A competitive volleyball algorithm to solve global optimization with the engineering problems

In real life, many engineering problems are nonlineble NP problems, in order to solve some of these problems, we put forward a competitive volleyball algorithm.The algorithm proposed in this paper is a metaheuristic technique based on swarm optimization. It is inspired by the competition between volleyball teams in a league and the improvement in players’ overall abilities in order to win the Most Valuable Player award. Several speciﬁc terms relating to competition, such as pre-match reinforcement, single round robin mechanism, optimal strategy constitute the structure of the algorithm. The sensitivity of several parameters of the algorithm is analyzed and tested for three types of benchmark functions: unimodal, high-dimensional multimodal and low-dimensional multimodal functions. Through the use of these three types of test functions, the performance of this algorithm is compared with nine classical metaheuristic algorithms: Genetic Algorithm (GA), Diﬀerential Evolution (DE), Harmony Search (HS), Artiﬁcial Bee Colony (ABC), Particle Swarm Optimization (PSO), Sine Cosine Algorithm (SCA), Soccer League Competition (SLC), League Championship Algorithm (LCA) and Volleyball Premier League (VPL). CVA has been used to solve three real-world engineering problems. The results show that the performance of the CVA is behaviorally promising and better than the other classical metaheuristic algorithms.


Introduction
The models solved by most algorithms require certain types of objective functions or constraints, while problems in different fields require different types of decision variables, objective functions and constraints. For this reason, a variety of intelligent algorithms have become effective choices in computing, and they have become an important stage in the development of metaheuristic algorithms. These methods have developed into general and powerful optimization algorithms and provide a reasonable tradeoff between the computation time and the quality of the achievable solutions.
The rest of this article is organized as follows. Section 2 describes the CVA algorithm in detail. The upper part of Section 3 analyzes the impact of several important parameters on the performance of the algorithm. The second half of this section uses 23 test functions to test the CVA, and compares the results with the results of the other 9 famous algorithms mentioned above. Finally, Section 4 concludes this article and prospects related research in the future.

The CV algorithm
CVA is used to find the global optimal solution in the continuous search space, and a better solution is obtained by using a variety of search mechanisms in a season.
Every season, the teams will try their best to win the championship. At the same time, every player wants to be the best player (BP) in the team. At the end of the league, the judges choose the best player (MVP). The coach should not only guide the players before the game, but also arrange the players in the field, and adjust the training program according to the results of the game, so the coach's ability plays an important role in a team. Volleyball team generally consists of two coaches, six players and six substitutes. The formation of the two teams on the field is shown in Fig.1.
Before describing the process of the algorithm designed, the following assumptions are stated: • Each athlete is graded by using the fitness function (objective function) to calculate the fitness value.
• The success or failure of the game depends on the average fitness value of the players in each team.
• The outcome of each game is unknown. In other words, a weaker team may beat a stronger team.
• During recovery, each team adjusts its strategy for the next game only based on last week's results.
• The result of the game is only win or lose, no draw.
• Each team can replace one or more players at any time in a game.
The basic steps of competitive volleyball algorithm can be represented by the schematic flow chart shown in Fig.2. In the following sections, we will introduce the main modules of CVA, especially the pre-match reinforcement, the judgment of winners and losers, and strategies of winners and losers after the competition.

Solution representation
Each volleyball team consists of coaches (head coach and assistant coach), players on the court and substitutes. These three types of members of a team constitute the representation structure of the solution. Every player on the court is the main part of finding the best solution, and the key object in the process of training and competition. Coaches and substitutes are the coordination part, and some adjustment strategies are inspired by this part, such as coach learning strategies and strategies for using substitutes. The rest of this section discusses only the players on the court who represent the solution. The relationship between team composition and solution is shown in Fig.3.
In CVA, the number of seasons is used to represent the number of iterations. A team represents a group of synchronized vector solutions, and each participant represents a vector solution. The composition of each vector solution (decision variable) represents the capabilities of the player. The team, players and abilities are represented as follows: (1) P layer j = Ab 1 Ab 2 Ab 3 · · · Ab P roblemSize (2) Figure 3. Relationships between team structure and the solution representation where P layerSize is the number of players on a team,SubstituteSize is the number of substitutes on a team, Ab P roblemSize stands for a player's ability, P roblemSize is the dimension of the problem, and f (P layer j ) is the objective function value (fitness value) of each player.

