We assume that the operator is interested in monitoring a multinomial process. In this case the items are classified into (

In the modern industrial context, operators are usually interested in evaluating process quality. Assuming that the quality depends on several correlated quality characteristics

In general, each item may be classified in

The most applied statistical method of monitoring the multinomial process is the chi-square control chart, originally proposed by Duncan [

So it is useful to define a two-sided control chart that uses, respectively, the upper control limit and the lower control limit to signal the deterioration or the improvement of the process quality. In the following, we define the sampling statistic that may be used as an index of the overall defectiveness level of the process and as a statistic necessary to determine a corresponding two-sided control chart with the approximate probabilistic control limits. In fact, the process is usually assumed to be capable if the proportion of nonconforming items is very small and remains low or declines over time. Therefore, because we have chosen to classify the nonconforming items into

For evaluating the overall defectiveness of the production, we draw from the process a sample of

Items are classified in each of the

If the operator has chosen to assign the same weight to every class of defects, then an overall defectiveness index may be the overall proportion of nonconforming items in the process; indirectly, the operator can use this index as a measure of the process capability. The process will be considered capable when conforming proportion is very high. Instead, it is reasonable to think that the operator assigns different weights for every type of defect, because they cause, in the process, different grades of dysfunction, dissatisfaction, economical loss, or demerit. In this case, we have to consider accurately this differentiation of weights if we are interested in evaluating and monitoring the real overall level of defectiveness in the process. This approach for classification of defective items was used in Lu et al. [

Let

Given

Nevertheless, since the parameter of global quality is defined as linear function of the components in the multinomial parameter vector

The null hypothesis in (

By Gold [

By the multivariate Lindeberg-Lévy Central Limit Theorem, the vector

Since we want to design a control chart with control limits such that the overall error rate is not much larger than the nominal level

If the operators are interested in monitoring a multinomial process

The control charts based on statistics (

The proposed control chart signals the deterioration of the process when the sampling statistic is plotted out the UCL or the improvement of the process when the sampling statistic is plotted out the LCL. Analysis of the pattern of plotted statistics may be used to identify significant changes in the process parameters. We note that, for a specified vector

The proposed statistical procedure is conservative because the overall coverage probability inside the control limits is at least

Since the

In this section, the performance of the control chart will be evaluated using three different cases of overall quality level: a

A first example is based on real data reported in Taleb and Limam [

Usually, the operator is interested in identifying a deterioration in process and considers only the null hypothesis to choose the size of samples, but in our approach we want to develop a procedure that will also be able to identify an improvement in process. Therefore, for defining the appropriate sample size we have to consider the minimum proportion indicated in the process-improvement hypothesis. In this first example the minimum proportion is 0.0204; then by Cochran’s rule we have to take a sample of size

For numerical evaluations we have simulated samples from multinomial processes with parameters

Two-sided multivariate

Two-sided multivariate

Two-sided multivariate

We note that if the parameter _{1} in function of different values of parameter _{1}: ARL_{1}(upper) and ARL_{1}(lower). The ARL_{1}(upper) indicates the control chart’s ability to identify correctly only deteriorations in the process; while ARL_{1}(lower) evaluates the control chart’s ability to identify significant improvements in process. Combining the two one-sided ARL_{1} we can estimate the ARL_{1}. The ARL_{1} curves in function of the parameter _{1}(upper) and ARL_{1}(lower) are, respectively, decreasing and increasing values of parameter

ARL comparisons (example one).

ARL for chart with simultaneous | ARL for chart with exact | ARL for chart with nonsimultaneous | ||||||||||

Šidák's approximation control limits | estimated control limits | approximation control limits | ||||||||||

ARL_{0} | ARL_{1} | ARL_{1} | ARL_{1} | ARL_{0} | ARL_{1} | ARL_{1} | ARL_{1} | ARL_{0} | ARL_{1} | ARL_{1} | ARL_{1} | |

upper | lower | upper | lower | upper | lower | |||||||

555 | 2.855 | 1.063 | 1.959 | 370 | 2.610 | 1.039 | 1.824 | 270 | 2.377 | 1.037 | 1.707 | |

417 | 1.963 | 1.118 | 1.541 | 370 | 2.109 | 1.073 | 1.591 | 286 | 1.734 | 1.077 | 1.406 | |

500 | 1.518 | 1.291 | 1.405 | 370 | 1.583 | 1.154 | 1.369 | 256 | 1.392 | 1.194 | 1.293 | |

555 | 1.406 | 1.505 | 1.456 | 370 | 1.440 | 1.231 | 1.336 | 303 | 1.313 | 1.347 | 1.330 | |

555 | 1.311 | 1.959 | 1.635 | 370 | 1.344 | 1.432 | 1.388 | 333 | 1.239 | 1.639 | 1.439 |

ARL_{1} comparisons (example one).

