Comparison of NSGA-II and MODE performances by using MCDM methods for multi-response experimental da

Abstract

Keywords: Multi-response Experimental Data; NSGA-II; MODE; MCDM

Introduction

Experimental data, which consist of input and response variables, can be produced by experimental design with

measurements in many field of science and engineering. If the experimental data contains more than one response variable, it is called multi-response experimental data. Generally, the multi-response experimental data sets can

*Corresponding author. *E-mail address: turksen@ankara.edu.tr This paper was recommended for publication in revised form by Editor-in-Chief Ahmet Selim Dalkilic Published by Yıldız Technical University Press, İstanbul, Turkey Copyright 2021, Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

be examined in three basic steps: (i) modeling, (ii) optimization, and (iii) decision making. In modeling stage, functional relationship between the responses and input variables are obtained by applying multivariate multiple linear regression analysis. Parameters of mathematical model are estimated by using Ordinary Least Squares (OLS) method to create predicted response models in polynomial type. During the modeling stage, it is necessary to define whether the responses are linearly correlated. In this case, Seemingly Unrelated Regression (SUR) method should be used to predict response functions by taking into account the linear relationship between the responses. The SUR method was first introduced by Zellner [1] for the analysis of multi-response experimental data sets in which the responses are correlated. Liu [2] showed that the SUR estimators have smaller variance than the OLS estimators in case the experimental data size is large enough. Multi-response studies on the SUR are available in the literature [3-6]. Optimization is as important as the modeling stage during the analyses of multi-response experimental data set. For this purpose, predicted response functions are considered as objective functions and optimized simultaneously. Simultaneous optimization of the predicted response functions is achieved in multi-objective perspective, called multi-objective optimization (MOO). There are several studies for incorporating the MOO approach into the solution of multi-response problems [2,7-12]. As a result of the MOO, it is not possible to obtain a single optimal solution for all objective functions. In this case, Pareto solution set is found which has many non-dominated alternative solutions. It is possible to get the Pareto solution set through mathematical programming methods [13]. However, these methods tend to obtain elements of the Pareto solution set one at a time. In recent years, multi-objective metaheuristic (MOM) methods are frequently applied to solve the MOO problems [14]. In this study, population-based MOM methods are preferred to use since these methods can generate the Pareto solution set in a single run. Two population-based MOM methods, (i) NSGA-II and (ii) MODE, are applied for obtaining the Pareto sets. The NSGA-II and MODE, also called Artificial Intelligence Optimization Algorithms, have stochastic search mechanism with genetic operators [15,16]. There have been several studies about the NSGA-II for the optimization of multi-response problems [17-21]. Some applications of the MODE for optimization stage of multi-response problems can be seen in the studies of [22,23]. There are different studies for comparison of these two algorithms [24-28]. In decision making stage, it is necessary to decide for a compromise solution among the Pareto solutions which are non-dominated to each other. Multi-criteria decision making (MCDM) methods are preferable to choose a compromise solution from Pareto solution set. The Pareto solution set is considered as decision matrix that is a tool to evaluate and to select the best alternative considering different

Sigma J Eng Nat Sci, Vol. 43, No. 1, pp. 133−147, February, 2025

criteria for the MCDM. The application of the MCDM to the multi-response problems can be handled in several studies: TOPSIS and AHP [20]; TOPSIS [29]; EDAS [30]; AHP and VIKOR [31]; SAW, WASPAS, TOPSIS, EDAS, VIKOR, MOORA, COPRAS, PIV, and PSI [32]; CODAS [33]; GRA [34]; COPRAS [35,36]; TOPSIS and GRA [37]; TOPSIS, VIKOR, MOORA, and AHP [38]; WASPAS and EDAS [39]. Further, combined MCDM methods are applied to multi-response studies, e.g. PSI-TOPSIS and PSI-EDAS [8]; GRA-TOPSIS [40]; AHP-TOPSIS-VIKOR [41]. The main aim of this study is to make comparison for the optimization performances of the NSGA-II and MODE by using various MCDM methods which are TOPSIS, COPRAS, WASPAS, EDAS, CODAS, GRA and MABAC. At first, Entropy method is applied to the response variables of the experimental data sets to define the weights objectively. Then, MCDM methods are applied to the Pareto sets to define a compromise solution considering the weights. The closeness of the compromise solution and the ideal solution vectors are calculated with respect to MAE and RMSE metrics. Finally, the MOM method with the smallest MAE and RMSE metric values is considered to have the best performance. The paper is organized as follows. In Section 2, multi-response experimenal data set and the SUR modeling are introduced briefly. In Section 3, structure of the MOO problem is defined and algorithmic steps of the NSGA-II and MODE are explained. In Section 4, considered MCDM methods are discussed for decision making. In Section 5, numerical example is carried out for three experimental data sets from different fields of engineering, chemistry, and food. The obtained results are also discussed in detailed. Finally, conclusion is given in Section 6.

