Size optimization of planar truss systems using the modified salp swarm algorithm

ABSTRACT This work evaluates the performance of the salp swarm algorithm (SSA) for truss system optimization problems and presents a novel method called the modified salp swarm algorithm (MSSA). Five truss structures, previously optimized by metaheuristics and containing discrete and continuous variables, were used for the evaluations. Size and size–shape optimization types were considered. Although the SSA performs poorly and has convergence issues in initial random solutions, it reaches comparable solutions to previously published results, particularly in continuous problems. In contrast, the MSSA achieves the best solutions for discrete problems and is relatively close to the best results in the reference literature on continuous problems. Moreover, the MSSA convergence curves exhibit a modest increase in convergence rates, especially for discrete problems. It is envisaged that the findings will contribute to improving solution performance, convergence speed and security for future real-world applications.


Introduction
Engineering design problems are quite challenging because of many constraints, such as strength, displacement limits and matrix computations, requiring complex solutions.These problems are also difficult to optimize and cannot be solved effectively using classical derivative-based mathematical optimization techniques.Metaheuristic techniques are overly critical in solving such problems.Novel metaheuristic methodologies, particularly swarm intelligence techniques, are phenomenally successful at tackling structural optimization problems.Although metaheuristic or swarm intelligence techniques are suitable for discrete engineering problems, one of the most significant drawbacks is that they employ stochastic solutions, which lack stability or constancy of performance in their results.Therefore, metaheuristic techniques yield reliable results for a specific mathematical problem but may have decreased performance when applied to another optimization problem.Thus, such techniques may require validation for various problems, and performing such analyses provides essential insight into the robustness of the solutions.As a result, applying recently developed optimization techniques to different problems is beneficial.
Since truss structures are a frequently used optimization problem in the literature, several metaheuristic algorithms have been developed to solve similar problems.The general categories of metaheuristic algorithms are (1) local search-based, (2) evolutionary search-based, and (3) swarm search-based algorithms (Abualigah et al. 2020).Local search-based algorithms consist of four different algorithms, namely simulated annealing, β-hill climbing, tabu search and hill-climbing.Some problems include truss optimization using the local search-based algorithms simulated annealing (Lamberti 2008;Millan-Paramo and Abdalla Filho 2020) and tabu search (Li and Lian 2018;Bennage and Dhingra 1995).To the best of the authors' knowledge, examples of β-hill climbing and hill-climbing algorithms are lacking in the literature for solving truss optimization problems.The evolutionary search-based algorithms, including genetic algorithms (Galante 1996;Lingyun et al. 2005;Xu et al. 2010), genetic programming (Rajeev and Krishnamoorthy 1997;Yang and Soh 2002;Assimi, Jamali, and Nariman-Zadeh 2017), differential evaluation algorithms (Wu and Tseng 2010;Bureerat and Pholdee 2016) and harmony search algorithms (Lee and Geem 2004;Degertekin 2012;Miguel and Miguel 2012), perform well in truss optimization.Finally, the third category is swarmbased algorithms, which include but are not limited to the krill herd algorithm (Gandomi and Alavi 2016), particle swarm algorithm (Gomes 2011;Perez and Behdinan 2007;Kaveh and Javadi 2014), cuckoo search algorithm (Gandomi et al. 2013;Cuong-Le et al. 2021), artificial bee colony algorithm (Sonmez 2011;Jawad et al. 2021), biogeography-based optimization (Jalili and Hosseinzadeh 2018;Jalili, Hosseinzadeh, and Taghizadieh 2016;Aydogdu, Carbas, and Akin 2017) and a few newly developed metaheuristic techniques, including electrostatic discharge algorithms (Aydogdu, Ormecioglu, and Carbas 2021) and search group algorithm (Gonçalves, Lopez, and Miguel 2015).Truss optimization is also projected as a test problem in many novel metaheuristic algorithms.Therefore, this work aims to evaluate truss optimization in benchmark problems, and a novel contribution to the literature will be offered by developing a modified metaheuristic algorithm.
As stated previously, metaheuristic methods can effectively solve many engineering problems.The salp swarm algorithm (SSA) is a recently developed metaheuristic algorithm that has never been applied to solving truss structure optimization, which is one of the conventional structural optimization problems.Mirjalili et al. (2017) developed the SSA, a population-based metaheuristic optimization method inspired by swarm intelligence.Although the SSA was developed recently, numerous applications of SSA have been documented in various fields.For instance, the SSA has been used to optimize benchmarks and has addressed real-world problems.Abualigah et al. (2020) conducted a comprehensive SSA survey.They summarized the SSA applications as machine learning applications (feature selection and training neural networks), engineering applications (scheduling and control of power systems), image processing, wireless networking and other salp swarm applications.Nevertheless, regarding these solutions, it is concluded that the SSA has not been used to optimize truss systems.Therefore, in this work, the performance of the SSA in truss optimization problems was investigated.In addition, when assessing the SSA, it was discovered that the performance of the algorithm was not highly effective.Therefore, a modified version of the SSA was also developed by reinforcing the deficient and weak characteristics of the original algorithm.While this study presents a new algorithm to the swarm-based algorithm literature, both algorithms are applied for the first time to optimize truss systems, one of the most well-studied structural optimization systems with discrete and continuous variables.For this purpose, a novel algorithm called the modified salp swarm algorithm (MSSA), which optimizes truss systems, was developed.The performance of the MSSA was assessed on five different benchmark problems.Then, the performance of the SSA and the MSSA was compared with metaheuristic methods previously analysed in the literature.

