Multiobjective Design Optimization of Grillage Systems according to LRFD-AISC

Both the entire weight and joint displacements of grid structures are minimized at the same time in this study. Four multiobjective optimization algorithms, NSGAII, SPEAII, PESAII, and AbYSS are employed to perform computational procedures related to optimization processes. The design constraints related to serviceability and ultimate strength of grid structure are implemented from Load and Resistance Factor Design-American Institute of Steel Constructions (LRFD-AISC Ver.13). Hence, while the computational performances of these four optimization algorithms are compared using different combinations of optimizerrelated parameters, the various strengths of grid members are also evaluated. For this purpose, multiobjective optimization algorithms (MOAs) employed are applied to the design optimization of three application examples and achieved to generate various optimal designations using different combinations of optimizer-related parameters. According to assessment of these optimal designations considering various quality indicators, IGD, HV, and spread, AbYSSS shows a better performance comparatively to the other three proposed MOAs, NSGAII, SPEAII, and PESAII.


Introduction
The grillage systems utilized in different structures like bridge or ship decks, building floors and space buildings, and so forth, contain traverse and longitudinal beams, which are made of available steel profiles with different crosssections.The optimal selection of steel cross-sections from a discrete set of practically available steel profiles provides a big contribution to constructing cost of a grid structure.Therefore, either weight of grid structure or deflection of its joints is minimized according to certain design limitations prescribed by any code of practice, such as LRFD.During the design optimization of grillage systems, designer is frequently faced with a problem related to making a decision about determination of the most appropriate one between these two conflicting and commensurable objective functions.Although a displacement-related constraint is imposed as a (max span/300) according to the provisions of LRFD-AISC specification, the safety margin on displacement constraint is large when taking into account the grid structures with higher sensitivity against displacement, such as ship decks and floors of industrial buildings which bears special machines required an horizontally balanced position for a regular work.This task has been easily overcome in a way of introducing the concept of multi-objective optimization to the design applications of grid systems.
Preliminary multi-objective optimization techniques, which their fundamentals were constituted on mathematical programming principles, were dated back to the 1950s.However, mathematical programming techniques adjust the decision variables of continuous type according to gradient information computed by use of objective functions.Furthermore, they fail when search space is concave and discontinuous.In order to deal with these tasks, alternative optimization procedures, which mimic various natural events, for example, evolutionary systems, immune systems, social behaviors of ants, insects, and animals, have been developed.The most preferable ones are the evolutionarybased algorithms.These biological evolutionary models utilize characteristic properties of nature, for example, heredity, selection, and so forth, to create populations with higher qualities.Particularly, genetic algorithms (GAs) are the most flexible multi-objective evolutionary tools due to allowing an implementation of various operators for its evolutionary Advances in Civil Engineering computation.Therefore, it has been hybridized with different local search techniques.
First study on the multiobjective evolutionary optimization, called Vector Evaluated Genetic Algorithm (VEGA) was performed by use of GA's principle [1].VEGA employs a number of subpopulations to search the solution space considering a modified selection mechanism.Following the emergence of VEGA, two featured approaches, linear aggregating and lexicographic ordering of objective functions were developed [2,3].They transformed all objective functions into a single objective function by way of optimizing each objective function without decreasing their solution qualities.
As an alternative to early attempts mentioned above, a pareto-based evolutionary approach was developed to increase the population diversity [4].Some of the preliminary pareto-based multi-objective optimization approaches are Nondominated Sorting Genetic Algorithm (NSGA) by Srinivas and Deb [5], a Niched Pareto Genetic Algorithm (NPGA) by Horn et al. [6], a Multi-objective Genetic Algorithm (MOGA) by Fonseca and Fleming [7], and a Multi-objective Evolutionary Algorithm (MOEA) by Tanaka and Tanino [8].In order to improve their optimal results, these algorithms have been developed by either enhancing their current optimization strategies, such as Nondominated Sorting Genetic Algorithm II (NSGA II), Improved Strength Pareto Evolutionary Algorithm II (SPEA II), Improved Pareto Envelope-Based Selection Algorithm (Region-Based Selection) II (PESA II), or adapting a competitive Search Technique, such as Adapting Scatter Search AbYSS.
Although it is clear that there are a number of evolutionary methods known in the field of evolutionary multiobjective optimization, in this paper, an exhaustive literature review on this field is omitted.Instead, representative works related to four multi-objective optimization techniques employed are summarized.Furthermore, the recent multi-objective approaches utilized in the field of structural engineering design are also reviewed.
In this regard, paper is organized as firstly introducing the first steps in MOAs including applications in the field of structural engineering after a brief introduction to the multi-objective optimization problem and concepts.The computational procedures of proposed MOAs, NSGAII, SPEA II, PESA II, and AbYSS are presented in Section 3. The design requirements prescribed by LRFD-AISC Ver. 13 and optimal design procedure are given in Section 4 prior to the introduction of search methodology located in Section 5. Following the discussion of results along presented in Section 6, final remarks are summarized in Section 7.

