Study of Phase Changes in Operational Risk for Trucks

Abstract

This study concerns the management of operational risk in truck transport using the reliability theory of risks. In this regard, the risk analysis of changes in the vehicle unavailability represents an important topic. In this study, the authors present their own method for analysing the phase changes in risk corresponding to successive sections (phases) of vehicle mileage. The presented risk analysis method is based on an integrated assessment of losses associated with the costs of incidental repairs and losses caused by lost income during vehicle downtime. This includes the following: assessment of differences in average risk and differences in the coefficient of variation in the time series of vehicle mileage phases, indicating the outliers and extremes of phase risk, identifying their physical causes and testing the statistical significance of phase risk differences. The proposed method is described mathematically and verified experimentally based on the operational data concerning trucks from two selected brands (20 trucks from each brand). We show that the method can be used to predict the continuity of transport services in the short term (one-year horizon). The method could also be useful to compare vehicles of different brands in the context of their sensitivity to operational risks.

1. Introduction

The dynamic growth of road freight transport companies, particularly in developed countries, has been accompanied by continuous improvements in management methods [1,2]. There has also been a widespread increase in customer expectations concerning the quality and reliability of transport services [3]. At the same time, the legal requirements for sustainable development and environmental protection standards in transport are increasing [4,5]. In this situation, it becomes increasingly important for transport companies to tackle the problem of predicting risks to the continuity of operation. These risks may be caused, for instance, by the insufficient reliability of vehicles [6] or management errors [7].

In recent years, a new trend has appeared in vehicle fleet management, based on the reliability theory of operational risks and risk modelling [8].

Relevant research can be used to adopt a general breakdown of the methods of risk assessment in road transport into qualitative and quantitative assessments [7]. The group of quantitative assessments includes studies of risks associated with the physical safety of motorists and studies connected with operational costs related to vehicle technical faults or fleet management errors. The risk associated with operational factors is referred to as operational risk.

In this study, the authors define “operational risk” as the “risk of all incidents occurring during road transport that had a negative effect on the provision of a transport service with specific operational costs and according to defined logistics and quality parameters” [9].

An important topic in the research on operational risk in transport is the analysis of phase risk variations. The analysis is based on the time series of risk phases, which corresponds to successive sections of the operational mileage of vehicles [10,11].

The results of risk analysis could be used to formulate a prediction and determine the conditions required to ensure uninterrupted transport services in the short-, medium- and long-term horizons (quarterly, annual and multi-annual) [10,12].

The existing research includes many studies on the dynamics of forecasting phenomena occurring in logistics and transport systems [13,14]. Numerous detailed case studies on preventing operational liquidity threats have also described, including managing potential workforce reserves [15], maintaining strategic reserve assets [16] and ensuring high-quality service procedures [17]. The authors of this paper previously analysed cumulative risk in the context of its usefulness for long-term forecasting [18,19,20]. In this study, however, the authors analysed changes in the risk of unavailability of a vehicle fleet in terms of their usefulness for short-term forecasting. Changes in the risk in successive sections (phases) of the operational mileage of vehicles with a length of 30 and 60 thousand km were analysed.

It was assumed that monitoring phase risk variations could be an effective method for predicting the unavailability of the fleet.

The authors assumed a general model of risk changes described with a time series, being the sum of the trend, a phase component of seasonal variations and a random component of incidental deviations.

The risk calculations were performed using the “model of the integrated risk of the operational incapacity of the vehicles” [18,20]. This model can be used to assess the impact of the technical failures of vehicles and undesirable human actions in the “carrier–vehicle–user” system, particularly with respect to the risks connected with vehicle reliability. The model covers, similarly to [21], only random threads. However, unlike in [22,23], where risk assessment was primarily based on environmental criteria and legal standards, the authors’ model is limited to a detailed assessment, consistent with the rules of mathematical description. In contrast to [24], which presented an econometric method for describing the vehicle operation process, the assessment model in this study has a technical–economic nature.

The content of this paper is organised as follows: Section 1 describes the subject of the research and the research problem. Section 2 concerns the primary assumptions for the probabilistic modelling of the integrated risk of the operational incapacity of the vehicle, including the phase risk of incapacity, as part of the description of the risk analysis method. Section 3 concerns the experimental tests and describes the research objects and conditions as well as the results of tests of the incapacity risk. Section 4 deals with the statistical examination of the significance of phase risk differences. The hypotheses were verified using the analysis of variance. A discussion of the quantitative results is included in Section 5. Finally, in Section 6, the authors include a summary and point out the advantages of the developed risk analysis method.

2. Risk Analysis Method

2.1. Model of the System of Transport Services

Considering vehicle reliability and referring to the definition of the anthropo-technical system presented in research papers, e.g., [8,25], the authors introduced their own definition of the system of transport services, consisting of the carrier (service provider)–vehicle–user (customer) (Figure 1). The purpose of this system is to provide the transport service ordered by the customer. It is important that the contract for the transport service between the carrier and the user has a specific guarantee of successful delivery.

If the service is not performed or is performed late or contrary to the order, this is equivalent to the incapacity of the system.

2.2. Integrated Risk of the Operational Incapacity of the Vehicle

To conduct the quantitative analysis of risk, the authors used “model of the integrated risk of the operational incapacity of the vehicles $R\left(l\right)$” [18,20], which aggregates two components referred to as partial risks.

$$R\left(l\right)=R_N\left(l\right)+R_G\left(l\right)$$

(1)

where:

$l$—operational mileage of the vehicle (km);

$R\left(l\right)$—integrated risk of operational incapacity;

$R_N\left(l\right)$—risk of losses connected with costs of incidental repairs;

$R_G\left(l\right)$—risk of losses connected with the absence of income during vehicle downtime.

After considering the fact that the risk is a product of the probability of its occurrence and severity of the risk, Formula (1) can be presented as a sum of the products of unreliability and risk:

$$R\left(l\right)=M_N\left(l\right)·N_N\left(l\right)+M_G\left(l\right)·N_G\left(l\right)$$

(2)

where:

$M_N\left(l\right)$—measure of unreliability due to technical failures of the vehicle [-];

$N_N\left(l\right)$—measure of the severity of the risk due to technical failures (damage) (PLN);

$M_G\left(l\right)$—measure of system unreliability due to vehicle unavailability [-];

$N_G\left(l\right)$—measure of the severity of the risk due to unavailability (PLN).

The first component in Formula (2) expresses the partial risk of system incapacity due to technical faults and vehicle repair. The measure of vehicle unreliability, in this case, is the ratio of the cumulative cost of incidental repairs to the expected threshold income, as in Formula (3).

$$M_N\left(l\right)=\frac{N\left(l\right)}{P\left(l\right)}$$

(3)

where:

$N\left(l\right)$—cumulative cost of incidental repairs of the vehicle at the $l$ mileage (PLN);

$P\left(l\right)$—required threshold income from the transport services using the vehicle at the $l$ mileage (PLN).

