Numerical design optimization of real size complex steel space frame structures using a novel adaptive cuckoo search method

In this study, real size complex steel space frame structures are numerically designed to achieve optimal design weight. For this aim, standard cuckoo search algorithm is rectified through a newly proposed adaptive method over two fundamental algorithmic parameters of alien egg probability detection (Pa) and step size control (α) to overcome incompetency of trapping into local optima. Besides, to strengthen exploitation phase of newly proposed algorithm, it is boosted with greedy selection (GS). So, novel algorithm has more promising exploration and exploitation capabilities. An 8‐story, 1024‐member and a 20‐story, 1860‐member real size complex steel space frame structures are selected as design examples. Also, initially to verify the supremacy of novel algorithm, a well‐known welded beam is optimized as a benchmark structural design problem. Afterwards, the steel space frame structures are optimally designed via novel algorithm. Since ready steel section lists are utilized as selection pool for design variables, discrete programming problem is come into existence. The dead, live, snow, and wind design loads acting on frame structures are calculated in direction of ASCE 7‐05 provisions. Furthermore, the structural design constraints are determined from LRFD‐AISC specifications. Eventually, the newly proposed adaptive cuckoo search algorithm boosted with GS presents outstanding algorithmic performance.


INTRODUCTION
Design optimization of the steel structures is absolutely essential for an economic design.Also, the reduction on amount of the utilized steel provides minimum weighted design namely low cost. 1,2In design optimization of steel space frames, some design loads such as snow and wind are calculated directly related with the geometry of the frame structure.Besides, the live loads are determined according to the intended usage purpose of the structure.However, earthquake and dead design loads are different than abovementioned loads.Both are directly associated with the structural design weight. 3hile the design loads are essentially needed to achieve a feasible structural design, yet it is required to calculate these loads accurately. 4But there are some vicious circles here.First, the selection of cross-sections to the structural members effects the structural rigidities.So, the distribution of the internal forces over the structural members are related with the cross-sections of the all members in a hyperstatic structural systems.However, the selection of the cross-sections is also related with the internal forces of the structural members.Second, as mentioned above, the selection of cross-sections for structural members directly affects to weight of the structure, namely the earthquake and dead design loads.However, the magnitudes of these design loads are related with the cross-sections of the structural members.6][7] Therefore, instead of trial-and-error, probabilistic iterative solution methods such as metaheuristic algorithms are needed.Metaheuristic algorithms are subtitled under the probabilistic optimization methods.As they are called probabilistic methods, the results obtained using these algorithms cannot be claimed to be exact or global optimum. 8Here, the algorithmic performance and robustness of these methods come into forefront. 9,102][13] Among them, genetic algorithms, 14 harmony search algorithm, 15 firefly algorithm 16 and so forth may be named as conventional probabilistic metaheuristic algorithms.Additionally, some researchers have focused on improving the more effective and robust optimization algorithms, recently.The backtracking search optimization algorithm, 17 black widow optimization algorithm, 18 gray wolf optimizer 19 and dandelion optimizer 20 may be given as examples of those recent metaheuristics.The metaheuristics are utilized in many engineering fields such as design of head shape of a high-speed train, 21 wave energy converters to maximize the power output, 22 and compact reactor radiation shielding design. 235][26][27] As instances from the current ones; for structural engineering field four modified versions of standard grasshopper optimization algorithm to solve large-scale real-size complex both truss and frame type steel structures are proposed, 28 the differential evolution with adaptive penalty method is used for structural multi-objective optimization, 29 the five different trusses are optimized via craziness based particle swarm optimization (CRPSO), which is an adaptive version of the particle swarm optimization (PSO), 30 and so forth.Furthermore, a developed metaheuristic algorithm may give very successful results in one design field, but may not show enough optimal solution performance in another design field. 31,32Actually, the number and type of the design variables (continuous and discrete), the number of objective(s), and the characteristics and amount of the design constraints directly affect the optimum solution performance of a metaheuristic. 33Besides, to verify algorithmic performance of a developed metaheuristic algorithm, various benchmark optimization problems are utilized at first. 34,35he gear train design problem, 36 the cantilever beam design problem, 37 and the concrete beam design problem 38 are some examples of these benchmark engineering design problems.Moreover, some specific field design problems have been started to be utilized as benchmark design problems in the process of time.This is to say, the design problems belonging to a specific engineering field have been considered as benchmark design problems in general over time. 39,40or example, the 18-bar truss, the 10-bar truss and the 2-story 2-bay frame design problems are handled as benchmark discrete optimization problems in the literature. 41he standard cuckoo search (sCS) algorithm has been encoded inspiring the reproduction strategy of some cuckoo species. 