standpoint of game theory and statistical decision theory. Some are inclined to deny the existence of "operations research" as an independent scientific discipline altogether, incorporating it into cybernetics (a term that is also insufficiently defined and understood differently by different people). Others, conversely, invest the concept of "operations research" with an excessively broad meaning, proclaiming this discipline to be an all-encompassing "science of sciences."
Time will tell in what forms this comparatively young science will continue to develop, which sections it will retain, and which will "branch off" as independent scientific disciplines. In particular, the future relationship between "operations research" and "systems theory" (or "theory of complex systems"), about which much has been said and written recently, remains unclear. In any case, it is undoubtedly true that in the most diverse areas of practice — organization of production and supply, transport operations, military operations and armaments, personnel deployment, consumer services, healthcare, communications, computing, and so on — problems are increasingly arising that are similar to one another in their formulation, possess a number of common features, and are solved by similar methods, which are conveniently grouped under the general heading of "operations research problems."
The typical situation is as follows: some purposeful course of action is organized in one way or another — that is, some "decision" can be made from a number of possible alternatives. Each alternative has advantages and disadvantages, and owing to the complexity of the situation, it is not immediately clear which is better (preferable) and why. To clarify the situation and compare the decision alternatives against a set of criteria, a series of mathematical calculations is performed. Their task is to help the people responsible for choosing the decision to perform a critical analysis of the situation and, ultimately, to settle on one alternative or another.
In our time, which is justly called the era of the scientific and technological revolution, science is paying ever greater attention to questions of organization and management. There are many reasons for this. The rapid development and growing complexity of technology, the unprecedented expansion in the scale of undertakings and the range of their possible consequences, the introduction of automated control systems into all areas of practice — all of this necessitates analyzing complex goal-directed processes from the standpoint of their structure and organization. Science is called upon to provide recommendations for the optimal (rational) management of such processes. Gone are the times when correct, effective management was found by organizers "by feel," through the method of "trial and error." Today, developing such management requires a scientific approach — the losses associated with errors are too great.
The needs of practice gave rise to special scientific methods that are conveniently grouped under the heading "operations research." By this term we shall understand the application of mathematical, quantitative methods for the justification of decisions in all areas of purposeful human activity.
Let us explain what is meant by a "decision." Suppose some undertaking is being carried out, directed toward achieving a specific goal. The person (or group of persons) organizing the undertaking always has some freedom of choice: they can organize it in one way or another — for example, they can choose the types of equipment to be used, distribute available resources in one way or another, and so on. A "decision" is precisely some choice from the range of possibilities available to the organizer. Decisions can be poor or good, well-considered or hasty, well-founded or arbitrary.
Operations research begins when mathematical methods are applied to justify decisions. Up to a certain point, decisions in any area of practice are made without special mathematical calculations, simply on the basis of experience and common sense. However, there are decisions that are far more consequential. Suppose, for example, that public transportation is being organized in a new city with a network of enterprises, residential districts, and so on. A number of decisions must be made: along which routes and with which transportation means should they be directed? At which points should stops be placed? How should the service frequency be adjusted based on the time of day? And so on.
These decisions are far more complex, and most importantly, a great deal depends on them. An incorrect choice can affect the working life of an entire city. Of course, in this case too, one can act intuitively when choosing a decision, relying on experience and common sense (and this is indeed frequently done). But decisions will be far more rational if they are supported by mathematical calculations. These preliminary calculations will help avoid a lengthy and costly search for the right decision "by feel."
The more complex, costly, and large-scale the planned undertaking, the less acceptable are arbitrary decisions, and the more important become the scientific methods that allow one to evaluate in advance the consequences of each decision, to reject in advance the inadmissible alternatives and recommend the most successful ones; to determine whether the information available to us is sufficient for a correct choice of decision and, if not, what additional information needs to be obtained. It is too dangerous in such cases to rely on one's intuition, on "experience and common sense." In our era of the scientific and technological revolution, technology and methods change so rapidly that "experience" simply does not have time to accumulate. Moreover, we are often dealing with unique undertakings being conducted for the first time. In such cases, "experience" is silent, and "common sense" can easily be deceived if it is not grounded in calculation. It is precisely such calculations, which help people make decisions, that operations research is concerned with.
An operation is any undertaking (system of actions), unified by a single plan and directed toward achieving some goal. An operation is always a managed undertaking — that is, it depends on us how to choose certain parameters characterizing its organization. "Organization" here is understood in the broad sense of the word, including the set of technical means employed in the operation.
