Multi-response Optimization Problem Research Articles

A novel TOPSIS linear programming model based on response surface methodology for determining optimal mixture proportions of lightweight concrete blocks containing sugarcane bagasse ash

The idea of utilizing waste from the agro-industrial sector to produce lightweight concrete is one of the good ideas for recycling and reusing waste materials. In a lightweight concrete production process, determining a material's optimal parameters is crucial, since it can help optimize the important properties of lightweight concrete blocks. This study introduces a novel TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) linear programming model, based on the Response Surface Methodology (RSM), to optimize the parameters of lightweight concrete blocks. The compressive strength, dry density, and water absorption are considered important responses, while black bagasse ash, cement with white bagasse ash, and aluminum powder are the factors considered. The proposed method successfully optimized the parameters, as confirmed by experimental results, showing a 7.22% increase in compressive strength, a 9.19% increase in dry density, and a 16.83% decrease in water absorption compared to the original condition. These improvements were achieved by using the optimal mixture ratio of 6:1:0.05 by weight, which consists of sugarcane bagasse ash, cement, and aluminum powder. The advantages of the proposed method are as follows: This paper presents a novel method called the TOPSIS linear programming model, which is a modified version of the original TOPSIS method, to calculate the closeness efficiency for each run. The proposed method is simpler and more practical, making it useful for solving multi-response optimization problems with large inputs. In addition, this research contributes to the advancement of sustainable materials and offers a practical solution for optimizing lightweight concrete block properties.

Multiple parameter optimization methodology by integrating a game theory principle into priority-based decision making

Mathematical model development for multivariate decision-making problems (i.e., multiple parameter and response optimization problems) has received significant attention and has been investigated by many researchers and practitioners. Although many different optimization models and methods, including priority and weight allocation problems, have been proposed in the literature, there is significant room for improvement. The majority of existing optimization methods may have difficulties in determining reliable priorities and weights for practical industrial situations. In addition, because they often focus on tradeoffs only among multiple output responses, these methods may not incorporate the multiple priority and/or weight effects directly from input factors to output responses (i.e., quality characteristics). To address these problems, the primary objective of this research is to propose a new multi-objective optimization approach by integrating a game theory principle (i.e., the Stackelberg leadership game) into a robust parameter design model to determine the optimal factor settings. First, a multi-response robust design optimization (RDO) problem was formulated using a mean squared error model and response surface methodology. A Stackelberg leadership game is then integrated into this RDO problem, where multiple responses play the role of game participants. Second, the integrated RDO model using the Stackelberg game-based multi-response (SGMR) formulation approach is analyzed by decomposition into various sequential optimization models in terms of different leader–follower relationships. Finally, non-dominated solutions are obtained from the proposed model by evaluating the overall quality loss (OQL). A pharmaceutical numerical example was performed to demonstrate the proposed integrated model and to examine whether this model can determine the most efficient solutions when the relative importance among multiple responses is unidentified. In addition, a comparative study associated with the conventional multi-objective optimization methods and the newly proposed RDO model is presented, and the proposed RDO model provided significantly better solutions than the conventional methods in the relevant comparative study.

Robust optimization for functional multiresponse in 3D printing process

Computer models are commonly used to simulate the functional relationships between inputs and outputs for quality design in 3D printing. However, the high-dimensional outputs of functional multiresponse make it challenging to develop the simulation model and perform robust optimization. This paper proposes a novel optimization method with an additive multiresponse Gaussian process model for dealing with functional multiresponse optimization problems. First, an additive covariance function is constructed to capture the correlation of the temporal inputs. Second, the Markov Chain Monte Carlo sampling technique is adopted to determine the simulation model and quantify the uncertainty. Finally, the optimization model is constructed by integrating the quality loss function and interval analysis method, and the Bayesian optimization algorithm is used to obtain the optimal solution. A numerical simulation example and a 3D printing case study are used to illustrate the effectiveness of the proposed method. The comparison results show that the responses of the proposed method are closer to the targets than the current ones, and all fall within the specified interval.

