In the standard SSA, the leader only searches according to the location of the food source, and the update mode of the follower is only related to its adjacent Levy flight [21] , proposed by French Mathematician Paul Pierre Lévy [21] , is a heavy-tailed distribution with a high probability of producing a large stride. Most animals follow this rule in their foraging behavior. The accidental long-distance step of Levy flight helps to expand the search range and jump out of the limit of local optimization, more short distance steps can carefully search the surrounding area and improve the local exploitation ability. The trajectory of Levy’s flight is shown in Figure 2.

The probability density function of Levy flight obeys the Levy distribution, which can be expressed by Equation (6)

Levy ( s ) = 1 π ∫ 0 ∞ exp ( − β | k | λ ) cos ( k s ) d k . (6)

According to Mantegna method [22] , the random step of Levy flight can be expressed as follows

l = u | v | 1 / β . (7)

where, u and v are Gaussian distributions that satisfy the following conditions

u ~ N ( 0 , δ u 2 ) , v ~ N ( 0 , δ v 2 ) , (8)

δ u = ( Γ ( 1 + β ) ⋅ sin ( β 2 π ) Γ ( 1 + β 2 ) ⋅ β ⋅ 2 β − 1 2 ) 1 / β , (9)

δ v = 1 , (10)

the value of β is generally 1.5, and Γ is a Gamma function.

Brownian motion is the irregular motion made by a suspended particle, a random wandering process without rules. Because the standard Brownian motion step change image is similar to a Gaussian distribution with mean 0 and variance 1, the Brownian motion random number is generated from Equation (11)

Brown ~ N ( 0, δ 2 ) . (11)

The crossover operator helped to increase the diversity of the population, thereby increasing the chance for the population to break out of local optimum food sources.

Different random integers were generated, each corresponding to the respective dimension of an individual, sorted based on their fitness values, and the positions of the corresponding dimensions were exchanged in sequence. The adjacent individuals underwent the crossover operator to generate the crossover individuals, and a greedy strategy was used to retain individuals with better fitness values for the next iteration. c random numbers satisfy

c = k D , 0 < s i ≤ D , s = { s 1 , s 2 , ⋯ , s c } , (12)

where, the random integers in s are different, and k is the proportion coefficient of the crossover operator.

In standard SSA, the position update of a follower is only related to the previous individual, which is blind and slows down the convergence of the overall algorithm. If the previous individual falls into a local optimum, then all the followers are unable to jump out of the local optimum.

Consider the position relationship between the follower and the average leader, and let the follower follow the average leader closely, with the following update method

x ¯ j = ∑ i = 0 N l x j i N l , (13)

where, x ¯ j is the average position of all leaders in the j-th dimension, and N l is the number of leaders.

In the standard SSA, the leader only searches according to the location of the food source, and the update mode of the follower is only related to its adjacent individual location. The global search or local search is determined according to c₁ value. If the food source is the local optimum, the leader will fall into it, and the follower will not jump out of the local optimum, so that the SSA cannot jump out of the local optimum. Therefore, in the early stage of the iteration, consider the location of random leader or food source to establish contact with the current individual to update the leader, and a small probability of Levy flight [21] random step size should be added to help jump out of the local optimal during the iteration process.

A random leader or food source establishes relationship with the current individual location to obtain the location update formula of leader

x j i ( t ) = { x j i ( t − 1 ) + c 1 ( F j − x j i ( t − 1 ) ) , r 2 ≥ 1 2 x j i ( t − 1 ) + c 1 ( x j i ( t − 1 ) − x j r a n d ) , r 2 < 1 2 (14)

among them, x j r a n d is the position of the randomly selected leader in the j-th dimension. In this way, the influence of the food source on the current individual and the influence of other individuals in the population are considered, the exploration and exploitation process of the algorithm is also considered.

Adding Levy flight random number with small probability is beneficial to better jump out of local optimum. When P < P 0 , calculate the Levy flight [21] random number through Equation (7) to Equation (10), and obtain the leader position update formula to add Levy flight random number

x j i ( t ) = { ( x j i ( t − 1 ) + c 1 ( F j − x j i ( t − 1 ) ) ) ⋅ l , r 2 ≥ 1 2 ( x j i ( t − 1 ) + c 1 ( x j i ( t − 1 ) − x j r a n d ) ) ⋅ l , r 2 < 1 2 (15)

where, P 0 is the probability of occurrence, l is the Levy flight [21] random number.

Later, in the iteration, as the value of c₁ continues to decrease, SSA tends to fine tune the search, and to improve the search accuracy, a random step of Brownian motion is used. Equation (11) is the position update method of leader adopting Brownian motion

x j i = { ( F j + c 1 ( r 1 ( u b j − l b j ) + l b j ) ) ⋅ Brown , r 2 ≥ 1 2 ( F j − c 1 ( r 1 ( u b j − l b j ) + l b j ) ) ⋅ Brown , r 2 < 1 2 (16)

where, Brown is the random step of Brownian motion.

The crossover operator can generate new individuals, expand the diversity of the population, improve the search efficiency of the population, and better jump out of the local optimum. Generate random dimensions according to Equation (12), and apply the crossover operator to the leader.

