The optimum preventive maintenance schedule at operating conditions can be determined by relating the reliability functions at accelerated conditions with that at normal conditions then utilize an optimization function that relates reliability to preventive maintenance schedule. In Section 7.6.1 we demonstrate these steps through an example. Another approach for determining the optimum preventive maintenance for degrading systems is to determine the optimum threshold degradation level at which maintenance actions are taken by minimizing the over-
all cost of maintenance or by ensuring a minimum acceptable system availability level (Liao et al. 2005). This will be illustrated in Section 7.6.2.
7.6.1 Optimum Preventive Maintenance Schedule at Operating Conditions The first step is to relate the accelerated testing results to stress conditions and obtain a reliability expression which is a function of the applied stresses. We then substitute the normal operating conditions in the expression to obtain a reliability function at normal conditions. We illustrate this by designing an optimum test plan then use its results to obtain the reliability expression.
Suppose we develop an accelerated life test plan for a certain type of electronic devices using two stresses: temperature and electric voltage. The reliability estimate at the design condition over a 10-year period of time is of interest. The design condition is characterized by 50 ºC and 5V. From engineering judgment, the highest levels (upper bounds) of temperature and voltage are pre-specified as 250 ºC and 10 V, respectively. The allowed test duration is 200 h, and the total number of devices placed under test is 200. The minimum number of failures at any test combination is specified as 10. The test plan is determined through the following steps:
1. According to the Arrehenius model, we use 1/(absolute temperature) as the first covariate z1 and 1/(Voltage) as the second covariate z2 in the ALT model.
2. The PH model is used in conducting reliability data analysis and designing the optimal ALT plan using the approach described in Section 7.4.1.1. The model is given by
( ) ( )
0 1 1 2 2
( ; )t t exp z z
λ z =λ β +β where λ0( )t =γ0+γ1t+γ2t2
3. A baseline experiment is conducted to obtain initial estimates for the model parameters. These values are:γˆ0= 0.0001, γˆ1= 0.5, γˆ2 = 0, βˆ1= −3800, and βˆ2= −10.
Approximating γˆ0 to zero we write the hazard rate function as
3800 10
( )
( ; , ) 0.5t T V = t e− T +V
λ (7.15)
The reliability and the probability density function (pdf) expressions are respec- tively given asf t( ;30 C,5 ) 0.5 exp[ (o V = t − e−3.6336 )t2 )]
0.25((3800 / ) 10 / ) )2
( ; , ) exp[ ( T V t )]
R t T V = − e− + (7.16)
0.25((3800 / ) 10 / ) )2
( ; , ) 0.5 exp[ ( T V t )]
f t T V = t −e− + (7.17)
Assume that the normal operating temperature is 30 oC and the normal operating voltage is 5 V. Substituting in Equations 7.16 and 7.17 yields
o 3.63362
( ) ( ;30 C,5 ) exp[ ( t )]
R tn =R t V = − e− (7.18)
o 3.63362
( ) ( ;30 C,5 ) 0.5 exp[ ( t )]
f tn = f t V = t −e− (7.19)
In the second step, we chose an appropriate preventive maintenance (PM) model and determine the optimum PM schedule.
Consider a simple preventive maintenance and replacement policy. Under this policy, two types of actions are performed. The first type is the preventive re- placement that occurs at fixed intervals of time. Components or parts are replaced at predetermined times regardless of the age of the component or the part being replaced. The second type of action is the failure replacement where components or parts are replaced upon failure. This policy is illustrated in Figure 7.3.
The most widely used criterion of maintenance models is to minimize the total expected maintenance and replacement cost per unit time. This can be accomplished by developing a total expected cost function per unit time as follows.
NEW ITEM FAILURE
REPLACEMENTS
PREVENTIVE REPLACEMENT
tp
0
ONE CYCLE
Figure 7.3. Constant interval replacement policy
Let c t( )p be the total replacement cost per unit time as a function of tp.Then
Total expected cost interval (0, ] Expected length of the interval
( )p tp
c t = . (7.20)
The total expected cost in the interval (0,tp] is the sum of the expected cost of failure replacements and the cost of the preventive replacement. During the interval (0,tp], one preventive replacement is performed at a cost of cpand M t( )p failure
replacements at a cost of cfeach, where M t( )p is the expected number of replacements (or renewals) during the interval (0,tp]. The expected length of the interval is tp. Equation 7.20 can be rewritten as
( )p p f ( )p
p
c c M t
c t t
= + . (7.21)
We apply the above model to determine the optimum preventive maintenance schedule for the example for the electronic devices whose reliability and pdf functions obtained from accelerated conditions and are expressed as given in Equations 7.18 and 7.19 respectively. Assuming cp=100 and cf=1200, we rewrite Equation 7.21 as:
0
10 1200 ( ) ( )
tp
n p
p
tf t dt
c t t
+
= ∫
(7.22)
Calculated values of the cost per unit time are shown in Table 7.1 and plotted in Figure 7.4. The optimum preventive maintenance schedule at normal operating conditions is 0.18 unit times.
Table 7.1. Time vs. cost per unit time values (bold numbers indicate optimum values) Time 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 Cost/unit time 918 885 862 847 839 836 840 848
0 500 1000 1500 2000 2500 3000 3500
0,03 0,13 0,23 0,33 0,43 0,53
Time
Cost per unit time
Figure 7.4. Optimum preventive maintenance schedule
7.6.2 Optimum Preventive Maintenance Schedule Based on Accelerated or Normal Degradation
Determining the optimum maintenance schedule for systems subject to degradation follows the same procedures described in Section 7.6.1. It begins by developing a degradation model (at normal operation conditions or at accelerated conditions then extrapolate to normal conditions as shown above). Liao et al. (2005) assume that the degradation is described by a gamma process and obtain the optimum degradation level accordingly. Ettouney and Elsayed (1999) obtain the reliability function for different threshold degradation levels. We demonstrate the determination of the degradation threshold level at normal stress levels using Ettouney and Elsayed (1999) results; then we utilize the optimum degradation level to determine the corresponding optimum preventive maintenance schedule as follows.
