DETAILS OF THE SERVICE AND FEATURES OF EACH DESTROYER POKOYNY (p. No. 701) Leningrad, building im. Zhdanov. Project 56. Enrolled in the lists of the Navy - August 19, 1952, laid down on the slipway - March 4, 1953 (official bookmark, with 33% readiness of the hull), launched - November 28, 1953, start of testing -

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Cars of the airfield service Back in the early 1950s, GAZ-51 cars served as the base for the first Soviet dual-purpose airfield service vehicles. A mobile hydraulic unit was used to check and test the onboard hydraulic systems of aircraft.

Fuel service vehicles On the GAZ-63 chassis, they installed both fairly simple water tankers AVTs-63 (1958) and fuel ATS-2-63, the first post-war tankers TZ-63 (1948) and oil tankers MZ-3904 (1958). ) dual-use, and the first domestic stations

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Fuel service vehicles ATsPT-4.1-130 is a simple dual-purpose tanker on the ZIL-130 chassis without its own pumping system. Designed for short-term storage and transportation drinking water and other liquid food products in regions with moderate

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5.3. Road Service Rules The Road Service Rules (RS) are mandatory for all ships, vessels and floating facilities. The following are the rules concerning boats. Boatmasters should know: - Signals regulating traffic in harbors and on

7.5. Organization of duty service on the shore To maintain internal order, protect the landing area and property of personnel, a daily duty is assigned. The composition of the order depends on the number of participants in the campaign, the time spent at the landing site and local conditions.

Determination of the residual life of machines and equipment based on probabilistic models

© Leifer L.A., Kashnikova P.M., 2007
CJSC "Privolzhsky Center
financial consulting and evaluation"

The determination of the remaining service life and residual life is an important element in the evaluation procedure market value machines and equipment.

Within the framework of the cost approach, the residual life (residual resource) is necessary to determine the residual value and, accordingly, the replacement cost of the object. When implementing the income approach, the residual term determines the period during which cash flows should be expected, and therefore its size significantly affects the estimated value of the market value. With a comparative approach, the residual service life serves as the basis for adjusting the prices of analogues that differ from the object being evaluated by the value of the operating time. Therefore, the accuracy of assessing the market value of machinery and equipment to a large extent depends on how correctly the residual service life (residual resource) of the object being evaluated is determined. Depending on what information the appraiser has, it is possible various methods determination of residual service life and residual resource. The most reliable forecast of the residual life can be made if a full-scale technical diagnostics of the machine is performed using appropriate diagnostic tools and introscopy. This approach is costly, and therefore, except for cases where single and expensive machines or production lines are evaluated, in general practice appraisal activities usually not applied. Methods for individual prediction of the residual life of machines and structures, based on models of physical processes of wear of machines and structures (accumulation of fatigue damage, wear of mechanisms, etc.), set out in various publications (see, for example, , ), also did not find practical application when evaluating the cost of machines due to their complexity and the need to use the complex mathematical apparatus of the theory of random processes.

The problem of estimating the cost of large arrays of equipment and machines has led to the need to create simplified technologies that provide a “flow” assessment using a minimum of input information about the object of assessment. These requirements are also met by residual life determination technologies based on linear or exponential wear models.

We will not consider the advantages and disadvantages of these methods. We only note that they are based on deterministic wear models. In this case, the residual service life (resource) within the framework of these models is usually defined as the difference between a certain standard service life and its effective age.

In recent years, in the practice of assessing machines and equipment, a different approach has begun to be used, based on the methodology developed in the framework of the theory of reliability of machines and complex structures. Unlike deterministic wear models, this methodology is based on the idea that the residual service life (resource) of a machine is a random variable that can only be described by probabilistic models. Such a methodology expands the possibilities of assessment methods and makes them more consistent with physical wear processes and common sense. Within the framework of such a methodology, it is possible to understand and take into account the fact that the actual service life may significantly exceed the standard one when calculating the cost of an object. At the same time, the service life (resource) established in the documentation has the meaning of the minimum service life (resource), during which the manufacturer guarantees normal operation with a high probability.

