On the distribution of the estimated process yield index Spk

This paper considers an asymptotic distribution for an estimate of the process yield index proposed by Boyles (1994). The asymptotic distribution of is useful in statistical inferences for . An illustrative example is given for hypothesis testing and for interval estimation on the yield index . Copyright © 2002 John Wiley & Sons, Ltd.     

The reliability analysis of balking and reneging in a repairable system with warm standbys

We study the reliability analysis of a repairable system with operating units, warm standby units and repairmen in which the balking and reneging of units are considered. It is assumed that failed units balk (refuse to join) with a constant probability ( ) and renege (leave the queue after joining) according to a negative exponential distribution with parameter . The failure times of the operating unit and of a standby unit are assumed to be exponentially distributed with parameters λ and α, respectively. The repair time distribution is also exponential. The repairable system is out of , in which . The reliability characteristics, such as the system reliability and the mean time to system failure (MTTF), in a repairable system are derived. Several cases are analysed graphically to study the effect of various parameters on the reliability and MTTF of the system. Copyright © 2002 John Wiley & Sons, Ltd.     

A fraction defective based capability index

Two problems greatly affect the use of capability indices such as , and : the lack of affinity with the process fraction defective π and the difficulty of dealing with the sampling distributions of their natural estimators. In this paper, a capability index which is in one‐to‐one correspondence with π is introduced and simple inferential procedures under a Bayesian perspective are developed to facilitate its use in industrial applications. Copyright © 2001 John Wiley & Sons, Ltd.     

On the reliability of the estimated incapability index

Greenwich and Jahr‐Schaffrath (1995) introduced the process incapability index , which provides an uncontaminated separation between information concerning the process precision and process accuracy . In this paper, we consider the three indices, and investigate the statistical properties of their natural estimators. For the three indices, we obtain their UMVUEs and MLEs, and compare the reliability of the two estimators based on the relative mean square errors. In addition, we construct 90%, 95%, and 99% upper confidence limits, and the maximum values of for which the process is capable 90%, 95%, and 99% of the time. The results obtained in this paper are useful to the practitioners in choosing good estimators and making reliable decisions on judging process capability. Copyright © 2001 John Wiley & Sons, Ltd.     

A Bayesian Approach to Obtain a Lower Bound for the Cpm Capability Index

The Taguchi capability index C_pm, which incorporates the departure of the process mean from the target value, has been proposed to the manufacturing industry for measuring manufacturing capability. A Bayesian procedure has been considered for testing process performance assuming , which was generalized without assuming . Statistical properties of the natural estimator of the index C_pm for normal processes have been investigated extensively. However, the investigation was restricted to processes with symmetric tolerances. Recently, a generalized C_pm, referred to as , was proposed to cover processes with asymmetric tolerances. Under the normality assumption, the statistical properties of the estimated including the exact sampling distribution, the rth moment, expected value, variance, and the mean‐squared error were obtained. In this paper, we use a Bayesian approach to obtain the interval estimation for the generalized Taguchi capability index . Consequently, the manufacturing capability testing can be performed for quality assurance. Copyright © 2005 John Wiley & Sons, Ltd.     

Three-dimensional crack analysis using singular boundary elements

A multi-domain method of solving three-dimensional elastic crack problems in an infinite elastic body using the boundary element method is proposed. The displacement and traction behaviours near a crack front are incorporated in special crack elements. The elimination of singularities arising from the term combined with Kelvin's kernel for displacement in the integrals is discussed in detail. Stress intensity factors of modes I, II and III are obtained directly from crack-front nodal values, without any extrapolation as in some other methods. No differentiation of conventional boundary integral equations (with Kelvin's tensor kernels) is necessary in the current approach. This method is applicable to cracks of arbitrary shape. Infinite bodies are modelled precisely as such, not approximated as large finite bodies. Numerical solutions of stress intensity factors are given for several problems involving a penny-shaped crack.     