To avoid a deficit, the authors assume that $N\left(l\right)\le P\left(l\right)$. It was assumed that the severity of the risk was defined by the formula $N_N\left(l\right)=P\left(l\right)$, and the required income $P\left(l\right)$ was equal to the decrease in the residual value of the vehicle at the $l$ mileage, i.e.,

$$P\left(l\right)=S_p\left(l\right)=C_0-C\left(l\right)$$

(4)

where:

$S_p\left(l\right)$—decrease in the residual value of the vehicle (PLN);

$C_0$—cost of acquisition of the vehicle;

$C\left(l\right)$—residual value of the vehicle at the operational mileage $l$ (PLN).

Assuming a measure of unreliability according to Formula (3) and a measure of the severity of the risk according to Formula (4), the partial risk of losses connected with the costs of faults and repairs $R_N\left(l\right)$ can be given by Formula (5).

$$R_N\left(l\right)=\frac{N\left(l\right)}{P\left(l\right)} ·P\left(l\right)$$

(5)

The measure of unreliability $M_g\left(l\right)$ of the vehicle at the mileage $l$ due to its unavailability was assumed to be technical unavailability $U\left(l\right)$ described with Formula (6).

$$M_g\left(l\right)=U\left(l\right)=\frac{U_0\left(l\right)}{T_0\left(l\right)+U_0\left(l\right)}$$

(6)

where:

$T_0\left(l\right)$—time of effective use of the vehicle at the mileage $l$;

$U_0\left(l\right)$—duration of vehicle downtime due to technical or organisational reasons at mileage $l$.

The measure of the severity of the risk $N_G\left(l\right)$ of losses connected with the absence of income during vehicle downtime was adopted as the lost threshold income $P\left(l\right)$.

$$N_G\left(l\right)=P\left(l\right)$$

(7)

Ultimately, the sub-risk of incapacity $R_G\left(l\right)$ due to the absence of income caused by unscheduled downtime can be expressed by Formula (8).

$$R_G\left(l\right)=U\left(l\right)·P\left(l\right)$$

(8)

Thus, the cumulative risk of operational incapacity $R\left(l\right)$ has the form of Formula (9):

$$R\left(l\right)=\frac{N\left(l\right)}{P\left(l\right)} ·P\left(l\right)+U\left(l\right)·P\left(l\right)=\left(\frac{N\left(l\right)}{P\left(l\right)}+U\left(l\right)\right)P\left(l\right)$$

(9)

The risk $R\left(l\right)$ defines the cumulative amount of financial expenses required to guarantee the continuity of operation (uninterrupted availability) of the vehicle until reaching operational mileage $l$.

2.3. Phase Risk of the Operational Incapacity of the Vehicle

The research problem in this study concerns the comparative analysis of the average risk due to the operational incapacity of the vehicle in the selected mileage phases. To experimentally clarify this problem, the authors examined the significance of differences in the phase risk of operational incapacity based on the databases of operational data for two vehicle brands. The design of the phase risk of operational incapacity of the vehicle was consistent with the assumptions shown in Section 2.1.

To this end, the total vehicle mileage $l$ was divided into $k$ mileage phases $\left(0, l_1\right), \left(l_1, l_2\right), \dots , (l_{k-1}, l_k)$ of equal length, $\Delta l$ and $l_k=l$. The assumption concerning the equal length of the phases is not necessary, but it was adopted to improve the transparency of the developed risk analysis method.

The integrated operational risk $R\left(l_{i-1},l_i\right), i=1, 2,\dots ,k$ in the $i$-th phase of vehicle mileage is a random value including two sub-risks and has the form of sum (10):

$$R\left(l_{i-1},l_i\right)=R_N\left(l_{i-1},l_i\right)+R_G\left(l_{i-1},l_i\right)$$

(10)

where:

$R_N\left(l_{i-1},l_i\right)$—random risk of losses connected with costs of incidental repairs in the $i$-th phase of vehicle mileage;

$R_G\left(l_{i-1},l_i\right)$—random risk of losses connected with the absence of income during vehicle downtime in the $i$-th phase of vehicle mileage.

After considering that the risk is a product of the probability and severity of the risk, Formula (10), similar to (2), can have the form of a sum of the products of unreliability and risk in the $i$-th mileage phase:

$$R\left(l_{i-1},l_i\right)=M_N\left(l_{i-1},l_i\right)·N_N(l_{i-1},l_i)+M_G\left(l_{i-1},l_i\right)·N_G\left(l_{i-1},l_i\right)$$

(11)

where:

$M_N\left(l_{i-1},l_i\right)$—measure of the random unreliability of the vehicle in the $i$-th mileage phase due to its technical faults [-];

$N_N(l_{i-1},l_i)$—measure of the severity of the risk in the $i$-th phase due to technical faults (damage) (PLN);

$M_G\left(l_{i-1},l_i\right)$—measure of random unreliability in the $i$-th phase due to vehicle unavailability [-];

$N_G\left(l_{i-1},l_i\right)$—measure of the severity of the risk in the $i$-th phase due to unavailability (PLN).

The measure of the random unreliability $M_N\left(l_{i-1},l_i\right)$ in the $i$-th vehicle mileage phase due to its technical faults [-] is defined as quotient (12) of the random cost of incidental repairs $N\left(l_{i-1},l_i\right)$ and the predicted threshold income $P\left(l_{i-1},l_i\right)$ in that phase.

$$M_N\left(l_{i-1},l_i\right)=\frac{N\left(l_{i-1},l_i\right)}{P\left(l_{i-1},l_i\right)}$$

(12)

To avoid a deficit in the considered mileage phases of the vehicle, the authors made the following assumption: $0\le N\left(l_{i-1},l_i\right)\le P\left(l_{i-1},l_i\right)$ for $i=1, 2,\dots k$.

The measure of random unreliability $M_g\left(l_{i-1},l_i\right)$ in the $i$-th vehicle mileage phase due to its unavailability was adopted by the authors as technical unavailability $M_G\left(l_{i-1},l_i\right)=U\left(l_{i-1},l_i\right)$ in that mileage phase defined by Formula (13).

$$M_G\left(l_{i-1},l_i\right)=\frac{U_0(l_{i-1},l_i)}{T_0(l_{i-1},l_i)+U_0(l_{i-1},l_i)}$$

(13)

where:

$T_0(l_{i-1},l_i)$—random time of effective use of the vehicle in the $i$-th phase of its mileage;

$U_0(l_{i-1},l_i)$—random duration of vehicle downtime due to technical or organisational reasons in the $i$-th phase of its mileage.

