Sampling Risks for Chain Sampling Plans with

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					 c Heldermann Verlag                                                    Economic Quality Control
ISSN 0940-5151                                                    Vol 22 (2007), No. 1, 117 – 126




         Sampling Risks for Chain Sampling Plans with
             Non-Constant Defective Probability

                     A.R. Sudamani Ramaswamy and A.R. Manju Priya



Abstract: This paper introduces modified producer ’s risk and consumer’s risk in the case of chain
sampling plans obtained by modelling the defective probability p not as a constant, but as a random
variable that follows a beta distribution.

Keywords: Acceptable Quality Level, Chain Sampling, Lot Tolerance Proportion Defective.


1        Introduction
The concept of chain sampling plans was introduced by Dodge [2] in 1955, to overcome the
problem of lack of discrimination for single sampling plans with zero acceptance number c = 0.
The procedure was developed to “chain” together the most recent inspections in a way that
would build up the shoulder of the operating characteristic (OC)curve of c = 0 plans. This is
especially desirable in a situation in which small samples are demanded because of economic or
physical difficulties for obtaining a sample.

The procedure of ChSP-1 is as follows:

    1.   From each lot, select a sample of n units.

    2.   Accept the lot, if

         (a)   no defects are found in the sample, or
         (b)   one defective is found in the sample, but no defective item was found in the previous
               i samples of n.

Soundararajan [3] has presented procedures and tables for the construction of chain sampling
plans and for selection of plans by specified properties under the conditions of the poisson model.
The design of chain sampling plan is based on the producer’s risk α and the consumer’s risk β.
Traditionally, α is defined as the probability of rejecting a lot in which the defective probability,
p, equals acceptable quality level (AQL) specified by the producer.
    α = P r(X > c | p = AQL)                                                                     (1)

Similarly, β is defined as the probability of accepting a lot in which p is equal to the lot tolerance
proportion defective (LTPD):
    β = P r(X ≤ c | p = LT P D)                                                                  (2)
118                                              A.R. Sudamani Ramaswamy and A.R. Manju Priya



The definitions of α and β are based on the assumption that incoming lots are formed from a
production process that is stable with a constant defective probability p. Chun and Rinks [1]
have assumed that the defective probability p is a random variable that follows beta distribution
and derived modified producer’s risk and consumer’s risk α and β for single sampling inspec-
tion plan. According to Chun and Rinks [1] the beta distribution represents the variations in
the defective probability p better than any other distribution. The intent of this paper is to
derive modified producer’s risk and consumer’s risk for chain sampling plans, assuming that the
defective probability is a random variable that follows beta distribution. The producer’s risk
and consumer’s risk displayed in Table 1 and Table 2 are derived on the assumption of perfect
inspection even though errors are inevitable in any inspection process.




2     Modified Producer’s and Consumer’s Risk
If p is considered constant, then for a chain sampling plan given by (n, i) the probability of
accepting a lot P a(p) is obtained by means of the binomial distribution:
    P a(p) = (1 − p)n + (np)(1 − p)n−1 (1 − p)ni                                             (3)

The producer’s risk α is given by
    α = 1 − P a(AQL) = 1 − E[P a(AQL)] = E[1 − P a(AQL)]
       = E[1 − P a(p) | p = AQL]                                                             (4)

Following the representation (4) Chun and Rinks [1] defined the modified producer’s risk α when
p is modelled by a random variable as the conditional expectation as follows:
    α = E[1 − P a(p) | 0 ≤ p ≤ AQL]
    ˜                                                                                        (5)

                             ˜
The modified consumer’s risk β is defined analogously as follows:
  ˜
  β = E[P a(p) | LT P D ≤ p ≤ 1]                                                             (6)

When p is modelled as a constant α and β for a chain sampling plan with parameters n and i
are calculated using the binomial distribution:
    α = 1 − (1 − AQL)n + nAQL(1 − AQL)n−1 (1 − AQL)ni                                        (7)
                          n                          n−1              ni
    β = (1 − LT P D) + nLT P D(1 − LT P D)                 (1 − LT P D)                      (8)

Let f (p | a, b) be the density function of the beta distribution with parameters a and b. Then
the following relation holds:
                     pa−1 (1 − p)b−1
    f (p | a, b) =                       for 0 < p < 1                                       (9)
                        B(a, b)
                                Γ[a]Γ[b]
       where         B(a, b) =
                                Γ[a + b]