In this article, a statistical approach to the problem of predicting the residual service life (resource) is developed on the basis of models that, according to the authors, may be the most appropriate in many real situations related to the evaluation of machines in conditions where the loss of value is mainly due to the physical degradation of the object. estimates. Basic concepts, terms and definitions

Since the problems related to the analysis of the service life and resource of technical devices and structures (hereinafter referred to as objects) are studied within the framework of the reliability methodology, the terms and definitions used in the article are taken mainly from the well-known standard.

Limiting state - the state of the object, in which its further operation is unacceptable or impractical, or the restoration of its operable state is impossible or impractical.

Notes:

1. An object can go into a limiting state, remaining operational, if, for example, its further use for its intended purpose becomes unacceptable according to the requirements of safety, economy and efficiency.

2. Reaching the limit state is not limited to physical wear and tear. As can be seen from the definition, the transition to the limiting state can also be due to the influence of functional obsolescence factors.

3. Usually, when the limit state is reached, the object is decommissioned. This, however, does not mean that the value of an object that has reached the limit state is equal to zero. As the analysis of the literature has shown (and this was confirmed by our research), the cost of an object that has reached the limit state is usually 10–20% of the initial cost. This cost may include the cost of remaining parts, materials, etc.

The service life of an object is a calendar time equal to the period of operation, counted from the commissioning of the object until the limit state is reached (decommissioning).

The resource of the object is the total operating time of the object, expressed in hours, kilometers, etc., counted from the commissioning of the object until the limit state is reached (decommissioning).

Notes:

1. During standard operation, usually the running time measured in hours or kilometers (for Vehicle), is proportional to the service life. Therefore, in the future we do not make a distinction between these concepts and will use one of these terms, understanding that all formulas, reasoning and conclusions related to one of them apply to the other to the same extent.

2. The actual moments of reaching the limit state by objects can vary significantly depending on the individual properties and operating conditions of the objects. Therefore, the service life, as well as the resource of the object, should be considered random variables. They can only be described by probabilistic models. As such a model, the distribution density or distribution law is usually used. In economic methodology, a closely related concept is used: the “survival curve”. More on probabilistic models in the next chapter.

Average service life (Average resource) - The average value of a random variable - the service life (resource), counted from the commissioning of the facility until the limit state is reached (decommissioning).

Established (Normative) service life (established resource) - the service life established in the technical documentation.

Notes:

1. The established (Normative) service life characterizes the durability of the object, its ability to maintain operational characteristics during the established period. Withdrawal of an object from operation due to the achievement of the limit state before the end of the established period of operation is considered unlikely. At the same time, the achievement of the normative term by the object does not mean that the object has reached the limit state and should be decommissioned. In order to ensure the reliable operation of the object during the established period, the object must have a certain margin of safety, which makes it possible to confidently operate the object during the standard period and for some time after the end of this period. The development and testing of the object carried out at the manufacturer's plant is aimed at ensuring reliable operation within the specified period (set resource) and at ensuring this reserve. From a probabilistic point of view, the period specified in the documentation is a quantile of the distribution of the expected service life.

2. It is necessary to distinguish between the average service life and the standard service life. The standard service life is not the average service life, but it can be used as input to determine the average service life and other statistical parameters that characterize the durability of the object.

3. If the design or operational documentation does not indicate the service life, then the value calculated on the basis of the depreciation rate of an object of this class can act as a standard period. In terms of meaning, this value also characterizes the durability of the object.

The age of an object is the period of time from the date of commencement of operation to the current moment.

Residual service life - The calendar duration of operation from the current moment until it reaches the limit state. It differs from the service life in that the current moment is taken as the starting point, until which it has been in operation for some time, and not the beginning of operation.

Residual resource of the object - the operating time of the object, expressed in hours, kilometers, etc., from the current moment until it reaches the limit state. It differs from the resource of the object in that the current moment is taken as the starting point, until which it has been exploited for some time, and has exhausted part of the initial resource.

