Mathematical methods and models in decision making. Mathematical methods of decision making Functions of mathematical methods of analysis and decision making

Efficiency in general terms is the effectiveness of something (production, labor, management, etc.). In economic theory, there are mainly two types of efficiency - economic and social. Economic efficiency characterizes the ratio of the obtained result to the costs, social – the degree of satisfaction of the population’s (consumers, customers) demand for goods and services. They are often combined under a single term - socio-economic efficiency, which is most relevant to the assessment of management decisions, since the latter are aimed at the state and behavior of people and thus have a high social significance and their assessment only from the standpoint of economic effect is not entirely correct. In recent decades, there has been a growing need for assessment of many management decisions. environmental efficiency, reflecting both the positive and negative impact of their implementation on the environmental situation. Here, as a rule, the possible costs of the organization to eliminate the negative impact on the environment, fines and other related payments or their savings with a positive impact on the environment are reflected.

Quality – from the standpoint of philosophy – expresses a set of essential features, 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 projects, management decisions, etc.) is associated with the concepts of “property” and “utility”. Property the result of labor determines objective aspects without assessing its importance for the consumer (for example, the technical level of a product, project); utility – the ability of a given result of labor to bring benefits and satisfy the requirements of a specific consumer. From here, quality of management decision – a set of properties that determine its ability to satisfy certain needs in accordance with its purpose. In the practice of organizations, efficiency and quality are inseparable and mutually determine each other. A solution cannot be highly effective if it is of low quality and, conversely, it cannot be of high quality if it is ineffective, i.e. efficiency – one of the characteristics of quality, and quality is an essential factor of efficiency.

The effectiveness and quality of a management decision are determined by the entire set of management processes that make up its relatively independent and interconnected stages in the technological cycle: development, adoption and implementation of decisions. In accordance with this, it is necessary to consider modifications of the management decision - the effectiveness and quality of the theoretically found, accepted decision-maker and practically implemented solution.

During the development and adoption stages of a management decision, its quality is the degree to which the parameters of the chosen solution alternative correspond to a certain system of characteristics, satisfying its developers and consumers and ensuring the possibility of effective implementation. At the implementation stage The quality of a management decision is expressed in its actual effectiveness and implementation efficiency.

The main characteristics that determine the quality of decisions include: validity, timeliness, consistency (coherence), reality, completeness of content, authority (authority), efficiency.

Validity of the decision is determined by: the degree of consideration of the patterns of functioning and development of the management object, trends in the development of the economy and society as a whole, the competence of its developing specialists and decision-makers. It should cover the entire range of issues, the entirety of the needs of the managed object. This requires knowledge of the features, development paths of the managed system and the external environment. A thorough analysis of resource provision, scientific and technical capabilities, target development functions, economic and social prospects of the company, region, industry, national and global economy is required. Comprehensive validity of decisions requires the search for new forms and ways of processing scientific, technical and socio-economic information, forms and methods of management, theory and practice of development and decision making, i.e. formation of advanced professional thinking, development of its analytical and synthetic functions. Only a decision that is made on the basis of reliable, systematized and scientifically processed information, which is achieved using scientific methods for developing and optimizing solutions, can be justified.

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

• taking into account the requirements of objective economic laws and patterns, current legislation and statutory documents;
• knowledge and use of patterns and trends in the development of the control object and its external environment;
• availability of complete, reliable, timely information;
• availability of special knowledge, education and qualifications of developers and decision makers;
• knowledge and application of decision-makers to the basic recommendations of management and decision-making theory;
• methods of analysis and synthesis of situations used.

The growing complexity and complexity of the problems being solved and their consequences require universal knowledge for the development and adoption of informed management decisions, which leads to the increasingly widespread use of collegial forms of decision-making.

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

• determination of conditions for the formation of acceptable options;
• compiling a list of indicators characterizing the essential properties of the found solution options, and developing scales for their measurement;
• screening out irrational options and determining the range of possible values ​​for each indicator using a variety of mathematical and heuristic methods;
• identifying the preference structure of decision makers;
• formation of criteria or rules for evaluating solution options;
• choosing the best option for a management decision or clarifying the structure of the decision maker’s preferences.

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

• 1. Multidimensional nature of assessments of the effectiveness of alternatives. When determining possible solution options, and even more so when choosing the most appropriate one, it is necessary to make economic, technical and technological, social, political, and environmental assessments. Moreover, each has several approaches. For example, valuation, according to international, European and Russian standards, uses cost, market (comparative) and income approaches, which use different methods depending on the object and valuation objectives. 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 influence various groups of people, which increases the number of possible assessments (both in relation to them and on their part). In many cases, it is necessary to take into account changes in estimates over time. At the same time, problems of taking into account new types of assessments that characterize the consequences of a decision at different points in the future arise more and more often.
• 2. Difficulties in identifying and comparing all aspects of comparison of alternatives. The existence of heterogeneous aspects of evaluating alternatives poses difficult problems of comparing them for developers and decision makers. It should be borne in mind here that such a comparison is subjective and therefore subject to criticism. This is aggravated many times over in collegial decision-making, where each member of the collective decision-making body may have different measures for comparing heterogeneous qualities. Some participants in development and decision-making may be interested mainly in economic criteria, others in political ones, others in environmental ones, etc.
• 3. Subjective nature of assessments of the effectiveness and quality of alternatives. Many estimates of the effectiveness and quality of alternatives can be obtained either by building special models or by collecting and processing expert opinions. Both methods involve the use of subjective assessments either by specialists developing the model or by experts. When choosing alternatives, it is necessary to take into account that the reliability of such subjective assessments cannot be absolute. Even with complete unanimity of experts, a situation is possible when their assessments turn out to be incorrect. It is also possible that there are different models or discrepancies in expert assessments. Consequently, several alternatives may have different evaluations, and the result of the choice depends on which of them will be used by the decision maker.

Timeliness management decision means that the decision made should neither lag behind nor be ahead of the needs for it in the development of the situation. Even the most optimal (of the ones that make sense for the decision maker) decision, designed to achieve the greatest socio-economic efficiency, may turn out to be useless if it is made late. It can even cause some damage. Premature decisions are no less harmful to the organization than late ones. They do not have the conditions necessary for implementation and development, and can give impetus to the development of negative trends, do not contribute to the solution of already “overripe” problems and further aggravate already painful processes.

Consistency ). A distinction is made between internal and external consistency of a solution. Under internal consistency solutions understand the correspondence of the goals and means of achieving them to the complexity of the problem being solved and the methods for developing a solution, individual provisions of the solution to each other and the meaning of the solution as a whole. Under external consistency decisions - their continuity, compliance with the strategy, company goals and previously made decisions (actions necessary to implement one decision should not interfere with the implementation of others). Achieving a combination of these two conditions ensures consistency and consistency of management decisions. Consistency with previously made decisions also means the need to maintain a clear cause-and-effect relationship of social development. Previously made decisions, if necessary, must be canceled or adjusted if they conflict with the new conditions of the managed system. The emergence of conflicting decisions is a consequence of poor knowledge and understanding of the laws of social development and a manifestation of a low level of management culture.

Reality. The decision must be developed and made taking into account the objective capabilities of the organization and its potential. In other words, the material, financial, informational and other resources and capabilities of the organization must be sufficient for the effective implementation of the chosen alternative.

Completeness of content decisions means that the decision must 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 management decision should reflect:

• the goal (set of goals) of the functioning and development of the managed object to which the decision is directed;
• resources used to achieve these goals;
• the main ways and means of achieving goals, the main methods of performing work that determine the implementation of the goals of the decision;
• deadlines for achieving goals, the beginning and end of their supporting work;
• the order of interaction between departments and individual employees.

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

The quality and effectiveness of a management decision are determined by many factors operating throughout the entire technological management cycle or at its individual stages, having an internal or external (environmental influence), objective or subjective nature. The most significant factors include:

• laws of the objective world related to the adoption and implementation of management decisions;
• goal formulation; why a management decision is made, what real results can be achieved, how to measure, correlate the set goal and achieved results;
• volume and value of available information - for successful management decision-making, the main thing is not so much the volume of information as its value, determined by the level of professionalism, experience, and intuition of personnel;
• time to develop a management decision - as a rule, a management decision is always made under conditions of time shortage and emergency circumstances (lack of resources, activity of competitors, market conditions, inconsistent behavior of politicians);
• organizational management structure, defined by organizational documents (formal) and actually existing (informal). In fact, the existing (current) management structure, almost in exceptional cases, coincides with that defined by the relevant organizational documents, within the framework of which all employees of the organization are required to act. The need to take this requirement into account is often a condition for making a decision that is not the most optimal;
• forms and methods of management activities, including the development and implementation of management decisions;
• the state of the control and managed systems (psychological climate, authority of the manager, professional and qualified personnel, etc.);
• system for assessing the level of quality and effectiveness of management decisions;
• the degree of risk associated with the consequences of implementing the decision. This factor requires the use of various risk assessment techniques (financial, economic, etc.); accordingly, the manager must have the skills to perform 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 decisions. It requires certain knowledge and skills in using modern information technologies in managing the activities of organizations;
• subjectivity of evaluation of the solution choice option. The decision-making process, the choice of a specific option, is creative in nature and depends on the individual and his state at the time of decision-making. The decision maker’s personal assessments act as a compass, pointing him in the desired direction when he has to choose between action alternatives. Each person has his own value system, which determines his actions and influences his decisions. Personal factors include:
• – psychological state of the decision maker at the moment of decision making. In a state of irritability, loaded with other decisions, the decision maker can make one decision on a given situation, and in a good mood, being relatively free, he can make another,
• – the measure of responsibility of the decision maker, determined both by the internal sense of responsibility for their actions and by the documents regulating their activities,
• – level of knowledge on this issue. The higher the level of knowledge of decision-makers about the object to which the decision is directed and its external environment, the greater the likelihood of them making a high-quality and effective decision,
• – experience, which, as the main resource for the development and implementation of decisions, is the determining factor in the adequate perception of the real assessment and effective response of decision-makers to what is happening, represents a certain bank of tested and adaptable options from which analogues and prototypes of developed, accepted and implemented decisions are drawn,
• – intuition, judgment (common sense) and rationality of the decision maker.

