ANN-based Lagrange optimization for RC circular columns having multiobjective functions

ABSTRACT Structural engineers must encounter multiobjective optimization (MOO) problems in designs including a design of reinforced concrete (RC) columns, where several design objectives must be satisfied to meet contractors’ interests and produce a sustainable design. MOO applications in structural engineering practice are uncommon despite their high demand. Multiobjective population methods have been mainly investigated. However, previous studies have mostly neglected the stopping criteria and convergence of these algorithms. This study proposes a novel gradient-based algorithm – ANN-based Lagrange optimization and its implementation in designs of circular reinforced concrete columns where three objectives, such as cost, CO2 emissions, and column weight, are minimized simultaneously. This study integrates each single-objective function extracted from an artificial neural network (ANN) into a global Lagrange function, a unified function of objectives (UFO), using the trade-off fractions. The Lagrange multiplier method is adopted to deal with constrained conditions by using Newton–Raphson iteration to solve Karush-Kuhn-Tucker (KKT) conditions. The proposed algorithm brought in a set of optimal results capturing multiple objectives, known as a Pareto frontier, which are compared with those obtained using the Nondominated Sorting Genetic Algorithm – II (NSGA-II), showing that two algorithms calibrated to each other. Overall, the ANN-based Lagrange optimization algorithm exhibited better convergence than NSGA-II. GRAPHICAL ABSTRACT


