Quality Control Designing of Variables Repetitive Group Sampling ...

Communications in Statistics—Simulation and Computation® , 34: 799–809, 2005 Copyright © Taylor & Francis, Inc. ISSN: 0361-0918 print/1532-4141 online DOI: 10.1081/SAC-200068424

Quality Control

Designing of Variables Repetitive Group Sampling Plan Involving Minimum Average Sample Number S. BALAMURALI, HEEKON PARK, CHI-HYUCK JUN, KWANG-JAE KIM, AND JAEWOOK LEE Division of Mechanical and Industrial Engineering, Pohang University of Science and Technology, Pohang, South Korea This article proposes the variables repetitive group sampling plan where the quality characteristic follows normal distribution or lognormal distribution and has upper or lower specification limit. The problem is formulated as a nonlinear programming problem where the objective function to be minimized is the average sample number and the constraints are related to lot acceptance probabilities at acceptable quality level (AQL) and limiting quality level (LQL) under the operating characteristic curve. Sampling plan tables are constructed for the selection of parameters indexed by AQL and LQL in the cases of known standard deviation and unknown standard deviation. It is shown that the proposed sampling plan significantly reduces the average sample number as compared with the single and double sampling plans. Keywords Average sample number; Compliance testing; Nonlinear programming; Sequential quadratic programming; Variables inspection. Mathematics Subject Classification 62P30; 62D05; 62L05.

1. Introduction Several sampling procedures are available in the literature of acceptance sampling for the application of attribute quality characteristics (see for example, Schilling, 1982). Suppose we have a lot of items of size N with fraction nonconforming, p which is unknown. The disposition of the lot is based on the inspection of sampled items drawn from that population. Suppose a random sample of size n is drawn and the number nonconforming items
d is observed in the sample. The number

Received March 12, 2004; Accepted February 18, 2005 Address correspondence to S. Balamurali, Division of Mechanical and Industrial Engineering, Pohang University of Science and Technology, Pohang 790 784, South Korea; E-mail: [email protected]

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of nonconforming items is then compared with a pre-determined number c. The lot is accepted if d ≤ c (p is small) and the lot is rejected (p is large) otherwise. If the decision about the lot is taken based on the inspection of only one sample, then the plan is called single sampling plan and if the decision is made from two samples then it is called double sampling plan. Further details about these sampling plans can be found in Duncan (1986). Repetitive group sampling (RGS) plan is one of the attribute sampling plans developed by Sherman (1965). The operation of this plan is similar to that of the sequential sampling plan. Sherman has pointed out that the RGS plan will give an intermediate in sample size efficiency between the single sampling plan and the sequential sampling plan. Variables sampling plans involve comparing a statistic, such as the mean, with an acceptance limit in much the same way that the number of nonconforming items is compared to an acceptance number in attributes plans. Whenever the quality characteristic of interest is measurable, a variables sampling plan can be applied. The main advantage of the variables sampling plan is that the same operating characteristic (OC) curve can be attained with a smaller sample size than would be required by an attributes sampling plan. Thus, a variables acceptance sampling plan would require less sampling. Also, when destructive testing is employed, the variables sampling is particularly useful in reducing the costs of inspection (Lieberman and Resnikoff, 1955). Another advantage is that measurements data usually provide more information about the manufacturing process or lot than do attributes data. Generally, numerical measurements of quality characteristics are more useful than simple classification of the item as conforming or nonconforming. It is also to be emphasized that when acceptable quality levels of a process are very small, the sample size required by attributes sampling plan is very large. Under these conditions, there may be significant advantages in switching to variables inspection. In this article, we consider the RGS plan for the measurable characteristics. As pointed out by Sherman (1965), the RGS plan is throwing some information away, but we will gain greatly in simplicity of design and operation and will pay only a modest price. Though the price paid for discarding the data is little bit more for variables sampling than the attributes sampling, this will be compensated with minimum inspection in terms of average sample number and this will be discussed in a separate section. An attempt is also made to design a variables repetitive group sampling (VRGS) plan involving minimum average sample number (ASN). For the selection of the parameters of the VRGS plan, the problem is formulated by nonlinear programming where the objective function to be minimized is the ASN and the constraints are related to lot acceptance probabilities at acceptable quality level (AQL) and limiting quality level (LQL). Tables are constructed for the VRGS plan parameters indexed by AQL and LQL. The advantages of this plan in terms of ASN are also discussed over the existing variables plans such as single and double sampling variables plans.

