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An inspection model with generally distributed restoration and repair times
~
Microelectron. Reliab., Vol. 37, No. 3, pp. 381-389, 1997 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0026-2714/97 $17.00+.00
PPergamon
PII: S0026-2714(96) 00067-4
AN INSPECTION MODEL WITH GENERALLY DISTRIBUTED RESTORATION AND REPAIR TIMES JAMES F U N G and VILIAM MAKIS Department of Industrial Engineering, University of Toronto, 4 Taddlecreek Road, Toronto, Ontario, Canada M5S 1A4 (Received for publication 28 March 1996) Abstract--In classical inspection models, it is assumed that the inspection time and the restoration time are either zero or fixed, and the production facility never breaks down. However, in real production, the production system is subject to random failures and the repair and restoration times are usually random. In this paper, we will study the effect of the exponential failure time and generally distributed restoration and repair time on the optimal inspection interval of a production unit subject to deterioration. Treating the process as a semi-regenerative process (SRP) and analyzing the SRP by Markov renewal theory, the formula for the long-run expected average cost per unit time and formulae for the steady-state probabilities of the SRt are obtained in an explicit form. The optimal inspection interval is obtained by minimizing the average cost function. Copyright © 1996 Elsevier Science Ltd.
l. INTRODUCTION Consider a production system consisting of a number of components, each component subject to deterioration with age and usage. A failure of any of the components may result in producing units with non-conforming quality, or even cause failure of the whole system and stop production until the system is repaired. The defective items will be reworked or replaced at some cost or, if passed to customers, warranty costs or loss of goodwill will be incurred. The state of the production process usually determines the product quality. When the process is in "good" condition, items produced may be of high or perfect quality. The process may deteriorate with time, as a result of which a proportion of the items produced is defective or of substandard quality. The shift of the production process from an "in-control" state to the "out-of-control" state can occur at a random time during production. In order to keep the product quality high, we may perform inspections on the deteriorating system periodically. The true state of the system can only be known through inspection, thus inspections are required to ensure that the system in an "out-of-control" state is restored to the normal condition in a timely manner. However, there is a tendency to reduce the number of inspections because their cost is added to the operating cost of the system. Therefore, we need to determine the optimal inspection schedules to minimize the total expected costs of delayed failure detection and the cost of inspections. Inspection models of systems subject to random deterioration have been widely studied since the 1960s. Barlow et al. [1"1 developed a basic inspection model and analyzed the tradeoff between inspection cost and the cost of undetected failure. Their objective was to minimize the total expected cost accumulated during 381
the time between system failure and its detection and the cost of the inspections. They also proposed an approximation of the optimal periodic inspection interval. The approximation is good as long as the cost of inspection is small relative to the cost of an undetected failure. Schultz [21 presented a new approximation of the optimal periodic inspection interval that performs well for a broader range of parameter settings. The new approximation is derived by characterizing the behavior of the optimal inspection interval as a function of the model parameters. For an increasing failure rate, Munford [3] showed that inspection policies with decreasing intervals between successive inspections are superior to periodic policies. He presented an alternative cost model for the inspection problem and proposed three suboptimal inspection policies for systems with increasing failure rates. Luss [4"1 studied an inspection model in which the duration of inspection and repair is not negligible. He assumed that the system was not operating during the fixed inspection and repair periods and obtained the optimal inspection schedule that minimizes the long-run expected cost per unit time. Nakagawa [5] presented a modified inspection model with a preventive maintenance. The system was checked periodically to see whether or not it needed to be replaced. If the system was not in good condition, it was repaired or replaced immediately. He obtained the mean time to failure and the expected number of checks before failure using renewal-type equations. He also derived the total expected cost and the expected cost per unit time until failure. He noted that it is very difficult to obtain the optimal inspection times analytically and suggested numerical search procedures to find them. The above inspection models assumed that the inspection and restoration times are either zero or
382
J. Fung and V. Makis
fixed, and that the production unit never breaks down. The effect of machine breakdown on the optimal inspection interval has generally been overlooked. In reality, production systems are subject to random deterioration and random breakdowns, and the time to restore or repair the failed unit is not usually predictable. In this paper, we will develop an inspection model with random machine failure and generally distributed restoration and repair times. The objective is to study the effect of the random machine failure on the optimal inspection interval and to generalize the usual assumption of instantaneous restoration times. Furthermore, we will provide an analysis of the steady-state behavior of the system. We assume exponentially distributed time to the "out-of-control" state and the time to failure, generally distributed restoration and repair times, and periodic inspections. The representation by a semi-regenerative process (SRP) and the analysis of this SRP by Markov renewal theory will be discussed in Sections 4 and 5. A special case of the model, in which the restoration and repair times are exponentially distributed, will be studied in Section 6. Numerical examples and discussion of the results will also be given in Section 6.
2. NOTATION I d ~-c ';~d Rq R~ R, ca c, d Td T, E T, Fr(t)
F~(t) v
state space of the semi-regenerative process Z state space of the Markov renewal process X machine out-of-control rate machine failure rate fixed cost of inspection ($) fixed cost of restoration ($) fixed cost of repair ($) cost per unit time incurred during a restoration period (S/time) cost per unit time incurred during a repair period (S/time) cost per unit time incurred during an out-of-control period (S/time) time to failure time to out-of-control time to repair the process restoration time cumulative distribution function of T~ cumulative distribution function of Ta length of an inspection interval (time)
3. MODEL DESCRIPTION Consider a single unit production system which operates continuously, is subject to random deterioration and random breakdown and is periodically inspected. The unit begins to operate at time zero. The time to the out-of-control state and the time to machine failure are exponentially distributed with parameters 2 c and 2 a, respectively. The machine condition can be classified as "in-control", "out-ofcontrol", and "breakdown". The status of whether the process is in control can only be learned through inspections. Successive checks are made at times k v (k = 1,2 . . . . ), where v ( > 0 ) is a constant. If the
process is found to be in an "out-of-control" state, the appropriate restoration procedure will be initiated. The time to restore the machine is assumed to be a random variable with density function fa(t). It is assumed that machine failure can be detected immediately and the time to repair is a random variable with density function fr(t). It is also assumed that the repair and the restoration renew the system to the initial working conditions. Inspections are assumed to be perfect and instantaneous, as is the case in many modern manufacturing systems adopting a computercontrolled inspection facility. We assume that a machine in the "out-of-control" state, which has not been discovered, may also break down. The total system cost includes the fixed inspection, restoration and repair cost, and the per unit time cost of restoration, repair, and machine out-of-control. An inspection model for the process described above will be developed in the next section. The objective is to determine the optimal inspection interval minimizing the average cost per unit time, and to study the steady-state performance characteristics of the system. 4. ANALYSIS The production system described in Section 2 can be characterized by two attributes: the condition of the production unit and the current maintenance activity, which may be inspection, restoration, repair, or inactive. We denote 0, e {0, 1, 2} and ~o, ~ {q, a, r, v} as the state variables representing machine condition and maintenance activity at time t, respectively. An operational machine has O, = 0, an out-of-control machine has 0 t = 1, and a failed machine has 0 t = 2. Maintenance activities are inspection (ogf = q), repair (09t = r), restoration (~o, = a), and idle (o9t = v). We denote Z~ as the system state at time t, hence Z t has state space I = {(09, 0): w ~ {q, a, r, v}, 0 e {0, 1, 2}}. The process Z makes transitions each time the machine state changes, for example, machine failure, repair or machine out-of-control, etc. It is noted that machine restoration times and repair times are non-exponential, therefore the process Z is not Markovian. To analyze this system, we introduce the following instants: Time instant of type 0: an inspection is just finished and the machine is found to Z t = (q, O) be in control. Time instant of type 1: an inspection is just finished and the machine is found to Z t = (q, 1) be out-of-control. Time instant of type 2: machine restoration is finished and the machine begins to Z t = (a, O) operate. Time instant of type 3: machine repair is finished and the machine begins to operate. Z, = (r, 0) The above time points are the instants of time immediately following completion of inspection,
An inspection model
383 The instant of machine faiha-e.
The instant machine gets out-of-control.
