The deTerminaTion procedure of The onseT of The objecT wear-ouT period based on moniToring of The empirical failure inTensiTy funcTion

The estimation of the number of failures of technical objects is of key importance throughout the object life cycle, particularly in the wear-out period when the number of failures begins to grow significantly. In the literature related to this problem, examples exist of solutions (mathematical models) that can assist the estimation of the number of failures. For the description of the life cycles of objects, functions are usually used of known forms of probability distribution of the number of object failures. The procedure presented in this paper assumes the use of statistical data related to the failures of uniform population of nonrenewable technical objects, recorded in the form of empirical function of failure intensity. It specifically serves the purpose of determining the characteristic point of life of these objects i.e. the onset of the wear-out period. Within the procedure, a model of fuzzy inference has been applied that reflects the human reasoning (expert of the system) observing/investigating the objects. The results of the developed procedure may constitute a basis for forecasting of failures of mechanical nonrenewable technical objects.


Introduction
In all periods of the object life cycle the number of failures needs to be forecasted.It is required by the demand estimation processes related to renewable objects, the need to configure maintenance systems that allow for the number of renewals as well as object proactive behavior, the lack of which may generate unacceptable hazard.
This problem is particularly conspicuous when analyzing the wear-out period and the usually surging number of failures in that period (assuming that the investigated population is sufficiently large).Relevant literature presents mathematical models that assist the process of reaction to object failure e.g.[2,3,10,12], procedures that describe and compare classes of forecast models (e.g.[4,17]) or assist in the forecasting of failures of nonrenewable technical objects in the wear-out period (e.g.[10,12]).In [2] a method is proposed of detecting the onset of the object wear-out period and determining the maintenance efficiency based on, as the authors of [2] would call it, a step model of aging and the Bayes techniques.In [3] a new reliability model is presented of complex repaired technical objects/systems based on the bathtub curve.
The procedure presented in this paper is dedicated to uniform nonrenewable mechanical objects.As a starting point, reference to the forecasting tool of the failure of nonrenewable technical objects was assumed.These tools were developed by one of the authors of this paper (i.a.[10,11,12]).The basis for the failure forecasting models is the estimation of the parameters of object operating time distribution until wear-out failure occurs.It was assumed that the parameters of this distribution are estimated based on statistical data related to: number of object failures occurring in the period between the onset of the wear-out period and the end of the observation time, number of objects that are forecasted to fail due to aging.-A troublesome point of the said models is the determination of the onset of the wear-out period that is necessary for the estimation of the sciENcE aNd tEchNology parameters of the operating time distribution until failure.The significance of this issue is also supported by other authors ( [1]) by mentioning the return point of the failure intensity function as useful in terms of maintenance and risk analysis related to failures.They present a certain way of solving problems basing on the modified function of Weibull distribution.
In relevant literature it is difficult to indicate formal algorithms allowing the determination of the onset of wear-out periods.Few publications in this matter pertain mainly to the attempts to find new forms of functions describing the processes of object operation [6,9] or focus on modeling of the entire course of the function of failure intensity ( [19]).A reliable solution is proposed by the authors of [2], but only for known continuous distributions.What is missing is the solution to the problem if the failure intensity function is a non-continuous characteristics and its form as a function is unknown.
The intention of the authors of this paper is to present the procedure of estimation of the onset of the wear-out period of objects based on the empirical function of failure intensity without having information on the reliability function.