Moreover, the incomplete beta function is given by:
Sampling Risks for Chain Sampling Plans with Non-Constant Defective Probability                                          119


                            y
                1
  Iy (a, b) =                   pa−1 (1 − p)b−1 dp                                                                      (10)
              B(a, b)
                        0



Proposition 1:
                                              ˜
Let A = AQL, then the modified producer’s risk α for a chain sampling plan with parameters
n and i is given by:
             B(a, n+b) IA (a, n+b)    B(a+1, n+ni+b−1) IA (a+1, n+ni+b−1)
  α=1−
  ˜                                +n                                                                                   (11)
              B(a, b)    IA (a, b)         B(a, b)           IA (a, b)

                                ˜
and the modified consumer’s risk β is given by:
  ˜ B(a, n+b) 1−IL (a, n+b) + n B(a+1, n+ni+b−1) 1−IL (a+1, n+ni+b−1)
  β=                                                                                                                    (12)
        B(a, b)   1−IL (a, b)             B(a, b)      1 − IL (a, b)


Proof:
                    ˜
The producer’s risk α is defined as:
  α = 1 − E[P a(p) | p ≤ A]
  ˜                                                                                                                     (13)

For a chain sampling plan the conditional expectation based on the binomial sampling model is
given as:
  E[P a(p) | p ≤ A] = E q n + npq n−1 q ni | p ≤ A
                        = E [q n | p ≤ A] + nE pq n−1 q ni | p ≤ A                                                      (14)

If p has beta distribution with parameters a and b then the conditional expectation becomes:
                                  A          a−1 b−1
                                                               A                 a−1 b−1
                                      q n p B(a,b) dp
                                               q
                                                                   pq n+ni−1 p B(a,b) dp
                                                                                  q
                                  0
  E[P a(p) | p ≤ A] =                 A
                                                            + n0        A
                                           pa−1 q b−1                       pa−1 q b−1
                                            B(a,b) dp                        B(a,b) dp
                                   0                                    0
                                  A                            A
                                          pa−1 q n+b−1             pa q n+ni+b−2
                                            B(a,b)     dp              B(a,b)    dp
                                  0
                        =             A
                                                            + n0   A
                                           pa−1 q b−1                  pa−1 q b−1
                                            B(a,b) dp                   B(a,b) dp
                                      0                            0
                                 B(a, n+b) IA (a, n+b)
                        =                              +
                                   B(a, b)   IA (a, b)
                                   B(a+1, n+ni+b−1) IA (a+1, n+ni+b−1)
                                 n                                                                                      (15)
                                           B(a, b)        IA (a, b)

Thus from equations (14) and (15) the modified producer’s risk (11) is obtained.

                            ˜
The modified consumer’s risk β as given by (12) is obtained analogously.

Corollary 1:
                                                                                         1−A
For a chain sampling plan with parameters n and i, let k =                                A    for a given acceptable quality
120                                                 A.R. Sudamani Ramaswamy and A.R. Manju Priya



                                                 ˜
level A = AQL. then the modified producer’s risks α converges to the classical producer’s risk
α as a → ∞ and b = ka:
       ˜
   lim α = α
  a→∞
                                                                                                           (16)
  b=ka


where a and b are the parameters of the considered beta distribution of p.


Proof:
                                                                                                   1−A
Consider the beta distribution with parameters a and b = ka for a fixed value k =                    A .   With
(7) we will show that
                B(a, n + b)
         lim
         a→∞
                            = (1 − A)n                                                                     (17)
         b=ka
                 B(a, b)
         and
                B(a + 1, n + ni + b − 1)
         lim
         a→∞
                                         = A(1 − A)n−1 (1 − A)ni                                           (18)
         b=ka
                        B(a, b)

Consider
       B(a, n + b)        Γ[a]Γ[n + b] Γ[a + b]
  lim
  a→∞
                     lim
                   = a→∞                                                                                   (19)
  b=ka
         B(a, b)     b=ka
                          Γ[a + n + b] Γ[a]Γ[b]

and
           Γ[a + b] Γ[n + b]                               Γ[a + b]                  (b + n − 1) · · · bΓ[b]
   lim                             =     lim
  a→∞
  b=ka
         Γ[a + b + n] Γ[b]             a→∞
                                       b=ka
                                               (a + b + n − 1) · · · (a + b)Γ[a + b]         Γ[b]
                                                    (b + n − 1) · · · b
                                   =     lim
                                       a→∞
                                       b=ka
                                               (a + b + n − 1) · · · (a + b)
                                              (ka + n − 1) · · · ka
                                   =     lim
                                     a→∞ (a + ka + n − 1) · · · (a + ka)

