Response Surface Exploration Research Articles

Optimization of ultrasound-assisted extraction of flavonoids from Emilia prenanthoidea DC. using response surface methodology and exploration of the ecological factors on total flavonoid and antioxidant activity

Optimization of ultrasound-assisted extraction of flavonoids from <i>Emilia prenanthoidea</i> DC. using response surface methodology and exploration of the ecological factors on total flavonoid and antioxidant activity

Appropriate Number of Center Points for Response Surface Exploration Using Small Box Behnken Design

Industrial exploration using the Box Behnken Design (BBD) has been faced with a serious setback due to the swift upsurge in the runs size as the number of factors increase. This, therefore, dissuades researchers and affects the application of the design. The Small Box Behnken Designs (SBBD) which achieve the research goal of BBD were proposed to overcome the setback. This paper aimed at recommending an appropriate number of center points suitable for response surface exploration and its applications in industries using the SBBD. The method adopted for assessing the center points is the prediction variance-based G-efficiency optimality criterion. The range of design factors, k, considered is 3 to 11, while comparing the designs at 0 - 5 number of center points. For each of the design factors considered, the result showed that increasing the center point, decreases the G-efficiency value. Hence, increasing the center point does not contribute significantly to the prediction variance capability of the designs considered. However, in other to test the model lack of fit and estimate pure error which are very important in experimental design analysis, this study recommends that at most two runs (center points) be replicated at the center. Since with this number, approximately 90% G-efficiency can be achieved for response surface exploration using the SBBD.

Multi-frequency power ultrasound green extraction of polyphenols from Pingyin rose: Optimization using the response surface methodology and exploration of the underlying mechanism

The effect of multi-frequency power ultrasound (MFPU) on the extraction of Pingyin rose polyphenols (PRP) and underlying mechanisms were studied. The MFPU included mono-, dual-, and tri-frequency ultrasound. Working modes for the dual- and tri-frequency were in sequential and simultaneous ways. The extraction parameter was firstly optimized by RSM (response surface methodology). It was found that the optimum conditions were 50% for ethanol concentration, 63 °C for extraction temperature, 31 min for extraction time, and 1:46 g/mL for solid-liquid ratio with a PRP extraction yield of 156.81 mg/g. The extraction yields of PRP assisted with ultrasound in different power levels and frequencies were further investigated. Compared to non-ultrasound, the extraction yields of PRP assisted with ultrasound were significantly (p < 0.05) increased. The tri-frequency ultrasound in sequential working mode (20/28/40 kHz) showed the most promising extraction efficiencies for PRP (187.13 mg/g), which was 19.33% higher than that of non-ultrasound. Besides, signal results of the MFPU field collected by a hydrophone and damage degree of aluminum foil paper showed that the ultrasonic fields with stronger noise signals, and more uniform broken holes had higher extraction yield of PRP. Conclusively, the MFPU is demonstrated as a green protocol for the valuable extraction of PRP.

Nanofiltration desalination of reverse osmosis concentrate pretreated by advanced oxidation with ultrafiltration: Response surface optimization and exploration of membrane fouling

Reverse osmosis (RO) technique has been widely used in the advanced treatment of industrial water and wastewater. However, this process generates a large amount of reverse osmosis concentrate (ROC) which contain high salinity and organic contaminants, and therefore pose serious environmental problems. This study explored the potential of advanced oxidation processes-ultrafiltration (AOPs-UF) pretreatment system and nanofiltration (NF90, Dow Chemicals) membrane system for ROC desalination. Under the optimal conditions, 78% of COD and 81% of UV254 pollutants were removed by using advanced oxidation processes with ultrafiltration pretreatment technology. Box-Behnken design (BBD) results showed that the maximum total soluble salts rejection of 97.75% was obtained under the optimized conditions of nanofiltration membrane (the 1.18 MPa operating pressure, 6.79 °C feed temperature, 64.33 min filtration time). A combined membrane cleaning method was finally determined via surface morphology characterization by the comparison of scanning electron microscope (SEM), X-ray energy dispersive analysis (EDS) and Fourier transform infrared (FTIR) technique analysis. The results confirmed the possibility of advanced oxidation coupled membrane treatment process to treat ROC and realize its reuse. Moreover, by evaluating the benefit calculation, it provides insights for the treatment of industrial wastewater.