Multiobjective Optimization: Problem and Concepts.
A general multiobjective optimization problem consisted of m objective functions and (J + K) constraints, defined by N decision variables, is represented as follows: min/max F(x) = f 1 (x) + p 1 , f 2 (x) + p 2 , . . ., f m (x) + p m , x ∈ DS, DS = x L n ≤ x n ≤ x L n , n = 1, 2, . . ., N , SS = x : g j (x) ≤ 0, h k (x) = 0, j = 1, 2, . . ., J, k = 1, 2, . . ., K X bounded by an upper and lower value, x U n and x L n are used to represent a decision variable set defined in decision variable space (DS) and computation of both objective functions f and constrains g j (x) and h k (x) in a solution space (SS).In order to explore the optimal solutions (designations) located in feasible region (FR) which contains the unpenalized solutions of SS, the obtained solutions are penalized when they do not satisfy the constraint conditions.Then, the penalized values denoted by p are included into their related objective functions f .
At each run of an evolutionary optimization algorithm, a random solutions set is obtained.Some of them are Nondominated solutions (none is better for all objectives) and referred to as "pareto solution" defined in a concept named as domination [9].Thus, the pareto solutions are used to form "pareto front" which determines bounds of Nondominated solutions.

First Steps in MOAs.
After introduction of the evolutionary mechanism to the multi-objective optimization problem, the development of new multi-objective approaches has been accelerated [4].Primary one of these approaches is Nondominated Sorting Genetic Algorithm (NSGA) proposed by Srinivas and Deb [5].NSGA determines the Nondominated solutions according to ranks of their reproductive potentials.
Although pareto-ranking procedure gives a guarantee for transmission of elite individuals to next generations, an excessive repetition of ranking procedure causes to lose promising genetic material due to the genetically distortion of migrated data.In order to diminish this negative effect, Niched Pareto Genetic Algorithm (NPGA) that determined the Nondominated solutions by tournament selection method was suggested [6].Afterwards, Fonseca and Fleming [7] suggested a penalization process that generated the promising pareto solutions considering their crowding densities.
Using an external archive to store Nondominated solutions leads to an increase in the capabilities of multiobjective optimization tools, such as in an evolutionary search proposed by Zitzler and Thiele, called Strength Pareto Evolutionary Algorithm (SPEA) [10].The members of external archive are chosen according to their closeness to pareto front.However, enlargement of the external archive makes the convergence speed of its evolutionary search to be poor due to a decrease in its selection pressure.In order to deal with this difficulty, Knowles and Corne [11] suggested a grid system, called Pareto Archived Evolutionary Strategy (PAES) to compute the optimization-related procedures.
Hence, distributing the entire population to this grid system by an adaptive mapping process, which of each node was used to represent an individual, makes it easier to maintain the diversity in pareto set.