                                               k + n−1 · · · k
                                                    a
                                   = lim
                                     a→∞ k + 1 + n−1 · · · (k + 1)
                                                   a
                                                                 n                    n
                                          kn              k                      1
                                   =            =                    =     1−                              (20)
                                       (k + 1)n          k+1                    k+1

Analogously we obtain:
                                                                     n−1                  ni
         B(a + 1, n + ni + b − 1)          1                1                     1
  lim
  a→∞
                                  =                   1−                    1−                             (21)
  b=ka
                 B(a, b)                  k+1              k+1                   k+1

For b = ka = 1−A a we obtain from the beta distribution:
              A
           a
  E[p] =       =A                                                                                          (22)
         a+b
                  1−A
Thus, for k =      A    in (20) and (21) the limiting value is given by:
                B(a, n + b)
         lim
         a→∞
                            = (1 − A)n                                                                     (23)
         b=ka
                 B(a, b)
                B(a + 1, n + ni + b − 1)
         lim
         a→∞
                                         = A(1 − A)n−1 (1 − A)ni                                           (24)
         b=ka
                        B(a, b)
Sampling Risks for Chain Sampling Plans with Non-Constant Defective Probability                     121



                            ˜
Thus, the limiting value of α is obtained:
                            B(a, n + b)          B(a + 1, n + ni + b − 1)
    lim α = 1 − a→∞
    a→∞
        ˜       lim                     − n a→∞
                                            lim
    b=ka             b=ka
                             B(a, b)        b=ka
                                                         B(a, b)
           = 1 − (1 − A)n + nA(1 − A)n−1 (1 − A)ni
           = α                                                                                      (25)

                                        1−L
In the same way, we obtain for k =       L    with L = LTPD the limit:
        ˜
    lim β = β                                                                                       (26)
    a→∞
    b=ka




3     Numerical Examples for the Modified Producer’s Risk
                            ˜
In Table 1 the values of α for various chain sampling plans are displayed . The defective
probability p is modelled by a beta distribution with parameters ((a, b). For comparison the
case of a constant defective probability with p = AQL is also included. Four different AQL
                                         ˜
values are considered. Comparing the α values in case of variable p with that of constant the
case, one can see that the risk is maximum when p is constant. Moreover, it is revealed that the
                          ˜
modified producer’s risk α has its lowest value when p has a beta distribution with parameter
(1, 1). Below the considered cases for the sampling plan, the beta distribution and the AQL are
listed.

                            sampling plan       n             10
                            parameters          i         1,2,3,4,5,6
                            beta distribution (a, b) (1,1), (2,18), (4,36)
                            parameters
                            AQL:                A 0.005, 0.025, 0.050, 0.150




4     Numerical examples for the Modified Consumer’s Risk
                                                            ˜
In Table 2 values for the modified consumer’s risk β for chain sampling plans with n = 10 and
i = 1, 2, . . . , 6 are displayed for the case that the defective probability is modelled by various beta
distributions. The considered LTPD values are set equal to 0.10, 0.15 and 0.20. Comparing the
        ˜
values β of the modified risks for the different beta distributions shows that the risk is maximum
for the case of a constant defective probability p.
122                                         A.R. Sudamani Ramaswamy and A.R. Manju Priya



                                         ˜
        Table 1: Modified Producer’s Risk α for Chain Sampling Plans with n = 10.

