An enhanced symbiotic organisms search algorithm for design optimization of trusses with frequency constraints

Abstract

An enhanced symbiotic organisms search (ESOS) algorithm is developed and presented. Modifications to the basic symbiotic organisms search algorithm are carried out in all three phases of the algorithm with the aim of balancing the exploitation and exploration capabilities of the algorithm. To verify validity and capability of the ESOS algorithm in solving general optimization problems, the CEC2014 set of 22 benchmark functions is first optimized and the results are compared with other metaheuristic algorithms. The ESOS algorithm is then used to optimize the sizing and shape of five benchmark trusses with multiple frequency constraints. The best (minimum) mass, mean mass, standard deviation of the mass, total number of function evaluations, and the values of frequency constraints are then compared with those of a number of other metaheuristic solutions available in the literature. It is shown that the proposed ESOS algorithm is generally more efficient in optimizing the shape and sizing of trusses with dynamic frequency constraints compared to other reported metaheuristic algorithms, including the basic symbiotic organisms search and its other recently proposed improved variants such as the improved symbiotic organisms search algorithm (ISOS) and modified symbiotic organisms search algorithm (MSOS).

Keywords

symbiotic organisms search algorithm truss structures structural optimization nonlinear dynamic optimization frequency constraints

Introduction

A major objective in finding an optimum design for a structure is to obtain the cheapest possible design within the specified design constraints. The frequency constraint weight optimization process is one significant method to achieve this objective. Optimum design of structures, such as trusses, with frequency constraints enables designers to control the chosen frequencies in a desired manner so that the dynamic properties of these structures can be enhanced (Grandhi, 1993). It is evident that the shape of a structure significantly influences its natural frequencies. In addition, the shape of the structure has a noticeable effect on its sizing optimization. The complexity of the optimization problem is enhanced when the sizing and shape optimizations are considered together along with the frequency constraints. Because the sizing and shape variables are of quite different natures, combining them increases the number of design variables, leading to greater numerical difficulties and even lack of convergence (Rozvany et al., 1995; Wang et al., 2004). On the other hand, with respect of design variable, multiple frequency constraints are highly nonlinear, non-convex, and implicit (Zuo et al., 2014). As a result, gradient-based optimization techniques are prone to converge to local optima since they cannot deal with such problems (Gomes, 2011). Also, for optimal solution they require a good starting point (Kaveh and Talatahari, 2010). Efficient global search optimization techniques are required to overcome these difficulties.

In recent decades, many researchers have used nature-based metaheuristic (MH) optimization algorithms to solve engineering problems. Some of these algorithms include tabu search (Glover, 1997), simulated annealing (Kirkpatrick et al., 1983), genetic algorithm (Goldberg and Holland, 1988), particle swamp optimization (Eberhart and Kennedy, 1995), ant colony algorithm (Dorigo et al., 1996), harmony search (HS) (Geem et al., 2001), big bang–big crunch (Erol and Eksin, 2006), the artificial bee colony algorithm (Karaboga and Basturk, 2007), cuckoo search (Yang and Deb, 2009), firefly algorithm (FA) (Yang, 2009), cuckoo optimization algorithm (Rajabioun, 2011), teaching-learning-based optimization (Rao et al., 2011), flower pollination algorithm (Yang, 2012), water cycle algorithm (Eskandar et al., 2012), krill herd algorithm (Gandomi and Alavi, 2012), ray optimization algorithm (Kaveh and Khayatazad, 2012), dolphin echolocation (Kaveh and Farhoudi, 2013), symbiotic organisms search (SOS) (Cheng and Prayogo, 2014), dragonfly algorithm (Mirjalili, 2016), Jaya algorithm (Rao, 2016), butterfly optimization algorithm (Qi et al., 2017), thermal txchange optimization (Kaveh and Dadras, 2017), focus group algorithm (Fattahi et al., 2018), squirrel search algorithm (Jain et al., 2019), Blue Monkey algorithm (Mahmood and Al-Khateeb, 2019), booster algorithm (Pakzad-Moghaddam et al., 2019), salmon migration algorithm (Deng and Zhu, 2019), sailfish optimizer algorithm (Shadravan, 2019), bear smell search algorithm (Ghasemi-Marzbali, 2020), most valuable player Algorithm (Bouchekara, 2020), and Newton MH algorithm (Gholizadeh et al., 2020). There are also numerous improved, modified, enhanced, and hybrid MH algorithms which improve on the above basic algorithms (Gholizadeh and Moghadas, 2014; Gholizadeh and Milany, 2018; Khatibinia and Khosravi, 2014; Khatibinia and Yazdani, 2018; Maheri and Narimani, 2014; Maheri and Talezadeh, 2018; Maheri et al., 2016; Seyedpoor et al., 2011).

Bellagamba and Yang (1981) were the first to study truss optimization, subject to dynamic frequency constraints. Later, Lin et al. (1982) proposed a bi-factor $α - β$ algorithm, based on the Kuhn–Tucker criteria for shape and sizing optimization of trusses with prohibited band frequency constraints. A two-step algorithm with a fundamental frequency constraint was later presented by Sadek (1986). The steps include frequency correction and weight reduction. Noilublao and Bureerat (2011) introduced an integrated design method for shape, sizing, and topology optimization of trusses with frequency constraints. They noted that population-based incremental learning MH algorithms are more suitable for optimization problems with frequency constraints. Gomes (2011) utilized the PSO algorithm to optimize shape and sizing of trusses with multiple natural frequency constraints. Kaveh and Zolghadr (2011) also developed an algorithm named charged system search (CSS) for optimization of trusses subject to frequency constraints. Later, Miguel and Miguel (2012) used FA and the HS method to solve this type of problem. Kaveh and Zolghadr (2012) furthered their previous work on CSS by hybridizing it with the big bang–big crunch algorithm using frequency constraints to enable the algorithm to escape from local optima traps. Baghlani and Makiabadi (2013) used the TLBO algorithm to optimize the shape and sizing of truss structures with dynamic frequency constraints. Khatibinia and Naseralavi (2014) introduced a hybrid approach based on a combination of multigravitational search algorithm and an orthogonal crossover, named orthogonal multigravitational search algorithm. Kaveh and Ghazaan (2015) hybridized aging leader and challengers algorithm with PSO algorithm and combined HS-based search method with PSO algorithm for optimization with natural frequency constraints leading to high rates of convergence. Farshchin et al. (2016) also proposed a multiclass TLBO solution (MC-TLBO) with frequency constraints to increase the exploration capacity of the algorithm. Savsani et al. (2016) presented an improved TLBO algorithm for topology optimization of trusses, having different static and dynamic constraints. Lieu et al. (2018) also proposed hybridization of the differential evolution algorithm and the firefly algorithm for shape and size optimization of truss structures under multiple frequency constraints. The comparative performance of a number of self-adaptive MHs for truss design has also been reported by Pholdee and Bureerat (2018). Degertekin et al. (2021) developed parameter free formulation of the Jaya algorithm for truss sizing-layout optimization under natural frequency constraints.

The SOS algorithm first introduced by Cheng and Prayogo (2014) has been examined for a large number of both the constrained and unconstrained engineering problems and is shown to have a better performance compared to many other MH algorithms (Cheng et al., 2016). Abdullahi et al. (2016) used an improved discrete SOS algorithm for task scheduling in cloud computing. They concluded that SOS algorithm is more effective than the PSO for task scheduling problems. Tran et al. (2016) used a modified version of SOS algorithm, termed MOSOS, to conduct a multiobjective optimization of time-cost-labor utilization tradeoff problem encountered in construction projects. Panda and Pani (2016) also used an adaptive penalty function to solve multiobjective constrained optimization problems using SOS. Later, Panda and Pani (2018) combined the SOS algorithm with augmented Lagrange multiplier method to solve constrained optimization problems. Prayogo et al. (2017) also explored the effectiveness of SOS algorithm in solving different civil engineering benchmark problems. They compared the results with those of other algorithms recently developed and demonstrated the superior performance of SOS compared to the other algorithms present in the literature. Tejani et al. (2016) introduced different adaptive benefit factors (BFs) to the basic SOS and presented three ABF1, ABF2, and ABF1&2 variants of the algorithm, which improve on the basic SOS. Tejani et al. (2018) furthered their work by presenting another improved variant of SOS, termed ISOS, by improving on the exploitation capacity of the algorithm. The feasibility and effectiveness of ISOS was examined through solving six benchmark trusses together with the CEC2014 30 functions test suite (2014) and comparing the results with other MHs. They concluded that ISOS is more efficient than the basic SOS algorithm. In another recent work, Kumar et al. (2019) presented a modified symbiotic organisms search (MSOS) algorithm. They introduced an adaptive BF and modified parasitism vector to the original SOS. It was shown that the MSOS considerably improves on the performance of the basic SOS and provides competitive results compared to other MH algorithms. Recently, Tejani et al. (2019) introduced a multiobjective modified adaptive symbiotic organisms search (MOMASOS) algorithm with two modified phases. They reported that the proposed MOMASOS outperforms the original MOSOS and other multiobjective MHs.

In the present article, the SOS optimizer is further enhanced by carrying out improvements in each of the three phases of mutualism, commensalism, and parasitism, so that a more suitable balance between the exploitation and exploration capabilities of the algorithm in searching for the optima is achieved. The rest of the article is organized as follows: A brief introduction to the SOS algorithm is provided in SOS algorithm. The proposed enhanced SOS (ESOS) is introduced in ESOS algorithm. The proposed algorithm is tested using the CEC2014 set of 22 benchmark functions in Testing ESOS through benchmark functions of the CEC2014. The efficiency of the proposed ESOS algorithm in solving shape and sizing optimization of truss problems with frequency constraints is verified in Application of ESOS algorithm to design optimization of trusses using five benchmark problems, and conclusions are made in Conclusions.