Reference. Intuition manifests itself as some kind of insight or instant understanding of a situation without the use of rational thinking. However, such insight is usually preceded by long and painstaking work of consciousness. First, through observation, information is accumulated in a person’s memory, systematized and arranged in a certain order. Often in this way they come to an expedient solution to the problem. If this does not happen, intuition and imagination come into play, generating numerous ideas and associations. One of the ideas can cause an intuitive insight, which, as it were, pushes the corresponding idea from the subconscious into consciousness. Intuition is a powerful decision-making tool that needs constant development and should be actively used in management activities.

When making a decision, the decision maker is often based on his own feeling that his choice is correct. Intuition develops with experience. Judgment-based decisions are based on knowledge and meaningful experiences from the past. Using them and relying on common sense, as adjusted for today, they choose the option that brought the greatest success in a similar situation in the past. However, from the author’s point of view, common sense among people is rare, so this method of decision-making is not very reliable, although it is captivating with its speed and cheapness. With this approach, the decision maker strives to act primarily in those directions that are familiar to him, as a result of which he risks missing out on good results in another area, consciously or unconsciously refusing to invade it;

The risk strategy criterion chosen by the decision maker: optimism, pessimism or indifference. The optimism criterion (maximax) determines the choice of the alternative that maximizes the maximum outcome for each alternative; pessimism (maximin) – an alternative that maximizes the minimum result for each alternative; indifference - an alternative with the maximum average result (in this case, there is an unspoken assumption that each of the possible states of the controlled system can occur with equal probability: as a result, the alternative that gives the maximum value of the mathematical expectation is selected).

At the implementation stage, the effectiveness of decisions is determined by the following factors:

• level of development and condition of the managed system, its equipment, technology, personnel (personnel), organization and economy. At a high level of development of all components of the managed system, when implementing a solution, greater efficiency can be obtained than that provided for by the solution, and vice versa, at a low level it is quite difficult to ensure the efficiency defined in the solution;
• socio-psychological climate in the team implementing the decision. The main criterion of the socio-psychological climate is the level of maturity of the team, which is understood as the degree of coincidence of individual and collective interests. The higher the level of maturity of the team, the more manageable it is, which is a necessary condition for its effective operation;
• the authority of managers ensuring the implementation of the decision. The higher the authority of managers, the more manageable the team and, accordingly, the higher the level of efficiency of its activities;
• the effectiveness of the mechanism for managing the activities of the team, which is expressed in the essence of management as the creation of conditions that encourage people to take the necessary actions to achieve goals;
• time to implement the solution. A timely, high-quality and effective decision, if implemented untimely, may turn out to be not only ineffective, but also unnecessary;
• compliance of the number and qualifications (education, skills and experience) of personnel with the volume and complexity of work to implement the solution. When the number of personnel is less than necessary to implement the solution, it is difficult to meet its deadlines. If the qualifications of workers are below the required level, the quality of work performance and, at the same time, the effectiveness of the implementation of the solution decreases;
• provision of necessary material, energy, labor, information and financial resources.

It was shown above that the effectiveness of a solution is determined at the stages of its development and implementation. At the first stage, it is determined by well-known methods for calculating the effectiveness of design decisions, at the second - as a rule, but using methods for calculating actual profit and profitability of activities. In recent years, to determine the effectiveness of strategic decisions at the stages of their development and implementation, calculation of the expected and actual changes in the market value of a business is often used, the results of which are the basis for assessing and choosing the organization’s strategy.

The effectiveness of management decisions at the stages of their development and adoption can be assessed using well-known indicators for assessing investment projects:

• net discounted (discounted, current) income (NPV) – NPV (Net Present Value ) – the current value of cash inflows (income) minus the cost of cash outflows (investment costs);
• internal rate of return (IRR) – IRR (Internal Rate of Return ) – the discount rate at which equality arises between the current value of projected cash inflows (income) and the current value of projected investment costs (cash outflows), i.e. net current income (NPV) is equal to zero;
• modified internal rate of return (MIRR) – MIRR (Modified Internal Rate of Return ) – an indicator characterizing the efficiency of capital investments (investments). If the current value of all investment

investments are considered as the initially invested capital, and the future value of all cash inflows - as the accumulated amount, then the discount rate for the accumulation factor is taken as the MVND;

• profitability index (RI) – P.I. (Profitability Index ) – the amount of net (discounted) cash flow per unit of investment;
• payback period - RR (Payback Period ) – the expected period of reimbursement of invested funds by net cash receipts;
• discounted payback period – DPP (Discounted Payback Period ) – the expected period of compensation (equality) of the current value of invested funds and the current value of net cash receipts;
• cost efficiency ratio – ARR (Accounting Rate of Return ) is equal to the ratio of the projected average annual net (balance sheet) profit to the average annual investment costs.

These indicators are widely used in practice, and the methods for their calculation are recognized as traditional. In numerous literature they are described in detail, examples are given illustrating their calculations for selecting projects (alternatives) for management decisions with different initial conditions.

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

• to determine the effectiveness of independent (no alternative) management decisions (the so-called absolute effectiveness), when a conclusion is made about whether to accept or reject it;
• to determine the effectiveness of mutually exclusive decision alternatives (comparative effectiveness), when a conclusion is made about which of them to accept as a management decision.

In assessing the effectiveness of management decisions, like any other activity, the results of its implementation (effect - Er) and the costs of its development, adoption and implementation (Zr) are involved. The effect of management decisions is manifested in the final results of the organization. Even in cases where the management decision is aimed at changing the technical, economic or socio-economic indicators of the organization’s activities (the level of condition and development of equipment and production technology, product range and range, quality of raw materials, design characteristics of work premises, social infrastructure, etc. ), the effect of its implementation is ultimately reflected in a change in the level of use of its potential and satisfaction of public needs for its products and services, i.e.

Er = f (P, Ip, Zr, Up)

at (P – IP), Zr min; Pack max,

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

This approach, called " resource-potential ", to assess the effectiveness of managing the activities of organizations, the product of which is management decisions and the results of their implementation, was proposed by Academician of the USSR Academy of Sciences V. A. Trapeznikov, substantiated and developed by professors F. M. Rusinov and V. I. Busov.

The development of an organization (its potential related to a particular goal, expressed in the desire for the maximum possible satisfaction of a certain type of social needs) has limitations determined by the ratio of supply and demand for products and services that a given organization is capable of producing. Exceeding the result of a particular function of an enterprise from its existing needs is a negative effect of its activities or an unhelpful result, tantamount to waste and loss of resources spent on it.

The second component of efficiency is the cost of resources for the development, adoption and implementation of management decisions. Increasing the level of return on these costs (their efficiency) is the most important task of managing the process of development, adoption and implementation of management decisions. An incorrect understanding of this task (especially in terms of development and decision-making) often leads in practice to reducing 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 them, and their reduction comes down to a reduction in the personnel involved in this process or the level of payment for their labor, as a result of which the quality of the management decision and the effect of its implementation, as well as the motivation of personnel, deteriorate. Reducing the costs of developing, making and implementing management decisions through a simple voluntaristic decision entails a decrease in the efficiency of the organization's activities associated with deterioration of control, an increase in the waiting time for a decision to be made in a given situation, deterioration in the quality of preparation, development and decision-making and other factors , affecting the level of resource losses.

An assessment of the effectiveness of the implementation of management decisions can be made for each major management decision or for the totality of those implemented in a certain period of time (for example, a quarter, half a year, a year). It consists of a system of indicators (Fig. 3.5), including:

• a generalizing integral indicator that specifies the effectiveness criterion;
• generalizing indicators reflecting the effectiveness of the implementation of groups of goals for the achievement of which a management decision was made (scientific, technical, economic, social, etc.);
• private indicators reflecting the efficiency of using certain types of resources at individual stages of the reproduction cycle.

When determining the effectiveness of the implementation of a management decision, the value used is not the potential of the organization’s resources in general, but its potential to perform the functions covered by this decision. To identify such a composition, you can use the matrices given in table. 1.2–1.5.

The level of potential utilization is defined as the difference between its value and losses. Moreover, the reserve part of the potential, necessary for the sustainable functioning and development of any division of the organization, does not apply to its losses.