An Overview of MOAs
Applied in Field of Structural Engineering.Although MOAs mentioned above are successful for a solution space represented by design variable of continuous type, they fail to explore optimal designations in nonconcave and discontinuous solution space of structural design problems represented by design variables of discrete type.Hence, the preliminary studies in the field of structural engineering with multiple objectives were developed using weighting [12], goal programming [13,14], and modified game theory methods [15].Sunar and Kahraman [16] compared the computational performances of these algorithms considering optimal designations of a space truss with 25 bars and a satellite system and showed that modified game theory and goal programming were superior to weighting approach.Although it was reported that the weighting approach failed to explore Nondominated solutions on nonconvex parts of pareto front [17][18][19], it has been improved due to its easy adaptable mechanism.One of these attempts was based on a systematic alteration of objective function weights [20].This adaptive approach was applied to design optimization of a truss problem with 3 bars and achieved to obtain a welldistributed pareto set in a nonconcave solution space.
Real-world engineering structures are represented by a large number of discrete design variables.Hence, the evolutionary search is easily misguided due to an increase in the computational effort.Therefore, new optimization algorithms inspired by some biological events, such as particle swarm, differential evolution [21], artificial immune system [22], ant colony [23], and some other evolutionary algorithms, such as microgenetic algorithm [24] have been utilized to solve optimization problems with multiple objectives.
Janga and Nagesh proposed an evolutionary technique, called elitist-mutated particle swarm optimization and applied to three test problems: design optimizations of twobar truss, I-beam, and welded beam [25].
Cooperate and coevolutionary strategies were introduced to the evolutionary search mechanism to increase diversity within the set of Nondominated solutions stored [26].For this purpose, chromosomes divided into different species are recombined and evolved to create a pareto set.However, the number of chromosomes chosen may exceed their predetermined number.In order to deal with this difficulty, the exceeded number of chromosomes is reduced according to the qualities of their crowding distances and redistributed using an elitism strategy.The fundamentals of NSGA [5], Niched Pareto Genetic Algorithm (NPGA) [27,28], and Controlled Elitist Nondominated Genetic Algorithm (CNSGA) [5] were constituted on this approach.The proposed evolutionary approach, which of main evolutionary operations has similarities to the optimization algorithms named NSGA, NPGA, and CNSGA, were utilized for topology optimization of 2D heat transfer structures.
It was demonstrated that using a decreased size of species increased the computational performance of the proposed optimization procedure.
The other promising approaches were developed as hybridizing these methods with each other.One of substantial attempts is the integration of neural network and fuzzy systems to provide a control mechanism for an evolutionary mechanism [29].While a neural network is utilized to predict an individual with higher quality, operator parameters of evolutionary algorithm are updated according to the rules of fuzzy logic.This hybrid system was applied for design optimization of several composite beams with three layers, piezoelectric bimorph beam, a truss structure, and airplane wing and displayed to obtain more converged optimal designations comparing to a pure and independent usage of each algorithms without any hybrid implementation.

Introduction of MOAs, NSGAII, SPEA II, PESA II, and AbYSS for Design Optimization of Grillage Systems Utilizing Multiple Objectives
Although it is displayed that the evolutionary algorithms have been successfully utilized as optimization tools, some computational bottlenecks cause to obtain a poordistributed pareto front.In order to increase their ability in generation of promising designations for a diverse pareto front, the conventional MOAs have been improved and/or developed.The promising versions are NSGA II, SPEA II, PESA II, and AbYSS.In this study, these improved or developed MOAs, are utilized for the optimal designations of grillage systems noting that the design optimization of grillage systems was carried out using a single objective by up to now [30][31][32] and a displacement-based matrix analysis approach [33].In order to execute their optimization procedures, JMETAL coded in a programming language Java is employed [34].Both JMETAL and the proposed four MOAs are introduced in the following subsections.
A Brief Introduction of JMETAL.JMETAL coded in a platform of object-oriented Java is used as an optimization tool to solve the multi-objective optimization problems [34].It contains a number of classes which represent the building blocks of various multiobjective algorithms.However, their basic evolutionary-related elements are the same.Therefore, the architecture of JMETAL is constituted on a simple but an interdependently higher framework.JMETAL is an open source project.The computational procedures of various MOAs including the ones employed in this study are extensively documented in [34].Therefore, the description of its base classes is not presented.Instead, the computational order of the preliminary classes that contain the fundamental parameters is briefly depicted by the pseudocodes.Whereas the class names are presented by use of a mark " ", their related parameters are defined by use of italic characters.In general, JMETAL contains six packages, named "base, experiments, metaheuristics, problems, quality indicator and util".While the package of "experiments" includes Equation ( 9) Equation ( 5) Equation (10) (0.7 * F y * S x ) configurations of various metaheuristic procedures, such as AbYSS and NSGAII, their default configurations are located in the "settings" package.Design problems with various complexities are comprised in the "problems" package.The run of JMETAL firstly begins by execution of a purposed metaheuristic class, such as AbYSS.Java, located in the "experiments" package.This class invokes the experiment class which is responsible for activation of proposed metaheuristic algorithm.The purposed metaheuristic class invokes a class of design problem to be solved.Basic computation processes, named "evaluate () and evaluateconstaint ()," are carried out in this class, named design problem.Due to the use of binary coding scheme for the evolutionary computations, generated individuals can exceed the predefined upper bounds of design variables.In order to deal with inappropriate binary strings, the purposed meta-heuristic class named AbYSS.Java, and its related subclasses are extended by including a new class named "repairing mechanism" which is responsible to correct the decoded values of binary strings according to the limits of design variables.

