Abstract

The krill herd (KH) algorithm is widely used for optimizing truss structures as no gradient information is necessary, and only a few parameters require adjustment. However, when the truss structure becomes discrete and complex, KH tends to fall into a local optimum. Therefore, a novel target-oriented KH (TOKH) algorithm is proposed in this study to optimize the design of discrete truss structures. Initially, a crossover operator is established between the “best krill” and “suboptimal krill” to generate a robust “cross krill” for global exploration. Additionally, an improved local mutation and crossover (ILMC) operator is introduced to fine-tune the “center of food” and candidate solutions for local exploitation. The proposed method and other optimization approaches are experimentally compared considering 15 benchmark functions. Then, the performance of the TOKH algorithm is evaluated based on four discrete truss structure optimization problems under multiple loading conditions. The obtained optimization results indicate that the proposed method presents competitive solutions in terms of accuracy, unlike other algorithms in the literature, and avoids falling into a local minimum.

TOKH algorithm flowchart

TOKH algorithm flowchart

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1 Introduction

Truss optimization has gained considerable attention because of its direct applicability to the design of structures. Designers and owners require optimized trusses to reduce building costs [1]. However, the implementation of discrete truss optimization problems is challenging as several truss designs generally entail a complex design space. Traditionally, researchers have used mathematical approaches that employ rounding-off techniques based on continuous solutions to solve discrete truss optimization problems. However, these methods may become infeasible or generate increasingly suboptimal solutions with numerous variables [2]. Therefore, simulation-based metaheuristic algorithms to solve truss optimization problems are required.

Metaheuristic algorithms combine rules and randomness to imitate natural phenomena and attempt to identify the optimum design in a reasonable amount of computing time using trial-and-error techniques [3]. The ability to balance exploitation (intensification) and exploration (diversification) during a search determines the efficiency of a specific metaheuristic algorithm. Exploration ensures the validity and breadth of the algorithm in the search area, which can be beneficial for global optimization. Exploitation expands the local search for the currently explored optimal area and further finds the minimum [4]. To address global search requirements, modern metaheuristic algorithms have evolved to incorporate three primary purposes, namely, solving problems faster, solving larger problems, and enhancing algorithm robustness [4]. Modern metaheuristic algorithms include the genetic algorithm (GA) [57], biogeography-based optimization [810], harmony search (HS) [1113], differential evolution [14,15], ant colony optimization [16], particle swarm optimization [1719], artificial bee colony [2022], teaching-learning based optimization (TLBO) [23], artificial fish swarm [24], firefly algorithm [2527], political optimizer (PO) [28], cuckoo search [29,30], bat algorithm [31,32], manta ray foraging optimization (MRFO) [33] and krill herd (KH) algorithm [34]. Among these algorithms, the KH algorithm is known for its powerful exploitation ability, adjustment of fewer parameters, and easy implementation.

The KH algorithm was inspired by the herding behavior of Antarctic krill swarms. During optimization, each krill is primarily affected by other krill individuals, foraging action, and physical diffusion. Foraging and other krill-induced motions include two global and two local strategies that operate in parallel, rendering KH a robust algorithm [34]. Compared to earlier metaheuristic algorithms, the KH algorithm imposes fewer mathematical requirements and can be easily adapted to solve various engineering optimization problems. Furthermore, rather than using a gradient search, the KH algorithm uses a stochastic search based on the krill population, which eliminates the need for derivative information. These features have increased the flexibility of the KH algorithm and produced better solutions. Consequently, it has been increasingly investigated and successfully employed in various practical structural optimization problems, such as truss structures [35], pin-jointed plane frames [36], and welded beams [37].

Nevertheless, the KH algorithm can cause a risk of stagnancy after the initial stage when applied to multi-extreme discrete optimization problems. Therefore, several approaches have been explored to increase the diversity of solutions, resulting in a few KH variations with improved performance. A chaotic KH (CKH) algorithm was presented to improve global optimization [38], wherein the three primary movements of a krill swarm were adjusted during the optimization process using various chaotic maps. An opposition KH (OKH) algorithm was proposed to increase the diversity of the population [39]. Herein, three operators, namely, opposition-based learning, position clamping, and Cauchy mutation were added to the normal KH algorithm to improve the global convergence. A multistage KH (MSKH) algorithm was introduced in Ref. [40] by adding a local mutation and crossover operator and an elite scheme in the exploitation stage. This resulted in the complete utilization of the global and local search capabilities of the krill swarm to solve the global numerical optimization problem. Guo et al. [41] proposed an improved KH (IKH) algorithm, wherein information could be exchanged between the top krill during movement. The IKH algorithm employed a novel Lévy flight distribution to extend the search range and added an elite scheme to update the krill-motion calculation, generate better candidate solutions, and accelerate global convergence. Laith et al. [42] proposed a modified KH algorithm (MKHA) to improve global exploration by modifying the genetic operator of the basic KH. After analyzing the influence of a step-size scaling factor ct on KH, Li et al. [43] advanced KH with a linear decreasing step (KHLD), wherein ct was linearly decreased over time to balance exploration and exploitation. Another study [44] used stud KH (SKH) algorithm for 22 benchmark functions. Furthermore, a levy-flight KH (LKH) algorithm [45] was proposed to improve the optimization performance of the KH algorithm; its effectiveness was verified using several benchmark functions.

Although several variants of KH algorithm enhance its optimization performance, their accuracy when handling discrete truss optimization problems remains unsatisfactory. The aforementioned algorithms ignore the effect of the “suboptimal krill” and “center of food” on the aggregation of KH, resulting in an inadequate balance between global exploration and local exploitation. Typically, the discrete truss optimization easily falls into a local minimum. To prevent this, a novel target-oriented KH (TOKH) algorithm is proposed in this study, which considers both the “suboptimal krill” and “center of food”. Initially, the “suboptimal krill” and “best krill” were crossed to generate a novel “cross krill” for better global exploration. Subsequently, an improved local mutation and crossover (ILMC) operator was applied to fine-tune the “center of food” and population to improve local exploitation for effectively solving truss optimization. Four discrete truss design problems are applied to verify the robustness of the developed TOKH algorithm. The optimization efficiency of TOKH algorithm increased by 20.90, 17.37, 53.53, and 88.01% for the four problems when compared to those of KH algorithm. The results verify that the proposed method is highly competitive with other optimization approaches reported in the literature and avoids falling into a local optimum.

The remainder of this paper is organized as follows: Section 2 presents the formulation of the discrete sizing optimization problem. A brief overview of the basic KH algorithm is provided in Sec. 3. Section 4 describes the proposed TOKH algorithm, and the superiority of TOKH algorithm is verified using 15 benchmark functions in Sec. 5. Section 6 explains the robustness of TOKH algorithm validated using four discrete truss optimization problems. Finally, the conclusions of the study and directions for future research are summarized in Sec. 7.

2 Formulation of Discrete Truss Optimization Problems

Discrete sizing optimization of the truss attempts to identify the optimal cross section of the system elements to minimize the structural weight. Moreover, the minimum design must satisfy the inequality constraints that limit the design variable sizes and structural responses [46].

The discrete structural optimization problem for a truss can be formulated as [28]
FindS=[S1,S2,,Sv],SiDi,Di=[di,1,di,2,,di,r(i)]TominimizeW(S)=i=1nmγi·Si·LiSubjecttoσminσiσmaxi=1,2,,nδminδjδmaxj=1,2,,m
(1)

Here, S represents the set of design variables, Di denotes an allowable set of discrete values for design variable Si, v indicates the number of design variables or member groups, r(i) represents the number of available discrete values for the i-th design variable, W(S) denotes the weight of the structure, n indicates the number of component members in the structure, m represents the number of nodes, γi denotes the material density of member I, Li indicates the length of member i, δj represents the nodal displacement/deflection at node j, σi denotes the stress developed in the i-th element, and δmin and δmax represent the lower and upper bounds, respectively.

The optimum design of truss structures must satisfy the optimization constraints stated in Eq. (1). This procedure comprises the following three rules:

  • Rule 1: Any feasible solution is better than any infeasible solution.

  • Rule 2: Between two feasible solutions, a solution with a better objective function value is preferable.

  • Rule 3: Between two infeasible solutions, the solution with the smallest constraint violation is preferred.

The first and third rules direct the search toward feasible regions, whereas the second rule directs the search to a feasible region with suitable solutions [47].

3 Krill Herd Algorithm

This section briefly introduces the principles of KH [34] algorithm. KH algorithm uses an optimization process to attain a global solution defined using an objective function similar to the process by which a krill swarm obtains food and gathers continuously. Over time, the location of an individual krill is determined by three primary movements, namely,

  • Motion induced by other individuals,

  • Foraging action, and

  • Physical diffusion.

These motion types can be expressed using a Lagrangian model in a n-dimensional decision space as follows [34]
dXidt=Ni+Fi+Di
(2)

where dXidt denotes the speed of each krill, and Ni, Fi, and Di indicate the motions induced by other krill individuals, foraging action, and physical diffusion of the i-th krill individuals, which can be obtained using Eqs. (3), (5), and (7), respectively.

The approximate value of the direction of induced motion (αi) can be calculated using a target swarm density (target effect), local swarm density (local effect), and repulsive swarm density (repulsive effect). For krill individual i, this movement can be formulated as [38]
Ninew=Nmaxαi+ωnNiold
(3)
where
αi=αilocal+αitarget
(4)

Here, Nmax denotes the maximum induced speed, ωn indicates the inertia weight of the motion induced in the range of [0,1], Niold represents the last motion induced, αilocal and αitarget denote the local and target effects, respectively.

The foraging action involves two primary components, namely, the current “center of food” location and previous experience with respect to the “center of food” position. The foraging action for the i-th krill can be expressed as follows [38]:
Fi=Vfβi+ωfFiold
(5)
where
βi=βifood+βibest
(6)

Here, Vf denotes the foraging speed, ωf indicates the inertia weight within the range [0,1], βifood represents the “center of food” attraction, βibest denotes the effect of the best fitness, and Fiold indicates the last foraging action.

The third part of physical diffusion is primarily affected by the maximum diffusion rate and random vector. The physical diffusion can be expressed as [38]
Di=Dmaxδ
(7)

where Dmax denotes the diffusion speed, and δ indicates a random vector within the range [−1,1].

Foraging actions and movements induced by other krill individuals involve two global and two local optimization strategies. The simultaneous operation of these optimization strategies renders the KH algorithm a powerful optimization method. During a specific period, different effective movement parameters of the movement can be used. The changes in the location of a krill individual from t to t + Δt can be expressed as follows [34]:
Xi(t+Δt)=Xi(t)+ΔtdXidt
(8)

where Δt is an important parameter that completely depends on the search space. Here, Δt can be increased appropriately when the search space is wide, and when the search space is small, it can be appropriately reduced. Additionally, the crossover and mutation mechanisms of GA [5] can be incorporated to improve the performance of KH algorithm.

4 Target-Oriented Krill Herd Algorithm

Local exploitation and global exploration are the two critical components of modern metaheuristic algorithms. Exploitation reinforces the local search for minimum or near-optimal solutions, whereas exploration involves a global search to ensure the efficient and effective exploration of the search space [4]. Excessive diversification causes solutions to jump around from one potentially optimal solution to another, increasing the convergence time required to reach optimum. However, excessive reinforcement may trap the algorithm in local optimum, as only a portion of the local space may be visited. Therefore, an effective algorithm requires an appropriate balance between these two components to ensure efficient convergence, avoid falling into a local optimum, and guarantee the solution accuracy.

Krill herd algorithm has demonstrated its ability to identify near-global regions in continuous-truss optimization problems [35]. However, the insufficient balance between global exploration and local exploitation causes the algorithm to easily fall into a local optimum when solving discrete truss optimization problems. The proposed TOKH algorithm intends to balance the associated exploitation and exploration components to solve the discrete truss optimization problem more efficiently.