Optimization methods: SSA and MSSA
In this section, the SSA and the newly developed MSSA methodology used in this work are introduced.First, a brief introduction to natural phenomena, including the swarm movement and the salp chain, is provided.Then, the SSA methodology, inspired by natural phenomena, is shared.Finally, the SSA and MSSA to be used for truss optimization problems are described in depth with a comprehensive description that includes the equations used by the algorithms and the definitions of each parameter.In addition, two novel improvements are made to enhance the performance of the SSA, which are vital aspects of this research that express the merits of this work.The flowchart and pseudo-code for SSA are provided, along with the necessary equation modifications for the MSSA.

Natural phenomena
More than 1.2 million marine species have already been catalogued in a centralized database, which serves as a resource for researchers (McCauley et al. 2015).Most of these species exhibit behaviours and features that are comparable to one another, such as communication strategies, locomotor performance and food-seeking behaviours.A salp is a kind of marine creature that is classified as a member of the Salpidae family and has a translucent, barrel-shaped body.The tissues, forms and movements of these creatures are very similar to those of jellyfish (Madin 1990).They have a cylindrical form with openings at the ends that allow water to flow through their gelatinous bodies, allowing them to move and feed via internal feeding filters.They often form a swarm in deep waters, known as a salp chain.This is one of the salps' fascinating behaviours, which may improve their mobility through quick, coordinated adjustments and foraging (Anderson and Bone 1980).

Methodology
The SSA was inspired and developed by mimicking the behaviours of salps in nature.To model the salp chains mathematically, the population is separated into two groups: a leader and followers.The salp at the head of the chain is called the leader, while the remaining salps are considered followers.As the name suggests, the leader of these salps directs the swarm while the followers follow one another.Since salps are defined in an n-dimensional search space, the positions of all salps are kept in a twodimensional matrix.In addition, the SSA defines a food supply in the search space as the swarm's target.The leader's position is solely updated with the food supply.The position of the followers is updated by Newton's law of motion.