Nondominated Sorting Genetic Algorithm II, NSGA II.
NSGA was firstly designed by Srinivas and Deb [5].The evolutionary computation of NSGA is managed by classified "population size" individuals.Classification process begins firstly ranking the population in order to determine Nondominated individuals.Then, fitness values of individuals are shared considering their niching measures.The individuals with shared fitness values are selected by use of a selection method, called "stochastic universal sampling" and regenerated using mutation operator with a probability mutation probablity and combination operator with a probability crossover probablity until completing a predetermined evolution number max evolution.The enhanced version of NSGA, NSGA II (see pseudocode in Algorithm 1) was developed to increase diversity among individuals [35,36].
Although NSGA II utilizes the evolutionary operators for generation of individuals as in the computational procedure of NSGA, the generated individuals stored in an archive with population size individuals are used to obtain a pareto front considering crowding distances of individuals.Furthermore, a hypercube, which is formed by use of the Nondominated individuals, is utilized to compute a hypervolume.

Improved Strength Pareto Evolutionary Algorithm, SPEA II.
The primary version of SPEA approach was firstly suggested by Zitzler and Thiele [10].Its evolutionary processes are managed by two populations.While one of these populations, called regular population is utilized to generate offspring, other population, called an archive with archive size individuals is employed to preserve the evolutionary information of pareto front.In the beginning of evolutionary process, the archive is empty and filled by promising individuals.The exceed number of individuals are reduced giving a higher chance to the individuals of archive.For this purpose, a clustering technique which is based on an assignment of strength value to the individuals and an assessment of these individuals according to their strength values is utilized to discard the related individuals.However, a decrease in variation among individuals of population causes to increase the randomness in the evolutionary search and hence decrease the accuracy degree of density estimation, which is utilized by clustering process [37].This leads to vanishing of the promising solutions located on the pareto front.In order to deal with this negativity, SPEA II is developed.SPEA II estimates density of strength values using kth nearest neighbor method (see Algorithm 2).Furthermore, a new selection mechanism, called "environmental selection", is used to update the archive and to preserve promising solutions during truncation of archive.

Improved Pareto Envelope-Based Selection Algorithm (Region-Based Selection), PESA II. Pareto Envelope-Based
Selection Algorithm (PESA) was firstly presented by Corne et al. [38].Having similarities to basic features of SPEA II, PESA uses two populations for its evolutionary processes.However, in the estimation of strength-value density, PESA uses a measure called "sequence factor."An extended version of PESA, PESA II, utilizes hyperboxes which are obtained by dividing the entire search space into small ones (see Algorithm 3).Thus, the number of individuals contained in hyperboxes is used to determine a sequence factor.Following the determination of subregion numbers defined by bisections, the archive is created in the module named "AdaptiveGridArchive."A population called "solution" is initialized and evolved by the application of evolutionary operators until being completed a fixed evolution number.
Although SPEA II was shown to be computationally faster than NSGA II and SPEA II, some complicating tasks, such as possibility of existing dominated individuals in any hyperboxes, keeping a fixed number of subregions throughout the evolutionary search, and so forth, must be overcome in order to obtain satisfactory optimal designations [39,40].

Adaptive Scatter Search for Multiobjective Optimization,
AbYSS.AbYSS can be categorized as an evolutionary-searchbased optimization algorithm derived by use of principle features of NSGA II, PESA II, and SEPA II [41].A pseudocode for ABYSS is presented in Algorithm 4. After initializing a population with population size individuals, the population is firstly regenerated using grid-based search technique named "diversification generation."Then, the regenerated population is maintained by discarding the mutated individuals with poor qualities according to a dominance-based comparison test.This elimination process named "improvement" is inspired from NSGA II approach.Then, a search process named "reference update" is invoked to firstly construct a sub-population RefSet2 with Ref Set size

Optimum Design of Grillage Systems
The proposed four MOAs are presented in the previous section.These MOAs are evaluated for the optimal designations of grillage systems in this study.For this purpose, the objective functions, their related constraints, and structural analysis procedures are coded in Java.In this regard, the design requirements used in constraints and design optimization procedure are introduced in Sections 4.1 and 4.2.