Some notes on the construction and evaluation of supersaturated designs

Supersaturated designs offer a potentially useful way to investigate many factors with few experiments. A super‐saturated design evaluates factors with experiments, where . In some supersaturated designs the number of factors being investigated may exceed the number of experiments by a factor of three or more. The problem of developing a supersaturated design is described in computational complexity terms and methods of construction are discussed. Using standard combinatorial formulation may permit researchers to use additional methods to develop additional designs. Designs are generated with a modified method and compared with published designs using existing criteria and a new criterion. Copyright © 2002 John Wiley & Sons, Ltd.     

Eigenspannungen als Parameter des röntgenographischen Bruchkriteriums und der Lebensdauervoraussage

Residual stresses as parameters of X-ray criterion of fracture and of fatigue life prediction Fatigue testing is notorious for considerable scatter in specimen life even if special measures are taken to precisely control parameters of stress cycling. The reason for that must be ascribed to variation in material microstructure existing prior to fatigue loading. Structural condition of material may be determined by the following function: Extensive fatigue tests on α-Ti and (α + β)-Ti alloy at temperatures 77 K and 295 K showed that failure did not occur until function V reached its critical value at a plane characteristic of each lattice type (failure plane). The condition V = K may be therefore regarded an X-Ray criterion and, once determined numerically, a sort of material constant. The life N_B of fatigue tested specimen was found to be clearly depended on structural condition of virgin material exhibited natural variation of structural characteristics. The finite fatigue life was found to conform with the relationship The expression may be used for predicting life N_B if prior to fatigue loading the material condition function V value is made known for a characteristic lattice plane.     

Determining the lifetime of CD‐RWs using efficient sampling on a production line

This work derives the intermittent batch size and sample size for product life testing on a production line. The sequential probability ratio test was adopted to determine the sample size, , and the expected number of samples (ENS). An inverse chi‐square approximation was used to determine the relationship between the degree of freedom, , and the estimated range of at various levels of confidence. Consequently, ENS and were used to determine the intermittent batch size, . Finally, rewritable compact disks (CD‐RWs) underwent testing to demonstrate the feasibility and effectiveness of the proposed efficient sampling approach. Copyright © 2003 John Wiley & Sons, Ltd.     

Distributional properties of estimated capability indices based on subsamples

Under the assumption of normality, the distribution of estimators of a class of capability indices, containing the indices , , and , is derived when the process parameters are estimated from subsamples. The process mean is estimated using the grand average and the process variance is estimated using the pooled variance from subsamples collected over time for an in‐control process. The derived theory is then applied to study the use of hypothesis testing to assess process capability. Numerical investigations are made to explore the effect of the size and number of subsamples on the efficiency of the hypothesis test for some indices in the studied class. The results for and indicate that, even when the total number of sampled observations remains constant, the power of the test decreases as the subsample size decreases. It is shown how the power of the test is dependent not only on the subsample size and the number of subsamples, but also on the relative location of the process mean from the target value. As part of this investigation, a simple form of the cumulative distribution function for the non‐central ‐distribution is also provided. Copyright © 2003 John Wiley & Sons, Ltd.     

Capability measures for processes with multiple characteristics

Process capability indices, such as , , and , have been widely used in the manufacturing industry providing numerical measures on process precision, process accuracy, and process performance. Capability measures for processes with a single characteristic have been investigated extensively. However, capability measures for processes with multiple characteristics are comparatively neglected. In this paper, we consider a generalization of the yield index proposed by Boyles, for processes with multiple characteristics. We establish a relationship between the generalization and the process yield. We also develop a control chart based on the proposed generalization, which displays all the characteristic measures in one single chart. Using the chart, the engineers can effectively monitor and control the performance of all process characteristics simultaneously. Copyright © 2003 John Wiley & Sons, Ltd.     