Notes:

1. The individual characteristics of an object (residual service life and residual resource) are random variables and can be accurately determined only after its limiting state has come. As long as these events have not occurred, we can only talk about predicting these values ​​with greater or lesser probability. Therefore, the residual service life is the predicted value of the expected time, after which the object will reach the limit state and be decommissioned. It should be emphasized that the remaining period in the general case is not equal to the remaining time until the standard period is reached. The same applies to the residual resource.

2. Since the residual service life (residual resource) is a random variable, it can only be described by probabilistic models. As such a model, just as in the case of the initial service life (resource), the survival curve can be used.

Average residual service life (Average residual resource) - the average value of a random variable - the residual service life (resource), counted from the current moment until the limit state is reached (decommissioning).

Notes:

1. It should be clearly understood that the average residual life does not indicate the exact period of time that the object being assessed will be in operation. It characterizes a certain center of dispersion of time moments, around which (some earlier, some later) objects of this class that have reached the limit state will be decommissioned. Since at the time of the assessment it is impossible to determine the exact time that the item is still capable of being used, the average remaining resource is the best guide to the expected life of the item being evaluated.

2. The average residual life depends on the initial characteristics of the durability of the object and its age. The older the object, the shorter its average residual life. Thus, the average residual period decreases as the age of the object of assessment increases. However, reaching the target life does not mean that the average remaining service life is zero.

Probabilistic models for describing the service life (resource)

Since the service life is a random variable, probabilistic models should be used to describe it. The probability that in time the object will not reach the limit state is defined as P(J ) = P (t ³ J )

The function P(J ) shows how many objects on average will "survive" until the time t . Therefore, it is called the "survival curve". The survival curve defined in this way is related to the probability distribution function F(J ) by the relation: F(J ) = 1- P(J )

The distribution density of time to the limit state f(J ) is a derivative of the distribution function: f(J ) = dF(J )/dJ = - dP(J )/dJ

In this case, if the time is counted from the current moment t characterizing the time until which the object has already been operated, then P(J /t ) characterizes the probability distribution of a random variable - the residual service life. In the language of probability theory, P(J /t) is the conditional probability that the remaining service life will be at least, provided that the object has been functioning properly up to the current moment - t. It is necessary to distinguish between the theoretical probability distribution and the empirical (or sample, i.e., built on sample data). It is not difficult to construct an empirical distribution based on statistical data. However, in order for the empirical distribution to be directly used to establish the theoretical distribution, large amounts of data are needed. Therefore, all conclusions regarding the theoretical distribution are made on the basis of an analysis of the nature of the data, the nature of the processes leading to the limit state, and the limited amount of sample data.

In the literature on the assessment of the market value of real estate, machinery and equipment, when discussing issues related to determining the residual life, the term borrowed from the theory of actuarial calculations [see, for example, 8, 16] - “survivor curve” has become widespread. . The survival curve is a graph showing the number of units from a given group of assets that remain functioning at some point in time from the forecast interval. In other words, it characterizes the process of decommissioning objects as they reach the limit state. This curve is a statistical analogue of the probability P(J ) introduced above. In the future, under the survival curve, we will understand the theoretical and empirical (statistical) version of the function P(J ).

Various distribution laws are used to describe the survival curve. Among the most commonly used tools for this purpose are the so-called Iowa-type survival curves. They were developed as a result of the study of empirical data relating to the characteristics of all types of machines and equipment that remained operational. Subsequently, they were used to assess the residual useful life of the property of trade and public utilities, electricity, water and gas supply, railways and others. With regard to the assessment of machines in Russian valuation practice, such models were considered in the works of Trishin V. N.). It should be especially noted that in these works the proposed methods are brought to specific solutions and, most importantly, the software system that implements these methods is based on input data available to the practicing Appraiser. In addition, probabilistic models for describing the useful life are used in the problems of estimating the value of intellectual property objects. In the cited work, known probability distributions are used to describe the useful life, in particular, the Weibull model and Iowa-type survival models. Along with the models proposed in the state of Iowa, for a probabilistic description of the service life of machines, mechanisms, complex technical systems the lognormal distribution can also be used, which, along with the Weibull distribution, has been widely used and developed in the theory of reliability of technical systems, machines and complex structures.