For the adopted assumptions, the integrated risk of operational incapacity $R\left(l_{i-1},l_i\right)$ in the $i$-th vehicle mileage phase is a random value and has the following form (14):

$$R\left(l_{i-1},l_i\right)=\left(\frac{N\left(l_{i-1},l_i\right)}{P\left(l_{i-1},l_i\right)}+U\left(l_{i-1},l_i\right)\right) P\left(l_{i-1},l_i\right) \mathrm{f}\mathrm{o}\mathrm{r} i=1, 2,\dots ,k$$

(14)

Formula (14) presents a probabilistic model of the integrated phase risk of the operational incapacity of the vehicle in the $i$-th mileage phase. The expected value of this risk given using the symbol $\mathbb{E}$ is defined in Formula (15):

$$\mathbb{E}R\left(l_{i-1},l_i\right)=\mathbb{E}N\left(l_{i-1},l_i\right)+P\left(l_{i-1},l_i\right)·\mathbb{E}U\left(l_{i-1},l_i\right) \mathrm{f}\mathrm{o}\mathrm{r} i=1, 2,\dots ,k$$

(15)

The variance ${\mathbb{D}}^2$ in this risk is defined by Formula (16):

$${\mathbb{D}}^{2}R\left(l_{i-1},l_i\right)={\mathbb{D}}^{2}N\left(l_{i-1},l_i\right)+P^2\left(l_{i-1},l_i\right)·{\mathbb{D}}^{2}U\left(l_{i-1},l_i\right) \mathrm{f}\mathrm{o}\mathrm{r} i=1, 2,\dots ,k$$

(16)

Assuming the arithmetic averages as the estimators of unknown expected values, the point score of the expected integrated risk of operational incapacity has the following form (17):

$$\overline{R}\left(l_{i-1},l_i\right)=\overline{N}\left(l_{i-1},l_i\right)+P\left(l_{i-1},l_i\right)·\overline{U}\left(l_{i-1},l_i\right) \mathrm{f}\mathrm{o}\mathrm{r} i=1, 2,\dots ,k$$

(17)

where the symbols with an upper line refer to the arithmetic averages of the individual values determined from a random sample.

3. Experimental Research

3.1. Research Objects and Conditions

The operational research was carried out on two samples of vehicles of selected brands, with 20 vehicles per brand. For the purposes of this study, the samples are referred to as I and M, and the individual vehicles in each sample are numbered with the number $j=1, 2, \dots n$. They were trucks with an average capacity and a gross vehicle weight of 12 tons. They were used by a single company, for similar functions in distribution transport. The technical specifications of the investigated vehicles are given in

Table 1. Specifications of the examined vehicles.

Characteristic	Vehicle Brand
	M	I
General specifications	two-axle, closed-box
Type-approval category	N2
Engine type	compression ignition
Engine displacement (dm3)	5.1	3.9
Max. engine power (kW)	130	134
Maximum grossweight (kg)	12,000
Load capacity (kg)	5.250	5.395
Cost of acquisition (PLN thousand)	208	175

.

The investigated vehicles were put into service on similar dates. The distribution of the initial operational mileage of both vehicle brands at the start of the research was small and did not exceed about a dozen kilometres. Consequently, it was found that the research sample met the homogeneity criteria.

During the research period, the operational mileage of the vehicles amounted to 242.6 thousand km for brand M, and they were driven (on average, in the sample of the particular brand) over 56 months. For vehicles of brand I, the total average mileage was 240.1 thousand km during 60 months of research. The detailed data are given in

Table 2. Descriptive statistics of the research sample of vehicles.

Parameter	Brand
	M	I
Size (numerousness) of the sample [-]	20	20
Average mileage in the examined period (thousand km)	242.6	240.1
Standard deviation (thousand km)	14.0	11.6
Maximum mileage (thousand km)	326.3	280.4
Average annual mileage (thousand km)	52.6	47.2
Average monthly mileage (thousand km)	4.4	3.8
Average observation time (calendar months)	56	60
Standard deviation of average observation time (calendar months)	2.2	2.1

.

The following parameters were recorded for each vehicle: mileage, repair description and repair cost. In line with the research problem specified in this paper, only unscheduled (incidental) repairs were considered in the observations.

Thus, the costs of fuel, fluids, periodic maintenance, insurance and taxes were omitted. Examples of entries in the repair records are given in

Table 3. Sample repair records for a vehicle of M brand.

Repair Description	Amount (PLN)	Mileage (km)
Tyre inspection	18.00	5585
Leaf spring pads, 2 pcs.—replacement	110.26	28,000
Overspeed limitation devices inspection	180.36	33,059
Flexible hose + hanger—replacement	605.18	36,410
Right direction indicator—replacement	100.55	46,834
Tyre inspection	144.00	47,528
Wheel balancing	90.00	47,528
Right side window—replacement	322.65	57,021
Brake system bleeding	40.00	99,185
Leaf spring pad (rear right)—replacement	195.11	99,800
Leaf spring pad (rear left)—replacement	195.11	105,978
Brake system—bleeding	40.00	106,352
Pneumatic hose—replacement	318.8	112,812
Repair of the electrical system of the lighting + replacing missing elements	104.3	128,532
Water pump + in-warranty replacement	0.00	128,532
Reverse sensor—replacement	262.98	128,532

.

Due to the absence of information about the duration of repair, it was decided that a single repair would result in an entire day of vehicle downtime. If more than one repair of the same vehicle was performed during a single day, the costs of the individual repairs were added together.

3.2. Research Results—Integrated Phase Risk

The integrated phase risk values determined experimentally using model (14) for each vehicle of brands M and I, broken down into phases (ranges) of the operational mileage with a length of $\Delta l=30$ thousand km, are given in

Table 4. Integrated phase risk in the sample of M vehicles for $\Delta l=30$ thousand km (thousand PLN).

Mileage Phase [thsnd. km]	${\mathit{l}}_{\mathit{i}}$	Vehicle Number j
		1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
[0–30]	$l_1$	0.66	0.39	1.06	0.87	0.81	2.94	0.62	1.03	2.33	0.76	0.76	1.26	2.10	1.50	0.94	0.78	2.27	0.92	0.87	1.31
[30–60]	$l_2$	2.49	2.57	3.00	2.22	2.64	3.67	4.73	3.63	0.88	3.18	3.78	3.71	3.21	0.87	5.14	3.39	8.83	3.95	3.67	3.12
[60–90]	$l_3$	3.63	2.18	3.35	5.39	2.96	0.62	0.00	0.00	4.54	0.00	0.00	1.17	0.00	5.19	1.80	8.17	3.41	0.00	0.00	0.00
[90–120]	$l_4$	1.32	0.00	1.99	0.00	0.49	2.79	2.38	1.37	3.72	0.20	1.08	5.17	0.00	4.64	5.81	0.00	0.00	7.75	15.24	0.00
[120–150]	$l_5$	8.74	5.56	0.00	1.16	0.45	5.82	5.84	6.25	4.26	3.29	8.29	5.86	3.47	3.38	1.93	0.28	0.65	7.80	2.24	7.48
[150–180]	$l_6$	2.52	3.31	2.96	0.00	3.18	19.69	1.44	3.35	1.87	1.51	5.98	11.84	6.54	1.39	4.83	3.46	5.23	9.64	3.29	5.43
[180–210]	$l_7$	11.06	17.01	12.93	8.47	5.06	7.88	8.34	10.71	9.68	4.50	5.67	7.69	8.38	1.41	3.63	0.81	1.62	3.91	3.33	4.66
[210–240]	$l_8$	5.61	6.02	2.96	1.22	16.05	0.89	7.15	3.34	7.06	2.44	1.98	9.77	12.45	1.12	3.55	5.78	4.25	1.72	1.65	1.25

,

Table 5. Integrated phase risk in the sample of I vehicles for $\Delta l=30$ thousand km (thousand PLN).