SOS algorithm

The SOS algorithm is a nature-inspired optimization algorithm mimicking the natural phenomena of symbiotic interactions between organisms. The symbiotic relationships in nature can be subdivided into three principal categories: mutualism, commensalism, and parasitism. Mutualism is an interdependence between two different species whereby both species benefit. Commensalism is a relationship whereby one species benefits while the other is not affected. Lastly, parasitism is a relationship whereby one species benefits and the other is harmed.

The detailed description of the SOS algorithm and its three phases of mutualism, commensalism, and parasitism can be found in work of Cheng and Prayogo (2014). A brief description of different phases of the original SOS algorithm is presented here as follows.

Mutualism phase

In the mutualism phase, a sequentially chosen organism (X_i) interacts with another random organism (X_j), where j ≠ i. The BFs and a mutual vector (MV) are used to produce new organisms X_{i new} and X_{j new}, as given in equations (1)–(5). The MV signifies the connection between organisms (X_i) and (X_j) to increase their survival advantage (see equation (1)). The values of BF₁ and BF₂ are randomly selected as 1 or 2, respectively (see equations (2) and (3)). Therefore, either of the BFs shows an instance of an organism enjoying partial or full benefits from the symbiotic relationship. The organism which has the best functional value and stands for the highest degree of adaptation is assumed as the best organism (X_best) of the ecosystem. It is noteworthy that an update is done only if the new calculated fitness function value (denoted by f(X_{i new}) and f(X_{j new})) is superior to the previous fitness functions (f(X_i) and f(X_j))

M V = \frac{X_{i} + X_{j}}{2}

(1)

B F_{1} = 1 + Round [rand (0,1)]

(2)

B F_{2} = 1 + Round [rand (0,1)]

(3)

X_{i n e w} = X_{i} + rand (0,1) \times (X_{b e s t} - MV \times B F_{1})

(4)

X_{j n e w} = X_{j} + rand (0,1) \times (X_{b e s t} - MV \times B F_{2})

(5)

where

i = 1, 2, \dots, n

and j is a randomly selected population,

j \neq i; j \in (1,2, \dots, n)

Commensalism phase

In the commensalism phase of the SOS algorithm, from the interaction between the organisms (X_i) and (X_j), a new organism (X_{i new}) is generated. From the relationship between these organisms, organism i benefits but organism j is not affected. Furthermore, organism (Xi) interacts with the best organism of the ecosystem. The old organism is replaced by the new organism only if its new functional value (f(X_{i new})) is better than that of the existing organism (f(X_i). The mathematical representation is specified as

X_{i n e w} = X_{i} + rand (- 1,1) \times (X_{b e s t} - X_{j})

(6)

where

i = 1, 2, \dots, n

and j is a randomly selected population,

j \neq i; j \in (1,2, \dots, n)

Parasitism phase

In the SOS algorithm, after the completion of the commensalism phase, the organism (X_i) randomly chooses a new organism (X_j) from the ecosystem again. Afterward, the organism (X_i) produces an artificial parasite termed the parasite vector (PV). PV is formed by reconstructing some randomly chosen elements of organism i within their specified boundary conditions (see equation (7))

P V^{k} = {\begin{cases} X_{i}^{k}, if rand (0,1) \leq rand (0,1) \\ L B + rand (0,1) \times (U B - L B), otherwise \end{cases}

(7)

where

i = 1,2, \dots, n

; k signifies design variables and

k \in (1,2, \dots, m)

; UB and LB are the upper and lower bounds of the problem, respectively. Therefore, PV is a clone of the original elements of organism i. Organism (X_j) works as a host to the parasite vector. If the fitness function value of the parasite vector is better than that of organism j, the parasite will destroy organism j and will replace it. If organism j is better, it will be immune to the parasite and the parasite will die.

ESOS algorithm

Performance of a MH optimization algorithm depends on its ability to explore and exploit. The exploration capability denotes the global search ability which affects the accuracy of the solution, whereas the exploitation capability characterizes the local search ability, affecting the rate of convergence and the speed of reaching the optimal solution. It is evident that an algorithm with a more superior exploration capacity than exploitation ability can reach the global optima, yet the convergence is slow. As a result, the ability of an optimization algorithm in searching for a global optima at a reasonably fast time depends on its effectiveness in setting a reasonable balance between exploring and exploiting the search space.

The SOS algorithm has been shown to have superior explorative capability compared to some other widely used algorithms such as GA, PSO, DE, and BA in solving different engineering optimization problems (Cheng et al., 2016). Although the original SOS algorithm is quite capable in global search, it does so at the expense of higher computational effort (Do and Lee 2017).

In this section, an enhanced version of the symbiotic organisms search algorithm, denoted by ESOS is presented. The aim is to improve the algorithm’s performance so that a reasonable balance is created between exploitation and exploration capacities of the algorithm in the search space. For this purpose, some features of the original SOS algorithm are modified. The modifications are carried out in all three phases of the algorithm, as follows.

Modification of mutualism phase

The BFs are the key parameters deciding the effect of the MV in the mutualism phase. In the original SOS algorithm, the BFs take values of either 1 or 2, depending on whether organisms (X_i) and (X_j) benefit partially or fully from the MV. When the lower value of the BF is in use, the algorithm conducts a fine search with small steps, resulting in a reduced convergence speed. Alternatively, if the larger value is considered, the algorithm speeds up the search, with a possibility of skipping over a nearby feasible value. This reduces the exploitation capacity of the algorithm. In addition, these BFs might not consistently be at their two ends in a real mutualism relationship and may vary in between. This leads us to propose replacing the benefit factors (BF₁ and BF₂) with new adaptive benefit factors (ABF₁ and ABF₂), which will give better convergence, superior search ability, and the required balance between exploitation and exploration capacities of the algorithm. The following adaptive BFs are proposed

A B F_{1} = {\begin{cases} 1 + \frac{f (X_{i}) - f (X_{b e s t})}{f_{a v g} - f (X_{b e s t})}, i f f (X_{i}) \leq f_{a v g} \\ 2, Otherwise \end{cases}

(8)

A B F_{2} = {\begin{cases} 1 + \frac{f (X_{j}) - f (X_{b e s t})}{f_{a v g} - f (X_{b e s t})}, i f f (X_{j}) \leq f_{a v g} \\ 2, Otherwise \end{cases}

(9)

It is noteworthy that in the above relations, the relative information of the objective values of the organisms is employed, including the objective value of the best organism $(f (X_{b e s t}))$ , the average objective value of the ecosystem $(f_{a v g})$ , and the objective values of the current organisms ( $f (X_{i})$ and $f (X_{j})$ ), in expressing the new ABFs. Guided by the ABFs, the BF is transformed according to the value of the objective function of each individual in such a way that good organisms tend to exploit the direction of the best organism by local fine-tuning, while substandard organisms undertake global exploration with larger steps. The ABFs make the algorithm to have a better exploration capacity and explore unvisited regions of the search space when an organism (i or j) is far from the best organism, whereas they set a good exploitation capacity for the algorithm and increase the speed of convergence, when an organism is close to the best organism. Therefore, it is expected that utilizing the proposed ABFs would result in a more reasonable balance between the exploitation and exploration capacities of the algorithm.

Modification of commensalism phase

In the commensalism phase of the original SOS algorithm, the convergence rate generally speeds up if the new solution is based on the best organism. The increased rate of convergence may lead to local solutions. To avoid this, a modification, in the form of equation (10), is also proposed for this phase

X_{i n e w} = {\begin{cases} X_{i} + rand (0,1) \times (X_{b e s t} - X_{i}) + rand (0,1) \times (X_{i} - X_{j}), i f f (X_{i}) \leq f (X_{j}) \\ X_{i} + rand (0,1) \times (X_{b e s t} - X_{i}) + r a n d (0,1) \times (X_{j} - X_{i}), Otherwise \end{cases}

(10)

In the above equation, the second term is in charge of local search and improves the exploitation capability of the algorithm, whereas the third term helps to expand the search space and improves the diversity of the ecosystem. It conducts global search and enhances the exploration capacity of the algorithm. Hence, by creating a balance between the exploitation and exploration capabilities, the stability of the algorithm, based on the value of the standard deviation, increases and the algorithm requires a smaller number of analyses to converge to a global optimal solution.

Modification in the parasitism phase

A great deal of studies have demonstrated that in the original SOS algorithm, the exploitation capability of the parasitism phase is noticeably low in comparison with its exploration capability (Kumar et al., 2019). In comparison with the previous phase, in this phase, a great number of new solutions are dismissed because of their inferior objective functional values. Increasing the number of function evaluations (FE) decreases the speed of convergence and increases the time needed for solving the problem.

Since in the parasitism phase of the original SOS algorithm, much attention is paid to exploration and a large number of FEs are wasted due to their inferior results, the rate of acceptance of a new solution is rapidly reduced as the algorithm progresses. In the current article, it is attempted to promote the exploitation potential of the parasitism phase of the algorithm without decreasing its global exploration capability. Therefore, the parasitism phase of the original SOS is enhanced to promote the convergence ability of the algorithm while maintaining the balance between its exploitation and exploration capabilities. The proposed approach permits the algorithm to explore different parts of the search space, while avoiding premature convergence due to population concentration in one region. In equation (11), the proposed modification in the parasite vector is demonstrated

E P V^{k} = {\begin{cases} X_{i}^{k}, i f rand (0,1) \leq rand (0,1) \\ X_{i}^{k} - rand (0,1) \times (X_{w o r s t}^{k} - X_{i}^{k}), Otherwise \end{cases}

(11)

where

X_{w o r s t}^{k}

is the value of the variable k for the worst organism and the negative term “

- rand (0,1) \times (X_{w o r s t}^{k} - X_{i}^{k})

” indicates the tendency of the enhanced parasite vector (EPV) to avoid the worst organism.

