Evaluation of thE procEss of milEagE growth during thE opEration of motor trucks , in sEvEral catEgoriEs of EnginE cubic capacity ocEna procEsu narastania przEbiEgu podczas

The intensity of use of motor vehicles and the range of the transport jobs performed change during the many-year operation of such vehicles. The changes are introduced in result of ongoing analyses of actual current vehicle operation costs, reliability, and performance characteristics. The changes in the vehicle operation process, taking place with vehicle’s age, were analysed on the grounds of the mileage of over 9 000 motor trucks. The analysis covered a 20-year period of vehicle operation. The analysis results were used to estimate the mathematical models that describe the basic characteristics of the mileage growth process, with changes in these characteristics as observed in the recent years along with an intensive development of the road transport in Poland. Models of vehicle mileage growth have been developed for seven categories of engine cubic capacity. The coefficients of model equations have been given and the accuracy of the mileage values calculated on these grounds has been comprehensively evaluated. The relative measures of the scatter in the mileage values obtained from these models do not exceed 12 % of the average values determined from experimental data. A procedure has been proposed that leads to evaluating the mileage growth process and is based on the experience having already been gained in this field. It has been shown that the mileage growth process is strongly related to the engine cubic capacity. The mileage growth process is an important source of information for the planning of vehicle operation, forecasting of costs, and estimating of exhaust emissions and energy consumption in the whole cycle of operation of motor vehicles by transport companies.


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(*) Tekst artykułu w polskiej wersji językowej dostępny w elektronicznym wydaniu kwartalnika na stronie www.ein.org.plLeon Prochowski Evaluation of thE procEss of milEagE growth during thE opEration of motor trucks, in sEvEral catEgoriEs of EnginE cubic capacity ocEna procEsu narastania przEbiEgu podczas Eksploatacji samochodów ciężarowych w kilku katEgoriach pojEmności silnika* The intensity of use of motor vehicles and the range of the transport jobs performed change during the many-year operation of such vehicles.The changes are introduced in result of ongoing analyses of actual current vehicle operation costs, reliability, and performance characteristics.The changes in the vehicle operation process, taking place with vehicle's age, were analysed on the grounds of the mileage of over 9 000 motor trucks.The analysis covered a 20-year period of vehicle operation.The analysis results were used to estimate the mathematical models that describe the basic characteristics of the mileage growth process, with changes in these characteristics as observed in the recent years along with an intensive development of the road transport in Poland.Models of vehicle mileage growth have been developed for seven categories of engine cubic capacity.The coefficients of model equations have been given and the accuracy of the mileage values calculated on these grounds has been comprehensively evaluated.
The relative measures of the scatter in the mileage values obtained from these models do not exceed 12 % of the average values determined from experimental data.A procedure has been proposed that leads to evaluating the mileage growth process and is based on the experience having already been gained in this field.It has been shown that the mileage growth process is strongly related to the engine cubic capacity.The mileage growth process is an important source of information for the planning of vehicle operation, forecasting of costs, and estimating of exhaust emissions and energy consumption in the whole cycle of operation of motor vehicles by transport companies.