I
I
°
I I T,
V
¢ ¢
I T~
v
T~ Fig. 1. An example of the system's behavior.
z,
(q,0)
(q,0)
.
(q,l)
(a,0)
(r,O)
(q,0)
(v,0)
(v,0)
(r,2)
Machine X.
(a,0)
(q,0)
Time Instant of
0
0
1
2
0
Type
oooooo
(q,O)
(r,O)
3
Operating, in-control
.l...
Out-of-control
xxxxxx Machine repair
Machine restoration
Fig. 2. Demonstration of the SRP and the embedded MRP. restoration, and repair. We may notice that Z possesses a strong Markov property at these selected time points. Let T = {To, 7"1, T2. . . . }, To = 0 be the sequence of the successive time instants of changes of state of the system after each inspection, restoration, and repair. We then define X. = Z(T.),
n=0,1,2 .....
It can be shown that {X., T., n = 0, 1,2 . . . . } is a homogeneous Markov renewal process (MRP) with state space J = {(q, 0), (q, 1), (a, 0), (r, 0)}, embedded at instants T. in the process Z. The process Z = {Z. y/> 0} is a SRP with state space I, and time instants of type 0, 1, 2, and 3 are the semi-regeneration points. Figures 1 and 2 demonstrate this system's behavior. Now, we may describe the system using a SRP on which a cost structure described in the previous section is imposed. The total cost of the system includes both fixed and variable portions. The system cost may be defined as the sum of a functional of the SRP and a functional of the embedded MRP. The system variable cost incurred at any time point is governed by the state of the SRP
Z. For example, the per unit time cost of producing defectives, d, is incurred so long as the machine is in the "out-of-control" state, that is whenever Z, equals (v, 1). The fixed cost of the system is governed by the state transitions made by the embedded Markov chain X. For instance, the fixed cost of restoration is incurred each time X, makes a transition from state (q, 1) to state (a, 0). We may obtain the longrun expected average cost per unit time using the results of the theory of semi-regenerative reward processes.
5. MATHEMATICAL FORMULATION We begin with the derivation of the semi-Markov kernel for the Markov renewal process (X,, T.), that is we find the probabilities Qo(t) defined by Q,j(t) = P { g , + t = j , T,+I - T~ ~< t l Sn = i}
i, j e J.
(1)
Using the fact that the time to machine out-of-control
384
J. Fung and V. Makis
and the time to failure are independent,
we have
tions is P,q. o,,r.o,
Q tv.o)w o,(f) = Qtr. o,cq.o,(t) = Q,,. o,cq,o,(t) e
(1, + Id)U
-
?L,rso)= %q.o,,q,
t>v
0
otherwise,
1 -
Qcq. om. I ,(t) = Qcr, o,,q, l,(t) = QI~. wq. I,@) ,-AldU
(1
=
_
,-A,u
1 _
t>v
1
0
(3)
otherwise.
Using the total probability theorem (see, e.g. Ref. [6]) and by conditioning on the time to machine failure, G, we obtain Qc. o,cr.o,@) = Q (0, o,,r, o,(t) = Q,r, o,,r, o,(t)
P( T, + & < t 1 & = u)fd(u) du
t <
u
P(T, + Td < t 1 Td = u)fJu)
t >
u
du
F,(t -
up, eeAd"du
f< v
c(t
u)&e-“d”du
t >
?q.O,,r.O)
+
G?*O),q.O)
e-b
,-,nc+i,,u
+
,-ido
,-(&+ld,u
Pk?.wl!O)
%o, = ___
n,,,o, = * _ ,~,&+id,u + ,-i,u
P,q*O,,r.O,
P(4.wq. 1) e _I‘j” _ ,-,&+i& n (I.0, = 1 _ e-‘“.+“d’u + e-ldu P(4.OW,0,
Z(q.1,
=
____
%I, 0,
=
%z.
(9)
1)’
We now impose the cost structure on the SRP 2 and the MRP X. The inspection cost is incurred whenever the process X makes a transition into states (q, 0) or (q, 1). The fixed restoration and repair costs are incurred when the process X makes a transition into state (a, 0) and (I, 0), respectively. We define Kj as the one-step transition cost which is governed by the embedded Markov renewal process X
=
S’
1, +
(2)
zz
0
=
K (.,o) = R,, K,,,o, = R, K,q,i, = Rq,
,
:
-
s 0
Q ,q,,j,a.Oj(t) = P(time to machine restoration is less than or equal to t) = F,(t),
(4)
All other Qij(t) = 0. Taking the limit of the probabilities (2)-(4), we have the following transition probabilities of the embedded Markov chain X P,q~o,,q.o)= PkLo,,q,o, = P,r.O),q.O,= e -,&+ldw, P,q,O,,q.l, =
4l.O,,q,l,kO,,q.L,
=
(1
-
emA=“)e- ‘d”,
P,q.ow.o,--P ,n.O,,r,O,=~r.0,,r,0)= 1 - e-
i = 0, 1. (10)
o
The variable costs of the system are governed by the state of the SRP. These cost rates are the cost per unit time of undetected machine out-of-control d, cost per unit time of machine restoration c,, and the cost per unit time of machine repair c,. The cost rate d is incurred whenever the SRP is in state (II, 1). The cost rates c, and c, are incurred whenever Z, is in state (a, 1) and (v, 2), respectively. These cost rates, j(j), are as follows: S(u,O)=O,
f(a,
l)=c,,
f(r,2)=c,.
(11)
(5)
f(u, l)=d,
(6)
We can calculate the expected cost accumulated between two successive semi-regeneration times for the MRP X using the formula (see L T,+, Ref. [7])
‘dL.,Fq, l,(a,Oj= 1. (7)
Since the embedded Markov chain X is irreducible and recurrent, we may find its steady-state probabilities 7tjby solving the following system of linear equilibrium equations and the normalizing equation
G,(co) = lim Gi(t) t-cc =
%J.O)= 7%.o?,,, oj,q.0, + ?I&o& o,,q.0)+ %.0,4r, ON&O)
s
omz, f(z)lCli(u, z)du+ 1 KjQij(a) jsJ
ieJ
1) %?,1, = %. o+L 0,,4.1,+ %. of,a. Ok 1,+ 5,. o+%.O,,%
$i(t, A) = c(Z, E A, Tl > t) i E J, A c I.