The subsequent subsections present the basic principle and time complexity of TOKH algorithm.

4.1 Basic Principle.

This subsection describes the basic principle of TOKH algorithm. The learning efficiency of the population oriented to the “suboptimal krill” and “best krill” should be improved to enhance the global optimization of KH algorithm. In the exploration phase, a crossover operator was established between the “suboptimal krill” and “best krill” to generate a “cross krill”. The crossover Lagrangian model can be expressed as
Xc=Xi+ω(XbestXi)λ(XiXsub)
(9)

where Xc, Xi, Xbest, and Xsub denote the “cross krill”, “current krill”, “best krill”, and “suboptimal krill” respectively; and ω and λ indicate different random numbers in the range of [0,1]. If the fitness value of Xc was better than that of Xbest, Xc was replaced by Xbest. Algorithm 1 presents the pseudo-code used to achieve the aforementioned crossover operator.

Algorithm 1

Pseudo-code of the crossover operator

Begin
Xc=Xi + ω* (Xbest-Xi)- λ*(Xi-Xsub)
if Xc is better than Xbestthen
Xbest = Xc
end if
End
Begin
Xc=Xi + ω* (Xbest-Xi)- λ*(Xi-Xsub)
if Xc is better than Xbestthen
Xbest = Xc
end if
End

After global exploration, the LMC operator was improved [40] to enhance the local exploitation. The LMC operator was inspired by the reproduction mechanism of GA; it prevented the premature fall into local optima by increasing the diversity of the population. However, the LMC operator only considered the “best krill” as the target for random crossover and mutation, which was not conducive to global optimization in the exploitation stage. The improved local mutation and crossover (ILMC) operator was developed to improve the local exploitation of TOKH. In ILMC operator, the optimum between the “center of food” and “best krill” served as the candidate solution, and the population learned from the candidate solution to accelerate aggregation to the optimum region in the exploitation stage. Algorithm 2 presents the pseudo-code of ILMC operator.

Algorithm 2

Pseudo-code of the ILMC operator

Begin
 if Xbest is better than Xfood
  Xcross=Xbest
 else
  Xcross=Xfood
 end if
 forj =1 to ddo
  ifrand 0.5 then
   Wi(j) =Xcross(j)
  else Wi(j) =Xcross(φ)
  end if
 end forj
  Obtain the individual fitness value Wi
 if Wi is better than Xithen
  Xi =Wi
 end
End
Begin
 if Xbest is better than Xfood
  Xcross=Xbest
 else
  Xcross=Xfood
 end if
 forj =1 to ddo
  ifrand 0.5 then
   Wi(j) =Xcross(j)
  else Wi(j) =Xcross(φ)
  end if
 end forj
  Obtain the individual fitness value Wi
 if Wi is better than Xithen
  Xi =Wi
 end
End

In the Algorithm 2, Xbest denotes the “best krill”, Xfood indicates the “center of food”. d represents the decision variables. Xi(j) denotes the j-th variable of the solution Xi. Wi indicates the offspring. φ represents a random integer number between 1 and d obtained from a uniform distribution, and rand denotes a random real number in the interval (0, 1) obtained from a uniform distribution.

Based on this analysis, the steps of the TOKH algorithm can be summarized as follows.

  • Step 1: Initialization: The generation counter was set to t=1, and the population P of NP krill individuals was randomly initialized. The foraging speed Vf, maximum diffusion speed Dmax, and maximum induced speed Nmax were also set.

  • Step 2: Krill population fitness evaluation: Each krill individual was evaluated according to its position.

  • Step 3: Motion calculation: The motion induced by the presence of other individuals was obtained using Eq. (3), the foraging motion was calculated using Eq. (5), and the physical diffusion was determined based on Eq. (7).

  • Step 4: The genetic operators [34] were implemented.

  • Step 5: Crossover operator: The “best krill” and “suboptimal krill” were crossed according to Algorithm 1.

  • Step 6: Population fine-tuning: The krill population was fine-tuned using ILMC operator in Algorithm 2. Each krill was evaluated considering its new position.

  • Step 7: Update the population position: The positions of krill individuals were updated in the search space.

  • Step 8: Repeat: steps 2–7 were repeated until a stop criterion was satisfied or a predefined number of iterations were completed.

Figure 1 illustrates the flowchart of the TOKH algorithm.

Fig. 1
TOKH algorithm flowchart
Fig. 1
TOKH algorithm flowchart
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4.2 Time Complexity Analysis of Target-Oriented Krill Herd Algorithm.

The time complexity of the algorithm was investigated to analyze the real-time efficiency of the TOKH algorithm in a better manner. The worst-case running time was expressed as a function of its input using a big Omicron (big-O) notation [48]. Typically, the big-O notation is used to denote an upper bound on the growth rate of a function and can be primarily applied to describe asymptotic behavior [48]. Logically, the big-O order is derived according to three rules:

  1. All addition and subtraction constants during runtime are replaced by constant 1,

  2. Only the highest order items are retained, and

  3. The highest term constant is removed.

Firstly, TOKH algorithm is started with parameters initialization which requires constant time complexity O(1). Secondly, in steps 2–4, three different movements of krill are implemented based on the rules of KH. The time complexity grows linearly according to the number of krill n, consequently, it requires O(3n). Thirdly, referring to crossover operator and ILMC operator, the time complexity in step 5 and step 6 are equal to O(1) and O(n), respectively. Moreover, to implement the population update, the time complexity in step 7 is equal to O(n). Finally, in step 8, the population evolves according to a specific number of iterations tmax. Therefore, the total time complexity of TOKH is expressed by O(1+(3n + 1+n + ntmax)O(n). In summary, it is obvious that most of the time complexity of TOKH comes from the basic KH, and the two operators introduced do not change the time complexity of TOKH.

The Rastrigin benchmark function [45] was selected for the verification. The functional dimensions were 30 and 100, and the population size was 50. All experiments were implemented and executed using matlab R2018a running on a personal computer with a 7-th generation core i7 CPU and Windows 10 OS. Figure 2 depicts the experimental results for the time complexity.

Fig. 2
Relationship between the running time and number of iterations
Fig. 2
Relationship between the running time and number of iterations
Close modal

As indicated in Fig. 2, the ordinate denotes the total time required in seconds for the algorithm to run 20 times on the Rastrigin benchmark function [45]. The abscissa represents the number of iterations, ranging from 0 to 1000; the running time was recorded once every 50 generations. The red and blue lines represent the fitting results of TOKH and KH algorithms, respectively. The running time and iteration of TOKH algorithm satisfied the linear relationship, and the time complexity of TOKH algorithm was O(n). The experimental results validated that the two optimization operators introduced do not change the time complexity.

5 Benchmark Mathematical Functions

Fifteen benchmark functions were used to compare the optimized performance of TOKH algorithm with nine other versions of KH algorithm, including KH [34], CKH [38], OKH [39], MSKH [40], IKH [41], MKH [42], KHLD [43], SKH [44], and LKH [45]. Table 1 summarizes the specific parameter settings of each algorithm.

Table 1

Specific parameter settings of each algorithm

Algorithm nameBasic parameters
KHThe maximum induced speed Nmax = 0.01, the foraging speed Vf = 0.02, the maximum diffusion speed Dmax = 0.005
KHLDCtmax = 2, Ctmin = 0, Other parameters are the same as KH.
TOKHAll parameters are the same as KH.
OKHReference [39], OKH7 is selected. δ1=0.4, δ2 =0.6. Other parameters are the same as KH.
CKHReference [38], M10 chaotic map is selected. Other parameters are the same as KH.
IKHReference [41], IKH2 is selected. Other parameters are the same as KH.
MSKHAll parameters are the same as KH.
SKHReference [44], SKH1 is selected. Crossover probability of single point crossover Pc =1. Other parameters are the same as KH.
LKHMax Levy-flight step size A = 1.0. Other parameters are the same as KH.
MKHElitism parameter Keep =2, Other parameters are the same as KH.
Algorithm nameBasic parameters
KHThe maximum induced speed Nmax = 0.01, the foraging speed Vf = 0.02, the maximum diffusion speed Dmax = 0.005
KHLDCtmax = 2, Ctmin = 0, Other parameters are the same as KH.
TOKHAll parameters are the same as KH.
OKHReference [39], OKH7 is selected. δ1=0.4, δ2 =0.6. Other parameters are the same as KH.
CKHReference [38], M10 chaotic map is selected. Other parameters are the same as KH.
IKHReference [41], IKH2 is selected. Other parameters are the same as KH.
MSKHAll parameters are the same as KH.
SKHReference [44], SKH1 is selected. Crossover probability of single point crossover Pc =1. Other parameters are the same as KH.
LKHMax Levy-flight step size A = 1.0. Other parameters are the same as KH.
MKHElitism parameter Keep =2, Other parameters are the same as KH.

Benchmark functions are widely used for evaluating algorithm performance owing to their ease of implementation and high reliability. In this study, 15 different dimensional benchmark functions were selected to compare the performances of the algorithms. The dimensions (n) of the benchmark functions were set to 10, 30, and 50, respectively; the optimization performance of the 30-dimension was particularly analyzed, which is a pretty representation of the average performance of the algorithm. The expressions and properties of these benchmark functions have been reported in a previous study [38].

The population size NP of all algorithms was 50. The maximum number of iterations was 50, and 30 simulations were independently performed by each benchmark function. Tables 25 summarize the simulation results of the algorithm considering 30-dimension. Table 2 lists the best-optimized performances of these algorithms, whereas Table 3 presents the general optimization performance, Table 4 indicates the worst optimization performance, and Table 5 describes the stability. Tables 6 and 7 summarize the general optimization performance when the dimensions of the benchmark functions were 10 and 50, respectively.

Table 2

Best values of the benchmark functions (the dimension of function is 30)

