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While Six Sigma metrics have been studied by researchers in detail for normal distribution-based data, in this paper, we have attempted to study the Six Sigma metrics for two-parameter Weibull distribution that is useful in many life test data analyses.

In the theory of Six Sigma, most of the processes are assumed normal and Six Sigma metrics are determined for such a process of interest. In reliability studies non-normal distributions are more appropriate for life tests. In this paper, a theoretical procedure is developed for determining Six Sigma metrics when the underlying process follows two-parameter Weibull distribution. Numerical evaluations are also considered to study the proposed method.

In this paper, by matching the probabilities under different normal process-based sigma quality levels (SQLs), we first determined the Six Sigma specification limits (Lower and Upper Six Sigma Limits- LSSL and USSL) for the two-parameter Weibull distribution by setting different values for the shape parameter and the scaling parameter. Then, the lower SQL (LSQL) and upper SQL (USQL) values are obtained for the Weibull distribution with centered and shifted cases. We presented numerical results for Six Sigma metrics of Weibull distribution with different parameter settings. We also simulated a set of 1,000 values from this Weibull distribution for both centered and shifted cases to evaluate the Six Sigma performance metrics. It is found that the SQLs under two-parameter Weibull distribution are slightly lesser than those when the process is assumed normal.

The theoretical approach proposed for determining Six Sigma metrics for Weibull distribution is new to the Six Sigma Quality practitioners who commonly deal with normal process or normal approximation to non-normal processes. The procedure developed here is, in fact, used to first determine LSSL and USSL followed by which LSQL and USQL are obtained. This in turn has helped to compute the Six Sigma metrics such as defects per million opportunities (DPMOs) and the parts that are extremely good per million opportunities (EGPMOs) under two-parameter Weibull distribution for lower-the-better (LTB) and higher-the-better (HTB) quality characteristics. We believe that this approach is quite new to the practitioners, and it is not only useful to the practitioners but will also serve to motivate the researchers to do more work in this field of research.

The objective of the paper is to consider the problem of the strength of a manufactured item against stress, when the component follows Weibull failure law. Different cases of stress and strength with varying parameters are discussed for the Weibull‐Weibull stress‐strength model considered in this paper. The application of the proposed technique will help in understanding the design methodology of the system and addressing the risks involved in perceived quality and reliability levels by eliminating or at least reducing the risk impact at the design phase.

Generalised Weibull‐Weibull stress‐strength models have been analysed for different cases of shape parameters for stress and strength to estimate the reliability of the system. The model is generalized using semi‐regenerative stochastic processes with the help of a state space approach to include a repair facility.

Different cases of stress and strength with varying parameters have been discussed for the Weibull‐Weibull stress‐strength models considered in this paper. The results show how the stress‐strength reliability model is affected by changes in the parameters of stress and strength. The application of the proposed technique will help in understanding the design methodology of the system, and also lead to the problem of addressing the risks involved in perceived quality and reliability levels by eliminating or at least reducing the risk impact in the design phase.

The present study is limited to a few special cases of Weibull‐Weibull stress‐strength models. The authors propose to continue to study the behaviour of general Weibull strength against exponential stress in particular and to identify the shape parameter that maximises the strength reliability.

The application of the proposed technique will help in understanding the design methodology of the system, and also lead to the problem of addressing the risks involved in perceived quality and reliability levels by eliminating or at least reducing the risk impact at the design phase. The model has been extended and generalized to include a repair facility under the assumption that all the random variables involved in the analysis are arbitrarily distributed (i.e. general).

In the Weibull‐Weibull stress‐strength model of reliability, different cases have been considered. In the first case, both parameters of stress‐strength have the same values and are independent of the distribution. In the second case, if the shape parameter of the strength is twice that of the stress, the probability will have a normal distribution with different parameter values. In the third case, if the shape parameter of the stress is twice that of the strength, then probability distribution is a parabolic cylindrical function. The study shows how to proceed in all cases. The model is generalized to include a repair facility, with all the random variables involved in the analysis being arbitrarily distributed using semi‐regenerative stochastic processes.

The purpose of this study is to select the most accurate and the most efficient method in estimating Weibull parameters at Agadir region in Morocco.

In this paper, Weibull distribution is used to model the wind speed in hourly time series format. Since several methods are used to adjust the Weibull distribution to the measured data, in reporting and analyzing the easiest and the most effective method, seven methods have been investigated, namely, the graphical method (GM), the maximum likelihood method (MLM), the empirical method of Justus (EMJ), the empirical method of Lysen (EML), the energy pattern factor method (EPFM), Mabchour’s method (MMab) and the method of moments (MM).

