An improved volleyball premier league algorithm based on sine cosine algorithm for global optimization problem Reza Moghdani, Mohamed Abd Elaziz, Davood Mohammadi & Nabil Neggaz Engineering with Computers An International Journal for SimulationBased Engineering ISSN 0177-0667 Engineering with Computers DOI 10.1007/s00366-020-00962-8 1 23 Your article is protected by copyright and all rights are held exclusively by Springer-Verlag London Ltd., part of Springer Nature. This eoffprint is for personal use only and shall not be self-archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”. 1 23 Author's personal copy Engineering with Computers https://doi.org/10.1007/s00366-020-00962-8 ORIGINAL ARTICLE An improved volleyball premier league algorithm based on sine cosine algorithm for global optimization problem Reza Moghdani1 · Mohamed Abd Elaziz2 · Davood Mohammadi3 · Nabil Neggaz4,5 Received: 19 June 2019 / Accepted: 22 January 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020 Abstract Volleyball premier league (VPL) simulating some phenomena of volleyball game has been presented recently. This powerful algorithm uses such racing and interplays between teams within a season. Furthermore, the algorithm imitates the coaching procedure within a game. Therefore, some volleyball metaphors, including substitution, coaching, and learning, are used to find a better solution prepared by the VPL algorithm. However, the learning phase has the largest effect on the performance of the VPL algorithm, in which this phase can lead to making the VPL stuck in optimal local solution. Therefore, this paper proposed a modified VPL using sine cosine algorithm (SCA). In which the SCA operators have been applied in the learning phase to obtain a more accurate solution. So, we have used SCA operators in VPL to grasp their advantages resulting in a more efficient approach for finding the optimal solution of the optimization problem and avoid the limitations of the traditional VPL algorithm. The propounded VPLSCA algorithm is tested on the 25 functions. The results captured by the VPLSCA have been compared with other metaheuristic algorithms such as cuckoo search, social-spider optimization algorithm, ant lion optimizer, grey wolf optimizer, salp swarm algorithm, whale optimization algorithm, moth flame optimization, artificial bee colony, SCA, and VPL. Furthermore, the three typical optimization problems in the field of designing engineering have been solved using the VPLSCA. According to the obtained results, the proposed algorithm shows very reasonable and promising results compared to others. Keywords Metaheuristic · Global optimization · Volleyball premier league · Sine cosine algorithm 1 Introduction * Mohamed Abd Elaziz abd_el_aziz_m@yahoo.com Reza Moghdani reza.moghdani@gmail.com Davood Mohammadi mohammady1366@yahoo.com Nabil Neggaz nabil.neggaz@univ-usto.dz 1 Industrial Management Department, Persian Gulf University, Boushehr, Iran 2 Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt 3 Industrial Engineering Department, Payam Noor University, Asalouyeh, Iran 4 Université des Sciences et de la Technologie d’Oran Mohamed Boudiaf, USTO-MB, BP 1505, EL M’naouer, 31000 Oran, Algeria 5 Faculté des Mathématiques et Informatique, Département d’Informatique, Laboratoire SIMPA, Oran, Algeria In recent years, human’s progress in the fields of physical and social science and especially in industrialization have emerged complex problems. This situation persuades scientists to use and develop new algorithms for solving these problems. Due to the nature and complexity of these problems, new algorithms have been developed in recent years to overcome the solving problems. In this matter, artificial intelligence and stochastic optimization approaches have been the center of attention to tackling these obstacles. The optimization is a process of finding the best solutions in a reasonable time for a problem to gain minimization or maximization of objective functions which are restricted by some constraints. The optimization is broadly executable in every field such as economics [1], engineering design [2], pattern recognition [3], chemistry and information technology [4, 5]. As the aforementioned, the complexity of problems needs to use a novel paradigm of optimization algorithms rather than traditional approaches. The new paradigm is inspired 13 Vol.:(0123456789) Author's personal copy Engineering with Computers by natural phenomena in terms of physical and biological. These algorithms are called Meta-heuristics (MH). These algorithms imitate natural processes such as natural selective or collective behavior in seeking the best solution. Generally speaking, evolutionary algorithms can be categorized into four groups. The first group, stochastic, uses randomness to explore in search space including Local Search (LS) [6], Adaptive Random Search (ARS) [7], Stochastic Hill Climbing (SHC), Iterated Local Search (ILS) [8], Variable Neighborhood Search (VNS) [9], Greedy Randomized Adaptive Search Procedure (GRASP) [9], Tabu Search (TS) [10]. The second group of EA named the population-based algorithm. The most well-known of MH, Genetic Algorithm (GA) [11], is dropped in this category. The most famous algorithms related to this group consist Evolution Strategies (ES) [12], Evolutionary Programming (EP) [13], Grammatical Evolution (GE) [14], Adaptive Differential Evolution (ADE) [15], Interior Search Algorithm (ISA) [16], and Stochastic Fractal Search (SFS) [17]. The third group of MH is called a Physical algorithm in which inspired a different range of physical systems combination neighborhood-based and global search techniques. The most well-known algorithms in this group are Ions Motion Algorithm (IMA) [18], Forest Optimization Algorithm (FOA), Water Wave Optimization (WWO) [19], Mine Blast Algorithm (MBA) [20], and Grenade Explosion Method (GEM) [21]. The swarm-based MH is the last group in which mimics the social and individual behavior of swarms, animals, and so on. The most prominent algorithms in this group include Particle Swarm Optimization (PSO) [22], Ant Colony Optimization (ACO) [23–25], Migrating Birds Optimization (MBO) [26], Grey Wolf Optimizer (GWO) [27], Bees Algorithm (BA) [28], Social Spider Optimization (SSO) [29], and Artificial Bee Colony (ABC) [30]. SCA is a new meta-heuristic algorithm that is inspired by mathematical concepts considering Sine and Cosine functions for enhancing the exploration and exploitation of the search space [31]. The SCA procedure consists of two parts, and the new position of each part could be inside or outside of the other part’s neighborhood [31]. To estimate the new position, the Sine and Cosine functions are used. In the same context, the VPL is taking place in the metaheuristic category [32]. This algorithm inspired by interaction and competition between teams within each season and imitating decision making procedure by coaches. VPL algorithm tries to solve the global optimization problem by utilizing a volleyball metaphor namely substitution, coaching, and learning. Like other meta-heuristic algorithms, VPL starts with generating random teams as an initial solution for a given problem. Each team consists of two configurations named formation and substitutes. To scheduling matches, VPL uses the single round-robin (SRR) method to specifying rivals. Afterwards, to determine the winner of each game, the algorithm uses a power factor that is applied in 13 a formulation to calculate the winning probability of each team. In the VPL algorithm coaching is presenting as a knowledge sharing strategy to extract information from the game and training players and substitutes during the match. Similar to any Evolutionary algorithm, VPL uses Repositioning Strategy and Substitution Strategy to change current player’s positions and change current players with substitutes, respectively, during the match based on their roles and match conditions to reach the supremacy in the match (generating new population). In the VPL, each team is located in the space search of the problem as a solution. Then each solution will evaluate respect to the objective function(s) at its contemporary positions. In this research, the advantages of SCA is used to enhance the space exploitation and exploration of the VPL algorithm. Generally speaking, the SCA will use to improve the updating stage of the Learning Phase in the VPL algorithm to provide a good diversity. Since the SCA has a high ability to improve the performance of other metaheuristic methods using its operators. The main contributions of this paper can be summarized as: 1. Proposed an alternative global optimization method based on a modified version of the recent MH method called VPL. 2. Using the operators of SCA to improve the performance of VPL so the proposed called VPLSCA. 3. Evaluate the performance of the proposed VPLSCA using a twenty-five benchmark function and three engineering problems. 4. Evaluate the results of VPLSCA with other similar MH algorithms The rest of the paper’s structure is constituted as follows. Section 2 will review the state of the art of related works in the Meta-heuristics with the focus on SCA, its variants, and applications. Section 3 takes a quick glance at the mechanisms of the VPL algorithm and the SCA. Section 4 devotes to introducing the mechanism of the proposed algorithm. Section 5 presents the experimental analysis and applications of the proposed algorithm on engineering test problems, respectively. In the last section, we will present a conclusion and future visions. 2 Related works In the past years, we have been witnessed the increasing need for developing many different meta-heuristic algorithms. Due to the no-free-lunch theory, it is not applicable to use all these algorithms for the same problem. On the other hand, the most suitable algorithm should be chosen Author's personal copy Engineering with Computers to solve the related problem. These algorithms are very versatile but commonly divided into two main fields named as swarm algorithms and evolutionary algorithms. Swarm algorithms inspired by the collective behavior of animal folks. The evolutionary algorithms refer to the methods which are inspired by Darwin’s evolutionary theory and use mutation and crossover operators. The major privileges of the swarm algorithms are robustness, fewer needing parameters and good efficiency in the exploitation of search space. As the aforementioned, due to the power of SCA, the performance of the PSO algorithm is enhanced using SCA to gain more exploration and exploitation in the field of search space in [33]. In this research paper, the proposed algorithm includes two levels named the bottom and the top. In the bottom level exploration rate will increase using SCA and the top layer gives more exploitation using PSO agents to find the best solutions. This approach provides a good diversity and also the best information of each position at each iteration. To the supremacy of premature convergence, a hybrid PSO with SCA is designed in [34]. The SCA and Differential Evolution algorithm are combined to avoid trapping in local optimum solutions and gain faster convergence rather than an original version of them. The new algorithm named hybrid sine cosine differential evolution algorithm is presented by [35]. The main purpose of developing this algorithm is designing a better framework in the field of