Multivariate Bayesian Control Chart Research Articles

Multivariate Bayesian Control Chart Based on Economic-Statistical Design with 2- and 3-Variable Sample Size

Multivariate Bayesian Control Chart Based on Economic-Statistical Design with 2- and 3-Variable Sample Size

Time to signal distribution of multivariate bayesian control chart with dual sampling scheme

The ability of a control chart to detect a shift in the process can be determined by the average run length (ARL), a widely used performance indicator for single sampling interval (SSI) control charts. When it comes to variable sampling interval (VSI) schemes, average time to signal (ATS) is utilized instead of ARL. Although analysis of ARL has been extensively studied for SSI schemes, calculation of the ATS for VSI control charts is still in its infancy due to several theoretical challenges. In this context, the paper derives closed-form expressions for time to signal distribution and the ATS for a spatial design of VSI multivariate Bayesian control chart referred to as the dual sampling scheme (DSS). In a DSS strategy, the process is being monitored initially with a longer sampling interval, which is changed to the shorter sampling interval at a designed switching point. Unlike previous approximate developments where the switching point is ignored, the paper derives exact expressions for computation of time to signal distribution and ATS within such a Bayesian framework. Derivations of the ATS are based on an intuitively pleasing definition of an artificial absorbing state and coded values of the posterior probability using Markov chain theory.

Multivariate Bayesian Control Chart Based on Economic-Statistical Design with 2 and 3-Variable Sample Size

Multivariate Bayesian Control Chart Based on Economic-Statistical Design with 2 and 3-Variable Sample Size

Economic and Economic-Statistical Design of Multivariate Bayesian Control Chart with Variable Sampling Intervals

Due to the proper performance of Bayesian control chart in detecting process shifts, it recently has become the subject of interest. It has been proved that on Bayesian and traditional control charts, the economic and statistical performances of the variable sampling interval (VSI) scheme are superior to those of the fixed ratio sampling (FRS) strategy in detecting small to moderate shifts. This paper studies the VSI multivariate Bayesian control chart based on economic and economic-statistical designs. Since finding the distribution of Bayesian statistic is t complicated, we apply Monte Carlo method and we employ artificial bee colony (ABC) algorithm to obtain the optimal design parameters (sample size, sampling intervals, warning limit and control limit). In the end, this case study is compared with VSI Hotelling’s T2 control chart and it is shown that this approach is more desirable statistically and economically.

A two-level Bayesian early fault detection for mechanical equipment subject to dependent failure modes

A two-level Bayesian control approach is presented to detect early fault for mechanical equipment subject to dependent degradation and catastrophic failures. The system degradation process is modeled using a continuous time stochastic process with three states. To model the dependence of two failure modes, we assume that the joint distribution of the time to catastrophic failure and sojourn time in the healthy state follows Marshall-Olkin bivariate exponential distribution. To avoid unnecessary sampling cost and to effectively detect impending failure, a two-level control policy, where longer sampling interval is applied for healthier state and shorter sampling interval is used in severe degradation state is proposed in Bayesian control chart framework for a multivariate observation process considering dependent failure modes. The optimization problem is formulated and solved in the semi-Markov decision process (SMDP) framework. A formula for the mean residual life (MRL) is also derived using the Bayesian approach. The validation of the proposed methodologies is carried out using real multivariate degradation data obtained from a milling machine. A comparison with the multivariate Bayesian control chart with a single sampling interval and a single control limit is given, which illustrates the effectiveness of the proposed approach.

Optimal Bayesian early fault detection for CNC equipment using hidden semi-Markov process

Abstract The goal of a computer numerically controlled (CNC) equipment is to guarantee high specified performance and to maintain it effectively over its life cycle time. Thus, the assessment of health condition is crucial for industrial systems. However, very few papers have dealt with the cost-optimal early fault detection and remaining useful life prediction of CNC equipment using multivariate positioning data. The novel approach presented here is based on vector autoregressive (VAR) degradation modeling and hidden semi-Markov modeling using the optimal Bayesian control technique. System condition is modeled using a continuous time semi-Markov chain with three states, i.e. unobservable healthy state 1, unobservable warning state 2 and observable failure state 3. Model parameter estimates are calculated using the expectation-maximization (EM) algorithm. The optimal control policy for the three-state model is represented by a Bayesian control chart for a multivariate observation process. The optimization problem is formulated and solved in the semi-Markov decision process (SMDP) framework. A formula for the mean residual life (MRL) is also derived based on Bayesian approach, which enables the estimation of the remaining useful life and early maintenance planning based on the observed data. The validation of the proposed methodologies is carried out using actual multivariate degradation data obtained from a CNC equipment. A comparison with the multivariate Bayesian control chart based on a hidden Markov model (HMM) is given, which illustrates the effectiveness of the proposed approach.

