Calculation of Spare Parts in a Car Service Enterprise Based on Evaluating the Average Time of Details Serviced Automobiles

The purpose of the work is the development of a technique for calculating spare parts in an auto service company based on an analysis of statistical information on the failure of details of each standard accumulated on daily information on the replacement of spare parts for serviced vehicles in previous planning periods. In the theory of system reliability, the calculation of the failure distribution function usually uses information on the main characteristic of failures the time between failures, and in the absence of such information, it is necessary to use information on the number of failures at the time of receipt for maintenance of cars in this auto service plant. In this connection, the relationship between the distribution functions of these random variables is established using the reciprocity of the distribution processes to failure and the number of failures. For each of the competing theoretical functions of the distribution of the operating time to failure, an estimate is made of the average operating time of the details of each standard in the auto service plant on the basis of actual demand. The estimates obtained make it possible to calculate the number of spare parts for the subsequent replenishment period of the SPIE.


Introduction
An analysis of previous work on the calculation of spare parts in car service stations (CSS) [1] shows that in recent years, most Russian auto service enterprises have been using the system for determining the need for spare parts, which is based on actual demand and individual details for the previous period Work.
The most important a priori information, which ultimately determines the amount of spare parts, is the theoretical model of failures that is taken into account in calculating the number of failures [2][3][4][5][6].
First of all, on the basis of failure statistics, the best model for the distribution of failures among a finite number of competing theoretical distribution functions is chosen, based on the Kolmogorov-Smirnov agreement, for the first part of each component model, including the diffusion (monotonic and no monotonic) distributions. For the best model for the distribution of failures of parts of each standard, an estimate of the mean time to failure (i.e., the average time between successive replacements) of the details of each standard is calculated taking into account the adequacy index of ensuring non-renewable spare parts [9]. According to these estimates, the expected size of the spare parts of each standard is determined, taken as the initial approximation of the proposed algorithm for calculating spare parts for the subsequent replenishment period SPIE.
It should be noted that the estimation of the mean time to failure of details is calculated from the statistics of their failures. At the same time, the relationship established between the distribution functions of the operating time to failure and the number of failures is substantially used in this paper, the use of which makes it possible to reduce the problem of estimating the mean time to failure to solve a nonlinear algebraic equation. The choice of the best model for the distribution of failures of the part and detail of each standard is carried out in accordance with Kolmogorov-Smirnov's consent criterion. The advantage of this criterion is that it is the only agreement criterion that, along with the choice of the theoretical distribution function, the least deviation from the empirical distribution function in the metric of the space of continuous functions, gives a confidence interval of deviation in the indicated metric.
Obviously, the higher the required level of reliability of providing non-renewable spare parts of a certain type and, correspondingly, the sufficiency index of this level of security, the higher the level of the required number of spare parts in the planned period.

Competing Theoretical Distribution
Functions (1). Exponential Distribution (E) is the mathematical expectation of random variable of operating time .
(2). Normal Distribution (N) where Φ is standardized normal distribution; = ! is the coefficient variation and is the dispersion of r.v. .
where Γ is gamma function.
It should be noted that among the competing hypotheses on the form of the distribution function of the operating time to failure, have included the diffusion no monotonic (DN) and diffusion monotonic (DM) distributions related to the probability-physical models representing the new technology of reliability research [10]. In our opinion, diffusion models for the distribution of failures must necessarily be considered among the competing ones, since each element of the system that is malfunctioning and replaced, not immediately after failure, may somehow affect the functioning of other elements (ie, a diffusion of faults from one element occurs point to another). Even the replacement of a malfunctioning element that has had time to influence other elements cannot result in regeneration (ie, complete restoration) of the system.
In Section 5, random variables (r.v) A B and C A B are taken into consideration for an arbitrary vehicle part, and for a part of the D -th type random values of E,A B and C E,A B are taken into consideration. In this case, which allows to pass from the distribution functions 1 0 -6 0 for the value of to the distribution functions of the quantity C.
Such a connection between the random variables A B and C A B plays an extremely important role, because, as a rule, the service center has a database on the number of failures (ie replaced) of each type of parts on the basis of cars received for maintenance Days (see, for example, Table 1 below).
As will be shown in Section 3, knowledge of the distribution of the random variable C A B makes it possible not only to determine the distribution of the random variable A B , but also to estimate the mean time to failure (ie, the average time between two successive failures) Which can estimate the number of spare parts of each type of par value for the entire replenishment period T RP .
The initial data on the failures of parts replaced at the auto service plant under review in the order of arrival of vehicles in each of the 27 working days are shown in Table 1. For the replenishment period T RP , one month is taken from 27 working days.  1  35  2  2  39  13  2  18  3  2  23  8  3  33  10  4  47  16  4  33  5  5  43  18  5  28  5  3  36  13  6  28  3  6  37  17  7  37  7  3  47  21  8  66  3  9  78  28  9  26  8  3  37  15  10  On the base of the initial data from table 1 Tables 2 and 3.  It is seen from table 2 and 3 that, for any part of the best distribution function is G R = [ G R , and for the parts of the nominal type it is