The choice of this or that distribution is determined by the nature of the prevailing physical processes, the availability of initial information, and the possibilities of computational procedures.

For the practical use of probabilistic models for the purposes of assessing market value, two main questions are:

1. How, based on the available information, to determine the parameters of the survival curve (the parameters of the distribution of the service life - random time until the limit state is reached)? 2. How to determine the characteristics of the residual service life, if the age of the object and the parameters of the distribution of time until reaching the limit state (survival curve) are known?

This article proposes a model that allows, under the assumptions made, to answer these questions and thereby create real prerequisites for the practical use of probabilistic models in the problems of determining the residual life of machines and equipment. As such a model, a log-normal distribution is used, which, according to the authors, is most adequate to the processes of physical wear, fatigue accumulation of damage and other mechanisms for the loss of efficiency of machines and mechanisms.

The log-normal distribution can be derived as a statistical model for a random variable whose values ​​are obtained by multiplying a large number of random factors. The log-normal distribution is used in fields ranging from economics to biology to describe processes in which an observed value is a random fraction of a previous value. The justification for the applicability of the log-normal distribution to describe life is also based on the property of multiplication of effects inherent in this distribution. Therefore, this distribution has been widely used and developed in works on the analysis of the degradation processes of mechanical systems.

Let us denote the dimensionless time equal to the ratio of the service life (t) to the standard service life (t x) by the letter t: t= t /t x

Then, in accordance with the accepted service life model, the distribution density of the random variable (t) has the form:

The distribution density contains all the information regarding the service life. However, directly for the assessment, it is necessary to know the main characteristics of this distribution (m and s).

Rice. 1. Distribution density of a random variable (t)

The mathematical expectation (T), variance (D) and coefficient of variation (r) of the random variable t (service life given in dimensionless form) are determined through the distribution parameters (m and s) as follows: (1)
(2)
(3)

From the standard service life to the distribution parameters of the actual service life

It is usually not possible to perform durability tests on objects similar to the object being evaluated during the evaluation process. Therefore, information available to the evaluator should be used to determine the distribution parameters. Such information can be used general information regarding the object of assessment and the standard service life specified in the operational documentation. As noted above, if there is no data on the service life, you can use depreciation rates, which also carry information about the object being valued.

Let's analyze the relevant information that allows us to determine the main characteristics of the lognormal distribution.

An analysis of the literature summarizing numerous studies on the reliability and durability of machines and equipment shows that the coefficient of variation for machines and equipment lies in the range: 0.3 - 0.4. This information allows you to determine the -D distribution option. In order for the standard service life related to this object to be used to determine the distribution parameters, we take into account that the standard service life is the calendar time during which the object must function properly (more precisely, it must not reach its limit state) . In essence, the standard service life indicates the minimum time during which the object must be operated if no abnormal situations occur. Thus, if we assume that an object with a high probability (for example, 0.9) should serve for a given period, then from the point of view of the accepted model, the standard period is a 10 percent quantile of the distribution. Using the above information and the corresponding assumptions, it is easy to calculate the parameters of the log-normal distribution and construct a survival curve that characterizes the process of disposal of the evaluated objects over the period of operation.

Let's set the level a , it will be the probability that the object of assessment will reach the limit state before the expiration of the standard period, which in turn is determined by the integral (4)

Using this equation (4) and relations (1), (2) and (3), it is possible to calculate the values ​​of the dimensionless average service life (T) from the given values ​​of a and r . Recall that the dimensionless average service life (T) is a value equal to the ratio of the average value of the actual service life to the standard service life.

Table 1 presents the results of such calculations for various values ​​of a and r.

Table 1. Dimensionless Average Life (T) Values

It is also possible to calculate the parameters of the log-normal distribution, which characterizes the probabilistic properties of the process of decommissioning objects of assessment from operation. On fig. 2 and 3 respectively show the distribution density of the service life of machines, equipment and structures and the survival curve (sometimes called the mortality curve), which describes the process of decommissioning of objects.

