APPROXIMATELY OPTIMAL SIMPLE
INSPECTION STRATEGIES
Hryniewicz O.
Systems Research Institute, Warsaw, Poland
Abstract: In the paper we present a very simple method for finding an approximately optimal
strategy of inspections of an equipment whose failures are not directly observable. The optimal inspection interval can be found with a limited amount of information using simple computational methods. We consider also the impact of imprecise input information on the computed values of optimal inspection periods. In such a case we describe imprecise numerical information by fuzzy sets, arriving at the imprecise (fuzzy) optimal inspection interval.
1. Introduction
For many technical objects it is difficult to predict, even approximately, the time of failure. For example, electronic components of technical systems do not signal in advance the possibility of their catastrophic failures; so such failures are usually unexpected. Moreover, in many practical situations failure of system's components, if they do not cause an immediate failure of the whole system, can be found only during certain inspection actions. Therefore, there is a need to plan such inspections in such a way that the total costs of inspection and costs related to failures should be minimized. There are many mathematical models for the optimization of inspection schedules. First papers devoted to the problem of optimal inspection policies were written more than forty years ago by Savage [18], Barlow et al. [2], Kamins [13] and Coleman and Abrams [6]. The seminal paper by Barlow, Hunter, and Proschan [3], who proposed a general model for the optimization of inspection policies in case of perfect inspections, started a long series of papers devoted to this problem. Usefull approximate solutions have been proposed in the papers by Munford and Shahani [15], [16], Tadikamalla [19], and Nakagawa and Yasui [17]. Interesting results have been presented, for example, in papers by Menipaz [14] and von Collani [7]. The problem of finding an optimal inspection interval is simplified when each inspection is followed by a renewal of a system (or, equivalently, when the time to failure is distributed exponentially). This problem was considered, for example, in papers by Baker [1], Chung [5], Vaurio [20], and Hariga [11]. However, nearly all of them are rather complicated, and thus difficult to be implemented in practice. For example,
sometimes they may require the usage of sophisticated computational techniques (e.g. dynamic programming). Moreover, complex mathematical models with a large number of parameters are very often difficult to understand for an average practitioner. Therefore there is a need to propose simple methods for the design of inspection strategies.
In the second section of the paper we present very simple approximately optimal strategies that can be found with a limited amount of information using simple computational methods. We consider a very simple inspection strategy: inspections are performed with a constant inspection period. In the third section of the paper we consider also the impact of imprecise input information on the computed values of optimal inspection periods. We describe imprecise numerical information by fuzzy sets, arriving at imprecise (fuzzy) optimal inspection interval.
2. Approximately optimal inspection intervals
For the construction of the approximately optimal inspection procedure we make the following assumptions:
1. The inspected equipment fails randomly in time, and its failures are not directly observable.
2. When the equipment remains in a non-failure state there exist certain profits from its exploitation.
3. After the failure the profits are smaller or may be even negative. 4. In order to detect a failure the equipment has to undergo an inspection. 5. There exists a constant inspection interval.
6. Inspection procedures may be not perfect, and are described by the probability of indicating a failure while the equipment remains in a non-failure state (we call this situation a false alarm or Type I error of inspection), and the probability of not finding an existing failure (Type II error of inspection). In the case of a perfect inspection these probabilities are equal to zero.
7. When the failure has been detected by an inspection procedure an error-free inspection procedure is applied in order to find the actual state of the inspected equipment. In the case of a false alarm the costs connected with the application of such a procedure are called the costs of a false alarm.
8. After revealing the failure the equipment is renewed (is as good as a new one). 9. The optimal inspection procedure maximizes the total expected profit per operational
time unit (i.e. per unit of time while the equipment operates).
10. The losses connected with inspection and renewal times are included in the average inspection costs.
To find the objective function for the optimization choice of an inspection procedure we use the approach originally proposed by von Collani [8], and generalized by Hryniewicz [12]. Let L be the expected total profit from the exploitation of the inspected equipment
during the time T between consecutive renewals. Then, the total expected profit per operational time unit can be found from the following formula:
T L
G (1)
Let's denote by h the length of the inspection interval (constant during the whole period of exploitation). The expected time between consecutive renewals T can be computed from the following formula:
A
h
A
h
T
1
2 (2)where A1
h is the expected number of inspections before the moment of failure of the monitored equipment, and A is the expected number of inspections between the failure2time and the beginning of renewal.