Experimental Data With The SUR

An experiment in which a number of responses are measured simultaneously for each setting of a group of input variables is called multi-response experiments [42]. The multi-response experimental data set with n observations, p input variables, and r response variables is shown in Table 1.

Sigma J Eng Nat Sci, Vol. 43, No. 1, pp. 133−147, February, 2025

Correlation of the responses should be considered to perform the functional relationship between the responses and input variables. If the responses are linearly correlated, the SUR method is proper to consider the linear correlation during modeling stage. Functional relationship between the response variables and input variables is expressed by linear regression model. The response model can be written as (1) In Equation (1),

. Solution set of the MOO problem, is called Pareto solution set. The Pareto solution set is obtained by using two population-based MOM methods, NSGA-II and MODE, which are quite successful to find Pareto solutions in a single run without getting stuck local solutions. Fast non-dominanted sorting and crowding distance approaches are two main principles for obtaining Pareto solution set by the MOM methods. The former ensures that solutions are sorted by separating them into surfaces according to their dominance as Rank-1, Rank-2, and etc. as shown in Figure 1, while the latter provides the measurement of the distances between the solutions on each surface and each other as illustrated in Figure 2.

. The assumptions of the response model are and . Parameter estimation with the SUR method consists of three main steps as follows [9]: Step 1: The predicted response model ( ) is obtained individually by using OLS method and residuals ( ) are calculated. Step 2: The residual covariance matrix, , for the responses is obtained. Calculate the covariance matrix components, . Step 3: The estimator vector of for the SUR is calculated as (2) The predicted response model can be written in polynomial form according to parameter estimates as

Figure 1. Surfaces determined by the fast non-dominated sorting.

Multi-Objective Optimization For Multi-Response Problem

Optimization of multi-response problem can be considered as a multi-objective optimization (MOO) problem. Mathematical formulation of the MOO problem can be written as

In Equations (4), is input variable vector and S is constraint set. Each predicted response model , is considered as an objective function,

Sigma J Eng Nat Sci, Vol. 43, No. 1, pp. 133−147, February, 2025

Non-dominated Sorting Genetic Algorithm-II (NSGA-II) The NSGA-II introduced by Deb et al. [43], is based on genetic operators. Simulations involving difficult problems have also shown that solutions are well spread and convergence is strong. Step 1: Determine the tuning parameters, population size ( ), maximum generation number ( ), genetic operators which are selection, crossover and mutation. Generate initial population (P) with size randomly and . Step 2: Create a new generation population (Q) with size by applying genetic operators to P. Step 3: Combine parent and new generation population, . Step 4: Obtain and sort non-dominated surface. Step 5: Calculate crowding distances of the solutions on each surface. Sort surfaces in ascending order and crowding distances within the surface in descending order. Step 6: Carry out crossover and mutation operators. Create new generation solutions (Q) according to crossover and mutation rates. Update number of generation, . Step 7: If number of generations is not reached, Return to Step 2, otherwise algorithm is terminated. Multi Objective Differential Evolution (MODE) The MODE proposed by Souza et al. [23], is an extension of the Differential Evolution (DE) algorithm developed by Storn and Price [44], for the single objective algorithm to multi-objective optimization. The MODE is a preferred method for difficult problems due to its ease of application. Step 1: Determine the tuning parameters, number of variables (D), population size ( ), maximum generation number ( ), scaling vector (F) and crossover ratio (CR). Step 2: Determine initial population.

: Randomly selected chromosomes from the pop) : jth gene of ith chromosome in G generation

Step 3: Compute mutation operator. : Mutated intermediate chromosome Step 4: Carry out crossover operator.

: Chromosome formed after crossover Step 5: Perform selection.

is reached. Step 6: Repeat Step 3–Step 5 until the Step 7: Apply non-dominated sorting. Step 8: Calculate crowding distance. Step 9: Sort surfaces in ascending order, crowding distances within the surface in descending order [16,22].

