RELIABILITY CONTROL
OF MEANS OF TRANSPORT
Okulewicz J., Salamonowicz T.
Warsaw University of Technology
Abstract: A method of means of transport maintenance with a requisite reliability is presented in this paper. Results are achieved by using redundancy of objects in a fleet and by making preventive replacement of objects’ elements. The acceptable level of a failure risk while executing transportation tasks has been taken as a criterion. Algorithm for selection of elements for preventive replacement has been developed and illustrated by an example.
1. INTRODUCTION
Nowadays, transportation tasks more often become unique projects in their character being a result of mixture of various conditions (requirements and limitations), combined by a logistician into one supply chain. A hereto considered possibility that one of the operating elements shall fail in the sequence – which could result in a failure to execute the earlier assumed function – requires for a logistical system to be designed, envisaging the failures of individual links. Therefore, a requirement for high reliability of objects being used is natural.
In transportation systems, those requirements first of all refer to the means of transport. In view of inevitable occurrence of various failures, a relevant redundancy level of those means of transport is essential to be maintained, which enables replacement of damaged objects and continuation of transportation tasks execution. A number of redundant means depends on the envisaged and therefore acceptable probability of failure during the task implementation period. In order to minimise the size of redundancy one should, on the one hand, be using objects of high reliability, and on the other hand, keep their reliability in the operating process at possibly high level.
Let us assume that n objects are essentially required for carrying out transportation tasks. If the entire fleet consists of n objects, then an assumption can be made that reliability structure of the fleet is in series. This imposes large requirements on reliability of each object, which is not always achievable. Then, in order to maintain reliability of the fleet at
its required level, redundant objects can be introduced into the fleet. Adding k redundant objects to the fleet allows for considering the fleet reliability structure as a threshold structure, in this case „n out of n+k”.
2. TYPES OF REDUNDANCIES
Redundant objects may play a role of the cold (unloaded) reserve, that is they passively wait for one of the objects to fail, or the hot (loaded) reserve, thus increasing the entire fleet capacity until one of the objects has failed. So, the fleet reliability model depends on the way the redundant objects are operating in.
In case of structure „n out of n+1” with the cold reserve, the fleet reliability function Rn+1(t) will be a sum of probabilities for occurrence of the following situations:
1) until moment t no object will fail out of n objects establishing a series reliability structure,
2) at any moment < t one out of n objects shall fail and will be replaced with a reserve object that will not fail along with the remaining objects in a range of (, t).
Probabilities for occurrence of the above situations are respectively:
P1 = Rn(t) (1)
where: R(t) – is each object’s reliability function,
t 0 n n 2 f τ R τ,t d P
(2)where: fn() – probability density function of a failure of one out of n identical
objects establishing a series reliability structure.
Rn(, t) – probability of a non-failure in the range of (, t) of the fleet consisting
of (n-1) objects aged and one new object.
R
t τ
τ R t R t τ, R 1 n n (3)
1 R
τ
nR
τ f τ dτ d τ f n n 1 n (4)
t 0 1 n 2 nR t f τ R t τ dτ P (5)
t 0 1 n 2 1 1 nt
P
P
R
t
R
t
n
f
τ
R
t
τ
dτ
R
(6)In case of structure „n out of n+2”, the analytical description becomes more complex, as there is the second reserve object. This means that in the fleet, established at the moment and consisting of (n-1) objects aged and one new object, one of the objects may fail and be replaced by the second reserve object before the moment t.
Probability density function of a failure of the fleet with a structure of „n out of n+1” with the cold reserve is expressed by relation:
t 0 1 n 1 n R t τ dF τ t R t f 1 -n -t f t R n t f
(7) and no recurrence formulas are known.In case of the fleet with structure of „n out of n+k” with the hot reserve, we may use the following relation:
ikn
i
kn
ni
k)n(n,
R)(1R
i
kn
R
(8) and the recurrence formula:n,n k R Rn 1,n k 1
1 R
Rn,n k 1R
Complexity of the analytical description, regardless of simplifying assumptions that have been made (i.e. identical objects, omission of the reliability structure of objects alone), indicates that there is a need for using a computer simulation for issues being considered here.
3. PREVENTIVE REPLACEMENTS
Besides maintaining redundant objects in the fleet, it is essential to keep up a high reliability level of technical objects in their operational use process. This purpose is served by preventive replacements envisaging upfront exchange of components being threatened by a failure.
A method that is known from literature and used for defining of a scope and deadlines of preventive replacements is to include the costs of upfront replacements and the costs generated by the occurring failures [1]. As a result of application of the method, minimum average costs per unit of time related to maintenance of objects in a proper reliability status are achievable. However, in order to benefit from that effect there is a need to replace individual elements in various time intervals, usually uncoordinated with the performance of tasks, which may wipe out advantages effecting from the implemented optimisation. Therefore, a possibility should be considered to make preventive replacements of selected elements of objects in the assumed time intervals whose scope is defined on the basis of assessment of reliability of the elements and the assumed reliability level of the entire fleet [3].
In case of complex objects, a failure commences whenever an element, which creates a series reliability structure with the others, has failed. A repair usually involves a replacement of the element for a brand new one. However, the replacement of the damaged element for the new one does not effect in recovery of such a reliability level as that before occurrence of the failure. Value of the reliability function of the damaged element was lower than one before the failure, and equal to one following the replacement. In effect, the condition of the object is slightly better after the repair than before the failure. So, practically, following the repair, there are no possibilities to recover such a status of the object as the one right before the failure. The process of repairing the object by replacement of its components is the process with step fluctuation of the reliability function at the moment when the component is being fixed.
