ABSTRACT

Modern acceptance sampling involves a system of principles and methods. Their purpose is to develop decision rules to accept or reject product based on sample data. Factors are:

1. The quality requirements of the product in the marketplacebest rolex clone 2021 audemars piguet watch replica www replica watch
2. The capability of the process
3. The cost and logistics of sample taking

This article uses the Two-Point method of sampling plan design, whereby the person designing the plan specifies two points on the operating characteristic curve. The following important visualization tools of this method will be explained:

1. Operating characteristic curves (OC-Curves)
2. Decision rules

It will show you how you can benefit by matching different types of sampling plan to the same oc-curve: attributes to variables, fixed-n to sequential, unknown to known .

USING MARKET REQUIREMENTS and PROCESS KNOWLEDGE

You can avoid the horrors of the product recall experience by applying the following principles. They involve methods of product acceptance and process control.

Rational Lots

Define homogeneous lots of product items. Use your knowledge of the patterns of the process to group the items into "natural lots". Lots based on purchase order alone must be subdivided into such natural lots for accurate acceptance decisions.

Known Standard Deviation

Apply known plans for variables where feasible. These plans use knowledge of the producer's process to reduce inspection cost. [ (sigma) = standard deviation].

Process Performance and Capability

Set an acceptable quality level (AQL) that "recognizes" the current process capability. This requires that you know the attainable levels of the process.

Market Requirement

Set a rejectable quality level (RQL) that enforces the customer requirement.

At rejection, you will know that:

1. The lot is unacceptable, and
2. The process has deviated from its known capability
   (Therefore start corrective action on the process.)

APPLYING OC-CURVE to EVALUATE SAMPLING PLANS

Figure 1. OC-Curve. (Operating Characteristic)

The Operating Characteristic (OC) curve shows the probability of acceptance, Pa, for any level of lot quality. See Figure 1. On the horizontal axis is the quality characteristic.

This oc-curve enables you to evaluate the probability of acceptance for any true lot quality level-on a what-if basis. This way, you can design sampling plans that perform the way you want.

Interpret the curve according to this example:

1. If the lot quality is 0.093 fraction defective, then the probability of acceptance, Pa, is 0.05.
2. If the lot quality is 0.018 fraction defective, then the probability of acceptance, Pa, is 0.95.
   

APPLYING OC-CURVES to COMPARE SAMPLING PLANS

Figure 2. Comparing Alternative Plans, A and B

You can use oc-curves to compare alternative plans. See Figure 2. Choose between the plans by their relative ability to detect rejectable lots. You should expect that the steeper the curve, the larger the sample size.

Complete this picture by comparing the costs of the sampling to the resulting performance.

 

WORKSHEET #1:
Develop an OC-Curve

APPLYING OC-CURVES to DESIGN -- THE TWO-POINT METHOD

The Two-Point method for developing acceptance sampling plans requires that you specify two points of the operating characteristic curve (oc-curve).

Producer's Point

Figure 3. The Two-Point Method

The producer's point controls the acceptance of lots that are at an acceptable quality level. (See figure 3) The goal: prevent good lots from being rejected.

Consumer's Point

The consumer's point controls the rejection of lots that are at a rejectable quality level. (See figure 3) The goal: prevent bad lots from being accepted.

Decision Table Defines the Two Points

Figure 4. Decision Table

An OC Curve helps you to address the producer's and consumer's points in a visual, right brain way. On the other hand, the decision table (See figure 4) defines the situation in a logical, left brain way. The Type I and Type II errors correspond logically to the two points on the oc-curve.

Type I error -- Wrongful Rejection

A type I error is associated with the producer's point -- to reject when the true value of the quality characteristic is AQL. The risk of rejecting an AQL lot is the producer's risk (alpha risk)

Type II error -- Wrongful acceptance

A Type II error is to accept when the true value of the quality characteristic is RQL -- at the consumer's point. The risk of accepting a lot, if it is an RQL lot, is the consumer's risk (ߠ= beta risk).

Summary of The Two-Point Method:

Choose alpha=0.05, beta=0.05, AQL, and RQL. These determine the sample size (n) and the decision limit(s). Then apply any process knowledge (within-lot standard deviation) and market requirements (ISLs).

WORKSHEET #2:
Develop a Fixed-n Sampling Plan

(Use of the program TP414 or the Variables Nomograph )

Determine the variables fixed-n sampling plan.

Product Requirement: Lower ISL=0
Process Capability: = 1 (known historical within-lot standard deviation)

Sampling Requirement

Producers Point: AQL=0.01, alpha=0.05
Consumers Point: RQL=0.10, beta=0.05

Your Task: use the program TP414 or the Variables Nomograph to determine:

Sample Size: n =_____
Decision Limit: K =_____

Useful definitions: K = (Xbar-LISL) / sigma
                     and: K = 3 * C_pk

Note: If you use the nomograph, use vertical n-lines for known sigma plans. (See scale at bottom of nomograph.)

SOME GUIDELINES FOR APPLICATION

No principle is without exceptions, but the following guidelines are generally useful to get you started in a sampling application.

How to choose the Producer's Point in practice

1. You should expect that lots at the producer's point quality level (AQL) will be accepted most of the time. Define AQL accordingly. Take into account historical quality levels.
2. Choose the producers risk of rejecting a lot that is of AQL quality. Typical: = 0.05.

How to choose the Consumer's Point in practice

1. You should expect that lots at the consumer's point quality level (RQL) will be rejected most of the time. Define RQL accordingly.
2. Choose the consumers risk of accepting a lot that is of RQL quality. Typical: = 0.05.

Classify Quality Characteristics

You do not have to inspect all quality characteristics to the same sampling requirement. Some characteristics need less inspection than others. Some will not require testing at all, and others might require 100% inspection.

Resources for Two-Point Calculation

The Appendix contains references and nomographs for developing two-point variables and attribute sampling plans. The bibliography below refers to the Two-Point software programs TP105, TP414, and TP781 that facilitate these calculations.

The Advantages of Two-Point Decision Rules

The Two-Point method develops plans based on desired performance of the decision rule. Unlike Mil-Standard AQL plans, their method of selection does not involve esoteric 'inspection levels' and 'code letters'. The underlying statistics is the same.

A Philosophical Issues with Mil-Standard AQL Plans

The Mil-Standard AQL plans require you only to choose the producers point-not the consumer's. Actually, the consumer's point is usually more important. As a producer your main purpose is to prevent acceptance of rejectable or recallable quality lots.

A Technical Issue with Lot Size

Many sampling plan techniques do not use lot size to calculate the oc-curve. This includes Mil-Standard 105 and 414, and the software programs TP105 and TP414.. This practice has negligible effect when sampling large lots with small samples. A rule of thumb for accuracy of the oc-curve calculation is: N>10*n. (n=sample size, N=lot size)

MATCHING the OC CURVES of DIFFERENT TYPES OF PLAN