Optimization designs of the combined Shewhart-CUSUM control charts

This article presents the optimization design of the combined Shewhart chart and CUSUM chart ( chart in short) used in Statistical Process Control (SPC). While the optimization design effectively improves the overall performance of the chart over the entire process shift range, it does not increase difficulties in understanding and implementing this combined chart. A new feature pertaining to an additional charting parameter w (the exponential of the sample mean shift) is also investigated, with the hope of further enhancing the detection effectiveness of the chart. Moreover, this article provides the SPC practitioners with a design table to facilitate the designs of the charts. From this design table, the users can directly find the optimal values of the charting parameters, according to the design specifications. The design table makes the design of an chart as simple as the design of the simplest chart. In general, this article will help to enhance the detection effectiveness of the chart, and facilitate and promote its applications in SPC.     

Notes on shift effects for T2-type charts on multivariate ARMA residuals

Statistical process control (SPC) needs to be aided by computers in order to deal with dynamic systems. Hence, more knowledge on the complexity of this issue is needed. This paper discusses in general the shift effects of residuals from vector autoregressive moving average process for Shewhart-type, i.e. Hotelling T²-type charts (we call it H charts). Three types of parameter shift were considered: mean shift, covariance shift, and coefficient shift. The estimation effects were addressed. The discussions begin with the shift effects for residuals then for T²-type chart on residuals. The out-of-control distributions of the chart statistic were provided in this paper.     

Performance of t control charts in short runs with unknown shift sizes

Recently, control charts plotting a statistic having a Student’s t distribution have been proposed as an efficient solution to perform Statistical Process Control (SPC) in short production runs where the shift size of the in-control process mean from μ₀ to μ₁ is known a priori. The shift size is usually measured as a multiple δ of the in-control process standard deviation σ₀: but in practice, at the beginning of the production run, both the value of next shift δ and σ₀ are unknown. As a consequence, when the actual shift size differs from the value assumed at the chart design stage, the performance of the control chart can be seriously affected. To overcome this problem, this paper investigates the statistical performance of the Shewhart, EWMA and CUSUM t charts for short production runs when the shift size is unknown and modeled by means of a statistical distribution. An extensive numerical analysis allows the properties of the three charts to be compared and discussed when uniform and triangular distributions are used by quality practitioners to fit the unknown shift size. An illustrative example is utilized to demonstrate a practical implementation of the best performing among the three investigated charts.     

A generalized Conforming Run Length control chart for monitoring the mean of a variable

The attribute Conforming Run Length (CRL) control chart has attracted increasing research interests in Statistical Process Control (SPC). It decides the process status based on the interval or distance between two nonconforming units. This article proposes a Generalized CRL chart (namely GCRL chart) for monitoring the mean of a measurable quality characteristic x under 100% inspection. To run a GCRL chart, each unit will be classified as a passing or nonpassing unit depending on whether the sample value of x falls within or beyond a pair of lower and upper inspection limits LIL and UIL. When a nonpassing unit is detected, the GCRL chart checks the distance between the current and last nonpassing units in order to determine the process status (in control or out of control). The inspection limits LIL and UIL are determined by an optimization design. The GCRL chart not only solves a dead-corner problem suffered by the conventional CRL chart, but also considerably outperforms the latter for detecting mean shifts. The most interesting finding is that the attribute GCRL chart excels the variable X chart to a significant degree in SPC for variables. It suggests that the simple attribute chart may replace the variable chart in some SPC applications. The design of the GCRL chart has to be carried out by a computer program, but the design can be completed almost in no time in a personal computer.     

On efficient use of auxiliary information for control charting in SPC

In Statistical Process Control (SPC), monitoring of the process dispersion has a major impact on the performance of processes like manufacturing, management and services. Control charts act as the most important SPC tool, used to differentiate between common and special cause variations in the process. The use of auxiliary information can enhance the detection ability of control charts and hence an efficient monitoring of process parameter(s) can be done. This study deals with the Shewhart type variability control charts based on auxiliary characteristics for the non-cascading processes, assuming stability of auxiliary parameters. The control chart structures of these variability charts are provided and their performance evaluations are carried out in terms of average run length (ARL), relative average run length (RARL) and extra quadratic loss (EQL) under the normal and t distributed process environments. The comparisons have been made among different variability charts and superiorities are established based on their detection abilities for different amounts of shifts in process dispersion. An illustrative example is also provided in support of the theory, and finally the study ends with concluding remarks and suggestions for future research.     

