ANALYSIS OF FIVE-PHASES MODEL
OF RELIABILITY OF INTERMODAL
TRANSPORTATION
Zając M.
Wroclaw University of Technology, Logistics and Transport Systems Division, Wrocław, Poland
Abstract: Paper presents five-phases model of reliability of combined transport system. All
phases and activities involved into them are separately described. There is assumption about exponential distribution of time to failure and repair applicated to system and services fluxes.
1. Introduction
Intermodal transport (IT) is transport of goods in integrated load units using at least two means of transportation [4]. According to the definition huge involvement of machines and complicated technology of transportation which needs transshipment of integrated load unit from one mean of transport to the other is seen. During the process, except carrying goods for far distance, activities like transshipment are also very important. Reliability of transportation system is one of basement, when choosing technology to done transportation task, so it is important to create model which can describe this feature of IT.
2. Mathematical assumption
If take into account rail-road IT with two transshipments, then model of reliability consist of five phases, which describe changes of carrying load unit machines. Base models of reliability are: model of exploitation of transport system [7] and tree phases model of reliability intermodal transportation [5]. The model concern rail-road IT, when:
during first phase transport from sender to terminal by road units is done, during second phase transshipment from road units to rail is done, during third phase rail transport is done,
during fourth phase transshipment from rail to road units is done,
As parameter of the process rates of time can be used. It is more important then length between departure and destination points, because on terminals units are not moved for far distance, but it takes pro rata much time.
Model of reliability of intermodal transport is based on Markov model defined [2],[11] as a stochastic process, in which for any sequence of parameters t0, t1, t2, ..., tn and any values x0, x1, ..., xn conditional probability: ] x ) t ( X x ) [X(t Pr ] x ) t ( X ,..., x ) t ( X x ) [X(t Pr n n n1 n1 0 0 n n n1 n1 (1) depends on state of process in prior moment.
Solution of states equitation is the item for assessment and estimation reliability of system described by reliability state graph. Probability of i - state can be characterized using probabilities of transition on the basis completed probability:
P
(
t
u
)
P
(
t
)
P
r,i(
u
);
i
0
,
1
,...,
N
.
N r 0 r r i
(2)That term is called as set of equations of stochastic process Z(t), and describes reliability behavior of system. The general method of solving is transformation equation of state to differential form, when u0. When substrate two-pronged Pi(t) and division by u it can
be received:
.
u
)
t
(
P
u
)
u
(
P
)
t
(
P
u
)
t
(
P
)
u
t
(
P
r N r,i i 0 r r i i
(3)General solving of this set of equation when u0 is possible when exist finite boundaries: j; i ), u ( P u 1 lim qi,j u 0 i,j and (4) ; ) 1 ) u ( P ( u 1 lim qi,i u 0 i,i
called intensities of transition from state Zi to state Zj and intensities stay in state Zi .
Because sum of probabilities of transition from state Zi to all of states is equal 1, then:
N j 0 j i,j N i 0 j i,j u 0u
(
P
(
u
)
1
)
0
1
lim
q
(5)If all intensities of transition qij exist, then set of states equations can be transformed to differential form:
N r 0 r r r,i i(
t
)
P
(
t
)
q
;
i
0,1,...,
N.
P
dt
d
(6) Then vector probabilities of states:P(t) [P0(t),P1(t),...,PN(t)]
(7) and square matrix intensities of transition:
N , N 1 , N o , N N , 2 1 , 2 o , 2 N , 1 1 , 1 o , 1 N , o 1 , 0 o , oq
...
q
q
q
...
q
q
q
...
q
q
q
...
q
q
Q
.Set of differential equations can be written in matrix form: P(t) P(t) Q dt d . (8)
To solve this set of equation it is necessary to know initial distribution of changes of states proces. The most general method to solv set with initial condition is transformation of Laplace:
0 st *(
s
)
[
P
(
t
);
s
]
P
(
t
)
e
dt
P
. (9) Solving has form:P*(s)P(0)[sIQ]1. (10)
To estimate solving depended on time reverse transformation must be done. Set of probabilities i - state reliability status Pi(t) is the basement to assess all reliability
measures. Along with the time process doesn’t depends on time. Because of it, according to ergodic theorem [11], it is known:
If for homogeneous Markov process with finite number of states, matrix of intensities of transition Q different from 0 exist, then bounders exist and are not dependent on initial distribution P(0+):
Pj t limPj(t)t limPi,j(t)
(11)
called stationary distribution or bounder distribution of the process.
