Learning Objectives

15.1 The Acceptance-Sampling Problem Acceptance sampling plan (ASP): ASP is a specific plan that clearly states the rules for sampling and the associated criteria for acceptance or otherwise. Acceptance sampling plans can be applied for inspection of (i) finish items, (ii) components, (iii) raw materials, (iv) materials in process, (v) supplies in storage, (vi) maintenance operations, (vii) data or records etc.

Three aspects of the sampling are important: 1. Purpose of acceptance sampling is to sentence lots, not to estimate the lot quality 2. Acceptance sampling plan simply accept or reject the lots. Do not provide any direct form of quality control. 3. Not use to inspect quality into product, instead use as a audit tool to ensure that the output of a process conforms to requirements. 15.1.1 Advantages and Disadvantages of Sampling Advantages of Sampling 1. Less expensive because of less inspection compare to entire lot. 2. Less handling of product and hence reduce damage 3. Applicable to destructive testing 4. Fewer personnel are involved in inspection activities 5. Reduces the amount of inspection error 6. The rejection of lot motivate the vendor for quality improvement Some Disadvantages of Sampling 1. Risk of accepting a bad lots and rejecting a good lots 2. Less information about the product 3. Need some plan and formulation compare to 100% inspection 15.1.2 Types of Sampling Plans Single Sampling Plan: Single sampling plan is the sampling inspection plan in which the lot disposition is based on the inspection of a single sample of size n . Double Sampling Plans: Following an initial sample, a decision based on the information in that sample is made either to (1) accept the lot, (2) reject the lot or (3) take a second sample . If the second sample is taken, the information from both the first and second sample is combined in order to reach a decision whether to accept or reject the lot.

Multiple Sampling Plan: Multiple sampling plan is an extension of the double sampling concept, where more than two samples are required in

order to reach a decision regarding the disposition of the lot. Sequential Sampling Plan: The ultimate extension of multiple sampling is sequential sampling, in which units are selected from the lot one at a time, and the following inspection of each units, a decision is made either to accept the lot, reject the lot or select another unit. 15.1.3 Lot Formation There are several important considerations in forming lots for inspection. 1. Lots should be homogeneous 2. Larger lots are preferred over smaller ones 3. Lots should be conformable to the materials-handling systems used in both the vendor and consumer facilities. 15.1.4 Random Sampling The selected units for inspection from the lot should be chosen at random, and they should be representative of all the items in the lot. The random sampling is an important concept in acceptance sampling plan. Without random sample, bias will be introduced in the results. 15.1.4 Guidelines for Using Acceptance Sampling The major types of acceptance-sampling procedures and their applications are shown in Table 15.1, page 681. In general, the selection of an acceptance-sampling procedure depends on both objective of the sampling organization and the history of the organization whose product is sampled. 15.2 Single Sampling Plan for Attributes 15.2.1 Definition of a Single-Sampling Plan The operating procedure of the single sampling attributes plan is as follows: 1. From a lot of size N , draw a random sample of size n and observe the number of nonconforming units d. 2. If d is less than or equal to the acceptance number c , which is the maximum allowable number of nonconforming units, accept the lot. If cd > , do not accept the lot.

15.2.2 OC Curve for Single Sampling Plan The OC curve reveals the performance of the acceptance sampling plan. We consider two types of OC curves: Type A OC curve: Type A OC curve is used to calculate the probabilities of acceptance an isolated lot of finite size N . Suppose the sample size is n and the acceptance number is c . Then the exact number of defective items in the sample has the Hypergeometric distribution. Type B OC curve: Type B OC curve is used to calculate the probabilities of acceptance a continuous stream of lots (infinite or large size). This is a curve showing the probability of accepting a lot as a function of the process average p. For very large lot size N , both curves give the same information. However, Type A OC curve will always be below the type B OC curve (see Figure 15.6, page 640). We will discuss about Type B OC curve only and will be used throughout the chapter. Suppose the lot size N is large (say infinity). Under this condition, the distribution of the number of defectives d in a random sample of n items is binomial with parameters n and p , where p is the fraction defective items in the lot. The probability of observing exactly d defectives is

dnd ppdnd

ndpdefectivesdP −−−

)(1)!(!

!=)(=}{ (1)

The probability of acceptance is that d is less than or equal to c . That is

dndc

da pp

dndncdP −−−

≤ ∑ )(1)!(!