Initialization
In the initialization phase, the total number of members of the team is randomly generated within the search area given by the problem. Let M l i,d be the l ability of the k member of the i team: where lb l and ub l are the lower bound and upper bound of the variable l, respectively, r is a uniformly distributed random number between 0 and 1.

Formation of team
Here, we assume that the comprehensive strength of the coaches is better than the players in the team. From a mathematical perspective, according to the results of the test function, the randomly generated initial solutions are distributed according to the results of the test function, and the best result is defined as the head coach. the second is deputy coach, and the rest of the initial solutions are randomly assigned to players and substitutes. When forming a team, the coaches need N c , the players and the substitutes need N p and N s respectively, and the number is equal .The following expression can be used to calculate the number of team members: where N is the total number of team members. The main parts of the initial solutions are demonstrated in a matrix P i , where the numbers in the lower corner represent the team code and player code, respectively, while the number in the upper corner represents a certain attribute or ability (dimension of the problem) of a player. Similarly, the matrix of coordination part (coaches and substitute) represents the same meaning as the matrix of players on the field. Eqs.7 -9 represent the coaches, players and substitutes of the team i, respectively, Players on each team are a major part of the solution. All players in the league, i.e., all candidate solutions generated at the same time, can be represented by composite matrix P , as follows: where N T is the total number of teams in the league.

Pre-match reinforcement
As the game is played under the conditions of high intensity and high speed, it is impossible for volleyball players to win without physical strength. At the same time, the atmosphere of the game is often tense, so how the player performs under pressure is a key to winning the game. This entails having the right physical and psychological qualities.
The activity that makes athletes perform better on the field by improving the overall quality is called prematch reinforcement strategy. We define team i s player on the court as X i,j and the substitute as Y i,k . The mathematical expressions of this strategy are as follows: where t indicates the current iteration, phy i,j indicates the effect of physical fitness training of the j player of team i, psy i,j indicates the effect of psychological guidance of the j player of team i, phy i,k indicates the effect of physical fitness training of the k substitute of team i, and psy i,k indicates the effect of psychological guidance of the k substitute of team i. The pseudocode of pre-match reinforcement is expressed in Algorithm 1.
Algorithm 1 pseudo-code of pre-match reinforcement for i = 1 to N T do for j = 1 to nP do Update physical ability of player j using Eq (11) Update psychological quality of player j using Eq (12) Update physical ability of substitute j using Eq (11) Update psychological quality of substitutej using Eq (12) end for end for

Match schedule
In the single round robin (SSR) (Jaramillo et al., 2016a;Kashan, 2014a;Moghdani and Salimifard, 2018) of each season, each team plays each other only once, and the final rankings are determined by the average strength of each team. In this paper, the SSR method of scheduling, also known as the counterclockwise rotation method, can maintain the excitement of the championship game.
If there are N T teams in the league, then each team will play (N T − 1) games. Then every season, the league has a total of (N T − 1)N T /2 games. Whether the number of participating teams is even or odd, it should be arranged according to an even number. If it is an odd number, you can add a team with the serial number of "0".When a team is scheduled to play against team "0",it takes a break.
To demonstrate how the SSR works, we drew a circle of eight teams in the league, keeping team A in the center of the circle. As mentioned above, the eight teams will play seven rounds (weeks) in a season (iteration). As shown in Fig.4, in the first week, team A competes with team H, and the other six teams, which are connected by horizontal lines, compete in pairs (B to G, C to F, and D to E). When the seven teams on the ring rotate in counterclockwise order, this becomes the next week's schedule. The competitive arrangement of the eight teams from the second week to the seventh week is shown in Fig.5 .
To better illustrate the competitive arrangement, corresponding to Fig.5, the league schedule is shown in Table.1.