According to ergodic theorem bounder distribution of the process can be designated. If: tlimPj(t)Pj, then P(t) 0 dt d lim j t . (12)
as a result can be achieved that when
t
set of differential states equations changes to linear equation. Consequently vector equation can be written in form:PQ 0 (13) Because set of equation is not designated, to solve set of equation any of equation must be replaced by normalizing condition:
N i 0 iP
i1
. (14)Basing on stationary distribution changes of the repairable process it is easy to estimate availability rate A. It goes from term:
Z} i Z :i {P
i {:iZi Z}P
jA
. (15)For systems which can’t be repaired, assessment of stationary characteristics doesn’t make sense, because R(t)0 when t.
3. Structure of five phases model of reliability of it system
3.1. Road transport phase
Road phase, which initiating IT includes load transportation from departure point to terminal and from terminal nearby destination to receiver. This part of transport of integrated load units can be done by trailers with semi-trailers. On figure 1 graph of states is shown. According to present states can be estimated when trucks work with loaded or
idle. Mathematic tool makes, that probability of intensities of transition have exponential distribution, in detail when:
failed during transportation with load,
failed during transportation without load, failed during transshipment.
Structure doesn’t allow to estimate intensities of failures during transshipment at sender’s or reciver’s place. If failures take place during this moments, then breakage is added to state directly before. Road phase includes states as follows: standby, idle running, load running, transshipment, preventive maintenance, failure.
Down states are preventive maintenance and failure, other are up states. Special case treats state of transshipment, because along with it vehicle doesn’t move, so it makes possibility to focus on problem reasons and effects of failures during transshipment.
3.2. Transshipment phase
Transshipment in clued load dislocation and transportation on terminal. The phase can be done by applicable machines and vehicles, ex. cranes, reach-stackers. On figure 2 graph of states is presented. Listed states allow to estimate:
total work-time during dislocation,
total work-time,
intensities of failures during transshipment, intensities of failures during other works.
Phase of transshipment includes states as follows: standby, dislocation works, transshipment, preventive maintenance, failure. Down states are preventive maintenance and failure, the other are up states.
Fig. 2. Transshipment phase, graph of states 3.3 Rail transport phase
Rail transport includes transportation between terminals. Rail part of CT can be done when rail trucks and locomotives are used. On figure 3 graph of states of rail phase is shown. This phase allows to estimate base operating rates of trains and assessment intensities of failures during running and transshipment. Possible states: standby, running, transshipment, preventive maintenance, failures. Down states are preventive maintenance and failure, other are up states.
4. Assumptions for computation
Computation is based on theoretic assumptions, which come from literature [1, 3, 6, 8, 9,10]. All probabilities have exponential distribution.
Terminal work’s assumption. One year’s shipment – 50 000 TEU. One year’s receiving – 40 000 TEU. 2/3 received and send containers are 1C model (20f). 1/3 received and send containers are 1A model (40f). Terminal is open 12hours/day, 25 days/moth.
Road phase assumption. 25 trailers with semi trailers, each carry 2 TEU. Each running is for distance of 30 km. Speed 40km/h. Time of transshipment outside of terminal 20min. Transshipment phase assumption. Transshipment done by four cranes. Load/Unload time 5min. Put back to field time –12 min. Put on from field time – 12 min.
Rail phase assumption.Number of rail-trucks – 20, each can carry 3 TEU. Length between terminal 500km. Train speed 30km/h. Time to form the train – 2hours.
5. Results
Each phase was computed to asses base operating and reliability rates. Results are shown in table 1.