!=}{=0=

(2)

To construct OC curve plot aP against the lot fraction defective p . The following Table give the OC curve calculation for 89=n and 2=c and the corresponding Figure is presented in Figure 15.2.

p 0.005 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080 0.090 aP

0.9897 0.9397 0.7366 0.4985 0.3042 0.1721 0.0919 0.0468 0.0230 0.0109

Figure 15.2. OC curve of the single-sampling plan 89=n , 2=c

Explanation: Consider the above sampling plan with 89=n and 2=c . If the lots have 1% defective, the probability of acceptance the lot is 0.9397. This means that if 100 lots from a process that manufactures 1% defective product are submitted to this plan, one would expect to accept 94 of the lots and reject 6 of them. Similarly, if the lots have 2% defective, the probability of acceptance the lot is 0.7366. This means that if 100 lots from a process that manufactures 2% defective product are submitted to this plan, one would expect to accept 74 of the lots and reject 26 of them. Exercise 15.2, page 669.

p f(d=0) f(d=1) f(d=2) Pr{d<=c}0.001 0.90479 0.09057 0.00449 0.99985 0.002 0.81857 0.16404 0.01627 0.99888 0.003 0.74048 0.22281 0.03319 0.99649 0.004 0.66978 0.26899 0.05347 0.99225 0.005 0.60577 0.30441 0.07572 0.98590 0.006 0.54782 0.33068 0.09880 0.97730 0.007 0.49536 0.34920 0.12185 0.96641 0.008 0.44789 0.36120 0.14419 0.95327 0.009 0.40492 0.36773 0.16531 0.93796 0.010 0.36603 0.36973 0.18486 0.92063 0.020 0.13262 0.27065 0.27341 0.67669 0.030 0.04755 0.14707 0.22515 0.41978 0.040 0.01687 0.07029 0.14498 0.23214 0.050 0.00592 0.03116 0.08118 0.11826 0.060 0.00205 0.01312 0.04144 0.05661 0.070 0.00071 0.00531 0.01978 0.02579 0.080 0.00024 0.00208 0.00895 0.01127 0.090 0.00008 0.00079 0.00388 0.00476 0.100 0.00003 0.00030 0.00162 0.00194 0.200 0.00000 0.00000 0.00000 0.00000

Type-B OC Curve for n=100, c=2

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.000 0.020 0.040 0.060 0.080 0.100 0.120 0.140

p

Pr{a

ccep

tanc

e}

Effect of n and c on OC curves A sampling plan that discriminated perfectly between good and bad lots would have an OC curve looks like Figure 15.3, page 639. The ideal OC curve in Figure 15.3 never be obtained in practice. Figure 15.4 shows that the OC curve becomes more like idealized OC curve shape as the sample size increases. Figure 15.5 shows how the OC curve changes as the acceptance number changes.

Figure 15.3: Ideal OC Curve

Specific Points on the OC curve Acceptable Quality Level (AQL): The AQL represents the poorest level of quality for the vender's process that the consumer would consider to be acceptable as a process average. Lot Tolerance Percent Defective (LTPD): The LTPD represents the poorest level of quality that the consumer is willing to accept in an individual lot. Note that both AQL and LTPD are not the properties of the sampling plan,

Figure 15.4. OC curve for different sample sizes

Figure 15.5. The effect of changing the acceptance number on OC curve

Exercise 15.7, page 668.

LTPD = 0.05 N1 = 5000 N2 = 10000 n1 = 500 n1 = 1000 pmax = 0.0200 pmax = 0.0200 cmax = 10 cmax = 20 binomial binomial p Pr{d<=10} Pr{reject} Pr{d<=20} Pr{reject} difference 0.0010 1.00000 0.0000 1.00000 0.0000 0.00000 0.0020 1.00000 0.0000 1.00000 0.0000 0.00000 0.0030 1.00000 0.0000 1.00000 0.0000 0.00000 0.0040 0.99999 0.0000 1.00000 0.0000 -0.00001 0.0050 0.99994 0.0001 1.00000 0.0000 -0.00006 0.0060 0.99972 0.0003 1.00000 0.0000 -0.00027 0.0070 0.99903 0.0010 0.99999 0.0000 -0.00095 0.0080 0.99729 0.0027 0.99991 0.0001 -0.00263 0.0090 0.99359 0.0064 0.99959 0.0004 -0.00600 0.0100 0.98676 0.0132 0.99850 0.0015 -0.01175 0.0200 0.58304 0.4170 0.55910 0.4409 0.02395 0.0250 0.29404 0.7060 0.18221 0.8178 0.11183 0.0300 0.11479 0.8852 0.03328 0.9667 0.08151 0.0400 0.00967 0.9903 0.00030 0.9997 0.00938 0.0500 0.00046 0.9995 0.00000 1.0000 0.00046 0.0600 0.00001 1.0000 0.00000 1.0000 0.00001 0.0700 0.00000 1.0000 0.00000 1.0000 0.00000