42The natural behaviors of the cuckoos such as the brood symbiosis and the exploration of the host bird's eggs are mimicked in algorithmic environment.The Gaussian or the uniform distribution methods are not the component of sCS.Alternatively, Levy flight distribution method is utilized to optimize the problem which has larger search space, and the most of structural design problems have wide search space.This is one of the main reasons for considering the sCS algorithm as optimizer tool in this study.The algorithmic performance of sCS algorithm stands out from all the rest in the literature as structural engineering design optimizer. 43Kaveh and Backhshpoori minimize design weight of a one-bay 10 story frame, a three-bay 15-story frame, and a three-bay 24-story frame by using sCS.So, the outstanding performance of the sCS algorithm is illustrated in planar steel frame design. 44The same sCS algorithm shows superior performance in 3D steel frame designs. 45][48] For instance, Cuong-Le et al. claims that taking the optimization parameters of the sCS algorithm as constant negatively effects the algorithmic performance.Thus, they suggest control parameters of sCS algorithm by combination between linear and nonlinear approach.The obtained novel version of sCS is proven operable on 23 typical benchmark functions. 49In another study, the time cost of the search process associated to the random-walk behavior in sCS algorithm is evaluated by chancing step size.So, an adaptive cuckoo search algorithm (ACSA) is generated, and satisfactory results are attained for some benchmark engineering design problems. 50Ong developed an sCS algorithm by implementing an adaptive step size rectification strategy and enabled faster convergence to the global optimal solutions of benchmark functional problems. 51Also, a modified cuckoo search (MACS) algorithm is proposed to improve the sCS algorithm without reducing high-efficiency searching feature of Lévy flights and it is verified on some well-known test problems. 52Zhang et al. present a self-adaptive multi strategy cuckoo search algorithm (MSACS) by proposing five different strategies for sCS algorithm.The performance of the MSACS was tested on 28 common benchmark functional problems. 53 Salgotra  et al. proposed a new version namely self-adaptive cuckoo search (SACS) algorithm to improve the performance of sCS algorithm by employing adaptive parameters and the SACS algorithm is tested on CEC2017 benchmark problems. 54Li and Yin suggested a novel mutation scheme based on rand and best entities amidst the whole population for standard CS algorithm.Further, for these two new parameters (rand and best entities), the self-adaptive uniform randomization was adjusted in order to increase the population diversity depending on their relative success.To prove the algorithmic capability of suggested novel self-adaptive CS algorithm, 16 benchmark functional design examples taken from literature are comparatively studied. 55Ahmid et al. present an adapted variant of discrete cuckoo search algorithm that utilizes a rank-value approach to turn real values of random Levy walks into the equivalent discrete values to solve the design problem of customized I-beam gantry crane. 56he novelty and originality of this study is that the proposed novel adaptation of the standard CS algorithm achieves outstanding algorithmic performance on life-size complex steel space frame structures.Although the sCS algorithm 57 is proven fruitful on many structural optimization problems in the literature, 43,45,58,59 it may not be ensured the rapid convergence relating with searching on a random walk and a small and/or a regular step size may cause trapping algorithm in a local optimal.In this study, in order to provide a rapid convergence rate and preventing to stack in local optimal of sCS algorithm, an adaptive version of which is proposed.In this newly proposed is proposed adaptive CS (aCS) algorithm, exponential equations to rectify P a and  parameters to avoid trapping in local optima and accelerating the convergence rate.Moreover, this novel adaptive cuckoo search (aCS) algorithm is boosted with greedy selection (GS) to improve the algorithmic exploitation performance of sCS algorithm.The algorithmic performance of newly proposed aCS algorithm boosted with GS is initially tested on widely utilized structural design benchmark problem that is known as welded beam design (WBD) problem. 60Afterwards, the supremacy of novel aCS boosted with GS is proved on obtaining the optimum structural designs of complex real size steel space frame design problems.For comparing the performance superiority of the aCS boosted with GS with other conventional metaheuristic algorithms, the 8-story, 1024-member steel space frame 1,61 and the 20-story, 1860-member steel space frame 59 problems are optimized to achieve minimum design weights.Here, Minimum Design Loads for Buildings and Other Structures, American Society of Civil Engineers 62 practical design specifications is used to calculate the design loads of the structures, and the provisions of Load and Resistance Factor Design Specification-American Institute of Steel Construction (LRFD-AISC) 63 are considered to determine the structural design constraints of the optimally designed structural problems.The attained optimal design results are contrasted with reference studies as abovementioned.The comparative results clearly indicate that the presented aCS boosted with GS has superior algorithmic performance for obtaining the design solutions of real size complex steel space frame structural design problems.
The general structure of the study is arranged as follows; • In Section 1, the motivation and innovative aspect of the study are examined.
• The proposed adaptive methodology is described from Sections 2 and 3.
• In Section 4, the descriptions and results of the handled structural design examples are given.Moreover, the attained optimal design results are evaluated in this section.
• Eventually, the discussions and fundamental concluding remarks of this study are presented in Sections 5 and 6, respectively.

