|
Article Excerpt
1. Introduction
Most manufacturing systems involve many process steps. For example, in manufacturing semiconductor devices, semiconductor wafers go through chemical deposition, photolithography, etching and several other machines numerous times (May and Spanos, 2006). The majority of the machines in these process steps are supplied by a few manufacturers. Thus, many machines have similar functionalities. With the increasing connectivity through data networks, sharing information from similar machines at the same factory or remote locations becomes possible. Since defects might not occur often and conducting experiments for improving process quality could be costly, it is natural to explore ways to integrate information from similar process steps and machines for root cause analysis and process monitoring and improvement.
In the literature, multistage manufacturing system research focuses on modeling data collected from machines connected in series (Ding et al. 2002; Zhou et al., 2003; Janakiram and Goernitz, 2005; Shi, 2006). Very few efforts have been made to explore how the data collected from similar machines or process steps can be used to improve process modeling and control. Although these similar machines or process steps might be connected in series or in parallel, this article focuses on developing procedures that deal with different degrees of similarities in data and leaves the data collection and alignment issues to future research. In particular, this research proposes a Bayesian framework for integrating data from various parallel sites.
The parallel sites could be several machines that accomplish the same process step, several industrial locations that produce the same product, or even several different time windows for the same machine (Fenner, 1999a; Shi, 2006). Huang and Shi (2004) presented a system-level methodology to model and analyze multiple variation streams in a series-parallel multistage manufacturing system (SP-MMS). They extended the state space modeling approach from a single process route to the SP-MMS with multiple routes. On the other hand, our Bayesian procedures allow modeling of data commonality through the same prior distribution. This is less restrictive than the approach that models them as the same data random variable.
The goal of parallel site modeling is to use the common ground between sites to facilitate better estimation and ultimately better modeling and/or control. Analyzing parallel site data facilitates a compromise between pooling and no pooling of the data. A pure pooled estimate assumes that data from various sources have the same distribution, which does not consider the situation that the data from each source are probably intrinsically different at least in some way at each source. Analyzing the data separately tacitly assumes that the data from the various sources have nothing in common. We propose a new Bayesian parallel site model for the compromise between pooling the data completely and treating sites as completely unrelated. This is done through the use of quasi-common parameters, which are defined to be parameters that differ from site to site but have some commonalities. The commonalities between quasi-common parameters are modeled by using common global hyperparameters for their distributions. The similarities between the individual quasi-common parameters are modeled through common global hyperparameters, which determine the prior distribution for quasi-common parameters.
As a case study, we present a Bayesian parallel site modeling approach to model uniformity in semiconductor manufacturing. The case study on the uniformity modeling (across the different dies on a silicon wafer, across slots in a furnace, etc.) illustrates how the hierarchy of a Bayesian model can be used to incorporate correlation structure. The approach is an alternative to the traditional single response surface or the newer multiple response surface approach (Guo and Sachs, 1993).
The proposed approach accurately estimates the site models in the case where the site models have some similarities, while modeling uniformity indirectly through site by site models results in several complex functions, which are difficult to interpret for an integrated conclusion. More importantly, the distributions of the estimates of uniformity and other decision quantities are difficult to obtain. Another advantage of the proposed method is that exact posterior intervals for the uniformity are available rather than the approximate confidence intervals that are used in the existing approaches. In addition to the direct contribution to the semiconductor industry applications of process control and uniformity modeling, the applications give insight into the flexibility of the Bayesian framework for handling many other possible parallel data-source scenarios that one might encounter.
Section 2 presents the proposed Bayesian parallel site model. Section 3 shows some illustrative examples with real-life case studies. Section 4 presents some concluding remarks and future work directions.
2. Parallel site model
2.1. General modeling structure
For case of discussion, parallel site modeling will be limited to the context of (transformed) linear models in this paper, but these ideas could be extended to more complicated non-linear modeling scenarios. The basic model for site k = 1, 2, ...., K and for observations j = 1, 2, ...., [n.sub.k] that will be considered is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
where [Y.sub.[j,k]] is the response, [X.sub.i,ss], [X.sub.i,cas] and [X.sub.[i,qc]] are the site-specific, common-across-sites and quasi-common input variables respectively, [N.sub.[j,k]] is the noisy error, [[beta].sub.[0,k]] are intercept parameters, which are site-specific; [[beta].sub.[i,k]] are site-specific parameters, [[gamma].sub.i] are common-across-site parameters; and [[delta].sub.[i,k]] are quasi-common parameters. The model can...
|