If we consider the results in Table _{1} curves in Figure

Besides, to exemplify the methods for calculating the ARL reported in Table _{0}) has been calculated; that is, the number of observations where the sampling statistic defined in (_{0} the mean of the 1000 RL_{0} simulation values has been calculated. Under the assumption of process in control the mean of the 1000 RL_{0} was equal to 18. The false-alarm probability _{0} and the sample size; _{1}) has been calculated; that is, the number of observations where the sampling statistic defined in (_{1} then the mean of the 1000 RL_{1} simulation values has been calculated. Under the assumption of process out control the mean of the 1000 RL_{1} was equal to 3503. The correct-alarm probability _{1} and the sample size;

A second example considers a process with a higher level of quality than that considered in case one; in fact, the defect-free proportion, that is, chosen is equal to 0.83. The following hypotheses, about the defect proportions in vector _{1} curves in Figures

ARL comparisons (example two).

ARL for chart with Šidák's | ARL for chart with exact | ARL for chart with normal | ||||||||||

approximation control limits | estimated control limits | approximation control limits | ||||||||||

ARL_{0} | ARL_{1} | ARL_{1} | ARL_{1} | ARL_{0} | ARL_{1} | ARL_{1} | ARL_{1} | ARL_{0} | ARL_{1} | ARL_{1} | ARL_{1} | |

upper | lower | upper | lower | upper | lower | |||||||

455 | 1.365 | 1.000 | 1.183 | 370 | 1.428 | 1.00000 | 1.214 | 256 | 1.271 | 1.000 | 1.136 | |

455 | 1.169 | 1.001 | 1.085 | 370 | 1.192 | 1.00020 | 1.096 | 256 | 1.124 | 1.000 | 1.062 | |

435 | 1.076 | 1.015 | 1.046 | 370 | 1.085 | 1.00370 | 1.044 | 270 | 1.056 | 1.007 | 1.032 | |

526 | 1.056 | 1.04 | 1.048 | 370 | 1.065 | 1.01060 | 1.038 | 312 | 1.037 | 1.022 | 1.030 | |

625 | 1.037 | 1.115 | 1.076 | 370 | 1.037 | 1.02838 | 1.033 | 294 | 1.024 | 1.070 | 1.047 |

Two-sided multivariate

Two-sided multivariate

Two-sided multivariate

ARL_{1} curves (example two).

Table

ARL_{1} comparisons.

If we consider the results in Table _{1} curves in Figure

A third example considers a process with a much higher level of quality than that considered in cases one and two; in fact, the defect-free proportion, that is, chosen is equal to 0.99 and the maximum defective proportion is 0.001, that is, 1000 ppm. This level of quality will be classified as very high. The following hypotheses, regarding the defect proportions in vector _{1} values, the chart works much better than it does at the low and high quality levels. In fact, the ARL_{1} in the example process with very high-quality level is always smaller than that in the example processes with high or low quality level. This indicates that our approach is more appropriate in the case of a process with a very high level of quality (see Figures

Two-sided multivariate

Two-sided multivariate

Two-sided multivariate

ARL_{1} curves (example three).

Table

ARL comparisons (example three).

ARL for chart with Šidák's | ARL for chart with exact | ARL for chart with normal | ||||||||||

approximation control limits | estimated control limits | approximation control limits | ||||||||||

ARL_{0} | ARL_{1} | ARL_{1} | ARL_{1} | ARL_{0} | ARL_{1} | ARL_{1} | ARL_{1} | ARL_{0} | ARL_{1} | ARL_{1} | ARL_{1} | |

upper | lower | upper | lower | upper | lower | |||||||

1250 | 1.000 | 1.080 | 1.040 | 370 | 1.000 | 1.034 | 1.017 | 385 | 1.000 | 1.042 | 1.021 | |

909 | 1.000 | 1.179 | 1.089 | 370 | 1.000 | 1.085 | 1.042 | 400 | 1.000 | 1.109 | 1.055 | |

769 | 1.000 | 1.460 | 1.230 | 370 | 1.000 | 1.211 | 1.106 | 526 | 1.000 | 1.302 | 1.151 | |

769 | 1.000 | 1.783 | 1.392 | 370 | 1.000 | 1.378 | 1.189 | 526 | 1.000 | 1.502 | 1.251 | |

909 | 1.000 | 2.539 | 1.770 | 370 | 1.000 | 1.586 | 1.293 | 455 | 1.000 | 1.999 | 1.500 |

Comparisons of ARL_{1} (example three).

The comparisons of the ARL, reported in the Tables _{1} is always better for the control chart with non simultaneous approximation control limits; (ii) ARL_{0} is always better for the control chart with simultaneous Šidák's approximation control limits; that is, the false alarm probability is always smaller in the simultaneous approach respect to the nonsimultaneous approach. In general, a good multiattribute quality control procedure is one that provides a method for identifying which subset of the classes of defects are responsible when the process is determined to be out of control. The control chart proposed in this paper and based on simultaneous confidence intervals meets this criterium. On the contrary the non simultaneous approach is not able to solve the identification problem (see [

In this article an index of weighted overall defectiveness of the process and a two-sided multivariate

The sampling statistic used to estimate the overall defectiveness of the process and to design the corresponding control chart is function of the weights associated to the vector of quality defect categories. Therefore, the performance of the control chart is influenced by these values. In this paper we propose to use weights that are in terms of the geometric progression; the parameter

An interesting methodological development is possible considering the multivariate binomial distribution as appropriate probabilistic model to monitor the process quality degree, because it is a more general distribution which contains the one used in this paper.

The author wishes to thank the editor and the anonymous referees for their thoughtful and detailed suggestions that improved the paper.