# operations research

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## operations research

## Operations Research

a scientific method for developing quantitatively well-grounded recommendations for decisionmaking. The importance of the quantitative factor in operations research and the purposefulness of the recommendations developed make it possible to define operations research as the theory of making optimum decisions. Operations research helps change the art of decision-making into a scientific, and also mathematical, discipline. (The Russian term *issledovanie operatsii* is a literal translation of the American expression, which in turn is a modification of the British “operational research,” which was introduced in the late 1930’s as a provisional name for one of the subdivisions of the British Air Force that dealt with questions of using radar equipment in the overall defense system.)

A description of any task of operations research includes the assignment of components of the decision, which can be understood as the decision’s immediate consequences (the components of a decision are usually, although not necessarily, numerical variables); the constraints imposed on the components, reflecting the limited nature of resources; and the system of goals. Any system of components of a decision that satisfies all constraints is called a permissible decision. For each goal there is a target function, which is given in the set of permissible decisions, whose values express the degree of achievement of the goal. The essence of the task of operations research is to find the most expedient, optimum decisions. Therefore, the tasks of operations research are usually called optimization tasks.

Some of the most important and highly developed problems of operations research have come to be called operations research models. They are usually distinguished by established terminology, and specific methods exist for their solution. Among them are the transportation (or traffic) problem, the location problem, reliability theory and the closely related replacement theory, scheduling theory (also called the theory of calendar planning), inventory theory, and critical-path planning theory. Queuing theory is also considered to be an operations research model, even though not all of its problems are of the optimization type.

Among operations-research problems, those that have one target function that assumes numerical values are singled out. The theory of such problems is called mathematical programming (or optimum programming). Opposed to such problems are problems with several target functions or with one target function that assumes vector values or values of an even more complex nature. Such problems are called multicriteria problems. They are solved on the basis of game theory or by reduction —often arbitrary—to problems with a single target function.

Decisions are made on the basis of information received by the subject making the decision. Therefore, it is natural to classify the problems of operations research according to their characteristics in terms of information theory. If the subject maintains his information state during decision-making—that is, neither acquires nor loses information—the decision-making may be considered an instantaneous event. The corresponding problems of operations research are called static problems. On the other hand if the subject changes his information state during decision-making, acquiring or losing information, in such a dynamic problem it is usually expedient to make the decision by stages (“multistep decisions”) or even to expand decision-making into a process that is continuous over time. A significant part of the theory of dynamic problems of operations research is included in dynamic programming.

Various relationships may exist between the information state of the subject and his true (“physical”) state. If the information state encompasses an entire set of true states (the subject knows that he is in one of the states of the set but cannot determine his own true state with greater precision), then the problem of decision-making is called indeterminate and is solved using methods of game theory. If the information state consists of several true states and the subject also knows the a priori probabilities of each of the true states, then the problem is called stochastic (probabilistic) and is solved using methods of stochastic programming. Finally, if the information state coincides with the true state, the problem is called deterministic.

The analytic form of the constraints and target function plays an important part in the solution of deterministic problems. Thus, if the target function is a linear form of the decision components and the constraints are described by linear inequalities, then the problem is one of linear programming. The remaining deterministic problems are considered in nonlinear programming, in which convex and quadratic programming are naturally distinguished. If, according to the conditions of the problem, the components of the decision can only assume integral values, then the problem is treated using integer (discrete) programming. The family of problems that depend on a parameter is sometimes brought together into a single program of parametric programming. Finding the minimax (and maximin) is a particular case of deterministic problems.

Operations research was originally involved with the solution of military problems, but even in the late 1940’s its sphere of use came to encompass various areas of human activity. Operations research is used to solve not only purely technical—particularly technological—and technical-economic problems but also problems of control at various levels. The use of operations research in practical optimization problems produces a significant economic benefit; if other expenditures are equal, the use of optimum decisions rather than traditional “intuitive” methods results in a gain of about 10 percent.

Only some of the problems of operations research can be solved analytically, and a comparatively small number are capable of manual numerical solution. Therefore, the growth of the potential of operations research has been closely tied to progress in computer technology. In turn, the need for solutions to problems in operations research influences the growth and composition of the supply of computers. Since problems in operations research are characterized by large quantities of numerical data, which constitute the conditions of the problems, computers with a large memory capacity are especially well suited for solving such problems. The practical application of operations research encounters numerous difficulties, which appear even during compilation of the problem as a model, and particularly in indicating the target function. The mathematical difficulties, especially computations, that arise in the process of finding the optimum solution may be serious.

Courses in operations research are offered in the USSR and other countries at many universities, higher technical schools, and institutes for advanced training.

Special journals of operations research include *Operational Research Quarterly* (London, since 1950), *Operations Research* (Baltimore, since 1952), *Naval Research Logistics Quarterly* (Washington, since 1954), and *Revue française de recherche opérationnelle* (Paris, since 1956).

