Method of determining time for preventive diagnosis in city bus operation systems

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Introduction
The process of degradation of technical systems occurs during the long period of use of a technical facility.As a result, damage occurs that negatively affects the operating profit of a given system and safety.Various preventive strategies are being introduced to reduce the number of damage to facilities.There are many studies in the literature presenting the results of research on technical objects of various classes.The presented results address a wide range of operational issues, including reliability, efficiency and safety [3,9,10,13,14,19,20,24,27].To also reduce system maintenance costs, effective repair strategies are developed.Exchange management and activities related to them are mainly related to the reliability, but also the availability of the technical system.These activities include preventive and corrective diagnosis and repairs.Corrective repairs of technical objects are carried out using two variants: minimal repair ("bad as old") and perfect repair ("good as new") [18,21].
Minimizing system maintenance costs is achieved by implementing effective preventive repair strategies.The repair schedule is most often set by the manufacturer or system designer.By definition, the costs of corrective repairs are more expensive than preventive actions.This also translates into the repair time of the technical facility, which is higher than in the case of preventive actions.
Repairing a damaged element in individual systems is possible without replacing it; we treat this type of repair as a minimal repair.This type of repair restores the damaged technical object to its pre-damage condition.Therefore, specific replacements may be called minimal repairs.Based on the literature, it can be seen that there are a number of models describing replacements using minimal repairs.This allows for optimization by reducing the costs of system operation, but also by minimizing the risk of side effects.In recent years, this has become an interesting topic in reliability engineering.[4] were the first to present the concept of minimal repair.In such a model, it is assumed that when an object is damaged, an exact repair is performed with probability p, and a minimal repair is performed with probability 1-p.If p = 0, it is assumed that this is a minimal repair, but when p = 1, we are dealing with an exact repair.In [21] Pham and Wang called this repair configuration an imprecise maintenance model with a rule (see q).According to Brown and Prochan, the probability of a precise repair depends on the age of the tested technical object at the time of failure.Various models of such repairs can be found in the literature.In the works [18,21], models that are used in the case of minimal repairs along with preventive actions according to the age of the object are presented.In the presented literature, only one cited work uses semi-Markov processes.Currently, results related to minimal repairs have been developed in the works [ 6,7,8,23,25,26,27].

Brown and Prochan in
The presented article analyzes the maintenance strategy of the tested system, taking into account the age-based replacement rule.The use of semi-Markov processes to build a preventive repair model with minimal repair is considered.In order to build the criterion function, the limit theorem for semi-Markov processes was used [11,12].In the works [17,18] the presented approach of building a criterion function was used.In this work, profit per unit time and system availability are analyzed as the criteria function.In both situations, it was assumed that the time until damage to the technical object follows a Weibull distribution.The considerations presented in this paper were developed on the example of a city bus operation system, but the proposed method can also be used for other types of technical facilities.

States of the public transport bus operation system model
The technical object considered in this work is the public bus operation system model.This object can reside in one of the five considered states of the operation process model: • state 1 -task implementation state -a state when the vehicle and the driver carry out the assigned passenger transport order on the route and are in full availability; • state 2 -preventive diagnosis state -a state in which the need for preventive repair is verified and the type and scope of preventive repair are determined; • state 3 -preventive repair state -a serviceable technical facility is subject to preventive maintenance after a specified hourly interval and in accordance with the previously adopted operational strategy; • state 4 -state of corrective repair by Technical Emergency Service; • state 5 -state of corrective repair at the Service Station. Figure 1 shows a directed graph of the reflection of changes in the states of the model of the operation process of the public transport bus operation under consideration.

Mathematical model of the public bus transport operation process
A mathematical model was built for the directed graph shown in Figure 1.At the same time, it was assumed to be a stochastic process X(t).The following mathematical model was developed on the basis of the theory of semi-Markov processes [12].
In this paper, a 5-state semi-Markov process model of the public bus transport operation process with state space S = {1, 2, 3, 4, 5} is considered.If X(t) = i, then the public bus transport shown is in state i at time t.The matrix of transitions inserted in the semi-Markov chain process for the considered model has the following form: where: pij, i, j = 1, 2, 3, 4, 5 -probability of transition from state i to state j.
To determine the boundary probabilities for a Markov chain, solve the following matrix system: where: πi, i = 1, 2, 3, 4, 5 -the boundary probability of the Markov chain inserted in the semi-Markov process.
The article considers a model in which a public bus transport undergoes diagnosis and preventive repair at age T or corrective repair, when the technical object is damaged.By T1(x) is defined the time to diagnosis and preventive repair or damage and subsequent corrective repair of the technical object.The variable T1(x) is defined as follows: The next stage is to build a system of linear equations: { π 1 + π 2 + π 3 + π 4 + π 5 = 1 p 12 π 1 = π 2 p 23 π 2 = π 3 p 14 π 1 = π 4 p 15 π 1 = π 5 (5) while the boundary probabilities determined for the Markov chain are shown below: where: It is assumed that after time x, when the bus has not damaged, it transitions to the state of diagnosis and preventive repair.The process of changing states i = 1, 2, 3, 4, 5 taking into account diagnosis and preventive repair after time x is a new semi-Markov process with a matrix P(x) of transition probabilities of the Markov chain inserted into the semi-Markov process.With respect to the matrix shown above as number 1, only the first row of the matrix P changes, then the matrix P(x) takes the form of: while the boundary probabilities determined for the Markov chain are shown below: where:

