International Journal of Industrial Engineering, 11(3), 261-272, 2004.
CHANGE PATTERN DISCOVERY IN MULTISTAGE STATISTICAL PROCESS CONTROL Ruixiang Sun and Fugee Tsung* Department of Industrial Engineering and Engineering Management Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong. *Email:
[email protected] Change pattern discovery and identification is one of the main challenges in multistage statistical process control, since it can provide operation engineers the information to diagnose the root cause once an abnormal operation occurs. This paper establishes a model for multistage processes in which the inertia of the process is emphasized. The analytical expressions for the change patterns are obtained by Z-transform. The properties of the change patterns are analyzed by partitioning into two parts: a steady state and a transient period. The final value can characterize the steady state and the relative information entropy can characterize the transient period. Significance: The discovery and analysis of change patterns will help in the quick identification of out-of-control sources by linking the current stage signal to information about earlier stages in the process. Keywords: Multistage processes, Change pattern analysis, SPC.
(Received 4 June 2001; Accepted in revised form 11 October 2002)
1. INTRODUCTION In today’s manufacturing industry, product quality plays an increasingly important role in improving competitive advantage in the global marketplace. Statistical process control (SPC) is a powerful tool that has been widely used to maintain high-quality production by reducing the variations in critical quality characteristics. Until recently, research work on and applications of SPC have been focused mainly on single-stage processes. It has been implemented effectively in a great number of manufacturing industries. However, the interactions among linked stages need greater attention, because defects in the semi-finished product after one stage can be transferred to succeeding stages. For example, a defect in a joint between two parts will possibly cause failure in a downstream assembly stage (Ding, Ceglarek and Shi 2000a; 2000b). Thus, the diagnosis of root causes cannot be restricted to within a certain individual stage. In order to improve SPC’s diagnostic accuracy and efficiency, it should be extended to all stages of the process that can possibly cause an out-of-control condition. A process change going through different stages will lead to different change patterns in the SPC data. The discovery and identification of change patterns in a multistage process will provide information on the diagnosis of special causes, which can narrow the scope of the search for the root cause to certain stage(s). Thus, an in-depth understanding of the nature of a multistage process can be obtained and more accurate corrective actions can be taken once the special causes have been identified. This paper focuses on change patterns in SPC in a multiple-stage manufacturing process. Some relevant literature is first reviewed. Then, a mathematical model is established for the discovery of change patterns. The analytical expression for different change patterns is obtained using a Z-transform, and the properties of change patterns are also analyzed. The last section concludes the paper with a discussion and suggestions for future research.
2. LITERATURE REVIEW Although literature is available for quality engineers on single-stage SPC, neither theoretical nor practical research on multiple stages is yet mature. Ding, Ceglarek and Shi (2000a, 2000b) investigated the modeling and diagnosis of multistage manufacturing processes in the automobile industry. The authors proposed to model dimensional error propagation. Faults were then diagnosed using a space state model. The proposed method was applied to a simulated 3-stage automobile body assembly process. This method’s application is limited to dimensional or geometrical variations, however, because it is based on coordinate system transformation. Lawless and Mackay (1999) discussed variation transmission in multistage ISSN 1072-4761
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manufacturing processes. The authors analyzed the measurements of individual parts as they pass through multiple stages using linear regression and analysis of variance (ANOVA) tools. Apley and Shi (1994) developed a method to combine fault detection, isolation, and identification using a generalized likelihood ratio test (GLRT). The form of the system analyzed was a multiple input – single output (MISO) linear system, which consisted of a number of subsystems. The interactions among the subsystems were considered, and a simulation example demonstrated the performance of the GLRT. Wade and Woodall (1993) studied the diagnosis of special causes in a multi-step process using cause-selecting control charts. An example of two stages involving measurements of a braking system in an automobile was provided to demonstrate the effectiveness of their method. There is also some other literature related to monitoring and diagnosis in SPC, although these studies do not focus on multiple stages. Roan and Hu (1995) presented a monitoring algorithm to detect, classify, and group process changes using statistical techniques and knowledge of the hierarchical structure. However, the sensitivity of the detection algorithm relies on the sample size of the moving window, and also on the sample mean and variance of each measurement point. For the identification of a change pattern or structure, Guo and Dooley (1992) studied process change detection, feature vector selection and error analysis using a neural network and Bayes decision approach. Monte Carlo simulation experiments showed that Guo and Dooley’s method can enhance the ability to diagnose special causes. Pham and Wani (1997) solved the problem of recognizing chart change patterns using nine chosen shape features. How to select the most efficient features or combination of features remains a problem. Hu and Roan (1996) have pointed out that removing autocorrelation from a time series would alter the change pattern. The change patterns of a single stage process were analyzed using different autoregressive moving average (ARMA) models, and theoretical results were provided.
