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Maritime University of Szczecin

Akademia Morska w Szczecinie

2010, 24(96) pp. 148–153 2010, 24(96) s. 148–153

The method of defining the number of substitute objects

on the basis of utilization system availability

Metoda wyznaczania liczby obiektów rezerwowych

na podstawie gotowości podsystemu wykonawczego

Maciej Woropay1_{, Klaudiusz Migawa}2

University of Technology and Life Sciences, Department of Machine Maintenance Uniwersytet Technologiczno-Przyrodniczy, Wydział Inżynierii Mechanicznej

85-789 Bydgoszcz, ul. Prof. S. Kaliskiego 7, e-mail: 1kem@utp.edu.pl, 2km@karor.com.pl Key words: transport system, threshold structure, operation process, Markov model

Abstract

The article presents a method of defining the required number of substitute technological objects used in the transport system, necessary for correct carrying out of the assigned transport task. The whole of the consideration was presented using the example of a chosen authentic system of the operation of transport means. On the basis of the identification of the authentic transport system and the means of transport operation process carried out in it, crucial operational states were designed as well as the possibilities of transferring between the particular states. Based on that, an event-centered model of the use of the means of transport was built, followed by a mathematical model of that process, assuming that its model is the homogenous Markov process. Then, for operational data obtained from research conducted in the authentic transport system, the values of the analyzed characteristics were defined. The presented results are a result of the research conducted as a part of a larger research project pertaining to the construction of a decision-making model of controlling transport system availability.

Słowa kluczowe: system transportowy, struktura progowa, proces eksploatacji, model Markowa Abstrakt

W artykule przedstawiono sposób wyznaczania wymaganej liczby technicznych obiektów rezerwowych, eksploatowanych w systemie transportowym, koniecznych do prawidłowej realizacji przydzielonego zadania przewozowego. Całość rozważań przedstawiono na przykładzie wybranego rzeczywistego systemu eksplo-atacji środków transportu. Na podstawie identyfikacji rzeczywistego systemu transportowego i realizowanego w nim procesu eksploatacji środków transportu wyznaczono istotne stany eksploatacyjne oraz możliwe przej-ścia między wyróżnionymi stanami. Na tej podstawie zbudowano zdarzeniowy model procesu eksploatacji środków transportu, a następnie matematyczny model tego procesu, zakładając, że jego modelem jest jedno-rodny proces Markowa. Następnie dla danych eksploatacyjnych, uzyskanych z badań przeprowadzonych w rzeczywistym systemie transportowym, wyznaczono wartości analizowanych charakterystyk. Prezentowa-ne wyniki są efektem prac realizowanych w ramach szerszego projektu badawczego, dotyczącego budowy decyzyjnego modelu sterowania gotowością systemu transportowego.

Introduction

The basic objective of the operation of transport systems is fulfilling the transportation needs as a result of the carrying out of transport on particular routes. The scope of the transport task assigned to a transport system is determined by the frequency of rides and the amount of the load carried on

a particular route in an assigned period of time. In general, the transport system consists of two main subsystems: utilization and logistics. In the utilization subsystem the assigned transport task is carried out and profits due to the carrying out of the task are generated. In the utilization subsystem, the direct carrying out of the transport task is performed by elementary subsystems such as

ope-rator – means of transport, linked by an appropriate structure. In the logistics subsystem, the processes carried out are supposed to assure task-based efficiency of the used technological objects.

The effectiveness of the operation of transport system depends on the possibility of correct carry-ing out of the assigned transport task. One of the factors strongly influencing the possibility of cor-rect carrying out of the transport task is the availa-bility of the utilization subsystem to carry out the task.

The technological object (element or system) availability is the object’s feature which is characte-ristic from the point of view of the possibility of timely obtaining or maintaining the state of effi-ciency (facilitating the realization of goals). The term ‘availability’ pertains to the systems characte-rized by the necessity of quick reaction in emergen-cy situations; such systems include: the military, police, ambulance service, fire department as well as transport systems. In such systems in cases of whenever there is a goal in an emergency situation, an individual or a team of individuals with their assigned technological objects attempt its imme-diate realization [1, 2, 3].

The subsystem availability for the carrying out of the assigned transport task depends on many factors and it is possible to shape it as a result of appropriate control decision making, such as:  choosing the appropriate number of

technologi-cal objects (means of transport) used in the transport system as well as the structure with which they are linked,

 choosing the appropriate technological objects of high reliability and receptivity to servicing and repair,

 assuring high availability and efficiency of the logistics subsystem as a result of choosing reliable and efficient intermediate devices as well as appropriate linking structure.