optimization problems solving techniques. An acceptable balance is guaranteed by the SCA amongst exploration and exploitation in the search space but unfortunately, like other meta-heuristics, it is in the habit of sticking in the suboptimal areas. To overcome this weakness, Abd Elaziz et al. [36], used the Opposition-Based Learning method to generate more best solutions by increasing the performance of the space searching. To omit the drawbacks of SCA such as the imparity of exploitation and trapping in local optimum areas, a combination of SCA and Multi-Orthogonal Search Strategy (MOSS) to the supremacy of these difficulties has presented by [37]. The binary version of SCA is presented by [38] in which they use a sigmoidal transformation function for binary mapping of continuous real-valued search space to the binary counterpart. This novel represented algorithm is used to solve electricity market problems. Also, the SCA is used for finding the best solution in the re-entry trajectory problem for space shuttle vehicles [39]. It is noticeable that the Multi-Objective version of SCA (MOSCA) is introduced by [40]. The MOSCA uses elitist’s non-dominated sorting and crowding distance attitude to gain non-dominated and provide diversity. Moreover, to test the abilities of SCA, it is used to design airfoil [31]. Furthermore, the SCA also applied in a different context. In [41], a handwritten of Arabic text is binarization using SCA. The SCA is applied to finding the best solution for unit commitment to generating energy [38]. A very amazing application of SCA is made by [42] in the galaxies discovery by applying image recovery. In the same context, the modified version of SCA is applied to finding the optimum solution for Multi-Objective problems [40]. The non-dominated sorting method amending non-dominated solutions achieved up to now by SCA, and the crowding distance part enhances the performance of the diversity of non-dominated solutions. The binary version of SCA benefiting from the rounding method for solving discrete and binary optimization problems introduced by Hafez et al. [43]. It is worth noticing that for testing this version, the feature selection problem is utilized. In the following, we will depict some advances in SCA, which are introduced by researchers recently. The Opposition-Based Learning (OBL) is a mean to evaluate the opposite position of each solution to boost the performance of the SCA algorithm in [36, 44]. Furthermore, a variety of operators are used to enhance the exploration and exploration behavior such as Levy flights, Chaotic maps, and weighted position updating in SCA in [45–48], respectively. Moreover, in terms of application, the SCA algorithm has been successfully adapted with Machine learning techniques to solve a wide variety of problems such as clustering, classification, regression, and prediction. From what has been discussed above, we may classified studies on SCA algorithm into hybrid algorithms and applications as shown in Table 1. 3 A brief review on volleyball premier league and sine cosine algorithm In this section, the general concepts of the VPL and SCA are discussed. 3.1 Volleyball premier league algorithm The VPL algorithm mimics the interactive behavior through the league teams of volleyball [32]. This algorithm has a certain peculiarity in the representation of the solution by comparing it with the evolutionary algorithms. The solution includes two different parts called the active and passive parts. The former section illustrates the typical team, which contains six players, i.e., the main formation where the fitness function is evaluated according to this part. The passive part represents a substitute player. The structure of solution representation is shown in Fig. 1. Figure 2 shows the different steps of the VPL and it is important to define a certain vocabulary dedicated to this algorithm. First, the term league means population. Second, the term team represents a solution and finally, a season 13 Author's personal copy Engineering with Computers Table 1 Review of SCA algorithm Contribution Title Reference Variants [34] [33], [49] [50], [35], [51], [52] [53] [54] [55] [37] [53] [56] [39] [57] [58] [59] [60] [61] [62] [63] [64] [65], [66] [67] [68] SCA with particle swarm optimization SCA with differential evolution SCA with ant lion optimizer SCA with whale optimization algorithm (WOA) SCA with grey wolf optimizer (GWO) SCA with water wave optimization algorithm SCA with multi-orthogonal search strategy SCA with crow search algorithm SCA with teaching learning based optimization Applications Re-entry trajectory optimization for a space shuttle Breast cancer classification Power distribution network Reconfiguration Temperature-dependent optimal power flow Pairwise global sequence alignment Tuning controller parameters for AGC of multi-source power systems Load frequency control of an autonomous power system Coordination of heat pumps, electric vehicles and AGC for efficient LFC in a smart hybrid power system Economic and emission dispatch problems Optimization of CMOS analog circuits Loss reduction in distribution system with unified power quality conditioner Capacitive energy storage with optimized controller for frequency regulation in realistic multisource deregulated power system Reduction of higher-order continuous systems Designing FO cascade controller in automatic generation control of multi-area thermal system incorporating dish-Stirling solar and geothermal power plants SSSC damping controller design in power system Selective harmonic elimination in five level inverter Short-term hydrothermal scheduling Optimal selection of conductors in Egyptian radial distribution systems Data clustering Loading margin stability improvement under a contingency Feature selection Designing vehicle engine connecting rods Designing a single sensor-based MPPT of partially shaded PV system for battery charging Handwritten Arabic manuscript image binarization Thermal and economical optimization of a shell and tube evaporator Forecasting wind speed Object tracking [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [35] represents an iteration and week represents the scheduling of the league, which will be explained in the following. 3.1.1 Initialization Substitutes 1 2 3 ... Formation i Fig. 1 Solution representation [32] 13 1 2 3 ... i In this step, NT teams are generated as mentioned above, each solution contains two parts: formation and substitutes. For each part and for each variable jth, random numbers are generated in the specified interval values using Eqs. (1) and (2). Author's personal copy Engineering with Computers Fig. 2 The framework of VPL algorithm Start Apply learning phase Yes Set Parameters Initialization Max num week=num week No Identify Best team Yes num_season= num_season+1, i=1; Remove top k worst team Generate league schedule Add new team to league Apply competition between team A and team B Apply transfer process Update Best Team Calculate power index for team A and team B Determine winner and loser team No Max num season=num season Apply different strategies Yes Update Best Team Determine Best solution End f Xj = lbj + Rand() × (ubj − lbj ) f Xj = lbj + Rand() × (ubj − lbj ) (1) (2) where lbj indicates the lower bound of the variable j and ubj represent its upper bound. A function that generates a distributed number which is randomly uniformed between 0 and 1 is shown by Rand(). To illustrate the formation and substitutes of teams we utilize the following matrices, respectively. f f f ⎡ X1,1 ⎢ f X F = ⎢ 2,1 ⎢ ⋮ ⎢ f ⎣ Xi,1 X1,2 f X2,2 ⋮ f Xi,2 ⋯ … ⋱ … X1,j ⎤ f ⎥ X2,j ⎥ ⋮ ⎥ f ⎥ Xi,j ⎦ s ⎡ X1,1 ⎢ Xs S = ⎢ 2,1 ⎢ ⋮ ⎢ Xs ⎣ i,1 s X1,2 s X2,2 ⋮ s Xi,2 ⋯ … ⋱ … s ⎤ X1,j s ⎥ X2,j ⎥. ⋮ ⎥ s ⎥ Xi,j ⎦ on the polygon method. The whole number of corporated teams within a tournament illustrated by N and N − 1 shows the number of games for a team which mean N(N − 1)∕2 matches will be played within a tournament. To better understand the SRR process, we will explain the concept based on Fig. 3. Each line determines the opposing teams which will be playing in the first round. For example, A plays with H, B with G, C with F and D with E. The polygon is rotated clockwise to assigning teams for the scheduling of the league for the next coming round (Fig. 4), and Table 2 shows generic scheduling of a league for eight teams. (3) Fig. 3 The first round of the SSR method H B G (4) 3.1.2 Match schedule In this section, we explain first the process of the SRR which provides the scheduling of the league. SRR relies A F C E D A BCD HGF E Round 1 13 Author's personal copy Engineering with Computers A E G E H F A B D D E F G F D HC A E C A G B D C A BC F C B A BD E B H A BCD GEDH F CHG Round 2 Round 3 C A FH H B D B G G C H A E G F A BCG A BCD F E A CD E E HG F DFEH HGF E B HG F Round 4 Round 5 Round 6 Round 7 D Fig. 4 Producing of scheduling a league for rounds 2 to 7 Table 2 Depicts the generic scheduling of a league for eight teams First week Second week Third week Fourth week Fifth week Sixth week Seventh week A–H B–G C–F D–E A–G B–E C–D F–H A–F B–C C–G D–F A–E B–H C–G D–F A–D B–F C–E G–H A–H B–G C–F D–E A–B C–H D–G E–F 3.1.3 Competition Table 3 Function of racing among team i and team j In this section, we will explain mathematically the chance to win and the winning probability in a competition. The power index 𝜑(i) of the formation of the team, i is repref senting by Xi in a week using the following Eqs. (5) and (6): f 𝜑(i) = Z= f (Xi ) (5) Z n ∑ f f (Xi ) (6) Function Competition (i, j) and using Eqs (5) and (6), Calculate using Eq (9), Calculate , Generate If team is the winner and team is the loser, Else team is the winner and team is the loser, End if apply winning strategy for the winning team, apply losing strategies for the loser team, End i=1 f where f (Xi ) represents the value of fittingness of the team i related to its configuration. The whole summation of the value of fittingness within a week is shown by Z. Higher value of fitness determines the stronger team. f We suppose teams j with the formation Xj and k with f the formation Xk . Therefore, the power index for each team is calculated as follows: f 𝜑(j) = f (Xj ) (7) Z f 𝜑(k) = f (Xk ) Z (8) The probability that the probability of team j wins the current game is defined by: 13 p(j, k) = 𝜑(j) 𝜑(j) + 𝜑(k) (9) According to the laws of probability, we find the following relation: p(j, k) + p(k, j) = 1 (10) After the determination of the winner team, we utilize its formation and strategies for the conqueror and loser teams. For the loser team, three strategies are considered such as knowledge sharing, repositioning, and substitution while the winner team utilizes the leading role strategy. Table 3 shows the function of racing among team i and j. Author's personal copy Engineering with Computers 3.1.4 Knowledge sharing strategy Bs = Xjs Knowledge sharing strategy is modeled by the following formulas: Afterwards, the vice versa of the aforementioned formulations from Eqs. (15)–(18) are presented as follows: f f (18) f Xj (t + 1) = Xj (t) + r1 𝜆f (ubj − lbj ) (11) Xi = Bf (19) Xjs (t + 1) = Xjs (t) + r2 𝜆s (ubj − lbj ) (12) Xis = Bs (20) where 𝜆f is the formation’s coefficient and 𝜆s is indicating a coefficient of substitutes. r1 and r2 are random numbers distributed uniformly in [0, 1]. The rate of sharing knowledge is represented by 𝛿ks and the number of knowledge sharing formulation is presented as follows: [ ] Nks = J𝛿ks (13) where the number of knowledge sharing positions represents Nks and each team has J positions. Table 4 shows the steps of the Knowledge sharing strategy. f Xj = Af (21) Xjs = As (22) Table 5 depicts the steps of the repositioning method. 