SSA for the truss optimization problem
In a truss optimization problem, the SSA attempts to determine the section area corresponding to the elements constituting the best-suited truss system to optimize it as much as possible.The terminology used in the SSA for the truss optimization problem is described as follows: According to the descriptions and definitions, the main steps in the SSA for the truss optimization problem are detailed as follows.
Step 1. Initialization: The SSA creates initial truss designs randomly, formulated as follows: where rnd is a function that returns floating-point numbers between 0 and 1 that are drawn from a uniform distribution.ub i and lb j are the upper and lower boundaries of the jth design variable.
The algorithm evaluates the initial designs and calculates their fitness values according to the criteria described in the online supplemental material (Altay, Çetindemir, and Aydoğdu 2022).Algorithm memory is created by recording the design variables, fitness and penalty values of the solutions.
Step 2. Movement of the leader: There are two movement models defined in the SSA: the movement of the leader and the movement of the followers.For leader movement, first, the algorithm sorts its solutions in descending order of their fitness values and assigns the first solution (truss design) in the ranking as the leader salp.Then, the SSA updates the position of the leader salp, as follows: where c 1 is the parameter that sets the salp swarm movement direction and size.The value of c 1 is particularly important to balance exploration and exploitation, and is determined as follows: where iter and iter max represent, respectively, the current iteration and maximum iteration number defined in the problem.The updated solution (truss design) is evaluated, and its fitness value is calculated.If the fitness value of the new truss design is greater than the fitness value of the old design, the new truss design takes the place of the old truss design.This procedure is named greedy selection.
Step 3. Movement of the followers: In this step, the follower salps (remaining truss designs) update their positions (change their design) relative to the leader salp.The movement is performed sequentially in order of Newton's law of motion.The SSA reformulates Newton's law of motion by considering the time as an iteration and setting the variety between iterations as 1 (Equation 4): Updated positions of the follower salps are evaluated, and their fitness value is calculated.The algorithm applies a greedy selection procedure for all follower salps.After Step 3, one iteration is completed, and the SSA goes back to Step 2. The SSA repeats the procedures between Steps 2 and 3 until the maximum iteration number is reached.

MSSA for the truss optimization problem
As a result of the tests (Section 3), it was seen that SSA could not adjust the balance of exploration and exploitation at the beginning and end of the optimization process.The following two improvements were made to address this shortcoming of SSA.

Modification of parameter c 1
While the classical SSA has achieved satisfactory results regarding solving mathematical problems in the literature, it has been demonstrated in tests (see Section 3) that the SSA has significant convergence problems in solving structural optimization problems.Convergence problems occur particularly in the initial stages of the solution process, and the SSA cannot improve the solutions.Figure 1 illustrates the search histories for the lowest and average values acquired by the current cycle of SSA for the 10bar truss example.According to Figure 1, the difference between the average and minimum values is quite large, demonstrating that the SSA produces highly dispersed solutions at the beginning of the optimization process.This problem persists for the first 600 cycles, and the SSA cannot improve the solutions sufficiently during this time.It was also stated in Section 2.3 that the value of c 1 is vital to prevent the generation of dispersed solutions.When the change in c 1 is investigated, c 1 variates close to linear, despite being formulated exponentially.To convert the change in c 1 into an exponential structure, the MSSA updates Equation (3) as follows.Changes in c 1 values calculated by the SSA and MSSA are shown in Figure 2.
Here, β is a constant parameter to prevent near-zero c 1 values in progressive iterations (when c 1 = 0, the leader salp cannot move).Preliminary parameter work was conducted to determine the convenient β parameter, using a discrete optimization problem (47-bar truss).The convergence curves and optimum solutions are shown in Figures 3 and 4, respectively.According to the figures, the best results are obtained when the β value is between 0.01 and 0.1.Therefore, it is recommended to consider the β constant parameter between these values.

Updating the movement of the followers' formula
Whereas the SSA method produces highly dispersed solutions at the beginning of the optimization process, exploitation becomes dominant in the last cycles.In this case, local convergences can be seen, especially in discrete problems.In the movement of the leader phase, parameter β in Equation ( 5) can help to overcome the stagnation problem.However, a new computation is needed to increase the exploration ability of SSA in the movement of the followers' phase.For this purpose, in the MSSA, the movement formula of followers is modified as follows: where subscript k represents a randomly selected salp from the algorithm memory.Figure 5 shows the pseudo-code of the SSA and MSSA.A detailed flowchart can be found in the online supplemental material.