KHKHLDTOKHOKHCKHIKHMSKHSKHLKHMKH
F11.950.290.000.0401.601.570.0121.361.840.051
F22.670.990.000.611.231.620.001.052.200.89
F36.71 × 1011.01 × 1014.58 × 10−38.38 × 10−12.15 × 1011.33 × 1017.59 × 10−41.78 × 1012.37 × 1013.09
F41.47 × 1033.67 × 1014.26 × 10−59.34 × 1011.33 × 1021.34 × 1021.10 × 10−55.06 × 1012.17 × 1023.33 × 101
F54.52 × 1037.822.32 × 10−51.671.89 × 1012.82 × 1018.57 × 10−61.64 × 1012.21 × 1013.11
F68.04 × 10−25.82 × 10−21.80 × 10−41.06 × 10−23.38 × 10−27.05 × 10−26.84 × 10−45.04 × 10−23.74 × 10−21.22 × 10−2
F71.60 × 1026.858.63 × 10−78.88 × 1019.24 × 1014.64 × 1015.37 × 10−36.24 × 1018.88 × 1015.05 × 101
F85.25 × 1012.79 × 1012.71 × 1011.13 × 1013.33 × 1013.61 × 1012.48 × 10−32.86 × 1013.82 × 1012.95 × 101
F98.66 × 1038.23 × 1031.21 × 10−12.20 × 1036.04 × 1032.20 × 1032.21 × 10−22.84 × 1035.20 × 1031.25 × 103
F102.35 × 1044.301.86 × 10−57.46 × 1031.44 × 1042.89 × 1031.266.19 × 1021.80 × 1048.67 × 103
F111.26 × 1049.45 × 1031.80 × 10−31.88 × 1013.62 × 1018.653.14 × 10−23.57 × 1013.39 × 1017.92
F126.942.268.94 × 10−42.78 × 10−16.006.533.57 × 10−27.365.283.81
F133.26 × 1022.242.03 × 10−55.03 × 10−15.04 × 1018.42 × 1011.49 × 10−23.466.73 × 1011.31
F144.41 × 1024.84 × 10−14.70 × 10−81.48 × 10−13.51 × 1011.43 × 1011.50 × 10−41.061.04 × 1028.65
F154.72 × 1023.986.67 × 10−11.353.99 × 1016.43 × 1019.73 × 10−24.341.09 × 1023.37
Total0090007000
KHKHLDTOKHOKHCKHIKHMSKHSKHLKHMKH
F11.950.290.000.0401.601.570.0121.361.840.051
F22.670.990.000.611.231.620.001.052.200.89
F36.71 × 1011.01 × 1014.58 × 10−38.38 × 10−12.15 × 1011.33 × 1017.59 × 10−41.78 × 1012.37 × 1013.09
F41.47 × 1033.67 × 1014.26 × 10−59.34 × 1011.33 × 1021.34 × 1021.10 × 10−55.06 × 1012.17 × 1023.33 × 101
F54.52 × 1037.822.32 × 10−51.671.89 × 1012.82 × 1018.57 × 10−61.64 × 1012.21 × 1013.11
F68.04 × 10−25.82 × 10−21.80 × 10−41.06 × 10−23.38 × 10−27.05 × 10−26.84 × 10−45.04 × 10−23.74 × 10−21.22 × 10−2
F71.60 × 1026.858.63 × 10−78.88 × 1019.24 × 1014.64 × 1015.37 × 10−36.24 × 1018.88 × 1015.05 × 101
F85.25 × 1012.79 × 1012.71 × 1011.13 × 1013.33 × 1013.61 × 1012.48 × 10−32.86 × 1013.82 × 1012.95 × 101
F98.66 × 1038.23 × 1031.21 × 10−12.20 × 1036.04 × 1032.20 × 1032.21 × 10−22.84 × 1035.20 × 1031.25 × 103
F102.35 × 1044.301.86 × 10−57.46 × 1031.44 × 1042.89 × 1031.266.19 × 1021.80 × 1048.67 × 103
F111.26 × 1049.45 × 1031.80 × 10−31.88 × 1013.62 × 1018.653.14 × 10−23.57 × 1013.39 × 1017.92
F126.942.268.94 × 10−42.78 × 10−16.006.533.57 × 10−27.365.283.81
F133.26 × 1022.242.03 × 10−55.03 × 10−15.04 × 1018.42 × 1011.49 × 10−23.466.73 × 1011.31
F144.41 × 1024.84 × 10−14.70 × 10−81.48 × 10−13.51 × 1011.43 × 1011.50 × 10−41.061.04 × 1028.65
F154.72 × 1023.986.67 × 10−11.353.99 × 1016.43 × 1019.73 × 10−24.341.09 × 1023.37
Total0090007000

Bold font in the table is used to highlight the “optimal values” among all methods.

Table 3

Mean values of the benchmark functions (the dimension of function is 30)

KHKHLDTOKHOKHCKHIKHMSKHSKHLKHMKH
F12.741.790.000.342.302.130.0222.652.670.72
F26.251.110.000.882.653.510.0201.663.791.08
F31.50 × 1022.64 × 1019.88 ×10−31.37 × 1015.28 × 1013.29 × 1013.11 × 10−36.71 × 1017.99 × 1017.34
F43.02 × 1051.30 × 1022.30 × 1043.84 × 1023.36 × 1021.47 × 1031.56 × 1011.49 × 1025.70 × 1021.09 × 102
F51.50 × 1052.47 × 1011.58 × 10−44.01 × 1028.76 × 1011.73 × 1033.93 × 10−53.63 × 1011.50 × 1039.94
F63.59 × 10−19.25 × 1018.26 × 10−46.78 × 10−21.11 × 10−12.09 × 10−14.04 × 10−31.84 × 10−19.38 × 10−24.08 × 10−2
F72.03 × 1025.39 × 1013.99 × 10−41.58 × 1021.49 × 1028.04 × 1013.001.24 × 1021.30 × 1027.12 × 101
F88.82 × 1012.92 × 1012.74 × 1013.07 × 1014.65 × 1016.04 × 1012.69 × 1013.34 × 1016.14 × 1013.76 × 101
F99.35 × 1039.48 × 1032.80 × 10−13.51 × 1036.82 × 1033.86 × 1031.18 × 10−14.37 × 1036.41 × 1032.24 × 103
F103.86 × 1047.85 × 1038.53 × 10−41.71 × 1042.41 × 1047.43 × 1032.10 × 1033.54 × 1032.83 × 1041.32 × 104
F119.55 × 1093.62 × 1091.12 × 10−23.96 × 1014.91 × 1011.76 × 1017.09 × 10−26.98 × 1014.82 × 1012.42 × 101
F121.01 × 1015.801.38 × 10−23.98 × 10−18.821.01 × 1012.83 × 10−11.24 × 1011.01 × 1016.66
F136.52 × 1029.668.86 × 10−41.741.62 × 1023.43 × 1023.56 × 10−25.71 × 1013.17 × 1029.67
F148.70 × 1023.251.16 × 10−51.88 × 1011.10 × 1027.11 × 1011.50 × 10−11.07 × 1013.05 × 1025.62 × 101
F152.27 × 1038.996.67 × 10−11.63 × 1021.77 × 1023.73 × 1024.63 × 10−11.70 × 1014.56 × 1021.83 × 101
Total00100005000
KHKHLDTOKHOKHCKHIKHMSKHSKHLKHMKH
F12.741.790.000.342.302.130.0222.652.670.72
F26.251.110.000.882.653.510.0201.663.791.08
F31.50 × 1022.64 × 1019.88 ×10−31.37 × 1015.28 × 1013.29 × 1013.11 × 10−36.71 × 1017.99 × 1017.34
F43.02 × 1051.30 × 1022.30 × 1043.84 × 1023.36 × 1021.47 × 1031.56 × 1011.49 × 1025.70 × 1021.09 × 102
F51.50 × 1052.47 × 1011.58 × 10−44.01 × 1028.76 × 1011.73 × 1033.93 × 10−53.63 × 1011.50 × 1039.94
F63.59 × 10−19.25 × 1018.26 × 10−46.78 × 10−21.11 × 10−12.09 × 10−14.04 × 10−31.84 × 10−19.38 × 10−24.08 × 10−2
F72.03 × 1025.39 × 1013.99 × 10−41.58 × 1021.49 × 1028.04 × 1013.001.24 × 1021.30 × 1027.12 × 101
F88.82 × 1012.92 × 1012.74 × 1013.07 × 1014.65 × 1016.04 × 1012.69 × 1013.34 × 1016.14 × 1013.76 × 101
F99.35 × 1039.48 × 1032.80 × 10−13.51 × 1036.82 × 1033.86 × 1031.18 × 10−14.37 × 1036.41 × 1032.24 × 103
F103.86 × 1047.85 × 1038.53 × 10−41.71 × 1042.41 × 1047.43 × 1032.10 × 1033.54 × 1032.83 × 1041.32 × 104
F119.55 × 1093.62 × 1091.12 × 10−23.96 × 1014.91 × 1011.76 × 1017.09 × 10−26.98 × 1014.82 × 1012.42 × 101
F121.01 × 1015.801.38 × 10−23.98 × 10−18.821.01 × 1012.83 × 10−11.24 × 1011.01 × 1016.66
F136.52 × 1029.668.86 × 10−41.741.62 × 1023.43 × 1023.56 × 10−25.71 × 1013.17 × 1029.67
F148.70 × 1023.251.16 × 10−51.88 × 1011.10 × 1027.11 × 1011.50 × 10−11.07 × 1013.05 × 1025.62 × 101
F152.27 × 1038.996.67 × 10−11.63 × 1021.77 × 1023.73 × 1024.63 × 10−11.70 × 1014.56 × 1021.83 × 101
Total00100005000

Bold font in the table is used to highlight the “optimal values” among all methods.

Table 4

Worst values of the benchmark functions (the dimension of function is 30)

KHKHLDTOKHOKHCKHIKHMSKHSKHLKHMKH
F13.392.760.0101.193.452.860.0313.363.341.65
F216.091.680.0101.234.656.000.0632.946.551.24
F32.40 × 1025.93 × 1011.67 × 10−28.43 × 1011.49 × 1027.51 × 1016.55× 10−32.59 × 1021.74 × 1021.89 × 101
F41.91 × 1063.31 × 1026.85 × 10−49.33 × 1025.06 × 1022.54 × 1049.35 × 1015.63 × 1021.07 × 1032.95 × 102
F57.95 × 1054.91 × 1015.12 × 10−47.30 × 1039.46 × 1021.31 × 1049.79× 10−55.19 × 1017.13 × 1032.33 × 101
F67.97 × 10−11.50 × 1022.77 × 10−31.74 × 10−11.80 × 10−16.17 × 10−11.79 × 10−24.28 × 10−12.77 × 10−17.70 × 10−2
F72.32 × 1021.86 × 1021.58 × 10−32.04 × 1021.84 × 1021.25 × 1022.99 × 1012.58 × 1021.61 × 1021.07 × 102
F81.40 × 1023.30 × 1012.77 × 1015.23 × 1016.40 × 1018.74 × 1012.81 × 1014.78 × 1011.11 × 1025.06 × 101
F91.01 × 1041.01 × 1043.82 × 10−15.61 × 1038.58 × 1034.91 × 1032.67 × 10−15.46 × 1037.19 × 1033.30 × 103
F106.09 × 1041.43 × 1045.16 × 10−32.73 × 1043.58 × 1041.28 × 1041.50 × 1041.23 × 1043.59 × 1041.87 × 104
F111.28 × 10118.28 × 10102.53 × 10−26.32 × 1017.61 × 1013.32 × 1011.04 × 10−11.02 × 1026.62 × 1015.85 × 101
F121.39 × 1019.944.85 × 10−27.74 × 10−11.34 × 1011.71 × 1013.481.99 × 1011.36 × 1011.11 × 101
F131.44 × 1033.33 × 1013.56 × 10−38.944.38 × 1027.61 × 1029.15 × 10−22.97 × 1025.89 × 1023.67 × 101
F141.42 × 1031.05 × 1018.07 × 10−57.34 × 1012.48 × 1021.66 × 1022.987.77 × 1015.74 × 1021.52 × 102
F157.37 × 1031.74 × 1016.67 × 10−12.74 × 1036.38 × 1021.19 × 1037.51 × 10−14.85 × 1011.07 × 1034.35 × 101
Total00120003000
KHKHLDTOKHOKHCKHIKHMSKHSKHLKHMKH
F13.392.760.0101.193.452.860.0313.363.341.65
F216.091.680.0101.234.656.000.0632.946.551.24
F32.40 × 1025.93 × 1011.67 × 10−28.43 × 1011.49 × 1027.51 × 1016.55× 10−32.59 × 1021.74 × 1021.89 × 101
F41.91 × 1063.31 × 1026.85 × 10−49.33 × 1025.06 × 1022.54 × 1049.35 × 1015.63 × 1021.07 × 1032.95 × 102
F57.95 × 1054.91 × 1015.12 × 10−47.30 × 1039.46 × 1021.31 × 1049.79× 10−55.19 × 1017.13 × 1032.33 × 101
F67.97 × 10−11.50 × 1022.77 × 10−31.74 × 10−11.80 × 10−16.17 × 10−11.79 × 10−24.28 × 10−12.77 × 10−17.70 × 10−2
F72.32 × 1021.86 × 1021.58 × 10−32.04 × 1021.84 × 1021.25 × 1022.99 × 1012.58 × 1021.61 × 1021.07 × 102
F81.40 × 1023.30 × 1012.77 × 1015.23 × 1016.40 × 1018.74 × 1012.81 × 1014.78 × 1011.11 × 1025.06 × 101
F91.01 × 1041.01 × 1043.82 × 10−15.61 × 1038.58 × 1034.91 × 1032.67 × 10−15.46 × 1037.19 × 1033.30 × 103
F106.09 × 1041.43 × 1045.16 × 10−32.73 × 1043.58 × 1041.28 × 1041.50 × 1041.23 × 1043.59 × 1041.87 × 104
F111.28 × 10118.28 × 10102.53 × 10−26.32 × 1017.61 × 1013.32 × 1011.04 × 10−11.02 × 1026.62 × 1015.85 × 101
F121.39 × 1019.944.85 × 10−27.74 × 10−11.34 × 1011.71 × 1013.481.99 × 1011.36 × 1011.11 × 101
F131.44 × 1033.33 × 1013.56 × 10−38.944.38 × 1027.61 × 1029.15 × 10−22.97 × 1025.89 × 1023.67 × 101
F141.42 × 1031.05 × 1018.07 × 10−57.34 × 1012.48 × 1021.66 × 1022.987.77 × 1015.74 × 1021.52 × 102
F157.37 × 1031.74 × 1016.67 × 10−12.74 × 1036.38 × 1021.19 × 1037.51 × 10−14.85 × 1011.07 × 1034.35 × 101
Total00120003000

Bold font in the table is used to highlight the “optimal values” among all methods.

