Chapter 15

Basic Acceptance Sampling Procedure

許湘伶

Statistical Quality Control(D. C. Montgomery)

The Acceptance-Sampling(允收抽樣) Problem I

I lot-by-lot(逐批) acceptance-sampling plans for attributes

acceptance sampling:I concerned with inspection(檢驗) and decision making

regarding product

The Acceptance-Sampling(允收抽樣) Problem II

A typical application of acceptance sampling

I a shipment (裝載的貨物) of product from a supplierI Product: a component; raw material used in the

company’s manufacturing processI A sample taken from the lot: quality characteristic

of the units in the sample is inspectedI A decision is made regarding lot disposition(傾向):

lot sentencing(貨批判定)I accept: put into productionI reject: returned to the supplier or to some lot

disposition action

The Acceptance-Sampling(允收抽樣) ProblemIII

Three aspects of sampling:

1. sentence lots, not to estimate the lot quality

2. simply accepts or rejects lots:Even if all lots are of the same quality, sampling will acceptsome lots and reject others

3. an audit(稽核) tool to ensure that the output of a processconforms to requirements

The Acceptance-Sampling(允收抽樣) Problem IV

Three approaches to lot sentencing:

1. accept with no inspection

2. 100% inspection (全檢):I inspect every item, removing all defective unitsI Situation: extremely critical component

3. acceptance sampling

The Acceptance-Sampling(允收抽樣) Problem V

Acceptance sampling is useful in the following situations:I Test is destructive(破壞的)I The cost of 100% inspection is extremely highI 100% inspection is not technologically feasible or 費時I many item to be inspected or the inspection error rate is

high than 100% inspectionI the supplier has an excellent quality historyI potentially serious product liability(責任) risks

Advantages and Disadvantages of Sampling I

Advantage: comparing with 100% inspectionI less inspectionI reduced damageI applicable to destructive testingI fewer personnel(人員) are involvedI reduces the amount of inspection errorI the simple return of defectives often provides a stronger

motivation to the supplier for quality improvements

Advantages and Disadvantages of Sampling II

Disadvantage:I risks of accepting "bad" lots and rejecting "good" lotsI less informationI Acceptance sampling requires planning and documentation

of the acceptance-sampling procedure

Types of Sampling Plans I

Variables: Chap. 16Attributes: Chap. 15

1. single-sampling plan:(sample size, acceptance number)=(n, c)if #{defective}>c ⇒ reject the lot

2. double-sampling plan

3. multiple-sampling plan: the extension is sequentialsampling

Lot Formation(組成) I

I Lots should be homogeneous: the same machine, the sameoperators, from common raw materials, at approximate thesame time

I Larger lots are preferred over smaller onesI Lots should be conformable(適合的) to the

materials-handling systems used in both the supplier andconsumer facilities(設備)

Single-Sampling Plan for Attributes I

I N : size of a lot; (Ex: N = 10, 000)I n: the sample size; (Ex: n = 89)I c: the acceptance number; (Ex: c = 2)I d: the number of observed defective items

d ≤ c ⇒ acceptes the lot

Operating characteristic(OC) curve

Single-Sampling Plan for Attributes II

I An important measure of the performance of anacceptance-sampling plan

I display(顯示) the discriminatory(有辨識力的) power of thesampling plan

I the probability that a lot submitted with a certainfraction(小部分) defective will be either accepted or rejected

Single-Sampling Plan for Attributes III

I B(n, p)

P{d defectives} =(

nd

)pd(1− p)n−d

I Probability of acceptance:

Pa = P{d ≤ c} =c∑

d=0

(nd

)pd(1− p)n−d

Ex: n = 89, c = 2, p = 0.01⇒ Pa = P{d ≤ 2} = 0.9397n = 89, c = 2, p = 0.02⇒ Pa = P{d ≤ 2} = 0.74100 lots: expect to accept 74 of the lots and reject 26 of them

Single-Sampling Plan for Attributes IV

(a) ideal (b) sample size (c) c

(a) discriminated perfectly between good and bad lots(b) c

n =constant: the greater the slope of the OC curve, thegreater is the discriminatory power