The pseudo code of the proposed enhanced symbiotic organisms search (ESOS) is presented in Figure 1.

Figure 1.

Pseudo code for the enhanced symbiotic organisms search algorithm.

Testing ESOS through benchmark functions of the CEC2014

To verify the strength and robustness of the proposed ESOS algorithm in solving general optimization problems, the CEC2014 set of benchmark functions (Liang et al., 2013) are optimized in this section. In Table 1, the 22 considered benchmark functions are listed. The functions are divided into three categories: unimodal (f1 to f3), multimodal (f4 to f16), and hybrid (f17 to f22). In this study, these functions were assumed to have 30 dimensions.

Table 1.

The CEC2014 benchmark functions.

Test function	Optimum	Test function	Optimum
F1: Rotated high conditioned Elliptic function	100	F12: Shifted and rotated Katsuura function	1200
F2: Rotated bent Cigar function	200	F13: Shifted and rotated HappyCat function	1300
F3: Rotated discus function	300	F14: Shifted and rotated HGBat function	1400
F4: Shifted and rotated Rosen-brock’s function	400	F15: Shifted and rotated Expanded Griewank’s plus Rosen-brock’s function	1500
F5: Shifted and rotated Ackley’s function	500	F16: Shifted and rotated Expanded Scaffer’s F6 function	1600
F6: Shifted and rotated Weierstrass function	600	F17: Hybrid function 1 ((F9, F8,F1)	1700
F7: Shifted and rotated Griewank’s function	700	F18: Hybrid function 2 (F2, F12, F8)	1800
F8: Shifted Rastrigin’s function	800	F19: Hybrid function 3 (F7, F6, F4, F14)	1900
F9: Shifted and rotated Rastrigin’s function	900	F20: Hybrid function 4 (F12, F3, F13, F8)	2000
F10: Shifted Schwefel’s function	1000	F21: Hybrid function 5 (F14, F12, F4, F9, F1)	2100
F11: Shifted and rotated Schwefel’s function	1100	F22: Hybrid function 6 (F10, F11, F13, F9, F5)	2200

The test on the ESOS was performed with a population of 50 members, the maximum FEs of 150,000, and a search range of −100 to 100. The results were extracted from 60 independent runs made on each function. The mean and standard deviation (SD) of the fitness values from the 60 runs on each of the test functions made using the proposed ESOS algorithm are compared with those of the corresponding functions obtained from other optimization algorithms in Tables 2 and 3, respectively. These algorithms include IWO, BBO, GSA, HuS, BA, WWO, SOS, ISOS, and MSOS. There are several statistical methodologies for comparing evolutionary and swarm intelligence algorithms (Derrac et al., 2011; Sheskin, 2003). The Friedman rank test is often used for such statistical comparisons (Kumar et al., 2019; Tejani et al., 2018). The Friedman rank test was performed on the mean of fitness values of each function category and the results are also compared in Table 2. Also listed in this table, are the overall Friedman values and ranks for the entire pool of 22 test functions. Table 2 indicates that amongst all the algorithms considered, the proposed ESOS algorithm ranks 2nd after the MSOS algorithm for the unimodal functions (f1 to f3), ranks 4th after WWO, IWO, and ISOS algorithms for the multimodal functions (f4 to f16), and ranks 2nd after WWO algorithm for the hybrid functions (f17 to f22). It can also be noted that the overall Friedman value for the proposed ESOS algorithm is 89 which is exactly the same as that for the ISOS, giving them an overall ranking in the third place, after the WWO and IWO algorithms. It should be noted that, although there is a notable difference between the overall Friedman value of the first ranked WWO and the second ranked IWO, there is no meaningful difference between the overall Friedman values of IWO (87) and the proposed ESOS algorithm (89). The results presented in Table 2 lead us to conclude that, regarding the exploration capability, only the WWO algorithm is superior to the proposed ESOS algorithm and that the ESOS algorithm outperforms all the other SOS variants.

Table 2.

Mean of fitness values of the CEC2014 benchmark functions.

Function	WWO (Zheng, 2015)	BA (Zheng, 2015)	Hus (Zheng, 2015)	GSA (Zheng, 2015)	BBO (Zheng, 2015)	IWO(Zheng, 2015)	SOS (Tejani et al., 2018)	ISOS (Tejani et al., 2018)	MSOS (Kumar et al., 2019)	ESOS
F1	628,064.73	316,593,399.33	5,555,804.77	14,413,625.13	27,262,607.18	1,463,430.67	1,026,753.82	982,235.85	467,711.355	270,276.8509
F2	330.4397	25,714,756,385.19	10,068.13	8771.2239	4,012,004.08	17,672.17	213.1503	205.2755	231.0785	219.5819
F3	526.8209	72,001.72	502.0203	45,384.25	13,100.31	8167.4657	938.939	779.0035	496.9134	818.8188
Friedman value of F1-F3	11	30	16	23	26	21	13	9	7	9
Friedman rank of F1-F3	4	10	6	8	9	7	5	2	1	2
F4	417.0105	3697.5439	506.9362	676.436	538.7936	500.3255	468.2918	459.8026	466.2221	447.9138
F5	519.9999	520.9716	520.7029	519.999	520.1556	520.014	520.5639	520.344	520.0303	520.5695
F6	605.9873	636.3693	623.065	619.5872	613.9623	602.2138	610.8746	610.4609	611.5147	615.9528
F7	700.0037	910.6678	700.0407	700.0001	701.0283	700.0337	700.0161	700.0156	700.0216	700.0131
F8	801.1436	1070.3076	940.1063	800.4991	877.4573	843.7475	852.1217	814.7215	824.6915	811.5573
F9	961.093	1250.0944	1011.9988	1059.7399	951.4286	946.0714	970.5093	954.3446	966.3686	975.4830
F10	1581.5778	6426.1095	2253.5001	4392.2443	1002.1744	2565.2591	2107.2343	1156.5243	1378.4436	1128.0388
F11	3349.4633	8152.1644	3302.9108	5099.2681	3247.3542	2887.3064	4017.4845	2882.3048	3636.8347	3604.8644
F12	1200.0995	1202.5771	1200.187	1200.0011	1200.2257	1200.0355	1200.6611	1200.2345	1200.3483	1200.3374
F13	1300.2617	1304.0199	1300.3921	1300.2972	1300.5091	1300.2789	1300.4233	1300.3774	1300.37	1300.3211
F14	1400.2169	1473.1361	1400.2377	1400.254	1400.4439	1400.236	1400.3309	1400.2711	1400.2643	1400.2557
F15	1503.2828	194,533.26	1517.0308	1503.2887	1514.6242	1503.6932	1517.6988	1510.6991	1510.9396	1513.2115
F16	1610.4351	1612.9981	1611.7074	1613.6691	1609.9125	1610.4324	1610.6564	1609.2194	1610.4357	1610.3376
Friedman value of F4-F16	34	129	91	67	74	47	89	51	69	64
Friedman rank of F4-F16	1	10	9	5	7	2	8	3	6	4
F17	26,618.68	4,641,277.77	198,099.04	578,588.76	4,299,306.67	86,437.00	143,235.17	176,709.85	81,339.7737	87,485.0145
F18	2026.3758	121,880,897.95	3780.558	2289.6856	28,418.23	5787.0752	8320.081	5689.7268	5807.7762	2662.4493
F19	1907.7291	2004.9297	1931.0413	1995.2919	1928.4718	1907.913	1923.3954	1907.7915	1920.0385	1907.5437
F20	5363.8611	19,356.89	38,657.34	22,421.91	31,411.18	2992.6053	5770.2949	6983.0969	5697.8995	4585.6921
F21	38,673.78	1,095,231.53	60,455.79	170,612.96	485,593.29	39,074.31	68,597.82	91,120.70	32,529.2852	31,993.8958
F22	2481.9864	3134.0717	3072.5807	3161.1458	2722.8879	2346.3986	2496.3689	2475.0983	2483.6063	2521.8974
Friedman value of F17-F22	13	56	42	45	50	19	35	29	24	17
Friedman rank of F17-F22	1	10	7	8	9	3	6	5	4	2
Overall Friedman value	58	215	149	135	150	87	137	89	101	89
Overall Friedman rank	1	10	8	6	9	2	7	3	5	3

Note: ESOS: enhanced symbiotic organisms search; MSOS: modified symbiotic organisms search; SOS: symbiotic organisms search.

Table 3.

Standard deviation (SD) of fitness values of the CEC2014 benchmark functions.