Introduction and analysis of the current state of knowledge in this field
Motor trucks are selected according to the transport tasks planned.There are many factors of considerable importance for the vehicle selection, such as load capacity, unladen mass, fuel consumption, or engine cubic capacity.They are interrelated, e.g. the engine capacity has an impact on the power output and fuel consumption.During the many-year operation of motor vehicles, changes take place in the intensity of use of such vehicles and in the range of the transport jobs performed.The intensity of vehicle use is chiefly measured by the rate of growth in vehicle mileage.The vehicle mileage values constitute a basis for vehicle operation planning and cost forecasting.The vehicle mileage is taken into account in various ways to estimate e.g.insur-ance risk [6,15], exhaust emissions [1,5,12,27], or costs of fuel, tyres, or spare parts [16,23].The publications concerning this subject matter are predominantly dedicated to the intensity of use of motor cars [5,6,9,12,23].However, the average annual mileage of heavy goods vehicles (HGV) is many times as high as that of motor cars (MC).As an example, the HGV to MC mileage ratio reported for the UK in 2006 exceeded 3.5 [25].Moreover, the intensity of MC operation has been recently decreasing [13,20].
In Poland, a high rate of growth in the motor truck traffic is now observed [14].The mileage of motor trucks with up to 3 500 kg (3.5 t) gross vehicle mass (GVM) is often compared with that of motor cars [23,26].The light motor trucks, popularly referred to as local transport vehicles (LTV) or light commercial vehicles (LCV), are chiefly used for distribution-type jobs.According to catalogues [7], the mileage of motor trucks with up to 3.5 t GVM, after 10 years in service, exceeds that of motor cars with comparable engine cubic capacity by 30 %.By contrast, the mileage values for HGVs (with GVM exceeding 3.5 t) are higher than those for motor cars are by more than 300 %.Increasingly often attention is paid to the negative effects of road transport, which include the external costs such as exhaust emission effects and high fatality of road accidents [4,18].In the UK, the number of fatal accidents with HGV drivers is 1.8 per 100 million km and it is twice as high as that with motor car drivers [2].The high intensity of motor truck operation determines the rate of replacement of the fleet of such vehicles in Poland.In consequence, the motor trucks aged up to 5 years perform 49.5 % of cargo transport jobs (in tonkilometres) and those aged up to 10 years carry out as much as 82.4 % of such jobs [22,24].Motor trucks having been operated for more than 20 years are rarely seen on roads and their share in the transport work is 0.5 % [19].
The objective of this work is to evaluate the process of growth in the mileage of motor trucks with successive years of the vehicles being in service.The evaluation will be based on indicators related to several measures of the deviations of model data from empirical data.The problem will be analysed in connection with the engine cubic capacity value.The models having been developed will facilitate the analysis of the mileage growth rate, recognition of the major tendencies, and identification of the development trends in the successive periods of vehicle operation.The models of the mileage growth process are often the main source of information in the prognostic calculations of the trends to develop and of the values of the indicators that help to manage the operation of a motor vehicle fleet.The research conducted in this field provides anticipating information about vehicles' mileage growth, rate of approaching the planned or target mileage, current advancement in reaching this mileage and the maximum time of cost-effective vehicle use, termination of the vehicle use, replacement of vehicle fleet, or modifications to the operation of the fleet [3,10,11,21].
This work consists of several successive stages, which include analysis of data and estimation of the models and model accuracy.Detailed and current data about the intensity of operation of motor trucks (i.e.HGVs and LTVs), based on a large dataset, are not easily available.There is a lack of data about the LTV mileage growth process and about the relation between this process and the engine cubic capacity.There is also a lack of current information how the increasing number and diversity of motor trucks influenced the intensity of operation of such vehicles in Poland.

Characterization of the dataset
The information about motor vehicles and their mileage was collected on the grounds of: surveys carried out among motor truck drivers; information about vehicles' mileage and characteristics re-corded during mandatory periodical inspections carried out at Vehicle Testing Stations (VTS); analysis of post-accident motor truck inspection records.-Apart from other information, the dataset includes basic technical data of the vehicle, date of first registration, and mileage covered in Poland.The data concerning special motor vehicles and vehicles with load capacity below 500 kg as well as vehicles with monthly mileage values below 100 km/m and above 25 000 km/m were excluded from the dataset; the upper limit was adopted because in practice the very high monthly mileage values are hardly achievable in the conditions of road traffic in Poland.The vehicle models that predominated in the dataset have been specified in Tables 1a and 1b.
Note: The GVM and engine cubic capacity values have been confirmed in catalogues [7].
In the dataset, the same vehicle makes can be seen that predominate in numerical summaries of the first registrations in Poland.For example: Citroen, Fiat, Ford, Mercedes-Benz, Peugeot, Renault, and Volkswagen predominate in the set of local transport vehicles.The same vehicle makes have been indicated as predominating among the first registrations of LTVs in the Automotive Industry Reports of 2008, 2014, and 2016 [22].In the group of vehicles with GVM exceeding 3.5 tons (metric), DAF, Iveco, MAN, Mercedes-Benz, Renault Trucks, Scania, and Volvo Trucks are the makes that predominate in the number of HGV sales in the Polish market and in the number of first registrations of such vehicles [22].The same makes predominate in Table 1b.

