Operations research- a science engaged in the development and practical application of mathematical, quantitative methods to substantiate decisions in all areas of purposeful human activity (effective organizational management).

General features of operations research

In each problem, we are talking about some kind of event that pursues a specific goal.

Some conditions have been set that characterize the situation (including the means that we can dispose of).

Within the framework of these conditions, it is required to make such a decision so that the planned event is in some sense the most profitable.

Features of Operations Research

A systematic approach to the analysis of the problem posed means that a particular problem should be considered from the point of view of its influence on the criterion of the functioning of the entire system.

The greatest effect can be achieved only with continuous research, ensuring continuity in the transition from one problem to another, arising in the course of solving the previous one.

Although the goal of operations research is to find an optimal solution, due to the complexity of calculating combinatorial problems, it is limited to finding a “good enough” solution.

Operational research is carried out in a comprehensive manner in many areas. To conduct the study, operating groups are created:

Basic Operations Research Concepts

OPERATION - any controlled (that is, depending on the choice of parameters) event, united by a single concept and aimed at achieving a goal.

SOLUTION - any definite choice of parameters depending on us.

OPTIMAL SOLUTIONS - decisions on one or another basis are preferable to others.

THE PURPOSE OF THE STUDY OF THE OPERATION is a preliminary quantitative substantiation of optimal decisions.

ELEMENTS OF A SOLUTION - parameters, the combination of which forms a solution.

PERFORMANCE INDICATOR (TARGET FUNCTION) is a quantitative criterion that allows you to compare different solutions in terms of efficiency and reflects the target orientation of the operation (W => max or W => min).

The best solution is the one that maximizes the achievement of the goal.

The concept of the mathematical model of the operation

A schematic, simplified description of the operation using one or another mathematical apparatus, reflecting the most important features of the operation, all the essential factors on which the success of the operation mainly depends.

Direct and inverse problems of operations research

DIRECT PROBLEMS answer the question of what will happen if, under the given conditions, we take some kind of solution x X, i.e. what will be the performance indicator W (or a number of indicators).

To solve such a problem, a mat is built. a model that allows one or more performance indicators to be expressed in terms of given conditions and decision elements.

INVERSE PROBLEMS answer the question of how to choose a solution x in order for the efficiency indicator W to turn to the maximum (minimum).

These tasks are more difficult. They can be solved by a simple enumeration of solutions to direct problems.

But when the number of variants of solutions is large, methods of directed enumeration are used, in which the optimal solution is found by performing successive "attempts" or approximations, each of which brings us closer to the desired optimal one.