Introduction
Multiobjective optimization (MOO) is a good tool for optimizing engineering design. MOO concepts and methods were conducted and summarized by Marler and Arora (2004), categorizing optimization algorithms into three groups, such as methods with a prior articulation of preferences, in which decision-makers preassigned relative importance among objective functions; methods with a posterior articulation of preference, in which the final solution is selected among a set of mathematical solutions generated from optimization algorithms; and methods that do not require any articulation of preferences. Overall, Marler stated that the second method (with a posterior articulation of preferences) was shown to be less efficient than methods with a priori articulation in terms of CPU time. Some multiobjective genetic algorithms have been widely developed in the last two decades, such as Nondominated Sorting Genetic Algorithm-II (NSGA-II) by Deb et al. (2002), Particle Swarm Optimization by Mohd Zain et al. (2018), and many others. These methods were referred to as multiobjective evolutionary algorithms (MOEAs). Zhou et al. (2011) performed a survey of these state-of-the-art MOEAs and summarized their applications to real-world problems. Applications of MOO algorithms, including population and gradient-based algorithms, are not commonly studied in the structural engineering field although this problem is extremely common. This is because singleobjective functions in the optimization design of structures are intricate due to various design parameters required to satisfy multiple constraints, such as design specifications and architectural needs, contributing to the complexity of MOO design problems. Sunar and Kahraman (2001) conducted an early study by investigating the efficiency of different MOO methods in structural design. The conventional weighting method was stated to be easily implemented. MOO problems in structural design are mainly bi-and triobjective, in which CO 2 emission and estimated cost index are minimized to attain structural design sustainability. Yoon et al. (2018) analyzed the relationship between cost and CO 2 emissions in designs of reinforced concrete columns in 2018. Gholami, Fathi, and Baghestani (2021) have recently studied a biobjective optimization design of a composite sandwich panel, in which the panel weight and cost are objective functions. Another study based on MOEAs is the topology optimization of spatially truss structures by Nan, Bai, and Wu (2020) in 2020, in which structural weight and maximum displacement are optimized objectives. Some other MOEA applications in structural designs of trusses (Kaveh and Mahdavi, 2019), reinforced concrete (RC) structures (Afshari, Hare, and Tesfamariam, 2019), conceptual designs (Brown et al., 2015;Brown 2016), and vibration suppression of building structures (Zheng and Hu, 2018) were studied. Liu and Reynolds (2016) investigated gradient-based MOO with applications to waterflooding optimization. Katrutsa et al. (2020) investigated the MOO gradient descent algorithm. Giacomini, Désidéri, and Duvigneau (2014) compared multiobjective gradient-based methods for structural shape optimization. Kayabekir et al. (2020) investigated eco-friendly design of RC retaining walls with harmony search applications. Previous studies presented applications of MOO in the structural engineering field. MOO algorithms must include good convergence criteria and diverse searching areas to obtain a converged and widespread set of optimal results. However, these aspects are not thoroughly investigated in previous studies and applications of MOO algorithms in structural engineering.
ANNs and machine learning (ML) have gained interests from researchers in the structural engineering field. Based on their outstanding learning features, ANNs and ML are favorable to giving predictions of structural behaviors by generating objective functions that are challenging when being analytically derived. For example, neural networks were investigated to predict concrete strength by Nguyen et al. (2019), Chou and Pham (2013), Chiew et al. (2017), Bui et al. (2018), andNaderpour, Rafiean, andFakharian (2018). ANN-based models for estimation of capacity of steel beams were investigated by Ferreira et al. (2022), Nguyen, Ly, and, Hosseinpour, Sharifi, and Sharifi (2020), and Sharifi et al. (2020). Some ML-based algorithms were studied to detect damage of high-rise building structures by Rafiei and Adeli (2017) and Oh et al. (2017). ; Yucel et al. (2019);Yucel, Nigdeli, and Bekdaş (2021) have investigated ANNs and ML to optimize tuned mass damper to lessen the effects of lateral forces such as winds and earthquake on structures. Design of structural systems via machine learning was estimated by Bekdaş, Yücel, and Nigdeli (2021). ANNs were also utilized to optimize design variables in civil engineering. Yucel et al. (2018) proposed ANN-based models for optimum design of tubular columns. Chen et al. (2018) cooperated ANN and Particle Swarm Optimization to obtain a hybrid model predicting shear strength of squat RC walls. Optimum designs of concrete structures such as carbon fiber-reinforced polymer beams and retaining walls were investigated by Yucel et al. (2020) and . Nguyen-Ngoc et al. (2021) and Ngoc-Nguyen et al. (2022) employed hybrid models such as ANN-GA (Genetic Algorithm) and ANN-FEM optimizing damaged detections in the truss bridge.
The present study proposed a novel MOO algorithm and its application in the design of RC circular columns. This algorithm is originally developed by  and  for optimization of single objectives such as cost minimization in design of RC columns and RC beams. The proposed algorithm implemented characteristics of artificial neural networks (ANNs) to derive objective functions, which demonstrated a good intercorrelated relationship between design parameters after training (Hong 2020). The single objective functions were globalized into a unified function of objective (UFO) by the weighting method. A set of MOO results is a Pareto frontier or a Pareto set. These results are solutions of Karush-Kuhn-Tucker (KKT) conditions from the Lagrange function, obtained using the Newton-Raphson iteration method. A design example of RC circular column was investigated in this study, in which three objectives, cost, CO 2 emissions, and column weight, are simultaneously minimized. A Pareto frontier offered by the proposed ANN-based Lagrange optimization algorithm is compared with that obtained by the NSGA-II algorithm. Convergence criteria and diversity of searching were discussed in this study. In addition, optimal results from the ANN-based Lagrange optimization algorithm display a particular trade-off among objectives, which is beneficial for engineers and decision-makers. Figure 1 illustrates the section of the RC circular column. Design parameters presented in this study include the column dimension (D), rebar ratio (ρ s ), material properties of concrete (f 0 c ) and steel (f y ), factored axial loads (P u ), and factored bending moments (M u ), based on conventional designs. Particularly, D and ρ s are selected as variables, whereas f 0 c ; f y ; P u ; and M u are fixed values as these four parameters are normally preassigned in a design problem. A design range of column dimension D is from 400 mm to 2000 mm based on practical demands, whereas that of rebar ratio ρ s is 0.01to 0.08, which is specified by design code . RC column design should satisfy strength requirements specified in ACI 318-19, reflected by a safety factor, SF ¼ ϕP n =P u ¼ ϕM n =M u � 1:0. α e=h is selected in this study to construct an axial load-bending moment interaction diagram (P-M diagram) (Figure 1), such that its design range is within 0-π=2. Cost index, CO 2 emission, and column weight denoted as CI c ; CO 2 , and W c , respectively, are determined for RC columns. The cost index of the RC column and column weight designs is estimated based on the Korean unit prices indicated in Table 2, whereas the determination of CO 2 emissions followed Hong et al. (2010). Table 1 presents notations and nomenclatures of RC column design parameters.