Rice. 3.3.

Shown in Fig. 3.5, the system of indicators reflects the structure of the “tree” of goals for increasing the efficiency of the organization.

The effectiveness of a management decision is defined as

where Entz and Entz, Epts and Epts, Ests and Ests, Eekts and Eekts are the effectiveness and effect of management decisions in achieving scientific, technical, production, social and environmental goals, respectively; Ei, is the effect of implementing a management decision in the t-th division of the organization (workplace of the division); Зр – costs of development and implementation of management decisions; P – the number of departments involved in the development and implementation of this management decision.

Participation effect i - department of the organization (workplace) in the development and implementation of a management decision is defined as the sum of the effects of changes in the level of use in the process to which this decision is directed, the existing potential of the department (workplace) - internal effect (Ev) - and the result of the implementation of the goals of the decision - external effect (Ec), i.e.

Ei = Ev + Ets.

The internal effect is determined by intensive (Ei) and extensive factors (Ee), i.e.

Ev = Ei + Ee.

Intensive factors determine changes in the productive use of potential due to the implementation of a given management decision, extensive factors determine changes in the unproductive use of potential and loss of resources.

The scheme for calculating efficiency indicators for managing the activities of an enterprise is shown in Fig. 3.6.

Since all resources arrive at the organization’s workplaces and are used here, the level of use of the potential of the enterprise’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 in the use of potential output (or labor productivity) in a given workplace before and after the implementation of a given management decision, i.e.

where and Вп – potential output at a given workplace, respectively, before and after the implementation of the management decision; , and Vf – actual output at a given workplace, respectively, before and after the implementation of the management decision.

Actual output (or labor productivity) in any production department (procurement, mechanical, foundry, assembly, etc.) is determined without much difficulty using generally accepted assessment methods.

Rice. 3.6.

Potential and actual output in the workplace form the basis for determining potential and actual output for a unit, function or type of activity of a unit. The volume of output at a workplace is influenced by: equipment productivity for a given technology of work performed at a given workplace; compliance of the employee’s qualifications with the level of complexity of the work; timely provision of the workplace with necessary materials, tools, organizational equipment, information and other resources; compliance of the quantity and quality of initial resources with technology requirements; rhythm of employee activity at the workplace. These factors reduce actual production compared to potential.

The potential output of a workplace (Vp(rm)) is determined by the output volume of the equipment installed on it with a maximum number of one hundred hours of work in a given period, taking into account the time for readjustment, repair, adjustment, i.e. according to the formula

Βп(рм) = (Фр – t m) P n ,

where Фр is the operating time of one unit (construction crane, bulldozer, concrete mixer, sanding machine, etc.) at the workplace per month; t n – standard time for setup and repair, reconfiguration of one unit; P – routine (technological) removal of products from a unit of equipment (unit) per unit of time; P – the number of similar units in the workplace during multi-machine service.

For workplaces with little mechanized and manual labor, including engineering and managerial workers, potential output is calculated based on the maximum shift output of the month, based on the fact that the maximum output for a given shift was achieved through the greatest use of the capabilities of the resources that make up this workplace, those.

Vp(rm) = Vs.max t r,

where Vs.max is the maximum shift output at the workplace in the billing month, standard hours; m – number of shifts in the billing month; R – cost of 1 standard hour, rub.

The initial data for the calculation is taken from production and wage accounting cards, which must be filled out in the departments of the enterprise.

A similar approach can be applied to any workplace, but for mechanized and automated workplaces, Vp should be calculated based on the productivity of the equipment.

Knowing the potential volume of output per month for all workplaces of the department, you can determine the potential volume of output of this department. It is calculated according to the technological chain of workplaces formed by the system of machines involved in the production of a given type of product, or determined by the sequence of execution of technological operations assigned to workplaces for the production of a given type of result of the unit’s activity.

Extensive use of economic potential through the internal effect of enterprise management system processes expresses losses and technologically unjustified waste of resources. The change in their value after the implementation of the management decision () in comparison with the base one (Pr) reflects the change in the internal effect of management on extensive factors, i.e.

.

The resources involved in the processes are used productively and unproductively.

Productive use of resources is also divided into two parts. The first part is the consumption of resources, calculated on the basis of unit costs, which are recognized as rational (technologically necessary). The second part is resource consumption that exceeds rational unit costs. Such costs represent wastage of resources.

Waste of resources occurs when products and services are not created. For example, unproductive use of resources includes the cost of working time of employees, the cost of production capacity of equipment and materials to correct defects, losses include absenteeism, whole-day and whole-shift downtime, unused capacity of installed equipment, irreparable defects, unused scientific and technical developments, damage to materials in the warehouse and etc.

The effect of implementing a management decision to achieve production goals is determined by an increase in the volume and quality of products and services, compliance with the deadlines for their provision to the consumer and is expressed in a change in the efficiency of their use by consumers; scientific and technical goals - in the efficiency of application of enterprise developments in innovative processes; social goals – saving time (increasing free time) and increasing social activity of the enterprise’s employees and consumers of the enterprise’s products and services; environmental goals - reducing waste and increasing the volume of recycling, landscaping, etc. The effect on social results is especially important for enterprises that provide various services to the population (utilities, transport, household services, postal services, catering, trade, etc.). Effect on environmental results - for enterprises in the fuel, petrochemical and chemical industries.

The costs of developing and implementing a management decision include the entire set of costs for performing work both in-house and by third parties (contractors), as well as for purchasing the necessary materials, equipment and other necessary resources.

The above approach is applicable only if the organization has the necessary initial data, provided by an organized system for monitoring and recording process parameters at workplaces and in departments, monitoring the needs and consumption of the company's products and services.

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

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

The main idea of ​​the value management concept is that the main financial goal of an organization is to increase its value (cost) not only for owners (shareholders), but also for all legal entities and individuals interested in the company's activities (company value management in the interests of stakeholders). The concept of “value” in this management concept is an internal category that characterizes the value and investment attractiveness of the company for owners, and is expressed as a monetary indicator of future growth opportunities.

Increase in value is an economic criterion that reflects the integral effect of the influence of management decisions implemented in an organization on all parameters by which its activities are assessed (market share and strength of competitive position, income, investment needs, operational efficiency, tax burden, regulation, cash flows and risk level ), allowing you to rank options in a multiple choice situation.

The value management system initially contains the premise that the command-and-administrative style of top-down management decision-making does not bring the desired results, especially in large multi-industry corporations. Lower-level managers need to learn to use cost indicators to make better and more effective management decisions. Cost management requires a reasonable balance of long-term and short-term performance goals. It, in essence, represents the development, adoption and implementation of management decisions that ensure continuous reorganization aimed at achieving maximum business value.

An important advantage of the cost approach to management is the fact that it offers management a single and understandable criterion for evaluating activities - cost. The business value increase parameter is a key tool for improving the quality and efficiency of management decisions, allowing you to create a universal coordinate system for determining the vector of business development, as well as create a unified scale for changing the achieved results in accordance with the established strategy.

The process of managing the market value of a company uses as a basis the income approach to valuing the company (business). Under this approach, the value of a company is the sum of the cash flows that will be generated by the company, adjusted for timing factors and associated risks, less all of the company's liabilities.

Assessing the effectiveness of a management decision using this method involves comparing two scenarios for the development of an organization “without the development and implementation of a management solution to a given situation-problem” and “subject to the development and implementation of a management solution to a given situation-problem.”

Estimating the value of an organization in the first option comes down to a forecast of cash flows for the enterprise as a whole, provided that nothing in it will fundamentally change during the billing period. This - discounted value business, which is determined by discounting cash flow at a rate that takes into account the existing risks of the organization as a whole:

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

The cost of the organization in the scenario of implementing a management decision (strategic value) is determined by discounting the project-adjusted cash flow at an adjusted rate that takes 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 organization’s expected flows, subject to the implementation of a management decision, i.e. The organization’s cash flows under two scenarios of its development are combined:

Where PV C – strategic value of the organization; CF c – strategic cash flow of the organization; CF pi – cash flow created by the implementation of a management decision.

Application capital market and transaction method to assess the increase in the value of an enterprise due to the implementation of a management decision, it is based on information about a similar company implementing a similar decision. In this case, the similarity of solutions is determined by the following factors:

• maximum similarity of the situations being solved in the compared organizations;
• general industry (functional) affiliation of the compared situations;
• use of similar resources;
• comparability of the scale of situations and the radicality of changes as a result of the implementation of management decisions.

To determine the increase in value created as a result of the implementation of a management decision, the capital market method uses the market coefficients of a similar company before and after its implementation of a solution to a similar situation, i.e.

where Δ CV – an increase in the market value of the company being valued as a result of the implementation of a management decision; E ok – current profit of the company being valued; – the price/earnings ratio for a similar company after implementing a solution to a similar situation; – the price/earnings ratio for a similar company before implementing a solution to a similar situation.