Competition
The aim of the proposed CVA is to solve the optimization problem through the analogy of volleyball competition, which is that the goal of every team is to win the championship, and that every player wants to be the best player (BP) or MVP. It should be stated that the BP is the best player on each team (a local optimum), and MVP is the best player in the league (a global optimum).
Usually, each team plays only one game a week. In order to quantify the comprehensive strength of the team every week and determine the winning team, we use the average strength of the players on the court to represent the strength of the entire team (assumption 2). The mathematical expression is where T (i) indicates the strength of the team i, f (X i,j ) is the fitness value of player j of team i, and N p is the number of players of team i. If there is no significant difference between the two T (i) values, the probability of individual selection is almost the same, which may weaken the CVA selection function. To ensure that the value of winning or losing is not very large, and that there is a certain gap between the two teams, we use regularization technology to calibrate T (i). The specific method is as follows: where φ(i) indicates the strength index for team i, and T min and T max represent the lowest and highest strength of the team in a season, respectively. ε is a disturbance (a very small number) so that even the worst team in the league still has a chance to win the game (assumption 3). This chance of winning the game decreases as the number of iterations increases, and indicates that the process of ε change can be represented by Eq.17. It is worth noting that the constant 0.1 is selected after many experiments, and good results are obtained for the optimization problems used in the experiments. However, for other specific instances, the constant 0.1 may not be the most suitable one. It is implied by Eq.(18) that the φ(i) value of each team is proportional to its fitness value, so the stronger the team, the higher the φ(i) value. Assuming that team k and team q play each other and their strengths are T (k) and T (q), respectively, the strength index values of the two teams can be calculated as follows: Let p(k, q) denotes the probability that team k beats team q. The specific formula is as follows: It follows that: It can be deduced from Eqs.19 and 20: To determine the winner (or loser) of each game, a random number r ∈ [0, 1] is introduced. If r ≤ p(k, q), team k wins; otherwise, team q wins. The manner of determining the winner/loser is consistent with assumption 3 and assumption 5 at the beginning of this section.
After the result of the competition, in order to better prepare for the next week's competition, the winning and losing sides adopt different strategies and training. The pseudocode of competition is shown in Algorithm 2.
The pseudo-code of competition is shown in Algorithm 2.
Algorithm 2 pseudo-code of competition function Competition(k, q) Calculate T(i) and φ(i) using Eqs. (15) and (16) Calculate p(k,q) using Eq. (19) Generate r ∈ [0, 1] if r ≤ p(k, q) then team k is the winner and team q is the loser else team q is the winner and team k is the loser end if apply winner promotion strategy for winner, apply loser strategy for loser, end function 2.7. Winner promotion strategy Every member of the winning team places high demands on themselves and wants to be the league MVP. Therefore, they try to learn from the MVP candidate in the league. We know that the learning ability of each participant is different, and so we introduce two evenly distributed random numbers, r 1 and r 2 , into this strategy. The overall strengths of player j and substitute k are defined as X i,j (t) and Y i,k (t), respectively.X * (t) is the MVP (Bouchekara et al., 2018;Bouchekara, 2020) candidate for the t season. The overall ability value of each member of the winning team after implementing the training strategy of the winner can be calculated by the following formulas: where r 1 and r 2 are uniformly distributed random numbers between 0 and 1. The pseudo-code of winner promotion strategy is shown in Algorithm 3.
Algorithm 3 pseudo-code of winner pseudo-code of winner promotion strategy for j = 1 to N p do Update the comprehensive strength of player j Eq.(22) end for for k = 1 to N s do Update the comprehensive strength of substitute k using Eq.(23) end for

Loser strategy
In order to achieve better results in their next game, the losing team should employ additional strategies and methods to improve its overall strength The coach's role is very important, because he not only strategizes to help his players improve their skills but also understands the skills of the best players on the other teams in the league and can formulate a strategy to overcome them.