Function	WWO(Zheng, 2015)	BA (Zheng, 2015)	Hus (Zheng, 2015)	GSA (Zheng, 2015)	BBO (Zheng, 2015)	IWO (Zheng, 2015)	SOS(Tejani et al., 2018)	ISOS(Tejani et al., 2018)	MSOS(Kumar et al., 2019)	ESOS
F1	244,526.81	104,690,309.26	2,620,084.80	13,187,933.16	16,720,012.88	571,747.01	732,930.23	705,470.86	255,076.7193	190,720.691
F2	202.2221	7,553,596,375.48	6012.6897	2903.3044	1,549,219.32	8673.4818	20.2822	17.1804	63.382	15.4990
F3	184.645	17,548.67	540.6109	10,432.65	12,764.87	2692.8884	527.5818	623.8666	249.8799	392.3175
F4	36.3636	1973.8532	36.6181	51.5149	38.3545	28.7968	31.7519	35.7188	40.9633	32.1939
F5	0.0007	0.0481	0.0783	0.0006	0.0422	0.0038	0.0801	0.0667	0.043	0.0817
F6	2.6204	1.5591	2.1784	1.8319	2.3542	1.1219	2.5681	2.3958	3.0635	2.8671
F7	0.0063	32.3193	0.0556	0.001	0.0264	0.0121	0.0214	0.0183	0.0271	0.0136
F8	2.3361	25.6476	12.7304	0.2063	20.6917	10.1117	12.3208	3.3488	6.9815	3.7793
F9	11.0977	44.1294	25.9919	17.4329	11.4372	11.3933	24.0796	13.4374	16.9302	17.0605
F10	361.6122	518.6548	433.1531	360.9861	0.68	380.019	344.1052	40.842	187.9776	106.4395
F11	289.218	362.2389	465.5429	567.3467	511.5523	447.716	835.0838	454.0984	506.0594	431.8235
F12	0.0561	0.3339	0.0777	0.001	0.0562	0.0148	0.1833	0.0573	0.1351	0.1964
F13	0.0641	0.5483	0.065	0.0665	0.1061	0.065	0.0864	0.071	0.0783	0.0597
F14	0.0441	13.9463	0.0474	0.0423	0.1992	0.1191	0.1296	0.0512	0.0538	0.0419
F15	0.7753	140,338.95	3.2695	0.7297	4.2976	0.8484	3.7981	3.7171	4.1851	4.5261
F16	0.4667	0.1904	0.7249	0.3428	0.5923	0.6144	0.6059	0.7314	0.7675	0.6231
F17	12,403.54	1,789,909.25	160,518.86	219,949.35	4,192,494.27	68,473.66	159,023.34	164,458.20	54,940.7907	204,516.3
F18	125.1962	100,285,357.35	2246.5148	377.9286	19,674.94	3690.0554	10,313.36	5150.4892	5702.6751	5500.236
F19	1.378	20.3164	33.1485	34.319	27.6885	1.6545	26.6142	1.7869	22.3884	1913.166
F20	3177.0847	10,283.63	8492.7252	13,860.36	17,604.90	700.4102	3295.2452	3393.2462	2466.1799	8580.707
F21	35,555.57	750,680.88	42,428.10	65,285.41	334,571.54	23,011.18	80,093.31	107,774.89	24,345.4802	96,953.18
F22	142.8952	205.4095	267.2685	250.0137	234.4393	73.3907	151.5147	145.4101	141.3538	2748.663
Overall Friedman value	64	180	144	114	159	83.5	141	116	124	84
Overall Friedman rank	1	10	4	4	9	2	8	6	7	3

Note: ESOS: enhanced symbiotic organisms search; MSOS: modified symbiotic organisms search; SOS: symbiotic organisms search.

Standard deviation is a good measure of the exploitation capability of an algorithm. To compare the algorithms regarding this capability, the Friedman values and ranks of the SD values for different test functions obtained from different optimization algorithms are compared in Table 3. This table shows that, with an overall Friedman value of 84, the proposed ESOS algorithm ranks a very close 3rd compared to the IWO (Friedman value of 83.5) algorithm; with the WWO algorithm again having the best performance amongst all the algorithms considered. Therefore, similar to the exploration capability, in exploiting a promising space, the proposed ESOS algorithm outperforms all the other algorithms (including all the SOS variants), except for the WWO algorithm.

Application of ESOS algorithm to design optimization of trusses

In the following, the effectiveness of the proposed ESOS algorithm in solving practical sizing and shape optimization of trusses with frequency constraints is tested, through five benchmark problems. First, the formulation of the design problem is presented, followed by the optimization of the truss benchmark problems.

Formulation of the design problem

In dealing with this type of optimization problems, it is assumed that the truss topology to be unchanged during the optimization process. However, the cross-sectional areas of elements and nodal coordinates are considered as design variables which should be optimized. The natural frequencies are considered as design constraints, so that in a dynamic environment resonance with external excitations may be avoided. The mass of truss should be minimized subject to the prescribed constraints. The mathematical representation of the problem is as follows

Find, X = {A, N}, wher e A = {A_{1}, A_{2}, . . ., A_{n e}} and N = {N_{1}, N_{2}, . . ., N_{m}}

(12)

To minimize, Mass of truss: $F (X) = \sum_{i = 1}^{n e} A_{i} ρ_{i} L_{i}$ .

Subjected to: $g_{1} (X) : f_{q} - f_{q}^{m i n} \geq 0$ , $g_{2} (X) : f_{r} - f_{r}^{m a x} \leq 0$ , $g_{3} (X) : A_{i}^{m i n} \leq A_{i} \leq A_{i}^{m a x}$ , $g_{4} (X) : N_{j}^{m i n} \leq N_{j} \leq N_{j}^{m a x}$ , $i = 1, 2, . . ., n e; j = 1, 2, . . ., n n$ .

where A_i, L_i, and $ρ_{i}$ are, respectively, the cross-sectional area, the length, and the material density of the ith element. $N_{j}$ stands for the nodal coordinates $(x_{j}, y_{j}, z_{j})$ of the jth node. Moreover, $f_{q}$ and $f_{r}$ are, respectively, the qth and rth natural frequencies. The superscripts “max” and “min” indicate the maximum and minimum allowable limits, respectively. As an analyzer, the finite element method is employed to compute the natural frequencies of the truss. It should be noted that the natural frequency is highly nonlinear and implicit with regard to the design variables.

To handle the constraints, a penalty function approach is employed. The penalty function is considered as follows (Kaveh and Zolghadr, 2013)

Penaltized F (X) = {\begin{matrix} F (X), if no violation of limits \\ F (X) \times F_{p e n a l t y}, otherwise \end{matrix}

(13)

F_{p e n a l t y} = {(1 + ε_{1} \times C)}^{ε_{2}}, C = \sum C_{q} + C_{r}, C_{q} = | 1 - \frac{| f_{q} - f_{q}^{m i n} |}{f_{q}^{m i n}} |, C_{r} = | 1 - \frac{| f_{r} - f_{r}^{m a x} |}{f_{r}^{m a x}} |

(14)

where

ε_{1}

and

ε_{2}

are constants. In the present study, values of

ε_{1}

and

ε_{2}

are both set as 3, based on the suggestion made by Kaveh and Zolghadr (2013).

Truss optimization

To investigate the effectiveness of the proposed ESOS algorithm in shape and sizing optimization of trusses with frequency constraints, five benchmark trusses have been solved. The results of ESOS optimization are then compared with corresponding results obtained through numerous other MH optimization algorithms, including PSO, DPSO, NHPGA, CSS, enhanced CSS, HS, FA, CSS-BBBC, hybrid OC-GA, TWO, TLBO, MC-TLBO, CBO, VPS, SOS, SOS-ABF1, SOS-ABF2, SOS-ABF1&2, ISOS, and MSOS. The design parameters for the five benchmark problems are given in Table 4.

Table 4.

Design parameters of the benchmark trusses.

	The 10-bar truss	The 37-bar truss	The 72-bar truss	The 52-bar truss	The 200-bar truss
Design variables	$A_{i}, i = 1,2, \dots, 10$	$A_{i}, i = 1,2, \dots, 14$ $y_{j}, j = 3,5,7,9,11$	$G_{i}, i = 1,2, \dots, 16$	$G_{i}, i = 1,2, \dots, 8$ $z_{A}, z_{B}, z_{F}, x_{A}, x_{F}$	$G_{i}, i = 1,2, \dots, 29$
Limits $f (H z)$	$f_{1} \geq 7,$ $f_{2} \geq 15,$ $f_{3} \geq 20$	$f_{1} \geq 20,$ $f_{2} \geq 40,$ $f_{3} \geq 60$	$f_{1} \geq 4,$ $f_{3} \geq 6$	$f_{1} \leq 15.916,$ $f_{2} \geq 28.648$	$f_{1} \geq 5,$ $f_{2} \geq 10,$ $f_{3} \geq 15$
Size variables $A_{i} (c m^{2})$	$[0.645, 50]$	$[1, 10]$	$[0.645, 30]$	$[1, 10]$	$[0.1, 30]$
Shape variables	_	$y_{j} \in [0.1, 3]$ m	_	$\pm 2 m$	_
Material density $ρ (k g / m^{3})$	2770	7800	2770	7800	7860
Young modulus $E (p a)$	$6.98 \times 10^{10}$	$2.1 \times 10^{11}$	$6.98 \times 10^{10}$	$2.1 \times 10^{11}$	$2.1 \times 10^{11}$

10-bar plane truss

The 10-bar planar truss illustrated in Figure 2 is considered as the first benchmark problem. In Table 4, the design parameters for this benchmark problem are presented. This problem has previously been solved by some investigators results of which are listed in Table 5. For this 10-bar planar truss, ten continuous design variables are considered for the sizing optimization. Moreover, as illustrated in Figure 2, a 454-kg nonstructural mass is attached to free nodes (nodes 1–4).

Figure 2.

Details of the 10-bar truss problem.

Table 5.

Optimal design parameters for the 10-bar truss problem (cross-sectional areas, Ai, are in cm²).

