Mathematical methods and models in decision-making. Mathematical theory of adoption of management decisions Mathematical model of decision-making

Mathematical methods and models in decision making

Introduction!

The purpose of modeling is the process of researching an object at different levels - from high-quality to exact quantitative, as the collection of information and development of the model.

In the mathematical area, methods and models are understood as comprehensive categories that include:

methods in decision making;

methods for researching operations;

economic and mathematical methods;

methods of economic cybernetics;

optimal control methods;

applied mathematics in the economy;

applied mathematics in the organization of production.

This list is not complete, which indicates a wide range of mathematical methods and models. In various sources, the content of which concerns the subject represented, mathematical models and methods are considered in certain combinations.

The practical proof of the designated thought is possible on the example of the well-known method of "probability theory", which is presented within the framework of mathematical models with a wide class and includes such concepts as "probability", "random event", "random value", "mathematical expectation (average meaning ) Random variable "," Dispersion (scattering) ", etc. In the late XIX - early XX centuries. A new object is allocated, which is a switched telephone system, implying such concepts as "an application application", "refusal", "Connection time", "switching" and the like elements.

The mathematical theoretical and probability model of processes in switched telephone networks was formed in the 20s. As a result of the connection of the presented method and object. The author of this operation became A.K. Erlang. As an example of the existing concepts of this model, it can be noted:

"Flow of applications";

"Average waiting time";

"The average length of service queue";

"Dispersion of waiting time";

"The likelihood of failure."

The subsequent development of this scientific direction demonstrated the effectiveness of the conceptual categories of the symbiosis model, revealed its large-scale constructive function.

This model in the process of its development was transformed into a method for studying complex systems. As an example, the "mass maintenance theory" can be distinguished, the categorical apparatus of which has ceased to be perceived as an integral component of telephone networks. Terminology and conceptual base acquired general theoretical character. Thus, the organization of new models can be carried out through the use of mass maintenance theory to such objects such as production processes, operating systems, computer, transport streams, etc.

As a result, the conclusion is obvious that the method is fully formed in the case of the development of a homogeneous set of models. The degree of study of the object directly depends on the number of developed models of the object. The dual essence of the model forms, in turn, the dualism of the categorical modeling apparatus, which integrates the concepts of common or specific, formed from the "method" and "object", respectively.

In other words, methods, models, objects organize a continuous sequence, which implies the presence of various groups of models formed in accordance with the specifics of its origin and applicability. Among such groups can be allocated:

models that involve the interaction of early developed methods and new objects;

models first created to implement a specific object describing, and new models can be applicable and relative to other objects.

Linear programming - Mathematical discipline devoted to theories and methods for solving extreme tasks on sets n.- the dimensional vector space defined by the systems of linear equations and inequalities.

Integer programming - A type of linear programming that implies that the desired values \u200b\u200bmust be integers.

The mathematical programming section in which the methods of finding extremes of functions in the parameter space are studied, where all or some variables are integers.

The simplest method of solving an integer programming problem is to reduce it to a linear programming task with a result of the result for integrity.

Threads in networks

The activities of modern society are closely related to different kinds of networks - take, for example, transport, communications, distribution of goods and the like. Therefore, a mathematical analysis of such networks has become the subject of fundamental importance.

Geometric programming- section , studies a certain class optimization tasksfound mainly in engineering and economic calculations. The main requirement of the method is to ensure that all the technical characteristics of the projected objects were quantified in the form of dependencies from regulated parameters. Geometric such type of programming is called because it effectively uses geometric average and a number of such geometric concepts like vector space, vectors, orthogonality and etc.

Nonlinear programming - section mathematical programmingstudying decision methods extreme tasks with nonlinear target function and (or) area of \u200b\u200bpermissible solutionsdefined nonlinear restrictions.

Optimal control - 1. Basic concept mathematical theory of optimal processes (owned by the section of mathematics under the same name - O. u.); means choice such control parametersthat would provide the best from the point of view of the specified criteria flow process or, otherwise, the best system behavior, its development to goal by optimal trajectory. These control parameters are usually considered as time functions, which means the possibility of their change in the course of the process to select at each stage of their best (optimal) values.

Mass maintenance theory- section studies of operationswho considers a variety of processes in the economy, as well as in telephone communication, healthcare and other areas as service processes, i.e. satisfaction of some requests, orders (eg, servicing ships in the port - their unloading and loading, maintenance of Tokarei in the instrumental storeroom - Issuance of cutters, customer service in the laundry - washing linen, etc.).

Theory of utility - Theoretical direction in the economic science, developed by representatives of the Austrian school in the XIX-XX centuries, based on the basic objective concept of "utility", perceived as pleasure, the satisfaction received by the person as a result of consumption of goods. The basic principle of the theory of utility - the law of decreasing utmost utilityAccording to which the utility increment obtained from one value added is continuously decreasing.

Theory of decision-making - Interdisciplinary research area, which is of interest to practitioners and related to mathematics, statistics, economy, philosophy, management and psychology; He examines how real decision makers choose solutions and how optimal solutions can be accepted.

Game theory - Mathematical method of studying optimal strategies in games. The game is understood as the process in which two or more of the parties are involved, leading the struggle for the implementation of their interests. Each party has its own goal and uses some strategy that can lead to a win or lose - depending on the behavior of other players. The theory of games helps to choose the best strategies taking into account the ideas about other participants, their resources and their possible aids.

Simulation modeling - A method that allows you to build models that describe the processes as they passed in reality. This model can "lose" in time for both the test and the specified set. In this case, the results will be determined by the random nature of the processes. According to these data, you can get sufficiently stable statistics.

Dynamic programming - This is a section of mathematics dedicated to theory and methods for solving multi-step optimal management tasks.

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Mathematical methods in decision making

Mathematics As a science from the very beginning is a tool in the process of finding truth, and therefore we can assume that any mathematical operations, even the simplest, are mathematical decision-making methods. Currently, the adoption of decisions means a special process of human activity, aimed at choosing the best option (alternatives) of actions. The decision-making processes underlie any targeted human activity. For example, when creating new equipment (machines, devices, devices), in construction in the design of new buildings, when organizing the functioning and development of social processes. In this regard, there is a need for decision-making guidance that would simplify this process and attached greater reliability solutions. In addition to the empirical perception of the situation and intuition in our time of complex economic situations and enterprise management processes, managers require some basis and the "proven guarantee" of the decision. Inevitably requires formalization of the decision-making process. As a rule, important decisions are made by experienced people, quite distant from mathematics, and especially from its new methods, and fascinating more to lose from formalization than to win.

Consequently, science requires recommendations for optimal decision-making. There was a while when the correct decisions were made to the touch, the method of "samples and errors". Today, a scientific approach is required to develop such a decision - the loss associated with errors is too large. Optimal solutions allow the enterprise the most favorable conditions for the production of products (maximum profits with minimal labor costs, material and labor resources).

Currently, the search for optimal solutions can be viewed using the sections of classical mathematics. For example, in mathematical statistics, in the section "Decision making", we study methods for making or not adopting some main hypothesis in the presence of a competing hypothesis, taking into account the function of losses. The theory of decision-making develops methods of mathematical statistics - methods of testing hypotheses. Different loss values \u200b\u200bwhen choosing different hypotheses lead to results other than those obtained by methods of statistical testing hypotheses. The choice of a less likely hypothesis may be more preferable if the loss in the event of the erroneousness of this choice will be less than the losses caused by the erroneousness of choosing a more likely competing hypothesis. Such tasks are called statistical decision-making tasks. To solve these tasks, it is necessary to find the minimum value of the risk function on the plurality of possible outcomes, i.e. Solve the task of finding the conditional extremum. As a rule, for these tasks, you can select the target and specify the conditions, i.e. Restrictions under which they should be solved. Similar tasks are engaged in the Mathematics "Mathematical Programming" section, which, in turn, is part of the section "Research operations".

The role of input data acts as a real task - randomly formulated a set of data on the problem situation. The first stage of solving the problem is its wording - bringing data to a convenient model to build a model. The model is an approximate (descriptive) display of reality. Further, on the constructed model, the search for optimal solutions and issuance of recommendations is carried out.

Models can be divided into 2 large groups:

Deterministic models:

Linear programming;

Integer programming and combinatorics;

Theory of graphs;

Threads in networks;

Geometric programming;

Nonlinear programming;

Mathematical programming;

Optimal control.

Stochastic models:

Mass maintenance theory;

Theory of utility;

Theory of decision making;

Game theory and game modeling;

Search theory;

Simulation;

Dynamic modeling.

In making decisions, it is necessary to find the optimum of some functional in deterministic or stochastic form. Two features should be noted. First, mathematical methods of making decisions for tasks related to various directions of human activity begin mutual penetration into each other, for example, optimization control objectives in the transition from continuous variables to discrete becomes the tasks of mathematical (linear) programming, the assessment of the separating function

in statistical decision-making methods, it can be carried out using linear or quadratic programming procedures, etc. Secondly, source numeric data as a result of measurements or observations

in decision-making tasks for real situations are not deterministic, and more often are random

with well-known or unknown distribution laws, therefore subsequent data processing requires the use of methods of mathematical statistics, the theory of fuzzy sets or theory of possibilities.

