Balancing the Subjective and Objective Weights for Correlated Multiresponse Optimization

Desirability function approach is very popular for multiresponse optimization problems. However, the approach ignores the correlations among multiple responses and does not consider how to reasonably determine the relative weights of multiple responses. In this paper, an integrative desirability function approach is proposed to simultaneously consider the correlations among the responses and the weight determination method. For the proposed approach, the root mean square error performance is regarded as a new response, and then the seemingly unrelated regression estimation is utilized to fit the models. Through balancing the subjective and objective information, the proposed approach can be used to make more reasonable decisions for correlated multiresponse optimization. Two examples are employed to validate the effectiveness of the proposed approach. Copyright © 2015 John Wiley & Sons, Ltd.     

基于主成分分析的函数型产品质量特性的优化方法研究

针对质量特性为轮廓（Profile）的输出响应的优化问题展开研究,提出一种基于主成分分析的双响应曲面法和满意度函数相结合的函数响应优化方法。将Profile的每个观测点看成一个独立响应,将Profile问题转化为多响应问题。求得多个观测点的均值和方差的满意度函数值,通过主成分分析法,将多个观测点的均值和方差的满意度函数值转化为主成分综合得分,并将这两者的加权和作为最终的优化指标。本文所提方法可以有效解决观测点之间存在的相关性的问题,并且优化过程同时考虑到每个观测点响应的均值和方差影响。实例证明,该方法简单易行,优化结果满意。     

A method of steepest ascent for multiresponse surface optimization using a desirability function method

Multiresponse problems are common in product or process development. A conventional approach for optimizing multiple responses is to use a response surface methodology (RSM), and this approach is called multiresponse surface optimization (MRSO). In RSM, the method of steepest ascent is widely used for searching for an optimum region where a response is improved. In MRSO, it is difficult to directly apply the method of steepest ascent because MRSO includes several responses to be considered. This paper suggests a new method of steepest ascent for MRSO, which accounts for tradeoffs between multiple responses. It provides several candidate paths of steepest ascent and allows a decision maker to select the most preferred path. This generation and selection procedure is helpful to better understand the tradeoffs between the multiple responses, and ultimately, it moves the experimental region to a good region where a satisfactory compromise solution exists. A hypothetical example is employed for illustrating the proposed procedure. The results of this case study show that the proposed method searches the region containing an optimum where a satisfactory compromise solution exists.     

Optimization of Correlated Multiple Quality Characteristics Using Desirability Function

A real problem in a product or process usually possesses multiple quality characteristics. For the multiple quality characteristics optimization problem, the most popular method for simultaneous quality characteristics optimization is the desirability function approach. However, the variation and correlation between quality characteristics are usually ignored in this approach. The variation reduction through robust design introduced by Taguchi is a major concept. This research presents an approach to optimizing the correlated multiple quality characteristics based on the modified double-exponential desirability function. The implementation and the effectiveness of the proposed approach are illustrated through two examples from previously published articles.     

An alternative desirability function for achieving ‘six sigma’ quality

To achieve high process yields or ‘six sigma’ quality, engineers often need to evaluate and optimize processes that are characterized by multiple quality characteristics. Existing desirability functions weigh together multiple objectives but they have a number of limitations. Most importantly, available desirability functions do not explicitly account for the combined effect of the mean and the dispersion of the quality characteristic. Therefore, it is easy to incur excessive expenditures or unknowingly to fail to achieve targeted yields. In this paper, a desirability function is proposed that addresses these limitations. This function conservatively estimates the ‘effective yield’ under assumptions described in the ‘six sigma’ literature. We use an arc‐welding application to illustrate how the proposed desirability function can yield a substantially higher level of quality as well as a more accurate assessment of the process capability than available alternatives. We suggest that the proposed desirability function should be used to facilitate multicriterion optimization when dispersion data are available. Copyright © 2003 John Wiley & Sons, Ltd.     