Economic design based on a multivariate Bayesian VSI chart with a dual control limit

ABSTRACTIn this paper, a multivariate Bayesian variable sampling interval (VSI) control chart for the economic design and optimization of statistical parameters is designed. Based on the VSI sampling strategy of a multivariate Bayesian control chart with dual control limits, the optimal expected cost function is constructed. The proposed model allows the determination of the scheme parameters that minimize the expected cost per time of the process. The effectiveness of the Bayesian VSI chart is estimated through economic comparisons with the Bayesian fixed sampling interval and the Hotelling's T2 chart. This study is an in-depth study on a Bayesian multivariate control chart with variable parameter. Furthermore, it is shown that significant cost improvement may be realized through the new model.

Multivariate Bayesian control chart with estimated parameters

In this paper, the effects of estimating the mean vector and covariance matrix on the performance of the multivariate Bayesian control chart are studied. Through using some indicative cases, we show that the economic performance is affected when the parameters are unknown compared to the known parameters case. We also show that when Mahalanobis distance (M-distance) is fixed, there are no significant differences between different shift directions. Furthermore, with the increase of the number of quality characteristics, the optimal expected average cost decreases in the case of estimated parameters. After investigating the sampling strategies of phase 2, we find the optimal expected average cost is significantly influenced by sampling interval and sample size. Finally, an example is given to show how to choose an enough the number of phase 1 samples.

Optimal Cost-Effective Maintenance Policy for a Helicopter Gearbox Early Fault Detection under Varying Load

Most of the existing fault detection methods rarely consider the cost-optimal maintenance policy. A novel multivariate Bayesian control approach is proposed, which enables the implementation of early fault detection for a helicopter gearbox with cost minimization maintenance policy under varying load. A continuous time hidden semi-Markov model (HSMM) is employed to describe the stochastic relationship between the unobservable states and observable observations of the gear system. Explicit expressions for the remaining useful life prediction are derived using HSMM. Considering the maintenance cost in fault detection, the multivariate Bayesian control scheme based on HSMM is developed; the objective is to minimize the long-run expected average cost per unit time. An effective computational algorithm in the semi-Markov decision process (SMDP) framework is designed to obtain the optimal control limit. A comparison with the multivariate Bayesian control chart based on hidden Markov model (HMM) and the traditional age-based replacement policy is given, which illustrates the effectiveness of the proposed approach.

Multivariate Bayesian control chart with estimated parameters

In this paper, the effects of estimating the mean vector and covariance matrix on the performance of the multivariate Bayesian control chart are studied. Through using some indicative cases, we show that the economic performance is affected when the parameters are unknown compared to the known parameters case. We also show that when Mahalanobis distance (M-distance) is fixed, there are no significant differences between different shift directions. Furthermore, with the increase of the number of quality characteristics, the optimal expected average cost decreases in the case of estimated parameters. After investigating the sampling strategies of phase 2, we find the optimal expected average cost is significantly influenced by sampling interval and sample size. Finally, an example is given to show how to choose an enough the number of phase 1 samples.

Economic design of multivariate Bayesian control chart with two sampling intervals

In this paper, the economic design of a multivariate Bayesian control chart with two sampling intervals for monitoring the process mean is developed. The economic design of traditional control charts such as T2, SPRT, EWMA, and CUSUM charts with variable sampling interval (VSI) has been developed in recent years to control the industrial processes by minimizing the average cost of the process, however, it has been proved that these traditional, non-Bayesian process control charts are not optimal. There has been some limited development of the economic design of Bayesian control charts with a single sampling interval (SSI), but no results regarding the economic design of the Bayesian control chart with two sampling intervals have been reported in the literature. The objective is to find the optimal control chart parameters, namely the sample size, sampling intervals and the control limits, minimizing the total long-run expected average cost per unit time. The proposed problem is formulated and solved in the semi-Markov decision process (SMDP) framework. A numerical example is provided to illustrate the whole procedure. The effectiveness of the newly proposed control chart is demonstrated through a comparison with the multivariate Bayesian control chart with a single sampling interval as well as with a recently developed MEWMA control chart. The results indicate that the proposed control chart has a considerably better performance.