(c) changing c does not dramatically change the slope of theOC curve

Designing a Single-Sampling Plan I

Type I and Type II errors:I H0 : the lot quality is good

State of the lotGood Quality of Lot Poor Quality of Lot(H0 is true) (H0 is false)

Accept H0 Correct β=Type II errorReject H0 α=Type I error Correct

I To the design of an acceptance-sampling plan: require thatthe OC curve pass through two designated points

Designing a Single-Sampling Plan II1− α =

c∑d=0

(nd

)pd

1 (1− p1)n−d

(1− α = the prob. of acceptance the lots with p1)

β =c∑

d=0

(nd

)pd

2 (1− p2)n−d

(β = the prob. of acceptance the lots with p2)

Designing a Single-Sampling Plan III

I 1− α, β: two simultaneous equations are nonlinearI there is no simple, direct solutionI (p1, 1− α) = (0.01, 0.95), (p2, β) = (0.06, 0.10)I Any two points on the OC curve could be used to define

the sampling planI p1 =AQL (producer’s risk), p2 =LTPD (consumer’s risk)I AQL: acceptable quality level (允收品質水準)I LTPD: lot tolerance percent defective (批拒收品質水準)=RQL(rejectable quality level)=LQL (limiting qualitylevel)

Designing a Single-Sampling Plan IV

Designing a Single-Sampling Plan V

I AQL: acceptable quality level (允收品質水準)the poorest level of quality for the supplier process (代表了顧客心目中所樂意於見到的品質水準，通常均標示於合約書中)

I LTPD: lot tolerance percent defective (批拒收品質水準)=RQL =LQLthe poorest level of quality that the consumer is willing toaccept in an individual lot (顧客心目中所能接受的最低品質水準，通常是顧客心中所認定最差但尚可維持產品正常使用的品質

水準)

Designing a Single-Sampling Plan VI

Rectifying Inspection(補正檢驗) I

I Rectifying inspection: 當送驗批被拒收之後，通常是對被拒收的貨批進行100%檢驗，並將不合格品剔除，以合格品取代。

Rectifying Inspection(補正檢驗) IIAverage outgoing quality(AOQ)(平均出廠品質): the evaluationof a rectifying sampling plan (評估補正抽樣計畫)

AOQ

AOQ = Pap(N − n)N

n/N→0' Pap

I n: after inspection, contain no defectivesI N − n: contain no defectives if the lot is rejectedI N − n: contain p(N − n) defectives if the lot is

acceptedAOQ: 貨批經過補正檢驗之後的品質，它是由一連串不合格率為 p 的貨批經過檢驗之後所獲得之平均品質水準。

Rectifying Inspection(補正檢驗) III

Rectifying Inspection(補正檢驗) IV

I AOQL: 經過補正檢驗之後，所可能產生之最差平均品質。AOQ之最大值。

I AOQL是一連串多批產品之平均不合格率，個別貨批的不合格率仍有可能高於AOQL

I ATI = n + (1− Pa)(N − n): the average total inspectionper lot

Double-Sampling Plans I

DSP

I DSP: a procedure; a second sample is requiredbefore the lot can be sentenced.

I four parameters:

n1 =sample size on the first samplec1 =acceptance number of the first samplen2 =sample size on the second samplec2 =acceptance number of the second sample

I di=the number of defectives in the ith sample

Double-Sampling Plans II

Double-Sampling Plans III

OC curve:Pa = PI

a + PIIa

I Ex: (n1,n2, c1, c2, p) = (50, 100, 1, 3, 0.05)

PIa =

1∑d1=0

(nd1

)(0.05)d1(0.95)50−d1 = 0.278

PIIa = P{d1 = 2, d2 ≤ 1}+ P{d1 = 3, d2 = 0}

= P{d1 = 2} · P{d2 ≤ 1}+ P{d1 = 3} · P{d2 = 0} = 0.0107Pa = PI

a + PIIa = 0.2897

Double-Sampling Plans IV

AOQ = [PIa (N − n1) + PII

a (N − n1 − n2)]pN

ATI = n1PIa + (n1 + n2)PII

a + N (1− Pa)