Dodge, H. F.

Stephens, K. S.

English

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The Statistics Center
RUTG_S--THE STATE UNIVERSITY
CHAIN SAMPLING INSPECTION PLANS--ChSP-0,4 AND ChSP-1,4
by
K. S. Stephens
GPO PRICE $ and
CFSTI PRICE(S) $- H. F. Dodge
Hard copy (HC) _/! _)_ -
Microfiche (MF)_
ff 653 July 65
TECHNICAL REPORT NO. N-24
February, 1966
Research supported by the Army, Navy, Air Force, an__ under Office of Naval
Research Contract No. Nonr 404(18) Task No. NR 042-241 w_th Rutgers--The State
University. Repeoduction In whole or in part is permitted for any purpose of
the United States Government.
N66 3797-7(ACCESSION NUMBER) |TH RU)
==o /
i0/--
(NASA GR OR I'MX OR AD NUMBER) ((:ATE )
_!
1966028637
https://ntrs.nasa.gov/search.jsp?R=19660028637 2020-03-16T17:28:10+00:00Z
_ne Statistics Center TECHNICAL R_PORT NO. N-24
RUTGERS--THE STATE UNIVERSITY Issue 1 - Page 1
CHAIN SAMPLING INSPECTION PLANS--ChSP-0,4AND CHSP-1,4
by
K. S. Stephens*
and
H. F. Dodge
--m
INTRODUCTION
This report extends the work on two-stage chain sampling inspection
plans to those included in the subsets designated ChSP-0,4 and Chsp-1,4;
i.e., plans with acceptance numbers of c1,C2 = 0,4 and 1,4. The format is
the same as the preceding Technical Report No. N-221, which presented similar
information on the subsets, ChSP-O,3 and ChSP-1,3. Earlier reports, Nos.
N-202 and N-213, contain the general description of the plans and the
theoretical development as well as results on the subsets, ChSP-O,1, and
ChSP-0,2, ChSP-1,2, respectively.
EVALUATING OPERATING CHARACTERISTICS
In this section Markov chains are presented for the following five
sampling plans selected from those that have been evaluated:
Plan kl, h; Cl' C2
(i) I, 2; O, 4
(2) 1, 2; 1, 4
(3) 1, 3; 0, 4
(4) 2, 3; O, 4
(5) 2, 3; I, 4
This report is based in part on work being done in preparation of a doctoral
thesis at Rutgers--The State University.
f
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TECHNICAL REPORT NO. N-24
Issue 1 - Page 2
An algebraic solution for Pa, the proportion of lots expected to be
accepted, is given for plan (1) by solving the Markov chain for the limiting
probabil_ y of the rejection (R) state.
Plan (I): I12 _ Or4
State at ith trial
0 1 2 3 4 R
0
Po I 4 i= I
i Po Pl P2 P3 " i- _0 pi= i
State 2 P0 P1 P2 " " 1 - _0Pii
at
(i-1)st
trial 3 P0 Pl " • • 1 - oPii=
4 Po .... I - PO
R Po .... I - PO
Fig. I. Markov Chain for ChSP-O,4 Plan: (kl,k2; CI,C 2 = 1,2; 0,4)
Proceeding as in former reports 1,2, the solution of the limiting
probability of state "R" from wb_tch _a is obtained is as follows:
PO + PI + P2 + e3 _ e4 + eR = I (I)
% = Po (2)
el=PFo PFIPie2Pie3(z)
% = Poeo  PFI* P_% (4)
es =%% +P_l (5)
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TECHNICAL RI_ORT NO. N-24
Issue I - Page 3
P4= P4% (6:_
Combining (2), (3), (4), and (5) results in the followlng:
PoPI+PoPlPz'PoPlP2P 3
P1 = (7)
1-PI-P2-P1P3+PlP2P3
Combining (I), (3), (6), and (7) leads to,
Po+PQP3-PoP2P3 (8)
eR = 1 - POP4 - I-PI-P2-PIPS+PIP2P3
from which we get,
_a = (Po+PoP3+PoP4+PoPIP4 + PoP2P3 + PoP2P4 + PoPIPzP4-PoPIP2P3P4)/
( 1-Pl-pz-PlP3+plp2p3 ) (9)
Plan (2): lr2; 1L4
State at ith trial
0 1 2 3 4 R
4
0 PO P1 P2 P3 P4 1 - E Pi=0 t
I Po Pl P2 P3 " 1 - l=0Pl
Stateat 2 Po Pl P2 " " 1 - t_=0pl
(l-l)st 1
trial 3 Po Pl " " " 1 - Z=oPii
4 PO .... 1 - PO
R Po Pl " " " 1 - _oPt£
Fig. 2. Markov Chain for ChSP-I,4 Plan: (kl,k2; Cl,C 2 = 1,2; 1,4)
/
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TECHNICAL REPORT NO. N-24
Issue 1 -Page 4
Plan (3): 1,3_ 014
State at tth trial
O0 Ol I0 II 02 20 12 21 22 03 30 13 31 04 40 RO R
,m ..