Coach training strategy
The technical performance level of players is strongly influenced by the guidance of the coaches. The head coach and assistant coach may individually as well as cooperatively train the players and substitutes. In the implementation of this strategy, the head coach and assistant coach guide the players on the court as well as the substitutes. The coaches should not only take into account each player's individual deficiencies but also consider the shortcomings common to all players.
Each of the players is instructed by the head or assistant coach. This process is called coach training, which may help losing teams perform better. The mathematical expressions of this strategy are where ub and lb are the lower and upper bound, respectively, of the variable vector. The pseudocode for the coach's training strategy is shown in Algorithm 4.
Algorithm 4 pseudo-code of coach training strategy for j = 1 to N p do Update the comprehensive strength of player j using Eq.(24) end for for k = 1 to N s do Update the comprehensive strength of substitute k using Eq.(25) end for

Optimal strategy
Under the guidance of the coach, the confidence of the players increases. Next, the players will improve their physical and psychological abilities. Generally speaking, other athletes should learn from the best players of the current season. At the same time, the best players in the league also serve as examples for the other players. Because the core idea of this strategy is to let players learn from the excellent players to achieve their goal of self-improvement, this strategy is called the optimal strategy.
Members of the losing team need to find the gap between themselves and others in order to improve. Let the gap between the player and the best player in the group be X BP i,j (t), and the gap between the player and the league's best player be X CM V P i,j (t) in the t season: In the t season, the gap between a particular substitute and the best player on the team i is defined as Y BP i,k (t), and the gap between that substitute and the best player in the league is defined as To search for different values, we introduce a common nonlinear activation function Tanh (Emad H.M. Zahran, 2016;Fréderic Godin, 2018) (the range is (-1,1)), so that the nonlinear part is introduced into the whole search space, and the search ability of the algorithm is improved. Next, in order to map (t) to the appropriate range, we improve Tanh by introducing the parameters aa,b and d. Here, aa is calculated from the constant a and the upper and lower bounds of variables, as shown in Eq.30: where a is constant, and ub and lb are the upper and lower bounds of the variable vector.
We use the gap between the players on the losing team and the best players on the team ( X BP i,j (t)) and in the league ( X CM V P i,j (t)) as the independent variable of the Tanh function. The specific equations are where η( X BP i,j (t)) and η( X CM V P i,j (t)) are coefficient values, and b, c and d are constant. In the next section, we will discuss the settings of the important parameters in the CVA.
The gap between the losing team's substitutes and the MVP is also mapped to the appropriate range through a variant of the activation function Tanh.
The overall abilities of the players are improved after adopting this strategy, and the skills and methods previously mastered will be partially retained, so in order to calculate the weight w of the retained skills in the improved overall abilities, w max and w min are used in the following formula: where w max refers to the maximum weight of previous ability, while w min refers to the minimum weight of retention. Using the various parameters discussed above, the equation of the overall abilities of the members of each losing team after training is defined as where t denotes the current iteration, and r 1 and r 2 are two uniformly distributed random numbers between 0 and 1.
A losing team can use this strategy to calculate the gap between its players with the team's best player and the best player in the league, and then use the variation of Tanh function to map it to the appropriate interval. Finally, combined with the items and weights discussed in this section, the overall ability value of the players after training according to this strategy is obtained. The pseudocode of the optimal strategy is shown in Algorithm 5.
Algorithm 5 pseudo-code of optimal strategy for j = 1 to N p (N s ) do Calculate the gap between the player j of team i and the best player in the team X BP i,j (t) and the league X CM V P i,j (t) respectively, using Eqs. (26)-(27) Calculate the gap between the substitute k of the team i and the best player in the team Y BP i,k (t) and the Y CM V P i,k (t) respectively, using Eqs. (28)-(29) Map the gap values to the appropriate range of values, using Eqs.(31)-(32) Update the comprehensive strength of player j and substitute k, using Eq.(34) end for

