2. DESCRIPTION OF UNCERTAINTIES IN DECISION-MAKING THEORY
2.2. Probabilistic and statistical methods for describing uncertainties in decision theory
2.2.1. Probability theory and mathematical statistics in decision making
How are probability theory and mathematical statistics used? These disciplines are the basis of probabilistic and statistical methods of decision making. To use their mathematical apparatus, it is necessary to express decision-making problems in terms of probabilistic-statistical models. The application of a specific probabilistic-statistical decision-making method consists of three stages:
The transition from economic, managerial, technological reality to an abstract mathematical and statistical scheme, i.e. construction of a probabilistic model of a control system, technological process, decision-making procedure, in particular based on the results of statistical control, etc.
Carrying out calculations and drawing conclusions using purely mathematical means within the framework of a probabilistic model;
Interpretation of mathematical and statistical conclusions in relation to a real situation and making an appropriate decision (for example, on the compliance or non-compliance of product quality with established requirements, the need to adjust the technological process, etc.), in particular, conclusions (on the proportion of defective units of product in a batch, on specific form of laws of distribution of controlled parameters of the technological process, etc.).
Mathematical statistics uses the concepts, methods and results of probability theory. Let's consider the main issues of constructing probabilistic models of decision-making in economic, managerial, technological and other situations. For the active and correct use of regulatory, technical and instructional documents on probabilistic and statistical methods of decision-making, preliminary knowledge is required. Thus, it is necessary to know under what conditions a particular document should be used, what initial information is necessary to have for its selection and application, what decisions should be made based on the results of data processing, etc.
Application examples probability theory and mathematical statistics. Let's consider several examples where probabilistic-statistical models are a good tool for solving management, production, economic, and national economic problems. So, for example, in A.N. Tolstoy’s novel “Walking through Torment” (vol. 1) it is said: “the workshop produces twenty-three percent of rejects, you stick to this figure,” Strukov told Ivan Ilyich.”
The question arises of how to understand these words in the conversation of factory managers, since one unit of production cannot be 23% defective. It can be either good or defective. Strukov probably meant that a large-volume batch contains approximately 23% defective units of production. The question then arises, what does “approximately” mean? Let 30 out of 100 tested units of production turn out to be defective, or out of 1000 - 300, or out of 100,000 - 30,000, etc., should Strukov be accused of lying?
Or another example. The coin used as a lot must be “symmetrical”, i.e. when throwing it, on average, in half the cases the coat of arms should appear, and in half the cases - a hash (tails, number). But what does "on average" mean? If you conduct many series of 10 tosses in each series, then you will often encounter series in which the coin lands as a coat of arms 4 times. For a symmetrical coin, this will happen in 20.5% of runs. And if after 100,000 tosses there are 40,000 coats of arms, can the coin be considered symmetrical? The decision-making procedure is based on probability theory and mathematical statistics.
The example in question may not seem serious enough. However, it is not. Drawing lots is widely used in organizing industrial technical and economic experiments, for example, when processing the results of measuring the quality indicator (friction torque) of bearings depending on various technological factors (the influence of the conservation environment, methods of preparing bearings before measurement, the influence of bearing loads during the measurement process, etc.). P.). Let's say it is necessary to compare the quality of bearings depending on the results of their storage in different preservation oils, i.e. in composition oils A And IN. When planning such an experiment, the question arises which bearings should be placed in the oil of the composition A, and which ones - in the oil composition IN, but in such a way as to avoid subjectivity and ensure the objectivity of the decision made.
The answer to this question can be obtained by drawing lots. A similar example can be given with quality control of any product. To decide whether the controlled batch of products meets or does not meet the established requirements, a sample is selected from it. Based on the results of the sample control, a conclusion is made about the entire batch. In this case, it is very important to avoid subjectivity when forming a sample, that is, it is necessary that each unit of product in the controlled batch has the same probability of being selected for the sample. In production conditions, the selection of product units for the sample is usually carried out not by lot, but by special tables of random numbers or using computer random number sensors.
Similar problems of ensuring objectivity of comparison arise when comparing various schemes for organizing production, remuneration, during tenders and competitions, selecting candidates for vacant positions, etc. Everywhere we need a draw or similar procedures. Let us explain with the example of identifying the strongest and second strongest teams when organizing a tournament according to the Olympic system (the loser is eliminated). Let the stronger team always defeat the weaker one. It is clear that the strongest team will definitely become the champion. The second strongest team will reach the final if and only if it has no games with the future champion before the final. If such a game is planned, the second strongest team will not make it to the final. The one who plans the tournament can either “knock out” the second-strongest team from the tournament ahead of schedule, pitting it against the leader in the first meeting, or provide it with second place by ensuring meetings with weaker teams right up to the final. To avoid subjectivity, a draw is carried out. For an 8-team tournament, the probability that the top two teams will meet in the final is 4/7. Accordingly, with a probability of 3/7, the second strongest team will leave the tournament early.
Any measurement of product units (using a caliper, micrometer, ammeter, etc.) contains errors. To find out whether there are systematic errors, it is necessary to make repeated measurements of a unit of product whose characteristics are known (for example, a standard sample). It should be remembered that in addition to systematic error, there is also random error.
Therefore, the question arises of how to find out from the measurement results whether there is a systematic error. If we only note whether the error obtained during the next measurement is positive or negative, then this task can be reduced to the previous one. Indeed, let’s compare a measurement to throwing a coin, a positive error to the loss of a coat of arms, a negative error to a grid (a zero error with a sufficient number of scale divisions almost never occurs). Then checking for the absence of systematic error is equivalent to checking the symmetry of the coin.
The purpose of these considerations is to reduce the problem of checking the absence of a systematic error to the problem of checking the symmetry of a coin. The above reasoning leads to the so-called “sign criterion” in mathematical statistics.
In the statistical regulation of technological processes, based on the methods of mathematical statistics, rules and plans for statistical process control are developed, aimed at timely detection of problems in technological processes and taking measures to adjust them and prevent the release of products that do not meet established requirements. These measures are aimed at reducing production costs and losses from the supply of low-quality units. During statistical acceptance control, based on the methods of mathematical statistics, quality control plans are developed by analyzing samples from product batches. The difficulty lies in being able to correctly build probabilistic-statistical models of decision-making, on the basis of which the questions posed above can be answered. In mathematical statistics, probabilistic models and methods for testing hypotheses have been developed for this purpose, in particular, hypotheses that the proportion of defective units of production is equal to a certain number p 0, For example, p 0= 0.23 (remember Strukov’s words from the novel by A.N. Tolstoy).