Mileage Phase [thsnd. km]	${\mathit{l}}_{\mathit{i}}$	Vehicle Number j
		1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
[0–30]	$l_1$	0.50	2.15	1.33	1.72	1.50	1.69	2.15	2.31	1.13	2.18	6.32	20.80	2.59	1.54	5.75	3.55	2.16	0.63	1.24	0.62
[30–60]	$l_2$	2.28	2.09	6.25	1.35	2.88	1.93	2.09	6.66	7.74	2.22	2.06	2.09	3.41	4.69	4.38	1.72	0.00	0.00	0.42	0.00
[60–90]	$l_3$	0.00	15.51	18.13	14.21	10.30	6.98	15.51	3.74	0.00	0.00	4.18	12.31	2.18	2.97	1.74	3.36	4.72	1.52	7.33	0.54
[90–120]	$l_4$	3.12	1.85	14.07	11.98	0.00	0.00	1.85	15.98	12.06	9.05	0.00	12.89	6.28	43.03	12.98	20.77	6.77	3.80	1.48	5.00
[120–150]	$l_5$	8.48	27.23	24.17	8.62	7.74	22.02	27.23	10.08	7.43	11.04	6.49	4.75	7.97	25.11	35.63	13.94	5.75	1.02	7.48	6.42
[150–180]	$l_6$	18.69	14.65	16.49	14.04	8.38	30.74	14.65	5.29	3.38	9.06	13.39	3.99	13.45	4.36	7.87	3.66	6.65	5.14	1.59	11.40
[180–210]	$l_7$	15.41	9.01	47.23	4.58	24.99	16.15	3.30	13.38	14.10	11.42	8.73	8.21	6.23	8.55	6.35	4.44	5.16	2.03	5.91	11.81
[210–240]	$l_8$	6.08	14.72	13.72	6.35	9.45	26.18	30.07	29.45	13.96	13.35	8.83	1.84	7.23	5.57	3.73	6.33	1.81	10.45	2.22	7.87

,

Table 6. Integrated phase risk in the sample of M vehicles for $\Delta l=60$ thousand km (thousand PLN).

Mileage Phase [thsnd. km]	${\mathit{l}}_{\mathit{i}}$	Vehicle Number j
		1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
[0–60]	$l_1$	3.15	2.97	4.06	3.09	3.45	6.61	5.35	4.66	3.20	3.94	4.53	4.96	5.31	2.37	6.08	4.16	7.42	4.87	4.53	4.43
[60–120]	$l_2$	4.94	2.18	5.35	4.35	3.45	3.41	2.38	1.37	8.26	0.20	1.08	6.34	0.00	9.83	7.62	8.17	3.41	7.75	15.24	0.00
[120–180]	$l_3$	11.26	8.87	2.96	1.16	3.62	25.51	7.28	9.60	6.13	4.79	14.27	17.70	10.02	4.77	6.77	3.74	5.88	17.43	5.53	12.91
[180–240]	$l_4$	16.67	23.03	15.89	9.69	21.10	8.77	15.49	14.05	16.74	6.94	7.66	17.45	20.83	2.52	7.18	6.60	5.87	5.63	4.98	5.92

and

Table 7. Integrated phase risk in the sample of I vehicles for $\Delta l=60$ thousand km (thousand PLN).

Mileage Phase [thsnd. km]	${\mathit{l}}_{\mathit{i}}$	Vehicle Number j
		1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
[0–60]	$l_1$	2.78	4.24	7.58	3.07	4.38	3.62	4.24	8.97	8.87	4.40	8.38	22.89	6.01	6.23	10.13	5.27	2.16	0.63	1.65	0.62
[60–120]	$l_2$	3.12	17.36	32.20	26.19	10.30	6.98	17.36	19.72	12.06	9.05	4.18	25.20	8.46	46.01	14.72	24.14	11.49	5.31	8.81	5.54
[120–180]	$l_3$	27.16	41.88	40.65	22.66	16.13	52.76	41.88	15.37	10.81	20.10	19.88	8.74	21.42	29.47	43.50	17.60	12.40	6.16	9.08	17.82
[180–240]	$l_4$	21.48	23.73	60.95	10.93	34.44	42.33	33.37	42.83	28.06	24.77	17.56	10.05	13.46	14.11	10.09	10.78	6.97	12.49	8.14	19.68

, broken down into phases with a length of $\Delta l=60$ thousand km. The numerousness of the datasets for the individual phases was the same and amounted to $n=20$.

In Section 4, the determined phase risk values for the examined vehicle fleets adopted as implementations of random samples were analysed in terms of the statistical significance of the differences in risk between the phases.

4. Phase Risk Statistical Analysis

4.1. Hypotheses

Based on model (14), the authors conducted a statistical comparative analysis of the risk for the identified mileage phases. For each mileage range, the random values were estimated independently based on the operational data concerning two fleets of homogeneous vehicles. The data were regarded as random samples used to estimate the values in Formula (14) and verify the following research hypotheses:

• First research hypothesis $H_{M,0}:$ there is no significant difference in the average integrated operational risk between the mileage phases for the vehicles of brand $M$, expressed in the following form (18):

  $$H_{M,0}:\mathbb{E}{R}_M\left(l_0,l_1\right)=\mathbb{E}{R}_M\left(l_1,l_2\right)=\dots =\mathbb{E}{R}_M\left(l_{k-1},l_k\right)$$

  (18)
• Second research hypothesis $H_{I,0}:$ there is no significant difference in the average integrated operational risk between the mileage phases for the vehicles of brand $I$, expressed formulaically in the following form (19):

  $$H_{I,0}:\mathbb{E}{R}_I\left(l_0,l_1\right)=\mathbb{E}{R}_I\left(l_1,l_2\right)=\dots =\mathbb{E}{R}_I\left(l_{k-1},l_k\right)$$

  (19)

The identified vehicle mileage phases create an ordered categorizing variable.

The verification of the hypotheses will enable us to determine the impact of the variable upon differentiating the integrated operational risk between the individual operational phases of the vehicles. The hypotheses were verified using the analysis of variance.

The result of each risk resulting from the operational incapacity of the vehicle was regarded as the result of the overlapping of two types of effects—a random effect and a systematic effect resulting from mileage phase, if there is such an effect. It was assumed that the main cause of the variability in the operational risk could be the identified $k$ mileage phases. Thus, $R_{ij}$ defined by Formula (20) is the probabilistic model of the $j$-th integrated risk in the $i$-th phase:

$$R_{ij}=\mu +{\eta }_i+{\epsilon }_{ij}$$

(20)

where:

$\mu$—expected value of the integrated operational risk of the vehicle within mileage $l_k/k$ without taking into account differentiating into mileage phases;

${\eta }_i$—impact of the $i$-th phase on the integrated risk;

${\mu }_i=\mu +{\eta }_i$—expected value of the integrated risk in the $i$-th phase;

${\epsilon }_{ij}$—random error of the model.