Multi-Criteria Decision Making Methods For Pareto Solution SET

Multi-Criteria Decision Making (MCDM) methods can be classified into three categories as sum-based, max/min value-based and range-based normalization techniques. In this study, TOPSIS and COPRAS for sum-based category; WASPAS, EDAS and CODAS for max/min value-based category; GRA and MABAC for range-based category are considered. These MCDM methods are applied to Pareto solution set to choose a compromise solution. The obtained Pareto solution set is considered as a decision matrix, , , . Each row of the decision matrix indicates non-dominated solution, called alternative and each column of the decision matrix indicates objective function value, called criteria. The criteria may not be of equal importance. The importance of the criteria is expressed by weights. In this study, the weight of the criteria , is calculated by using Entropy method which is one of the MCDM methods. The Entropy method is applied to response values which is presented as in Table 1. The weight vector of the criteria is represented as, , for each MCDM method. Entropy method is widely used for determining weights for MCDM problems. Mathematical theory of the Entropy was proposed by Shannon [45]. Subjective, objective, and integral weighting methods are the three main weighting methods. Objective entropy weighting method is applied to evaluate weights for MCDM. Advantage of the Entropy method is that it avoids subjective evaluations of decision makers in finding criterion weights, therefore increasing the objectivity of comprehensive evaluation results which are reliable and effective [21,46,47]. The algorithmic steps of the MCDM, called TOPSIS, COPRAS, WASPAS, EDAS, CODAS, GRA and MABAC, can be seen below, respectively. Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) The TOPSIS method developed by Hwang and Yoon [48] defines a closeness coefficient to determine ranking order of all alternatives by calculating distances to both the positive ideal solution and negative ideal solution simultaneously. Algorithm of the TOPSIS method for Pareto solution set of multi-response problems are given as follows: Step 1: Construct decision matrix as, , , . Step 2: Obtain weight vector of the criteria as, , by Entropy method.

Sigma J Eng Nat Sci, Vol. 43, No. 1, pp. 133−147, February, 2025

Step 2: Obtain weight vector of the criteria as, , by Entropy method. Step 3: Determine normalized decision matrix, , , , by using (12)

In Equation (5), B and C denote benefit criteria and cost criteria, respectively. Step 4: Construct weighted normalized decision matrix values of as follows:

Step 4: Construct weighted normalized decision matrix as follows: values of

Step 5: Determine positive ideal solution and negative ideal solution as follows:

Step 6: Compute relative importance values alternative by using:

Step 7: Calculate performance index of each alternative by using as follows: Step 8: Rank the alternatives according to descending [51]. order of

Step 7: Calculate closeness coefficients of each alternatives as follows: (11) Step 8: Rank the alternatives according to descending order of [49,50]. Complex Proportional Assessment (COPRAS) The COPRAS method, developed by Zavadskas et al. [51], has noticeable advantages among the other methods. Calculation time is very short and it can be easily implemented to any program source code. Algorithm of the COPRAS method for Pareto solution set of multi-response problems are given as follows: Step 1: Construct decision matrix as, , , .

Weighted Aggregated Sum Product Assessment (WASPAS) The WASPAS method combines weighted sum model (WSM) and weighted product model (WPM). Due to its mathematical simplicity and capability, WASPAS provides more accurate results as compared to WSM and WPM methods. Algorithm of the WASPAS method for Pareto solution set of multi-response problems are given as follows: , Step 1: Construct decision matrix as, , . Step 2: Obtain weight vector of the criteria as, , by Entropy method. Step 3: Determine normalized decision matrix, , , , by using

Sigma J Eng Nat Sci, Vol. 43, No. 1, pp. 133−147, February, 2025

Step 4: Calculate total relative importance of each alternative based on WSM method, as follows: (23)

(18) Step 5: Calculate total relative importance of each alternative based on WPM method, as follows:

Step 6: Compute joint generalized criterion of weighted aggregation of additive and multiplicative methods for each alternative by: (20) Step 7: Rank the alternatives according to decreasing values of [52].

(25) and Step 6: Calculate normalized each alternative as follows, respectively:

(26) Evaluation Based on Distance from Average Solution (EDAS) The EDAS method introduced by Keshavarz-Ghorabaee et al. [53] uses average solution for appraising the alternatives. Two measures which called positive distance from average (PDA) and negative distance from average(NDA) are considered for the appraisal which are calculated according to type of criteria, benefit or cost. Algorithm of the EDAS method for Pareto solution set of multi-response problems are given as follows: Step 1: Construct decision matrix as, , , . Step 2: Obtain weight vector of the criteria as, , by Entropy method. Step 3: Determine average values of the criteria, by, (21) Step 4: Compute positive distance from average (PDA) and negative distance from average (NDA) of each alternative with respect to each criteria as,

(27) Step 7: Calculate the assessment score native as follows:

(28) Step 8: Rank the alternatives according to decreasing [53]. values of Combinative Distance-based Assessment (CODAS) The CODAS method was introduced by KeshavarzGhorabaee et al. [54] as an efficient method to solve MCDM problems. It is used to obtain the desirability of attributes by a ranking which is estimated by two measures, Euclidean distance and Taxicab distance, from a negative ideal solution. Algorithm of the CODAS method for Pareto solution set of multi-response problems are given as follows: , Step 1: Construct decision matrix as, , .

