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The purpose of this paper is to develop an economic statistical design based on the concepts of adjusted average time to signal (AATS) and ANF for X ¯ control chart under a Weibull shock model with multiple assignable causes.

The design used in this study is based on a multiple assignable causes cost model. The new proposed cost model is compared with the same cost and time parameters and optimal design parameters under uniform and non-uniform sampling schemes.

Numerical results indicate that the cost model with non-uniform sampling cost has a lower cost than that with uniform sampling. By using sensitivity analysis, the effect of changing fixed and variable parameters of time, cost and Weibull distribution parameters on the optimum values of design parameters and loss cost is examined and discussed.

This research adds to the body of knowledge relating to the quality control of process monitoring systems. This paper may be of particular interest to practitioners of quality systems in factories where multiple assignable causes affect the production process.

The cost functions for uniform and non-uniform sampling schemes are presented based on multiple assignable causes with AATS and ANF concepts for the first time.

The purpose of this paper is to develop a double-objective economic statistical design (ESD) of (X ‾) control chart under Weibull failure properties with the Linex asymmetric loss function. The authors have expressed the probability of type II error (β) as the statistical objective and the expected cost as the economic objective.

The design used in this study is based on a double-objective economic statistical design of (X ‾) control chart with Weibull shock model via applying Banerjee and Rahim's model for non-uniform and uniform schemes with Linex asymmetric loss function. The results in the least average cost and β in uniform and non-uniform schemes by Linex loss function, compared with the same schemes without loss function.

Numerical results indicate that it is not possible to reduce the second type of error and costs at the same time, which means that by reducing the second type of error, the cost increases, and by reducing the cost, the second type of error increases, both of which are very important. Obtained based on the needs of the industry and which one has more priority has the right to choose. These designs define a Pareto optimal front of solutions that increase the flexibility and adaptability of the X ‾ control chart in practice. When the authors use non-uniform schemes instead of uniform schemes, the average cost per unit time decreases by an average and when the authors apply loss function, the average cost per unit time increases by an average. Also, this quantity for double-objective schemes with loss function compared to without loss function schemes in cases uniform and non-uniform increases. The reason for this result is that the model underestimated the costs before using the loss function.

This research adds to the body of knowledge related to flexibility in process quality control. This article may be of interest to quality systems experts in factories where the choice between cost reduction and statistical factor reduction can affect the production process.

The cost functions for double-objective uniform and non-uniform sampling schemes with the Weibull shock model based on the Linex loss function are presented for the first time.

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.

Studies the economic design of x^‐ control charts in a situation in which the duration time that the process remains in the in‐control state follows a general distribution which has an increasing hazard rate. In this situation, the active and persistent action for quality control is to design a process in which a preventive maintenance procedure is performed periodically. Addresses first the relationship between preventive maintenance and x^‐ control charts. A cost function which is opposed to those given by Banerjee and Rahim and by Hu is derived. The computational results indicate that the proposed model under a preventive maintenance policy has a lower expected total cost per hour than have those of Banerjee and Rahim’s and Hu’s Weibull shock models. Numerical examples also demonstrate that the model has great flexibility when applied in the situation previously mentioned. Presents the advantages of the combination of a preventive maintenance policy and x^‐ control charts and concludes that a preventive maintenance policy performed under a certain condition can be particularly instrumental in reducing the expected total cost per hour.

The purpose of this paper is to find the proper statistical distribution function, which can cover the failure time of a single machine or a group of machines. To this end, an innovative program is written in an Excel software, capable of assessing at least six statistical distribution functions. This research study intends to show the advantages of applying statistical distribution functions in an integrated model format to create or increase productive reliability machines. Productive reliability is a simultaneous combination of efficiency and effectiveness in reliability.

The method of theoretical research methodology comprises data collection tools, reference books and articles in addition to exploiting written reports of the Iranian Center for Defence’s Standards. The practical research method includes deploying and assessing the proposed model for a selected machine (in this case a computerized numerical control machine).

A comprehensive program in an Excel software having the capability of assessing at least six statistical distribution functions was developed to find the most efficient option for covering the failure times of each machine in the shortest time with the highest precision. This is regarded as the most important achievement of the present study. Furthermore, the advantages of applying the developed model are discussed and a large group of which have direct influences on the productivity of equipment reliability.