Table 5

Standard deviations of the benchmark functions (the dimension of function is 30)

KHKHLDTOKHOKHCKHIKHMSKHSKHLKHMKH
F10.340.490.000.370.460.340.000.480.350.56
F22.720.140.000.140.961.200.0100.541.260.080
F35.00 × 1011.20 × 1013.27 × 10−32.02 × 1012.87 × 1011.36 × 1011.67 × 10−35.90 × 1012.86 × 1013.75
F44.58 × 1057.71 × 1011.31 × 10−41.88 × 1029.86 × 1014.78 × 1033.54 × 1011.02 × 1022.15 × 1026.24 × 101
F51.85 × 1057.919.96 × 10−51.41 × 1031.73 × 1022.99 × 1032.35 × 10−58.192.22 × 1035.55
F62.07 × 10−13.05 × 1015.88 × 10−44.06 × 10−24.66 × 10−21.30 × 10−14.07 × 10−39.48 × 10−24.72 × 10−22.04 × 10−2
F71.85 × 1015.57 × 1014.17 × 10−42.50 × 1012.12 × 1011.90 × 1019.114.77 × 1012.00 × 1011.34 × 101
F82.27 × 1011.011.39 × 10−15.718.831.48 × 1015.084.511.55 × 1015.61
F93.67 × 1024.13 × 1027.16 × 10−25.18 × 1025.58 × 1027.37 × 1025.56 × 10−26.88 × 1025.09 × 1024.59 × 102
F109.81 × 1032.67 × 1031.30 × 10−34.66 × 1035.62 × 1032.21 × 1033.43 × 1032.67 × 1034.69 × 1033.08 × 103
F112.58 × 10101.52 × 10106.17 × 10−31.12 × 1017.965.881.55 × 10−21.80 × 1019.941.10 × 101
F121.931.841.00 × 10−21.13 × 10−11.572.396.51 × 10−13.572.131.57
F132.69 × 1027.669.11 × 10−41.579.20 × 1011.86 × 1021.81 × 10−26.51 × 1011.52 × 1027.71
F142.71 × 1022.561.65 × 10−51.82 × 1014.93 × 1014.20 × 1015.52 × 10−11.58 × 1011.13 × 1023.81 × 101
F151.79 × 1033.615.61 × 10−55.05 × 1021.34 × 1022.77 × 1022.32 × 10−11.00 × 1012.73 × 1029.04
Total00120003000
KHKHLDTOKHOKHCKHIKHMSKHSKHLKHMKH
F10.340.490.000.370.460.340.000.480.350.56
F22.720.140.000.140.961.200.0100.541.260.080
F35.00 × 1011.20 × 1013.27 × 10−32.02 × 1012.87 × 1011.36 × 1011.67 × 10−35.90 × 1012.86 × 1013.75
F44.58 × 1057.71 × 1011.31 × 10−41.88 × 1029.86 × 1014.78 × 1033.54 × 1011.02 × 1022.15 × 1026.24 × 101
F51.85 × 1057.919.96 × 10−51.41 × 1031.73 × 1022.99 × 1032.35 × 10−58.192.22 × 1035.55
F62.07 × 10−13.05 × 1015.88 × 10−44.06 × 10−24.66 × 10−21.30 × 10−14.07 × 10−39.48 × 10−24.72 × 10−22.04 × 10−2
F71.85 × 1015.57 × 1014.17 × 10−42.50 × 1012.12 × 1011.90 × 1019.114.77 × 1012.00 × 1011.34 × 101
F82.27 × 1011.011.39 × 10−15.718.831.48 × 1015.084.511.55 × 1015.61
F93.67 × 1024.13 × 1027.16 × 10−25.18 × 1025.58 × 1027.37 × 1025.56 × 10−26.88 × 1025.09 × 1024.59 × 102
F109.81 × 1032.67 × 1031.30 × 10−34.66 × 1035.62 × 1032.21 × 1033.43 × 1032.67 × 1034.69 × 1033.08 × 103
F112.58 × 10101.52 × 10106.17 × 10−31.12 × 1017.965.881.55 × 10−21.80 × 1019.941.10 × 101
F121.931.841.00 × 10−21.13 × 10−11.572.396.51 × 10−13.572.131.57
F132.69 × 1027.669.11 × 10−41.579.20 × 1011.86 × 1021.81 × 10−26.51 × 1011.52 × 1027.71
F142.71 × 1022.561.65 × 10−51.82 × 1014.93 × 1014.20 × 1015.52 × 10−11.58 × 1011.13 × 1023.81 × 101
F151.79 × 1033.615.61 × 10−55.05 × 1021.34 × 1022.77 × 1022.32 × 10−11.00 × 1012.73 × 1029.04
Total00120003000

Bold font in the table is used to highlight the “optimal values” among all methods.

Table 6

Mean values of the benchmark functions (the dimension of function is 10)

KHKHLDTOKHOKHCKHIKHMSKHSKHLKHMKH
F12.112.00 × 10−20.001.20 × 10−15.40 × 10−14.10 × 10−14.50 × 10−21.371.426.10 × 10−2
F26.20 × 10−11.40 × 10−10.005.30 × 10−15.00 × 10−12.20 × 10−13.40 × 10−22.70 × 10−14.10 × 10−12.60 × 10−1
F39.689.00 × 10−20.003.50 × 10−13.422.069.40 × 10−12.15 × 1013.159.10 × 10−2
F42.56 × 1011.561.63 × 10−44.30 × 10−14.482.093.461.19 × 1015.393.10 × 10−1
F57.069.00 × 10−21.44 × 10−42.00 × 10−15.00 × 10−15.90 × 10−15.43 × 10−61.242.011.33 × 10−4
F67.00 × 10−31.70 × 10−31.00 × 10−43.79 × 10−48.41 × 10−45.18 × 10−42.00 × 10−22.40 × 10−31.20 × 10−33.14 × 10−4
F75.30 × 10−29.90 × 10−11.03 × 10−59.80 × 10−21.182.981.00 × 10−34.972.051.20 × 10−1
F88.125.776.561.425.661.30 × 10−19.08 × 10−54.405.462.01
F91.99 × 1031.65 × 1031.50 × 10−28.55 × 1026.74 × 1024.60 × 10−21.60 × 10−33.36 × 1027.38 × 1021.50 × 10−2
F104.47 × 1021.72 × 1018.77 × 10−62.35 × 1011.67 × 1026.116.99 × 10−53.80 × 10−11.27 × 1026.10 × 10−1
F111.81 × 1016.275.18 × 10−41.945.00 × 10−12.90 × 10−31.82 × 10−31.85 × 10−34.964.56 × 10−3
F124.40 × 10−21.70 × 10−21.70 × 10−35.90 × 10−25.30 × 10−27.00 × 10−31.10 × 10−25.00 × 10−31.80 × 10−26.90 × 10−3
F131.10 × 10−22.50 × 10−33.14 × 10−58.20 × 10−37.90 × 10−35.33 × 10−61.50 × 10−33.05 × 10−63.85 × 10−42.77 × 10−5
F146.452.20 × 10−31.37 × 10−98.08 × 10−42.00 × 10−22.29 × 10−44.04 × 10−71.87 × 10−45.60 × 10−12.55 × 10−5
F157.10 × 10−16.60 × 10−14.08 × 10−46.70 × 10−11.00 × 10−11.00 × 10−11.01 × 10−47.41 × 10−51.30 × 10−12.20 × 10−3
Total00100013100
KHKHLDTOKHOKHCKHIKHMSKHSKHLKHMKH
F12.112.00 × 10−20.001.20 × 10−15.40 × 10−14.10 × 10−14.50 × 10−21.371.426.10 × 10−2
F26.20 × 10−11.40 × 10−10.005.30 × 10−15.00 × 10−12.20 × 10−13.40 × 10−22.70 × 10−14.10 × 10−12.60 × 10−1
F39.689.00 × 10−20.003.50 × 10−13.422.069.40 × 10−12.15 × 1013.159.10 × 10−2
F42.56 × 1011.561.63 × 10−44.30 × 10−14.482.093.461.19 × 1015.393.10 × 10−1
F57.069.00 × 10−21.44 × 10−42.00 × 10−15.00 × 10−15.90 × 10−15.43 × 10−61.242.011.33 × 10−4
F67.00 × 10−31.70 × 10−31.00 × 10−43.79 × 10−48.41 × 10−45.18 × 10−42.00 × 10−22.40 × 10−31.20 × 10−33.14 × 10−4
F75.30 × 10−29.90 × 10−11.03 × 10−59.80 × 10−21.182.981.00 × 10−34.972.051.20 × 10−1
F88.125.776.561.425.661.30 × 10−19.08 × 10−54.405.462.01
F91.99 × 1031.65 × 1031.50 × 10−28.55 × 1026.74 × 1024.60 × 10−21.60 × 10−33.36 × 1027.38 × 1021.50 × 10−2
F104.47 × 1021.72 × 1018.77 × 10−62.35 × 1011.67 × 1026.116.99 × 10−53.80 × 10−11.27 × 1026.10 × 10−1
F111.81 × 1016.275.18 × 10−41.945.00 × 10−12.90 × 10−31.82 × 10−31.85 × 10−34.964.56 × 10−3
F124.40 × 10−21.70 × 10−21.70 × 10−35.90 × 10−25.30 × 10−27.00 × 10−31.10 × 10−25.00 × 10−31.80 × 10−26.90 × 10−3
F131.10 × 10−22.50 × 10−33.14 × 10−58.20 × 10−37.90 × 10−35.33 × 10−61.50 × 10−33.05 × 10−63.85 × 10−42.77 × 10−5
F146.452.20 × 10−31.37 × 10−98.08 × 10−42.00 × 10−22.29 × 10−44.04 × 10−71.87 × 10−45.60 × 10−12.55 × 10−5
F157.10 × 10−16.60 × 10−14.08 × 10−46.70 × 10−11.00 × 10−11.00 × 10−11.01 × 10−47.41 × 10−51.30 × 10−12.20 × 10−3
Total00100013100

Bold font in the table is used to highlight the “optimal values” among all methods.