Procedure of evaluation of the mileage growth process
Very wide scatter can be observed in the motor truck mileage values after vehicle operation periods of the same duration.This scatter results from very diverse transport jobs performed and from variations in the intensity of vehicle operation.This characteristic feature of the motor truck mileage is confirmed by multiannual statistics.For illustration: the average annual mileage of such vehicles in the UK has been determined as 79 000 km with a standard deviation of 63 000 km, according to [17].
To find a relatively representative image of the process of growth in the motor truck mileage in Poland with taking into account the above as well as the results of the calculations carried out previously [19], a procedure consisting of the following stages was prepared: data collection; analysis of the data, with dividing the dataset into subsets; estimation of the average mileage values with determining the coefficient of variation; removal of outliers; approximation of the relation between the mileage and the ve-hicle operation time; estimation of the mileage growth models; evaluation of the accuracy of the estimators determined for the mileage growth models.
In the calculations, the following notation was adopted for the basic quantities: L i -mileage covered by the i th vehicle during time t i ; L 10 , L 20 -total vehicle mileage after 10 and 20 years of vehicle operation, respectively; X, M, Q -arithmetic mean, median, and quantile, respectively; S -standard deviation; W -relative coefficient of variation; R, φ -coefficient of determination and fraction of variance unexplained (FVU), respectively.
The scope of the work done at individual stages of the procedure, denoted by A1, A2, etc., has been described below.

A1.
The data collection stage pertains to the motor trucks that are currently participating in the road cargo transport.In result of this stage, a representative dataset should be obtained that would include at least the technical specifications, mileage, and period of operation of individual vehicles.At this stage, it is important that various information sources should be used for the dataset formed to be representative.
A2.The analysis of the data collected should make it possible to clear the dataset, in a reasonable way, of the data that might adversely affect the calculation results, e.g. to remove the data of motor cars and car-based vehicles, which may be registered in Poland as accepted for cargo transport (or, more precisely, as passenger/cargo vehicles).At this stage, certain subsets are separated, which consist of uniformcategory vehicles (e.g.vehicles for local, interurban, or international transport).The vehicles of individual categories are used for different transport jobs, which have an impact on the vehicle mileage.The division of a dataset depends on the dataset size and on the exact objective of the calculations.
The data analysis was based on an assumption made that it would cover a 20-year period of vehicle operation and the distance travelled by the end of such a period was referred to as "target mileage" (expressed in kilometres).The vehicles that were operated within this period for less than 3 months or that were more than 242 months old were not taken into account.The vehicles operated for only the first two months of the period under analysis were ignored because of very wide scatter in the values of the mileage covered by them during that time.
At this stage, the motor trucks of each category were divided into n subgroups.The vehicles whose operation time t i was within the interval: were counted in the k th subgroup.The limits of disjoint time intervals were defined with a monthly step, i.e.: 12( 1) and, additionally, T 1 = 2 months and T 21 = 242 months. ( The separation of subgroups in which the numbers of vehicles would be adequate for research purposes (i.e. would be not less than a required minimum denoted by m 0 ) makes a basis for further calculations.In many cases, vehicles are divided into subgroups that cover successive vehicle operation periods whose length is T 0 = 12 months.
A3.At the stage of estimation of the average mileage values in individual subgroups, the scope of calculations was defined with taking into account, inter alia, the following premises: the arithmetic mean is an estimator rather insusceptible to an error in the evaluation of the properties of a dataset but it is susceptible to outliers (extreme values).the median is rather insusceptible to outliers, but it is unsuitable for statistical calculations.the average monthly mileage value calculated by dividing the total mileage value by the number of months of the vehicle operation period is burdened with an error arising from changes in the mileage growth rate with vehicle operation time.
In each subgroup, the vehicle mileage L was treated as a random variable and L i was the i th value of this variable.In the k th subgroup that consisted of m ≥ m 0 vehicles, the following operations were carried out: B1.
Forming of a series of mileage values 0 1 { ,..., ,..., } L L L L consisting of m members arranged in ascending order.
B2. Calculation of the estimators: arithmetic mean X 0k (τ k ), median M 0k (τ k ), quantiles Q 1k (τ k ) and Q 3k (τ k ), and relative coefficient of variation W 0k (τ) as the following quotient: where S 0k is standard deviation, τ k is the central value of the vehicle operation period in the k th subgroup, and Q 1k (τ k ) and Q 3k (τ k ) are the lower and upper quantile, respectively.
In result of these calculations, the following series of discrete values were obtained for every vehicle category: A4. Results of the calculations carried out at stage A3 make it possible to estimate the coefficient of mileage variation, the value of which shows whether the removal of outliers is necessary.High W 0k values indicate excessive scatter in the mileage values in the k th subgroup.If the scatter is wide, the average mileage value in a specific subgroup not always adequately characterizes the mean mileage level.In such a case, efforts should be made to reduce the impact of the values that most differ from the average in the subgroup on the results of further calculations.The steps of this kind taken in the k th subgroup may be divided into two sub-stages.At the first one, a series denoted by L Ek was formed by removing 10 % of the first and last members of L 0k , i.e. the lowest and highest mileage values.Thus, the L Ek series consisted of m − 2m E members, i.e.
where operator E has the meaning of rounding off to the nearest integer.
At the second sub-stage, a series denoted by L Qk was formed by clearing L 0k of the members where the L i values were covered by the criterion: Following this operation, the L Qk series consisted of members situated centrally in L 0k and falling within the range After the removal of the outlying mileage values, the numbers of vehicles in individual subgroups decreases.In consequence, the numbers of vehicles in some subgroups may become m < m 0 .If the k th subgroup consists of less than m 0 vehicles, it should be merged with subgroup k+1.The subgroup thus combined will consist of vehicles whose operation time would fall within the interval: which is twice as wide as those considered previously.
Therefore, an operation was carried out to reduce the width of such "double" intervals to T 0 months with maintaining the natural scatter in the vehicle mileage values.With this objective in view, the mean mileage value X (k) and the central value of the vehicle operation time τ (k) for the combined subgroup was calculated.Now, the X (k) and τ (k) values were used to determine the deviation of the monthly mileage (PM i ) for the i th vehicle of the subgroups combined together: For the vehicles where |t i − τ (k) | ≥ 0.5T 0 (i.e. whose operation time was longer or shorter than τ (k) by more than 0.5T 0 months), a reduced mileage value L (i) and a reduced vehicle operation time t (i) were calculated as follows: In result of the above calculations, new parameters denoted by L (i) and t (i) were obtained for the i th vehicle.They were substituted for the parameters L i and t i used previously and they already fell within the time interval: The new subgroup, formed by merger of subgroups k and k+1, has characteristics similar to those of the subgroups that consisted of at least m 0 vehicles when they were initially defined.After these calculations, the diversity of the vehicle mileage and operation time values remained close to that recorded originally, although the said values were reduced to fall within the interval as defined by (12).The diversity of the mileage values L i must be maintained in the subgroup merging process for the same procedure, with two sub-stages of removing the mileage outliers, to remain applicable when forming the L Ek and L Qk series.
The values of members of the L Ek and L Qk series were used to determine series of the following discrete values for each vehicle category: where: X E,Q represents series X E and X Q , respectively, and n 0 < n in result of the merger of some subgroups.

