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zB1Ý{ { mathcad/worksheet.xml ( Converted from "Manufacturing Performance Curves in Mathcad" created by Mathcad Staff Industrial Engineering:Operating Characteristic CurvesThis model plots the acceptance probability, average outgoing quality, and average total inspection for a single sampling plan.This application uses a single sampling plan: Extract a sample of size n and accept the lot if c or fewer nonconforming parts are found.In the model illustrated here, the lot size N is many times larger than the sample size, and the number of defectives in a sample of size n is approximately Poisson distributed with mean n×p.If the lot size is small then the hypergeometric distribution provides a good model for the sampling process. This is discussed in the section Using the Application.BackgroundWhen the output of a manufacturing process is sampled to control the quality of the items shipped, the performance of the sampling plan is often described by plotting three performance curves:·the lot acceptance probability·the average outgoing quality·the average total inspectionConsider the following typical single sampling plan: From each lot of size N a sample of size n is taken. The lot is accepted for shipment if c or fewer defective items are found. If more than c defective items are found, the entire lot is inspected, and all defective items are replaced before the lot is shipped.Lot Acceptance Probability The probability that a lot will be accepted is a function of the actual fraction of defectives in the output from the manufacturing process. This fraction, p, is unknown. For a given sample size n and acceptance criterion c, the operating characteristic curve shows PA(p), the probability that the lot will be accepted, as a function of p.In the model illustrated here, the lot size N is many times larger than the sample size, and the number of defectives in a sample of size n is approximately Poisson distributed with mean n×p. The acceptance probability, PA(p), is the sum of the probabilities of finding each possible number of defectives between 0 and c.Average Outgoing Quality (AOQ)The outgoing product consists of lots that were accepted, which contain a fraction p of defectives, and lots that were rejected, which, after replacement, have no defectives. Thus the average fraction of defectives in the shipped product is PA(p)×p. This quantity, which is again a function of p, is called the average outgoing quality or AOQ. The maximum of the AOQ curve is called the AOQ limit, and represents the worst-case quality for the outgoing product.If p is very small, most lots are accepted, and the quality in all lots is high. If p is close to 1, most lots will be rejected, and all defectives in these lots are replaced, so again the outgoing product will have few defectives. Thus the AOQ limit typically occurs at some intermediate value of p.Average Total InspectionAnother quantity of interest is the average number of items in each lot that are inspected. This average total inspection is one measure of the cost of the sampling procedure. A sample of size n is drawn from every lot, but if the lot is rejected, the rest of the lot must also be inspected. Thus the average total inspection ranges between n and N, and takes on these extreme values when p is 0 or 1.Mathcad ImplementationThis application generates the three standard performance curves for the following single sampling plan: Extract a sample of size n and accept the lot if c or fewer nonconforming parts are found.Define the sampling plan by choosing values for the sample size n and the acceptance criterion c.The operating characteristic curve calculated below shows the probability of acceptance versus the fraction of defectives (p), for all possible values of p. A Poisson distribution for the number of defective parts is assumed.Note: The if construction in the equation for PA(p) insures that the function will take the value 1 when p is equal to 0.lines11011101---- Probability of acceptance v. defect rateThe average outgoing quality represents the average fraction of defectives in the shipped product. The AOQ curve is calculated below.lines1101110.3---- Average outgoing quality v. defect rateThe AOQ limit is given by the maximum of this curve. Mathcad's numerical derivative operator and the root function are used to find pql, the value of p at which the AOQ curve reaches a maximum. The corresponding AOQ value is the AOQ limit.The AOQ limit isAssume that, for a lot of size N, n components are sampled in those lots that pass and N components must be inspected for those lots that fail. The average total inspection curve is calculated below.Define lot size:lineslines1101110N---- Average total inspection curve---- sample size nUsing the ApplicationSince the sampling is done without replacement, the probability of finding a given number of defectives is not well modeled by the Poisson distribution when the sample size is an appreciable fraction of the lot size. In this "small lot" case, the hypergeometric distribution provides a good model for the sampling process.The application below has been modified to model sampling from small lots using the hypergeometric distribution. Define the sampling plan (n = sample size, c = acceptance criterion):Define a lot to be sampled ( N = lot size, D = number of defectives):Generate the Operating Characteristic Curvelines11111---- Probability of acceptance v. defect rateGenerate the Average Outgoing Quality (AOQ) curvelines11113---- Average outgoing quality v. defect rateThe AOQ limit is:Generate the Average Total Inspection curvelineslines111110N---- Average total inspection curve---- sample size nAdapted from the MathSoft Electronic Book Topics in Mathcad: Statistics.PK
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