The expected number of inspections before the moment of failure obviously depends upon the probability distribution F(t) of the time to failure. It can be calculated from the following formula (see von Collani [8]):
1 i
1
h
1
F
ih
A
(3)The expected number of inspections between the failure time and the beginning of renewal depends only on the efficiency of the inspection procedure. In the case of perfect inspection it is equal to one. However, the inspection might not be absolutely perfect. Let us assume now, that there exists a certain probability 01of not detecting an existing failure. Thus, the number of inspections that are needed to detect an existing failure is distributed according to the geometrical distribution, and its expected value is calculated from the following formula:
1 1 A2 (4)
Now, let us calculate the expected total profit L from the exploitation of the inspected equipment during the time T between consecutive renewals. It consists of the following five components:
the expected profit from the correctly working equipment Pw;
the expected profit from the equipment in the case of its failure Pf (possibly negative);
the expected costs of possible false alarms (when the inspection erroneously indicates
a non-existing failure) Cf ;
the expected costs of inspections Ci;
The expected profit from the correctly working equipment Pw depends on the expected
time to failure , and is given as follows:
1
w g
P , (5)
where g1 is an average profit per time unit from a correctly working equipment (i.e. in a
non-failure state).
Similarly, the expected profit from equipment being in a failure state is proportional to the expected time between the failure and the renewal. Thus, it can be computed as follows:
g TPw 2 , (6)
where g2 is an average profit per time unit from the equipment being in a failure state.
Let us also assume that there exists a certain probability 01of a false alarm. Denote by ethe average costs of the consequences of a false alarms (for example, the
costs of looking for a non-existing failure and the loss of profit from a non-working equipment, when it is stopped during such search). The expected total costs of false alarms can be now calculated from the following formula:
A he
Cf 1 (7)
The expected cost of inspections depends, of course, on the expected number of inspections and their unit costs. In general, costs of inspections of a correctly working equipment may be different than the cost of inspection of a failed one. However, in order to simplify the model, we assume that the average unit cost of inspection does not depend upon the state of the inspected equipment, and is equal to S . Thus, the expected cost of
inspections is given by the following expression:
A
h
A
S
C
i 1 2 (8)Now, we can calculate the expected total profit L from the following expression:
r i f f w P C C C P L (9)
Following von Collani [8], let us introduce new transformed parameters
g
g
C
/
e
b
1 2 r (10) e S S (11)Parameter b has an economic interpretation as the standardized average profit from one renewal, and in the majority of practical situations has large values (otherwise the renewals are not profitable).
The objective function (1) can be now transformed in the following way:
2 2 1 2 1 1g
h
A
h
A
S
A
h
A
h
A
b
e
G
(12) Hence, the optimal inspection procedure can be found by the maximization of:
A
h
A
h
S
A
h
A
h
A
b
G
2 1 2 1 1
. (13)The optimal inspection interval h can found by maximization of (13) using a numerical
procedure. In practice, however, it is useful to have a simple ready-to-use approximate formula. We can find it using the approach proposed by Hryniewicz [12].
When the inspection interval is small in comparison to the expected time to failure the following approximation, valid for a wide class of probability distributions, has been proposed in Hryniewicz [12]:
0,5h h
A1 (14)
Hence, the optimal solution approximately depends only on the expected value of the time-to-failure, and does not depend on the form of its probability distribution. This result facilitates dramatically a possible application of the procedure because the functional form of the time-to-failure distribution is generally unknown, but the average time can be estimated from field data.
When the probability of not detecting an existing failure is small, and the length of the inspection interval is small in comparison to the expected time to failure it is possible to show that the maximization of (13) is approximately equivalent to the minimization of the following goal function:
h
S
5,
0
A
bh
G
2
(15) The new objective function can be expanded in the Taylor series around h=0, and the solution of the minimization problem can be reduced to the problem of solving a quadraticequation. Hence, the approximately optimal inspection interval can be expressed as follows:
2A 1
b S 2 h 2 . (16)It is easy to notice that the application of this approximation significantly simplifies the optimization procedure. From a practical point of view it is also very important that the number of input parameters is relatively small, and some of them may be estimated separately. For example, the value of the expected profit from the renewal b may be found not necessarily from the formula (10), but from other considerations as well.