	Gomes (2011)	Wei et al. (2011)	Kaveh and Zolghadr (2011)		Miguel and Miguel (2012)		Zuo et al. (2014)	Kaveh and Ghazzan (2015)	Tejani et al. (2016)				Tejani et al. (2018)	Kumar et al. (2019)	Present study
		PSO	NHPGA	CSS	Enhanced CSS	HS	FA	Hybrid OC-GA	VPS	SOS	SOS-ABF1	SOS-ABF2	SOS-ABF1&2	ISOS	MSOS	ESOS
$A_{1}$	37.712	36.63	38.811	39.569	34.282	36.198	37.810	35.1471	35.3794	34.4523	35.3013	36.4206	35.2654	35.2834	35.1763
$A_{2}$	9.959	13.043	9.0307	16.74	15.653	14.03	9.518	14.6668	14.8826	14.9767	14.8119	14.301	14.6803	14.4487	14.6041
$A_{3}$	40.265	34.229	37.099	34.361	37.641	34.754	36.463	35.6889	35.7321	36.1675	34.9522	34.1835	34.4273	34.5268	34.9451
$A_{4}$	16.788	15.289	18.479	12.994	16.058	14.9	19.095	15.0929	14.3069	14.6638	14.9436	15.5395	14.9605	14.6773	14.6604
$A_{5}$	11.576	0.645	4.479	0.645	1.069	0.645	2.851	0.645	0.645	0.668	0.645	0.645	0.645	0.645	0.6450
$A_{6}$	3.955	4.8472	4.205	4.802	4.74	4.672	5.526	4.6221	4.7142	4.5484	4.5828	4.6247	4.5927	4.5878	4.5725
$A_{7}$	25.308	22.14	20.842	26.182	22.505	23.467	19.463	23.5552	24.1569	23.9613	23.5712	22.2793	23.3417	23.5452	23.7313
$A_{8}$	21.613	27.983	23.023	21.26	24.603	25.508	26.400	24.468	23.6047	23.4914	23.5602	24.8589	23.8236	24.1081	23.8040
$A_{9}$	11.576	15.034	13.763	11.766	12.867	12.707	14.346	12.7198	12.159	12.0449	11.9314	12.9163	12.8497	12.7202	12.3893
$A_{10}$	11.186	10.216	11.414	11.392	12.099	12.351	10.643	12.6845	12.0061	12.4632	13.0401	11.8151	12.5321	12.4136	12.5258
$f_{1} (H z)$	7.000	7.0003	7.000	7.000	7.0028	7.0002	7.0007	7.000	7.0005	7.0001	7.0003	7.0007	7.0001	7.000	7.0000
$f_{2} (H z)$	17.786	16.08	17.442	16.238	16.7429	16.164	17.03	16.1599	16.2484	16.2437	16.1997	16.2072	16.1703	16.1666	16.1830
$f_{3} (H z)$	20.000	20.002	20.031	20	20.0548	20.0029	20.156	20.000	19.9999	20.0064	20.0022	19.9996	20.0024	20.0012	20.0000
BestMass (kg)	537.98	535.14	531.95	529.25	534.99	531.28	535.61	530.77	525.2789	524.9274	524.8289	525.2702	524.7341	524.5747	524.4697
Avg. Mass (kg)	540.89	–	536.39	538.53	537.68	535.07	–	535.64	531.4033	528.6291	528.5501	528.7075	530.0286	527.797	526.159
SD	6.84	–	3.32	5.97	2.49	3.64	–	2.55	4.2243	3.4999	2.9827	2.8779	3.4763	2.9121	2.4646
$FE$	20,000	–	4000	4000	20,000	5000	8000	30,000	4000	4000	4000	4000	4000	4000	4000

Note: ESOS: enhanced symbiotic organisms search; MSOS: modified symbiotic organisms search; SOS: symbiotic organisms search; ABFs: adaptive benefit factors; HS: harmony search; FA; firefly algorithm; CSS: charged system search.

This simple frame was solved using the proposed ESOS algorithm by considering a population size of 20 and the maximum number of FEs was set to be 4000. The optimal solutions of the cross-sectional areas (A_i), frequency responses (f_i), minimum (best) mass, mean (avg.) mass, SD of mass, and the number of FEs for 100 independent runs, are listed in Table 5. The results show that, with the optimal mass of 524.4697 kg, the ESOS has obtained a structure which is lighter than that obtained by the MSOS algorithm, which is the best previously-reported solution. The best mass saving for ESOS is 13.5103, 10.6703, 7.4803, 4.7803,10.5203, 6.8103, 11.1403, 6.0003, 0.8092, 0.4577, 0.3592, 0.8005, 0.2644, and 0.105 kg as compared to masses obtained from the PSO, NHPGA, CSS, enhanced CSS, HS, FA, hybrid OC-GA, VPS, SOS, SOS-ABF1, SOS-ABF2, SOS-ABF1&2, ISOS, and MSOS algorithms, respectively.

Also, with the mean mass of 526.159, the proposed ESOS algorithm gives the lowest mean mass among the considered algorithms. The mean mass benefit for ESOS is evaluated as 14.731, 10.231,12.371, 11.521, 8.911, 9.481, 5.2443, 2.4701, 2.3911, 2.5485, 3.8696, and 1.638 kg in comparison with those calculated from the PSO, CSS, enhanced CSS, HS, FA, VPS, SOS, SOS-ABF1, SOS-ABF2, SOS-ABF1&2, ISOS, and MSOS algorithms, respectively. As can be observed, in general, the SOS and its variant algorithms generate better mean masses compared to the other MH algorithms.

Standard deviation is a measure of robustness of an algorithm in searching for the optima. A lower value of SD indicates more near optimal solutions. Table 5 shows that, with a SD value of 2.4646, the proposed ESOS algorithm appears to be also more robust than the other MHs considered. The other well performing algorithms in this regard are the HS and VPS algorithms. Regarding the FEs, it can be noted in Table 5 that all the SOS and CSS variant algorithms, including the proposed ESOS, have a fixed maximum value of 4000, which is far less than the FEs for the other algorithms. This shows that the SOS and CSS variants, including the proposed ESOS, have reached their optimal values with much less effort compared to the other algorithms such as HS and VPS.

Finally, as it may be seen in Table 5, for the best solution obtained using the ESOS algorithm, all frequency constraints are satisfied. It is worth noting that, similar to the results found by PSO, enhanced CSS and VPS algorithms, the first and the third frequencies obtained by ESOS are exactly the lower bounds of frequency constraints. However, the optimal masses found by these algorithms are much more than the optimal mass found by the proposed ESOS algorithm. Results of this benchmark problem show that the SOS and its variants generally perform better than the other algorithms. To further compare the performance of the ESOS with other SOS variants, their convergence histories are plotted in Figure 3. As it is noted in this figure, the proposed ESOS algorithm is more efficient in finding the optimum mass compared to both the basic SOS and the more efficient modified SOS (MSOS).

Figure 3.

Convergence histories of ESOS and other SOS variants for the 10-bar plane truss problem.

37-bar planar truss

The initial configuration of the second truss problem; a 2D, 37-bar truss, is shown in Figure 4. Wang et al. (2004) were the first researchers to consider this benchmark problem. The problem was solved later by a number of other researchers, some of which are mentioned in Table 6. In Table 4, the design variable bounds and constraints and the mechanical characteristics of the material for this problem are presented. The truss is optimized to minimize its mass considering the shape and member sizes (as design variables) and the frequency constraints given in Table 4. In this problem, the design variables are the nodal coordinates in the upper chord (y_i) and the cross-sectional areas of the members (A_i). The cross section area of all the members on the lower chord is 0.4 cm², while the remaining bars have an initial cross section area of 1 cm². Taking into account the symmetry of the truss, the members are divided into 14 groups. A 10-kg mass is connected to each of the nodes on the lower chord of the structure (Figure 4). In this benchmark problem, the nodes on the upper chord are allowed to move only vertically. Finally, to provide the symmetry of the structure, there are only five shape and 14 sizing variables in this optimization problem.

Figure 4.

Details of the 37-bar truss problem.

Table 6.

Optimal design parameters for the 37-bar truss problem (y_i are in m and A_i are in cm²).

