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Product control in statistical quality control is the methodology that deals with procedures for taking decisions about one or more lots of finished products manufactured by production processes. Sampling inspection by variables is one of the major classifications of product control and comprises procedures for deciding about the disposition of a lot of individual units based on sample measurements of units on a quality characteristic under study. These procedures are defined under the assumption that the quality characteristic is measurable on a continuous scale, and the functional form of the probability distribution must be known. Inspection procedures which have been developed under the implicit assumption that the quality characteristic is distributed as normal with the related properties are found in the literature of product control. The assumption of normality may not be realized often in practice, and it becomes indispensable to investigate the properties of sampling plans based on nonnormal distributions. In this paper, a single sampling plan by variables is formulated and evaluated when the quality characteristic is assumed to be distributed according to a Pareto distribution. A procedure is developed for determining the parameters of the plan for specified requirements to ensure protection to both the producer and consumer.

Sampling inspection is an activity for taking decisions on one or more lots of finished products which have been submitted for inspection. The decision of either acceptance or rejection of the lots is usually taken by adopting suitable sampling inspection procedures called sampling plans. Sampling plans are generally categorized into two types, namely, lot-by-lot sampling by attributes and lot-by-lot sampling by variables. In lot-by-lot inspection by attributes, one or more samples of items are drawn from a given lot of manufactured items; each item in the sample(s) is classified as conforming or nonconforming, and the decision of acceptance or rejection of the lot is made based on a specific rule. In lot-by-lot inspection by variables, one or more samples of items are drawn from a given lot; the measurement of a quality characteristic in each sampled item is recorded, and the decision of acceptance or rejection of the lot is made as a function of such measurements. The theory of inspection by variables is applicable when the quality characteristic of sampled items is measurable on a continuous scale and the functional form of the probability distribution is assumed to be known. A variables sampling is advantageous in the sense that it generates more information from each item inspected, requires small sample, and provides same protection when compared to attributes sampling. (See Bowker and Goode [

On the basis of the implicit assumption that the quality characteristic is distributed according to normal with mean

When the quality characteristic

In this paper a study on single sampling plans by variables is formulated under the assumption that the quality characteristics have a Pareto distribution. A procedure for determining the parameters of the proposed plan for specified requirements in terms of producer’s and consumer’s protection is also developed.

A single sampling inspection plan by variables is defined under the following assumptions.

The quality characteristic, denoted by

Each individual unit in a submitted lot has a one-sided specification say, lower specification,

The operating procedure of a variable sampling plan is as follows.

Draw a random sample of

When

Thus, a single sampling plan by variables is designated by two parameters, namely, the sample size,

An important measure of the performance of a variables sampling plan is its operating characteristic function, which is a function of the proportion

Assume that

The mean and the variance of the underlying distribution are given by

The producer’s risk

When

In the industrial practice the unknown standard deviation variables plans are more realistic than the known standard deviation variables plans. If the distribution is non-normal, then the designing of unknown

The methodology proposed in [

In the case of unknown sigma plan, the determination of

It is known that

The operating characteristic function

On substituting the expressions (

The mathematical expressions for

Suppose that a set of measurements yields

It is known that Pareto distribution is a very heavy-tailed distribution for which the

Sampling inspection by variables is the methodology used for deciding about the disposition of a lot based on sample measurements of units on a quality characteristic under study. Such a procedure is defined under the assumption that the quality characteristic is measurable on a continuous scale and the functional form of the probability distribution is known. Though sampling plans by variables under the implicit assumption that the quality characteristic is distributed as normal have been developed with the related properties, it should be realized that the assumption of normality is seldom hold in practice. Hence, it is inevitable to develop sampling plans by variables when the underlying quality characteristics follow non-normal distributions, which may be skewed or symmetric. In this paper a single sampling plan by variables is formulated and evaluated when the quality characteristic is distributed according to a Pareto distribution. The approximation of normal distribution for a linear combination of independent variables even when the individual variables depart from normality is effectively used in developing the procedure for determining the parameters of the plan when requirements for producer’s and the consumer’s protection are specified.

The authors are grateful to the Editor/Editorial Board Member of the journal and anonymous referees who have made significant suggestions for the improvements in the content of the paper. They also acknowledge University Grants Commission, New Delhi, India, for providing financial support to carry out this work.