On-line SPC with consideration of learning curve

This work identifies a link between on-line statistical process control (SPC) and the learning effect for the process standard deviation (PSD) caused by the quality improvement (QI) program. The learning curve (LC) is used to describe and forecast, and the exponentially weighted root mean square control chart is used to monitor the progress in reducing PSD. A modification of the quality control chart (QCC) that considers LC of PSD is proposed. The reduction rate of PSD may be large during the initial stage of the QI program, and influences QCC construction. Simulation is used to compare the shift-detecting ability of the Shewhart- control chart and EWMA- control chart, without- and with- consideration of LC. The EWMA- chart with consideration of LC performs best. In comparison, the Shewhart- chart without LC consideration has almost no shift-detecting ability when the shift magnitude of the process mean is small, leading to rendering quality control ineffective.     

Monitoring the software development process using a short-run control chart

Techniques for statistical process control (SPC), such as using a control chart, have recently garnered considerable attention in the software industry. These techniques are applied to manage a project quantitatively and meet established quality and process-performance objectives. Although many studies have demonstrated the benefits of using a control chart to monitor software development processes (SDPs), some controversy exists regarding the suitability of employing conventional control charts to monitor SDPs. One major problem is that conventional control charts require a large amount of data from a homogeneous source of variation when constructing valid control limits. However, a large dataset is typically unavailable for SDPs. Aggregating data from projects with similar attributes to acquire the required number of observations may lead to wide control limits due to mixed multiple common causes when applying a conventional control chart. To overcome these problems, this study utilizes a Q chart for short-run manufacturing processes as an alternative technique for monitoring SDPs. The Q chart, which has early detection capability, real-time charting, and fixed control limits, allows software practitioners to monitor process performance using a small amount of data in early SDP stages. To assess the performance of the Q chart for monitoring SDPs, three examples are utilized to demonstrate Q chart effectiveness. Some recommendations for practical use of Q charts for SDPs are provided.     

Economic design of two-stage non-central chi-square charts for dependent variables

Non-central chi-square charts are more effective than the joint and R charts in detecting small mean shifts or variance changes of a performance variable. However, the cost may be high to monitor a primary quality characteristic, such as the weight of each bag in a cement filling process. It is more economical to monitor a surrogate variable, for example, the milliampere of the load cell. When the correlation of the performance variable of surrogate variable exists, this article proposes a two-stage charting design to monitor either the performance variable or its surrogate variable in an alternating fashion rather than monitoring the performance variable alone. The proposed method simplifies process monitoring when users only concern about whether a process is in control or not. The application of the proposed method and the advantages of the proposed chart over the existing methods are presented through an example. Numerical results show that the proposed chart is insensitive on the correlation of the performance variable and surrogate variable even when the historical information on the correlation coefficient is not very accurate.     

On domination number of Cartesian product of directed cycles

Let γ(G) denote the domination number of a digraph G and let Cm□Cn denote the Cartesian product of Cm and Cn, the directed cycles of length m,n?2. In this paper, we determine the exact values: γ(C₂□Cn)=n; γ(C₃□Cn)=n if , otherwise, γ(C₃□Cn)=n+1; if , otherwise, .     