Table 7

Mean values of the benchmark functions (the dimension of function is 50)

KHKHLDTOKHOKHCKHIKHMSKHSKHLKHMKH
F11.11 × 1017.458.60 × 10−33.831.05 × 1011.02 × 1018.03 × 10−21.07 × 1011.06 × 1018.48
F24.41 × 1012.05 × 1011.90 × 10−31.033.43 × 1014.52 × 1019.29 × 10−22.99 × 1013.50 × 1011.87 × 101
F34.91 × 1018.991.30 × 10−33.332.45 × 1012.86 × 1017.11 × 10−22.16 × 1012.52 × 1019.93
F48.02 × 1062.86 × 1034.60 × 10−33.28 × 1021.98 × 1067.56 × 1066.78 × 1016.37 × 1032.88 × 1062.51 × 105
F52.07 × 1061.70 × 1042.20 × 10−35.148.07 × 1051.54 × 1061.571.13 × 1056.99 × 1052.05 × 105
F62.514.09 × 1023.00 × 10−35.80 × 10−21.522.012.84 × 10−21.781.598.52 × 10−1
F74.48 × 1022.17 × 1022.00 × 10−33.03 × 1023.37 × 1022.61 × 1021.40 × 1013.22 × 1022.75 × 1022.04 × 102
F81.95 × 1062.45 × 1054.88 × 1017.64 × 1021.21 × 1061.66 × 1066.79 × 1015.10 × 1051.11 × 1064.13 × 105
F91.75 × 1041.73 × 1049.00 × 10−17.40 × 1031.39 × 1041.05 × 1041.19 × 1021.07 × 1041.36 × 1048.23 × 103
F101.56 × 1053.53 × 1042.314.12 × 1047.81 × 1044.98 × 1045.17 × 1043.55 × 1042.17 × 1045.88 × 104
F114.20 × 10222.76 × 10202.46 × 10−21.50 × 1091.07 × 10108.99 × 1011.71 × 10−11.71 × 1029.83 × 1071.26 × 102
F122.48 × 1011.42 × 1012.10 × 10−23.32 × 10−12.29 × 1012.44 × 1011.162.54 × 1012.02 × 1012.11 × 101
F134.78 × 1032.03 × 1031.70 × 10−31.45 × 1014.19 × 1034.72 × 1031.13 × 10−13.14 × 1034.44 × 1031.87 × 103
F143.27 × 1031.14 × 1026.29 × 10−54.42 × 1016.15 × 1027.68 × 1022.45 × 1012.14 × 1025.27 × 1023.35 × 102
F153.02 × 1043.02 × 1038.42 × 10−16.051.57 × 1042.08 × 1042.547.15 × 1031.60 × 1046.28 × 103
Total00150000000
KHKHLDTOKHOKHCKHIKHMSKHSKHLKHMKH
F11.11 × 1017.458.60 × 10−33.831.05 × 1011.02 × 1018.03 × 10−21.07 × 1011.06 × 1018.48
F24.41 × 1012.05 × 1011.90 × 10−31.033.43 × 1014.52 × 1019.29 × 10−22.99 × 1013.50 × 1011.87 × 101
F34.91 × 1018.991.30 × 10−33.332.45 × 1012.86 × 1017.11 × 10−22.16 × 1012.52 × 1019.93
F48.02 × 1062.86 × 1034.60 × 10−33.28 × 1021.98 × 1067.56 × 1066.78 × 1016.37 × 1032.88 × 1062.51 × 105
F52.07 × 1061.70 × 1042.20 × 10−35.148.07 × 1051.54 × 1061.571.13 × 1056.99 × 1052.05 × 105
F62.514.09 × 1023.00 × 10−35.80 × 10−21.522.012.84 × 10−21.781.598.52 × 10−1
F74.48 × 1022.17 × 1022.00 × 10−33.03 × 1023.37 × 1022.61 × 1021.40 × 1013.22 × 1022.75 × 1022.04 × 102
F81.95 × 1062.45 × 1054.88 × 1017.64 × 1021.21 × 1061.66 × 1066.79 × 1015.10 × 1051.11 × 1064.13 × 105
F91.75 × 1041.73 × 1049.00 × 10−17.40 × 1031.39 × 1041.05 × 1041.19 × 1021.07 × 1041.36 × 1048.23 × 103
F101.56 × 1053.53 × 1042.314.12 × 1047.81 × 1044.98 × 1045.17 × 1043.55 × 1042.17 × 1045.88 × 104
F114.20 × 10222.76 × 10202.46 × 10−21.50 × 1091.07 × 10108.99 × 1011.71 × 10−11.71 × 1029.83 × 1071.26 × 102
F122.48 × 1011.42 × 1012.10 × 10−23.32 × 10−12.29 × 1012.44 × 1011.162.54 × 1012.02 × 1012.11 × 101
F134.78 × 1032.03 × 1031.70 × 10−31.45 × 1014.19 × 1034.72 × 1031.13 × 10−13.14 × 1034.44 × 1031.87 × 103
F143.27 × 1031.14 × 1026.29 × 10−54.42 × 1016.15 × 1027.68 × 1022.45 × 1012.14 × 1025.27 × 1023.35 × 102
F153.02 × 1043.02 × 1038.42 × 10−16.051.57 × 1042.08 × 1042.547.15 × 1031.60 × 1046.28 × 103
Total00150000000

Bold font in the table is used to highlight the “optimal values” among all methods.

When the dimension of the benchmark function was 30, TOKH and MSKH exhibited the best optimization performance for nine and seven benchmark functions, respectively (Table 2). TOKH algorithm performed better than the other KH algorithms on F1–F2, F6–F7, and F10–F14; MSKH algorithm performed best on F2–F5, F8–F9, and F15. About the general optimization performance (Table 3), TOKH algorithm performed best on ten benchmark functions (F1–F2, F4, F6–F7, and F10–F14), and MSKH algorithm performed best on five benchmark functions (F3, F5, F8–F9, and F15). Furthermore, TOKH algorithm performed best on twelve benchmark functions (F1–F2, F4, F6–F8, and F10–F15) with respect to the worst optimization performance (Table 4). In terms of stability (Table 5), TOKH algorithm exhibited the best stability for twelve benchmark functions (F1–F2, F4, F6–F8, and F10–F15). As indicated in Table 6, when the dimension of the benchmark function was 10, TOKH algorithm performed best on ten benchmark functions (F1–F4, F6–F7, F10–F12, and F14), MSKH algorithm performed best on F5 and F8–F9, and SKH exhibited the best performance on F15. When the dimension of the benchmark function was 50, TOKH algorithm performed best for all 15 benchmark functions (Table 7). Figure 3 illustrates the convergence process of several benchmark functions for 30-dimension. As indicated in the figure, TOKH algorithm was significantly superior to all other algorithms with respect to the optimization process.

Fig. 3
Performance comparison considering the 30-dimension benchmark functionsPerformance comparison considering the 30-dimension benchmark functions
Fig. 3
Performance comparison considering the 30-dimension benchmark functionsPerformance comparison considering the 30-dimension benchmark functions
Close modal

Based on the data presented in Tables 27 and Fig. 3, we concluded that the developed metaheuristic TOKH algorithm was more robust and stable than other metaheuristic search methods.

6 Truss Optimization Problems

To further investigate the robustness of TOKH algorithm for truss optimization, we solved the weight minimization problems of four truss structures under multiple loading conditions using discrete variables. The algorithms were coded in MATLAB and the structures were analyzed using the direct stiffness method. The optimization results were compared to the results obtained from other optimization methods (including TLBO [23], PO [28], MRFO [33], KH [34], CKH [38], OKH [39], IKH [41], KHLD [43], and LKH [45].) in the literature to evaluate the robustness of TOKH algorithm. Twenty independent runs were performed for each design problem with the population size of each algorithm set to 30.

6.1 Planar 10-Bar Truss.

The planar 10-bar truss structure is one of the most popular test problems in structural optimization, previously solved in Ref. [23]. Figure 4 depicts the geometry and support conditions followed for this two-dimensional, cantilevered truss under loading conditions. As indicated in the figure a static load of 100 kips was applied downward to two nodes. To satisfy the stress and displacement constraints, the minimum weight of the 10-bar truss was obtained by adjusting the cross-sectional area of each member. A set of 41 discrete values were used for the possible cross-sectional areas for each member, as follows: S ={1.62, 1.80, 1.99, 2.13, 2.38, 2.62, 2.88, 2.93, 3.09, 3.13, 3.38, 3.47, 3.55, 3.63, 3.84, 3.87, 3.88, 4.18, 4.22, 4.49, 4.59,4.80, 4.97, 5.12, 5.74, 7.22, 7.97, 11.5, 13.5, 13.9, 14.2, 15.5, 16.0, 16.9, 18.8, 19.9, 22.0, 22.9, 26.5, 30.0, 33.5} (in2). The combined size of all feasible solutions was (41)10. The Young's modulus of the material was 107 psi, and the weight density was 0.1 lb/in3. The displacement constraints in the X and Y directions at each node were limited to 2 in. The restraining stress of each member was less than 25 ksi.

Fig. 4

Table 8 summarizes the optimal designs and their corresponding number of structural analyses (NSA) using the TOKH with nine other methods. TOKH, TLBO, and MRFO achieved the lightest designs. However, statistical results after 20 runs also demonstrate that TLBO and MRFO exhibited less stability compared to TOKH. Based on the average of 20 independent runs, the NSA is 3100 for TOKH and 1580 required for KH. Although TOKH has 1520 more structural analyses than KH, the optimization efficiency of TOKH algorithm is 20.90% higher than that of KH algorithm. Figure 5 shows the average convergence curves obtained for TOKH and the nine different methods when the planar 10-bar truss is applied. This is a relatively simple structural sizing optimization problem, and its optimal sizing is easy to find. It is obvious that through efficient global exploration, TOKH quickly converges to the minimum with fewer iterations. The TOKH algorithm has the highest global convergence rate. For other algorithms, MRFO works very well, because it ranks 2 among ten methods.

Fig. 5
Algorithm optimization process of the 10-bar truss
Fig. 5
Algorithm optimization process of the 10-bar truss
Close modal
Table 8

Optimized designs for the 10-bar truss

KHKHLDCKHLKHIKHTOKHOKHTLBOPOMRFO
S130.0033.5033.5030.0030.0033.5028.5033.5033.5033.50
S21.623.555.942.131.801.6213.501.621.621.62
S328.5028.5022.9028.5022.9022.9030.0022.9022.9022.90
S415.5011.5015.5011.5018.8014.2018.8014.2015.5014.20
S51.621.621.621.621.621.625.941.621.621.62
S61.625.941.621.991.621.625.941.621.621.62
S719.9018.8022.0016.9022.9022.9022.9022.9018.8022.90
S87.9711.507.9716.0011.507.977.227.977.227.97
S92.133.551.624.181.621.625.941.621.621.62
S1022.0018.8022.0022.9018.8022.9018.8022.9028.5022.90
NSA1580.001580.001580.003080.001930.003100.006080.004410.001780.003030.00
Computational time (s)1.331.421.511.811.641.803.011.540.500.58
Optimal value (lb)5.63 × 1035.83 × 1035.65 × 1035.78 × 1035.59 × 1035.49 × 1036.54 × 1035.49 × 1035.62 × 1035.49 × 103
Mean (lb)7.13 × 1037.19 × 1036.13 × 1037.15 × 1036.85 × 1035.64 × 1036.83 × 1035.91 × 1036.56 × 1035.66 × 103
Std (lb)1.51 × 1039.66 × 1022.53 × 1021.31 × 1031.14 × 1034.84 × 1012.86 × 1022.50 × 1028.01 × 1021.09 × 102
Increased efficiency−0.84%14.03%−0.28%3.93%20.90%4.21%17.11%7.99%20.62%
KHKHLDCKHLKHIKHTOKHOKHTLBOPOMRFO
S130.0033.5033.5030.0030.0033.5028.5033.5033.5033.50
S21.623.555.942.131.801.6213.501.621.621.62
S328.5028.5022.9028.5022.9022.9030.0022.9022.9022.90
S415.5011.5015.5011.5018.8014.2018.8014.2015.5014.20
S51.621.621.621.621.621.625.941.621.621.62
S61.625.941.621.991.621.625.941.621.621.62
S719.9018.8022.0016.9022.9022.9022.9022.9018.8022.90
S87.9711.507.9716.0011.507.977.227.977.227.97
S92.133.551.624.181.621.625.941.621.621.62
S1022.0018.8022.0022.9018.8022.9018.8022.9028.5022.90
NSA1580.001580.001580.003080.001930.003100.006080.004410.001780.003030.00
Computational time (s)1.331.421.511.811.641.803.011.540.500.58
Optimal value (lb)5.63 × 1035.83 × 1035.65 × 1035.78 × 1035.59 × 1035.49 × 1036.54 × 1035.49 × 1035.62 × 1035.49 × 103
Mean (lb)7.13 × 1037.19 × 1036.13 × 1037.15 × 1036.85 × 1035.64 × 1036.83 × 1035.91 × 1036.56 × 1035.66 × 103
Std (lb)1.51 × 1039.66 × 1022.53 × 1021.31 × 1031.14 × 1034.84 × 1012.86 × 1022.50 × 1028.01 × 1021.09 × 102
Increased efficiency−0.84%14.03%−0.28%3.93%20.90%4.21%17.11%7.99%20.62%