O0 PO PI " " P2 .... P3 " " " P4 " • I -I_=oPf
01 " " PO Pl " " P2 .... P3 .... 1-i___oPi
Po Pl P2 P3 - _0 p!0 ............ I 1= t
II • " Po Pl " " P2 ......... I-i_oPt
Po P1 P2 - _oPt02 .............. 1 t=
20 Po Pl " " P2 ........... I - _oPli=
12 ..... PO" Pl ........ 1-t_oPt_
21 PO Pl ' 1 i_oPt
State 22 ..... PO .......... 1 - PO
at i_oP i(i-1)st 03 .......... PO " P1 " " • 1- =
trial i___opt30 PO P1 ............... 1 - m
13 .......... PO ...... 1 - PO
31 • " Po ............. I - PO
04 .............. PO " 1 - PO
40 PO ............... 1 - PO
RO PO Pl " " P2 .... P3 " " " P4 " • 1 -t___oPt
R "............. ' " " Po I - PO
Fig. 3. Markov Chain for ChSP-0,4 Plan: (kl,k2; CI,C 2 = 1,3; 0,4)
1966028637-005
TECHNICAL REPORT NO. N-24
Issue 1 - Page 6
Plan 44): 2r3t; 0L4
State at lth trial
O0 O1 10 11 02 20 12 21 22 03 30 13 31 04 40 RO R
4
O0 Po Pl " " P2 .... P3 " " " P4 " " 1 -i_oPl
• " PO P1 " " P2 .... P3 ..... I -i _-Pl=O-
O1
10 Po P1 " " P2 .... P3 ...... 1 -i_oPi_
Po PI P2 -'_oPl11 ............. I i=
02 ..... Po " P1P2....... I -ioPi=
Po PI " " P2 ........... I -i_oPi=
20
..... Po " Pl ........ 1 -l_oPl=
12
State 1
21 " " Po P1 ............ 1 -i_oPi=
at
(i-1)st L
trial 22 ...... PO ........... 1 - P3
.......... PO " Pl " " " 1 -i _-Pi=O
O3
1
30 Po Pl " I i_oPi
13 .......... PO ..... 1 - PO
31 " " PO ............. 1 - PO
04 .............. Po " 1 - Po
40 Po I Po
RO Po ............... 1 -- Po
R ............... Po 1 - Po
Fig. 4. Markov Chain for ChBP-O,4 Plan: (kl,k2; Cl,C2 = 2,3; 0,4)
/
1966028637-006
((
TECHNICAL REPORT NO. N-24
Issue 1 - Page 6
Plan (5): 2_3; It4
State at tth t-ial
00 01 I0 II 02 20 12 21 22 03 30 13 31 04 40 R0 R1 R
4
00 P0 Pl " " P2 .... P3 " " " P4 " " • 1 -i_=0Pi
P0 P1 " " -i_0Pl01 .... P2 " " " P3 .... 1 =
P0 P1 P2 P3 - _0Pi10 ............. 1 i=
ii " " P0 P1 " " P2 .......... 1 - _0Pii =
02 ..... P0 " Pl P2 ........ I -1_,0Pi.=
20 P0 P1 " " P2 ............ 1 -i_oPi=
12 ..... P0 " P1 ......... I -i__0Pi_
I
• " P0 Pl ............. 1 -i_0Pi
21
State =
at 22 ..... P0 ........... 1 - P0(i-1)st
trial
03 .......... Po " P1 .... 1 -1_,0Pi.=
30 P0 Pl 1 -1_0Pi
13 .......... P0 ...... 1 - P0
31 " " P0 .............. 1 - P0
04 .............. P0 " • 1 - P0
40 PO ................ 1 - Po
1
RO PO Pl ............... 1 -i_=oP l
R1 " " Po ............... 1 - PO
1
R ............... P0 Pl 1 - Z Pli=O
Fig. 5. Markov Chain for ChSP-I,4 Plan: (kl,k2; CI,C 2 = 2,3; 1,4)
1966028637-007
TECHNICAL REPORT NO. N-24
Issue I - Page 7
DISCUSSION OF OPERATING CHARACTERISTICS
Again, as in N-22 I, operating characteristlc_ for four different
sample sizes (n = 10,20,50 and 100) for a wide range of ChSP-O,4 and
ChSP-I,4 sampling plans are pcesented in the Appendix . The OC curves
for each sample size are presented on a single page. Each of Figs. 6,7,8
and 9 contains three sets of OC curves, one for each of three values of
k2(k 2 = 2,3 and 5). The OC curvew are of Type B*, based oJl probabilities of
sampling from a process.