                                        Average Quality Level(AQL)
                    i Betaparameters 0.005     0.025    0.050    0.150
                           (a,b)        Modified Producer’s Risk, α
                    1      (1,1)    0.001160 0.024679 0.081225 0.361239
                          (2,18)    0.001715 0.034487 0.105681 0.356257
                          (4,36)    0.002283 0.045504 0.137535 0.423776
                         constant   0.003432 0.069135 0.212586 0.734726
                    2      (1,1)    0.001923 0.038141 0.116344 0.426683
                          (2,18)    0.002841 0.053044 0.150438 0.427417
                          (4,36)    0.003775 0.069521 0.193477 0.502790
                         constant   0.005654 0.103700 0.288295 0.789660
                    3      (1,1)    0.002658 0.049362 0.141066 0.455137
                          (2,18)    0.003922 0.068337 0.181371 0.459815
                          (4,36)    0.005205 0.089000 0.230819 0.536124
                         constant   0.007768 0.130534 0.333625 0.800474
                    4      (1,1)    0.003367 0.058740 0.158691 0.469281
                          (2,18)    0.004961 0.080967 0.202975 0.476258
                          (4,36)    0.006575 0.104817 0.255919 0.551552
                         constant   0.009779 0.151366 0.360766 0.802604
                    5      (1,1)    0.004049 0.066601 0.171425 0.477155
                          (2,18)    0.005960 0.091420 0.218231 0.485392
                          (4,36)    0.007888 0.117679 0.272918 0.559294
                         constant   0.011691 0.167538 0.377016 0.803023
                    6      (1,1)    0.004707 0.073210 0.180754 0.481947
                          (2,18)    0.006920 0.100091 0.229130 0.490849
                          (4,36)    0.009146 0.128153 0.284522 0.563455
                         constant   0.013509 0.180093 0.386745 0.803105




5     Numerical Examples for Sample Size and Producer’s Risk
For designing a chain sampling plan, the sample size n and the number i are specified for
attaining a certain level of the two risks. Table 3 shows the sample size n required to obtain
                     ˜
the producer’s risk α = 0.05 for given i = 1, 2, . . . , 6. As in Chun and Rinks [1], we observe
that the required sample size increases with decreasing AQL. Again, the defective probability is
modelled by beta distributions with parameters (1, 1), (2, 18) or (4, 36) or that p is a constant
equal to the AQL value.
Sampling Risks for Chain Sampling Plans with Non-Constant Defective Probability             123



         Table 2: Modified Consumer’s Risk for Chain Sampling Plans with n = 10.

                        i Beta parameters           LTPD
                                (a,b)       0.10     0.15     0.20
                        1       (1,1)     0.041347 0.022240 0.011477
                               (2,18)     0.269898 0.150922 0.079159
                               (4,36)     0.316003 0.181848 0.096393
                             constant     0.483764 0.265274 0.136197
                        2       (1,1)     0.033724 0.018429 0.009878
                               (2,18)     0.216980 0.121555 0.066042
                               (4,36)     0.253804 0.145553 0.079714
                             constant     0.395780 0.210340 0.110469
                        3       (1,1)     0.032199 0.017973 0.009770
                               (2,18)     0.204084 0.117319 0.064986
                               (4,36)     0.237794 0.139962 0.078275
                             constant     0.365102 0.199526 0.107706
                        4       (1,1)     0.031833 0.017909 0.009762
                               (2,18)     0.200662 0.116664 0.064896
                               (4,36)     0.233416 0.139060 0.078146
                             constant     0.354405 0.197396 0.107410
                        5       (1,1)     0.031736 0.017900 0.009761
                               (2,18)     0.199704 0.116558 0.064888
                               (4,36)     0.232167 0.138910 0.078134
                             constant     0.350675 0.196977 0.107378
                        6       (1,1)     0.031709 0.017898 0.009761
                               (2,18)     0.199426 0.116540 0.064887
                               (4,36)     0.231800 0.138850 0.078133
                             constant     0.349375 0.196895 0.107375



Example:

Consider the case of AQL=0.005 and i = 2. For a constant p, the chain sampling plan with
the sample size n = 32 gives a producer’s risk of α = 0.05. If p is assumed to follow a beta
distribution with parameters (4, 36), then n should be 40 to achieve the same value of the
producer’s risk.

If p is assumed to be a beta random variable with parameters (2, 18), then the sample size value
must be increased to n = 47 to achieve the same value of the producer’s risk.If p is assumed
to follow beta distribution with parameters (1, 1) the sample size should be even larger, namely
n = 58.

Thus we can see that when the variance of p increases with the mean fixed at a certain value,
the sample size n for a fixed value of i has to be increased in order to guarantee the same level
of the producer’s risk.
124                                         A.R. Sudamani Ramaswamy and A.R. Manju Priya



                                                                                    ˜
Table 3: Sample Size Requirements for chain sampling plans and a producer’s risk of α = 0.05.