Design examples
Since this article aims to assess the performance of the SSA for optimization problems of truss systems, five different benchmark problems are investigated in this research to compare the performance of the SSA and the MSSA.These are the 10-bar, 18-bar, 47-bar, 52-bar and 200-bar truss problems, which have previously been solved using various methodologies described in the literature.Two are selected as continuous (10-bar and 200-bar trusses) and the other two as discrete (47-bar and 52bar trusses) for size optimization.Moreover, the 18-bar truss problem is analysed using discrete and continuous variables for size and shape optimization.Figures 6-9 give the graphical illustrations of these examples.Information about the design examples (load conditions, material and connection properties, and strength and displacement limitations) is given in the online supplemental material (Altay, Çetindemir, and Aydoğdu 2022).Thirty independent tests were conducted using truss structures, the SSA and MSSA methodologies, and a variety of seed values.The area values of the designs with the lowest weight derived from 30 independent analyses, averages and standard deviation values are included in a detailed table as supplemental material, along with published data references (Altay, Çetindemir, and Aydoğdu 2022).The search histories, box-plot diagrams and tables for detailed optimum solutions can also be found in the supplemental material.

A 10-bar truss
As seen in Table 1, the results obtained with the SSA and MSSA remain between the reference results.The SSA's solution is about 0.4% heavier than the best result (Asl, Aslani, and Panahi 2013), which is more significant than the weight difference between the MSSA and Asl, Aslani, and Panahi's (2013) method, which is only 0.05%.Statistically, the MSSA has a lower average value and standard deviation than the SSA.Detailed search histories and convergence trends indicate that the MSSA has a higher  convergence rate than the standard SSA.In addition, the MSSA reaches the optimal solution when it reaches 20% of the optimization process, whereas the standard SSA reaches the optimal solution when it reaches around 95% of the optimization process.Based on the SSA and MSSA box-plot diagrams, the range of optimum solutions for the classical SSA is wider than that for the MSSA in the test results, and the MSSA achieves comparably close optimum solutions within a much narrower range.This demonstrates that the MSSA performs more consistently than the classical SSA.
Table 2 shows that the SSA and MSSA performed, respectively, about 5% and 1% worse (heavier) than one of the best-yielding methods in the literature, sequential cellular particle swarm optimization (SCPSO).The SSA and MSSA box-plot diagrams indicate that the range of optimum solutions for both the classical SSA and the MSSA have similar consistency behaviours.Figure 10 illustrates the optimum layout of the 18-bar truss for the SSA and MSSA.Demand/capacity (D/C) ratios are shown according to the elements' allowable stresses, and the elements' cross-sectional areas are illustrated through line thickness.In addition, the initial layout is shown as dashed lines.

A 47-bar truss
Table 3 shows the lowest structure weight acquired during 30 analyses.To the best of the authors' knowledge, the MSSA achieved the lowest result (2107.245lb) among the reference studies presented.While SSA developed a more optimal (lower) solution (2218.483lb) than harmony search (HS) (2396.8lb), colliding bodies optimization (CBO) (2386.0lb) and the discrete advanced Jaya algorithm (DAJA) (2376.019lb), it performed worse than the switching teams algorithm (STA) approach (2172.49lb).The difference between the results of the SSA and MSSA methods is about 5%.On the other hand, the difference between the MSSA method and the method (STA) that performs closest to it is 3%.As in the first example, statistically, the average value of the MSSA is much lower than the SSA, and the standard deviation value is also lower than the SSA.In addition, the standard deviation of the SSA is significantly greater than the reference values.
As illustrated in the supplemental material, the MSSA has an exceptionally smooth convergence curve and achieves the optimal solution without encountering any issues during the optimization process.In contrast, the SSA has significant convergence problems; notably, the first 30-40% of the optimization process has difficulty developing optimal results, but it can make significant progress after 40%, and optimization convergence appears to be approaching a better solution after 60-70%.
As in the other representative examples, based on box-plot diagrams of the SSA and MSSA, it is observed that the range of optimum solutions obtained by the classical SSA is broader than that obtained by the MSSA.In addition, the MSSA provides extremely close optimum solutions for each cross-section within a relatively narrow range, indicating that the MSSA performs more reliably than the classical SSA.However, unlike the 10-bar truss, the difference between the average and best solution intervals and the average and worst solution intervals for the SSA was more uneven.This indicates that the solution has the potential to generate extremely poor solutions in the interval, reducing the reliability of the SSA approach.On the other hand, the MSSA's average, best and worst solutions are gathered in a balanced and very narrow range, which makes the MSSA a much more reliable and stable method than the SSA.