Substitute strategy
When the team is behind in the score, or if one or more of the starting players are injured, the coach should consider putting in a substitute or substitutes (Purnomo and Wee, 2015). If necessary, the coach will choose the substitutes according to the opposing team's playing style and the substitutes' skills, so that the substitutes and the other starting players can form a new overall balance. This strategy is shown in Fig.1 In this paper, there is no limit to the number of substitutes (assumption 6).
We will use this strategy for the losing team and define δ as the team's substitution rate. The following formula calculates the number of team substitutions during the competition: 100] where N sub denotes the number of substitutes. Suppose the coach puts substituteY i,k in because starting player X i,j is injured or tired. We define two variables A and B as intermediate variables, and assign the properties of X i,j and Y i,k to A and B, respectively. The strategy is expressed by the following equations: The pseudocode of the strategy for using substitutes is shown in Algorithm 6.
Algorithm 6 pseudo-code of substitute strategy for j = 1 to N sub do Randomly select a player j and a substitute k Reverse two members j and k using Eq.(36) end for 2.9. One-to-one guidance strategy Prior to the next week's game, the head coach adopted a one-on-one guidance strategy for the players on the court. In general, the coach has the most experience and the deepest understanding of the game, so the best solution in a group represents the head coach. To show that the ability of players will improve as the season goes on, we introduce a dynamic parameter λ, which changes with the number of iterations. The dynamic equation is where t represents the current number of iterations, and T represents the maximum number of iterations. The function is depicted in Fig.6. With the increase in the number of iterations, the value rapidly decreases to the lowest value, and then gradually increases with the increase in the number of iterations. The change in the value of λ can accelerate the convergence process of the solutions and avoid falling into local optimum. The improvement of a player's ability depends not only on the guidance of the coach but also on the continuing efforts of the player to improve. At this stage, it is expressed by where Z ( i, 1) denotes the head coach of team i, and r represents a uniformly distributed random number with values in the range of [-1,1].
The pseudocode of the one-to-one guidance strategy is shown in Algorithm 7.
Algorithm 7 Pseudo-code of one-to-one guidance strategy for i = 1 to N T do for j = 1 to N p do Update the comprehensive strength of player j using Eq.(38) end for end for 2.10. Inter-team learning strategy The main idea of the inter-team learning strategy is that before the game, each team should learn from the other team. The coach should discuss the characteristics of the strongest team in the league to the players so that they can learn why that team is the strongest. For the purposes of this paper, the best team is the one with the highest average overall ability. Players from other teams should learn from the best team. To represent learning from the best team, we define the following equations: where θ and ν are coefficient values, h is constant, and r 1 and r 2 are uniformly distributed random numbers from 0 to 1. m is a linearly decreasing coefficient, as shown in Fig.7. Eq.41 is defined to determine the value of m: where α is constant, t represents current iteration, and T denotes the maximum number of iterations. Using the parameters proposed above, the equation for describing this strategy is From Eq.42, it can be seen that the players on other teams learn from the players in corresponding positions on the best team. As can be seen from Fig.7, as the number of iterations increases, the value of m gradually changes from α to 0, and each player of X gradually tends to the corresponding best player.

Algorithm 8 pseudo-code of Inter-team learning strategy
Selected the best team in the league this season. for i = 1 to N T do for j = 1 to N p do Update the comprehensive strength of player j using Eq.(42) end for end for The pseudocode that describes this process is shown in the Algorithm 8.

Promotion and demotion mechanism
After the competition stage, the top Q teams of the league took part in the next season, and the (N T − Q) teams were demoted. In order to keep the number of teams in each season unchanged, teams with similar strength to those entering the next season will be promoted (Moghdani et al., 2020;Purnomo et al., 2017;Qiao et al., 2020).
If the number of eliminated teams N e p is calculated according to the elimination rate ξ, the number of eliminated teams is calculated by Eq.43. To make the strength of the N ep advancing teams equal to that of the existing teams, the method of this algorithm is to randomly select coaches, players and substitutes from the team that advances to the second stage of competition: The pseudocode explaining the promotion and demotion strategy is shown in Algorithm 9.
Algorithm 9 pseudo-code of promotion and demotion strategy Remove N ep worst teams of league Define N ep empty teams P T with three properties: coaches,players and substitutes for i = 1 to N T do for j = 1 to N p do s = randomly selected from current available teams (N T − N ep ) pl = randomly selected from the players of the s team sub = randomly selected from the substitutes of the s team co = randomly selected from the coaches of the s team X Nep,j = X s,pl Y Nep,j = Y s,sub Z Nep,1 = Z s,co Z Nep,2 = Z s,co end for end for Add N ep teams to league