	Wang et al. (2004)	Wei et al. (2011)	Kaveh and Zolghadr (2011)		Miguel and Miguel (2012)		Gomes (2011)	Baghlani and Makiabadi (2013)	Tejani et al. (2016)				Tejani et al. (2018)	Kumar et al. (2019)	Present study
		OC	NHPGA	CSS	Enhanced CSS	HS	FA	PSO	TLBO	SOS	SOS-ABF1	SOS-ABF2	SOS- ABF1&2	ISOS	MSOS	ESOS
$y_{3}, y_{19}$	1.2086	1.09693	0.8726	1.0289	0.8415	0.9392	0.9637	0.9357	0.9598	0.9168	0.9413	0.9060	0.9257	1.0111	0.9596
$y_{5}, y_{17}$	1.5788	1.45558	1.2129	1.3868	1.2409	1.327	1.3978	1.2941	1.3867	1.298	1.3393	1.2665	1.3188	1.4030	1.3472
$y_{7}, y_{15}$	1.6719	1.59539	1.3826	1.5893	1.4464	1.5063	1.5929	1.4758	1.5698	1.4777	1.5434	1.4834	1.4274	1.6095	1.5249
$y_{9}, y_{13}$	1.7703	1.76551	1.4706	1.6405	1.5334	1.6086	1.8812	1.6173	1.6687	1.6046	1.6744	1.6004	1.5806	1.7610	1.6545
$y_{11}$	1.8502	1.87413	1.5683	1.6835	1.5971	1.6679	2.0856	1.7001	1.7203	1.6596	1.7571	1.6397	1.6548	1.8513	1.7287
$A_{1}, A_{27}$	3.2508	2.62463	2.9082	3.4484	3.2031	2.9838	2.6797	2.8471	2.9038	2.8448	2.9344	3.3609	2.6549	2.9619	2.8986
$A_{2}, A_{26}$	1.2364	1.0000	1.0212	1.5045	1.1107	1.1098	1.1568	1.0023	1.0163	1.0785	1.0256	1.0203	1.0383	1.0202	1.0299
$A_{3}, A_{24}$	1.0000	1.00176	1.0363	1.0039	1.1871	1.0091	2.3476	1.0000	1.0033	1.0000	1.0095	1.0169	1.0000	1.0000	1.0082
$A_{4}, A_{25}$	2.5386	2.07586	3.9147	2.5533	3.3281	2.5955	1.7182	2.5114	3.1940	2.8906	2.5838	2.6772	3.0083	2.3282	2.6784
$A_{5}, A_{23}$	1.3714	1.22071	1.0025	1.0868	1.4057	1.261	1.2751	1.0684	1.0109	1.0335	1.1569	1.0166	1.0024	1.1719	1.2001
$A_{6}, A_{21}$	1.3681	1.48922	1.2167	1.3382	1.0883	1.1975	1.4819	1.2712	1.5877	1.2119	1.2548	1.2244	1.4499	1.2374	1.1850
$A_{7}, A_{22}$	2.429	2.30847	2.7146	3.1626	2.1881	2.4264	4.6850	2.9509	2.4104	3.1886	2.5104	2.7056	3.1724	2.1430	2.4667
$A_{8}, A_{20}$	1.6522	1.43236	1.2663	2.2664	1.2223	1.3588	1.1246	1.3501	1.3864	1.3435	1.4626	1.5535	1.2661	1.5308	1.4161
$A_{9}, A_{18}$	1.8257	1.64678	1.8006	1.2668	1.7033	1.4771	2.1214	1.5152	1.6276	1.7247	1.5245	1.4833	1.4659	1.4839	1.4859
$A_{10}, A_{19}$	2.3022	2.87072	4.0274	1.7518	3.1885	2.5648	3.8600	2.8262	2.3594	2.698	2.4586	2.4032	2.9013	2.4001	2.4664
$A_{11}, A_{17}$	1.3103	1.50405	1.3364	2.7789	1.0100	1.1295	2.9817	1.2135	1.0293	1.1401	1.1888	1.0000	1.1537	1.1678	1.2455
$A_{12}, A_{15}$	1.4067	1.31328	1.0548	1.4209	1.4074	1.3199	1.2021	1.3549	1.3721	1.284	1.3765	1.4982	1.3465	1.5085	1.2753
$A_{13}, A_{16}$	2.1896	2.32277	2.8116	1.01	2.8499	2.9217	1.2563	2.4864	2.0673	2.3044	2.2341	2.7480	2.6850	2.0768	2.5249
$A_{14}$	1.0000	1.04258	1.1702	2.2919	1.0269	1.0004	3.3276	1.0001	1.0000	1.0000	1.0007	1.0072	1.0000	1.0075	1.0000
$f_{1} (H z)$	20.085	20.0819	20.0000	20.0028	20.0037	20.0024	20.0001	20.0000	20.0366	20.023	20.0052	20.0023	20.0119	20.0017	20.0006
$f_{2} (H z)$	42.0743	40.0961	40.0693	40.0155	40.005	40.0019	40.0003	40.0020	40.0007	40.0394	40.0048	40.0363	40.0964	40.0018	40.0016
$f_{3} (H z)$	62.9383	60.0321	60.6982	61.2798	60.0082	60.0043	60.0001	60.0003	60.0138	60.0339	60.0077	60.0065	60.0066	60.0164	60.0016
Best Mass (kg)	366.5	363.032	362.84	362.38	361.5	360.05	377.2	359.997	360.8658	360.426	359.905	360.5007	360.7432	360.3018	359.8367
Avg. Mass (kg)	–	–	366.77	365.75	362.04	360.37	381.2	361.08	364.8521	363.3662	363.0816	363.6336	363.3978	362.9610	360.6852
SD	–	–	3.742	3.461	0.52	0.26	4.26	1.06	2.965	2.1704	1.8304	2.0771	1.5675	1.7265	0.48558
$FE$	–	–	4000	4000	20,000	5000	12,500	11,400	4000	4000	4000	4000	4000	4000	4000

For this truss, the ESOS algorithm was also run 100 times using a 20-member population and taking the maximum number of FEs as 4000. The results, in the form of optimal solutions of the upper cord coordinates (y_i), member cross-sectional areas (A_i), frequency responses (f_i), minimum (best) mass, mean (Avg.) mass and standard deviation (SD), and the number of FEs obtained for the 100 runs, are presented in Table 6. Table 6 indicates that the optimal (best) mass of 359.8367 kg, found by the proposed ESOS algorithm is also the best mass among all the solutions by other stated MH algorithms.

The average minimum mass of 360.6852 obtained by ESOS is also the lowest amongst all the solutions presented in Table 6. The SD of masses for the ESOS solution, calculated as 0.4856 is the second best after the FA solution. It is noted that the ESOS achieves better solutions regarding the average and the SD of mass among other MH solutions for the 4000 FEs. Furthermore, it is noted that the number of FEs used in the proposed algorithm and the other SOS variants and CSS variants is far less than the number of FEs used in the PSO, HS, FA, and TLBO algorithms. It is also worth noting that for the best solution obtained using the ESOS algorithm, all frequency constraints are not only satisfied, but they are almost at their minimum bounds.

72-bar space truss

Figure 5 shows details of this benchmark truss. Similar to the previous truss case studies, this large-scale optimization problem has also been studied by many researchers. In Table 4, the design parameters for this problem was presented. Utilizing the symmetry of the structure, the cross-sectional areas of the members, are divided into 16 groups. Furthermore, as illustrated in Figure 6, 2770-kg nonstructural lumped masses are attached to all top-level nodes (nodes 1–4).

Figure 5.

Details of the 72-bar truss problem.

Figure 6.

Details of the 52-bar truss problem.

This problem was solved using the proposed ESOS algorithm by considering a population of 20-members and a maximum number of FEs of 4000. From the results shown in Table 7, it may be noted that the best mass found by the proposed ESOS algorithm (323.9748 kg) is again the lowest among all the reported best masses found using other algorithms, which indicates that in solving this problem the ESOS performs better than the other MH solutions. The best mass benefits for the ESOS compared to the CSS, enhanced CSS, CSS-BBBC, CBO, TLBO, MC-TLBO, TWO, SOS, SOS-ABF1, SOS-ABF1&2, ISOS, and MSOS algorithms are 4.8392, 4.4182, 3.5322, 0.7804, 3.5932, 3.6002, 4.8552, 1.5837, 1.1112, 0.7149, 1.2569, 1.0934, and 0.3712 kg, respectively.

Table 7.

Optimal design parameters for the 72-bar truss problem (cross-sectional areas, A_i, are in cm²).

	Kaveh and Zolghadr (2011)		Kaveh and Zolghadr (2012)	Kaveh and Mahdavi (2015a)	Farshchin et al. (2016)		Kaveh and Zolghadr (2017)	Tejani et al. (2016)				Tejani et al. (2018)	Kumar et al. (2019)	Present study
		CSS	Enhanced CSS	CSS_BBBC	CBO	TLBO	MC-TLBO	TWO	SOS	SOS-ABF1	SOS-ABF2	SOS-ABF1&2	ISOS	MSOS	ESOS
$A_{1} - A_{4}$	2.528	2.522	2.854	3.3699	3.5491	3.4188	3.38	3.6957	4.182	3.6273	3.8745	3.3563	3.3177	3.5222
$A_{5} - A_{12}$	8.704	9.109	8.301	7.3428	7.9676	7.9263	8.086	7.1779	7.899	7.9416	7.6185	7.8726	8.0232	7.8578
$A_{13} - A_{16}$	0.645	0.648	0.645	0.6468	0.645	0.645	0.647	0.645	0.645	0.646	0.645	0.645	0.6458	0.6450
$A_{17} - A_{18}$	0.645	0.645	0.645	0.6457	0.645	0.645	0.646	0.6569	0.645	0.645	0.6957	0.645	0.645	0.6450
$A_{19} - A_{22}$	8.283	7.946	8.202	8.0056	8.1532	8.0143	8.89	7.7017	8.0149	7.5653	8.4112	8.5798	8.1354	8.0061
$A_{23} - A_{30}$	7.888	7.703	7.043	8.0185	7.9667	7.9603	8.136	7.9509	8.1772	8.0171	7.7833	7.6566	7.891	7.8453
$A_{31} - A_{34}$	0.645	0.647	0.645	0.6458	0.645	0.645	0.654	0.645	0.645	0.6714	0.645	0.7417	0.6535	0.6450
$A_{35} - A_{36}$	0.645	0.6456	0.645	0.6457	0.645	0.645	0.647	0.645	0.645	0.645	0.645	0.645	0.645	0.6450
$A_{37} - A_{40}$	14.666	13.465	16.328	12.4585	12.9272	12.7903	13.097	12.3994	12.4516	13.4781	12.0976	13.0864	12.3358	12.663
$A_{41} - A_{48}$	6.793	8.25	8.299	8.1211	8.1226	8.1013	8.101	8.6121	7.729	7.6531	7.7086	8.0764	7.9314	7.9784
$A_{49} - A_{52}$	0.645	0.645	0.645	0.646	0.6452	0.645	0.663	0.645	0.6525	0.645	0.645	0.645	0.645	0.6455
$A_{53} - A_{54}$	0.645	0.646	0.645	0.6459	0.645	0.6473	0.646	0.645	0.645	0.645	0.645	0.6937	0.6454	0.6454
$A_{55} - A_{58}$	16.464	18.368	15.048	17.3636	17.0524	17.4615	16.483	17.4827	16.8203	16.6583	16.9516	16.2517	17.4473	16.974
$A_{59} - A_{66}$	8.809	7.053	8.268	8.3371	8.0618	8.1304	7.873	8.1502	7.9846	8.1609	8.7289	8.1703	7.9102	8.0449
$A_{67} - A_{70}$	0.645	0.645	0.645	0.646	0.645	0.645	0.651	0.674	0.6742	0.645	0.645	0.645	0.6452	0.6450
$A_{71} - A_{72}$	0.645	0.646	0.645	0.6476	0.645	0.6451	0.657	0.655	0.645	0.645	0.645	0.645	0.645	0.6450
$f_{1} (H z)$	4.000	4.000	4.000	4.000	4.000	4.000	4.000	4.0023	4.0045	4.0013	4.0016	4.000	4.001	4.000
$f_{3} (H z)$	6.006	6.004	6.004	6.000	6.000	6.000	6.000	6.002	6.0019	6.0002	6.0003	6.0008	6.001	6.000
Best Mass (kg)	328.814	328.393	327.507	324.7552	327.568	327.575	328.83	325.5585	325.086	324.6897	325.2317	325.0682	324.346	323.9748
Avg. Mass (kg)	337.7	335.77	–	330.4154	328.684	327.693	336.1	331.1228	328.6582	328.4621	334.9979	329.4699	326.7847	324.5336
SD	5.42	7.2	–	7.7063	0.73	0.125	5.8	4.2278	2.7948	2.46	6.0566	2.6642	2.1968	0.4118
$FE$	4000	4000	4000	6000	15,000	15,000	–	4000	4000	4000	4000	4000	4000	4000

Note: ESOS: enhanced symbiotic organisms search; MSOS: modified symbiotic organisms search; SOS: symbiotic organisms search; ABFs: adaptive benefit factors; CSS: charged system search.