PROVISION-BASED DISCRETE DESIGN OPTIMIZATION OF REAL-SIZED SPACE STEEL FRAMES
In real-size space steel frame design, the structural designer aims at minimum weighted design of structure because of reaching minimum cost, namely economical design.Moreover, the loads on the lower floor columns and general earthquake loads on the steel frame structures are directly related to the design weight of the structure.So, obtaining minimum weighted real-sized space steel frames is the main objective of such kind of optimum design problems.In building type steel structures, the standard ready-section steel profiles are generally assigned to structural members for practical purposes.Thus, in the design of real-sized space steel frame structures the optimum cross-sections, in which each one is numbered with integer numbers, are chosen from a ready list of profiles.The practical structural specifications impose certain design rules on designers.Thus, the structural design constraints and limit states must be satisfied by final design of an optimizer.Here, the utilized mathematical expressions to minimize the design weight of a real size space steel frame structures is presented and discussed as follows.
The objective function created by multiplying the total structural volume by the unit volume weight is given below Equation (1). ( In Equation ( 1), the total weight of a real size space steel frame is given as W(x).The A k is cross-section area of the kth section group, ng stands for number of groups, mk is the number of the members in a group (kth), and L i represents the length of the ith member, respectively.The density of the frame members is given as  i .
In the design of real size space steel frame structures, finding the solitary minimum weight is not enough.Additionally, the feasibility of boundary design conditions must also be determined.For these, the stress and the displacement limits can be considered.The code specifications of the LRFD-AISC 63 are operated to check the limit states and design constraints of the optimum structural design problems.The utilized design constraint expressions related to stress and displacements are given as follows.
The inter-story drift of a real size space steel frame structure is expressed in Equation ( 2).The  j and  j−1 are two lateral deflections of ensuing story levels, and h j represents the height of the jth story.Total story number is given as ns.
The displacement limitation must also be checked, besides inter-story drift of the structure.For example, beam deflections are given in these design constraints.In Equation (3),  iu is the allowed ultimate lateral displacement.This value can be obtained by dividing the total height of a real sized space steel frame structure (H) by 400 and, the limit of a drift in any story is calculated by dividing the story height by 400.
Utilizing Equations ( 4) and ( 5), all steel members which are exposed to combined stress regarding to bending and axial forces are evaluated.Here, the nominal and axial strengths of kth member are symbolized with P uk and P nk .Here, P nk is obtained by multiplying yield stress of the steel material and gross area of the member of the frame structure.The factor of resistance in comparison is , and it is taken as 0.85.The M uxk and M uyk are flexural strengths in x-and y-directions, and M nxk and M nyk are the corresponding nominal strengths.In case of gross section yielding, the reduction factor of strength ( b ) taken as 0.9, otherwise it is considered as 0.75.
According to F2 section of the LRFD-AISC practice code provisions, the necessary strength of the beams is shown in Equation ( 6).The M uxt and M nxt are required and nominal strengths of the beam in major axis.The  is taken as 0.9 as flexural factor of resistance, and nb is the total number of beams.
The notations of a beam-to-column connection node.
There are geometric constraints beside strength and displacement constraints in connection nodes of the real size space steel frame structures.The flange width of a beam in the same story should not be greater than those of the columns.The notations of a beam to column connection mentioned in Equation ( 7) is shown in Figure 1.Here, b fb and b ′ fb are flange with of the B1 and B2 beams and b fc is the flange width of the column.The d c is height of the W section and, t f is thickness of the flanges of the column.As seen Figure 1, beams can be connected to web of the columns, as well.In this case, the clear distance between flanges of the columns should not be less than flange width of the B2 beam.Moreover, the flange width of the B1 should be less than the flange width of the column.