operations research) I. о. - comparatively new area, Short story a cut goes back to the beginning of the Second World War. This exact mate. science contains a well-defined set general principles, to-rye provide researchers with a plan for the implementation of scientific research operations. It includes the following stages. 1. Formulation of the problem. 2. Construction of mate. a model representing the system under study. 3. Obtaining a solution from a given model. 4. Verification of the model and the solution obtained from it. 5. Establishing control over the decision. 6. Practice. solution implementation: implementation. Formulating the problem Serious attention must be paid to defining general problems and, more importantly, the objectives of the research. These goals should be formulated in behavioral terms in order to minimize or eliminate ambiguity and ambiguity. Time should also be allocated to correctly prioritize realistically achievable goals. Too long a list of goals can create potential difficulties in achieving them, especially if these goals are not clearly linked in a logical sequence. Building a mathematical model The second phase of research with t. Sp. And about. assumes a description of the model. The purpose of the model is to represent the real world... In I. about. such models are symbolic, expressed in mat. terms. The classical equation E = mc2 is a typical example of math. models. Traditional forms for such models are algebraic equations, to-rye not only mean. are more economical than verbal formulations, but also entail the thoroughness and precision of definition necessary for a clear expression and understanding of individual elements and their interrelationships. The most important task in building such a model is a clear and precise development and definition of the target function. This function expresses the relationship between independent and dependent variables. Getting a solution from a given model The third phase is to find a solution. As a rule, it is desirable to find the optimal or the best solution, but it should be borne in mind that such a solution will have value only in the context of the model in question. Since the model is only a representation of a real-world problem, there are many situations in which the optimal solution may not be associated with the best choice ... However, when the optimal solution is combined with less optimal or more realistic alternative solutions, with the possibility of subsequent testing against a real problem, there are certain benefits to using the optimal solution. One of these benefits is associated with the definition at the end of the study. the relative distance between this ideal solution and the accepted alternative. A by-product of this methodology of using I. o. is the assumption that less optimal solutions can be considered as stepping stones on the way to achieving the goal. This method of successive approximations can lead the operations researcher to more fruitful results. There are many checkmates. procedures for obtaining solutions in the I.O. model. These procedures are based on applications of probability theory. Validation of the model and the solution derived from it Validation of the model and the solution involves the implementation of two steps. The first is a thorough analysis of all elements of the model, incl. rechecking of its algebraic factors for the presence of simplistic cosmetic errors, which can affect the validity. Dr. an even more important step involves redefining the relationships between the model and the prerequisites that were originally used to develop this model. A more systematic verification plan also includes the use of ist. data that can be easily entered into the model so that an experimental (prototype) solution can be obtained. These data should be carefully reviewed to ensure the validity of the check by the operations investigator. It should be noted that as long as this model is practically developed on the basis of the previous history. data and needs, she may behave very differently in the future. Dr. a common mistake is the introduction of factors into the model, to-rye were not represented in the ist. database. Establishing Control The fifth stage, establishing control over the decision, appears in the course of repeated use of the model. Control over the model is established in cases where the operations researcher admits discrepancies in the values ​​of ist. data and recognizes that these discrepancies can influence the relationships between model elements and the resulting solutions. Dr. an important step can be the development of constraints on the selected bases. parameters of the model to establish a range of acceptable values ​​based on real data. Implementing the Model The final step is to introduce the real data model. Practice. implementation of the model is associated with an obvious step of entering real data and obtaining a solution to a real problem. In addition, it seems also important to assess the proximity of the real solution to the truth. solutions obtained earlier, as well as the consequences of this decision for improving the ways of operating the model. These steps provide an important connection between the mat. nature I. o. and practical. research results. Ultimately, these decisions and their managerial implications are used by an experienced I.O. to fine-tune the model for possible future use. See also Research Methodology R.S. Endrulis

Under operation means any event, united by a single idea and direction to achieve a specific goal.

An operation is always a managed event, i.e. the choice of parameters that characterize the way of its organization depends on us.

Any definite choice of parameters depending on us will be called decision.

Optimal solutions are those that, for one reason or another, are preferable to others.

The main task of operations research is preliminary quantitative justification of optimal solutions... Operations Research does not aim to fully automate decision making. The decision is always made by a person. The mission of operations research is to produce quantitative data and recommendations that make it easier for a person to make decisions.

Along with the main task - justification of optimal decisions - Operations research includes other tasks:

Comparative assessment of various options for organizing the operation,

Assessment of the influence of various parameters on the operation,

Research on bottlenecks, ie. elements, the malfunction of which has a particularly strong effect on the success of the operation, etc.

These auxiliary tasks become especially important when the given operation is considered not in isolation, but as constituent element whole systems operations. A “systematic” approach to the tasks of operations research requires taking into account the mutual dependence and conditionality of a whole range of activities, ie. make the final decision taking into account the role and place of this operation in the system.

Under efficiency operation is understood as the degree of its adaptability to the performance of the task facing it.

In order to judge the effectiveness of an operation and to compare differently organized operations in terms of efficiency, one needs to have some numerical evaluation criterion or performance indicator.

The sequence of actions in the research of operations.

1. The research goal is formulated and the problem statement is developed.

2. To apply quantitative methods in any field, it is always required to construct a mathematical model of the phenomenon. Based on the analysis of the properties of the original, this model is built.