2. Variables Repetitive Group Sampling Plan The VRGS plan can be applied to the measurable characteristics under the conditions of normal distribution and lognormal distributions. Suppose that the quality characteristic of interest has the upper specification limit
U  and follows a

Variables RGS Plan

801

normal distribution with mean  and known standard deviation . In addition, the usual conditions for application of single sampling variables plans should also be valid. Then the following procedure of the VRGS plan is employed. Step 1: Take a random sample of size n , say
X1  X2      Xn  and compute
n ¯ v =
U − X , where  X = n1 i=1 Xi . 

Step 2: Accept the lot if v ≥ k2 and reject the lot if v < k1
k1 < k2 . If k1 ≤ v < k2 , then repeat Steps 1 and 2. It is also worthwhile to note that the same VRGS can be applied to the case where the quality characteristic follows a log-normal distribution since the logarithm transformation leads to a normal distribution. Suppose that the quality characteristic of interest follows a lognormal distribution. Let ˆ be an estimate of mean which is calculated from the logarithmic transformation of the data. Then the VRGS plan is employed as follows. Step 1:

Take a random sample of size n and compute v =

log
U−ˆ . 

Step 2: Accept the lot if v ≥ k2 and reject the lot if v < k1 . If k1 ≤ v < k2 , then repeat the Steps 1 and 2. Thus, the VRGS plan has the parameters of the sample size n , and the acceptable criterion k1 and k2 . Note that if k1 = k2 , then VRGS plan becomes variables single sampling plan. When the quality characteristic has the lower specification limit L instead of U , the same VRGS plan can be applied with the  ˆ modification in Step 1 as v =
X−L in the case of normal distribution and v = −log
L   in the case of lognormal distribution. If the quality characteristic has both the lower and upper specification limits, the design problem would be quite different and it will be left for a future study. If the population standard deviation  is unknown, then it is estimated from the sample by the sample standard deviation S. Hamaker (1979) used the following approximation for finding the parameters of unknown sigma single sampling variables plan.   k2 n 1+ 2

 nS =

(1)

In the case of double sampling variables plan, the following Wallis approximation is used to find the parameters of unknown sigma plans.   kt2 N  1+ 2

 NS =

(2)

Here N = n1 + n2 and kt is the rejection constant for the combined sample (see Sommers, 1981). However, the parameters of unknown sigma VRGS plan can be determined in a different manner which will be explained in Sec. 3. The unknown sigma VRGS plan is operated as a known sigma VRGS plan but the parameters are
nS  k1S , and k2S .

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3. Operating Characteristic Function of the VRGS Plan 3.1. Known Sigma Plan The fraction nonconforming in a lot will be defined as   U − p = PX > U   = 1 − 

(3)

where
yis given by
y =



y

−

 2 1 −z dz √ exp 2 2

(4)

Let Pa and Pr , respectively, be the probability of accepting and rejecting a lot based on a single sample under VRGS plan with parameters
n  k1  k2  when the fraction nonconforming is p. That is, Pa = Pv ≥ k2  p

(5)

Pr = Pv < k1  p

(6)

Therefore, the probability of repeating the sampling becomes R = Pk1 ≤ v ≤ k2  p = 1 − Pa − Pr 

(7)

Then, the probability of accepting a lot under the VRGS plan can be derived as L
p = Pa + RPa + R2 Pa + · · · =

Pa  Pa + P r

(8)

The probabilities in (5) and (6) can be expressed by Pa = PX ≤ U − k2   p =
w2  Pr = PX > U − k1   p = 1 −
w1  √ √   where w1 = U − − k1 n and w2 = U − − k2 n . So, the lot acceptance probability in (8) is given by L
p =


w2   1 −
w1  +
w2 

(9)

