Cumulative conformance count charts with variable sampling intervals for correlated samples

The cumulative conformance count (CCC) control chart is often employed to monitor the fraction nonconforming of high-yield processes. Traditional CCC chart is used when the items from a process are inspected one-at-a-time following the production order. In recent years, the CCC chart has been generalized to accommodate some industrial practices where items from a process are inspected sample by sample and not according to the production order. In order to increase the sensitivity of the generalized CCC (GCCC) chart to changes in fraction nonconforming, the variable sampling interval (VSI) scheme is used in this study. The output characteristic within each sample is assumed with correlation. The statistical properties of the GCCC chart with the VSI scheme are deduced using the Markov chain method. In evaluating the usefulness of the VSI feature, GCCC charts with VSI and fixed sampling interval (FSI) schemes are compared in terms of their statistical properties. The comparison results show that using the VSI scheme can improve the speed of GCCC chart in detecting changes in fraction nonconforming. Finally, according to the comparison results, a design procedure is applied to an industrial example to validate its practicability.     

A teaching–learning based optimization approach for economic design of X-bar control chart

Being simple to use X-bar control chart has been most widely used in industry for monitoring and controlling manufacturing processes. Measurements of a quality characteristic in terms of samples are taken from the production process at regular interval and the sample means are plotted on this chart. Design of a control chart involves the selection of three parameters, namely the sample size (n), the sampling interval (h) and the width of control limits (k). In case of economic design, these three control chart parameters are selected in such a manner that the total cost of controlling the process is the least. The effectiveness of this design depends on the accuracy of determination of these three parameters. In this paper, a new efficient and effective optimization technique named as teaching–learning based optimization (TLBO) has been used for the global minimization of a loss cost function expressed as a function of three variables n, h and k in an economic model of X-bar chart based on unified approach. In this work, the TLBO algorithm has been modified to simplify the tuning of teaching factor. A MATLAB computer program has been developed for this purpose. A numerical example has been solved and the results are found to be better than the earlier published results. Further, the sensitivity analysis using fractional factorial design and analysis of variance have been carried out to identify the critical process and cost parameters affecting the economic design.     

A multivariate synthetic double sampling T2 control chart

In this article, we propose a multivariate synthetic double sampling T² chart to monitor the mean vector of a multivariate process. The proposed chart combines the double sampling (DS) T² chart and the conforming run length (CRL) chart. On the whole, the proposed chart performs better than its standard counterparts, namely, the Hotelling’s T², DS T², and synthetic T² charts, in terms of the average run length (ARL) and average number of observations to sample (ANOS). The proposed chart also outperforms the multivariate exponentially weighted moving average (MEWMA) chart for moderate and large shifts but the latter is more sensitive than the former towards small shifts. For a variable sample size chart, like the synthetic DS T² chart, ANOS is a more meaningful performance measure than ARL. ANOS relates to the actual number of observations sampled but ARL merely deals with the number of sampling stages taken. Interpretation based on ARL is more complicated as either n₁ or n₁ + n₂ observations are taken in each sampling stage.     

Economic statistical design of non-uniform sampling scheme X bar control charts under non-normality and Gamma shock using genetic algorithm

This paper presented an approach which simultaneously considered the properties of cost and quality based on the Burr distribution and the non-unifampling scheme. The objective was to determine three parameters, namely, sample size, sampling interval between successive samples, and control limits, when an X bar chart monitors a manufacturing process with Gamma (λ, 2) failure characteristic and non-normal data. The design parameters of the X bar control charts can be obtained through the genetic algorithm (GA) method. An example was also adopted to indicate the solution procedure and sensitivity analyses. The results show that an increase of skewness coefficient (α₃) results in a slight decrease for sample size (n) while an increase of kurtosis coefficient (α₄) leads to a wider control limit width.     

Adaptive CUSUM control chart with variable sampling intervals

The standard cumulative sum chart (CUSUM) is widely used for detecting small and moderate process mean shifts, and its optimal detection ability for any pre-specified mean shift has been demonstrated by its equivalence to continuous sequential tests. In real practice, the assumption of knowing the true mean shift in prior cannot be always met. So it is desirable to design a procedure that is efficient for detecting a range of future expected but unknown mean shifts. Adaptive CUSUM control chart, which can continuously adjust itself by a one-step forecasting operator, has been proposed to detect efficiently and robustly for a range of mean shifts in the early literature. Moreover, in terms of sampling time to signal, control chart with the VSI (variable sampling intervals) feature can detect the process changes more quickly than the traditional FSI (fixed sample intervals) chart. In this paper, a new CUSUM control chart which is based on both adaptive and VSI features is discussed. Also, a two-dimensional Markov chain model is developed to evaluate its run-time performance.     