3. After building the model, the results are obtained on it.

4. They are interpreted in terms of the original and transferred to the original.

5. By means of comparison, the simulation results are compared with the results obtained by direct examination of the original.

If the results obtained using the model are close to the results obtained in the study of the original, then with regard to these properties, the model can be considered adequate to the original.

During the design and operation of ACS, tasks often arise related to the analysis of both quantitative and qualitative patterns of their functioning, determination of their optimal structure, etc.

Direct experimentation on objects for solving these problems has a number of significant disadvantages:

1. The set operating mode of the object is violated.

2. In a full-scale experiment, it is impossible to analyze all the alternative options for constructing a system, etc.

It is advisable to solve these problems on a model separated from the object and implemented on a computer.

When modeling information systems, mathematical models are widely used.

The method of mathematical modeling is a way to study various objects by compiling an appropriate mathematical description and calculating, on its basis, the characteristics of the object under study.

It is necessary to build a mathematical model. It formally reflects the process of functioning of the original and describes the basic laws of its behavior. In this case, all secondary, insignificant factors are excluded from consideration.

Complex systems are the object of mathematical modeling. A complex system is called a specifically organized and purposeful functioning set. a large number information related and interacting elements under the influence of external factors.

There are 4 main stages of modeling systems on a computer:

Building a conceptual model of the system and its formalization;

Algorithmization of the system model and development of a simulation program;

Obtaining and interpretation of preliminary simulation results;

Checking the adequacy of the model and system; model correction

The main calculation of the quality indicators of the functioning of the system based on the results of modeling, the implementation of the model.

Lecture 3. Basic concepts of the method expert assessments... Formation of expert groups. Polling procedures. Ranking methods, pairwise comparisons, relative scale assessment.

The basic concepts and definitions of operations research should be learned.

Operation - any controlled event aimed at achieving a goal. The result of the operation depends on the way it is carried out, the organization, otherwise - on the choice of some parameters. Any specific choice of parameters is called a solution. The optimal solutions are those that, for one reason or another, are preferable to others. Therefore, the main task of operations research is a preliminary quantitative substantiation of optimal solutions.

Remark 1

Attention should be paid to the statement of the problem: decision-making itself goes beyond the framework of operations research and belongs to the competence of a responsible person or a group of persons who may take into account other considerations other than mathematically justified.

Remark 2

If in some problems of operations research the optimal solution is that the selected criterion of efficiency takes the maximum or minimum value, then in other problems this is not at all necessary. So, in the problem, the optimal one can be considered, for example, such a number of retail outlets and personnel in them, at which the average time of customer service does not exceed, for example, 5 minutes, and the length of the queue on average at any time will be no more than 3 people (1, p. . 10-11).

The efficiency of production and commercial activities is largely determined by the quality of decisions made by managers on a daily basis different levels... Concerning great importance acquire the tasks of improving decision-making processes, which can be solved by operations research. The term "operations research" was first used in 1939-1940. in the military field. By this time, military equipment and its control became fundamentally more complicated due to the scientific and technological revolution. And therefore, by the beginning of World War II, there was an urgent need for scientific research in the effective use of new military equipment, quantitative assessment and optimization of decisions taken by the command. In the post-war period, the successes of the new scientific discipline were in demand in peaceful areas: in industry, business and commercial activities, in government agencies, in educational institutions.

Operations Research is a methodology for applying mathematical quantitative methods to substantiate problem solutions in all areas of purposeful human activity. Operations research methods and models provide solutions that best serve the purposes of the organization.

Operations Research is the science concerned with the development and practical application of methods for the most effective (or optimal) management of organizational systems.

The main postulate of operations research is as follows: the optimal solution (control) is a set of values ​​of variables at which the optimal (maximum or minimum) value of the efficiency criterion (objective function) of the operation is achieved and the specified constraints are met.

The subject of operations research is the problem of making optimal decisions in a system with control based on assessing the efficiency of its functioning. Typical concepts of operations research are: model, mutable variables, constraints, objective function.