Assessment tasks. In a number of managerial, production, economic, and national economic situations, problems of a different type arise - problems of assessing the characteristics and parameters of probability distributions.
Let's look at an example. Let a batch of N electric lamps From this batch, a sample of n electric lamps A number of natural questions arise. How to determine the average service life of electric lamps based on the test results of sample elements and with what accuracy can this characteristic be assessed? How will the accuracy change if we take a larger sample? At what number of hours T it can be guaranteed that at least 90% of electric lamps will last T and more hours?
Let us assume that when testing a sample size n electric lamps turned out to be defective X electric lamps Then the following questions arise. What boundaries can be specified for a number? D defective light bulbs in a batch, for the level of defectiveness D/ N and so on.?
Or, when statistically analyzing the accuracy and stability of technological processes, it is necessary to evaluate such quality indicators as the average value of the controlled parameter and the degree of its scatter in the process under consideration. According to probability theory, it is advisable to use its mathematical expectation as the average value of a random variable, and dispersion, standard deviation or coefficient of variation as a statistical characteristic of the spread. This raises the question: how to estimate these statistical characteristics from sample data and with what accuracy can this be done? There are many similar examples that can be given. Here it was important to show how probability theory and mathematical statistics can be used in production management when making decisions in the field of statistical management of product quality.
What is "mathematical statistics"? Mathematical statistics is understood as “a branch of mathematics devoted to mathematical methods of collecting, systematizing, processing and interpreting statistical data, as well as using them for scientific or practical conclusions. The rules and procedures of mathematical statistics are based on probability theory, which allows us to evaluate the accuracy and reliability of the conclusions obtained in each problem based on the available statistical material.” In this case, statistical data refers to information about the number of objects in any more or less extensive collection that have certain characteristics.
Based on the type of problems being solved, mathematical statistics is usually divided into three sections: data description, estimation, and hypothesis testing.
Based on the type of statistical data processed, mathematical statistics is divided into four areas:
Univariate statistics (statistics of random variables), in which the result of an observation is described by a real number;
Multivariate statistical analysis, where the result of observing an object is described by several numbers (vector);
Statistics of random processes and time series, where the result of observation is a function;
Statistics of objects of a non-numerical nature, in which the result of an observation is of a non-numerical nature, for example, it is a set (a geometric figure), an ordering, or obtained as a result of a measurement based on a qualitative criterion.
Historically, some areas of statistics of objects of a non-numerical nature (in particular, problems of estimating the proportion of defects and testing hypotheses about it) and one-dimensional statistics were the first to appear. The mathematical apparatus is simpler for them, so their example is usually used to demonstrate the basic ideas of mathematical statistics.
Only those data processing methods, i.e. mathematical statistics are evidence-based, which are based on probabilistic models of relevant real phenomena and processes. We are talking about models of consumer behavior, the occurrence of risks, the functioning of technological equipment, obtaining experimental results, the course of a disease, etc. A probabilistic model of a real phenomenon should be considered constructed if the quantities under consideration and the connections between them are expressed in terms of probability theory. Correspondence to the probabilistic model of reality, i.e. its adequacy is substantiated, in particular, using statistical methods for testing hypotheses.
Non-probabilistic methods of data processing are exploratory; they can only be used in preliminary data analysis, since they do not make it possible to assess the accuracy and reliability of conclusions obtained on the basis of limited statistical material.
Probabilistic and statistical methods are applicable wherever it is possible to construct and justify a probabilistic model of a phenomenon or process. Their use is mandatory when conclusions drawn from sample data are transferred to the entire population (for example, from a sample to an entire batch of products).
In specific areas of application, both probabilistic and statistical methods of general application and specific ones are used. For example, in the section of production management devoted to statistical methods of product quality management, applied mathematical statistics (including design of experiments) are used. Using its methods, statistical analysis of the accuracy and stability of technological processes and statistical quality assessment are carried out. Specific methods include methods of statistical acceptance control of product quality, statistical regulation of technological processes, reliability assessment and control, etc.
Applied probabilistic and statistical disciplines such as reliability theory and queuing theory are widely used. The content of the first of them is clear from the name, the second deals with the study of systems such as a telephone exchange, which receives calls at random times - the requirements of subscribers dialing numbers on their telephone sets. The duration of servicing these requirements, i.e. the duration of conversations is also modeled by random variables. A great contribution to the development of these disciplines was made by Corresponding Member of the USSR Academy of Sciences A.Ya. Khinchin (1894-1959), Academician of the Academy of Sciences of the Ukrainian SSR B.V. Gnedenko (1912-1995) and other domestic scientists.
Briefly about the history of mathematical statistics. Mathematical statistics as a science begins with the works of the famous German mathematician Carl Friedrich Gauss (1777-1855), who, based on probability theory, investigated and justified the least squares method, created by him in 1795 and used for processing astronomical data (in order to clarify the orbit of a small planet Ceres). One of the most popular probability distributions, the normal one, is often named after him, and in the theory of random processes the main object of study is Gaussian processes.
At the end of the 19th century. - early 20th century Major contributions to mathematical statistics were made by English researchers, primarily K. Pearson (1857-1936) and R. A. Fisher (1890-1962). In particular, Pearson developed the chi-square test for testing statistical hypotheses, and Fisher developed analysis of variance, the theory of experimental design, and the maximum likelihood method for estimating parameters.
In the 30s of the twentieth century. Pole Jerzy Neumann (1894-1977) and Englishman E. Pearson developed the general theory of testing statistical hypotheses, and Soviet mathematicians Academician A.N. Kolmogorov (1903-1987) and corresponding member of the USSR Academy of Sciences N.V. Smirnov (1900-1966) laid the foundations of nonparametric statistics. In the forties of the twentieth century. Romanian A. Wald (1902-1950) built the theory of sequential statistical analysis.
