An optimization method based on chaotic multi-strategy fusion to improve whale algorithm
By introducing improved Sinusoidal chaotic mapping, boundary detection, Chebyshev chaotic mapping and reverse learning strategies into the whale optimization algorithm, the problem that the algorithm is prone to fall into local optimality in stretching/compression spring design is solved, achieving more efficient and accurate optimization effects.
Patent Information
- Application Number
- CN202411593872.3
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2024-11-08
- Publication Date
- 2025-02-28
- Estimated Expiration
- 2044-11-08
AI Technical Summary
When solving the tensile/compression spring design problems, whale optimization algorithm is prone to falling into problems such as local optimal solutions, poor optimization accuracy, and slow convergence speed.
The improved whale algorithm (ICWOA) based on chaotic multi-strategy fusion is adopted to improve the initialization of populations by improving Sinusoidal chaotic mapping, improving boundary detection methods, and introducing Chebyshev chaotic mapping and reverse learning strategies to improve the algorithm's global search ability and convergence speed.
Effectively avoiding the algorithm from falling into local optimal solutions, improving the optimization accuracy and convergence speed, and significantly improving the optimization effect of tensile/compression spring design.
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Abstract
Description
Technical Field
[0001] The invention relates to an optimization method for improving a whale algorithm based on chaos multi-strategy fusion, and belongs to the technical field of mechanical engineering. Background Art
[0002] With the rapid development of science and technology, the complexity of engineering design and optimization problems is increasing. Traditional optimization algorithms often have difficulty in effectively solving these complex challenges. The design of tension / compression springs is very important in the field of engineering applications. The flexibility and versatility of springs make them an indispensable component of mechanical systems. They can optimize system performance, control vibration, store energy, and play a key role in various fields, thereby ensuring the stability and reliability of the system.
[0003] Due to the complexity of spring design and its profound impact on system performance, it is worthwhile to seek efficient and accurate optimization algorithms. Swarm intelligence optimization algorithms can solve optimization problems by simulating the behavioral habits of animal groups in nature. When dealing with tension / compression spring design problems, swarm intelligence optimization algorithms can automatically explore and approximate the optimal design parameter combination in a complex design space to meet engineering requirements and constraints, bringing higher efficiency and accuracy to spring design.
[0004] The Whale Optimization Algorithm (WOA) is a swarm intelligence optimization algorithm proposed by Australian scholars Mirjalili et al. in 2016 to simulate the humpback whale's predation behavior in nature. The algorithm is inspired by the humpback whale's unique predation strategy - the bubble net predation method. By simulating the behavior of humpback whales forming a bubble net around their prey and gradually shrinking the encirclement, it realizes the intelligent search of the solution space of the optimization problem. The WOA algorithm has the advantages of simple principle, few parameters, and strong optimization ability. However, like most swarm intelligence optimization algorithms, the WOA algorithm still has the disadvantages of being easy to fall into local optimal solutions, low convergence accuracy, slow convergence speed, and strong dependence on the initial population when solving complex engineering design problems. Therefore, in view of the specific needs of tension / compression spring design, it is necessary to further explore the improvement strategy of the whale optimization algorithm and improve the application effect of the whale optimization algorithm in spring design optimization. Summary of the invention
[0005] The technical problem to be solved by the present invention is as follows: In view of the shortcomings of the whale optimization algorithm, an optimization method for improving the whale algorithm based on chaotic multi-strategy fusion is provided to solve the problems that the tension / compression spring design method is prone to fall into the local optimal solution, has poor optimization accuracy, and has slow convergence speed.
[0006] The technical solution of the present invention is: the present invention proposes an optimization method based on the improved whale algorithm (ICWOA) of chaotic multi-strategy fusion. First, the Sinusoidal chaotic mapping is improved to initialize the population, which can effectively improve the quality and diversity of the initial population; then, the boundary detection method is improved so that individuals beyond the feasible solution range can be adjusted to a position closer to the current optimal solution, thereby accelerating the convergence speed of the algorithm and enabling the algorithm to approach the global optimal solution more quickly; then, the Chebyshev chaotic mapping is used to make the key parameter l chaotic, thereby improving the algorithm's exploration ability in the feasible solution space and increasing the chance of discovering a better solution; finally, the reverse learning strategy is introduced to increase the diversity of the population by generating new individuals that are opposite to the individuals in the current population, thereby avoiding the algorithm from falling into the local optimal solution, thereby exploring the solution space more comprehensively.
[0007] The specific steps of an optimization method based on chaotic multi-strategy fusion to improve the whale algorithm are as follows:
[0008] Step 1: Construct a mathematical model of the tension / compression spring design problem;
[0009] Step 1.1: Determine the variable parameters of the tension / compression spring design problem, which are the wire diameter d, the average coil diameter D, and the number of effective coils L;
[0010] Specifically, in the tension / compression spring design problem, the main optimization goal is to minimize the weight of the spring by finding the best combination of three variable parameters (wire diameter d, average coil diameter D, and number of effective coils L) under a series of constraints.