6.2 Spatial 25-Bar Truss.

Figure 6 illustrates the 25-bar transmission tower spatial truss, which has been analyzed by several researchers [23]. All structural elements were organized into eight groups, where the members of each group shared the same material and cross-sectional properties. Table 9 presents each element group according to the member number; (each member is defined based on its start and end node numbers). Table 10 lists the coordinates of the 25-bar truss nodes. The Young's modulus of the material was 107 psi, and the weight density was 0.1 lb/in3. A single load case was applied to the structure in the design of the 25-bar truss (Table 11). The allowable stresses for each member were ±40 ksi and the allowable displacements for each node in the X, Y and Z directions were ±0.35 in. Discrete values for each cross-sectional area were obtained from the available set S = {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 3.0, 3.1, 3.2, 3.3, 3.4} (in2). The size of the design search space was (34)8.

Fig. 6
Table 9

Element information for the 25-bar truss

Design variablesMembers
S1(1,2)
S2(1,4), (2,3), (1,5), (2,6)
S3(2,5), (2,4), (1,3), (1,6)
S4(3,6), (4,5)
S5(3,4), (5,6)
S6(3,10), (6,7), (4,9), (5,8)
S7(3,8), (4,7), (6,9), (5,10)
S8(3,7), (4,8), (5,9), (6,10)
Design variablesMembers
S1(1,2)
S2(1,4), (2,3), (1,5), (2,6)
S3(2,5), (2,4), (1,3), (1,6)
S4(3,6), (4,5)
S5(3,4), (5,6)
S6(3,10), (6,7), (4,9), (5,8)
S7(3,8), (4,7), (6,9), (5,10)
S8(3,7), (4,8), (5,9), (6,10)
Table 10

Nodal coordinates for the 25-bar truss

NodeX (in.)Y (in.)Z (in.)
1−37.50.0200.0
237.50.0200.0
3−37.537.5100.0
437.537.5100.0
537.5−37.5100.0
6−37.5−37.5100.0
7−100.0100.00.0
8100.0100.00.0
9100.0−100.00.0
10−100.0−100.00.0
NodeX (in.)Y (in.)Z (in.)
1−37.50.0200.0
237.50.0200.0
3−37.537.5100.0
437.537.5100.0
537.5−37.5100.0
6−37.5−37.5100.0
7−100.0100.00.0
8100.0100.00.0
9100.0−100.00.0
10−100.0−100.00.0
Table 11

Loading conditions for the 25-bar truss

NodePx (kips)Py (kips)Pz (kips)
11.0−10.0−10.0
20.0−10.0−10.0
30.50.00.0
60.60.00.0
NodePx (kips)Py (kips)Pz (kips)
11.0−10.0−10.0
20.0−10.0−10.0
30.50.00.0
60.60.00.0

Table 12 compares the final optimum design and the corresponding results calculated by the ten methods. TOKH algorithm still has the robustness compared with the other nine methods. The average weight gained by TOKH algorithm is 488.34 lb, 17.37% lighter than that of KH algorithm. The standard deviation with the TOKH algorithm is 2.59, which is the lowest among all methods. Figure 7 depicts the average convergence curves for the 25-bar truss. TOKH quickly converges to a better global region in the early iterations and continues to search the minimum in about 10–20 iterations through local exploitation. MRFO algorithm converges slowly in the early stage and gradually overtakes TLBO and PO in the late iteration.

Fig. 7
Algorithm optimization process of the 25-bar truss
Fig. 7
Algorithm optimization process of the 25-bar truss
Close modal
Table 12

Optimized designs for the 25-bar truss

KHKHLDCKHLKHIKHTOKHOKHTLBOPOMRFO
S11.601.400.501.800.200.101.200.900.100.20
S21.500.700.502.800.100.501.200.300.401.10
S33.003.403.402.003.403.403.003.203.403.20
S40.100.700.100.400.100.100.900.100.100.20
S50.701.602.800.400.601.901.002.502.401.30
S60.900.701.101.001.001.000.900.901.100.90
S70.601.000.200.401.200.400.900.800.300.40
S83.403.403.403.403.403.403.203.403.403.40
NSA1580.001580.001580.003080.001930.003100.006080.004410.001780.003030.00
Computational time (s)0.720.770.861.400.981.901.382.180.821.13
Optimal value (lb)520.66531.48494.30540.05503.38485.05529.56501.86487.33493.83
Mean (lb)590.98558.37530.87571.76549.41488.34579.05523.36509.75500.76
Std (lb)39.8919.7218.5416.7436.362.5937.3914.3225.595.94
Increased efficiency5.52%10.17%3.25%7.03%17.37%2.02%11.44%13.74%15.27%
KHKHLDCKHLKHIKHTOKHOKHTLBOPOMRFO
S11.601.400.501.800.200.101.200.900.100.20
S21.500.700.502.800.100.501.200.300.401.10
S33.003.403.402.003.403.403.003.203.403.20
S40.100.700.100.400.100.100.900.100.100.20
S50.701.602.800.400.601.901.002.502.401.30
S60.900.701.101.001.001.000.900.901.100.90
S70.601.000.200.401.200.400.900.800.300.40
S83.403.403.403.403.403.403.203.403.403.40
NSA1580.001580.001580.003080.001930.003100.006080.004410.001780.003030.00
Computational time (s)0.720.770.861.400.981.901.382.180.821.13
Optimal value (lb)520.66531.48494.30540.05503.38485.05529.56501.86487.33493.83
Mean (lb)590.98558.37530.87571.76549.41488.34579.05523.36509.75500.76
Std (lb)39.8919.7218.5416.7436.362.5937.3914.3225.595.94
Increased efficiency5.52%10.17%3.25%7.03%17.37%2.02%11.44%13.74%15.27%

Bold font in the table is used to highlight the “optimal values” among all methods.

6.3 Spatial 72-Bar Truss.

The optimization of the 72-bar, four-level tower (Fig. 8) was reported in Ref. 23. The material density and modulus of elasticity of this 72-bar truss were identical to those of the 10- and 25-bar truss structures. The allowable stress of each member was ±25 ksi, and the permissible displacement of each point on the top layer in all directions was ±0.25 in. The 72-bar space truss was divided into 16 groups. Table 13 lists the two independent load cases applied to the spatial 72-bar space truss. The range of the discrete cross-sectional areas was 0.1–3.0 in2 with an increment of 0.1 in2 for each of the 16 element groups, resulting in 30 discrete cross section. The size of the resulting search space was (30)16 designs.

Fig. 8
72-bar truss: (a) side view and (b) typical story
Fig. 8
72-bar truss: (a) side view and (b) typical story
Close modal
Table 13

Load cases for the 72-bar truss

CaseNodePx (kips)Py (kips)Pz (kips)
117.00.00.0−5.0
18.00.00.0−5.0
19.00.00.0−5.0
20.00.00.0−5.0
217.05.05.0−5.0
CaseNodePx (kips)Py (kips)Pz (kips)
117.00.00.0−5.0
18.00.00.0−5.0
19.00.00.0−5.0
20.00.00.0−5.0
217.05.05.0−5.0

Table 14 compares the final optimum design and the corresponding results identified by the ten methods. The lightest weight and average weight achieved by TOKH algorithm, which are 387.94 lb and 402.30 lb, respectively. Based on the average weight of 20 independent runs, although TOKH algorithm has 1520 more structural analyses than KH, the optimization efficiency of TOKH is 53.53% higher than KH. These results indicate that TOKH algorithm has an apparent advantage in search ability compared with the other nine methods. Figure 9 shows the optimization results for the spatial 72-bar truss, which is a complex structural sizing optimization problem. From Fig. 9, different from the planar 10-bar truss as shown in Fig. 5, the figure shows that TOKH quickly converges to a better global region in the early iterations by global exploration and continues to find the minimum in about 10–30 iterations through efficient local exploitation. MRFO algorithm still has the disadvantage of slow convergence in the early stage, and as the iterations increase, the convergence speed gradually surpasses that of TLBO and PO.

Fig. 9
Algorithm optimization process of the 72-bar truss
Fig. 9
Algorithm optimization process of the 72-bar truss
Close modal
Table 14

Optimized designs for the 72-bar truss

KHKHLDCKHLKHIKHTOKHOKHTLBOPOMRFO
S11.101.601.800.901.902.001.200.501.701.70
S20.400.700.600.800.400.501.000.600.700.60
S30.300.600.500.200.200.100.100.100.100.10
S40.701.301.000.200.600.100.101.800.100.30
S50.601.801.201.701.901.101.300.401.301.60
S60.800.800.500.400.800.500.800.700.600.40
S70.901.200.701.600.100.100.100.300.100.20
S80.901.600.201.500.200.100.400.600.100.20
S91.401.001.401.000.800.600.800.501.400.70
S101.600.600.500.500.400.601.100.100.400.50
S110.800.400.401.700.200.100.100.400.100.10
S121.700.400.501.600.600.100.701.600.100.30
S130.300.600.401.500.200.201.100.500.200.40
S140.600.300.601.000.600.501.100.700.500.90
S150.501.300.301.300.300.500.101.900.300.40
S161.800.700.600.900.700.600.500.500.800.40
NSA1580.001580.001580.003080.001930.003100.006080.004410.001780.003030.00
Computational time (s)1.811.901.823.332.274.817.125.581.963.09
Optimal value (lb)739.63681.36520.59785.15461.01387.94644.58449.94412.66419.00
Mean (lb)865.71782.99584.17821.81502.50402.30705.74523.17463.53442.38
Std (lb)67.6165.8349.7833.3824.936.0440.2048.8151.4318.21
Increased efficiency9.56%32.52%5.07%41.96%53.53%18.48%39.57%46.46%48.90%
KHKHLDCKHLKHIKHTOKHOKHTLBOPOMRFO
S11.101.601.800.901.902.001.200.501.701.70
S20.400.700.600.800.400.501.000.600.700.60
S30.300.600.500.200.200.100.100.100.100.10
S40.701.301.000.200.600.100.101.800.100.30
S50.601.801.201.701.901.101.300.401.301.60
S60.800.800.500.400.800.500.800.700.600.40
S70.901.200.701.600.100.100.100.300.100.20
S80.901.600.201.500.200.100.400.600.100.20
S91.401.001.401.000.800.600.800.501.400.70
S101.600.600.500.500.400.601.100.100.400.50
S110.800.400.401.700.200.100.100.400.100.10
S121.700.400.501.600.600.100.701.600.100.30
S130.300.600.401.500.200.201.100.500.200.40
S140.600.300.601.000.600.501.100.700.500.90
S150.501.300.301.300.300.500.101.900.300.40
S161.800.700.600.900.700.600.500.500.800.40
NSA1580.001580.001580.003080.001930.003100.006080.004410.001780.003030.00
Computational time (s)1.811.901.823.332.274.817.125.581.963.09
Optimal value (lb)739.63681.36520.59785.15461.01387.94644.58449.94412.66419.00
Mean (lb)865.71782.99584.17821.81502.50402.30705.74523.17463.53442.38
Std (lb)67.6165.8349.7833.3824.936.0440.2048.8151.4318.21
Increased efficiency9.56%32.52%5.07%41.96%53.53%18.48%39.57%46.46%48.90%