Analytical model, describing the process of changes in a technical condition of the object (failures and replacements of elements), is analogical to the presented above fleet reliability model with redundant objects with the cold reserve. In view of complexity of models, which include a large number of components in objects, a criterion is needed to enable application of the assumed preventive replacements strategy in practice.
Both the objects and their components are considered when developing the preventive replacements strategy. Properties of the components are more predicable than those of objects which they are part of.
Dynamic determination of a scope of preventive replacements envisages a statistical assessment of present status an object’s elements. In order to do that, data is required about a distribution of time to failure and its parameters as well as about its operational use so far (since being new or from the moment of its replacement). On that basis, a set of those elements is defined the replacement of which will effect in a situation that a failure probability will not exceed its assumed value in the duration of the scheduled task. For any moment t the following conditions has to be met:
d (t)
qp (10) where: d – task implementation period,
qp(t) – quantile residual lifetime function, order p.
Function qp(t) shall be defined as follows [2]:
p inf
x:F (x) p
F (t) q 1 t t p (11) where: , x,t 0 R(t) x) R(t (x) R (x) F 1 t t ,Ft(x) – cumulative distribution function of the residual lifetime,
Rt(x) – conditional reliability function.
The actual technical condition of the object is not taken into consideration here as that would require for the object to be excluded from its operational use. Statistical control can be performed at any moment because it retrieves data gathered in the informational area of the means-of-transport maintenance management system. Having reliability characteristics of elements, updated working time of individual elements, a period for execution of the transportation task, then it is possible to define elements that require preventive replacement in order for the project implementation probability not to decline below its assumed value. Probability of a failure in the project implementation period can be determined in both cases, that is, when the replacements either have or have not been made. Additionally, the assessment may refer to the entire fleet of objects that have been assigned for execution of the transportation task.
4. ALGORITHM FOR SELECTING ELEMENTS TO BE REPLACED
A preventive replacement of elements is made if the value of function (11) of a given order, which has been calculated for the entire set of objects, is lower than the duration of the scheduled task for that set of objects. In order to select a set of elements to be replaced, an updated value of the reliability function is calculated including operational time of each and every one of them. Then a quantile of a given order is calculated for a distribution of the residual lifetime of each element. The elements are put in order according to the growing quantile value.
Subsequent elements, starting from an element of the lowest quantile value until the quantile of the entire fleet of objects – calculated by having included the replacement of assigned elements for brand new ones – is not lower than the duration of the scheduled task (algorithm on Fig.1), are assigned for replacement. The replacement of elements that have been assigned in that way ensures the assumed probability that the object will not fail during implementation of the transport task.
Fig.1. Algorithm for selecting elements for preventive replacement
Example. Three objects are required for execution of tasks. Allowable (acceptable)
probability of occurrence of the fleet unserviceability is at p = 0.1. The object consists of 3 elements with the same distribution of time to failure: Weibull distribution = 2.5, = 50.
The following operational use strategies are feasible:
3 objects are being used (fleet without redundancy), 4 objects are being used (one redundant object, hot reserve), 5 objects are being used (two redundant objects, hot reserve).
In every case, the required reliability is maintained by preventive replacements of elements. Reliability model of the structure „n out of n+k” with the cold reserve at the level of objects, and the hot reserve at the level of the fleet, has been applied in the solution. Simulation results for time 500:
Tab.1. Simulative Experiment Results
Variant
Statistical controls 3 out of 3 3 out of 4 3 out of 5 Average number
of failures by 5 units of time 9.9 17.4 24.6
by 3 units of time 14.6 24.7 34.8 without preventive actions 101.2 116.2 130.9
Average number of preventive replacements by 5units of time 397.6 351.8 331.5 by 3 units of time 281.3 249.1 233.8 Quantile of order 0.1 8.4 15.2 19.5
Based on the performed simulative experiment, the following conclusions can be made: 1) a considerable reduction in a number of incidental failures of elements, compared
to a use without statistical controls, is achievable through application of the statistical control,
2) the shorter the interval between controls the higher is the number of allowable failures at the required level of reliability,
3) adding a redundant object effects in a higher number of damages of elements, which is related to a reduced number of preventive replacements.
5. CONCLUSION
Based on the comparison of the operational use strategy, an assumption can be made that application of redundant objects in the fleet allows for achievement of the required reliability level by using a lower number elements being preventively replaced.
The hereto presented method for setting a scope of preventive replacements, based on reliability properties of individual components of objects being used, allows for matching the parameters of replacements for applied reliability parameters of the objects.
Reliability analysis with respect to preventive replacements can be performed with reference to elements being of critical importance for a task that is executed. The analysis can be carried out for any set of objects that will jointly be used for execution of the task.
[1] Barlow R.E., Proschan F.: Mathematical Theory of Reliability. SIAM Philadelphia 1996.
[2] Joe H., Proschan F.: Percentile residual life functions. Operations Research, vol. 32, 3; pages 668-679, 1983.
[3] Okulewicz J., Salamonowicz T.: Porównanie wybranych strategii odnów
profilaktycznych. Materiały XXXIV Zimowej Szkoły Niezawodności, pages
218-227, Szczyrk 2006.
[4] Salamonowicz T.: Model niepełnej odnowy przy naprawach wymuszonych i
profilaktycznych. Materiały XXXIII Zimowej Szkoły Niezawodności, pages