Mathematical methods in the economy and decision-making can be divided into several groups:

1. Optimization methods.

2. Methods that take into account the uncertainty, first of all, probabilistic statistical.

3. Methods for building and analyzing imitation models,

4. Methods for analyzing conflict situations (game theory).

Optimization methods

Optimization in mathematics is an operation of finding an extremum (minimum or maximum) of the target function in some region of the vector space limited by a set of linear or nonlinear equalities (inequality).

The theory and methods for solving optimization problem studies mathematical programming.

Mathematical programming is a matter of mathematics, developing the theory, numerical methods for solving multidimensional tasks with restrictions. Unlike classical mathematics, mathematical programming is engaged in mathematical methods for solving the tasks of finding the best options from all possible.

Setting the optimization problem

In the design process, the task of determining the best, in a sense, structures or values \u200b\u200bof object parameters is set. This task is called optimization. If optimization is associated with the calculation of the optimal parameter values \u200b\u200bat a given object structure, it is called parametric optimization. The task of choosing the optimal structure is a structural optimization.

The standard optimization mathematical problem is formulated in this way. Among the elements of h, forming many hours, find such an element of h *, which provides the minimum value f (h *) \u200b\u200bof a given function f (h). In order to correctly put the task of optimization, you must set:

1. A permissible set - a lot

decision mathematics game

2. Target feature - mapping;

3. Search criteria (MAX or MIN).

Then to solve the task means one of:

1. Show what.

2. Show that the target function is not limited to below.

If, then find:

If the minimized function is not convex, then often restricted by searching for local lows and maxima: points such that everywhere in some of their surroundings for a minimum and for a maximum.

If the permissible set is, such a task is called the problem of unconditional optimization, otherwise - the problem of conditional optimization.

Classification of optimization methods

The overall recording of optimization tasks sets a wide variety of their classes. The method of the method depends on the task class (efficiency of its solution). The classification of tasks is determined by: the target function and the permissible area (set by the system of inequalities and equalities or a more complex algorithm).

Optimization methods are classified according to optimization tasks:

1. Local methods:

converge to some local extremum of the target function. In the event of an unimodal target function, this extremum is unique, and will be a global maximum / minimum.

2. Global methods:

deal with multi-letter target functions. With a global search, the main task is to identify the tendencies of the global behavior of the target function.

Existing search methods can be divided into three large groups:

1. Determined;

2. Random (stochastic);

3. Combined.

According to the dimension criterion of the permissible set, the optimization methods are divided into methods of one-dimensional optimization and methods of multidimensional optimization.

According to the target function and the permissible set, optimization tasks and methods for their solution can be divided into the following classes:

Optimization tasks in which the target function and restrictions are linear functionsare allowed so-called linear programming methods.

Otherwise they deal with the task nonlinear programming and apply the appropriate methods. In turn, two private tasks are distinguished:

if both are convex functions, then such a task is called the task of convex programming;

if, then deal with the task of integer (discrete) programming.

According to smoothness and the presence of private derivatives in the target function, they can also be divided into:

· Direct methods requiring only the calculations of the target function at the points of approximations;

· First-order methods: require the calculation of the first private derived functions;

· Second order methods: require the calculation of second private derivatives, that is, the hessian of the target function.

In addition, optimization methods are divided into the following groups:

Analytical methods (for example, Lagrange Multiplier Method and Karusha-Kun Tracker Conditions);

Numerical methods;

Graphic methods.

Depending on the nature of the set X, the tasks of mathematical programming are classified as:

· Discrete programming tasks (or combinatorial optimization) - if X is of course or counting;

· Objectives of integer programming - if X is a subset of many integers;

· Nonlinear programming tasks, if limitations or target function contain nonlinear functions and X is a subset of the finite-dimensional vector space.

If all the limitations and target feature contain only linear functions, this is a linear programming task.

In addition, the sections of mathematical programming are parametric programming, dynamic programming and stochastic programming.

Mathematical programming is used in solving optimization tasks of research operations.

The method of finding an extremum is completely determined by the task class. But before getting a mathematical model, you need to perform 4 stages of modeling:

1. Definition of the borders of the optimization system

Discard the links of an optimization object with an external world that cannot strongly affect the optimization result, and, more precisely, those without which the decision is simplified

2. Selection of managed variables

"Freeze" the values \u200b\u200bof some variables (unmanaged variables). Others leave to accept any values \u200b\u200bfrom the subject of permissible solutions (managed variables)

3. Determination of restrictions on managed variables (equality and / or inequality).

Choosing a numerical optimization criterion (for example, performance indicator)

4. Create a target function.

Probabilistic statistical methods

The essence of probabilistic statistical methods of decision making

How are the approaches, ideas and results of probability theory and mathematical statistics are used when making decisions?

The base is a probabilistic model of a real phenomenon or process, i.e. The mathematical model in which the objective ratios are expressed in terms of probability theory. Probability is used primarily to describe the uncertainties that need to be considered when making decisions. Meant as unwanted features (risks) and attractive ("happy case"). Sometimes the chance is made to the situation consciously, for example, with a draw, random selection of units to control, carry out lotteries or consumer surveys.

The theory of probabilities allows one probabilities to calculate the other interested researchers. For example, according to the likelihood of the emblence, it is possible to calculate the likelihood that at least 3 coins will fall out of 10 strokes. Such a calculation relies on a probabilistic model according to which the challenges of coins are described by the scheme of independent tests, in addition, the deposition of the coat of arms and the lattice is equal, and therefore the probability of each of these events is equal to ѕ. A more complex is a model in which instead of throwing a coin considers the quality of the product quality. The corresponding probability model relies on the assumption that the quality control of various units of products is described by the scheme of independent tests. In contrast to the model with throwing coins, you must enter a new parameter - the likelihood of the fact that the unit of products is defective. The model will be fully described if you make that all units of products have the same probability to be defective. If the latter assumption is incorrect, then the number of model parameters increases. For example, it can be assumed that each unit of production has its own probability to be defective.

Let us discuss the quality control model with a common product unity for all units. The probability of defectiveness R. so that when analyzing the model to "get to the number", it is necessary to replace P to some specific value. To do this, it is necessary to exit the probabilistic model frames and refer to the data obtained when monitoring quality. Mathematical statistics solves the opposite task in relation to the theory of probability. Its goal is based on the results of observations (measurements, analyzes, testing, experiments) to obtain conclusions about probabilities underlying the probabilistic model. For example, based on the frequency of the appearance of defective products during control, conclusions can be drawn about the probability of defectiveness (see Bernoulli theorem above). On the basis of Chebyshev inequality, conclusions were made on compliance with the frequency of the appearance of defective products of the hypothesis that the probability of defectiveness takes a certain value.

Thus, the use of mathematical statistics is based on a probabilistic phenomenon model or process. Two parallel rows of concepts are used - relating to theory (probabilistic model) and related to practice (sample of observation results). For example, the theoretical probability corresponds to the frequency found by the sample. Mathematical expectation (theoretical series) corresponds to a selective arithmetic (practical range). As a rule, selective characteristics are estimates of theoretical. At the same time, the values \u200b\u200brelating to the theoretical series "are in the heads of researchers" belong to the world of ideas (according to the ancient Greek philosopher Platon), are not available for direct measurement. Researchers have only selective data with which they try to establish their properties of the theoretical probability model.

Why do you need a probabilistic model? The fact is that only with its help you can transfer properties installed on the results of the analysis of a particular sample, to other samples, as well as for the entire so-called general population. The term "general aggregate" is used when it comes to a large, but the ultimate aggregate of the units studied. For example, about the combination of all residents of Russia or the totality of all consumers of soluble coffee in Moscow. The purpose of marketing or sociological surveys is that the statements obtained in a sample of hundreds or thousands of people will be transferred to the general aggregate of several million people. When monitoring quality in the role of the general population, a batch of products.

To transfer conclusions from a sample to a more extensive set, you need certain assumptions about the connection of the sample characteristics with the characteristics of this more extensive aggregate. These assumptions are based on an appropriate probabilistic model.

Of course, you can process selective data without using one probabilistic model. For example, you can calculate the selective arithmetic average, count the frequency of performing certain conditions and the like. However, the results of the calculations will only be applied to a specific sample, the transfer of the conclusions obtained with their help to any other combination of incorrect. Sometimes such activities are called "data analysis". Compared to probabilistic statistical methods, data analysis has limited cognitive value.

So, the use of probabilistic models based on estimation and testing hypotheses using selective characteristics is the essence of probabilistic-statistical decision-making methods.

We emphasize that the logic of using selective characteristics for making decisions based on theoretical models involves the simultaneous use of two parallel rows of concepts, one of which corresponds to the probabilistic models, and the second - selective data. Unfortunately, in a number of literary sources, usually outdated or written in the prescription spirit, there are no distinction between selective and theoretical characteristics, which leads readers to adequate and errors in the practical use of statistical methods.

The use of a specific probability statistical method consists of three stages:

1. Transition from the economic, managerial, technological reality to the abstract mathematic-statistical scheme, that is, the construction of a probabilistic model of the management system, a technological process, decision-making procedures, in particular according to the results of statistical control, and the like.

2. Conducting the calculations and obtaining conclusions is purely mathematical means in the framework of a probabilistic model.