7%.0, = %?.oG%. O,,r.O) + %I. o,P,o, O,,r*O) + Tr. o,h. oe. 0)
The quantity t+bi(t,A) is the probability that, given the initial state of the MRP X0 = i, the transition time is greater than t and the Z, E A. We obtain the following formula for tii(t, A):
%I.01= %I. 1IP(4.1KO. 0,
The solution
to the above system of linear equa-
+,,. ,,(t, (0, 1)) = 1 - F,(t) = ~a’.(o.
(12)
(13)
An inspection model
By Theorem 4.4 of Ref. I-7], the long-run expected average cost per unit time g(v) can be bound by using the formula
For i E {(r, 0), (q, 0), (a, 0)}, we have "e ~i(t, (V, 0)) =
(ae + 2a)t
t ~ V
0
385
t>v
u))2 d e - aau du
f i ( l -- F,(t
f f (1 -- Fr(t -- u))2 a e- ~au du
{(01 - e - ~ ' ) e-~d' ¢i(t, (V, 1)) =
t ~< V otherwise.
~-~keJ ~ k m k
t <<.v
Substituting Eqs (15-18) into (19), we obtain
O,(t, (r, 2)) =
t>v
g(v) =
%E(T~) + R o + Rq - R, -
(
(14)
+ cr
;o fo
R~ - %E(Ta)) e -(~+ ~ '
d
+ \,~ + ~
Then, for t/> 0 G(q, j i(t) =
f ( a , 1) O(q, 1)(u, (a, 1)) du
(;o;oF,(t
- u)2 a e xdu du dt
+ i ~ jlv f r ( t - u ) 2 a e
+ K~.o}Q¢q, 1)~a,O)(t)
= %
(19)
g(v) = ~'j~J n j G j ( ~ )
ft.(u) du + R, F,(t).
(15)
q 2d
Next, for t > v
d"dudt)
2c + 2d + R,
/(e -a~° -- e-¢a°+a~lV)E(Ta) + v
G(q.o) = G{,,o~(t)= G(a,o)(t ) = J'i [f(v, 1)~k,q,o)(U,(v, 1))
-
+ f ( r , 2)@~,o)(U, (r, 2))
+
+ f(v, 0)~klq,o)(U, (v, 0))] du
+ R,Q(q.o)(,,o)(t)
+
c,
F,(t-u)2de-~,'dudt
fo f:
By Proposition 7.6 of Ref, [8], the steady-state probabilities of the SRP Zt can be obtained using the following:
F~(u -- z)2 d e -a"~ dr du
+ R,
G
. (20)
5.1. Performance measures (1 -_/~+~e-¢a°+a~)~)]_j
7"/:k~OC° ~,~(t, 0
P ( i ) = lim Pt(i)= (~k~J +
]
(1-F,(t-u))2de-~'dudt
We can find the optimum value of v by minimizing the function g@).
+ Rq[Q~q,o)~a,o~(t)+ Q~q,o}tq,1}(t)]
- d [ (1 -2ae-~'")
fof
t~ ~
.(u -- z)2a e- a.~ dz du
, (21)
~k~J ~kmk
where
f ? q%,~}(t,(a, 1)) dt = fo~ FAt) dt
F~(t- u)Ane- ~" du + Rq e- a.~. (16)
and for i e {(q, 0), (a, 0), (r, 0)}
We next determine the mean sojourn times m~ for each state i
f f ¢~(t,(v,O))dt= f~e-'a°*a~"dt 1 - e -ta`+aa)v
m, = E[T.+ , -- T. I X . = i] -
fo°
~ki(t, (r, 2)) dt =
m~q.1~ = j ~ ff,(t)dt = E(T,)
dr)
(17)
f:fo
2~ + 2a
'
(22)
ff,(t - u)2 a e -a~" du dt
(t - U)2d e - ~ " du dt
+ mtr, o) = m{a. o} = re(q, o}
fo +
(1 - g(t - U))2d e - a " du dt.
(18)
Oi(t, (v, 1)) d t - -
fo
(1 - e - ~ ' ) e - ~ '
dt
_ 2 c + 2 a e-(~+a")~--(2~ + 2a) e-~,~ 2d(2~ + 2n) (23)
386
J. Fung and V. Makis
Hence, the steady-state probability that the machine is down is P(r, 2) -
long-run average cost per unit time: g(v) =
(Zk~L nk ~ tPk(t, (r, 2)) dt)
c + ,~
~k~J 7Zkmk
where
+
L = {(q, 0), (a, 0), (r, 0)}.
(
m
R q - - R t q:- R a +
;)..
c,
(24) + R~ + c,. + d #r 2a
The steady-state probability that the machine is being restored is
d I 2c + 2a J e - .lay
P(a, l) = (Tt(q.1)~0 ¢{q, 1)(t ' (a, 1)) dr)
+
(25)
laa
ff~.k~J 7[kmk
~.a/
#~
+
The steady-state probability that the machine is out-of-control is
+
.
(29)
The optimal inspection interval can be obtained by minimizing the cost function g(v).
oo
(~,k~L k ~0 ~bk(t, (v, 1)) dt)
P(v, 1)
(26)
~ k e J 7[km k
6. N U M E R I C A L
EXAMPLES
In this section, we provide an analysis of a special case and present some numerical results. We assume that the restoration and repair times are exponentially distributed with parameters p, and p~, respectively. From eqns 05) and (16), we have
6.1. Performance measures The steady-state probabilities of the SRP {Zz} for this special case can be obtained using the formulae (24)-(26) derived in the previous section. The formula for the steady-state probability that the machine is down has the form P(r, 2)=
(1 - e -a~) #'[e-adv(1
)1Y , ~a
e-(~°+z~)~ L ] ' ~/ ' 1~~+ .
(30)
Ca G(q, 1)(ov) = - - + R ~ ,
The steady-state probability that the machine is being restored is given by
Pa
6,(oo) = a[(,+
(,-
+
J
a[0
P(a, 1)---
+R, (1-e-aa~)+Rqe-a"v, i e {(q, 0), (a, 0), (r, 0)}.
(31) (27)
The expected sojourn times are obtained from eqns (17) and (18), re(q,
1 pa
m(q,o} = m(a.O) ~ m(r.O)
+
and the steady-state probability that the machine is out-of-control is P(v, 1) =
2~ + 2de -(~°+~)~ - (2~ + 2a)e -a~
(32)
e -(a°+aa)v 1 1 ] ' + + #, /#
1) --
=v-
e-~v(1 _e-acv)
;L Lf:
(1 - e-"~"-~))2a e- ~" du dt
e-"'(t-"}2a e - ~ " du dt
= (1 - e -a"~) x
+
.
(28)
By using the formula in eqn (20), we obtain the
Using the above steady-state probabilities, we may find the various system long-run characteristics analytically. The long run fraction of time = P(r, 2) machine is in failure state The long run fraction of time machine is producing quality = P(v, O) products The long run fraction of time = P(v, 1) machine is producing defectives The long run fraction of time = P(r, 2) + repairman is busy
P(a,
1).