An economic design of cumulative sum charts to control non-normal process means

This paper proposes a method for the economic design of Cumulative sum (Cusum) control charts to maintain the current control of the process means, where the observations are independently, but non-normally distributed. The economic design of the Cusum charts involves the determination of the design parameters that minimize a relevant cost function. The design parameters are the sample size n, sampling interval s, the reference value jk, and the decision interval h. Approximating the non-normal probability density function of the process by an Edgeworth series, and deriving the average run lengths in Cusum control schemes by the use of a system of linear algebraic equations, an expression for the expected loss-cost function for the process is identified. An algorithm for near-optimal determination of the design parameters, by minimizing the loss-cost function, is presented, and its application is demonstrated through a numerical example. Finally, the effects of changes in various parameters and the cost coefficients on the loss-cost function are discussed.     

Statistical computation and analyses for attribute events

This article studies the monitoring of the attribute events based on statistical computation and analyses. The size of an attribute event is an integer rather than a continuous variable. For example, the detection of a product lot containing defectives is an attribute event, the size of which is the number of defectives found in this lot. While many control charts have been developed for monitoring the time interval (T) between the occurrences of an event, many other attribute charts may be employed to examine the size (C) of the event. However, these two types of control charts have been investigated and applied separately with limited syntheses in Statistical Process Control (SPC). This article presents a single SPC chart (called the rate chart for attribute, or rate chart in short) for simultaneously monitoring the time interval T and size Cof an attribute event based on the ratio between C and T. Our studies show that the new chart is more effective for detecting the out-of-control status of the attribute event compared with an individual t chart or an individual c chart, as well as a combined chart. More profound is that the rate chart performs more uniformly than other charts for detecting both T shift and C shift, as well as the joint shift in T and C. The rate chart has demonstrated its potential for both manufacturing systems and non-manufacturing sectors (e.g., supply chain management, office administration and health care industry), especially for the latter.     

The variable sampling interval run sum X‾ control chart

Traditional control charts for process monitoring are based on taking samples from the process at fixed length sampling intervals. More recently, research works focused on the use of variable sampling intervals (VSIs), where the lengths of the sampling intervals are varied according to the process quality. A short sampling interval is considered when the process quality indicates a possible out-of-control situation while a long sampling interval is considered, otherwise. In this paper, the VSI run sum (RS) $\bar{X}$ chart is proposed with its optimal scores and parameters determined using an optimization technique to minimize the out-of-control average time to signal (ATS) or the adjusted average time to signal (AATS). A Markov-chain method is used to evaluate both the ATS and AATS of the proposed chart, for the zero and steady state cases, respectively. Results show that the VSI RS $\bar{X}$ chart is considerably more efficient than the basic RS $\bar{X}$ chart. The VSI RS $\bar{X}$ chart performs generally well compared with other competing charts, such as the standard $\bar{X}$, synthetic $\bar{X}$, exponentially weighted moving average (EWMA) $\bar{X}$, VSI $\bar{X}$ and VSI EWMA $\bar{X}$ charts. The sensitivity of the VSI RS $\bar{X}$ chart can be enhanced further by adding more scoring regions or a head-start feature. An illustrative example is presented to explain the implementation of the proposed VSI RS $\bar{X}$ chart.     

Multi-criteria design of an X̄ control chart

Control chart designs are widely studied because control charts are not only costly used but also play an important role in improving firms' quality and productivity. Design of control charts refers to the selection of parameters, including sample size, control-limit width, and sampling frequency. In this paper, a possible combination of design parameters is considered as a decision-making unit; it is characterized by three attributes: hourly expected cost, average run length of process being controlled, and detection power of the chart designed with the selected parameters. Accordingly, optimal design of control charts can be formulated as a multiple criteria decision-making (MCDM) problem. To solve the MCDM problem, a solution procedure on the basis of data envelopment analysis is proposed. Finally, an industrial application is presented to illustrate the solution procedure. Also, adjustment to control chart design parameters is suggested when there are process improvements or process deteriorations.     