According to Equation (13), the position updating method for the follower is derived.

x j i ( t ) = 1 2 ( x ¯ j + x j i ( t − 1 ) ) , (17)

where, x ¯ j is the mean position of the leaders calculated by Equation (13).

Followers are led purposefully, and the population is able to move faster towards the optimal food source, allowing the algorithm to converge more quickly.

Each iteration of the algorithm uses a greedy strategy to select the optimal fitness-valued individuals.

The flowchart of MSD-SSA is shown in Figure 3.

The pseudo code of the MSD-SSA is described as follows:

Time complexity is an important indicator of the amount of work required to run the algorithm and to evaluate the time consumption of the algorithm. The time complexity is usually denoted by O. The MSD-SSA algorithm consists of three parts: population initialization, updating the leader position and updating the follower position. Assume that the iteration number of the algorithm is T, the population size is N, and the dimension is D.

1) Initialize the population, which needs to be run ND time.

2) To calculate the individual fitness value and select the best individual as food, it needs to run N ( N − 1 ) 2 times.

3) The algorithm parameters are updated and need to be run 4 times.

4) Leaders need to run D times to update in space.

5) Followers update ( N − 1 ) D times in space.

6) Levy flight update needs to run 1 2 N D times.

7) Brownian motion update needs to run 1 2 N D times.

8) It takes ND time to find out the food source from the population and export it.

Each of the above operation units goes through T iteration, so the total time complexity of MSD-SSA is Equation (18)

O ( MSD-SSA ) = T [ N D + N ( N − 1 ) 2 + 4 + D + ( N − 1 ) D ] . (18)

The 10 benchmarks test functions [23] are shown in Table 1. The benchmark test functions used are all 30-dimensional, the population of the algorithm has 30 individuals and the algorithm is iterated 500 times. Comparison is made using (Wolf Pack Algorithm) WPA, SSA and MSD-SSA. The parameter settings for WPA are α = 4 , β = 6 , ω = 30 . The number of leaders and followers is half of the population respectively, the parameter value in MSD-SSA is P 0 = 0.2 , k = 0.2 . The test functions in Table 1 were solved using WPA, SSA and MSD-SSA respectively, and the python was used to make 30 independent experiments and compare the mean, standard deviation and convergence speed (time) of the three algorithms to avoid the bias caused by the randomness of the algorithms and to ensure the reasonableness of the algorithms. The experimental results are shown in Table 2 and the resulting convergence curves are plotted in Figure 4.

The CEC2017 test functions [20] are shown in Table 3. The test functions are in 10 and 30 dimensions, respectively. The population of the algorithm has 100 individuals, the number of leaders and followers is half of the population respectively, and the algorithm is iterated 1000 times independently. The parameter value in MSD-SSA is P 0 = 0.2 , k = 0.2 . The test functions in Table 3 were solved using MSD-SSA and SSA respectively, and the python was used to make 30

independent experiments and compare the mean, standard deviation and convergence speed (time) of the two algorithms to avoid the bias caused by the randomness of the algorithms and to ensure the reasonableness of the algorithms. Table 4 and Table 5 show the experimental results in 10 and 30 dimensions, respectively, and the obtained convergence curves are shown in Figure 5 and Figure 6, respectively.

For benchmarks test functions, the experimental results in Table 2 show that MSD-SSA has significant advantages over SSA in optimizing most functions. In terms of convergence precision, the convergence accuracy of MSD-SSA for f 1 , f 2 , f 3 , f 4 , f 6 , f 7 , f 8 , f 9 is significantly better than that of SSA. For f 5 and f 10 , the final convergence is not optimal, but the convergence results are greatly improved compared with SSA. In terms of algorithm robustness, except f 8 , the stability is better than SSA. Although the stability of f 8 is slightly weaker than

SSA, its standard deviation also has reaches 10⁻⁷. In terms of convergence speed, MSD-SSA converges to the optimal point faster than SSA.

For CEC2017 test functions, the CEC2017 test functions [20] include single and multi-modal problems, which can effectively evaluate the performance of algorithms. Table 4 tests 16 CEC2017 test functions [20] with 10-dimensions. The experimental results show that under the same environment, MSD-SSA has

a better convergence effect and faster convergence speed than SSA. For most CEC2017 test functions, MSD-SSA is more stable. However, Table 4 shows that the standard deviation of CEC10 and CEC12 is too large, indicating that the robustness of the improved algorithm is poor for CEC10 and CEC12.

Table 5 tests 16 CEC2017 test functions [20] with 10 dimensions. Compared with the CEC2017 10-dimensionals test function, the 30-dimensionals test function is more complex. According to Table 5, MSD-SSA is superior to SSA in terms of convergence accuracy, manipulation speed and robustness. However, the standard deviations of CEC10, CEC11, CEC12, CEC15 and CEC16 are too large, indicating that the robustness of MSD-SSA is poor and needs to be improved.

In summary, the improved algorithm has a stronger search capability and faster convergence to the target value for most functions.

