A dynamic economic model for an x¯-control chart design

Parkhideh and Case (HE Transactions, 21, 313-323 (1989)) developed an economic model for a dynamic jc-control chart. In developing the model they considered six decision variables in their design methodology. It therefore became very complicated to obtain the optimal combination that resulted in the minimum loss-cost. This note proposes an alternative and simplified design methodology that reduces the number of design variables from six to three. The optimal values are obtained by imposing the following constraints. The optimal sampling interval h_i, (i = 1,2,⃛) is chosen such that the integrated hazard rate over each sampling interval is constant. The optimal sample size n_i (i = 1,2,⃛) is chosen such that the relative sample size per unit time during each sampling interval is constant. Analogously, the optimal control limit coefficient k_i (i = 1,2,⃛) is chosen so that the power of the control charts remains constant over each sampling interval. The process failure mechanism is assumed to follow a Weibull shock model and the product quality characteristic is considered to be normal. Computational experience indicates that the proposed dynamic nonuniform control chart design is much simpler and provides a lower cost than that of Parkhideh and Case's dynamic model.     

A more accurate formulation for a class of modelsfor the economic design of control charts

Some of the published models for the economic design of control charts use the expected cost per unit of output as the objective function to be minimized. In these models, the computation of steady-state probabilities that the process is in each possible state does not take into account the effect of the corrective action that may follow a signal from the control chart. As a result, the expected cost per unit is overestimated and the selection of the chart parameters is not optimal. The purpose of this paper is to eliminate this inaccuracy by proposing the exact formulation and to estimate the magnitude of errors resulting from the inaccurate formulation of the objective function. By solving numerical examples of joint design of X and R charts, it is shown that these errors are typically very large and consequently it is imperative to use the exact formulation, proposed in this paper, to avoid inefficient and costly control chart designs. Finally, an additional opportunity for further cost improvements in process control is identified and discussed.     

Asymptotic Properties of the Restricted Bayesian Double Sampling Plan

We consider double sampling inspection plans for a given lot size, cost function, and prior distribution of the number of defectives. The restriction we impose on the sampling plan is that the size and acceptance number of the second sample do not depend on the outcome of the first sample. Hence, a double sampling plan is determined by five parameters: two sample sizes, two acceptance numbers, and one rejection number. The optimal plan is defined as the plan which minimizes the expected cost. Our results consist of finding optimal relations between the sample sizes, acceptance and rejection numbers and the lot size as the lot size tends to infinity. An asymptotic expansion of the regret function shows that for the optimal plan it increases as the lot size raised to the power 2/5 times the logarithm of the lot size raised to the power 1/5.     

Choice of Double Sampling Plans Based on Prior Distributions and Costs

Approximate models for the optimum economic design of double sampling plans for attributes are presented. Total cost is assumed to consist of sampling costs, the cost of accepting defective items, and the cost of rejecting good items. This cost is minimized relative to a specified prior distribution of process fraction defective. Models are developed for situations in which rejected lots are either scrapped or subjected to 100 percent inspection with defective items removed. Other variations of the basic model include the incorporation of restrictions among the sampling plan parameters, which may be helpful in improving administrative efficiency, and the use of curtailment on the second sample. Numerical examples for the various models are presented. Model sensitivity to the cost coefficients and to potential misspecification of the parameters of the prior distribution is investigated.     

Constrained optimization model for the design of an adaptive X chart

An economic-statistical model is developed for variable parameters (VP) X charts in which all design parameters vary adaptively, that is, each of the design parameters (sample size, sampling interval and control-limit width) vary as a function of the most recent process information. The cost function due to controlling the process quality through a VP X chart is derived. During the optimization of the cost function, constraints are imposed on the expected times to signal when the process is in and out of control. In this way, required statistical properties can be assured. Through a numerical example, the proposed economicstatistical design approach for VP X charts is compared to the economic design for VP X charts and to the economic-statistical and economic designs for fixed parameters (FP) X charts in terms of the operating cost and the expected times to signal. From this example, it is possible to assess the benefits provided by the proposed model. Varying some input parameters, their effect on the optimal cost and on the optimal values of the design parameters was analysed.     