Comparison of standard and individual limits Phase I Shewhart $\overline{X}$, $R$, and $S$ charts

Phase I control charts are used as aids in bringing a process into and defining the meaning of a process being in a state of statistical in‐control. This is done by looking at the data in retrospect to answer the question ‘were these data taken from an in‐control process?’ Typically the data is collected as independent random samples taken periodically from the output of the process. Commonly it is recommended that each sample be compared via a statistic(s) to a control limit(s) that is (are) function(s) of all the data. We refer to these charts as ‘standard limits’ charts. An alternative method is to compare each sample with all or some subcollection of the remaining samples. These charts are referred to as ‘individual’ limits charts. The individual limits charts appear to be attractive in detecting a sample from an out‐of‐control process when the other samples were taken when the process was in‐control. We demonstrate using a simple probability analysis that a standard limits Phase I Shewhart chart performs better than the individual limits Phase I Shewhart chart. Furthermore, it is shown that the individual limits Phase I Shewhart charts can be designed to be equivalent. This also holds for the individual and standard limits Phase I Shewhart charts. Copyright © 2003 John Wiley & Sons, Ltd.     

Exponential fitting using integral equations

The fitting of a function y =\documentclass{article}\pagestyle{empty}\begin{document}$ \sum\nolimits_{i = 1}^n {A_i {\rm{e}}^{\lambda ix}} $\end{document} to experimental data is considered. Integral equations are developed which have the functions \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\nolimits_{i = 1}^n {A_i {\rm{e}}^{\lambda ix}} $\end{document} as their solutions for n = 1, 2, 3, These integral equations are used to find the frequencies λ_i. Examples are worked out to illustrate the method. The method is shown to be capable of extension to other functions.     

A sensitivity analysis of an integrated model for joint determination of economic design of $\overline{x}$‐control charts,economic production quantity and production run length for a deteriorating production system

This study presents an integrated model for the joint economic design of ‐control charts and maintenance schedules and, simultaneously, determines the economic production quantity and production run length for a deteriorating production system. The operating state of the production process is classified as either in control or out of control. In the latter state, the process produces some defective items. An ‐control chart is used to monitor the process mean. Both uniform and non‐uniform inspection schemes are adopted. Inspection and maintenance are performed simultaneously. Replacement cost is assumed to be very high. The process failure mechanism is assumed to follow a general probability distribution with an increasing failure rate. The concept of a truncated production cycle is introduced. The production cycle begins when a new component is installed and ends with a repair after the detection of a failure or after a specified number of inspection intervals, , whichever occurs first. The effects of preventive maintenance on quality control are discussed. Numerical examples are provided to evaluate the performance of the model. Sensitivity analyses are conducted to study the effects of various model parameters. Copyright © 2002 John Wiley & Sons, Ltd.     

Estimated incapability index: reliability and decision making with sample information

The process incapability index , which provides an uncontaminated separation between information concerning the process precision and process accuracy, has been proposed to measure process performance for industry applications. In this paper, we investigate the reliability of the natural estimator computationally, based on the ‐level confidence relative error for various sample sizes. We also develop a decision‐making procedure for judging if the process satisfies the preset quality requirement. The investigation is useful to the practitioners in determining the sample sizes required in their applications for the decisions reliable to the desired level. Copyright © 2002 John Wiley & Sons, Ltd.     

Construction of double sampling s‐control charts for agile manufacturing

Double sampling (DS) ‐control charts are designed to allow quick detection of a small shift of process mean and provides a quick response in an agile manufacturing environment. However, the DS ‐control charts assume that the process standard deviation remains unchanged throughout the entire course of the statistical process control. Therefore, a complementary DS chart that can be used to monitor the process variation caused by changes in process standard deviation should be developed. In this paper, the development of the DS s‐charts for quickly detecting small shift in process standard deviation for agile manufacturing is presented. The construction of the DS s‐charts is based on the same concepts in constructing the DS ‐charts and is formulated as an optimization problem and solved with a genetic algorithm. The efficiency of the DS s‐control chart is compared with that of the traditional s‐control chart. The results show that the DS s‐control charts can be a more economically preferable alternative in detecting small shifts than traditional s‐control charts. Copyright © 2002 John Wiley & Sons, Ltd.     

