| Summary: | 碩士 === 中原大學 === 工業工程研究所 === 91 === This thesis considers the economical-statistical design of Xbar-control chart assuming that the quality characteristic measurement (i.e. observations) are nonnormal and the in-control time is Weibull. When designing a control chart, three parameters-the sample size n, time h between successive samples, and control-limit factor k-must be determined. In economic-statistical design, the three parameters are chosen so that the expected cost per hour is minimized under constraints on Type I and Type II error probabilities. The cost function is computed by the McWilliams cost model.
Although some nonnormal literature exists, the assumption is made on the distribution of sample average, which depends on the unknown sample size. We assume that the quality characteristic measurement are sampled independently from a Johnson distribution. The Johnson distribution is general in that it can be modeled to fit all possible values of the skewness and kurtosis. This is a stochastic optimization problem because Type I and Type II error probabilities is difficult to compute and need to be estimated via simulation experiment. Hence, a stochastic optimization algorithm is needed.
We propose an algorithm of stochastic optimization, revised retrospective optimization flexible tolerance method, to solve our optimization problem including one discrete and two continuous decision variable vector X ={n, h, k}. Empirical results show that the solution of Revised RO-FT is very close to the true optimum in our testing problems. Finally, we perform the sensitivity analysis of the nonnormality and Weibull effect on the optimal values of {n, h, k}. Conclusion of sensitivity analysis are as follows.
When skewness is constant and equal to zero, an increase on kurtosis leads to increases on sample size n, decreases on time h between successive samples, as well as a wider control-limit factor k. When skewness is constant and not only greater than but also close to zero (as well as far away zero), an increase on kurtosis leads to increases on n, decreases on h, as well as a wider k. When kurtosis is constant and close to normal (as well as a little far away normal), an increase on skewness leads to decreases on n, increases on h, as well as a narrower k. When kurtosis is constant and far away normal, an increase on skewness leads to no signidicant effect on n, increases on h, as well as a narrower k. When an increase both on skewness and kurtosis leads to increases on n and h, as well as a wider k.
Weibull shape parameter heta and scale parameter lambda are sensitive when shape parameter heta<=1 on the decision variable time h between successive samples. When an increase on shape parameter heta leads to increases on h.
Keywords: Economic-Statistical Design; Expected Cost Per Hour; Revised Retrospective Optimization Flexible Tolerance Method; Nonnormality; Xbar Control Chart.
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