An inspection model Table 1. Parameter values for numerical analysis Parameter
Value
Re ($) Rr (S) Rq ($) ca (S/time) cr (S/time) d (S/time)
20 50 20 5 10 5
6.2. Numerical results The p a r a m e t e r values in Tables 1 a n d 2 are used in the examples. F r o m Fig. 3, we can see that the total cost curve becomes more fiat as the failure rate increases. This means that for larger 2 d there is more flexibility to choose the optimal inspection interval, v. In fact, any value of v greater t h a n 20 time units is optimal for the case of ;'d = 0.04. The total cost for all three different values of 2d tends to a positive c o n s t a n t as v increases. This may not be the case in the classical
387
inspection models which d o not assume system failure. Their total cost would go to infinity because the cost of defects goes to infinity with v. In o u r inspection model with machine failure, the long-run fraction of time the machine stays " o u t - o f - c o n t r o l " will not increase indefinitely with v as illustrated in Fig. 5). Hence, the cost of defects becomes steady as v increases. The reason for this is t h a t the occurrence of machine failures will renew the deteriorated system occasionally. In Fig. 4, it is interesting to note that the long-run fraction of time the machine stays " i n - c o n t r o l " , P(v, 0), for 2d = 0.08 is greater than that for smaller 2d when the inspection interval is large. The explanation for this is that the machine with a higher failure rate is likely to break d o w n more often a n d hence the process is renewed more frequently. We also observe that v has lesser effect o n P(v, 0) for larger 2d. We c a n n o t improve P(v, 0) for 2a = 0.08 as m u c h as we can improve P(v, 0) for 2d = 0.01 by shortening the inspection interval. The reason is t h a t the long-run fraction of time the machine is u n d e r repair, P(r, 2),
Table 2. Results of numerical analysis Case
)~,.
/1
2d
v*
g(v*)
P(v, 0)
P(r, 1)
P(r, 2)
P(a, 1)
1
0.006
/x, = 0.5 ~r = 0.4
2
0.06
p, = 0.5 1~ = 0.8
3
0.06
/~ = 1.0 #r = 0.4
4
0.006
/~ = 0.5 ~r = 0.4
0.001 0.01 0.04 0.08 0.16 0.01 0.04 0.08 0.16 0.01 0.04 0.08 0.01 0.04 0.08
19.57 20.97 28.03 45.04 31.15 21.01 28.38 46.33 35.45 20.73 28.26 47.86 44.56 62.95 64.62
3.91 4.25 5.37 6.79 9.55 4.19 5.15 6.50 9.47 4.23 5.37 6.80 1.74 3.30 5.29
0.550 0.535 0.493 0.487 0.522 0.541 0.513 0.530 0.607 0.553 0.500 0.485 0.858 0.811 0.780
0.383 0.386 0.391 0.344 0.192 0.391 0.412 0.376 0.225 0.394 0.400 0.347 0.110 0.100 0.056
0.002 0.023 0.088 0.166 0.285 0.012 0.046 0.091 0.167 0.024 0.090 0.166 0.024 0.091 0.167
0.065 0.056 0.028 0.003 0.001 0.057 0.029 0.003 0.001 0.029 0.014 0.001 0.008 0.002 0.000
6.5 $
20 = 0 . 0 4
i;
o 0
§
&=o.ol ~r D e. 0
"
20=o.ooi 4
I
I
I
I
t
I
I
10
2O
3O
40
60
60
7O
Inll~
Inlevll v
Fig. 3. Long-run average cost as a function of inspection interval for case 1.
I
388
J. Fung and V. Makis
_e J~ 2 o.s Ilk
~-d = 0.08 ~,d = 0.04
~0.25
"gd = 0.01
I 20
I 40
I 60
I 80
I 100
mlm(:uo. Intwv~l v
Fig. 4. Steady-state probability as a function of inspection interval for case 1.
~
= 0.01
~ - d
0.76
.Zd = 0.04 2 ~ Q.
~,d ----"0.08
l I
I
I
I
I
20
40
60
80
100
Inspectl~ Intenral v
Fig. 5. Steady-state probability as a function of inspection interval for case 1.
~0.76
~=0.8 i.o II.
0.5
/a, = 0.4
J I
I
I
I
I
I
I
0.06
0.1
0.16
0.2
0.28
0.3
0.35
M a c h i n e Failure Rate
Fig. 6. Steady-state probability as a function of machine failure rate 2d for cases 1 and 2. is independent of the inspection interval, v (as can be seen from Fig. 7). While P(r, 2) stays high for all v, P(v, 0) cannot be improved much by shortening the length of the inspection interval nor can it be worsened
much by lengthening the interval. This is also the reason for the behavior of P(v, 1) in Fig. 5. Since, for a system subject to higher failure rate, the time the machine stays "in-control" becomes more
An inspection model
389
0.76 a .ID
.o
P(v,l)
0.
~
o.5
m,
P(v,O)
"0
~
0.25
P(r,2) 0 0
10
20
30
40
60
60
70
80
Inspection Interval v Fig. 7. Steady-state probabilities as a function of inspection interval for case 1 with 2~ = 0.04. reliant on 2d, we may improve P(v, 0) by shortening the repair time. This is demonstrated in Fig. 6. The higher 2 e is the larger is the difference between the steady-state probabilities P(v, 0) for the two cases.
inspection interval does not matter much (as long as it is at least as large as the optimal inspection interval for a lower failure rate).
7. CONCLUSIONS
REFERENCES
In this paper, we have presented an inspection model with machine failure and generally distributed restoration and repair times. Using Markov renewal theory, we have obtained the formula for the long-run expected average cost per unit time and formulae for the steady-state system characteristics. Computational results indicate that the long-run fraction of time the machine stays in-control is dependent on both the length of the inspection interval and the machine failure rate. The dependency of P(r,, 0) on v weakens as 2 a increases. One way of improving P(v, 0) for the system subject to high failure rate is to upgrade the repair facility and increase the rate of repair. We have also observed that when the failure rate is high the average cost function flattens out so much that the
1. Barlow, R. E., Hunter, L. C. and Proschan F., Optimum checking procedures. J. Soc. Ind. Appl. Math., 1963, 11, 1078- 1095. 2. Schultz, C. R., A note on computing periodic inspection policies. M#mt Sci., 1985, 31, 1592-1596. 3. Munford, A. G., Comparison among certain inspection policies. M#mt Sci., 1981, 27, 260-267. 4. Luss, H., Inspection policies for a system which is inoperative during inspection periods. AI1E Trans., 1977 9, 189-194. 5. Nakagawa, T., Periodic inspection policy with preventive maintenance. Naval Res. Lo#istics Q., 1984, 31, 33-40. 6. Tijms, H. C., Stochastic Modellin# and Analysis: a Computational Approach. John Wiley, New York. 7. Schellhaas, H., Semiregenerative processes with unbounded rewards. Math. Operations Res., 1979, 4 (1), 70-78. 8. (~inlar, E., Introduction to Stochastic Processes. PrenticeHall, Englewood Cliffs, NJ, 1975.
