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Improved inspection scheme with a loss-based capability index

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Abstract

Acceptance sampling is a practical quality tool for lot sentencing and is used primarily for incoming or outgoing inspections. The repetitive group sampling plan (RGSP) has been developed and shown to be superior to the traditional single sampling plan. However, RGSP’s lot disposition does not consider the valuable information from preceding lots, which may reduce its discriminatory power and sampling efficiency. Therefore, we propose a modified RGSP for variables inspection using the loss-based capability index. It uses not only the current sample information but also the results of preceding lots to sentence the current lot. The proposed plan’s operating characteristic (OC) function is derived using the estimated index’s exact sampling distribution. An optimization model is formulated to solve the plan parameters by minimizing the average sample number (ASN) required for inspection. The proposed plan’s advantages over the traditional sampling plans in terms of the OC curve and the ASN are addressed. Moreover, plan parameters are tabulated under various conditions and an application example is presented to illustrate its applicability.

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Funding

This work was partially supported by the Ministry of Science and Technology of Taiwan under Grant No. MOST 103-2221-E-007-103-MY3.

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Authors and Affiliations

  1. Department of Industrial Engineering and Systems Management, Feng Chia University, Taichung, Taiwan

    Zih-Huei Wang

  2. Department of Industrial Engineering and Engineering Management, National Tsing Hua University, Hsinchu, Taiwan

    Chien-Wei Wu

Authors
  1. Zih-Huei Wang
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  2. Chien-Wei Wu
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Correspondence to Chien-Wei Wu.

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Appendices

Appendix 1. Derivation of π A(C pm) in Eq. (12)

According to the proposed plan’s operational procedure, the eventual probability of accepting a lot (shown in Eq. (12)), πA(Cpm), can be derived as follows:

$$ {\displaystyle \begin{array}{l}{\pi}_A\left({C}_{pm}\right)={P}_1\left({C}_{pm}\right)+{P}_2\left({C}_{pm}\right){P}_a\left({C}_{pm}\right)+{P}_2\left({C}_{pm}\right)\left[1-{P}_a\left({C}_{pm}\right)-{P}_r\left({C}_{pm}\right)\right]{P}_a\left({C}_{pm}\right)\kern0.3em \\ {}\kern4.899998em +{P}_2\left({C}_{pm}\right){\left[1-{P}_a\left({C}_{pm}\right)-{P}_r\left({C}_{pm}\right)\right]}^2{P}_a\left({C}_{pm}\right)+...\kern0.4em \\ {}\kern4.099998em ={P}_1\left({C}_{pm}\right)+\sum \limits_{r=0}^{\infty }{P}_2\left({C}_{pm}\right){\left[1-{P}_a\left({C}_{pm}\right)-{P}_r\left({C}_{pm}\right)\right]}^r{P}_a\left({C}_{pm}\right)\\ {}\kern4.099998em ={P}_1\left({C}_{pm}\right)+\frac{P_a\left({C}_{pm}\right){P}_2\left({C}_{pm}\right)}{1-\left[1-{P}_a\left({C}_{pm}\right)-{P}_r\left({C}_{pm}\right)\right]}\\ {}\kern3.999998em =\frac{P_a\left({C}_{pm}\right)+\left\{{P}_r\left({C}_{pm}\right)\left[1-{P}_a\left({C}_{pm}\right)-{P}_r\left({C}_{pm}\right)\right]{\left[{P}_a\left({C}_{pm}\right)\right]}^m\right\}}{P_a\left({C}_{pm}\right)+{P}_r\left({C}_{pm}\right)}.\end{array}} $$

Appendix 2. Derivation of ASN function in Eq. (15)

The ASN function for the proposed plan expressed in Eq. (15) can be derived as follows:

$$ {\displaystyle \begin{array}{l}\mathrm{ASN}\left({C}_{pm}\right)=n+n{P}_2\left({C}_{pm}\right)+n{P}_2\left({C}_{pm}\right)\left[1-{P}_a\left({C}_{pm}\right)-{P}_r\left({C}_{pm}\right)\right]\kern0.2em \\ {}\kern5.599997em +n{P}_2\left({C}_{pm}\right){\left[1-{P}_a\left({C}_{pm}\right)-{P}_r\left({C}_{pm}\right)\right]}^2+\kern0.3em ...\kern0.3em \\ {}\kern5.199997em =n+\sum \limits_{r=0}^{\infty }n{P}_2\left({C}_{pm}\right){\left[1-{P}_a\left({C}_{pm}\right)-{P}_r\left({C}_{pm}\right)\right]}^r\\ {}\kern4.999998em =n+\frac{n{P}_2\left({C}_{pm}\right)}{1-\left[1-{P}_a\left({C}_{pm}\right)-{P}_r\left({C}_{pm}\right)\right]}=\frac{n\left\{1-\left[1-{P}_a\left({C}_{pm}\right)-{P}_r\left({C}_{pm}\right)\right]{\left[{P}_a\left({C}_{pm}\right)\right]}^m\right\}}{P_a\left({C}_{pm}\right)+{P}_r\left({C}_{pm}\right)}.\end{array}} $$

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Wang, ZH., Wu, CW. Improved inspection scheme with a loss-based capability index. Int J Adv Manuf Technol 104, 1321–1331 (2019). https://doi.org/10.1007/s00170-019-03980-z

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