MOO design of RC columns
Multiobjective optimization results in a design with a minimal column weight, which is advantageous from both architectural and assembly perspectives, while its cost index (CI c ) and estimated CO 2 emissions are minimized. This problem is described in Eq 1-3, in which f CI c x ð Þ; f CO 2 x ð Þ; and f W c x ð Þ are single-objective optimization functions of cost, CO 2 , and column weight, respectively. x is an input vector containing design parameters of RC columns presented in Section 2.1. RC column's designs in practice are associated with constrained conditions, which are presented as equality functions, c m x ð Þ = 0, and inequality functions, v l x ð Þ � 0. These conditions represent restrictions of a column design, such as a limitation of rebar ratio (specified in design code), architectural restriction for column dimension, preassigned material properties, and factored loads,

Establishment of ANN
An ANN is developed to map five input parameters to six output parameters. Input and output parameters are selected to perform a forward design of RC columns under axial loads and flexural moments. Input para- , denote the column dimension, rebar ratio, material properties (concrete and steel), factored axial load, and factored moment.
selected to represent safety factors, rebar strain, cost index, CO 2 emission, and column weight, in which the output parameter safety factor, A hundred thousand data sets are generated. Notably, α e=h is neglected in the training in this study. This is because this parameter is only required to construct an interaction P-M diagram of RC columns, which can be determined in design after training using P u ; M u ; and SF. Therefore, α e=h is requisite for data generation, but its training is trivial. The parallel training method (PTM) is then used to train ANN using 5 and 10 layers with 20, 50, and 80 neurons in each hidden layer. Figure 2 illustrates the topology of neural networks used in this study. Table 3 presents training results and training accuracies.

Unified function of objectives (UFO)
UFO shown in Eq. 4 is derived based on separate single- shown in Eq. 1 using weight vector w ¼ w CI c ; w CO 2 ; w W c f g T to optimize multiobjective functions. All weights are nonnegative and summed to one. Each weight varies in a range of 0-1 to generate a Pareto frontier that contains nondominated design points. Cost, CO 2 emissions, and column weight are the objectives, they are calculated in different units, and hence, all objectives are normalized to be unit-free before being applied into UFO. Eq. 5 presents the normalization of each objective based on the maximum and minimum of a single-objective function. The maximum and minimum of a single-objective function are obtained by an ANN-based Lagrange optimization, (4)

Description of optimization
Design optimizing UFO of SRC columns is a multiple constrained problem, such that solving for saddle points optimizing multiple objective functions is  difficult. The Lagrange multipliers are used to transfer constrained problems into nonboundary problems to optimize multiple objective functions. Figure 3 shows a MOO based on UFO proposed in this study, which is described in five steps as follows.
Step 1: ANNs are established based on three substeps, which are described as follows.
Step 1.2: Generation of large structural data sets.
Step 1.3: Derivation of single-objective functions Step 2: Constrained conditions of design problems are developed based on equality equations {c i x ð Þ ¼ 0} shown in Eq. 2 and inequality equations fv i x ð Þ � 0} shown in Eq. 3.
Step 3: This step maximizes and minimizes each objective function to define their boundaries for the normalizations discussed in Step 4. The Lagrange functions of each objective function (CI b , CO 2 , and W b Þ of SRC columns shown in Eq. 6a-6c are separately minimized and maximized as presented in Table 5, satisfying equalities and inequalities shown in Table 4. The singleobjective optimization Lagrange function is formulated in Eq 6a-6c. Lagrange functions are solved following firstorder necessary conditions (KKT conditions) based on the Newton-Raphson iteration. Eq. 6a-6c are optimized to yield CI max c ; CI min c ; CO max 2 ; CO min 2 ; W max c ; gtandW min c for each single-objective function, Step 4    L UFO ðx; λ c ; λ v Þ ¼ UFOðxÞ À λ T c cðxÞ À λ T v vðxÞ: (7)