Reliability Evaluation of the of Parts and Sufficiency Indicators of SPIA
The required level of the "reliability" indicator of the products (nodes) at the end of the replenishment period J_ will be established: the probability of failure-free operation the C`A ab Acd , for example, C`A ab Acd = 0,9 (this means that at the end of the J_ period there at least 0,1 • # details must remain from the planned store of spare parts (#). The sufficiency indicators of the SPIE g h is determined on the base of the analysis of expected 0C H J_ 2 and required C`A ab Acd reliability indicators 0g i ≥ C`A ab It should be noted that the sufficiency index of SPIA g h can be set by the customer independently of the expected reliability of the product.
To evaluate the sufficiency of single complex of SPIA-S products with non-recoverable spare elements, g h is used as the probability so that during the operating time J_ , of the product will occur in no failures of SPIA. The probability s π is used to estimate the sufficiency of a set of SPIA provided that, all store in this set are replenished periodically with identical periods and the reliability indicators of the product is the probability of failure-free operation. The initial data for calculating the sufficiency index g h of the SPIA-S are is the expectal probability of failure-free operation on the final replenishment period of SPIA and the requirements for the reliability index of the product C`A ab Acd .
If it is not known C H J_ , then it is calculated where * G is the most adequate theoretical model of the failures distribution of any type of products, the choice of which is described in section 2. For g h , the number rounding to the nearest values of the series (11) is adopted. If C`A ab Acd C H J_ ⁄ = 0,9, then g h : = 0,9.

Evaluation of the Required Probability of Failure-Free Operation of the Elements
A value of C`A ab Acd is adopted satisfying the relation C`A ab Acd ≤ C H J_ and according to the value of the series (11), the sufficiency index of a set of SPIE-O is determined by the ratio g h ≥ C`A ab Acd C H J_ ⁄ If C H J_ is less than 0.9, then the value g h = C`A ab Acd is accepted from reasoning on importance of feasible functions and the importance of the economic expediency. Further, the required probability level of fail-free operation is calculated C E Acd for the elements of the D nominal type >C E Acd = 0C`A ab Acd 2 ) l @ , rounding to the nearest value of the series (11), where m is the total number of car's nominal types (nodes) (in our case m = 3).

Probability Evaluation of Fail-Free Operation and Sufficiency of Any Type of Spare Elements
The calculation of the probability of failure-free operation of D type elements C E at the considered period 0, n + J_ will be carried out by the formula: where E * G E R is the most adequate theoretical distribution model of the failure of D-th type of elements; n is the total operating time of the products at the beginning of the replenishment period, taking into account the intensity coefficient at the operation during this period.
According to the values of C E and C E Acd , the sufficiency index g h for the guarantee of to provide D type nonrecoverable spare parts is determined (rounding to the nearest larger value from the recommended series (11) by the formula (by the condition C E < C E Acd ) If C E ≥ C E Acd , then the elements of this type are n't included in the nomenclature of SPIA (in this case the number of the spare parts v E in the planned period is accepted equal v E = 0); m is the total number of D type (in our case m = 3).

The Calculation of Mean Operating Time Until the Failures of Constituent Elements
Suppose that is a random variable (abbreviated r.v) of operating time until failure (i.e. the time between two consistent failures) with the values , but A -r.v. operating time until G-th failure (beginning from the initial moment of the time) with the values A . Suppose, further C is r.v. of the failure member with values r and C A B is r.v. with values G ≤ G H , where G H is some fixed natural number. Suppose that, wG L x, M = 1, … , O is a given sample of observations random variable C and w L x is the corresponding (unobservable) sequence of a values random variable with mathematical expectation = H , where 0 T is the quantity to be determined.
It is obvious that,  Assuming v E c•`c¬ for the initial approximation of the number of spare parts a_i, we can get a more accurate estimate for a_i from the condition of satisfying the required level of reliability g i,E J_ to ensure the predicted demand using the following algorithm.
The results of calculating v E from this algorithm are shown in Table 4.

Conclusion
The primary task in the calculation of spare parts for the period of replenishment of spare parts SPIA (in the Englishlanguage literature -SPIA) is the problem of identifying the distribution of applications for spare parts. Usually it is believed that the flow of applications for vehicle maintenance is Poisson and use the normal law of distribution of applications in practice, since this law is a good approximation for Poisson distribution for sufficiently large quantities of applications. However, often the requirements for Poisson flow (stationary, absence of consequences and ordinariness) can be violated, which calls into question the advisability of using the normal law of distribution of applications (for replacement of spare parts of failed parts).
In this regard, we previously solve the problem of identifying the best model for the distribution of requests for maintenance. Using Kolmogorov-Smirnov's criterion, the set of known parametric distributions (exponential, normal, lognormal, Weibull, diffusion no monotonic and monotonic) is chosen consistent with the empirical function of the distribution of applications.
Using the mutual-reversibility process B of the run-tofailure distribution established in the work, with a fixed number of failures r 0 and the process A B for the distribution of the number of failures for a fixed running time H , the best model for the distribution of failures of the -th type details yields an estimate of the expected number of failures v E c•`c¬ .
On the basis of these estimates, an algorithm has been developed for calculating the number of spare parts required for individual standard sizes, while fulfilling two conditions: a) the sufficiency of spare elements of type i is not less than The required sufficiency index g i,E J_ of elements of a given type; b) the average time between failures of the required number of spare parts of each type is not less than the entire replenishment period T RP .
The proposed methodology can be used with an arbitrary finite number of competing theoretical functions for the distribution of applications and is applicable to the calculation of spare parts in other technical branches.