Assume that F is a nonlinear operator on a real Hilbert space H which is strongly monotone and Lipschitzian on a nonempty closed convex subset C of H. Assume also that C is the intersection of the fixed point sets of a finite number of nonexpansive mappings on H. We make a slight modification of the iterative algorithm in Xu and Kim (Journal of Optimization Theory and Applications, Vol. 119, No. 1, pp. 185-201, 2003), which generates a sequence {xn} from an arbitrary initial point x0 in H. The sequence {xn} is shown to converge in norm to the unique solution u* of the variational inequality, under the conditions different from Xu and Kim¡¦s ones imposed on the parameters. Applications to constrained generalized pseudoinverse are included. The results presented in this paper are complementary ones to Xu and Kim¡¦s theorems (Journal of Optimization Theory and Applications, Vol. 119, No. 1, pp. 185-201, 2003).

由近期的研究中發現變動所有參數的管制圖在偵測小幅度偏移時的速度比起傳統的舒華特管制圖來的快，許多文獻也討論到利用調適性管制技術同時監控製程的平均數和變異數。而在這份研究中，為了改善現有管制圖的偵測效率，依序提出了U-V管制圖以及Max-M管制圖來偵測單一製程與兩相依製程的平均數和變異數。採用AATS及ANOS來衡量管制圖的偵測績效，並利用馬可夫鏈推導計算得之。透過兩階段的範例來介紹所提出的管制圖的應用方法並將VP U-V管制圖、VP Max-M管制圖與FP Z(X-bar)-Z(Sx^2)管制圖加以比較。從所研究的數值分析中發現VP Max-M管制圖比另兩種管制圖的表現來的好，再加上只需要單一管制圖在使用上對工程師來說也較為簡便，因此建議Max-M管制圖値得在實務上被使用。 / Recent studies have shown that the variable parameters (VP) charts detect small process shifts faster than the traditional Shewhart charts. There have been many papers discussed adaptive control schemes to monitor process mean and variance simultaneously. In the study, to improve the efficiency and performance of the existing control charts, the U-V control charts and Max-M control charts are respectively proposed to monitor the process mean and variance for a single process and two dependent process steps. The performance of the proposed control charts is measured by using adjusted average time to signal (AATS) and average number of observations to signal (ANOS). The calculation of AATS and ANOS is derived by Markov chain approach. The application of the proposed control charts is illustrated by a numerical example for two dependent process steps, and the performance of VP U-V control charts, VP Max-M control charts and FP Z(X-bar)-Z(Sx^2) control charts is compared. From the results of data analyses, it shows that the VP Max-M control charts have better performance than VP U-V control charts and FP Z(X-bar)-Z(Sx^2) control charts. Furthermore, using a single chart to monitor a process is easier than using two charts for engineers. Hence, Max-M control charts are recommended in real industrial process.

近年來有許多研究發現，適應性管制圖在偵測製程或產品幅度偏移時的速度比傳統的舒華特管制圖來的快，許多文獻也討論到利用適應性管制技術同時監控製程的平均數和變異數。隨著科技的發達，許多產品在製造上更加精密，現今普遍使用的固定參數管制圖並無法有效率的偵測出製程失控，導致巨大的成本損失。為了改善現有管制圖的偵測效率與有效控制製程失控下的損失，我們提出了三種適應性加權損失管制圖，包括變動抽樣間隔(VSI)、變動樣本數與抽樣間隔(VSI)、變動管制參數(VP)來偵測單一製程與兩相依製程的平均數和變異數。採用製程發生變動後到管制圖偵測出異常訊息所需的平均時間(AATS)與所需的總觀測數(ANOS)來衡量管制圖的偵測績效，並利用馬可夫鏈推導計算得之。從數值分析中發現，適應性加權損失管制圖在「偵測小偏移幅度時的偵測效率」與「成本的控制」明顯比傳統管制圖表現的更好，再加上每一個製程僅需採用單一管制圖，對使用者也較為簡便並且容易理解，因此適應性加權損失管制圖在實務上是值得被推薦使用的。 / Recent research has shown that control charts with adaptive features detect process shifts faster than traditional Shewhart charts. In this article, we propose three kinds of adaptive weighted loss (WL) control charts, variable sampling intervals (VSI) WL control charts , variable sample sizes and sampling intervals (VSSI) WL control charts and variable parameters (VP) WL control charts, to monitor the target and variance on a single process step and two dependent process steps simultaneously. These adaptive WL control charts may effectively distinguish which process step is out-of-control. We use the Markov chain approach to calculate the adjusted average time to signal (AATS) and average number of observations to signal (ANOS) in order to measure the performance of the proposed control charts. From the numerical examples and data analyses, we find the adaptive WL control charts have better detection abilities and performance than fixed parameters (FP) WL control charts and FP Z(X-bar)-Z(Sx^2) and Z(e-bar)-Z(Se^2) control charts. We also proposed the optimal adaptive WL control charts using an optimization technique to minimize AATS when users cannot specify the values of the variable parameters. In addition, we discuss the impact of misusing weighted loss of outgoing quality control chart. In conclusion, using a single chart to monitor a process is inherently easier than using two charts. The WL control charts are easy to understand for the users, and have better performance and detection abilities than the other charts, thus, we recommend the use of WL control charts in the real industrial process.