The components of model (20) are assessed based on empirical data. To this end, using the least squares method, i.e., minimising the sum

$$S=\sum _{i=1}^k\sum _{j=1}^{n_l}{\left(x_{ij}-\mu -{\eta }_i\right)}^2$$

(21)

successively due to $\mu$ and ${\eta }_i$, the estimators of parameters $\mu$ and ${\eta }_i$ are determined. They are the total mean $\overline{x}$ and the difference between means ${\overline{x}}_i-\overline{x}$.

The following symbols are introduced:

$\mathrm{S}\mathrm{S}=\sum _{i=1}^k\sum _{j=1}^{n_l}{\left(x_{ij}-\overline{x}\right)}^2$—total sum of squares;

$\mathrm{S}\mathrm{S}\mathrm{T}=\sum _{i=1}^k{n}_l{\left({\overline{x}}_i-\overline{x}\right)}^2$—sum of squares of deviations between phases (sum of squares for treatments);

$\mathrm{S}\mathrm{S}\mathrm{E}=\sum _{i=1}^k\sum _{j=1}^{n_l}{\left(x_{ij}-{\overline{x}}_i\right)}^2$—sum of squares inside the groups (sum of squares for error).

For these symbols, the basic identity of the analysis of variance can be formulated shortly:

$$\mathrm{S}\mathrm{S}=\mathrm{S}\mathrm{S}\mathrm{T}+\mathrm{S}\mathrm{S}\mathrm{E}$$

(22)

If the means in the groups are too “close” to each other, they will also be close to the total mean, and, consequently, the sum $\mathrm{S}\mathrm{S}\mathrm{T}$ will turn out to be “small”. If the differences between the means for the phases are large, certain means for the phases will significantly differ from the general mean, resulting in a “large” sum $\mathrm{S}\mathrm{S}\mathrm{T}$. This will be grounds to reject the null hypothesis concerning the equality of the expected values of integrated operational risk of the vehicle for all mileage phases. The sum $\mathrm{S}\mathrm{S}$ is calculated based on deviations of $n=kl$ observations from the general mean, thus having $kl-1$ degrees of freedom. The sum $\mathrm{S}\mathrm{S}\mathrm{T}$ is calculated from $k$ deviations of independent phase means from the general mean, thus having $k-1$ degrees of freedom. The sum $\mathrm{S}\mathrm{S}\mathrm{E}$ is calculated from deviations of $kl$ observations from $k$ group means, thus having $k(l-1)$ degrees of freedom. Dividing the sums $\mathrm{S}\mathrm{S}\mathrm{T}$ and $\mathrm{S}\mathrm{S}\mathrm{E}$ by the corresponding degrees of freedom produces the following mean squares:

$\mathrm{M}\mathrm{S}\mathrm{T}=\frac{\mathrm{S}\mathrm{S}\mathrm{T}}{k-1}$ is the mean square for the groups;

$\mathrm{M}\mathrm{S}\mathrm{E}=\frac{\mathrm{S}\mathrm{S}\mathrm{E}}{k(l-1)}$ is the mean square for error;

$\mathrm{M}\mathrm{S}\mathrm{E}$ is an estimator of the unknown variance in the error of the model ${\sigma }^2$.

The formulated research hypotheses come down to an answer to the following question:

Are the differences between the expected values of phase risks only due to random reasons or are they also systematic?

The following criteria were used to determine the answer:

• If the former is true, the mean square for phases $\mathrm{M}\mathrm{S}\mathrm{T}$ will be an estimator of homogeneous variance for all $k$ phases.
• If, on the other hand, the differences between the means are systematic, it is necessary to consider the contribution of this systematic difference in the inter-phase variability.
• The authors verified Equations (18) and (19) using statistic (23):

$$F=\frac{\mathrm{M}\mathrm{S}\mathrm{T}}{\mathrm{M}\mathrm{S}\mathrm{E}}$$

(23)

which has Snedecor’s $F$ distribution, with $\left(k-1,k(l-1)\right)$ degrees of freedom.

The formal conditions for applying model (20) require that the following assumptions are met:

• The random components of the model should be independent random variables with normal distribution $\mathcal{N}(\mu , \sigma )$;
• Variances for the phases should be equal, i.e., ${\sigma }_1^2={\sigma }_2^2=\dots ={\sigma }_k^2$;
• Random variables ${\eta }_i$ and $i=1, 2,\dots ,k$ should be independent.

The normality of distribution was verified using the Kolmogorov–Smirnov and Shapiro–Wilk tests. It was assumed that if the normality condition was met, the homogeneity of variance for the phases, i.e., null Equation (24),

$${\mathrm{H}}_0: {\sigma }_1^2={\sigma }_2^2=\dots ={\sigma }_k^2$$

(24)

would be tested using Bartlett’s, Hartley’s and Cochrane’s parametric tests. The results of the analysis of variance are given in the ANOVA table.

4.2. Analysis of Descriptive Statistics of Phase Risk ($\Delta l=30$ Thousand km)

The initial analysis of risk tendencies was performed based on the calculation results given in

Table 8. Descriptive statistics of the phase risk for M vehicles for $\Delta l=30$ thousand km.

Statistic	$\mathbf{Phase} {\mathit{l}}_{\mathit{i}}$
	${\mathit{l}}_1$	${\mathit{l}}_2$	${\mathit{l}}_3$	${\mathit{l}}_4$	${\mathit{l}}_5$	${\mathit{l}}_6$	${\mathit{l}}_7$	${\mathit{l}}_8$
Average (thousand PLN)	1.21	3.25	2.07	2.70	4.14	4.87	6.84	4.81
Median (thousand PLN)	0.93	3.30	1.49	1.34	3.87	3.33	6.68	3.45
Std. deviation (thousand PLN)	0.68	1.14	2.32	3.74	2.86	4.50	4.16	4.09
Minimum (thousand PLN)	0.39	0.87	0.00	0.00	0.00	0.00	0.81	0.89
Maximum (thousand PLN)	2.94	5.15	8.17	15.24	8.74	19.69	17.01	16.05
Interquartile range (thousand PLN)	0.69	1.17	3.57	4.41	4.80	3.81	5.68	5.14
Variation coefficient (%)	56.2	35.1	112.1	138.5	69.1	92.4	60.8	85.0

and

Table 9. Descriptive statistics of the phase risk for I vehicles for $\Delta l=30$ thousand km.

Statistic	$\mathbf{Phase} {\mathit{l}}_{\mathit{i}}$
	${\mathit{l}}_1$	${\mathit{l}}_2$	${\mathit{l}}_3$	${\mathit{l}}_4$	${\mathit{l}}_5$	${\mathit{l}}_6$	${\mathit{l}}_7$	${\mathit{l}}_8$
Average (thousand PLN)	3.09	2.71	6.26	9.15	13.43	10.34	11.35	10.96
Median (thousand PLN)	1.93	2.09	3.96	6.53	8.55	8.72	8.64	8.35
Std. deviation (thousand PLN)	4.43	2.22	5.95	10.03	9.69	6.95	10.05	8.57
Minimum (thousand PLN)	0.50	0.00	0.00	0.00	1.02	1.59	2.03	1.81
Maximum (thousand PLN)	20.80	7.74	18.13	43.03	35.63	30.74	47.23	30.07
Interquartile range (thousand PLN)	1.26	2.69	10.24	11.10	16.90	9.94	8.57	8.20
Variation coefficient (%)	143.4	81.9	95.0	109.6	72.1	67.2	88.6	78.2

and in box plots (Figure 2 and Figure 3).