The International Federation of Operations Research Societies (IFORS) holds international congresses every three years; the first was conducted in London in 1957.

### REFERENCES

Morse, P. M., and G. E. Kimball.*Metody issledovaniia operatsii.*Moscow, 1956. (Translated from English.)

Kaufmann, A., and R. Faure.

*Zaimemsia issledovaniem operatsii.*Moscow, 1966. (Translated from French.)

Churchman, C. W., R. Ackoff, and L. Arnoff.

*Vvedenie v issledovanie operatsii.*Moscow, 1968. (Translated from English.)

Ackoff, R., and M. V. Sasieni.

*Osnovy issledovaniia operatsii.*Moscow, 1971. (Translated from English.)

Venttsel’, E. S.

*Issledovanie operatsii.*Moscow, 1972.

Wagner, H. M.

*Osnovy issledovaniia operatsii*, vols. 1–3. Moscow, 1972–73. (Translated from English.)

*Operationsforschung: Mathematische Grundlagen, Methoden und Modelle*, vols. 1–3. Edited by W. Dück and M. Bliefernich. Berlin, 1971–73.

N. N. VOROB’EV

## operations research

[‚äp·ə′rā·shənz ri‚sərch]## Operations research

The application of scientific methods and techniques to decision-making problems. A decision-making problem occurs where there are two or more alternative courses of action, each of which leads to a different and sometimes unknown end result. Operations research is also used to maximize the utility of limited resources. The objective is to select the best alternative, that is, the one leading to the best result.

To put these definitions into perspective, the following analogy might be used. In mathematics, when solving a set of simultaneous linear equations, one states that if there are seven unknowns, there must be seven equations. If they are independent and consistent and if it exists, a unique solution to the problem is found. In operations research there are figuratively “seven unknowns and four equations.” There may exist a solution space with many feasible solutions which satisfy the equations. Operations research is concerned with establishing the best solution. To do so, some measure of merit, some objective function, must be prescribed.

In the current lexicon there are several terms associated with the subject matter of this program: operations research, management science, systems analysis, operations analysis, and so forth. While there are subtle differences and distinctions, the terms can be considered nearly synonymous. *See* Systems engineering

#### Methodology

The success of operations research, where there has been success, has been the result of the following six simply stated rules: (1) formulate the problem; (2) construct a model of the system; (3) select a solution technique; (4) obtain a solution to the problem; (5) establish controls over the system; and (6) implement the solution.

The first statement of the problem is usually vague and inaccurate. It may be a cataloging of observable effects. It is necessary to identify the decision maker, the alternatives, goals, and constraints, and the parameters of the system. A statement of the problem properly contains four basic elements that, if correctly identified and articulated, greatly eases the model formulation. These elements can be combined in the following general form: “Given (the system description), the problem is to optimize (the objective function), by choice of the (decision variable), subject to a set of (constraints and restrictions).”

In modeling the system, one usually relies on mathematics, although graphical and analog models are also useful. It is important, however, that the model suggest the solution technique, and not the other way around.

With the first solution obtained, it is often evident that the model and the problem statement must be modified, and the sequence of problem-model-technique-solution-problem may have to be repeated several times. The controls are established by performing sensitivity analysis on the parameters. This also indicates the areas in which the data-collecting effort should be made.

Implementation is perhaps of least interest to the theorists, but in reality it is the most important step. If direct action is not taken to implement the solution, the whole effort may end as a dust-collecting report on a shelf.

#### Mathematical programming

Probably the one technique most associated with operations research is linear programming. The basic problem that can be modeled by linear programming is the use of limited resources to meet demands for the output of these resources. This type of problem is found mainly in production systems, but is not limited to this area. *See* Linear programming

#### Stochastic processes

A large class of operations research methods and applications deals with stochastic processes. These can be defined as processes in which one or more of the variables take on values according to some, perhaps unknown, probability distribution. These are referred to as random variables, and it takes only one to make the process stochastic.

In contrast to the mathematical programming methods and applications, there are not many optimization techniques. The techniques used tend to be more diagnostic than prognostic; that is, they can be used to describe the “health” of a system, but not necessarily how to “cure” it.

#### Scope of application

There are numerous areas where operations research has been applied. The following list is not intended to be all-inclusive, but is mainly to illustrate the scope of applications: optimal depreciation strategies; communication network design; computer network design; simulation of computer time-sharing systems; water resource project selection; demand forecasting; bidding models for offshore oil leases; production planning; classroom size mix to meet student demand; optimizing waste treatment plants; risk analysis in capital budgeting; electric utility fuel management; optimal staffing of medical facilities; feedlot optimization; minimizing waste in the steel industry; optimal design of natural-gas pipelines; economic inventory levels; optimal marketing-price strategies; project management with CPM/PERT/GERT; air-traffic-control simulations; optimal strategies in sports; optimal testing plans for reliability; optimal space trajectories. *See* GERT, Inventory control, PERT