Microelectron. Reliab., Vol. 37, No. 3, pp. 381-389, 1997 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0026-2714/97 $17.00+.00
PPergamon
PII: S0026-2714(96) 00067-4
AN INSPECTION MODEL WITH GENERALLY DISTRIBUTED RESTORATION AND REPAIR TIMES JAMES F U N G and VILIAM MAKIS Department of Industrial Engineering, University of Toronto, 4 Taddlecreek Road, Toronto, Ontario, Canada M5S 1A4 (Received for publication 28 March 1996) Abstract--In classical inspection models, it is assumed that the inspection time and the restoration time are either zero or fixed, and the production facility never breaks down. However, in real production, the production system is subject to random failures and the repair and restoration times are usually random. In this paper, we will study the effect of the exponential failure time and generally distributed restoration and repair time on the optimal inspection interval of a production unit subject to deterioration. Treating the process as a semi-regenerative process (SRP) and analyzing the SRP by Markov renewal theory, the formula for the long-run expected average cost per unit time and formulae for the steady-state probabilities of the SRt are obtained in an explicit form. The optimal inspection interval is obtained by minimizing the average cost function. Copyright © 1996 Elsevier Science Ltd.
l. INTRODUCTION Consider a production system consisting of a number of components, each component subject to deterioration with age and usage. A failure of any of the components may result in producing units with non-conforming quality, or even cause failure of the whole system and stop production until the system is repaired. The defective items will be reworked or replaced at some cost or, if passed to customers, warranty costs or loss of goodwill will be incurred. The state of the production process usually determines the product quality. When the process is in "good" condition, items produced may be of high or perfect quality. The process may deteriorate with time, as a result of which a proportion of the items produced is defective or of substandard quality. The shift of the production process from an "in-control" state to the "out-of-control" state can occur at a random time during production. In order to keep the product quality high, we may perform inspections on the deteriorating system periodically. The true state of the system can only be known through inspection, thus inspections are required to ensure that the system in an "out-of-control" state is restored to the normal condition in a timely manner. However, there is a tendency to reduce the number of inspections because their cost is added to the operating cost of the system. Therefore, we need to determine the optimal inspection schedules to minimize the total expected costs of delayed failure detection and the cost of inspections. Inspection models of systems subject to random deterioration have been widely studied since the 1960s. Barlow et al. [1"1 developed a basic inspection model and analyzed the tradeoff between inspection cost and the cost of undetected failure. Their objective was to minimize the total expected cost accumulated during 381
the time between system failure and its detection and the cost of the inspections. They also proposed an approximation of the optimal periodic inspection interval. The approximation is good as long as the cost of inspection is small relative to the cost of an undetected failure. Schultz [21 presented a new approximation of the optimal periodic inspection interval that performs well for a broader range of parameter settings. The new approximation is derived by characterizing the behavior of the optimal inspection interval as a function of the model parameters. For an increasing failure rate, Munford [3] showed that inspection policies with decreasing intervals between successive inspections are superior to periodic policies. He presented an alternative cost model for the inspection problem and proposed three suboptimal inspection policies for systems with increasing failure rates. Luss [4"1 studied an inspection model in which the duration of inspection and repair is not negligible. He assumed that the system was not operating during the fixed inspection and repair periods and obtained the optimal inspection schedule that minimizes the long-run expected cost per unit time. Nakagawa [5] presented a modified inspection model with a preventive maintenance. The system was checked periodically to see whether or not it needed to be replaced. If the system was not in good condition, it was repaired or replaced immediately. He obtained the mean time to failure and the expected number of checks before failure using renewal-type equations. He also derived the total expected cost and the expected cost per unit time until failure. He noted that it is very difficult to obtain the optimal inspection times analytically and suggested numerical search procedures to find them. The above inspection models assumed that the inspection and restoration times are either zero or
382
J. Fung and V. Makis
fixed, and that the production unit never breaks down. The effect of machine breakdown on the optimal inspection interval has generally been overlooked. In reality, production systems are subject to random deterioration and random breakdowns, and the time to restore or repair the failed unit is not usually predictable. In this paper, we will develop an inspection model with random machine failure and generally distributed restoration and repair times. The objective is to study the effect of the random machine failure on the optimal inspection interval and to generalize the usual assumption of instantaneous restoration times. Furthermore, we will provide an analysis of the steady-state behavior of the system. We assume exponentially distributed time to the "out-of-control" state and the time to failure, generally distributed restoration and repair times, and periodic inspections. The representation by a semi-regenerative process (SRP) and the analysis of this SRP by Markov renewal theory will be discussed in Sections 4 and 5. A special case of the model, in which the restoration and repair times are exponentially distributed, will be studied in Section 6. Numerical examples and discussion of the results will also be given in Section 6.
2. NOTATION I d ~-c ';~d Rq R~ R, ca c, d Td T, E T, Fr(t)
F~(t) v
state space of the semi-regenerative process Z state space of the Markov renewal process X machine out-of-control rate machine failure rate fixed cost of inspection ($) fixed cost of restoration ($) fixed cost of repair ($) cost per unit time incurred during a restoration period (S/time) cost per unit time incurred during a repair period (S/time) cost per unit time incurred during an out-of-control period (S/time) time to failure time to out-of-control time to repair the process restoration time cumulative distribution function of T~ cumulative distribution function of Ta length of an inspection interval (time)
3. MODEL DESCRIPTION Consider a single unit production system which operates continuously, is subject to random deterioration and random breakdown and is periodically inspected. The unit begins to operate at time zero. The time to the out-of-control state and the time to machine failure are exponentially distributed with parameters 2 c and 2 a, respectively. The machine condition can be classified as "in-control", "out-ofcontrol", and "breakdown". The status of whether the process is in control can only be learned through inspections. Successive checks are made at times k v (k = 1,2 . . . . ), where v ( > 0 ) is a constant. If the
process is found to be in an "out-of-control" state, the appropriate restoration procedure will be initiated. The time to restore the machine is assumed to be a random variable with density function fa(t). It is assumed that machine failure can be detected immediately and the time to repair is a random variable with density function fr(t). It is also assumed that the repair and the restoration renew the system to the initial working conditions. Inspections are assumed to be perfect and instantaneous, as is the case in many modern manufacturing systems adopting a computercontrolled inspection facility. We assume that a machine in the "out-of-control" state, which has not been discovered, may also break down. The total system cost includes the fixed inspection, restoration and repair cost, and the per unit time cost of restoration, repair, and machine out-of-control. An inspection model for the process described above will be developed in the next section. The objective is to determine the optimal inspection interval minimizing the average cost per unit time, and to study the steady-state performance characteristics of the system. 4. ANALYSIS The production system described in Section 2 can be characterized by two attributes: the condition of the production unit and the current maintenance activity, which may be inspection, restoration, repair, or inactive. We denote 0, e {0, 1, 2} and ~o, ~ {q, a, r, v} as the state variables representing machine condition and maintenance activity at time t, respectively. An operational machine has O, = 0, an out-of-control machine has 0 t = 1, and a failed machine has 0 t = 2. Maintenance activities are inspection (ogf = q), repair (09t = r), restoration (~o, = a), and idle (o9t = v). We denote Z~ as the system state at time t, hence Z t has state space I = {(09, 0): w ~ {q, a, r, v}, 0 e {0, 1, 2}}. The process Z makes transitions each time the machine state changes, for example, machine failure, repair or machine out-of-control, etc. It is noted that machine restoration times and repair times are non-exponential, therefore the process Z is not Markovian. To analyze this system, we introduce the following instants: Time instant of type 0: an inspection is just finished and the machine is found to Z t = (q, O) be in control. Time instant of type 1: an inspection is just finished and the machine is found to Z t = (q, 1) be out-of-control. Time instant of type 2: machine restoration is finished and the machine begins to Z t = (a, O) operate. Time instant of type 3: machine repair is finished and the machine begins to operate. Z, = (r, 0) The above time points are the instants of time immediately following completion of inspection,
An inspection model
383 The instant of machine faiha-e.
The instant machine gets out-of-control.