According to the statistical analysis tools (the coefficient of determination, root mean square error and chi-square test), it was found that for five months, the MLM presents more efficiency, and for four months, EMJ is ranked first and it is ranked second for February. To select only one method, the selected methods (MLM and EMJ) were compared by calculating the error in estimating the power density using Weibull distribution adjusted by those methods. The average error was found to be −0.51 and −4.56 per cent for MLM and EMJ, respectively.

This investigation is the first of its kind for the studied region. To estimate the available wind power at Agadir in Morocco, investors can directly use MLM to determine the Weibull parameters at this site.

The purpose of this paper is to investigate wind power potential of site using wind speed, wind direction and other meteorological data including temperature and air density collected over a period of one year.

The site-specific air density, wind shear, wind power density, annual energy yield and capacity factors have been calculated at 30 and 10 m above the ground level (AGL). The Weibull parameters have been calculated using empirical, maximum likelihood, modified maximum likelihood, energy pattern and graphical methods to determine the other dependent parameters. The accuracies of these methods are determined using correlation coefficient (R²) and root mean square error (RMSE) values.

The site-specific wind shear coefficient was found to be 0.18. The annual mean wind speeds were found to be 5.174 and 4.670 m/s at 30 and 10 m heights, respectively, with corresponding standard deviations of 2.085 and 2.059. The mean wind power densities were found to be 59.50 and 46.75 W/m² at 30 and 10 m heights, respectively. According to the economic assessment, the wind turbine A is capable of producing wind energy at the lowest value of US$ 0.034/kWh.

This assessment provides the sustainable solution of energy which minimizes the dependence on continuous supply of oil and gas to run the conventional power plants that is a major cause of increasing load shedding in the significant industrial and thickly populated city of Pakistan. Also, this will minimize the quarrel between the local power producer and oil and gas supplier during the peak season.

This wind resource assessment has some important social implications including decreasing the environmental issues, enhancing the uninterrupted supply of electricity and decreasing cost of energy per kWh for the masses of Karachi.

The results are showing that the location can be used for installing the wind energy power plant at the lower cost per kWh compared to other energy sources. The wind energy is termed as sustainable solution at the lowest cost.

The purpose of this paper is to analyze and compare four numerical methods to estimate the most suitable one which describes wind speed distribution of M’Sila, a province of northern Algeria.

The site chosen in this investigation is characterized by calm winds; in this case, the appropriate wind speed distribution is that of hybrid Weibull.

The four numerical methods used in the present paper are the maximum likelihood method, the graphical method, the moment method and the energy pattern factor method. The hybrid Weibull distributions using the abovementioned approaches are compared with the measured data via three statistical parameters, namely, the correlation coefficient, the root mean square error and the Chi-square error.

The obtained results showed that the moment method is the suitable one in describing month and annual wind speed hybrid Weibull parameters of this region.

Describes a probabilistic methodology for fracture assessments of flawed structures constructed of ferritic steels using the research code WSTRESS. The probabilistic formulation for cleavage fracture implements a multiaxial form of the weakest link model which couples the macroscopic fracture behavior with a micromechanics model based on the statistics of microcracks. The Weibull stress, σ_w, emerges as a suitable near‐tip parameter to provide a connection between the microregime of failure and remote loading (J). WSTRESS builds on an iterative procedure to incorporate a 3‐D finite element description of the crack‐tip stress field and measured values of fracture toughness to calibrate the Weibull modulus, m, and the scale parameter, σ_u. Specific features of the code include statistical inference of Weibull parameters based on uncensored and censored models (with maximum likelihood method), construction of confidence intervals, several definitions for the near‐tip fracture process zone and other general facilities such as spatial integration of the stress field (to incorporate the random orientation of microcracks) and stochastic simulation of fracture data using the Monte Carlo method. The code also includes a convenient free‐form command language and a seamless interface with finite element results files stored in Patran binary or ASCII format.

The purpose of this paper is contribution to estimate the potential of wind energy in Douala in Cameroon, by modeling and predicting the regime of wind. The paper deals with the analysis and comparison of seven numerical methods for the assessment of effectiveness in determining the parameters for the Weibull distribution, using wind speed data collected at Douala International Airport in Cameroon, in the period from September 2011 to May 2013, obtained by meteorological equipment belonging to the Laboratory of Energy Research of the Institute of Geological and Mining Research.

By using ANOVA, root mean square error and chi-square tests to compare the proposed methods, this study aims to determine which methods are effective in determining the parameters of the Weibull distribution for the available data, in an attempt to establish acceptable criteria for better usage of wind power in Douala, which is the economic capital and ought to have prominence in the use of renewable sources for electricity generation in Cameroon.

