An important topic in Quality Control is the study of tolerance limits. In Quality Control, another important topic which is closely related to tolerance limits is the acceptance sampling plans. There are two kinds of acceptance sampling plans: attribute sampling plans and variable sampling plans. For an attribute sampling plan, the quality of an item is measured by the attribute of the item, defective and non-defective say. Otherwise, if the quality of an item is measured by a random variable, it is a variable sampling plan. Perhaps, a more crucial problem in sampling inspection is the design of a sampling plan, i.e. the determination of the sample size and the specification limit(s). Many schemes have been studied for the design of a sampling plan. There are: the producer's and consumer's risk point schemes, the defence sampling schemes, Dodge and Romig's schemes, and the decision theory schemes.

In this article, a sequential variable sampling plan is studied. Suppose that the quality of an item in a batch is measured by a random variable with exponential distribution; its parameter is unknown having a gamma prior distribution. Then by using Bayesian approach and considering a Markov decision process, the optimality equations for the minimum total expected cost are formulated. We show that an optimal decision rule will have a control limit structure and monotonicity. A backward induction method is suggested that is a finite algorithm for the numerical solution of the sequential sampling plan.

Purpose The purpose of this paper is to study a geometric process (GP) maintenance model and policy for a repairable system. Design/methodology/approach Lam first introduced the GP and its application to maintenance model. Assume that a replacement policy N is applied by which the system will be replaced by a new, identical one following the N th failure. Findings For a deteriorating system, an optimal replacement policy is determined analytically, and the monotonicity properties of the optimal replacement policy are then studied. Originality/value For an improving system, the paper shows that the optimal replacement policy is the ∞ policy, i.e., the policy without replacement.

In this article, a general geometric process model with δ-shock for a deteriorating system and an improving system is studied. A system will fail as soon as a shock arrives with interarrival time being inside a real set. Assume that a replacement policy N is adopted by which the system will be replaced by a new, identical one at the time following the N-th failure. Then the optimal replacement policy N* for minimising the long-run average cost per unit time is determined analytically. Finally, the sensitivity analysis is studied.

Geometric process modeling is a useful tool to study repairable deteriorating systems in maintenance problems. This model has been used in a variety of situations such as the determination of the optimal replacement policy and the optimal inspection-repair-replacement policy for standby systems, and the analysis of data with trend. In this article, Bayesian inference for the geometric process with several popular life distributions, for instance, the exponential distribution and the lognormal distribution, are studied. The Gibbs sampler and the Metropolis algorithm are used to compute the Bayes estimators of the parameters in the geometric process. Simulation results are presented to illustrate the use of our procedures.

A geometric process delta -shock maintenance model for a repairable system is introduced. If there exists no shock, the successive operating time of the system after repair will form a geometric process. Assume that the shocks will arrive according to a Poisson process. When the interarrival time of two successive shocks is smaller than a specified threshold, the system fails, and the latter shock is called a deadly shock. The successive threshold values are monotone geometric. The system will fail at the end of its operating time, or the arrival of a deadly shock, whichever occurs first. The consecutive repair time after failure will constitute a geometric process. A replacement policy N is adopted by which the system will be replaced by a new, identical one at the time following the N th failure. Then, for the deteriorating system, and the improving system, an optimal policy N* for minimizing the long-run average cost per unit time is determined analytically.

Geometric Process Other versions of this articleLATEST VERSIONGeometric ProcessYeh Lam, Published online:  15 October 2008DOI: 10.1002/0471667196.ess6025.pub3Full textPDFReferencesRequest permissionsPREVIOUS VERSION 2Geometrical ProcessPublished online:  15 August 2006DOI: 10.1002/0471667196.ess6025.pub2AbstractFull textPDFReferencesRequest permissionsPREVIOUS VERSION 1Geometrical ProcessYeh Lam, Published online:  15 October 2004DOI: 10.1002/0471667196.ess6025AbstractFull textReferencesRequest permissions Yeh Lam, Yeh Lam ylam@saas.hku.hk The University of Hong Kong, Hong Kong The Northeastern University at Qmhuangdao, ChinaSearch for more papers by this author Yeh Lam, Yeh Lam ylam@saas.hku.hk The University of Hong Kong, Hong Kong The Northeastern University at Qmhuangdao, ChinaSearch for more papers by this author First published: 15 October 2008 https://doi.org/10.1002/0471667196.ess6025.pub3 Read the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Encyclopedia of Statistical SciencesBrowse other articles of this reference work:BROWSE BY TOPICBROWSE A-Z RelatedInformation

In this article, an integral equation satisfied by the expected number of events occurred by time t in a geometric process is considered; this equation is a generalization of the renewal equation. A numerical solution based on the trapezoidal integration rule to the integral equation is introduced. To explain the method used in the article, four numerical examples with the exponential distribution, gamma distribution, Weibull distribution and lognormal distribution, respectively, are studied.

The sampling inspection problem is one of the main research topics in quality control. In this paper, we employ Bayesian decision theory to study single and double variable sampling plans, for the Weibull distribution, with Type II censoring. A general loss function which includes the sampling cost, the time-consuming cost, the salvage value, and the after-sales cost is proposed to determine the Bayes risk and the corresponding optimal sampling plan. Explicit expressions for the Bayes risks for both single and double sampling plans are derived, respectively. Numerical examples are given to illustrate the effectiveness of the proposed method. Comparisons between single and double sampling plans are made, and sensitivity analysis is performed.

Geometric Process Geometric Function Statistical Inference of Geometric Process Application to Data Analysis Geometric Process Maintenance Model Application to Analysis of System Reliability Application to Operational Research.

