Haim Shore

General control charts for attributes

Traditional Shewhart-type control charts ignore the skewness of the plotted statistic. Occasionally, the skewness is too large to be ignored, and in such cases the classical Shewhart chart ceases to deliver satisfactory performance. In this paper, we develop a general framework for constructing Shewhart-like control charts for attributes based on fitting a quantile function that preserves all first three moments of the plotted statistic. Furthermore, these moments enter explicitly into the formulae for calculating the limits. To enhance the accuracy of these limits the value of the skewness measure used in the calculations is inflated by 44%. This inflation rate delivers accurate control limits for diversely-shaped attribute distributions like the binomial, the Poisson, the geometric and the negative binomial. A new control chart for the M/M/S queueing model is developed and its performance evaluated.

Profit Maximizing Warranty Period with Sales Expressed by a Demand Function

The problem of determining the optimal warranty period, assumed to coincide with the manufacturers lower specification limit for the lifetime of the product, is addressed. It is assumed that the quantity sold depends via a Cobb–Douglas-type demand function on the sale price and on the warranty period, and that both the cost incurred for a non-conforming item and the sale price increase with the warranty period. A general solution is derived using Response Modeling Methodology (RMM) and a new approximation for the standard normal cumulative distribution function. The general solution is compared with the exact optimal solutions derived under various distributional scenarios. Relative to the exact optimal solutions, RMM-based solutions are accurate to at least the first three significant digits. Some exact results are derived for the uniform and the exponential distributions. Copyright

General control charts for variables

When the distribution of the monitoring statistic used in statistical process control is non-normal, traditional Shewhart charts may not be applicable. A common practice in such cases is to normalize the data, using the Box-Cox power transformation. In this paper, we develop an inverse normalizing transformation (INT), namely, a transformation that expresses the original process variable in terms of the standard normal variable. The new INT is used to develop a general methodology for constructing process control schemes for either normal or nonnormal environments. Simplified versions of the new INT result in transformations with a reduced number of parameters, allowing fitting procedures that require only low-degree moments (second degree at most). The new procedures are incorporated in some suggested SPC schemes, which are numerically demonstrated. A simple approximation for the CDF of the standard normal distribution, with a maximum error (for z > 0) of +/-0.00002, is a by-product of the new transformations.

Process Capability Analysis When Data are Autocorrelated

When routine indices are used to assess process capability, autocorrelated data may lead to a biased estimate of the true capability and ultimately to wrong claims regarding the process performance. In this article, some of the undesirable effects that ..

Setting safety lead-times for purchased components in assembly systems: a general solution procedure

In a recent paper, Hopp and Spearman (1993) presented a model to determine optimal lead-times for purchasing components where the only manufacturing operation is the final assembly. It is assumed that delivery times of the different components are normal variates. An approximate iterative solution procedure is presented that facilitates derivation of the optimal lead-times. Based on an approximation for the delivery-time distribution, an alternative solution procedure is proposed that results in closed-form expressions for the decision variables of the problem. Both simplicity and ease of calculation are achieved. Furthermore, the new procedure is extended to non-normal delivery-time distributions. For Hopp and Spearmans numerical examples, the new procedure is shown to yield better accuracy.

Fitting a distribution by the first two moments (partial and complete)

Given a sample of observations from an unknown population, a common practice to derive distributional representation for the given data is to fit a four-parameter distribution via matching of the first four moments. However, third and fourth sample moments are notorious for their large standard errors, which require sample sizes that in a typical industrial setting are rarely available. In this paper we propose an alternative approach that employs only the first two moments (partial and complete) to fit a certain four-parameter distribution to the given sample data. The fitted distribution is a mixture of two components, where each is a linear transformation of a symmetrically distributed standardized variable. Separate transformations are used for each half of the distribution. Estimation of the parameters is carried out by matching of the mean, the variance, and the first and second partial moments. This fitting procedure is shown to be approximately a least squares solution, that provides good-estimates for the fractiles of the approximated distribution. Moreover, the linear transformations may provide mathematically manageable solutions to stochastic optimization problems (like inventory problems) that would otherwise require complex solution procedures. Some numerical examples and a simulation study attest to the effectiveness of the new approach when sample data are scarce.

Optimal solutions for stochastic inventory models when the lead-time demand distribution is partially specified

Abstract Stochastic optimization models used in operations management require that the underlying distributions be completely specified. When this requirement cannot be met, a common approach is to fit a member of some flexibly shaped four-parameter family of distributions (via four-moment matching), and thence use the fitted distribution to derive the optimal solution. However, sample estimates of third and fourth moments tend to have large mean-squared errors, which may result in unacceptable departure of the approximate solution from the true optimal solution. In this paper we develop an alternative approach that requires only first- and second-degree moments in the solution procedure. Assuming that only the first two moments, partial and complete, are known, we employ a new four-parameter family of distributions to derive solutions for two commonly used models in inventory analysis. When moments are unknown and have to be estimated from sample data, the new approach incurs mean-squared errors that are appreciably smaller relative to solution procedures based on three- or four-moment fitting.

Control charts for the queue length in a G/G/S system

Statistical-process-control-based monitoring of performance measures of queuing systems has so far eluded mainstream practices within the quality engineering discipline. There are two reasons for this. First, most measures have highly skewed distributions, and, secondly, no general theoretical formulae exist that may provide a platform to calculate control limits for the general G/G/S queue. In this paper, Shewhart-like general control charts for attributes, recently introduced, are combined with existent highly accurate approximations for the steady-state probabilities of the G/G/S queue, to develop control charts for the queue size in a G/G/S system. The effectiveness of this approach is discussed and demonstrated by applying the proposed control charts to a sample of queues with various inter-arrival time and service time distributions.

Comparison of Generalized Lambda Distribution (GLD) and Response Modeling Methodology (RMM) as General Platforms for Distribution Fitting

Distribution fitting is widely practiced in all branches of engineering and applied science, yet only a few studies have examined the relative capability of various parameter-rich families of distributions to represent a wide spectrum of diversely shaped distributions. In this article, two such families of distributions, Generalized Lambda Distribution (GLD) and Response Modeling Methodology (RMM), are compared. For a sample of some commonly used distributions, each family is fitted to each distribution, using two methods: fitting by minimization of the L 2 norm (minimizing density function distance) and nonlinear regression applied to a sample of exact quantile values (minimizing quantile function distance). The resultant goodness-of-fit is assessed by four criteria: the optimized value of the L 2 norm, and three additional criteria, relating to quantile function matching. Results show that RMM is uniformly better than GLD. An additional study includes Shores quantile function (QF) and again RMM is the best performer, followed by Shores QF and then GLD.

Determining measurement error requirements to satisfy statistical process control performance requirements

Measurement error (ME) is a source of variation that may considerably affect the performance of control charts applied within a statistical process control scheme. While the consequences of ME on the actual performance of various control charts has been studied in recent publications, the more important inverse problem of how to specify ME requirements to achieve desirable control chart performance characteristics has not been addressed in the literature. In this paper, we develop guidelines regarding the formulation of specification limits to distribution-related ME characteristics. Both the Shewhart X control chart and the S 2 control chart are addressed. The main results of this paper are delivered in the form of expressions, from which permissible values for measurement-error bias and the standard deviation, required to achieve specified average run length characteristics, may be easily identified. Some related issues are addressed.