Numerical integration of structural dynamics equations including natural damping and periodic forcing terms

This paper shows that it is comparatively simple to analyse algorithms for the numerical integration of the Space discretized equations from structural dynamics when applied to \documentclass{article}\pagestyle{empty}\begin{document}$\ddot x + \mu \dot x\omega ^2 x = p{\rm e}^{ist} $\end{document}, instead of the usual \documentclass{article}\pagestyle{empty}\begin{document}$ \ddot x + \omega ^2 x = 0 $\end{document}, and suggests that this should be done in order to gain some insight into the effect with natural damping and/or a periodic forcing term. The method is illustrated on some three- and four-time-level schemes.     

A new 12 DOF quadrilateral element for analysis of thick and thin plates

This paper presents a simple quadrilateral 12 DOF plate bending element based on a modified version of the hybrid-Trefftz approach. This element makes use of two independent fields of generalized displacements:

• i a non-conforming field (11 Trefftz terms for transverse displacement w and the corresponding rotations Θ_x, Θ_y) satisfying the governing differential equations of Reissner-Mindlin theory;
• ii an auxiliary conforming field with displacements w? linked to rotations $ \tilde \Theta _x,\tilde \Theta _y$, by the requirement of constant boundary distribution of the corresponding tangential component S?_t, of the transverse shear. This allows quadratic w? and linear $ \tilde \Theta _x,\tilde \Theta _y $, at the element boundary to be obtained with only 3 DOF at the corner nodes.

The resulting element, denoted by Q?21–11, is robust and free of shear locking in the thin limit. The numerical assessment involves comparison with several recently presented 12 DOF thick plate quadrilaterals as well as with the standard 16 DOF hybrid-Trefftz quadrilateral, Q21-15S, with 15 Trefftz terms and independently interpolated w? and $ \tilde \Theta _x,\tilde \Theta _y $.     

A Comparative Study of Joint Monitoring Schemes for APC Processes

To monitor processes under automatic process control (APC), several joint monitoring schemes have recently been proposed, such as the U statistics, and , that originate from the uniformly most powerful test. It is known that the chart is sensitive to large shift detection whereas the chart is sensitive to small shift detection. To take advantage of both charts, this paper examines a combined use of the two U charts. A comparative study indicates that the combined procedure improves over a single U chart and performs better than other joint monitoring schemes, including Hotelling's chart for APC processes. Copyright © 2006 John Wiley & Sons, Ltd.     

Measuring process capability based on Cpmk with gauge measurement errors

Due to their effectiveness and simplicity of use, the process capability indices , , and have been popularly accepted in the manufacturing industry as management tools for evaluating and improving process quality. Combining the merits of those indices, a more advanced index, , is proposed that takes into account process variation, process centering, and the proximity to the target value, and has been shown to be a very useful index for manufacturing processes with two‐sided specification limits. Most research works related to assume no gauge measurement errors. However, such an assumption inadequately reflects real situations even when highly advanced measurement instruments are employed. Conclusions drawn regarding process capability are therefore unreliable and misleading. In this paper, we conduct a sensitivity investigation for the process capability index in the presence of gauge measurement errors. We consider the use of capability testing of as a method for obtaining lower confidence bounds and critical values for true process capability when gauge measurement errors are unavoidable. The results show that using the estimator with sample data contaminated by measurement errors severely underestimates the true capability, resulting in an imperceptibly smaller test power. To measure the true process capability, three methods for the adjusted confidence bounds are presented and their performances are compared using computer simulation. Copyright © 2006 John Wiley & Sons, Ltd.