3. Interpretation of mathematic-statistical conclusions in relation to the real situation and the adoption of the appropriate solution (for example, on the compliance or non-compliance of product quality by established requirements, the need to set up the technological process), in particular, conclusions (on the proportion of defective units of products in the party, about the specific form of laws Distribution of controlled process parameters and similar).

Mathematical statistics applies concepts, methods and results of probability theory. Then we consider the main issues of constructing probabilistic models in a variety of cases. We emphasize that for the active and proper use of regulatory and instructive methodological documents on probabilistic statistical methods, preliminary knowledge is needed. So, it is necessary to know under what conditions one or another document should be applied which initial data is needed for its choice and application, which solutions must be taken from the results of data processing, and so on.

Consider several examples when probabilistic statistical models are a good task solving means.

In the novel, Alexei Nikolayevich Tolstoy "Walking on the flour" (Volume 1) says: "The workshop gives twenty-three percent of the marriage, you are kept this figure," said Ivan Ilychu. How to understand these words in the conversation of the factory leaders? Feeds products cannot be defective by 23%. It can be either suitable or defective. Probably, Podkov thought that in the large volume party contained approximately 23% of the defective units of products. Then the question arises: what does "approximately" mean? Let from 100 proven products of production 30 will be defective, or out of 1000 - 300, or out of 100,000 - 30,000 ... Is it necessary to blame the strugency in lies?

The coin used as a lot must be "symmetrical": on average, in half cases of throwing, an eagle should fall out, and half of the cases - the rush. But what does "on average" mean? If you have a lot of series of 10 throws in each series, then there will often be a series in which the coin 4 times falls out an eagle. For a symmetric coin, this will occur in 20.5% of the series. And if 40,000 eagles turn out to be 100,000 rubles, then it is possible to consider a symmetric coin? The decision-making procedure is based on the theory of probabilities and mathematical statistics.

An example may seem non-serious. This is not true. The draw is widely used in the organization of industrial feasibility experiments. For example, when processing the results of measuring the quality indicator (friction torque) of bearings, depending on the various technological factors (the effects of the conservation medium, the methods of preparing bearings before measuring, the effects of the loads of bearings during the measurement process and the like). Suppose you need to compare the quality of bearings depending on the results of storing them in different conservation oils. When planning such an experiment, the question arises which bearings should be placed in the oil of one composition, and which in the other, but to avoid subjectivism and ensure the objectivity of the decision. The answer can be obtained by lot.

A similar example can be brought with the quality control of any products. To decide, the product consistent or does not match the requirements of the established requirements, the representative part is selected from it: on this sample, the entire party is judged. Therefore, it is desirable that each unit in the controlled batch has the same probability of being selected. In production conditions, the choice of products of products is usually done not by lot, but according to special tables of random numbers or with the help of computer sensors of random numbers.

Similar problems of ensuring comparison objectivity arise in comparison of various production organization schemes, wages, when conducting tenders and contests, selection of candidates for vacancies. Everywhere you need a draw or similar measures.

Let it be necessary to identify the strongest and second strength team when organizing a tournament in the Olympic system (the loser is dropped). Suppose that a stronger team always wins more weak. It is clear that the strongest team will definitely become a champion. The second, the team will be released in the final only when she does not have games with a future champion before the final. If such a game is planned, then the second team will not fall into the final. The one who is planning a tournament can either "knock out" the second largest team from the tournament, bringing it in the first meeting with the leader, or provide her second place, providing a meeting with more weak teams up to the final. To avoid subjectivism, carry out the draw. For the tournament of 8 teams, the likelihood that two strongest teams will meet in the final, equal to 4 out of 7. Accordingly, with a probability of 3 of the 7th second, the team will leave the tournament ahead of schedule.

With any measurement of products of products (using a caliper, micrometer, ammeter ...) there are errors. To find out if there are systematic errors, it is necessary to measure the units of products repeatedly, the characteristics of which are known (for example, a standard sample). It should be remembered that in addition to a systematic error there is a random error.

The question arises, as for measurements to identify a systematic error. If you note only whether the error obtained during the next measurement is positive or negative, then this task can be reduced to the considered. Indeed, we compare the measurement with the throwing of the coin: a positive error - with an eagle falling, negative - the rush (zero error with a sufficient number of divisions of the scale almost never occurs). Then checking the absence of a systematic error is equivalent to checking the symmetry of the coin.

So, the task of checking on a systematic error is reduced to the task of checking the symmetry of the coin. Conducted arguments lead to the so-called "criterion of signs" in mathematical statistics.

With statistical regulation of technological processes, based on methods of mathematical statistics, rules and plans for statistical control of processes are developed, aimed at timely detection of the folding of technological processes and take measures to adjust them and prevent the production of products that are not relevant to the established requirements. These measures are aimed at reducing production costs and losses from the supply of poor-quality products. With statistical acceptance control on the basis of methods of mathematical statistics, quality control plans are developed by analyzing samples from product batches. Difficulty is to be able to properly build probabilistic-statistical solutions. In mathematical statistics, probabilistic models and methods for testing hypotheses are developed, in particular, the hypotheses that the proportion of defective product units is equal to a certain number, for example,.

Game theory

The theory of games is a mathematical method of studying optimal strategies in games. The game is understood as a process in which each of the Parties (two or more) is struggled for their interests. Each side pursues its goals and enjoys a certain strategy that may in turn lead to win or lose (the result depends on other players. Game theory provides the ability to choose the best strategy with ideas about other players, their capabilities and possible actions.

The theory of games is the section of applied mathematics, more precisely - research operations. Most often, methods of game theory are used in the economy, a little less often in other social sciences - sociology, political science, psychology, ethics, jurisprudence and others. Since the 1970s, it was assigned biologists to study the behavior of animals and theories of evolution. It has very important importance for artificial intelligence and cybernetics, especially with the manifestation of interest in intellectual agents.

Optimal solutions or strategies in mathematical modeling were offered in the XVIII century. The tasks of production and pricing in the conditions of oligopoly, which became later by the textbook examples of the game theory, were considered in the XIX century. A. Kurto and J. Bertran. At the beginning of the XX century. E. Lasker, E. Cermelo, E. Borel put forward the idea of \u200b\u200bmathematical theory of conflict of interest.

The mathematical theory of games takes its origin from the neoclassical economy. For the first time, mathematical aspects and applications of the theory were set forth in the classic book of 1944 by John von Neumanan and Oscar Morgettern "Theory of Games and Economic Behavior" (Eng. Theory of Games and Economic Behavior).

This region of mathematics found some reflection in public culture. In 1998, American writer and journalist Sylvia Nazar published a book about the fate of John Nash, the Nobel laureate in economics and scientist in the field of game theory; And in 2001, the film "Mind Games" was filmed based on the book. Some American television shows, for example, "Friend or Foe", "Alias" or "Numbers", periodically refer to theory in their episodes.

J. Nash in 1949 writes the dissertation on the theory of games, in 45 years he receives the Nobel Prize in the economy. J. Nash after the end of the Polytechnic Institute Carnegie with two diplomas - Bachelor and Magista - entered Princeton University, where John Von Neuman lecture visited. In his writings, J. Nash has developed the principles of "management dynamics". The first concepts of the game theory analyzed antagonistic games when there are losers and players who won their account. Nash develops methods of analysis in which all participants or won, or suffer defeat. These situations received the names of the Nash Equilibrium, or "Neooperative Equilibony", in the situation of the parties use the optimal strategy, which leads to the creation of a steady equilibrium. Players favorably maintain this balance, since any change will worsen their position. These works of J. Nash did a serious contribution to the development of the theory of games, mathematical instruments of economic modeling were revised. J. Nash shows that the classic approach to competition of A. Smith, when everyone for himself is neopped. More optimal strategies when everyone tries to do better for themselves, making better for others.

Although the theory of games was originally and considered economic models, until the 1950s, it remained a formal theory within the framework of mathematics. But since the 1950s. Attempts begin to apply the methods of game theory not only in the economy, but in biology, cybernetics, technique, anthropology. During World War II and immediately after her game theory, military were seriously interested in the military who saw a powerful apparatus in it to study strategic decisions.

In 1960-1970 Interest in the theory of games is fading, despite significant mathematical results obtained by that time. From the mid-1980s. The active practical use of the theory of games begins, especially in economics and management. Over the past 20 - 30 years, the importance of game theory and interest is growing significantly, some directions of modern economic theory cannot be set out without the use of game theory.

The work of Tomas Solling, the Nobel laureate in the economy of 2005, "Conflict Strategy" was a great contribution to the use of game theory. T. Shelling considers various "strategies" of the behavior of participants in conflict. These strategies coincide with the conflict management tactics and the principles of conflict analysis in conflictology (this is psychological discipline) and in conflict management in the organization (theory of management). In psychology and other sciences, the word "game" in other senses is used than in mathematics. Some psychologists and mathematics are skeptical about the use of this term in other senses prevailing earlier. The cultural concept of the game was given in the work of Yohan Hewing "Homo Ludens" (articles on the history of culture), the author speaks of the use of games in justice, culture, ethics, that the game is older than the person himself, since the animals also play. The concept of the game is found in the concept of Eric Burna "Games in which people play, people who play games." These are purely psychological games based on transactional analysis. The concept of playing Y. Hözing differs from the interpretation of the game in the theory of conflicts and the mathematical theory of games. Games are also used to study in business cases, seminars G.P. Shchedrovik, the founder of the organizational and activity approach. During the restructuring in the USSR G.P. Shchedrovitsky spent many games with Soviet managers. According to psychological inhabitants, Odi (organizational and activity) were so strong that they served as a powerful catalyst for changes in the USSR. Now in Russia there is a whole movement one. Critics celebrate artificial uniqueness Odi. The main methodological circle (MMK) became the basis of the Oda.