OPTIMUM DESIGN METHODOLOGY
There are specific design constraints and particular objective or objectives besides design variables in the optimization process.In design optimization process of a real size steel space frame structure, standard-section steel profiles are used as design variables.So, the design variables of such a structural design problem must be discrete.The selected steel profiles effect to distribution of the inner forces on the structural members.Moreover, these kind of design problems are too complex and hard for computing via deterministic mathematical based optimization algorithms.Consequently, the probabilistic optimization algorithms, which mimics the natural phenomena, are more useful to compute such design problems.5][66][67][68] Among these, the sCS algorithm is selected as the optimizer.

Standard cuckoo search (sCS) algorithm
The sCS algorithm was first encoded by Yang and Deb. 57The aggressive reproduction strategy in the breeding behavior of the cuckoo species is mimicked as a natural phenomenon in this algorithm.In nature, cuckoos lay their eggs on nests of other bird species.Correspondingly, each set of solutions is represented as a cuckoo, eggs, and host nest in this algorithm.More feeding opportunity is considered in the selection of nest and, there can be only a single egg in each selected nest.
The sCS algorithm has following main rules; Rule 1: A cuckoo lays only a single egg at a time.The egg is deposited in a randomly chosen nest.

Rule 2:
The candidate solutions, which represent the best nests with high-quality eggs, are considered in the next iteration.

Rule 3:
The number of hosts is constant.The hosts can detect the alien egg.The probability of detection is based on pa ∈ [0, 1].So, the new nest is entirely rebuilt in a new location or the alien egg can be thrown out of the nest by the host.

Lévy flight mechanism
In nature, the behavior of some fly species is considered, and they inspire the mechanism of the Lévy flight.The sharp movement of these based on a random walk with a random step size is given in Equations ( 8) and ( 9). Levy Here, t represents the number of iteration and i is the number of solution vector.The η is step size which should be positive and, ⊕ represents the entry-wise multiplication.

Steps of sCS algorithm
To initialize computation, the assignations of sCS algorithm parameters and randomly generated initial population are a necessity.The initial population is defined as Equation (10).
Here, each row of the Pop represents a randomly generated solution vector.So, Pop consists of N Pop numbered candidate solution vectors.After the production of the initial population, cost of each randomly generated candidate solution is computed via cost function and, the rows of the Pop are sorted by considering these computed cost values.One another detail is that new solution vector is generated via Equation (11).
Here, the  is step size, and its value should be selected greater than 1.0.The length of the step size is represented with λ.
The previous and new candidate solution vectors are illustrated as (x i ) v and (x i ) v+1 .Mantegna suggests an approach to compute step size based on random selection control parameter generating probabilistic variable (0.3 ≤  ≤ 1.99). 69The , which represents the length of the step size, can be attained by utilizing Equation (12).
Here, x and y named two probabilistic variants are presented utilizing the standard deviation.
The standard deviation is given as  x and  y in above Equation (13).In case of  equals to 1.0,  y also equals to 1.0.The function of Gamma (G) can be computed via following Equation (14).
In the above equation, if z and k is taken as same value on condition that k is positive and integer value, the G(k) would equal to (k − 1).Then, a random nest is chosen, and this nest is named as j.In case of the fitness of Obj ( X j ) is more suitable than the fitness of Obj(X i ), X i is replaced with X j .Otherwise, X i is kept in memory.The abandon of the worst nest and generation of the new one is based on P a probability.Thence, the nest owners discover each nest to catch utilizing Equation (15).
Here, the r is an arbitrary number generated between 0 and 1 for each detection process.In this manner, the R matrix consists of Boolean type variables which can just be 0 and 1.After the generation of the R, Equation (16). is put in process.
Here, the p1 i and p2 i represent the permutation of the two rows of the nest.
All designs in the Pop matrix are rankly reorganized in the light of the computed Obj(X) values.The optimization procedure continue until the maximum iteration criteria is satisfied as given in the flowchart of the sCS algorithm in Figure 2.