6.4 A 942-Bar Truss Tower.

The final example considered in this study is that of a 26-story space truss tower consisting of 942 bars and 244 nodes schematically depicted in Fig. 10. This problem aims to identify the lightest design with the design variables defined as the member cross-sectional areas and divided into 59 group. A single load case is considered such that it consists of lateral loads of 5.0 KN (1.12 kips) applied in both x- and y-directions and a vertical load of –30 KN (–6.74 kips) applied in the z-direction at all nodes of the tower. The density and elastic modulus of the material are 2767.99 kg/m3 (0.1 lb/in3) and 69 GPa (1.0 × 104 ksi), respectively. The constraint conditions include allowable stresses and displacements for the truss tower. The maximum allowable stress in each member under tension and compression equals 172.37 MPa (25 ksi) while the maximum allowable displacement in x, y, z directions for the all the nodes is 38.1 cm (15.0 in). A discrete set of 137 economical standard steel sections selected from W-shape profile list based on area and radii of gyration properties is used to size the variables. The lower and upper bounds on size variables are taken as 6.16 in2 and 215.0 in2, respectively Ref. [49]. The size of the design search space is (137)59. Further details regarding member grouping and design constraints can be found in Ref. [50].

Fig. 10
A 942-bar truss tower
Fig. 10
A 942-bar truss tower
Close modal

Compared with the above examples, this truss tower includes more elements and load cases. Table 15 lists the final optimum design and the corresponding results calculated by the ten methods. Even though the lightest weight gained by TOKH algorithm is marginally heavier than PO algorithm, TOKH algorithm has the smallest average weight and standard deviation of the optimum weight among all the approaches. Regarding computational burden, due to a more discrete and nonlinear search space, TOKH requires 12,330 structural analyses and KH requires 6,230 structural analyses. Although TOKH requires a greater NSA iteration compared to KH algorithm, the truss weight is 88.01% lighter than KH. While considering both the search robustness and the optimize efficiency, the proposed achieves markedly better performance. Figure 11 shows the optimization process for the 942-bar truss tower, which is a complex structural sizing optimization problem with high dimensional design and nonlinear buckling constraints. When attempting to solve this optimization problem, methods may easily trap into a local optimum. Hence, a method capable of maintaining both efficient global exploration and local exploitation is likely to produce better results. As can be seen in Fig. 11, in the early iterations, TOKH can quickly converge to a better global region by global exploration. Then, the minimum is further found in about 10–100 iterations through effective local exploitation. We can draw the conclusion that, TOKH is superior to the other algorithms during the process of optimization, while PO and MRFO performs the second and the third best in this complex truss sizing optimization, respectively.

Fig. 11
Algorithm optimization process of the 942-bar truss
Fig. 11
Algorithm optimization process of the 942-bar truss
Close modal
Table 15

Optimized designs for the 942-bar truss

KHKHLDCKHLKHIKHTOKHOKHTLBOPOMRFO
S117.6017.60101.0038.2035.106.1623.2059.2031.1011.50
S248.8020.0018.3042.1038.806.4811.8091.506.1616.80
S316.2032.7051.3061.8017.906.1620.8051.807.6815.60
S468.8021.8050.0015.6083.306.169.1348.806.167.69
S58.2515.606.166.4917.606.1611.2035.906.4913.00
S68.2525.90118.0010.6020.806.1614.70147.0010.609.71
S7178.0019.1056.806.1618.308.2510.306.166.168.25
S8196.0024.107.6110.6012.606.169.13162.006.1614.10
S926.5068.8035.10101.0032.0027.3025.3032.9013.2029.80
S107.0813.3074.1017.908.2511.207.617.086.1617.10
S1114.70101.0083.3088.2020.807.6123.206.166.169.71
S1243.2028.2044.7015.8027.308.8417.9059.206.168.25
S1344.7055.8038.808.7931.106.1638.806.166.1614.10
S1429.809.716.1611.509.137.6920.8059.206.167.08
S1544.7015.8081.9026.506.498.257.08117.006.1614.10
S1643.2015.806.1610.6039.907.0825.3023.206.168.85
S1761.8050.00109.0019.7018.306.4931.1074.1013.3011.20
S1832.9017.906.1610.0042.706.1610.606.166.168.85
S1935.1088.20125.0035.9021.806.4811.80162.006.4911.70
S2022.6035.9028.5046.7051.306.1611.2072.806.1611.80
S2142.1035.3038.206.1644.7020.0010.3053.6031.1014.70
S227.698.796.1615.6024.106.1612.606.166.169.12
S23162.0011.2074.1013.5014.708.7914.1080.806.4816.20
S2420.8016.707.0811.80118.007.0814.10178.0014.1028.20
S257.0817.6022.3015.6017.6023.2013.20178.0029.4029.10
S2655.8031.1091.5011.2015.6023.209.719.716.1612.60
S2761.8083.3055.808.8514.409.1312.6018.306.1621.80
S28162.0072.8039.9053.606.166.1625.3056.806.4819.70
S2998.8031.106.1615.608.256.1621.1089.607.6913.00
S3059.2016.20134.0029.1010.606.1614.7046.706.1619.70
S3144.7010.3050.0025.6019.1035.3018.30178.0031.1020.80
S3242.1031.2039.9020.0031.1010.3011.206.166.169.71
S3313.5016.806.1611.2014.406.4831.2062.106.1614.70
S347.6114.4043.2032.0018.309.7113.5043.206.1614.70
S3526.2023.20125.0010.3031.107.659.7114.106.1612.60
S3639.9028.506.1611.808.257.0825.3051.806.1610.00
S3729.1018.3017.1032.9018.3039.9024.3015.6031.1039.90
S3829.8032.906.498.7914.106.1611.2020.006.167.08
S3935.9021.5016.2011.208.796.4827.30125.006.1610.60
S4042.106.4935.9035.9016.8011.8014.70101.006.167.69
S4119.7011.8020.8026.5011.206.169.7128.506.168.25
S4220.80134.0075.9014.4024.106.1623.206.166.498.25
S4322.3025.3059.2028.5042.1048.8035.10125.0051.8032.00
S4417.1011.506.169.1317.606.169.719.716.1610.60
S4544.7032.0014.1016.708.256.1644.7080.806.169.71
S4680.8021.506.167.6953.606.169.1214.706.168.25
S4721.807.086.1623.2012.607.087.08101.006.1617.90
S4835.1025.3021.8011.808.796.4832.9048.806.1610.00
S4946.3038.2068.8048.8028.5065.4039.9051.3075.9035.90
S507.697.696.166.4816.207.0811.5023.206.1613.20
S5129.8059.2025.9022.6016.706.4810.306.166.1611.20
S5261.8010.60101.0010.6046.308.2524.1038.206.4911.80
S5310.0020.0035.1014.7021.8010.3032.70196.0016.2011.20
S5438.8020.8026.2014.7011.206.489.7168.8017.9035.10
S5519.7072.80162.0026.5038.2031.1027.30162.0031.1026.20
S5639.9022.6043.2035.9010.006.1627.3074.106.1618.30
S5762.1051.806.1631.1074.1055.8029.8017.1032.9029.80
S5821.5013.50147.006.1653.6010.3012.6097.906.1615.60
S59215.0032.9038.8016.2031.106.499.13107.006.169.13
NSA6230.006230.006230.0012230.007630.0012330.0024230.0017910.007030.0012030.00
Computational time (s)120.11120.64132.06236.92156.79242.04471.50311.18203.98214.48
Optimal value (lb)7.52 × 1054.47 × 1056.11 × 1053.29 × 1054.57 × 1051.70 × 1052.92 × 1059.09 × 1051.63 × 1052.06 × 105
Mean (lb)1.51 × 1061.41 × 1067.75 × 1051.38 × 1061.42 × 1061.81 × 1053.50 × 1059.63 × 1052.08 × 1052.33 × 105
Std (lb)3.70 × 1054.47 × 1057.11 × 1045.13 × 1054.37 × 1054.70 × 1032.68 × 1043.76 × 1045.41 × 1041.57 × 104
Increased efficiency6.62%48.68%8.61%5.97%88.01%76.82%36.22%86.23%84.57%
KHKHLDCKHLKHIKHTOKHOKHTLBOPOMRFO
S117.6017.60101.0038.2035.106.1623.2059.2031.1011.50
S248.8020.0018.3042.1038.806.4811.8091.506.1616.80
S316.2032.7051.3061.8017.906.1620.8051.807.6815.60
S468.8021.8050.0015.6083.306.169.1348.806.167.69
S58.2515.606.166.4917.606.1611.2035.906.4913.00
S68.2525.90118.0010.6020.806.1614.70147.0010.609.71
S7178.0019.1056.806.1618.308.2510.306.166.168.25
S8196.0024.107.6110.6012.606.169.13162.006.1614.10
S926.5068.8035.10101.0032.0027.3025.3032.9013.2029.80
S107.0813.3074.1017.908.2511.207.617.086.1617.10
S1114.70101.0083.3088.2020.807.6123.206.166.169.71
S1243.2028.2044.7015.8027.308.8417.9059.206.168.25
S1344.7055.8038.808.7931.106.1638.806.166.1614.10
S1429.809.716.1611.509.137.6920.8059.206.167.08
S1544.7015.8081.9026.506.498.257.08117.006.1614.10
S1643.2015.806.1610.6039.907.0825.3023.206.168.85
S1761.8050.00109.0019.7018.306.4931.1074.1013.3011.20
S1832.9017.906.1610.0042.706.1610.606.166.168.85
S1935.1088.20125.0035.9021.806.4811.80162.006.4911.70
S2022.6035.9028.5046.7051.306.1611.2072.806.1611.80
S2142.1035.3038.206.1644.7020.0010.3053.6031.1014.70
S227.698.796.1615.6024.106.1612.606.166.169.12
S23162.0011.2074.1013.5014.708.7914.1080.806.4816.20
S2420.8016.707.0811.80118.007.0814.10178.0014.1028.20
S257.0817.6022.3015.6017.6023.2013.20178.0029.4029.10
S2655.8031.1091.5011.2015.6023.209.719.716.1612.60
S2761.8083.3055.808.8514.409.1312.6018.306.1621.80
S28162.0072.8039.9053.606.166.1625.3056.806.4819.70
S2998.8031.106.1615.608.256.1621.1089.607.6913.00
S3059.2016.20134.0029.1010.606.1614.7046.706.1619.70
S3144.7010.3050.0025.6019.1035.3018.30178.0031.1020.80
S3242.1031.2039.9020.0031.1010.3011.206.166.169.71
S3313.5016.806.1611.2014.406.4831.2062.106.1614.70
S347.6114.4043.2032.0018.309.7113.5043.206.1614.70
S3526.2023.20125.0010.3031.107.659.7114.106.1612.60
S3639.9028.506.1611.808.257.0825.3051.806.1610.00
S3729.1018.3017.1032.9018.3039.9024.3015.6031.1039.90
S3829.8032.906.498.7914.106.1611.2020.006.167.08
S3935.9021.5016.2011.208.796.4827.30125.006.1610.60
S4042.106.4935.9035.9016.8011.8014.70101.006.167.69
S4119.7011.8020.8026.5011.206.169.7128.506.168.25
S4220.80134.0075.9014.4024.106.1623.206.166.498.25
S4322.3025.3059.2028.5042.1048.8035.10125.0051.8032.00
S4417.1011.506.169.1317.606.169.719.716.1610.60
S4544.7032.0014.1016.708.256.1644.7080.806.169.71
S4680.8021.506.167.6953.606.169.1214.706.168.25
S4721.807.086.1623.2012.607.087.08101.006.1617.90
S4835.1025.3021.8011.808.796.4832.9048.806.1610.00
S4946.3038.2068.8048.8028.5065.4039.9051.3075.9035.90
S507.697.696.166.4816.207.0811.5023.206.1613.20
S5129.8059.2025.9022.6016.706.4810.306.166.1611.20
S5261.8010.60101.0010.6046.308.2524.1038.206.4911.80
S5310.0020.0035.1014.7021.8010.3032.70196.0016.2011.20
S5438.8020.8026.2014.7011.206.489.7168.8017.9035.10
S5519.7072.80162.0026.5038.2031.1027.30162.0031.1026.20
S5639.9022.6043.2035.9010.006.1627.3074.106.1618.30
S5762.1051.806.1631.1074.1055.8029.8017.1032.9029.80
S5821.5013.50147.006.1653.6010.3012.6097.906.1615.60
S59215.0032.9038.8016.2031.106.499.13107.006.169.13
NSA6230.006230.006230.0012230.007630.0012330.0024230.0017910.007030.0012030.00
Computational time (s)120.11120.64132.06236.92156.79242.04471.50311.18203.98214.48
Optimal value (lb)7.52 × 1054.47 × 1056.11 × 1053.29 × 1054.57 × 1051.70 × 1052.92 × 1059.09 × 1051.63 × 1052.06 × 105
Mean (lb)1.51 × 1061.41 × 1067.75 × 1051.38 × 1061.42 × 1061.81 × 1053.50 × 1059.63 × 1052.08 × 1052.33 × 105
Std (lb)3.70 × 1054.47 × 1057.11 × 1045.13 × 1054.37 × 1054.70 × 1032.68 × 1043.76 × 1045.41 × 1041.57 × 104
Increased efficiency6.62%48.68%8.61%5.97%88.01%76.82%36.22%86.23%84.57%

