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Lot sizing and testing for items with uncertain quality
Mathl. Comput.
Vol. 22, No. 10-12, pp. 35-44, 1995 Copyright@1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved
Modelling
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Lot Sizing and Testing for Items with Uncertain Quality I. DJAMALUDIN Department of Mechanical Engineering The University of Queensland, QLD 4072, Australia
R. J. WILSON Department of Mathematics The University of Queensland, QLD 4072, Australia D. N. P. MURTHY Technology Management Centre The IJniversity of Queensland, QLD 4072, Australia Abstract-we consider the problem where items axe producedin lots and sold with warranty. Due to manufacturingvariability, some items do not conform to the design speciiications and their performanceis inferior (for example, have higher failure rate). The warrantyservicing c& for these is much higher than for those which conform. Two approacheshave been advocated for reducing the wsxranty cost per item released, and in both it is achieved at the expense of increased manufacturing cost. The first involves life testing to weed out nonconforming items and the second involves strategies to reduce nonconforming items being produced. In this paper, the authors develop a model which combines both approaches, and quality control decisions are made optimally to minimize the total (manufacturing and warranty) cost. It extends the earlier models of the authors which deals with quality decisions based solely on either the first or the second approach.
Keywords-&
sizing,Life testing, Quality control, Warranty.
1. INTRODUCTION Due to variability in manufacturing processes, some items produced will not conform to specifications and their performance will be inferior to conforming items. Nonconforming items may be inoperable (due to errors in assembly or a component being defective and inoperable) or op erational with a shorter expected time to failure. The latter of these can only be differentiated from conforming items by life testing (or burn-in). In order to protect consumers against defective or poor quality items, manufacturers often offer a warranty-a contract which requires the manufacturer to rectify all failures occurring within the warranty period. The rectification can include repair, cash refund, or replacement. Manufacturers often use warranty as a promotional tool. Warranty also protects manufacturers against unreasonable claims from consumers. Offering warranty results in additional cost to the manufacturer. This cost depends on the failure characteristics of the item (determined by design and manufacturing decisions), the type of warranty (free replacement and pro-rata are the most common), the warranty period, and other associated costs. The design decisions determine the reliability of the product (for example, whether it operates and its mean time to failure) which affects the warranty servicing cost. This cost can be reduced by better design, but this results in increased design and development cost. Obviously, the expected warranty servicing cost for nonconforming items reaching the consumer is considerably more than that for conforming items. This cost can be reduced by a “weeding out” process, using life testing to reduce the number of nonconforming items being released, or
35
I. DJAMALUDIN et al.
36
by developing
a scheme (such as control
of lot size) to reduce the number
of nonconforming
items
produced. Many
models
have been developed
to characterize
and Porteus
[2] independently
concept
“the smaller lot size gives the better
that
The Porteus’
proposed
model has been extended
models
Throughout shorter (that
this
expected
paper,
time to failure.
is, it has stabilized),
probability
we shall
of conforming
assume
Djamaludin
and
Lee
can be found earlier in, for example,
uncertain
that
variations
time)
burn-in
Fbsenblatt
demand
[I]
with lot size, while the
[4] and to incorporate
[3].
learn-
review of this literature. nonconforming are purely
to the design specification.
of time (called burn-in
items for sale. The optimal
quality
items
For the case where the manufacturing
the quality
lot sizing plays no role, and the optimal a length
variations.
product
quality”
to include
ing effects [5,6]. See [7] for a more complete
quality
linking
quality
random
are operational process is in steady
strategy
state
et
al.
[8] show that
is either to life test all items for
and scrap those which fail or do no testing
time for different
a
and each item has the same
In this case, Djamaludin
control
with
types of warranties
and release all
are examined
in [9].
et al. [lo] deal with a model similar to that proposed by Rosenblatt and Lee [l] where the process is in-control at the start of production for each lot and can [2],
and Porteus change from in-control to out-of-control during production. If it changes, then it stays out-ofcontrol until the lot is completed. The probability of obtaining a nonconforming item is much higher when the process is out-of-control. In [lo], all the items produced are released with no life testing. The lot size is selected optimally to reduce the total expected cost per item, smaller lot size implies smaller warranty cost but higher manufacturing cost.
since
Here, we consider the case where the process changes from in-control to out-of-control as in [lo]. The process at the end of the production of a lot can be either in-control or out-of-control. We assume that, through inspection, the true state at the end of each production can be determined. If the process is in-control, then all items in the lot were produced with the process in-control If the process is out-of-control, then the items and are released for sale with no life testing. towards the end of the lot are produced with the process out-of-control and are more likely to be nonconforming. To remove these, a specified number of the items at the end of the lot are tested and those which fail are scrapped. The earlier items (not tested) and those which survive the test are released for sale. Thus, the quality control involves lot sizing (to reduce nonconforming items being produced) and life testing (to remove any nonconforming items produced). The decision variables (lot size, number of items tested, and burn-in time) are selected optimally to minimize the expected
cost per item sold.
The type of warranty
also affects the warranty servicing cost. Many types of warranty policies Murthy and Blischke [ll] define a taxonomy to put the and analyzed.
have been formulated Repairable items are often sold with free replacedifferent types of warranties into groupings. repaired or replaced by a new item. ment warranty (FRW) with failed items either minimally Nonrepairable items sre usually sold with either prorata warranty (PRW) with a refund or free replacement warranty with failed items replaced by new items. A variety of models have been developed to obtain the expected warranty cost per unit, and reviews of these can be found in (12-141. In this paper, we consider FRW with minimal repair only. The outline of the paper is ES follows. In Section 2, we give the details of the model formulation, including quality variations, the effects of lot sizing and life testing on the quality of released items, and the warranty policy. Section 3 deals with the optimal selection of the decision variables (lot size, number of items tested and the duration of testing), the manufacturing cost and the warranty cost for repairable items covered by FRW with minimal repair. A numerical example is given in Section 4. We conclude with topics for further research in Section 5.
2. MODEL In this section,
we give the details
FORMULATION
of the model formulation.