Multiple response optimization using mixture-designed experiments and desirability functions in semiconductor scheduling

Today's highly competitive semiconductor markets place a great emphasis on responsiveness to customers. In the past, competition has primarily focused on the product design arena. More recently, short lead times and good on-time delivery performance have become equally important to winning customer satisfaction. To meet these criteria, a recent thrust of manufacturing management has focused on the use of effective scheduling techniques to manage wafer movement. Dabbas and Fowler (1999) proposed an approach that combines multiple dispatching criteria into a single rule with the objective of maximizing multiple response measures simultaneously. This is accomplished using a linear combination with relative weights. The weights identify the contribution of the different criteria. This paper details the use of experimental design methodology as well as a desirability function approach in the optimization of the weights' assignment to the different criteria. The basic idea of the desirability function approach is to transform a multi-response problem into a single-response problem by means of a mathematical transformation. The responses of interest are on time delivery, variance of lateness, mean cycle time and variance of cycle time. Results demonstrate that the proposed approach is superior to the use of single-dispatching criteria with an average of 20% improvement for all responses. All data presented in this paper have been normalized to disguise actual performance results as the raw data are considered to be Motorola confidential data.     

IP‐MRSO: An iterative posterior preference articulation method to multiple response surface optimization

One of the most important issues in multiple response surface optimization (MRSO) is obtaining a satisfactory “compromise” solution considering a decision maker (DM)'s preference information on the tradeoffs among multiple responses. A promising alternative to incorporate the DM's preference information into the problem is the posterior preference articulation approach, which first generates all (or most) of the nondominated solutions and then makes the DM select the best one from the set of nondominated solutions a posteriori. However, it has an inefficiency problem in that it generates an excessive number of nondominated solutions in which almost all are not used for the DM's selection. This paper proposes a new posterior method called “IP‐MRSO” to overcome the limitation of the existing posterior method. The proposed IP‐MRSO is illustrated through a well‐known MRSO case problem.     

An Integrative Loss Function Approach to Multi‐Response Optimization

Loss function approach is effective for multi‐response optimization. However, previous loss function approaches ignore the dispersion performance of squared error loss and model uncertainty. In this paper, a weighted loss function is proposed to simultaneously consider the location and dispersion performances of squared error loss to optimize correlated multiple responses with model uncertainty. We propose an approach to minimize the weighted loss function under the constraint that the confidence intervals of future predictions for the multiple responses should be contained in specification limits of the responses. An example is illustrated to verify the effectiveness of the proposed method. The results show that the proposed method can achieve reliable optimal operating condition under model uncertainty. Copyright © 2013 John Wiley & Sons, Ltd.     

Optimization of multiple responses using a fuzzy-rule based inference system

The optimization of multiple responses (or performance characteristics) has received increasing attention over the last few years in many manufacturing organizations. Many Taguchi practitioners have employed past experience and engineering knowledge or judgement when dealing with multiple responses. This approach brings an element of uncertainty to the decision-making process and therefore is not recommended for optimization of multiple responses. The approach presented in this paper takes advantage of both the Taguchi method and a fuzzy-rule based inference system, which forms a robust and practical methodology in tackling multiple response optimization problems. The paper also presents a case study to illustrate the potential of this powerful integrated approach for tackling multiple response optimization problems. The variance analysis is also an integral part of the study, which identifies the most critical and statistically significant parameters.     

A semiparametric technique for the multi‐response optimization problem

Multi‐response optimization (MRO) in response surface methodology is quite common in applications. Before the optimization phase, appropriate fitted models for each response are required. A common problem is model misspecification and occurs when any of the models built for the responses are misspecified resulting in an erroneous optimal solution. The model robust regression (MRR) technique, a semiparametric method, has been shown to be more robust to misspecification than either parametric or nonparametric methods. In this study, we propose the use of MRR to improve the quality of model estimation and adapt its fits of each response to the desirability function approach, one of the most popular MRO techniques. A case study and simulation studies are presented to illustrate the procedure and to compare the semiparametric method with the parametric and nonparametric methods. The results show that MRR performs much better than the other two methods in terms of model comparison criteria in most situations during the modeling stage. In addition, the simulated optimization results for MRR are more reliable during the optimization stage. Copyright © 2010 John Wiley & Sons, Ltd.     

Generating experimental designs involving control and noise variables using genetic algorithms

Efficient estimation of response variables in a process is an important problem that requires experimental designs appropriated for each specific situation. When we have a system involving control and noise variables, we are often interested in the simultaneous optimization of the prediction variance of the mean (PVM) and the prediction variance of the slope (PVS). The goal of this simultaneous optimization is to construct designs that will result in the efficient estimation of important parameters. We construct new computer‐generated designs using a desirability function by transforming PVM and PVS into one desirability value that can be optimized using a genetic algorithm. Fraction of design space (FDS) plots are used to evaluate the new designs and six cases are discussed to illustrate the procedure. Copyright © 2009 John Wiley & Sons, Ltd.     

