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A new approximation for the inspection period of systems with different failure rates
European Journal of Operational Research 45 (1990) 219-223 North-Holland
219
A new approximation for the inspection period of systems with different failure rates J. R O D R I G U E S
DIAS
Department of Mathematics, Universityof Evora, 7000 Evora, Portugal Abstract: Consider a system whose working state is only known if it is inspected. Suppose that each inspection has a fixed cost and also suppose that there is a fixed cost for unit time of bad working. In this paper we present a new approximate solution for the inspection period, which is a generalization of a previous one. We verify that when the failure rate is constant (exponential distribution) then the approximate solution is a (nearly) optimal solution. We also analyse the validity of that new approximation when the failure rate is: (a) decreasing; (b) increasing; (c) first increasing and then decreasing; and, finally, (d) when the failure rate has a 'bathtub' shape. Generally, we verify that in these cases this approximate solution is better than a previous one. Keywords: Inspection policy, inspection period, approximation, failure rate, cost comparison
1. Introduction
Let us consider a system whose lifetime is a continuous random variable T with a reliability function R(t) and a failure rate h(t) given by
R ( t ) = l - F ( t ) , t>O, h(t)=f(t)/R(t), t>0,
(l) (2)
where f(t) and F(t) are the density and the distribution functions of T. Let E(T) be the expected value of 7". We are going to consider the next model: (a) The system has only two working states: a good one concerning his lifetime and a bad one as the result of a failure. (b) This bad working state is only known if the system is inspected. (c) The inspections are periodic, perfect, instantaneous and do not interfere in the system state. (d) There is a fixed cost C 1 for each inspection and a fixed cost C 2 for each unit time of not detected bad working.
Many other models have been (and are being) studied in the literature. In Pierskalla and Voelker (1976) and in Sherif (1982) we can find a lot of different models and references. In the model we are considering here, the problem is to obtain the period P between inspections so that the expected total cost E(C) is a minimum during a cycle, with
E( C) = CIE( N) + CzE( D),
where E(N) is the expected number of inspections during a cycle and E(D) is the expected time between the system failure and its detection. We consider that a cycle begins when the system is new (or like a new one, after a repair) and ends when the failure is detected. This problem can find applications in quality control, medicine, nuclear energy, defense, etc. In this model (using periodic inspections), the solution for the inpection period P is obtained from the equation (Rodrigues Dias, 1983)
R ( k P ) - ( r + P ) ~ kf(kP)=O, k~O
Received June 1989
(3)
r = C,/C
0377-2217/90/$3.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
k~l
z.
(4)
J. Rodrigues Dias / An approximation for the inspectionperiod of systems
220
For the constant failure rate (exponential distribution, with R ( t ) = e x p ( - t / a ) , E ( T ) = a) we obtain
exp( P / a ) - (1 + P / a + r / a ) = 0.
(5)
I n this case, the periodic inspection is the optimal one. Generally, Barlow et al. (1965) showed that for increasing failure rates the o p t i m a l inspection policies are n o t periodic, having o b t a i n e d a recurrence f o r m u l a to calculate the inspection times. However, some authors consider difficult that calculation. So, M u n f o r d a n d S h a h a n i (1972), N a k a g a w a a n d Yasui (1980) a n d Rodrigues Dias (1987a), a m o n g others, proposed a p p r o x i m a t e solutions n o t periodic b u t easier to obtain. C o n s i d e r i n g the fact that the periodic inspections are very simple in practice (simple models are very i m p o r t a n t to the enterprises, as it can be observed in Saniga a n d Shirland (1977) a b o u t quality control), specially when P can be easily obtained, so in this p a p e r we are going to study periodic inspections. N a k a g a w a a n d Yasui (1979) had o b t a i n e d the a p p r o x i m a t e result for the i n s p e c t i o n period:
e*=
[2rE(T)]1/2
(6)
This result was obtained considering an approximation obtained by Schneeweiss (1976), which is analysed in Rodrigues Dias (1983), but in the case of the exponential distribution it can be easily obtained if in (5) we consider exp( P / a ) ~ 1 + P / a + ½ ( P / / a ) 2.
(7)
Rodrigues Dias (1983) studied that a p p r o x i m a tion (6) in terms of costs for different lifetime d i s t r i b u t i o n s (not only for the W e i b u l l distribution), having c o n c l u d e d that it was m u c h better t h a n their authors had referred (in terms of the relative errors of P * ).
2. The new approximation for the constant failure rate If in expression (5) we consider
e'= P/E(T),
r' = r / E ( T ) ,
(8)
then we can write e x p ( P ' ) - (1 + P ' + r ' ) = O.