Multi-criteria design of an control chart

Control chart designs are widely studied because control charts are not only costly used but also play an important role in improving firms' quality and productivity. Design of control charts refers to the selection of parameters, including sample size, control-limit width, and sampling frequency. In this paper, a possible combination of design parameters is considered as a decision-making unit; it is characterized by three attributes: hourly expected cost, average run length of process being controlled, and detection power of the chart designed with the selected parameters. Accordingly, optimal design of control charts can be formulated as a multiple criteria decision-making (MCDM) problem. To solve the MCDM problem, a solution procedure on the basis of data envelopment analysis is proposed. Finally, an industrial application is presented to illustrate the solution procedure. Also, adjustment to control chart design parameters is suggested when there are process improvements or process deteriorations.     

Soluzione delle equazioni della cinetica di un reattore nucleare per mezzo di approssimazioni successive di tipo monotono

The solution of the one-energy group space-independent reactor kinetics equations is obtained in the form of the limit of two monotone bounded sequences of functions {N _j ^?} and {N _j ⁺}, non decreasing and non increasing respectively, defined as $$\begin{gathered} N_{j + 1}^ - = T_1 N_j^ + + T_2 N_j^ - + f \hfill \\ N_{j + 1}^ + = T_1 N_j^ - + T_2 N_j^ + + f \hfill \\ \end{gathered} $$ whereT ₁ andT ₂ are monotone-type operators, precisely antitone and isotone. In this work a procedure for obtaining the two initial elements,N ₀ ^? andN ₀ ⁺, satisfying the required properties to assure the convergence of the two sequences {N _j ^?} and {N _j ⁺}, is described; moreover, it is proved that the two sequences converge uniformely to the same limit. In addition, some numerical results are presented.     

Repeated detection of conjunctive predicates in distributed executions

Given a conjunctive predicate ? over a distributed execution, this paper gives an algorithm to detect all interval sets, each interval set containing one interval per process, in which the local values satisfy the Definitely(?) modality. The time complexity of the algorithm is O(n³p), where n is the number of processes and p is the bound on the number of times a local predicate becomes true at any process. The paper also proves that unlike the Possibly(?) modality which admits O(pn) solution interval sets, the Definitely(?) modality admits O(np) solution interval sets. The paper also gives an on-line test to determine whether all solution interval sets can be detected in polynomial time under arbitrary fine-grained causality-based modality specifications.     

A microcontroller system for measurement of three independent components in impedance sensors using a single square pulse

A novel time domain method and its implementation in a simple smart impedance sensor controlled by an 8-bit microcontroller is presented in the paper. The method is based on stimulation of a voltage divider consisting of a resistor working as a current-to-voltage converter and the impedance sensor by a single square-voltage pulse with a duration time T directly generated by the output of the microcontroller and on sampling the resulting voltage on the sensor at three different selected moments T/8, T/2 and 7T/8 by the internal ADC of the microcontroller. The sensor is modeled by a three-components circuit. The duration time T is determined by the first timer and the ADC is triggered by the second timer of the microcontroller. The measurement procedure takes less than 1 ms and the determination of model component values is based on basic calculus. Thanks to this, smart sensors basing on this solution are energy-saving, they can work on the same batteries by a few years, and low cost on the level tens euros. Hence, they can be used in wireless sensor networks, especially basing on the ZigBee protocol. The results of the simulation investigation and the experimental verification of the method are included in this paper.     

A variable parameter Shewhart control scheme for joint monitoring of process mean and variance

This paper presents a new Statistical Process Control model for the economic optimization of a variable-parameter control chart monitoring a process operation where two assignable causes may occur, one affecting the mean and the other the variance of the process. Therefore, it is possible for the process to operate in statistical control, when none of the two assignable causes has occurred, or under the effect of one or both the assignable causes. By making the assumption that the occurrence rate of each assignable cause is exponential, a Markov chain approach is utilized to determine the probabilities that the process operates at any of the above possible states. The model uses an economic (or an economic/statistical) optimization criterion for the time to the next sampling instance, the size of the next sample, as well as the control limits of the inspection. That is, all design parameters of the control scheme are selected so as to minimize the total expected quality-related costs. The superiority of the proposed model is estimated by comparing its expected quality control cost vs. the outcome of the Fp (Fixed-parameter) Shewhart control chart, the Variable Sample Size (VSS) control chart, the Variable Sampling Interval (VSI) and the Variable Sample Size and Sampling Interval (VSSI) control chart, for a benchmark of examples. The numerical investigation indicates that the economic improvement of the proposed model may be significant.     