Mathematical statistics is developing rapidly at the present time. Thus, over the past 40 years, four fundamentally new areas of research can be distinguished:
Development and implementation of mathematical methods for planning experiments;
Development of statistics of objects of non-numerical nature as an independent direction in applied mathematical statistics;
Development of statistical methods that are resistant to small deviations from the probabilistic model used;
Widespread development of work on the creation of computer software packages designed for statistical data analysis.
Probabilistic-statistical methods and optimization. The idea of optimization permeates modern applied mathematical statistics and other statistical methods. Namely, methods of planning experiments, statistical acceptance control, statistical regulation of technological processes, etc. On the other hand, optimization formulations in decision-making theory, for example, the applied theory of optimization of product quality and standard requirements, provide for the widespread use of probabilistic statistical methods, primarily applied mathematical statistics.
In production management, in particular, when optimizing product quality and standard requirements, it is especially important to apply statistical methods at the initial stage of the product life cycle, i.e. at the stage of research preparation of experimental design developments (development of promising product requirements, preliminary design, technical specifications for experimental design development). This is due to the limited information available at the initial stage of the product life cycle and the need to predict the technical capabilities and economic situation for the future. Statistical methods should be used at all stages of solving an optimization problem - when scaling variables, developing mathematical models of the functioning of products and systems, conducting technical and economic experiments, etc.
In optimization problems, including optimization of product quality and standard requirements, all areas of statistics are used. Namely, statistics of random variables, multivariate statistical analysis, statistics of random processes and time series, statistics of objects of non-numerical nature. It is advisable to select a statistical method for analyzing specific data in accordance with the recommendations.
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Methods for making decisions under risk conditions are also developed and justified within the framework of the so-called theory of statistical decisions. Statistical decision theory is the theory of making statistical observations, processing these observations, and using them. As is known, the task of economic research is to understand the nature of an economic object and to reveal the mechanism of the relationship between its most important variables. This understanding allows us to develop and implement the necessary measures to manage this object, or economic policy. To do this, we need methods adequate to the task that take into account the nature and specificity of economic data that serves as the basis for qualitative and quantitative statements about the economic object or phenomenon being studied.
Any economic data represents quantitative characteristics of any economic objects. They are formed under the influence of many factors, not all of which are accessible to external control. Uncontrollable factors can take on random values from some set of values and thereby cause the data they define to be random. The stochastic nature of economic data necessitates the use of special statistical methods adequate to them for their analysis and processing.
Quantitative assessment of business risk, regardless of the content of a specific task, is possible, as a rule, using the methods of mathematical statistics. The main tools of this assessment method are dispersion, standard deviation, and coefficient of variation.
Typical designs based on measures of variability or probability of risk conditions are widely used in applications. Thus, financial risks caused by fluctuations in the result around the expected value, for example, efficiency, are assessed using dispersion or the expected absolute deviation from the average. In capital management problems, a common measure of the degree of risk is the probability of losses or loss of income compared to the predicted option.
To assess the magnitude of the risk (degree of risk), we will focus on the following criteria:
- 1) average expected value;
- 2) fluctuation (variability) of the possible result.
For statistical sampling
Where Xj - expected value for each observation case (/" = 1, 2,...), l, - number of observation cases (frequency) value l:, x=E - average expected value, st - variance,
V - coefficient of variation, we have:
Let's consider the problem of assessing risk under business contracts. Interproduct LLC decides to enter into an agreement for the supply of food products from one of three bases. Having collected data on the terms of payment for goods by these bases (Table 6.7), it is necessary, after assessing the risk, to select the base that pays for the goods in the shortest possible time when concluding a contract for the supply of products.
Table 6.7
|
Payment terms in days |
Number of cases observed P |
HP |
(x-x) |
(x-x ) 2 |
(x-x) 2 p |
|
For the first base, based on formulas (6.4.1):
For second base
For third base
The coefficient of variation for the first base is the smallest, which indicates the advisability of concluding a product supply agreement with this base.
The considered examples show that risk has a mathematically expressed probability of loss, which is based on statistical data and can be calculated with a fairly high degree of accuracy. When choosing the most acceptable solution, the rule of optimal probability of the result was used, which consists in choosing from among the possible solutions the one at which the probability of the result is acceptable for the entrepreneur.
In practice, the application of the rule of optimal probability of a result is usually combined with the rule of optimal variability of the result.
As is known, the variability of indicators is expressed by their dispersion, standard deviation and coefficient of variation. The essence of the rule of optimal fluctuation of the result is that from the possible solutions, the one is selected in which the probabilities of winning and losing for the same risky investment of capital have a small gap, i.e. the smallest amount of variance, the standard deviation of the variation. In the problems under consideration, the choice of optimal solutions was made using these two rules.
How are the approaches, ideas and results of probability theory and mathematical statistics used in decision making?
The basis is a probabilistic model of a real phenomenon or process, i.e. a mathematical model in which objective relationships are expressed in terms of probability theory. Probabilities are used primarily to describe the uncertainties that must be taken into account when making decisions. This refers to both undesirable opportunities (risks) and attractive ones (“lucky chance”). Sometimes randomness is deliberately introduced into a situation, for example, when drawing lots, randomly selecting units for control, conducting lotteries or conducting consumer surveys.
Probability theory allows one probabilities to be used to calculate others of interest to the researcher. For example, using the probability of getting a coat of arms, you can calculate the probability that in 10 coin tosses you will get at least 3 coats of arms. Such a calculation is based on a probabilistic model, according to which coin tosses are described by a pattern of independent trials; in addition, the coat of arms and the hash marks are equally possible, and therefore the probability of each of these events is equal to ½. A more complex model is one that considers checking the quality of a unit of production instead of tossing a coin. The corresponding probabilistic model is based on the assumption that the quality control of various units of production is described by an independent testing scheme. Unlike the coin toss model, it is necessary to introduce a new parameter - the probability p that a unit of production is defective. The model will be fully described if we assume that all units of production have the same probability of being defective. If the last assumption is incorrect, then the number of model parameters increases. For example, you can assume that each unit of production has its own probability of being defective.
Let us discuss a quality control model with a probability of defectiveness p common for all units of production. In order to “get to the number” when analyzing the model, it is necessary to replace p with some specific value. To do this, it is necessary to move beyond the probabilistic model and turn to data obtained during quality control.
