EVALUATION OF IMPACT OF THE OPERATIONAL AND TECHNICAL FACTORS ON DOWNTIME OF MUNICIPAL BUSES BASED ON A LINEAR REGRESSION MODEL

Resume The objective of this study was to assess the effect of selected operational and technical factors on downtime of vehicles. The sample consisted of buses from a municipal transport company (Poland). Estimation of parameters of a linear regression model was performed. Month of failure (downtime event) and its type were used as predictors. Failures were divided into three categories: events related to the company’s operations, including vehicle failures (1) and other (organizational) problems (2), as well as failures caused by external factors unrelated to the operations of the transport company (3). The downtime was found to be significantly associated with failure type and month of failure. A linear regression model of downtime with a reduced number of impact factors, taking into account two main failure types and two main periods of their occurrence during the year, was developed. Article info Received 18 September 2020 Accepted 16 March 2021 Online 16 September 2021


ISSN 1335-4205 (print version) ISSN 2585-7878 (online version)
can be limited to improve the regularity of headways [8][9][10]. Berrebi et. al. have studied the practical effects of implementation of these corrective strategies. They demonstrated that the "bus holding" system reduced bunching, thus also decreasing average passenger waiting time [2]. Adamski and Turnau have presented a transport system control strategy in which buses were sent at specific times to critical bus stops with high numbers of passengers [11]. However, it is worth mentioning that the "bus holding" method also had negative effects, such as disturbances in traffic flow and an increase in average waiting time [1,[12][13][14][15][16][17][18].
An important component of the "bus holding" strategy is prediction of fleet availability. One prediction method involves simulation of readiness based on a regression model developed with use of the retrospective data.
In this paper, a linear regression model is proposed, which links the bus downtime (not-ready time) with the month of the year in which a bus was stationary and type of downtime. A regression analysis of downtime was performed based on data obtained from the municipal transport company in Lublin, Poland. The main objective was to develop a regression model with a reduced number of factors, which could be used to effectively predict bus downtime and ensure continuity of system operation.

Introduction
In public transport systems, vehicle failures and organizational shortcomings often substantially increase passenger waiting time. In this paper, disablement of a vehicle caused by technical or operational factors has been referred to with the umbrella term "failure". Vehicle failures are of concern to both drivers and fleet managers. In public transport systems, randomness of bus departure times and travel times has an influence on the quality of transport services [1]. In a situation when compliance with the timetable is the major requirement, the real travel time in the whole transport system is adjusted to a vehicle with the lowest operational speed. In the literature, this phenomenon is known as "bunching". Bunching forces passengers to arrive early at stations and to budget long travel time [2][3][4][5].
The literature describes several corrective strategies to reduce bus bunching. Hickman has proposed a stochastic model of vehicle operations based on recursive equations for expected values of headways and bus loads [6]. His strategy of improving transport services consists in holding operating buses along the service line, in order to regulate the system on an ongoing basis.
Daganzo and other authors have developed a mixed strategy in which passenger boarding and alighting status of the vehicles. Unfortunately, the data were not detailed enough to allow to discriminate which failure was caused by which factor. Type B were failures related to events outside the company's control (e.g. collision with another vehicle, freezing of the pneumatic system, vehicle trapped in the snow, a blocked route, an incident inside the vehicle).

Results
Downtime observation results were divided into "monthly" groups. The number of all the failures recorded in 2018-2019, broken down by month, is given in Table 2. Downtime data for the years 2018-2019, also broken down by month, are shown in Table 3. Monthly downtime duration is presented in Table 2. Figure 1 shows a box plot of downtime per month. As seen in the graph, median values of downtime are different for different months. The highest mean values were recorded in January, September, October and November. It is worth stressing that the number of failures is different for each month ( Table 3). The largest number of failures occurred in March, however, the mean and median downtime values for this month were the lowest ( Table 2), which means that the failures were short-term ones.
Significance of differences in downtime between months was assessed using the non-parametric Kruskal-Wallis test.
The Chi-squared statistic was χ 2 = 243.98 and the p-value < p <0.0001, which indicated that, at the significance level α = 0.05, the null hypothesis of equality of means was rejected. This demonstrates that there were significant differences between at least two monthly downtime groups.
Another factor that was analysed was the type of failure. The observed failures were classified as one of the three categories (types), designated here as A1, A2, B. The largest group were type A2 failures, which occurred 1338 times in the whole study period.