The transaction method differs from the capital market method in that the price/earnings ratio for the peer company(ies) is calculated by taking into account only the stock prices of the peer company(ies) that were observed in the near past based on actual transactions for the purchase and sale of large blocks of shares or with a corresponding quotation of shares. At the same time, large packages are considered to be those whose purchase makes it possible to acquire at least participation in control over the company by introducing a representative (or yourself) to its board of directors, which allows you to control the company’s management. Hence, finding a similar company that implements a management solution for a similar situation, information on which is publicly available, is an extremely difficult task and sometimes simply impossible. In practice, this makes it significantly difficult or impossible to use capital market and transaction methods to assess the effectiveness of management decisions.

Features of the application of mathematical theory in making management decisions

Note 1

Methods that are based on the use of mathematics allow making management decisions that can be formalized or fully describe the relationship and interdependence of their conditions, factors and results.

The use of mathematical theory is typical for making tactical and partially operational decisions.

The application of mathematical theory is effective in the presence of a number of parameters of a management decision:

• the goal or optimization criterion is clearly known in advance;
• the main limitations are obvious - the conditions for achieving this goal;
• The management problem is well structured.

Algorithm of mathematical theory

The peculiarity of the mathematical theory of substantiation of management decisions is the presence in it of a certain algorithm, which precisely prescribes the execution of a certain system of operations in an established sequence to solve a certain class of problems.

The algorithm of the mathematical theory of management decision making must meet a number of requirements:

• certainty, i.e. accuracy and unambiguity, leaving no room for arbitrariness;
• mass scale and universality - applicability for solving a specific class of problems when the initial data varies within known limits;
• effectiveness, i.e. the ability to solve a given problem in a limited number of operations.

Mathematical methods for making management decisions

The main methods for solving typical management problems within the framework of mathematical theory are:

1. The method of mathematical analysis is used in calculations to justify resource requirements, cost accounting, project development, etc.
2. The method of mathematical statistics is convenient to use when the change in the indicators under study is a random process.
3. The econometric method involves the use of an economic model - a schematic representation of an economic process or phenomenon.
4. Linear programming is the solution of a system of equations when there is a strictly functional relationship between the phenomena under study.
5. Dynamic programming is used to solve optimization problems where the constraints or objective function have a nonlinear relationship.
6. Queuing theory is used to find the optimal number of service channels for a given level of demand for them. An example of such a situation is the choice of the optimal option for organizing work with clients so that service time is minimal and quality is high without additional costs.
7. Operations research method is the use of mathematical probabilistic models that represent the process, activity or system under study. Optimization comes down to a comparative study of numerical estimates of those parameters that cannot be estimated by conventional methods.
8. Situational analysis is a complex technology for making and implementing management decisions, which is based on analyzing a separate management situation. Such an analysis is based on a specific situation, a problem that arises in the activities of the organization, which requires making a management decision.
9. Methods of game theory - modeling a situation in which, when justifying decisions, it is necessary to take into account the conflict or divergence of interests of various individuals.
10. Break-even points are a method in which total revenues are equalized with total expenses to find the point that brings the minimum profit to the enterprise.
11. Trend projection is a time series analysis based on the assumption that what happened in the past provides a good approximation when estimating the future. This method is used to identify past trends and extend them into the future.

Of the various methods of making economic decisions, the most common can be identified: mathematical programming; game theory; statistical decision theory; queuing theory; method of cause-and-effect analysis; using the model

Mathematical programming represents theoretical principles and analytical methods for solving problems in which the search for the extremum (minimum or maximum) of a certain function in the presence of restrictions imposed on unknowns occurs. A special place in mathematical programming is occupied by linear programming, which is the most developed and widely used in practice. Linear programming includes analytical methods for solving problems in which the objective function and constraints are expressed in linear form, that is, the unknowns included in the objective function and constraints must be the first stage. Problems in which the maximum and minimum values ​​of a linear function under linear constraints are found are called linear programming problems.

Depending on the type of objective function and system of restrictions, mathematical programming methods are divided into

linear programming - the objective function and the constraint functions included in the constraint system are linear (first order equation)

nonlinear programming - the objective function or one of the constraint functions included in the constraint system is nonlinear (higher order equation)

Integer (discrete) programming - if at least one variable is subject to an integer condition;

dynamic programming - if the parameters of the objective function and / or the system of restrictions change over time or the objective function has an additive / multi-step form or the mass decision-making process itself is multi-step in nature.

Depending on the information about the process in advance, mathematical programming methods are divided into

Stochastic programming - not all information about the process is known in advance: the parameters included in the objective function or the constraint function are random or decisions have to be made under risk conditions

Deterministic programming - all information about the process is known in advance.

Depending on the number of target functions, tasks are divided into:

Single-criteria;

Rich criteria.

Linear programming combines theory and methods for solving a class of problems in which a set of values ​​of variables is determined that satisfy a given linear constraint and maximize (or minimize) some linear function. That is, linear programming problems are optimization problems in which the objective function and functional constraints - linear functions - take any values ​​from a certain set of values.

For linear programming problems, numerous solution methods and corresponding software have been developed for various situations. To solve linear programming problems, several methods are used, among which the most common are the simplex method and the graphical method.

The most convenient method for solving such problems is the simplex method, which allows, starting from the initial solution to the problem, to obtain the optimal option in a certain number of steps. Each of these steps (iterations) consists of finding a new option that corresponds to the largest (when solving maximum problems) or less (when solving minimum problems) values ​​of the linear function than the value of the same function in the previous option. The process is repeated until the optimal solution is obtained, which has an extreme value.

Thus, we can assume that the optimal plan is the one that provides the maximum production effect for a given volume of material, raw materials, and labor resources. The maximum production effect is determined by the optimization criterion, which determines the objective function.

The most typical problems for which the simplex method is used are: optimal planning at enterprises (planning assortment production), optimal set of raw materials, efficient use of raw materials, material, labor, financial and energy resources, problems of optimizing the organization of production (transport problem ).

Optimization of the production program (assortment tasks) at enterprises is a group of tasks in which the production program is determined taking into account the influence on the enterprise of internal factors (equipment capabilities, raw material limits, labor factors) and some external requirements (demand for commercial products in general or individual assortment groups and types, average price of the assortment that is produced, etc.).

The main stages of formulating and solving the production program optimization problem:

1) building an economic and mathematical model: collecting information, preparing it for building the model; selection of optimization criterion; selection of restrictions and their construction in general form; analytical and tabular view of the model with real coefficients;

2) finding the optimal solution to the problem;

3) analysis of the solution results and practical recommendations.

In an optimal production plan, the selection of optimization criteria is carried out in accordance with the goal of solving the problem. The optimization criterion can be different cost and natural indicators. In addition to the goal function, the model uses restrictions, since the resources available to the enterprise are in most cases limited, and the assortment output must be calculated taking into account the demand for the product. Restrictions are selected depending on the resources that are used to release the enterprise's production program.

The effectiveness of the task and the optimality of the resulting assortment is assessed using systems of economic indicators (changes in production volumes in physical and value terms, reduction in production costs, increase in profits and profitability, reduction in costs per 1 ruble, use of raw materials, etc.).

Game theory studies quantitative patterns in conflict situations. The main goal of game theory is to develop or quantitatively substantiate recommendations for choosing the most rational solution in conflict situations. In economic research, conflict situations are those situations when it becomes necessary to choose a rational solution from two or more mutually exclusive options.

The theory of statistical decisions, which uses methods for studying processes and phenomena that are highly influenced by random, uncertain factors, is based on the theory of probability.

Queuing theory studies the patterns of queuing processes and, on their basis, develops effective methods for managing service systems. Methods of queuing theory make it possible to rationally organize the service process and ensure the most efficient functioning of the queuing system (reducing waiting time for service, reducing service costs). The basis of queuing theory is probability theory and mathematical statistics.

Decision tree (can also be called classification trees or regression trees) - used in the field of statistics and data analysis for predictive models. The tree structure contains the following elements: "leaves" and "branches". The attributes on which the objective function depends are written on the edges (“branches”) of the decision tree, the values ​​of the objective function are written in the “letter,” and the attributes on which cases differ are written in other nodes. To classify a new case, you need to go down the tree to a leaf and produce the corresponding value. Such decision trees are widely used in data mining. The goal is to create a model that predicts the value of a target variable based on several input variables.

Each leaf represents the value of the target variable as it changes as it moves from the root through the leaf. Each internal node corresponds to one of the input variables. The tree can also be "learned" by dividing the output variable sets into subsets based on testing the attribute values. This is a process that is repeated on each of the resulting subsets. The recursion ends when the subset at the node has the same values ​​of the target variable, so it adds no predictive value. The top-down process, Decision Tree Induction (TDIDT), is an example of an ingestive greedy algorithm, and is by far the most common decision tree strategy for data, but it is not the only possible strategy. In data mining, decision trees can be used as mathematical and computational techniques to help describe, classify and summarize a set of data, which can be written as follows:

The dependent variable Y is the target variable that needs to be analyzed, classified and summarized. The vector x consists of input variables X1, x2, x3, etc., which are used to perform this task.

In decision analysis, decision trees are used as a visual and analytical decision support tool where the expected values ​​(or expected utilities) of competing alternatives are calculated.

The decision tree consists of three types of nodes.