The subject of operations research in reality is the systems of organizational management (organizations), which consist of a large number of interacting units, and the interests of the units are not always consistent with each other and may be opposite.

The purpose of operations research is to quantitatively substantiate the decisions taken on the management of organizations.

The solution that is most beneficial for the entire organization is called the optimal solution, and the solution that is most beneficial for one or more departments will be suboptimal.

As an example of a typical organizational management problem, where conflicting interests of departments collide, consider the problem of enterprise inventory management.

The production department strives to produce as many products as possible at the lowest cost. Therefore, he is interested in the longest and most continuous production, that is, in the production of products in large batches, because such production reduces the cost of equipment changeover, and hence the overall production costs. However, the release of products in large batches requires the creation of large volumes of stocks of materials, components, etc.

The sales department is also interested in large stocks of finished products to satisfy any customer's request at any given time. By concluding each contract, the sales department, striving to sell as many products as possible, must offer the consumer the widest possible range of products. As a result, there is often a conflict between the production department and the sales department over the product range. At the same time, the sales department insists on the inclusion in the plan of many products that are produced in small quantities even when they do not bring much profit, and the production department requires the exclusion of such products from the product range.

The financial department, seeking to minimize the amount of capital required for the operation of the enterprise, is trying to reduce the amount of "tied" working capital. Therefore, he is interested in reducing stocks to a minimum. As you can see, the requirements for the size of stocks for different departments of the organization are different. The question arises as to which inventory strategy will be most beneficial for the entire organization. This is a typical organizational management task. It is connected with the problem of optimizing the functioning of the system as a whole and affects the conflicting interests of its divisions.

Key features of operations research:

1. A systematic approach to the analysis of the problem. The systems approach, or systems analysis, is the main methodological principle of operations research, which is as follows. Any task, no matter how particular it may seem at first glance, is considered from the point of view of its influence on the criterion of the functioning of the entire system. Above, the systematic approach was illustrated by the example of the inventory control problem.

2. It is characteristic of operations research that with each problem solving, more and more new tasks arise. Therefore, if narrow, limited goals are first set, the application of operational methods is not effective. The greatest effect can be achieved only with continuous research, ensuring continuity in the transition from one task to another.

3. One of the essential features of operations research is the desire to find the optimal solution to the problem. However, such a solution is often unattainable due to the limitations imposed by the available resources (money, machine time) or the level of modern science. For example, for many combinatorial problems, in particular scheduling problems with the number of machines n> 4, the optimal solution with the modern development of mathematics turns out to be possible to find only by a simple enumeration of options. Then one has to confine oneself to the search for a “good enough” or suboptimal solution. Therefore, one of its founders, T. Saaty, defined operations research as "... the art of giving bad answers to those practical questions to which even worse answers are given by other methods."

4. The peculiarity of operational research is that it is carried out in a complex, in many directions. To conduct such a study, an operating group is created. It consists of specialists from different fields of knowledge: engineers, mathematicians, economists, sociologists, psychologists. The task of creating such operational groups is a comprehensive study of the entire set of factors influencing the solution of the problem, and the use of ideas and methods of various sciences.

Each operational study goes through the following main stages in sequence:

1) a description of the planning task,

2) building a mathematical model,

3) finding a solution,

4) checking and adjusting the model,

5) implementation of the found solution in practice.

Description of the planning task:

Network planning and management tasks

consider the relationship between the timing of the completion of a large complex of operations (works) and the moments of the beginning of all operations of the complex. These tasks consist in finding the minimum duration of a complex of operations, the optimal ratio of the values ​​of cost and the timing of their implementation.

Queuing problems are devoted to the study and analysis of service systems with queues of claims or requests and consist in determining the performance indicators of the systems, their optimal characteristics, for example, in determining the number of service channels, service time, etc.