Material and methods
Twenty one buses (8 different makes and models) were studied. The vehicles were between 10 and 16 years old. Observations were conducted in standard public transport conditions over 2 years of operation (2018-2019). The dates and times of bus arrival to and departure from the depot and the vehicle downtime were registered. Source documentation included the company's daily internal reports on the operational status of the fleet. The basic descriptive statistics of the buses are presented in Table 1.
The buses serviced standard routes in municipal traffic. The average monthly mileage was approximately 4637 km. The lowest average monthly mileage of 4313 km was recorded for a bus make 5 (Standard Deviation-SD 1711 km) and the highest mileage of 6041 km for a bus make 2 (SD 1699 km).
Fleet downtime data for the years 2018-2019 were analysed. The impact of two factors, month of failure and failure type, on the dependent variable (downtime) was considered.
Month of failure was analysed repeatedly in each year of observation and was thus an indicator of seasonality related to seasonal changes in weather and vehicle loads (number of passengers) over the year. In winter, many downtime events were caused by door freezing, failures of driver's cabin and passenger compartment heating and power outages. During the summertime, downtime was mainly due to high temperatures, i.e. engine overheating and lack of air conditioning in the vehicle.
The second factor that has been analysed was the failure type. Three types of most frequent failures (downtime events) were considered. Type A1 were failures related to a vehicle damage (e.g. broken/jammed door lock, fluid leakage, broken brakes, no ignition, engine overheating). Type A2 were organizational failures and other technical problems (e.g. a damaged wind shield, mirror, tyre). Some of these failures were related to weather conditions and some to the general  Table 4 and a box-plot of downtime versus failure type is shown in Figure 2.
As Figure 2 shows, downtime duration differed    studied category. Variables with extreme average values were selected: type A1 failure and the month of April. In this way, the effect of single-signedness of the remaining parameters with regards to the level of the omitted variable was obtained. A statistical analysis was conducted to determine the structure of the linear regression model with binary variables, which resulted from the quantitative nature of the dependent variable and the qualitative character of the independent variables. The estimated model parameters are given in Table 5.
Four of the estimated parameters were statistically insignificant. The AIC (Akaike Information Criterion) = 34011 and corrected R 2 = 48 % indicate that the model does not fully explain the observed phenomena. This is also indicated by the residual distribution, which is different from the normal distribution (Lilliefors test statistics D = 0.088 and p-value < 0.001). Additionally, an analysis of the autocorrelation function ( Figure 3) demonstrated significant dependencies not described by the model. This means that bus downtime is dependent on factors which have not been included in the model.
The regression equation is given by: y= 18 (2)

Linear regression model of downtime
Based on the bus downtime data discussed in Section 3, a multi-regression model describing the relationship of downtime duration with month and type of failure was developed. The general formula for the linear regression model is as follows: where y is dependent variable, β 0 is intercept, x k are independent variables, β k are model parameters, ε is random parameter. The regression coefficient β k describes by how much the average value of the independent variable y will change if the value of the independent variable x k changes by a unit, all the other independent variables being constant. The random component in the model reflects an incomplete fit to empirical data.
Due to the fact that the independent variables had a qualitative character and formed closed sets (24 months and 3 types of failures), they had to be recoded as binary variables. Then, each variable took either the value of 1 -when the phenomenon does occur or 0 -when it does not occur. Parameters of the regression function were estimated using the least-squares method after initial elimination of a selected variable in each of the is that the differences are statistically significant. The test statistic is given by: where sgn is the sign function, R R i i j j k = | , R ij is the rank of observation, x 1, x 2 are study groups, N is sample size (number of study groups). Test results are presented in Table 6.
Based on the results, three groups of months were selected for which the downtime distributions were not significantly different. Additionally, the lack of significance of differences in each group was confirmed using the Kruskal-Wallis test. The first group of months (group I) included August, September, October and November. For this group, the chi-squared statistics χ 2 = 1.245 and p-value = 0.742. The second group (group II) included February, March, April, May, June, July The quality of fit of the regression model was evaluated using AIC ( Figure 3). The value of AIC was found from the following equation: where k is the number of model parameters and L is the reliability function.
In accordance with the objective of the study, in the next stage of the calculations, the model was simplified. To limit the number of predictors, it was proposed that months with similar regression coefficients should be aggregated. Statistically similar months were grouped with Pairwise Wilcoxon Rank Sum Test, which is a nonparametric test with multi-testing correction\ used to compare pairs in groups. The null hypothesis for Pairwise Wilcoxon Rank Sum Test is that there are no differences between distributions, while the alternative hypothesis  The model also permits to predict the availability of a transportation system as part of the strategy of ensuring the continuity of transportation services, e.g. by introducing the «bus holding» control strategy. From among the three types of downtime events, considered as independent variables, the model includes type B failures (events outside the company's control) as the dominant type and type A2 (operational and organizational) failures (which have four times less impact than type B events). Among the selected month groups, the reduced model presents the summer-autumn season, including August, September, October and November, as well as the winter season, which is represented by a single month -January. The effects of the two seasons on downtime duration are comparable and similar to the impact of A2 type failures. 0.117 0.910 0.701 0.005 0.000 0.000 0.001