I
I
°
I I T,
V
¢ ¢
I T~
v
T~ Fig. 1. An example of the system's behavior.
z,
(q,0)
(q,0)
.
(q,l)
(a,0)
(r,O)
(q,0)
(v,0)
(v,0)
(r,2)
Machine X.
(a,0)
(q,0)
Time Instant of
0
0
1
2
0
Type
oooooo
(q,O)
(r,O)
3
Operating, in-control
.l...
Out-of-control
xxxxxx Machine repair
Machine restoration
Fig. 2. Demonstration of the SRP and the embedded MRP. restoration, and repair. We may notice that Z possesses a strong Markov property at these selected time points. Let T = {To, 7"1, T2. . . . }, To = 0 be the sequence of the successive time instants of changes of state of the system after each inspection, restoration, and repair. We then define X. = Z(T.),
n=0,1,2 .....
It can be shown that {X., T., n = 0, 1,2 . . . . } is a homogeneous Markov renewal process (MRP) with state space J = {(q, 0), (q, 1), (a, 0), (r, 0)}, embedded at instants T. in the process Z. The process Z = {Z. y/> 0} is a SRP with state space I, and time instants of type 0, 1, 2, and 3 are the semi-regeneration points. Figures 1 and 2 demonstrate this system's behavior. Now, we may describe the system using a SRP on which a cost structure described in the previous section is imposed. The total cost of the system includes both fixed and variable portions. The system cost may be defined as the sum of a functional of the SRP and a functional of the embedded MRP. The system variable cost incurred at any time point is governed by the state of the SRP
Z. For example, the per unit time cost of producing defectives, d, is incurred so long as the machine is in the "out-of-control" state, that is whenever Z, equals (v, 1). The fixed cost of the system is governed by the state transitions made by the embedded Markov chain X. For instance, the fixed cost of restoration is incurred each time X, makes a transition from state (q, 1) to state (a, 0). We may obtain the longrun expected average cost per unit time using the results of the theory of semi-regenerative reward processes.
5. MATHEMATICAL FORMULATION We begin with the derivation of the semi-Markov kernel for the Markov renewal process (X,, T.), that is we find the probabilities Qo(t) defined by Q,j(t) = P { g , + t = j , T,+I - T~ ~< t l Sn = i}
i, j e J.
(1)
Using the fact that the time to machine out-of-control
384
J. Fung and V. Makis
and the time to failure are independent,
we have
tions is P,q. o,,r.o,
Q tv.o)w o,(f) = Qtr. o,cq.o,(t) = Q,,. o,cq,o,(t) e
(1, + Id)U
-
?L,rso)= %q.o,,q,
t>v
0
otherwise,
1 -
Qcq. om. I ,(t) = Qcr, o,,q, l,(t) = QI~. wq. I,@) ,-AldU
(1
=
_
,-A,u
1 _
t>v
1
0
(3)
otherwise.
Using the total probability theorem (see, e.g. Ref. [6]) and by conditioning on the time to machine failure, G, we obtain Qc. o,cr.o,@) = Q (0, o,,r, o,(t) = Q,r, o,,r, o,(t)
P( T, + & < t 1 & = u)fd(u) du
t <
u
P(T, + Td < t 1 Td = u)fJu)
t >
u
du
F,(t -
up, eeAd"du
f< v
c(t
u)&e-“d”du
t >
?q.O,,r.O)
+
G?*O),q.O)
e-b
,-,nc+i,,u
+
,-ido
,-(&+ld,u
Pk?.wl!O)
%o, = ___
n,,,o, = * _ ,~,&+id,u + ,-i,u
P,q*O,,r.O,
P(4.wq. 1) e _I‘j” _ ,-,&+i& n (I.0, = 1 _ e-‘“.+“d’u + e-ldu P(4.OW,0,
Z(q.1,
=
____
%I, 0,
=
%z.
(9)
1)’
We now impose the cost structure on the SRP 2 and the MRP X. The inspection cost is incurred whenever the process X makes a transition into states (q, 0) or (q, 1). The fixed restoration and repair costs are incurred when the process X makes a transition into state (a, 0) and (I, 0), respectively. We define Kj as the one-step transition cost which is governed by the embedded Markov renewal process X
=
S’
1, +
(2)
zz
0
=
K (.,o) = R,, K,,,o, = R, K,q,i, = Rq,
,
:
-
s 0
Q ,q,,j,a.Oj(t) = P(time to machine restoration is less than or equal to t) = F,(t),
(4)
All other Qij(t) = 0. Taking the limit of the probabilities (2)-(4), we have the following transition probabilities of the embedded Markov chain X P,q~o,,q.o)= PkLo,,q,o, = P,r.O),q.O,= e -,&+ldw, P,q,O,,q.l, =
4l.O,,q,l,kO,,q.L,
=
(1
-
emA=“)e- ‘d”,
P,q.ow.o,--P ,n.O,,r,O,=~r.0,,r,0)= 1 - e-
i = 0, 1. (10)
o
The variable costs of the system are governed by the state of the SRP. These cost rates are the cost per unit time of undetected machine out-of-control d, cost per unit time of machine restoration c,, and the cost per unit time of machine repair c,. The cost rate d is incurred whenever the SRP is in state (II, 1). The cost rates c, and c, are incurred whenever Z, is in state (a, 1) and (v, 2), respectively. These cost rates, j(j), are as follows: S(u,O)=O,
f(a,
l)=c,,
f(r,2)=c,.
(11)
(5)
f(u, l)=d,
(6)
We can calculate the expected cost accumulated between two successive semi-regeneration times for the MRP X using the formula (see L T,+, Ref. [7])
‘dL.,Fq, l,(a,Oj= 1. (7)
Since the embedded Markov chain X is irreducible and recurrent, we may find its steady-state probabilities 7tjby solving the following system of linear equilibrium equations and the normalizing equation
G,(co) = lim Gi(t) t-cc =
%J.O)= 7%.o?,,, oj,q.0, + ?I&o& o,,q.0)+ %.0,4r, ON&O)
s
omz, f(z)lCli(u, z)du+ 1 KjQij(a) jsJ
ieJ
1) %?,1, = %. o+L 0,,4.1,+ %. of,a. Ok 1,+ 5,. o+%.O,,%
$i(t, A) = c(Z, E A, Tl > t) i E J, A c I.