[pt] Os gráficos de controle de processo criados por Shewhart na década de 20 e em uso até hoje são eficientes para sinalizar alterações de grande magnitude na característica de qualidade de um processo (por exemplo, desvios da ordem de mais de 2 desvios-padrão, no caso do gráfico de médias); já para alterações de menos magnitude, ele são mais lentos. Para estas últimas, são sabidamente mais eficientes os esquemas CUSUM e EWMA, bem como os gráficos adaptativos, de desenvolvimento bem mais recente, também chamados de gráficos de parâmentros variáveis, porque alguns ou todos os seus parâmetros (tamanho de amostra, intervalo de tempo entre amostras, e limites de controle) passam a variar durante a operação, em função da informação fornecida pela última amostra. Nesta pesquisa, é prposta a incorporação da estratégia de gráficos adaptativos (usando um intervalo de tempo entre amostras variável) ao esquema EWMA na busca de melhorias no desempenho de gráficos de controle por atributos. O esquema proposto é aplicado a gráficos de c para detecção de alterações de pequena magnitude no número médio de não-conformidades em um processo de produção. É desenvolvido o modelo matemático para cálculo das medidas de desempenho do gráfico, e é realizada a análise de desempenho do esquema para diversos valores de c0 e c1 (número médio em controle e fora de controle de não- conformidades), com comparação com outros gráficos de controle por atributos. Resultados mostram, na maioria das situações analisadas, a vantagem do esquema proposto, em termos de uma maior rapidez de detecção de alterações de diversas magnitudes. / [en] The process control charts created by Shewhart in the 20 s and still in use today are efficient in signaling large shifts in the quality characteristics of a process (e.g. shifts greater than two standard deviations, in the case of the chart for means); they are however slower in the case of small and moderate shifts, in which case CUSUM and EWMA schemes are known to be more efficient, as are the recently developed adaptive charts, also called variable parameter charts because some or all of their design parameters (sample size, sampling interval and control limits) are allowed to vary during the operation, according to the information of the latest sample. In this thesis, looking for an enhancement in the performance of control charts for attributes, the strategy of adaptive charts (using a variable sampling interval) is incorporated to the EWMA scheme. The proposed scheme is applied to c charts for detecting small shifts in the number of nonconformities in a production process. A mathematical model is developed for calculation of the performance measures of the chart, and a performance analysis is carried out for several values of c0 and c1 (in- and out-of-control number of nonconformities), together with a comparison with other control charts for nonconformities. The results show the advantage of the proposed scheme in the majority of the analyzed situations, through faster detection of a range of shifts.

This article proposes the algorithm of a new control chart (loss function control chart) based on the Taguchi loss function with an adaptive scheme for binomial data. The loss function control chart is able to monitor cost variation from the process by applying loss function in the design. This new angle economically explores production cost. This research provides designs of the loss function control chart with specified VSI, optimal VSI, VSS and VP, respectively. Numerical analyses show that the specified VSI loss function chart, the optimal VSI loss function chart, the optimal VSS loss function chart and the optimal VP loss function chart outperform the Fp loss function chart significantly and show costs can be controlled systematically.