The mathematical theory of games is now rapidly developing, dynamic games are considered. However, the mathematical apparatus of game theory is cost. It is used for justified tasks: politics, monopolies and distribution of market power, etc. A number of famous scientists became Nobel laureates in economics for their contribution to the development of game theory, which describes socio-economic processes. J. Nash, thanks to his research in the theory of games, became one of the leading experts in the field of holding the "Cold War", which confirms the scale of the tasks that the theory of games is engaged.

Nobel laureates for the economy for achievements in the field of game theory and economic theory of steel: Robert Aumman, Reinhard Zelten, John Nash, John Harsania, William Varia, James Mirlis, Thomas Schelling, George Aerlof, Michael Spence, Joseph Stiglitz, Leonid Gurvitz, Eric Miskin , Roger Maerson, Lloyd Shepley, Alvin Roth, Jean Tyrol.

Presentation of games

Games are strictly defined mathematical objects. The game is formed by players, a set of strategies for each player and indication of winnings, or payments, players for each combination of strategies. Most cooperative games are described by a characteristic function, while for other species more often use a normal or extensive form. Characterizing signs of games as a mathematical model of the situation:

1. Availability of several participants;

2. The uncertainty of the behavior of participants associated with the presence of several options for each of them;

3. Difference (incomprehension) of participants' interests;

4. The interconnectedness of the behavior of participants, since the result obtained by each of them depends on the behavior of all participants;

5. The presence of rules of behavior known to all participants.

Extensive form

The game " Ultimatum"In extensive form

Games in extensive, or extended, form are presented in the form of a oriented tree, where each vertex corresponds to the choice of the player's choice of its strategy. Each player is associated with a whole level of vertices. Payments are recorded at the bottom of the tree, under each leaf vertex.

In the picture on the left - a game for two players. Player 1 goes first and selects a strategy F or U. Player 2 analyzes his position and decides - choose a strategy A or R. Most likely the first player will choose U, and the second - a (for each of them is optimal strategies); Then they will receive 8 and 2 points, respectively.

The extensive form is very visual, it is especially convenient to represent games with more than two players and games with successive strokes. If the participants make simultaneous moves, then the corresponding vertices are either connected by a dotted line, or are reduced by a solid line.

Normal shape of the game

In normal, or strategic, the game is described by a payment matrix. Each side (more precisely, the measurement) of the matrix is \u200b\u200ba player, strings define the strategies of the first player, and the columns are the second. At the intersection of two strategies, you can see winnings that players will receive. In the example of the right, if the player 1 selects the first strategy, and the second player is the second strategy, then we see (? 1,? 1) at the intersection, this means that as a result of the course, both players lost one point.

Players chose strategies with maximum result for themselves, but lost, because of the ignorance of the course of another player. Usually, in a normal form, games are presented in which the moves are made simultaneously, or at least it is assumed that all players do not know what other participants do. Such games with incomplete information will be discussed below.

Characteristic function

In cooperative games with transferable utility, that is, the possibility of transferring funds from one player to another, it is impossible to apply the concept of individual payments. Instead, use the so-called characteristic function defining the winnings of each coalition of players. It assumes that the winnings of the empty coalition is zero.

The foundations of this approach can be found in the book of Nymanan and Morgettern. Studying a normal form for coalition games, they judged that if the Caliation C is formed in the game with two sides, then the coalition n \\ c against it is. A game for two players is formed. But since there are many options for possible coalitions (namely, 2N, where n is the number of players), then the win for C will be some characteristic value depending on the coalition composition. Formally the game in this form (also called TU-play) is presented by a pair (N, V), where N is the set of all players, and V: 2N\u003e R is a characteristic function.

Such a form of representation can be applied to all games, including without transferable utility. Currently, there are ways to translate any game from the normal form to the characteristic, but the conversion in the opposite direction may not be in all cases.

Application of game theory

The theory of games, as one of the approaches in applied mathematics, is used to study human and animal behavior in various situations. Initially, the theory of games began to develop within the framework of economic science, allowing you to understand and explain the behavior of economic agents in various situations. Later, the area of \u200b\u200bapplication of the game theory was expanded on other social sciences; Currently, the theory of games is used to explain the behavior of people in political science, sociology and psychology. The theoretical gaming analysis was first used to describe the behavior of animals by Ronald Fisher in the 30s of the 20th century (although even Charles Darwin used the ideas of the game theory without a formal justification). In the work of Ronald Fisher, the term "theory of games" does not appear. However, work is essentially completed in the direction of the theoretical analysis. Developments made in the economy were applied by John Maynard Smith in the book "Evolution and theory of Games". The game theory is used not only for prediction and explanation of behavior; Attempts were made to use the theory of games to develop theories of ethical or reference behavior. Economists and philosophers applied game theory for a better understanding of good (worthy) behavior. Generally speaking, the first theoretical and gaming arguments explaining the correct behavior, they were expressed by Plato.

Description and modeling

Initially, the game theory was used to describe and model the behavior of human populations. Some researchers believe that with the help of determining equilibrium in the appropriate games, they can predict the behavior of human populations in a situation of real confrontation. This approach to the game theory has recently been criticized for several reasons. First, the assumptions used in modeling are often disturbed in real life. Researchers may assume that players choose behaviors maximizing their total benefits (model of an economic person), however, in practice, human behavior often does not correspond to this premise. There are many explanations of this phenomena - irrationality, discussion modeling, and even different motifs of players (including altruism). The authors of the theoretical and game models object to this, saying that their assumptions are similar to such assumptions in physics. Therefore, even if their assumptions are not always fulfilled, the game theory can use as a reasonable ideal model, by analogy with the same models in physics. However, a new critique shaft hit the theory of games when as a result of experiments it was revealed that people do not follow the equilibrium strategies in practice. For example, in the Games "Sorry-naval", "dictator" participants often do not use a strategy profile that makes a balance on Nash. Disputes continue about the meaning of such experiments. According to another point of view, there is no predictation of the expected behavior on Nash, it only explains why populations already in equilibrium on Nash remain in this state. However, the question of how these populations come to the equilibrium Nash remains open. Some researchers in search of an answer to this question switched to the study of evolutionary game theory. Models of evolutionary game theory suggest limited rationality or irrationality of players. Despite the name, the evolutionary theory of games is engaged not only and not so many issues of natural selection of biological species. This section of the theory of games is studied by the model of biological and cultural evolution, as well as the models of the learning process.

Regulatory analysis (detection of best behavior)

On the other hand, many researchers consider the theory of games not as a tool for predicting behavior, but as a tool for analyzing situations in order to identify the best behavior for a rational player. Since the balance of Nash includes strategies that are the best response to the behavior of another player, the use of the Nash equilibrium concept to select behavior looks quite reasonable. However, such use of theoretical and game models has been criticized. First, in some cases, the player favorably choose a strategy that is not included in the balance, if it expects other players will also not follow the equilibrium strategies. Secondly, the famous "Dilemma Prisoner" game allows you to bring another counterexample. In the "Prisoner Dilemma" following personal interests leads to the fact that both players find themselves in the worst situation in comparison with the one in which they would donate personal interests.

Cooperative and neooperative

The game is called cooperative, or coalition, if players can unite into groups, taking some obligations to other players and coordinating their actions. This is different from non-optimitive games in which everyone is obliged to play for themselves. Entertainment games are rarely cooperative, but such mechanisms are not uncommon in everyday life.

It often suggests that cooperative games are distinguished by the possibility of communication of players with each other. In general, it is incorrect. There are games where communication is allowed, but players pursue personal goals, and vice versa.

Of the two types of games, uncooperative describes situations in the smallest details and give more accurate results. Cooperative consider the game process as a whole. Attempts to combine two approaches gave considerable results. The so-called Nash program has already found solutions to some cooperative games as a situation of equilibrium of non-optherapy games.

Hybrid games include elements of cooperative and non-optherapy games. For example, players can form groups, but the game will be conducted in a non-optherapy style. This means that each player will chase the interests of their group, at the same time trying to achieve personal gain.

Symmetric and asymmetrical




Asymmetric game

The game will be symmetrical when the relevant strategies for players will be equal, that is, have the same payments. In other words, if players can change places and at the same time their winnings for the same moves will not change. Many studied games for two players are symmetrical. In particular, those are: "The dilemma of the concluded", "deer hunting", "hawks and pigeons". As an asymmetric games, you can cite "ultimatum" or "dictator".

In the example of the right, the game at first glance may seem symmetric due to similar strategies, but this is not the case - after all, the winning of the second player under strategies profiles (A, A) and (B, B) will be greater than that of the first.

With zero amount and with a nonzero amount

Games with zero amount - a special kind of games with a permanent amount, that is, such where players cannot increase or reduce available resources, or the game fund. In this case, the sum of all winns is equal to the sum of all losers at any course. Look to the right - numbers mean payments to the players - and their sum in each cell is zero. Examples of such games can serve poker, where one wins all bets of others; Reversal, where the chips of the enemy are captured; either banal theft.