A Novel TOPSIS Linear Programming Model Based on the Taguchi Method for Solving the Multi-Response Optimization Problems: A Case Study of a Fish Scale Scraping Machine

A Novel TOPSIS Linear Programming Model Based on the Taguchi Method for Solving the Multi-Response Optimization Problems: A Case Study of a Fish Scale Scraping Machine

Hybrid of "Intersection" Algorithm for Multi-Objective Optimization with Response Surface Methodology and its Application

Recently, a new "intersection" method for multi-objective optimization was developed in the points of view set theory and probability theory, which introduces a new idea of favorable probability to reflect the favorable degree of the utility of performance indicator in multi-objective optimization, and the product of all partial favorable probabilities of entire utilities of performance indicators makes the overall / total favorable probability of the candidate. Here, in this paper, the new "intersection" algorithm for multi-objective optimization is combined effectively with response surface methodology (RSM) by taking each response as one objective, which transfers the multi-response optimization problem into a single response one with the help of the overall / total favorable probability of each scheme. The overall / total favorable probability is the uniquely decisive index of the scheme in the optimization. Applications of the hybrid approach with two examples in material technology are given, proper predictions are obtained.

A PDCA Framework towards a Multi-Response Optimization of Process Parameters Based on Taguchi-Fuzzy Model

Multi-response optimization problems investigation is a crucial element in initiatives designed to enhance quality and overall productivity for manufacturing processes. Since no particular algorithm can be employed for all multi-response problems, defining the method that is utilized as a problem-solving technique is a vital step in the process factors optimization. Identifying a formal procedure of implementing the improvement approach in a multi-criteria decision-making problem is a critical need to ensure the consistency and sustainability of the enhancement methods. In this study, a Plan–Do–Check–Act (PDCA) framework is implemented for a case study in the food industry under which a multi-response optimization problem is investigated. The design of experiment (DOE) is used to examine the effect of process parameters on the quality responses by using the Taguchi method to find the optimal setting for each parameter. An orthogonal array (OA) and signal-to-noise (SNR) ratio is employed to investigate the performance characteristics. Each performance characteristic is then converted into a signal-to-noise ratio, and all the ratios are then fed into a fuzzy model to produce a single comprehensive output measure (COM). The average COM values for various factor levels are calculated, and the level that maximizes the COM value for each factor is identified as the optimal level. Results indicated the effectiveness of the applied method to find the optimal factor levels for the multi-response optimization problem under study. The global optimal factor levels that are driven from the fuzzy logic for the studied parameters are 1250, 40, 7.5, and 1:2, for the speed, frying time, cooking time, and the coating ratio, respectively. Means of all the studied quality characteristics were closer to the target values when compared with the initial factors’ settings.

Parameter Optimization and Machining Performance of Inconel 625 with Nanoparticles Dispersed in Biolubricant

Productivity and cost-effectiveness are essential components of any long-term manufacturing system. While quantity and quality are linked to productivity, the economy focuses on energy-efficient processes that produce a high output-to-input ratio. Hard-to-cut materials have always been difficult to machine because of more significant tool wear and power losses. Inconel 625 is a hard material used in aerospace and underwater applications and is milled using biolubricants with nanoparticles. Palm oil is considered a biolubricant, and titanium dioxide (TiO2) and copper oxide (CuO) are selected as nanoparticles. When the combination of biolubricants and nanoparticles is added to the workpiece’s surface, it enhanced some properties while machining. Experiments involving four factors with four levels were carried out using the Taguchi design of experiments (DoE). The feed, depth of cut, speed, and coolant with nanoparticle additives were all factors. The responses were surface roughness, spindle vibration along X, Y, and Z axes, and material removal rate. Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) was used to alter the multiresponse optimization problem to a single-response optimization problem. The S/N of TOPSIS closeness coefficients was calculated, and the optimal machining conditions were determined. Surface roughness, material removal rate, and spindle vibration were reduced by 3.10%, 6.14%, 7.54% (Vx), and 6.78% (Vz), respectively, due to the TOPSIS optimization.

A scenario-based desirability function for correlated multi-response optimization problems considering modeling and implementation errors

Simultaneous improvement of multiple quality characteristics is one of the main concerns in today's complex industrial systems. Multiresponse optimization enables us to address this concern and find a compromise solution of design factors that leads to desirable quality outputs. This problem becomes complicated when some degree of uncertainty affects the result of the experiments. In this paper, two sources of uncertainty are included in multi-response optimization problems. The first is experimental errors related to unknown variables, also called environmental nuisances, creating response surface modeling errors. The second source is errors in implementation. In the experimentation, the controllable factors are mainly set at some predefined levels, but after the optimal design has been found, it oscillates around that value during the implementation phase. To handle this problem, a scenario-based two-stage stochastic programming is developed in which desirability functions are aggregated to form the final objective function. Scenarios are generated by sampling from probability distribution functions of the response surfaces' coefficients under normality assumptions. The numerical results show that joint consideration of the error sources leads to a smaller range of desirability values among the different scenarios and a higher value of the total desirability index.