Lot Sizing
2.1.The Manufacturing
37
Process
The process is in-control at the start of production of each lot. If the process is in-control at the start of the production of an item, then during the production, it can go out-of-control with probability (1 - q) or stay in-control with probability q. Once out-of-control, it stays outof-control until the end of the lot. After the lot is completed, the process is checked and set to in-control if it is out-of-control. If the process is in-control at the end of the production of an item, the probability that the item produced is conforming [nonconforming] is Br [(l - &)I. If the process is out-of-control at the end of the production of an item, the probability that the item produced is conforming [nonconforming] is 02 [(l - es)]. W e assume 131> 02, so that there is a higher probability of obtaining conforming items when the process is in-control. The number of conforming items in each lot is random. Items are characterized by their time to failure. Suppose that Fr(t)[F~(t)] denotes the failure distribution function for conforming [nonconforming] items. Let fj@)
Fj(t) = 1 - q(t),
= y,
q(t) = _fjo Fj(t)
denote the density function, survivor function, and failure rate associated with Fj (t) for j = 1,2. The inferiority of nonconforming items to conforming items is characterized by assuming that nonconforming items have a higher failure rate than conforming items:
Thus, Fz(t) > Fl(t) fort L 0, and 00 s0
F,(t)
dt >
00 F,(t) dt s0
(that is, the mean time to failure for nonconforming forming items). 2.2. Lot Sizing and Inspection
items is much smaller than that for con-
Policy
We assume that there is a constant demand (per unit time) for the product. This demand is met by producing items in lots of size L. We assume that the time horizon is sufficiently large so that it can be approximated as being infinite. Hence, we consider the asymptotic case where the number of lots goes to infinity. Let C’s denote the cost of checking the process at the end of a lot and q the cost of bringing the process to in-control if it is out-of-control. The cost to manufacture a lot is comprised of this fixed setup cost and a variable cost depending on lot size. Thus, the manufacturing cost for lot i, c,(L)i, is given by C,(L), = cs + $i?j + C,L, where C,,, is the material and labor cost to produce a single item and +i = 1 if the setup for lot i involves bringing the process to in-control and $Q = 0 if no change is involved. (& is random.) The inspection scheme is as follows. If the process is in-control at the end of the production of a lot, then all items in the lot are released for sale untested. If the process is out-of-control, the last K items are life tested for a duration of time T. Items which fail during the test are scrapped and those which survive are released for sale with the untested items. For lot i, let Ei denote the number of items which fail the test. If the process is in-control at the end of production, then ei = 0. The number of items released for sale from lot i is (L - Q). If G(T) and C,, denote the testing and scrapping cost per item, respectively, then, with $i as before, the inspection cost for lot i, Cinsp(K,T)iy is given by
I. DJAMALUDIN
38
et at.
2.3. Warranty Policy
We confine attention to nonrenewing failure free warranty (FRW) policy with a warranty period W. Under this policy, the manufacturer agrees to either repair or replace (by a new item) should failure occur within the warranty period W subsequent to the sale, with no cost to the consumer. The repaired (or replacement) item is covered by the remainder of the original warranty. We shall assume that failed items are repairable and the manufacturer always repairs items which fail under warranty. The failure characterization of repaired items depends on the type of repair. (See [13] for different types of repair.) Here, we assume that all failures are rectified by minimal repair (see, for example, [15]). U n d er minimal repair, the failure rate after repair is the same as before failure. We assume that the cost of each repair is CR and includes the handling cost for each warranty claim. Let &v(L, K, T)i denote the total warranty cost to service items released for sale from lot i. As well, we assume the following: (a) All failures under warranty result in warranty claims. (b) A claim is made ss soon as an item fails and all claims are valid. (c) The time to repair is relatively short compared to the mean time between failures so that it can be treated as zero. (d) The inventory holding cost is ignored. formulation.)
(One can easily incorporate
this into the model
(e) The lot size is bounded above by Lu (to reflect physical limitations).
3. ANALYSIS
OF MODEL
For the model given in Section 2, we wish to choose the lot size (L), the number of items tested (K), and the burn-in time (2’) which gives the lowest cost. If TC(L, K,T)i denotes the total cost for manufacturing, life testing (if needed), and servicing of items released from lot i for sale, then TC(L, KYT)i = Cp(L)i + Ci”sp(KyT)i + Cw(L, KYT)i
fori= 1,2,..., I, where I denotes the number of lots produced. The total cost and the number of items released for sale for all lots are given by cf, TC(L, K, T)i and ‘&(L -Q), respectively, with the cost per item released given by their ratio. As this is a random cost, we consider the asymptotic csse by taking the limit in probability of this ratio as I --) 00. Suppose that J(L, K, T) denotes this asymptotic cost per item released. Since the lots are statistically similar, it is easily shown that J(L, K, T) is given by J(L K T) = E [TC(L, K,T)il 9 7
L - E[E~]
= r(L, K, T) + Cw(L, K, T)
(1)
where E [&Lfi -G,K,T)
=
is the sum of the asymptotic manufacturing
+ Gnap(KT T)i]
(L - ENI and testing costs per item released, while
is the asymptotic warranty cost per item released. J(L, K,T) is obviously nonrandom. The optimization problem is to find L, K and T which minimize J(L, K, T) subject to the constraints 1LL
Lot Sizing
3.1. Asymptotic
Manufacturing
39
Cost
In this section, we derive an expression for r(L, K, 2’). To do so, we need to characterize the number of nonconforming items in a lot and the output quality. Let Ni denote the number of items produced in lot i (of size L) while the process is in-control, so that the number of items produced with the process out-of-control is (L - Ni). Ni has probability function given by qn(l-q) O
ENI =
(1
”
q)
(1- q=).
Let p1((t) [4(t)] denote the failure distribution function for a conforming [nonconforming] item which survives the test and is released for use. It is easily seen that j$+) = R(t
3
for j = 1,2. The failure density fj(t)
Fjm1 Fj CT)
+ T) -
and the failure rate +j((t) are then given by fj(t) = hct + T, Fj (T)
and 4(t)
=
fjct+T) = rj(t+T) Fj(t+T)
for j = 1,2. The number of items scrapped in lot i is random, with expected value E[Q] = PI(T)EKL +pz(T){KPr(Ni
(4
I L - 1) - EKL},
where E[NJ is given by (3)) Pr{Ni I L - 1) = Pr{Gi = 1) = 1_ Pr{qi = 0) = I _ q~, E
{QL-K-QL}
-KQ= 7
and pi(T) [p2(T)]is the probability that an item produced while the process is in-control [outof-control] fails the test, with
Pj(T) = ejJ’l(T) + (l- 0,) J’2(T) for j = 1,2. The derivation is similar to that in [lo]. From the analysis carried out so far, is easily shown that E [c&Q
+ &,(KT)i]
= [LG
+ Cs + (I-
and so, with E[Q] given by (4), the manufacturing 7(L
T)
K
7
,
=
qL) (77+ KG(T)}
+
EhlGc] ,
and testing cost per unit released is given by
LCm+ Cs + (1- q=){rl + KG(T)) L - E[Q]
+
EhlGc
(6)
I. DJAMALUDIN et al.
40
3.2. Asymptotic
Warranty Cost
We need to consider conforming and nonconforming items which are tested and not tested separately to evaluate the warranty cost per item released. The expected number of conforming items not tested and for sale is given by &{E(Ni)-
EKL}+@~{L-E(N~)-K(~
-Q~)+EKL},
(7)
and the expected number of tested conforming items which survive and are released for sale is given by
9”)
B~F~(T)EKL+~~~~~(T){K(~
-EKL}.