Implementation of Newton-Raphson iteration
The stationary points of the Lagrange function shown in Eq. 6a-c and 7 are saddle points is implemented if this criterion for the first iteration is not satisfied. The convergence criterion is now checked with The same calculation is repeated until convergence is satisfied (L x; λ c ; λ v ð Þ � 1e À 15 ), followed by Eq. 8 and 9 for the k th iteration. Another terminating criterion is defined by a maximum number of iterations (as 50 iterations in this study) to save computation effort. Notably, the Newton-Raphson method depends on initial vectors, and hence, the proposed ANN-based Lagrange optimization adopts multiple initial vectors to ensure that saddle points are captured. A number of initial trial vectors are determined as 5 n , where n is the number of variables in the design problem. In RC column designs, an input vector x ¼ fD; ρ s ; f 0 c ; f y ; P u ; M u g T contains six parameters, four of which are typically predefined in design problems: f 0 c ; f y ; P u and M u . Thus, only two parameters, D and ρ s , vary, resulting in n ¼ 2 and 25 initial trial vectors. The Newton-Raphson algorithm to solve KKT conditions of Lagrange functions utilized in Steps 3 and 5 of Section 3.3.1 of the optimization process is summarized in Figure 4,

MOO design of RC columns based on NSGA-II
NSGA-II developed by K. Deb et al. (Deb et al. 2002) is a popular algorithm of MOO problems. This algorithm is part of the MOO's posterior articulation of preference methods, providing a set of multiobjective design alternatives (Pareto frontier). The optimum set provides engineers and decision-makers with a comprehensive evaluation of a design project. A final design option is selected among the optimal set. An algorithm following NSGA-II to generate a Pareto set of three-objective optimal design in this study ( Figure 5) is described by the following steps: Step 1: Design parameters are selected as D; ρ s ; f 0 c ; f y ; P u ; M u ; SF; ε s ; CI c ; CO 2 ; W c ; , and α e=h described in Section 2.1 and Table 1. Ranges of these parameters, variables, and preassigned parameters are defined based on constrained conditions of design problems. D and ρ s are selected as variables, whereas four parameters, f 0 c ; f y ; P u ; , and M u are fixed.
Step 2: Initialization. This step is performed to generate the first population set, involving two substeps as follows.
Step 2.1: Randomly generating a first-generation, assigned as first-parent P 0 with a size of N populations.
Step 3: Producing the first offspring generation, Q 0 , with a size of N populations from its parent P 0 by natural selection, crossover, and mutation. This study implemented tournament selection and simulated binary crossover with a 90% probability and polynomial mutation to generate an offspring from its parent with a 10% probability.
Step 4: Formulation of a combined set of population, R (size of 2 N populations), by merging parent, P (size of N), and offspring, Q (size of N).
Step 5: Ranking populations of combined set R based on the nondomination level (F R 1 ; F R 2 ; . . . ).
Step 6.1: F R 1 ; F R 2 ; . . . ; are added to P tþ1 until its size reaches N populations.
Step 6.2: F R last , the last nondominated set, can be added to P tþ1 . Sorting F R last populations by the crowding distance of each population. Choose the best solutions until N populations are satisfied in P tþ1 .
Step 7: The termination criterion is determined as the first domination level (F R 1 ) population size reaches N populations (defined as Pareto frontier). If this stopping criterion is not satisfied, regenerate offspring Q (following a similar procedure in Steps 2-3) and repeat Steps 3-5. A maximum number of iterations are defined for convergence as 10 generations, producing a Pareto frontier having N populations continuously.