3.1.6 Substitution strategy Within a match, the substitution number is defined by the following formula: 3.1.5 Repositioning strategy Ns = [rJ] This step is led by the coach who determines the best position of each player. This procedure is called a repositioning strategy. In the volleyball game, the role of each player is presented as its position. The amount of repositioning method within a team is illustrated by 𝛿rs. [ ] Nrs = J𝛿rs (14) where Ns illustrates the substitution number within a team, r is a random number distributed uniformly between 0 and 1, and J indicates the positions number. Within a competition, we determine the loser team, and we select randomly an index of a position called h . All members of the formation set F and substitutions set S are exchanged randomly. The pseudo-code is shown in Table 6. where the number of repositioning methods within a match is shown by Nks. We select randomly two positions i and j . A and B represent active and passive players. The allocating attributes of positions i to A and j to B are shown as follows: 3.1.7 Winner strategy f Af = Xi (15) As = Xis (16) f Bf = Xj (23) We determine the position of the winning team, and we combine with a random position to generate a new position using the following formulas: Table 5 Steps repositioning method (17) Table 4 Steps of Knowledge sharing strategy For k=1 : Select randomly a position For j=1 to Update position j of formation property using Eq (11), Update position j of substitutes property using Eq (12), End For End For For k=1 to Select randomly two locations i and j Define A and B Use Eqs. (15) -(18), Reverse two positions i and j using Eqs. (19)- (21) End for Table 6 Pseudo-code for the substitution process Compute the number of substitution process using Eq (14) Define sets ℎ, , and For k=1 to End for 13 Author's personal copy Engineering with Computers X f (t + 1) = X f (t) + r1 𝜓 f (X f (t)∗ − X f (t)) (24) X s (t + 1) = X s (t) + r2 𝜓 s (X s (t)∗ − X s (t)) (25) In Eq. (24), 𝜓 f represent the weights of inertia of formation, while, in Eq. (25), the 𝜓 s is the weights of inertia of substitutes. Moreover, r1, r2 are random numbers distributed uniformly in [0, 1]. 3.1.8 Learning phase where g represents a set, which contains substitutes and formation (g = {s, f }) and the index 𝛷 takes a value from 1 to 3 which means the best team (1), the second (2) and the third g team (3), respectively. Xj (t + 1)𝛷 represents the worth of the location j of attribute g in the field of the supreme solution g 𝛷. Xj (t) is the value of position j of the current iteration t . 𝜃 and 𝜗 are coefficient values. 𝜃 = dbr1 − b (27) 𝜗 = dr2 (28) where r1 and r2 are random numbers distributed uniformly in [0, 1]. b is linearly decreased from 𝛽 to 0 using the following equation: b = 𝛽 − (t(𝛽∕T)) (29) The coaches seek to understand the gameplay of teams on the podium to find the best combination of active (formation) and passive players (substitutes). The following equations are given to capture the learning phase for formation and substitutes properties: ( ) | | f f f f Xj (t + 1)1 = (Xj (t))1 − 𝜃 |𝜗(Xj (t))1 − Xj (t)| (30) | | + 1)2 = f (Xj (t))2 ( ) | | f f − 𝜃 |𝜗(Xj (t))2 − Xj (t)| | | ( ) | | f f f f Xj (t + 1)3 = (Xj (t))3 − 𝜃 |𝜗(Xj (t))2 − Xj (t)| | | f f Xj (t + 1) = ( ) | | Xjs (t + 1)2 = (Xjs (t))2 − 𝜃 |𝜗(Xjs (t))2 − Xjs (t)| | | ( ) | | Xjs (t + 1)3 = (Xjs (t))3 − 𝜃 |𝜗(Xjs (t))2 − Xjs (t)| | | Xjs (t + 1) = The main formula to explain the learning phase is depicted as follow: ( ) | g g | g g Xj (t + 1)𝛷 = (Xj (t))𝛷 − 𝜃 |𝜗(Xj (t))𝛷 − Xj (t)| (26) | | f Xj (t ( ) | | Xjs (t + 1)1 = (Xjs (t))1 − 𝜃 |𝜗(Xjs (t))1 − Xjs (t)| | | f (31) (32) 3 (33) (36) (37) 3.1.9 Season transfers We select randomly H teams for the transfer process if r is greater than 0.5 where r is a random number distributed uniformly in [0, 1]. Hence, the following formulation represented the number of participated teams within the transferring season: ] [ Nst = N𝛿st (38) where 𝛿st represents the percentage of teams that participate to transfer process. The process of season transfer is shown in Table 7. 3.1.10 Promotion and relegation process In the VPL algorithm, we consider only one league, therefore, we remove Npr worst teams and replaced by new teams that are generated randomly. the number of transferred teams to other leagues is shown by Npr and the whole number of teams is indicated by N. Table 7 Steps of transferring season method For k=1 to | End For For k=1 to For j=1 to r=rand() If r>0.5 w=select randomly from current available teams f Xj (t + 1)1 + Xj (t + 1)2 + Xj (t + 1)3 3 (35) Generally speaking, we have used these equations to improve the exploitation process of the proposed algorithm. End If End For End For 13 Xjs (t + 1)1 + Xjs (t + 1)2 + Xjs (t + 1)3 (34) Author's personal copy Engineering with Computers [ ] Npr = N𝛿pr (39) where 𝛿pr represents the percentage of teams that are relegated and promoted. Table 7 depicts the pseudo-code of the promotion and relegation process. 3.2 Sine cosine algorithm The SCA algorithm is a new method that belongs to the class of population-based optimization techniques. This algorithm is introduced by [31]. The particularity of this algorithm lies in the movement of search agents that uses two mathematical operators based on the sine and cosine functions as in Eqs. (40) or (41), respectively: ( ) | | [Xit+1 = Xit + r1 × sin r2 × |r3 Bestpost − Xit | if i | | r4 < 0.5] ( ) | | = Xit + r1 × cos r2 × |r3 Bestpost − Xit | if i | | r4 ≥ 0.5] [Xit+1 (40) (41) where Bestpost is the target solution in i th dimension at t th i iteration, Xit is the current solution in i th dimension at ith iteration, || indicates the absolute cost. r1 , r2 , r3 and r4 are random numbers (Table 8). The parameter r1 controls the balance between exploration and exploitation. This parameter is modified during the iterations using the following formula: r1 = a − t a T (42) where t is the current iteration, T is the maximum number of iterations and a is a constant which is equal to 2. r2 Table 8 Pseudo-code for the promotion and relegation method Remove N worst teams of the league. Define empty teams with formation and substitutes For k=1 to For j=1 to s=select randomly from currently available teams ( End For End For teams to the league Add Table 9 Pseudo-code SCA determines the direction of the movement of the next solution if it towards or outwards target. r3 indicates the weight for the best solution to stochastically emphasize ( r3 > 1) or de-emphasize (r3 < 1) the effect of destination in defining the distance [36]. The parameter r4 allows switching between sine and cosine or vice versa using (Eqs. (40) and (41)). Then the general frame work of the SCA is depicted in Table 9. 4 Proposed algorithm This paper aims to propose an improvement for the VPL algorithm, employs some strong exploitation mechanisms to enhancing its learning phase. This enhancement is performed using the SCA, so the proposed algorithm is called VPLSCA. In general, the proposed VPLSCA algorithm begins by constructing a population of teams that represents the solutions for the given problem, this process performed using Eqs. (1) and (2). The next step is to generate the league schedule and apply the competition between the teams and find the winner and loser teams using the fitness function as in Eqs. (5)–(10). Thereafter, the knowledge sharing strategy is applied to the loser teams followed by the repositioning strategy, then the substitution strategy; while the winning team will apply the leading role operators to update its behavior. The next step is to improve the behaviors of teams using the learning phase, however, this phase is different from the original VPL algorithm. In which the operators of SCA and traditional strategy in learning phase are used together to learn the teams through computing the probability of the fitness function as the following: f Probi = ∑n i i=1 fi ) (43) Based on the value of the Probi the current team can update its behavior using the SCA or the traditional process in VPL. If the value of Probi ≥ rpr (it is a random number which determined based on our experiments and its value is equal to 0.7) then the traditional learning phase (Sect. 3.1.8) is used; otherwise, the SCA operators are used. Then apply 1. Initialize N solutions 2. Repeat 3. Evaluate each solution and we determine the best solution 4. Update random parameters r1 , r2 , r3 and r4 5. Update the position of search using Eqs. (40) and (41) 6. Until t < T 7. Return the best solution obtained as the global optimum solution. 13 Author's personal copy Engineering with Computers the promotion and relegation process, also apply the season transfer process. The previous steps are performed until the terminal conditions are met. For more clarification, Table 9 shows the general framework of proposed the approach. 4.1 The complexity of VPLSCA Computation complexity is a crucial factor for measuring its performance, which can be expressed based on the structure of the proposed algorithm. Theoretically speaking, the computational complexity of the proposed algorithm can be grasped from different factors such as the size of the population, dimension size, maximum number of iterations, and sorting mechanism, which is applied in all iterations. According to [83, 84], the quicksort algorithm, with the complexity of O(nlogn) and O(n2) in the best and worst case, has been utilized in both algorithm. Since our proposed compromises of two different algorithms, VPL and SCA, so we have: O(VPLSCA) = O(VPL)O(f ) + O(SCA)O(f ) (44) where O(f ) is the complexity of the objective function. The complexity of VPL is defined as follows: O(VPL) = (O(T(O(qs)) + O(pu))O(f ) O(VPL) = (O(T(2n)2 ) + O(nd))O(f ) = (O(T(4n2 )) + O(2nd))O(f ) (46) it is worth mention here that VPL uses specific position including passive part and the active part which changes n to 2n . Another significant part of the complexity of the proposed algorithm is related to SCA, which is defined as: (47) And finally, we have the following formula for the computational complexity of our proposed algorithm. O(VPLSCA) = (O(T(5n2 )) + O(3nd))O(f ) 5.1 The definition of the tested Functions In this section, the definition of test functions is given where these functions include three different categories (1) unimodal, (2) multimodal, (3) fixed dimension. The description of these functions is illustrated in Table 10, in which in this table the unimodal functions have only a single extreme maximum or minimum in the specified domain. These functions were applied to evaluate the quality of exploitation for the optimization method (an example of these functions are F1–F10). While the multimodal functions have many local minima and they are applied to evaluate the ability of the methods to avoid the stagnation at these optimal points (an example of these functions are F11–F25). In Table 10, the Dim and fmin represent the dimension of the test function and its corresponding optimum value of fitness function. (45) where T denotes the number of iterations, qs is the quicksort algorithm, O(f ) states the complexity of the objective function. Let