Many the games studied by mathematicians, including the already mentioned "file of the concluded", of a different kind: in games with the nonsense sum of the winning of some player does not necessarily mean losing another, and vice versa. The outcome of such a game may be less or more zero. Such games can be converted to zero amount - this is done by the introduction of a fictitious player who "assigns themselves" surplus or fills the lack of funds.

Another game with an excellent amount from zero is trade where each participant benefits. A widely known example, where it decreases is the war.

Parallel and consistent

In parallel games, players go at the same time, or at least they are not aware of the choice of others until everyone is done. In consecutive, or dynamic, games, participants can make moves in a predetermined or random order, but at the same time they receive some information about the preceding actions of others. This information may not even be completely complete, for example, a player can find out that his opponent out of ten strategies did not choose the fifth, without learning about others.

Differences in the presentation of parallel and consistent games were considered above. The first is usually presented in a normal form, and the second is in extensive.

With full or incomplete information

An important subset of consistent games make up games with full information. In such a game, participants know all the moves made before the current moment, as well as possible strategies of opponents, which allows them to some extent to predict the subsequent development of the game. Full information is not available in parallel games, as the current moves of opponents are unknown. Most of the games are studied in mathematics - with incomplete information. For example, the whole "salt" of the "lifting dilemma" or "coin comparison" is in their incompleteness.

At the same time, there are interesting examples of games with full information: "Ultimatum", "Multonozza". This also includes chess, checkers, go, mankala and others.

Often the concept of complete information is confused with similar - perfect information. For the latter, only the knowledge of all strategies available to opponents, knowledge of all their moves is optional.

Games with an infinite number of steps

Games in the real world or the game studied in the economy, as a rule, the final number of moves. Mathematics is not so limited, and in particular, games are considered in the theory of sets that can continue indefinitely. And the winner and his winnings are not defined until the end of all moves.

The task that is usually placed in this case is not in finding an optimal solution, and in the search for at least a winning strategy. Using axiom of choice, it is possible to prove that sometimes even for games with full information and two outcomes - "won" or "lost" - none of the players has such a strategy. The existence of winning strategies for some specially designed games has an important role in the descriptive theory of sets.

Discrete and continuous games

Most of the games studied are discrete: they have a finite number of players, moves, events, outcomes, etc. However, these components can be expanded on a plurality of real numbers. Games including such elements are often called differential. They are associated with a real scale (usually - time scale), although the events occurring in them can be discrete in nature. Differential games are also considered in the theory of optimization, they find their use in technique and technologies, physics.

Metaire

These are games, the result of which is a set of rules for another game (called target or game object). The purpose of the meta game is to increase the utility of the set set of rules. The Meta Theory is associated with the theory of optimal mechanisms.

Methods for constructing and analyzing imitation models (simulation).

Simulation modeling (Situational modeling) is a method that allows you to build models describing the processes as they passed in reality. This model can "lose" in time for both the test and the specified set. In this case, the results will be determined by the random nature of the processes. According to these data, you can get sufficiently stable statistics.

Simulation modeling is a research method in which the system studied is replaced by the model, with sufficient accuracy of the actual system with which experiments are carried out in order to obtain information about this system. Experimentation with the model is called imitation (imitation - this comprehension of the essence of the phenomenon without resorting to experiments on the real object).

Simulation modeling is a special case of mathematical modeling. There is a class of objects for which, for various reasons, analytical models have not been developed, or methods for solving the obtained model have not been developed. In this case, the analytical model is replaced by a simulator or simulation model.

Simulation modeling is sometimes called obtaining private numerical solutions of a formulated problem based on analytical solutions or using numerical methods.

Imitation model - a logic-mathematical description of an object that can be used to experiment on a computer for design, analysis and evaluation of the object functioning.

Application of imitation modeling.

It is resorted to simulation modeling when:

· Expensive or impossible to experiment in a real object;

· It is impossible to build an analytical model: there are time in the system, causal links, consequences, nonlinearity, stochastic (random) variables;

· It is necessary to simulate the behavior of the system in time.

The purpose of imitation modeling is to reproduce the behavior of the system under study based on the results of the analysis of the most significant relationships between its elements or other words - the development of the simulator (Eng. Simulation Modeling) of the subject area under study for various experiments.

Types of simulation

Three simulation approaches

Simulation approaches on the scale of abstraction

· Agent modeling is relatively new (1990-2000) direction in imitation modeling, which is used to study decentralized systems, the dynamics of which are determined not by global rules and laws (as in other modeling paradigms), but on the contrary, when these global rules and Laws are the result of the individual activity of the members of the group. The purpose of the agent models is to obtain an idea of \u200b\u200bthese global rules, the overall behavior of the system, based on the assumptions about the individual, private behavior of its individual active objects and the interaction of these objects in the system. The agent is a certain essence that has an activity, autonomous behavior, can make decisions in accordance with some set of rules, interact with the environment, and also change themselves.

· Discrete-event modeling is an approach to modeling, offering abstract from the continuous nature of events and consider only the main events of the simulated system, such as: "Waiting", "Order Processing", "Movement with cargo", "Unloading" and others. Discrete-event modeling is most developed and has a huge sphere of applications - from logistics and mass maintenance systems to transport and production systems. This type of modeling is most suitable for modeling production processes. Based on Jeffrey Gordon in the 1960s.

Algorithm of mathematical theory

The peculiarity of the mathematical theory of justification of management decisions is the presence of a certain algorithm in it, which accurately prescribes to perform a certain system of operations in the established sequence to solve a specific class of tasks.

The algorithm of the mathematical theory of making management decisions must comply with a number of requirements:

certainty, i.e. accuracy and unambiguity, not leaving spaces for arbitrariness;
mass and versatility - applicability to solve a specific class of tasks, when initial data vary in known boundaries;
performance, i.e. The ability to solve the established task for a limited number of operations.

Mathematical methods of making management decisions

The main methods of solving typical management tasks within the mathematical theory are:

The method of mathematical analysis is used in the calculations to substantiate the needs for resources, accounting for the cost, project development, etc.
The method of mathematical statistics is convenient to use when the change in the studied indicators is a random process.
The econometric method involves the use of an economic model - a schematic representation of the economic process or phenomenon.
Linear programming is a solution of a system of equations when there is a strictly functional dependence between the studied phenomena.
Dynamic programming is used to solve optimization tasks, where limitations or target function have a nonlinear dependence.
The queue theory is used to search for the optimal number of service channels at a given level of need for them. An example of such a situation is the choice of the optimal organization of working with clients so that the service time is minimally, and the quality is highly at no additional cost.
The operation method of operations is the use of mathematical probabilistic models that represent the process under study, type of activity or system. Optimization is reduced to a comparative study of numerical estimates of those parameters that cannot be assessed by conventional methods.
Situation analysis is a comprehensive technology for making and implementing a management decision, which is based on the analysis of a separate management situation. Such an analysis is repelled from a specific situation, the problems arising in the organization's activities that requires a management decision.
Methods of game theory - modeling of a situation in which, in justifying the decisions, it is necessary to take into account the conflict or inappropriate interests of various persons.
Break-sufficiency points - a method in which total income is equalized with the total costs to search for a minimum profit-generating point.
Projections of the trend - analysis of temporary rows based on the assumption that the incident that occurred in the past gives a good approximation in the event of an assessment of the future. This method is used to identify the trends of the past and their extension for the future.

Efficiency in general form is the effectiveness of something (production, labor, management, etc.). In economic theory, they differ mainly two types of efficiency - economic and social. Economic efficiency characterizes the ratio of the result obtained to the costs social - the degree of satisfaction of the supply of the population (consumers, customers) on goods and services. Often they are combined with a single term - socio-economic efficiency Which most applies to the evaluation of management decisions, since the latter are aimed at the state and behavior of people and in this way have high social importance and their assessment is not entirely correct from the positions of the economic effect. In recent decades, the need for evaluation on many managerial decisions environmental efficiency Reflecting both positive and negative impact of their implementation on an ecological situation. Here, as a rule, it is reflected in the possible cost of organizing negative environmental impact, fines and other fees related payments or their savings with positive impact on the environment.

Quality - From the standpoint of philosophy - expresses a set of essential signs, features and properties that distinguish one object or phenomenon from others and give it certainty. The quality of the result of labor (products, services, investment project, managerial solution, etc.) is associated with the concepts of "property" and "utility". Property The result of labor determines the objective sides without assessing its importance for the consumer (for example, the technical level of products, project); utility - The ability of this result of labor to benefit and meet the requirements of a specific consumer. Hence quality of management solution - A set of properties that cause its ability to satisfy certain needs in accordance with the appointment. In the practice of organizing organizations, efficiency and quality are inseparable and mutually determine each other. The decision cannot be highly efficient if it has low quality and, on the contrary, it cannot be qualitative if it is ineffective, i.e. efficiency – one of the characteristics of quality, and the quality is a significant factor of efficiency.

The effectiveness and quality of management decisions are determined by the entire combination of the management processes that constitute it relatively independent and interrelated stages in the technological cycle: development, adoption and implementation of decisions. In accordance with this, it is necessary to consider modifications of management solution - the effectiveness and quality of the theoretically found, adopted by the LPR and a practically implemented solution.