Newly proposed adaptive cuckoo search (aCS) algorithm
In this study, the sCS algorithm is strengthened to efficiently overcome the deficiencies on exploration and exploitations phases on obtaining optimum design of real size space steel frame structures via adaptive rectifications of equations for P a and  parameters. 70In the sCS algorithm both P a and  parameters are selected as constant values for exploitative and explorative search phases during research operation.Contrariwise, in proposed aCS algorithm the P a and  parameters adaptively improvised in fact forms the basis for the algorithm to gain adaptation to varying characteristics of the design search space.Originally, it was stated that P a gives good results for most optimization problems at a value of 0.25, 57 while  should be taken as 0.01, 0.1, or 1.0, depending on the size of the handled optimization problem. 51In this context, in novel aCS algorithm in order to maintain a better environment exploration and supplying a good exploitation ability for algorithm new adaptations are proposed for the P a and  as presented in Equations ( 17) and (18).Hence, duration of the optimum search process, the control parameters of P a and  are adaptively determined.Moreover, to improve computational performance of the proposed novel aCS algorithm, it is boosted with GS.
The flowchart of the sCS algorithm.

3.4.1
Novel adaptive rectification of standard cuckoo search (sCS) algorithm In optimum design problems, in case of the number of design pool and design variables are small, the satisfactory designs can be attained via sCS algorithm. 43,44In fact, if a sufficient number of iterations is exceeded, global optimum results can be achieved for relatively complex design problems. 45However, if the handled optimization problem is too complex and having large size, a better convergence rate and exploitation ability becomes necessary.So, an adaptive version of the sCS algorithm is needed.Here, there are two algorithmic parameters of sCS discussed.In this study, starting from the abovementioned recommended values of P a and  51,57 are gradually decreased towards zero to tolerate the slight infeasible and improvable solutions in the optimum design search process.This supplies the sCS algorithm with a rapid convergence rate and a more efficient exploitative search capability.First, the probability of detection of the alien egg is based on P a is adaptively tuned through Equation ( 17) for each iteration.
Here, iter represents number of current iteration and iter max is the maximum number of iterations which designated before initialization of algorithm.
Second, the calculation of the length of the step size via Equation ( 12) is linearly based on the  parameter.So, the step size of the random-walk behavior is adaptively updated in each iteration via Equation (18).
The flowchart of the aCS algorithm is illustrated in Figure 3.

Boosting with greedy selection
The metaheuristic algorithms are based on stochastic computation as mentioned before.This means that the solution method of these algorithms includes probabilistic processes.These can overcome computation of the extremely challenging design problems with high complexity.However, it cannot be claimed that the obtained results from stochastic calculation is definitely a global optimum because stochastic optimization method has probabilistic process.At this point, The flowchart of the aCS algorithm.
the examination of the success of the optimization is important.The obtained candidate solutions during the iterative process are precisely related with the utilized selection method.Namely, the selection process in optimization effects the convergence performance of the metaheuristic algorithm.In this study, the GS is utilized as one of the popular selection strategies.The solution candidates are generated considering their fitness.In algorithmic process, feasible candidate solutions are selected more effectively.So, the convergence performance of the executed metaheuristic algorithm is improved.
In complex optimization problem, the GS presents rapid selection strategy than other selection methods.The steps of GS are based on current case.It does not consider any future effect of its selection.Ultimately, achieving the optimum solution can always be achieved by choosing the best possible solution. 71n this study, the fitness values of each generated candidate solution are compared with previous one during the optimization process.The more feasible solutions are kept in memory, and it is updated in each iteration by this way.Thereby, optimum solution is obtained.The GS is used in selection process that is operated between fitness values.The operating of the GS for boosting newly proposed aCS algorithm is given as following pseudo code (Algorithm 1).
Algorithm 1.The pseudo code of novel aCS algorithm boosted with GS.
Step 1. Initially generate a population matrix.
Step 2. Compute the fitness values of each nests.
Step 3. Adaptively update P a and  parameters.
Step 4. Get a cuckoo randomly with Levy flight.
Step 6. Due to the nature of the greedy selection, if the fitness of the cuckoo is greater than any fitness value of the nest, replace the cuckoos to worst nest, and sort the all nest.
Step 7. Abandon P a of the worst nest, and generate new ones at new locations via Levy flights.
Step 8. Save the current best.
Step 9.If termination criterion is satisfied, end the optimization procedure.Otherwise go to the Step 3.