A 52-bar truss
The MSSA approach obtained the best (lowest) solution; the SSA method also performed well, outperforming seven of the eight reference solutions taken from the literature, with a difference of around 0.033% more than (lower performance) the hybrid particle swarm optimization and genetic algorithm (HPSOGA) method (Table 4).The difference between the two strategies (SSA and MSSA) is around 0.12%, and both methods perform exceptionally well in general.As illustrated in the figures of the supplemental material, convergence curves and trends for the current 52-bar truss problem reveal that the MSSA exhibits superior convergence behaviour to SSA.Initial solutions of the SSA have trouble advancing, resulting in a low convergence rate.Box-plot diagrams of the SSA and MSSA show that the range of optimum solutions obtained by the classical SSA is broader than that obtained by the MSSA, and that the MSSA provides extremely close optimum solutions for each cross-section within a relatively narrow range.This demonstrates that the MSSA performs more reliably than the classical SSA.In addition, as in the 42-bar truss problem, the difference between the average and best solution intervals and the average and worst solution intervals for the SSA is uneven.This indicates that the SSA has the potential to generate extremely poor solutions in the interval.In contrast, the MSSA's average, best and worst solutions are gathered in a balanced and very narrow range, making the MSSA a much more reliable and stable method than the SSA, and demonstrating good performance.
and Geem 2004;Lamberti 2008;Kim and Byun 2020;Pierezan et al. 2021;Degertekin and Hayalioglu 2013).According to the figures in the supplemental material, although the SSA has a decent convergence curve, the MSSA appears superior.The MSSA's solution interval is narrower than the SSA's solution interval, as seen in the box-plot diagrams.In addition, the MSSA's mean and standard deviation values are smaller than those for the SSA.Moreover, although the SSA appears to be more stable than the prior solutions, it was noticed that the solution intervals are larger, and the values obtained for the solution interval (average, maximum and minimum values) are nearly twice as large as those obtained for the MSSA.

Determination of search parameters
One of the biggest advantages of metaheuristic methods is the ease of adaptation to problems of several types and sizes.However, the search parameters of metaheuristic techniques need to be calibrated to enable them to be adapted to different problems.Since the SSA and MSSA have not been adapted to structural optimization problems in the literature, sensitivity analysis was performed, and the most suitable method parameters were determined in this study.In this context, all samples in the study were analysed according to different search agent values.Since all parameters are generated randomly in the formulae, the number of search agents is taken as the only variable for each benchmark problem.The minimum and standard deviation values were determined by performing 1000 iteration analyses 30 times, beginning with 10, the number of salps, and ending with the value 100 (including 100) at intervals of 10, to identify the optimal search agent values.A detailed table of the search parameters is provided in the supplemental material, with the minimum values (the best results) determined by these investigations highlighted in bold.

Discussion
The findings in Section 3 are discussed in terms of the differences in continuous and discrete problems regarding the best results for weight, average, standard deviation and convergence performance of the two optimization methods, the SSA and the MSSA.The SSA and MSSA have been shown to perform better in discrete problems than in continuous problems.In both discrete problems (47-bar and 52-bar trusses), MSSA demonstrated superior performance and found the best solutions.The differences between the literature results and the MSSA's optimal solution are 3-12% and 0.08-3.6%for the 47-bar and 52-bar trusses, respectively.Although the SSA could not find the optimal solution, it obtained the third-best solution for discrete problems.
Even though the MSSA cannot produce the best solution for continuous problems, the differences between the best solutions are 0.05% and 2.36% for the 10-bar and 200-bar trusses, respectively.In contrast, the SSA did not perform well in continuous problems, and it found 0.37% and 47.78% heavier designs for the 10-bar and 200-bar trusses, respectively.
As the problem sizes increase, the decrease in the performance of the SSA becomes much more evident, and the more significant the difference between the SSA and MSSA.
It is seen that the SSA performs in a more dispersed manner, has a broader solution range, has higher standard deviation values and has a more unreliable structure in all solutions compared to the MSSA.In contrast, the MSSA seems to produce a more stable, more balanced and lower solution range than the SSA.
The convergence trends for all benchmark problems are defined by power functions, proving mathematically that the power coefficient (absolute value) for the MSSA is greater than that for the SSA, indicating a higher convergence rate.
The 18-bar truss is adopted as the second benchmark problem to assess whether the SSA and the modified methodology (MSSA) can simultaneously address continuous and discrete design variables and optimize the size and shape of truss structures subject to service loads.The MSSA performs comparably to one of the best solutions in the literature (SCPSO), but about 1% worse (heavier).In contrast, the SSA yields 5% worse than the SCPSO method.In addition, the MSSA obtains the best solution after 30% of the optimization process, whereas the standard SSA does so just before 90%.