The values of integrated phase risk for M vehicles are highly scattered. Most phases included outliers on the high end. However, outliers below the minimum values were recorded for the phase ending with the mileage $l_2$. Observations with zero values were recorded in phases $l_3,{ l}_4,{ l}_6$. In phase $l_6$, there was an extreme observation, with a risk value 3-times higher than the interquartile range. For most phases, the interquartile range was fairly stable. Beginning with phase $l_1$, the coefficient of variation of $56.2\mathrm{\%}$ indicates moderate variation, suggesting that the vehicles at such an early stage of operation have fairly stable repair costs. During the phase $l_3$, the variation increases significantly to $112.1\mathrm{\%}$, suggesting high variation in repair costs in that period. The phase $l_5$ variation coefficient is $69.1\mathrm{\%}$, indicating that the risk is stabilising to a certain extent but on a fairly high level. This is indicated by the last two phases $l_7$ and $l_8$, whose variation is $60.8\mathrm{\%}$ and $85.0\mathrm{\%}$, respectively.

A high scattering of the phase risk value was also observed for the group of vehicles of brand I. Six outliers and three extremes were identified. For vehicles of brand I, the maximum risk was $47.23$ PLN thousand (point 123, Figure 3). For phase $l_1$, there is a high level of variation, $143.4\%$, indicative of intra-phase risk variations at an early stage of operation. This high variation persists until phase $l_4$. In the later phases, although slightly decreased, the variation coefficient is still fairly high, at, respectively, $72.1\%$, $67.2\%$, $88.6\%$ and $78.2\%$.

Table 10. Description of the risk outliers and extremes for M vehicles.

${\mathit{l}}_{\mathit{i}}$	Point in Figure 2	Causes of the Outliers
$l_1$	point 6	road collision
$l_2$	point 29	virtually no repairs for vehicles in this phase
$l_2$	point 34	virtually no repairs for vehicles in this phase
$l_4$	point 79	the vehicle required repair and towing
$l_5$	point 106	many repairs connected with the driving system (wheels, hubs, bearings)
$l_6$	point 112	many different individual repairs
$l_8$	point 145	unscheduled tyre change

and

Table 11. Description of the risk outliers and extremes for I vehicles.

${\mathit{l}}_{\mathit{i}}$	Point in Figure 3	Causes of the Outliers
$l_1$	point 11	vehicle towing
$l_1$	point 12	repairs indicating a road collision
$l_1$	point 15	many repairs of the exhaust system and exhaust gas treatment system
$l_2$	point 29	numerous repairs
$l_4$	point 74	numerous repairs of the exhaust system and exhaust gas treatment system, body and paint repair
$l_6$	point 106	numerous expensive repairs
$l_7$	point 123	numerous expensive repairs, gearbox replacement
$l_7$	point 146	vehicle repair due to a road collision
$l_7$	point 147	expensive gearbox repair and another expensive repair
$l_7$	point 148	expensive gearbox repair

contain a synthetic description of the technical causes for the outliers and extremes of phase risk.

4.3. Analysis of Phase Risk Differences’ Significance ($\Delta l=30$ Thousand km)

Normality in mileage phases $l_k$ was tested using the Kolmogorov–Smirnov and Shapiro–Wilk tests. The results of the analysis of normality in $k$ phases for $\Delta l=30$ thousand km for vehicles of brands M and I are given in

Table 12. Analysis of the distribution normality of the phase risk for M vehicles.

$\mathbf{Phase} {\mathit{l}}_{\mathit{i}}$	Kolmogorov–Smirnov			Shapiro–Wilk
	Statistic	df	Significance p	Statistic	df	Significance p
$l_1$	0.238	20	0.004	0.837	20	0.003
$l_2$	0.122	20	0.200 *	0.939	20	0.232
$l_3$	0.214	20	0.017	0.846	20	0.005
$l_4$	0.235	20	0.005	0.738	20	<0.001
$l_5$	0.140	20	0.200 *	0.936	20	0.205
$l_6$	0.223	20	0.010	0.775	20	<0.001
$l_7$	0.116	20	0.200 *	0.957	20	0.486
$l_8$	0.171	20	0.126	0.845	20	0.004

* Lower bound of true significance. With the Lilliefors significance correction.

and

Table 13. Analysis of the distribution normality of the phase risk for I vehicles.

$\mathbf{Phase} {\mathit{l}}_{\mathit{i}}$	Kolmogorov–Smirnov			Shapiro–Wilk
	Statistic	df	Significance p	Statistic	df	Significance p
$l_1$	0.345	20	<0.001	0.506	20	<0.001
$l_2$	0.228	20	0.008	0.896	20	0.035
$l_3$	0.202	20	0.031	0.873	20	0.013
$l_4$	0.181	20	0.085	0.787	20	<0.001
$l_5$	0.248	20	0.002	0.849	20	0.005
$l_6$	0.123	20	0.200 *	0.892	20	0.029
$l_7$	0.217	20	0.015	0.717	20	<0.001
$l_8$	0.181	20	0.086	0.839	20	0.003

* Lower bound of true significance. With the Lilliefors significance correction.

. Significance results $p (<0.05)$ indicate that the distribution normality is not met in most cases for the phases of both brands M and I. The failure to meet the normality criterion prevents the use of parametric tests. In this situation, further statistical analysis was performed using nonparametric Kruskal–Wallis (K-W) and Kolmogorov–Smirnov tests.

The nonparametric K-W test performed for the phases of M vehicles indicates that the distribution of all $k$ populations is not the same, and the null hypothesis should be rejected (significance $p<0.001$). The result was the same for vehicles of brand I (

Table 14. Summary of the Kruskal–Wallis test for M and I vehicles.

Brand	M	I
Total N	160	160
Test statistics	44.599	49.493
Degree of freedom	7	7
Asymptotic significance (two-sided test)	<0.001	<0.001

).

The compatibility of the distributions in the integrated phase risk of the vehicles was tested for two successive mileage phases using the Kolmogorov–Smirnov test (

Table 15. Results of the Kolmogorov–Smirnov for M and I vehicles.

Tested Phase Pair	Significance a,b
	Brand
	M	I
$l_1-l_2$	<0.001	0.819
$l_2-l_3$	0.013	0.172
$l_3-l_4$	0.978	0.819
$l_4-l_5$	0.172	0.172
$l_5-l_6$	0.819	0.560
$l_6-l_7$	0.082	0.978
$l_7-l_8$	0.172	1.000

^a The significance level is 0.050. ^b Asymptotic significance was presented.

). The test showed that the null hypothesis of the homogeneity of risks between adjacent phases should be rejected only for pairs $l_1-l_2$ and $l_2-l_3$ of brand M. All other pairs showed no significant differences.

Due to the absence of differences between the adjacent phases in most pairs, the authors conducted multiple comparisons using the Tukey test (

Table 16. Results of the Tukey test for M and I vehicles.

