Chapter 15Lot-by-Lot Acceptance Sampling

for Attributes

The Acceptance-Sampling Problem

• Acceptance sampling is concerned with inspection and decision making regarding products.

Three aspects of sampling

• The purpose of acceptance sampling is to sentence lots, not to estimate the lot quality– Although, some plans do this

• Acceptance sampling is not quality control– Reject or accept lots only– Even if lots are of the same quality, sampling

will accept some lots and reject others

Three aspects of sampling

• Quality cannot be inspected into the product– Acceptance sampling is an audit tool that

insures that the output of a process conforms to requirements

The Acceptance-Sampling Problem

• Three approaches to lot sentencing:

1. Accept with no inspection

2. 100% inspection

3. Acceptance sampling

The Acceptance-Sampling Problem

Why Acceptance Sampling and Not 100% Inspection?

• Testing can be destructive• Cost of 100% inspection is high• 100% inspection is not feasible

– Requires too much time– Can be inaccurate

• If vendor has excellent quality history

The Acceptance-Sampling Problem

Advantages and Disadvantages of Sampling Advantages• Less expensive• Reduced damage• Reduces the amount of inspection errorDisadvantages• Risk of accepting “bad” lots, rejecting “good” lots• Less information generated• Requires planning and documentation

The Acceptance-Sampling Problem

Types of Sampling Plans • There are variables sampling plans and attribute

sampling plans (this chapter is about attributes)

1. Single sampling plan

2. Double-sampling plan

3. Multiple-sampling plan

4. Sequential-sampling

The Acceptance-Sampling Problem

Lot Formation

• Considerations before inspection:– Lots should be homogeneous

• Produced by the same machine, same operators, common raw materials, approximately the same time

Lot Formation

• Considerations before inspection:– Larger lots more preferable than smaller lots

• More economical

– Lots should be conformable to the materials-handling systems used in both the vendor and consumer facilities.

The Acceptance-Sampling Problem

Random Sampling• The units selected for inspection should be

chosen at random.

• If random samples are not used, bias can be introduced.

• If any judgment methods are used to select the sample, the statistical basis of the acceptance-sampling procedure is lost.

Guidelines for Using acceptance Sampling

• It is a statement of the sample size to be used and the associated acceptance or rejection criteria.

• Sampling scheme is defined as the set of procedures consists of acceptance sampling plans in which lot sizes, sample sizes, and acceptance criteria along with the 100% inspection and sampling are related.

• Acceptance sampling procedure depends upon the objectives and the history of the organization.

• Application of methodology is not static. It keep on moving from one level to another. e.g, we might begin with attribute sampling, and as our experience with the supplier increase then move to much less inspection and variable sampling

Single-Sampling Plans For Attributes

Definition of a Single-Sampling Plan• A single sampling plan is defined by sample size, n, and the

acceptance number c. Say there are N total items in a lot. Choose n of the items at random. If more than c of the items are unacceptable, reject the lot.

• N = lot size

• n = sample size

• c = acceptance number

• d = observed number of defectives

• The acceptance or rejection of the lot is based on the results from a single sample - thus a single-sampling plan.

Example

• N = 10000, n = 89, c = 2– From a lot of 10,000, take a sample of size 89– Observe the number of defectives, d– If d < 2, accept– Otherwise, reject

Single-Sampling Plans For Attributes

The OC Curve• The operating-characteristic (OC) curve measures

the performance of an acceptance-sampling plan.

• The OC curve plots the probability of accepting the lot versus the lot fraction defective.

• The OC curve shows the probability that a lot submitted with a certain fraction defective will be either accepted or rejected.

Example

• If p = .01, Pa = .9397

• If p = .02, Pa = .7366 means that 73.66% of lots will be expected to be accepted and 26.34% will be rejected

Effect of n and c on OC curves

• Fig. 15.3 is the ideal OC curve– Pa = 1.0 until a level of quality that is

considered ‘bad’ is reached– But it can never be attained in practice.

Effect of n and c on OC curves

OC curve for different values of n– By increasing the sample size, we get closer to

the ideal OC curve

Effect of n and c on OC curves

OC curve for different values of c– As c is decreased, the OC curve shifts to the left– When c = 0, it is very hard on the vendor

• Type A or Type B OC curves– In the type B OC curve, it is assumed that the samples

come from the large lot or from a stream of lots selected at random.

– Binomial distribution is used as p is constant– In the Type A OC curve, isolated lot of finite size is

used with size N– The exact probability distribution is ‘hypergeometric’

as p is not constant.– Type A OC curve will lie below Type B

Rectifying inspection

• Acceptance sampling require corrective action when lots are rejected– 100% screening of rejected lots

– Defective items are removed, returned to the supplier, or replaced.