Conclusions
The recently developed SSA and MSSA methods are evaluated on five different benchmark problems.The significant findings of this article are listed as follows: (1) The SSA method performs poorly, particularly in continuous problems (10-bar and 200-bar trusses), and produces results that are inferior to those obtained in benchmark problems, whereas the MSSA method yields solutions that are superior to those obtained by the SSA and close to the best results presented in the literature, used as reference.
(2) Both methods produce acceptable results in discrete problems (47-bar and 52-bar trusses), although the MSSA outperforms the SSA.(3) The MSSA shows a reasonable improvement in convergence rate, as seen in the convergence curves, whereas the SSA has significant convergence issues in discrete problems.This is due to the previously specified parameter (c 1 ), and the MSSA was developed because of changes in the parameter.Therefore, the beneficial effect of the changed parameter on the convergence rate of the MSSA was shown.Thus, the MSSA achieves a significantly higher rate of convergence when solving all benchmark problems.(4) As the problem size increases, especially in continuous problems, the performance of the SSA decreases, whereas better and more stable solutions are obtained in the MSSA.In addition, the difference between the performance of the SSA and MSSA increases as the problem size increases.Statistically, the MSSA exhibits more consistent and more stable behaviours than the SSA, resulting in the MSSA being a safer approach.(5) The SSA and MSSA are also tested on a problem that has both discrete and continuous variables (18-bar truss).The performance of the MSSA is remarkably close to the best results in the references.Like in continuous problems, the best solution of the SSA is much worse than the reference results.
In summary, it is demonstrated that the modifications to the MSSA yield outstanding results in terms of solution performance, convergence speed and solution security; it is also demonstrated that the MSSA is a robust and efficient technique that is effectively applicable to several similar problems.There may be issues in the performance of algorithms such as the MSSA in real-size problems, which could be considered in future studies.Because of certain drawbacks experienced in their solutions, optimization studies to improve the performance of the SSA and MSSA by incorporating well-known mathematical techniques such as Lévy flight and chaos theories may be considered in future studies.

Disclosure statement
No potential conflict of interest was reported by the authors.
Algorithm memory ([X]): A matrix that contains design variables, fitness values and objective functions of the truss designs that are generated from the SSA.Salps (x): A candidate truss design/a truss design.Number of salps (N ss ): The number of truss designs stored in the [X].Position of the salp swarm (x j ; j ∈ [1, 2, . . .ndv]): Any design variable value of the truss.Position change of the salp: Modification of the truss design.Performance of the salp (fit i ; i ∈ [1, 2, . . ., N ss ]): The fitness value of the truss design.Salp having the best performance (fit best ): The global best solution in the optimization problem.

Figure 1 .
Figure 1.Comparison of the lowest and average values acquired by the current cycle of the salp swarm algorithm (SSA) for the 10-bar truss example.

Figure 2 .
Figure 2. Changes in c 1 values calculated by the salp swarm algorithm (SSA) and modified salp swarm algorithm (MSSA).

Figure 3 .
Figure 3. Convergence curves for the 47-bar truss based on various β parameters.

Figure 4 .
Figure 4. Minimum weight for the 47-bar truss based on the β parameter on a logarithmic scale.

Table 1 .
Optimum solution details for the 10-bar truss example (weight in lb).

Table 2 .
Optimum solution details for the 18-bar truss example.

Table 4 .
Optimum solution details for the 52-bar truss example (weight in kg).

Table 5 .
Optimum solution details for the 200-bar truss example (weight in lb).