(8)
Similarly, the expected number of nonconforming items not tested and for sale is given by (7) with& and82replacedby(1-01) and(l-82), respectively, and the expected number of tested nonconforming items which survive and are released for sale is given by (8) with 8i,& and pi(T) replaced by (1 - @I), (1 - 02) and Fz (T), respectively. As a result, the expected warranty cost for lot i is given by E
pW(L,K,T),]
= CR[%(W)E(&)
+'~2(W)tL-E(4)1
(9)
+ {~(T,W)-?'I(W))EKL +
{Pz(T,W)
-yz(W)}{K(l
-qL)
-EKL}]~
where E[&] is given by (3), EKL is given by (5) and W 'yj(W)=
f3j w n(t)& J0
+
(1 -
f3j)
7.2(t)dt J
0
and w +1(t) dt + (1 - ej)Fz(T)
pj(T, W) = $Fi(T) J
for j = 1,2. The asymptotic using (4) and (9).
0
1”
Pz(t) dt,
0
warranty cost per item released, Cw(L, K, T), is given by (2),
3.3. Optimal Solution The asymptotic total cost per item, J(L, K,T), is given by (1) with y(L, K,T) given by (6) and C&L, K, T) given by (2) using (9) and (4). Let L’, K* and T* be the optimal values of L, K and T which minimize J(L, K,T) subject to the constraint 0 < L 5 L~J, 0 5 K 5 L and T 2 0. It is impossible to find analytical expressions for L*, K* and T*. As a result, a numerical procedure is needed to obtain them. The authors are currently researching into analytical results for (i) K* to take on the two extreme values, i.e., K* = 0 or L*; (ii) O
4. NUMERICAL
EXAMPLE
In this section, an example is given. We assume that the failure distributions for both conforming and nonconforming items are exponential with parameters r$r and 42, respectively, (42 > 41) and that Lu = 100 (the upper bound of L). The nominal values of the parameters are given in Table 1. The cost parameters are in dollars ($) and oz in dollars/year. The mean time to failure is ten years for a conforming item and 0.1 years for a nonconforming item. We consider four different values for the warranty period W, ranging from 1 to 4 years. We also consider the case W = 0, which corresponds to the product being sold with no warranty.
Lot Sizing
41
Table 1. Nominal values for parameters.
Cs
Cm
C,
al
a2
C,,
CR
500
10
100
1.0
1.0
0.5
3.0
Q
e1
e2
41
42
0.95
0.95
0.15
0.1
10.0
4.1.Optimal Values The optimal values for L, K,T and the asymptotic total cost per released item (that is, L*,K*,T* and J(L*,K*,T*)) were obtained by evaluating J(L, K, T) for L = 1, . . . , Lu, K = 1 7‘.‘7 L, T incremented in steps of 0.001 from 0 to 1, and for W equal to 0, 1,2,3and 4. We also consider two other cases: Suppose that the lot size is not a variable and is set equal to Lu (the upper bound). In this case, the quality can be improved only through K and T. Let KG and T; denote the optimal number of items tested and the optimal duration of the life testing and J(Lu, KG, Tc) the corresponding
asymptotic cost per item released.
When no testing is employed, then K = 0 and T = 0. In this case, the only way to improve quality is through lot sizing. Let Lf, denote the optimal L under this condition and let J(Lt , 0,O) denote the corresponding asymptotic cost per item released. (This case corresponds to the model formulation of Djamaludin et al. [lo].) Let RC1, RC2 and RC3 denote the following percentage reductions in the costs: RCl
RC2
=
=
loo[J(W,O) - J(L*,K*,T*)l J(Lu,O,O)
loo[JWW)
’
- J(Lu~K;~T;)l ’
J(Lu,O,O) and
Table 2 shows L’, K*, T’, K&,T;, totic costs per item released. For vri = 0,l
Lf, along with RC1, RCp., RC3 and the corresponding asymp-
(year), no lot sizing (that is, L’ = Lu) or testing is carried out for the three
cases. However, as W increases from 2 to 4, the number of items tested and the testing time increase. A longer warranty period increases the warranty costs for any nonconforming items released and so, more items are tested and for longer to decrease the expected number of released nonconforming items. The optimal J( L*, K*, T*) increases as the warranty period increases. For the cases where lot sizing or testing are carried, the asymptotic cost per released item is less than the appropriate value of J(Lu, 0,0), as should be expected. The percentage reductions (RG, RC2, RC 3 ) sh ow the advantage of using lot sizing and testing, just testing, and just lot sizing over using neither lot sizing or testing. From Table 2, it can be seen that RC1 > RC2 and RC1 > RC3 and hence, the saving in cost from employing both life testing and lot sizing is greater than that from employing just lot sizing or just life testing. This is expected, since optimization over all variables will give at least as good a result ss optimization over a subset of all variables. In this example, RC2 > RC3 for W = 3,4. This will not always be the case, since RCs and RC3 depend upon the choice of LU (FMwell as the other parameter values). As Lu is increased, lot sizing is more likely to be important. All of the percentage reductions (RCI, RC2, RC 3) increase with W. The increases indicate that lot sizing and/or life testing become more effective as the warranty period increases.