Rice. 2. Lifetime distribution density (r =0.3, a =0.1)

Rice. 3. Survival curve (r =0.3, a =0.1)

In this case, the distribution density and the survival curve are constructed based on the conditions: r =0.3, a =0.1. The choice of such initial data was based on two circumstances:

1. The limiting state of mechanical systems occurs mainly due to the processes of physical wear and fatigue accumulation of damage. Therefore, based on numerous studies in the theory of reliability (see, for example, ), a value equal to 0.3 - 0.4 can be taken as the coefficient of variation.

2. The normative period (appointed), specified in the design or operational documentation, is nothing more than the minimum allowable period of operation of the object, during which the object should not reach its limit state. Since, nevertheless, such a possibility cannot be completely ruled out, we proceed from the fact that the object is decommissioned and written off in no more than 10% of cases. As a result, the survival curve mainly characterizes the process of disposal of objects in the period of time after the standard service life. Naturally, in accordance with this assumption, the average service life of the object, which is used in further evaluation calculations, exceeds the standard service life, which is quite justified from the point of view of the real market picture.

Remaining service life.

If an object has reached a certain age, then it is natural to expect that the residual life for it will decrease somewhat. At the same time, the higher the age of the object (assuming the same life history of the objects), the shorter its residual period. This statement is consistent with all known loss of value models and common sense.

In this case, the distribution of the residual life of the estimated object and, accordingly, the survival curve, which characterizes the probabilistic process of retirement of objects of a given class that have survived to a given age, can be calculated based on the conditional probability distribution. The conditional density of the log-normal distribution of the residual life, expressed in relative units, corresponding to the condition that the object lived to the age t, is determined as follows: (5)

Further calculations and corresponding graphs are based on the assumption that the coefficient of variation r = 0.3 and the allowable level of retirement of objects from operation before they reach the standard period a = 0.1

Rice. 4. Conditional distribution density of the residual service life, provided that the object has been operated up to the current moment.

Note that n is the age of the object at the time of assessment in relative units, numerically equal to the actual operating time divided by the standard service life:

n \u003d t / t n

Knowing the distribution density of the residual service life (5), it is possible to determine the average value of the residual service life T (in relative units), provided that the object has already been in operation for some time (t). Below is the dependence of the average value of the remaining service life on the actual service life preceding the assessment date. This dependence is constructed by statistical modeling of random variables generated by the mentioned distribution density and subsequent calculation of the mean and median. The results obtained reflect the probabilistic nature of the durability of machines and are more consistent with reality than deterministic models. In particular, they take into account that the achievement of the normative term by the object does not mean that the resource is completely exhausted. With the parameters included in the above calculations, an object that has worked out its standard period retains the possibility of further operation, on average, for up to 40% of the standard period. The remaining period takes into account the pledged margin for the resource of the machine, since the standard period is not the period of complete exhaustion of the resource. The graph also shows that with an increase in the previous service life, the average value of the residual service life decreases, and an object that has worked significantly more than its standard service life expects to reach the limit state soon.

The examples below show how the stated theory can be used in practical calculations in the process of assessing the market value of machinery and equipment.

Rice. 5. Dependence of the average value of the residual life (T) on the previous service life (n).

Examples of calculating the residual life of movable property.

In conclusion, we give examples of determining the average residual life, illustrating the process of estimating the residual service life when evaluating machinery and equipment using a graph for the average residual life (Fig. 5).

Example 1

1. The object of assessment is a complex technological line with a given standard service life of 20 years.

2. The equipment was purchased from dealers and put into operation 14 years ago. The line was operated under normal conditions in compliance with all the requirements of operational documentation (scheduled preventive maintenance, preventive maintenance, etc.). Currently, it is in working order.

3. Degradation processes occurred under the influence of physical wear and fatigue damage accumulation. The coefficient of variation can therefore be taken equal to 0.3.

4. Determination of the average residual life is required to establish the period during which the object should be expected to generate cash flows. This value is required to implement the income approach.