Concept and main assumptions
The solution to the problem is the analysis of the data on the number of failures in subsequent periods of time of their investigation/observation and then selection of the moment when the number of failures begins to grow significantly.If information in the form of non-continuous functions is used, the top or bottom limit of a given interval group is assumed (interval in which the number of failures grows significantly and its growth continues in further intervals).
The simplest but least accurate and informal method to solve the problem is intuitive choice of the onset of the wear-out period by the researcher (expert) based on the analysis of the course of selected reliability functions.It is advantageous in the case of functions of untypical courses and allows (due to lack of other tools) a quick obtainment of a satisfactory result.
This enabled an adoption of the following concept of the procedure: a man well acquainted with the modeled system/object, i.e. system expert, may correctly indicate the onset of the wear-out period of objects even if he infers having limited (partial) information on the object failures.Such a subjective choice is usually made because of experience and knowledge about the object.From the observations conducted by the authors we know that a man who is not a system expert but has all the necessary needed information related to the time of the loss of object worthiness (data in the form of courses of functions of failure intensity) is also capable of deciding about the onset of the wear-out period.
This paper reproduces (through fuzzy inference models) the system expert's reasoning that leads to the indication of the onset of the wear-out period based on observations of the course of the empirical function of failure intensity.Literature mentions applications of elements of fuzzy inference to solve a variety of problems related to reliability of objects.For example [18] presents the application of fuzzy sets in the problem of matching curves to the reliability data, [20] describes its use in reliability analysis of elements while [7,13] discuss the application of fuzzy inference in methods designed to determine the measures of reliability.
The concept of the procedure consists in determining moment t p -the onset of the wear-out period of technical objects through mathematical models that reproduce (simulate) the reasoning of the system expert.The mathematical model was developed based on the following main assumptions: Uniform population of technical objects is analyzed, - The number of failures of technical objects in time is known -(observation time intervals) and the statistical data on the failures are stored in the form of stemplot, The type of probability distribution of the operating time of ob-jects until failure is unknown, Fuzzy inference is possible based on the results of monitoring of the empirical value of the function of failure intensity λ N (t), The onset of the wear-out period -t p falls between moment t p0 of the first increase of function λ N (t) and moment t k -the end of object observation, The failure intensity function is a constant interval non-decreas-ing function in the wear-out period.

General mathematical model
The structure of the inference models is formed by properly written rules of inference i.e. fuzzy implications R k (k = 1,2,…,l) [8,14,15].These are the if-then type of rules that in a general form can be written as follows: x is 1 j A and (2) x is 2 j A and … and ( ) where: -input variables of the inference model forming the m-dimensional input vector x.It has been assumed that at the first stage of the calculations, variables x (i) assume the values of the empirical function of failure intensity of objects λ(t i ) in subsequent i-th intervals ( ) -fuzzy sets of the conclusion of the inference rules.
Graphic interpretations of the assumptions and the understanding of some of the elements of the presented procedure have been shown in Figure 1.In further considerations, the constructive model of inference was applied [8].In models of this type the value of µ A ( ) x of the function of membership related to the degree of rule activation, is interpreted in the form of a logical product of fuzzy sets.In the fuzzy sets the product operation (as well as the sum of these sets) can be performed in different ways.In the literature, many different relations for each of these operations have been presented.For a logical sum of the fuzzy sets there is a group of relations referred to as the s-norm operators and for the logical products -a group of t-norm operators.For example, to calculate the logical product of fuzzy sets a minimum (MIN) operator can be used -relation (2): The MIN operator has many disadvantages (as described in detail in [15]), which is why the product operator is more frequently used.The calculation of the function of membership of the product of fuzzy sets with the use of this operator is done according to the following formula: It was assumed that the aggregation on the implication level is realized as an algebraic product of the degrees of membership of the fuzzy sets (relation ( 3)) for both the implication premise and the consequent.
The output of the inference models is made by the superposition of the outputs of individual inference rules.It consists (based on the R k rules) in the reproduction of the realization of the input variables x (i) into a certain output quantity y representing moment ˆp t .The first step of this procedure consists in combining (for certain input data) the premises (antecedents) of the k-th fuzzy rule.We may use the operation of the product of sets -relation (2) or (3).In this way we determine ζ -degree of rule activation (activity) of the rule.Since, the inputs are non-fuzzy values the degree of rule activation ζ of each of the rules forming the database of inference rules can be determined as follows: Assuming that the database of inference rules is composed of l-th number of inference rules, another step of the procedure is the determination of the fuzzy sets C k (k = 1, 2,…, l) derived by the k-th rule.Let sets C k be determined in certain space y in the following way: where: ζ k -denotes degree of rule activation (activity) of the k-th inference rule determined according to relation (4).
By performing the aggregation of sets C k we may obtain value C for the output value y as a relation: wheras C is a fuzzy subset determined in space y.
The aggregation of fuzzy sets C k can be performed in many ways [8,15,16].For example, one may use the operation of logical sum of the fuzzy sets i.e.: ...

