Monitoring profiles in multistage processes using the multivariate multiple regression model

Abstract

Most recent production processes comprise multiple stages, where the final quality of products is determined by the upstream effects of output quality at each stage. Moreover, the quality of each stage is characterized by a functional relationship, referred to as a profile, between input variables and output quality variables. Therefore, the effective profile monitoring of the multiple stages is crucial in maintaining and improving the final output quality. A multistage process is separated into a series of single-stage processes, and each single-stage process is treated as a profile structure. In each stage, we implement the multivariate multiple linear regression (MMLR) model using the orthogonal design coding. The coefficient estimators of the MMLR are obtained as a matrix, whose column vectors indicate the coefficient vectors for given output quality variables and row vectors indicate the coefficients of output quality variables for given regression coefficients. Since the use of orthogonal design coding makes the regression coefficient estimator vectors for the intercept and the input variables mutually independent, it is possible to monitor the process using Hotelling's 𝑇 2 charts separately. We explain the process of estimating and monitoring the regression coefficients from Phase I samples, and evaluate the performance of the proposed procedure in Phase II.

A new non‐parametric control chart for monitoring general linear profiles based on log‐linear modelling

“…Lately, Ahmadi et al 24 suggested a new control chart for multiple linear profiles monitoring by means of two robust estimate methods. Park and Lee 25 proposed both Phase I and Phase II study of profile monitoring in multistage processes based on the multivariate multiple regression profile. The aforementioned control charts for Phase I and/or Phase II study of the general linear profiles are briefly summarized in Table 1.…”

“…Phase I: Mahmoud and Woodall, 3 Mahmoud et al, 6 Mahmoud, 17 Kalaei et al 12 Phase II: Kang and Albin, 1 Zou et al, 4 Gupta et al, 5 Zou et al, 7 Zou et al, 16 Eyvazian et al, 18 Zou et al, 19 Xu et al, 8 Huwang et al, 20 Amiri et al, 9 Kazemzadeh et al, 10 Khedmati and Niaki, 11,21 Zhang et al, 22 Ghashghaei and Amiri, 23 Abbas et al, 13 Yeganeh et al, 14 Ahmadi et al 24 Both Phase I and Phase II: Kim et al, 2 Moheghi et al, 15 Park and Lee. 25 used to estimate this functional relationship and hence control charts are developed based on the functional relationship estimated to check whether the relationship is stable or not.…”

Robust Surveillance Schemes Based on Proportional Hazard Model for Monitoring Reliability Data

“…To analyze a two-stage dependent process, EWMA and CUSUM charts are suggested by Nabeel et al [25] using logistic distribution. Park and Lee [26] considered each stage of a multistage process as a profile structure and suggested a multivariate multiple linear regression model implemented in each stage using the orthogonal design.…”