Modeling and optimization for multiple correlated responses with distribution variability

Abstract

In production design processes, multiple correlated responses with different distributions are often encountered. The existing literature usually assumes that they follow normal distributions for computational convenience, and then analyzes these responses using traditional parametric methods. A few research papers assume that they follow the same type of distribution, such as the t-distribution, and then use a multivariate joint distribution to deal with the correlation. However, these methods give a poor approximation to the actual problem and may lead to the recommended settings that yield substandard products. In this article, we propose a new method for the robust parameter design that can solve the above problems. Specifically, a semiparametric model is used to estimate the margins, and then a joint distribution function is constructed using a multivariate copula function. Finally, the probability that the responses meet the specifications simultaneously is used to obtain the optimal settings. The advantages of the proposed method lie in the consideration of multiple correlation patterns among responses, the absence of restrictions on the response distributions, and the use of nonparametric smoothing to reduce the risk of model misspecification. The results of the case study and the simulation study validate the effectiveness of the proposed method.

KEYWORDS:

• Rrobust parameter design
• multiple responses
• semiparametric model
• copula
• correlation pattern

Additional information

Funding

This work is supported by the National Natural Science Foundation of China (NSFC) [grant numbers: 71931006, 72171118, 71771121], and the Natural Sciences and Engineering Research Council of Canada (NSERC) [grant number: RGPIN-2018-03862].

Notes on contributors

Shijuan Yang

Shijuan Yang is a PhD candidate in Quality Management and Quality Engineering at Nanjing University of Science and Technology, China. She is now supported by the China Scholarship Council as a visiting PhD student at University of Calgary, Canada. Her research interests include applied statistics and quality engineering.

Jianjun Wang

Jianjun Wang is a professor at the Department of Management Science and Engineering, Nanjing University of Science and Technology, China. He is a senior member of the Chinese Society of Optimization, Overall Planning, and Economical Mathematics. He is a reviewer of several international journals such as JQT, EJOR, IJPR, CAIE, and QTQM. His current research interests include quality engineering and quality management, robust parameter design, Bayesian modeling and optimization, and industrial statistics.

Jiawei Wu

Jiawei Wu is a full lecturer at the School of Information Management, Jiangxi University of Finance and Economics, Nanchang, China. He has held visiting appointments at the University of Toronto in Canada. He has authored or coauthored more than 15 journal papers in the fields of quality and reliability engineering, optimization design, and product development.

Yiliu Tu

Yiliu Tu is a professor at the Department of Mechanical and Manufacturing Engineering, University of Calgary, Canada. He received his BSc in electronic engineering and MSc in mechanical engineering, both from Huazhong University of Science and Technology (HUST), China; PhD in production engineering from Aalborg University (AU), Denmark. His current research interests are OKP (One-of-a-Kind Production) product design and manufacture, ultra-fast laser micro-machining technology, project management, supply chain integration, and complex product life cycle quality assurance and control. He is a senior member of SME (Society of Manufacture Engineers) and a professional engineer of APEGA (The Association of Professional Engineers, Geologists, and Geophysicists of Alberta).