I. DJAMALUDINet al.
42
Table 2. Optimal values and related costs. W ‘VW,
O*O)
0
1
2
3
4
15.994
37.051
58.109
79.166
100.223
100
43
42
43
L’
100
K’
0
0
29
41
43
T’
0.00
0.00
0.247
0.410
0.471
JCL’, K*,T*)
15.994
37.051
55.461
57.926
59.023
I
41.11 100 100 0.493 64.393 RC2 @Jo)
0.00
0.00
0.00
20.06
LC
100
100
41
27
21
.J(L& O,O)
15.994
37.051
56.093
70.327
82.426
RC3 (rn)
0.00
0.00
3.47
11.17
17.76
35.75
]
For the case of just life testing, there is a sudden change in strategy from no testing (for W = 2) to almost 100% testing (for W = 3). This is due (in part) to the nature of the geometric distribution which governs when the process goes out-of-control and which has a “heavy” tail. For this example (Q = 0.95), the distribution of the number of items produced with the process in-control has a median of about 12 and the 95th percentile is about 57. As a result, it is more likely that most of the (Lu = 100) items are produced while the process is out-of-control. Hence, if it is sensible to test some items produced while the process is out-of-control, then it is probably sensible to test most of the items. 4.2. Sensitivity Analysis The sensitivity of the values of L*, K* , T‘ and J(L* , K*, T*) to the model parameters is discussed in this section. Tables and plots of these can be found in [17]. It is important to observe that lot sizing applies to all lots, whereas life testing is only carried out if the process is found to be out-of-control at the end of the production of a lot. In addition, the model for N, the number of items produced when the process is in-control, is a geometric distribution, which is very “flat.” This results in a high variability for N, which affects the choice of the optimal strategy. Since a longer warranty period implies a higher warranty cost, the optimal cost per item released (J(L*, K*, T*)) increases as the warranty period (W) increases. Consequently, the optimal lot size (L’) tends to decrease initially from L* = La with no testing, until 100% testing occurs, after which L’ tends to increase. As well, the number of items tested (K*) and the testing period (T’) start at zero (that is, no items are tested) and then increase as W increases, with K’ increasing to L*. For larger W, the warranty cost is much greater for nonconforming items and so it is more important to remove any nonconforming items produced before they are sold. Since lot sizing is less costly than life testing, lot sizing is used first to remove nonconforming items. As W increases and it becomes more critical to remove these items, life testing is also used. Life testing is more effective at removing nonconforming items produced while the process is out-of-control, and can be made still more so by increasing T*. Hence, once 100% testing is being carried out, L* can be increased ss W increases in order to compensate for the higher warranty cost. Note that, as W increases, the changes in strategy below are more sudden, since it becomes more imperative to remove the nonconforming items before sale. For a fixed warranty period (W), the optimal cost per item released decreases as the quality (that is, the expected number of conforming items) increases. This happens if any of q (the
Lot Sizing
43
probability the process stays in-control during the production of an item), 81 (the probability of a conforming item while the process is in-control) and 0z (the probability of a conforming item while the process is out-of-control) increase. In general, lot sizing and life testing are used if (a) the process spends enough time in-control and enough time out-of-control, (b) there is enough difference between the proportion of nonconforming items produced while the process is in-control and the proportion of nonconforming items produced while the process is out-of-control, or (c) there are reasonable proportions of conforming and nonconforming items produced. Consequently, for “low” q, no lot sizing is used (no lot can be reduced enough to remove all items produced while the process is out-of-control) and no life testing (life testing would leave very few items to be released) is carried out. (This is partly due to the high variability of N.) As q increases, it is worthwhile lot sizing as there will be some items produced while the process is in-control. However, little or no life testing should be carried out (for the nominal values of 191= 0.95 and 132= 0.15), due to the high variability of N and the undesirability of removing too many items when many are produced while the process is out-of-control. As q increases still further, lot sizing is still beneficial and life testing becomes more important. This occurs because there is a reasonable number of items which are produced while the process is in-control and it is worth removing those produced while the process is out-of-control. Life testing increases up to 100% as q increases since it is only done for those lots for which the process goes out-of-control. For large q, there is less benefit from lot sizing as all the items in most lots should be produced with the process in-control (and so, they should be conforming). Again, since life testing is only done for those lots for which the process goes out-of-control and this occurs rarely, 100% life testing is worth carrying out on all items. If 8i is “low” or 82 is “high” (with q set at the nominal value of 0.95), there is not a great deal of difference between what happens when the process is in-control and when it is out-of-control. Consequently, lot sizing is not effective. However, if there are reasonable proportions of both conforming and nonconforming items, 100% life testing should be used; otherwise, life testing is less effective and so few items should be tested. As 81 increases (02 decreases), lot sizing becomes more effective and life testing can become less effective (depending on the value of q). Further discussion of the effects of the other parameters can be found in [16]. 5. EXTENSIONS The model discussed in this paper can be extended in several ways. Some of the following are currently being investigated by the authors. 1. We have discussed only the FRW policy with minimal repair. This approach can be used to study other types of policies, such as pro-rata or combination warranties, in a similar manner. 2. We assumed a testing scheme where the last K items are tested if the process at the end of the lot production is out-of-control. If the items are numbered sequentially, then an alternate scheme for testing is as follows. If the process at the end is out-of-control, test item L - KI, with K1 < K, for a period T. If it fails, test all items from L - K1 to L (i.e., test only the last K1 items in the lot). If not, test all items numbered from L - K to L (i.e., test the last K items in the lot). Many other testing schemes can be formulated. 3. We have assumed that inspection at the end of a lot production reveals the true state of the process. An interesting extension is where the true state is either unknown or known with uncertainty. 4. The change in process can be modelled differently, such as the process deteriorating gradually as opposed to a sudden change.
44
I. DJAMALUDIN et a(.
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5. 6. 7.
8. 9.
10. 11. 12. 13. 14. 15. 16.