Design Requirements of Grillage Systems according to
LRFD-AISC Ver. 13.Grillage systems comprise a number of lateral braced beams.If the beams loaded in plane of lateral system have no sufficient lateral stiffness, then they are buckled out of plane of loading.This case is called lateraltorsional buckling.The lateral-torsional buckling strength varies depending on the unbraced length and compactness of beam plays an important role in the load carrying capacity of beam.If a compact beam determined according to its web and flange dimensions has a sufficient unbraced length, then nominal flexural strength is calculated in an elastic domain, otherwise an inelastic one.In the inelastic case, a short and unbraced beam length causes to yield its outer fibers before attaining elastic buckling load.The formulation of nominal flexural strength M n that is managed by limit states of yielding, lateral torsional, and flange local buckling is presented in the following part as defined in AISC-LRFD Ver.13 (see Figure 1).For simplicity, two distinct figures used to depict the lateral-torsional and flange local buckling depicted in AISC-LRFD Ver.13 is coarsely combined, but equation numbers corresponding to formulations for limit states are presented in a separate parenthesis.The limit states of yielding of beam cross-section are written as ( In elastic and inelastic domains, two unbraced lengths L p and L r are used to determine the compactness of sections manage the strength of lateral-torsional buckling.Nominalflexural strength M n is computed as follows.
For inelastic-torsional buckling limited by where, For elastic-torsional buckling occurred in a segment limited by L r < L b (see Figure 1), where, In (6), moment modification factor C b is utilized to consider the effect of lower torsional buckling arisen from a nonuniform distribution of moment.Therefore, associating with moment diagrams, it is computed as follows.
The local buckling of flanges for noncompact section is governed by two parameters, λ p f , λ r f .Depending on these parameters, nominal-flexural strength M n is computed as follows.
For inelastic-flange buckling occurred in a segment limited by For elastic-flange buckling occurred in a segment limited by λ r f < b f /2t f , In addition, nominal-shear strength V n is computed as follows (see the limit states of shear strength in Figure 2): where, In (11), web-plate buckling coefficient k v is equal to 5 for unstiffened webs with (h/t w < 260).

Design Procedure for Optimal Grillage Systems.
In this work, a design problem of grillage system is represented by two objectives, entire weight of grillage system, and joint deflection, and expressed as follows: Subject to The term W is total weight of all grid members and computed using w and l which are unit weight to be selected from W-sections list of LRFD-AISC Ver.13 and length of a grid member.While d is termed as a joint displacement corresponding to related degree of freedom which is denoted i, total numbers of joint and grid member are indicated by n and m. d max is taken as (max span/300).In constraint inequalities, while displacements of joints are constrained by an upper limit f max , bending-moment strength of grid members M u is limited by allowable nominal-moment strength M n .Shear strength of grid members V u is limited by allowable nominal-shear strength V n (see V n in (11)).In (14) and (15), φ b and φ s are resistance factors for moment and shear and taken as 0.9.Furthermore, it must be noted that M n indicated in (14) manages total three strength requirements: yielding (2), flange-local buckling ( 9) and (10), and lateraltorsional buckling (3) and (5).

Search Methodology
Due to the fact that application problems are chosen from real-world engineering design problems with design variables of discrete type, a reliable and consistent search Figure 6: Strengths and displacements of random penalized solutions corresponding to minimum displacement (a1)-(a5), unpenalized solutions corresponding to maximum displacement (b1)-(b5), and maximum weight (c1)-(c5) (see Figure 5).
strategy must be established for an evaluation of proposed MOAs' search capabilities.After generating optimal designations, the computational performances of MOAs are assessed according to the closeness of these designations to a pareto front known beforehand thereby using a number of quality measuring metrics.Furthermore, accuracy of assessment of multiobjective optimization algorithms must be confirmed by outcomes of statistical tests.The other important difficulty is how to adopt a common methodology for a conventional mathematical model, which is performed its computational procedures by usage of design variables of continuous type.Therefore, a reasonable approach is Advances in Civil Engineering Hyper volume ratio (HVR) is an indicator which shows a ratio of current hyper volume to the true hyper volume computed by use of true pareto front and nondominated solutions true pareto front (see (18)), Higher values of HVR indicate a large coverage of Nondominated solutions in a solution space.
Inverted Generational Distance.Inverted generational distance (IGD) estimates the far of Nondominated solutions included in current pareto front generated by the proposed MOA, from those included in true pareto front (see (19)), where n is number of Nondominated solutions found by proposed MOA and d i is Euclidian distance between each of these and nearest member of true pareto front.A lower value of IGD indicates an increase in approximation of current pareto front obtained to the true pareto front in terms of convergence.
Spread.This quality metric abbreviated as (S) is used to measure an expanding spread exhibited by Nondominated solutions obtained and computed as where d i is Euclidian distance between consecutive Nondominated solutions, d is mean of these distances, and d f and d l are distances to extreme solutions of current pareto Pareto front of a set of random penalized solutions (design example 2) 14 front.A lower S value points out a better distribution among Nondominated solutions.In other words, it is implied that Nondominated solutions are located in different positions.