Different sample sizes offer different levels of protection. For N = 5,000, Pa(p = 0.025) = 0.294; while for N = 10,000, Pa(p = 0.025) = 0.182. Also, the consumer is protected from a LTPD = 0.05 by Pa(N = 5,000) = 0.00046 and Pa(N = 10,000) = 0.00000, but pays for the high probability of rejecting acceptable lots like those with p = 0.025. 15.2.3 Designing a Single Sampling Plan with a specified OC Curve Suppose that we wish to construct a sampling plan such that the probability of acceptance is α−1 for lots with fraction defective 1p , and the probability of acceptance is β for lots with fraction defective 2p . Suppose that binomial sampling (with Type -B OC curves) is appropriate, then the sample size n and acceptance number c are the solution to

1 1

=0

2 2=0

!1 = (1 )!( )!

!= (1 )!( )!

cd n d

d

cd n d

d

n p pd n d

n p pd n d

α

β

−

−

− −−

−−

∑

∑ (3)

The two simultaneous equations in (3) are nonlinear, and there is no simple or direct solution. The nomograph in Figure 15.9, page 643 can be used for solving theses equations in (3).

Figure 15.9. Bionomial nomograph

15.2.4 Rectifying Inspection Accepting sampling programs usually require corrective action when lots are rejected. This generally takes the form of 100% inspection or screening of rejected lots, with all discovered defectives items either removed for subsequent rework or return to the supplier or replaced from a stock of known good items. Such sampling programs are called rectifying inspection programs, because the inspection activity affetcs the final quality of the outgoing product. This has been illustrtaed in the following Figure 15.10.

Average outgoing quality (AOQ) Those lots not accepted by a sampling plan will usually be 100% inspected or screened for nonconforming or defective units. After screening, nonconforming units may be rectified or discarded or replaced by good units, usually taken from accepted lots. Such a programmed of inspection is known as a rectifying or screening inspection. For those lots accepted by the sampling plan, no screening will be done and the outgoing quality will be the same as that of the incoming quality p . For those lots screened, the outgoing quality will be zero, meaning that they contain no nonconforming items. Since the probability of accepting a lot is aP , the outgoing lots will contain a proportion of apP defectives. If the nonconforming units found in the sample of size n are replaced by good ones, the average outgoing quality (AOQ) in lot size of N will be

( )=

=

a

a

N nAOQ pPN

pP for latge N

− (4)

In short, one defines the average outgoing quality as the expected quality of outgoing product following the use of an acceptance sampling plan for a given value of the incoming quality. Figure 15.11, gives a typical AOQ curve as a function of the incoming quality.

Figure 15.11. Average outgoing quality curve for 89=n If the incoming quality is good, then a large proportion of the lots will be accepted by the sampling plan and only a smaller fraction will be screened and hence the outgoing quality will be small (good). Similarly, when the incoming quality is not good, a large proportion of the lots will go for screening inspection and in this case also, the outgoing quality will be good since defective items will be either replaced or rectified. Only for intermediate quality levels, lot acceptance will be at a moderate rate and hence the AOQ will rise (see Figure 15.11). The maximum ordinate of the AOQ curve represents the worst possible average for the outgoing quality and is known as the average outgoing quality limit (AOQL). In other words, the AOQL is defined as the maximum AOQ over all possible levels of the incoming quality for a known acceptance sampling plan. The AOQL of a rectifying inspection plan is very important characteristic. It is possible to design a rectifying inspection program that have specified value of AOQL.

Exercise 15.10, page 669. N = 3000, n = 150, c = 2

p Pa=Pr{d<=2} AOQ ATI 0.001 0.99951 0.0009 151 0.002 0.99646 0.0019 160 0.003 0.98927 0.0028 181 0.004 0.97716 0.0037 215 0.005 0.95991 0.0046 264 0.006 0.93769 0.0053 328 0.007 0.91092 0.0061 404 0.008 0.88019 0.0067 491 0.009 0.84615 0.0072 588 0.010 0.80948 0.0077 693 0.015 0.60884 0.0087 AOQL 1265 0.020 0.42093 0.0080 1800 0.025 0.27341 0.0065 2221 0.030 0.16932 0.0048 2517 0.035 0.10098 0.0034 2712 0.040 0.05840 0.0022 2834 0.045 0.03292 0.0014 2906 0.050 0.01815 0.0009 2948 0.060 0.00523 0.0003 2985 0.070 0.00142 0.0001 2996 0.080 0.00036 0.0000 2999 0.090 0.00009 0.0000 3000 0.100 0.00002 0.0000 3000