In this paper, a geometric process maintenance model with preventive repair is studied. A maintenance policy (T,N) is applied by which the system will be repaired whenever it fails or its operating time reaches T whichever occurs first, and the system will be replaced by a new and identical one following the Nth failure. The long-run average cost per unit time is determined. An optimal policy (T∗,N∗) could be determined numerically or analytically for minimizing the average cost. A new class of lifetime distribution which takes into account the effect of preventive repair is studied that is applied to determine the optimal policy (T∗,N∗).

In this paper, we study a geometric process model for M/M/1 queueing system with a repairable service station. By introducing a supplementary variable, some queueing characteristics of the system and reliability indices of the service station are derived. Then a replacement policy N for the service station by which the service station will be replaced following the Nth failure is applied. An optimal replacement policy N∗ for minimizing the long-run average cost per unit time for the service station is then determined.

In this paper, a sequential variable sampling plan is studied. Suppose that the quality of an item in a batch is measured by a normally distributed random variable with a known variance, but the mean is unknown with a normal prior distribution. Then by using Bayesian approach and considering a Markov decision process, the optimality equations for the minimum total expected cost are formulated. We show that an optimal decision rule will have a control limit structure. An algorithm for a sequence of ϵ-optimal decisions is introduced. Then, the statistical procedure for conducting the sequential sampling plan is presented.

In this paper, a δ-shock maintenance model for a deteriorating system is studied. Assume that shocks arrive according to a renewal process, the interarrival time of shocks has a Weibull distribution or gamma distribution. Whenever an interarrival time of shocks is less than a threshold, the system fails. Assume further the system is deteriorating so that the successive threshold values are geometrically nondecreasing, and the consecutive repair times after failure form an increasing geometric process. A replacement policy N is adopted by which the system will be replaced by an identical new one at the time following the Nth failure. Then the long-run average cost per unit time is evaluated. Afterwards, an optimal policy N* for minimizing the long-run average cost per unit time could be determined numerically.

During the outbreak of an epidemic disease, for example, the severe acute respiratory syndrome (SARS), the number of daily infected cases often exhibit multiple trends: monotone increasing during the growing stage, stationary during the stabilized stage and then decreasing during the declining stage. Lam first proposed modelling a monotone trend by a geometric process (GP) [X(i), i=1,2,...] directly such that [a(i-1)X(i), i=1,2,...] forms a renewal process for some ratio a>0 which measures the direction and strength of the trend. Parameters can be conveniently estimated using the LSE methods. Previous GP models limit to data with only a single trend. For data with multiple trends, we propose a moving window technique to locate the turning point(s). The threshold GP model is fitted to the SARS data from four regions in 2003.

In this paper, we study a monotone process maintenance model for a multistate system with k working states and ℓ failure states. By making different assumptions, we can apply the model to a multistate deteriorating system as well as to a multistate improving system. We show that the monotone process model for a multistate system is equivalent to a geometric process model for a two-state system. Then, for both the deteriorating and the improving system, we analytically determine an optimal replacement policy for minimizing the long-run average cost per unit time.

A stochastic process {Xi} is a geometric process if there exists a positive real number a such that {ai−1Xi} generates a renewal process. Under the assumption that X1 follows a Gamma distribution, the statistical inference problem for the geometric process is studied. The parameters a,μ and σ2, where μ and σ2, are respectively, the mean and variance of X1, are estimated by parametric methods including maximum likelihood method along with some nonparametric methods previously proposed by Y. Lam such as the modified moment method. Limiting distributions for the maximum likelihood estimators are derived and this enables us to construct confidence intervals and perform hypothesis testing on parameters. Then some suggestions on the choice of methods are made based on simulation experiments and real data analysis.

In this paper, a shock model for the maintenance problem of a repairable system is studied. Assume that shocks will arrive according to a Poisson process. If the interarrival time of two successive shocks is less than a threshold, then the system will fail. For a deteriorating system, we assume that the successive threshold values are geometrically nondecreasing after repair, and the consecutive repair times after failure form an increasing geometric process. For an improving system, we assume that the successive threshold values are geometrically decreasing after repair, and the consecutive repair times after failure form a decreasing geometric process. A replacement policy N is adopted by which we shall replace the system by an identical new one at the time following the Nth failure. Then for each of the deteriorating system and improving system, an optimal policy N∗ for minimizing the long-run average cost per unit time is determined explicitly.

Geometric process was first introduced by Lam [10,11] . A stochastic process { X i , i = 1, 2, · · ·} is called a geometric process (GP) if, for some a > 0, { a i -1 X i , i = 1, 2, · · ·} forms a renewal process. In this paper, the GP is used to analyze the data from a series of events. A nonparametric method is introduced for the estimation of the three parameters in the GP. The limiting distributions of the three estimators are studied. Through the analysis of some real data sets, the GP model is compared with other three homogeneous and nonhomogeneous Poisson models. It seems that on average the GP model is the best model among these four models in analyzing the data from a series of events.

In this paper, an inspection–repair–replacement (IRR) model for a deteriorating system with unobservable state is studied. Assume that the system state can only be diagnosed by inspection and an inspection is imperfect. After inspection, if the system is diagnosed as being in a down state, a minimal repair will be undertaken, otherwise we do nothing. Assume further that the system lifetime is a random variable having increasing failure rate. A feasible IRR policy is studied. An algorithm is then suggested for determining an optimal feasible IRR policy for minimizing the long-run average cost per unit time after a finite-step search.