THE COST OF RANDOMIZATION IN WORK SAMPLING: AN ILLUSTRATION

The effects on work sampling statistics of randomization and of sampling delays, minimum spacings between otherwise random observations, are illustrated with a process modeled as an alternating Poisson process, (APP). The relative efficiency (which can be viewed as a sample size ratio) of any sampling plan, to systematic sampling, is formed and plotted for random sampling, and for random sampling following delays of 10%, 20% and 50% of the average observation spacing.

Guidelines for sufficient spacing of systematic observations are modified for random sampling. Imposition of cost constraints on time span of a study yields a simple rule for cost optimum spacing of random observations which is shown to approximate the iteratively generated cost optimal spacings of all restricted random sampling plans.     

Economic–statistical Design of Maximum Exponentially Weighted Moving Average Control Charts

The maximum exponentially weighted moving average (MaxEWMA) control chart effectively combines the two EWMA charts into one chart and monitors both increases and decreases in the mean and/or variability. In this paper, we develop the economic–statistical design of the MaxEWMA control chart in which the Taguchi's quadratic loss function is incorporated into the Duncan's economical model. Numerical simulations are executed to minimize the expected total cost model and determine the optimal decision variables, including the sample size, sampling interval, control limit width, and the smoothing constant of the MaxEWMA control chart. It is shown that the optimal control limit width and smoothing constant increase as the optimal cost value increases and that both the optimal sample size and sampling interval always decrease as the magnitudes of mean and/or variance shifts increase. Copyright © 2012 John Wiley & Sons, Ltd.     

质量和成本约束下CSP-T和Spk集成过程控制方案

多水平连续抽样检验方案(CSP-T)是在线过程质量控制工具，但该方案的第一类风险和第二类风险都较高，且不能满足成本约束。提出了CSP-T和过程良率指数(S_pk)集成过程控制方案，该集成方案在满足质量约束的同时，以最小成本运行，并将两类风险控制在既定水平。建立了极限检验能力下的最优CSP-T方案，依据数据的计数特征驱动质量控制方案运行。基于S_pk估计的精确分布建立了风险控制方案，依据数据的计量特征驱动风险控制方案运行。质量控制方案和风险控制方案是独立互补关系。相比于CSP-T方案，检验工作量没有增加。企业案例验证了集成控制方案的可行性和有效性。     

An integrated EPQ model based on a control chart for an imperfect production process

The notion of quality control is introduced into the classic economic production quantity (EPQ) model. Based on previous research, this paper contributes to an integrated EPQ model combining the concepts of statistical process control and maintenance. An imperfect production process involving three different conditions is considered. A control chart is adopted to monitor the whole process with Taguchi's loss function estimating the quality cost of each condition. A corresponding maintenance policy is planned for the machine depending on what condition it runs in. Thus the proposed EPQ model takes quality-related costs and maintenance-related costs into account other than storage costs and ordering costs already considered in the classic model. The objective of this model is to minimise the total expected production cost per production cycle while simultaneously determining the optimal parameters of control chart design, the interval between sampling, the sample size and the maintenance decision policy. The pattern search method is used to solve the problem using the MATLAB toolbox. In addition, a case study, sensitivity analysis and comparison analysis are presented to demonstrate the application of the model.     

Availability maximization under partial observations

In this paper, we propose a new model for availability maximization under partial observations for maintenance applications. Compared with the widely studied cost minimization models, few structural results are known about the form of the optimal control policy for availability maximization models. We consider a failing system with unobservable operational states. Only the failure state is observable. System deterioration is driven by an unobservable, continuous-time homogeneous Markov process. Multivariate condition monitoring data which is stochastically related to the unobservable state of the system is collected at equidistant sampling epochs and is used to update the posterior state distribution for decision making. Preventive maintenance can be carried out at any sampling epoch, and corrective maintenance is carried out upon system failure. The objective is to determine the form of the optimal control policy that maximizes the long-run expected average availability per unit time. We formulate the problem as an optimal stopping problem with partial information. Under standard assumptions, we prove that a control limit policy is optimal. A computational algorithm is developed, illustrated by numerical results.     