(9)
In Table 1 we present, for different values of r ' , the relative errors, in percentage, of the inspection period P * a n d of the expected total cost E * (C), given b y
Q1 = ( P * - P ) / P " Q2 = [ E * ( C )
(10)
100 %,
- E(C)]/E(C).
100 %,
(11)
where P a n d P * are o b t a i n e d from (5) a n d (6). E ( C ) a n d E * ( C ) are the c o r r e s p o n d e n t expected total costs per cycle c o n c e r n i n g to P a n d P * in the case of the c o n s t a n t failure rate. The very interesting a n d i m p o r t a n t p o i n t that we previously have not seen (Rodrigues Dias, 1983) is that the relative errors Q1 can be very well fitted b y a f u n c t i o n of the type QI* = a + b y e - .
(12)
If we write x = v;/-, then we have the linear regression
(13)
O~ = a + bx.
Table 1 Relative errors (%) of the inspection periods and the expected total costs for the constant failure rate
r'
Q1
Q2
Q~ *
Q~ *
0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.75 1.05 1.29 1.49 1.67 1.83 1.97 2.11 2.24 2.36 3.33 4.08 4.71 5.27 5.77 6.23 6.66 7.06 7.45 10.52 12.87 14.85 16.59 18.16 19.60 20.94 22.20 23.38
< 0.01 < 0.01 < 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.03 0.05 0.08 0.11 0.13 0.16 0.18 0.21 0.23 0.26 0.49 0.71 0.92 1.13 1.32 1.51 1.69 1.86 2.03
< 0.01 < 0.01 < 0.01 0.01 0.01 0.01 0.01 0.01 0.02 0.02 0.02 0.03 0.03 0.03 0.04 0.04 0.04 0.04 0.04 0.05 0.05 0.05 0.04 0.03 0.02 0.01 > - 0.001 - 0.01
< 0.o00001 < 0.000001 < 0.000001 < 0.000o01 < 0.000001 < 0.000001 < 0.000001 < 0.0o001 < 0.00o01 < 0.00001 < 0.00001 < 0.00001 < 0.00001 < 0.00001 < 0.00001 < 0.00001 < 0.00001 < 0.00001 < 0.00001 < 0.0001 < 0.0001 < 0.0001 < 0.00001 < 0.00001 < 0.00001 < 0.000001 < 0.00000001 < 0.000001
J. Rodrigues Dias / An approximation for the inspection period of systems
Using the well known least-squares method we can obtain a and b: a = 0.000255,
b = 0.2340,
r 2 (determination coef.) = 0.999994.
It is obvious that the fitting is excellent. If we consider a -- 0, so we can write Q~* = 0.234vf~.
(14)
1 + 0.234v~-
-
1 + 0.234vc~- '
100%,
Q~'*=[E**(C)-E(C)]/E(C)-100%.
exp(P') ~ 1 + P' + ½(p,)2 + (~)(p,)3 + (~)(p,)4,
(17)
Note that E * * ( C ) is the expected total cost per cycle concerning the inspection period P * * estimated from (15). As we can see, the relative errors of P * * are very small, never greater than 0.05% and almost smaller, for the big range of values of r'. On other hand, if we analyse the relative errors of the expected total cost E * * ( C ) , we conclude that in practical applications they are irrelevant, which means that this new approximation can be considered a (nearly) optimal one in the case of the constant failure rate. If we compare Ql* * and Q~' * with Qa and Q2, these ones concerning the approximation P * obtained by Nakagawa and Yasui (1979), we conclude that P * * is always much better, in special for greater values of r ' (just when the approximation E ( D ) ~ ½P is worse). In this case the factor K = 1/(1 + 0.234v/~) can be interpreted as a correction factor for P * . So, the new approximation P * * can be seen as a generalization of P * We have tried to obtain directly this new approximate solution from (5), such as we have done
(19)
so we can write from (9), r,
(1 + y x / ~ ) 2
(16)
(18)
and if we consider
(15)
Just now we have obtained a new approximation for the inspection period P * *, it is important to analyse how good it is in fact. We will begin to see the case when the failure rate is constant (Rodrigues Dias, 1987b). In Table 1 we also have the relative errors of the approximations P * * and E * * (C), in percentage, given by Q¢'* = ( P * * - P ) / P .
to obtain (6), considering more terms in (7), but we think it is not analytically easy. However, if we suppose, from (15), that the approximate solution for the inspection period has the form P = P*/[1 +yV~-],
If we have in mind (6), (10) and (14), then we can finally obtain the new approximation for the inspection period when the failure rate is constant: p**
221
- r
t
r,3/2vl ~
+
+ 3(1 + yv/rV) 3
= o.
