Biometrika Trust
Serial Sampling Acceptance Schemes for Large Batches of Items where the Mean Quality
has a Normal Prior Distribution
Author(s): G. E. G. Campling
Source: Biometrika, Vol. 55, No. 2 (Jul., 1968), pp. 393-399
Published by: Oxford University Press on behalf of Biometrika Trust
Stable URL: https://www.jstor.org/stable/2334882
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Biometrika
(1968),
55,
2,
p.
393
393
Printed in Great Britain
Serial sampling acceptance schemes for
large batches of items where the mean quality has
a normal prior distribution
BY G. E. G. CAMPLING
Brighton College of Technology
SUMMARY
Often a positive correlation exists between the quality of sequentially produced batches
of items. The present paper considers whether, for a model where the mean batch quality
is assumed to have a normal prior distribution, an estimate of this correlation and informa-
tion on some or all of the preceding batches can be used advantageously when designing
an acceptance sampling scheme.
1. INTRODUCTION
Most of the theories discussed in papers on acceptance sampling techniques consider
only single-batch sentencing rules even though a positive correlation may exist between
sequentially produced batches of items; see, however, Cox (1960). The present paper sets
out to show whether this correlation can be used to obtain a more worthwhile acceptance
sampling scheme by employing sentencing rules which take into account information
relating to the quality of some or all of the preceding batches.
2. THE MODEL
2@ 1. General statements
Suppose that random samples from sequentially produced batches of items are subjected
to inspection in order that the mean quality of the products in each batch can be estimated.
For the general batch Bj denote the qualities of the items sampled by xjl, ... , xjn and the
true mean quality of the batch by inj. There are two possible decisions on each batch, i.e.
to accept it or to reject it. Let the value of batch Bj be k'(mj - nO), where k' is measured in
monetary units and m0 is the break-even quality level. The cost of sampling each item is
taken to be constant and equal to the unit of cost. The variance of the observations is
denoted by T2 and the unknown mean mj is assumed to have a continuous prior distribut
N(,t + up, wj2).
If we write yj = in1-, f1 = X- t, t2= _ 2/n, then yj has the prior distribution N(uj, w
and fj is conditionally N(y, t2), so that yj has the posterior distribution N(v, Vj2) where
V. = (t2Uj + wj2fj)/(t2 + w2), V2 = w2t2/(w2 + t2). (1)
We note here that if (y, v) are jointly normal variates and y has the marginal distribution
N(u, W2) and if, given v, y is N(v, V2), then it follows that v has the marginal distribution
N(u, 02), where
02
=
W2_
V2.
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(2)
394 G. E. G. CAMPLING
Without loss of generality let mo = 0. The
accept batch B1 if and only if vj +4 > 0. (3)
Since the return on batch B; is, conditionally on v;,
- n + k' max (0, v; + It),
the expected utility is given by
U*(n) =-n + k'E{max (0, vj + ,u)}. (4)
We shall discuss the formulation of single batch and serial sampling schemes and consider the circumstances in which it is advantageous to use serial schemes.
2*2. Single-batch sampling scheme
In this scheme no attention is paid to the observed qualities of previous batches. Putting
uj = 0 and wj2 = C2, we have that the return on batch Bj is
-n + k'O m'ax (o ~j+ 1) = - n +k'oi max(o4 it+ )
ii ( i t j)i (
where g is N(O, 1), since vj is N (u1, O1).
U*(n) =-n + k'Oj E(max (O, + g) (5)
= n +io (n + z) , exp (-_ Z2) dz
-n + k'Oj{(u/10j) (D (jt/10) + q (u/10j)}, (6)
where 02= wj2- VI = nC4 (-2+ no2),
and 56(x) and 4D(x) are the standard normal density and distribution function respectively.
This expression for the expected utility assumes that the correct decision rule is used.
It is easily seen that v- = no2(Zj- b)/ (T2 2+nC2) (7)
Wetherill & Campling (1966, ?2.1) have shown that for an arbitrary decision rule of the
form accept batch B1 if and only if zj > d, (8)
the expected utility is given by
U*(n) = - n + k',tAD(A) + k1'o2{n/(r2 + no-2)}k 0S(A), (9)
where A = (It - d) {n/(r2 + no-2)}1 and, optimally,
d = - (,ur2)/(nO-2). (10)
It is obvious that (6) and (9) are identical if (3), (7) and (10) are satisfied.