Brand	$\mathbf{Phase} {\mathit{l}}_{\mathit{i}}$		Difference in Means (I–J)	Significance	95% Confidence Interval
					Lower Bound	Upper Bound
M	$l_1$	$l_2$	−2.0403	0.488	−5.1819	1.1012
		$l_3$	−0.8602	0.990	−4.0017	2.2814
		$l_4$	−1.4889	0.829	−4.6304	1.6527
		$l_5$	−2.9296	0.087	−6.0712	0.2120
		$l_6$	−3.6635	0.011	−6.8051	−0.5219
		$l_7$	−5.6283	<0.001	−8.7699	−2.4867
		$l_8$	−3.6045	0.013	−6.7460	−0.4629
	$l_2$	$l_3$	1.1802	0.943	−1.9614	4.3217
		$l_4$	0.5515	0.999	−2.5901	3.6930
		$l_5$	−0.8893	0.988	−4.0308	2.2523
		$l_6$	−1.6232	0.757	−4.7647	1.5184
		$l_7$	−3.5880	0.013	−6.7295	−0.4464
		$l_8$	−1.5642	0.790	−4.7057	1.5774
	$l_3$	$l_4$	−0.6287	0.999	−3.7702	2.5129
		$l_5$	−2.0694	0.469	−5.2110	1.0721
		$l_6$	−2.8033	0.118	−5.9449	0.3382
		$l_7$	−4.7681	<0.001	−7.9097	−1.6266
		$l_8$	−2.7443	0.135	−5.8859	0.3973
	$l_4$	$l_5$	−1.4407	0.852	−4.5823	1.7008
		$l_6$	−2.1746	0.402	−5.3162	0.9669
		$l_7$	−4.1394	0.002	−7.2810	−0.9979
		$l_8$	−2.1156	0.439	−5.2572	1.0259
	$l_5$	$l_6$	−0.7339	0.996	−3.8754	2.4077
		$l_7$	−2.6987	0.150	−5.8402	0.4429
		$l_8$	−0.6749	0.998	−3.8164	2.4667
	$l_6$	$l_7$	−1.9648	0.538	−5.1064	1.1768
		$l_8$	0.0590	1.000	−3.0826	3.2006
	$l_7$	$l_8$	2.0238	0.499	−1.1178	5.1654
I	$l_1$	$l_2$	0.3796	1.000	−7.1276	7.8868
		$l_3$	−3.1691	.899	−10.6763	4.3381
		$l_4$	−6.0557	0.212	−13.5629	1.4515
		$l_5$	−10.3367	0.001	−17.8439	−2.8295
		$l_6$	−7.2511	0.067	−14.7583	0.2562
		$l_7$	−8.2580	0.020	−15.7652	−0.7508
		$l_8$	−7.8674	0.033	−15.3746	−0.3602
	$l_2$	$l_3$	−3.5487	0.831	−11.0559	3.9585
		$l_4$	−6.4353	0.152	−13.9426	1.0719
		$l_5$	−10.7163	<0.001	−18.2235	−3.2091
		$l_6$	−7.6307	0.043	−15.1379	−0.1235
		$l_7$	−8.6376	0.012	−16.1448	−1.1304
		$l_8$	−8.2470	0.020	−15.7542	−0.7398
	$l_3$	$l_4$	−2.8866	0.936	−10.3939	4.6206
		$l_5$	−7.1676	0.073	−14.6748	0.3396
		$l_6$	−4.0820	0.706	−11.5892	3.4252
		$l_7$	−5.0889	0.430	−12.5961	2.4183
		$l_8$	−4.6983	0.537	−12.2055	2.8089
	$l_4$	$l_5$	−4.2810	0.653	−11.7882	3.2262
		$l_6$	−1.1953	1.000	−8.7025	6.3119
		$l_7$	−2.2023	0.986	−9.7095	5.3049
		$l_8$	−1.8116	0.996	−9.3189	5.6956
	$l_5$	$l_6$	3.0857	0.911	−4.4215	10.5929
		$l_7$	2.0787	0.990	−5.4285	9.5859
		$l_8$	2.4693	0.972	−5.0379	9.9766
	$l_6$	$l_7$	−1.0070	1.000	−8.5142	6.5003
		$l_8$	−0.6163	1.000	−8.1235	6.8909
	$l_7$	$l_8$	0.3906	1.000	−7.1166	7.8978

). It was shown that there were significant differences for brand M between phase $l_1$ and phases $l_6$, $l_7$, $l_8$ and between phase $l_7$ and phases $l_2$, $l_3$, $l_{4.}$. For brand I, significant differences were observed between phase $l_1$ and the last phases of the observed mileage ($l_5$, $l_7$, $l_8$). A similar pattern was observed for phase $l_2$.

4.4. Analysis of Phase Risk Descriptive Statistics ($\Delta l=60$ Thousand km)

During the next stage of calculations, the authors analysed the difference in risk with consideration of phases $\Delta l=60$ thousand km. The results of the calculations are given in

Table 17. Phase risk descriptive statistics for $\Delta l=60$ thousand km.

Statistic	$\mathbf{Phase} {\mathit{l}}_{\mathit{i}}$
	Brand M				Brand I
	${\mathit{l}}_1$	${\mathit{l}}_2$	${\mathit{l}}_3$	${\mathit{l}}_4$	${\mathit{l}}_1$	${\mathit{l}}_2$	${\mathit{l}}_3$	${\mathit{l}}_4$
Average (thousand PLN)	4.46	4.77	9.01	11.65	5.81	15.41	23.77	22.31
Median (thousand PLN)	4.48	3.90	7.02	9.23	4.39	11.78	19.99	18.62
Std. deviation (thousand PLN)	1.28	3.90	6.06	6.29	4.89	10.92	13.58	14.32
Minimum (thousand PLN)	2.37	0.00	1.16	2.52	0.62	3.12	6.16	6.97
Maximum (thousand PLN)	7.42	15.24	25.51	23.03	22.89	46.01	52.76	60.95
Interquartile range (thousand PLN)	1.96	6.15	7.72	10.64	5.33	15.69	24.71	21.23
Coefficient of variation (%)	28.80	81.75	67.22	54.02	84.26	70.86	57.10	64.19

and Figure 4 and Figure 5.

4.5. Analysis of Phase Risk Differences’ Significance ($\Delta l=60$ Thousand km)

The results of the normality analysis in $k$ phases for $\Delta l=60$ thousand km for vehicles of brands M and I are given in

Table 18. Analysis of the phase risk distribution normality of the vehicles.

Brand	$\mathbf{Phase} {\mathit{l}}_{\mathit{i}}$	Kolmogorov–Smirnov			Shapiro–Wilk
		Statistic	df	Significance p	Statistic	df	Significance p
M	$l_1$	0.345	20	<0.001	0.506	20	<0.001
	$l_2$	0.228	20	0.008	0.896	20	0.035
	$l_3$	0.202	20	0.031	0.873	20	0.013
	$l_4$	0.181	20	0.085	0.787	20	<0.001
I	$l_5$	0.248	20	0.002	0.849	20	0.005
	$l_6$	0.123	20	0.200 *	0.892	20	0.029
	$l_7$	0.217	20	0.015	0.717	20	<0.001
	$l_8$	0.181	20	0.086	0.839	20	0.003

* Lower bound of true significance. With the Lilliefors significance correction.

