Multiple Response Surface Optimization Research Articles

A process capability function approach to multiple response surface optimization based on a posterior procedure

Multiple response surface optimization (MRSO) aims to determine a setting of input variables that simultaneously optimizes multiple responses. The process capability function (PCF) approach is an attractive alternative to MRSO because it considers both the mean and variance of the responses systematically through the process capability index. In MRSO, the responses are often in conflict. Thus, the preference information of a decision maker (DM) on the tradeoffs among the responses should be considered. However, the existing PCF methods are based on an unrealistic assumption that all DM preference information is given in advance. In this article, a posterior preference articulation procedure based on the PCF approach, called P-PCF, is proposed. P-PCF first generates nondominated solutions using the ε-constraint method and then allows the DM to select his/her preferred cluster or final solution using the K-means clustering method. The proposed method is illustrated using a well-known MRSO problem.

A pairwise comparison-based interactive procedure for the process capability approach to multiple-response surface optimization

Multiple-response surface optimization (MRSO) aims to find a setting of the input variables that simultaneously optimizes the multiple responses. The process capability approach incorporates a decision maker’s (DM’s) preference information in the problem-solving process while considering the dispersion effects of the responses systematically through the framework of the process capability index. However, the existing process capability methods are based on an unrealistic assumption that all the preference information of the DM is given in advance. Although a few methods incorporate the DM’s preference information while solving the problem, they require too much preference information from the DM. In this article, a new interactive capability method, the pairwise comparison-based interactive process capability approach (PC-IPCA), is proposed to alleviate the DM’s burden in the solving process. The proposed method is illustrated through a well-known MRSO problem. The advantages of the proposed method compared with existing process capability methods are discussed.

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

AbstractOne 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.

Optimization and in vitro antiproliferation of Curcuma wenyujin's active extracts by ultrasonication and response surface methodology.

BackgroundCurcuma wenyujin, a member of the genus Curcuma, has been widely prescribed for anti-cancer therapy. Multiple response surface optimization has attracted a great attention, while, the research about optimizing three or more responses employing response surface methodology (RSM) was very few.ResultsRSM and desirability function (DF) were employed to get the optimum ultrasonic extraction parameters, in which the extraction yields of curdione, furanodienone, curcumol and germacrone from C. wenyujin were maximum. The yields in the extract were accurately quantified using the validated high performance liquid chromatography method with a good precision and accuracy. The optimization results indicated that the maximum combined desirability 97.1 % was achieved at conditions as follows: liquid–solid ratio, 8 mL g−1; ethanol concentration, 70 % and ultrasonic time, 20 min. The extraction yields gained from three verification experiments were in fine agreement with those of the model’s predictions. The surface morphologies of the sonication-treated C. wenyujin were loose and rough. The extract of C. wenyujin presented obvious antiproliferative activities against RKO and HT-29 cells in vitro.ConclusionResponse surface methodology was successfully applied to model and optimize the ultrasonic extraction of four bioactive components from C. wenyujin for antiproliferative activitiy.Graphical abstract.Electronic supplementary materialThe online version of this article (doi:10.1186/s13065-016-0177-9) contains supplementary material, which is available to authorized users.

A confidence region for the ridge path in multiple response surface optimization

Ridge analysis allows the analyst to explore the optimal operating conditions of the experimental factors. A confidence region is desirable for the estimated ridge path. Most literature concentrates on the univariate response situation. Little is known for the confidence region of the ridge path for the multivariate response; only a large-sample confidence interval for the ridge path is available. The simultaneous coverage rate for the existing interval is typically too conservative in practice, especially for small sample sizes. In this paper, the ridge path (via desirability function) is estimated based on the seemingly unrelated regression (SUR) model as well as standard multivariate regression (SMR) model, and a conservative confidence interval suitable for small sample sizes is proposed. It is shown that the proposed method outperforms the existing methods. Real-life examples and simulative study are given for illustration.

Multiple response surface optimization for effects of processing parameters on physicochemical and bioactive properties of apple juice inoculated with Zygosaccharomyces rouxii and Zygosaccharomyces bailii

Abstract In this study, response surface methodology was used to evaluate effect of some processing parameters; namely, sodium benzoate (SB, 0–0.1%), mint essential oil (MEO, 0–1000 ppm), storage temperature (4–20 °C) and time (1–41 days) on physicochemical and bioactive properties of apple juice inoculated with Zygosaccharomyces rouxii and Zygosaccharomyces bailii. A multiple-response optimization was conducted to determine simultaneous effects of the processing variables that maximize and minimize the characterized properties of apple juice. The results showed that MEO had a remarkable anti-yeast effect on Z. rouxii and Z. bailii, which revealed that it had a comparable effect with that of SB in terms of anti-yeast activity. In addition, synergistic/antagonistic effects were investigated; however, it was revealed that there were antagonistic effects between SB and MEO in terms of bioactive properties. The effect of MEO on the bioactive properties was more pronounced in apple juice samples inoculated with Z. bailii than in those inoculated with Z. rouxii. Multiple-response optimization analysis revealed that minimum yeast count (YC) values (0.017 or 0.271 log cfu/ml) would occur at storage time = 15.78 or 15.72 days, storage temperature = 9.28 or 7.47 °C, SB concentration = 0.00% and MEO concentration = 1,000 ppm.