1. Solution nodes - usually represented by squares.

2. Probabilistic nodes - represented as a circle.

3. Closing nodes - represented in the form of a triangle.

In Fig. 4.1 below, the decision tree should be read from left to right. A decision tree cannot contain cyclic elements, that is, each new leaf can subsequently only split, there are no paths that converge. Thus, when constructing a tree manually, we may encounter the problem of its dimension, therefore, as a rule, we can obtain a decision tree using specialized programs. Typically, a decision tree is presented as a symbolic diagram, which makes it easier to understand and analyze.

Rice. 4.1. decision tree

Decision trees used in Data Mining come in two main forms:

Classification tree analysis, when the predicted result is the class to which the data belongs;

Regression tree analysis, where the predicted outcome can be treated as a real number (for example, the price of a house, or the length of a patient's stay in a hospital).

The terms mentioned above were first used by Breiman et al. The types listed have some similarities as well as some differences, such as the procedure used to determine where to split. Some methods allow you to build more than one decision tree:

The bag decision tree, the earliest decision tree, builds multiple decision trees that repeatedly interpolate data with replacement, and voting trees to predict consensus. The random forest classifier uses a number of decision trees to improve classification rates;

"Promoted" trees can be used for regression type and type classification of problems.

"Forest rotation" - trees in which each decision tree is analyzed by first applying principal component analysis (PCA) to random subsets of the input functions.

The general scheme for constructing a decision tree based on test examples is as follows (according to the algorithm in Fig. 4.2):

Rice. 4.2. Algorithm for constructing a decision tree

The main question: how to choose the next attribute? There are different ways to select the next attribute:

IDZ algorithm, where the attribute is selected based on the increase in information (English Gain), or based on the Gini coefficient.

Algorithm C4.5 (improved version of ID3), where the attribute is selected based on the normalized gain of information (English Gain Ratio).

The CART algorithm and its modifications - IndCART, DB-CART.

Automatic Chi-square (force) interaction detector. Performs multi-level partitioning when calculating tree classification.

MARS: Extends decision trees to improve digital data processing.

In practice, the results of these algorithms often result in trees that are too detailed, which, when further applied, produce many errors. This is due to the phenomenon of overfitting. To prune trees, pruning is used.

Tree depth adjustment is a technique that allows you to reduce the size of a decision tree by removing sections of the tree that have little weight.

One of the questions that arises in the decision tree algorithm is the optimal size of the final tree. Thus, a small tree may not cover some important information about the sample space. However, it is difficult to say when the algorithm should stop because it is impossible to predict which node adding will significantly reduce the error. This problem is known as the "horizon effect". However, the general strategy of limiting the tree is preserved, that is, removing nodes is implemented if they do not provide additional information.

It should be noted that adjusting the tree depth should reduce the size of the training tree model without reducing its prediction accuracy or by using cross-validation. There are many methods for adjusting tree depth, which differ in how they measure performance optimization.

Tree pruning can be done from top to bottom or from bottom to top. From top to bottom - pruning begins from the root, from bottom to top - the number of leaves of the tree is reduced. One of the simplest control methods is to reduce the tree constraint error. Starting with leaves, each node is replaced by the most popular class. If the prediction accuracy is not affected, then the change is retained.

When making decisions, a manager can use one of the above methods. The best decisions are made as a group. The effectiveness of group decisions largely depends on the leader. Taking into account the skills, character and mood of the leader, his teaching abilities, attention to people and other qualities, psychologists identify five types of leaders: dictator, democrat, pessimist, organizer and manipulator.

The method, based on a scientific-practical approach, requires the use of modern technical means and, above all, electronic computer technology.

In general, the problem of a manager’s choice of a solution is one of the most important in modern management science and practice.

Modern measurement theory and expert assessments. How to analyze the expert responses collected by the working group? For a more in-depth consideration of the problems of expert assessments, we will need some concepts of the so-called representative theory of measurement(Chapter 2.1), which serves as the basis for the theory of expert assessments, primarily that part of it that is associated with the analysis of expert opinions expressed in qualitative (rather than quantitative) form.

Representative (i.e. related to presentation relations between real objects in the form of relations between numbers) measurement theory (hereinafter abbreviated as RTI) is one of the components of econometrics. Namely, it is part of statistics of objects of non-numerical nature. We are interested in RTI primarily in connection with the development of the theory and practice of expert assessment, in particular in connection with the aggregation of expert opinions and the construction of generalized indicators (they are also called ratings).

Opinions received from experts are often expressed in ordinal scale, i.e. an expert can say (and justify) that one type of product will be more attractive to consumers. Than another, one indicator of product quality is more important than another, the first technological object is more dangerous than the second, etc. But he is unable to say how many times or for how long more important, therefore more dangerous. Therefore, experts are often asked to give a ranking (ordering) of the objects of examination, i.e. arrange them in order of increasing (or, more precisely, non-decreasing) intensity of the characteristics of interest to the organizers of the examination.

Rank is the number (of the object of examination) in an ordered series. Formally, ranks are expressed by the numbers 1, 2, 3, ..., but it is very important that you cannot do the usual arithmetic operations with these numbers. For example, although 2 + 3 = 5, it cannot be argued that for an object in third place in the ordering (in another terminology - ranking), the intensity of the characteristic being studied is equal to the sum of the intensities of objects with ranks 1 and 2. Thus, one of the types of expert assessments - student assessments. It is unlikely that anyone will seriously argue that the knowledge of an excellent student is equal to the sum of the knowledge of a D student and a C student (although 5 = 2 + 3), a good student corresponds to two D students (2 + 2 = 4), and between an excellent student and a C student there is the same difference as between a good student and a poor student (5 - 3 = 4 - 2). Therefore, it is obvious that to analyze this kind of qualitative data, it is not ordinary arithmetic that is needed, but another theory that provides the basis for the development, study and application of specific calculation methods. This other theory is RTI. The basics of RTI are discussed in Chapter 2.1.

Let us consider, as an example of the application of the results of measurement theory related to average values ​​on an ordinal scale, one story related to rankings and ratings.

GPA methods. Currently, expert, marketing, qualimetric, sociological and other surveys are common in which respondents are asked to give points to objects, products, technological processes, enterprises, projects, applications for research work, ideas, problems, programs, policies, etc. P. Then the average scores are calculated and considered as integral (i.e. generalized, final) assessments, exhibited by a team of interviewed experts. What formulas should be used to calculate averages? After all, as we know, there are many different types of average sizes.

Usually used average. Measurement theorists have known for about 30 years that this method is incorrect, since scores are usually measured in ordinal scale (see above). It is reasonable to use medians as average scores. However, completely It is not advisable to ignore arithmetic averages because of their familiarity and prevalence. Therefore, it seems rational to use both methods simultaneously - the method of arithmetic average ranks (scores) and the median method. ranks. This recommendation is in agreement with general scientific concept of sustainability, which recommends using different methods to process the same data in order to highlight the conclusions obtained simultaneously with all methods. Such conclusions apparently correspond to reality, while conclusions that vary from method to method depend on the subjectivity of the researcher who chooses the method for processing initial expert assessments.

An example of a comparison of eight projects. Let's look at a specific example of applying the approach just formulated.

On instructions from the company's management, eight projects proposed for inclusion in the company's strategic development plan were analyzed. They are designated as follows: D, L, M-K, B, G-B, Sol, Steph, K (by the names of the managers who proposed them for consideration). All projects were sent to 12 experts included in the expert commission, organized by decision of the Board of Directors of the company. Table 1 below shows the ranks of eight projects assigned to them by each of the 12 experts in accordance with the experts’ views on the advisability of including the project in the company’s strategic plan. In this case, the expert assigns rank 1 to the best project that must be implemented. Rank 2 is received from the expert by the second most attractive project, ... and finally, rank 8 is the most dubious project, which should be implemented only as a last resort.

Table 1.

Ranks of 8 projects by degree of attractiveness

for inclusion in the company’s strategic development plan

Expert No.

Note. Expert No. 4 believes that projects M-K and B are equivalent, but inferior to only one project - project Sol. Therefore, projects M-K and B should be in second and third places and receive points 2 and 3. Since they are equal, they receive an average score of (2+3) / 2 = 5 / 2 = 2.5.

Analyzing the results of the experts’ work (i.e. the mentioned table), members of the analytical unit of the Working Group, who analyzed the experts’ answers on the instructions of the Board of Directors of the company, were forced to state that there was no complete agreement between the experts, and therefore the data presented in the table should be subjected to more careful mathematical analysis.

Method of arithmetic average ranks. First, the method of arithmetic average ranks was used to obtain the group opinion of experts. To do this, first of all, the sum of ranks assigned to projects was calculated (see Table 1). Then this amount was divided by the number of experts, as a result the arithmetic average rank was calculated (it was this operation that gave the name to the method). Based on the average ranks, the final ranking is built (in other terminology - ordering), based on the principle - the lower the average rank, the better the project. Project B has the lowest average rank, equal to 2.625, and therefore receives a rank of 1 in the final ranking. Project M-K has the next highest amount, equal to 3.125, and receives a final rank of 2. Projects L and Sol have the same sums (equal to 3.25), which means that from the point of view of experts they are equivalent (with the considered method of bringing together the opinions of experts), and therefore they should be in 3rd and 4th places and receive an average score (3+4) / 2 = 3 ,5. Further results are given in table. 2 below.

