THE EQUABLE MAINTENANCE STRATEGY
Drobiszewski J., Smalko Z.
Technical University of Warsaw Air Force Institute of TechnologyAbstract: Method of control of the risk of failure of the objects to which preventive renewal method is applied under the zero option maintenance strategy.
1. Introduction
Application of technical objects always involves a risk of failure. A failure of the object may lead to undesirable effects. A commonly used method, by which the risk of failure can be reduced is to treat the object in a way preventing its failure. This requires, that preventive period, or life of the object must be determined, on completion of which, a preventive renewal has to be performed. To this effect, various strategies are applied for the objects which have to be maintained in required technical state. These include, among others, a strategy of renewal at fixed life periods (preventive periods).
To determine the preventive period, economical criteria are used. Conditions which provide for the minimum unit maintenance costs are considered as optimum. The fixed preventive period method makes it possible to determine the risk of failure, which is measured by the probability of the object failing before the preventive renewal has been performed. It is suggested, that the risk of failure might be reduced by ignoring the criterion of the minimum unit maintenance cost, but instead, applying the criterion of the so called zero option. The zero option is such an arrangement, that the maintenance costs would be equal whether or not preventive renewal treatments have been performed. It should be presumed, that the costs of the lost opportunities of saving money on maintenance expenditures (due to the preventive renewals being abandoned) will be compensated by reduction of fines (for not keeping to the job deadlines) and insurance breakdown damages (for injuries and/or property damages).
By suitable comparative review of the maintenance strategy considering these two criteria we expect to reveal: the maximum possible declines of the risk of failure, extension of the life period before failure and reduction of preventive period of the object between any two renewals (whether preventive or corrective). Subject to the review will be an example of triangular distribution of probability of failure. This corresponds to a situation, where output decisions are undertaken in a progressively changing technical state of an object in response to many independent inputs.
2. Specific features of fixed periodical strategy of maintenance
Cost of maintenance of an object maintained to a rule of periodical renewal within time interval (0,t] is a random variable, the expected value of which can be expressed as follows:
k1 R(t) + k2 F(t) = k1+ (k2 - k1) F(t), k2.>k1 (1)
where:
R(t), F(t): function of reliability and function of unreliability respectively k2, k1: costs of preventive and corrective renewals respectively.
Expected unit cost of maintenance within the limits (0,x] can be expressed as:
x x F k k k x K
) ( ) ( ) ( 1 2 1 (2) where:K(x): expected unit cost of maintenance in the predicted life period between renewals, x = R t dt x ) ( 0
: predicted life period between any two subsequent renewals (whether preventive or emergency),x: fixed preventive period or fixed period of operation of an object before preventive renewal has been performed.
Expected cost of maintenance in the case of preventive renewals being abandoned (policy of emergency shutdowns), can be expressed as follows:
2 limK x K b kt (3)
where:
K(b): expected unit cost of maintenance during period of operation before corrective renewal,
= R(t)dt
0
: predicted life period before actual corrective (emergency) renewal
Expected relative unit cost of maintenance is a ratio of costs of preventive to corrective renewal maintenance treatments.
1
0
,
,
)
(
)
1
(
)
(
)
(
)
(
2 1
k
k
x
F
b
K
x
K
x
k
x (4) where:k(x): expected relative unit cost of maintenance with preventive renewal method applied,
: coefficient of economic discouragement1
An indispensable condition for existence of the optimum preventive period under criterion of the minimum maintenance costs is, that a value of x must be found to meet the following requirement:
1
1
)
(
)
(
)
(
0 0 xox
R
x
R
x
R
(5) where:x: optimum preventive period under criterion of minimum costs of maintenance An indispensable condition for existence of the admissible preventive period under criterion of the zero option is, that a value of x must be found to meet the following requirement: K(xo) = K(b)
(
1
)
(
)
1
x ox
F
(6)3. Minimum risk of failure of an object in the zero option strategy
A layout of triangular distribution of probability will be further considered to demonstrate the game with Nature under condition of many independent circumstances, which have an impact on inadmissible changes of technical state of objects.
The following forms can be assumed by the triangular distribution functional features:
b
t
b
t
b
b
t
b
b
t
b
t
t
t
F
1
2
)
(
2
1
2
0
2
0
0
2 2 2 2 (7) where:1 Coefficient of economic discouragement will change within the interval of (0,1).
When rising, this coefficient reflects increase of preventive renewal costs in the cost of maintenance of an object in a required technical state.
b: parameter of distribution, or upper limit of survival of an object before failure (for convenience of comparison of numerical values .this parameter is considered further to assume value of one)
A position of the two preventive periods xo and x. on the time axis is illustrated in Fig 1
by practical examples of the expected relative unit costs of maintenance.
0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 x0 x . k(x) x
Fig. 1. Practical examples of relative costs of maintenance.
Now, we shall trace the changes of the function of non-reliability F(x) which illustrates the risk of failure and some of the related functional features, such as: predicted period of operation before corrective renewal vx , or predicted operation period of object
between subsequent renewals x.