Mathematical statistics solves the inverse problem in relation to probability theory. Its goal is, based on the results of observations (measurements, analyses, tests, experiments), to obtain conclusions about the probabilities underlying the probabilistic model. For example, based on the frequency of occurrence of defective products during inspection, conclusions can be drawn about the probability of defectiveness (see Bernoulli's theorem above).
Based on Chebyshev’s inequality, conclusions were drawn about the correspondence of the frequency of occurrence of defective products to the hypothesis that the probability of defectiveness takes a certain value.
Thus, the application of mathematical statistics is based on a probabilistic model of a phenomenon or process. Two parallel series of concepts are used - those related to theory (probabilistic model) and those related to practice (sampling of observation results). For example, the theoretical probability corresponds to the frequency found from the sample. The mathematical expectation (theoretical series) corresponds to the sample arithmetic mean (practical series). As a rule, sample characteristics are estimates of theoretical ones. At the same time, quantities related to the theoretical series “are in the heads of researchers”, relate to the world of ideas (according to the ancient Greek philosopher Plato), and are not available for direct measurement. Researchers have only sample data with which they try to establish the properties of a theoretical probabilistic model that interest them.
Why do we need a probabilistic model? The fact is that only with its help can the properties established from the analysis of a specific sample be transferred to other samples, as well as to the entire so-called general population. The term "population" is used when referring to a large but finite collection of units being studied. For example, about the totality of all residents of Russia or the totality of all consumers of instant coffee in Moscow. The goal of marketing or sociological surveys is to transfer statements obtained from a sample of hundreds or thousands of people to populations of several million people. In quality control, a batch of products acts as a general population.
To transfer conclusions from a sample to a larger population requires some assumptions about the relationship of the sample characteristics with the characteristics of this larger population. These assumptions are based on an appropriate probabilistic model.
Of course, it is possible to process sample data without using one or another probabilistic model. For example, you can calculate a sample arithmetic mean, count the frequency of fulfillment of certain conditions, etc. However, the calculation results will relate only to a specific sample; transferring the conclusions obtained with their help to any other population is incorrect. This activity is sometimes called “data analysis.” Compared to probabilistic-statistical methods, data analysis has limited educational value.
So, the use of probabilistic models based on estimation and testing of hypotheses using sample characteristics is the essence of probabilistic-statistical methods of decision making.
We emphasize that the logic of using sample characteristics for making decisions based on theoretical models involves the simultaneous use of two parallel series of concepts, one of which corresponds to probabilistic models, and the second to sample data. Unfortunately, in a number of literary sources, usually outdated or written in a recipe spirit, no distinction is made between sample and theoretical characteristics, which leads readers to confusion and errors in the practical use of statistical methods.
Give the concept of statistical decisions for one diagnostic parameter and for making a decision in the presence of a zone of uncertainty. Explain the decision-making process in various situations. What is the connection between decision boundaries and the probabilities of errors of the first and second types? The methods under consideration are statistical....
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Lecture 7
Subject. METHODS OF STATISTICAL SOLUTIONS
Target. Give the concept of statistical decisions for one diagnostic parameter and for making a decision in the presence of a zone of uncertainty.
Educational. Explain the decision-making process in various situations.
Developmental. Develop logical thinking and a natural - scientific worldview.
Educational . Cultivate interest in scientific achievements and discoveries in the telecommunications industry.
Interdisciplinary connections:
Supporting: computer science, mathematics, computer technology and MP, programming systems.
Provided: Internship
Methodological support and equipment:
Methodological development for the lesson.
Syllabus.
Training program
Working programm.
Safety briefing.
Technical teaching aids: personal computer.
Providing jobs:
Workbooks
Progress of the lecture.
Organizing time.
Analysis and checking of homework
Answer the questions:
- What allows you to determine Bayes formula?
- What are the basics of Bayes' method?Give the formula. Give a definition of the exact meaning of all quantities included in this formula.
- What does it mean thatimplementation of a certain set of features K* is determining?
- Explain the principle of formationdiagnostic matrix.
- What does it mean deciding rule of acceptance?
- Define the method of sequential analysis.
- What is the relationship between decision boundaries and the probabilities of errors of the first and second types?
Lecture outline
The methods considered are statistical. In statistical decision methods, the decision rule is selected based on certain optimality conditions, for example, the minimum risk condition. Originating in mathematical statistics as methods for testing statistical hypotheses (the work of Neyman and Pearson), the methods under consideration have found wide application in radar (detection of signals against a background of interference), radio engineering, general communication theory and other fields. Statistical solution methods are successfully used in technical diagnostics problems.
STATISTICAL SOLUTIONS FOR ONE DIAGNOSTIC PARAMETER
If the state of the system is characterized by one parameter, then the system has a one-dimensional feature space. The division is made into two classes (differential diagnosis or dichotomy(bifurcation, sequential division into two parts that are not interconnected.) ).
Fig.1 Statistical probability density distributions of diagnostic parameter x for serviceable D 1 and defective D 2 states
It is important that areas of serviceable D 1 and defective D 2 states intersect and therefore it is fundamentally impossible to choose the value of x 0, at which there was no would be wrong decisions.The task is to choose x 0 was in some sense optimal, for example, it gave the least number of erroneous decisions.
False alarm and missed target (defect).These previously encountered terms are clearly related to radar technology, but they are easily interpreted in diagnostic tasks.
A false alarm is calledthe case when a decision is made about the presence of a defect, but in reality the system is in good condition (instead of D 1 is accepted as D 2 ).
Missing a target (defect)making a decision about a working condition, while the system contains a defect (instead of D 2 is accepted as D 1 ).
In control theory, these errors are calledsupplier risk and customer risk. It is obvious that these two types of errors can have different consequences or different goals.
The probability of a false alarm is equal to the probability of two events occurring: the presence of a serviceable state and the value x > x 0 .
Medium risk. The probability of making an erroneous decision consists of the probabilities of a false alarm and missing a defect (mathematical expectation) of risk.
Of course, the cost of an error is relative, but it must take into account the expected consequences of a false alarm and missing a defect. In reliability problems, the cost of missing a defect is usually significantly greater than the cost of a false alarm.