So, the ranking by sums of ranks (or, what is the same, by arithmetic average ranks) has the form:

B< М-К < {Л, Сол} < Д < Стеф < Г-Б < К. (1)

Here is a record like "A"<Б" означает, что проект А предшествует проекту Б (т.е. проект А лучше проекта Б). Поскольку проекты Л и Сол получили одинаковую сумму баллов, то по рассматриваемому методу они эквивалентны, а потому объединены в группу (в фигурных скобках). В терминологии математической статистики ранжировка (1) имеет одну связь.

Method of median ranks. This means that science has had its say, the result of the calculations is ranking (1), and on its basis a decision must be made? This is how the question was posed when discussing the results obtained at a meeting of the company’s Management Board. But then the member of the Board most familiar with modern econometrics remembered what was discussed above. He remembered that the experts’ answers were measured on an ordinal scale, and therefore it was unlawful for them to carry out averaging using the method of arithmetic averages. We must use the median method.

What does it mean? You need to take the experts' answers corresponding to one of the projects, for example, project D. These are ranks 5, 5, 1, 6, 8, 5, 6, 5, 6, 5, 7, 1. Then they need to be arranged in non-decreasing order (easier It would be possible to say “in ascending order,” but since some answers are the same, we have to use the unusual term “non-decreasing”). We get the sequence: 1, 1, 5, 5, 5, 5, 5, 6, 6, 6, 7, 8. In the central places - the sixth and seventh - there are 5 and 5. Therefore, the median is 5.

Table 2.

Calculation results using the arithmetic average method

and the median method for the data given in Table 1.

Sum of ranks
Arithmetic mean of ranks
Final rank based on arithmetic mean
Medians of ranks
Final rank by medians

The medians of sets of 12 ranks corresponding to certain projects are given in the penultimate row of Table 2. (In this case, medians are calculated according to the usual rules of statistics - as the arithmetic mean of the central members of the variation series.) The final ordering of the expert commission using the median method is given in the last row of the table. The ranking (i.e. ordering - the final opinion of the expert commission) by medians has the form:

B< {М-К, Л} < Сол < Д < Стеф < К <Г-Б. (2)

Since projects L and M-K have the same median scores, according to the ranking method under consideration they are equivalent, and therefore are combined into a group (cluster), i.e. from the point of view of mathematical statistics, ranking (4) has one connection.

Comparison of rankings using the method of arithmetic means and the method of medians. Comparison of rankings (1) and (2) shows their closeness (similarity). It can be assumed that projects M-K, L, Sol are ordered as M-K< Л < Сол, но из-за погрешностей экспертных оценок в одном методе признаны равноценными проекты Л и Сол (ранжировка (1)), а в другом - проекты М-К и Л (ранжировка (2)). Существенным является только расхождение, касающееся упорядочения проектов К и Г-Б: в ранжировке (3) Г-Б < К, а в ранжировке (4), наоборот, К < Г-Б. Однако эти проекты - наименее привлекательные из восьми рассматриваемых, и при выборе наиболее привлекательных проектов для дальнейшего обсуждения и использования на указанное расхождение можно не обращать внимания.

The considered example demonstrates the similarities and differences between the rankings obtained using the method of arithmetic average ranks and the method of medians, as well as the benefits of their combined use.

Method for matching clustered rankings. The problem is to identify a general non-strict order from a set of clustered rankings (in statistical language - rankings with connections). This set may reflect the opinions of several experts or be obtained by processing expert opinions using various methods. A method for reconciling clustered rankings is proposed, which makes it possible to “drive” contradictions inside specially constructed clusters (groups), while the ordering of the clusters corresponds simultaneously to all the original orderings.

In various application areas, there is a need to analyze several clustered rankings of objects. Such areas include primarily ecology, engineering business, management, economics, sociology, forecasting, scientific and technical research, etc., especially those sections that are associated with expert assessments (see, for example,). The objects can be product samples, technologies, mathematical models, projects, candidates for positions, etc. Clustered rankings can be obtained both with the help of experts and in an objective way, for example, by comparing mathematical models with experimental data using one or another quality criteria. The method described below was developed in connection with the problems of chemical safety of the biosphere and environmental insurance.

This paragraph discusses a method for constructing a clustered ranking that is consistent (in the sense disclosed below) with all the clustered rankings under consideration. In this case, the contradictions between the individual initial rankings turn out to be contained within the clusters of the agreed ranking. As a result, the ordering of the clusters reflects the general opinion of experts, or more precisely, the general opinion contained in the original rankings.

Clusters contain objects for which some of the original rankings contradict each other. New research needs to be conducted to sort them out. These studies can be either formal mathematical (for example, calculating the Kemeny median (more about it below), ordering by average ranks or medians, etc.), or require the involvement of new information from the relevant applied area, possibly carrying out additional scientific or applied works.

Let us introduce the necessary concepts, then formulate an algorithm for matching clustered rankings in a general form and consider its properties.

Let there be a finite number of objects, which for simplicity of presentation we will depict as natural numbers 1,2,3,..., k and call their totality “carrier”. By clustered ranking, defined on a given medium, we understand the following mathematical construction. Let the objects be divided into groups, which we will call clusters. There can be only one element in a cluster. Objects included in one cluster will be enclosed in curly braces. For example, objects 1,2,3,...,10 can be divided into 7 clusters: (1), (2,3), (4), (5,6,7), (8), (9) , (10). In this partition, one cluster (5,6,7) contains three elements, another (2,3) contains two, the remaining five contain one element each. Clusters do not have common elements, and their union (as sets) is the entire set of objects under consideration (the entire medium).

The second component of clustered ranking is a strict linear order between clusters. It is specified which one is first, which one is second, etc. We will represent order using the sign< . При этом кластеры, состоящие из одного элемента, будем для простоты изображать без фигурных скобок. Тогда кластеризованную ранжировку на основе введенных выше кластеров можно изобразить так:

A = [ 1 < {2,3} < 4 < {5,6,7} < 8 < 9 < 10 ] .

We will enclose specific clustered rankings in square brackets. If, for simplicity of speech, the term “cluster” is applied only to a cluster of at least 2 elements, then we can say that in a clustered ranking A includes two clusters (2,3) and (5,6,7) and 5 individual elements.

The clustered ranking introduced in the described way is a binary relation on the carrier - the set (1,2,3,...,10). Its structure is as follows. An equivalence relation is given with 7 equivalence classes, namely, (2,3), (5,6,7), and the remaining 5 classes consist of the remaining 5 individual elements. A strict linear order between equivalence classes is then introduced.

The introduced mathematical object is known in the literature as "ranking with connections"(M. Hollender, D. Wolfe), "ordering"(J. Kemeny, J. Snell), "quasi-series"(B.G. Mirkin), "perfect quasi-order"(Y.A. Schrader). Given the difference in terminology, it was found useful to introduce our own term "clustered ranking" since it explicitly names the main elements of the mathematical object being studied - clusters, considered at the stage of coordination of rankings as equivalence classes, and ranking - a strict perfect order between them (in the terminology of Yu. A. Schrader).

The next important concept is inconsistency. It is defined for a quadruple - two clustered rankings on the same medium and two different objects - elements of the same medium. In this case, we will connect two elements from the same cluster with the equal symbol = as equivalent.

Let A And IN- two clustered rankings. We call a pair of objects (a,b) “contradictory” with respect to clustered rankings A and B, if these two elements are ordered differently in A and B, i.e. a< b V A and a > b in B (the first version of inconsistency) or a > b in A and a< b в В (второй вариант противоречивости). Note that according to this definition, a pair of objects ( a,b), equivalent in at least one clustered ranking, cannot be inconsistent: equivalence a =b does not constitute a “contradiction” with either a < b, neither with a > b. This property turns out to be useful in identifying contradictory pairs.

As an example, consider, except A, two more clustered rankings

IN = [{1,2} < { 3,4, 5} < 6 < 7 < 9 < {8, 10}],

C = .

We will call the set of contradictory pairs of objects for two clustered rankings A and B the “core of contradictions” and denote S(A,B). For the three clustered rankings considered above as examples A, IN And WITH, defined on the same carrier (1, 2, 3,..., 10), we have

S(A,B) = [(8, 9)], S(A,C) = [(1, 3), (2,4)],

S(B,C) = [(1, 3), (2, 3), (2, 4), (5, 6), (8,9)].

Both when manually and programmatically finding a kernel, you can look for pairs (1,2), (1,3), (1,4), .... , (1, k), then (2,3), (2,4), ..., (2, k), then (3,4), ..., (3, k), etc., up to the last pair ( k-1, k).

Using the concepts of discrete mathematics, the “core of contradictions” can be depicted count with vertices at points of the support. Wherein contradictory pairs define the edges of this graph. Graph for S(A,B) has only one edge (one connected component from more than one point), for S(A,C) - 2 edges (two connected components from more than one point), for S(B,C) - 5 edges (three connected components from more than one point, namely, (1, 2, 3, 4), (5, 6) and (8, 9)).