Multi-Response Optimization of Friction Stir Welding of AA2050 Using Response Surface Methodology Coupled with Grey Relational Analysis and Principal Component Analysis

The third generation aluminium-lithium alloy AA2050 finds wide applications in defence and aircraft industries by virtue of its high strength-to-weight ratio and excellent corrosion resistance. Friction stir welding (FSW), relatively novel technique, is more suitable to join this alloy compared to other conventional fusion welding techniques. In this work, the overall quality of the weld joint was decided from the higher values of tensile strength, yield strength, percentage elongation, hardness of weld zone, hardness of heat affected zone, bending load and lower value of width of heat affected zone. The optimal (combined) design was used to design the experiments with four numeric factors (traverse speed, rotational speed, tilt angle and shoulder diameter) and a categoric factor (tool pin profile). The multi-response optimization problem was reduced into a single-response optimization problem using grey relational analysis (GRA); principal component analysis (PCA) was used to assign optimal weighting values for the responses in the process of dimensionality reduction. Mathematical model for the reduced single response, which can be perceived as overall weld quality, was developed by the response surface methodology (RSM) and the optimization of process parameters was also carried by the RSM. Analysis of variance (ANOVA) was carried to evaluate the significance of each parameter on the overall weld quality and the adequacy of the developed model. The confirmation tests conducted at optimum levels of parameters proved the effectiveness and robustness of the method.

Multi-Response Optimization of Surface Grinding Process Parameters of AISI 4140 Alloy Steel Using Response Surface Methodology and Desirability Function under Dry and Wet Conditions

The effect of four controllable input process parameters of AISI 4140 steel, cross-feed, workpiece velocity, wheel velocity, and the depth of cut were experimentally investigated under dry and wet conditions. Three responses, contact temperature, material removal rate (MRR), and machining cost during surface grinding of AISI 4140 steel, were considered. The process was optimized using a recently developed combined methodology based on response surface methodology (RSM) and desirability functional approach (DFA). RSM generated the models of the responses for prediction while DFA solved these multi-response optimization problems. The DFA approach employed an objective function known as the desirability function, which converts an estimated response into a scale-free value known as desirability. The optimum parameter was attained at the maximum overall desirability. An analysis of variance (ANOVA) was conducted to confirm the model adequacy. From the results of the study, for equal weights of responses, the corresponding optimal values of the input parameters cross-feed, workpiece velocity, the wheel or cutting velocity and the depth of cut were found to be 6 mm/pass, 12 m/min, 15 m/s, and 0.095 mm respectively in wet conditions. The corresponding predicted output responses were: 134.55 °C for the temperature, and 7.366 BDT (Taka, Currency of Bangladesh) for the total cost with an overall desirability of 0.844. Confirmation testing of optimized parameters, i.e., checking the validity of optimal set of predicted responses with the real experimental run were conducted, and it was found that the experimental value for temperature and total cost were 140.854 °C and 8.36 BDT, respectively, with an overall desirability of 0.863. Errors of the predicted value from the experimental value for equal weightage scheme were 4.47% for the temperature and 7.37% for the total cost. It was also found that if the temperature was prioritized, then the wet condition dominated the overall desirability, which was expected. However, if the cost was given high weightage, dry condition achieved the highest overall desirability. This can be attributed to the cutting in the wet condition which was more expensive due to the application of cutting fluid. The proposed model was found to be new and highly flexible in the sense that there was always an option at hand to focus on a particular response if needed.

Multi-objective optimization of a diesel engine fueled with different fuel types containing additives using grey-based Taguchi approach.