Coach improvement strategy
One of the key factors in improving a team's level of play is that the team have coaches who are proficient in coaching and highly knowledgeable about volleyball skills. Therefore, the self-improvement of coaches is very important.
Coaches should have a full understanding not only of the ability of their players but also of the playing methods, tactics and habits of the players on the other teams in the league. These coaching skills are accumulated over a period of time. This process can be expressed mathematically as where r 1 and r 2 represent uniformly distributed random numbers with values in the range of [0, 1]. Z i,m represents the head coach or assistant coach of team i, where m is 1 or 2. X * is a candidate for MVP, and X i,j denotes any player j of team i. The procedure of a coach improvement strategy is as shown in Algorithm 10.
Algorithm 10 pseudo-code of coach promotion strategy Selected the candidate for MVP in the league this season for i = 1 to N T do for m = 1 to 2 do Update the comprehensive strength of coach m using Eq.(44) end for end for

Comparison between algorithms inspired by sports tournaments
In order to better understand our proposed algorithm, compare CVA with VPL(Moghdani and Salimifard, 2018),SLC (Jaramillo et al., 2016b) and LCA (Alizadeh and Kashan, 2019;Kashan, 2014b). Table 2 shows the similarities and differences of various features between the four algorithms.

Experiments
To evaluate the performance of CVA, we used the CEC(Behzad Javidy, 2015) for testing purposes. We compared the performance of the CVA with that of nine other classical metaheuristic algorithms (Moghdani and Salimifard, 2018). The convergence speed, global optimization ability and search accuracy of the algorithm were verified by 23 test functions, as shown in Table 3. These test functions have different characteristics.
f 1 ∼ f 7 are unimodal functions, which can test the optimization accuracy and performance of the algorithm.
f 8 ∼ f 23 are multimodal functions with multiple local minima, which can be used to test the ability of the algorithm to jump out of the local optimal solution. Multimodal functions are either high-dimensional or lowdimensional. f 8 ∼ f 13 are high-dimensional multimodal functions, and f 14 ∼ f 23 are low-dimensional, allowing the search ability of the algorithm to be verified in different dimensions. The algorithm was tested by CEC in MATLAB. To make the maximum number of function evaluations equal, we set it to 100,000, and to reduce the impact of accidents on the accuracy of the results, we repeated each test experiment 30 times for each algorithm. Finally, the performance of the algorithm was evaluated through the experimental results.

Effects of various parameter settings in CVA
In order to test and evaluate the proposed algorithm, we performed the empirical study of selecting six benchmark functions from three types of test functions (f 1 , f 2 , f 7 , f 12 , f 20 , f 22 ) (Andrea Tangherloni, 2019). f 1 is a Sphere Model function with a unique global minimum, which is mainly used to test the optimization accuracy of the algorithm. f 2 is the Schwefel 2.22 function, and its independent variable is epistatic, so its gradient direction will not change along the axis, which means it is difficult to optimize. f 7 is a Quartic noise function with a uniformly distributed random variable in a variation range of [0,1). This function is used to measure the performance of the optimization algorithm when dealing with unimodal functions that are mixed with many noises. f 12 is a Generalized Penalized Function, which is used to detect the ability of an algorithm to jump out of the local optimum. f 20 is the Hartmann 6-D function, with six local minimums. f 22 is the Shekel function, which has seven local minimums. Fig.8 shows the graphs of these six benchmark functions.
Since all the parameters in the algorithm cannot be adjusted at the same time, the method of controlling variables is used when adjusting the parameter values, that is, keeping the other parameter values unchanged and adjusting only the studied parameter values. Results for the optimal values of the different benchmark functions were obtained under different parameter values.