Each clustered ranking, like any binary relation, can be specified by the matrix || x(a,b)|| from 0 and 1 order k x k. Wherein x(a,b) = 1 if and only if a< b or a = b. In the first case x(b,a) = 0, and in the second x(b,a) = 1. Moreover, at least one of the numbers x(a,b) And x(b,a) is equal to 1. From the definition of inconsistency of the pair ( a, b) it follows that to find all such pairs it is enough to multiply two matrices element by element || x(a,b)|| and || y(a,b)|| corresponding to two clustered rankings, and select those and only those pairs for which x( a,b)y(a,b) = x(b,a)y(b,a)=0.

The proposed algorithm for reconciling a certain number (two or more) clustered rankings consists of three stages. On the first contradictory pairs stand out objects in all pairs of clustered rankings. At the second stage, clusters of the final clustered ranking are formed (i.e., equivalence classes - connected graph components, corresponding to the union of pairwise contradiction kernels). At the third stage these clusters (equivalence classes) are ordered. To establish the order between the clusters, one object is randomly selected from the first cluster and the second from the second, the order between the clusters is established the same as it would be between the selected objects in any of the clustered rankings under consideration. (If one of the initial clustered rankings has equality, and the other has inequality, then the inequality is used when constructing the final clustered ranking.)

The correctness of such ordering, i.e. its independence from the choice of a particular pair of objects follows from the corresponding theorems proven in the article.

Two objects from different clusters of the matching clustered ranking may turn out to be equivalent in one of the original clustered rankings (i.e., be in the same cluster). In this case, it is necessary to consider the ordering of these objects in some other of the original clustered rankings. If in all the initial clustered rankings the two objects in question were in the same cluster, then it is natural to assume (and this is a clarification to stage 3 of the algorithm) that they are in the same cluster and in the matching clustered ranking.

Result of matching clustered rankings A, IN, WITH,... denote f( A, B, C,...). Then

f(A, B) = ,

f(A, C) = [{1,3}<{2, 4}<6<{5,7}<8<9<10],

f(B, C) = [{1,2,3,4}<{5,6}<7<{8,9}<10],

f(A, B, C) = f(B, C) = [{1,2,3,4} <{5,6}<7<{8, 9}<10].

So, in case f(A, B) only objects 8 and 9 require additional study for the purpose of ordering. In the case f(A, WITH) cluster (5,7) appeared not because there is a contradiction regarding objects 5 and 7, but because in both initial rankings these objects do not differ. In the case f( IN, WITH) four objects 1,2,3,4 merged into one cluster, i.e. The clustered rankings turned out to be so contradictory that the reconciliation procedure did not allow for a sufficiently complete decomposition of the problem of finding the final expert opinion.

Let's consider some properties of matching algorithms.

1. Let D = f(A, IN, C,...). If a in matching clustered ranking D, That a or a=b in each of the original rankings A, IN, C, ..., and in at least one of them a strict inequality holds.

2. The construction of matching clustered rankings can be carried out in stages. In particular, f(A,B,C) = f(f(A,B), f(A,C), f(B,C)). It's clear that the contradiction kernel for a set of clustered rankings is the union of such kernels for all pairs of rankings under consideration.

3. The construction of matching clustered rankings is aimed at identifying the general ordering in the original clustered rankings. However, in this case, some general properties of the original clustered rankings may be lost. Thus, when coordinating rankings B and WITH discussed above, there was no contradiction in the ordering of elements 1 and 2 - in ranking B these objects were included in one cluster, i.e. 1 = 2, while 1<2 в кластеризованной ранжировке WITH. This means that when considering them separately, we can accept the ordering 1<2. Однако в f(B,C) they fell into one cluster, i.e. the possibility of their ordering disappeared. This is due to the behavior of object 3, which "jumped" into WITH to the first place and “carried along into contradiction” the pair (1, 2), forming contradictory pairs with both 1 and 2. In other words, the connected component of the graph corresponding to the core of contradictions is not itself always a complete graph. The missing edges correspond to pairs of type (1, 2), which in themselves are not contradictory, but are “dragged into contradiction” by other pairs.

4. The need to harmonize clustered rankings arises, in particular, when developing a methodology for using expert assessments in the problems of environmental insurance and chemical safety of the biosphere. As already mentioned, a popular method is the ordering by average ranks, in which the final ranking is based on the arithmetic average ranks set by individual experts. However, it is known from measurement theory (see Chapter 2.1) that it is more reasonable to use medians rather than arithmetic means. At the same time, the method of average ranks is very well known and widely used, so it is not advisable to simply discard it. Therefore, it was decided to use both methods simultaneously. The implementation of this solution required the development of a methodology for harmonizing the two indicated clustered rankings.

5. The scope of application of the method under consideration is not limited to expert assessments. It can be used, for example, to compare the quality of mathematical models of the process of liquid evaporation. There were experimental data and calculation results using 8 mathematical models. Models can be compared according to various quality criteria. For example, by the sum of the modules of the relative deviations of the calculated and experimental values. You can also act differently: at each experimental point, order the models by quality, and then obtain uniform estimates using the methods of average ranks and medians. Other methods were also used. Then, methods were used to harmonize the clustered rankings obtained in various ways. As a result, it turned out to be possible to organize models by quality and use this ordering when developing a bank of mathematical models used in problems of chemical safety of the biosphere.

6. The considered method for harmonizing clustered rankings is built in accordance with methodology of stability theory, according to which the result of data processing, invariant with respect to the processing method, corresponds to reality, and the result of calculations, depending on the processing method, reflects the subjectivity of the researcher, and not objective relationships.

Basic mathematical problems of analyzing expert assessments. It is clear that when analyzing expert opinions, a wide variety of statistical methods can be used; describing them means describing almost all applied statistics. Nevertheless, we can highlight the main currently widely used methods of mathematical processing of expert assessments: checking the consistency of expert opinions (or classifying experts if there is no consistency) and averaging the opinions of experts within an agreed group.

Since the answers of experts in many expert survey procedures are not numbers, but objects of a non-numerical nature, such as gradations of qualitative characteristics, rankings, partitions, results of paired comparisons, fuzzy preferences, etc., methods of statistics of objects of a non-numerical nature are useful for their analysis .

Why are experts' answers often non-numerical? The most common answer is that people don't think in numbers. Human thinking uses images, words, but not numbers. Therefore, demanding an answer from an expert in the form of numbers means raping his mind. Even in economics, entrepreneurs make decisions only partially based on numerical calculations. This is evident from the conditional (i.e., determined by arbitrarily accepted agreements, usually formalized in the form of instructions) nature of balance sheet profit, depreciation charges and other economic indicators. Therefore, a phrase like “the company strives to maximize profits” cannot have a strictly defined meaning. It is enough to ask: “Profit maximization - over what period?” And it will immediately become clear that the degree of optimality of the decisions made depends on the planning horizon (at the economic and mathematical level, this subject is discussed in the monograph).

An expert can compare two objects, say which of the two is better (paired comparison method), give them ratings like “good”, “acceptable”, “bad”, order several objects by attractiveness, but usually cannot answer how many times or How much is one object better than another? In other words, the expert's responses are usually measured on an ordinal scale, or are rankings, pairwise comparisons, and other non-numerical objects, but not numbers. A common misconception is that they try to consider the answers of experts as numbers, they are engaged in “digitizing” their opinions, assigning numerical values ​​to these opinions - points, which are then processed using methods of applied statistics as the results of ordinary physical and technical measurements. If “digitization” is arbitrary, the conclusions obtained as a result of data processing may not be relevant to reality. In connection with “digitization,” it is appropriate to recall the classic parable about a man who is looking for his lost keys under a lantern, although he lost them in the bushes. When asked why he does this, he answers: “It’s lighter under the lantern.” This is of course true. But, unfortunately, the chances of finding lost keys under a lamp are very low. So it is with the “digitization” of non-numeric data. It makes it possible to imitate scientific activity, but not the opportunity to find the truth.

Checking the consistency of expert opinions and classifying expert opinions. It is clear that the opinions of different experts vary. It is important to understand how big this difference is. If it is not enough, averaging the opinions of experts will allow us to highlight what all experts have in common, discarding random deviations in one direction or another. If it is large, averaging is a purely formal procedure. So, if we imagine that the experts’ answers evenly cover the surface of the donut, then formal averaging will indicate the center of the donut hole, but not a single expert holds this opinion. From the above, the importance of the problem of checking the consistency of expert opinions is clear.

A number of methods for such verification have been developed. Statistical methods for testing consistency depend on the mathematical nature of the experts' responses. The corresponding statistical theories are quite difficult if the answers are rankings or partitions, and quite simple if the answers are the results of independent pairwise comparisons. This leads to a recommendation for organizing an expert survey: do not try to immediately get a ranking or breakdown from an expert, it is difficult for him to do this, and the available mathematical methods do not allow one to go far in analyzing such data. For example, it is recommended to check the consistency of rankings using the Kendall-Smith rank concordance coefficient. But let's remember what statistical model is used. The null hypothesis is tested, according to which the rankings are independent and uniformly distributed over the set of all rankings. If this hypothesis is accepted, then, of course, we cannot talk about any consistency of expert opinions. What if it is rejected? It's also not possible. For example, there may be two (or more) centers around which expert responses are grouped. The null hypothesis is rejected. But can we really talk about consistency?