Constraint handling
In novel aCS boosted with GS, the candidate solutions are generated via typical procedure based on randomization.So, in some cases, the algorithm generates some slight constraint violating solutions.Hence, the attained impracticable designs are penalized utilizing a penalty approach.The values of the objective function of infeasible designs are computed via Equation (19).
In Equation (19), f equals to constrained objective function value, W is the weight of the candidate design, nc stands for number of constraints, and g i is the value of the any constraint.Here, if value of the computed constraint is smaller than 0, its value is considered as 0. Else, its original value is used. 8,28,71

DEFINITIONS OF DESIGN PROBLEMS AND EVALUATIONS OF ACHIEVED RESULTS
In this study, one popular structural benchmark design problem and two real size complex steel space frame structural design problems are considered as design examples.For this purpose, the WBD problem, the eight-story, 1024-member steel space frame structure and the 20-story, 1860-member steel space frame structure are designed via ACSA boosted with greedy selection (aCS boosted with GS).Besides, it is also worthy to mention that in order for making a fair and correct comparison, all the design parameters of the design examples are taken into account exactly same as in the original resource studies.So, the population size is taken as 25 for WBD problem and 100 and 75 for 1024-member and 1860-member steel space frame design problems, respectively. 59,61And the obtained results are compared with results of sCS algorithm as well as those obtained via other conventional metaheuristic optimization algorithms for same design problems previously declared in literature.

Welded beam design problem
3][74][75][76] In WBD problem, the end of a rectangular beam is welded to a rigid object.The vertical point force of 600 lb is applied to free end of the beam.So, WBD carrying the P load in an economical way with minimum cost is primarily aimed.The general scheme of the WBD is shown in Figure 4.
There are four design variables as illustrated in Figure 4.The x 1 and x 4 can be selected between 0.1 and 2.0 and, the x 2 and x 3 can be selected between 0.1 and 10.0.Design parameters of the problem consist of the maximum vertical deflection as 0.25 inch, the design normal stress of the material of the beam as 30 ksi, the design stress of the material of the weld as 13.6 ksi, the young modulus of the beam material as 30,000 ksi, the shear modulus of the beam material as 12,000 ksi, and length of the welded beam as 14 in, respectively.The equation of the objective function can be simplified utilizing all of these parameters as in Equation ( 20); The minimum coasted WBD should satisfy seven constraints presented as in Equation ( 21); where, ) .
The detail of the attained optimum results of the WBD problem via newly proposed aCS boosted with GS are tabulated in Table 1.The final values of the design constrained are also given in the same table.Minimum weight of the WBD problem is computed as about 1.7137 with proposed aCS boosted with GS.The design constraints are found to be at an acceptable tolerated level in terms of structural engineering.Moreover, the previously announced some other metaheuristic algorithm solutions as well as the sCS algorithm is represented in Table 2.If these results are compared with the one attained via proposed aCS boosted with GS, it can be seen that the achieved optimum design of aCS boosted with GS is 1.035% lighter than the final design obtained by sCS 49 as tabulated in Table 2.

Eight-story, 1024-member steel space frame
The 3D view, front view, plan view and column orientation plans of the eight-story, 1024-member steel space frame structure are depicted in Figure 5.The design of this real size complex structural problem was previously considered utilizing a biogeography-based optimization (BBO) algorithm, 1 using a biogeography-based optimization algorithm with Levy flight (BBO with LF), 61 and using a sCS algorithm. 59