Impact of ξ value
Elimination rate ξ refers to the proportion of eliminated teams in the league. To maintain the number of teams in the league, we make the elimination rate equal to the promotion rate. At the same time, in order to make the strength of the team members from the promoted team equal to remaining teams in the next season, we randomly select members from other teams to form the promoted team. Therefore, the size of the promotion rate determines the number of promoted teams, which indirectly affects the quality of candidate solutions. To determine the parameter value, we select six equations from three benchmark functions and test each test function with a different set of values for ξ. The results are shown in Table 4.
To better illustrate how the value of parameter ξ is determined, we look at the data in Table 4 (as shown in visual format in Fig.3.5) to see which parameter value is the best. Our rankings are based on the optimal solutions obtained by the different parameter values. The better the solution, the higher the ranking.

Impact of g value
In order to analyze the influence of g value setting on CVA, different parameter values are introduced into the same test function, and the results are shown in Table.5 . As can be seen from this table, it is not that the larger the g, the better the results. In other words, the performance of CVA is not proportional to the g value. As can be seen from Fig.3.5, when g = 3.5, except for the third in f 1 and f 2 , the ranking is the first in all other test functions, and the average ranking is the highest when g = 3.5. So, g = 3.5 is the most appropriate one. It can also be seen from Fig.3.5 that the value of g has no effect on the results of two low-dimensional multimodal functions f 20 and f 22 .  Table 6. It can be seen that when N T ≥ 20, the optimal values of unimodal and multimodal functions can be searched. For each test function, the ranking of different values for N T is shown in Fig.3.5. To balance the relationship between the number of iterations in each test and the number of groups, N T is set to 20. We can also see that when N T = 20, the optimal value of most of the functions can be found with high accuracy, except for in f 2 , when it is in fourth place.  The last value to be examined is w max . To determine whether different values of w max affect the performance of CVA, we set the values for the parameter as 1, 2, 3, 4 and 5. The results obtained, as shown in Table 7, show that after determining the three parameters discussed above, w max has little influence on the performance of the algorithm, and there are only some differences in the accuracy of the optimal solutions. According to the ranking of different w max values shown in Fig.3.5, the value is w max = 4. Now we summarize the parameter settings: ξ = 0.5,g = 3.5,N T = 20 and w max = 4. The control variable method used in parameter settings in this paper may not be perfect, and more in-depth research may be needed to determine these parameters, but this is not the focus of this paper.  To ensure the fairness of the comparative experiment, the number of teams of the algorithm was set at 20, the number of iterations in a single test was 50, and the other algorithms also ensured that the product of the number of groups and the number of iterations was 1000.
In Table 3, f 1 ∼ f 7 are unimodal functions with only a strict local minimum (peak) in the considered interval. Table 8 shows the statistical results of these 10 algorithms on unimodal functions. In Fig. 10, the bar chart represents the average ranking of algorithm results in f 1 ∼ f 7 , and the line chart is the average ranking of algorithm standard deviation in f 1 ∼ f 7 . Both the average ranking of the results and the average ranking of standard deviation have the highest rankings.
The multimodal functions have many local extremes within the constraint range, which can be used to investigate the ability of jumping out of the local optimum and searching efficiency. Table 9 lists the best, worst, and average optimal solutions, as well as the variance of these 10 algorithms in 30 iterations (Chen, Wang, and Zhao, 2020;Gupta and Deep, 2020). It can be seen that except for f 10 , the average optimal solution of the algorithm proposed ranks in the top three. The same conclusion can be drawn from Fig. 11.

Convergence property of the algorithm
To explore the convergence of CVA, we selected two benchmark functions from each group. The convergence of different functions is shown in Fig.13.

Application of CVA in Engineering Optimization
This section is dedicated to evaluating the performance of CVA in practical engineering applications, including design of a pressure vessel,minimization of the weight of a tension / compression string and Himmelblau's non-linear optimization problem.The results are compared with the results calculated by other classical metaheuristic algorithms. In this paper, the penalty function is used to solve the constrained nonlinear programming problem, and the appropriate penalty function is made by using the constraint functions in the problem, from which the objective function with parameters is constructed, and the problem is transformed into an unconstrained nonlinear programming problem as follows (Coello, 2002): Where M is a sufficiently large positive number, g i (x), h i (x) and k i (x) denote the violation from constraint i , the sum of r, s and t is the total number of constraints.