n and d be a number of population (team) and dimension space, so we have: O(SCA) = (O(T(n2 )) + O(nd))O(f ) series, the performance of the VPLSCA method is compared with the traditional VPL and SCA using different optimization problems at different conditions such as variant population size, and dimension. Finally, the proposed VPLSCA is applied to different engineering problems in the third experimental series. (48) 5.2 Parameter setting In this study, the results of the proposed VPLSCA are compared with other approaches including cuckoo search (CS) [85], Social-Spider Optimization (SSO) algorithm [86], Ant Lion Optimizer (ALO) [87], Grey Wolf Optimizer (GWO) [27], Salp Swarm Algorithm (SSA) [88], Whale Optimization Algorithm (WOA) [89], Moth Flame Optimization (MFO) [83], Artificial Bee Colony (ABC) [30], SCA [31], and VPL [32]. Where the parameter value of each method is given as mentioned in the original reference. In addition, for a fair comparison between these methods and the proposed method, the common parameters are set the same value for all these methods. For example, the population size is 30, the maximum number of iterations is set to 150 and for providing a suitable statistical analysis each method was run 30 times. All the methods are implemented using Matlab R2017b that installed over windows 10 64 bit, the system of 3.40 GHz processor with 4 GB RAM. 5 Experimental analysis To show the validation and capability of the proposed algorithm, we have used a set of experiments to explore the quality of the proposed approach. In this regard, first, the performance of the proposed method is compared against the other state-of-the-art methods. In the second experimental 13 5.2.1 Measures of performance To evaluate the ability of each method as a global optimization method, a set of performance metrics is used. For example, average and standard deviation of the fitness function, Author's personal copy Engineering with Computers Success rate, number function calling and they are defined as [45]: ∙ Mean of fitness values: Nr 1 ∑ F Mean = Nr i=1 i (49) Standard deviation (STD): √ √ Nr √ 1 ∑ ( )2 Fi − mean STD = √ Nr − 1 i=1 (50) Success rate (SR): SR = NVTR Nr (51) where Nr represents the total number of runs, whereas and NVTR is the total number that the algorithm reached to value-to-reach (VTR). Table 10 Steps of the proposed approach 5.3 Experimental series 1: Comparison with state‑of‑the‑art approaches The aim of this experimental series is to assess the performance of the proposed VPLSCA against some of the stateof-the-art algorithms such as SSO, SSA, CS, ALO, GWO, WOA, MFO, ABC, VPL, and SCA. Since all these kinds of algorithms have been proposed recently, and most of them are considered as the most prominent algorithms in the evolutionary computation context, it would be quite fair to compare the proposed algorithm with these state-of-the-art methods. It is worth mention here that all used algorithms in this study are considered continuous metaheuristic optimization methods. In this study, for the more convenient, ten search agents are applied to find the best solutions over 150 iterations for all mentioned algorithms. The comparison results are given in Table 11, in which one can observe that the results of the proposed VPLSCA method are better than other methods in general. However, the proposed method achieves the best performance in eleven functions (i.e., F2, F4, F6, F7, F15, F16, F17, F18, F20, F24, F25). While, VPL has a better average of the fitness function in two functions namely F22, and F23. As well as, the WOA can reach the best value at the functions F12, F13, and F19. Meanwhile, for the functions F15, the SSO has a better value overall for Input: (Generation)=0, parameters, cost function OutPut: the best solution Initialization stage While < Generate a league schedule For i=1: ( -1)×2 Best team =Select Best team according to Cost Functions Apply Competition procedure between team A, and B Determine winner and loser teams Apply different strategies for winner and loser teams For j=1: number of teams Compute Probj using Eq. (43). ≥ If Update the position of the team(j) by Eqs. (30) to (37). Else Generate a random number from 0 to 1 ( 4 ). If 4 <rand (rand represents a random number belong to [0,1]) Update the position of the team(j) by Eq. (40). Else Update the position of the team(j) by Eq. (41). End if End if End for End for Apply Promotion and relegation process Apply season transfer process = +1 End While 13 Author's personal copy Engineering with Computers Table 11 The definition of the functions Objective function Dim Search range 30 [− 100, 100] 0 30 [− 10, 10] 0 30 [− 100,100] 0 30 [− 100,100] 0 30 [− 30,30] 0 ��2 xi + 0.5 30 [− 100,100] 0 ixi4 + rand[0, 1) 30 [− 1.28,1.28] 0 ixi2 30 [− 10,10] 0 ixi4 30 [− 1.28,1.28] 0 30 [− 1,1] 0 30 [− 500,500] � � � xi2 − 10 cos 2𝜋xi + 10 30 [− 5.12,5.12] 0 � 30 [− 32,32] 0 30 [− 600,600] 0 30 [− 50,50] 0 30 [− 50,50] 0 30 [− 10,10] 0 30 [− 10,10] 0 30 [− 1,1] 0 30 [− 5,10] 0 30 [− 100,100] 0 30 [− 5,5] 0 n ∑ F1(x) = xi2 i=1 n n ∑� � ∏ � � �xi � + �xi � F2(x) = i=1 f3 (x) = n ∑ fmin i=1 � i=1 i ∑ �2 xj j−1 { } f4 (x) = maxi ||xi ||, 1 ≤ i ≤ n n−1 � �2 � �2 ∑ f5 (x) = [100 xi+1 − xi2 + xi − 1 ] i=1 f6 (x) = n �� ∑ i=1 n f7 (x) = ∑ i=1 n f8 (x) = ∑ i=1 n f9 (x) = ∑ i=1 n ∑ � �(i+1) �xi � �� � n ∑ �xi � f11 (x) = −xi sin � � f10 (x) = i=1 − 418.9829 × 5 i=1 f12 (x) = n � ∑ i=1 � 1 n f13 (x) = −20exp −0.2 f14 (x) = 1 4000 n ∑ i=1 n xi2 − ∏ cos i=1 n ∑ i=1 � � � n � � � ∑ − exp 1n cos 2𝜋xi + 20 + e xi2 x √i i � i=1 +1 n � � � n−1 �2 � � �� � �2 � ∑� ∑ f15 (x) = 𝜋n {10 sin 𝜋y1 + yi − 1 1 + 10sin2 𝜋yi+1 + yn − 1 } + u xi , 10, 100, 4 i=1 i=1 �m ⎧ � � � ⎪ k xi − a xi > a xi+1 yi = 1 + 4 , u xi , a, ak, m = ⎨ � 0 − a <� xi < a ⎪ k −xi − a m xi < −a ⎩ � � n n � � � ∑ �2 � � �� � �2 � � �� � ∑ � f16 (x) = 0.1 sin2 3𝜋x1 + xi − 1 1 + sin2 3𝜋xi + 1 + xn − 1 1 + sin2 2𝜋xn + u xi , 5, 100, 4 i=1 i=1 f17 (x) = n � ∑ i=1 n f18 (x) = �2 � � �� � � � �� � xi − 1 1 + sin2 3𝜋xi + 1 + sin2 3𝜋x1 + ��xn − 1�� 1 + sin2 3𝜋xn � � ∑� � �xi . sin xi + 0.1.xi � � i=1 � f19 (x) = 0.1n − (0.1 n ∑ i=1 f20 (x) = n ∑ i=1 f21 (x) = n ∑ x2i + � 0.5 + i=1 n ∑ n ∑ cos(5𝜋xi ) − i=1 0.5ixi i=1 �2 + � n ∑ i=1 x2i 0.5ixi �4 √ sin2 ( 2 100xi−1 +xi2 −0.5 1+.001(xi2 −2xi−1 xi +xi2 ) 2 � � n−1 �2 � � �� � �2 � � �� ∑� f22 (x) = 0.1sin2 3𝜋x1 + xi − 1 1 + sin2 3𝜋xi+1 + xn − 1 1 + sin2 2𝜋xn i=1 13 Author's personal copy Engineering with Computers Table 11 (continued) Objective function n f23 (x) = ∑� 106 �(i−1)∕(n−1) i=1 f24 (x) = (−1)n+1 n ∏ i=1 f25 (x) = 0.5 + xi2 � n � � � �2 ∑� xi − 𝜋 cos xi .exp − Dim Search range 30 [− 100,100] 0 30 [− 100,100] 0 30 [− 100,100] 0 fmin i=1 √∑n sin2 ( 2 i=1 xi −0.5 ∑ 2 (1+0.001 ni=1 x2i ) other algorithms. In addition, it can be noticed from this table that the proposed VPLSCA and VPL have the same average at seven functions namely F1, F3, F5, F8–F10 and F14. However, the VPLSCA and VPL cannot reach to the optimal value at the functions F11, in which the WOA and ALO have the better value. As shown in the following table in which the numbers in boldface specifies the best average of the fitness found in various test functions, VPLSCA has obtained better results in comparison with other methods. Moreover, to investigate the stability of these algorithms, the standard deviation of the fitness function value is computed overall the number of runs (i.e., 25 in this study) as illustrated in Table 12. It can be concluded from this table that the MFO is the method that allocates the first rank overall the other methods, followed by the SSO. Also, the proposed VPLSCA and VPL achieve the third and fourth rank, respectively. However, by taking the results in Table 12, we can observe that the superiority of the MFO and SSO is a negative effect since both of them cannot reach the best solution at any function except the SSO algorithm achieves only the best value at F21. The results of the success rate (SR) for each algorithm are given in Table 13, it can be concluded from this table that the SSO and SCA algorithms cannot reach to the specified value (1E − 5) overall the tested functions. Meanwhile, the proposed VPLSCA allocates the first rank in SR followed by VPL and WOA in the second and third rank, respectively. In addition, GWO, ALO, MFO, CS are allocating the followed ranks, which is the same order. The convergence curve of each algorithm along each function is given in Figs. 5, 6, 7 and 8, in which we can observe that the WOA algorithm can convergence faster than other algorithms at functions F12, F13, F19, and F21. However, the convergence of the traditional VPL and the proposed VPLSCA to the best solution is faster than the other methods at the rest of the test functions. As well as, by analysis, the convergence of the VPL and the VPLSCA at some functions like F1 and F2 we can see that at F1 the proposed VPLSCA can converge to the best solution after around 80 iterations but the VPL needs the all iterations to reach the best value for F1. In addition, at function F2, we can see the convergence of the VPL is better than the VPLSCA at the first of 95 iterations, however, after the 95th iteration the proposed can reach the optimal solution of F2 due to the behaviors of the SCA and VPL are combined. The previous behaviors of VPLSCA are common over the most functions. In addition, the Wilcoxon’s rank-sum test (WRST) is applied to provide more statistical evaluation of the performance of the VPLSCA method. This test is a non-parametric which used to provide a statistical value to indicate there is a significant difference between the VPLSCA and the other approaches or not. In this test there are two hypotheses, the first one is called null which means that no significant difference between the proposed and other methods, while the second one is called alternative which means there is a significant difference between the VPLSCA and others. The results of the WRST are given in Table 14, in which according to these results it can be noticed that there is a significant difference between the VPLSCA and the other methods all tested functions. However, this significant difference at some functions is positive, which indicates the VPLSCA is better, and at other functions is negative, which indicates the VPLSCA is the worst, for example, F11, F12, F13, F19, and F22. In those functions, the VPLSCA fails to find the best solution and it provides results worse than the other functions (these worst results are represents using the negative sign). Moreover, by analysis the WRST of comparison the proposed VPLSCA and the traditional VPL method, it can be concluded that there is a significant difference between them at functions F2, F4, F6, F7, F15, F16, F17, F18, F13, F22, F24, and F24. 