The use of probability limits for process control based on geometric distribution

Synthetic-type control charts for time-between-events monitoring

This paper proposes three synthetic-type control charts to monitor the mean time-between-events of a homogenous Poisson process. The first proposed chart combines an Erlang (cumulative time between events, T_r) chart and a conforming run length (CRL) chart, denoted as Synth-T_r chart. The second proposed chart combines an exponential (or T) chart and a group conforming run length (GCRL) chart, denoted as GR-T chart. The third proposed chart combines an Erlang chart and a GCRL chart, denoted as GR-T_r chart. By using a Markov chain approach, the zero- and steady-state average number of observations to signal (ANOS) of the proposed charts are obtained, in order to evaluate the performance of the three charts. The optimal design of the proposed charts is shown in this paper. The proposed charts are superior to the existing T chart, T_r chart, and Synth-T chart. As compared to the EWMA-T chart, the GR-T chart performs better in detecting large shifts, in terms of the zero- and steady-state performances. The zero-state Synth-T₄ and GR-T_r (r = 3 or 4) charts outperform the EWMA-T chart for all shifts, whereas the Synth-T_r (r = 2 or 3) and GR-T₂ charts perform better for moderate to large shifts. For the steady-state process, the Synth-T_r and GR-T_r charts are more efficient than the EWMA-T chart in detecting small to moderate shifts.

The demerit‐based control chart for trinomial distribution

Purpose

The purpose of this paper is to investigate the properties of the classical goodness of fit test statistics X2, G2, GM2, and NM2 in testing quality of process represented as the trinomial distribution with dip null hypothesis and to devise a control chart for the trinomial distribution with dip null hypothesis based on demerit control chart.

Design/methodology/approach

The research involves the linear form of the test statistics, the linear function of counts since the marginal distribution of the counts in any category is binomial or approximated Poisson, in which the uniformly minimum variance unbiased estimator is the linear function of counts. A control chart is used for monitoring student characteristics in Thailand. The control chart statistics based on an average of the demerit value computed for each student as a weighted average lead to a uniformly most powerful unbiased test marginally. The two‐sided control limits were obtained using percentile estimates of the empirical distribution of the averages of the demerit.

Findings

The demerit control chart of the weight set (1, 25, 50) shows a generally good performance, robust to direction of out‐of‐control, mostly outperforms the GM2 and is recommended. The X2, NM2 are not recommended in view of inconsistency and bias. The performance of the demerit control chart of the weight set (1, 25, 50) does not dramatically change between both directions.

Practical implications

None of the multivariate control charts for counts presented in the literature deals with trinomial distribution representing the practical index of the quality of the production/process in which the classification of production outputs into three categories of “good”, “defective”, and “reworked” is common. The demerit‐based control chart presented here can be applied directly to this situation.

Originality/value

The research considers how to deal with the trinomial distribution with dip null hypothesis which no research study so far has presented. The study shows that the classical Pearson's X2, Loglikelihood, modified Loglikelihood, and Neyman modified X2 could fail to detect an “out‐of‐control”. This research provides an alternative control chart methodology based on demerit value with recommended weight set (1, 25, 50) for use in general.

Geometric probability distribution for modeling of error risk during prescription dispensing

PURPOSE

The relationship between the number of prescriptions dispensed by individual pharmacy staff during a single workday and the probability of committing at least one dispensing error during that same workday period was evaluated using a geometric probability distribution.

SUMMARY

A cross-sectional descriptive study involving 50 pharmacies located in six cities across the United States was conducted. A pharmacist trained to detect dispensing errors recorded the number of prescriptions filled by each pharmacy staff member and noted which prescription represented the staff member's first dispensing error (defined as any deviation from the prescriber's order) made during the observation period. The Kolmogorov-Smirnov tests for discrete distributions revealed that the observed cumulative distribution of dispensing errors could have come from a geometric probability distribution that assumed dispensing error rates of 2-3%. In terms of risk analysis, this study's findings suggest that there can be a quantifiable statistical relationship between a measure of workload and the risk of committing at least one dispensing error. The ability to model dispensing errors using a geometric probability distribution enables the safety and health care practitioner to directly assess dispensing error risk as a function of a pharmacy's accuracy rate and the number of prescriptions a pharmacy staff member should dispense during a work shift.

CONCLUSION

A geometric probability distribution effectively modeled the relationship between the number of prescriptions filled and the occurrence of the first dispensing errors.