On other hand, if we powers of 1/(1 + y ~ ) in only the terms in r ' , ( r ' ) can write for small values
r ,2
6(1 + y v / ~ ) 4 (20)
use the series for the (20), and if we consider 3/2 and ( r ' ) 2, then we of r ' ,
r ' ( 1 - 2 y v ~ + 3y2r ') + -~q~(r')3/2(1 - 3yx/r7) + -~r ' 2 - r ' = O.
(21)
We can verify easily that the left member of (21) is equal to zero if y = ~ - - ~ 0 . 2 3 5 7 . So, replacing y in (18), we obtain a quite similar solution for the inspection period: P** ~ P*/[1 + (lv~)¢~].
(22)
This approximation seems a little better than (15) when r ' is very small, but it becomes worse when r ' increases. Also, if we note that 2/(,~e) -- 0.234199 = 0.234, the approximate solution P * * can be written using the very interesting form P * * = P * / [ 1 + (2/,~e)v~7].
(23)
The results obtained from (23) are very little better than those obtained from (15). This fact can be easily explained if we have in mind that we have considered a = 0.000255 = 0 to obtain (14). In conclusion, as the differences are irrelevant, we think that for practical purposes the simplest factor is 0.234, instead of the interesting factors -~v~- and 2 / ( r e ) .
J. Rodrigues Dias / An approximation for the inspection period of systems
222
3. Validity of the new approximation for other failure rates
(b) The Hjorth (1980) distribution, with reliability function
In the previous section we have obtained a (nearly) optimal solution for the inspection period of systems with a constant failure rate. Now we are going to analyse it for different failure rates. To do that we are going to consider the next distributions: Weibull, Hjorth (1980) and log-normal. We choose these distributions, because: (a) The Weibull distribution, with reliability function
R ( T ) = e x p ( - 6 t 2 / 2 ) / ( 1 + fit) a/È
(24)
R ( T ) = e x p [ - ( t / a ) a]
(25)
presents different types of failure rates depending on the values of the positive parameters a, fl and 0. Here we are going to consider 0 = fl = 1 and 8 = 0.01, just a case with a bathtub-shaped failure rate. (c) The log-normal distribution, with density function
f ( t ) = e x p [ - (log t - / 1 ) 2 / ( 2 o 2 ) ] / [
2~ot] (26)
has a decreasing failure rate when the shape parameter fl < 1; it has a constant failure rate when fl = 1 (exponential distribution); and it has an increasing failure rate when fl > 1. In this paper we will consider fl = 0.7, 1.0, 1.5, 2.0, 2.5 and 3.0. In Rodrigues Dias (1987a) other values of fl were considered.
has an increasing failure rate at the beginning and then a decreasing one. With these distributions of the system lifetime T, we are considering the different and important cases that we can find in practice for the failure rates.
Table 2 Relative errors (%) of the inspection periods and the expected total costs for different failure rates Distribution
r ' = 0.0125
Q1
r ' = 0.0500
Q{" *
Q2
Q~ *
Q1
Q~" *
Q2
Q~ *"
WeibuH ( fl ) 0.7 1.0 1.5 2.0 2.5 3.0
7.3 2.6 0.4 0.0 0.0 0.0
4.5 0.0 -2.2 -2.5 -2.6 -2.6
0.3 0.0 0.0 0.0 0.0 0.0
0.1 0.0 0.0 0.0 0.0 0.0
11.3 5.3 1.1 0.0 -0.1 -0.1
5.8 0.0 -3.9 -5.0 -5.1 -5.1
0.6 0.1 0.0 0.0 0.0 0.0
0.2 0.0 0.1 0.1 0.1 0.1
Hjorth
6.0
3.3
0.2
0.1
11.1
5.6
0.6
0.2
Log-normal
-0.4
-2.9
0.0
0.0
1.3
-3.8
0.0
0.1
Distribution
r'=0.2000
r'=0.8000
QI
QI* *
Q2
Q~' *
Q1
QI* *
Q2
Q~ *
17.5 10.5 3.4 0.0 -1.0 -1.0
6.4 0.1 --6.4 --9.5 -10.4 -10.4
1.3 0.5 0.1 0.0 0.0 0.0
0.2 0.0 0.2 0.4 0.5 0.5
27.0 ~.9 11.8 1.3 -11.2 -13.7
5.1 0.0 --7.5 --16.2 -26.5 -28.6
2.5 1.7 0.6 0.0 1.0 3.1
0.1 0.0 0.3 1.3 4.7 10.6
19.0
7.7
1.5
0.3
30.0
7.5
3.0
0.2
9.0
-1.3
0.4
0.0
24.8
3.2
2.5
0.1
Weigh ( fl ) 0.7 1.O 1.5 2.0 2.5 3.0
Worth ~g-normal
J. Rodrigues Dias / An approximation for the inspection period of systems
