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## Chapter 9

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### Chapter 9

Lot-by-Lot Acceptance Sampling for Attributes

Acceptance-Sampling Problem

- Acceptance samplingis concerned with inspection and decision making regarding products.
- Three aspects of sampling are important:
- Involves random sampling of an entire “lot”
- Accept and Reject Lots (does not achieve quality improvement) “Lot sentencing”
- Audit tool

Acceptance-Sampling Problem

- Three approaches to lot sentencing:
- Accept with no inspection
- 100% inspection
- Acceptance sampling

Acceptance-Sampling Problem

Why Acceptance Sampling and Not 100% Inspection?

- Testing is destructive
- Cost of 100% inspection is high
- 100% inspection is not feasible (require too much time)
- If vendor has excellent quality history

Acceptance-Sampling Problem

Advantages and Disadvantages of Sampling

Advantages

- Less expensive
- Reduced damage
- Reduces the amount of inspection error

Disadvantages

- Risk of accepting “bad” lots, rejecting “good” lots.
- Less information generated
- Requires planning and documentation

Acceptance-Sampling Problem

Types of Sampling Plans

- There are variables sampling plans and attribute sampling plans (this chapter concentrates on attributes)
- Single sampling plan
- Double-sampling plan
- Multiple-sampling plan
- Sequential-sampling

Acceptance-Sampling Problem

Lot Formation

- Considerations before inspection:
- Lots should be homogeneous
- Larger lots more preferable than smaller lots
- Lots should be conformable to the materials-handling systems used in both the vendor and consumer facilities.

Acceptance-Sampling Problem

Random Sampling

- The units selected for inspection should be chosen at random.
- 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.

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 at least 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.

Single-Sampling Plans For Attributes

The OC Curve

- The operating-characteristic (OC) curvemeasures the performance of an acceptance-sampling plan.
- The OC curve plots the probability of accepting the lot versus the lot fraction defective.

Single-Sampling Plans For Attributes

Designing a Single-Sampling Plan with a Specified OC Curve

- Let the probability of acceptance be 1 - for lots with fraction defective p1.
- Let the probability of acceptance be for lots with fraction defective p2.
- Assume binomial sampling (with type-B OC curves) is appropriate.

Single-Sampling Plans For Attributes

Designing a Single-Sampling Plan with a Specified OC Curve

- The sample size n and acceptance number c are the solution to

Single-Sampling Plans For Attributes

Example

- Consider constructing a sampling plan for which
- p1 = 0.01
- = 0.05
- p2 = 0.06
- = 0.10
- N = 1000
- Using computer software or a graphical approach (using an appropriate binomial nomograph) it can be shown that the necessary values of n and c are 85 and 2, respectively.

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:
- Single
- Double
- Multiple
- Provisions for each type of sampling plan include
- Normal inspection
- Tightened inspection
- Reduced inspection

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

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 provide are used to determine the appropriate sampling scheme.

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

Description of the Standard

- Switching Rules
- Normal to tightened
- Tightened to normal
- Normal to reduced
- Reduced to normal
- Discontinuance of inspection

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

Procedure

- Choose the AQL
- Choose the inspection level
- Determine the lot size
- Find the appropriate sample size code letter from Table 14-4
- Determine the appropriate type of sampling plan to use (single, double, multiple)
- Enter the appropriate table to find the type of plan to be used.
- Determine the corresponding normal and reduced inspection plans to be used when required.

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

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 14-4 indicates the appropriate sample size code letter is K.
- From Table 14-5 for single-sampling plans under normal inspection, the normal inspection plan is n = 125, c = 2.

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

Discussion

- There are several points about the standard that should be emphasized:
- MIL STD 105E is AQL-oriented
- The sample sizes selected for use in MIL STD 105E are limited
- The sample sizes are related to the lot sizes.
- Switching rules from normal to tightened and from tightened to normal are subject to some criticism.
- A common abuse of the standard is failure to use the switching rules at all.

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

Discussion

- ANSI/ASQC Z1.4 or ISO 2859 is the civilian standard counterpart of MIL STD 105E.
- Differences include:
- Terminology “nonconformity”, “nonconformance”, and “percent nonconforming” is used.
- Switching rules were changed slightly to provide an option for reduced inspection without the use of limit numbers
- Several tables that show measures of scheme performance were introduced
- A section was added describing proper use of individual sampling plans when extracted from the system.
- A figure illustrating switching rules was added.

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