The originality of the research was ascertained by managers and experts working in maintenance issues at the different levels of the Defense Industries Organization.

The purpose of this paper is to investigate a new sampling methodology previously proposed for systems with a known lifetime distribution: the Predetermined Sampling Intervals (PSI) method.

The methodology is defined on basis of system hazard cumulative rate, and is compared with other approaches, particularly those whose parameters may change in real time, taking into account current sample information.

For different lifetime distributions, the results obtained for adjusted average time to signal (AATS) using a control chart for the sample mean are presented and analysed. They demonstrate the high degree of statistical performance of this sampling procedure, particularly when used in systems with an increasing failure rate distribution.

This PSI method is important from a quality and reliability management point of view.

This methodology involves a process by which sampling instants are obtained at the beginning of the process to be controlled. Also this new approach allows for statistical comparison with other sampling schemes, which is a novel feature.

Sets out to develop an integrated model that considers simultaneously inventory production decisions, PM schedule, and warranty policy for a deteriorating system that experiences shifts to an out of control state. The time to shift follows a general probability distribution with increasing hazard rate, so that time‐based PM is effective in improving the system reliability.

A profit function is used to model the production system. Optimization techniques are used to generate optimal solutions for the problem. Although global optimality cannot be guaranteed, empirical results show that global optimal solutions are obtained.

The integrated model provides decisions on inventory levels, production run length, and PM schedule simultaneously. It is illustrated through numerical examples that investment in PM can lead to savings in warranty claims for repairable products. As a result, the overall profit per unit, in certain cases, is higher with PM than without PM.

The production system is taken, numerical examples are presented and a sensitivity analysis is conducted to gain more insight into the developed model. In particular, the numerical analysis shows that a better PM program reduces warranty claims.

In addition to the joint optimization of production/inventory decisions and PM schedule, such models can be very useful in making resource allocation decisions between warranty and PM programs. It is clear from the numerical analysis that a better PM program reduces warranty claims.

The paper provides a joint optimization of production inventory decisions and the PM schedule for a system subject to a time to shift that follows a general probability distribution. Previous research considered only an exponential distribution and did not consider PM.

Develops the joint economic designs of • and S² control charts under four operating policies to monitor the process in a situation where the occurrence of the assignable cause follows a general distribution with an increasing hazard rate. The four operating policies can be chosen by quality controllers to cope with the specific process situation. Policy I and policy II assume that the process performs the preventive maintenance programme at equal and decreasing sampling time intervals, respectively. Policy III and policy IV in turn merely take samples using the non‐uniform and uniform sampling interval schemes without preventive maintenance. The derivation of the four models is not very difficult, so it can be used to derive another model. Offers numerical examples to compare the economic designs and the total expected costs per hour of the four models. Finds, from the computational results, policy II is the best for adoption in the design of • and S² control charts. The results also show that the proposed solution procedure is more accurate and better than Rahim et al.’s and Chung and Chen’s procedures. Concludes with remarks and some advantages of introducing the periodic preventive maintenance policy into a process.

The purpose of this paper is to measure the performance of a sampling method through the average number of samples drawn in control.

Matching the adjusted average time to signal (AATS) of sampling methods, using as a reference the AATS of one of them the paper obtains the design parameters of the others. Thus, it will be possible to obtain, in control, the average number of samples required, so that the AATS of the mentioned sampling methods may be equal to the AATS of the method that the paper uses as the reference.

A more robust performance measure to compare sampling methods because in many cases the period of time where the process is in control is greater than the out of control period. With this performance measure the paper compares different sampling methods through the average total cost per cycle, in systems with Weibull lifetime distributions: three systems with an increasing hazard rate (shape parameter β=2, 4 and 7) and one system with a decreasing failure rate (β=0, 8).

In a usual production cycle where the in control period is much larger than the out of control period, particularly if the sampling costs and false alarms costs are high in relation to malfunction costs, the paper thinks that this methodology allows us a more careful choice of the appropriate sampling method.

To compare the statistical performance between different sampling methods using the average number of samples need to be inspected when the process is in control. Particularly, the paper compares the statistical and economic performance between different sampling methods in contexts not previously considered in literature. The paper presents an approximation for the average time between the instant that failure occurs and the first sample with the process out of control, as well.