At the stages of development and adoption His quality management solution is the degree of compliance of the parameters of the selected alternative to solving a specific characteristics system, satisfying its developers and consumers and ensures the ability to effectively implement. At the implementation stage The quality of the managerial solution is expressed in its actual efficacy, implementation efficiency.

The main characteristics that determine the quality of solutions include: validity, timeliness, consistency (consistency), reality, completeness of detention, authority (power), efficiency.

Radiation solution It is determined: the degree of accounting for the laws of the functioning and development of the management facility, the trends in the development of the economy and society as a whole, the competence of its developing professionals and the LPR. It should cover the whole range of issues, all completeness of the managed object needs. For this, it is necessary to know the characteristics, ways to develop a managed system and an external environment. A thorough analysis of resource support, scientific and technical capabilities, targeted development functions, economic and social prospects of the company, region, industry, national and global economy are required. The comprehensive validity of decisions requires the search for new forms and ways to process scientific and technical and socio-economic information, forms and methods of management, theory and practices of development and decision-making, i.e. The formation of advanced professional thinking, the development of its analytical synthetic functions. Only the solution that is based on reliable, systematized and scientifically processed information can be reasonable, which is achieved by using scientific methods for developing and optimizing solutions.

Thus, the validity of the solution is ensured by the following main factors:

accounting of the requirements of objective economic laws and patterns of existing legislation and statutory documents;
knowledge and use of patterns and trends in the development of the management facility and its external environment;
the presence of complete, reliable, timely information;
the presence of special knowledge, education and qualifications of developers and LPR;
Knowledge and application of the LPR of the main recommendations of management and theory of decision-making;
used methods for analyzing and synthesizing situations.

The increasing complexity and complexity of solved problems and their consequences require universal knowledge to develop and adopt informed management decisions, which necessitates the increasing distribution of collegial forms of decision-making.

The validity of management decisions can be achieved by the following actions:

definition of conditions for the formation of permissible options;
drawing up a list of indicators characterizing the essential properties of the solutions found, and the development of the scales for measurement them;
exploded by irrational options and determining the range of possible values \u200b\u200bof each indicator using a variety of mathematical and heuristic methods;
identifying the structure of the preferences of the LPR;
forming a criterion or rules for evaluating solutions;
Select the best option to manage or clarify the structure of the preferences of the LPR.

The implementation of these actions does not always guarantee the high quality and efficiency of solutions, since the choice of alternatives is significantly hampered by the following factors.

1. The multidimensional nature of the effectiveness estimates of the alternative. When determining possible solutions and especially when choosing from them, the most appropriate is to produce economic, technical and technological, social, political, environmental assessment. At the same time, each has several approaches. For example, the value estimate, according to international, European and Russian standards, uses costly, market (comparative) and profitable approaches that use various methods depending on the object and objectives of the assessment. When choosing options for the development of an open joint-stock company, it is necessary to take into account the entire set of stakeholders, since the decisions made can significantly affect various groups of people, which increases the number of possible estimates (both in relation to them and on their part). In many cases, it is necessary to take into account changes in time estimates. At the same time, problems of recording new types of ratings arise, which characterize the consequences of the decision made in different moments of the future.
2. Difficulties of detection and comparison of all aspects of comparison alternatives. The existence of heterogeneous aspects of the assessment of the alternative places the difficult problems of their comparison before developers and LPR. Here it should be borne in mind that such a comparison is subjective and therefore can be criticized. It is aggravated by many times with collegial decision-making, where each of the members of the collective authority's decision makers may have different measures to compare heterogeneous qualities. Some participants in the development and decision-making may be interested in mainly in economic criteria, others in political, third - in environmental, etc.
3. The subjective nature of evaluating performance and quality alternatives. Many estimates of efficiency and quality alternatives can be obtained either by building special models or by collecting and processing expert opinions. Both methods are associated with the use of subjective assessments or specialists developing models or experts. When choosing alternatives, it is necessary to take into account that the reliability of such subjective assessments cannot be absolute. Even with the full unanimity of experts, a situation is possible when their assessments are incorrect. There is also an existence of various models or a mismatch of expert assessments. Consequently, several alternatives may have different estimates, and the result of the choice depends on which of them will be used.

Timeliness A managerial solution means that the decision made should not fall behind, neither ahead of the need for the development of the situation. Even the most optimal (of expedient for LPR), the solution designed to receive the greatest socio-economic efficiency may be useless if it is customary late. It can even bring certain damage. Premature decisions are no less harmful to the organization than captivated. They do not have the conditions necessary for implementation and development, and can give impulses for the development of negative trends, do not contribute to the solution of already "overwhelming" tasks and further exacerbate the already painful processes.

Consistency (consistency ). Distinguish the inner and external consistency of the solution. Under internal consistency The decisions are meant the compliance of the goals and means to achieve the complexity of the problem and the methods of developing a solution, certain provisions of solving each other and the meaning of the decision as a whole. Under external consistency Solutions are their continuity, compliance with the strategy, company goals and previously accepted solutions (the actions necessary to implement one solution should not interfere with the fulfillment of others). Achieving a combination of these two conditions and ensures consistency and consistency of the management decision. Consistency with previously accepted decisions also mean the need to respect for a clear causal relationship of social development. Decisions taken earlier, if necessary, should be canceled or adjusted if they conflict with new conditions for the activity of the managed system. The emergence of contradictory solutions - a consequence of the poor knowledge and understanding of the laws of social development, manifestations of a low level of management culture.

Reality. The decision should be developed and taken into account the objective possibilities of the organization, its potential. In other words, material, financial, information and other resources, the possibilities of the organization should be sufficient to effectively implement the selected alternative.

Fullness of content Decisions means that the decision should cover the entire set of parameters of the managed object necessary to ensure the achievement of goals, all areas of its activity, all directions of development. The content of the managerial solution should reflect:

The goal (the set of goals) of the functioning and development of a managed object to which the decision is directed;
resources used to achieve these goals;
Major paths and ways to achieve goals, the main methods of performance of work that determine the implementation of the objectives of the decision;
Terms of achieving goals, the beginning and end of their providing works;
The procedure for interaction between divisions and individual employees.

So, the managerial decision can be considered high-quality if it meets all the requirements listed above. Moreover, we are talking about the system of requirements, since non-compliance with at least one of them leads to a decrease in the quality of the decision and, consequently, to loss of efficiency, difficulties, or even the impossibility of its implementation.

The quality and efficacy of management decisions are determined by the set of factors operating during the entire management cycle or on its separate stages having an intrasystem or external (influence of the environment), objective or subjective in nature. The most essential factors include:

the laws of objective world related to the adoption and implementation of the management decision;
Object formulation; For which the management decision is made, what real results can be achieved, how to measure, relate the set goal and the results achieved;
The volume and value of the disposable information - for the successful adoption of the management decision, the main thing is not so much the amount of information as its value determined by the level of professionalism, experience, personnel intuition;
The development time of the management solution - as a rule, the management solution is always accepted in the conditions of time deficit and emergency (resource deficit, the activity of competitors, market conditions, inconsistent behavior of politicians);
The organizational structure of the management defined by organizational documents (formal) and in fact existing (informal). In fact, an existing (valid) management structure, in almost exceptional cases, coincides with the relevant organizational documents, which requires all employees of the organization. The need to take into account this requirement is often a condition for accepting not the most optimal solution;
Forms and methods of management activities, including the development and implementation of the management decision;
state of managing and managed systems (psychological climate, authority of the head, professional qualification composition of personnel, etc.);
The quality assessment system of the quality and efficiency of the management solution;
The degree of risk associated with the consequences of the implementation of the decision. This factor requires the use of various risk assessment techniques (financial, economic, etc.); Accordingly, the manager must have the skills of performing such an analysis;
Office equipment, including IVS. The use of modern information systems is a powerful factor in activating the process of developing, making and implementing solutions. It requires certain knowledge and skills of using modern information technologies in managing the activities of organizations;
The subjectivity of the evaluation of the solution choice. The decision-making process, the choice of a specific option is creative and depends on the specific personality, its state at the time of decision. PDR personal estimates act as a compass indicating the desired direction when you have to choose between alternatives actions. Each person has its own system of values, which determines its actions and affects decisions made. Personal factors include:
- PRP psychological state at the time of decision. In the irritability state, the workload of other solutions of the LPR can take one decision on this situation, and in a good mood, being relatively free - the other,
- the measure of the responsibility of the LPR, defined as an internal sense of responsibility for their actions, and regulating its activities by documents,
- Level of knowledge on this issue. The higher the level of knowledge of the LPR on the object to which the solution is directed, and its external environment, the greater the likelihood of their adoption of a qualitative and effective solution,
- Experience, which as the main resource for the development and implementation of decisions is the determining factor in the adequate perception of the real estimate and effective reaction of the LPR on what is happening, is a specific bank of tested and adaptable options in which the analogues of the analogues and prototype of developed, accepted and implemented decisions are drawn,
- intuition, judgment (common sense) and rationality of the LPR.