This paper presents the way to define the required number of substitute technological objects in order to assure the correct carrying out of the assigned transport task for the given availability of a single technological object as well as the given structure (threshold structure) which links the individual technological objects.

Event model of the operational use process

The operational use process model was built on the basis of analysis of the space of states and events concerning technical objects (city buses) used in the analyzed real transport system. In result of identification of the analyzed transport system,

there have been determined multi-state process of technical object operation and maintenance realized within it, significant states of the exploitation process and possible transitions between these states. On this basis a graph of the operational use process state changes, has been built and presented in figure 1. S1 S3 S4 S5 S6 S7 S9 S10 S11 S12 S13 S14 S16 S15 S2 S8

Fig. 1. Directed graph representing means of transport opera-tion and maintenance process

Rys. 1. Skierowany graf odwzorowania procesu eksploatacji środków transportu

Below, there have been presented names of the operational use states for the process of opera-tion and maintenance of city bus transport means (Fig. 1):

S1 – awaiting the carrying out of the task at the bus

depot parking space, S2 – repair at the bus depot

parking space without losing a ride, S3 – carrying

out of the transport task, S4 – waiting for the

deci-sion of the traffic controller after occurrence of the vehicle damage, S5 – diagnosing by the emergency

service unit, S6 – repair by technical support unit

without losing a ride, S7 – repair by the emergency

service with losing a ride, S8 – awaiting the start of

task realization after technical support repair, S9 –

emergency exit, S10 – waiting for action of the

S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 S16

maintenance subsystem, S11 – refueling, S12 –

main-tenance check on the operation day, S13 –

reali-zation of periodical servicing, S14 – prior to repair

diagnosing in the serviceability assurance subsys-tem, S15 – repair in the serviceability assurance

subsystem, S16 – diagnosing after the repair in the

serviceability assurance subsystem.

The method of defining the number of substitute technological objects for the assigned transport task

The required availability of the utilization subsystem

The required availability of the utilization sub-system for the carrying out of the assigned transport task depends on the required number of ready tech-nological objects Lwym = const (means of transport)

defined by the scope of the assigned transport task as well as the number of all technological objects used in the transport system N  const and is defined by the following relation:

 

N L N GPW wym wym  (1)

The number of all technological objects used in the transport system N is the sum of the required number of objects ready for the apponted task Lwym

as well as the number Lrez of the other objects not

defined for the carrying out of the transport task which are part of the transport system. These objects serve as substitute objects available as substitutes for the damaged objects carrying out of the transport task.

rez

wym L

L

N   (2)

Then the required availability of the utilization subsystem for the carrying out of the defined task is defined by the relation (3) and is the function of the number of substitute objects which are part of the transport system.

 

rez wym wym rez wym L L L L GPW   (3)

On the basis of the identification of the tested transport system, a number of used technological objects N = 182 as well as the required number of technological objects ready the carrying out of the assigned transport task Lwym = 159 were both

defined. Then the required availability of the utilization subsytem for the assigned transport task, in accordance with (1), amounts to:



182



0.8736

wym N 

GPW

Actual availability of the utilization subsystem

The actual availability of the utilization sub-system is defined depending on the availability of a single technological object (means of transport), the number of substitute technological objects as well as the structure which links technological objects used in the tested authentic transportation system. In order to define the availability of a sin-gle technological object, a mathematical model of the process of the technical objects use was built.

Due to the random nature of the factors influen-cing the running and efficiency of the transport means operation process introduced in a complex system, most often in the process mathematical mo-deling of the operation process, stochastic processes are used. From among the random processes, both Markov and semi-Markov processes [4, 5] are widely used in the modeling of technological objects. The implementation of model research while using the above-mentioned operation process models makes it possible to, on the one hand, analyze detailed problems connected with the operation of technological objects, and on the other hand, the relations between the designed number of model parameters [6, 7, 8, 9, 10].

As a result of the conducted analysis of the assumptions and limits it was decided that the model of the process of the technical objects use is Markov X(t) process [11, 12]. Using Markov processes to mathematically model, the following assumptions were made:

 Markov X(t) process projects the modelled authentic use process well enough from the point of view of the research objective;

 The modelled use process has a finite number of states Si, i = 1, 2 … 16;

 The random X(t) process as the mathematical model of the process of use is a homogenous process;

 At t = 0 moment the process finds itself in the S1

state (the initial state is state S1).

Based on the directed graph shown in figure 1, a matrix of the changes intensity of  state of X(t) process was built (4), where: pij – probability of

transfer from state Si to state Sj, i – intensity of

remaining at state Si of the X(t) process, ij –

intensity of transfer from state Si to state Sj of the

X(t) process.