Bayesian learning of orthogonal embeddings for multi-fidelity Gaussian Processes

Uncertainty propagation in complex engineering systems often poses significant computational challenges related to modeling and quantifying probability distributions of model outputs, as those emerge as the result of various sources of uncertainty that are inherent in the system under investigation. Gaussian Processes regression (GPs) is a robust meta-modeling technique that allows for fast model prediction and exploration of response surfaces. Multi-fidelity variations of GPs further leverage information from cheap and low fidelity model simulations in order to improve their predictive performance on the high fidelity model. In order to cope with the high volume of data required to train GPs in high dimensional design spaces, a common practice is to introduce latent design variables that are typically projections of the original input space to a lower dimensional subspace, and therefore substitute the problem of learning the initial high dimensional mapping, with that of training a GP on a low dimensional space. In this paper, we present a Bayesian approach to identify optimal transformations that map the input points to low dimensional latent variables. The “projection” mapping consists of an orthonormal matrix that is considered a priori unknown and needs to be inferred jointly with the GP parameters, conditioned on the available training data. The proposed Bayesian inference scheme relies on a two-step iterative algorithm that samples from the marginal posteriors of the GP parameters and the projection matrix respectively, both using Markov Chain Monte Carlo (MCMC) sampling. In order to take into account the orthogonality constraints imposed on the orthonormal projection matrix, a Geodesic Monte Carlo sampling algorithm is employed, that is suitable for exploiting probability measures on manifolds. We extend the proposed framework to multi-fidelity models using GPs including the scenarios of training multiple outputs together. We validate our framework on three synthetic problems with a known lower-dimensional subspace. The benefits of our proposed framework, are illustrated on the computationally challenging aerodynamic optimization of a last-stage blade for an industrial gas turbine, where we study the effect of an 85-dimensional shape parameterization of a three-dimensional airfoil on two output quantities of interest, specifically on the aerodynamic efficiency and the degree of reaction.

Bus fleet selection analysis for a BRT concession in the Mexico valley metropolitan zone

The present paper addresses the problem of determining the bus fleet composition (size and mix) for a Bus Rapid Transit (BRT) transportation service, where revenue, demand, and passenger-km index are considered for a sensitivity analysis. The proposed method relies on financial engineering methods to produce several cost and income inputs over time by applying Net Present Value (NPV) concepts. The sensitivity analysis is developed by solving different instances with a mixed-integer linear programming (MILP) model structured as an investment selection problem. The model is capable of determining the optimal size and mix selection under relevant operational requirements as restrictions. The method is applied for a case study in Mexico Valley Metropolitan Zone (MVMZ), illustrating the valuable insights obtained with the sensitivity analysis (response surface exploration) and providing the set of potential solutions and their estimated financial and operational outcomes, information that is relevant for potential operators to endorse a discussion and negotiation with the government authorities (concessioners).

Statistical applications of optimization methods and mathematical programming

Abstract Optimization techniques perform an important role in different domains of statistic. Examples of parameter estimation of different distributions, correlation analysis (parametric and nonparametric), regression analysis, optimal allocation of resources in partial research, exploration of response surfaces, design of experiments, efficiency tests, reliability theory, survival analysis are the most known methods of statistical analysis in which we find optimization techniques. The paper contains a synthetic presentation of the main statistical methods using classical optimization techniques, numerical optimization methods, linear and nonlinear programming, variational calculus techniques. Also, an example of applying the “simplex” algorithm in making a decision to invest an amount on the stock exchange, using a prediction model..

Optimization of the Extraction Conditions for Buddleja officinalis Maxim. Using Response Surface Methodology and Exploration of the Optimum Harvest Time.

The Box-Behnken design was used to evaluate the effects of the methanol concentration (60–100%), liquid to solid ratio (20:1 to 40:1 mL/g) and extraction time (20–40 min) on the yield of 11 constituents from Buddleja officinalis Maxim using ultrasound-assisted extraction. The Derringer’s desirability function approach showed that the modified optimum extraction conditions were: 76% methanol concentration, 33 min extraction time and a 34:1 mL/g solvent to solid ratio. Under these conditions, the experimentally measured yields of the compounds were in good agreement with the predicted values. An accurate and sensitive method was also established using high-performance liquid chromatography with diode-array detection for the simultaneous determination of the 11 compounds in Buddleja officinalis. The newly developed method was used to determine the amounts of bioactive components in Buddleja officinalis during four different growth stages. According to these results, we recommend that the full blossom stage is the best time for harvesting this plant to obtain the highest yield of crude materials.

Modified Exact Single-Value Criteria for Partial Replications of the Central Composite Design

Replication of the factorial (cube) and/or axial (star) portions of the central composite design (CCD in response surface exploration has gained great attention recently. Some well known metrics (called single-value functions or criteria) and graphical methods are utilized in evaluating the regression based response surface design. The single-value functions considered here are the A-efficiency, and the D-efficiency, , where , k is number of factors, is the kth design measure, is the design’s information matrix, is its inverse and N is the total number of experimental runs. These two functions are very popular in parameter estimation in response surface methodology. The exact measures of these two design criteria will be developed analytically in this work to account for partial replication of the cube and/or star components of the CCD. This will alleviate the burden of manual computation of these metrics when there are partial replications and reduce over reliance on software values which, often, are approximate values and maybe inaccurate.