Minimum Risk Method. The probability of making an erroneous decision is defined as minimizing the extremum point of the average risk of erroneous decisions at a maximum likelihood, i.e. the minimum risk of an event occurring is calculated at availability of information about as many similar events as possible.
rice. 2. Extremum points of the average risk of erroneous decisions
Rice. 3. Extremum points for double-humped distributions
The ratio of the probability densities of the distribution of x under two states is called the likelihood ratio.
Let us remember that the diagnosis D 1 corresponds to good condition, D 2 defective condition of the object; WITH 21 cost of a false alarm, C 12 cost of missing the goal (the first index accepted state, the second valid); WITH 11 < 0, С 22 < 0 цены правильных решений (условные выигрыши). В большинстве практических задач условные выигрыши (поощрения) для правильных решений не вводятся.
It is often convenient to consider not the likelihood ratio, but the logarithm of this ratio. This does not change the result, since the logarithmic function increases monotonically with its argument. The calculation for normal and some other distributions when using the logarithm of the likelihood ratio turns out to be somewhat simpler. The minimum risk condition can be obtained from other considerations that will turn out to be important later.
Method of the minimum number of erroneous decisions.
Probability of an erroneous decision for a decision rule
In reliability problems, the method under consideration often gives “careless decisions”, since the consequences of erroneous decisions differ significantly from each other. Typically, the cost of missing a defect is significantly higher than the cost of a false alarm. If the indicated costs are approximately the same (for defects with limited consequences, for some control tasks, etc.), then the use of the method is completely justified.
The minimax method is intendedfor a situation where there is no preliminary statistical information about the likelihood of diagnoses D 1 and D 2 . The “worst case” is considered, i.e. the least favorable values of P 1 and P 2 , leading to the greatest value (maximum) of risk.
It can be shown for unimodal distributions that the risk value becomes minimax (i.e., the minimum among the maximum values caused by the “unfavorable” value Pi ). Note that for P 1 = 0 and P 1 = 1 there is no risk of making an erroneous decision, since the situation has no uncertainty. At P 1 = 0 (all products are faulty) leaks x 0 → -oo and all objects are indeed recognized as faulty; at P 1 = 1 and P 2 = 0 x 0 → +оо and in accordance with the existing situation, all objects are classified as serviceable.
For intermediate values 0< Pi < 1 риск возрастает и при P 1= P 1* becomes the maximum. The method under consideration is used to select the value x 0 in such a way that for the least favorable values Pi losses associated with erroneous decisions would be minimal.
rice . 4. Determination of the limit value of a diagnostic parameter using the minimax method
NeymanPearson method. As already indicated, estimates of the cost of errors are often unknown and their reliable determination is associated with great difficulty. At the same time, it is clear that in all s l u In teas, it is desirable, at a certain (acceptable) level of one of the errors, to minimize the value of the other. Here the center of the problem shifts to a reasonable choice of an acceptable level errors with using previous experience or intuitive considerations.
The NeymanPearson method minimizes the probability of missing a target at a given acceptable level of false alarm probability.Thus, the probability of a false alarm
where A is the specified acceptable level of probability of a false alarm; R 1 probability of good condition.
Note that usually This the condition is referred to as the conditional probability of a false alarm (factor P 1 absent). In technical diagnostic tasks, the values of P 1 and P 2 in most cases are known from statistical data.
Table 1 Example - Calculation results using statistical solution methods
|
No. |
Method |
Limit value |
Probability of false alarm |
Probability of missing a defect |
Medium risk |
|
|
Minimum Risk Method |
7,46 |
0,0984 |
0,0065 |
0,229 |
||
|
Minimum number of errors method |
9,79 |
0,0074 |
0,0229 |
0,467 |
||
|
Minimax method |
Basic option |
5,71 |
0,3235 |
0,0018 |
0,360 |
|
|
Option 2 |
7,80 |
0,0727 |
0,0081 |
0,234 |
||
|
NeymanPearson method |
7,44 |
0,1000 |
0,0064 |
0,230 |
||
|
Maximum likelihood method |
8,14 |
0,0524 |
0,0098 |
0,249 |
||
From the comparison it is clear that the method of the minimum number of errors gives an unacceptable solution, since the costs of errors are significantly different. The limit value of this method leads to a significant probability of missing a defect. The minimax method in the main version requires a very large decommissioning of the devices under study (approximately 32%), since it is based on the least favorable case (the probability of a faulty state P 2 = 0.39). The use of the method may be justified if there are no even indirect estimates of the probability of a faulty state. In the example under consideration, satisfactory results are obtained using the minimum risk method.
- STATISTICAL SOLUTIONS IN THE PRESENCE OF A ZONE OF UNCERTAINTY AND OTHER GENERALIZATIONS
Decision rule in the presence of a zone of uncertainty.
In some cases, when high reliability of recognition is required (high cost of errors in missing a target and false alarms), it is advisable to introduce a zone of uncertainty (zone of refusal of recognition). The decision rule will be as follows
at refusal of recognition.
Of course, failure to recognize is an undesirable event. It indicates that the available information is not enough to make a decision and additional information is needed.
rice. 5. Statistical solutions in the presence of a zone of uncertainty
Determination of average risk. The value of the average risk in the presence of a zone of refusal of recognition can be expressed by the following equality
where C o the cost of refusing recognition.
Note that C o > 0, otherwise the task loses its meaning (“reward” for failure to recognize). In the same way C 11 < 0, С 22 < 0, так как правильные решения не должны «штрафоваться».
Minimum risk method in the presence of a zone of uncertainty. Let us determine the boundaries of the decision-making area based on the minimum average risk.
If you don't encourage good decisions (C 11 = 0, C 22 = 0) and do not pay for refusing recognition (C 0 = 0), then the region of uncertainty will occupy the entire region of parameter change.
The presence of a zone of uncertainty makes it possible to ensure specified error levels by refusing recognition in “doubtful” cases
Statistical solutions for multiple states.Cases were considered above when statistical decisions were made d To distinguish between two states (dichotomy). In principle, this procedure makes it possible to separate n states, each time combining the results for the state D 1 and D 2. Here under D 1 refers to any states that meet the condition “not D 2 " However, in some cases it is of interest to consider the question in a direct formulation: statistical solutions for classification n states.