Statistical Tests.
The quality indicators mentioned above are utilized in comparison of MOAs' distribution qualities.Therefore, after computing means and standard deviations of quality indicators obtained by in the end of independent 10 executions, consistency of these results are checked through performing a statistical analysis in a certain level of confidence.If a probability value resulted from a statistical testing procedure satisfies a user-defined significance level, then it is said that distribution of current MOAs approximation set is acceptable.
Computational procedures of the statistical analysis are performed in MATLAB [42].Firstly, lillie test is carried out to check quality indicator values for whether to be exhibited a normal distribution (if the null distribution is completely specified, then Kolmogorov-Smirnov test is more appropriate).Then, existence of a variance homogeneity is controlled through Levene' test.If homogeneity in the variance exists, Welch test is performed, otherwise Anova test.In order to compare the statistical outputs acquired from different algorithms, a post hoc testing is performed through "multicompare" function coded in MATLAB.

Discussion of Results
It is known that the computational performances of MOAs vary depending on the values of their interacting parameters.Therefore, in order to provide an accurate evaluation for their computational performances, some combination sets of parameter values are chosen.Due to a variety on the operator parameters of MOAs, various parameter combinations and the assigned values for their evolutionary operators are summarized in Table 1.The combination numbers (C.No.) are denoted by superscripts attached to related parameter values.In order to provide unbiased competition for the performance evaluation of MOAs employed, the parameter values of evolutionary operators are kept for each MOA.In this regard, the parameter combinations are sorted into two main groups denoted by numbers (1-4) and (5)(6)(7)(8).Furthermore, each of main groups contains both upper and lower value sets of mutation and crossover distribution indexes in order to provide an intensive mutation or crossover effect for evolutionary search.For example, in first case of AbYSS (shown by italic characters in The penalty-related parameters used by penalty functions of weight and displacement are taken as r 0 = 0.05 and 0.005, ϕ = 3 and 2, and f = 2 and 1 for P 1 and P 2 (see (17) and ( 18)), respectively.
Using these different parameter combinations, an optimal design of grillage systems is carried out by four MOAs, NSGA II, SPEA II, PESA II, and AbYSS according to optimum design procedure mentioned previously.The yield stress, elastic modulus, and shear modulus of steel material used to construct the grillage system are taken as 50 ksi (345 MPa), 29,000 ksi (200 MPa), and 14,500 ksi (100 MPa).Cross-sectional properties of grid members are chosen from a discrete set with 274 W-sections.Sequence number of each cross-sections included in this profile list database is the same as given in LRFD-AISC Ver.13 and contains all sectional properties (such as area, inertia moments in all directions).Design variables are represented by binary strings.Thus, a binary length of l = 9 with 2 9 = 512 possible gene combinations will be adequate to represent 274 ready sections.Profile list database, finite element attributes of design examples and a structural analysis formulation for the grillage system are coded in Java in order to appropriately compile with the optimization tool named JMETAL.In this regard, a user-defined input data coded in Java Language and related parts are presented for the design optimization of example 1 by utilizing AbYSS algorithm-based optimizer (see Algorithm 5).Details about class names and their related packages in Algorithm 4 are given in Section 3.  Design examples of real-world engineering structures are presented in an order of increasing size of their elements and joints.In order to display the change in the strength of grid members, both penalized and unpenalized optimal results corresponding to its maximum joint displacement or weight of grid structure are presented considering the joint and member numbers.Furthermore, convergence history obtained in the end of a complete evolutionary search is also presented.In order to provide an easier visualization for both weight and displacement values, the weight and displacement values are displayed for each of 100 equal segments obtained by a division of the maximum evolution number.Thus, the optimal designation located in one of these segments is easily determined.Generation no (×50) .
The pareto fronts of a set of random Nondominated solutions obtained by both use of an increased population size and evolution number and AbYSS, NSGAII, PESAII, and SPEAII are depicted in Figures 4(a) and 4(b1)-4(b4).The means of quality indicator values, standard deviations values corresponding to best means, and assessment of significance levels according to statistical tests are presented in Tables 2  and 3. Considering Tables 2 and 3, an increase in the evolution number and population size leads to an improvement in the values of quality indicator, in other words, an increase in the convergence degree of optimal designations.According to assessment of statistical analysis results corresponding to the lower evolution number and population size, it is clear that there is not a statistical confidence among proposed MOAs.The best values of quality indicators are obtained by AbYSS.SPEAII and PESAII exhibits better computational performance compared to NSGAII.It is also observed that an increase in the evolution number and population size forces MOAs to use higher values of operator parameter for improvement of their optimal designation qualities.This result is confirmed by examining the values of quality indicators with bold and italic characters.These higher quality indicator values are obtained by Cases 1-4, which indicate a usage of lower evolution number and population size and Cases 5-8, which indicate a usage of higher evolution number and population size.Considering the best spread values, the pareto fronts obtained by four MOAs employed are compared with both each other and the pareto front obtained by use of higher population size and evolution number (Figures 4(b1)-4(b4)).The convergence history of the optimizer AbYSS (Case 3) with the best spread value 0.8067 is presented in Figure 5 with two axes.A designation with (weight = 5446.1415lb (2470.3282kg) and displacement = 0.0406 ft (12.3748 mm)) is obtained in a generation of no.= 4638 located in a segment of no.92 with an interval (90 × 50 = 4500 and 100 × 50 = 5000).

