On the Monitoring of Linear Profiles in Multistage Processes Masoumeh Eghbali Ghahyazi, a Seyed Taghi Akhavan Niaki b * † and Paria Soleimani a In most modern manufacturing systems, products are often the output of several correlated stages. Nevertheless, quality of a product or process in both single and multistage processes is usually expressed by a single quality characteristic, two or more characteristics, or profiles. Although there are many studies in univariate and multivariate-multistage process monitoring, fewer works focus on profile monitoring of multistage processes. This paper addresses the problem of monitoring a simple linear profile that is going through a multistage process in phase II. Using a first-order autoregressive correlation model, the relationship between the stages is first modeled. Then, the cascade effect of multistage processes on the performance of T 2 control chart is studied. We show that the cascade property has a significant impact on the performances of the chart in downstream stages. Next, a U statistic is used to eliminate the cascade effect, and the control scheme is modified accordingly. Simulation studies reveal that the modified control chart performs well. Copyright © 2013 John Wiley & Sons, Ltd. Keywords: profile monitoring; multistage processes; cascade property; phase II monitoring; average run length (ARL) 1. Introduction D ue to advances in technology, statistical control charts were adopted to monitor more complicated products and processes. Knowledge of process or product structure is useful to employ appropriate control charts; otherwise, they may end up with misleading results and interpretations of a process state. Nowadays, many manufacturing processes consist of several successive stages so that quality characteristics in subsequent stages are affected by the quality characteristics of preceding stages. In other words, quality of an item in a specific stage not only depends on the operation condition of its current stage but also is affected by the condition of its previous stage(s). This is called the cascade property of multistage processes, for which proper remedial measures have to be adopted in order to reduce or eliminate it and make control chart interpretations easy. Model-based monitoring procedures are useful for the cascade effects in multistage processes. Similar to the regression control chart proposed by Mandle, 1 Zhang’s 2 control chart that is called the cause-selecting control chart (CSC) is another method of monitoring multistage processes. Various extensions of CSC were also proposed by Zhang. 3–9 Wade and Woodall 10 reviewed Zhang’s studies and proposed CSC with prediction limits as a modification of the common CSC. Other types of model-based regression adjustment named model-void and model-fixed were proposed by Hawkins, 11,12 respectively. Moreover, Shu et al. 13 introduced a multiple cause-selecting chart, and Asadzadeh et al. 14 proposed a robust cause-selecting control scheme. In some applications, the quality characteristics in a stage of multistage processes are correlated. Hauk et al. 15 extended the work of Hawkins 12 to handle the problem of multivariate-multistage process monitoring. Furthermore, Niaki and Davoodi 16 presented another multivariate-multistage quality control system by designing a single neural network. Other studies in this area include economic design of multistage control charts (Yang, 17–19 Yang and Yang 20 ), multistage process monitoring with autocorrelated observations (Leordo et al., 21 Shu and Tsung, 22 Yang and Yang 23 ), and adaptive cause selecting control charts (Yang and Shu 24–26 and Yang and Chen. 27 ) Further, while normal observations are assumed in all the aforementioned works, Skinner et al. 28 and Jearkpaporn et al. 29–31 investigated situations in which quality characteristics in multistage processes are not normal. In addition to monitoring the mean of multistage processes in all the above works, Zeng and Zhou 32 studied the properties of a regression- adjustment-based method for monitoring the variation propagation in multistage processes. The concentration of most of the studies in multistage process monitoring is mainly to use linear regression models to describe multistage processes. However, some researchers such as Ding et al. 33 and Xiang and Tsung 34 considered state-space engineering models to model multistage processes. Engineering models were also used by some researches including Zhou et al. 35 and Li and Tsung 36 to monitor variation in multistage processes. a Department of Industrial Engineering, South Tehran Branch, Islamic Azad University, Tehran, Iran b Department of Industrial Engineering, Sharif University of Technology, P.O. Box 11195-9414 Azadi Ave., Tehran, 1458889694, Iran *Correspondence to: S. T. A. Niaki, Department of Industrial Engineering, Sharif University of Technology, P.O. Box 11195-9414 Azadi Ave., Tehran 1458889694 Iran. † E-mail: niaki@sharif.edu Copyright © 2013 John Wiley & Sons, Ltd. Qual. Reliab. Engng. Int. 2013 Research Article (wileyonlinelibrary.com) DOI: 10.1002/qre.1531 Published online in Wiley Online Library