An Individuals Generalized Likelihood Ratio Control Chart for Monitoring Linear Profiles

This paper investigates a generalized likelihood ratio (GLR) control chart for detecting sustained changes in the parameters of linear profiles when individual observations are sampled. The control charts usually used for monitoring linear profiles are based on taking a sample of n observations at each sampling time point, where n is large enough that a regression model can be fitted at each sampling point using these n observations. For this sampling scenario, it has been shown that a GLR control chart has many advantages over other control chart schemes in terms of convenience of design, fast detection of process changes, and useful diagnostic aids. However, in many applications, it may not be convenient or possible to take a sample larger than n = 1. Therefore, it is desirable to develop some control chart to monitor profile data with individual observations (n = 1) at each sampling point. In this paper, we consider a GLR control chart based on individual observations and show that it has certain advantages compared with the GLR chart based on groups of observations. An important advantage of GLR control charts is that the only design parameter that needs to be specified in order to use a GLR chart is the control limit, and here, control limits for linear profiles up to eight regression coefficients are provided for convenient use by practitioners. Copyright © 2013 John Wiley & Sons, Ltd.     

A charged particle-inspired sampling scheme for improved surrogate model quality

The Monte Carlo simulation (MCS) is an established tool for probabilistic analyses. It allows to estimate statistics of output values, to obtain sensitivities, and to build surrogate models. The quality of these results is increased with increasing sample size, however, this comes with additional computational cost. In practice, a compromise must be found between accuracy and time. Hence, the goal is to extract as much information as possible with one single sample.In this paper, Latinized Particle Sampling (LPS) is introduced as a new sampling method, which distributes the samples uniformly in the sample space. For this, the samples are considered as charged particles, which repel each other. In an iterative process a force equilibrium is obtained. In order to obtain the desired marginal distributions, the sample is latinized, giving a valid Latin Hypercube design. Additionally, a correlation control algorithm is applied to obtain a desired target correlation. Due to the uniform space filling, the quality of surrogate models is increased in comparison to regular Latin Hypercube sampling (LHS). Compared to an optimized LHS design, the surrogate model quality of LPS are lower, but LPS samples can be created much faster.     

Optimal process mean and quality improvement in a supply chain model with two-part trade credit based on Taguchi loss function

The two-part trade credit policy is developed to accelerate cash inflow that can avoid bad debt risk in the earlier economic order quantity (EOQ) models allowing only one period of time for delay in payment. Taguchi loss function has proved to be a more realistic function for fitting the actual quality loss cost in economic product quantity (EPQ) model. To minimise quality loss, optimal process mean setting shifts process mean to balance the cost outside the specification limits, quality improvement applies investment to reduce process variation. Supply chain integration has been proved that it can be used to minimise the entire cost more effectively than independent EOQ or EPQ models. This paper improves the earlier studies by incorporating the above research topics that have not been simultaneously discussed before, develops a supply chain model based on the Taguchi loss function, which combines the trade strategy from the retailer’s perspective and the quality adjustments from the supplier’s perspective to maximise total supply chain profit. We find that the trade credit terms definitely affect suppliers and retailers’ optimal decisions, and numerical examples can provide decision references for supply chain managers to set a trade credit policy and control quality.     

Economic design of a variable sampling rate EWMA chart

A Variable Sampling Rate (VSR) control chart is a control chart whose sampling scheme is to vary the sampling interval and the sample size for the next sample depending on the current chart statistic. A VSR EWMA chart is an EWMA chart with the VSR sampling scheme. An economic model, which was developed for a VSR chart, is also applied here to evaluate the efficiency of the VSR EWMA chart. The properties of the VSR EWMA chart are obtained by using a Markov chain approach. The model contains cost parameters which allow the specification of the costs associated with sampling, false alarms and operating off target as well as search and repair. This economic model can be used to quantify the cost saving that can be obtained by using a VSR chart instead of a Fixed Sampling Rate (FSR) chart and can also be used to gain insight into the way that a VSR chart should be designed to achieve optimal economic performance. It is shown that with some design parameter combinations the economically optimal VSR chart has a lower false alarm rate than the FSR chart.     