Sigma J Eng Nat Sci, Vol. 43, No. 1, pp. 133−147, February, 2025

Step 2: Obtain weight vector of the criteria as, , by Entropy method. Step 3: Determine normalized decision matrix, , , , by using

Step 4: Construct weighted normalized decision matrix as follows: values of

Grey Relational Analysis (GRA) Grey system theory developed by Deng [55] deals with poor, incomplete, and uncertain information. The GRA method based on grey system theory is appropriate for solving MCDM problems with complicated interrelationships between multiple factors and variables. Algorithm of the GRA method for Pareto solution set of multi-response problems are given as follows: Step 1: Construct decision matrix as, , , . Step 2: Obtain weight vector of the criteria as, , by Entropy method. Step 3: Determine normalized decision matrix, , by using

Step 6: Compute Euclidean distance (Ei) and Taxicab distance (Ti) of each alternative from the negative ideal solution by using, (32)

(38) In Equation (38), Step 5: Compute grey relational coefficient matrix as, by following: (39)

Step 7: Define relative assessment matrix, as follows: (34) indicates a threshold function In Equation (34), according to the threshold parameter set by decision maker, defined as

Step 7: Rank the alternatives according to descending [42,55,56]. order of

Multi-Attributive Border Approximation Area Comparison(MABAC) The MABAC method, developed by Pamucar and Ćirović [57], is based on distance of the criterion function of each alternative from the border approximation area. Algorithm of the MABAC method for Pareto solution set of multi-response problems are given as follows: Step 1: Construct decision matrix as, , , .

(36) Step 9: Rank the alternatives according to decreasing . The alternative with the highest values of assessment score is the most desirable alternative [54].

Sigma J Eng Nat Sci, Vol. 43, No. 1, pp. 133−147, February, 2025

Step 2: Obtain weight vector of the criteria as, , by Entropy method. Step 3: Determine normalized decision matrix, , by using

Modeling stage was achieved by using the SUR method since responses are correlated for each data set. Pareto solution sets are obtained by using the NSGA-II and MODE in the optimization stage. In decision making stage, compromise solutions are determined via seven MCDM methods. The MAE and RMSE metrics are calculated to evaluate these compromise solutions as follows: (47)

Step 4: Construct weighted normalized decision matrix values of as follows: (42) Step 5: Compute Border Approximation Area for each criterion as, by: (43) Step 6: Calculate distance of the alternative from border approximation area for matrix elements (Q) as follows: (44) Step 7: Determine belonging of each alternative approximation area ( , or ) as follows:

In Equation (45), indicates border approximation area, indicates upper approximation area which contains the ideal alternative ( ) while indicates lower approximation area which contains the anti-ideal alternative( ). Step 8: Compute final values of the criterion function for each alternative which is based on the sum of the distance of the alternatives from the border approximation areas as: (46) Step 9: Rank the alternatives according to decreasing values of . The alternative with the highest assessment score is the most desirable alternative [57].

Numerical Examples

Three real data sets related to multiple-response experiments are given to illustrate for the performance comparison of the NSGA-II and MODE. These are (i) engineering data set, (ii) chemistry data set, and (iii) food data set.

(48) R 3.6.1 and MATLAB R2021b were used in all steps and computations. Example 1 (Engineering data set) Engineering data set which consist of three input and two response variables is shown in Table 2 [5]. The input variables are number of wings (X1), wing height (X2) and wing thickness (X3). The response variables are friction coefficient multiplied by the Nusselt number (Y1) and the Reynolds number (Y2), wanted to be maximized and minimized, respectively. The experimental data set is composed with Box-Behnken design. It should be noted that input variables are given by coded values in Table 2. The coded levels of the input variables and corresponding real values are presented in [5].