Inventory management tasks are to find the optimal values ​​for inventory levels (points of order) and order sizes. The peculiarity of such tasks is that with an increase in the level of stocks, on the one hand, the costs of their storage increase, but, on the other hand, losses due to a possible shortage of the stored product decrease.

Resource allocation problems arise with a certain set of operations (work) that must be performed with limited available resources, and it is required to find the optimal allocation of resources between operations or the composition of operations.

The tasks of repairing and replacing equipment are relevant due to wear and tear of equipment and the need to replace it over time. The tasks boil down to determining the optimal timing, the number of preventive repairs and inspections, as well as the moments of replacing the equipment with modernized ones.

The tasks of scheduling (scheduling) are to determine the optimal sequence of operations (for example, parts processing) on ​​various types of equipment.

The tasks of planning and placement are to determine the number and location of new objects, taking into account their interaction with existing objects and among themselves.

Route selection problems, or network problems, are most often encountered in the study of various problems in transport and in the communication system and consist in determining the most economical routes (1, p. 15).

By operation, we mean any event (or system of actions), united by a single concept and aimed at achieving a specific goal.

Examples of operations.

• 1. A system of measures aimed at improving the reliability of the technical device.
• 2. Reflection of air raid by means of air defense.
• 3. Placing orders for the production of equipment.
• 4. Reconnaissance search for a group of aircraft behind enemy lines.
• 5. Launching a group of artificial earth satellites to establish a television communication system.
• 6. A transportation system that supplies a number of points of a certain type with goods.

An operation is always a controlled event, that is, it depends on us to choose in one way or another some parameters that characterize the way it is organized. "Organization" is understood here in the broad sense of the word, including the choice of technical means used in the operation. For example, organizing the reflection of an air raid by air defense, we can, depending on the situation, choose the type and properties of the technical means (missiles, installations) used, or, with given technical means, solve only the problem of rational organization of the very procedure of reflection on a ^ eta (distribution targets between installations, the number of missiles directed at each target, etc.).

We will call any definite choice of parameters depending on us a solution.

Decisions can be good or bad, reasonable or unreasonable. Optimal solutions are those that, for one reason or another, are preferable to others.

The main task of operations research is preliminary quantitative substantiation of optimal solutions.

Note that the very decision-making goes beyond the research of operations and belongs to the competence of the person in charge (or a group of persons) who have been given the right to make the final choice. In this choice, the persons responsible for it can take into account, along with the recommendations arising from the mathematical calculation, a number of considerations (of a quantitative and qualitative nature) that were not taken into account by the calculation.

Thus, operations research does not set itself the task of completely automating decision-making, completely excluding the thinking, evaluating, criticizing human consciousness from this process. Ultimately, the decision is always made by a person (or a group of people); the task of operations research is to prepare quantitative data and recommendations that make it easier for a person to make a decision *).

*) Even in those cases when the decision-making seems to be completely automated (for example, in the process of automatic control of an enterprise or a spacecraft), the role of a person is not eliminated, because, ultimately, the choice of the algorithm by which control is carried out depends on him. ...

Along with the main task of justifying optimal decisions, other tasks are also related to the field of operations research, such as

• - a comparative assessment of various options for organizing the operation;
• - assessment of the impact on the result of the operation of various parameters (elements of the solution and specified conditions);
• - research of the so-called "bottlenecks", that is, elements of the controlled system, the disruption of which especially strongly affects the success of the operation, etc.

These "auxiliary" tasks of operations research acquire special importance when we consider a given operation not in isolation, but as a constituent element of a whole system of operations. The so-called "systemic" approach to the tasks of operations research requires taking into account the mutual dependence and conditionality of a whole range of measures. Of course, in principle, it is always possible to combine a system of operations into one complex operation of a higher order, but in practice this is not always convenient (and not always desirable), and in some cases it is advisable to single out individual elements of the system as “operations”, and the final the decision to take taking into account the role and place of this operation in the system.