Above, we considered cases when the state of the system (product) was characterized by one parameter x and the corresponding (one-dimensional) distribution. The system state is characterized by diagnostic parameters x 1 x 2, ..., x n or vector x:
x= (x 1 x 2,...,x n).
M Minimum risk method.
The methods of minimum risk and its special cases (the method of the minimum number of erroneous decisions, the maximum likelihood method) are most easily generalized to multidimensional systems. In cases where the statistical solution method requires determining the boundaries of the decision area, the calculation side of the problem becomes significantly more complicated (Nayman-Pearson and minimax methods).
Homework: § notes.
Fixing the material:
Answer the questions:
- What is a false alarm?
- What does missing a target (defect) mean?
- Give an explanationsupplier's risk and customer's risk.
- Give the formula for the method of the minimum number of erroneous decisions. Define a careless decision.
- For what cases is the minimax method intended?
- NeymanPearson method. Explain its principle.
- For what purposes is the zone of uncertainty used?
Literature:
Amrenov S. A. “Methods for monitoring and diagnostics of communication systems and networks” LECTURE NOTES -: Astana, Kazakh State Agrotechnical University, 2005.
I.G. Baklanov Testing and diagnostics of communication systems. - M.: Eco-Trends, 2001.
Birger I. A. Technical diagnostics. M.: “Mechanical Engineering”, 1978.240, p., ill.
ARIPOV M.N., DZHURAEV R.KH., DZHABBAROV S.YU.“TECHNICAL DIAGNOSTICS OF DIGITAL SYSTEMS” - Tashkent, TEIS, 2005
Platonov Yu. M., Utkin Yu. G.Diagnostics, repair and prevention of personal computers. -M.: Hotline - Telecom, 2003.-312 p.: ill.
M.E.Bushueva, V.V.BelyakovDiagnostics of complex technical systems Proceedings of the 1st meeting on the NATO project SfP-973799 Semiconductors . Nizhny Novgorod, 2001
Malyshenko Yu.V. TECHNICAL DIAGNOSTICS part I lecture notes
Platonov Yu. M., Utkin Yu. G.Diagnostics of computer freezes and malfunctions/Series “Technomir”. Rostov-on-Don: “Phoenix”, 2001. 320 p.
PAGE \* MERGEFORMAT 2
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METHODS OF MAKING MANAGEMENT DECISIONS
Areas of training
080200.62 "Management"
is the same for all forms of education
Graduate qualification (degree)
Bachelor
Chelyabinsk
Methods for making management decisions: Work program of the academic discipline (module) / Yu.V. Podpovetnaya. – Chelyabinsk: Private Educational Institution of Higher Professional Education “South Ural Institute of Management and Economics”, 2014. – 78 p.
Methods of making management decisions: The work program of the academic discipline (module) in the direction 080200.62 “Management” is the same for all forms of training. The program is compiled in accordance with the requirements of the Federal State Educational Standard for Higher Professional Education, taking into account the recommendations and PropOPOP of Higher Education in the direction and profile of training.
The program was approved at a meeting of the Educational and Methodological Council on August 18, 2014, protocol No. 1.
The program was approved at a meeting of the academic council on August 18, 2014, protocol No. 1.
Reviewer: Lysenko Yu.V. – Doctor of Economics, Professor, Head. Department of Economics and Enterprise Management of the Chelyabinsk Institute (branch) of the Federal State Budgetary Educational Institution of Higher Professional Education "REU named after G.V. Plekhanov"
Krasnoyartseva E.G. - Director of the private educational institution "Center for Business Education of the South Ural Chamber of Commerce and Industry"
© Publishing house of the Private Educational Institution of Higher Professional Education "South Ural Institute of Management and Economics", 2014
I Introduction………………………………………………………………………………………...4
II Thematic planning……………………………………………………...8
IV Assessment tools for ongoing monitoring of academic performance, intermediate certification based on the results of mastering the discipline and educational and methodological support for students’ independent work………………..…………………………………………………….38
V Educational, methodological and information support of the discipline ...........76
VI Logistical support of discipline………………………...78
I INTRODUCTION
The work program of the academic discipline (module) “Methods of making management decisions” is intended for the implementation of the Federal State Standard of Higher Professional Education in the direction 080200.62 “Management” and is uniform for all forms of education.
1 Purpose and objectives of the discipline
The purpose of studying this discipline is:
Formation of theoretical knowledge about mathematical, statistical and quantitative methods for developing, making and implementing management decisions;
Deepening the knowledge used for research and analysis of economic objects, developing theoretically based economic and management decisions;
Deepening knowledge in the field of theory and methods for finding the best solutions, both in conditions of certainty and in conditions of uncertainty and risk;
Formation of practical skills in the effective use of methods and procedures for selection and decision-making to perform economic analysis and find the best solution to a given problem.
2 Entrance requirements and the place of the discipline in the structure of the undergraduate OPOP
The discipline “Methods of making managerial decisions” belongs to the basic part of the mathematical and natural science cycle (B2.B3).
The discipline is based on the knowledge, skills and competencies of the student obtained from studying the following academic disciplines: “Mathematics”, “Innovative Management”.
The knowledge and skills obtained in the process of studying the discipline “Methods of making managerial decisions” can be used in studying the disciplines of the basic part of the professional cycle: “Marketing Research”, “Methods and Models in Economics”.
3 Requirements for the results of mastering the discipline “Methods of making management decisions”
The process of studying the discipline is aimed at developing the following competencies presented in the table.