Multivariate Bayesian control chart based on economic and economic-statistical design using Monte Carlo method and ABC algorithm

Recent studies on multivariate control charts are based on Hotelling's T2 control chart. Due to the fact that the Bayesian control charts in a univariate can be effective on performance of control charts, this paper has proposed the multivariate Bayesian control chart based on economic and economic-statistical design. As finding distribution of Bayesian statistic is too difficult, the Monte Carlo method has been employed in order to determine design optimum parameters (sample size, sampling interval and control limit). Also, artificial bee colony (ABC) algorithm has been applied to search for the optimal design parameters. Ultimately, the performance of the proposed control chart and Hotelling's T2 were compared for more in-depth investigation.

Optimal Maintenance Policy for a Partially Observable Two-Unit System

In this paper, we present a maintenance model of a two-unit series system with economic dependence. Unit#1 which is considered to be more expensive and more important, is subject to condition monitoring (CM) at equidistant, discrete time epochs and unit#2, which is not subject to CM has a general lifetime distribution. The multivariate observation vectors obtained through condition monitoring carry partial information about the hidden state of unit#1, which can be in a healthy or a warning state while operating. Only the failure state is assumed to be observable for both units. The objective is to find an optimal opportunistic maintenance policy minimizing the long-run expected average cost per unit time. The problem is formulated and solved in the partially observable semi-Markov decision process framework. An effective computational algorithm for finding the optimal policy and the minimum average cost is developed, illustrated by a numerical example. Keywords—Condition-Based Maintenance, Semi-Markov Decision Process, Multivariate Bayesian Control Chart, Partially Observable System, Two-unit System.

Economic optimisation in automotive body manufacturing quality control based on multivariate Bayesian control chart

The present paper is a case study of quality control cost optimisation in an automobile body production system, where the multivariate Bayesian control chart is applied. Based on the enterprise’s historical data, the applicability of the proposed Bayesian control method is analysed. The cost function of engine compartment front axle vertical beam welding is clarified, and the parameters of the cost function are determined. The Monte Carlo simulation method is used to testify the improvement of the multivariate Bayesian control chart. Based on the result, it is believed that decreasing the rate of false alarm is an important factor for the economic improvement of the multivariate Bayesian control chart.

A simulation-based multivariate Bayesian control chart for real time condition-based maintenance of complex systems

When complex systems are monitored, multi-observations from several sensors or sources may be available. These observations can be fused through Bayesian theory to give a posterior probabilistic estimate of the underlying state which is often not directly observable. This forms the basis of a Bayesian control chart where the estimated posterior probability of the state can be compared with a preset threshold level to assess whether a full inspection is needed or not. Maintenance can then be carried out if indicated as necessary by the inspection. This paper considers the design of such multivariate Bayesian control chart where both the transition between states and the relationship between observed information and the state are not Markovian. Since analytical or numerical solutions are difficult for the case considered in this paper, Monte Carlo simulation is used to obtain the optimal control chart parameters, which are the monitoring interval and the upper control limit. A two-stage failure process characterised by the delay time concept is used to describe the underlying state transition process and Bayesian theory is used to compute the posterior probability of the underlying state, which is embedded in the simulation algorithm. Extensive examples are shown to demonstrate the modelling idea.

A Bayesian model and numerical algorithm for CBM availability maximization

In this paper, we consider an availability maximization problem for a partially observable system subject to random failure. System deterioration is described by a hidden, continuous-time homogeneous Markov process. While the system is operational, multivariate observations that are stochastically related to the system state are sampled through condition monitoring at discrete time points. The objective is to design an optimal multivariate Bayesian control chart that maximizes the long-run expected average availability per unit time. We have developed an efficient computational algorithm in the semi-Markov decision process (SMDP) framework and showed that the availability maximization problem is equivalent to solving a parameterized system of linear equations. A numerical example is presented to illustrate the effectiveness of our approach, and a comparison with the traditional age-based replacement policy is also provided.

Economic and economic-statistical design of a multivariate Bayesian control chart for condition-based maintenance

Economic and economic-statistical design of a multivariate Bayesian control chart for condition-based maintenance

Multivariate Bayesian Control Chart

A multivariate Bayesian control chart for monitoring process mean under the assumption that the vector of process observations follows a multivariate normal distribution is considered. Traditional control charts such as Hotelling's T2, EWMA, and CUSUM charts have been applied to control industrial processes characterized by several measurable variables. It is well known that these traditional, non-Bayesian process control techniques are not optimal, but very few results regarding the structure of the Bayesian control policy have been reported in the literature, all dealing with the univariate, finite-horizon case. In this paper, we formulate the multivariate Bayesian process control problem in the optimal stopping framework. The objective is to find a stopping rule under partial observations, minimizing the long-run expected average cost per unit time for a given sample size and sampling interval. Under standard operating and cost assumptions, it is proved that a control limit policy is optimal, and an algorithm is presented to find the optimal control limit and the minimum average cost.