                                        Average Quality Level (AQL)
                       i Betaparameters 0.005 0.025 0.050   0.100
                              (a,b)         Required sample size n
                       1      (1,1)       74    14    7        3
                             (2,18)       59    12    6        3
                             (4,36)       51    10    5        3
                            constant      41     8    4        2
                       2      (1,1)       58    11    5        2
                             (2,18)       47    9     5        2
                             (4,36)       40    8     4        2
                            constant      32     6    3        1
                       3      (1,1)       50    10    5        2
                             (2,18)       40    8     4        2
                             (4,36)       34    7     3        2
                            constant      27     5    2        1
                       4      (1,1)       45    9     4        2
                             (2,18)       36    7     3        2
                             (4,36)       30    6     3        1
                            constant      24     4    2        1
                       5      (1,1)       41    8     4        2
                             (2,18)       33    6     3        1
                             (4,36)       28    5     2        1
                            constant      22     4    2        1
                       6      (1,1)       38    7     3        1
                             (2,18)       30    6     3        1
                             (4,36)       26    5     2        1
                            constant      21     4    2        1




6     Numerical Examples for Sample Size and Consumer’s Risk
                                                                      ˜
Table 4 shows the sample size n required to achieve a consumer’s risk β ≈ 0.10 without exceeding
it. A The LTPD values are taken as 0.10,0.15 and 0.20, which are the same as in Chun and
Rinks [1]. From the table it is seen that required sample size decreases with increasing LTPD
value for i = 1, 2, . . . , 6.


Example:
Consider a LTPD=0.15 and i = 2, then n = 15 is necessary for achieving β ≈ 0.10 when p is a
constant.

When p is a random variable following a beta distribution, the required sample size is smaller
than in the case of a constant defective probability. For illustration let LTPD=0.15 and i = 2,
then a sample size n = 12 is necessary for the beta random variable with parameters (4, 36) and
                              ˜
n = 11 for (2, 18) to achieve β ≈ 0.10.
Sampling Risks for Chain Sampling Plans with Non-Constant Defective Probability             125



From the numerical results we conclude that, as the variance of p increases, the required sample
size n gets smaller for achieving a fixed consumer’s risk.


    Table 4: Sample Size Requirements for Chain Sampling Plans and a Consumer’s Risk of
                                          ˜
                                         β ≈ 0.10.
                          i Beta parameters          LTPD
                                  (a,b)     0.10 0.15    0.20
                                            Required sample size n
                          1       (1,1)      7    6       5
                                 (2,18)      17 12        10
                                 (4,36)      18 13        10
                               constant      24 16        12
                          2       (1,1)      6    5       4
                                 (2,18)      15 11        9
                                 (4,36)      17 12        10
                               constant      23 15        11
                          3       (1,1)      6    5       4
                                 (2,18)      15 11        9
                                 (4,36)      17 12        10
                               constant      22 15        11
                          4       (1,1)      6    5       4
                                 (2,18)      15 11        9
                                 (4,36)      17 12        10
                               constant      22 15        11
                          5       (1,1)      6    5       4
                                 (2,18)      15 11        9
                                 (4,36)      17 12        10
                               constant      22 15        11
                          6       (1,1)      6    5       4
                                 (2,18)      15 11        9
                                 (4,36)      17 12        10
                               constant      22 15        11




7     Conclusion
In the design of acceptance sampling plan , random variations in the defective probability p
must be taken appropriately accounted for. If this is done, it is shown by means of a chain
sampling plan that the resulting modified producer’s risk and consumer’s differ considerably
from the classical producer’s risk and consumer’s risk. The related problem can be solved by
reducing the variations in p.
126                                       A.R. Sudamani Ramaswamy and A.R. Manju Priya



References
[1]   Chun, Y.H. and Rinks, D.B. (1998): Three Types Of Producer’s and Consumer’s Risks in
      the Single Sampling Plan. Journal of Quality Technology 30, 254-268.

[2]   Dodge, H.F.(1955): Chain Sampling Inspection Plans. Industrial Quality Control 11, 10-
      13.

[3]   Soundararajan, V.(1978): Producer’s and Tables for the Construction and Selection Of
      Chain Sampling Plans (ChSP-1), Part-1. Journal of Quality Technology 10, 56-60.



A.R. Sudamani Ramaswamy                  A.R. Manju Priya
Department of Mathematics                PSG College of Technology
Avinashilingam Deemed University         Coimbatore-641004
Coimbatore-641043                        Tamilnadu
Tamilnadu                                India
India