Reference. Intuition is manifested as some insight or instant understanding of the situation without using rational thinking. However, such insight is usually preceded by a long and painstaking work of consciousness. First, by observing, the information accumulates in the person's memory, is systematized and is located in a certain order. Often this way come to the appropriate solution to the problem. If this does not occur, intuition and imagination are connected, generating numerous ideas and associations. One of the ideas can cause an intuitive insight that, as if pushing the corresponding idea from the subconscious consciousness. Intuition is a powerful decision-making tool that needs continuous development and should be actively used in managerial activities.

When making a decision of the LPR, it is often based on its own sense that its choice is correct. Intuition develops as experience gained. The basis of decisions based on judgment lie knowledge and meaningful experience of the past. Using them and relying on common sense, with the amendment today, choose the option that has brought the greatest success in a similar situation in the same time. However, common sense in humans, from the point of view of the author, it is rare, therefore, this method of making decisions is not very reliable, although bribes with its speed and low cost. With this approach, the LPR seeks to act preferably in those directions that he knows well, as a result of which it risks to miss a good result in another area, consciously or unconsciously refusing the invasion into it;

Selected LPR criterion risk strategy criterion: optimism, pessimism or indifference. The optimism criterion (MAXIMAX) determines the choice of an alternative that maximizes the maximum result for each alternative; Pesssimism (Maximin) is an alternative that maximizes the minimum result for each alternative; Indifference - an alternative with a maximum average result (in this case there is a secret assumption that each of the possible states of the controlled system may occur with equal probability: as a result, an alternative is chosen that gives the maximum value of the mathematical expectation).

At the implementation stage, the effectiveness of solutions define the following factors:

The level of development and status of a managed system, its equipment, technology, personnel (personnel), organization and economics. With a high level of development of all components of the managed system, a greater efficiency can be obtained in the implementation of the solution than the decision, and vice versa, with a low level, it is rather difficult to ensure efficiency defined in the decision;
Socio-psychological climate in the decision to solve the team. The main criterion of the socio-psychological climate is the level of maturity of the team, under which the degree of coincidence of individual and collective interests is understood. The higher the level of the maturity of the team, the more manageable, which is a necessary condition for its effective activities;
Authority of managers to ensure the implementation of the decision. The higher the authority of the leaders, the more managed the team and, accordingly, the level of efficiency of its activities;
The effectiveness of the management mechanism of the team, which is expressed in the essence of management as creating conditions that encourage people to be necessary to achieve the objectives of actions;
Decision implementation time. The timely adopted qualitative and effective solution in the untimely implementation of its implementation may not be only ineffective, but unnecessary;
Compliance of the number and qualifications (education, skills and experience) frames of volume and complexity of work on the implementation of the decision. With the number of personnel, less necessary to implement the solution is difficult to comply with its terms. When qualifying employees below the required level, the quality of work is reduced and together with this, the effectiveness of the implementation of the decision;
Providing the necessary material, energy, labor, information and monetary resources.

It has been shown above that the effectiveness of the solution is determined at the stages of its development and implementation. At the first stage, it is determined by the well-known methods for calculating the effectiveness of design solutions, on the second - as a rule, but methods for calculating the actual profit and profitability of activities. In recent years, to determine the effectiveness of strategic decisions at the stages of their development and implementation, the calculation of the intended and actual change in the market value of the business is often used, the results of which are the basis for the assessment and choice of the organization's strategy.

An assessment of the effectiveness of management decisions at the stages of their development and adoption can be carried out on well-known indicators for assessing investment projects:

Clean discounted (reduced, current) income (CDD) - NPV (Net Present Value ) - the current value of cash tributaries (income) less the cost of cash outflows (investment costs);
Internal yield rate (GNI) - Irr. (INTERNAL RATE OF RETURN ) - the discount rate, in which the equality of the current value of the projected cash inflows (income) and the current value of the projected investment costs (cash outlines), i.e. Clean current revenue (NPV) It is zero;
Modified internal rate of return (MVND) - Mirr (Modified Internal Rate Of Return ) - an indicator characterizing the effectiveness of investments (investments). If the current value of all investment

attachments to consider as initially embedded capital, and the future value of all cash tributaries - as an extensive amount, the binding of the discount rate of the increasing coefficient is adopted;

Profitability index (IR) - PI (Profitability Index ) - the amount of pure (discounted) cash flow occurring per unit of investment investments;
payback period - Pp. (Payback Period. ) - the expected period of compensation of nested funds with net cash receipts;
Discounted payback period - DPP. (Discounted Payback Period. ) - the estimated period of compensation (equality) of the current value of the invested funds and the current value of net cash revenues;
Cost efficiency coefficient - Arr. (Accounting Rate Of Return ) It is equal to the ratio of the forecast average annual pure (book) profit to the average annual investment costs.

These indicators are widely applied in practice, and their calculation methods are recognized as traditional. In numerous literature, they are described in detail, examples illustrating their calculations to select projects (alternatives) of management decisions with various baseline conditions.

These indicators, as well as the methods corresponding to them, are used in two versions:

To determine the effectiveness of independent (non-alternative) management solutions (the so-called absolute efficiency), when the conclusion is concluded, to accept it or reject;
To determine the effectiveness of mutually exclusive alternatives to solutions (comparative efficiency), when the conclusion is made about which one to accept as a managerial solution.

In assessing the effectiveness of management decisions, as well as any other activity, the results of its implementation (Effect - ER) and the costs of its development, adoption and implementation (SR) are involved. The effect of management solutions is manifested in the final results of the organization. Even in cases where the management decision is aimed at changes in the technical and economic or socio-economic indicators of the organization (the level of condition and development of equipment and production technology, the nomenclature and product range, the quality of the initial raw materials, the design characteristics of the work premises, social infrastructure, etc. ), The effect of its implementation is ultimately reflected in the change in the level of use of its potential and meet the social needs for its products and services, i.e.

Er \u003d. f. (P, IP, SP, UE)

at (P - IP), SP Š min; UP Š MAX,

where P is the potential of the organization; IP is its use; UE - the level of satisfaction of social needs for its products and services.

This approach called " respo-Pottencial ", to an assessment of the effectiveness of the management of organizations, the product of which is the management decisions and the results of their implementation, was proposed by Academician of the USSR Academy of Sciences V. A. Trapeznikov, justified and developed by Professors F. M. Rusinov and V. I. Busov.

The development of the organization (its potential, attributed to a particular purpose expressed in the desire to maximize the possible satisfaction of a certain type of social needs) has limitations determined by the ratio of supply and demand for products and services that can produce this organization. Exceeding the result on one or another function of the enterprise available in it is a negative effect of its activity or an uncommon result, equivalent to waste and loss of resources spent on it.

The second component of efficiency is the costs of resources for the development, adoption and implementation of the management decision. Increasing the level of recoil of these costs (their effectiveness) is the most important task of managing the process of developing, adopting and implementing management decisions. Incorrect understanding of this problem (especially in terms of development and decision-making), it often leads to a reduction in these costs even to the detriment of the effectiveness of management decisions. This is due to the fact that the main share of costs is often wages and accruals on it and the reduction of them is reduced to a reduction in personnel participating in this process or the level of payment of its labor, resulting in the quality of the management decision and the effect of its implementation, personnel motivation. Reduction of the costs of development, adoption and implementation of management decisions by simply voluntaristic solution entails a decrease in the effectiveness of the organization's activities related to the deterioration of control, an increase in the waiting time for a decision on a particular situation, deterioration in the quality of training, developing and making decisions and with other factors affecting the level of resource loss.

An assessment of the effectiveness of implementing management decisions can be made for each major managerial decision or for a set of realized at a certain period of time (for example, a quarter, half a year, year). It consists of a system of indicators (Fig. 3.5), including:

generalizing integral indicator specifizing efficiency criterion;
generalizing indicators reflecting the effectiveness of the implementation of groups of goals to achieve which the management decision (scientific and technical, economic, social, etc.) is adopted;
Private indicators reflecting the efficiency of using certain types of resources at individual stages of the reproduction cycle.

In determining the effectiveness of the management solution, the magnitude of the organization's resources is not used in general, and its potential for the performance of functions that covers this decision. To identify such a composition, you can use the matrices shown in Table. 1.2-1.5.

The level of capacity use is defined as the difference in its magnitude and loss. Moreover, the reserve part of the potential necessary for the sustainable functioning and development of any division of the organization does not relate to its losses.

Fig. 3.3.

Shown in fig. 3.5 The system of indicators reflects the structure of the "tree" of the objectives of improving the effectiveness of the organization's activities.

The effectiveness of the managerial solution is defined as

where ETC and Entz, EEC and EPC, ESC and ESC, ECC and EECTs are the effectiveness and effect of a management decision in achieving scientific and technical, industrial, social and environmental purposes, respectively; EI, - the effect of implementing a managerial solution in the T-M division of the organization (workplace of the unit); SP - the cost of developing and implementing a managerial decision; p - Number of divisions involved in the development and implementation of this management decision.

Effect of participation i. -to units of the organization (workplace) in the development and implementation of the managerial decision is defined as the sum of the effects of changes in the level of use in the process, which this decision is sent to the potential of the unit (workplace) - the internal effect (eV) - and the result of the implementation of the objectives of the decision - External effect (EC), i.e.

Ei \u003d EV + EC.

The internal effect is determined by intense (EI) and extensive factors (EE), i.e.

EV \u003d EI + EE.

Intensive factors determine the implementation of this managerial decision to change productive capacity, extensive - changes in non-production capacity and resource loss.