Whenever two points on the OC curve namely p1 , the acceptable quality level (AQL), and p2 , the limiting quality level (LQL), with corresponding producer’s and consumer’s risks  and , respectively, are specified, the following should hold: L
p1  =


w21  =1− 1 −
w11  +
w21 

(10)

L
p2  =


w22  = 1 −
w12  +
w22 

(11)

Variables RGS Plan

803

where √ √ w11 =
zp1 − k1  n  w21 =
zp1 − k2  n √ √ w12 =
zp2 − k1  n  w22 =
zp2 − k2  n

(12)

and zp is the standard normal variate having upper tail probability of p. 3.2. Unknown Sigma Plan Whenever the standard deviation is unknown, we should use the sample standard deviation S instead of . So, the determination of parameters
nS , k1S , k2S  is slightly different from the known sigma case. It is known that X ± k2S S is approximately 2 2 normally distributed with mean  ± k2S E
S and variance n + k2S Var
S (see S Duncan, 1986). That is,  2 2 2   X + k2S S ∼ N  + k2S  + k2S nS 2nS 

Then Pa can be written as 



 U − k2S  −   Pa = PX ≤ U − k2S S  p =   =
w2S  √ k2
/ nS  1 + 22S Similarly, Pr may be written as 



 U − k1S  −   1 − Pr = 1 − PX + k1S S > U  p =   =
w1S  √ k2
/ nS  1 + 21S where w1S =

 U − 

− k1S



nS 2 1+05k1S

and w2S =

 U − 

− k2S



nS 2 . 1+05k2S

Then the lot acceptance probability is given by L
p =


w2S   1 −
w1S  +
w2S 

(13)

Under the condition of the two points on the OC curve, the following should hold: L
p1  =


w21S  =1− 1 −
w11S  +
w21S 

(14)

L
p2  =


w22S  = 1 −
w12S  +
w22S 

(15)

where √ √ w11S =
zp1 − k1S  nS  w21S =
zp1 − k2S  nS √ √ w12S =
zp2 − k1S  nS  w22S =
zp2 − k2S  nS 

(16)

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We may find the design parameters
n  k1  k2  for known sigma plan by solving nonlinear equations in (10) and (11) and for the unknown sigma plan the parameters
nS  k1S  k2S  are determined by solving the nonlinear equations given in (14) and (15). However, there may exist multiple solutions since there are three unknowns while there are only two equations. A sampling plan would be desirable if the required number of units sampled is as small as possible. So, we introduce the average sample number (ASN) as the objective function to be minimized with the above two equations as constraints. A similar idea has been introduced in Kuralmani and Govindaraju (1995), where the problem of minimizing the sum of ASN at AQL and LQL for double sampling attributes plan was investigated.

4. Design of VRGS Plan for Minimizing Average Sample Number The average sample number (ASN), by definition, means the expected number of sampled units per lot used for making decisions. The concept of ASN is meaningful under Type B sampling situations. The ASN for the known sigma VRGS plan is calculated as follows when the fraction nonconforming is p. ASN
p = n
1 − R

  i=0


i + 1Ri =

n n = 1−R 1 −
w1  +
w2 

(17)

The ASN given in (17) can be used as an objective function to solve for the parameters
n  k1  k2 . Since the ASN is a function of p there are several choices in selecting the objective function. The usual choice is to minimize the ASN at AQL or to minimize the sum of ASN at both AQL and LQL. We will adopt the both cases and compare with each other. In either case the constraints will be lot acceptance probabilities given in (10) and (11). We rather use inequalities in the following formulation to increase the feasibility region. If the objective is to minimize the ASN at AQL, then the problem will be reduced to the following nonlinear optimization problem:   1 (18) Min ASN
p1  = n n k1 k2 1 −
w11  +
w21  s.t.  L
p1  =


w21  1 −
w11  +
w21 

 ≥1−

(18a)

≤

(18b)

n ≥ 2 k1 ≥ 0 k2 ≥ 0 k2 > k1 

(18c)

 L
p2  =


w22  1 −
w12  +
w22 



On the other hand, if the objective is to minimize the sum of ASN at AQL and LQL, then the following nonlinear optimization problem should be considered:   1 1 + Min ASN
p1  + ASN
p2  = n n k1 k2 1 −
w11  +
w21  1 −
w12  +
w22  (19)