2*3. The serial sampling scheme
Let
where
yi
zj
=
is
ry11+z1,
N(O,
oaS)
and
(11)
is
indepen
then var (y-) = o-2/(l-r2), cov(yx., y1) = ro-2/1(lr2). (12)
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Serial sampling acceptance schemes for large batches of items 395
In a single-batch sampling scheme it is natural to use this stationary distribution as the
prior distribution for y1. In general let Yj-, have the prior distribution N(uj-, wj2-,) an
f-1L and fj be the observations from Bj11 and Bj, conditionally independent. Then fjconditionally N(yj1, t2) and, givenf11, - is N(v1_1, V_L); see (1). Thus the prior distri
tion for yj is N(uj, wj), where
uj = rvj-1, w2 = r2VI_3i+ o2. (13)
Take as the ith sampling scheme on batch Bj that scheme whe
Bj involves xj, . . ., x>_i+l, i.e. the 'memory' is of length i. Now for
distribution N{O, o2/(l - r2)} and yj has a posterior distribution N
meters vj and Vj are obtained by i applications of (1) and (13).
Now if L2is the posterior variance of y1 when using the ith sampl
then from (1), if o.2 = o2/(l -r2),
2 = t20-2/(t2 + a-2) = _r2c2/(r2+ nco2) (14)
and, from (13),
- t2(r2L2+ x2)/(t2+ r2LD + o02). (15)
If
0-
=o2-
L,
(16)
then, by (2), the argument of ?2*2 shows that the utility associated with scheme i is simply
U* (n) =-n + I'jt(D/iO ) + k'OsO(#//0j). (17)
If r = 0, i.e. there is no correlation between batches, then LD = L1, since o2 = o2, as we
would expect, and (17) reduces to (6), i.e. the case where i = 1.
If now i = 2, it is easily seen from (14)-(17) that
U2*(n) =-n + k'#t(ID(,u/O2) + k'020q(It/02), (18)
where 02 = {no4/( 1-r2)} [{t2 + r272 + no2}/{(72 + no2) 2 _r2T4}].
This expression for the utility assumes that all parameter values are known exactly.
In practice this is unlikely and it is therefore of interest to obtain an expression for the
utility when incorrect parameter values are used. Suppose that the following decision rule
is employed:
accept batch Bj if and only if Xj + AxT_j > D. (19)
The correct decision rule is given when
A - rr2/(72?+ no2)
D =- -(/t7r2)/(nO) + {jtrT2(rr2 + no-2)}/{no2(72 + no2)}. (20)
In terms of this arbitrary decision rule,
U2*(n) =-n + k'uJ(C) + {k'o2(j + rA)}/{T(l - r2)} 0(C) (21)
where C = {,u(l + A )-D}/T,
2 = {o.2/(l -r2)} (1 +rA)2+ (r2/n) (1 +A2) +A2o2,
with the true parameter values inserted.
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396 G. E. G. CAMPLING
Suppose now that i -* xo, so that results on a large number of previous batches are avail-
able. From (14) and (15) it follows that {LJ} is strictly decreasing and has a limit L given by
-2 = t2(r2L2 + o2)/(t2+ r2L2 + -2). (22)
There are, in fact, two limits but only this one is relevant. For
values see Bather (1963).
For all i, the expected utility, U*, if the batch is accepted or rejected without samp
so that n = 0 and hence Oi = 0, is, from (17),
UO = t (It ),} (23)
The absolute maximum to the expected utility is obtained for perfect information
without cost, with n = 0, Li = 0, and is
U*ax = k',uJ)(#/4o) + k'4(jt/lo). (24)
2*4. Numerical results
Table 1 gives the optimum sample sizes and expected utilities for various sampling
schemes assuming the stated parameter values. The optimum sample sizes were obtained
by calculating the utilities for sample sizes 1, 2, 3,... and selecting that size for which
the utility was a maximum; it can be seen that, with respect to sample size, the value of the
expected utility usually increases, passes through a maximum, then decreases steadily.
The terms L2 given in the table are the posterior variances of yj and it is easily verified from
the preceding sections that
Di = (S2-r2)/(r2
+ nS2), (25)
21~ ~ ~ ~ ~ ~~~~~~~~(5
where S2 = 2 in the single batch scheme (i = 1) and S2 = r2L2 + 02 in the all-batches-back
scheme (i = oo). Note from Table 1 that as r2 increases, with the other parameter values
fixed, the utility achieved by any sampling scheme decreases while the gain produced by
employing serial schemes increases; this result can be verified by expressions (14)-(17).
