# Comparison of Rigorous Design Procedure with Approximate Design Procedure for Variable Sampling Plans Indexed by Quality Loss

• Ishii, Yoma (Graduate School of Natural Science and Technology, Okayama University) ;
• Arizono, Ikuo (Graduate School of Natural Science and Technology, Okayama University) ;
• Tomohiro, Ryosuke (Graduate School of Natural Science and Technology, Okayama University) ;
• Takemoto, Yasuhiko (Faculty of Management and Information Systems, Prefectural University of Hiroshima)
• Received : 2016.03.31
• Accepted : 2016.08.08
• Published : 2016.09.30
• 55 15

#### Abstract

Traditional acceptance sampling plans have focused on the proportion of nonconforming items as an attribute criterion for quality. In today's modern quality management under high quality production environments, the reduction of the deviation from a target value in a quality characteristic has become the most important purpose. In consequence, various inspection plans for the purpose of reducing the deviation from the target value in the quality characteristic have been investigated. In this case, a concept of the quality loss evaluated by the deviation from the target value has been accepted as the variable evaluation criterion of quality. Further, some quality measures based on the quality loss have been devised; e.g. the process loss and the process capability index. Then, as one of inspection plans based on the quality loss, the rigorous design procedure for the variable sampling plan having desired operating characteristics (VS-OC plan) indexed by the quality loss has been proposed by Yen and Chang in 2009. By the way, since the estimator of the quality loss obeys the non-central chi-square distribution, the rigorous design procedure for the VS-OC plan indexed by the quality loss is complicated. In particular, the rigorous design procedure for the VS-OC plan requires a large number of the repetitive and complicated numerical calculation about the non-central chi-square distribution. On the other hand, an approximate design procedure for the VS-OC plan has been proposed before the proposal of the above rigorous design procedure. The approximate design procedure for the VS-OC plan has been constructed by combining Patnaik approximation relating the non-central chi-square distribution to the central chi-square distribution and Wilson-Hilferty approximation relating the central chi-square distribution to the standard normal distribution. Then, the approximate design procedure has been devised as a convenient procedure without complicated and repetitive numerical calculations. In this study, through some comparisons between the rigorous and approximate design procedures, the applicability of the approximate design procedure has been confirmed.

#### Keywords

Non-Central Chi-Square Distribution;Operating Characteristics;Patnaik Approximation;Wilson-Hilferty Approximation

#### Acknowledgement

Grant : Proposal of quality management system based on the information and communication technology (ICT) towards the next generation production system

Supported by : JSPS

#### References

1. Arizono, I., Kanagawa, A., Ohta, H., Watakabe, K., and Tateishi, K. (1997), Variable Sampling Plans for Normal Distribution Indexed by Taguchi's Loss Function, Naval Research Logistics, 44, 591-603. https://doi.org/10.1002/(SICI)1520-6750(199709)44:6<591::AID-NAV5>3.0.CO;2-Z
2. Arizono, I., Miyazaki, T., and Takemoto, Y. (2014), Variable Sampling Inspection Plans with Screening Indexed by Taguchi's Quality Loss for Optimising Average Total Inspection, International Journal of Production Research, 52, 405-418. https://doi.org/10.1080/00207543.2013.828175
3. Arizono, I., Okada, Y., Tomohiro, R., and Takemoto, Y. (2016), Rectifying Inspection for Acceptable Quality Loss Limit Based on Variable Repetitive Group Sampling Plan, International Journal of Advanced Manufacturing Technology, 85, 2413-2423. https://doi.org/10.1007/s00170-015-8090-2
4. Morita, M., Arizono, I., and Takemoto, Y. (2009), Variable Sampling Inspection Plans with Screening for Assuring Average Outgoing Surplus Quality Loss Limit Indexed by Taguchi's Loss, International Journal of Advanced Manufacturing Technology, 41, 908-915. https://doi.org/10.1007/s00170-008-1549-7
5. Patnaik, P. B. (1949), The Non-central ${\chi}^2$ - and F-Distributions and Their Applications, Biometrika, 36, 202-232.
6. Seifi, S. and Nezhad, M. S. F. (2016), Variable Sampling Plan for Resubmitted Lots Based on Process Capability Index and Bayesian Approach, International Journal of Advanced Manufacturing Technology, DOI: 10.1007/s00170-016-8958-9, First online: 03 June 2016. https://doi.org/10.1007/s00170-016-8958-9
7. Sharman, R. E. (1965), Design and Evaluation of a Repetitive Group Sampling Plan, Technometrics, 7, 11-21. https://doi.org/10.1080/00401706.1965.10490222
8. Subramani, J. and Balamurali, S. (2016), A Modified Single Sampling Plan for the Inspection of Attribute Quality Characteristics, Industrial Engineering and Management Systems, 15, 41-48. https://doi.org/10.7232/iems.2016.15.1.041
9. Suzuki, Y., Takemoto, Y., and Arizono, I. (2009), Variable Sampling Inspection with Screening When Lot Quality Follows Mixed Normal Distribution, Industrial Engineering and Management Systems, 8, 131-138.
10. Taguchi, G. (1985), A Tutorial on Quality Control and Assurance-the Taguchi Methods, Annual Meeting, Las Vegas, Nevada.
11. Tomohiro, R. Arizono, I., and Takemoto, Y. (2013), Design of Variable Repetitive Group Sampling Plan on Operating Characteristics Indexed by Quality Loss, Proc. of the 14th Asian Pacific Industrial Engineering and Management Systems Conference (APIEMS).
12. Tomohiro, R., Arizono, I., and Takemoto, Y. (2016), Proposal of Variable Sequential Sampling Plan Having Desired Operating Characteristics Indexed by Quality Loss, International Journal of Production Research, 54, 5742-5760. https://doi.org/10.1080/00207543.2016.1173249
13. Wald, A. (1947), Sequential Analysis, John Wiley and Sons, New York, NY.
14. Wald, A. and Wolfowitz, J. (1948), Optimum Character of the Sequential Probability Ratio Test, The Annals of Mathematical Statistics, 19, 326-339. https://doi.org/10.1214/aoms/1177730197
15. Wilson, E. B. and Hilferty, M. M. (1931), The Distribution of Chi-square, Proceedings of the National Academy of Sciences of the United States of America, 17, 684-688. https://doi.org/10.1073/pnas.17.12.684
16. Yen, C.-H. and Chang, C.-H. (2009), Designing Variable Sampling Plans with Process Loss Consideration, Communications in Statistics-Simulation and Computation, 38, 1579-1591. https://doi.org/10.1080/03610910903046809