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805

s.t. 


w21  1 −
w11  +
w21 

 ≥1−

(19a)

≤

(19b)

n ≥ 2 k1 ≥ 0 k2 ≥ 0 k2 > k1 

(19c)

L
p1  =  L
p2  =


w22  1 −
w12  +
w22 



To determine the parameters
nS  k1S  k2S  of the unknown sigma plan, similar kind of nonlinear optimization problem can be proposed, but
w11S  w12S  w21S and w22S  are used in the places of
w11  w12  w21 and w22 , respectively. To solve the above nonlinear optimization problems given in (18) and (19), the sequential quadratic programming (SQP) proposed by Nocedal and Wright (1999) can be used. The SQP is implemented in Matlab Software using the routine “fmincon”. By solving the nonlinear problem mentioned above, the parameters
n  k1  k2  are determined. We assume that  = 005 and = 01 for all cases. These parametric values are tabulated in Table A1 for specified values of p1 and p2 . Similarly, the parameters of unknown standard deviation VRGS plans are determined for the same combinations of p1 and p2 and these values are tabulated in Table A2. Example 4.1. Suppose we want to determine the known sigma VRGS plan for given p1 = 003, p2 = 006,  = 5%, and = 10%. From Table A1, one can find the parametric values as n = 306602 ≈ 31, k1 = 15414, and k2 = 18479. Thus the VRGS plan with (31, 1.5414, 1.8479) will have minimum ASN at AQL. For same values AQL and LQL, one can find the values as n = 325588, k1 = 15560, and k2 = 18335. This plan with (33, 1.556, 1.8335) will have minimum sum of ASN at both AQL and LQL. Similarly, we can find the unknown sigma VRGS plans using Table A2. We may compare the performance of several sampling plans in terms of their ASN. Obviously, a sampling plan having smaller ASN would be more desirable. Table 1 shows the ASNs for two cases in VRGS plan together with the sample size in the single sampling plan, ASN at AQL in double sampling plan, and ASN of sequential sampling plan for some selected combinations of (AQL, LQL) when the sigma is known. We may construct similar tables from Table A2 when the sigma is

Table 1 Comparison of ASNs among different plans (for known sigma) VRGS plan p1

p2

0.001 0.002 0.004 0.006 0.008 0.010

ASN
p1 

ASN
p1  (1/2)[ASN
p1  + ASN
p2 ]

Single

12029 2818 1619 1165 927

19042 4461 2563 1845 1468

12916 3026 1738 1251 995

Double Sequential 1549 368 209 151 120

8867 2077 1194 859 684

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Balamurali et al.

unknown. The sample size in the single sampling plan and the ASN in the double sampling plan can be found in Sommers (1981). The ASN of sequential sampling plan is calculated by the approximation given in Schilling (1982). As can be seen in Table 1, the VRGS plan reduces the ASN significantly as compared with the single or double sampling plan. The ASN at AQL in the VRGS plan is about 77% of that in the double sampling plan and just 63% of the sample size required for the single sampling plan. Similar reduction in ASN can be achieved for any combination of AQL and LQL values. This implies that the VRGS plan will give desired protection with minimum inspection so that the cost of inspection will greatly be reduced. However, it is to be pointed out that the VRGS plan has higher ASN than the variables sequential sampling plan. Comparing the ASN for the sequential variables sequential sampling plan with the ASN for the VRGS plan, it is observed that the sequential plan’s ASN is about 74% of the VRGS plan. Thus the VRGS plan provides an intermediate protection between the single, double sampling plans and the sequential sampling plan.

5. Conclusions In this article, the problem of minimizing average sample number in variable RGS plan has been considered. It has been demonstrated that the VRGS plan has smaller average sample number than the existing single and double sampling variables plans. Because of the minimum inspection involved in VRGS plan, the cost of inspection is also reduced with desired amount of protection. Moreover, the proposed plan is not as complicated to administer as the double sampling variables plan. This study can also be extended for the double specification limits. Further, the VRGS plan is expected to apply to reliability sampling where the quality characteristic follows a nonnormal distribution such as Weibull or Gamma.