Modelling and optimization of a GMA welding process by genetic algorithm and response surface methodology

The welding process, due to its complexity, has relied on empirical and experimental data to determine its welding conditions. However, trial-and-error methods to determine optimal conditions incur considerable time and cost. In order to overcome these problems, a genetic algorithm and response surface methodology have been suggested for determining optimal welding conditions. First, in a relatively broad region, near-optimal conditions were determined through a genetic algorithm. Then, the optimal conditions for welding were determined over a relatively small region around these near-optimal conditions by using response surface methodology. In order to give different objective function values according to the positive or negative response from the set target value in the optimization problem, a desirability function approach was used. Application of the method proposed in this paper revealed a good result for finding the optimal welding conditions in the gas metal arc (GMA) welding process.     

Set theoretic formulation of performance reliability of multiple response time‐variant systems due to degradations in system components

This paper presents a design stage method for assessing performance reliability of systems with multiple time‐variant responses due to component degradation. Herein the system component degradation profiles over time are assumed to be known and the degradation of the system is related to component degradation using mechanistic models. Selected performance measures (e.g. responses) are related to their critical levels by time‐dependent limit‐state functions. System failure is defined as the non‐conformance of any response and unions of the multiple failure regions are required. For discrete time, set theory establishes the minimum union size needed to identify a true incremental failure region. A cumulative failure distribution function is built by summing incremental failure probabilities. A practical implementation of the theory can be manifest by approximating the probability of the unions by second‐order bounds. Further, for numerical efficiency probabilities are evaluated by first‐order reliability methods (FORM). The presented method is quite different from Monte Carlo sampling methods. The proposed method can be used to assess mean and tolerance design through simultaneous evaluation of quality and performance reliability. The work herein sets the foundation for an optimization method to control both quality and performance reliability and thus, for example, estimate warranty costs and product recall. An example from power engineering shows the details of the proposed method and the potential of the approach. Copyright © 2006 John Wiley & Sons, Ltd.     

Finding and optimising the key factors for the multiple-response manufacturing process

With the advent of modern technology, manufacturing processes became so sophisticated that a single quality characteristic cannot reflect the true product quality. Thus, it is essential to perform the key factor analysis for the manufacturing process with multiple-input (factors) and multiple-output (responses). In this paper, an integrated approach of using the desirability function in conjunction with the Mahalanobis-Taguchi-Gram Schmit (MTGS) system is proposed in order to find and optimise the key factors for a multiple-response manufacturing process. The aim of using the MTGS method is to standardise and orthogonalise the multiple responses so that the Mahalanobis distance for each run can be calculated and the multi-normal assumption for the correlated responses can be relaxed. A realistic example of the solder paste stencil printing process is then used to demonstrate the usefulness of our proposed approach in a practical application.     

Simultaneous Optimization of Robust Parameter and Tolerance Design Based on Generalized Linear Models

Robust parameter design (RPD) and tolerance design (TD) are two important stages in design process for quality improvement. Simultaneous optimization of RPD and TD is well established on the basis of linear models with constant variance assumption. However, little attention has been paid to RPD and TD with non‐constant variance of residuals or non‐normal responses. In order to obtain further quality improvement and cost reduction, a hybrid approach for simultaneous optimization of RPD and TD with non‐constant variance or non‐normal responses is proposed from generalized linear models (GLMs). First, the mathematical relationship among the process mean, process variance and control factors, noise factors and tolerances is derived from a dual‐response approach based on GLMs, and the quality loss function integrating with tolerance is developed. Second, the total cost model for RPD‐TD concurrent optimization based on GLMs is proposed to determine the best control factors settings and the optimal tolerance values synchronously, which is solved by genetic algorithm in detail. Finally, the proposed approach is applied into an example of electronic circuit design with non‐constant variance, and the results show that the proposed approach performs better on quality improvement and cost reduction. Copyright © 2012 John Wiley & Sons, Ltd.     