M. J. RosenbIatt and H.L. Lee, Economic production cycles with imperfect production processes, Z.I.E. !f+unsactions 18, 48-55 (1986). E.L. Porteus, Optimal lot sizing, process quality improvement and setup cost reduction, Opemtions Research 84, 137-144 (1986). R.J. Schonberger, Japanese Manufacturing Techniques: Nine Hidden Lessons in Simplicity, The Free Press, New York, (1982). G. Keller and H. Noori, Impact of investing in quality improvement on the lot size model, Omega Intern. Jr. of Management Science 10, 595-601 (1988). C. Fine, Quality improvement and learning in productive systems, Management Science 32, 1301-1315 (1986). S. Chand, Lot sizes and setup frequency with learning in setups and process quality, Eum. Jr. of Oper. Res. 42, 190-202 (1989). D.N.P. Murthy and I. Djamaludin, QuaIity control in single stage production system, International Journal of Production Research 28, 2219-2242 (1990). I. Djamaludin, D.N.P. Mm-thy and R.J. Win, Product warrantyand quality control, ORSA/TZMS Meeting, Anaheim, CA, (1991). D.N.P. Murthy, R.J. Wilson and I. Djamaludin, Product warranty and quality control, Quality and Reliab. Eng. Intern. 9, 431443 (1993). I. Djamaludin, R.J. Wilson and D.N.P. Murthy, Quality control through lot sizing for items sold with warranty, Intern. Jr. of Prod. Econ. 33, 97-107 (1994). W.R. BIischke and D.N.P. Murthy, Product warranty management-I: A taxonomy for warranty policies, Euro. Jr. of Oper. Res. 62, 127-148 (1992). W.R. Blihke, Mathematical models for analysis of warranty policies, Mathl. Comput. Modelling 13 (7), 1-16 (1990). D.N.P. Murthy and W.R. Blichke, Product warranty management-II: An integrated framework for study, Euro. Jr. of Oper. Res. 62, 261-281 (1992). D.N.P. Murthy and W.R. Blischke, Product warranty management-III: A review of mathematical models, Euro. Jr. of Oper. Res. 63, l-34 (1992). D.N.P. Murthy, A note on minimal repair, IEEE ‘Bunsactions on Reliability 40, 245-246 (1991). I. Djamaludin, Quality control schemes for items sold with warranty, Ph.D. Theeis, The University of Queensland,, (1993).
Vol. 22, No. 10-12, pp. 35-44, 1995 Copyright@1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved
Modelling
0895-7177(95)00178-6
089571?7/95-$9.50 + 0.00
Lot Sizing and Testing for Items with Uncertain Quality I. DJAMALUDIN Department of Mechanical Engineering The University of Queensland, QLD 4072, Australia
R. J. WILSON Department of Mathematics The University of Queensland, QLD 4072, Australia D. N. P. MURTHY Technology Management Centre The IJniversity of Queensland, QLD 4072, Australia Abstract-we consider the problem where items axe producedin lots and sold with warranty. Due to manufacturingvariability, some items do not conform to the design speciiications and their performanceis inferior (for example, have higher failure rate). The warrantyservicing c& for these is much higher than for those which conform. Two approacheshave been advocated for reducing the wsxranty cost per item released, and in both it is achieved at the expense of increased manufacturing cost. The first involves life testing to weed out nonconforming items and the second involves strategies to reduce nonconforming items being produced. In this paper, the authors develop a model which combines both approaches, and quality control decisions are made optimally to minimize the total (manufacturing and warranty) cost. It extends the earlier models of the authors which deals with quality decisions based solely on either the first or the second approach.
Keywords-&
sizing,Life testing, Quality control, Warranty.
1. INTRODUCTION Due to variability in manufacturing processes, some items produced will not conform to specifications and their performance will be inferior to conforming items. Nonconforming items may be inoperable (due to errors in assembly or a component being defective and inoperable) or op erational with a shorter expected time to failure. The latter of these can only be differentiated from conforming items by life testing (or burn-in). In order to protect consumers against defective or poor quality items, manufacturers often offer a warranty-a contract which requires the manufacturer to rectify all failures occurring within the warranty period. The rectification can include repair, cash refund, or replacement. Manufacturers often use warranty as a promotional tool. Warranty also protects manufacturers against unreasonable claims from consumers. Offering warranty results in additional cost to the manufacturer. This cost depends on the failure characteristics of the item (determined by design and manufacturing decisions), the type of warranty (free replacement and pro-rata are the most common), the warranty period, and other associated costs. The design decisions determine the reliability of the product (for example, whether it operates and its mean time to failure) which affects the warranty servicing cost. This cost can be reduced by better design, but this results in increased design and development cost. Obviously, the expected warranty servicing cost for nonconforming items reaching the consumer is considerably more than that for conforming items. This cost can be reduced by a “weeding out” process, using life testing to reduce the number of nonconforming items being released, or
35
I. DJAMALUDIN et al.
36
by developing
a scheme (such as control
of lot size) to reduce the number
of nonconforming
items
produced. Many
models
have been developed
to characterize
and Porteus
[2] independently
concept
“the smaller lot size gives the better
that
The Porteus’
proposed
model has been extended
models
Throughout shorter (that
this
expected
paper,
time to failure.
is, it has stabilized),
probability
we shall
of conforming
assume
Djamaludin
and
Lee
can be found earlier in, for example,
uncertain
that
variations
time)
burn-in
Fbsenblatt
demand
[I]
with lot size, while the
[4] and to incorporate
[3].
learn-
review of this literature. nonconforming are purely
to the design specification.
of time (called burn-in
items for sale. The optimal
quality
items
For the case where the manufacturing
the quality
lot sizing plays no role, and the optimal a length
variations.
product
quality”
to include
ing effects [5,6]. See [7] for a more complete
quality
linking
quality
random
are operational process is in steady
strategy
state
et
al.
[8] show that
is either to life test all items for
and scrap those which fail or do no testing
time for different
a
and each item has the same
In this case, Djamaludin
control
with
types of warranties
and release all
are examined
in [9].
et al. [lo] deal with a model similar to that proposed by Rosenblatt and Lee [l] where the process is in-control at the start of production for each lot and can [2],
and Porteus change from in-control to out-of-control during production. If it changes, then it stays out-ofcontrol until the lot is completed. The probability of obtaining a nonconforming item is much higher when the process is out-of-control. In [lo], all the items produced are released with no life testing. The lot size is selected optimally to reduce the total expected cost per item, smaller lot size implies smaller warranty cost but higher manufacturing cost.
since
Here, we consider the case where the process changes from in-control to out-of-control as in [lo]. The process at the end of the production of a lot can be either in-control or out-of-control. We assume that, through inspection, the true state at the end of each production can be determined. If the process is in-control, then all items in the lot were produced with the process in-control If the process is out-of-control, then the items and are released for sale with no life testing. towards the end of the lot are produced with the process out-of-control and are more likely to be nonconforming. To remove these, a specified number of the items at the end of the lot are tested and those which fail are scrapped. The earlier items (not tested) and those which survive the test are released for sale. Thus, the quality control involves lot sizing (to reduce nonconforming items being produced) and life testing (to remove any nonconforming items produced). The decision variables (lot size, number of items tested, and burn-in time) are selected optimally to minimize the expected
cost per item sold.