Optimization of operating conditions for a double-row tapered roller bearing

This paper proposes a methodology that combines the Finite Element Method and multiple response surface optimization to search for the optimal operating conditions of a double-row Tapered Roller Bearing (TRB) that has a Preload (P), radial load (Fr), axial load (Fa) and torque (T). Initially, FE models based on a double-row TRB are built and validated in the basis of experimental data and theoretical models. Three of the most important parameters used in the design of TRB were obtained from a simulation of the FE models with a combination of several operating conditions that were previously selected in accordance with a design of experiments. The design parameters are: contact stress radio for both rows of rollers (S1 and S2), maximum deformation of the outer raceway (αmax), and the difference between the gaps of the inner raceways (Δδ) or misalignment. Based on the results of the FE simulations, quadratic regressions models are generated that use the response surface method to predict the design parameters when new operating condition are applied. Then, a multi-response optimization study based on these models and using desirability functions is conducted. It is concluded that the accuracy of the results demonstrates that this methodology may be used to search for the optimal operating condition in a double-row TRB.

Multiobjective Optimization of Cryogenic Freezing Conditions of Chicken Drumstick via Compromise Simplex Method

Because frozen foods have various quality characteristics, an optimization of a cryogenic freezing process is a complex task. This study proposes a compromise simplex method (CSM) using regression analysis to determine four optimal controllable process parameters corresponding to multiresponses for the level of three product quality characteristics. The results of the experimental process for the modified simplex method and multiple response surface optimization model both outperformed the results of the existing operating conditions, thus verifying that the CSM result in food preservation with the product quality measures close to customer satisfaction. The four recommended optimal conditions were as follows: the freezer set temperature of −100C, exhaust fan speed at 10 Hz, retention time of 12 min and counter-current nitrogen flow. Practical Applications This study optimized multiple responses for chicken drumsticks in a cryogenic freezing process. This optimization method has potential for practical application. All compromised quality characteristics are of topmost importance to the frozen food industry. This method, with some modifications, can also be used in further studies of related food products.

다중반응표면 최적화를 위한 가중평균제곱오차

본 다중반응표면 최적화는 다수의 반응변수(품질특성치)를 동시에 고려하여, 입력변수의 최적 조건을 찾는 것을 목적으로 한다. 지금까지 다중반응표면 최적화를 위하여 다양한 방법이 제안되어 왔는데, 그 중 평균제곱오차 최소화법은 다수의 반응변수의 평균과 표준편차를 동시에 고려하여 최적화하는 방법이다. 이 방법은 기본적으로 평균과 표준편차가 동일한 가중치를 가지고 있다는 것을 전제로 하고 있다. 그러나 문제의 상황에 따라 평균과 표준편차에 서로 다른 가중치를 부여해야 하는 경우도 있다. 이에 본 논문에서는 기존의 평균제곱오차를 확대하여 평균과 표준편차에 서로 다른 가중치도 부여할 수 있도록 가중평균제곱오차 최소화법을 제안하고자 한다. Multiple response surface optimization (MRSO) aims at finding a setting of input variables which simultaneously optimizes multiple responses. The minimization of mean squared error (MSE), which consists of the squared bias and variance terms, is an effective way to consider the location and dispersion effects of the responses in MRSO. This approach basically assumes that both the terms have an equal weight. However, they need to be weighted differently depending on a problem situation, for example, in case that they are not of the same importance. This paper proposes to use the weighted MSE (WMSE) criterion instead of the MSE criterion in MRSO to consider an unequal weight situation.

A robust interactive approach to optimize correlated multiple responses

Setting of process variables to meet the required specification of quality characteristics is a common problem in the process quality control. To obtain the most satisfactory solution, a decision maker's (DM) preference information on the trade-offs among the quality characteristics should be incorporated into the optimization procedure. In this regard, several multiple response surface optimization (MRO) techniques have been proposed in recent years. Most of these techniques require that all the preference information is specified in advance which is very difficult in practice. Furthermore, most of them assume independency of quality characteristics where their variances are constant over the experimental space. An interactive approach to optimize multiple responses is presented that does not require any information about DM's preference before solving process. This method aims to identify process variables to consider correlation among quality characteristics and minimize the variation in deviation of responses from their targets. It also accommodates dispersion effects and specification limits as well as location effects in a unified framework based on desirability functions. The features of the proposed method are investigated and the results are compared with some existing techniques through a real numerical example. Obtained results indicate the superiority of proposed methodology with respect to the major existing MRO techniques.