Table 2 shows the effect on the sample size and utility of errors in the estimation of r;
the parameter values have been chosen so that differences in utilities are relatively large;
for cases where r2 is approximately equal to 0.2, the differences are much smaller.
3. CoNCLusIoNs
The following general conclusions can be drawn.
(i) If r2 is approximately equal to o.2, the economic advantages, in terms of expected
utility, of using a serial sampling scheme rather than a single-batch sampling scheme
seem to be negligible, for the type of model considered here, unless the correlation, r, is
high, say greater than 0 90. There may, however, be some reduction in optimum sample
sizes.
If (r2/cr2) is large, say greater than 5, the economic advantages as well as optimum sample
size reductions may be considerable; see Table 1.
Similar comments can be applied to the posterior variance, LD. If r2 is approximately equal
to 0.2, the posterior variances resulting for each of the sampling schemes are nearly equal.
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Sertial sampling acceptance schemes for large batches of items 397
Table 1. Optimum sample sizes and expected utilities for a
variety of parameter values (0-2 = 1)
The value a* is that sample size giving the same posterior variance, Li, as that for all-batches-back scheme.
Optimum sampling plans
Parameter
k'
values
One
batch
All
batches
back
Single-batch scheme back scheme scheme
Uo
*
'---N
1
r , 1000 72 1000 n n U*(n) L n U*(n) a U* (n) L U*
0 70 0 1 1 0 14 13 371-64 0.07 13 372 48 13 372 48 0 07 398-94
10 39 37 317-16 *21 33 323-14 32 323-50 -21
100 93 75 186-31 *57 67 208-89 62 215-00 -52
5 1 0 31 31 1,933-39 03 30 1,934-29 30 1,934-29 03 1,994-71
10 93 92 1,803-46 -10 86 1,811-27 85 1,811-40 -10
100 260 244 1,436-88 *29 212 1,485-84 205 1,490-89 -28
10 1 0 44 44 3,903-04 *02 43 3,903-96 43 3,903-96 02 3,989-42
10 135 134 3,716-55 -07 126 3,724-86 126 3,724-93 *07
100 387 374 3,171-67 -21 329 3,231-37 323 3,235-07 -21
0-70 2 1 1 2 4 3 2,000-06 -25 3 2,000-64 3 2,000-68 -22 2,008-49
10 - - 2,000-00 - - 2,000-00 - 2,000-00
100 - - 2,000-00 - - 2,000-00 - 2,000-00
-
5 1 10 10 10 10,021-93 -09 9 10,022-71 9 10,022-72 -09 10,042-46
10 - - 10,000-00 - - 10,000-00 - 10,000-00 100 - - 10,000-00 - - 10,000-00 - 10,000-00 10 1 20 15 15 20,055-82 -06 14 20,056-66 14 20,056-67 -06 20,084-91
10 35 - 20,000-00 - 29 20,006-40 29 20,006-80 -22
100 - - 20,000-00 - - 20,000-00 - 20,000-00 0-90
0
1
1
0
15 13 371-64 -07 11 374-52 11 374-72 -07 398-94
10 45 37 317-16 -21 29 331-42 25 335-72 -18
100 133 75 186-31 -57 65 225-05 47 254-58 -43
5 1 0 31 31 1,933-39 -03 28 1,936-96 27 1,937-03 -03 1,994-71
10 98 92 1,803-46 -10 74 1,827-99 70 1,830-91 -09
100 317 244 1,436-88 -29 192 1,540-65 153 1,583-47 -24
10 1 0 44 44 3,903-04 -02 40 3,906-81 40 3,906-85 -02 3,989-42
10 139 134 3,716-55 -07 111 3,745-22 108 3,747-29 -07
100 426 374 3,171-67 -21 291 3,314-24 246 3,357-25 -19
0-90 2 1 1 2 4 3 2,000-06 -25 3 2,001-39 2 2,001-81 -21 2,008-49
10
-
-
2,000-00
-
-
2,000-00
-
2,000-00
-
100 - - 2,000-00 - - 2,000-00 - 2,000-00 -
5 1 10 11 10 10,021-93 -09 8 10,024-47 8 10,024-73 -08 10,042-46
10 35 - 10,000-00 - - 10,000-00 12 10,001-06 -28
100 - - 10,000-00 - - 10,000-00 - 10,000-00 -
10 1 20 16 15 20,055-82 -06 12 20,058-78 12 20,058-98 -06 20,084-91
10 43 - 20,000-00 - 27 20,014-05 23 20,018-38 -19
100 - - 20,000-00 - - 20,000-00 - 20,000-00 -
0-99
0
1
1
0
21 13 371-64 -07 10 378-62 6 383-56 -05 398-94
10 81 37 317-16 -21 28 338-65 12 363-98 -11
100 317 75 186-31 -57 65 234-03 25 323-28 -24
5 1 0 39 31 1,933-39 -03 22 1,947-38 17 1,953-27 *03 1,994-71
10 157 92 1,803-46 -10 67 1,853-45 36 1,895-68 -06