The type of warranty
also affects the warranty servicing cost. Many types of warranty policies Murthy and Blischke [ll] define a taxonomy to put the and analyzed.
have been formulated Repairable items are often sold with free replacedifferent types of warranties into groupings. repaired or replaced by a new item. ment warranty (FRW) with failed items either minimally Nonrepairable items sre usually sold with either prorata warranty (PRW) with a refund or free replacement warranty with failed items replaced by new items. A variety of models have been developed to obtain the expected warranty cost per unit, and reviews of these can be found in (12-141. In this paper, we consider FRW with minimal repair only. The outline of the paper is ES follows. In Section 2, we give the details of the model formulation, including quality variations, the effects of lot sizing and life testing on the quality of released items, and the warranty policy. Section 3 deals with the optimal selection of the decision variables (lot size, number of items tested and the duration of testing), the manufacturing cost and the warranty cost for repairable items covered by FRW with minimal repair. A numerical example is given in Section 4. We conclude with topics for further research in Section 5.
2. MODEL In this section,
we give the details
FORMULATION
of the model formulation.
Lot Sizing
2.1.The Manufacturing
37
Process
The process is in-control at the start of production of each lot. If the process is in-control at the start of the production of an item, then during the production, it can go out-of-control with probability (1 - q) or stay in-control with probability q. Once out-of-control, it stays outof-control until the end of the lot. After the lot is completed, the process is checked and set to in-control if it is out-of-control. If the process is in-control at the end of the production of an item, the probability that the item produced is conforming [nonconforming] is Br [(l - &)I. If the process is out-of-control at the end of the production of an item, the probability that the item produced is conforming [nonconforming] is 02 [(l - es)]. W e assume 131> 02, so that there is a higher probability of obtaining conforming items when the process is in-control. The number of conforming items in each lot is random. Items are characterized by their time to failure. Suppose that Fr(t)[F~(t)] denotes the failure distribution function for conforming [nonconforming] items. Let fj@)
Fj(t) = 1 - q(t),
= y,
q(t) = _fjo Fj(t)
denote the density function, survivor function, and failure rate associated with Fj (t) for j = 1,2. The inferiority of nonconforming items to conforming items is characterized by assuming that nonconforming items have a higher failure rate than conforming items:
Thus, Fz(t) > Fl(t) fort L 0, and 00 s0
F,(t)
dt >
00 F,(t) dt s0
(that is, the mean time to failure for nonconforming forming items). 2.2. Lot Sizing and Inspection
items is much smaller than that for con-
Policy
We assume that there is a constant demand (per unit time) for the product. This demand is met by producing items in lots of size L. We assume that the time horizon is sufficiently large so that it can be approximated as being infinite. Hence, we consider the asymptotic case where the number of lots goes to infinity. Let C’s denote the cost of checking the process at the end of a lot and q the cost of bringing the process to in-control if it is out-of-control. The cost to manufacture a lot is comprised of this fixed setup cost and a variable cost depending on lot size. Thus, the manufacturing cost for lot i, c,(L)i, is given by C,(L), = cs + $i?j + C,L, where C,,, is the material and labor cost to produce a single item and +i = 1 if the setup for lot i involves bringing the process to in-control and $Q = 0 if no change is involved. (& is random.) The inspection scheme is as follows. If the process is in-control at the end of the production of a lot, then all items in the lot are released for sale untested. If the process is out-of-control, the last K items are life tested for a duration of time T. Items which fail during the test are scrapped and those which survive are released for sale with the untested items. For lot i, let Ei denote the number of items which fail the test. If the process is in-control at the end of production, then ei = 0. The number of items released for sale from lot i is (L - Q). If G(T) and C,, denote the testing and scrapping cost per item, respectively, then, with $i as before, the inspection cost for lot i, Cinsp(K,T)iy is given by
I. DJAMALUDIN
38
et at.
2.3. Warranty Policy
We confine attention to nonrenewing failure free warranty (FRW) policy with a warranty period W. Under this policy, the manufacturer agrees to either repair or replace (by a new item) should failure occur within the warranty period W subsequent to the sale, with no cost to the consumer. The repaired (or replacement) item is covered by the remainder of the original warranty. We shall assume that failed items are repairable and the manufacturer always repairs items which fail under warranty. The failure characterization of repaired items depends on the type of repair. (See [13] for different types of repair.) Here, we assume that all failures are rectified by minimal repair (see, for example, [15]). U n d er minimal repair, the failure rate after repair is the same as before failure. We assume that the cost of each repair is CR and includes the handling cost for each warranty claim. Let &v(L, K, T)i denote the total warranty cost to service items released for sale from lot i. As well, we assume the following: (a) All failures under warranty result in warranty claims. (b) A claim is made ss soon as an item fails and all claims are valid. (c) The time to repair is relatively short compared to the mean time between failures so that it can be treated as zero. (d) The inventory holding cost is ignored. formulation.)
(One can easily incorporate
this into the model
(e) The lot size is bounded above by Lu (to reflect physical limitations).
3. ANALYSIS
OF MODEL
For the model given in Section 2, we wish to choose the lot size (L), the number of items tested (K), and the burn-in time (2’) which gives the lowest cost. If TC(L, K,T)i denotes the total cost for manufacturing, life testing (if needed), and servicing of items released from lot i for sale, then TC(L, KYT)i = Cp(L)i + Ci”sp(KyT)i + Cw(L, KYT)i
fori= 1,2,..., I, where I denotes the number of lots produced. The total cost and the number of items released for sale for all lots are given by cf, TC(L, K, T)i and ‘&(L -Q), respectively, with the cost per item released given by their ratio. As this is a random cost, we consider the asymptotic csse by taking the limit in probability of this ratio as I --) 00. Suppose that J(L, K, T) denotes this asymptotic cost per item released. Since the lots are statistically similar, it is easily shown that J(L, K, T) is given by J(L K T) = E [TC(L, K,T)il 9 7
L - E[E~]
= r(L, K, T) + Cw(L, K, T)
(1)
where E [&Lfi -G,K,T)
=
is the sum of the asymptotic manufacturing
+ Gnap(KT T)i]
(L - ENI and testing costs per item released, while
is the asymptotic warranty cost per item released. J(L, K,T) is obviously nonrandom. The optimization problem is to find L, K and T which minimize J(L, K, T) subject to the constraints 1LL
Lot Sizing
3.1. Asymptotic
Manufacturing
39
Cost
In this section, we derive an expression for r(L, K, 2’). To do so, we need to characterize the number of nonconforming items in a lot and the output quality. Let Ni denote the number of items produced in lot i (of size L) while the process is in-control, so that the number of items produced with the process out-of-control is (L - Ni). Ni has probability function given by qn(l-q) O
ENI =
(1
”
q)
(1- q=).