Design of a pressure vessel
In this problem, there is a special cylindrical container with hemispherical heads at both ends. The purpose of this problem is to minimize the total cost, including material cost, forming cost and welding costKashan (2011). A schematic diagram of the problem and its characteristics is shown in Fig.14.
There are four design variables in this problem: shell thickness (x 1 ), container head thickness (x 2 ), inner diameter (x 3 ) and cylindrical section length (x 4 ). The engineering optimization problem can be expressed as follows: 10 ≤ x 4 ≤ 200 (55) In the above formula, the objective function Eq.47 means to minimize the total cost. Eqs.48-51 are all the constraints of the problem, and the range of variables can be expressed by Eqs.52-55 .
Various methods have been applied to solve the Pressure vessel design problem. These methods include , improved ACO (Kaveh and Talatahari, 2010a), CSS (Kaveh and Talatahari, 2010b), different genetic algorithms (Coello, 2000;,CPSO (He and Wang, 2007), improved HS (Shayanfar and Gharehchopogh, 2018),DE (Li et al., 2007a),ES (Mezura-Montes and Coello Coello, 2008a),and branch and bound method (Sandgren, 1990). The results of the best solutions found using different methods are shown in Table11. Compared with other algorithms, CVA has found a better solution.

Minimization of the weight of a tension / compression string
In this problem, the aim is to minimize the weight of the tension / compression spring, which is subject to minimum deflection, shear stress, surge frequency and limits on outside diameter. The problem consists of three variables: wire diameter (x 1 ), average coil diameter (x 2 ) and number of movable coils (x 3 ) (Coello Coello, 2000). The schematic diagram is shown in Fig.15. The problem can be stated as follows: 0.05 ≤ x 1 ≤ 2.00, Figure 15. Tension / compression string design and its features.
In the above formulas, Eq.56 is the objective function of the problem, Eqs.57-60 are the limiting conditions of the problem, and Eqs. 61-63 define the range of variables.

Himmelblau's non-linear optimization problem
This question was originally raised by Him-melblau (Himmelblau, 1972). It has previously been used as a benchmark for several other GA-based algorithmsCoello Coello (2000). In this problem, there are five variables, six nonlinear inequality constraints and ten boundary conditions. This problem can be expressed in mathematical expressions as: M inimizef (x) = 5.3578547x 2 3 + 0.8356891x 1 x 5 + 37.29329x 1 − 40, 792.141.

Conclusions
This paper presents a swarm-based optimization algorithm. The main idea behind the algorithm is to simulate that all the teams in a volleyball league want to win the championship, and every player wants to be the best player on their team, or even the MVP of the league. A team's strength is defined as the average of the strengths of its players. In order to improve the overall ability of the team and the ability of each individual player, every team and every team member should employ strategies for improving their ability. Each team uses the same pre-game training method, and in a single round robin tournament, the post-match training used will depend on whether the team won or lost. Lastly, each team takes part in a final series of intensive training prior to the next week's game.
The highlight of this algorithm is that both teams and players are constantly improving themselves, and this interactive process of improvement makes the candidate solutions tend to the global optimal solution. In addition, coaches not only lead the team, but also continue to accumulate experience in every season to improve their teaching ability, so that the abilities of players are also constantly being improved. From a mathematical perspective, the algorithm uses the variant of the activation function to strengthen the global search ability of the algorithm. In addition, the algorithm also uses self-adaptability to make the algorithm jump out of the local optimum. Like other sports-inspired algorithms, this algorithm also uses a single round robin system in the competition phase and a promotion and demotion mechanism at the end of each season. These common processes in the field of sports can improve the performance of the algorithm.
Through a series of experimental tests, the value of the important parameters of the algorithm is set, which makes the algorithm more suitable for the specific optimization problem to be solved. The CVA and the other nine algorithms are tested by three groups of benchmark functions (unimodal, multimodal and fixeddimensional multimodal functions), and then compared against each other. The results show that the CVA algorithm obviously outperforms the other nine algorithms.