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Article Excerpt
1. Introduction
In modern manufacturing industries, operations have become very complicated. In most cases, a product cannot be produced by a single operational stage. Multistage manufacturing processes can be found in various industries, such as automobile assembly and semiconductor manufacturing. Multistage processes have a unique property that is different from single-stage processes: the quality that the current stage's output not only depends on the current stage itself, but also on the quality characteristics of previous stages. In other words, there is a relationship between different stages that impacts the quality of each process's output. The quality control of the process is thus complicated.
Conventional Statistical Process Control (SPC) and statistical quality control techniques have been well developed and widely applied in many industries (see Woodall and Montgomery (1999) and Stoumbos et al. (2000) for detailed surveys). However, most SPC methods assume a single-stage process and do not take multistage complexity into consideration. Ignoring inherent structural information (stage-wise correlations) among stages could make the conventional SPC less effective and efficient.
In order to detect out-of-control conditions in multistage processes, people may monitor process outputs at the final stage using SPC techniques (Xiang and Tsung, 2008), such as Shewhart charts, Cumulative Sum (CUSUM) charts and Exponentially Weighted Moving Average (EWMA) charts for univariate cases, and Hotelling's [T.sup.2] charts, multivariate EWMA (MEWMA) and multivariate CUSUM (MCUSUM) charts for multivariate cases. However, as discussed by Xiang and Tsung (2008). the final-stage charting may not perform well due to overlooking stage-wise correlations thereby missing the opportunity of early-stage detection. Alternatively, quality measurements may be monitored at each individual stage. However, it is usually too costly to perform all-stage monitoring when the number of stages is large, such as in automobile manufacturing. How to efficiently and effectively allocate the charts at appropriate stages through consideration of a stage-wise correlation structure remains a challenge.
No matter which control chart we use, where to allocate the charts in a multistage process is always an important issue. This is because cost and resources are always limited in reality and it is not always possible to set up control charts to measure and monitor the process outputs at every single stage in a complex multistage process. When dealing with such cases, decisions about chart allocation are usually made based on common sense and experience. Because of the lack of a systematic way to support the decisions, many of the decisions cause the monitoring performance to be inefficient and costly. However, little attention has been given to this problem in SPC research.
The remainder of this paper is organized as follows. We first introduce a Stream-Of-Variance (SOV) model in a state space form to describe a general multistage process in Section 2. Based on that, chart allocation strategies for both the output monitoring and residual monitoring methods are proposed in Sections 3 and 4. We aim at finding the stage in which the critical fault can be detected most quickly according to the criterion of non-centrality parameters and its corresponding Average Time to Signal (ATS). In Section 5, two real examples are used to demonstrate how the chart allocation strategies can be applied and illustrate the efficiency of the strategies. Section 6 investigates the impact of uncertainty in structural parameters on the proposed method. An extension to multiple fault cases is tackled via dynamic programming optimization in Section 7. Section 8 concludes this paper with a summary of contributions and some future extensions of the research.
2. Multistage process modeling and charting methods
2.1. Multistage modeling
The modeling of multistage manufacturing processes is more complicated than a single-stage process because of the complex interrelationships among stages. For example, the quality of downstream stages can be affected by upstream stages. In such a case, a SOV model in a state space form has been applied successfully to describe such interrelationships and variation propagation at the process level of a multistage process (Ding, Shi and Ceglarek, 2002; Ding, Ceglarek and Shi, 2002). With this model, physical and engineering knowledge can help to make the inherent structural information explicit. Djurdjanovic and Ni (2001), Huang et al. (2002) and Zhou, Huang and Shi (2003) considered applications of this model in machining processes. Tsung et al. (2008) also reviewed how the state space model was applied in multistage processes.
Here, we apply the popular model represented in Equation (1) to describe a multistage process and to develop our chart allocation strategies.