TA B L E 1
The best solution result attained by aCS boosted with GS for WBD problem.
x This steel space frame structure has 1024 structural members connected each other through 384 joints.All structural members are divided into 40 groups in total.A section assignment is made for each member groups.So, there are 40 individualistic design variables in this design problem as given in Table 3. ASCE7-05 is utilized to determine acting loads on the structure.Thus, the dead load of the 2.88 kN/m 2 and live load of 2.39 kN/m 2 are considered.Moreover, the basic wind speed of 38 m/s (85 mph) is taken into consideration in design.Through the provision-based combinations of dead, live, wind, and snow loads, the load combinations of 1.2D+1.6L+0.5S,1.2D+1.6WX+L+0.5S,and 1.2D+0.5L+1.6Sare utilized.Here, D, L, S, WX, and WZ respectively symbolize the dead load, live load, snow load, wind load through global x-and z-axis.The same code provisions include some deflection limits for the structures.The upper limit of inter-story and top-story drifts are taken as 0.875 and 7.0 cm, respectively.Similarly, the deflections of the all beams are restricted with 2.0 cm.
The design history achieved by aCS boosted with GS for eight-story, 1024-member steel space frame is illustrated in Figure 6.The attained minimum design weight of the steel space frame yields as 6030.72 kN in 1750th iteration.The softness of the design history curve shows that the proposed adaptive approach boosted with GS for cuckoo search optimization algorithm is running very promisingly.The acquired optimum design of the 8-story, 1024-member steel space frame is tabulated in Table 4.Moreover, same table includes accomplished maximum inter-story drift, maximum strength ratio, maximum top-story drift, maximum number of iterations, and statistical results of the five different optimal design attempts.Here, the maximum strength ratio of 0.973 and maximum top-and inter-story drifts show that the accomplished final optimum design of real size 8-story, 1024-member steel space frame structure is acceptable according to structural code specifications of LRFD-AISC.Additionally, the fact that the ultimate value of yielded maximum strength ratio approaches its upper bound of 1.0 increases the probability of the achieved optimal design results being the global optimum.
The statistical results of the final optimal design clearly illustrated that newly proposed aCS boosted with GS utilized to minimize weight of the real size 8-story, 1024-member steel space frame shows stable performances.The comparison of the attained minimum design weight with those acquired by the previously declared metaheuristic algorithms in the literature is tabulated in Table 5.In present study, the minimum design weight of 6030.72 kN yielded by aCS boosted with GS is 1.031% lighter than those yielded by LFBBO, 61 7.164% lighter than those yielded by BBO 1 and 10.077% lighter than those yielded by sCS. 59

Twenty-story, 1860-member steel space frame
Twenty-story, 1860-member steel space frame is selected as second design problem of the present study because it has real size and complex structural characteristics. 59,77The 3D and plan views of the second structural design problem of this study is illustrated in Figure 7.There are 1860 structural member connected to each other with 820 joints and combined into 86 member groups.As in the previous real size steel space frame structural design problem, the design loads acting on the structure is determined according to the provisions of ASCE 7-05.The dead load of 2.88 kN/m 2 and the live load of 2.39 kN/m 2 is assigned to 20-story, 1860-member steel space frame structure.The basic wind load of 38 m/s (85 mph) is

Story
Here, D, L, S, WX, and WZ symbolize the dead load, live load, snow load, wind load through global x-and z-axis, respectively.The top-story and inter-story drift limits are taken as 15 and 0.75 cm, respectively.Additionally, the limit of the deflection of all beams are assumed as 1.67 cm.The design history of real size 20-story, 1860-member steel space frame structure derived through newly proposed aCS boosted with GS is depicted in Figure 8.The attained minimum design weight of this steel space frame yields as 5439.40 kN during the first 1380 iterations.When the design history curve is examined, it is seen that the proposed aCS boosted with GS optimization approach is running successfully for this real size high level complex steel space frame structure.The optimal design weight for designated steel sections, maximum inter-story and top-story drifts, maximum strength ratio, maximum number of iterations, and statistical results of the five independent attempts to optimally design the 20-story, 1860-member steel space frame via aCS boosted with GS is tabulated in Table 6.Here, it is obvious that the maximum strength ratio, maximum top-and inter-story drifts are less from upper bound of 1.0 proving that the accomplished optimum design is applicable according to code specifications.
The statistical results obtained demonstrate that from aCS boosted with GS utilized to minimize weight of the real size 20-story, 1860-member steel space frame structure have robust performances.The attained minimum design weights and optimum design result acquired by other so-called conventional metaheuristic algorithms previously reported in the literature are tabulated Table 7.The aCS boosted with GS yields minimum design weight of 5439.40 kN that is 1.604% lighter than sCS and 2.402% lighter than standard ACO.