In the era of big data, solving the constrained optimization problem is crucial to engineering. Although benchmark functional testing was addressed in the previous section, the issues in actual projects are within specific limitations. Optimize practical engineering problems using SSA and MSD-SSA, and verify the feasibility of MSD-SSA for practical engineering problems.

The goal of CBD [24] is to minimize the weight of cantilever beam and square section, see Figure 7. The variables of this problem are composed of five hollow square members. The CBD problem [24] is expressed as follows:

min f ( x ) = 0.0624 ( x 1 + x 2 + x 3 + x 4 + x 5 ) s .t . g ( x ) = 61 x 1 3 + 27 x 2 3 + 19 x 3 3 + 7 x 4 3 + 1 x 5 3 − 1 ≤ 0 0.01 ≤ x 1 , x 2 , x 3 , x 4 , x 5 ≤ 100 (19)

Do 30 independent experiments to verify the effectiveness of the algorithm. See Table 6 for the experimental results.

The purpose of TCSD [26] is to minimize the weight of the spring. This problem has three variables, namely wire diameter, mean coil diameter, and the number of dynamic coils. Equation (20) describes the TCSD [26] problem.

Do 30 independent experiments to verify the effectiveness of the algorithm. See Table 7 for the experimental results.

min f ( x ) = ( x 3 + 2 ) x 2 x 1 2 s .t . g 1 ( x ) = 1 − x 2 3 x 3 71785 x 1 4 ≤ 0 g 2 ( x ) = 4 x 2 2 − x 1 x 2 12566 ( x 2 x 1 3 − x 1 4 ) + 1 5180 x 1 2 ≤ 0 g 3 ( x ) = 1 − 140.45 x 1 x 2 3 x 3 ≤ 0 g 4 ( x ) = x 1 + x 2 1.5 − 1 ≤ 0 0.05 ≤ x 1 ≤ 2 , 0.25 ≤ x 2 ≤ 1.3 , 2 ≤ x 3 ≤ 15 (20)

The goal is to minimize the weight of a speed reducer so that the engine and propeller can rotate efficiently. This problem involves constraints on stresses in the shafts, transverse deflection of the shafts, surface stress and bending stress of the gear teeth (see Figure 8). Equation (21) describes the SRD problem [27] .

Do 30 independent experiments to verify the effectiveness of the algorithm. See Table 8 for the experimental results.

min f ( x ) = 0.7854 x 1 x 2 2 ( 3.3333 x 3 2 + 14.9334 x 3 − 43.0934 ) − 1.508 x 1 ( x 6 2 + x 7 2 ) + 7.4777 ( x 6 3 + x 7 3 ) + 0.7854 ( x 4 x 6 2 + x 5 x 7 3 ) s .t . g 1 ( x ) = 27 x 1 x 2 2 x 3 − 1 ≤ 0 , g 2 ( x ) = 397.5 x 1 x 2 2 x 3 2 − 1 ≤ 0 , g 3 ( x ) = 1.93 x 4 3 x 2 x 6 4 x 3 − 1 ≤ 0 , g 4 ( x ) = 1.93 x 5 3 x 2 x 7 4 x 3 − 1 ≤ 0 , g 5 ( x ) = [ ( 745 x 4 / x 2 x 3 ) 2 + 16.9 × 10 6 ] 1 / 2 110 x 6 3 − 1 ≤ 0 ,

g 6 ( x ) = [ ( 745 x 5 / x 2 x 3 ) 2 + 157.5 × 10 6 ] 1 / 2 85 x 7 3 − 1 ≤ 0 , g 7 ( x ) = x 2 x 3 40 − 1 ≤ 0 , g 8 ( x ) = 5 x 2 x 1 − 1 ≤ 0 , g 9 ( x ) = x 1 12 x 2 − 1 ≤ 0 , g 10 ( x ) = 1.5 x 6 + 1.9 x 4 − 1 ≤ 0 , g 11 ( x ) = 1.1 x 7 + 1.9 x 5 − 1 ≤ 0 2.6 ≤ x 1 ≤ 3.6 , 0.7 ≤ x 2 ≤ 0.8 , 17 ≤ x 3 ≤ 28 , 7.3 ≤ x 4 ≤ 8.3 , 7.3 ≤ x 5 ≤ 8.3 , 2.9 ≤ x 6 ≤ 3.9 , 5.0 ≤ x 7 ≤ 5.5 (21)

For the CBD problem [24] , the optimization results of MSD-SSA and SSA are 1.313731 and 1.339961, the optimal solution is shown in Table 6.

For the TCSD problem [26] , the optimization results of MSD-SSA and SSA are 0.0126917 and 0.0128271, the optimal solution is shown in Table 7.

For the SRD problem [27] , the optimization results of MSD-SSA and SSA are 2997.0889 and 3041.6166, the optimal solution is shown in Table 8.

The average operation time(s) obtained from the above practical engineering problems are as follows:

It can be seen from Table 9 that MSD-SSA takes less time to solve the same

practical engineering problems than SSA, which shows that the improved algorithm is superior to the original algorithm in terms of convergence speed.