A variable parameter Shewhart control scheme for joint monitoring of process mean and variance

This paper presents a new Statistical Process Control model for the economic optimization of a variable-parameter control chart monitoring a process operation where two assignable causes may occur, one affecting the mean and the other the variance of the process. Therefore, it is possible for the process to operate in statistical control, when none of the two assignable causes has occurred, or under the effect of one or both the assignable causes. By making the assumption that the occurrence rate of each assignable cause is exponential, a Markov chain approach is utilized to determine the probabilities that the process operates at any of the above possible states. The model uses an economic (or an economic/statistical) optimization criterion for the time to the next sampling instance, the size of the next sample, as well as the control limits of the inspection. That is, all design parameters of the control scheme are selected so as to minimize the total expected quality-related costs. The superiority of the proposed model is estimated by comparing its expected quality control cost vs. the outcome of the Fp (Fixed-parameter) Shewhart control chart, the Variable Sample Size (VSS) control chart, the Variable Sampling Interval (VSI) and the Variable Sample Size and Sampling Interval (VSSI) control chart, for a benchmark of examples. The numerical investigation indicates that the economic improvement of the proposed model may be significant.     

A run sum Hotelling’s χ2 control chart

A run sum Hotelling’s χ² control chart is proposed and its average run length (ARL) performance is evaluated using the Markov chain approach. A fast initial response (FIR) feature of this chart is also considered. In the optimization of the run sum χ² chart, computer programs are used to compute the chart’s optimal parameters. It is shown that the run sum χ² chart is superior to the various χ² charts with runs rules and the synthetic χ² chart, for all sizes of shifts in the mean vector, but less sensitive than the multivariate EWMA (MEWMA) chart toward small shifts. The sensitivity of the run sum χ² chart in detecting small shifts can be further enhanced by adding more regions and scores, so that this chart is as competitive as the MEWMA chart. We reckon that the run sum χ² chart is a relatively easy and effective tool for practitioners, as the χ² chart’s statistics can be plotted in its original scale of measurement, in contrast to the MEWMA chart which plots the transformed measurements.     

An integration of NSGA-II and DEA for economic–statistical design of T2-Hotelling control chart with double warning lines

Control charts are the most applicable tools for monitoring the quality of processes. The day-to-day changes in industrial processes and customers’ expectations motivate the process engineers to monitor multiple correlated quality characteristics, simultaneously. Hence, in this paper, the design of a “double warning lines T²-Hotelling” control chart is studied because of the advantages of this multivariate control chart in detecting moderate and small shifts in a process. In this regard, this research aims to optimize a multi-objective economic–statistical design model that considers monitoring costs and statistical features of control chart, concurrently. The non-dominated sorting genetic algorithm II is utilized to obtain a suitable Pareto set for the model. Since it is difficult for the decision makers to select the most efficient solution among the Pareto set, three different methods of data envelopment analysis consisting of Charnes–Cooper–Rhodes model, cross-efficiency technique and aggressive formulation are used to rank the members of Pareto set and to select the most efficient one. Also, in this research the performance of these three methods in discriminating between the efficient solutions is compared to each other. Eventually, a comparative study is conducted to show the better performance of the suggested model in comparison with the corresponding economic design model.

     

A hybrid fuzzy adaptive sampling - Run rules for Shewhart control charts

In crisp run control rules, usually it is stated that a process moves very sharply from in-control condition to out-of-control act. This causes an increase in both false-alarm rate and control chart sensitivity. Moreover, the classical run control rules are not implemented on an intelligent sampling strategy that changes control charts’ parameters to reduce error probability when the process appears to have a shift in parameter values. This paper presents a new hybrid method based on a combination of fuzzified sensitivity criteria and fuzzy adaptive sampling rules, which make the control charts more sensitive and proactive while keeping false alarms rate acceptably low. The procedure is based on a simple strategy that includes varying control chart parameters (sample size and sample interval) based on current fuzzified state of the process and makes inference about the state of process based on fuzzified run rules. Furthermore, in this paper, the performance of the proposed method is examined and compared with both conventional run rules and adaptive sampling schemes.     

Development of a soft computing-based framework for engineering design optimisation with quantitative and qualitative search spaces

Most real world engineering design optimisation approaches reported in the literature aim to find the best set of solutions using computationally expensive quantitative (Q^T) models without considering the related qualitative (Q^L) effect of the design problem simultaneously. Although, the Q^T models provide various detailed information about the design problem, unfortunately, these approaches can result in unrealistic design solutions. This paper presents a soft computing-based integrated design optimisation framework of Q^T and Q^L search spaces using meta-models (design of experiment, DoE). The proposed approach is applied to multi-objective rod rolling problem with promising results. The paper concludes with a detailed discussion on the relevant issues of integrated Q^T and Q^L design strategy for design optimisation problems outlining its strengths and challenges.