Economic design of an attributes control system for a multistage serial production process

A model for the economic design of an np-control system integrated within a multiple stage serial production process is presented. The total expected quality control cost includes the costs of sampling, the costs of investigating an out-of-control alarm and possibly correcting an assignable cause(s), and the costs associated with the production of non-conforming items. The model is represented as a directed network with decision variables of sample size, rejection number and frequency of sampling occurring at each stage of the process. A combination of dynamic programming and direct search techniques is applied to determine the set of sampling policies which yield minimum total expected cost. Numerical examples and results of a sensitivity analysis are reported.     

Scheduling methods for the statistical setup adjustment problem

Feedback control methods have been proposed in recent literature to regulate the quality characteristic of parts or products in a manufacturing process. Depending on the costs involved, adjustments may not be needed at each time instant (i.e. for every part or product). This paper presents scheduling methods to determine the optimal time instants for adjusting a process. The focus is on the set-up adjustment problem, in which it is necessary to adjust the process in order to compensate for an initial offset that occurs due to an incorrect set-up operation. The performance of three scheduling methods are compared in terms of the expected manufacturing cost and computational effort of each method. The adjustment methods considered are based on estimates of the process variance and the size of the offset. The robustness of these methods with respect to biased estimates of the process variance and of the set-up error or offset are discussed. One simple method, a backward implementation of the Silver-Meal heuristic used for inventory control is recommended based on a performance analysis.     

Sample Sizes for Comparison of Proportions

This paper examines the problem of determining sample sizes for comparing sample proportions from n independent binomial populations under the null hypothesis that the population proportions are different. For specified precisions and sampling costs, it is shown that the problem of determining optimal sample sizes can be formulated as a convex programming problem. In addition, the problem of obtaining the best precision within a budgeted cost of sampling is examined. This methodology is illustrated with an example application.     

Optimal Design of VSI ―X Control Charts for Monitoring Correlated Samples

This paper develops an economic design of variable sampling interval (VSI)―X control charts in which the next sample is taken sooner than usual if there is an indication that the process is off‐target. When designing VSI―X control charts, the underlying assumption is that the measurements within a sample are independent. However, there are many practical situations that violate this hypothesis. Accordingly, a cost model combining the multivariate normal distribution model given by Yang and Hancock with Bai and Lee's cost model is proposed to develop the design of VSI charts for correlated data. An evolutionary search method to find the optimal design parameters for this model is presented. Also, we compare VSI and traditional ―X charts with respect to expected cost per unit time, utilizing hypothetical cost and process parameters as well as various correlation coefficients. The results indicate that VSI control charts outperform the traditional control charts for larger mean shift when correlation is present. In addition, there is a difference between the design parameters of VSI charts when correlation is present or absent. Copyright © 2005 John Wiley & Sons, Ltd.     

Economically optimal sample size,frequency and capability indices using X¯ charts

In determining inspection plans for quality control, two important variables are the frequency of sampling and the sample size. Typical values are once per hour, with a sample of size 4 or 5. However, this may be far from the most appropriate or efficient scheme in a given setting. Also, the capability index necessary to produce zero defects under the historical maximum and average shift in the process is essential information in providing motivation for process improvement. This paper presents microcomputer-based software using readily available information for determining economically optimal sampling plans and zero-defects capability indices. Great economic benefits can result from more careful attention to sampling plans, and the attainment of zero defect process capability indices.     

Economic production quantity (EPQ) model with partial backordering and a discount for imperfect quality batches

Economic production quantity (EPQ) models are traditionally used in operations management. Despite the large number of papers that describe the models, the classic EPQ model does not consider either imperfect quality batches or shortages. However, some industries may be able to sell imperfect items for a lower price, reducing the total production cost. This paper proposes an EPQ model with partial backordering and discount for imperfect quality batches and an algorithm that returns optimal values for the problem. From a numerical example, it is possible to analyse how the changes in the variables affect each part of the total cost function, which provides a useful tool for strategic decision-making. We conclude that it is better to sell imperfect items as soon as possible because the savings in holding costs results in a total cost reduction. It is more profitable for the producer to have planned shortages considering that some costumers are willing to wait. Furthermore, the reduction of the goodwill cost does not necessarily reduce the total cost.