Based on the matrix  shown in relation (4), simultaneous equations were built fulfilling condition (5):



    16 1 * _{0, 1,2...16} i ij i j p  (5)

hence:                                                                                                                               0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 * 16 16 , 16 * 15 16 , 15 * 16 15 , 16 * 15 15 , 15 * 14 15 , 14 * 13 15 , 13 * 12 15 , 12 * 11 15 , 11 * 10 15 , 10 * 14 14 , 14 * 13 14 , 13 * 12 14 , 12 * 11 14 , 11 * 13 13 , 13 * 12 13 , 12 * 12 12 , 12 * 11 12 , 11 * 16 11 , 16 * 15 11 , 15 * 11 11 , 11 * 10 11 , 10 * 10 10 , 10 * 9 10 , 9 * 3 10 , 3 * 9 9 , 9 * 5 9 , 5 * 4 9 , 4 * 8 8 , 8 * 7 8 , 7 * 7 7 , 7 * 5 7 , 5 * 6 6 , 6 * 5 6 , 5 * 5 5 , 5 * 4 5 , 4 * 4 4 , 4 * 3 4 , 3 * 8 3 , 8 * 7 3 , 7 * 6 3 , 6 * 3 3 , 3 * 2 3 , 2 * 1 3 , 1 * 2 2 , 2 * 1 2 , 1 * 13 1 , 13 * 12 1 , 12 * 11 1 , 11 * 1 1 , 1 p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p                                                  (6) The analitical result of simultaneous linear equations (6) is theoretically possible, hower it is difficult and arduous, whereas the formulas obtained as a result, defining limit probabilities pi*,

are very complicated and expanded. For data ob-tained as a result of the research of the use process cunducted in an authentic transport system, the values of the intensity of process state change ij

were established. Them with the use of Mathe-matica software, limit probabilites pi* of remaining

at process states were set. The results are shown in table 1.

Operational data pertained to 182 technological objects (municipal buses) used in a chosen authentic municipal bus transport system in the period between April, 2009 and December, 2009. The operational tests were made using the method of passive experiment in natural operational condi-tions of the use of tested technological objects.

Table 1. Values of limit probabilities pi* of remaining at Markov process states

Tabela 1. Wartości prawdopodobieństw granicznych pi* pozo-stawania w stanach procesu Markowa

p1* p2* p3* p4* 0.3214 0.0032 0.5316 0.0011 p5* p6* p7* p8* 0.0027 0.0019 0.0015 0.0027 p9* p10* p11* p12* 0.0002 0.0896 0.0059 0.0046 p13* p14* p15* p16* 0.0032 0.0009 0.0288 0.0007

In the general case, availability of a single technological object defined on the basis of the Markov model of operational process is determined as the sum of limit probabilities pi* of remaining at

states belonging to the availability states set:



 

 

i i i G

OT _{p S S i}

G_rzecz , for , 1,2...16 (7)

In order to define availability of technological objects (means of transport) based on the Markov model of operational process, the operational states

                                                                      16 , 16 15 , 16 11 , 16 16 , 15 15 , 15 11 , 15 15 , 14 14 , 14 15 , 13 14 , 13 13 , 13 1 , 13 15 , 12 14 , 12 13 , 12 12 , 12 1 , 12 15 , 11 14 , 11 12 , 11 11 , 11 1 , 11 15 , 10 11 , 10 10 , 10 10 , 9 9 , 9 8 , 8 3 , 8 8 , 7 7 , 7 3 , 7 6 , 6 3 , 6 9 , 5 7 , 5 6 , 5 5 , 5 9 , 4 5 , 4 4 , 4 10 , 3 4 , 3 3 , 3 3 , 2 2 , 2 3 , 1 2 , 1 1 , 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0                                                  (4)

of the technological object should be divided into availability states SG and non-availability states SNG

of the object for the carrying out of the assigned task. The states of availability of the technological objects are such states, when the object, including the operator remains in the operational system, is efficient and supplied, or will be repaired and/or supplied in a period of time shorter than the period of the time reserve assigned for the task. Non-availability states are such states, when the object or the operator remain outside of the operational system (efficient or not) as well as when an inefficient and/or unsupplied object remains within the operational system.

In the presented model, the following technolo-gical object availability states were enummerated: • state S1 – awaiting the carrying out of the task at

the bus depot parking space,

• state S2 – repair at the bus depot parking space

without losing a ride,

• state S3 – carrying out of the transport task,

• state S6 – repair by technical support unit

with-out losing a ride,

• state S8 – awaiting the start of task realization

after technical support repair.