The OC curves for ChSP-O,4 and ChSP-I,4 are completely analogous to
those of ChSP-0,3 and ChSP-1,3 as well as ChSP-0,2 and ChSP-I,2 presented
in Technical Report Nos. M-221 and N-212 respectively. Again there are
close similarities in the effects of changes in the parameters kl, k2, and
n; hence they will only be summarized here.
Effect of k 1
H, .
This is shown for each k 2 within each of the individual charts e.g.,
Figs. 6.1, 6.2 and 6.3 for n = 10. The range in k 1 is normally from 0 to
k2-1. However the curve for k 1 = 0 has been shown for k 2 = 2 only. These
curves consistently result in poor discrJ.mlnation between good and bad
quality. The k 1 = 1, k 2 = 5 curve for C1 = 1, C2 = 4 has also been omitted.
Increasing k 1 has the effect of tightening (i.e. loweriug) the GC curves,
again more so for ChSP-0,4 than for ChSP-1,4.
See Reference (4), pp. 56-60.
/
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TECHNICAL R_PORT NO. N-24
Issue 1 -Page 6
Effect of k 2
This effect is shown between the three charts on a single page for
each _, e.g. between Figs. 6.1, 6.2 and 6.3 for n = 10. _hree values of
k 2 i.e., k 2 = 2,3 and 5 are shown. In general, increasing k 2 also has the
effect of tightening the OC curves.
Effect of n
Again as in ChSP-0,2 and 1,2 as well as ChSP-0,3 and 1,3, there
is a close similarity in the patterns of the ChSP-0,4 and 1,4 plans for
different sample sizes as can be seen among Firs. 6, 7, 8 and 9. In
presentlng the OC curves for n = 100 in Fig. 9, a double horizontal scale,
fraction defective, p, add number defectiCe, pn, has again been used.
For binomial sampling the_e OC curves provide suitabl_ approximations for
sample sizes of n = 50 and larger by using the lower ov pn scale. For
Poisson sampling Fig_ 9 #_ applicable for all sample _s.
COMPARISONOF ChKr-0,4, ChflP-1,4AND OTK_R rh_.i' PLANS
: On each of the charts o_ the four figures _r_ shown the Ch_-0,3, -"
ChSP-0,2 and ChSP-O,1 plans having the stated c_t.-values of k 2 and having
k 1 = k2-1. In each case these curves are show_ by solid lines and are the
la_ three numbered curves of each set, ChSP-0,3, ChBP-0,2 and Ch_-0,1
re:spectively. Also shown on each chart, with a thin dashed line, is the
OC curve for the single sampling plan: given sample _ize, n, and c ffi 0.
This is again shown for reference purposes.
These new charts indicate that the ChSP-O,4 plans are even more
effective l_ adding a swelling on the underlying OC curve of the c = 0 plan
than the corresp_nding 0,3, 0,2 and 0,1 plans. How this swelling increases
by going successively from a 0,1 to a 0,2 and to a 0,3 and now to a 0,4 plan,
1966028637-009
TECHNICAL RKPORT NO. N-24
Issue 1 - Page 9
can be seen, for example on each chart by comparing the plans having k 1'=
k2-1. The improvement in the "operating ratio," O.R., is again roughly
comparable to that obtained in single sampling by going to a larger
acceptance number, c; iv this case from c = 1, c = 2 and c = 3 to c = 4.