Let p1((t) [4(t)] denote the failure distribution function for a conforming [nonconforming] item which survives the test and is released for use. It is easily seen that j$+) = R(t
3
for j = 1,2. The failure density fj(t)
Fjm1 Fj CT)
+ T) -
and the failure rate +j((t) are then given by fj(t) = hct + T, Fj (T)
and 4(t)
=
fjct+T) = rj(t+T) Fj(t+T)
for j = 1,2. The number of items scrapped in lot i is random, with expected value E[Q] = PI(T)EKL +pz(T){KPr(Ni
(4
I L - 1) - EKL},
where E[NJ is given by (3)) Pr{Ni I L - 1) = Pr{Gi = 1) = 1_ Pr{qi = 0) = I _ q~, E
{QL-K-QL}
-KQ= 7
and pi(T) [p2(T)]is the probability that an item produced while the process is in-control [outof-control] fails the test, with
Pj(T) = ejJ’l(T) + (l- 0,) J’2(T) for j = 1,2. The derivation is similar to that in [lo]. From the analysis carried out so far, is easily shown that E [c&Q
+ &,(KT)i]
= [LG
+ Cs + (I-
and so, with E[Q] given by (4), the manufacturing 7(L
T)
K
7
,
=
qL) (77+ KG(T)}
+
EhlGc] ,
and testing cost per unit released is given by
LCm+ Cs + (1- q=){rl + KG(T)) L - E[Q]
+
EhlGc
(6)
I. DJAMALUDIN et al.
40
3.2. Asymptotic
Warranty Cost
We need to consider conforming and nonconforming items which are tested and not tested separately to evaluate the warranty cost per item released. The expected number of conforming items not tested and for sale is given by &{E(Ni)-
EKL}+@~{L-E(N~)-K(~
-Q~)+EKL},
(7)
and the expected number of tested conforming items which survive and are released for sale is given by
9”)
B~F~(T)EKL+~~~~~(T){K(~
-EKL}.
(8)
Similarly, the expected number of nonconforming items not tested and for sale is given by (7) with& and82replacedby(1-01) and(l-82), respectively, and the expected number of tested nonconforming items which survive and are released for sale is given by (8) with 8i,& and pi(T) replaced by (1 - @I), (1 - 02) and Fz (T), respectively. As a result, the expected warranty cost for lot i is given by E
pW(L,K,T),]
= CR[%(W)E(&)
+'~2(W)tL-E(4)1
(9)
+ {~(T,W)-?'I(W))EKL +
{Pz(T,W)
-yz(W)}{K(l
-qL)
-EKL}]~
where E[&] is given by (3), EKL is given by (5) and W 'yj(W)=
f3j w n(t)& J0
+
(1 -
f3j)
7.2(t)dt J
0
and w +1(t) dt + (1 - ej)Fz(T)
pj(T, W) = $Fi(T) J
for j = 1,2. The asymptotic using (4) and (9).
0
1”
Pz(t) dt,
0
warranty cost per item released, Cw(L, K, T), is given by (2),
3.3. Optimal Solution The asymptotic total cost per item, J(L, K,T), is given by (1) with y(L, K,T) given by (6) and C&L, K, T) given by (2) using (9) and (4). Let L’, K* and T* be the optimal values of L, K and T which minimize J(L, K,T) subject to the constraint 0 < L 5 L~J, 0 5 K 5 L and T 2 0. It is impossible to find analytical expressions for L*, K* and T*. As a result, a numerical procedure is needed to obtain them. The authors are currently researching into analytical results for (i) K* to take on the two extreme values, i.e., K* = 0 or L*; (ii) O
4. NUMERICAL
EXAMPLE
In this section, an example is given. We assume that the failure distributions for both conforming and nonconforming items are exponential with parameters r$r and 42, respectively, (42 > 41) and that Lu = 100 (the upper bound of L). The nominal values of the parameters are given in Table 1. The cost parameters are in dollars ($) and oz in dollars/year. The mean time to failure is ten years for a conforming item and 0.1 years for a nonconforming item. We consider four different values for the warranty period W, ranging from 1 to 4 years. We also consider the case W = 0, which corresponds to the product being sold with no warranty.
Lot Sizing
41
Table 1. Nominal values for parameters.
Cs
Cm
C,
al
a2
C,,
CR
500
10
100
1.0
1.0
0.5
3.0
Q
e1
e2
41
42
0.95
0.95
0.15
0.1
10.0
4.1.Optimal Values The optimal values for L, K,T and the asymptotic total cost per released item (that is, L*,K*,T* and J(L*,K*,T*)) were obtained by evaluating J(L, K, T) for L = 1, . . . , Lu, K = 1 7‘.‘7 L, T incremented in steps of 0.001 from 0 to 1, and for W equal to 0, 1,2,3and 4. We also consider two other cases: Suppose that the lot size is not a variable and is set equal to Lu (the upper bound). In this case, the quality can be improved only through K and T. Let KG and T; denote the optimal number of items tested and the optimal duration of the life testing and J(Lu, KG, Tc) the corresponding
asymptotic cost per item released.
When no testing is employed, then K = 0 and T = 0. In this case, the only way to improve quality is through lot sizing. Let Lf, denote the optimal L under this condition and let J(Lt , 0,O) denote the corresponding asymptotic cost per item released. (This case corresponds to the model formulation of Djamaludin et al. [lo].) Let RC1, RC2 and RC3 denote the following percentage reductions in the costs: RCl
RC2
=
=
loo[J(W,O) - J(L*,K*,T*)l J(Lu,O,O)
loo[JWW)
’
- J(Lu~K;~T;)l ’
J(Lu,O,O) and
Table 2 shows L’, K*, T’, K&,T;, totic costs per item released. For vri = 0,l
Lf, along with RC1, RCp., RC3 and the corresponding asymp-
(year), no lot sizing (that is, L’ = Lu) or testing is carried out for the three
cases. However, as W increases from 2 to 4, the number of items tested and the testing time increase. A longer warranty period increases the warranty costs for any nonconforming items released and so, more items are tested and for longer to decrease the expected number of released nonconforming items. The optimal J( L*, K*, T*) increases as the warranty period increases. For the cases where lot sizing or testing are carried, the asymptotic cost per released item is less than the appropriate value of J(Lu, 0,0), as should be expected. The percentage reductions (RG, RC2, RC 3 ) sh ow the advantage of using lot sizing and testing, just testing, and just lot sizing over using neither lot sizing or testing. From Table 2, it can be seen that RC1 > RC2 and RC1 > RC3 and hence, the saving in cost from employing both life testing and lot sizing is greater than that from employing just lot sizing or just life testing. This is expected, since optimization over all variables will give at least as good a result ss optimization over a subset of all variables. In this example, RC2 > RC3 for W = 3,4. This will not always be the case, since RCs and RC3 depend upon the choice of LU (FMwell as the other parameter values). As Lu is increased, lot sizing is more likely to be important. All of the percentage reductions (RCI, RC2, RC 3) increase with W. The increases indicate that lot sizing and/or life testing become more effective as the warranty period increases.