So, let us consider a separate operation O. Reflecting on the organization of the operation, we strive to make it the most effective. The efficiency of an operation is understood as the degree of its adaptability to the performance of the task facing it. The better the operation is organized, the more effective it is.

To judge the effectiveness of an operation and to compare differently organized operations in terms of efficiency, you need to have some numerical criterion or performance indicator (in some manuals, the performance indicator is called the "objective function").

In the future, we will denote the efficiency indicator by the letter W.

Specific type of performance indicator W, which should be used in the numerical assessment of efficiency depends on the specifics of the operation under consideration, its target orientation, as well as on the research task, which can be posed in one form or another.

Many operations are carried out in conditions that contain an element of chance (for example, operations related to fluctuations in supply and demand, with population movements, morbidity, mortality, and all military operations). In these cases, the outcome of the operation, even if it is organized in a strictly defined way, cannot be accurately predicted, it remains random. If so, then as an indicator of efficiency W not just a characteristic of the outcome of the operation is selected, but its average value (mathematical expectation). For example, if the task of the operation is to maximize profit, then the average profit is taken as an indicator of efficiency. In other cases, when the task of the operation is to implement a well-defined event, the probability of this event is taken as an indicator of efficiency (for example, the probability that a given group of targets will be hit as a result of an air raid).

Choosing the right performance indicator - necessary condition the usefulness of the research used to inform the decision.

Let's look at a number of examples, each of which has a performance indicator W selected in accordance with the target orientation of the operation.

Example 1. The work of an industrial enterprise is considered from the point of view of its profitability, and a number of measures are taken to increase this profitability.Efficiency indicator is the profit (or average profit) brought by the enterprise for the financial year

Example 2 A group of fighters rises into the air to intercept a single enemy aircraft. The goal of the operation is to shoot down the aircraft. Efficiency indicator - the probability of hitting (shooting down) an aircraft

Example 3. The repair shop is engaged in the maintenance of machines; its profitability is determined by the number of cars serviced during the day. Efficiency indicator - the average number of cars serviced per day ("average" because the actual number is random)

Example 4. A group of radar stations in a certain area is monitoring the airspace. The task of the group is to detect any aircraft, if it appears in the area. Performance indicator - the probability of detecting any aircraft that appeared in the area.

Example 5. A number of measures are being taken to improve the reliability of an electronic digital computer (EDS). The purpose of the operation is to reduce the frequency of occurrence of malfunctions ("failures") of the electronic computer, or, which is equivalent, to increase the average time interval between failures ("mean time between failures"). The efficiency indicator is the average time of computer uptime (or the average relative time of correct operation).

Example 6. There is a struggle for cost savings in the production of a certain type of goods. The performance indicator is the amount (or average amount) of savings.

In all the examples considered, the efficiency indicator, whatever it may be, had to be turned to the maximum ("the more, the better"). In general, this is not necessary: ​​in operations research often use indicators that need to be turned not to the maximum, but to the minimum ("the less, the better"). For example, in example 4, it would be possible to take as an indicator of efficiency "the probability that the aircraft that appears will not be detected" - it is desirable to make this indicator as small as possible. In example 5, the performance indicator could be taken as the "average number of failures per day", which it is desirable to minimize. If a system is being evaluated that ensures the guidance of a projectile at a target, then as an indicator of efficiency, you can choose the average value of the missile "miss" (the distance from the trajectory to the center of the target), which is desirable to make as small as possible. It is also desirable to make the amount of funds allocated for the implementation of a task minimal, as well as the cost of the system of measures undertaken. Thus, in many problems of operations research, a reasonable solution should provide not a maximum, but a minimum of a certain indicator.

Obviously, the case when the efficiency indicator W must be minimized, it can be easily reduced to the problem of maximization (for this it is sufficient, for example, to change the sign of the quantity W). Therefore, in what follows, considering in general the problem of investigating operations, for simplicity we will only talk about the case when W it is required to turn into maksimum. As for practical specific tasks, we will use both the performance indicators that need to be maximized and those that need to be minimized.