3. Fuzzy optimal inspection intervals
Calculation of the approximately optimal inspection interval is very simple indeed. However, in practice its implementation is faced with significant difficulties. Despite a small number of parameters there exist problems with their precise estimation. Practitioners are usually not able to present precise values of all these parameters. Therefore, there is a need to propose a method which enables the users to define the values of parameters in an imprecise way.
The theory of fuzzy sets introduced by Lotfi A. Zadeh is the most widely used methodology for dealing with imprecise notions and imprecise data. According to this theory every element of a set has an assigned value of its membership in the considered set. In the problem considered in this paper we can use fuzzy sets for the description of the imprecisely known values of parameters involved in the computation of the optimal inspection interval.
Let us introduce two general concepts of the theory of fuzzy sets that will be used for the calculation of the fuzzy optimal inspection interval. For the description of imprecisely defined numerical values we use the concept of a fuzzy number.
Definition 1. The fuzzy subset A of the real line R, with the membership function
0,1 :R A
, is a fuzzy number iff(a) A is normal, i.e. there exists an element x , 0 such that
A
x
0
1
;(b) A is fuzzy convex, i.e.
A
x
1
1
x
2
A
x
1
A
x
2 ,
0,1 , , 2 1 x x R
; (c) Ais upper semi-continuous; (d) supp(A) is bounded.This definition is due to Dubois and Prade (see [9]). It is easily seen from this definition that if A is a fuzzy number then its membership function has the following general form:
4 4 3 u 3 2 2 1 l 1 Aa
x
for
0
a
x
a
for
x
r
a
x
a
for
1
a
x
a
for
x
r
a
x
for
0
x
.
(17) where a1,a2,a3,a4R,a1 a2 a3 a4,r
l:
a
1,
a
2
0
,
1
is anon-decreasing upper semi-continuous and
r
u:
a
3,
a
4
0
,
1
is a non-increasing uppersemi-continuous function. Functions r and l r are called sometimes the left and theu
right arms (or sides) of the fuzzy number, respectively. By analogy to classical arithmetic we can add, subtract, multiply and divide fuzzy numbers (for more details we refer the reader to [9, 10] ).
A useful tool for dealing with fuzzy numbers is the concept of −cut or −level set. The −cut of a fuzzy number A
is a non-fuzzy set defined as
x
:
x
A
R
.
(18)A family
A
:
0
,
1
is a set representation of the fuzzy number A. Basing on the
resolution identity introduced by L. Zadeh, we get:
x
sup
I
A
x
:
0
,
1
A
,
(19)where
I
A
x
denotes the characteristic function of A. From Definition 1 we can seethat every −cut of a fuzzy number is a closed interval. Hence we have
A
A
L,
A
U
,
where
x
:
x
.
sup
A
,
x
:
x
inf
A
R L
R
R
(20) Hence, by (20) we get
1 u U 1 l L r ,A r A.
We will use the concept of fuzzy numbers and their −cuts for the description of imprecise information about the parameters the optimal inspection interval, calculated according to (16) depends on . The fuzzy equivalents of the input parameters b, A2, S, ,
and we describe using the fuzzy sets b~,A~2,S~,~ and ~ , respectively. These fuzzy sets can be defined by their respective −cuts:
b
,
b
b
:
b
b max minR
(21)
2 2 A 2 max , 2 min , 2,
A
A
:
A
A
R
(22)
S
min,
S
max
S
R
:
S
S
(23)
,
0
,
1
:
max min (24)
,
:
max minR
(25)where:
b
b
,
A2
A
2,
SS
,
,
are the membership functions of thefuzzy sets b~,A~2,S~,~ and ~ , respectively.
In order to find a fuzzy version of the optimal inspection interval we propose to apply the result described in the paper of Canestrelli and Giove [4] that may be directly applied in the considered case. According to this result, the fuzzy-optimal solution to the unconstrained optimization problem may be found by the application of the well known in the fuzzy set theory Zadeh’s extension principle to the non-fuzzy solution of the equivalent crisp optimization problem. In the considered case the fuzzy-optimal solution can be obtained by applying the extension principle to (16). In our case let us assume that fuzzy parameters ~b,A~2,S~,~ and ~ are not interactive, i.e. that their respective membership functions are independent
.