Design scenario
A three-objective optimization example is investigated on RC circular column design, where the cost index (CI c ), CO 2 emission, and column weight (W c ) are minimized simultaneously. The design of RC circular columns is based on the configuration shown in Figure 1, with design parameters presented in Table 1. This example considers RC column design under a single load combination of factored axial load, P u ¼ 1500kN, factored moment, j ¼ 1; 2; :::; 10 indicating 10 hidden layers (10a) 40MPa and f y ¼ 500MPa, which are governed by the strength requirement,

Optimization by the ANN-based Lagrange algorithm
Four parameters, f 0 c ; f y ; P u ; and M u are preassigned and regarded as equality constraints in the Lagrange function. Inequality constraints are established based on a restriction of column dimension (400mm � D � 2000mm), rebar ratio (0:01 � ρ s � 0:08), and safety factor (SF � 1:0). These constrained conditions are developed into equations as shown in Table 4. The Lagrange function is obtained as described in Eq. 6 to illustrate Step 2 (Section 3.3) to optimize the threeobjective function, simultaneously, where f x ð Þ is a single-objective function as f CI c x ð Þ; f CO 2 x ð Þ; and f W c x ð Þ extracted from ANN ( Figure 2) after training, indicated in Eq. 10a-10c. c x ð Þ and v x ð Þ are equalities and inequalities described in Table 4. An objective function f x ð Þ subjected to these constraints is transformed to a nonboundary Lagrange function by Lagrange multipliers λ c ¼ � T for four equalities and five inequalities, respectively. The maxima and minima of each objective function are then separately determined (Step 3) using the ANN-based Lagrange. Consequently, the maxima and minima of each objective function presented in Table 5 are used to normalize the UFO function. The UFO equation is developed based on the maxima and minima of each objective, as shown in Eq. 11, illustrating Step 4. The weight of each objective (w CI c ; w CO 2 ; andw W c Þ varies from 0 to 1, whose sum is 1 (Eq. 12). The Lagrange function using the UFO equation (Eq. 7) and its KKT conditions are determined to optimize threeobjective functions, simultaneously, w CI c 2 ½0; 1�; w CO 2 2 ½0; 1�; w W c 2 ½0; 1� w CI c þ w CO 2 þ w W c ¼ 1 :

Comparison between ANN-based Lagrange optimization and NSGA-II
RC circular column design (Example 1) is optimized based on three-objective functions, based on the proposed algorithm -ANN-based Lagrange optimization, which are illustrated by red dots in Figure 6. The RC circular column design (Example 1) is compared with a set of optimal results provided by the NSGA-II algorithm (shown as black dots in Figure 6) and found that two Pareto sets yield. Particularly, Figure 6a illustrates that 200 fractions are performed using the ANN-based Lagrange optimization algorithm, whereas the NSGA-II algorithm generates 200 (11) j ¼ 1; 2; :::; 5 indicating 5 hidden layers j ¼ 1; 2; :::; 5 indicating 5 hidden layers populations. Similarly, 300 iterations and populations are used by two algorithms to provide optimal results in Figure 6b. Notably, a Pareto frontier by the ANN-based Lagrange algorithm (red dots) provides a smoother curve compared with the one provided by the NSGA-II algorithm in both figures (Figures 6a and 6b). In the case of 200 fractions performed using the ANN-based Lagrange algorithm, the Pareto frontier is noticeably sparse in some areas (shown as red circles in Figure 6a). The sparse areas are filled with data when 300 fractions are used. However, there is no observable change between the two figures regarding the NSGA-II algorithm where 200 and 300 populations are generated. In addition, a number of fractions of objective functions used in the ANN-based Lagrange algorithm are determined depending on a particular design example.