Appendix Table A1 A VRGS plan having minimum ASN for specified values of p1 and p2 with  = 005, = 01 ( known) Min[ASN
p1  + ASN
p2 ]

Min ASN(p1 ) p1

p2

n

k1

k2

ASN

n

k1

k2

ASN

0001

0002 0004 0006 0008 0010 0006 0008 0010 0012 0014

72.4613 16.9745 9.7517 7.0205 5.5849 803.513 116.972 52.3579 32.0851 22.7424

28695 26342 24885 23811 22951 25095 24021 23161 22441 21818

30688 30460 30319 30215 30132 25694 25590 25507 25437 25376

120.289 28.1785 16.1883 11.6544 9.2711 1333.9 194.180 86.9184 53.2628 37.7535

76.9477 18.0256 10.3553 7.4552 5.9307 853.265 124.214 55.6007 34.0718 24.1505

28789 26537 25143 24114 23292 25124 24095 23273 22583 21987

30595 30267 30064 29914 29795 25666 25516 25397 25296 25209

129.156 30.2555 17.3815 12.5135 9.95450 1432.20 208.493 93.3250 57.1890 40.5364

0005

(continued)

Variables RGS Plan

807

Table A1 Continued Min[ASN
p1  + ASN
p2 ]

Min ASN(p1 ) p1

p2

n

k1

k2

ASN

n

k1

k2

ASN

003

004 006 008 010 012 006 008 010 012 014 006 008 010 012 014 008 010 012 014 016 008 010 012 014 016 010 012 014 016 018

192.512 30.6602 14.3998 9.0753 6.5417 84.9068 27.2824 14.8072 9.8328 7.2518 401.615 56.6805 24.6907 14.7612 10.2254 145.416 43.6553 22.5937 14.4770 10.3801 651.612 86.3758 36.0163 20.8369 14.0662 213.600 61.5598 30.9003 19.3285 13.5892

17454 15414 13856 12571 11461 15468 13909 12624 11514 10529 15511 13953 12667 11558 10572 13990 12704 11595 10609 09715 14022 12736 11627 10641 09748 12765 11656 10670 09777 08953

18677 18479 18328 18203 18096 17309 17158 17033 16926 16830 16358 16207 16082 15974 15879 15397 15272 15165 15069 14982 14687 14562 14454 14359 14272 13926 13819 13723 13636 13557

319.583 50.8979 23.9046 15.0655 10.8597 140.950 45.2902 24.5806 16.3229 25.8511 666.701 94.0924 40.9876 24.5041 16.9749 241.397 72.4697 37.5067 24.0325 17.2315 1081.7 143.388 59.7888 34.5905 23.3505 354.579 102.191 51.2960 32.0864 22.5588

204.434 32.5588 15.2913 9.6374 6.9468 90.1639 28.9716 15.7239 10.4416 7.7007 426.481 60.1897 26.2193 15.6752 10.8586 154.420 46.3581 23.9927 15.3735 11.0228 691.956 91.7237 38.2462 22.1272 14.9370 226.821 65.3704 32.8135 20.5253 14.4306

17512 15560 14068 12838 11776 15555 14063 12833 11771 10828 15551 14060 12829 11767 10824 14056 12826 11764 10821 09965 14053 12823 11761 10818 09962 12820 11758 10815 09960 09172

18619 18335 18118 17939 17784 17223 17006 16827 16672 16535 16318 16101 15922 15767 15630 15331 15152 14997 14860 14735 14655 14476 14322 14184 14060 13872 13717 13580 13455 13341

343.139 54.6495 25.6666 16.1760 11.6601 151.339 48.6285 26.3925 17.5260 12.9256 715.850 101.028 44.0088 26.3103 18.2261 259.191 77.8115 40.2713 25.8040 18.5016 1161.45 153.957 64.1960 37.1402 25.0717 380.715 109.724 55.0770 34.4515 24.2216

004

005

006

007

008

Table A2 A VRGS plan having minimum ASN for specified values of p1 and p2 with  = 005, = 01 ( unknown) Min[ASN
p1  + ASN
p2 ]

