Robust fuzzy programming method for MRO problems considering location effect, dispersion effect and model uncertainty
Introduction
Response surface methodology (RSM) is a powerful technique consisting of a collection of tools for statistical design of experiments (DOEs), empirical model-building and numerical optimization. It has been widely used in optimizing process and product designs (Vining, 1998, Box and Draper, 2007, Myers et al., 2009, Vining, 2011). If multiple responses are involved, then what we need to solve is multiple response optimization (MRO) problems (Khuri & Cornell, 1996). Kim and Lin (2006) presented the optimization method of multiple responses considering both location and dispersion effects. He, Zhu, and Park (2012) presented a robust desirability function method for multi-response surface optimization considering model uncertainty.
In product or process development, if the model uncertainty, location and dispersion effects of the responses should be considered simultaneously, then the final solutions obtained in He et al. (2012) and Kim and Lin, 1998, Kim and Lin, 2006 may be optimal in terms of overall desirability but not robust. Thus the purpose of this paper is to present a new method for MRO problems considering location effect, dispersion effect and model uncertainty simultaneously to make the final solutions are not only optimal but also robust.
The conventional RSM approach focuses on a single response. Montgomery (2012) and Ryan (2007) presented regression models for fitting a surface to given set of data to determine optimum factor levels. However, in product or process development, it is common to have several responses of interest that are often in conflict with each other, which means the determination of optimum conditions on the input variables would require simultaneous consideration of all the responses. Derringer and Suich, 1980, Khuri and Conlon, 1981, Khuri and Cornell, 1996 obtained the optima for multiple response surface (MRS) problems with the consideration of all the responses simultaneously. Jeong and Kim (2009) developed an interactive desirability function method to multiple responses optimization. Lee, Kim, and Koksalan (2011) proposed a posterior preference articulation approach to multiple responses surface optimization.
Most aforementioned approaches for multiple responses optimization (MRO) assume that the response models have been fitted reasonably well, which does not always hold in practice. Myers (1999) stressed that the variability of the predicted responses is a critical issue for practitioners. And then many creative ideas have been put forth in the literature for MRO considering the uncertainties of the response models and the robustness of the solutions. He et al. (2012) pointed out that the solution reached by traditional desirability function method may be optimal in terms of overall desirability but not robust if model uncertainty is considered. Xu and Albin (2003) considered uncertainties of model coefficients and developed a robust optimization method for experimentally derived objective functions. He et al., 2009, He et al., 2010 developed the robust multiple response surface methods, considering the optimality and robustness of the solution simultaneously. Yadav, Bhamare, and Rathore (2010) combined the reliability function-based design optimization with robust design and developed a hybrid quality function-based multi-objective model. Wang, Ma, and Su (2011) presented an integrated grey relational grade index and employed the TOPSIS to find optimal robust parameter design for dynamic multiple response systems. Goethals and Cho (2011) proposed a robust design method for time-oriented dynamic quality characteristics with a target profile. Some other significant applications considering robustness can refer to Hu, Cao, and Hong (2012), Ben-Tel, Hertog, Waegenaere, Mertrand, and Rennen (2013), Wiesemann, Kuhn, and Sim (2014), and Gorissen, Blanc, Hertog, and Ben-Tal (2014).
Considering that the membership function is suitable to reflect the uncertainty of the objects belonging to a concept which can refer to fuzzy set theory (Bellman and Zadeh, 1970, Zadeh, 1965), Kim and Lin (1998) extended RSM to fuzzy environments and constructed the membership functions of the responses involved to interpret the degrees of responses satisfying the targets on different input variable values, which is a creative idea to consider uncertainties for MRO. Lai and Chang (1994) developed a fuzzy multiple response optimization procedure to search an appropriate combination of parameter settings. Akpan, Koko, Orisamolu, and Gallant (2001) used the fuzzy parameters to analyze the structures of response. De Munck, Moenens, Desmet, and Vandepitte (2008) proposed the interval and fuzzy finite element method and applied it to the calculation of envelop frequency response function of imprecisely structures. Bashiri and Hosseininezhad (2012) developed a fuzzy decision support systems based on fuzzy inference systems for MRO problem.