Verification of optimal results by ANN-based Lagrange MOO
ANN-based Lagrange optimization having multiple objectives using UFO functions is verified based on errors of an ANN and a structural mechanics-based nonlinear analytical software package, "AutoCC," which is developed using Matlab code based on strain compatibility. An input vector,    Figure 7 illustrates computational errors of five output parameters, SF; ε s ; CI c ; CO 2 ; and W c , which are less than 2% for all 300 fractions. Two million structural data sets are generated to verify the proposed method in this study, where ranges of design parameters are similar to constraint conditions presented in Table 4 ð400mm � D � 2000mm; 0:01 � ρ s � 0:08; f 0 c ¼ 40MPa; f y ¼ 500MPa; P u ¼ 1500kN; M u ¼ 2500kN:mÞ: From these two million data sets, 140,000 with safety factors from 1.0 to 1.5 (1:0 � SF � 1:5) are sorted. Figure 8 compares ANN-based Lagrange optimization with 140,000 data sorted from two million structural data sets having safety   factors from 1.0 to 1.5, where three objective functions are minimized at the same time. A Pareto frontier based on NSGA-II is also compared.

Discussion of the diversity of optimal results
Because NSGA-II belongs to multiobjective evolutionary algorithms and is a population method, it can provide several optimal outcomes, indicating spreadout populations in a searching area. However, a searching point being trapped at the local extremum is a common gradient-based optimization algorithm challenge. The proposed algorithm in this study -ANN-based Lagrange MOO algorithm -is gradientbased, which necessitates further investigation of its results diversity. Optimal results provided by the ANNbased Lagrange algorithm are solutions of KKT conditions obtained using the Newton-Raphson iteration method, which depends on its initial vectors. This study used 25 initial trial vectors, as presented in Section 3.3.2, to achieve a wide search. The fractions of each objective in optimal results obtained by ANNbased Lagrange optimization are plotted in Figure 9 to investigate the density of Pareto frontier by the proposed algorithm, demonstrating that a searching area is covered with values of three weights ranging from 0 to 1. This plot shows 300 fractions for an RC column design that can withstand a factored axial load of 1500kN and a factored moment of 2500kN:m. For example, point A 0:37; 0:31; 0:32 ð Þ and point B 0:03; 0:35; 0:62 ð Þ represent objective functions (CI c , CO 2 emissions, and W c ) optimized based on weight ratios of three objectives w CI c : w CO 2 : w W c as 0:37 : 0:31 : 0:32 and 0:03 : 0:35 : 0:62, respectively.