It is much easier for an expert to compare only two objects at each step. Let him do pairwise comparisons. Nonparametric theory of paired comparisons (Lucian theory) allows you to solve more complex problems than statistics of rankings or partitions. In particular, instead of the uniform distribution hypothesis, one can consider the homogeneity hypothesis, i.e. instead of the coincidence of all distributions with one fixed (uniform) one, one can check only the coincidence of the distributions of experts’ opinions among themselves, which is naturally interpreted as the consistency of their opinions. Thus, it is possible to get rid of the unnatural assumption of uniformity.

In the absence of agreement between experts, it is natural to divide them into groups of similar opinions. This can be done using various methods of statistics of objects of a non-numerical nature related to cluster analysis, after first introducing the metric into the space of expert opinions. The idea of ​​the American mathematician John Kemeny about the axiomatic introduction of metrics (see below) found numerous successors. However, cluster analysis methods are usually heuristic. In particular, it is impossible from the standpoint of statistical theory to justify the “legality” of combining two clusters into one. There is an important exception - for independent pairwise comparisons (Lucians), methods have been developed that allow testing the possibility of combining clusters as a statistical hypothesis. This is another argument for considering the Lucian theory as the core of mathematical methods of expert assessments.

Finding the final opinion of the expert commission. Let the opinions of the expert commission or some part of it be recognized as agreed. What is the final (average, general) opinion of the commission? According to John Kemeny's idea, one should find the middle opinion as a solution optimization problem. Namely, it is necessary to minimize the total distance from the average candidate to the opinions of experts. The average opinion found in this way is called the “Kemeny median.”

The mathematical difficulty lies in the fact that expert opinions lie in a certain space of objects of a non-numerical nature. The general theory of such averaging has been constructed in a number of works; in particular, it has been shown that, due to the generalization of the law of large numbers, the average opinion with an increase in the number of experts (whose opinions are independent and equally distributed) approaches a certain limit, which is naturally called mathematical expectation(a random element having the same distribution as the experts' answers).

In specific spaces of non-numerical expert opinions, calculating the Kemeny median can be quite difficult. In addition to the properties of space, the role of specific metrics is important. Thus, in the ranking space, when using a metric associated with the Kendall rank correlation coefficient, it is necessary to carry out quite complex calculations, while using a difference indicator based on the Spearman rank correlation coefficient leads to ordering by average ranks.

Binary relations and Kemeny distance. As is known, the binary relation A on a finite set Q = (q 1 , q 2 ,..., q k )- this is a subset Cartesian square Q 2 = ((q m , q n), m,n = 1,2,…,k). At the same time, the couple (q m , q n) included in A if and only if between q m And qn there is a relationship in question.

Recall that each clustered ranking, like any binary relation, can be specified by a square matrix || x(a,b)|| from 0 and 1 order k x k. Wherein x(a b)= 1 if and only if a< b or a = b. In the first case x(b a)= 0, and in the second x(b a)= 1. Moreover, at least one of the numbers x(a b) And x(b,a) equals 1.

Expert methods use, in particular, such binary relations as rankings (orderings, or divisions into groups between which there is a strict order), equivalence relations, tolerance relations (similarity relations). As follows from the above, every binary relation A can be described by the matrix || a(i,j)|| from 0 and 1, and a(i,j)= 1 if and only if qi And qj are in a relationship A, And a(i,j)= 0 otherwise.

Definition. Kemeny distance between binary relations A and B, described by matrices ||a(i,j)|| and || b(i,j)|| accordingly, the number is called

D (A, B) = ∑ │a(i,j) - b(i,j) │,

where the summation is performed over all i,j from 1 to k, those. The Kemeny distance between binary relations is equal to the sum of the moduli of the differences of elements located in the same places in their corresponding matrices.

It is easy to see that the Kemeny distance is the number of mismatched elements in the matrices || a(i,j)|| and || b(i,j)||.

The Kemeny distance is based on some system of axioms. This system of axioms and the derivation from it of the formula for the Kemeny distance between orderings is contained in the book, which played a large role in the development in our country of such a scientific direction as the analysis of non-numerical information. Subsequently, under the influence of Kemeny, various systems of axioms were proposed to obtain distances in certain spaces needed for socio-economic research, for example, in the spaces of sets.

Kemeny median and laws of large numbers. Using the Kemeny distance, the final opinion of the expert committee is found. Let A 1, A 2, A 3,…, A r- answers from p experts, presented in the form of binary relations. To average them, the so-called Kemeny median

Arg min ∑ D (A i ,A) ,

where Arg min - this or those values A, at which the specified sum of Kemeny distances from the experts’ answers to the current variable reaches a minimum A, along which the minimization is carried out. Thus,

∑ D (A i ,A) = D (A 1 ,A) + D (A 2 ,A) + D (A 3 ,A) +…+D(A p,A) .

In addition to the Kemeny median, they use Kemeny average, in which instead of D (A i ,A) costs D 2 (A i ,A) .

The Kemeny median is a special case of determining the empirical mean in spaces of non-numerical nature. The law of large numbers is valid for it, i.e. the empirical average approaches as the number of components increases (i.e. R- number of terms in the sum), to the theoretical average:

Arg min ∑ D (A i ,A)→ Arg min M D (A 1 , A) .

Here M is the symbol of mathematical expectation. It is assumed that the answers R experts A 1, A 2, A 3,…, A r there are grounds to consider as independent identically distributed random elements (i.e., as a random sample) in the corresponding space of an arbitrary nature, for example, in the space of orderings or equivalence relations. Empirical and theoretical averages and the corresponding various versions of the laws of large numbers have been systematically studied in a number of works (see, for example,).

The laws of large numbers show, firstly, that the Kemeny median has sustainability in relation to a slight change in the composition of the expert commission; secondly, with an increase in the number of experts, it approaches a certain limit. It is natural to regard it as true opinion experts, from which each of them deviated somewhat for random reasons.

The law of large numbers considered here is a generalization of the “classical” law of large numbers known in statistics. It is based on a different mathematical basis - optimization theory, while the "classical" law of large numbers uses summation. Orderings and other binary relations cannot be added, so different mathematics must be used.

Calculating the Kemeny median is an integer programming problem. In particular, to find it, various algorithms of discrete mathematics are used, in particular, those based on the branch and bound method. Algorithms based on the idea of ​​random search are also used, since for each binary relation it is not difficult to find many of its neighbors.

Let's look at an example of calculating the Kemeny median. Let a square matrix (of order 9) of pairwise distances be given for a set of binary relations of 9 elements A 1, A 2, A 3,..., A 9(see table 3). Find in this set median for a set of 5 elements ( A 2, A 4, A 5, A 8, A 9}.

Table 3.

Pairwise distance matrix

In accordance with the definition of the Kemeny median, the function

WITH(A) = ∑ D(A i ,A) = D(A 2 ,A)+D(A 4 ,A)+D(A 5 ,A)+D(A 8 ,A)+D(A 9 ,A),

WITH(A 1 ) = D (A 2 ,A 1) + D (A 4 ,A 1) + D (A 5 ,A 1) + D (A 8 ,A 1) + D (A 9 ,A 1) =

= 2 + 1 +7 +3 +11 = 24,

WITH(A 2 ) = D (A 2 ,A 2) + D (A 4 ,A 2) + D (A 5 ,A 2) + D (A 8 ,A 2) + D (A 9 ,A 2) =

= 0 + 6 + 1 + 5 + 1 = 13,

WITH(A 3 ) = D (A 2 ,A 3) + D (A 4 ,A 3) + D (A 5 ,A 3) + D (A 8 ,A 3) + D (A 9 ,A 3) =

= 5 + 2 + 2 + 5 +7 = 21,

WITH(A 4 ) = D (A 2 ,A 4) + D (A 4 ,A 4) + D (A 5 ,A 4) + D (A 8 ,A 4) + D (A 9 ,A 4) =

= 6 + 0 + 5 + 8 + 8 = 27,

WITH(A 5 ) = D (A 2 ,A 5) + D (A 4 ,A 5) + D (A 5 ,A 5) + D (A 8 ,A 5) + D (A 9 ,A 5) =

= 1 + 5 + 0 +3 + 7 = 16,

WITH(A 6 ) = D (A 2 ,A 6) + D (A 4 ,A 6) + D (A 5 ,A 6) + D (A 8 ,A 6) + D (A 9 ,A 6) =

= 3 + 4 + 10 + 1 + 5 = 23,

WITH(A 7 ) = D (A 2 ,A 7) + D (A 4 ,A 7) + D (A 5 ,A 7) + D (A 8 ,A 7) + D (A 9 ,A 7) =

= 2 + 3 +1 + 6 + 3 = 15,

WITH(A 8 ) = D (A 2 ,A 8) + D (A 4 ,A 8) + D (A 5 ,A 8) + D (A 8 ,A 8) + D (A 9 ,A 8) =

= 5 + 8 + 3 + 0 +9 = 25,

WITH(A 9 ) = D (A 2 ,A 9) + D (A 4 ,A 9) + D (A 5 ,A 9) + D (A 8 ,A 9) + D (A 9 ,A 9) =

= 1 + 8 + 7 + 9 + 0 = 25.

Of all the calculated sums, the smallest is 13, and it is achieved when A=A 2, therefore, the Kemeny median is the set ( A 2), consisting of one element A 2.

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