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The purpose of this paper is to develop a cost model by the variable sampling interval and optimization of the average cost per unit of time. The paper considers an economic–statistical design of the X̅ control charts under the Burr shock model and multiple assignable causes were considered and compared with three types of prior distribution for the mean shift parameter.

The design of the modified X̅ chart is based on the two new concepts of adjusted average time to signal and average number of false alarms for X̅ control chart under Burr XII shock model with multiple assignable causes.

The cost model was examined through a numerical example, with the same cost and time parameters, so the optimal of design parameters were obtained under uniform and non-uniform sampling schemes. Furthermore, a sensitivity analysis was conducted in a way that the variability of loss cost and design parameters was evaluated supporting the changes of cost, time and Burr XII distribution parameters.

The economic–statistical model scheme of X̅ chart was developed for the Burr XII distributed with multiple assignable causes. The correlated data are among the assumptions to be examined. Moreover, the optimal schemes for the economic-statistic chart can be expanded for correlated observation and continuous process.

The economic–statistical design of control charts depends on the process shock model distribution and due to difficulties from both theoretical and practical aspects; one of the proper alternatives may be the Burr XII distribution which is quite flexible. Yet, in Burr distribution context, only one assignable cause model was considered where more realistic approach may be to consider multiple assignable causes.

This study presents an advanced theoretical model for cost model that improved the shock model that presented in the literature. The study obviously indicates important evidence to justify the implementation of cost models in a real-life industry.

The purpose of this paper is to apply fuzzy spectrum in order to collect the vague and imprecise data and to employ the fuzzy U control chart in variable sample size using fuzzy rules. This approach is improved and developed by providing some new rules.

The fuzzy operating characteristic (FOC) curve is applied to investigate the performance of the fuzzy U control chart. The application of FOC presents fuzzy bounds of operating characteristic (OC) curve whose width depends on the ambiguity parameter in control charts.

To illustrate the efficiency of the proposed approach, a practical example is provided. Comparing performances of control charts indicates that OC curve of the crisp chart has been located between the FOC bounds, near the upper bound; as a result, for the crisp control chart, the probability of the type II error is of significant level. Also, a comparison of the crisp OC curve with OC_avg curve and FOC_α curve approved that the probability of the type II error for the crisp chart is more than the same amount for the fuzzy chart. Finally, the efficiency of the fuzzy chart is more than the crisp chart, and also it timely gives essential alerts by means of linguistic terms. Consequently, it is more capable of detecting process shifts.

This research develops the fuzzy U control chart with variable sample size whose output is fuzzy. After creating control charts, performance evaluation in the industry is important. The main contribution of this paper is to employs the FOC curve for evaluating the performance of the fuzzy control chart, while in prior studies in this area, the performance of fuzzy control chart has not been evaluated.