Sigma J Eng Nat Sci, Vol. 43, No. 1, pp. 133−147, February, 2025

The normality assumption for the responses was checked. It was seen that the responses were normally distributed (p-value of Y1= 0.52 >0.05, p-value of Y2= 0.57>0.05). Pearson correlation coefficient between the responses is calculated as 0.98 which is statistically significant. Due to relationship between the responses, predicted response models were obtained by SUR method. In order to obtain predicted response model with the SUR, variance-covariance matrix estimation is calculated from residuals of OLS response estimations. Predicted response models are written as

Nsga-Ii MODE

According to ANOVA results belonging to (p-value = 0.0001<0.05) and (p-value = 0.0001<0.05), models obtained by the SUR method were found to be statistically significant. Predicted response models are considered as objective functions, f1 and f2. Ideal solution vector is calculated as by optimizing objective functions f1 and f2, individually. The MOO problem can be given as follows:

The MOO is achieved through the NSGA-II and the MODE. The tuning parameters of the NSGA-II are and the tuning parameters of the . Pareto MODE are solutions are obtained by the NSGA-II and the MODE as shown in Figure 3. Size of Pareto solution set is 53 which is number of Rank-1 non-dominated solutions. Entropy method was applied to obtain weights of the response variables Y1 and Y2 through experimental response values given in Table 2. The weight vector is calcu. Each MCDM method lated as was applied to Pareto solutions to determine compromise solutions. The closeness of the compromise solution to the

Table 3. Compromise solutions of NSGA-II and MODE with metrics for engineering data set MCDM

Nsga-Ii

Figure 3. Pareto solutions obtained with the NSGA-II and the MODE for engineering data set.

Mabac

Sigma J Eng Nat Sci, Vol. 43, No. 1, pp. 133−147, February, 2025

ideal solution is calculated by the MAE and RMSE metrics as given in Table 3. It was seen that compromise solution with the smallest MAE and RMSE values belongs to MODE. This compromise solution was chosen by TOPSIS method. These results show that the MODE can find a better compromise solution closer to ideal solution than NSGA-II. Example 2 (Chemistry data set) Chemistry data set which consist of three input and two response variables is shown in Table 4 [12]. The input variables are temperature (X1), pressure (X2) and co-solvent (X3). The response variables are extraction efficiency of carotenoids (Y1) and the extraction efficiency of chlorophylls (Y2) which are wanted to be maximized. The experimental data set is composed with the Box-Behnken design. Coded values and real values for chemistry data set are shown in [13]. The normality assumption for the responses was checked. It was seen that the responses were normally distributed (p-value of Y1= 0.55 > 0.05, p-value of Y2= 0.67> 0.05). Pearson correlation coefficient between the response variables is determined as 0.56 and it is found to be statistically significant. Due to relationship between the responses, predicted response models were used by SUR method. In order to achieve predicted response model with the SUR, variance-covariance matrix estimation is calculated from the residuals of the OLS response estimations. The predicted response models can be written as (52)

According to ANOVA results belonging to (p-value = 0.009 < 0.05) and (p-value = 0.002 < 0.05), models obtained by the SUR method were found to be statistically significant. Ideal solution vector is calculated as . MOO problem is presented in Equation (54) as follows:

(54) The MOO is achieved through the NSGA-II and the MODE. The tuning parameters of the NSGA-II are and the tuning parameters of the MODE are . Pareto solutions obtained with the NSGA-II and MODE for chemistry data set are illustrated in Figure 4. Entropy method was applied to obtain weights of the response variables Y1 and Y2 through experimental response values given in Table 4. The weight vector is calculated as . Each MCDM method was applied to Pareto solutions to determine compromise solutions. The closeness of the compromise solution to the ideal solution is calculated by the MAE and RMSE metrics as given in Table 5. It was seen that compromise solution with the smallest MAE and RMSE values belongs to the MODE. This compromise solution was chosen by COPRAS method. These results show that the MODE can find a better compromise solution closer to ideal solution than the NSGA-II.

Nsga-Ii MODE

Figure 4. Pareto solutions obtained with the NSGA-II and MODE for chemistry data set.

Sigma J Eng Nat Sci, Vol. 43, No. 1, pp. 133−147, February, 2025

Table 5. Compromise solutions of NSGA-II and MODE with metrics for chemistry data set MCDM

Mabac

Example 3 (Food data set) Food data set which consist of three input variables and two response variables for 17 observations is shown in Table 6 [58]. The input variables are temperature (X1), washing time (X2) and washing rate (X3). The response variables are thiobarbituric acid number (Y1) and percent cooking loss (Y2) which are wanted to be maximized. The Table 6. Food data set Input variables

experimental data set is composed with Central Composite Design (CCD). Real values and coded values of input variables are shown in [58]. The normality assumption for the responses was checked. It was seen that one of the responses was normally distributed (p-value of Y1= 0.01 < 0.05, p-value of Y2= 0.41> 0.05). However, it was assumed that the Y1 has normally distributed. Then, Pearson correlation between the response variables is determined as 0.68 and it is found to be statistically significant. The predicted response models obtained by the SUR method can be written as

(56) According to ANOVA results for (p-value = 0.001< 0.05) and (p-value = 0.001< 0.05), models obtained by the SUR method were found to be statistically significant. Ideal . solution vector is calculated as MOO problem is presented in Equation (57) as follows: (57) The MOO is achieved through the NSGA-II and the MODE. The tuning parameters of the NSGA-II are and the tuning parameters of the . Pareto MODE are

Sigma J Eng Nat Sci, Vol. 43, No. 1, pp. 133−147, February, 2025

results show that the MODE can find a better compromise solution closer to ideal solution than the NSGA-II.