Calculation

The following are used as initial data:
normative term - 20 years,
the current age is 14 years (in relative units 14/20 = 0.7).
From the graph, we determine the average residual service life in relative units, which will be 0.6.
Hence the average residual term is 0.6 * 20 = 12 years.

Example 2

1. The object of assessment is an agricultural tractor, the standard service life according to the design documentation is 12 years

2. The tractor was purchased in trading network and was operated in normal mode for a full service life of 12 years.

3. At the moment, the tractor is operational, i.e. capable of performing the specified functions in accordance with the requirements of regulatory, technical and design documentation. The resource parameters are within acceptable limits.

5. The determination of the residual service life is required to determine the value of the loss in the value of an object that has served its full service life and has not reached the limit state, within the framework of the cost approach.

Calculation

Initial data:
normative term - 12 years,
the current age is 12 years (in relative units 12/12 = 1).

From the graph, we determine the average residual service life in relative units: 0.4.

Thus, the average remaining life: 0.4 * 12 = 4.8 years.

Hence, if we consider the amount of depreciation by the method of economic life, we get: Depreciation = current age / current age + average residual resource. Wear = 12/ (12+4.8) = 0.7. Using the resulting depreciation value as input, you can calculate the current value of the object.

Example 3

1. The object of appraisal is an imported passenger car manufactured in 1993, purchased on the secondary market. The car is currently 11 years old.

2. There is no standard service life in the operational documentation. However, some idea of ​​it is given by depreciation rates that reflect the average life of objects of this class. Based on depreciation rates, the standard life of a car of this class is 7 years.

3. At the moment, the car is operational, i.e. capable of performing the specified functions in accordance with the requirements of regulatory, technical and design documentation. The resource parameters are within acceptable limits.

4. Degradation processes related to resource parameters (clearances in interfaces, wear in bearings, gears, shafts, etc.) occurred mainly under the influence of physical wear. Therefore, the coefficient of variation of the service life can be taken equal to 0.3.

5. Despite the fact that the car has served the standard service life, since the car is in good condition, it was decided to continue its operation. This should be reflected in the assessment of the market value of fixed assets of the enterprise. To do this, it is necessary to determine the remaining service life.

Calculation

We use as initial data:
normative period - 7 years,
the current age is 11 years (in relative units 11/7 = 1.5). From the graph, we determine the average residual service life (in relative units): - 0.3

Thus, the average residual term is 0.3 * 7 = 2.1 years.

Findings.

1. The article describes an approach that allows predicting the remaining service life with a minimum of initial information. The initial data for predicting the average value of the residual service life are: the standard service life of the object and the actual service life preceding the moment of assessment.

2. In an implicit form, the described method takes into account information about wear mechanisms. This information is contained in the value of the coefficient of variation of the service life, incorporated in the calculation formulas. This increases the information content of the method, giving it additional advantages compared to the simplified model.

3. The approach outlined in the article is based on probabilistic models and develops methods for determining the statistical characteristics of the residual life, based on the use of survival curves that are successfully used in actuarial calculations.

4. Fundamental in the proposed model is the recognition that the normative service life is not equal to the expected lifetime, during which the object reaches the limit state. The method is based on the assumption that in the overwhelming majority of cases (for example, not less than 90%), the object must successfully work without reaching the limiting state during the entire standard period.

5. As a basic probabilistic model, a log-normal distribution is used, which, together with the Weibull distribution and survival curves, called the Iowa curves, make it possible to describe the process of decommissioning objects from operation as they reach the limit state.

6. Within the framework of the above method, an individual analysis of the technical condition of the object being evaluated is not supposed, which would certainly improve the accuracy of the forecast of the residual service life (residual resource) of each specific object. Therefore, the described method can be used for mass valuation of machines, when it is required to minimize the costs of assessing a large number of machines and equipment.

7. The description of the method and its interpretation refer to the assessment of machinery and equipment. However, with minor refinements, the method can be applied to determine the residual life of real estate, intellectual property and other objects of assessment, for which the life or useful life can be considered a random variable.

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