Data trend analysis of three objectives
The data trends of three objectives are investigated based on sorted data from two million structural data sets for fixed design parameter values (f 0 c ¼ 40MPa; f y ¼ 500MPa; P u ¼ 1500gtkN; M u ¼ 2500gtkN:m), and SF is filtered in a range of 1.0 to 1.01. Figure 10(a) shows a proportional trend between the cost index (CI c ) and CO 2 emissions of data sets projected to the YZ plane for a value of W c ranging from 13 to 14 kN/m. Similarly, the data trend between CI c and W c is obtained by projecting data to the XZ plane, with CO 2 ranging from 0.44 to 0.46 ton/m, as illustrated in Figure 10(b). The optimized design based on three-objective functions by ANN-based Lagrange and data sets on the lower boundary line (purple dotted line) exhibits a reverse trend between the cost (CI c ) and the weight (W c ). There are several data points shown in blue that do not follow this trend. For example, the cost and weight of Points A and B exhibit a proportional trend since SFs of the two points are not the same, leading to the difference in optimized diameters and rebar ratio so that the amount of concrete and steel of rebar used in these designs is different. Table 6 shows the design parameters of these two points, which indicates that point B presents a design with a higher SF value of 1.009 compared with 1.0005 of point A, resulting in a 0.06% increase in cost (CI c ) and 1% growth in weight (W c ), following the trend illustrated in Eq. (13) explaining the data trends of two objective functions when the third objective function is fixed. The CI c trend has been opposite when W c increases from 12.99 to 13.11 because SF is based on 1.0005 to 1.009, which is inconsistent. Overall, these two design points show that an increase in column dimension (more concrete) decreases the rebar ratio (less rebar) based on cost comparison. Figure 10(c) shows a similar case where CO 2 and weight (W c ) exhibit a reverse trend, illustrated by red lines in Figure 10. However, few blue points show an unusual data trend between CO 2 and the weight (W c ). Eq. 13 shows that CI c increases when increasing W c for fixed CO 2 which is in the case of Point A and Point B, for example, whereas an increase in W c leads to a reduction in CO 2 for fixed CI c . In contrast, CI c and W c of optimized designs have a reverse trend because they have different CO 2 , illustrated by red lines in Figure 10. Rebar decreases when the concrete weight increases. However, cost increment resulting from the increased concrete weight is more than the cost reduction resulting from decreased rebars. In other words, a decreasing rate of rebar has a less impact than an increasing rate of concrete when CO 2 emissions are constant. This increases the total cost.

Conclusions
This study proposed a novel gradient-based algorithm to solve MOO problems in RC circular column design. A hybrid network of ANNs based on Lagrange optimization is proposed to design RC columns sustaining multiple biaxial loads based on multiobjective functions. ANNs are shown to offer universally generalized functions of objective Table 6. Design parameters of Points A and B presented in Figure 10(b) (CO 2 is fixed at 0.46 ton/m). functions and constraints, which are wellbehaving. It is challenging to derive analytical objective functions and constraints for conventional Lagrange optimization of some complex design problems. This study will aid engineers in making final decisions by providing them with an overall assessment of a design project, in which three objectives (cost, CO 2 , and column weight) can be optimized simultaneously. Some conclusions drawn from this study for practical design applications are as follows.
(1) A unified function of objectives (UFO) is established with respect to three objective parameters, which are cost index (CI c ), CO 2 emissions, and column weights (W c ). These single-objective functions are then globalized and scalarized into UFO, allowing optimization algorithms to transform an optimization design problem with multiple constraints into a nonboundary problem by implementing the Lagrange multipliers.
(2) MOO based on the UFO function is solved under KKT conditions for Lagrange functions. A set of multiobjective optimal designs known as a Pareto frontier is established by varying weight fractions among objective functions (CI c , CO 2 emissions, and W c ). The proposed MOO algorithm maintains diversity in searching for optimizations. (3) The design accuracy of optimal results provided by ANN-based Lagrange optimization is witnessed less than 2% compared with structural mechanics-based software, confirming the proposed algorithm's reliability. (4) The optimal results provided by the ANN-based Lagrange algorithm, as a set of Pareto optimality, are compared with those presented by the NSGA-II algorithm. The Pareto frontier obtained by the ANN-based Lagrange MOO algorithm is calibrated to those offered by the NSGA-II algorithm. Results of ANN-based Lagrange optimization exhibited better convergence than those of NSGA-II. (5) A Pareto frontier based on the ANN-based Lagrange MOO algorithm shows the weight reactions of each objective and target. Engineers and decision-makers are informed of a particular trade-off between objectives to assess a design project holistically. The NSGA-II algorithm may not provide a specific trade-off of objectives. Thus, ANN-based Lagrange optimization provides a set of optimal designs that capture multiple targets, which engineers are more likely to use than the NSGA-II algorithm.
(6) This study evaluates trends among objectives (cost index, CO 2 emissions, and column weight), revealing a proportional trend of cost index and CO 2 emissions. In contrast, a reverse data trend is observed for cost index and column weight. However, a proportional trend of cost and weight is observed when CO 2 values are fixed in large data sets.