I. DJAMALUDINet al.
42
Table 2. Optimal values and related costs. W ‘VW,
O*O)
0
1
2
3
4
15.994
37.051
58.109
79.166
100.223
100
43
42
43
L’
100
K’
0
0
29
41
43
T’
0.00
0.00
0.247
0.410
0.471
JCL’, K*,T*)
15.994
37.051
55.461
57.926
59.023
I
41.11 100 100 0.493 64.393 RC2 @Jo)
0.00
0.00
0.00
20.06
LC
100
100
41
27
21
.J(L& O,O)
15.994
37.051
56.093
70.327
82.426
RC3 (rn)
0.00
0.00
3.47
11.17
17.76
35.75
]
For the case of just life testing, there is a sudden change in strategy from no testing (for W = 2) to almost 100% testing (for W = 3). This is due (in part) to the nature of the geometric distribution which governs when the process goes out-of-control and which has a “heavy” tail. For this example (Q = 0.95), the distribution of the number of items produced with the process in-control has a median of about 12 and the 95th percentile is about 57. As a result, it is more likely that most of the (Lu = 100) items are produced while the process is out-of-control. Hence, if it is sensible to test some items produced while the process is out-of-control, then it is probably sensible to test most of the items. 4.2. Sensitivity Analysis The sensitivity of the values of L*, K* , T‘ and J(L* , K*, T*) to the model parameters is discussed in this section. Tables and plots of these can be found in [17]. It is important to observe that lot sizing applies to all lots, whereas life testing is only carried out if the process is found to be out-of-control at the end of the production of a lot. In addition, the model for N, the number of items produced when the process is in-control, is a geometric distribution, which is very “flat.” This results in a high variability for N, which affects the choice of the optimal strategy. Since a longer warranty period implies a higher warranty cost, the optimal cost per item released (J(L*, K*, T*)) increases as the warranty period (W) increases. Consequently, the optimal lot size (L’) tends to decrease initially from L* = La with no testing, until 100% testing occurs, after which L’ tends to increase. As well, the number of items tested (K*) and the testing period (T’) start at zero (that is, no items are tested) and then increase as W increases, with K’ increasing to L*. For larger W, the warranty cost is much greater for nonconforming items and so it is more important to remove any nonconforming items produced before they are sold. Since lot sizing is less costly than life testing, lot sizing is used first to remove nonconforming items. As W increases and it becomes more critical to remove these items, life testing is also used. Life testing is more effective at removing nonconforming items produced while the process is out-of-control, and can be made still more so by increasing T*. Hence, once 100% testing is being carried out, L* can be increased ss W increases in order to compensate for the higher warranty cost. Note that, as W increases, the changes in strategy below are more sudden, since it becomes more imperative to remove the nonconforming items before sale. For a fixed warranty period (W), the optimal cost per item released decreases as the quality (that is, the expected number of conforming items) increases. This happens if any of q (the
Lot Sizing
43
probability the process stays in-control during the production of an item), 81 (the probability of a conforming item while the process is in-control) and 0z (the probability of a conforming item while the process is out-of-control) increase. In general, lot sizing and life testing are used if (a) the process spends enough time in-control and enough time out-of-control, (b) there is enough difference between the proportion of nonconforming items produced while the process is in-control and the proportion of nonconforming items produced while the process is out-of-control, or (c) there are reasonable proportions of conforming and nonconforming items produced. Consequently, for “low” q, no lot sizing is used (no lot can be reduced enough to remove all items produced while the process is out-of-control) and no life testing (life testing would leave very few items to be released) is carried out. (This is partly due to the high variability of N.) As q increases, it is worthwhile lot sizing as there will be some items produced while the process is in-control. However, little or no life testing should be carried out (for the nominal values of 191= 0.95 and 132= 0.15), due to the high variability of N and the undesirability of removing too many items when many are produced while the process is out-of-control. As q increases still further, lot sizing is still beneficial and life testing becomes more important. This occurs because there is a reasonable number of items which are produced while the process is in-control and it is worth removing those produced while the process is out-of-control. Life testing increases up to 100% as q increases since it is only done for those lots for which the process goes out-of-control. For large q, there is less benefit from lot sizing as all the items in most lots should be produced with the process in-control (and so, they should be conforming). Again, since life testing is only done for those lots for which the process goes out-of-control and this occurs rarely, 100% life testing is worth carrying out on all items. If 8i is “low” or 82 is “high” (with q set at the nominal value of 0.95), there is not a great deal of difference between what happens when the process is in-control and when it is out-of-control. Consequently, lot sizing is not effective. However, if there are reasonable proportions of both conforming and nonconforming items, 100% life testing should be used; otherwise, life testing is less effective and so few items should be tested. As 81 increases (02 decreases), lot sizing becomes more effective and life testing can become less effective (depending on the value of q). Further discussion of the effects of the other parameters can be found in [16]. 5. EXTENSIONS The model discussed in this paper can be extended in several ways. Some of the following are currently being investigated by the authors. 1. We have discussed only the FRW policy with minimal repair. This approach can be used to study other types of policies, such as pro-rata or combination warranties, in a similar manner. 2. We assumed a testing scheme where the last K items are tested if the process at the end of the lot production is out-of-control. If the items are numbered sequentially, then an alternate scheme for testing is as follows. If the process at the end is out-of-control, test item L - KI, with K1 < K, for a period T. If it fails, test all items from L - K1 to L (i.e., test only the last K1 items in the lot). If not, test all items numbered from L - K to L (i.e., test the last K items in the lot). Many other testing schemes can be formulated. 3. We have assumed that inspection at the end of a lot production reveals the true state of the process. An interesting extension is where the true state is either unknown or known with uncertainty. 4. The change in process can be modelled differently, such as the process deteriorating gradually as opposed to a sudden change.
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