Due to the reduction of fossil fuels' resources and their contribution to environmental problems, biodiesel fuels have attracted significant attention as substitutes for diesel fuels. However, since their NOx emissions are higher than that of diesel fuels in most cases and also because of their higher viscosity than diesel, fuel additives are used to enhance their properties and reduce emissions. In this study, the effect of n-hexane and n-hexadecane addition to biodiesel and diesel fuels on exhaust emissions and performance of a single-cylinder diesel engine was investigated by using grey-based Taguchi method. Fuel additive, the additive amount, and fuel type were considered as the operating parameters. Three fuel types including diesel, rapeseed oil biodiesel, and cottonseed oil biodiesel were used in this investigation, while n-hexane and n-hexadecane were considered as the two fuel additives. As well as, three levels were assigned to the additive amount which were 4, 8, and 12%. Based on the operating parameters and their levels, the plan of experiments was generated according to L18 orthogonal array. Using grey relational analysis, this multi-response optimization problem was first transformed into a single response optimization. Then, this single system response, which is known as grey relational grade, was utilized in Taguchi approach for statistical evaluations. The results demonstrated that rapeseed was the best selection for fuel type compared to cottonseed and diesel in order to have the optimum system responses and hexadecane gave better results for system optimization in comparison with hexane additive. As well as, the analysis of variance showed that fuel type was the predominant operating factor influencing the grey relational grade which means fuel type was the most important parameter in the simultaneous optimization of exhaust emissions and engine performance. The Taguchi results also revealed that the optimum condition of engine performance and exhaust emissions happened when engine was fueled with rapeseed biodiesel containing 12% hexadecane as an additive. The confirmation test result validated the reliability of Taguchi approach in this investigation.

A multiobjective portfolio optimization for energy assets using D-Optimal design and mixture design of experiments

Paper aims Frequently, researchers try to find a better way to allocate assets in order to have maximum return and low variability in a portfolio as diverse as possible. This paper aims to apply D-Optimal Design in the context of Mixture Design and portfolio optimization to efficiently select the runs of the proposed experimental design. Originality A new approach to find the optimal weights that maximize the returns and minimize the risk using D-Optimal Design was used. A multi-response optimization problem considering returns, variability and entropy as functions of the weights was proposed. However, as there is a significant correlation between the objective functions, a Factor Analysis combined with FMSE to dimensionality reduction was used. Research method All the steps for both stages of the methodology applied in this paper are presented below: select real time series; predict one step ahead; generate a D-Optimal mixture design; apply weights and generate mathematical models; solve the optimization problem. Main findings Using the desirability method, the optimal values were determined, obtaining approximately 79% for the compound desirability function. The proposed method presented a 16.80% higher return with a 4.98% higher risk exposure if compared against Naïve method. Implications for theory and practice The proposed methodology can be applied to any portfolio optimization study. Mixture Design studies have already been proposed for modeling portfolio optimization problems. However, the D-Optimal Design proved to be adequate, which represents less computational effort.

Solving Multi-Response Problem Using Goal Programming Approach and Quantile Regression

Response surface methodology (RSM) is a group of mathematical and statistical techniques helpful for improving, developing and optimizing processes. It also has important uses in the design, development and formulation of new products. Moreover, it has a great help in the enhancement of existing products. (RSM) is a method used to discover response functions, which meet and fulfill all quality diagnostics simultaneously. Most applications have more than one response; the main problem is multi-response optimization (MRO). The classical methods used to solve the Multi-Response Optimization problem do not guarantee optimal designs and solutions. Besides, they take a long time and depend on the researcher's judgment. Therefore, some researchers used a Goal Programming-based method; however, they still do not guarantee an optimal solution. This study aims to form a goal programming model derived from a chance constrained approach using quantile regression to deal with outliers not normal and errors. It describes the relationship between responses and control variables at distinctive points in the response conditional distribution; it also considers the uncertainty problem and presents an illustrative example and simulation study for the suggested model.

Grey-based Taguchi Method to Optimize the Multi-response Design of Product Form Design

This paper presents a multi-response optimization method that uses the grey-based Taguchi method as the integrative product form design optimization method, and it serves as a tool for product form design to determine the optimal combination of design parameters in Kansei engineering (KE). This method is unique in that it combines the Taguchi method (TM) and grey relational analysis (GRA), allowing it to take advantage of the benefits of both methods. The TM is used to design experiments and generate combinative product form design samples which can be used to improve product quality. The GRA is applied to multi-response optimization problems. Factor effect analysis and analysis of variance (ANOVA) are used to determine which combinations of design parameters will result in the optimal product design. To demonstrate the applicability of the grey-based TM, a case study of a car form design is presented, and a confirmation test is performed to verify the performance of the optimal product design. The results show that the grey-based TM can deal with optimization problems with multiple Kansei responses and determine an optimal car form design that is representative of the consumers' perception in a systematic manner. The confirmation test results also show that the optimal product design generated by the grey-based TM can be used to improve the overall quality of a product form.