Table 7 also shows that the ESOS produces the best result for the mean or average mass (324.5336 kg) compared to all the other algorithms. It is also noted that the ESOS with a minimum mass SD of 0.4118 in 100 runs is the second most robust algorithm (after the MC-TLBO algorithm with SD = 0.125) to solve this benchmark problem. However, ESOS has achieved its optimal result with only 4000 FEs compared to the 15,000 FEs carried out by the MC-THBO. Furthermore, it is worth noting that for the best solution obtained using the ESOS algorithm, all frequency constraints are exactly at their minimum bounds.

52-bar dome structure

Figure 6 illustrates the initial configuration of a 52-bar spatial truss, selected as the fourth benchmark problem. This problem was first solved by Lin et al. (1982) using the Bi-factor algorithm. Several other scholars later optimized this truss problem by employing other algorithms mentioned in Table 8. In Table 4, the design parameters for this problem were presented. The sizing and shape optimization of this truss with the objective to minimize its mass using multiple natural frequency constraints (see Table 4) is carried out using the proposed ESOS method. A 50-kg lumped mass is connected to all free nodes. Utilizing structural symmetry, the 52 bars are divided into eight groups. In addition, to maintain the symmetry of the dome truss, the free nodes are permitted to move ± 2 m in the vertical direction. Consequently, 13 independent design variables exist including five shape and eight sizing variables.

Table 8.

Optimal design parameters for the 52-bar truss problem (shape variable, z_i and x_i are in m and cross-sectional areas, A_i, are in cm²).

	Lingyun et al. (2005)	Gomes (2011)	Kaveh and Zolghadr (2011)		Miguel and Miguel (2012)		Lin et al. (1982)	Kaveh and Zolghadr (2014)	Tejani et al. (2016)				Tejani et al. (21018)	Kumar et al. (2019)	Present study
	GA	PSO	CSS	Enhanced CSS	HS	FA	Bi-factor algorithm	DPSO	SOS	SOS-ABF1	SOS-ABF2	SOS-ABF1&2	ISOS	MSOS	ESOS
$z_{A}$	5.8851	5.5344	5.2716	6.1590	4.7374	6.4332	4.3201	6.1123	5.7624	5.965	6.012	5.895	6.1631	5.8551	5.9441
$x_{B}$	1.7623	2.0885	1.5909	2.2609	1.5643	2.2208	1.3153	2.2343	2.3239	2.324	2.425	2.4237	2.4224	2.3651	2.2494
$z_{B}$	4.4091	3.9283	3.7093	3.9154	3.7413	3.9202	4.174	3.8321	3.7379	3.7002	3.700	3.703	3.8086	3.7000	3.7260
$x_{F}$	3.4406	4.0255	3.5595	4.0836	3.4882	4.0296	2.9169	4.0316	3.9842	3.9636	4.0201	3.9926	4.108	3.9960	3.9564
$z_{F}$	3.1874	2.4575	2.5757	2.5106	2.6274	2.52	3.2676	2.5036	2.5121	2.5000	2.5000	2.5000	2.5018	2.5000	2.5000
$A_{1} - A_{4}$	1.0000	0.3696	1.0464	1.0335	1.0085	1.005	1.0000	1.0001	1.0988	1.0000	1.0000	1.0000	1.0074	1.0000	1.0000
$A_{5} - A_{8}$	2.1417	4.1912	1.7295	1.0960	1.4999	1.3823	1.33	1.1397	1.0031	1.1797	1.0000	1.0000	1.0003	1.0459	1.1197
$A_{9} - A_{16}$	1.4858	1.5123	1.6507	1.2449	1.3948	1.2295	1.58	1.2263	1.1956	1.2109	1.128	1.0000	1.1982	1.2388	1.1999
$A_{17} - A_{20}$	1.4018	1.562	1.5059	1.2358	1.3462	1.2662	1.0000	1.3335	1.4563	1.48	1.4466	1.5759	1.2787	1.4634	1.4547
$A_{21} - A_{28}$	1.9110	1.9154	1.7210	1.4078	1.6776	1.4478	1.71	1.4161	1.3773	1.3977	1.4298	1.4046	1.4421	1.407	1.3842
$A_{29} - A_{36}$	1.0109	1.1315	1.002	1.0022	1.3704	1.0000	1.54	1.0001	1.0055	1.0229	1.0032	1.0000	1.0000	1.0028	1.0000
$A_{37} - A_{44}$	1.4693	1.8233	1.7415	1.6024	1.4137	1.5728	2.65	1.575	1.7397	1.6747	1.7686	1.6494	1.4886	1.5907	1.6168
$A_{45} - A_{52}$	2.1411	1.0904	1.2555	1.4596	1.9378	1.4153	2.87	1.4357	1.3084	1.3033	1.277	1.5664	1.499	1.3675	1.3718
$f_{1} (H z)$	12.81	12.751	9.246	11.849	12.2222	11.3119	15.22	11.315	12.7144	11.8992	12.2594	12.814	12.5459	12.5185	11.45
$f_{2} (H z)$	28.65	28.649	28.648	28.649	28.6577	28.6529	29.28	28.648	28.654	28.6478	28.6576	28.7301	28.6518	28.6546	28.6482
Best Mass (kg)	236.046	228.381	205.237	197.337	214.94	197.53	298.0	195.351	195.4969	194.8089	195.173	198.263	194.7483	193.773	193.1647
Avg. Mass (kg)	-	234.3	213.101	205.617	229.88	212.8	-	198.71	214.6676	210.7033	211.5683	224.505	207.5498	204.4279	196.546
SD	37.462	5.22	7.391	6.924	12.44	17.98	–	13.85	15.1499	11.8339	12.7871	17.8552	8.7354	8.4014	3.3878
$FE$	–	11,270	4000	4000	20,000	10,000	–	6000	4000	4000	4000	4000	4000	4000	4000

The ESOS algorithm was run by considering a 20-member population and a total number of FEs of 4000. Table 8 lists the results obtained through 100 independent runs by the ESOS and other MH algorithms. The results show that the ESOS algorithm again ranks first in producing the best mass (193.1647 kg) amongst all the MH solutions, whereas the MSOS ranks second. The mass benefit for the ESOS algorithm is 42.8813, 35.2163, 12.0723, 4.1723, 21.7753, 4.3653, 104.8353, 2.1863, 2.3322, 1.6442, 2.0083, 5.0983, 1.5836, and 0.6083 kg compared to the NGHA, PSO, CSS, enhanced CSS, HS, FA, Bi-factor, DPSO, SOS, SOS-ABF1, SOS-ABF2, SOS-ABF1&2, ISOS, and MSOS algorithms, respectively.

Comparing the values of average mass listed in Table 8, also reveals that the proposed ESOS performs better than all the other MH solutions considered here. The mean mass benefit for ESOS is 37.754, 16.555, 9.071, 33.334, 16.254, 2.164, 18.1216, 14.1573, 15.0223, 27.959, 11.0038, and 7.8819 kg compared to the PSO, CSS, enhanced CSS, HS, FA, DPSO, SOS, SOS-ABF1, SOS-ABF2, SOS-ABF1&2, ISOS, and MSOS solutions, respectively. Regarding the SD of mean masses for this problem, Table 8 shows that with an SD = 3.3878, the proposed ESOS ranks first in performance compared to the other solutions, followed by the PSO solution with SD = 5.22. In addition, the total FEs utilized in the proposed algorithm (4000) is considerably less than the FEs used in the PSO, HS, FA, and DPSO solutions.

It is clear from Table 8 that the results from the ESOS algorithm solution are superior to the results for this benchmark problem using other MHs. It is worth noting that none of the natural frequency constraints were violated using the ESOS. The small value of SD of the mean mass values for the 100 independent runs also shows the robustness of the proposed algorithm for solving this type of complex problem.

The superior performance of the ESOS algorithm is further verified by comparing its convergence history with those of the other SOS variants in Figure 7. It can be noted that the proposed ESOS algorithm is more efficient in finding the optimal mass of this problem compared to both the basic and modified SOS variants.

Figure 7.

Convergence histories of ESOS and other SOS variants for the 52-bar truss problem.

200-bar planar truss structure

In Figure 8, the topology, node pattern, and element numbering of a 200-bar planar truss problem are shown. In Table 4, the design variables and frequency constraints for this problem were given. A constant 100-kg lumped mass is attached to nodes 1 to 5. To maintain geometrical symmetry, the 200 elements are divided into 29 groups.

Figure 8.

Details of the 200-bar truss problem.

Similar to the previous trusses, the proposed ESOS algorithm was run 100 times with a population size of 20. Nevertheless, because of the large size of the problem, the maximum number of FEs was increased to 10,000. The results for the ESOS and other MH algorithms are compared in Table 9. The best mass solution using the ESOS algorithm is found to be 2159.718 kg. This mass is again less than all the best masses produced by the other algorithms. The best mass of ESOS algorithm is 100.142, 138.892, 43.494, 29.362, 20.603, 5.166, 6.083, 48.17, 9.741, and 4.752 kg less than the best masses reported for the CSS, enhanced CSS, CBO, 2D-CBO, SOS, SOS-ABF1, SOS-ABF2, SOS-ABF1&2, ISOS, and MSOS algorithms.