5.4 Experimental series 2: comparison with traditional SCA and VPL In this experimental series, the proposed method is compared with the SCA and the VPL at variant dimensions (i.e., 60, 100, and 1000) using fifteen optimization problems. The comparison results are given in Table 15, one can be observed from this table that the VPLSCA approach is better 13 13 Table 12 The average of the fitness function for each algorithm ALO GWO SSA WOA MFO ABC SCA VPL VPLSCA 2.27E + 03 4.35E + 01 1.45E + 04 3.23E + 01 4.35E + 05 2.33E + 03 9.57E − 02 2.46E + 02 2.77E − 01 8.90E − 04 − 1.16E + 03 1.89E + 02 1.49E + 01 1.86E + 01 1.99E + 03 3.37E + 05 6.99E + 01 2.04E + 01 2.50E + 00 3.39E + 02 − 4.71E + 00 2.22E + 01 8.54E + 06 7.63E + 00 4.99E − 01 3.44E + 00 3.42E + 00 1.33E + 03 1.23E + 01 2.26E + 02 5.62E + 00 1.20E − 01 2.45E + 01 4.95E − 01 3.15E − 03 − 6.26E + 02 9.82E + 01 3.02E + 00 1.10E + 00 2.71E + 00 4.12E + 00 5.28E + 00 2.41E + 00 2.08E + 00 8.46E + 01 1.59E + 00 3.11E + 00 6.64E + 05 2.30E + 00 3.73E − 01 2.75E + 03 7.93E + 01 1.41E + 04 3.00E + 01 6.42E + 05 3.72E + 03 7.22E − 02 4.03E + 02 8.45E − 02 1.48E − 03 − 1.63E + 03 9.00E + 01 1.23E + 01 1.96E + 01 5.40E + 01 3.42E + 05 1.34E + 02 9.69E + 00 2.21E + 00 4.09E + 02 − 2.70E + 01 3.14E + 01 3.70E + 07 8.90E + 00 4.99E − 01 6.57E − 06 4.38E − 04 4.00E + 01 1.88E − 01 2.88E + 01 2.18E + 00 8.75E − 03 1.85E − 06 3.14E − 15 5.18E − 23 − 8.01E + 02 1.79E + 01 6.26E − 04 2.15E − 02 8.20E − 02 1.68E + 00 5.56E + 00 9.27E − 03 2.51E − 08 4.75E + 00 3.94E + 00 4.10E + 00 2.37E − 02 4.00E − 01 7.82E − 02 3.07E + 02 1.51E + 01 7.16E + 03 1.82E + 01 2.38E + 04 2.90E + 02 1.34E − 02 6.74E + 01 1.66E − 02 3.68E − 03 − 1.08E + 03 5.92E + 01 6.83E + 00 5.39E + 00 2.40E + 01 8.62E + 01 4.89E + 01 6.10E + 00 1.19E + 00 2.93E + 02 − 1.05E + 00 1.82E + 01 1.30E + 07 5.48E + 00 4.87E − 01 1.67E − 18 5.84E − 15 8.69E + 04 3.35E + 01 2.88E + 01 1.58E + 00 1.82E − 02 1.58E − 22 6.31E − 35 2.52E − 24 -1.63E + 03 3.41E − 14 4.25E − 11 1.01E − 01 8.17E − 02 1.16E + 00 7.91E + 00 1.01E − 15 8.88E − 17 5.16E + 02 − 2.87E + 01 2.40E + 00 4.35E − 15 2.00E − 01 4.81E − 02 2.51E + 03 5.57E + 01 4.05E + 04 7.12E + 01 2.01E + 06 2.99E + 03 1.57E − 01 8.77E + 02 6.54E − 01 3.68E − 04 -1.41E + 03 2.24E + 02 1.89E + 01 3.40E + 01 6.74E + 06 9.57E + 05 1.27E + 02 1.31E + 01 2.63E + 00 4.72E + 02 − 1.04E + 01 3.19E + 01 4.19E + 07 8.53E + 00 4.99E − 01 1.41E + 01 4.34E − 01 2.63E + 04 6.68E + 01 2.05E + 02 1.50E − 01 2.09E − 01 1.45E − 02 3.53E − 06 1.51E − 03 − 1.23E + 03 5.08E + 01 9.10E + 00 4.40E − 01 9.26E − 04 1.33E − 02 3.52E − 01 2.95E + 00 6.48E − 01 3.11E + 02 − 1.06E + 01 1.09E − 01 1.24E + 04 1.12E + 01 5.00E − 01 2.13E + 03 3.21E + 00 3.01E + 04 6.75E + 01 9.44E + 06 1.77E + 03 5.43E − 02 1.04E + 02 2.93E + 00 6.82E − 02 − 7.21E + 02 9.50E + 01 1.75E + 01 1.34E + 01 1.53E + 07 4.99E + 07 6.29E + 01 6.25E + 00 7.56E − 01 1.21E + 02 4.83E + 00 3.12E + 01 6.31E + 05 5.56E + 00 4.81E − 01 0.00E + 00 7.10E − 292 0.00E + 00 3.57E − 57 0.00E + 00 1.51E − 07 3.83E − 03 0.00E + 00 0.00E + 00 0.00E + 00 6.55E + 04 − 1.62E + 02 − 3.30E + 11 0.00E + 00 7.95E − 09 1.38E − 07 2.03E − 05 2.52E − 294 − 1.35E + 00 1.05E − 33 − 2.61E + 01 1.52E − 06 0.00E + 00 2.71E − 02 7.17E − 03 0.00E + 00 0.00E + 00 0.00E + 00 1.18E − 112 0.00E + 00 0.00E + 00 0.00E + 00 0.00E + 00 0.00E + 00 0.00E + 00 − 7.28E + 02 − 1.60E + 02 − 1.38E + 11 0.00E + 00 0.00E + 00 0.00E + 00 0.00E + 00 0.00E + 00 − 1.29E + 00 0.00E + 00 − 2.87E + 01 − 2.87E + 01 2.09E − 270 4.79E − 153 0.00E + 00 Bold values indicate that the best value Author's personal copy SSO Engineering with Computers F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15 F16 F17 F18 F19 F20 F21 F22 F23 F24 F25 CS Author's personal copy Engineering with Computers Table 13 The standard deviation of each algorithm F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15 F16 F17 F18 F19 F20 F21 F22 F23 F24 F25 CS SSO ALO GWO SSA WOA MFO ABC SCA VPL VPLSCA 470.5189 7.321595 2193.915 2.621772 187162.1 662.1452 0.029892 55.59725 0.151267 0.000751 78.37489 14.70913 1.743181 3.111753 3667.794 197646 14.92508 1.613329 0.203527 51.30038 5.841165 5.200115 3076595 0 0.000511 0 0 0 0 3.18E − 14 0 0 0 6.21E − 17 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1335.196 44.83635 3816.533 4.992455 492,583.1 2259.513 0.015361 128.7383 0.050645 0.001133 0 24.46471 2.264061 13.74987 27.61479 721,653 31.43881 3.340244 0.357598 167.3644 3.265889 10.62915 21,298,086 0 0.000997 6.43E − 06 0.000219 27.5037 0.110942 0.111073 0.651926 0.004321 1.42E − 06 3.42E − 15 1.03E − 22 127.09 7.066993 0.000274 0.029628 0.05027 0.386406 1.805604 0.002601 2.05E − 08 2.746084 0.493572 2.581385 0.02092 0 1.37E − 08 174.8933 4.965003 4273.52 4.488419 11279.46 67.09181 0.006918 46.69334 0.014584 0.001537 51.80483 12.80315 1.985675 2.021409 5.911447 41.99036 34.95171 1.060625 0.197541 74.31802 2.991278 6.258481 6,786,806 0 0.004627 3.72E − 18 1.05E − 14 23926.06 24.03471 0.04695 0.514979 0.013054 1.89E − 22 8.8E − 35 5.5E − 24 0 5.08E − 14 2.99E − 11 0.22559 0.018293 0.227589 2.656536 1.05E − 15 1.99E − 16 136.7639 0.422347 1.30332 9.17E − 15 0 0.029667 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 31.20456 0.090987 8773.53 5.823474 119.866 0.07342 0.048019 0.005351 6.38E − 06 0.000801 30.71497 12.29266 0.93617 0.260479 0.001066 0.017531 0.365191 0.531883 0.317798 40.8132 1.939147 0.108599 8334.345 0 0.000113 2225.367 2.989914 9745.905 7.441104 14,148,718 1685.431 0.017026 70.49272 1.689006 0.119225 36.51013 33.81461 6.017207 12.28756 20,818,361 55,721,913 37.95476 4.205247 0.620385 55.65405 0.884634 4.301878 469,185.2 0 0.016262 0 0 0 7.24E − 57 0 7.68E − 08 0.002389 0 0 0 0.005864 24.37028 1.77E + 11 0 2.98E − 09 4.73E − 08 1.01E − 05 0 0.204641 2.35E − 33 3.882627 6.63E − 07 0 8.29E − 06 0.00372 0 0 0 2.6E − 112 0 0 0 0 0 0 35.410 26.87455 5.41E + 10 0 0 0 0 0 0.249273 0 0.289588 0.289588 0 0.000172 0 than the other two methods on most of all test functions. For example, at the dimension 60, the VLP is better than the SCA algorithm, while the performance of the VPL and the VPLSCA nearly the same as in functions (F1–F3, and F14). However, at the functions F4–F7, F9, F15–F16, the proposed method has the smallest average and standard deviation than the traditional VPL. Moreover, the three algorithms have the same characteristics as the other dimension (100, and 1000), but the VPLSCA provides better results than VPL at the function F2 at dimension 1000. In addition, from this table, it can observe that the quality of the proposed method not changed with a variant of the dimension in all functions (used in this experimental) except only two functions namely F4, and F16. Figure 9 depicts the diversity of solutions for each of the three methods (i.e., VPL, SCA, VPLSCA). From this figure, it can be noticed that the proposed VPLSCA has high diversity value than the other two methods. Further analysis of the diversity can be noticed that SCA improves the diversity of the traditional VPL which observed from the diversity curve of VPLSCA. In addition, by comparing the diversity of SCA and the proposed VPLSCA it can be seen that the diversity of the SCA decreases with increasing the number of iterations. In contrast, the solutions of the proposed VPLSCA maintain their diversity during the iterations. However, the diversity of SCA is better than VLP and VPLSCA at dimension 1000, but by analysis the behaviors of SCA we observed its diversity is decreased and this means if the number of iterations is increased the diversity of SCA become very small that will effect on the quality of the final solution. 5.5 Experimental series 3: engineering application In this experimental series, the quality of the results of the VPLSCA algorithm is evaluated to solve different real engineering problems. These engineering problems are tested with different conditions, namely, tension/compression spring design, welded beam design, and pressure vessel design. In which to handle various inequality constraints, the easiest way, called the death penalty function is used, in which the objective function is given a large constant value if any violated constraint. 13 Author's personal copy Engineering with Computers Fig. 5 The convergence curve for functions F1–F6 13 Author's personal copy Engineering with Computers Fig. 6 The convergence curve for functions F7–F12 13 Author's personal copy Engineering with Computers Fig. 7 The convergence curve for functions F13–F18 13 Author's personal copy Engineering with Computers Fig. 8 Convergence curve for functions F19–F25 13 Author's personal copy Engineering with Computers Table 14 The average of SR for each algorithm F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15 F16 F17 F18 F19 F20 F21 F22 F23 F24 F25 Average CS SSO ALO GWO SSA WOA MFO ABC SCA VPL VPLSCA 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.4453 0 0 0 0 0.0178 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.9866 0 0 0 0 0.0394 0 0 0 0 0 0 0 0 0 0 0 0.6993 0.8365 0 0 0 0 0 0.5302 0 0 0 0 0 0 0.0826 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.1666 0 0 0 0 0.0066 0.3963 0.5096 0 0 0 0 0 0.3861 0 0 0.548936 0.702041 0.66205 0 1 1 0 0.4331 1 0 0.9882 0 0.3642 0 0 0.3196 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.7333 0 0 0 0 0.0293 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.1333 0 0 0 0 0.0053 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.7436 0.7430 0.7369 0.7089 0.7373 0.1453 0 0.7383 0 0 0.71773 0.737415 0.750693 0.8885 0.6282 0.7272 0 0.7426 0.4624 0.3712 0.7373 0 0.7368 0.1444 0 0.4879 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.8418 0.8264 1 1 0.7941 1 1 1 1 1 1 0.9785 5.5.1 Welded beam design The objective function of the welded beam design problem is to minimize total fabrication cost subject to some constraints such as bending stress in the beam ( 𝜎 ), buckling load on the bar ( Pc ), shear stress (𝜏 ), end deflection of the beam (𝛿) in which four variables including the width, ( )length of the welded area, the depth, and the thickness b x4 of the main beam are computed. The diagram of this problem is clarified in Fig. 10. The formulation of this problem can be expressed as follows: [ ] Consider x⃗ = x1 x2 x3 x4 = [hltb] (52) ( ) ( ) Minimize f x⃗ = 1.10471x2 x12 + 0.04811x3 x4 14.0 + x2 (53) ( ) ( ) g1 x⃗ = 𝜏 x⃗ − 𝜏max ≤ 0 (54) 13 ( ) ( ) g2 x⃗ = 𝜎 x⃗ − 𝜎max ≤ 0 (55) ( ) ( ) g3 x⃗ = 𝛿 x⃗ − 𝛿max ≤ 0 (56) ( ) g4 x⃗ = x1 − x4 ≤ 0 (57) ( ) ( ) g5 x⃗ = P − Pc x⃗ ≤ 0 (58) ( ) g6 x⃗ = 0.125 − x1 ≤ 0 (59) ( ) g7 x⃗ = 1.10471x12 + 0.04811x3 x4 (14.0 + x2 ) − 5.0 ≤ 0 (60) 0.10 ≤ x1 ≤ 2.00, (61) 0.10 ≤ x2 ≤ 10.00, (62) Author's personal copy Engineering with Computers Table 15 The results of the Wilcoxon’s rank-sum test for comparison between VPLSCA and other algorithms VPLSCA vs# CS SSO ALO GWO SSA WOA MFO ABC SCA VPL F1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (−)1 0.0079 (−)1 0.0079 (−)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (−)1 0.0079 (+)1 0.0079 (+)1 0.0079 (−)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (−)1 0.0079 (−)1 0.0079 (−)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (−)1 0.0079 (+)1 0.8413 0 0.0079 (−)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (−)1 0.0079 (−)1 0.0079 (−)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (−)1 0.0079 (+)1 0.0079 (+)1 0.0079 (−)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (−)1 0.0079 (−)1 0.0079 (−)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (−)1 0.0079 (+)1 0.0079 (+)1 0.0079 (−)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (−)1 0.0079 (−)1 0.0079 (−)1 0.0476 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (−)1 0.0079 (+)1 0.6905 0 0.0079 (−)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (−)1 0.0079 (−)1 0.0079 (−)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (−)1 0.0079 (+)1 0.0079 (+)1 0.0079 (−)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (−)1 0.0079 (−)1 0.0079 (−)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (−)1 0.0079 (+)1 0.0079 (+)1 0.0079 (−)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (−)1 0.0079 (−)1 0.0079 (−)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (−)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 1 0 0.007937 1 (+)1 0 0.007937 (+)1 1 0 0.007937 (+)1 0.007937 (+)1 (+)1 0 1 0 1 0 1 0 1 0 0.1507 0 1 0 0.007 (+)1 0.0079 1 0.0079 (+)1 0.0079 (+)1 0.8412 0 0.0079 (+)1 0.2222 0 0.0079 (−)1 1 0 0.0158 (+)1 0.0079 (+)1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15 F16 F17 F18 F19 F20 F21 F22 F23 F24 F25 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (−)1 0.0079 (−)1 0.0079 (−)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 0.0079 (−)1 0.0079 (+)1 0.0079 (+)1 0.0079 (−)1 0.0079 (+)1 0.0079 (+)1 0.0079 (+)1 13 Author's personal copy Engineering with Computers Fig. 9 Diversity curve for functions F3, F5, and F13 at dimension 60, 100, and 100 0.10 ≤ x3 ≤ 10.00, (63) 0.10 ≤ x4 ≤ 2.00, (64) P , 𝜏� = √ 2x1 x2 √ where ( ) 𝜏 x⃗ = R= √ 13 x (𝜏 � )2 + 2𝜏 � 𝜏 �� 2 + (𝜏 �� )2 2R x22 4 ( + 𝜏 �� = x1 + x3 2 MR , J )2 � x � M =P L+ 2 2 Author's personal copy Engineering with Computers x2 x1 Load x3 A B x4 5.5.2 Tension/compression spring design Fig. 10 The structure of welded beam design � � � � �� √ x22 x1 + x3 2 + J=2 2x1 x2 4 2 ( ) 6PL ( ) 6PL3 , 𝛿 x⃗ = 𝜎 x⃗ = x4 x32 Ex4 x32 √ 4.013E ( ) Pc x⃗ = x32 x46 36 L2 ( x 1− 3 2L √ Table 16. According to this table, some methods (e.g. SCA, Improved HS GSA, WOA, CBO, MCSS, and ACO) indicate better cost values than our proposed algorithm. With respect to analyses the results scrupulously, we have identified the methods in which has the better cost function than our proposed algorithm violating constraints of the model. To show the efficacy of the presented approach, we have added a new column in Table 16 to show feasible solutions of the different method. So that, we can simply conclude that VPLSCA has better performance in comparison with its rivals. ) E 4G The schematic of the problem is exposed in Fig. 11. This problem aims to minimize the weight of a tension/compression spring (TCS). There are many studies considering this problem in the literature. As shown in figure, there are four variables consisting of wire diameter (d ), mean coil diameter ( D ), and the number of active coils ( N ) (Table 17). The problem comprises three constraints including surge frequency, minimum deflection, and shear stress. The problem formalization is shown in the following: [ ] Consider x⃗ = x1 x2 x3 = [dDN] (65) ( ) ( ) Minimize f x⃗ = x3 + 2 x2 x12 P = 6000 lb, L14In, (66) 𝛿max = 0.25In, E = 30 × 106 psi, G = 30 × 106 psi 𝜏max = 13600psi, 𝜎max = 30000psi As stated above, the objective function is simply revealed in Eq. (53), associated seven constraints are reflected in Eqs. (54)–(60), and finally, related variables are shown in Eqs. (61)–(64). VPLSCA is applied to this problem and also compared with some most well-known studies in the literature. In this regard, many methods such as simplex, random, Davidon–Fletcher–Powell method (DFP) [90], co-evolutionary differential evolution (CEDE) [91], genetic algorithm (GA) [92], co-evolutionary particle swarm optimization (CPSO) [93], evolution strategy (ES) [94], ant colony optimization (ACO) [95], gravitational search algorithm (GSA) [96], CSS [95], multi-verse optimization (MVO) [84], harmony search (HS) [97], improved harmony search (IHS) [98], Reinforced Cuckoo Search Algorithm (RCSA) [99], grouping particle swarm optimizer (GPSO) [100], whale optimization algorithm(WOA) [89], ray optimization (RO) [101] and magnetic charged system search (MCSS) [102]. The results of applying different methods can be seen in ( ) g1 x⃗ = 1 − ( ) g2 x⃗ = x23 x3 71785x14 ≤0 4x22 − x1 x2 1 −1≤0 )+ ( 5108x12 12566 x2 x13 − x14 (67) (68) ( ) 140.45x1 g3 x⃗ = 1 − ≤0 x22 x3 (69) ( ) x + x2 −1≤0 g4 x⃗ = 1 1.5 (70) 0.05 ≤ x1 ≤ 2.00, (71) 0.25 ≤ x2 ≤ 1.30, (72) 2.00 ≤ x3 ≤ 15.00, (73) Equation (66) shows the function of objective considering the weight minimization of a tension/compression spring. Equations (67)–(70) expresses all constraints and Eqs. (71)–(73) describe variable ranges. 13 Author's personal copy Engineering with Computers Table 16 The average and standard deviation of fitness value for VPL, SCA, and VPLSCA algorithms Function F1 F2 F3 F4 F5 F6 F7 F8 F9 F14 F15 F16 Measure Avg STD Avg STD Avg STD Avg STD Avg STD Avg STD Avg STD Avg STD Avg STD Avg STD Avg STD Avg STD Dim = 60 Dim = 100 Dim = 1000 SCA VPL VPLSCA SCA VPL VPLSCA SCA VPL VPLSCA 16,441.98 9269.677 18.11929 8.052529 131144.2 28,975.32 85.61415 4.964163 85,168,409 55,356,994 14,075.26 8263.1 0.335055 0.145676 2944.409 1737.702 75.23315 60.48956 114.7424 91.1922 2.62E + 08 1.53E + 08 4.43E + 08 1.8E + 08 0 0 0 0 0 0 1.12E − 64 2.5E − 64 0 0 9.36E − 07 7.3E − 07 0.000173 4.16E − 05 1.21E − 14 0 1.25E − 43 0 0 0 2.9E − 14 4.21E − 12 3.37E − 12 4.6E − 12 0 0 0 0 0 0 9E − 280 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 33,188.84 12,736.53 32.05697 19.20775 403,038.5 105,043.2 93.46759 1.904676 3.43E + 08 1.15E + 08 32947.32 16,299.74 1.38401 0.468815 17,304.54 8409.813 470.7525 142.1045 361.3009 142.9304 6.65E + 08 2.34E + 08 1.55E + 09 5.08E + 08 0 0 0 0 0 0 2.44E − 60 5.45E − 60 0 0 0.000224 0.000367 0.000203 0.00015 1.46E − 12 3.26E − 12 4.53E − 28 1.01E − 27 0 0 8.79E − 12 1.16E − 11 3.08E − 12 3.56E − 12 0 0 0 0 0 0 1.3E − 160 2.6E − 160 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 911,983.1 174,962.1 65,535 65,535 38,446,881 8,007,180 99.6176 0.111 7.99E + 09 6.5E + 08 878,432.3 232,821.2 296.2154 32.61321 4,170,860 1,156,112 135,779.7 17,014.49 8303.409 1656.362 2.35E + 10 2.46E + 09 3.99E + 10 4.81E + 09 0 0 8.42E − 278 0 0 0 7.35E − 64 1.63E − 63 0 0 2.183286 0.531683 7.59E − 05 5.39E − 05 2.43E − 09 6.19E − 08 9.32E − 20 2.79E − 24 0 0 1.25E − 12 1.25E − 12 8.17E − 13 8.23E − 13 0 0 0 0 0 0 3.15E − 124 7E − 124 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.894479 0.169847 Bold values indicate that the best value x3 x2 x1 Fig. 11 Structure of the TCS design There are many scholars working on different approaches to solving this problem. These researchers have used a variety of techniques to reach a solution to this problem. Ray optimization (RO) [101], Genetic algorithm (GA) [11], A 13 novel particle swarm optimizer [105], Reinforced Cuckoo Search Algorithm (RCSA) [99], grouping particle swarm optimizer (GPSO) [100], GA [92], evolution strategies (ES) [12], WOA [89], SES [107], CEDE [91], co-evolutionary particle swarm optimization approach (CPSO) [104], new swarm algorithm with information sharing strategy [108] Improved harmony search algorithm (HIS) [98], and others performed different approaches to handle the current problem, which are shown in Table 18. As shown in this table, the result of VPLSCA shows superior performance in comparison with others. 5.5.3 Pressure vessel design The final instance represents the minimization of total cost in designing a pressure vessel, including the cost of the materials, forming and welding, is considered as an objective function. Figure 12 describes this problem and its features. To solve this problem, variables including the shell thickness (x1), the head thickness (x2), the radius of the interior (x3), and the cylindrical part length of the vessel (x4) are Author's personal copy Engineering with Computers Table 17 Results of VPLSCA and other methods to solve the welded beam design problem Algorithm VPLSCA VPL [32] SCA [31] WOA [89] RCSA [99] GPSO [100] GSA [96] Improved HS [98] Simplex [90] CBO [103] CEDE [91] ACO [25] MCSS [102] DFP [90] CPSO [104] RO [101] Random [90] CSS [95] GSA [96] APPROX [90] GA [92] MVO [84] GA [11] ES [12] GA [92] HS [97] GA [92] Optimum variables l h b t 6.898941 6.898945 3.47111 3.484293 0.20572 0.206 3.856989 3.47049 5.6256 3.47041 3.542998 3.471131 3.470493 6.2552 3.544214 3.528467 4.7313 3.468109 3.856979 6.2189 3.420500 3.473193 3.544214 3.612060 6.1730 6.2231 3.471328 0.215231 0.215235 0.20581 0.205396 3.47041 7.092 0.182129 0.20573 0.2792 0.205722 0.203137 0.205700 0.205729 0.2434 0.202369 0.203687 0.4575 0.205820 0.182129 0.2444 0.2489 0.205463 0.202369 0.199742 0.2489 0.2442 0.205986 0.215253 0.216253 0.2157 0.206276 9.03727 9.037 0.202376 0.2057 0.2796 0.205735 0.206179 0.205731 0.205729 0.2444 0.205723 0.207241 0.6600 0.205723 0.202376 0.2444 0.21000 0.205695 0.205723 0.206082 0.2533 0.2443 0.206480 8.811012 8.815033 9.037125 9.037426 0.20573 0.206 10.000000 9.03662 7.7512 9.037276 9.033498 9.036683 9.036623 8.2915 9.04821 9.004233 5.0853 9.038024 10 8.2915 8.997500 9.044502 9.048210 9.037500 8.1789 8.2915 9.020224 Optimum cost Feasibility (yes/ no) 2.25998 2.26973 1.800885 1.730499 1.7246 2.218 1.879952 1.7248 2.5307 1.724663 1.733461 1.724918 1.724853 2.3841 1.72802 1.735344 4.1185 1.724866 1.87995 2.3815 1.748309 1.72802 1.728024 1.737300 2.4331 2.3807 1.728226 Yes Yes No No No Yes No No No No No No No Yes No No Yes No No Yes No No No No Yes No No given which are written in the mathematical formulation as follows: [ ] ] [ Consider x⃗ = x1 x2 x3 x4 = Ts Th RL (74) 0 ≤ x1 ≤ 99, (80) 0 ≤ x2 ≤ 99, (81) ( ) Minimize f x⃗ = 0.62224x1 x3 x4 + 1.7781x2 x32 10 ≤ x3 ≤ 200, (82) 10 ≤ x4 ≤ 200, (83) + 3.1661x4 x12 + +19.84x3 x12 (75) ( ) g1 x⃗ = −x1 + 0.0193x3 ≤ 0 (76) ( ) g2 x⃗ = −x3 + 0.00954x3 ≤ 0 (77) ( ) 4 g3 x⃗ = −𝜋x32 x4 − 𝜋x33 + 1, 296, 000 ≤ 0 3 (78) ( ) g4 x⃗ = x4 − 240 ≤ 0 (79) Equation (75) is defined as an objective function in which minimizing the total cost of the problem, Eqs. (76)–(79) are related to all constraints, and the ranges of variables are shown in Eqs. (80)–(83). Like the above-mentioned problems, there are some approaches implemented in this problem. We can mention branch and bound method [109], WOA, improved ACO [95], augmented Lagrangian multiplier approach [110], CEDE, wind-driven water wave optimization (WDWWO) [111], RCSA [99], grouping particle swarm optimizer (GPSO) [100] improved HS [98], different GAs [112], [113] [92], CPSO [104], MVO [84], ES 13 Author's personal copy Engineering with Computers Table 18 Comparison of the proposed approach with other methods for TCS design Algorithm VPLSCA VPL [32] SCA [31] IHS [98] CEDE [91] RO [101] ES [12] [108] GA [11] PSO [105] ACO [25] GPSO [100] SES [107] GA [92] WOA [89] DE [91] MCSS [102] RCSA [99] x1 x4 x3 Optimum variables D N d 0.331580 0.331680 0.343215 0.349871 0.354714 0.349096 0.363965 0.050417 0.351661 0.357644 0.361500 0.0517 N/A 0.355360 0.345215 0.354714 0.356496 0.051688 12.744269 12.834269 11.994032 12.076432 11.410831 11.76279 10.890522 3.979915 11.632201 11.244543 11.00000 0.3573 N/A 11.397926 12.004032 11.410831 11.271529 0.356710 0.0501550 0.0501910 0.050905 0.051154 0.051609 0.051370 0.051989 0.321532 0.051480 0.051728 0.051865 11.2540 N/A 0.051643 0.051207 0.051609 0.051645 11.289398 x2 x3 Fig. 12 Pressure vessel design and its features [94], CSS [95], PSO-DE [114], and DE [105] as the main approaches considering in the literature to cope with this problem. The comparison of the results given by different approaches is shown in Table 19 which is described as the competitive result of the proposed approach in comparison with other approaches. From the experimental results obtained from different comparisons between the proposed VPLSCA method and other methods. It can be observed that the VPLSCA has shown good behavior at the level of convergence for global optimization problems and engineering problems. It can be seen that VPLSCA outperforms other optimizers (CS, SSO, ALO, GWO, WOA, MFO, ABC, SCA, VPL) for 44% of the total number of functions and provides the same performance with 28%. The main reason for this high performance of the proposed VPLSCA is using the SCA as local search operators to the traditional VPL. This leads to providing the 13 Optimum cost Feasibility (yes/ no) 0.012298157 0.0123947 0.012446006 0.0126706 0.0126702 0.0126788 0.0126810 0.013060 0.0127048 0.0126747 0.0126432 0.0127 0.012732 0.0126698 0.0126763 0.0126702 0.0126192 0.0126652 Yes Yes Yes Yes Yes No Yes Yes Yes Yes Yes No N/A Yes Yes Yes No No VPL with suitable operators that avoiding the stagnation at the local optima as well as maintain the diversity of the solutions during the optimization process. However, it still needs more improvement especially in the CPU time this the original VPL is time-consuming and this can be solved by applying parallel procedure. Another limitation can be highlighted, the VPLSCA contains several parameters that are randomly generated. This factor influences deeply on the convergence of the algorithm. Therefore, the chaotic maps can be used to overcome this limitation. 