Construction of Second Order Rotatable Simplex Designs

Rotatable designs are mainly for the exploration of response surfaces. These designs provide the preferred property of constant prediction variance at all points that are equidistant from the design center, thus improving the quality of the prediction. Initially, they were constructed through geometrical configurations and several second order designs were obtained. Full Factorial Design of Experiment provides the most response information about factor main effects and interactions, the process model’s coefficients for all factors and interactions, and when validated, allows process to be optimized. On the other hand, mixture designs are a special case of response surface designs where prediction and optimization are the main goals. These designs usually predict all possible formulations of the ingredients however, little or no research has been done incorporating rotatability with the mixture designs. This paper therefore aims at constructing Rotatable Simplex Designs (RSDs) using the properties of Simplex - Lattice Designs (SLDs) in connection with Full Factorial Designs (FFDs).

Optimization of Processing Technology of Instant Sea Cucumber with Fuzzy Mathematic Comprehensive Evaluation by Response Surface Methodology and Exploration on Nutritional Value of Instant Sea Cucumber

Optimization of Processing Technology of Instant Sea Cucumber with Fuzzy Mathematic Comprehensive Evaluation by Response Surface Methodology and Exploration on Nutritional Value of Instant Sea Cucumber

An approach toward optimization of the influential growth determinants of opportunistic yeast isolate Pichia guilliermondii

ABSTRACTThe present study reports statistical optimization of growth conditions of an opportunistic fungal strain Pichia guilliermondii, isolated from the blood of patients suffering from bancroftian filariasis. Seven key determinants, namely, primary inoculums size (%), volume (mL) and pH of media, serum proportion, temperature (°C), incubation time (hr), and agitation speed (rpm) that influence in vitro growth of the pathogen were optimized statistically using response surface methodology (RSM). RSM with seven factors and two-level Box–Behnken design was employed for designing experimental run, prediction of case statistics, suitable exploration of quadratic response surfaces, and constructing a second-order polynomial equation. Analysis of variance (ANOVA) showed that primary inoculums size, volume of culture media, temperature, incubation time, and agitation speed exert most significant influence over fungal growth. The RSM study predicted that optimum fungal growth can be obtained using 10% primary inoculums size in 100 mL culture media with pH 6.0, 6.28% serum, 32.5°C temperature, and 24 hr of incubation, alongside agitation speed at 400 rpm. The desirability of the optimized growth model for P. guilliermondii is 99.123%, which indicated its accuracy and acceptability. Finally, the optimized growth module illustrated in the study could be useful in improving in vitro growth of clinically important P. guilliermondii.

Competition between two cities using cordon tolls: an exploration of response surfaces and possibilities for collusion

Extending the literature on competition in the presence of cordon tolls, this article explores the implications of competition between two cities. The city authorities are assumed to maximise the welfare of their own residents whilst taking advantage of tax export mechanisms by charging traffic from the competing authority. The problem is posed as an Equilibrium Problem with Equilibrium Constraints (EPEC) which is a special form of a Nash game with a hierarchical structure. Employing a grid search to explore the response surfaces and to determine the Nash Equilibrium toll levels, we demonstrate the possibility that there may exist multiple local Nash solutions and that competition may lead to a sub-optimal outcome for one or both authorities depending on whether there exists a stronger player. We then consider the impact of elasticity of demand and other parameter assumptions on the potential number of Nash solutions. Finally incentives for collusion are studied.

Combining two-level and three-level orthogonal arrays for factor screening and response surface exploration

Experimental design and analysis is an effective and commonly used tool in scientific investigations and industrial applications. Orthogonal arrays, such as factorial and fractional factorial designs, are popular experimental plans for identi- fying important factors. Motivated by an antiviral drug experiment, we introduce a new class of composite designs based on a two-level factorial design and a three- level orthogonal array. These designs have many desirable features and are effective for factor screening and response surface modeling. Some advantages are that they can use resolution IV designs in the screening stage, they can perform in-depth analyses, and they can be used in either a single or a sequential experiment. We study the construction method and compare the new composite designs with exist- ing ones. We illustrate the methodology with data from an experiment that studies the effects of five antiviral drugs on the Herpes simplex virus type 1.

Fractional Box–Behnken Designs for One-Step Response Surface Methodology

In contrast with the usual sequential nature of response surface methodology (RSM), recent literature has proposed both screening and response surface exploration using a single three-level design; this approach is known as “one-step RSM”. We discuss and illustrate two shortcomings of the current one-step RSM designs and analysis. Subsequently, we propose a class of three-level response surface designs, constructed by taking subsets of Box–Behnken designs, that will address these shortcomings and aid the user in being appropriately advised as to factor importance. We illustrate the designs and analysis with simulated and real data.