Bold font in the table is used to highlight the “optimal values” among all methods.

From above-analyses about the Figs. 5, 7, 9, and 11, the intrinsic property of TOKH, which distinguishes it from the other methods in the literature, is that in optimizing the sizing of complex trusses, TOKH quickly converges to a better global region in the early iterations by global exploration, and then the local exploitation can be used to further find the minimum.

6.5 Robustness and Computational Cost Analysis in Truss Sizing Optimization.

To examine the robustness and computational cost of TOKH, this section compares the statistical results of the four trusses examples yielded by TOKH with those gained by nine other widely used metaheuristic algorithms. Table 16 lists the statistical results of the optimized designs for the four truss examples from 20 runs using TLBO, PO, MRFO, KH, CKH, OKH, IKH, KHLD, LKH, and TOKH. Among ten methods, TOKH identifies the lightest designs in the 10-, 25- and 72-bar trusses. As the structure becomes more complex, the performance of TOKH degrades, ranking it second in the 942-bar truss. The average weight and standard deviation obtained by TOKH ranks first among 10-, 25-, 72-, and 942-bar trusses. These results indicate that TOKH possesses robustness and stability compared to the methods mentioned in the literature. Results also indicate that in the optimization of 10-, 25-, 72-, and 942-bar trusses, the optimization efficiency of TOKH algorithm has improved by 20.90, 17.37, 53.53, and 88.01%, respectively, compared to KH algorithm. With the higher the structural dimension and the stronger the discretization, the more prominent the robustness of TOKH algorithm.

Table 16

Statistical results of the optimized designs for the four trusses using different algorithms

ExampleDesign VariableKHKHLDCKHLKHIKHTOKHOKHTLBOPOMRFORanking
Planar 10-bar trussNSA1580.001580.001580.003080.001930.003100.006080.004410.001780.003030.008
Computational time (s)1.331.421.511.811.641.803.011.540.500.588
Optimal value (lb)5.63 × 1035.83 × 1035.65 × 1035.78 × 1035.59 × 1035.49 × 1036.54 × 1035.49 × 1035.62 × 1035.49 × 1031
Mean (lb)7.13 × 1037.19 × 1036.13 × 1037.15 × 1036.85 × 1035.64 × 1036.83 × 1035.91 × 1036.56 × 1035.66 × 1031
Std (lb)1.51 × 1039.66 × 1022.53 × 1021.31 × 1031.14 × 1034.84 × 1012.86 × 1022.50 × 1028.01 × 1021.09 × 1021
Spatial 25-bar trussNSA1580.001580.001580.003080.001930.003100.006080.004410.001780.003030.008
Computational time (s)0.720.770.861.400.981.901.382.180.821.138
Optimal value (lb)520.66531.48494.30540.05503.38485.05529.56501.86487.33493.831
Mean (lb)590.98558.37530.87571.76549.41488.34579.05523.36509.75500.761
Std (lb)39.8919.7218.5416.7436.362.5937.3914.3225.595.941
Spatial 72-bar trussNSA1580.001580.001580.003080.001930.003100.006080.004410.001780.003030.008
Computational time (s)1.811.901.823.332.274.817.125.581.963.098
Optimal value (lb)739.63681.36520.59785.15461.01387.94644.58449.94412.66419.001
Mean (lb)865.71782.99584.17821.81502.50402.30705.74523.17463.53442.381
Std (lb)67.6165.8349.7833.3824.936.0440.2048.8151.4318.211
942-bar truss towerNSA6230.006230.006230.0012230.007630.0012330.0024230.0017910.007030.0012030.008
Computational time (s)120.11120.64132.06236.92156.79242.04471.50311.18203.98214.488
Optimal value (lb)7.52 × 1054.47 × 1056.11 × 1053.29 × 1054.57 × 1051.70 × 1052.92 × 1059.09 × 1051.63 × 1052.06 × 1052
Mean (lb)1.51 × 1061.41 × 1067.75 × 1051.38 × 1061.42 × 1061.81 × 1053.50 × 1059.63 × 1052.08 × 1052.33 × 1051
Std (lb)3.70 × 1054.47 × 1057.11 × 1045.13 × 1054.37 × 1054.70 × 1042.68 × 1043.76 × 1045.41 × 1041.57 × 1041
ExampleDesign VariableKHKHLDCKHLKHIKHTOKHOKHTLBOPOMRFORanking
Planar 10-bar trussNSA1580.001580.001580.003080.001930.003100.006080.004410.001780.003030.008
Computational time (s)1.331.421.511.811.641.803.011.540.500.588
Optimal value (lb)5.63 × 1035.83 × 1035.65 × 1035.78 × 1035.59 × 1035.49 × 1036.54 × 1035.49 × 1035.62 × 1035.49 × 1031
Mean (lb)7.13 × 1037.19 × 1036.13 × 1037.15 × 1036.85 × 1035.64 × 1036.83 × 1035.91 × 1036.56 × 1035.66 × 1031
Std (lb)1.51 × 1039.66 × 1022.53 × 1021.31 × 1031.14 × 1034.84 × 1012.86 × 1022.50 × 1028.01 × 1021.09 × 1021
Spatial 25-bar trussNSA1580.001580.001580.003080.001930.003100.006080.004410.001780.003030.008
Computational time (s)0.720.770.861.400.981.901.382.180.821.138
Optimal value (lb)520.66531.48494.30540.05503.38485.05529.56501.86487.33493.831
Mean (lb)590.98558.37530.87571.76549.41488.34579.05523.36509.75500.761
Std (lb)39.8919.7218.5416.7436.362.5937.3914.3225.595.941
Spatial 72-bar trussNSA1580.001580.001580.003080.001930.003100.006080.004410.001780.003030.008
Computational time (s)1.811.901.823.332.274.817.125.581.963.098
Optimal value (lb)739.63681.36520.59785.15461.01387.94644.58449.94412.66419.001
Mean (lb)865.71782.99584.17821.81502.50402.30705.74523.17463.53442.381
Std (lb)67.6165.8349.7833.3824.936.0440.2048.8151.4318.211
942-bar truss towerNSA6230.006230.006230.0012230.007630.0012330.0024230.0017910.007030.0012030.008
Computational time (s)120.11120.64132.06236.92156.79242.04471.50311.18203.98214.488
Optimal value (lb)7.52 × 1054.47 × 1056.11 × 1053.29 × 1054.57 × 1051.70 × 1052.92 × 1059.09 × 1051.63 × 1052.06 × 1052
Mean (lb)1.51 × 1061.41 × 1067.75 × 1051.38 × 1061.42 × 1061.81 × 1053.50 × 1059.63 × 1052.08 × 1052.33 × 1051
Std (lb)3.70 × 1054.47 × 1057.11 × 1045.13 × 1054.37 × 1054.70 × 1042.68 × 1043.76 × 1045.41 × 1041.57 × 1041

However, to improve the robustness of optimized truss sizing, TOKH requires more computational cost than KH. In the 10-, 25-, and 72-bar trusses, after 50 iterations, TOKH requires 3100 structural analyses while KH requires 1580 structural analyses. But considering the computational time, TOKH takes only 0.47 s, 1.18 s, and 3 s longer than KH, respectively. In the 942-bar truss, due to the higher dimensional design space, TOKH requires more structural analyses, and its computational time takes 121.93 s longer than that of KH. As the truss becomes more complex, the computational cost of TOKH increases more significantly, which means that the computational efficiency is sacrificed to improve the robustness. But the computational time of TOKH increases within an acceptable range.

7 Conclusions and Future Work

A novel variant of KH algorithm, referred to as TOKH algorithm, was developed for the sizing optimization of discrete truss structures. First, a crossover operator was established between the “best krill” and “suboptimal krill” to produce a robust “cross krill”. Second, an ILMC operator was introduced to fine-tune the “center of food” and candidate solutions. The objective of TOKH algorithm was to optimize the balance between exploration and exploitation. Therefore, the crossover operator was used to focus on the global exploration, whereas ILMC operator was used for the local exploitation. It is found that:

  1. The time complexity experiment performed using the Rastrigin benchmark function demonstrated that the running time of TOKH algorithm was linear based on the number of iterations, which was consistent with the inference of big-O. The two operators introduced do not change the time complexity of TOKH.

  2. TOKH algorithm was compared with nine other algorithms using 15 benchmark functions. The performance of TOKH algorithm for most functions and different dimensions, particularly for different types of high-dimensional functions, was statistically superior to those of the other metaheuristic algorithms.

  3. TOKH algorithm was applied to four discrete truss optimization problems under multiple loading conditions. We compared the numerical results for various trusses obtained using TOKH algorithm with other methods in the literature to verify the effectiveness, efficiency, and robustness. The results indicated that, among the ten algorithms, TOKH algorithm is competitive in terms of optimal weight, average weight, and stability. Furthermore, TOKH algorithm demonstrated significantly faster convergence to the optimal solution compared to other methods. Compared to KH algorithm, although TOKH required slightly more computational cost, its optimization efficiency improved by 20.90, 17.37, 53.53, and 88.01%, respectively. As the complexity of the truss increased, the advantage of TOKH became more evident. The proposed TOKH algorithm can serve as an ideal method for handling discrete truss problems.

Although TOKH algorithm takes on better global optimization capability, it has a high computational cost. Therefore, further research is needed on TOKH algorithm to improve robustness while reducing the computational cost.

Funding Data

  • The National Natural Science Foundation of China (Grant No. 52278135; Funder ID: 10.13039/501100001809).

  • The Alexander von Humboldt Foundation (Funder ID: 10.13039/100005156).

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

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