7%.0, = %?.oG%. O,,r.O) + %I. o,P,o, O,,r*O) + Tr. o,h. oe. 0)
The quantity t+bi(t,A) is the probability that, given the initial state of the MRP X0 = i, the transition time is greater than t and the Z, E A. We obtain the following formula for tii(t, A):
%I.01= %I. 1IP(4.1KO. 0,
The solution
to the above system of linear equa-
+,,. ,,(t, (0, 1)) = 1 - F,(t) = ~a’.(o.
(12)
(13)
An inspection model
By Theorem 4.4 of Ref. I-7], the long-run expected average cost per unit time g(v) can be bound by using the formula
For i E {(r, 0), (q, 0), (a, 0)}, we have "e ~i(t, (V, 0)) =
(ae + 2a)t
t ~ V
0
385
t>v
u))2 d e - aau du
f i ( l -- F,(t
f f (1 -- Fr(t -- u))2 a e- ~au du
{(01 - e - ~ ' ) e-~d' ¢i(t, (V, 1)) =
t ~< V otherwise.
~-~keJ ~ k m k
t <<.v
Substituting Eqs (15-18) into (19), we obtain
O,(t, (r, 2)) =
t>v
g(v) =
%E(T~) + R o + Rq - R, -
(
(14)
+ cr
;o fo
R~ - %E(Ta)) e -(~+ ~ '
d
+ \,~ + ~
Then, for t/> 0 G(q, j i(t) =
f ( a , 1) O(q, 1)(u, (a, 1)) du
(;o;oF,(t
- u)2 a e xdu du dt
+ i ~ jlv f r ( t - u ) 2 a e
+ K~.o}Q¢q, 1)~a,O)(t)
= %
(19)
g(v) = ~'j~J n j G j ( ~ )
ft.(u) du + R, F,(t).
(15)
q 2d
Next, for t > v
d"dudt)
2c + 2d + R,
/(e -a~° -- e-¢a°+a~lV)E(Ta) + v
G(q.o) = G{,,o~(t)= G(a,o)(t ) = J'i [f(v, 1)~k,q,o)(U,(v, 1))
-
+ f ( r , 2)@~,o)(U, (r, 2))
+
+ f(v, 0)~klq,o)(U, (v, 0))] du
+ R,Q(q.o)(,,o)(t)
+
c,
F,(t-u)2de-~,'dudt
fo f:
By Proposition 7.6 of Ref, [8], the steady-state probabilities of the SRP Zt can be obtained using the following:
F~(u -- z)2 d e -a"~ dr du
+ R,
G
. (20)
5.1. Performance measures (1 -_/~+~e-¢a°+a~)~)]_j
7"/:k~OC° ~,~(t, 0
P ( i ) = lim Pt(i)= (~k~J +
]
(1-F,(t-u))2de-~'dudt
We can find the optimum value of v by minimizing the function g@).
+ Rq[Q~q,o)~a,o~(t)+ Q~q,o}tq,1}(t)]
- d [ (1 -2ae-~'")
fof
t~ ~
.(u -- z)2a e- a.~ dz du
, (21)
~k~J ~kmk
where
f ? q%,~}(t,(a, 1)) dt = fo~ FAt) dt
F~(t- u)Ane- ~" du + Rq e- a.~. (16)
and for i e {(q, 0), (a, 0), (r, 0)}
We next determine the mean sojourn times m~ for each state i
f f ¢~(t,(v,O))dt= f~e-'a°*a~"dt 1 - e -ta`+aa)v
m, = E[T.+ , -- T. I X . = i] -
fo°
~ki(t, (r, 2)) dt =
m~q.1~ = j ~ ff,(t)dt = E(T,)
dr)
(17)
f:fo
2~ + 2a
'
(22)
ff,(t - u)2 a e -a~" du dt
(t - U)2d e - ~ " du dt
+ mtr, o) = m{a. o} = re(q, o}
fo +
(1 - g(t - U))2d e - a " du dt.
(18)
Oi(t, (v, 1)) d t - -
fo
(1 - e - ~ ' ) e - ~ '
dt
_ 2 c + 2 a e-(~+a")~--(2~ + 2a) e-~,~ 2d(2~ + 2n) (23)
386
J. Fung and V. Makis
Hence, the steady-state probability that the machine is down is P(r, 2) -
long-run average cost per unit time: g(v) =
(Zk~L nk ~ tPk(t, (r, 2)) dt)
c + ,~
~k~J 7Zkmk
where
+
L = {(q, 0), (a, 0), (r, 0)}.
(
m
R q - - R t q:- R a +
;)..
c,
(24) + R~ + c,. + d #r 2a
The steady-state probability that the machine is being restored is
d I 2c + 2a J e - .lay
P(a, l) = (Tt(q.1)~0 ¢{q, 1)(t ' (a, 1)) dr)
+
(25)
laa
ff~.k~J 7[kmk
~.a/
#~
+
The steady-state probability that the machine is out-of-control is
+
.
(29)
The optimal inspection interval can be obtained by minimizing the cost function g(v).
oo
(~,k~L k ~0 ~bk(t, (v, 1)) dt)
P(v, 1)
(26)
~ k e J 7[km k
6. N U M E R I C A L
EXAMPLES
In this section, we provide an analysis of a special case and present some numerical results. We assume that the restoration and repair times are exponentially distributed with parameters p, and p~, respectively. From eqns 05) and (16), we have
6.1. Performance measures The steady-state probabilities of the SRP {Zz} for this special case can be obtained using the formulae (24)-(26) derived in the previous section. The formula for the steady-state probability that the machine is down has the form P(r, 2)=
(1 - e -a~) #'[e-adv(1
)1Y , ~a
e-(~°+z~)~ L ] ' ~/ ' 1~~+ .
(30)
Ca G(q, 1)(ov) = - - + R ~ ,
The steady-state probability that the machine is being restored is given by
Pa
6,(oo) = a[(,+
(,-
+
J
a[0
P(a, 1)---
+R, (1-e-aa~)+Rqe-a"v, i e {(q, 0), (a, 0), (r, 0)}.
(31) (27)
The expected sojourn times are obtained from eqns (17) and (18), re(q,
1 pa
m(q,o} = m(a.O) ~ m(r.O)
+
and the steady-state probability that the machine is out-of-control is P(v, 1) =
2~ + 2de -(~°+~)~ - (2~ + 2a)e -a~
(32)
e -(a°+aa)v 1 1 ] ' + + #, /#
1) --
=v-
e-~v(1 _e-acv)
;L Lf:
(1 - e-"~"-~))2a e- ~" du dt
e-"'(t-"}2a e - ~ " du dt
= (1 - e -a"~) x
+
.
(28)
By using the formula in eqn (20), we obtain the
Using the above steady-state probabilities, we may find the various system long-run characteristics analytically. The long run fraction of time = P(r, 2) machine is in failure state The long run fraction of time machine is producing quality = P(v, O) products The long run fraction of time = P(v, 1) machine is producing defectives The long run fraction of time = P(r, 2) + repairman is busy
P(a,
1).