So, considering now these distributions, in Table 2 we present for r ' = 0.0125, 0.0500, 0.2000 and 0.8000 the values of the relative errors Q1 and Q2 concerning the approximation P * obtained by Nakagawa and Yasui (1979). In Rodrigues Dias (1983) we analyse these values. Now, also in Table 2, we present the relative errors Ql* * and Q2 , concerning the new approximation P * * If we compare the values of Qa with QI** and, on other hand, the values of Q2 with Q~' *, we can point out the next conclusions: (a) For the decreasing failure rate (Weibull distribution, /3 = 0.7) and for the bathtub-shaped failure rate the results concerning the approximation P * * are always much better than those ones concerning P *. (b) For the log-normal distribution, P * * is specially better than P * for greater values of r'. (c) For the increasing failure rate (Weibull distribution, fl > 1) P * is better than P * * . But in this case we can use a not periodic inspection policy that is much better in terms of costs and in which approximate inspection times can be easily calculated (Rodrigues Dias, 1986, 1987a). Generally, if we have in mind the different types of failure rates, so we can say that the new approximation P * * is better than that one P * obtained by Nakagawa and Yasui (1979). On the other hand, if we analyse the relative errors Q~" *, which are almost smaller than 0.5%, we can say that P * * is a very good approximate solution.
4. Conclusions In a previous paper (Rodrigues Dias, 1983) we have shown, for different failure rates, that the approximate inspection period P * obtained by Nakagawa and yasui (1979), if we analyse it in terms of costs, is a better solution than their authors have referred. In this paper, using the method of least squares, we obtain another approximation P * * for the inspection period, also easy to calculate, which we justify analytically and which we analyse for the main types of failure rates. We conclude that this new approximation is a (nearly) optimal one for the constant failure rate (exponential distribution), what did not occur with the approximation of Nakagawa and Yasui (1979), specially for big values of r'.
223
For other types of failure rates (decreasing, increasing, bathtub-shaped and, finally, increasing and after decreasing), generally we conclude that P * * is better than P * . Only for the case when the failure rate is increasing, we have obtained worse results. However, this is a case in which not periodic inspections are preferable. Generally, if we analyse P * * in terms of costs, as it should be, we conclude that the relative errors are almost smaller than 0.5%, which means that P * * is a very good solution. Finally, we can consider P * * as a generalization of P * , introducing a correction factor K = 1/(1 + 0.234~7), which depends on the costs C 1 and C 2 and on the expected system lifetime E(T).
References Barlow, R.E., and Proschan, F. (1965), Mathematical Theory of Reliability, Wiley, New York. Hjorth, U. (1980), "A reliability distribution with increasing, decreasing, constant and bathtub-shaped failure rates", Technometrics 22/1, 99-107. Munford, A. G., and Shahani, A.K. (1972), "A nearly optimal inspection policy", Operational Research Quarterly 23/3, 373-379. Nakagawa, T., and Yasui, K. (1979), "Approximate calculation of inspection policy with Weibull failure times", IEEE Transactions on Reliability R28/5, 403-404. Nakagawa, T., and Yasui, K. (1980), "Approximate calculation of optimal inspection times", Journal of the Operational Research Society 31/9, 851-853. Pierskalla, W.P., and Voelker, J.A. (1976), "A survey of maintenance models: The control and surveillance of deteriorating systems", Naval Research Logistics Quarterly 23/3, 353-388. Rodrigues Dias, J. (1983), "Influence de la prriode d'inspection sur les cofits dans l'inspection prriodique de systrmes", Revue de Statistique Appliqu~e 31/4, 5-15. Rodrigues Dias, J. (1986), "Correspondence between results of a periodic and a not periodic inspection policies", Proceedings of the X I Spanish-Portuguese Conference on Mathematics, Vol. 1L 533-542, University of Extremadura, Badajoz (in Portuguese). Rodrigues Dias, J. (1987a), "Systems inspection policies", PhD thesis, University of Evora (in Portuguese). Rodrigues Dias, J. (1987b), "A solution for the inspection period of systems", X I I Portuguese-Spanish Conference on Mathematics, University of Minho (in Portuguese). Saniga, E.M., and Shirland, L.E. (1977), "Quality control in practice--A survey", Quality Progress 10/5, 30-33. Schneeweiss, W.G. (1976), "On the mean duration of hidden faults in periodically checked systems", IEEE Transactions on Reliability R-25/5, 346-348. Sherif, Y.S. (1982), "Reliability analysis: Optimal inspection and maintenance schedules of failing systems", Microelectronic Reliability 22/1, 59-115.