A5.
At this stage, a transition takes place from the series of discrete averaged values M 0 , X E , and X Q in vehicle subgroups to a continuous function.This is done by approximating the dependence of the averaged vehicle mileage values on the vehicle operation time.In this work, the approximation was based on regression lines.In result of the approximation process, continuous functions in the form as follows were obtained, describing the vehicle mileage growth with vehicle operation time: In consideration of the properties of the process of approximation based on regression lines, including strong dependence of the course sciENcE aNd tEchNology of such lines on the degree of the approximating polynomial and on the number of the data available and on their nature, the approximating functions were determined in two ways: by using models with a 3 -rd degree polynomial based on three series, i.e.M 0 , X E , and X Q , the following was calculated: , , , , ˆ( ) by selecting models with 2 -nd and 4 th degree polynomials based on the X Q series of values, the following was obtained: ˆ( ) where subscripts W2 and W4 correspond to the degree of the polynomial.
In total, five approximating functions were calculated for each vehicle category.
A6.The estimation of the vehicle mileage growth models so that they were burdened with the smallest possible error was based on the combined use of approximating functions ( 15) and ( 16) for determining averaged functions on these grounds.The method of moving average was used to calculate the values: that provided a basis for the estimation of the averaged approximating function: The averaged approximating function (17) was treated as an estimator of the mileage growth model prepared for motor trucks.
A7.The evaluation of the accuracy of this estimator was based on absolute and relative indicators.The following absolute indicators were used (based on [8,28]): standard error of estimation: -0 2 , , , , The relative indicators used included: coefficient of determination related to the values of -X E , X Q , and M 0 , i.e. ) coefficient of variation in the error of estimation: - , ,  , , , , 100% where: , , EQM y -average value of series X E , X Q , and M 0 , respectively; fraction of variance unexplained (FVU): - ) The symbol S E,Q,M covers three quantities, i.e. S E , S Q , and S M , which are calculated from X E , X Q , or M 0 , as appropriate.The same subscripts have been used in equations ( 18)- (22).
The determination of the estimator evaluation indicators is the final step of the procedure, which was followed in practice at the evaluation of the process of growth in the motor truck mileage during the 20-year vehicle operation period.