Design Example 2:
A Grid System with Five Bays.A grillage system with 160 grid members and 152 nodes is shown in Figure 7.This grillage system is almost 10times larger than first design example.Grid members are linked in 8 separate groups, resulting in total four design variables in x direction and total four design variables in y direction.According to this linkage system, members (1-      4 and 5. Examining the values of quality indicators tabulated in Tables 4 and 5, it is seen that the lower evolution number and population size decrease the quality degree of optimal designations leading to obtain an inconsistent statistical confidence among quality indicators.However, the quality degree of optimal designations is elevated associated by an increase in the evolution number and population size.According to the quality indicator values, AbYSS (Case 1) succeeded to obtain a lower spread 0.8356 and a higher Advances in Civil Engineering W18×106, W18×97, W21×122, W21×122, and W18×97), and (W40×503, W12×305, W14×500, W14×730, W14×550, W40×278, W16×100, W36×361, W14×665, W40×593, and W14×311), respectively.Examining Figure 14(a1)-14(a5), it is seen that torsional and yielding strengths play a bigger and more important role in the bearing capacity of grid structure compared to the other strengths.A similar result as in the application of design examples 1 and 2 is obtained in this application example: a decrease in the displacement values causes the flange bucking, torsional, and yielding-related strengths to obtain highly near to their related yielding limit state values.Using the steel profiles with larger cross-sectional properties leads to an increase in both weight of grid structure and limit state values.

Conclusion
In this study, the designs of grillage systems are optimized using the optimization tools named NSGAII, SPEAII, PESAII, and AbYSSS according to the design provisions of LRFD-AISC Ver.13 specification.Hence, the computational performances of the MOAs employed are not only evaluated but the change in the strength of gird members in conjunction with joint displacements is also displayed.In order to assess the computational performance of MOAs, the various quality indicators, IGD, HV, and spread are computed considering the application of three design examples with an order of increasing complexity degree and evaluated their statistical confidences according to outcomes of statistical tests.According to the evaluation of quality indicators and optimal designations, the following results are observed.(iii) An increase in the parameter values of evolutionary operators in conjunction with the evolution number and population size leads to an improvement in the convergence degree of optimal designations.
(iv) In conjunction with increasing the evolution number and populations size, the degrees of quality indicators are improved by use of (higher crossover probability and crossover distribution index-higher mutation probability and mutation distribution index-higher reference sets and archive size) for the evolutionary operators of AbYSS, and (lower crossover probability and crossover distribution index-lower mutation probability and mutation distribution index-higher archive size) or (higher crossover probability and crossover distribution index-lower mutation probability and mutation distribution index-higher archive size) for the evolutionary operators of SPEAII and PESAII.(v) A decrease in the joint displacement correspondingly causes the values of flange buckling, torsional, and yielding strengths to obtain near to their related limit state values.In this regard, the bearing capacity of grid structure has to be kept in the upper levels against any drastic decrease in joint displacements.Therefore, using a multi-objective optimization algorithm becomes a reasonable approach for the optimal design of grillage systems.
In the next work, the effect of using the different evolutionary parameter values on the quality degree of optimal designations will be evaluated.Furthermore, the proposed MOAs will be hybridized with both each other and the other evolutionary optimization algorithms in order to both reduce the evolutionary computational cost spent to obtain Nondominated solutions and improve the curves of pareto fronts.

