In this study, we have conducted comparative analysis between false alarm rate (FAR) and average run length (ARL) based control charts with runs rules. In this regard, we have considered various univariate and multivariate control charts which include mean, standard deviation, variance, Hotelling, and generalized variance. For evaluation purpose, we have used actual false alarm rate, power, in-control actual average run length, and out-of-control average run length as performance indicators. Furthermore, the performance indicators are calculated through Monte Carlo simulation procedures. Results revealed that performance order of runs rules with FAR based control charts are persistent whereas, performance order of runs rules with ARL based control charts are dependent on the circumstances, that is, sample size, size of shift, type of control chart, and side of control limit (upper-sided and lower-sided). Besides, we have provided a real life example using the data on electrical resistance of insulation. In this approach, we have determined that behaviour of FAR and ARL based control charts using the real data is recorded similar to the behavior using the statistical performance indicators.

The prospects of meeting the future's high energy demands lie in the exploration of unconventional hydrocarbon reservoirs, of which the shale gas and the tight gas are two important resources. The deep understanding of such reservoirs is crucial to the economical recovery of such energy resources. With the advancement in the technological sides, such as, hydraulic fracturing and horizontal drilling, new mathematical models are needed that can precisely capture the complexity of the physical phenomena and can describe the flow of gas through the natural and induced fractures. The performance and future behavior of such reservoirs can be enhanced through careful modeling. We develop a new mathematical model based on time fractional derivative combined with the consideration of various flow regimes and a nonlinear treatment of reservoir parameters. The model describes the transport of gas in tight porous media (such as shale formations). The derivation of the model is done by using the mass balance equation and momentum conservation equation (basically modified time-fractional form of Darcy's law) which incorporates the properties of tight porous media and accounts on the previous behavior. We find the pressure equation by considering that the rock properties, such as, permeability, viscosity, porosity, are pressure dependent. The pressure equation can be used to study the pressure distribution in the reservoir.

We propose an efficient preconditioning strategy to accelerate the convergence of Krylov subspace methods, specifically for solving complex nonlinear systems with a block five-by-five structure, commonly found in cell-centered finite difference discretizations for image deblurring using mean curvature techniques. Our method introduces two innovative preconditioned matrices, analyzed spectrally to show a favorable eigenvalue distribution that accelerates convergence in the Generalized Minimal Residual (GMRES) method. This technique significantly improves image quality, as measured by peak signal-to-noise ratio (PSNR), and demonstrates faster convergence compared to traditional GMRES, requiring minimal CPU time and few iterations for exceptional deblurring performance. The preconditioned matrices’ eigenvalues cluster around 1, indicating a beneficial spectral distribution. The source code is available at https://github.com/shahbaz1982/Precondition-Matrix.

In this study, efficient techniques are utilized to design the multivariate Hotelling control chart with sensitizing rules for detecting small-to-moderate variations. The control limit of the proposed chart is derived relative to probability of a single point and number of process characteristics. To calculate probability of a single point for sustained in-control average run length, a generalized single polynomial equation is derived. For evaluation, performance measures are considered based on the average, the median, and the percentile run length. These measures are calculated using Monte Carlo simulation and numerical integration. The results indicate that the proposed control chart has consistent behavior when a process is in control. The in-control average run length is obtained equal to prefixed level which remains valid for all choices of sensitizing rules. This implies that the proposed control chart can resolve the issue of existing control chart in terms of sustained behavior. The effectiveness of sensitizing rules is dependent on process characteristics and variations of mean vector. A comparative analysis of different choices of sensitizing rules is conducted to locate optimal choices of process characteristics. Real-life example, dowel-pin manufacturing, shows that proposed control chart with sensitizing rules is efficient for diagnosing small variations.