[y.sub.k] = [C.sub.k][x.sub.k] + [w.sub.k], [X.sub.k] = [A.sub.k][X.sub.[k-1]] + [U.sub.k] - [v.sub.k], (1)
Two kinds of quality information are described in this equation. The first is the state vector, [x.sub.k], such as the dimensional deviations of parts in an assembly process. The second is the observed quality information, [y.sub.k], which is the quality measurement of the process output at the kth stage. Through a cascade process, these two kinds of information are transferred when a product is passed to its downstream stage. In addition, [A.sub.k] denotes how the quality information in stage k-1 transfers to the quality information in stage k. [C.sub.k] indicates the relationship between the quality measurement, [y.sub.k], and the state vector, [X.sub.k], in stage k. In practice, both [A.sub.k] and [C.sub.k] can be obtained from engineering knowledge and product information. Moreover, inherent process noise is considered: [v.sub.k] represents the process noise such as background disturbance and unmodeled error, and [W.sub.k] is the measurement error, such as sensor noise in the process.
Here [U.sub.k] represents a process fault or an out-of-control condition, such as an unacceptable fixture deviation. It is natural to assume that the process fault has an additive effect on [x.sub.k]. This is consistent with common practice in quality control, in which the process faults (e.g., the fixture error, machining error or thermal errors in machining processes, etc.) are considerd as system inputs (Zhou et al., 2004). We also assume a prior knowledge about the fault patterns, i.e., their magnitudes and directions. Many researchers have discussed how to obtain and estimate the fault patterns in various industries and processes (Ceglarek et al., 1994; Apley and Shi, 1998; Ceglarek and Shi, 1996). For example, in an auto assembly process, the set of potential tooling faults can be prefixed and limited to certain major elements of the fixture according to the CAD data and mechanical structure of the machines. The relationship between the faults and output measurements can closely fit a linear model, so that the directions and magnitudes of the faults can be estimated accordingly (Ceglarek and Shi, 1996).
In the following, the EWMA and MEWMA charts are applied for multistage process control because of their popularity. We first consider the situation with the occurrence of a single fault in the monitoring of a multistage process. Extensions to cases with multiple faults will be discussed in a later section.
2.2. Charting methods
Nowadays, most multistage processes also involve multiple dimensional quality characteristics. Therefore, in this paper, multivariate control charts are used. Hotelling's [T.sup.2] charts (Hotelling's, 1947), MEWMA (Lowry et al 1992) and MCUSUM (Crosiers, 1988) are the most popular and well-known multivariate control charts in SPC. If the number of variables is large, more advanced methods such as latent structure methods including principal components analysis and partial least squares techniques can be applied (MacGregor et al, 1994; Smilde et al, 2000). Recently, the MEWMA chart has received considerable attention from researchers because of its excellent detection power and flexibility (Prabhu and Runger, 1997, Molnau, Montgomery, and Runger, 2001, Stoumbos and Sullivan, 2002, Testik et al., 2003). Thus, we will particularly emphasize the use of MEWMA charts in this paper.
Lowry et al, (1992) developed the MEWMA control chart based on the EWMA control chart. Suppose that we monitor p variables simultaneously. We have the variables in a p x 1 vector [y.sub.[i,k]], where i is the time point and k is the stage number. Then, the EWMA statistic, [Z.sub.[i,k]], is obtained as:
[Z.sub.[i,k]] = [lambda][y.sub.[i,k]] + (1 - [lambda])[Z.sub.[i-1,k]] (2)
where < [lambda] [less than or equal to] 1 and [Z.sub.0] = 0. Since [Z.sub.i] is a vector and is difficult to plot directly on a control chart, [T.sup.2] is used to construct the charting statistic:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
where the covariance matrix is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
and [SIGMA] is the covariance matrix of the original data, [y.sub.[i,k]] Here, we can see that the univariate EWMA control chart is a special case of the MEWMA chart when there is only one variable. In the univariate case, process shifts are often presented in terms of standard deviation...
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