219
A new approximation for the inspection period of systems with different failure rates J. R O D R I G U E S
DIAS
Department of Mathematics, Universityof Evora, 7000 Evora, Portugal Abstract: Consider a system whose working state is only known if it is inspected. Suppose that each inspection has a fixed cost and also suppose that there is a fixed cost for unit time of bad working. In this paper we present a new approximate solution for the inspection period, which is a generalization of a previous one. We verify that when the failure rate is constant (exponential distribution) then the approximate solution is a (nearly) optimal solution. We also analyse the validity of that new approximation when the failure rate is: (a) decreasing; (b) increasing; (c) first increasing and then decreasing; and, finally, (d) when the failure rate has a 'bathtub' shape. Generally, we verify that in these cases this approximate solution is better than a previous one. Keywords: Inspection policy, inspection period, approximation, failure rate, cost comparison
1. Introduction
Let us consider a system whose lifetime is a continuous random variable T with a reliability function R(t) and a failure rate h(t) given by
R ( t ) = l - F ( t ) , t>O, h(t)=f(t)/R(t), t>0,
(l) (2)
where f(t) and F(t) are the density and the distribution functions of T. Let E(T) be the expected value of 7". We are going to consider the next model: (a) The system has only two working states: a good one concerning his lifetime and a bad one as the result of a failure. (b) This bad working state is only known if the system is inspected. (c) The inspections are periodic, perfect, instantaneous and do not interfere in the system state. (d) There is a fixed cost C 1 for each inspection and a fixed cost C 2 for each unit time of not detected bad working.
Many other models have been (and are being) studied in the literature. In Pierskalla and Voelker (1976) and in Sherif (1982) we can find a lot of different models and references. In the model we are considering here, the problem is to obtain the period P between inspections so that the expected total cost E(C) is a minimum during a cycle, with
E( C) = CIE( N) + CzE( D),
where E(N) is the expected number of inspections during a cycle and E(D) is the expected time between the system failure and its detection. We consider that a cycle begins when the system is new (or like a new one, after a repair) and ends when the failure is detected. This problem can find applications in quality control, medicine, nuclear energy, defense, etc. In this model (using periodic inspections), the solution for the inpection period P is obtained from the equation (Rodrigues Dias, 1983)
R ( k P ) - ( r + P ) ~ kf(kP)=O, k~O
Received June 1989
(3)
r = C,/C
0377-2217/90/$3.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
k~l
z.
(4)
J. Rodrigues Dias / An approximation for the inspectionperiod of systems
220
For the constant failure rate (exponential distribution, with R ( t ) = e x p ( - t / a ) , E ( T ) = a) we obtain
exp( P / a ) - (1 + P / a + r / a ) = 0.
(5)
I n this case, the periodic inspection is the optimal one. Generally, Barlow et al. (1965) showed that for increasing failure rates the o p t i m a l inspection policies are n o t periodic, having o b t a i n e d a recurrence f o r m u l a to calculate the inspection times. However, some authors consider difficult that calculation. So, M u n f o r d a n d S h a h a n i (1972), N a k a g a w a a n d Yasui (1980) a n d Rodrigues Dias (1987a), a m o n g others, proposed a p p r o x i m a t e solutions n o t periodic b u t easier to obtain. C o n s i d e r i n g the fact that the periodic inspections are very simple in practice (simple models are very i m p o r t a n t to the enterprises, as it can be observed in Saniga a n d Shirland (1977) a b o u t quality control), specially when P can be easily obtained, so in this p a p e r we are going to study periodic inspections. N a k a g a w a a n d Yasui (1979) had o b t a i n e d the a p p r o x i m a t e result for the i n s p e c t i o n period:
e*=
[2rE(T)]1/2
(6)
This result was obtained considering an approximation obtained by Schneeweiss (1976), which is analysed in Rodrigues Dias (1983), but in the case of the exponential distribution it can be easily obtained if in (5) we consider exp( P / a ) ~ 1 + P / a + ½ ( P / / a ) 2.
(7)
Rodrigues Dias (1983) studied that a p p r o x i m a tion (6) in terms of costs for different lifetime d i s t r i b u t i o n s (not only for the W e i b u l l distribution), having c o n c l u d e d that it was m u c h better t h a n their authors had referred (in terms of the relative errors of P * ).
2. The new approximation for the constant failure rate If in expression (5) we consider
e'= P/E(T),
r' = r / E ( T ) ,
(8)
then we can write e x p ( P ' ) - (1 + P ' + r ' ) = O.
(9)
In Table 1 we present, for different values of r ' , the relative errors, in percentage, of the inspection period P * a n d of the expected total cost E * (C), given b y
Q1 = ( P * - P ) / P " Q2 = [ E * ( C )
(10)
100 %,
- E(C)]/E(C).