The values assumed by the function of non-reliability at the points of performance of preventive renewal F(xo) and (Fx.) can be considered as a measure of risk of failure.
Thus, a curve of a risk of failure versus coefficient of economic discouragement can be shown as in Fig 2. 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 F(x0) F(x .) F(x)
Fig. 2. Curve of changes of the risk of failure versus coefficient of economic discouragement
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 F(x)
Fig 3. Practical reduction of number of failures versus coefficient of economic discouragement.
Please note, that there exists such a value of coefficient of discouragement = 0.6, at which the difference F(x) = F(x.) - F(xo) will assume the maximum of F(x) 0.30.
This is illustrated in Fig. 3. However, in this case, we have to cope with a relatively high risk of failure F(x.) 0.35. A small deviation of the admissible value of coefficient of discouragement down to a level of = 0.4 0.5 would be just advantageous. This would immediately reduce the risk of failure down to a level of F(x.) = 0.125 0.200.
Thus, the zero option strategy of maintenance will offer considerable reduction of the risk of failure. This, however, would not eliminate necessity to undertake decisions on the period of operation before preventive renewal in the intermediate circumstances, i.e. the situations when the cost of maintenance is included between the minimum and emergency values.
An interesting exercise will be to trace the changes of functional features resulting from application of the zero function strategy, such as: predicted period of operation before corrective renewal vx and predicted period of operation between any subsequent
renewals x.
Predicted period of operation before corrective renewal vx can be traced as follows:
b
x
b
x
b
b
b
b
x
b
bx
x
b
x
x
xb
x
x
x2
)
(
2
6
12
12
4
2
0
6
3
2
0
2 2 2 3 2 2 3
(8)where: vx - predicted period of operation before emergency failure occurs, when
preventive renewals are practised
x - fixed period of operation before preventive renewal treatment, x (x.,xo)
For changes in the predicted period of operation before corrective renewal treatment, see curves shown in Fig. 4.
0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 x 0 x . x
Fig. 4. Changes in the predicted period of operation before corrective renewal versus coefficient of economic discouragement
0.0 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 x
Fig. 5. Practical extension of the predicted period of operation before corrective renewal versus coefficient of economic discouragement.
Please, consider again, that similar to the case of reduction of the risk of failure, considerable increase of the predicted period of operation before corrective renewal occurs at the same level of economic discouragement ranging between 0.4 and 0.5, which may be as much as four to five times higher, than the predicted period of operation before the first emergency failure, i.e. Vx = (3 4), see curve in Fig. 5.
Predicted period of operation between any subsequent renewals x may be expressed as below:
b
x
b
b
b
x
b
bx
x
b
x
b
xb
x
x
x
x2
6
12
12
4
2
0
3
3
2
0
3 2 2 3 2 2 3
(9) where:x : predicted period of operation between subsequent renewals
x : fixed period of operation before preventive renewal, x (x.,xo).
For curves illustrating changes in predicted period of operation between subsequent renewals see Fig. 6.
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.4 0.5 x 0 x. x
Fig. 6. Changes in predicted period of operation between subsequent renewals versus coefficient of economic discouragement.
0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.05 0.10 0.15 0.20
Fig. 7. Practical reduction of the predicted period of operation versus coefficient of economic discouragement.
The most rapid decline in the predicted period of operation between subsequent renewals can be observed, when the coefficient of economic discouragement assumes value of = 0.20. This, can be attached to a considerable reduction of the predicted period of operation, down to a value of x. = 0.34. Thus, with the coefficient
assuming a value between 0.4 and 0.5 improvement of the results can be observed. In effect the predicted period of operation will arrive at a range of x. = 0.46 0.60.
Extreme reduction of the predicted period of operation can impose certain strict limits in application of the zero option strategy.
Conclusions
In the situations of severe consequences of emergency shutdowns, the risk of failure can be reduced by application of the zero option strategy of maintenance. Additionally, predicted period of operation before corrective renewal can be extended by the same. An important limitation in application of the zero option strategy can be imposed by extreme reduction of the predicted period of operation between subsequent renewals. Numeric examples relate to the act of making decisions in the circumstances of gradual progress of destructive processes as described by the triangular distribution of probability of failure.
References
[1] Bobrowski D., Modele i metody matematyczne teorii niezawodności. WNT, Warszawa 1985,
[2] Jaźwiński J., Smalko Z., Wykorzystanie rozkładu trójkątnego prawdopodobieństwa w ocenie gotowości i niezawodności obiektów technicznych ,XXV Zimowa Szkoła Niezawodności , Szczyrk 1996r..
[3] Raiffa, H.: Decision Analisis. Addisson-Wesley, Massachusetts, London, Ontario 1968.
[4] Smalko Z. Risk analysis of transport systems. Archives of Transport, Vol. 7, issue 1-4, Warsaw 1995, pp 23-28
[5] Smalko Z., Modelowanie eksploatacyjnych systemów transportowych, ITE, Radom 1996.
