## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 26, Number 4 (1955), 686-704.

### Multi-Level Continuous Sampling Plans

Gerald J. Lieberman and Herbert Solomon

#### Abstract

In 1943 Dodge [1] published a sampling inspection plan for a continuous production line. He assumed the production process to be in statistical control and also assumed the items were classified, after measurement, as "defective" or "non-defective". Dodge derived the Average Outgoing Quality (AOQ) function for his plan, obtained the Average Outgoing Quality Limit (AOQL), and provided a graphical procedure for choosing the parameters of the plan which guarantee a specified AOQL. Wald and Wolfowitz [2], in 1945, discussed a sampling inspection plan for continuous production which insures a prescribed limit on the outgoing quality even when production is not in statistical control. However, they demonstrate an awareness of the penalty involved in accomplishing this end and discuss other desirable features an optimal plan should enjoy, namely, a minimum amount of inspection to reduce inspection costs, and protection to insure what they term "local stability", i.e., the ability to detect quickly "too many long sequences" of poor quality. Dodge in his paper also discusses minimum inspection and an idea similar to "local stability" which he calls "protection against spotty quality". An inconvenient feature of both plans is the abrupt change between partial inspection and 100% inspection. This can lead to hardships in personnel assignments in the administration of an inspection program. For example, in an item such as aircraft engines, a smoother transition to 100% inspection is needed. Both plans also tend to produce a form of tightened inspection when the process average may not warrant it. In a later paper [3] Dodge considers two modifications of his plan which delay the beginning of 100% inspection and also add some insurance for local stability. He derived the AOQ function for each of the two plans. The primary purpose of this paper is to consider an extension of Dodge's first plan which (a) allows for smoother transition between sampling inspection and 100% inspection, (b) requires 100% inspection only when the quality submitted is quite inferior, and (c) allows for a minimum amount of inspection when quality is definitely good. This aim is accomplished by the introduction of a multi-level sampling plan which specifically allows for any number of sampling levels subject to the provision that transitions can only occur between adjacent levels. This inspection plan will be recognized as a random walk model with reflecting barriers. The first Dodge plan is easily recognized as a special case containing only one sampling level. The AOQ function for the plan is derived and contours of constant AOQL are developed for a two-level and an infinite-level plan. These are added to the contours of constant AOQL for Dodge's single-level plan to present a picture in Figs. 1, 2 and 3 reflecting the relationship between a fixed AOQL contour and the number of sampling levels used in the plan. In addition an approximation procedure is presented for determining contours of constant AOQL when the number of sampling levels lies between three and infinity. For a desired AOQL and a given process average, criteria for selecting a specific multi-level plan are discussed.

#### Article information

**Source**

Ann. Math. Statist., Volume 26, Number 4 (1955), 686-704.

**Dates**

First available in Project Euclid: 28 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177728428

**Digital Object Identifier**

doi:10.1214/aoms/1177728428

**Mathematical Reviews number (MathSciNet)**

MR75496

**Zentralblatt MATH identifier**

0065.12502

**JSTOR**

links.jstor.org

#### Citation

Lieberman, Gerald J.; Solomon, Herbert. Multi-Level Continuous Sampling Plans. Ann. Math. Statist. 26 (1955), no. 4, 686--704. doi:10.1214/aoms/1177728428. https://projecteuclid.org/euclid.aoms/1177728428