Determination of the criterion functions
This paper considers an age-based semi-markov model of diagnosis and preventive repairs.
Based on the literature [12], for the considered model of the preventive repair system, the criterion function describing the total profit per unit time is expressed by the formula: 9) where: ET1(x) -the average value of the dwell time in state 1, calculated based on the formula: ET2, ET3, ET4, ET5 -average values of dwell times in states 2, 3, 4, 5, respectively.
In particular, based on the literature [12], it can be written: where: F1j(x), j = 2, 4, 5 -conditional distributions of the residence time in state 1, provided that the next state will be state j [9]; R1(x) = 1 -F1(x) -reliability function of the random variable T1.In order to simplify further considerations, it was assumed that the equations are true: Taking the above into account, the criterion function ( 9) describing the unit profit is expressed by the formula: Taking the above into account, the criterion function (9) describing the availability of the facility is expressed by the formula:   2) the average values of the residence times of the technical object in the states of the exploitation process model in [h]: ET2 = 0.620, ET3 = 1.783,ET4 = 0.717, ET5 = 3.621; for the uptime T1 a Weibull distribution was adopted, for which the value of the scale parameter = 10; three cases were analysed when the value of the shape parameter of the Weibull distribution is shape {9, 11.5, 14}, respectively.

Summary
Public transport buses get damaged while carrying out the task of transporting passengers, therefore corrective repairs are carried out by the Technical Emergency Service or the buses have to pull over in an emergency or be towed to Service Stations, where corrective repairs are carried out.This causes disruptions in the execution of tasks and additional losses related to the corrective repair itself, but also to loss of income and the costs of penalties for transport not carried out in accordance with the concluded contracts.Due to the specific functioning of this type of systems, buses should be diagnosed preventively to reduce the number of corrective repairs.The method presented in the article allowed for the development of a mathematical model taking into account the preventive replacement of the public transport bus system.The presented model was built based on semi-Markov processes.Based on the presented graph, a criteria function describing the unit profit and the availability of the technical object was developed.
For individual values of the shape parameter of the Weibull distribution, the described criterion functions.Based on the graph, it can be concluded that the considered criterion function z(x) -unit profit -reaches the extreme (maximum) -then the optimal time for preventive repair x is approximately 9 ÷ 9.5 hours for a profit for unit time of 6.5 ÷ 7.5 [PLN/h] and the second considered criterion function g(x) -availability -reaches extreme (maximum) -then the optimal time for preventive repair x is approximately 9.5 ÷ 10 hours for a availability of 0.804 ÷ 0.808.

ET 1 (Example 1 . 2 )
x)+F 1 (x)[(p 12 −1)⋅(ET 2 +p 23 ET 3 )+p 14 ET 4 +p 15 ET 5 ]+ET 2 +p 23 ET 3 Figure 2, 3 and 4, presented below, shows the graphs of the criterion function z(x) -profit per unit time [PLN/h] as a function of time to preventive diagnosis x [h].The calculations were performed for the following data: 1) the values of the probability matrix of changes of states of the model P: the average values of the residence times of the technical object in the states of the exploitation process model in [h]: ET2 = 0.620, ET3 = 1.783,ET4 = 0.717, ET5 = 3.621; for the uptime T1 a Weibull distribution was adopted, for which the value of the scale parameter = 10; three cases were analyzed when the value of the shape parameter of the Weibull distribution is shape {9, 11.5, 14}, respectively; 3) the average values of profits (costs) per unit time in each state of the model in [PLN/h]: z1 = 38, z2 = -98, z3 = -121, z4 = -84, z5 = -143.

Fig. 2 .Fig. 3 .
Fig. 2. Graph of the function z(x) -profit per unit time [PLN/h] as a function of time to preventive diagnosis x [h], determined for a Weibull distribution with parameter values scale = 10 and shape = 9.

Fig. 4 .Example 2 .
Fig. 4. Graph of the function z(x) -profit per unit time [PLN/h] as a function of time to preventive diagnosis x [h], determined for a Weibull distribution with parameter values scale = 10 and shape = 14.Example 2. Figure 5, 6 and 7, presented below, shows the graphs of the criterion function g(x) -availability as a function of time to preventive diagnosis x [h].The calculations were performed for the following data:1) the values of the probability matrix of changes of states of the model P:

Fig. 5 .Fig. 6 .
Fig. 5. Graph of the function g(x) -availability as a function of time to preventive diagnosis x [h], determined for a Weibull distribution with parameter values scale = 10 and shape = 9.

Fig. 7 .
Fig. 7. Graph of the function g(x) -availability as a function of time to preventive diagnosis x [h], determined for a Weibull distribution with parameter values scale = 10 and shape = 14.