Table 9.

Optimal design parameters for the 200-bar truss problem (cross-sectional areas, G_i, are in cm²).

Group no	Bars	Kaveh and Zolghadr (2011)		Kaveh and Mahdavi (2015b)		Tejani et al. (2016)				Tejani et al. (2018)	Kumar et al. (2019)	Present study
Group no	Bars	CSS	Enhanced CSS	CBO	2D-CBO	SOS	SOS- ABF1	SOS-ABF2	SOS-ABF1&2	ISOS	MSOS	ESOS
$G_{1}$	1,2,3,4	1.2439	0.2934	0.3268	0.4460	0.4781	0.2822	0.3058	0.3845	0.3072	0.3951	0.3005
$G_{2}$	5,8,11,14,17	1.1438	0.5561	0.4502	0.4556	0.4481	0.5014	0.5196	0.8524	0.5075	0.4399	0.4540
$G_{3}$	19,20,21,22,23,24	0.3769	0.2952	0.1000	0.1519	0.1049	0.1071	0.1000	0.1130	0.1001	0.1000	0.1000
$G_{4}$	18,25,56,63,94,101,132,139,170,177	0.1494	0.1970	0.1000	0.1000	0.1045	0.1002	0.1092	0.1000	0.1000	0.1000	0.1000
$G_{5}$	26,29,32,35,38	0.4835	0.8340	0.7125	0.4723	0.4875	0.5277	0.5238	0.5084	0.5893	0.4422	0.5095
$G_{6}$	6,7,9,10,12,13,15,16,27,28,30,31,33,34,36,37	0.8103	0.6455	0.8029	0.7543	0.9353	0.8248	0.7956	0.8885	0.8328	0.813	0.8246
$G_{7}$	39,40,41,42	0.4364	0.1770	0.1028	0.1024	0.1200	0.1300	0.1003	0.1000	0.1431	0.1000	0.1000
$G_{8}$	43,46,49,52,55	1.4554	1.4796	1.4877	1.4924	1.3236	1.4016	1.3119	1.217	1.3600	1.4808	1.4326
$G_{9}$	57,58,59,60,61,62	1.0103	0.4497	0.1000	0.1000	0.1015	0.1000	0.1056	0.1356	0.1039	0.1000	0.1000
$G_{10}$	64,67,70,73,76	2.1382	1.4556	1.0998	1.6060	1.4827	1.4657	1.6178	1.5477	1.5114	1.4549	1.5949
$G_{11}$	44,45,47,48,50,51,53,54,65,66,68,69,71,72,74,75	0.8583	1.2238	0.8766	1.2098	1.1384	1.1327	1.1954	1.0568	1.3568	1.2452	1.1607
$G_{12}$	77,78,79,80	1.2718	0.2739	0.1229	0.1061	0.102	0.1196	0.1615	0.4552	0.1024	0.1397	0.1250
$G_{13}$	81,84,87,90,93	3.0807	1.9174	2.9058	3.0909	2.9943	3.0262	2.9102	3.4433	2.9024	3.0458	2.9857
$G_{14}$	95,96,97,98,99,100	0.2677	0.1170	0.1000	0.7916	0.1562	0.2527	0.1134	0.1000	0.1000	0.1000	0.1000
$G_{15}$	102,105,108,111,114	4.2403	3.5535	3.9952	3.6095	3.433	3.3267	3.5156	3.606	3.412	3.3282	3.2300
$G_{16}$	82,83,85,86,88,89,91,92,103,104,106,107,109,110, 112,113	2.0098	1.3360	1.7175	1.4999	1.6816	1.5963	1.6227	1.446	1.4819	1.5484	1.5819
$G_{17}$	115,116,117,118	1.5956	0.6289	0.1000	0.1000	0.1026	0.2417	0.3687	0.1893	0.2587	0.3122	0.2604
$G_{18}$	119,122,125,128,131	6.2338	4.8335	5.9423	5.2951	5.0739	4.8557	4.6196	5.1791	4.8291	5.154	5.0668
$G_{19}$	133,134,135,136,137,138	2.5793	0.6062	0.1102	0.1000	0.1068	0.1001	0.1543	0.2666	0.1499	0.1032	0.1011
$G_{20}$	140,143,146,149,152	3.052	5.4393	5.8959	4.5288	6.0176	5.4975	5.6545	5.875	5.509	5.3092	5.3907
$G_{21}$	120,121,123,124,126,127,129,130,141,142,144,145,147,148,150,151	1.8121	1.8435	2.1858	2.2178	2.0340	2.0829	2.2106	2.5624	2.2221	2.1574	2.1087
$G_{22}$	153,154,155,156	1.2986	0.8955	0.5249	0.7571	0.6595	0.8522	0.6688	0.7535	0.6113	0.7208	0.7775
$G_{23}$	157,160,163,166,169	5.881	8.1759	7.2676	7.7999	6.9003	7.5480	7.4241	7.9706	7.3398	7.8337	7.6649
$G_{24}$	171,172,173,174,175,176	0.2324	0.3209	0.1278	0.3506	0.202	0.1279	0.1187	0.3324	0.1559	0.3241	0.1050
$G_{25}$	178,181,184,187,190	7.7536	10.98	7.8865	7.8943	6.8356	7.6278	7.5955	7.3386	8.6301	7.9443	7.8761
$G_{26}$	158,159,161,162,164,165,167,168,179,180,182,183,185,186,188,189	2.6871	2.9489	2.8407	2.8097	2.6644	3.0233	2.7572	3.0958	2.8245	2.9322	2.8397
$G_{27}$	191,192,193,194	12.5094	10.5243	11.7849	10.422	12.143	10.3024	11.1467	9.1512	10.8563	10.3005	10.3838
$G_{28}$	195,197,198,200	29.5704	20.4271	22.7014	21.2576	22.2484	21.4034	21.4328	20.723	20.9142	20.7009	21.2356
$G_{29}$	196,199	8.291	19.0983	7.884	11.9061	8.9378	10.481	9.869	12.1258	10.5305	11.2426	10.8558
$f_{1} (H z)$		5.000	5.010	5.001	5.0016	5.0001	5.0001	5.0000	5.0000	5.0000	5.0000	5.0000
$f_{2} (H z)$		15.961	12.911	12.5247	13.3868	13.4306	12.1388	12.3327	13.3064	12.4477	12.8238	12.1931
$f_{3} (H z)$		16.407	15.416	15.1845	15.1981	15.2645	15.1284	15.1454	15.5397	15.2332	15.0104	15.0312
Best Mass (kg)		2259.86	2298.61	2203.212	2189.08	2180.321	2164.884	2165.801	2207.888	2169.459	2164.47	2159.718
Avg. Mass (kg)		–	–	2481.492	2308.443	2303.303	2186.574	2187.252	2405.348	2244.637	2185.431	2161.232
SD				250.8259	132.5148	83.5897	15.2711	16.9436	128.1578	43.4808	15.4062	2.3521
$FE$		10,000	10,000	10,000	10,000	10,000	10,000	10,000	10,000	10,000	10,000	10,000

Note: ESOS: enhanced symbiotic organisms search; MSOS: modified symbiotic organisms search; SOS: symbiotic organisms search; ABFs: adaptive benefit factors; CSS: charged system search.

Table 9 shows that with a mean (average) mass of 2161.232 kg, the ESOS algorithm again outperforms all the other reported MH solutions. Furthermore, with a mass SD of 2.3521, the ESOS appears to be far more robust than all the other algorithms. The results also show that in the optimal solution given by the ESOS algorithm all frequency constraints are satisfied.

Conclusions

An enhanced SOS (ESOS) was developed and presented. Modifications to SOS algorithm are carried out in all three phases: mutualism, commensalism, and parasitism. The modifications improve the efficiency of search by creating a good balance between the exploration and exploitation capabilities of the algorithm. The proposed algorithm can be easily implemented because no tuning parameter needs to be calibrated in the algorithm. The proposed algorithm was first evaluated using a set of 22 CEC2014 benchmark functions. The ESOS algorithm was also applied to solving five benchmark truss size and shape optimization problems subjected to multiple natural frequency constraints. The results of the above investigations lead us to the following conclusions:

To evaluate the performance of the proposed ESOS algorithms, a set of 22 unimodal, multimodal, and hybrid mathematical benchmark functions, proposed in the CEC2014 competition, were solved and the results were compared with those of the SOS, ISOS, MSOS, WWO, BA, HuS, GSA, BBO, and IWO algorithms. The Friedman rank test performed on the mean fitness values and SD of the fitness values for the 22 CEC2014 standard functions considered indicates that the overall ranking of the proposed ESOS algorithm amongst all the MH algorithms considered is 2nd in both the mean fitness and the SD values. Therefore, it is concluded that the proposed ESOS algorithm outperforms all the other algorithms (including all the SOS variants), except for the WWO algorithm, in optimizing this set of standard functions.

To evaluate the performance of the proposed algorithm in solving truss optimization problems with frequency constraints, the results of the ESOS algorithm were compared with those available for the NHGA, PSO, NHPGA, CSS, enhanced CSS, HS, FA, CSS-BBBC, hybrid OC-GA, TWO, SOS, SOS-ABF1, SOS-ABF2,SOS-ABF1&2, ISOS, and MSOS algorithms. A review of the results obtained from solving the five benchmark truss problems shows that the SOS algorithm and its variants are generally superior and more reliable compared to the other MH solutions reported in the literature. It is also found that the proposed ESOS algorithm is more efficient and robust compared to the basic SOS algorithm and its other recently proposed ISOS and MSOS variants.

The numerical results (such as those depicted in Figures 3 and 7) indicate that by using the proposed ESOS to solve truss optimization problems, not only better solutions may be obtained but also a significant reduction in computational cost may be achieved in comparison with other optimization algorithms.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Mahmoud R Maheri

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