100 614 244 1,436-88 -29 187 1,580-56 75 1,773-11 -14
10 1 0 51 44 3,903-04 -02 32 3,921-19 26 3,926-88 -02 3,989-42
10 190 134 3,716-55 -07 97 3,786-26 57 3,835-76 -05
100 809 374 3,171-67 -21 280 3,386-53 121 3,639-80 -11
0-99 2 1 1 2 8 3 2,000-06 -25 3 2,002-08 1 2,004-62 -12 2,008-49
10 15 - 2,000-00 - - 2,000-00 2 2,000-83 -26
100
-
-
2,000-00
-
-
2,000-00
-
2,000-00
-
5 1 10 16 10 10,021-93 -09 7 10,027-31 4 10,031-76 -06 10,042-46
10 61 - 10,000-00 - - 10,000-00 8 10,018-47 -14
100 - - 10,000-00 - - 10,000-00 - 10,000-00 -
10 1 20 22 15 20,055-82 -06 11 20,063-49 7 20,068-64 -04 20,084-91
10 81 - 20,000-00 - 28 20,020-95 13 20-046-73 -11
100 270 - 20,000-00 - - 20,000-00 19 20,008-30 -27
26
Biom.
55
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398 G. E. G. CAMPLING
Table 2. Sample sizes and utilities (minus 10
estimation of r, the true value of which is 0-40; A = 1, r2 = 50, ki = 10,000, o21
Estimated value One-batch-back scheme
of
r
A
n
0-10
195
U*
(n)
396-11
*20
194
398*32
-60
-70
180
172
394*98
383*36
*30 192 399*88
*40 189 400*49
*50 185 399-40
*80
*90
163
154
Optimum
356-84
298-84
single
batch plan 195 393-25
UO-
U.ax
-
0
833-16
When T2 iS large, the value of L2 for the all-batches-back scheme is considerably less than
that for the single batch scheme provided r is at least about 0 -90; see Table 1.
(ii) As T2 increases, with the other parameters held constant, the expected utility of any
sampling scheme decreases from the constant maximum expected utility while the gain
produced by serial schemes increases; see Table 1.
(iii) It seems that it is not essential to have very precise knowledge of the true value of the
correlation, r, for a serial scheme to be more advantageous than a single batch scheme.
In his unpublished Ph.D. thesis Campling, using asymptotic results, suggests a general
rule that, provided |,t/lo- is not greater than about 3, the true value of r must be gr
than or equal to 75 % of the estimated value. Table 2 suggests that it is more serious to o
estimate r than to underestimate it.
(iv) Cox (1960) considered serial sampling inspection schemes for a Poisson model where
the batch quality is assumed to be the realization of a simple Markov chain. An implication
in this paper (?7, paragraph 3) is that there is little to be gained by employing a sampling
scheme which considers observations on more than the past three batches; this conclusion
is confirmed here subject to the condition that r2 and o2 are approximately equal. In the
following paragraph of his paper, Cox suggests that if information relating to succeeding
batches as well as preceding batches is used in the sampling scheme, then this is more
advantageous than simply considering past batches. This point has not been examined for
the model discussed here and must be the subject of future work.
I am deeply indebted to Dr G. B. Wetherill for his valuable advice and guidance during
the preparation of this paper, which is part of my thesis submitted for the degree of Ph.D.
in the University of London. I should also like to thank the referee for helpful criticisms of
earlier versions of this paper.
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Serial sampling acceptance schemes for large batches of items 399
REFERENCES
BATHER, J. A. (1963). Control charts and the minimisation of costs. J. R. Statist. Soc. B 25, 49-80.
Cox, D. R. (1960). Serial sampling acceptance schemes derived from Bayes's Theorem. Technometrics 2, 353-60.
WETHERILL, G. B. & CAMPLING, G. E. G. (1966). The decision theory approach to sampling inspection. J. R. Statist. Soc. B 28, 381-416.
[Received April 1967. Revised January 1968]
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