The study helps to determine that moment, empirical and energy pattern factor methods used to determine the shape parameter k and the scale parameter c of the Weibull distribution present a better curve fit with the histogram of the wind speed. This fact is clearly validated by means of the statistical tests. But, all the seven methods gave excellent performance. Then, k reaching levels ranging from 3.5 to 5.5 and c range from 1.7 to 2.4.

Then as far as we are concerned, for a significant contribution, it could be more effective to have a model for prediction of wind characteristics using wind data collected per hour, one at least three years. A comparison of results obtained from lots of other methods (seven in this case) is necessary before an efficient discussion. Standard deviations and errors between measured and predicted data must also be presented.

This paper deals with the estimation of the stress–strength reliability R = P(X < Y), when X and Y follow (1) independent generalized gamma (GG) distributions with only a common shape parameter and (2) independent Weibull random variables with arbitrary scale and shape parameters and generalize the proposal from Kundu and Gupta (2006), Kundu and Raqab (2009) and Ali et al. (2012).

First, a closed form expression for R is derived under the conditions (1) and (2). Next, sufficient conditions are given for the convergence of the infinite series expansions used to calculate the value of R in case (2). The models GG and Weibull are fitted by maximum likelihood using Broyden–Fletcher–Goldfarb–Shanno (BFGS) quasi-Newton method. Confidence intervals and standard errors are calculated using bootstrap. For illustration purpose, two real data sets are analyzed and the results are compared with the existing recent results available in the literature.

The proposed approaches improve the estimation of the R by not using transformations in the data and flexibilize the modeling with Weibull distributions with arbitrary scale and shape parameters.

The proposals of the paper eliminate the misestimation of R caused by subtracting a constant value from the data (Kundu and Raqab, 2009) and treat the estimation of R in a more adequate way by using the Weibull distributions without restrictions in the parameters. The two cases covered generalize a number of distributions and unify a number of stress–strength probability P(X < Y) results available in the literature.

Pakistan is an energy starving country that needs continuous supply of energy to keep up its economic speed. The aim of this paper is to assess the wind resource and energy potential of Quaidabad site for minimizing the dependence on fuels and improving the environment.

The Quaidabad site wind shear coefficient and turbulence intensity factor are investigated. The two-parameter k and c Weibull distribution function is used to analyze the wind speed of site. The standard deviation of the site is also assessed for a period of a year. The wind power density and energy density are assessed for a period of a year. The economic assessment of energy/kWh is investigated for selection of appropriate wind turbine.

The mean wind shear coefficient was observed to be 0.2719, 0.2191 and 0.1698 at 20, 40 and 60 m, respectively, for a period of a year. The mean wind speed is found to be 2.961, 3.563, 3.907 and 4.099 m/s at 20, 40, 60 and 80 m, respectively. The mean values of k parameters were observed to be 1.563, 2.092, 2.434 and 2.576 at 20, 40, 60 and 80 m, respectively, for a period of a year. The mean values of c m/s parameter were found to be 3.341, 4.020, 4.408 and 4.625 m/s at 20, 40, 60 and 80 m, respectively, for a period of a year. The major portion of values of standard deviation was found to be in between 0.1 and 2.00 at 20, 40, 60 and 80 m. The wind power density (W/m²) sum total values were observed to be 351, 597, 792 and 923 W/m² at 20, 40, 60 and 80 m, respectively, for a period of a year. The mean coefficient of variation was found to be 0.161, 0.130, 0.115 and 0.105 at 20, 40, 60 and 80 m, respectively. The sum total energy density was observed to be 1,157, 2,156, 2,970 and 3,778 kWh/m² at 20, 40, 60 and 80 m, respectively. The economic assessment is showing that wind turbine E has the minimum cost US$0.049/kWh.

The Quaidabad site is suitable for installing the utility wind turbines for energy generation at the lowest cost.

The purpose of this paper is to develop an economic statistical design based on the concepts of adjusted average time to signal (AATS) and ANF for X ¯ control chart under a Weibull shock model with multiple assignable causes.

The design used in this study is based on a multiple assignable causes cost model. The new proposed cost model is compared with the same cost and time parameters and optimal design parameters under uniform and non-uniform sampling schemes.

Numerical results indicate that the cost model with non-uniform sampling cost has a lower cost than that with uniform sampling. By using sensitivity analysis, the effect of changing fixed and variable parameters of time, cost and Weibull distribution parameters on the optimum values of design parameters and loss cost is examined and discussed.

This research adds to the body of knowledge relating to the quality control of process monitoring systems. This paper may be of particular interest to practitioners of quality systems in factories where multiple assignable causes affect the production process.

The cost functions for uniform and non-uniform sampling schemes are presented based on multiple assignable causes with AATS and ANF concepts for the first time.