This assumption is not exactly true for b~ , and ~. The mathematical model described by (10) provides the link between these parameters in a crisp case. Using that model we may evaluate the connection between b~ , and ~ . However, in order to keep the model as simple as possible, we assume that both these fuzzy values are evaluated independently. Now, the fuzzy-optimal solution to our optimization problem is given as a fuzzy number h~ defined by the set of -cuts
max min
,
h
max max , 2 min min min min1
2
2
b
A
S
h
. (26)
min min , 2 max max max max1
2
2
b
A
S
h
. (27)If a crisp (non-fuzzy) value of the optimal inspection interval is needed we may be find it by applying one of many defuzzification methods that are described in the literature devoted to the theory and practice of fuzzy sets (see [10]).
Bibliography
1. Baker, M.J.C.: How often should a machine be inspected?, International Journal of Quality and Reliability Management, vol.4 (1990), No.4, 14-18.
2. Barlow, R.E., L.C. Hunter, F. Proschan: Optimum checking procedures. In: Proc. of the Seventh National Symposium on Reliability and Quality Control (1960), vol.9, 485-495.
3. Barlow, R.E., L.C. Hunter, F.Proschan: Optimum checking procedures. Journal of SIAM, vol. 11 (1963), 1078-1095.
4. Canestrelli E., Giove S. (1991). Minimizing a fuzzy function. In: Interactive Fuzzy Optimization, M.Fedrizzi, J.Kacprzyk, and M.Roubens (Eds.), Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin, 1991, 30-35. 5. Chung, K.-J.: A note on the inspection interval of a machine. International Journal of
Quality and Reliability Management, vol.10 (1993), No.3, 71-73.
6. Coleman, J.J., I.J. Abrams: Mathematical models for operational readiness. Operations Research, vol. 10 (1962), 126-138.
7. Collani von, E.: On the choice of optimal sampling intervals to maintain current control of a process. In: Frontiers in Statistical Quality Control, Lenz, H.-J., et al. (Eds.), Physica Verlag, Wuerzburg, 1981, 38-44.
8. Collani von, E.: The economic design of control charts. B.G.Teubner, Stuttgart, 1989.
9. Dubois, D., Prade, H.: Operations on Fuzzy Numbers. Int. J. Syst. Sci., vol.9 (1978), 613-626.
10. Dubois, D., Prade, H.: Fuzzy Sets and Systems. Theory and Applications. Academic Press, New York, 1980.
11. Hariga, M.A.: A maintenance inspection model for a single machine with general failure distribution. Microelectronics and Reliability, vol.36 (1996), 353-358. 12. Hryniewicz, O.: Approximately optimal economic process control for a general class
of control procedures. In: Frontiers in Statistical Quality Control IV. Eds. H.J. Lenz, et al., Physica Verlag, 1992, 201-215.
13. Kamins, M.: Determining checkout intervals for systems subject to random failures. RAND Corporation, RM-2578, Santa Monica, CA (1960).
14. Menipaz, E.: On economically based quality control decisions. European Journal of Operational Research, vol.2 (1978), 246-256.
15. Munford, A.G., Shahani, A.K.: A nearly optimal inspection policy. Operational Research Quarterly, vol.23 (1972), 373-379.
16. Munford, A.G., Shahani, A.K.: An inspection policy for the Weibull case. Operational Research Quarterly, vol.24 (1973), 453-458.
17. Nakagawa, T., Yasui, K.: Approximate calculation of optimal inspection times. Journal of the Operational Research Society, vol.31, 851-853.
18. Savage, I.R.: Cycling. Naval Research Logistic Qurterly, vol.3 (1956), 163 -175. 19. Tadikamalla, P.R.: An inspection policy for the gamma failure distribution.
Operational Research Quarterly, vol.30, p. 77-80, 1979
20. Vaurio, J.K.: A note on optimal inspection intervals. International Journal of Quality and Reliability Management, vol.11, No.6, p. 65-68, 1994