Research of multi-response optimization of milling process of hardened S50C steel using minimum quantity lubrication of Vietnamese peanut oil

This study aims to build a regression model when surveying the milling process on S50C steel using Minimum Quantity Lubrication (MQL) of Vietnamese peanut oil-based on Response Surface Methodology. The paper analyses and evaluates the effect of cutting parameters, flow rates, and pressures in minimum quantity lubrication system on cutting force and surface roughness in the milling process of S50C carbon steel materials after heat treatment (reaching a hardness of 52 HRC). The Taguchi method, one of the most effective experimental planning methods nowadays, is used in this study. The statistical analysis software, namely Minitab 19, is utilized to build a regression model between parameters of the cutting process, flow rates and pressures of the minimum quantity lubrication system and the cutting force, surface roughness of the part when machining on a 5-axis CNC milling machine. Thereby analyzing and predicting the effect of cutting parameters and minimum quantity lubrication conditions on the surface roughness and cutting force during machining to determine the influence level them. In this work, the regression models of Ra and F were achieved by using the optimizer tool in Minitab 19. Moreover, the multi-response optimization problem was solved. The optimum cutting parameters and lubricating conditions are as follows: Cutting velocity Vc=190.909 m/min, feed rate fz=0.02 mm/tooth, axial depth of cut ap=0.1 and nozzle pressure P=5.596 MPa, flow rate Q=108.887 ml/h. The output parameters obtained from the above parameters are Ra=0.0586 and F=162.035 N, respectively. This result not only provides the foundation for future research but also contributes reference data for the machining process

Multi-response optimization of process parameters in friction stir welded aluminum 6061-T6 alloy using Taguchi grey relational analysis

PurposeFriction stir welding (FSW) is considered an environmentally sound process compared to traditional fusion welding processes. It is a complex process in which various parameters influence weld strength. Therefore, it is essential to identify the best parameter settings for achieving the desired weld quality. This paper aims to investigate the multi-response optimization of process parameters of the FSWed 6061-T6 aluminum (Al) alloy.Design/methodology/approachThe input process parameters related to FSW have been sorted out from a detailed literature survey. The properties of weldments such as yield strength, ultimate tensile strength, percentage elongation and microhardness have been used to evaluate weld quality. The process parameters have been optimized using the Taguchi-based grey relational analysis (GRA) methodology. Taguchi L16 orthogonal array has been considered to design the experiments. The effect of input parameters on output responses was also determined by the analysis of variance (ANOVA) method. Finally, to corroborate the results, a confirmatory experiment was carried out using the optimized parameters from the study.FindingsThe ANOVA result indicates that the tool rotation speed was the most significant parameter followed by tool pin profile and welding speed. From the confirmation test, it was observed that the optimum FSW process parameters predicted by the Taguchi method improved the grey relational grade by 13.52%. The experimental result also revealed that the Taguchi-based GRA method is feasible in finding solutions to multi-response optimization problems in the FSW process.Originality/valueThe present study is unique in the multi-response optimization of FSWed 6061-T6 Al alloy using the Taguchi and GRA methodology. The weld material having better mechanical properties is essential for the material industry.

MULTI-RESPONSE OPTIMIZATION OF END-MILLING PARAMETERS FOR INCONEL 625 USING TAGUCHI COUPLED WITH TOPSIS

Using Taguchi design of experiments (DoE), experiments were conducted with 3 factors and 3 levels. The factors were the depth of cut, spindle speed, and feed. The responses were surface roughness, flank wear, material removal rate, and spindle vibration along [Formula: see text] ([Formula: see text]), [Formula: see text] ([Formula: see text]), and [Formula: see text] ([Formula: see text]) axis. To convert the multi-response optimization problem into a single response optimization problem, the technique for order of preference by similarity to ideal solution (TOPSIS) was applied. The S/N of the closeness coefficients from TOPSIS was calculated and optimum machining conditions were obtained. Further, analysis of variance (ANOVA) was performed to verify which input parameter significantly affects the output responses. From TOPSIS optimization, the responses like surface roughness and flank wear were decreased by 0.99% and 2.55%. The vibration in [Formula: see text], [Formula: see text], and [Formula: see text]-axis decreased by 3.84%, 16.87% and 12.48% respectively. This optimization significantly enhances the machining characteristics.