100 %,
(11)
where P a n d P * are o b t a i n e d from (5) a n d (6). E ( C ) a n d E * ( C ) are the c o r r e s p o n d e n t expected total costs per cycle c o n c e r n i n g to P a n d P * in the case of the c o n s t a n t failure rate. The very interesting a n d i m p o r t a n t p o i n t that we previously have not seen (Rodrigues Dias, 1983) is that the relative errors Q1 can be very well fitted b y a f u n c t i o n of the type QI* = a + b y e - .
(12)
If we write x = v;/-, then we have the linear regression
(13)
O~ = a + bx.
Table 1 Relative errors (%) of the inspection periods and the expected total costs for the constant failure rate
r'
Q1
Q2
Q~ *
Q~ *
0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.75 1.05 1.29 1.49 1.67 1.83 1.97 2.11 2.24 2.36 3.33 4.08 4.71 5.27 5.77 6.23 6.66 7.06 7.45 10.52 12.87 14.85 16.59 18.16 19.60 20.94 22.20 23.38
< 0.01 < 0.01 < 0.01 0.01 0.01 0.02 0.02 0.02 0.02 0.03 0.05 0.08 0.11 0.13 0.16 0.18 0.21 0.23 0.26 0.49 0.71 0.92 1.13 1.32 1.51 1.69 1.86 2.03
< 0.01 < 0.01 < 0.01 0.01 0.01 0.01 0.01 0.01 0.02 0.02 0.02 0.03 0.03 0.03 0.04 0.04 0.04 0.04 0.04 0.05 0.05 0.05 0.04 0.03 0.02 0.01 > - 0.001 - 0.01
< 0.o00001 < 0.000001 < 0.000001 < 0.000o01 < 0.000001 < 0.000001 < 0.000001 < 0.0o001 < 0.00o01 < 0.00001 < 0.00001 < 0.00001 < 0.00001 < 0.00001 < 0.00001 < 0.00001 < 0.00001 < 0.00001 < 0.00001 < 0.0001 < 0.0001 < 0.0001 < 0.00001 < 0.00001 < 0.00001 < 0.000001 < 0.00000001 < 0.000001
J. Rodrigues Dias / An approximation for the inspection period of systems
Using the well known least-squares method we can obtain a and b: a = 0.000255,
b = 0.2340,
r 2 (determination coef.) = 0.999994.
It is obvious that the fitting is excellent. If we consider a -- 0, so we can write Q~* = 0.234vf~.
(14)
1 + 0.234v~-
-
1 + 0.234vc~- '
100%,
Q~'*=[E**(C)-E(C)]/E(C)-100%.
exp(P') ~ 1 + P' + ½(p,)2 + (~)(p,)3 + (~)(p,)4,
(17)
Note that E * * ( C ) is the expected total cost per cycle concerning the inspection period P * * estimated from (15). As we can see, the relative errors of P * * are very small, never greater than 0.05% and almost smaller, for the big range of values of r'. On other hand, if we analyse the relative errors of the expected total cost E * * ( C ) , we conclude that in practical applications they are irrelevant, which means that this new approximation can be considered a (nearly) optimal one in the case of the constant failure rate. If we compare Ql* * and Q~' * with Qa and Q2, these ones concerning the approximation P * obtained by Nakagawa and Yasui (1979), we conclude that P * * is always much better, in special for greater values of r ' (just when the approximation E ( D ) ~ ½P is worse). In this case the factor K = 1/(1 + 0.234v/~) can be interpreted as a correction factor for P * . So, the new approximation P * * can be seen as a generalization of P * We have tried to obtain directly this new approximate solution from (5), such as we have done
(19)
so we can write from (9), r,
(1 + y x / ~ ) 2
(16)
(18)
and if we consider
(15)
Just now we have obtained a new approximation for the inspection period P * *, it is important to analyse how good it is in fact. We will begin to see the case when the failure rate is constant (Rodrigues Dias, 1987b). In Table 1 we also have the relative errors of the approximations P * * and E * * (C), in percentage, given by Q¢'* = ( P * * - P ) / P .
to obtain (6), considering more terms in (7), but we think it is not analytically easy. However, if we suppose, from (15), that the approximate solution for the inspection period has the form P = P*/[1 +yV~-],
If we have in mind (6), (10) and (14), then we can finally obtain the new approximation for the inspection period when the failure rate is constant: p**
221
- r
t
r,3/2vl ~
+
+ 3(1 + yv/rV) 3
= o.