Multiple response surface optimization with correlated data

Setting of process variables to meet the required specification of quality characteristics is an important problem in the process quality control. There are often several conflicts in quality characteristics, which should be simultaneously satisfied. These types of problems are called “Multiple Response Optimization” (MRO). When quality characteristics are correlated, MRO problems may become increasingly difficult. In design of experiments, identifying covariates effects could reduce error and uncovered variances as well as give more insight about the process. This study aims to identify process variables to consider correlated covariates and correlated quality characteristics. It also accommodates dispersion effects and specification limits as well as location effects in a unified framework based on desirability functions. The features of the proposed method are investigated and the results are compared with some existing techniques by applying two numerical examples.

Multiple Response Surface Optimization Considering the Decision Maker’s Preference Information

Most of the studies in Response Surface Methodology commonly involve one response or quality characteristics, whereas in most industrial applications considering all responses simultaneously is required. Multiple Response Surface (MRS) Optimization Problems often deal with responses that are conflicting. In dealing with incommensurate responses, incorporating a decision maker’s preference information into the problem has lots of advantages although a few researches in MRS literature have taken this into attention. The purpose of this paper is to take a detailed look at the most prominent approaches that has been suggested so far in MRS, and also to review and discuss the classifications with a special focus on the decision maker’s preference information. In today’s competitive market satisfying the customer is of high importance. The DM can be a customer and reaching a compromise with an interactive method would help the firm to succeed in having loyal customers. A case study is applied to show that interactive method with existing MRS approach leads to better results.

Finding Design Space and a Reliable Operating Region Using a Multivariate Bayesian Approach with Experimental Design

The posterior predictive approach for multiple response surface optimization presented by Peterson [7] is used to identify a region of process operating conditions where all quality attributes of the product are highly likely to meet specifications. The approach consists of calculating the probability that future responses will meet specification over a multidimensional grid of operating conditions. Examples from the pharmaceutical industry are used to show how the method is applied to statistically designed experiments and the results are used to generate reliability surface plots. The approach supplements traditional analysis and optimization techniques with calculated values that capture the maturity of the process under development, and provide a useful figure of merit in the definition of Design Space [5]. Also considered is the distinction between determining a Design Space to meet the specifications of critical quality attributes (CQA’s) [2] for the active pharmaceutical ingredient (API), and a reliable operating region (ROR) that also satisfies desirable manufacturing attributes, such as cost, yield, or throughput. A Bayesian posterior predictive approach offers benefits over traditional frequentist approaches to optimization. The traditional approaches, such as desirability functions or overlapping contours, do not account for model parameter uncertainty and the correlation of the responses at fixed operating conditions.

A Posterior Predictive Approach to Multiple Response Surface Optimization

In this paper we present an approach to multiple response surface optimization that not only provides optimal operating conditions, but also measures the reliability of an acceptable quality result for any set of operating conditions. The most utilized multiple response optimization approaches of “overlapping mean responses” or the desirability function do not take into account the variance-covariance structure of the data nor the model parameter uncertainty. Some of the quadratic loss function approaches take into account the variance-covariance structure of the predicted means, but they do not take into account the model parameter uncertainty associated with the variance-covariance matrix of the error terms. For the optimal conditions obtained by these approaches, the probability that they provide a good multivariate response, as measured by that optimization criterion, can be unacceptably low. Furthermore, it is shown that ignoring the model parameter uncertainty can lead to reliability estimates that are too large. The proposed approach can be used with most of the current multiresponse optimization procedures to assess the reliability of a good future response. This approach takes into account the correlation structure of the data, the variability of the process distribution, and the model parameter uncertainty. The utility of this method is illustrated with two examples.

A Bayesian Approach for Multiple Response Surface Optimization in the Presence of Noise Variables

An approach for the multiple response robust parameter design problem based on a methodology by Peterson (2000) is presented. The approach is Bayesian, and consists of maximizing the posterior predictive probability that the process satisfies a set of constraints on the responses. In order to find a solution robust to variation in the noise variables, the predictive density is integrated not only with respect to the response variables but also with respect to the assumed distribution of the noise variables. The maximization problem involves repeated Monte Carlo integrations, and two different methods to solve it are evaluated. A Matlab code was written that rapidly finds an optimal (robust) solution in case it exists. Two examples taken from the literature are used to illustrate the proposed method.

Automated enzymatic assay for plasma ammonia.

An enzymatic method for determining plasma ammonia with the Du Pont Automatic Clinical Analyzer (aca) is described. The assay requires a sample volume of 500 muL for a kinetic ammonia measurement. The reaction is initiated with glutamate dehydrogenase and the rate of depletion of NADPH is monitored with two measurements, 17 s apart, at 340 nm. Reaction conditions have been optimized for maximum sensitivity through both one-factor-at-a-time and multiple variable response surface optimization techniques. Linearity to 1000 mumol of ammonia per liter of plasma has been achieved. No significant interferences were observed from anticoagulants or endogenous blood components, including pyruvate and oxalacetate. Use of the coenzyme NADPH (instead of NADH) in this aca procedure eliminates the lengthy pre-incubation otherwise required for endogenous dehydrogenase reactions.