Vining and Myers (1990) and Kim and Lin (2006) pointed out that the major focus of the aforementioned existing approaches to multiresponse optimization is on the location effect only, ignoring the dispersion effect of the responses. As a result, all observations are assumed to have equal variation. However, evidence from real examples indicates that the equal variation assumption may not be valid in practice. In such a case, the existing multiresponse optimization approaches can be misleading, thus they presented the optimization method of multiple responses considering both location and dispersion effects simultaneously.
Given a comprehensive consideration of all issues discussed above, in product or process development, we may encounter the following cases for MRO problems.
- (1)
Several responses of interest in MRO problems are in conflict with each other, and thus the determination of optimum conditions on the input variables would require simultaneous consideration of all the responses.
- (2)
The response models in MRO problems may have not been fitted reasonably well, and the equal variation assumption may not be valid in practice, thus both the location and dispersion effects of multiple responses should be taken account.
- (3)
Kim and Lin (1998) extended RSM to fuzzy environments which is a creative idea for solving MRO problems. However, the solution reached by traditional fuzzy method may be optimal in terms of overall desirability but not robust if model uncertainty is considered.
Then the weaknesses in the most related previous papers can be concluded as that, (1) He et al. (2012) only considered the location effect of the responses, ignoring the dispersion effect; (2) Kim and Lin (1998) constructed the membership functions of the estimated mean and standard deviation of the responses in fuzzy environment, but the model uncertainty of the responses is not considered. Which lead to the result that the solutions obtained in He et al. (2012) and Kim and Lin (1998) may be optimal in terms of overall desirability but not robust.
To consider the above three cases and solve the weaknesses in He et al. (2012) and Kim and Lin (1998), we consider the model uncertainty, location and dispersion effects of the responses simultaneously, and propose the robust fuzzy programming approach for MRO problems to ensure the robustness of the obtained solutions. We develop the robust fuzzy desirability sets and construct the robust membership functions of the responses, considering the uncertainty associated with the fitted response surface models and the satisfaction degrees of the response values with respect to the given targets. We translate the feasible region into the -level sets and incorporate the model uncertainty with the confidence intervals simultaneously to ensure the robustness of the feasible regions. Finally the example from literatures shows the practicality and effectiveness of the proposed algorithm, and some comparisons and discussions are given to further illustrate the developed approach.
To systematically introduce the topic, the remainder of this paper is organized as follows. Section 2 gives preliminary introduction of some basic concepts. Section 3 constructs the robust desirability membership functions and develops the robust fuzzy programming method to solve the multiple responses optimization (MRO) problems. Section 4 gives a numerical example to show the feasibility and validity of the new approach and gives some comparisons and discussions to further illustrate the developed algorithm. Finally, Section 5 concludes the paper.
Section snippets
Preliminaries
This section provides the basic concepts that are needed for the rest of this paper, including fuzzy sets and some basic operations (Bellman and Zadeh, 1970, Zadeh, 1965), the traditional desirability function method (Derringer & Suich, 1980) and the robust desirability function (He et al., 2012).
Robust fuzzy programming method for MRO considering location effect, dispersion effect and model uncertainty
In this section, we present the robust fuzzy programming (RFP) method for multiple responses optimization considering location effect, dispersion effect and model uncertainty simultaneously.
Illustrated example and comparisons
In this section, an example from literatures is illustrated to show the practicality and effectiveness of the proposed algorithm, and some comparisons and discussions are given to further illustrate the developed approach.
Conclusions
This paper proposes the robust desirability membership functions and the robust fuzzy programming method to solve the multiple responses optimization (MRO) problems, taking the location effect, dispersion effect and model uncertainty of the involved responses into consideration simultaneously. The practicality and effectiveness of the proposed algorithm is illustrated by an example from the literatures and the comparisons and discussions are given to further illustrate the developed algorithm.