A Prediction Region‐based Approach to Model Uncertainty for Multi‐response Optimization

Multi‐response optimization methods rely on empirical process models based on the estimates of model parameters that relate response variables to a set of design variables. However, in determining the optimal conditions for the design variables, model uncertainty is typically neglected, resulting in an unstable optimal solution. This paper proposes a new optimization strategy that takes model uncertainty into account via the prediction region for multiple responses. To avoid obtaining an overly conservative design, the location and dispersion performances are constructed based on the best‐case strategy and the worst‐case strategy of expected loss. We reveal that the traditional loss function and the minimax/maximin strategy are both special cases of the proposed approach. An example is illustrated to present the procedure and the effectiveness of the proposed loss function. The results show that the proposed approach can give reasonable results when both the location and dispersion performances are important issues. Copyright © 2015 John Wiley & Sons, Ltd.     

Robust structural damage identification based on multi‐objective optimization

Inverse analysis for structural damage identification often involves an optimization process that minimizes the discrepancies between the computed responses and the measured responses. Conventional single‐objective optimization approach defines the objective function by combining multiple error terms into a single one, which leads to a weaker constraint in solving the identification problem. A multi‐objective approach is proposed, which minimizes multiple error terms simultaneously. Its non‐domination‐based convergence provides a stronger constraint that enables robust identification of damages with lower false‐negative detection rate. Another merit of the proposed approach is quantified confidence in damage detection through processing Pareto‐optimal solutions. Numerical examples that simulate static testing are provided to compare the proposed approach with conventional formulation based on single‐objective optimization. Copyright © 2009 John Wiley & Sons, Ltd.     

An integration design optimization framework of robust design,axiomatic design,and reliability‐based design

Robust design, axiomatic design, and reliability‐based design provide effective approaches to deal with quality problems, and their integration will achieve better quality improvement. An integration design optimization framework of robust design, axiomatic design, and reliability‐based design is proposed in this paper. First, the fitted response model of each quality characteristic is obtained by response surface methodology and the mean square error (MSE) estimation is given by a second‐order Taylor series approximation expansion. Then the multiple quality characteristics robust design model is developed by the MSE criteria. Finally, the independence axiom constraints for decoupling and reliability constraints are integrated into the multiple quality characteristics robust design model, and the integration design optimization framework is formulated, where the weighted Tchebycheff approach is adopted to solve the multiple objective programming. An illustrative example is presented at the end, and the results show that the proposed approach can obtain better trade‐offs among conflicting quality characteristics, variability, coupling degree and reliability requirements. Copyright © 2011 John Wiley & Sons, Ltd.     

Compromise ascent directions for multiple‐response applications

A process or system under study often requires the measurement of multiple responses. The optimization of multiple response variables has received considerable attention in the literature with the majority focusing on locating optimal operating conditions within the current experimental region and thus often occurs in the later stages of experimentation. This article focuses instead on the initial experiment and the location of additional experimental runs if the region of interest shifts. Considerable trade‐off is often required in the multiple response context. In order to account for uncertainty in the model parameters and correlations among the responses, we propose the computation of Bayesian reliabilities to determine optimal factor settings for future experimental runs. The approach will be described in detail for two common design follow‐up strategies: steepest ascent (descent) and shifting factor levels. Illustrative examples are provided for each application. Copyright © 2011 John Wiley & Sons, Ltd.     

Process Optimization via Robust Parameter Design when Categorical Noise Factors are Present

When categorical noise variables are present in the Robust Parameter Design (RPD) context, it is possible to reduce process variance by not only manipulating the levels of the control factors but also by adjusting the proportions associated with the levels of the categorical noise factor(s). When no adjustment factors exist or when the adjustment factors are unable to bring the process mean close to target, a popular approach for determining optimal operating conditions is to find the levels of the control factors that minimize the estimated mean squared error of the response. Although this approach is effective, engineers may have a difficult time translating mean squared error into quality. We propose the use of a parts per million defective objective function. Furthermore, we point out that in many situations the levels of the control factors are not equally desirable due to cost and/or time issues. We have termed these types factors non‐uniform control factors. We propose the use of desirability functions to determine optimal operating conditions when non‐uniform control factors are present and illustrate this methodology with an example from industry. Copyright © 2006 John Wiley & Sons, Ltd.