The scheme for calculating the performance of the management efficiency of the enterprise is shown in Fig. 3.6.

Since all resources come to the workplaces of the organization and are used here, the level of use of the potential of the company's resources is determined by the processes at its workplaces. The change in the level of productive use of resources in the workplace is determined by the difference between the use of potential production (or labor productivity) at this workplace before and after the implementation of this management decision, i.e.

where and the potential development in this workplace, respectively, before and after the implementation of the management decision; , and VF - actual production at this workplace, respectively, before and after the implementation of the managerial solution.

The actual development (or labor productivity) in a production unit (procurement, mechanical, foundry, assembly, etc.) is determined without special difficulties on generally accepted assessment methods.

Fig. 3.6.

Potential and actual workplace production form the basis for determining the potential and actual development of the division, function or type of activity of the unit. The amount of production in the workplace is influenced by: equipment performance with this technology of work performed at this workplace; compliance of the employee's qualifications level of complexity of work; the timeliness of providing the workplace with the necessary materials, tools, organizing, information and other resources; compliance with the amount and quality of source resources with the requirements of technology; The rhythm of the work of the employee in the workplace. These factors reduce actual production compared to potential.

The potential working of the workplace (VP (PM)) is determined by the production of equipment installed on it at the maximum number of hours of work in this period, taking into account the time for reference, repair, commissioning, i.e. according to the formula

ΒP (PM) \u003d (FR - t. n) P. n. ,

where the FR is a regime flow of the operation of one unit (building crane, bulldozer, concrete mixers, a squabble machine, etc.) in the workplace per month; t. n - regulatory time on adjustment and repair, reference of one unit; P - regime (technological) products from equipment (aggregate) per unit of time; p - The number of same type of units in the workplace with multi-service maintenance.

For jobs with low-level and manual labor, including engineering and managerial workers, potential development is calculated on the maximum replacement of the month's development, based on the fact that the maximum development of this shift was achieved due to the most use of the resources that make up this workplace, those.

VP (PM) \u003d Sun.max T p,

where Sun.max is the maximum replacement workplace workplace in the settlement month, normal hours; m. - the number of shifts in the estimated month; r - cost of 1 norm-hour, rub.

The initial data for calculation is taken from the accounting and wages cards, which should be filled into enterprise units.

A similar approach can be applied to any workplace, but for mechanized and automated workplaces VP should count on the performance of the equipment.

Knowing the potential volume of development per month for all working places of the unit, one can determine the potential amount of production of this unit. It is calculated according to the technological chain of workplaces formed by the system of machines involved in the production of this type of product, or the sequence of the implementation of the result of the operation of the division, which is determined by the technological operations of the production operations.

Extensive use of economic potential on the internal effect of the processes of the enterprise management system expresses losses and technologically unfounded waste of resources. The change in their value after the implementation of the managerial solution () compared to the basic (PR) reflects the change in the internal effect of the control on extensive factors, i.e.

The resources participating in the processes are used productivity and unproductive.

The productive use of resources is also subdivided into two parts. The first part is the cost of resources, calculated on the specific costs, which are recognized as rational (technologically necessary). The second part is resource costs exceeding rational specific costs. Such costs are loss of resources.

Unproductive use of resources is observed in the case when products and services are not created. For example, unproductive use of resources includes the costs of working time workers, the costs of the production capacity of equipment and materials for the correction of marriage, to the loss - absenteeism, native and intelligent downtime, unused power facilities, an incorrigible marriage, unused scientific and technical development, damage for materials in stock and etc.

The effect of implementing a management decision to achieve production goals is determined by the increase in the amount and quality of products and services, complying with the timing of their provision to the consumer and is expressed in changing the effectiveness of their use in consumers; scientific and technical purposes - in the effectiveness of the application of enterprise development in innovation processes; social goals - in saving time (increasing free time) and increasing the public activity of employees of the enterprise and consumers of products and services of the enterprise; Environmental purposes - in reducing waste and increase their disposal, landscaping of territory, etc. The effect on social results is especially important for enterprises producing various services to the population (utilities, transport, household, postal, catering, trade, etc.). Effect on environmental results - for enterprises of the fuel, petrochemical and chemical industry.

The costs of developing and implementing managerial solutions include the entire set of costs for working both by their own and third-party organizations (contractors), as well as to acquire the necessary materials, equipment and other necessary resources.

The foregoing approach is applicable only in the conditions of the presence in the organization of the necessary source data provided by the organized control system and accounting parameters of processes in workplaces and in divisions, monitoring the needs and consumption of products and services of the company.

In countries with a developed economy, it has long been a textbook cost approach in management of organizations and, accordingly, in assessing the effectiveness of management decisions.

Reference.In the US capital market, the value concept is widespread in practice and the only adopted in the scientific literature. In May 2010, KPMG in collaboration with the State University of Higher School of Economics (GU-HSE) conducted a study on the use of cost management methods by Russian companies. It showed a high relevance of value management for Russian companies in the current market situation and interest for managers, since the growth value of the business necessitates an increase in the investment attractiveness and competitiveness of the organization.

The main idea of \u200b\u200bthe Cost Management Concept is that the main financial goal of the organization is the growth of its value (value) not only for owners (shareholders), but also for all interested in the company's legal entities and individuals (the company's cost in the interests of stakeholders). The concept of "cost" in this management concept is an internal category that characterizes the value of the company's investment attractiveness, and is expressed in the monetary indicator of future growth opportunities.

Grease value - This is an economic criterion that reflects the integral effect of the impact of implemented in the organization of management decisions to all parameters for which its activities (market share and strength of the competitive position, income, investment needs, operational efficiency, tax burden, regulation, cash flows and risk levels ), allowing you to rank options in a multiple selection situation.

In the cost management system, the premise is initially laid out that the command-administrative style of adopting control solutions "top down" does not bring due results, especially in large multidisciplinary corporations. Low link managers need to learn how to use the cost indicators for making better and efficient management decisions. Cost management requires a reasonable equilibrium of long-term and short-term activities. It is essentially the development, adoption and implementation of management decisions that ensure continuous reorganization aimed at achieving the maximum cost of business.

An important advantage of the cost approach is the fact that it offers management a single and understandable criterion for assessing activities - cost. The cost of the business cost is a key tool for improving the quality and efficiency of management solutions, which allows you to create a universal coordinate system to determine the business development vector, as well as create a single scale of changing the results achieved in accordance with the established strategy.

The company's market value management process uses a profit approach to the Company's assessment (business). Within the framework of this approach, the cost of the company is the amount of cash flows that will be created by the company corrected taking into account the factors of time and relevant risks, minus all the obligations of the company.

Evaluation of the effectiveness of the management solution by this method involves a comparison of two scenarios for the development of the organization "without developing and implementing the management decision of this situation" and "subject to the development and implementation of the management solution to this problem."

An assessment of the cost of the organization in the first version is reduced to the forecast of cash flows on the enterprise as a whole, provided that nothing in it will not be fundamentally changing in it. It - discounted cost Business, which is determined by discounting cash flow at a rate, taking into account the available risks of the organization as a whole:

where PV 0 - the discounted value of the organization during its development without solving existing problem situations; CF. 0i - expected cash flow in the period of g; r. - discount rate; p - The number of periods during which the organization will generate cash flows (in years).

The cost of the organization with a management decision implementation scenario (strategic value) It is determined by discounting a corrected cash flow project according to the adjusted rate, taking into account both the risk of the organization as a whole and the risks of the management decision. It will be equal to the residual current value of the expected flow streams, subject to the implementation of the management decision, i.e. The monetary flows of the organization for two scenarios of its development are combined:

where PV C is the strategic value of the organization; CF. c is the organization's strategic cash flow; CF. pI - cash flow created by implementing a managerial solution.

Application method of capital and transactions To assess the growth of the value of the enterprise, through the implementation of the managerial decision, it is based on information about the company analogue that implements a similar solution. At the same time, the similarity of solutions is determined by the following factors:

Maximum similarity of solved situations in compared organizations;
General sectoral (functional) affiliation of compared situations;
use of similar resources;
Compariability of the scale of situations and radicality of changes as a result of the implementation of the management solution.

To determine the cost of value created as a result of the implementation of the management solution, the market coefficients of the company-analogue are used by the capital market coefficients before and after implementing it to solve a similar situation, i.e.

where Δ. CV - an increase in the market value of the company assessed due to the implementation of the management decision; E. oK - current profit of the estimated company; - the ratio of "price / profit" for a similar company after implementing a similar situation; - The ratio of "price / profit" for a similar company to the implementation of a similar situation.

The transaction method differs from the capital market method by the fact that the coefficient "price / profit" for the analogue company (self-containing companies) is calculated, taking into account only the prices of the shares of the company's company (analog companies), which were observed in the next past on the actual past Transactions of the sale of large packages or with the corresponding stock quotation. At the same time, large packages are considered to be the purchase of which makes it possible to acquire at least participation in the management of the company through the introduction of its representative (or himself) to its board of directors, which allows you to control the management of the company. From here, find a company analogue that implements a management solution for a similar situation, information on which is in public access - the task is extremely complex and sometimes simply impracticable. In practice, this makes it much difficult or makes it impossible to apply the capital market methods and transactions to assess the effectiveness of management decisions.

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