Detailed mathematical model and the method algorithm
The first stage of determining of moment t p is evaluating the input value x (i) of the inference models.It was assumed that this evaluation can be done by a minimum number (two) linguistic terms (w = 2).A finite set Φ of these terms takes the form: Linguistic terms are written in the form of fuzzy sets A j (j = 1, 2) of polygonal [8,15] (triangular and trapezoidal) functions of membership.These sets are regular convex fuzzy sets [16] of the support limited with values a, b, c, d.
The database of inference rules, with a relatively large number of λ Ν (t) function monitoring intervals may have an excess number of rules.In order to reduce this number, it is proposed to search for moment t p in a limited range.This is referred to as the life cycle analysis range (marked in Fig. 1).The range covers the period between moment t p0 -of the first increase of function λ Ν (t) and moment t k -end of object observation.It was assumed that moment t p0 equals the onset of the monitoring interval where the first positive increment of function λ Ν (t) takes place.
In the range of analysis (t p0 , t k ) four inference intervals are then introduced Τ ι = (τ ι−1 ; τ ι ), (ι = 1, 2, 3, 4).They are created by a combination (Fig. 1) of the subsequent monitoring intervals (t i-1 ; t i ) (i = 1,2,…, m).In such a case number s must be determined i.e. the number of monitoring intervals that compose a single inference interval (τ ι−1 ; τ ι ).The preliminary number s of monitoring intervals is obtained from relation: where: Such a method of creating inference intervals results in a situation when, in some cases, the sum of the lengths of these intervals ∆τ ι−1,ι exceeds the end of the range of life cycle analysis.If such a situation occurs, a shortening of each of intervals Τ ι is admissible by length ∆t i−1,i , i.e. by the length of one monitoring interval.It is also proposed that the shortening be realized starting from the last (ι = 4) of intervals Τ ι .As a result of such an operation, lengths ∆τ ι−1,ι of intervals Τ ι that will eventually be used in the inference model, may differ from one another.Number s (relation (10)) will thus be dependent on the Τ ι interval number, which is further marked as s (ι) .
In further calculations, the values of functions Λ(τ ι ) obtained according to relation (11) were assumed as input variables of the inference model: (11) sciENcE aNd tEchNology where p0 is the subsequent number of interval (t i-1 ; t i ) in which the first increase of function λ Ν (t) was observed (monitoring).
Moment t p was described with fuzzy numbers L ι w related to individual inference intervals Τ ι .Fuzzy numbers L ι w were written as follows: and expressed through appropriate fuzzy sets B j (y), (j = 1, 2,…, 8): To describe fuzzy sets B j (y), j = 1, 2,…, 8, triangular forms of the membership function were used.The sets support points (a, b, c) are within the limits of relevant monitoring intervals, which, using earlier adopted symbols, can be written as follows: In the case of the fuzzy number "near Τ ι " = L 12 , trapezoidal extreme function of membership was applied.
The general algorithm of the procedure in a graphical form has been shown in Figure 2.

Example of procedure realization
The example of the monitoring of the course of the function of object failure intensity was performed for 100 nonrenewable railroad objects (locomotives).The objects were observed for the time corresponding to the mileage of 600.000 km.During the investigations, in the subsequent intervals ∆l i-1,i = 50000 km of the locomotive mileage, the number of failures was recorded.The results (number of failures and the value of the empirical function of failure intensity) have been shown in Table 1.
In further part of the paper the results of the realization of selected steps of the procedure algorithm have been presented.Figure 3 shows the course of the function of object failure intensity.
In the initial stage of the calculations the life cycle analysis range is determined based on the value of function λ Ν (t).To this end, the increment of the function must be determined.Its first increase was recorded in the intervals from 300000 to 350000 km of the object operation i.e. 7 th (p0 = 7) failure record interval.For this interval the increment of function λ Ν (t) was 2.22222E-07.The subsequent increments of the function were: 7.57576E-07 (for the 10th interval), 1.71369E-06 (for the 11 th interval) and 2.81859E-06 (for the 12 th interval).The beginning of the life cycle analysis range was thus assumed to be 300000 km.
The output value of the inference model (fuzzy onset of the wearout period of nonrenewable railroad vehicle objects that were subjected to analysis) has been shown in figure 4 in a graphical form.
In order to determine the non-fuzzy ˆp t , an operation of defusification of the model output value by the COA (Center Of Area) method was performed.The non-fuzzy onset of the wear-out period obtained according to the said procedure is 428215 km.

Fig. 1 .
Fig. 1.Graphic interpretation of the assumptions of the determination procedure of the onset of the wear-out period based on the monitoring of the failure intensity function

Fig. 2 .
Fig. 2. General algorithm of the determination procedure of the onset of the wear-out period based on the monitoring of the failure intensity function

Table 1 .
Record of failure information of nonrenewable railroad objects [5]rce: based on[5]