. The results of the Kolmogorov–Smirnov significance test showed that the condition for the distribution normality was met both for brands M and I. According to the Shapiro–Wilk test, the normality condition was not met for some phases. Finally, the authors conducted ANOVA and found significant differences in risk between the phases both for M and I vehicles (significance $p<0.001$) (

Table 19. Analysis of the phase risk variance.

Brand	Source	Sum of Squares	Degrees of Freedom	Mean Squares	F Ratio	Significance
M	Inter-phase	724.600	3	241.533	10.376	<0.001
	Intra-phase	1769.218	76	23.279
	Total	2493.818	79
I	Inter-phase	4035.914	3	1345.305	10.105	<0.001
	Intra-phase	10,117.931	76	133.131
	Total	14,153.845	79

).

5. Discussion of the Results

A comparison of the phase risk scattering of phase risk for $\Delta l=30$ thousand km and $\Delta l=60$ thousand km (Table 8, Table 9 and Table 17) indicates that the variation coefficient decreased accordingly from $V=(80–90)\%$ to $V=(60–70)\%.$ The reduction in the coefficient $V$ in this case is due to the doubling of the amount of source data in each calculation cycle. In particular,

$V_M\left(30\right)\subset \left(35.1÷138.5\right)\mathrm{\%},$ mean ${\overline{V}}_M=81.15\mathrm{\%}$

$V_I\left(30\right)\subset \left(67.2÷143.4\right)\mathrm{\%},$ mean ${\overline{V}}_I=91.99\mathrm{\%}$

$V_M\left(60\right)\subset \left(28.80÷81.75\right)\mathrm{\%},$ mean ${\overline{V}}_M=57.95\mathrm{\%}$

$V_I\left(60\right)\subset \left(57.10÷84.26\right)\mathrm{\%},$ mean ${\overline{V}}_I=69.10\mathrm{\%}$

It should also be noted that:

for $\Delta l=30$ thousand km (six-month vehicle mileage), phase risks $R_M\left(30\right)$, $R_I\left(30\right)$ are within the following ranges:

$R_M\left(30\right)\subset \left(1.21;6.84\right) \mathrm{t}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{s}\mathrm{a}\mathrm{n}\mathrm{d} \mathrm{P}\mathrm{L}\mathrm{N},$ mean ${\overline{R}}_M\left(30\right)=3.74$ thousand PLN

$R_I\left(30\right)\subset \left(2.71;13.43\right) \mathrm{t}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{s}\mathrm{a}\mathrm{n}\mathrm{d} \mathrm{P}\mathrm{L}\mathrm{N}$, mean ${\overline{R}}_I\left(30\right)=8.41$ thousand PLN

For $\Delta l=60$ thousand km, in turn (annual vehicle mileage), the phase risks $R_M\left(60\right)$, $R_I\left(60\right)$ are within the following ranges:

$R_M\left(60\right)\subset \left(4.46;11.65\right) \mathrm{t}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{s}\mathrm{a}\mathrm{n}\mathrm{d} \mathrm{P}\mathrm{L}\mathrm{N}$, mean ${\overline{R}}_M\left(60\right)=7.47$ thousand PLN

$R_I\left(60\right)\subset \left(5.81;22.31\right) \mathrm{t}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{s}\mathrm{a}\mathrm{n}\mathrm{d} \mathrm{P}\mathrm{L}\mathrm{N}$, mean ${\overline{R}}_I\left(60\right)=16.82$ thousand PLN

Research analysing the dynamic of vehicle maintenance includes examples of similar assessments of the statistical scattering of empirical results in papers [26,27].

According to studies by Andrzejczak and Selech [27], describing the costs of tram repairs (a sample of 45 pcs.) in annual periods of 7 × 30 thousand km, the interquartile interval was $\Delta V=Vmax - Vmin$ $=19.94\%-8.44\%=115\%$, and the mean value was $\overline{V}=11.24\%$.

According to a study by Rymarz 2021 [26], the variation coefficient of the repair costs of city buses (sample of 2 × 20 pcs.) in successive years of operation from year 1 to year 6 converted to a single month and a single vehicle was within the range of $V_1=(100÷307)\%$ with a mean ${\overline{V}}_1=154\%$ for vehicles made domestically and in the range of $V_2=(107÷411)\%$ with a mean ${\overline{V}}_2=165\%$ for imported vehicles.

In line with the purpose of this study, in analysing the phase risk variations, the authors tested the differences’ significance in phases with a length of $\Delta l=30$ thousand km and $\Delta l=60$ thousand km.

In the case of $\Delta l=30$ thousand km, it was shown that significant differences in risk occurred only for M vehicles between phase $l_1$ and phases $l_6, { l}_7, { l}_8$ and between phase $l_7,$ and phases $l_2, l_3, { l}_4$. Similarly, the authors of [21] did not find any systematic and significant differences in the average costs of vehicle repair, and this was justified by the high variability in the individual costs of damage repair. Considering the natural impact of the physical wear of vehicle parts during prolonged use, the costs of repairs and downtime should be expected to increase, which means that the phase risk should increase as well. However, no significant risk changes were found in K–S and Tukey tests in this study. This indicates shortcomings in the risk analysis method, including the insufficient amount of data in the intra-phase calculation cycle or insufficient precision of the observations. The reproducibility of the observations can be increased by eliminating outliers and extremes. However, this requires an accordingly large total dataset.

The significance of risk differences was also tested for a doubled phase length of $\Delta l=60$ thousand km, approximately corresponding to the annual mileages of the vehicles. It was demonstrated that from this perspective, the differences between the phases were significant both for M and I vehicles. Thus, in the context of the presented research for trucks, the adoption of annual mileages as separate phases for a sample with a population of 40 observations in each phase should be regarded as correct.

6. Conclusions

The original operational risk analysis method presented in this study based on an integrated assessment of losses associated with the costs of incidental repairs and losses caused by lost profits during vehicle downtime includes the following:

• Analysis of the phase risk descriptive statistics of vehicles’ operational incapacity, including, in particular, the assessment of differences in average risk and differences in the variation coefficient in the time series of vehicle mileage phases;
• Indicating the outliers and extremes of phase risk and identifying their physical (technical) causes;
• Analysis of the differences’ statistical significance in phase risk.

The method developed enables the effective prediction of the continuity of transport services in the short term (one-year horizon). Current business management based on the analysis of phase risk is particularly needed in the transport sector, where small- and medium-sized companies predominate, heavily dependent on periodic market fluctuations.

This method has the following advantages:

• It focuses on the practical needs of the carrier (service provider) regarding the methods of ensuring the continuity of operation;
• There is a short calculation time thanks to simple calculation procedures.

The method could also be useful to compare vehicles of different brands in the context of the reliability theory of risk. Further research and method development should focus on public transport enterprises, especially urban transportation, as well as car-sharing companies.