Conclusion

Figure 5. Pareto solutions obtained with the NSGA-II and MODE for food data set.

In this study, analysis of multi-response experimental data was achieved in three basic stages, modeling, optimization and decision making. The SUR method was used for modeling of the responses since the responses were linearly correlated. The predicted response models were considered as objective functions. Ideal solution vector is provided by optimizing objective functions individually. Multi-response problem was handled as a MOO problem considering the objective functions simultaneously. The NSGA-II and the MODE, two commonly used MOM methods, were applied to obtain Pareto solution sets which were the solution of the MOO problem.

Table 7. Compromise solutions for NSGA-II and MODE with metrics for food data set MCDM

Mabac

solutions obtained with the NSGA-II and the MODE for chemistry data set are illustrated in Figure 5. Entropy method was applied to obtain weights of the response variables Y1 and Y2 through experimental response values given in Table 4. The weight vector is calcu. Each MCDM method lated as was applied to Pareto solutions to determine compromise solutions. The closeness of the compromise solution to the ideal solution is calculated by MAE and RMSE metrics as given in Table 7. It was seen that compromise solution with the smallest MAE and RMSE values belongs to MODE. This compromise solution was chosen by EDAS method. These

In multi response studies each response may have different importance. Thus, in this study, Entropy method was applied to obtain weights of the response variables objectively over experimental results for each dataset. Each MCDM method, considering weights, was applied to the Pareto solutions to determine a compromise solution. The main aim of this study is to compare the performance of the NSGA-II and MODE. For this purpose, firstly, the MCDM methods applied to the Pareto sets to define a compromise solution. Secondly, closeness of the compromise solution and the ideal solution vectors were calculated with respect to MAE and RMSE metrics. Three data sets related to engineering, chemistry and food fields were considered for

Sigma J Eng Nat Sci, Vol. 43, No. 1, pp. 133−147, February, 2025

application. Calculation results show that the compromise solution of the MODE is closer to the ideal solution than the compromise solution of the NSGA-II. The MODE has the smallest MAE and RMSE values according to the at least one of the MCDM methods, for instance TOPSIS for engineering data set, COPRAS for chemistry data set, EDAS for food data set. Accordingly, the MODE performs better than NSGA-II to find Pareto solution set that contains a compromise solution closer to the ideal solution.

Acknowledgement

This study is supported by Ankara University Scientific Research Projects (Project Number: 21L0430015).