Consider the case of corrosion in reinforced concrete bridges which is a major concern to professional engineers because of both public safety and cost which associated with needed repairs and replacement. Prediction of bridge functional degradation due to corrosion conditions is investigated below.
The two main corrosion parameters which affect the reinforcing bars in reinforced concrete bridges are the corrosion rate, rcorr , and the time it takes to initialize corrosion, T1. Enright and Frangopol (1998) present several mean and variance test measurements for both rcorr and T1. In a typical case, they show that the mean and variance of rcorr are between 0.005 in/year and 3 10× −6 in/year, respectively. The mean and standard deviation of T1 are 10 years and 0.4 years, respectively.
In order to estimate the time-variant strength of a reinforced concrete corroded beam, the corrosion effects on the diameter of the reinforcing bars is evaluated first. After corrosion initiation time, T1, the diameter of a reinforcing bar, D(t), can be evaluated as
( ) i corr( 1)
D t D r= − t T− (7.23)
where Di = 1.41 in. is the initial reinforcing bar diameter and t is the elapsed time.
Note that t ≥ T1 and D(t) ≥ 0. For more details of Equation 7.23 the reader is referred to Enright and Frangopol (1998).
The time-variant reinforced concrete strength, Mp(t), can now be evaluated using the conventional design equations in Enright and Frangopol (1998):
p s y 2a
M =nA f ⎛⎜⎝d− ⎞⎟⎠ (7.24)
( s y) (0.85 c` )
a nA f= f b (7.25)
Note that As=πD t( ) 42 . The reinforcing steel and the concrete strengths are fy and fc`, respectively. The number of reinforcing bars is n. The effective depth and the width of the beam are d and b, respectively. For the current example, the
values of different parameters are chosen as fy=40 ksi, fc`=3 ksi, d = 27 in. and b = 16. Using Equation 7.23 through Equation 7.25 the random time-variant strength, Mp(t), can be estimated. Using the previously mentioned values of rcorr
and T1 and a Monte Carlo simulation technique, different strength values for different reinforced concrete beams can be simulated. Thus, a discrete time-variant reinforced concrete strength, xij can be evaluated from the Monte Carlo simulation of the continuous strength Mp(t).
Eghbali and Elsayed (2001) show that the reliability function for a specified failure threshold degradation x is expressed as
( ) ( ; ) exp[ ]
exp( )
x x
R t P X x t
b at
− γ
= > =
− (7.26)
where X is a random variable represents the degradation measure, a, b and γ are
constants. The Maximum Likelihood method was utilized to estimate the para- meters of Equation 7.26:
1
1 1 1
( , , , ) ( ex p ( )) ex p ( ex p ( ))
i i
m n m n ij
i ij i
i i j
L a b t b a t x b x a t
γ γ
γ γ −
= = =
=∏ − ∏ ∏ − − (7.27)
where m is the number of years, ni is the total number of degradation data in a year i and xij is the strength of unit j in year i. Taking the logarithm of Equation 7.27 we obtain
1 1 1 1 1 1 1
ln ln ln ( 1)ln
exp( )
i i
n n
m m m m m
ij
i i i i ij
i i i i j i j i
L n n b n at x x
b at
γ
γ γ
= = = = = = =
= − + + − −
∑ ∑ ∑ ∑∑ ∑∑ − (7.28)
Equating the partial derivatives of Equation 7.28 with respect to γ , a and b to zeros and solving the resulting equations using a modified Powell hybrid algorithm and a finite difference approximation to the Jacobian yields: a = 0.12, b = 1.1346×107 and γ = 1.49. The resulting reliability function is
( ) ( ; ) exp[ ]
exp( )
x x
R t P X x t
b at
− γ
= > =
− or
1.49
( ) exp[ 7 ]
1.1346598 10 exp(-0.12 )
x x
R t t
= −
× × .
The reliability for different threshold values of the strength is shown in Figure 7.5. The time to failure for threshold values of 4800, 4000, 3500, 3000, and 2500 are 25.04, 27.25, 28.88, 30.76, and 33.0 years respectively.
Figure 7.5. Reliability for different threshold levels
The next step is to determine the optimum preventive maintenance schedule for every threshold level and select the schedule corresponding to the smallest cost among all optimum cost values. This will represent both the optimum threshold level and the corresponding optimum preventive maintenance schedule.
We demonstrate this for two threshold levels (S = 4800 and S = 2500) assuming cp=10 and cf=1200; we utilize Equation 7.21 as follows:
0
10 1200 ( ; ) ( )
tp
p
p
tf x t dt
c t t
+
= ∫
(7.29)
where
( ; ) 1exp( ), 0, ( )
( ) ( ) at
f x t x x t t be
t t
γ γ
γ θ
θ − θ −
= − > = (7.30)
As shown in Figure 7.6, the optimum tp values for S=400 and S=2500 are 17 and 16 years respectively. The minimum of the two is the one corresponding to S = 2500. Therefore, the optimum threshold is 2500 and the corresponding optimum maintenance schedule is 16 years.
0 0.2 0.4 0.6 0.8 1
0 10 20 30 40 50 60
Time (Years)
Reliability
s=2500 s=3000
s=3500 s=4000
s=4800
0 0,5 1 1,5 2 2,5 3
2 12 22 32
Time
Cost / Unit Time S=4800S=2500
Figure 7.6. Total cost per unit time vs. time