Example implementation of the procedure developed
At the initial data collection stage, the vehicles were divided into two groups, based on the GVM value, i.e. up to and above 3.5 tons (metric).Thus, two separate datasets were formed, which covered 5 084 vehicles of the first group and 3 855 vehicles of the other one.The distribution of the number of vehicles, by age, in both groups has been shown in Fig. 1.Both in the dataset and in the real road cargo transport, the vehicles having been operated for up to 10 years predominate.
In Fig. 3, the engine cubic capacity values have been grouped in ranges of up to 1 000 cm 3 , 1 001-1 200 cm 3 , and so on.In consideration of the above, the distribution of engine capacity values as shown in Fig. 2, and classification adopted in [22,24], the vehicles under analysis have been divided into engine capacity categories as follows: four categories in the group of local transport (delivery) vehi-cles (LTV); three categories in the group of heavy goods vehicles (HGV, with more than 3.5 metric tons GVM).
The limits of individual engine capacity categories and percentage of vehicles of individual categories in both groups have been presented in Table 2.
In each category, the vehicles were divided into n = 20 subgroups, based on the vehicle operation time, which was determined with 1 month accuracy, according to (1)-( 3).The subgroups varied in size, with 63 vehicles per subgroup, on average, in the group of vehicles with GVM of up to 3.5 t and 64 vehicles per subgroup, on average, in the group of vehicles with GVM of above 3.5 t.The first subgroup consisted of vehicles whose operation time ranged from 3 months to 12 months; in the other one, there were vehicles having been operated for 13-24 months.When forming the subgroups, an assumption was made that a subgroup should consist of not less than m 0 = 20 vehicles.However, there were more than 10 subgroups where the number of vehicles remained below this minimum, in spite of a long data collection time (3 years).For these subgroups, the calculations to be done at stage A3 were not carried out.Only when these subgroups were merged with the next ones (at stage A4), the data about the vehicles of these subgroups were taken for further computations.
In the subgroups, the estimators of arithmetic mean X 0k (τ k ), median M 0k (τ k ), quantiles Q 1k (τ k ) and Q 3k (τ k ), and relative coefficient of variation W 0k (τ k ) were calculated.Example results of calculations carried out for the subgroups of vehicles belonging to category S02 have been presented in Fig. 4. Gradual growth in X 0k and M 0k with increasing τ k as well as high values of the coefficient of variation W 0k can be seen in the graphs.High values of the relative coefficient of variation (W > 0.4) can be seen in Fig. 4 and for many vehicle subgroups, especially for k ≤ 4.This shows that the average value not always adequately characterizes the mean mileage level.Therefore, 10 % of the first and last members were removed at stage A4 from L 0k in the subgroups and thus the L Ek series were formed.Then, on the grounds of results of calculation of quantiles Q 1k (τ k ) and Q 3k (τ k ), the L 0k series were cleared of the vehicles whose mileage was covered by the following criterion: In result of the above, the series L Ek and L Qk of mileage values were formed, whose members were situated centrally in L 0k .
When these operations were completed, all the 20 × 7 = 140 subgroups under analysis included 37 ones with m < 20 (i.e. which consisted of less than 20 vehicles).Therefore, the undersized subgroups were merged with the next ones.The new series of mileage values L Ek and L Qk were used for determining discrete estimator values, based on (4) and (13), i.e.: where, as an example: S Ek and S Qk are standard deviations and W E and W Q are coefficients of variation in mileage values in series L Ek and L Qk .
Examples of the calculation results have been presented in Table 3.
These calculation results were used to evaluate the effectiveness of the outliers-removal operation.Attention was paid to the impact of the outliers-removal operation used on the average mileage values in individual subgroups and on the values of the coefficient of variation.The X 0k , X Ek , and X Qk values in Table 3 do not differ very much from each other.They insignificantly changed with the removal of outliers in individual subgroups, in particular: X -0k > X Ek > X Qk , when the average values were chiefly affected by very high mileage values L i ; X -0k < X Ek < X Qk , when a predominating impact was exerted by very low values L i .
An analysis of values W{W 0 , W E , W Q }, based on calculation results obtained for all the vehicle categories, revealed beneficial effects of the removal of outliers from L E and L Q .This is confirmed by the W E and W Q values being definitely lower than W 0 (see Table 3).Desirable effects of the outliers-removal operation has also been illustrated in Fig. 5 by the relations between the extreme L i values in series L 0 , L E , and L Q and the X 0k , X Ek , and X Qk values.The calculation results have been presented in percentage terms, where the average values in individual subsets were assumed as 100 %.Fig. 6 shows example curves representing the approximating functions determined according to (15) and (16).Figs 6a and 6c shows values X Ek , X Qk , and M 0k (points in the graphs) and the functions that approximate the mileage growth process for vehicles of categories S03 and S12, based on models with a 3 rd degree polynomial and marked  In Figs 6b and 6d, the X Qk values and the approximating functions based on models with 2 nd and 4 th degree polynomials and marked as Polynomial XQ2 and Polynomial XQ4, respectively, have been presented.
In result of an analysis of the approximating functions, the following findings have been formulated: The comparison of the approximating functions calculated ac-cording to (15) showed small differences between them within the range of 6-14 years of vehicle operation (e.g. in Fig. 6a).
In several cases, the use of a 3 -rd degree polynomial resulted in the showing of an excessive growth in the mileage value in the 18 th , 19 th , and 20 th year of vehicle operation (Fig. 6c).The use of 2 -nd degree polynomials eliminated the deviations mentioned above but the value of the coefficient of determination became considerably lower than that for polynomials of higher degrees.The use of a 4 -th degree polynomial resulted in a formal improvement in the model quality, manifested in a growth in the coefficient of determination; however, the approximating function showed excessively dynamic changes in the final part of the vehicle operation period (e.g.fluctuations in the function values, see Figs 6b and 6d).These findings give reasons for a need of going through the next stage of the procedure presented, i.e. the averaging of the approximating functions calculated as described above.