The relationship between ChSP-I,4 and ChSP-1,3 (See Reference 1) is
shown graphically for one case only--n = 100, k 1 = 1, k 2 = 2 in Fig. 9.1.
REFERENCE8
1. Stephens, K. 8. and H. F. Dodge, **Chain Sampling Inspection Plans--ChSP-0,3
and Ch_-l,3, "Rutgers University St_tlstics Center, Technical
Report No. N-22, December, 1965.
2° Dodge, H. F. and K° S. Stephens, "A General Family of Chain Sampling
Inspection Plans," Rutger_ University Statistics Center,
Technical Report No. N-20, December 30, 1964.
3. 8tephens, K. S° and H. F° Dodge, "Chain Sampling Inspection Plans--ChSP-0,2
and ChaP-I,2," Rutgers University Statistics Center, Technical
Report No. N-21, May_ 1965.
4° Dodge, H. F° and" H. G. Romig, Sampling InspectionTabl-s, John
Wiley and Sons, Inc., New York, 2nd Edltiol_, 1959.
/
1966028637-010
TECHNICAL REPORT NO. N-24
Issue 1 - Page 10
Appendix: Operating Characteristic Curves for ChSP-Ot4 and ChSP-lt4 Plans
OC curves for the following chain sampling plans are presented here.
Figs. 6-9 give OC curves for s,mple sizes, 10, 20, 50 and 100 respectively.
pigs. 6.1-9.1 Fi_s. 6.2-9.2 Fi_s. 6.3-9.3
kl k2 C1 C2 kl k2 C1 C2 kl k2 C1 C2
0 2 0 4 1 3 1 4 2 5 1 4
1 2 1 4 2 3 1 4 4 5 1 4
1 2 1 3* 1 3 0 4 1 5 0 4
1 2 0 4 2 3 0 4 2 5 0 4
1 2 0 3** 2 3 0 3** 4 5 0 4
1 2 0 2** 2 3 0 2** 4 5 0 3**
1 2 0 1"* 2 3 0 1"* 4 5 0 2**
4 5 0 1"*
* Thls ChSP-I,3 plan is presented in Fig._l only for n = I00 in order to
show the typical effect of changing from a 1,3 to a 1,4 plan.
** These CbSP-O,3, _2 and 0,i plans along with the single sampling plans
for c = 0 are presented for comparison with the ChSP-0,4 and ChSP-I,4
plans.
1966028637-011
TI_CHNICAL REPORT NO. N-24
_tttttlft _44,,_,++*
,4_, ,_ ,, :IIi+*
'''{_'_"*t''_ * *'_:+'*_1-+ '-4+'i'-t*_%,I 3 1 2 0 4 t _
T:Tl::tt.t':lT',_,j"'_t '7_t t W,tt_il:ttff:_fN "' '
i .',21111r' 2111::i
"22.1 . ..i. ,2 ._2;'i:_" _4,
................. ; _ L.fli LLL iI!+_-L!4; . ;.g_i;
o
0 .05 I0 .15 .20 .25 .30 .35 .40
Fraction Defy{tire, p
1966028637-012
1.00 _/_
i_i -_
• bO _
.+ T.4o :2,+ _,
0
0 .02 .0,1 .06 .08 .10 .12 .14 .16
Fl-a(ttt,n D_iecti_c, p
1.00
:'£g. 7.3, k 2 = 5
Curve
n # kl kL CI C2
.80 20 1 2 5 _ 4
2 4 5 I 4
3 i 5 0 4
• 60 4 2 _ 0 4
5 4 5 0 4
Pa
6 4 5 0 3
7 4 5 0 2
.,o ::: i'::: ::i::t! 't: ii_.;Ii ,_
4 _0
........ : :;"[";**I_I _+_+t............. :, _ , 'it
0
o .02 , o I .06 .08 . I0 ,12 .14 .16
Fraction Deto(ttve, p
t'IG. 7 OC CUR\ ES- .- n - 20, k 2 = 2, 3 AND 5
1966028637-01:3

.2O
0
0 ,005 .010 .015 .020 .025 .030 .035 ,040
Fraction Defective, p
0 ,5 1.0 1.5 2,0 2.5 3.0 3.5 4.0
pn
1966028637-015


Chain sampling inspection plans - ChSP-0,4 AND ChSP-1, 4 Technical report no. n-24

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