Min ASN(p1 ) p1

p2

ns

k1s

k2s

ASNs

ns

k1s

k2s

ASNs

0001

0002 0004 0006 0008 0010

40715 9292 5260 3749 2960

28797 26764 25604 24794 24171

30660 30329 30079 29869 29684

66568 14903 8325 5870 4593

41293 9014 4980 3491 2723

28818 26680 25414 24514 23816

30637 30435 30336 30269 30215

69516 15136 8321 5803 4502

(continued)

808

Balamurali et al. Table A2 Continued Min[ASN
p1  + ASN
p2 ]

Min ASN(p1 ) p1

p2

ns

k1s

k2s

ASNs

ns

k1s

k2s

ASNs

0005

0006 0008 0010 0012 0014 004 006 008 010 012 006 008 010 012 014 006 008 010 012 014 008 010 012 014 016 008 010 012 014 016 010 012 014 016 018

343650 49304 21829 13260 9331 52191 8092 3737 2327 1662 20848 6571 3519 2313 1693 93070 12853 5516 3262 2241 31347 9256 4733 3005 2139 134490 17504 7203 4126 2764 41595 11816 5868 3640 2543

25106 24092 23318 22693 22170 17503 15711 14416 13511 12734 15579 14246 13226 12398 11699 15535 14120 13040 12166 11430 14057 12921 12003 11231 10563 14037 12848 11888 11083 10386 12811 11812 10973 10250 09612

25691 25571 25462 25361 25267 18663 18384 18109 17836 17565 17277 17047 16815 16579 16342 16351 16155 15955 15749 15538 15378 15203 15022 14834 14641 14683 14528 14368 14200 14026 13913 13770 13619 13462 13297

567740 80766 35495 21419 14982 85614 12987 5880 3597 2527 33949 10508 5532 3580 2581 153220 20814 8791 5120 3466 51277 14920 7520 4709 3307 221800 28481 11563 6536 4322 68220 19137 9384 5749 3967

359230 50283 21846 13072 9085 53474 7904 3542 2162 1523 20987 6387 3339 2158 1560 96325 12795 5341 3096 4349 31919 9141 4570 2853 2005 139920 17620 7072 3975 2624 42652 11793 5735 3501 2415

25126 24115 23319 22663 22105 17525 15662 14313 13259 12396 15588 14184 13078 12167 11394 15557 14113 12967 12018 13022 14075 12898 11919 11082 10351 14057 12857 11853 10991 10236 12833 11809 10926 10150 09459

25670 25546 25460 25397 25347 18639 18444 18316 18202 18085 17267 17124 17014 16910 16802 16328 16164 16048 15948 13550 15357 15229 15128 15035 14941 14662 14519 14411 14319 14229 13890 13773 13678 13591 13503

603860 84629 36763 21976 15250 89963 13245 5878 3543 2462 35287 10681 5532 3536 2524 161990 21483 8913 5120 3429 53677 15321 7608 4707 3275 235240 29602 11828 6601 4319 71719 19783 9570 5798 3964

003

004

005

006

007

008

Acknowledgments The authors thank the editor and the anonymous referees for their constructive comments and suggestions leading to a significantly improved presentation of the article. This research was supported by the Advanced Product and Production Technology Center at POSTECH.

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References Duncan, A. J. (1986). Quality Control and Industrial Statistics. 5th ed. Homewood, Illinois: Richard D. Irwin. Hamaker, H. C. (1979). Acceptance sampling for percent defective by variables and by attributes. J. Qual. Technol. 11:139–148. Kuralmani, V., Govindaraju, K. (1995). Modified tables for the selection of double sampling attribute plan indexed by AQL and LQL. Commun. Statist. Theor. Meth. 24:1897–1921. Lieberman, G. J., Resnikoff, G. J. (1955). Sampling plans for inspection by variables. J. Amer. Stat. Assoc. 50:457–516. Nocedal, J., Wright, S. J. (1999). Numerical Optimization. New York: Springer. Schilling, E. G. (1982). Acceptance Sampling in Quality Control. New York: Marcel Dekker. Sherman, R. E. (1965). Design and evaluation of repetitive group sampling plan. Technometrics 7:11–21. Sommers, D. J. (1981). Two-point double variables sampling plans. J. Qual. Technol. 13(1):25–30.