References

1. Zellner A. An efficient method of estimating seem- Uygulamaları. Ankara: Seçkin Yayıncılık; 2021. ingly unrelated regressions and tests of aggregation [Turkish] bias. J Am Stat Assoc 1962;57:348–368. [CrossRef] [16] Karaboğa D. Yapay Zeka Optimizasyon
2. Liu A. Efficient estimation of two seemingly Algoritmaları. Ankara: Nobel Yayınları; 2020. unrelated regression equations. J Multivar Anal [Turkish] 2002;82:445–456. [CrossRef] [17] Esfe MH, Hajmohammad H, Moradi R, Arani AAA.
3. Bera S, Barman G, Mukherjee I. Effect of seemingly Multi-objective optimization of cost and thermal unrelated regression-based modeling approach on performance of double walled carbon nanotubes/ solution quality for correlated multiple response water nanofluids by NSGA-II using response sur- optimization problems. Proceedings of the 2011 face method. Appl Therm Eng 2017;112:1648–1657. IEEE IEEM. 2011. p. 1490–1494. [CrossRef] [CrossRef]
4. Costa N, Lourenço J, Pereira ZL. Responses model- ing and optimization criteria impact on the optimi- Comparison of NSGA-II,MAOLA and MODA for zation of multiple quaility characteristics. Comput multi-objective optimization of micro-machining Ind Eng 2012;62:927–935. [CrossRef] processes. Materials (Basel) 2021;14:5109. [CrossRef]
5. Rout SK, Hussein AK, Mohanti CP. Multi-objective optimization of a three-dimensional internally SG, Jeba E. Cordinated controller design of grid finned tube based on response surface methodolo- connected DFIG based wind turbine using response gy(RSM). Therm Eng 2015;1:131–145. [CrossRef] surface methodology and NSGA II. Sustain Energy
6. Shah HK, Montgomery DC, Carlyle WM. Response Technol Assess 2014;8:120–130. [CrossRef] surface modeling and optimization in multiresponse [20] Thirumalai R, Seenivasan M, Panneerselvam K. experiments using seemingly unrelated regression. Experimental investigation and multi response opti- Qual Eng 2004;16:387–397. [CrossRef] mization of turning process parameters for inconel
7. Akçay H, Anagün AS. Multiresponse optimiza- 718 using TOPSIS approach. Mater Today Proc tion application on a manufacturing factory. Math 2021;45:467–472. [CrossRef] Comput Appl 2013;18:531–538. [CrossRef] [21] Torabi SHR, Alibabaei S, Bonab BBM, Sadeghi H,
8. Shivakoti I, Kalita K, Kibria G, Sharma A, Pradhan Faraji G. Design and optimization of turbine blade BB, Ghadai RK. Parametric analysis and multi preform forging using RSM and NSGA II. J Intell response optimization of laser surface texturing Manuf 2017;28:1409–1419. [CrossRef] of titanium super alloy. J Braz Soc Mech Sci Eng [22] Babu B V, Anbarasu B. Multi-objective differential 2021;43:400. [CrossRef] evolution (MODE): an evolutionary algorithm for
9. Tunçel S. Çok yanıtlı deneysel verilerin görünüşte multi-objective optimization problems (moops). ilişkisiz regresyon analizi ile modellenmesi ve opti- Proceedings of International Symposium and 58th mal değişken değerlerinin belirlenmesi (Yüksek Annual Session of IIChE, 2005. lisans tezi). Ankara: Ankara Üniversitesi; 2022. [23] Souza MN, Silva MA, Machado AR, Lobato FS.
10. Türkşen Ö. Optimization and decision making Treatment of multi-response surface applied to stages for multiple responses: an application of machinability of stainess steel using multi-objective NSGA-II and FCM clustering algorithm. Gazi Univ optimization differential evolution. 21. Brazilian J Sci 2015;28:321–330. Congress of Mechanical Engineering, 2011. 146 Sigma J Eng Nat Sci, Vol. 43, No. 1, pp. 133−147, February, 2025
11. Basu M.: Economic environmental dispatch using multi-objective differential evolution. Appl Soft optimization of EDM process parameters for bio- Comput 2011;11:2845–2853. [CrossRef] degradable AZ31 magnesium alloy using TOPSIS
12. Gaitonde VN, Manjaiah M, Maradi S, Karnik SR, and grey relational analysis. Sādhanā 2022;47:136. Petkar PM, Davim JP. Multiresponse optimization [CrossRef] in wire electric discharge machining (WEDM) of [38] Divya C, Raju LS, Singaravel B. Application of HCHCR steel by integrating response surface meth- mcdm methods for process parameter optimiza- odology (RSM) with differential evolution(DE). tion in turning process-a review. Recent Trends in Comput Methods Eng 2017;199–221. [CrossRef] Mechanical Engineering. Singapore: Springer; 2021.
13. Monsef H, Naghashzadegen M, Jamali A, Farmani p. 199–207. [CrossRef] R. Comparison of evolutionary multi objective opti- [39] Bagal DK, Giri A, Pattanaik AK, Jeet S, Barua A, mization algorithms in optimum design of water Panda SN. MCDM optimization of characteristics distribution network. Ain Shams Eng 2019;10:103– in resistance spot welding for dissimilar meterials