– Such sampling programs are known as “rectifying inspection programs”

– Affects the outgoing quality

– Fraction defective =po, average fraction defective = p1

Rectifying inspection

Inspection activity

Rejected lots

Accepted lots

Incoming lots

Fraction defective p0

Fraction defective

0

Fraction defective

p0

Outgoing lots

Fraction defective p1<p0

Average outgoing quality

• AOQ is the quality in the lot resulting from applying rectifying inspection– In a lot of size N, there will be

• n items in the sample that, after inspection, contain no defectives (all of the defectives were replaced)

• N-n items that, if the lot is rejected, also contain no defectives (the balance of the lot was inspected 100%)

• N-n items that, if the lot is accepted, contain p(N-n) defectives

Average outgoing quality

• AOQ = [Pa p (N-n)]/N

• Example– N = 10000, n = 89, c = 2, p = .01

– Previously determined that Pa = .9397

– AOQ [(.9397)(.01)(10000-89)]/10000– AOQ = .0093

– Since (N-n)/N 1, AOQ ~ Pap

AOQ curve for rectifying inspection

for n = 89, c = 2

• When incoming quality is very good, average fraction defective of outgoing lots is low

• When incoming quality is very poor, average fraction defective of outgoing lots is low

Average outgoing quality limit

• AOQL = .0155– No matter how bad the incoming lots are, the

outgoing quality level will never be worse than 1.55% fraction defective

Military Standard 105E (ANSI/ASQC Z1.4 ISO 2859)

Description of the Standard• Developed during World War II• MIL STD 105E is the most widely used

acceptance-sampling system for attributes• Gone through four revisions since 1950.• MIL STD 105E is a collection of sampling

schemes making it an acceptance-sampling system

Military Standard 105E (ANSI/ASQC Z1.4 ISO 2859) Description of the Standard• Three types of sampling are provided for:

1. Single2. Double3. Multiple

• Provisions for each type of sampling plan include

1. Normal inspection2. Tightened inspection3. Reduced inspection

Military Standard 105E

Description of the Standard• The acceptable quality level (AQL) is a primary focal

point of the standard• The AQL is generally specified in the contract or by the

authority responsible for sampling.• Different AQLs may be designated for different types of

defects.• Defects include critical defects, major defects, and

minor defects.• Tables for the standard provided are used to determine

the appropriate sampling scheme.

Military Standard 105E

Description of the Standard• Switching Rules

– Normal to tightened– Tightened to normal– Normal to reduced– Reduced to normal– Discontinuance of inspection

Military Standard 105E

Procedure1. Choose the AQL2. Choose the inspection level3. Determine the lot size4. Find the appropriate sample size code letter from Table

15-45. Determine the appropriate type of sampling plan to use

(single, double, multiple)6. Enter the appropriate table to find the type of plan to be

used.7. Determine the corresponding normal and reduced

inspection plans to be used when required.

Military Standard 105E

Example• Suppose a product is submitted in lots of size

N = 2000. The AQL is 0.65%. Say we wanted to generate normal single-sampling plans.

• For lots of size 2000, (and general inspection level II) Table 15-4 indicates that the appropriate sample size code letter is K.

• From Table 15-5 for single-sampling plans under normal inspection, the normal inspection plan is n = 125, c = 2.

Military Standard 105E

Discussion• There are several points about the standard that

should be emphasized:1. MIL STD 105E is AQL-oriented2. The sample sizes selected for use in MIL STD 105E

are limited3. The sample sizes are related to the lot sizes.4. Switching rules from normal to tightened and from

tightened to normal are subject to some criticism.5. A common abuse of the standard is failure to use the

switching rules at all.

Normal TightenedReduced

2 out of 5 consecutive lots

rejected

5 consecutivelots accepted

O Production steadyO 10 consecutive lots acceptedO Approved by responsible authority

O Lot rejectedO Irregular productionO Lot meets neither accept nor reject criteria

O Other conditions warrant return to normal inspection

“And” conditions

“Or” conditions

Start

10 consecutivelots remain on

tightened inspection

Discontinue

inspection

Switching rules

Dodge-Romig Plans

• For rectifying inspection

• See Table 15-8 for an example for

AOQL = 3%– Indexed by lot size (N) and process average (p)

Example

• N = 5000, p = .01

• Want a single sampling plan (w/rectifying inspection) with AOQL = 3%

• Read n = 65, c = 3 from the table

• These plans minimize ATI– Pa = .9957 at p = .01 (determined as previously)

– ATI = 65 + (1 - .9957)(5000 – 65) = 86.22

Example, cont.

• Also, note that LTPD = 10.3%

• This is the point on the OC curve for which Pa = .10

– That is, this plan provides that 90% of incoming lots that are as bad as 10.3% defective will be rejected

LTPD plans

• Can also develop a plan for a specified LTPD

• Table 14-9 is for LTPD = 1%

Example

• N = 5000, p = .25%

• We want a single sampling plan (w/rectifying inspection) with LTPD of 1%

• Find n = 770, c = 4