On other hand, if we powers of 1/(1 + y ~ ) in only the terms in r ' , ( r ' ) can write for small values
r ,2
6(1 + y v / ~ ) 4 (20)
use the series for the (20), and if we consider 3/2 and ( r ' ) 2, then we of r ' ,
r ' ( 1 - 2 y v ~ + 3y2r ') + -~q~(r')3/2(1 - 3yx/r7) + -~r ' 2 - r ' = O.
(21)
We can verify easily that the left member of (21) is equal to zero if y = ~ - - ~ 0 . 2 3 5 7 . So, replacing y in (18), we obtain a quite similar solution for the inspection period: P** ~ P*/[1 + (lv~)¢~].
(22)
This approximation seems a little better than (15) when r ' is very small, but it becomes worse when r ' increases. Also, if we note that 2/(,~e) -- 0.234199 = 0.234, the approximate solution P * * can be written using the very interesting form P * * = P * / [ 1 + (2/,~e)v~7].
(23)
The results obtained from (23) are very little better than those obtained from (15). This fact can be easily explained if we have in mind that we have considered a = 0.000255 = 0 to obtain (14). In conclusion, as the differences are irrelevant, we think that for practical purposes the simplest factor is 0.234, instead of the interesting factors -~v~- and 2 / ( r e ) .
J. Rodrigues Dias / An approximation for the inspection period of systems
222
3. Validity of the new approximation for other failure rates
(b) The Hjorth (1980) distribution, with reliability function
In the previous section we have obtained a (nearly) optimal solution for the inspection period of systems with a constant failure rate. Now we are going to analyse it for different failure rates. To do that we are going to consider the next distributions: Weibull, Hjorth (1980) and log-normal. We choose these distributions, because: (a) The Weibull distribution, with reliability function
R ( T ) = e x p ( - 6 t 2 / 2 ) / ( 1 + fit) a/È
(24)
R ( T ) = e x p [ - ( t / a ) a]
(25)
presents different types of failure rates depending on the values of the positive parameters a, fl and 0. Here we are going to consider 0 = fl = 1 and 8 = 0.01, just a case with a bathtub-shaped failure rate. (c) The log-normal distribution, with density function
f ( t ) = e x p [ - (log t - / 1 ) 2 / ( 2 o 2 ) ] / [
2~ot] (26)
has a decreasing failure rate when the shape parameter fl < 1; it has a constant failure rate when fl = 1 (exponential distribution); and it has an increasing failure rate when fl > 1. In this paper we will consider fl = 0.7, 1.0, 1.5, 2.0, 2.5 and 3.0. In Rodrigues Dias (1987a) other values of fl were considered.
has an increasing failure rate at the beginning and then a decreasing one. With these distributions of the system lifetime T, we are considering the different and important cases that we can find in practice for the failure rates.
Table 2 Relative errors (%) of the inspection periods and the expected total costs for different failure rates Distribution
r ' = 0.0125
Q1
r ' = 0.0500
Q{" *
Q2
Q~ *
Q1
Q~" *
Q2
Q~ *"
WeibuH ( fl ) 0.7 1.0 1.5 2.0 2.5 3.0
7.3 2.6 0.4 0.0 0.0 0.0
4.5 0.0 -2.2 -2.5 -2.6 -2.6
0.3 0.0 0.0 0.0 0.0 0.0
0.1 0.0 0.0 0.0 0.0 0.0
11.3 5.3 1.1 0.0 -0.1 -0.1
5.8 0.0 -3.9 -5.0 -5.1 -5.1
0.6 0.1 0.0 0.0 0.0 0.0
0.2 0.0 0.1 0.1 0.1 0.1
Hjorth
6.0
3.3
0.2
0.1
11.1
5.6
0.6
0.2
Log-normal
-0.4
-2.9
0.0
0.0
1.3
-3.8
0.0
0.1
Distribution
r'=0.2000
r'=0.8000
QI
QI* *
Q2
Q~' *
Q1
QI* *
Q2
Q~ *
17.5 10.5 3.4 0.0 -1.0 -1.0
6.4 0.1 --6.4 --9.5 -10.4 -10.4
1.3 0.5 0.1 0.0 0.0 0.0
0.2 0.0 0.2 0.4 0.5 0.5
27.0 ~.9 11.8 1.3 -11.2 -13.7
5.1 0.0 --7.5 --16.2 -26.5 -28.6
2.5 1.7 0.6 0.0 1.0 3.1
0.1 0.0 0.3 1.3 4.7 10.6
19.0
7.7
1.5
0.3
30.0
7.5
3.0
0.2
9.0
-1.3
0.4
0.0
24.8
3.2
2.5
0.1
Weigh ( fl ) 0.7 1.O 1.5 2.0 2.5 3.0
Worth ~g-normal
J. Rodrigues Dias / An approximation for the inspection period of systems
So, considering now these distributions, in Table 2 we present for r ' = 0.0125, 0.0500, 0.2000 and 0.8000 the values of the relative errors Q1 and Q2 concerning the approximation P * obtained by Nakagawa and Yasui (1979). In Rodrigues Dias (1983) we analyse these values. Now, also in Table 2, we present the relative errors Ql* * and Q2 , concerning the new approximation P * * If we compare the values of Qa with QI** and, on other hand, the values of Q2 with Q~' *, we can point out the next conclusions: (a) For the decreasing failure rate (Weibull distribution, /3 = 0.7) and for the bathtub-shaped failure rate the results concerning the approximation P * * are always much better than those ones concerning P *. (b) For the log-normal distribution, P * * is specially better than P * for greater values of r'. (c) For the increasing failure rate (Weibull distribution, fl > 1) P * is better than P * * . But in this case we can use a not periodic inspection policy that is much better in terms of costs and in which approximate inspection times can be easily calculated (Rodrigues Dias, 1986, 1987a). Generally, if we have in mind the different types of failure rates, so we can say that the new approximation P * * is better than that one P * obtained by Nakagawa and Yasui (1979). On the other hand, if we analyse the relative errors Q~" *, which are almost smaller than 0.5%, we can say that P * * is a very good approximate solution.