Table - Structure of competencies formed as a result of studying the discipline
| Competency code | Name of competency | Characteristics of competence |
| OK-15 | master methods of quantitative analysis and modeling, theoretical and experimental research; | know/understand: be able to: own: |
| OK-16 | understanding the role and importance of information and information technologies in the development of modern society and economic knowledge; | As a result, the student must: know/understand: - basic concepts and tools of algebra and geometry, mathematical analysis, probability theory, mathematical and socio-economic statistics; - basic mathematical models of decision making; be able to: - solve standard mathematical problems used in making management decisions; - use mathematical language and mathematical symbols when constructing organizational and management models; - process empirical and experimental data; own: mathematical, statistical and quantitative methods for solving typical organizational and management problems. |
| OK-17 | master the basic methods, methods and means of obtaining, storing, processing information, skills in working with a computer as a means of information management; | As a result, the student must: know/understand: - basic concepts and tools of algebra and geometry, mathematical analysis, probability theory, mathematical and socio-economic statistics; - basic mathematical models of decision making; be able to: - solve standard mathematical problems used in making management decisions; - use mathematical language and mathematical symbols when constructing organizational and management models; - process empirical and experimental data; own: mathematical, statistical and quantitative methods for solving typical organizational and management problems. |
| OK-18 | ability to work with information in global computer networks and corporate information systems. | As a result, the student must: know/understand: - basic concepts and tools of algebra and geometry, mathematical analysis, probability theory, mathematical and socio-economic statistics; - basic mathematical models of decision making; be able to: - solve standard mathematical problems used in making management decisions; - use mathematical language and mathematical symbols when constructing organizational and management models; - process empirical and experimental data; own: mathematical, statistical and quantitative methods for solving typical organizational and management problems. |
As a result of studying the discipline, the student must:
know/understand:
Basic concepts and tools of algebra and geometry, mathematical analysis, probability theory, mathematical and socio-economic statistics;
Basic mathematical models of decision making;
be able to:
Solve typical mathematical problems used in making management decisions;
Use mathematical language and mathematical symbols when constructing organizational and management models;
Process empirical and experimental data;
own:
Using mathematical, statistical and quantitative methods for solving typical organizational and management problems.
II THEMATIC PLANNING
SET 2011
DIRECTION: "Management"
DURATION OF STUDY: 4 years
Full-time form of education
| Lectures, hour. | Practical lessons, hour. | Laboratory classes, hour. | Seminars | Coursework, hour. | Total, an hour. | ||
| Topic 4.4 Expert assessments | |||||||
| Topic 5.2 Game models of PR | |||||||
| Topic 5.3 Positional games | |||||||
| Exam | |||||||
| TOTAL |
Laboratory workshop
| No. | Labor intensity (hours) | ||
| Topic 1.3 Target orientation of management decisions | Laboratory work No. 1. Search for optimal solutions. Application of optimization in PR support systems | ||
| Topic 2.2 Main types of decision theory models | |||
| Topic 3.3 Features of measuring preferences | |||
| Topic 4.2 Method of paired comparisons | |||
| Topic 4.4 Expert assessments | |||
| Topic 5.2 Game models of PR | |||
| Topic 5.4 Optimality in the form of equilibrium | |||
| Topic 6.3 Statistical games with a single experiment |
Recruitment 2011
DIRECTION: "Management"
FORM OF STUDY: correspondence
1 Scope of discipline and types of academic work
2 Sections and topics of the discipline and types of classes
| Name of sections and topics of the discipline | Lectures, hour. | Practical lessons, hour. | Laboratory classes, hour. | Seminars | Independent work, hour. | Coursework, hour. | Total, an hour. |
| Section 1 Management as a process of making management decisions | |||||||
| Topic 1.1 Functions and properties of management decisions | |||||||
| Topic 1.2 Management decision-making process | |||||||
| Topic 1.3 Target orientation of management decisions | |||||||
| Section 2 Models and simulation in decision theory | |||||||
| Topic 2.1 Modeling and analysis of action alternatives | |||||||
| Topic 2.2 Main types of decision theory models | |||||||
| Section 3 Decision making under multi-criteria conditions | |||||||
| Topic 3.1 Non-criteria and criteria methods | |||||||
| Topic 3.2 Multicriteria models | |||||||
| Topic 3.3 Features of measuring preferences | |||||||
| Section 4 Ordering of alternatives based on taking into account the preferences of experts | |||||||
| Topic 4.1 Measurements, comparisons and consistency | |||||||
| Topic 4.2 Method of paired comparisons | |||||||
| Topic 4.3 Principles of group selection | |||||||
| Topic 4.4 Expert assessments | |||||||
| Section 5 Decision making under conditions of uncertainty and conflict | |||||||
| Topic 5.1 Mathematical model of the PR problem under conditions of uncertainty and conflict | |||||||
| Topic 5.2 Game models of PR | |||||||
| Topic 5.3 Positional games | |||||||
| Topic 5.4 Optimality in the form of equilibrium | |||||||
| Section 6 Decision making under risk conditions | |||||||
| Topic 6.1 Theory of statistical decisions | |||||||
| Topic 6.2 Finding optimal solutions under conditions of risk and uncertainty | |||||||
| Topic 6.3 Statistical games with a single experiment | |||||||
| Section 7 Decision making under fuzzy conditions | |||||||
| Topic 7.1 Compositional models of PR | |||||||
| Topic 7.2 Classification models of PR | |||||||
| Exam | |||||||
| TOTAL |
Laboratory workshop
| No. | Module (section) number of the discipline | Name of laboratory work | Labor intensity (hours) |
| Topic 2.2 Main types of decision theory models | Laboratory work No. 2. Decision making based on economic and mathematical models, queuing theory models, inventory management models, linear programming models | ||
| Topic 4.2 Method of paired comparisons | Laboratory work No. 4. Method of paired comparisons. Ordering of alternatives based on pairwise comparisons and taking into account expert preferences | ||
| Topic 5.2 Game models of PR | Laboratory work No. 6. Construction of the game matrix. Reducing a zero-sum game to a linear programming problem and finding its solution | ||