3. MODELING A MULTISTAGE PROCESS In most multistage manufacturing processes, the products are fabricated in a pipelining mode, so the sequential model is as shown in figure 1. The output of the i-th stage will be the input of the succeeding stage, the (i+1)th stage.
Initial Input
Stage 1
Stage i+1
Stage 2
...
...
Stage i
Stage n-1
Stage n
End Output
Figure 1. Sequential model for multistage processes
Lawless and Mackay (1999) employed a linear regression model to analyze the transmission of variation in an n-stage process, expressed as
Y1 ~ N ( µ1 , σ 12 ) , Yi = α i + β i Yi −1 + ei , i = 2,..., n,
…
(1)
…
(2)
where Yi is a measurement of the characteristic at the i-th stage, and the ei s are independent normally distributed random variables with mean 0 and variance
σ iA2 .
In this model, Yi depends only on Yi −1 , i.e., only the previous stage has a direct
effect on the current one. Considering the inertia of most manufacturing processes, Eq. (2) can be modified to
Yit = α i + β i Yi −1,t −1 + γ i Yi ,t −1 + eit ,
…
(3)
where the first subscript i is the stage number and the second subscript t is a time sequence number. Compared with Eq.(2), the altered model can consider inertia through the introduction of the new term,
γ i Yi ,t −1 , where γ i < 1 to keep the process
stable. Compare this with Box’s first-order dynamic model (1992, 1997),
Yt = α + δYt −1 + g (1 − δ ) X t −1 , 0 ≤ δ < 1 ,
…
(4)
Change Pattern Discovery in Multistage SPC - 263
where α is the process level,
δ
is a constant to measure the process inertia and g is the process gain. Note that the model of
Eq.(4) is a special case of Eq.(3). Consider a two-stage process with α1 =0. The situation based on the model of Eq.(3) with no disturbances is first investigated. The situation with disturbances will be discussed later in Section 5. In the first stage, the relationship between the input and the output is
Y1t =
β1 B Y0t , 1 − γ 1B
...
(5)
…
(6)
where B is a backward shift operator. Similarly, in the second stage, the relationship between the input and the output is
Y2t =
β2B Y1t . 1− γ 2B
The connection between the first and the second stages is that the output of the first stage, Y1t , is the input to the second stage. Any change in the first stage that alters its output will cause changes in the second stage.
4. CHANGE PATTERN DISCOVERY For the model established in Section 3, it is clear that for a given type of fault at the first or the second stage, the end output will be correspondingly subject to a two-stage or one-stage dynamic effect. In order to identify the fault accurately and in time for corrective action, it is crucial to distinguish the change patterns from different stages’ outputs. This section addresses this issue and presents theoretical results characterizing the change patterns by applying the Z-transform as an analytical tool. The change patterns of two types of disturbances are analyzed here: a step disturbance and an impulse function disturbance, each of unit magnitude. The mathematical expressions for each disturbance are
step function:
impulse function:
⎧1, u (t − k ) = ⎨ ⎩0, ⎧1, δ (t − k ) = ⎨ ⎩0,
t≥k t