(a)

OC Curve for n=150, c=2

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0.60

0.80

1.00

0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080

p

Pr{a

ccep

t}

(b) AOQ Curve for n=150, c=2

0.0000

0.0010

0.0020

0.0030

0.0040

0.0050

0.0060

0.0070

0.0080

0.0090

0.0100

0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080

p

AO

Q

AOQL ≅ 0.0087

(c)

ATI Curve for n=150, c=2

0

500

1000

1500

2000

2500

3000

3500

0.000 0.010 0.020 0.030 0.040 0.050 0.060 0.070 0.080

p

ATI

Average total inspection If the lot quality is 1<<0 p , the average amount of inspection per lot will vary between the sample size n and the lot size N . If the lot is of quality p and the probability of lot acceptance is aP , then the average total inspection per lot will be

))((1= nNPnATI a −−+ (5) The ATI curves for the sampling plan 89=n , 2=c and lot sizes of 10000 , 5000 and 1000 are shown in Figure 15.12.

Figure 15.12. Average outgoing quality curve for 89=n

15.3 Double, Multiple, and Sequential Sampling 15.3.1 Double-Sampling Plan A double -sampling plan is an procedure in which, under certain circumstances, a second sample is required before the lot can be sentenced. Application of double sampling requires that a first sample of size 1n is taken at random from the (large) lot. The number of defectives is then counted and compared to the first sample's acceptance number 1c . Denote the number of defectives in sample 1 by 1d and in sample 2 by 2d . If 11 cd ≤ , the lot is accepted on the first sample. Also 2c denotes the acceptance number for both sample. If 21 > cd , the lot is rejected on the first sample. If 211 < cdc ≤ , a second sample of size 2n is drawn from the lot. Then the combined number of observed defective from both sample is 21 dd + . If 221 cdd ≤+ , the lot is accepted. However, if 221 > cdd + the lot is rejected. The operation of double sampling plan is presented in Figure 15.13, page 647.

Figure 15.13: Operation of the double-sampling plan, 1 1 2 2n =50, c =1, n =100 and c =3.

The OC Curve for Double Sampling Plan The double sampling OC curve has a primary OC curve that gives the probability of acceptance as a function of lot or process quality. It also has supplementary OC curves that show the probability of lot acceptance and rejection on the first sample. The OC curve for the plan 50=1n , 1=1c ,

100=2n and 3=2c are shown in Figure 15.14.

Average Sample Number Curve for a Double Sampling Plan Since when using a double sampling plan the sample size depends on whether or not a second sample is required, an important consideration for this kind of sampling is the Average Sample Number (ASN) curve. This curve plots the ASN versus lot fraction defective p . The general formula for the ASN in double sampling, if we assume complete inspection of the second sample, is

1 1 2

1 2

= ( )(1 )= (1 )

I I

I

ASN n P n n Pn n P

+ + −+ −

(6)

where IP is the probability of making a lot-dispositioning decision on the first sample. That is

}{}{= samplefirsttheonrejectedislotPsamplefirsttheonacceptaedislotPPI +

The ASN curve for formula for a double-sampling plan with curtailment on the second sample is

⎥⎦

⎤⎢⎣

⎡+−+

+−+−+ ∑

+

2)1,(1),(),(= 222

2221

2

11=1 jcnP

pjcjcnPnjnPnASN ML

c

cj (7)

where ),( 1 jnP is the probability of observing exactly j defectives in a sample of size 1n , ),( 22 jcnPL − is the probability of observing exactly jc −2 or fewer defectives in a sample of size 2n , and 2)1,( 22 +−+ jcnPM probability of observing exactly 22 +− jc defectives in a sample of size 12 +n . Figure 15.15, page 650 compares the average sample number curves of complete and curtailed inspection for the double-sampling plan, 60=1n , 2=1c ,

120=2n , 3=2c and the average sample number that would be used in the single-sampling plan with 89=n and 2=c . For more details page 650-651.

Figure 15.15. Average sample number curves for single double sampling Rectifying Inspection When rectifying inspection is performed with double sampling, the AOQ curve is given by

N

pnnNPnNPAOQII

aI

a )]()([= 211 −−+− (8)

assuming that all defective items discovered, either in sampling or 100% inspection, are replaced with good ones. The average total inspection curve is give by

)(1)(= 211 aII

aI

a PNPnnPnATI −+++ (9) Note that

IIa

Iaa PPP +=

is the probability of final lot acceptance and that the acceptance probabilities depend on the level of lot or process quality p .