Process Design and Optimization of end Milling Parameters of Al 7075 Metal Matrix Composite

This paper investigates the machinability characteristics of end milling operation to yield minimum tool wear with the maximum material removal rate using RSM. Twenty-seven experimental runs based on Box-Behnken Design of Response Surface Methodology (RSM) were performed by varying the parameters of spindle speed, feed and depth of cut in different weight percentage of reinforcements such as Silicon Carbide (SiC-5%, 10%,15%) and Alumina (Al2O3-5%) in alluminium 7075 metal matrix. Grey relational analysis was used to solve the multi-response optimization problem by changing the weightages for different responses as per the process requirements of quality or productivity. Optimal parameter settings obtained were verified through confirmatory experiments. Analysis of variance was performed to obtain the contribution of each parameter on the machinability characteristics. The result shows that spindle speed and weight percentage of SiC are the most significant factors which affect the machinability characteristics of hybrid composites. An appropriate selection of the input parameters such as spindle speed of 1000 rpm, feed of 0.02 mm/rev, depth of cut of 1 mm and 5% of SiC produce best tool wear outcome and a spindle speed of 1838 rpm, feed of 0.04 mm/rev, depth of cut of 1.81 mm and 6.81 % of SiC for material removal rate.

Effect of process parameters on sliding wear performance of red brick dust-filled glass–epoxy composites

This research is focused on the study of the tribological properties of epoxy composites reinforced with red brick dust and glass fiber. Wear tests are conducted on a pin-on-disk apparatus. The wear rate and coefficient of friction are measured after tribological tests. The relative effect of process parameters such as sliding velocity, normal load, filler content on specific wear rate, and coefficient of friction is also examined using the Taguchi model and analysis of variance technique. Furthermore, an innovative optimization approach is adopted by combining grey relational analysis with the metaheuristic firefly algorithm to obtain the desired response values. A nonlinear regression model is generated to cater to the relationship between the grey relational grade and the process parameters. This model is further employed in the firefly algorithm to move the firefly to the neighboring brighter and attractive firefly. The most influencing factors for wear rate and friction coefficient are sliding velocity and normal load, respectively. This work not only opens up an opportunity for value-added utilization of waste material such as red brick dust but also the proposed metaheuristic approach can be customized and applied for any multi-response optimization problem.

Multivariate-Based Technique for Solving Multi-Response Surface Optimization (MRSO) Problems: The Case of a Maximization Problem

Multi-response surface optimization (MRSO) is a problem that is peculiar to an industrial setting, where the aim of a process engineer is to set his process at operating conditions that simultaneously optimize a set of process responses. In Statistics, several methods have been proffered for tackling problems of this nature. Some of such methods are that of: overlapping contour plots, constrained optimization problem, loss function approach, process capability approach, distance function approach, game theory approach, and the desirability function approach. These, methods are however, not without teething flaws as they are either too problem specific, or require very complex and inflexible routines; little wonder, the method of desirability function has gained popularity especially because it overcomes the latter limitation. In this article, we have proposed and implemented a multivariate-based technique for solving MRSO problems. The technique fused the ideas of response surface methodology (RSM), multivariate multiple regression and Pareto optimality. In our technique, RSM was implemented on an all-maximization problem as a case-study process; in which case, first-order models (FOMs) for the responses were fitted using 2k factorial designs until the FOMs proved to be inadequate, while uniform precision rotatable central composite design was used to obtain second-order models (SOMs) for the respective responses in the event of model inadequacy of the FOMs. With the implementation of the proposed technique to the case study, optimal operating conditions were obtained, with observations stemming thereof summarized as axioms. The first, second and third axioms respectively stated that: (1) the mid-point of all optimal operating conditions obtained via the proposed technique is Pareto optimal, (2) the mid-point of all optimal responses at the Pareto optimal operating condition is Pareto optimal, and (3) the region bounded by each of the optimal operating conditions from each second-order model (SOM) is a Pareto front.