Estimation of the mileage growth models
An example of using the approximating functions ( 15) and ( 16) to determine their average values Y Ak and to define an averaged approximating function has been presented in Fig. 7.In the graphs, the average mileage values directly calculated from the initial data (points X Ek , X Qk , and M 0k ) have been compared with the final result shown in the form of a mileage growth model ( ) ˆA y τ (τ) or two vehicle categories S02 and S10.It can be seen in Fig. 7 that the curves obtained from the mileage growth model are situated close to the points that represent the average values directly calculated from the experimental data.
Fig. 8 shows a comparison between the curves representing the averaged approximating functions (17)  The values of coefficients 123 ( , , ) r a a a a of equation ( 23) have been given in Table 4.

Evaluation of the conformity of models with empirical (initial) data
The mileage growth process described by the estimation of the averaged approximating function (17) was evaluated in several steps.Within this evaluation, the mileage values calculated from the models were compared with average values X E , X Q, and median M 0 , directly based on experimental data.Thus, the evaluation of the accuracy of the estimators determined for the mileage growth models was based on indicators of conformity of model values with empirical data.Results of the calculation of values of the indicators of conformity have been given in Table 5.
The values of standard errors of estimation S E , S Q , and S M are the basic measure of the scatter in mileage values around the values determined from the model equations; the V E , V Q , and V M values define this scatter in percentage terms.The scatter in values X E , X Q , and M 0 around the mileage values determined from the model does not exceed 12 %.The values of the coefficient of determination R 2 of regression equations (23) for the coefficients specified in Table 4 are 0.99, i.e. these equations represent 99 % of the information contained in the sets of values Y Ak (cf.Fig. 7).Figures of special importance for

Fig. 1 .
Fig. 1.Age (years of operation) of motor trucks in groups of up to and above 3.5 t GVM

Fig. 4 .
Fig. 4. Results of calculations of the X 0k , M 0k , and W 0k values in the subgroups of vehicles belonging to category S02

Fig. 5 .Fig. 6 .
Fig. 5. Extreme (MIN and MAX) and mean mileage values in series L 0 , L E , and L Q , calculated for the S02 category vehicles (in percentage terms) and compared with each other for τ k = 2.5, 10.5, and 18.0 years of vehicle operation determined at stage A6 of this procedure.The mileage growth models have been based here on polynomials: -vehicle mileage [km] and τ -vehicle operation time [years].

Table 1b .
Motor trucks belonging to the group of GVM > 3.5 t

Table 2 .
Limits of individual engine capacity categories [cm 3 ] and percentage of vehicles of individual categories in both groups

Table 3 .
(15)es of the mileage and coefficient of variation for a few values of τ k Average values of coefficient W, calculated for 20 years of vehicle operation; the L 20 values were calculated from the approximating functions ŷ according to(15). *

Table 4 .
Values of coefficients a r of the equations that model the vehicle mileage growth process for seven categories of engine cubic capacity as well as positions of points X Ek , X Qk , M 0k , and average values Y Ak ; the example pertains to vehicles of categories S02 and S10Fig.8.Models of the process of vehicle mileage growth with vehicle operation time