DISCUSSION REMARKS
The main aim of this study is to adaptively rectify the sCS algorithm via tuning of P a and  parameters which effectively improves the performance of the standard algorithm.Additionally, these adaptations are boosted with GS.This novel algorithm is so-called aCSA boosted with greedy selection (aCS boosted with GS).The newly proposed aCS boosted with GS establishes the better balance between exploration and exploitation phases of the sCS algorithm.To check the supereminence of the proposed novel adaptive design methodology, the WBD problem as a well-known popular structural benchmark design problem is optimally solved at first.The minimum costed design of a welded beam (WBD) is aimed in this benchmark problem.In the design optimization of WBD problem, it is seen that the encoded aCS boosted with GS are accurately and overwhelmingly executed.After then, the optimal design weights of the eight-story, 1024-member steel space frame structure and 20-story, 1860-member steel space frame structure are minimized.So, the superior algorithmic performances of aCS boosted with GS are unearthed by this way.The followings are the main deducted discussions of this study: • Minimum cost of the WBD is ended with 1.714.The attained optimum design is 1.035% lighter than the final design cost obtained by sCS algorithm mentioned in the literature.
• The attained minimum design weight of eight-story, 1024-member steel space frame structure yields as 6030.72 kN in 1750th iteration.The softness of the design history curve shows that the aCS boosted with GS approach is running very successfully on this real size complex structural design problem.The statistical results acquired through five independent optimization runs are clearly illustrated the stable performance of proposed aCS boosted with GS in minimization of the design weight.In present study, minimum design weight of 6030.72 kN obtained via aCS boosted with GS is 7.164% lighter than BBO, and 10.077% lighter than sCS that were previously declared in the literature for the first steel space frame design example.
• The accomplished minimum design weight of 20-story, 1860-member steel space frame structure is yielded as 5439.40 kN via aCS boosted with GS during the first 1380 iterations.When the design history curve is examined, it is seen that newly proposed aCS boosted with GS is running very promisingly for this real size high level very complex steel space frame structural design problem.The statistical results accomplished from five different runs of design optimization illustrate that proposed aCS boosted with GS utilized to minimize the design weight have robust algorithmic performance.In present study, the minimum design weight of 5439.40 kN achieved via aCS boosted with GS is 1.604% lighter than those obtained with sCS algorithm and 2.402% lighter than those obtained with sACO that were previously reported in the literature for this structural design problem.• The attained design constraints for both abovementioned structural design problems are less than upper bounds reveal that the yielded optimum designs are applicable according to practice structural code specifications.
• Finally, the usage of the newly proposed aCS boosted with GS approach in structural design presents economic solutions and it is easy to usage in real size high level complex structural design problems even with discrete design variables.Here, the minimum design weight of the high-level steel space frames is considered as a main indication of algorithmic success.

CONCLUSIONS
In literature, to check the algorithmic performance of the proposed metaheuristics, the relatively small-sized benchmark design problems are generally used.But the algorithmic performances of metaheuristic algorithms depend on the type and size of the handled design problem.Yet in the minimum weighted optimal design of real size complex steel space frame structures, the more exploration and exploitation performance of a metaheuristic is generally required.For this reason, in this study, the tuning P a and α parameters of sCS algorithm is adaptively rectified to supply a rapid convergence rate and to prevent stacking the algorithm in a local optimum.Also, the GS is used to effectively boosts the performance of the sCS algorithm.The newly proposed aCS boosted with GS is fundamentally tested on real size complex steel space frame structures.It is obviously illustrated in the design examples that the novel adaptation strategies boosted with a GS scheme rectify the algorithmic performance of the standard CS significantly.The optimum designs obtained by the aCS with boosted GS algorithm in the design examples display faster convergence and attentive exploration and exploitative search performance than those attained by the standard CS algorithm.As a future study, the handled real size complex steel space frame structures are subjected to additionally exposed to earthquake seismic loads besides existing design loads.In this way, the more complex structural design optimization problems can be tackled by newly proposed aCS boosted with GS to find solutions for extended structural requirements.

F I G U R E 6
Design history of 8-story, 1024-member steel space frame structure.

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Design history of 20-story, 1860-member steel space frame structure.
The comparisons of the best solution results attained by aCS boosted with GS and other conventional metaheuristic optimization algorithms for WBD problem.
Optimal design of eight-story, 1024-member steel space frame structure.
Optimal designs of eight-story, 1024-member steel space frame structure.
Optimal design of 20-story, 1860-member steel space frame structure.Optimal designs of 20-story, 1860-member steel space frame structure.
TA B L E 6