14. Singh GK, Yadava V, Kumar R. Multiresponse opti- pled CoCoSo, EDAS and WASPAS method. Next mization of electro-discharge diamond face grind- Generation Meterials and Processing Technologies. ing process using robust design of experiments. Singapore: Springer; 2020. [CrossRef] Mater Manuf Process 2013;25:851–856. [CrossRef] [40] Abhilash P, Chakradhar D. Multi-response optimi-
15. Zhao W, Ma A, Ji J, Chen X, Yao T. Multiobjective zation of wire EDM of inconel 718 using a hybrid optimization of a double-side linear vernier PM entropy weighted GRA-TOPSIS method. Process motor using response surface method and differ- Integr Optim Sustain 2022;6:61-72. [CrossRef] ential evolution. IEEE Trans Ind 2020;67:80–90. [41] Singh A, Prapoorna Biligiri K, Venkatesh Sampath [CrossRef] P. Development of framework for ranking per-
16. Reddy VV. Turning process parameters opti- vious concrete pavement mixtures: application mization of al7075 hybrid mmc’s composite of multi-criteria decision-making methods. Int J using TOPSIS method. Sigma J Eng Nat Sci Pavement Eng 2022;24:2021406. [CrossRef] 2020;38:2043–2055. [42] Kuo Y, Yang T, Huang GW. The use of grey relational
17. Yalçın N, Uncu N. Applying EDAS as an applicable analysis in solving multiple attribute decision-mak- mcdm method for industrial robot selection. Sigma ing problems, Comput Ind Eng 2008;55:80–93. J Eng Nat Sci 2019;37:779–796. [CrossRef]
18. Bölükbaş U, Güneri AF. A fuzzy multi-criteria decision approach for measuring technology com- elitist multiobjective genetic algorithm: NSGA-II, petency performance of smes. Sigma J Eng Nat Sci IEEE Trans Evol Comput 2002;6:182–197. [CrossRef] 2017;8:31–40. [44] Storn R, Price K. Differential evolution- a simple and
19. Trung DD. A combination method for multi-criteria efficient adaptive scheme for global optimization decision making problem in turning process. Manuf over continuous spaces. J Glob Optim 1997;11:341– Rev. 2021;8:1–17. [CrossRef] 359. [CrossRef]
20. Pandiyan GK, Prabaharan T, James DJD, Sivalingam V. Machinability Analysis and Optimization cation. Bell Syst Tech J 1948;27:379–423. [CrossRef] of Electrical Discharge Machining in AA6061- [46] Zavadskas EK, Podvezko V. Integrated determina- T6/15wt.% SiC Composite by the Multi-criteria tion of objective criteria weights in MCDM. Int J Inf Decision-Making Approach. J Mater Eng Perform Technol Decis Mak 2016;15:267–283. [CrossRef] 2022;31:3741–3752. [CrossRef] [47] Gül Ş, Fırat M. Determination of priority regions for
21. Khan MM, Dey A. Hybrid MCDM approach for rehabilitation in water networks by multiple crite- examining the high-stress abrasive wear behaviour ria decision making methods. Sigma J Eng Nat Sci of in situ ZA-27/TiCp MMCs. Mater Chem Phys 2020;38:1481–1494. 2022;277:125319. [CrossRef] [48] Hwang CL, Yoon K. Multiple Attribute Decision
22. Patil SB, Patole TA, Jadhav RS, Suryawanshi SS, Making: Methods and Applications. New York: Raykar SJ. Complex proportional assessment Springer- Verlag; 1981. [CrossRef] (COPRAS) based multiple-criteria decision making [49] Chen C. Extensions of the TOPSIS for group deci- (MCDM) paradigm for hard turning process param- sion-making under fuzzy environment. Fuzzy Sets eters. Mater Today Proc 2022;59:835–840. [CrossRef] Syst. 2000;114:1–9. [CrossRef]
23. Rahman Z, Siddiquee AN, Khan AZ, Ahmad S. Multi-response optimization of FSP parameters on MCDM methods: a comparative analysis of VIKOR mechanical properties of surface composite, Mater and TOPSIS. Eur J Oper Res 2004;156:445–455. Today Proc 2022;62:5–8. [CrossRef] [CrossRef] Sigma J Eng Nat Sci, Vol. 43, No. 1, pp. 133−147, February, 2025 147
24. Zavadskas EK, Kaklauskas A, Sarka V. The new multi-criteria decision-making. Econ Comp and method of multi-criteria Complex Proportional Econ Cyb Stud Res 2016;50:25–44. Assessment (COPRAS) of Projects. Technol Econ [55] Deng J. Control problems of grey systems. Syst Dev Econ 1994;1:131-139. Control Lett 1982;5:288-294. [CrossRef]
25. Chakraborty S, Zavadskas EK, Antucheviciene J. Applications of WASPAS method as a multi-criteria tional analysis to optimize turning operations with decision-making tool. Econ Comput Econ Cyber multiple performance characteristics. Mater Manuf Studi Res 2015;49:5–22. Process 2004;19:209–220. [CrossRef]
26. Keshavarz-Ghorabaee M, Zavadskas EK, Olfat L, Turkis Z. Multi-criteria ınventory classification handling resources in logistics centers using Multi- using a new method of evaluation based on dis- Attributive Border Approximation area Comparison tance from average solution (EDAS). Informatica (MABAC). Expert Syst Appl 2015;42:3016–3028. 2015;26:435–451. [CrossRef] [CrossRef]
27. Keshavarz-Ghorabaee M, Zavadskas Turkis Z. Antucheviciene, J. A new combinative dis- and Analyses. New York: Marcel Dekker; 1996. p. tance-based assessment (CODAS) method for 510.

Share and Cite