A new search procedure of steepest ascent in response surface exploration

The method of steepest ascent direction has been widely accepted for process optimization in the applications of response surface methodology (RSM). The procedure of steepest ascent direction is performed on experiments run along the gradient of a fitted linear model. Therefore, the RSM practitioner needs to decide a suitable stopping rule such that the optimum point estimate in the search direction can be determined. However, the details of how to deflect and then halt a search in the steepest ascent direction are not thoroughly described in the literature. In common practice, it is convenient to use the simple stopping rules after one to three response deteriorations in a row after a series of fitted linear models used for exploration. In the literature, there are two formal stopping rules proposed, that is, Myers and Khuri's [A new procedure for steepest ascent, Comm. Statist. Theory Methods A 8(14) (1979), pp. 1359–1376] stopping rule and del Castillo's [Stopping rules for steepest ascent in experimental optimization, Comm. Statist. Simul. Comput. 26(4) (1997), pp. 1599–1615] stopping rule. This paper develops a new procedure for determining how to adjust and then when to stop a steepest ascent search in response surface exploration. This proposal wishes to provide the RSM practitioner with a clear-cut and easy-to-implement procedure that can attain the optimum mean response more accurately than the existing procedures. Through the study of simulation optimization, it shows that the average optimum point and response returned by using the new search procedure are considerably improved when compared with two existing stopping rules. The number of experimental trials required for convergence is greatly reduced as well.

A comparison of designs for one‐step screening and response surface estimation

AbstractThe sequential design approach to response surface exploration is often viewed as advantageous as it provides the opportunity to learn from each successive experiment with the ultimate goal of determining optimum operating conditions for the system or process under study. Recent literature has explored factor screening and response surface optimization using only one three‐level design to handle situations where conducting multiple experiments is prohibitive. The most straightforward and accessible analysis strategy for such designs is to first perform a main‐effects only analysis to screen important factors before projecting the design onto these factors to conduct response surface exploration. This article proposes the use of optimal designs with minimal aliasing (MA designs) and demonstrates that they are more effective at screening important factors than the existing designs recommended for single‐design response surface exploration. For comparison purposes, we construct 27‐run MA designs with up to 13 factors and demonstrate their utility using established design criterion and a simulation study. Copyright 2011 © John Wiley &amp; Sons, Ltd.

Steepest Ascent for Multiple-Response Applications

The path of steepest ascent proposed by Box and Wilson consists of points that maximize the predicted response for a fitted first-order model among all points with the same standard error of prediction. When there are multiple responses or additional constraints (on, e.g., cost or throughput), the standard steepest ascent is of limited use, because it often optimizes one response to the detriment of others. We answer three pertinent questions: (1) For multiple-response applications, what search directions are inferior and can be excluded from consideration?; (2) What graphical tools can assist in our understanding the tradeoffs and choosing a compromise direction for exploration?; and (3) Do compromise directions exist that ensure improvement in all responses? We prove that only search directions that are convex combinations of the paths of steepest ascent/descent need be considered. We also construct confidence bounds for improvement of each response relative to its performance at the center of the design. Whereas the literature often deals with multiple response optimization in the later stages of response surface exploration, our article addresses how to proceed after a successful initial screening experiment.

Two-level experiments for binary response data

Information in a statistical experiment is often measured through the determinant of its information matrix. Under first order normal linear models, the determinant of the information matrix of a two-level factorial experiment neither depends on where the experiment is centered, nor on how it is oriented, and balanced allocations are more informative than unbalanced ones with the same number of runs. In contrast, under binary response models, none of these properties hold. The performance of two-level experiments for binomial responses is explored by investigating the dependence of the determinant of their information matrix on their location, orientation, range, presence or absence of interactions and on the relative allocation of runs to support points, and in particular, on the type of fractionating involved. Conventional wisdom about two-level factorial experiments, which is deeply rooted on normal response models, does not apply to binomial models. In binary response settings, factorial experiments should not be used for screening or as building blocks for binary response surface exploration, and there is no alternative to the optimal design theory approach to planning experiments.

Identifying Rising Ridge Behavior in Quadratic Response Surfaces

Canonical analysis is a common method for exploring and exploiting fitted quadratic response surfaces. Much attention in canonical analysis is given to identifying ridge behavior in these surfaces in order to achieve optimal response at minimum cost. However, little attention has been given to classifying the identified ridge as a stationary ridge or a rising ridge. Knowing whether a ridge is stationary or rising is critical for making decisions about how to continue the response surface exploration or for setting process parameters. This article presents two methods that allow for identification, classification and confirmation of ridge behavior. The first method is based on linear regression and though easily implemented, can be imprecise. The second method is more precise and is based on a new parameterization of the canonical form. It uses nonlinear regression techniques that are becoming increasingly accessible through software packages.