Acknowledgment
The work was supported in part by the National Natural Science Foundation of China (Nos. 71225006, 71532008), in part by the National Research Foundation of Korea (No. 2015R1C1A1A01051952), and in part by the project of the National Natural Science Foundation of China and the National Research Foundation of Korea.
Yingdong He is currently working towards the Ph.D. degree at the College of Management & Economics, Tianjin University, Tianjin, China. He has contributed about twenty articles to professional journals as the first author, such as IEEE Transactions on Fuzzy Systems, IEEE Transactions on Cybernetics, Information Sciences, Applied Soft Computing, Computers & Industrial Engineering, Journal of the Operational Research Society, Expert Systems with Applications, Soft Computing, International Journal
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Yingdong He is currently working towards the Ph.D. degree at the College of Management & Economics, Tianjin University, Tianjin, China. He has contributed about twenty articles to professional journals as the first author, such as IEEE Transactions on Fuzzy Systems, IEEE Transactions on Cybernetics, Information Sciences, Applied Soft Computing, Computers & Industrial Engineering, Journal of the Operational Research Society, Expert Systems with Applications, Soft Computing, International Journal of Fuzzy Systems, etc. His current research interests include fuzzy mathematics, information fusion, group decision making, quality management and control, Six Sigma and the multiple response analysis. Mr. He received the first prize in the tenth National Postgraduate Mathematical Contest in Modeling in 2013, the National Scholarship for Postgraduates of China Awards in 2013 and the National Scholarship for Ph.D. Students of China Awards in 2014, 2015 and 2016. The China Youth Science and Technology Innovation Award in 2016.
Zhen He received the Ph.D. degree at the College of Management & Economics Tianjin University, Tianjin, China, in 2001. He is a professor with the College of Management & Economics, Tianjin University, Tianjin, China. His main research interests include quality management and control, Six Sigma and Lean Production. He has published over 200 journal papers in IEEE Transactions on Fuzzy Systems, IEEE Transactions on Cybernetics, European Journal of Operational Research, Quality Progress, Total Quality Management & Business Excellence, International Journal of Production Research, International Journal of Production Economics, Quality and Reliability Engineering International, Applied Soft Computing, Journal of the Operational Research Society and other international and domestic academic journals. Mr. He is an Academician of the International Academy for Quality, the recipient of NSFC (National Natural Science Foundation of China) Outstanding Young Scholars and the recipient of the New Century Excellent Talents program, Ministry of Education of China.
Dong-Hee Lee is an assistant professor with the College of Interdisciplinary Industrial Studies at Hanyang University in Korea. He received his Ph.D. in Industrial and Management Engineering in 2011 from Pohang University of Science and Technology (Korea). He worked as a senior researcher in quality team of semiconductor division at Samsung Electronics for four years (2011–2014) and received CRE (certified reliability engineer) from ASQ (American Society for Quality). His research interests include quality improvement methods such as response surface methodology, quality function deployment, statistical process control, and so on. He has published several research papers about multiresponse surface optimization.
Kwang-Jae Kim is Professor in the Department of Industrial and Management Engineering at Pohang University of Science and Technology (POSTECH), Korea. His research interests include quality assurance in product and process design, new product/service development, and service engineering. His work has been applied in various areas including semiconductor manufacturing, steel manufacturing, automobile design and manufacturing, and healthcare, telematics, and telecommunications services. He was the founding chairman of KIIE Service Science Research Group and serves as a vice president of Korean Society of Quality Management.
Lin Zhang is currently working towards the Ph.D. degree at the College of Management & Economics, Tianjin University, Tianjin, China.
Xiaoxi Yang is currently working towards the Ph.D. degree at the College of Management & Economics, Tianjin University, Tianjin, China. Her current research interests include quality management, service quality and product-service system. She has published journal papers in Production Planning & Control, Total Quality Management & Business Excellence.