Figure 4 :
Figure 4: Pareto fronts of a set of random nondominated solutions obtained by use of an increased population size and evolution number (a) and AbYSS, NSGAII, PESAII, and SPEAII (b1)-(b4).

Figure 8 :
Figure 8: Pareto fronts of a set of random nondominated solutions obtained by use of an increased population size and evolution number (a) and AbYSS, NSGAII, PESAII, and SPEAII (b1)-(b4).

Figure 11 :
Figure 11: Geometry of 236-member grid structure including locations of point loads.

Figure 12 :
Figure 12: Pareto fronts of a set of random nondominated solutions obtained by use of an increased population size and evolution number (a) and AbYSS, NSGAII, PESAII, and SPEAII (b1)-(b4).

( i )
According to the values of quality indicators, AbYSS shows better computational performance compared to SPEAII, PESAII, and NSGAII.Moreover, NSGAII obtains the worst values of quality indicators than SPEAII and PESAII, whose computational performances are almost equal.(ii) An efficient exploration of search space is only provided by usage of an increased evolution number and population size.Otherwise, the outcomes from statistical tests point out an inconsistent relation between quality indicators obtained by use of a decreased evolution number and population size.
b f : Flange width C b : Lateral-torsional buckling modification factor for non-uniform moment diagrams E: Elasticity modules F cr : Critical stress F y : Specified minimum yield stress h: Clear distance between flanges h 0 : Distance between flange centroids I y : Out-of-plane moment of Inertia J c : Torsional constant k c : Coefficient for slender unstiffened elements k v : Web-plate buckling coefficient L p : Limiting laterally unbraced length for the limit state of yielding L b : Distance between braces L r : Limiting laterally unbraced length for the limit state of inelastic lateral-torsional buckling M n : Nominal flexural strength M p : Plastic bending moment M A : Absolute value of monet at quarter point of unbraced segment M B : Absolute value of monet at centerline point of unbraced segment M C : Absolute value of monet at three-quarter point of unbraced segment r y : Radius of gyration about y-axes r ts : Effective radius of gyration used in the determination of L r for lateral-torsional buckling limit state S x : Elastic section modules about principal axes t f : Thickness of flange t w : Web thickness Z x : Plastic section modules about principal axes λ p : Limiting slenderness parameter for compact element λ p f : Limiting slenderness parameter for compact flange λ r f : Limiting slenderness parameter for compact web λ: Slenderness parameter.
Figure 3: Geometry of 68-members grid structure including locations of point loads.
individuals from a sub-population RefSet1 with Ref Set2 size individuals and then, update the sub-population RefSet2.The sub-population RefSet2 is built by the individuals of RefSet1 with minimum euclidian distance.After generation of reference sets, the Nondominated individuals extracted from the reference sets are stored in an external population called archive with archive size individuals using density estimation of individuals.In this regard, the existence of individuals in a densest region is identified considering their niching measures.Its features of utilizing the niching measures and density estimation for the evolutionary search are inspired from adaptive grid method by PAES II and selection strategy by SPEA II, respectively.
Figure 7: Geometry of 160-members grid structure including locations of point loads.
(17)btain a pareto front by running current discrete optimization model in bigger and repeated generation numbers.In this regard, independent 10 runs of proposed MOAs are executed by use of both increased and decreased size of generation and population.Then, optimal designations obtained are utilized in computation of quality-measuring metrics, such as hyper volume and generational distance.(see(17)).For this purpose, a reference point chosen among Nondominated solutions with worst objective is utilized in computation of hypercube of each Nondominated solutions,

Table 1 :
Cases used to represent the parameter combination for evolutionary operators of MOAs and their assigned values.

Table 1
), Crossover and Mutation Probability values are 1.00 and 1.00; the values of Crossover and Mutation Distribution Indexes are 30 and 30; the values of Reference set I-II and archive size are 20, 20, and 40.Hence, reproducibility of related MOAs by use of these parameters is ensured.The detailed descriptions of these parameters are found in Section 3.

Table 7 :
Values of measurement quantities obtained by "Eval.Number = 5,000" and "Pop.Size = 100" for design example 3. The pareto fronts of a set of random Nondominated solutions obtained by both use of an increased population size and evolution number and AbYSS, NSGAII, PESAII, and SPEAII are depicted in Figures8(a) and 8(b1)-8(b4).