For the values of limit probalities pi* of

remaining at Markov X(t) process states shown in table 1, actual availability value was defined for a single technological object used in the tested transport system: 8608 . 0 * 8 * 6 * 3 * 2 * 1 rzecz p p p p p  GOT

In the tested transport system the individual technological object (means of transport) are linked by a threshold structure of the k of n type, where k = Lwym marks the required number of

techno-logical objects ready for the carrying out of the assigned transport task, while n = N the number of all technological objects used in the system. Then the availability of the system with a threshold structure is marked by the relation:

 



 

wym



wym wym rzecz rzecz rzecz 1 L N OT L OT N L i PW _{G G} i N N G            



(8) Taking into consideration (2) as well as the fact that the required number of the objects ready for the carrying out of the assigned transport task Lwym is

constant for the concrete task, the availability of the utilization subsystem determined in the description of the transport task depends on the number of substitute technological objects Lrez as well as the

availability of a single technological objectGrzeczOT .

Then the availability of the utilization subsystem PW

G_rzecz described by the relation (9) and for

operational data obtained by testng of the authentic transport system amounts to:

 

  

wym



rez rez wym wym rzecz rzecz rez wym rez rzecz 1 OT L L OT L L L i PW G G i L L L G            



  (9)



182



rzecz



rez 23



0.3540 rzecz N G L   GPW PW

The required number of substitute technological objects

From the obtained values of the required and actual availability of the utilization subsytem one observes that actual availability is much lower than the required one:



182



_wym



182



rzecz N G N 

GPW PW



_rez 23



_wym



_rez 23



rzecz L  G L  GPW PW 8736 . 0 3540 . 0 

In order to assure the correct carrying out of the assigned transport tasks it is necessary to increase the value of actual availability of the utilization subsystem. Obtaining a higher value of actual availability of the utilization subsystem for the constant value of single technological object availability is possible also through choosing the appropriate number of substitute technological objects, so that the values of actual availability of the value of actual availability of the utilization subsystem is higher than the value of the required availability. The criterion of choosing the required number of substitute technological objects may be represented by the relation:

 

rez wym

 

rez rzecz

wym rez

rez L minG L G L

L _{ } PW _ PW ₍₁₀₎

In figure 2 the comparison of the value of the required and actual availability of the utilization subsystem with threshold structure is shown, determined for various numbers of substitute technological objects as well as a given number of objects required for the realization of transport task Lwym = 159 and given value of a single

technolo-gical object G_rzeczOT = 0.8608. Based on the obtained chart one may conclude that the minimal number of substitute objects in the tested transport system is Lrez wym = 31. Then:



rez 31



wym



rez 31



rzecz L  G L 

GPW PW

Fig. 2. The values of the required and actual availability of the utilization subsystem with threshold structure in the function of the number of substitute technological objects as well as the given number of objects required for the carrying out of the transport task Lwym = 159 and the given value of a single

technological object G_rzeczOT = 0.8608

Rys. 2. Wartości gotowości wymaganej oraz rzeczywistej podsystemu wykonawczego o strukturze progowej w funkcji liczby obiektów technicznych rezerwowych oraz danej liczby obiektów wymaganych do realizacji zadania przewozowego

Lwym = 159 i danej wartości gotowości pojedynczego obiektu

technicznego G_rzeczOT = 0.8608

Conclusion

The presented method facilitates defining of the required number of substitute technological objects needed for the carrying out of the assigned transport task, depending on the values of the availability of a single technological object, the type of structure which links the technological objects as well as the parameters characteristic of the scope of transport task. The number of substitute technological objects is defined so that the value of actual availability of the utilization subsystem amounts to at least the value of required availability which the utilization subsystem should have for proper carrying out of the assigned transport task in given conditions.

The objective of the individual stages of the conducted research is the construction of a model of evaluation and shaping of the availability and efficiency of logistics subsystem posts, which will enable one to define the number of technological objects serviced at the posts of this subsystem over

a given period of time. The analysis of the obtained results pertaining to the number of non-ready technological objects (requiring supplies and / or damaged during the carrying out of the transport task) as well as serviced at the logistics subsystem posts over a given period of time will serve as basis for creating a method of defining the required number of technological objects used in the tran-sport system, the number of substitute objects, as well as the required number of objects repaired in the period of time assigned for such activities, and, at further stages, the construction of a decision- -making model of transport system availability.

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Recenzent: dr hab. inż. Zbigniew Matuszak, prof. AM Akademia Morska w Szczecinie 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 5 10 15 20 25 30 35 40 45 GPW L_rez GPWwym.G_wymPW GPWrzecz.G_rzeczPW