• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.1 no.1
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• pp.105-125
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• 1997

In this paper we consider multigrid algorithms for solving elasticity problems by using Wilson's nonconforming finite element method. We consider two types of intergrid transfer operators which is needed to define the multigrid algorithm and prove convergence of $\mathcal{W}$-cycle mutigrid algorithm and uniform condition number estimates for the variable $\mathcal{V}$-cycle multigrid preconditioner.

• Industrial Engineering and Management Systems
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• v.15 no.3
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• pp.231-238
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• 2016

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.