Tab. 1. Comparison of rates for different phases
Lifetime 1 2 3 4 5 Road 0.732 0.685 0.676 0.674 0.673 Transhipment 0.952 0.934 0.924 0.918 0.914 Rail 0.849 0.847 0.84 0.846 0.846 Road 0.559 0.314 0.177 0.099 0.0557 Transshipment 0.859 0.739 0.636 0.547 0.47 Rail 0.244 0.076 0.024 0.007 0.002 Road 0.827 1.53 2.21 2.88 3.56 Transshipment 0.969 1.91 2.84 3.76 4.68 Rail 0.85 1.7 2.54 3.39 4.24
Computations for five phases are also done. Current phases changes as follows: road phase I, transshipment I, rail phase, transshipment II, road phase II.
In table 2 two variants of computation for different lifetime are shown. First variant total lifetime is 3.4. Rate of rail phase in all CT process is 58%, road 15%. In variant II road
phase is cut, so rail phase last almost 80% of total time of the process. Results are presented using graphs.
Tab. 2. Variants of phases phases
Phase Road Transshipment Rail Transshipment Road Variant I Lifetime 0.5 0.2 2 0.2 0.5 3.4 Rate 14.7 % 5.8 % 58 % 5.8 % 14.7 % 100 % Variant II Lifetime 0.2 0.1 2 0.1 0.2 2.6 Rate 15.4 % 3.8 % 77 % 3.8 % 15.4 % 100 %
5.1. Results for variant I
Figure 4 shows stabilization availability value on level A = 0.85 during rail phase. Figure 5 shows availability of next three phases, where fluctuation of function of availability is visible.
Figure 6 presents reliability profile where each phase is visible. They are different because of folling angle values. However there aren’t determined in rail and road phases. Phase of transshipment brings stabilization.
Fig.4. Availability profile for variant I
Transshipment II Road Ph. II
Rail Ph. Transshipment
Fig.5. Availability profile for 3 cycles, variant I
Fig.6. Reliability profile for 1 cycle, variant I 5.2. Results for variant II
Computations for variant II: one cycle, three cycles, for lifetime 10,2 (length of 3 cycles by variant I). Figure 7 shows stabilization of availability values during rail phase; it is on the level of A = 0.87. On figure 8 availability for 3 cycles is presented.
Transshipment II Road Ph. II Rail Ph. Transshipment I Road Ph. I Transshipment II Road Ph. II Rail Ph. Transshipment I Road Ph. I
Fig.7. Availability profile for 1 cycle, variant II
Fig.8. Availability profile for 3 cycle, Fig.9. Availability profile for lifetime 10,2; variant II variant II
During second cycle decrease of availability is visible, particularly during rail phase. However on next figure (lifetime 10.2 – fig.9.) repeatability is also seen. Maximum value achieved for rail phase is higher then in variant I. Value of reliability after one cycle is unacceptably low, R = 0.28 (figure 10). In variant I is close to 0.2. When rail phase rate increases in transportation process, then value of reliability in the end of the process is higher. Analysis for each phase separately showed, that reliability of rail phase is the lowest (tab.1.). However lifetime for variant I is 3.4, for variant II is 2.6. Value of reliability in variant I, when lifetime is 2.6 is close to 3. When cycle is shorter, then reliability increases in the end.
6. Conclusions
Five phases model represents approximation to real system, thus mathematical method (Markov Model) allows only for academic speculation. Model includes 27 states in 5 phases, 1/3 of them are down states.
Fig.10. Reliability profile for 1 cycle, variant II
Machines design and operation is necessary to asses properly each intensity of transition in theoretical way. Before application of the model, data (ex. truck time from terminal to sender/receiver, work time of cranes, number of failures, etc.) must be prepared, however it bring difficulty. Verification of assumption and collecting data from real system are in progress.
Obtained results show differences between phases. All availability profiles show, that phase of transshipment has soft angle of folling, in comparison to other phases not so sheer. Rail phase presents low value of reliability. Reliability profile folls rapidly and definitely pursues to R = 0. Availability profile for road phase shows, that the phase achieved the lowest rate of parameters. For this phase stabilization is present for values A = 0.7, when for rail phase A = 0.85, and transshipment A = 0.92. When transportation process is shorter (ex. shorter phases of road transport and transshipment) then functions of reliability achieve better values.
References
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