Any specific choice of the parameters that depend on us is called a decision. Decisions can be successful or unsuccessful, rational or irrational. Decisions are called optimal if, by one attribute or another, they are preferable to others. The goal of operations research is to provide quantitative support for choosing optimal decisions.
Sometimes (relatively rarely) a study yields a single strictly optimal decision; far more often it identifies a range of practically equivalent optimal (rational) decisions, within which the final choice can be made.
It should be noted that the actual making of a decision lies outside the scope of operations research and falls within the competence of a responsible individual, or more often a group of individuals, who are granted the right of final choice and bear responsibility for that choice. In making their choice, they may take into account, alongside the recommendations resulting from mathematical calculations, a number of additional considerations (of both quantitative and qualitative character) that were not accounted for in the calculations.
The indispensable presence of a human being (as the final authority making the decision) is not nullified even in the presence of a fully automated control system that would appear to make decisions without human involvement. One must not forget that the very creation of the control algorithm, the choice of one of its possible variants, is itself a decision — and a highly consequential one.
The parameters whose totality forms a decision are called decision variables. Various numbers, vectors, functions, physical attributes, and so on may serve as decision variables. In the simplest operations research problems, the number of decision variables may be comparatively small. But in most problems of practical significance, the number of decision variables is very large.
In addition to the decision variables, which we can dispose of within certain limits, every operations research problem also contains given, "disciplining" conditions that are fixed from the outset and cannot be violated (for example, the load capacity of a vehicle, the size of a planned assignment, the weight characteristics of equipment, and the like). In particular, such conditions include the resources (material, technical, human) that we are entitled to dispose of, and other constraints imposed on the decision. Taken together, they form the so-called "feasible set."
To compare different decisions with one another in terms of effectiveness, one needs some quantitative criterion, a so-called measure of effectiveness (often called the "objective function"). This measure is chosen so as to reflect the goal-directedness of the operation. The "best" decision will be considered the one that contributes to the greatest extent to achieving the stated goal. To choose, to "name" the measure of effectiveness W, one must first ask oneself: what do we want, what are we striving for in undertaking the operation? In choosing a decision, we naturally prefer the one that turns the measure of effectiveness W into a maximum (or a minimum). For example, one would like to maximize the income from an operation; if the measure of effectiveness is costs, one would like to minimize them.
Very often the conduct of an operation is accompanied by the action of random factors. In such cases, the measure of effectiveness is usually taken to be not the quantity itself that one would like to maximize (minimize), but its average value (expected value).
In some cases, an operation accompanied by random factors pursues a quite definite goal that can only be either fully achieved or not achieved at all (the "yes–no" scheme), and no intermediate results are of interest. In such cases, the probability of achieving this goal is chosen as the measure of effectiveness. For example, if fire is being directed at some target with the mandatory condition of destroying it, then the probability of destroying the target will be the measure of effectiveness.
An incorrect choice of the measure of effectiveness is very dangerous. Operations organized from the standpoint of a poorly chosen criterion can lead to unjustified expenditures and losses. Unfortunately, in most problems of practical significance, the choice of the measure of effectiveness is not simple and is resolved ambiguously. For any reasonably complex problem, it is typical that the effectiveness of an operation cannot be exhaustively characterized by a single number — other measures must be brought in to supplement it.
To apply quantitative research methods in any field, some kind of mathematical model is always required. In constructing a model, the real phenomenon (in our case, the operation) is inevitably simplified and schematized, and this schema (a schematic representation of the phenomenon) is described using mathematical tools. The more successfully the mathematical model is chosen, the better it reflects the characteristic features of the phenomenon, the more successful the study will be and the more useful the resulting recommendations.
There are no general methods for constructing mathematical models. In each specific case, the model is chosen based on the type of operation, its goal-directedness, and with consideration of the research task (which parameters need to be determined and which factors' influence needs to be reflected). It is also necessary in each specific case to balance the precision and detail of the model with: (a) the precision with which we need to know the solution, and (b) the information that we have or can acquire. If the initial data needed for calculations are known imprecisely, then there is obviously no point in going into subtleties, building a very detailed model, and spending time (one's own and the computer's) on fine and precise optimization of the solution. Unfortunately, this principle is often neglected, and overly detailed models are chosen to describe phenomena.
A mathematical model must reflect the most important features of the phenomenon, all the essential factors on which the success of the operation principally depends. At the same time, the model should be as simple as possible, not "cluttered" with a mass of minor, secondary factors: accounting for them complicates the mathematical analysis and makes the research results harder to survey. Two dangers always lie in wait for the model builder: the first is to get bogged down in details ("not see the forest for the trees"), and the second is to oversimplify the phenomenon ("throw the baby out with the bathwater"). The art of building mathematical models is indeed an art, and experience in it is acquired gradually.