4. Conclusions In a previous paper (Rodrigues Dias, 1983) we have shown, for different failure rates, that the approximate inspection period P * obtained by Nakagawa and yasui (1979), if we analyse it in terms of costs, is a better solution than their authors have referred. In this paper, using the method of least squares, we obtain another approximation P * * for the inspection period, also easy to calculate, which we justify analytically and which we analyse for the main types of failure rates. We conclude that this new approximation is a (nearly) optimal one for the constant failure rate (exponential distribution), what did not occur with the approximation of Nakagawa and Yasui (1979), specially for big values of r'.
223
For other types of failure rates (decreasing, increasing, bathtub-shaped and, finally, increasing and after decreasing), generally we conclude that P * * is better than P * . Only for the case when the failure rate is increasing, we have obtained worse results. However, this is a case in which not periodic inspections are preferable. Generally, if we analyse P * * in terms of costs, as it should be, we conclude that the relative errors are almost smaller than 0.5%, which means that P * * is a very good solution. Finally, we can consider P * * as a generalization of P * , introducing a correction factor K = 1/(1 + 0.234~7), which depends on the costs C 1 and C 2 and on the expected system lifetime E(T).
References Barlow, R.E., and Proschan, F. (1965), Mathematical Theory of Reliability, Wiley, New York. Hjorth, U. (1980), "A reliability distribution with increasing, decreasing, constant and bathtub-shaped failure rates", Technometrics 22/1, 99-107. Munford, A. G., and Shahani, A.K. (1972), "A nearly optimal inspection policy", Operational Research Quarterly 23/3, 373-379. Nakagawa, T., and Yasui, K. (1979), "Approximate calculation of inspection policy with Weibull failure times", IEEE Transactions on Reliability R28/5, 403-404. Nakagawa, T., and Yasui, K. (1980), "Approximate calculation of optimal inspection times", Journal of the Operational Research Society 31/9, 851-853. Pierskalla, W.P., and Voelker, J.A. (1976), "A survey of maintenance models: The control and surveillance of deteriorating systems", Naval Research Logistics Quarterly 23/3, 353-388. Rodrigues Dias, J. (1983), "Influence de la prriode d'inspection sur les cofits dans l'inspection prriodique de systrmes", Revue de Statistique Appliqu~e 31/4, 5-15. Rodrigues Dias, J. (1986), "Correspondence between results of a periodic and a not periodic inspection policies", Proceedings of the X I Spanish-Portuguese Conference on Mathematics, Vol. 1L 533-542, University of Extremadura, Badajoz (in Portuguese). Rodrigues Dias, J. (1987a), "Systems inspection policies", PhD thesis, University of Evora (in Portuguese). Rodrigues Dias, J. (1987b), "A solution for the inspection period of systems", X I I Portuguese-Spanish Conference on Mathematics, University of Minho (in Portuguese). Saniga, E.M., and Shirland, L.E. (1977), "Quality control in practice--A survey", Quality Progress 10/5, 30-33. Schneeweiss, W.G. (1976), "On the mean duration of hidden faults in periodically checked systems", IEEE Transactions on Reliability R-25/5, 346-348. Sherif, Y.S. (1982), "Reliability analysis: Optimal inspection and maintenance schedules of failing systems", Microelectronic Reliability 22/1, 59-115.