An inspection model Table 1. Parameter values for numerical analysis Parameter
Value
Re ($) Rr (S) Rq ($) ca (S/time) cr (S/time) d (S/time)
20 50 20 5 10 5
6.2. Numerical results The p a r a m e t e r values in Tables 1 a n d 2 are used in the examples. F r o m Fig. 3, we can see that the total cost curve becomes more fiat as the failure rate increases. This means that for larger 2 d there is more flexibility to choose the optimal inspection interval, v. In fact, any value of v greater t h a n 20 time units is optimal for the case of ;'d = 0.04. The total cost for all three different values of 2d tends to a positive c o n s t a n t as v increases. This may not be the case in the classical
387
inspection models which d o not assume system failure. Their total cost would go to infinity because the cost of defects goes to infinity with v. In o u r inspection model with machine failure, the long-run fraction of time the machine stays " o u t - o f - c o n t r o l " will not increase indefinitely with v as illustrated in Fig. 5). Hence, the cost of defects becomes steady as v increases. The reason for this is t h a t the occurrence of machine failures will renew the deteriorated system occasionally. In Fig. 4, it is interesting to note that the long-run fraction of time the machine stays " i n - c o n t r o l " , P(v, 0), for 2d = 0.08 is greater than that for smaller 2d when the inspection interval is large. The explanation for this is that the machine with a higher failure rate is likely to break d o w n more often a n d hence the process is renewed more frequently. We also observe that v has lesser effect o n P(v, 0) for larger 2d. We c a n n o t improve P(v, 0) for 2a = 0.08 as m u c h as we can improve P(v, 0) for 2d = 0.01 by shortening the inspection interval. The reason is t h a t the long-run fraction of time the machine is u n d e r repair, P(r, 2),
Table 2. Results of numerical analysis Case
)~,.
/1
2d
v*
g(v*)
P(v, 0)
P(r, 1)
P(r, 2)
P(a, 1)
1
0.006
/x, = 0.5 ~r = 0.4
2
0.06
p, = 0.5 1~ = 0.8
3
0.06
/~ = 1.0 #r = 0.4
4
0.006
/~ = 0.5 ~r = 0.4
0.001 0.01 0.04 0.08 0.16 0.01 0.04 0.08 0.16 0.01 0.04 0.08 0.01 0.04 0.08
19.57 20.97 28.03 45.04 31.15 21.01 28.38 46.33 35.45 20.73 28.26 47.86 44.56 62.95 64.62
3.91 4.25 5.37 6.79 9.55 4.19 5.15 6.50 9.47 4.23 5.37 6.80 1.74 3.30 5.29
0.550 0.535 0.493 0.487 0.522 0.541 0.513 0.530 0.607 0.553 0.500 0.485 0.858 0.811 0.780
0.383 0.386 0.391 0.344 0.192 0.391 0.412 0.376 0.225 0.394 0.400 0.347 0.110 0.100 0.056
0.002 0.023 0.088 0.166 0.285 0.012 0.046 0.091 0.167 0.024 0.090 0.166 0.024 0.091 0.167
0.065 0.056 0.028 0.003 0.001 0.057 0.029 0.003 0.001 0.029 0.014 0.001 0.008 0.002 0.000
6.5 $
20 = 0 . 0 4
i;
o 0
§
&=o.ol ~r D e. 0
"
20=o.ooi 4
I
I
I
I
t
I
I
10
2O
3O
40
60
60
7O
Inll~
Inlevll v
Fig. 3. Long-run average cost as a function of inspection interval for case 1.
I
388
J. Fung and V. Makis
_e J~ 2 o.s Ilk
~-d = 0.08 ~,d = 0.04
~0.25
"gd = 0.01
I 20
I 40
I 60
I 80
I 100
mlm(:uo. Intwv~l v
Fig. 4. Steady-state probability as a function of inspection interval for case 1.
~
= 0.01
~ - d
0.76
.Zd = 0.04 2 ~ Q.
~,d ----"0.08
l I
I
I
I
I
20
40
60
80
100
Inspectl~ Intenral v
Fig. 5. Steady-state probability as a function of inspection interval for case 1.
~0.76
~=0.8 i.o II.
0.5
/a, = 0.4
J I
I
I
I
I
I
I
0.06
0.1
0.16
0.2
0.28
0.3
0.35
M a c h i n e Failure Rate
Fig. 6. Steady-state probability as a function of machine failure rate 2d for cases 1 and 2. is independent of the inspection interval, v (as can be seen from Fig. 7). While P(r, 2) stays high for all v, P(v, 0) cannot be improved much by shortening the length of the inspection interval nor can it be worsened
much by lengthening the interval. This is also the reason for the behavior of P(v, 1) in Fig. 5. Since, for a system subject to higher failure rate, the time the machine stays "in-control" becomes more
An inspection model
389
0.76 a .ID
.o
P(v,l)
0.
~
o.5
m,
P(v,O)
"0
~
0.25
P(r,2) 0 0
10
20
30
40
60
60
70
80
Inspection Interval v Fig. 7. Steady-state probabilities as a function of inspection interval for case 1 with 2~ = 0.04. reliant on 2d, we may improve P(v, 0) by shortening the repair time. This is demonstrated in Fig. 6. The higher 2 e is the larger is the difference between the steady-state probabilities P(v, 0) for the two cases.
inspection interval does not matter much (as long as it is at least as large as the optimal inspection interval for a lower failure rate).
7. CONCLUSIONS
REFERENCES
In this paper, we have presented an inspection model with machine failure and generally distributed restoration and repair times. Using Markov renewal theory, we have obtained the formula for the long-run expected average cost per unit time and formulae for the steady-state system characteristics. Computational results indicate that the long-run fraction of time the machine stays in-control is dependent on both the length of the inspection interval and the machine failure rate. The dependency of P(r,, 0) on v weakens as 2 a increases. One way of improving P(v, 0) for the system subject to high failure rate is to upgrade the repair facility and increase the rate of repair. We have also observed that when the failure rate is high the average cost function flattens out so much that the
1. Barlow, R. E., Hunter, L. C. and Proschan F., Optimum checking procedures. J. Soc. Ind. Appl. Math., 1963, 11, 1078- 1095. 2. Schultz, C. R., A note on computing periodic inspection policies. M#mt Sci., 1985, 31, 1592-1596. 3. Munford, A. G., Comparison among certain inspection policies. M#mt Sci., 1981, 27, 260-267. 4. Luss, H., Inspection policies for a system which is inoperative during inspection periods. AI1E Trans., 1977 9, 189-194. 5. Nakagawa, T., Periodic inspection policy with preventive maintenance. Naval Res. Lo#istics Q., 1984, 31, 33-40. 6. Tijms, H. C., Stochastic Modellin# and Analysis: a Computational Approach. John Wiley, New York. 7. Schellhaas, H., Semiregenerative processes with unbounded rewards. Math. Operations Res., 1979, 4 (1), 70-78. 8. (~inlar, E., Introduction to Stochastic Processes. PrenticeHall, Englewood Cliffs, NJ, 1975.