Since a mathematical model does not follow with necessity from the problem description, it is always useful not to trust any single model blindly, but to compare results obtained from different models, arranging a kind of "contest of models." In this process, the same problem is solved not once but several times, using different systems of assumptions, different apparatus, and different models. If the scientific conclusions change little from model to model, this is a strong argument in favor of the objectivity of the study. If they diverge significantly, one must reconsider the concepts underlying the various models, determine which of them is more adequate to reality, and if necessary, conduct a control experiment. Also characteristic of operations research is the repeated return to the model (after the first round of calculations has already been carried out) to introduce corrections into it.
Creating a mathematical model is the most important and responsible part of the study, requiring deep knowledge not so much of mathematics as of the essence of the phenomena being modeled. As a rule, "pure" mathematicians (without the help of specialists in the field to which the problem belongs) cope poorly with model construction. Their attention is focused on the mathematical tools and their subtleties, rather than on the real practical problem.
Experience shows that the most successful models are created by specialists in the given area of practice who have received, in addition to their primary training, a deep mathematical preparation, or by teams that bring together practical specialists and mathematicians. Great benefit comes from consultations given by a mathematician well versed in operations research to practitioners — engineers, biologists, medical professionals, and others — who encounter in their work the need for scientific justification of decisions.
It must be especially emphasized that knowledge of probability theory is necessary — not so much extensive and deep as informal and effective, along with a habit of working with statistical data and probabilistic concepts. The special requirements in precisely this area of mathematical knowledge are explained by the fact that most operations are conducted under conditions of incomplete certainty, and their course and outcome depend on random factors. Unfortunately, among the broad circles of specialists — engineers, biologists, medical professionals, chemists — good command of probability theory is rarely encountered. Its propositions and rules are often applied formally, without a genuine understanding of their meaning and spirit. Not infrequently, probability theory is viewed as a kind of "magic wand" that allows one to obtain information "from nothing," from complete ignorance. This is a misconception: probability theory only allows one to transform information — that is, from data about certain phenomena accessible to observation, to draw conclusions about others that are inaccessible.
In constructing a mathematical model, mathematical tools of varying complexity may be used (depending on the type of operation, the research tasks, and the precision of the initial data). In the simplest cases, the phenomenon is described by simple algebraic equations. In more complex cases, when it is necessary to examine the phenomenon in its dynamics, the methods of differential equations (ordinary or partial) are employed. In the most complex cases, when the development of an operation and its outcome depend on a large number of intricately intertwined random factors, analytical methods fail entirely, and the method of statistical modeling (Monte Carlo) is employed.
In a first, rough approximation, the idea of this method can be described as follows: the process of the operation's development, with all the accompanying randomness, is in effect "copied," reproduced on a computer. The result is a single specimen (a "realization") of the random process of the operation's development with a random course and outcome. A single such realization by itself provides no basis for choosing a decision, but having obtained many such realizations, we process them as ordinary statistical material (hence the term "statistical modeling"), find the average characteristics of the process, and gain an understanding of how, on average, the conditions of the problem and the decision variables influence them.
In operations research, both analytical and statistical models are widely used. Each of these types has its own advantages and disadvantages. Analytical models are cruder, account for fewer factors, and always require certain assumptions and simplifications. On the other hand, their computational results are easier to survey and more clearly reflect the fundamental patterns inherent in the phenomenon. And most importantly, analytical models are better suited for searching for optimal solutions.
Statistical models, compared with analytical ones, are more precise and detailed, do not require such crude assumptions, and allow one to account for a large (in theory, unlimited) number of factors. But they too have their disadvantages: cumbersomeness, poor surveyability, and most importantly, the extreme difficulty of searching for optimal solutions, which must be sought "by feel," through guesses and trials.
The best works in the field of operations research are based on the combined use of analytical and statistical models. An analytical model makes it possible to gain a general understanding of the phenomenon, to sketch out, as it were, the "contour" of the fundamental patterns. Any refinements can be obtained with the help of statistical models.
In conclusion, let us say a few words about so-called "simulation" modeling. It is applied to processes into whose course human will may intervene from time to time. The person (or group of persons) directing the operation can, depending on the situation that has developed, make one decision or another, much as a chess player, looking at the board, chooses their next move. Then the mathematical model is set in motion, showing what change in the situation is expected in response to this decision and what consequences it will lead to after some time. The next "current decision" is made with the new, actual situation already taken into account, and so on. As a result of the repeated application of this procedure, the manager effectively "gains experience," learns from their own and others' mistakes, and gradually learns to make correct decisions — if not optimal, then nearly optimal.