| Topic 6.3 Statistical games with a single experiment | Laboratory work No. 8. Choice of strategies in a game with an experiment. Using posterior probabilities |
DIRECTION: "Management"
DURATION OF STUDY: 4 years
Full-time form of education
1 Scope of discipline and types of academic work
2 Sections and topics of the discipline and types of classes
| Name of sections and topics of the discipline | Lectures, hour. | Practical lessons, hour. | Laboratory classes, hour. | Seminars | Independent work, hour. | Coursework, hour. | Total, an hour. |
| Section 1 Management as a process of making management decisions | |||||||
| Topic 1.1 Functions and properties of management decisions | |||||||
| Topic 1.2 Management decision-making process | |||||||
| Topic 1.3 Target orientation of management decisions | |||||||
| Section 2 Models and simulation in decision theory | |||||||
| Topic 2.1 Modeling and analysis of action alternatives | |||||||
| Topic 2.2 Main types of decision theory models | |||||||
| Section 3 Decision making under multi-criteria conditions | |||||||
| Topic 3.1 Non-criteria and criteria methods | |||||||
| Topic 3.2 Multicriteria models | |||||||
| Topic 3.3 Features of measuring preferences | |||||||
| Section 4 Ordering of alternatives based on taking into account the preferences of experts | |||||||
| Topic 4.1 Measurements, comparisons and consistency | |||||||
| Topic 4.2 Method of paired comparisons | |||||||
| Topic 4.3 Principles of group selection | |||||||
| Topic 4.4 Expert assessments | |||||||
| Section 5 Decision making under conditions of uncertainty and conflict | |||||||
| Topic 5.1 Mathematical model of the PR problem under conditions of uncertainty and conflict | |||||||
| Topic 5.2 Game models of PR | |||||||
| Topic 5.3 Positional games | |||||||
| Topic 5.4 Optimality in the form of equilibrium | |||||||
| Section 6 Decision making under risk conditions | |||||||
| Topic 6.1 Theory of statistical decisions | |||||||
| Topic 6.2 Finding optimal solutions under conditions of risk and uncertainty | |||||||
| Topic 6.3 Statistical games with a single experiment | |||||||
| Section 7 Decision making under fuzzy conditions | |||||||
| Topic 7.1 Compositional models of PR | |||||||
| Topic 7.2 Classification models of PR | |||||||
| Exam | |||||||
| TOTAL |
Laboratory workshop
| No. | Module (section) number of the discipline | Name of laboratory work | Labor intensity (hours) |
| Topic 1.3 Target orientation of management decisions | Laboratory work No. 1. Search for optimal solutions. Application of optimization in PR support systems | ||
| Topic 2.2 Main types of decision theory models | Laboratory work No. 2. Decision making based on economic and mathematical models, queuing theory models, inventory management models, linear programming models | ||
| Topic 3.3 Features of measuring preferences | Laboratory work No. 3. Pareto optimality. Building a Tradeoff Scheme | ||
| Topic 4.2 Method of paired comparisons | Laboratory work No. 4. Method of paired comparisons. Ordering of alternatives based on pairwise comparisons and taking into account expert preferences | ||
| Topic 4.4 Expert assessments | Laboratory work No. 5. Processing of expert assessments. Expert Agreement Ratings | ||
| Topic 5.2 Game models of PR | Laboratory work No. 6. Construction of the game matrix. Reducing a zero-sum game to a linear programming problem and finding its solution | ||
| Topic 5.4 Optimality in the form of equilibrium | Laboratory work No. 7. Bimatrix games. Application of the principle of equilibrium | ||
| Topic 6.3 Statistical games with a single experiment | Laboratory work No. 8. Choice of strategies in a game with an experiment. Using posterior probabilities |
DIRECTION: "Management"
DURATION OF STUDY: 4 years
FORM OF STUDY: correspondence
1 Scope of discipline and types of academic work
2 Sections and topics of the discipline and types of classes
| Name of sections and topics of the discipline | Lectures, hour. | Practical lessons, hour. | Laboratory classes, hour. | Seminars | Independent work, hour. | Coursework, hour. | Total, an hour. |
| Section 1 Management as a process of making management decisions | |||||||
| Topic 1.1 Functions and properties of management decisions | |||||||
| Topic 1.2 Management decision-making process | |||||||
| Topic 1.3 Target orientation of management decisions | |||||||
| Section 2 Models and simulation in decision theory | |||||||
| Topic 2.1 Modeling and analysis of action alternatives | |||||||
| Topic 2.2 Main types of decision theory models | |||||||
| Section 3 Decision making under multi-criteria conditions | |||||||
| Topic 3.1 Non-criteria and criteria methods | |||||||
| Topic 3.2 Multicriteria models | |||||||
| Topic 3.3 Features of measuring preferences | |||||||
| Section 4 Ordering of alternatives based on taking into account the preferences of experts | |||||||
| Topic 4.1 Measurements, comparisons and consistency | |||||||
| Topic 4.2 Method of paired comparisons | |||||||
| Topic 4.3 Principles of group selection | |||||||
| Topic 4.4 Expert assessments | |||||||
| Section 5 Decision making under conditions of uncertainty and conflict | |||||||
| Topic 5.1 Mathematical model of the PR problem under conditions of uncertainty and conflict | |||||||
| Topic 5.2 Game models of PR | |||||||
| Topic 5.3 Positional games | |||||||
| Topic 5.4 Optimality in the form of equilibrium | |||||||
| Section 6 Decision making under risk conditions | |||||||
| Topic 6.1 Theory of statistical decisions | |||||||
| Topic 6.2 Finding optimal solutions under conditions of risk and uncertainty | |||||||
| Topic 6.3 Statistical games with a single experiment | |||||||
| Section 7 Decision making under fuzzy conditions | |||||||
| Topic 7.1 Compositional models of PR | |||||||
| Topic 7.2 Classification models of PR | |||||||
| Exam | |||||||
| TOTAL |
Laboratory workshop
| No. | Module (section) number of the discipline | Name of laboratory work | Labor intensity (hours) |
| Topic 2.2 Main types of decision theory models | Laboratory work No. 2. Decision making based on economic and mathematical models, queuing theory models, inventory management models, linear programming models | ||
| Topic 4.2 Method of paired comparisons | Laboratory work No. 4. Method of paired comparisons. Ordering of alternatives based on pairwise comparisons and taking into account expert preferences | ||
| Topic 5.2 Game models of PR | Laboratory work No. 6. Construction of the game matrix. Reducing a zero-sum game to a linear programming problem and finding its solution | ||
| Topic 6.3 Statistical games with a single experiment | Laboratory work No. 8. Choice of strategies in a game with an experiment. Using posterior probabilities |
DIRECTION: "Management"
DURATION OF TRAINING: 3.3 years
FORM OF STUDY: correspondence
1 Scope of discipline and types of academic work
2 Sections and topics of the discipline and types of classes