Approximating the dense set-cover problem

We study the set-cover problem, i.e. given a collection C of subsets of a finite set U, find a minimum size subset C′⊆C such that every element in U belongs to at least one member of C. An instance (C,U) of the set-cover problem is k-bounded if the number of occurrences in C of any element is bounded by a constant k?2.We present an approximation algorithm for the k-bounded set-cover problem, that achieves the ratio , where ε is defined as . If ε is relatively high, we say that the problem is dense, and this ratio in this case is better than k, which is the best known constant ratio for this problem. In the case that the number of occurrences in C of any element is exactly k=2 the problem is known as the vertex-cover problem. For dense graphs, our algorithm achieves an approximation ratio better than that of Nagamochi and Ibaraki (Japan J. Indust. Appl. Math. 16 (1999) 369), and the same approximation ratios as Karpinski and Zelikovsky (Proceedings of DIMACS Workshop on Network Design: Connectivity and Facilities Location, Vol. 40, Princeton, 1998, pp. 169-178). In our algorithm we use a combinatorial property of the set-cover problem, which is based on the classical greedy algorithm for the set-cover problem. We use this property to define a “greedy-sequence”, which is defined over a given instance of the set-cover problem and its cover.In addition, we show evidence that the ratio we achieve for the ε-dense k-bounded set-cover problem is the best constant ratio one can expect. We do this by showing that finding a better constant ratio is as hard as finding a constant ratio better than k for the k-bounded set-cover problem in which the optimal cover is known to be of size at least . (k is the best known constant ratio for this version of the k-bounded set-cover problem.) We show a similar lower bound for the approximation ratio for the vertex-cover problem in ε-dense graphs.     

The size and depth of layered Boolean circuits

We consider the relationship between size and depth for layered Boolean circuits and synchronous circuits. We show that every layered Boolean circuit of size s can be simulated by a layered Boolean circuit of depth . For synchronous circuits of size s, we obtain simulations of depth . The best known result so far was by Paterson and Valiant (1976) [17], and Dymond and Tompa (1985) [6], which holds for general Boolean circuits and states that , where C(f) and D(f) are the minimum size and depth, respectively, of Boolean circuits computing f. The proof of our main result uses an adaptive strategy based on the two-person pebble game introduced by Dymond and Tompa (1985) [6]. Improving any of our results by polylog factors would immediately improve the bounds for general circuits.     

The Updatable \overline{X} & S Control Charts

The tools for Statistical Process Control (SPC) should be continuously improved in order to continuously improve the product quality. This article proposes a scenario for continuously improving the & S control charts during the use of the charts. It makes use of the information collected from the out-of-control cases in a manufacturing process to update the charting parameters (i.e. the sample size, sampling interval and control limits) step-by-step. Consequently, the resultant control charts (called the updatable charts) become more and more effective to detect the mean shift δ_μ and standard deviation shift δ_σ for the particular process. The updatable charts are able to considerably reduce the average value of the loss function due to the occurrences of the out-of-control cases. Noteworthily, unlike the designs of the economic control charts, the designs of the updatable charts only require limited number of specifications that can be easily decided.     

A phase II nonparametric control chart based on precedence statistics with runs-type signaling rules

Nonparametric control charts do not require knowledge about the shape of the underlying distribution and can thus be attractive in certain situations. Two new Shewhart-type nonparametric control charts are proposed for monitoring the unknown location parameter of a continuous population in Phase II (prospective) applications. The charts are based on control limits given by two specified order statistics from a reference sample, obtained from a Phase I (retrospective) analysis, and using some runs-type signaling rules. The plotting statistic can be any order statistic in a Phase II sample; the median is used here for simplicity and robustness. Exact run length distributions of the proposed charts are derived using conditioning and some results from the theory of runs. Tables are provided for practical implementation of the charts for a given in-control average run length between 300 and 500. Comparisons of the average run length ARL, the standard deviation of run length (SDRL) and some run length percentiles show that the charts have robust in-control performance and are more efficient when the underlying distribution is t (symmetric with heavier tails than the normal) or gamma (1, 1) (right-skewed). Even for the normal distribution, the new charts are quite competitive. An illustrative numerical example is given. An added advantage of these charts is that they can be applied before all the data are collected which might lead to savings in time and resources in certain applications.     

List edge and list total colorings of planar graphs without short cycles

Let G be a planar graph with maximum degree Δ(G). We use and to denote the list edge chromatic number and list total chromatic number of G, respectively. In this paper, it is proved that and if Δ(G)?6 and G has neither C₄ nor C₆, or Δ(G)?7 and G has neither C₅ nor C₆, where Ck is a cycle of length k.