6 Conclusion and future works This paper provides a modified version for the VPL algorithm to enhance its ability to find the best solution. In general, the VPL algorithm emulates the rules of the volleyball game such as competition, the interaction between teams during a season, as well as simulates the coaching process. The VPL is applied to a different number of optimization and engineering problems and the results established that it has a high ability to reach the best solution than other methods. However, the accuracy of the VPL algorithm still requires to improve, especially the learning phase which can pressure the VPL towards the local optimal point. Therefore, this paper used the SCA which has a high ability to explore the search space. In the proposed method called VPLSCA, the SCA is used during the Learning Phase which gives the VPL a high ability to search for the solution at this phase. To Author's personal copy Engineering with Computers Table 19 Comparison of the proposed approach results with literature for a pressure vessel design problem Algorithm VPLSCA VPL [32] GA [92] WOA [89] ACO [25] improved HS [98] MVO [84] GA [11] SCA [31] ES [12] PSO-DE [114] DE [91] ALM [115] CSS [95] GA [113] B & B [109] CEDE [91] WDWWO [111] RCSA [99] GAS [112] CPSO [104] GPSO [100] Optimum variables Ts Th R L 0.8152 0.815200 0.937500 0.812500 0.812500 1.125000 08125 0.812500 0.8125 0.812500 0.812500 0.812500 1.125000 0.812500 0.812500 1.125000 0.812500 0.9803 0.9803 0.937500 0.812500 0.778 0.4265 0.426500 0.437500 0.437500 0.437500 0.625000 0.4375 0.437500 0.4378 0.437500 0.437500 0.437500 0.625000 0.437500 0.437500 0.625000 0.437500 0.4854 0.4854 0.50000 0.437500 0.385 42.0851245 42.0912541 42.097398 42.0982699 42.098353 58.29015 42.090738 40.323900 42.0883699 42.098087 42.098446 42.098411 58.2910 42.103624 48.329000 47.7000 42.098411 50.7236 50.7236 48.329 42.091266 40.321 176.73154 176.742314 176.654050 176.638998 176.637751 43.69268 176.73869 200.000000 176.648998 176.640518 176.636600 176.637690 43.69 176.572656 112.679000 117.7010 176.637690 92.7062 92.7062 112.679 176.746500 200.000 investigate the performance of the proposed VPLSCA algorithm, a set of experimental series is performed using a set of different twenty-five CEC2005 optimization and three engineering problems. The results of these experimental series show that the proposed VPLSCA algorithm has higher performance than that from other algorithms such as CS, SSA, ALO, MFO, WOA, and the classic types of SCA, and VPL. Based on the superiority of the proposed VPLSCA algorithm, it can be used in future works in different fields using it as (1) feature selection method by converting it to a binary version, (2) a multi-level thresholding image segmentation through finding the optimal threshold value, (3) reducing the energy consumption of virtual machine placement in cloud computing. 2. 3. 4. 5. 6. 7. Compliance with ethical standards 8. Conflict of interest The authors declare no conflict of interest. 9. 10. References 1. Mousavi-Avval SH et al (2017) Application of multi-objective genetic algorithms for optimization of energy, economics and 11. Optimum cost Feasibility (yes/ no) 6042.711935 6043.986 6059.9463 6059.7410 6059.7258 7197.730 6060.8066 6288.7445 6058.2907 6059.7456 6059.71433 6059.7340 7198.0428 6059.0888 6410.3811 8129.1036 6059.7340 6335.4270 6335.4270 6410.3811 6061.0777 5885.703 Yes Yes Yes Yes Yes No Yes Yes Yes No Yes Yes No No No No Yes No No Yes Yes Yes environmental life cycle assessment in oilseed production. J Clean Product 140:804–815 Chou J-S, Pham A-D (2017) Nature-inspired metaheuristic optimization in least squares support vector regression for obtaining bridge scour information. Inf Sci 399:64–80 Shamir J et al (1992) Optimization methods for pattern recognition. In: Critical reviews. SPIE, Bellingham Ghaedi AM et al (2016) Adsorption of Triamterene on multiwalled and single-walled carbon nanotubes: artificial neural network modeling and genetic algorithm optimization. J Mol Liq 216:654–665 Wang Z et al (2016) A modified ant colony optimization algorithm for network coding resource minimization. IEEE Trans Evol Comput 20(3):325–342 Voudouris C, Tsang EP, and Alsheddy A (2010) Guided local search. In: Handbook of metaheuristics. Springer, New York, pp 321–361 Baba N, Shoman T, Sawaragi Y (1977) A modified convergence theorem for a random optimization method. Inf Sci 13(2):159–166 Lourenço HR, Martin O, Stützle T (2001) A beginner’s introduction to iterated local search. In: Proceedings of MIC Mladenović N, Hansen P (1997) Variable neighborhood search. Comput Oper Res 24(11):1097–1100 Burke EK, Kendall G, Soubeiga E (2003) A tabu-search hyperheuristic for timetabling and rostering. J Heuristics 9(6):451–470 Goldberg D (1989) Genetic algorithms in search, optimization, and machine learning. In: Ohno K, Esfarjani K, Kawazoe Y (eds) Computational materials and science. Addison-Wesley, Reading 13 Author's personal copy Engineering with Computers 12. Beyer H-G, Schwefel H-P (2002) Evolution strategies—a comprehensive introduction. Nat Comput 1(1):3–52 13. Yao X, Liu Y, Lin G (1999) Evolutionary programming made faster. IEEE Trans Evol Comput 3(2):82–102 14. O’Neill M, Ryan C (2001) Grammatical evolution. IEEE Trans Evol Comput 5(4):349–358 15. Cui L et al (2016) Adaptive differential evolution algorithm with novel mutation strategies in multiple sub-populations. Comput Oper Res 67:155–173 16. Gandomi AH (2014) Interior search algorithm (ISA): a novel approach for global optimization. ISA Trans 53(4):1168–1183 17. Salimi H (2015) Stochastic fractal search: a powerful metaheuristic algorithm. Knowl-Based Syst 75:1–18 18. Javidy B, Hatamlou A, Mirjalili S (2015) Ions motion algorithm for solving optimization problems. Appl Soft Comput 32:72–79 19. Zheng Y-J (2015) Water wave optimization: a new natureinspired metaheuristic. Comput Oper Res 55:1–11 20. Sadollah A et al (2013) Mine blast algorithm: a new population based algorithm for solving constrained engineering optimization problems. Appl Soft Comput 13(5):2592–2612 21. Ahrari A, Atai AA (2010) Grenade explosion method—a novel tool for optimization of multimodal functions. Appl Soft Comput 10(4):1132–1140 22. Eberhart RC, Kennedy J (1995) A new optimizer using particle swarm theory. In: Proceedings of the sixth international symposium on micro machine and human science, New York 23. Dorigo M, Maniezzo V, Colorni A (1996) Ant system: optimization by a colony of cooperating agents. IEEE Trans Syst Man Cybern Part B Cybern 26(1):29–41 24. Dorigo M et al (2008) Ant colony optimization and swarm intelligence. In: Proceedings of the 6th international conference, ANTS 2008, vol 5217, Springer, Brussels, 22–24 Sep 2008 25. Dorigo M, Stützle T (2010) Ant colony optimization: overview and recent advances. In: Handbook of metaheuristics 26. Dorigo M, Gambardella LM (1997) Ant colony system: a cooperative learning approach to the traveling salesman problem. IEEE Trans Evol Comput 1(1):53–66 27. Mirjalili S, Mirjalili SM, Lewis A (2014) Grey wolf optimizer. Adv Eng Softw 69:46–61 28. Pham D et al (2011) The Bees algorithm–a novel tool for complex optimisation. In: Intelligent production machines and systems—2nd I* PROMS virtual international conference, 3–14 Jul 2006, Elsevier 29. Cuevas E et al (2013) A swarm optimization algorithm inspired in the behavior of the social-spider. Expert Syst Appl 40(16):6374–6384 30. Karaboga D, Ozturk C (2011) A novel clustering approach: artificial bee colony (ABC) algorithm. Appl Soft Comput 11(1):652–657 31. Mirjalili S (2016) SCA: a sine cosine algorithm for solving optimization problems. Knowl-Based Syst 96:120–133 32. Moghdani R, Salimifard K (2018) Volleyball premier league algorithm. Appl Soft Comput 64:161–185 33. Issa M et al (2018) ASCA-PSO: adaptive sine cosine optimization algorithm integrated with particle swarm for pairwise local sequence alignment. Expert Syst Appl 99:56–70 34. Chen K et al (2018) A hybrid particle swarm optimizer with sine cosine acceleration coefficients. Inf Sci 422:218–241 35. Nenavath H, Jatoth RK (2018) Hybridizing sine cosine algorithm with differential evolution for global optimization and object tracking. Appl Soft Comput 62:1019–1043 36. Abd Elaziz M, Oliva D, Xiong S (2017) An improved oppositionbased sine cosine algorithm for global optimization. Expert Syst Appl 90:484–500 13 37. Rizk-Allah RM (2018) Hybridizing sine cosine algorithm with multi-orthogonal search strategy for engineering design problems. J Comput Des Eng 5(2):249–273 38. Reddy KS et al (2018) A new binary variant of sine–cosine algorithm: development and application to solve profit-based unit commitment problem. Arab J Sci Eng 43(8):4041–4056 39. Banerjee A, Nabi M (2017) Re-entry trajectory optimization for space shuttle using sine–cosine algorithm. In: 2017 8th international conference on recent advances in space technologies (RAST) 40. Tawhid MA, Savsani V (2017) Multi-objective sine–cosine algorithm (MO-SCA) for multi-objective engineering design problems. Neural Comput Appl 41. Mohammed Mudhsh SX, El Aziz MA, Hassanien AE, Duan P (2017) Hybrid swarm optimization for document image binarization based on Otsu function. CASA 42. Abd El Aziz M, Selim IM, Xiong S (2017) Automatic detection of galaxy type from datasets of galaxies image based on image retrieval approach. Sci Rep 7(1):4463 43. Hafez AI et al (2016) Sine cosine optimization algorithm for feature selection. In: 2016 international symposium on innovations in intelligent systems and applications (INISTA). IEEE, New York 44. Bairathi D, Gopalani D (2017) Opposition-based sine cosine algorithm (OSCA) for training feed-forward neural networks. In: 2017 13th international conference on signal-image technology & internet-based systems (SITIS). IEEE, New York 45. Li N, Li G, Deng Z (2017) An improved sine cosine algorithm based on levy flight. In: Ninth international conference on digital image processing (ICDIP 2017). International Society for Optics and Photonics 46. Qu C et al (2018) A modified sine–cosine algorithm based on neighborhood search and greedy levy mutation. Comput Intell Neurosci 47. Zou Q et al (2018) Optimal operation of cascade hydropower stations based on chaos cultural sine cosine algorithm. In: IOP conference series: materials science and engineering. IOP Publishing 48. Meshkat M, Parhizgar M (2017) A novel weighted update position mechanism to improve the performance of sine cosine algorithm. In: 2017 5th Iranian joint congress on fuzzy and intelligent systems (CFIS). IEEE, New York 49. Bureerat S, Pholdee N (2017) Adaptive sine cosine algorithm integrated with differential evolution for structural damage detection. In: International conference on computational science and its applications. Springer, New York 50. Elaziz MEA et al (2017) A hybrid method of sine cosine algorithm and differential evolution for feature selection. In: International conference on neural information processing. Springer, New York 51. Zhou C et al (2017) A sine cosine mutation based differential evolution algorithm for solving node location problem. Int J Wirel Mobile Comput 13(3):253–259 52. Oliva D et al (2018) Context based image segmentation using antlion optimization and sine cosine algorithm. Multimed Tools Appl 77(19):25761–25797 53. Pasandideh SHR, Khalilpourazari S (2018) Sine cosine crow search algorithm: a powerful hybrid meta heuristic for global optimization. arXiv preprint: arXiv:1801.08485 54. Singh N, Singh S (2017) A novel hybrid GWO-SCA approach for optimization problems. Eng Sci Technol Int J 20(6):1586–1601 55. Zhang J, Zhou Y, Luo Q (2018) An improved sine cosine water wave optimization algorithm for global optimization. J Intell Fuzzy Syst 34(4):2129–2141 56. Nenavath H, Jatoth RK (2019) Hybrid SCA–TLBO: a novel optimization algorithm for global optimization and visual tracking. Neural Comput Appl 31(9):5497–5526 Author's personal copy Engineering with Computers 57. Majhi SK (2018) An efficient feed foreword network model with sine cosine algorithm for breast cancer classification. Int J Syst Dyn Appl (IJSDA) 7(2):1–14 58. Raut U, Mishra S (2019) Power distribution network reconfiguration using an improved sine–cosine algorithm-based meta-heuristic search. In: Soft computing for problem solving. Springer, New York, pp 1–13 59. Ghosh A, Mukherjee V (2017) Temperature dependent optimal power flow. In: 2017 international conference on technological advancements in power and energy (TAP energy). IEEE, New York 60. Issa M et al (2018) Pairwise global sequence alignment using sine–cosine optimization algorithm. In: International conference on advanced machine learning technologies and applications. Springer, New York 61. SeyedShenava S, Asefi S (2018) Tuning controller parameters for AGC of multi-source power system using SCA algorithm. Delta 2(B2):B2 62. Rajesh K, Dash S (2019) Load frequency control of autonomous power system using adaptive fuzzy based PID controller optimized on improved sine cosine algorithm. J Ambient Intell Hum Comput 10(6):2361–2373 63. Khezri R et al (2018) Coordination of heat pumps, electric vehicles and AGC for efficient LFC in a smart hybrid power system via SCA-based optimized FOPID controllers. Energies 11(2):420 64. Mostafa E, Abdel-Nasser M, Mahmoud K (2017) Performance evaluation of metaheuristic optimization methods with mutation operators for combined economic and emission dispatch. In: 2017 nineteenth international middle east power systems conference (MEPCON). IEEE, New York 65. Singh PP et al (2017) Comparative analysis on economic load dispatch problem optimization using moth flame optimization and sine cosine algorithms 2:65–75 66. Majeed MAM, Rao PS (2017) Optimization of CMOS analog circuits using sine cosine algorithm. In: 2017 8th international conference on computing, communication and networking technologies (ICCCNT) 67. Ramanaiah ML, Reddy MD (2017) Sine cosine algorithm for loss reduction in distribution system with unified power quality conditioner. i-Manag J Power Syst Eng 5(3):10 68. Dhundhara S, Verma YP (2018) Capacitive energy storage with optimized controller for frequency regulation in realistic multisource deregulated power system. Energy 147:1108–1128 69. Singh V (2017) Sine cosine algorithm based reduction of higher order continuous systems. In: 2017 international conference on intelligent sustainable systems (ICISS). IEEE, New York 70. Tasnin W, Saikia LC (2017) Maiden application of an sine– cosine algorithm optimised FO cascade controller in automatic generation control of multi-area thermal system incorporating dish-Stirling solar and geothermal power plants. IET Renew Power Gener 12(5):585–597 71. Rout B, Pati BB, Panda S (2018) Modified SCA algorithm for SSSC damping Controller design in Power System. ECTI Trans Electric Eng Electron Commun 16(1):46–63 72. Sahu N, Londhe ND (2017) Selective harmonic elimination in five level inverter using sine cosine algorithm. In: 2017 IEEE international conference on power, control, signals and instrumentation engineering (ICPCSI). IEEE, New York 73. Das S, Bhattacharya A, Chakraborty AK (2018) Solution of short-term hydrothermal scheduling using sine cosine algorithm. Soft Comput 22(19):6409–6427 74. Ismael SM, Aleem SHA, Abdelaziz AY (2017) Optimal selection of conductors in Egyptian radial distribution systems using sine– cosine optimization algorithm. In: 2017 nineteenth international middle east power systems conference (MEPCON). IEEE, New York 75. Kumar V, Kumar D (2017) Data clustering using sine cosine algorithm: data clustering using SCA. In: Handbook of research on machine learning innovations and trends.IGI Global, pp 715–726 76. Mahdad B, Srairi K (2018) A new interactive sine cosine algorithm for loading margin stability improvement under contingency. Electr Eng 100(2):913–933 77. Sindhu R et al (2017) Sine–cosine algorithm for feature selection with elitism strategy and new updating mechanism. Neural Comput Appl 28(10):2947–2958 78. Yıldız BS, Yıldız AR (2018) Comparison of grey wolf, whale, water cycle, ant lion and sine–cosine algorithms for the optimization of a vehicle engine connecting rod. Mater Test 60(3):311–315 79. Kumar N et al (2017) Single sensor-based MPPT of partially shaded PV system for battery charging by using cauchy and gaussian sine cosine optimization. IEEE Trans Energy Convers 32(3):983–992 80. Abd Elfattah M et al (2017) Handwritten Arabic manuscript image binarization using sine cosine optimization algorithm. In: Genetic and evolutionary computing. Springer, Cham 81. Turgut OE (2017) Thermal and economical optimization of a shell and tube evaporator using hybrid backtracking search— sine–cosine algorithm. Arab J Sci Eng 42(5):2105–2123 82. Wang J et al (2018) A novel hybrid forecasting system of wind speed based on a newly developed multi-objective sine cosine algorithm. Energy Convers Manag 163:134–150 83. Mirjalili S (2015) Moth-flame optimization algorithm: a novel nature-inspired heuristic paradigm. Knowl-Based Syst 89:228–249 84. Mirjalili S, Mirjalili SM, Hatamlou A (2016) Multi-verse optimizer: a nature-inspired algorithm for global optimization. Neural Comput Appl 27(2):495–513 85. Yang XS, Deb S (2009) Cuckoo search via Levy flights. In: Proceedings of world congress on nature & biologically inspired computing, pp 210–225 86. Yu JJQ, Li VOK (2015) A social spider algorithm for global optimization. Appl Soft Comput 30:614–627 87. Mirjalili S (2015) The ant lion optimizer. Adv Eng Softw 83:80–98 88. Mirjalili S et al (2017) Salp swarm algorithm: a bio-inspired optimizer for engineering design problems. Adv Eng Softw 114:163–191 89. Mirjalili S, Lewis A (2016) The whale optimization algorithm. Adv Eng Softw 95:51–67 90. Ragsdell K, Phillips D (1976) Optimal design of a class of welded structures using geometric programming. J Eng Ind 98(3):1021–1025 91. Huang F-Z, Wang L, He Q (2007) An effective co-evolutionary differential evolution for constrained optimization. Appl Math Comput 186(1):340–356 92. Coello CAC, Montes EM (2002) Constraint-handling in genetic algorithms through the use of dominance-based tournament selection. Adv Eng Inform 16(3):193–203 93. Krohling RA, Hoffmann F, Coelho LS (2004) Co-evolutionary particle swarm optimization for min-max problems using Gaussian distribution. In: Proceedings of the 2004 congress on evolutionary computation (IEEE cat. no. 04TH8753) 94. Mezura-Montes E, Coello CAC (2008) An empirical study about the usefulness of evolution strategies to solve constrained optimization problems. Int J Gen Syst 37(4):443–473 95. Kaveh A, Talatahari S (2010) Optimal design of skeletal structures via the charged system search algorithm. Struct Multidiscip Optim 41(6):893–911 96. Rashedi E, Nezamabadi-pour H, Saryazdi S (2009) GSA: a gravitational search algorithm. Inf Sci 179(13):2232–2248 13 Author's personal copy Engineering with Computers 97. Lee KS, Geem ZW (2005) A new meta-heuristic algorithm for continuous engineering optimization: harmony search theory and practice. Comput Methods Appl Mech Eng 194(36):3902–3933 98. Mahdavi M, Fesanghary M, Damangir E (2007) An improved harmony search algorithm for solving optimization problems. Appl Math Comput 188(2):1567–1579 99. Thirugnanasambandam K et al (2019) Reinforced cuckoo search algorithm-based multimodal optimization. Appl Intell 100. Zhao X, Zhou Y, Xiang Y (2019) A grouping particle swarm optimizer. Appl Intell 101. Kaveh A, Khayatazad M (2012) A new meta-heuristic method: ray optimization. Comput Struct 112–113:283–294 102. Kaveh A, Motie Share M, Moslehi M (2013) A new meta-heuristic algorithm for optimization: magnetic charged system search. Acta Mech 224(1):85–107 103. Kaveh A, Mahdavi VR (2014) Colliding bodies optimization: a novel meta-heuristic method. Comput Struct 139:18–27 104. He Q, Wang L (2007) An effective co-evolutionary particle swarm optimization for constrained engineering design problems. Eng Appl Artif Intell 20(1):89–99 105. Li L et al (2007) A heuristic particle swarm optimizer for optimization of pin connected structures. Comput Struct 85(7):340–349 106. Belegundu AD (1983) Study of mathematical programming methods for structural optimization. Diss Abstr Int Part B Sci Eng 43(12):1983 107. Mezura-Montes E, Coello CAC, Landa-Becerra R (2003) Engineering optimization using simple evolutionary algorithm. In: Proceedings of the 15th IEEE international conference on tools with artificial intelligence 13 108. Ray T, Saini P (2001) Engineering design optimization using a swarm with an intelligent information sharing among individuals. Eng Optim 33(6):735–748 109. Sandgren E (1990) Nonlinear integer and discrete programming in mechanical design optimization. J Mech Des 112(2):223–229 110. Kannan B, Kramer SN (1994) An augmented Lagrange multiplier based method for mixed integer discrete continuous optimization and its applications to mechanical design. J Mech Des 116(2):405–411 111. Zhang J, Zhou Y, Luo Q (2019) Nature-inspired approach: a wind-driven water wave optimization algorithm. Appl Intell 49(1):233–252 112. Deb K (1997) GeneAS: A robust optimal design technique for mechanical component design. In: Evolutionary algorithms in engineering applications. Springer, New York, pp 497–514 113. Coello CAC (2000) Use of a self-adaptive penalty approach for engineering optimization problems. Comput Ind 41(2):113–127 114. Liu H, Cai Z, Wang Y (2010) Hybridizing particle swarm optimization with differential evolution for constrained numerical and engineering optimization. Appl Soft Comput 10(2):629–640 115. Souza E, Nikolaidis I, Gburzynski P (2010) A new aggregate local mobility (ALM) clustering algorithm for VANETs. In: 2010 IEEE international conference on communications. IEEE, New York Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.