[0011] Step 1.2: Determine the objective function and constraints of the tension / compression spring design problem;
[0012] Specifically, the variable parameters wire diameter d, average coil diameter D and effective coil number L are respectively denoted as , under the constraints of minimum deflection, shear, impact frequency and outer diameter, minimizing the weight of the tension / compression spring is defined as the objective function, and the expression is:
[0013]
[0014] The constraints for the tension / compression spring design problem are:
[0015]
[0016] in, is the minimum deflection constraint, is the shear constraint, is the impact frequency constraint, is the outer diameter constraint, and the variable value range is .
[0017] Step 2: Use a variety of improvement strategies to improve the traditional whale optimization algorithm;
[0018] Step 3: Use the improved whale optimization algorithm to optimize the parameters of the mathematical model of the tension / compression spring design problem to obtain the global optimal solution and optimal parameter combination of the tension / compression spring design problem.
[0019] The location update strategy of the traditional WOA algorithm is as follows:
[0020] (1) Encirclement strategy:
[0021]
[0022]
[0023]
[0024]
[0025] in Indicates the current optimal whale position, Indicates the updated whale position, Indicates the current location of the whale. and is the coefficient, is the linear convergence factor, which decreases from 2 to 0 as the number of iterations decreases. is the maximum number of iterations, t is the current number of iterations, , is a random number between 0 and 1.
[0026] (2) Bubble net predation strategy:
[0027]
[0028] Where b is a constant that defines the shape of the logarithmic spiral, is a random number in the range (−1,1).
[0029] (3) Random search strategy:
[0030]
[0031] in Represents a random selection of a whale position vector.
[0032] The method of selecting the position update strategy during the algorithm iteration process: set a random number P between [0,1]. When P>=0.5, select the bubble net predation strategy to update the position; when P<0.5, judge according to the value of |A|. If |A|>=1, select the random search strategy for position update; if |A|<1, use the prey encirclement strategy for position update.
[0033] In step 2, the whale optimization algorithm is improved using multiple strategies as follows:
[0034] Improvement strategy 1: Introduce improved Sinusoidal chaotic mapping to initialize the population and improve the population quality and diversity;
[0035] The traditional WOA algorithm population initialization method is usually a random initialization method. This may lead to problems such as uneven initial distribution of the population and failure to fully cover the solution space, thus affecting the subsequent optimization effect of the algorithm. Chaotic mapping is a nonlinear dynamic system with characteristics such as nonlinearity, randomness, and ergodicity. Researchers usually apply chaotic mapping to cryptography, communication encryption, optimization algorithms and other fields. Sinusoidal chaotic mapping combines the characteristics of sinusoidal functions and nonlinear dynamic systems, and has periodicity and complexity. The expression of Sinusoidal chaotic mapping is:
[0036]
[0037] The parameter a is usually set to 2.3. The chaotic value distribution of the Sinusoidal chaotic map is mainly concentrated between [0.5, 0.9]. Therefore, it is necessary to improve the Sinusoidal chaotic map to make its distribution more uniform. The improved Sinusoidal chaotic map expression is:
[0038]
[0039] in, is the value of the nth iteration, is the value of the next iteration, and a is a control parameter.
[0040] Using the improved Sinusoidal chaotic mapping (I-Sinusoidal) to initialize the population helps increase the diversity of the population, thereby enhancing the exploration ability of individual whales in the population in the feasible solution space, effectively preventing the algorithm from falling into the local optimal solution, and improving the global search ability of the algorithm.
[0041] Improvement strategy 2: Improve the boundary detection method and increase the convergence speed of the algorithm;
[0042] In swarm intelligence optimization algorithms, boundary detection is a common processing method to ensure that the individual positions generated by the optimization algorithm are always within the solution space. When the position of a population individual exceeds the boundary of the solution space, a strategy is usually adopted to correct it to the boundary value. This avoids the negative impact of searching for positions outside the solution space on the performance of the optimization algorithm. However, the strategy of correcting the individual position to the boundary value simply ensures that the individual position does not exceed the solution space, without considering whether the boundary position of the solution space is beneficial to the optimization of the optimization algorithm. The boundary position of the solution space may not be the optimal solution, so simply correcting the individual position to the boundary value does not necessarily lead to the best optimization result.
[0043] The present invention improves the traditional boundary detection method so that individuals beyond the solution space can more effectively approach the optimal solution of the problem, thereby improving the convergence speed of the optimization algorithm. Specific improvement method: adjust the position of individuals beyond the upper limit of the solution space to a value between the current optimal solution and the upper limit of the solution space; adjust the position of individuals below the lower limit of the solution space to a value between the current optimal solution and the lower limit of the solution space.
[0044] Specific improved boundary detection formula:
[0045]
[0046] in, is the current optimal position, is the current individual position, is the corrected individual position, ub is the upper bound of the solution space, lb is the lower bound of the solution space, and r is a random number in [0,1]. Individuals that are beyond the scope of the solution space are randomly adjusted near the current optimal solution to help individuals get closer to the optimal solution.
[0047] Improvement strategy 3: Introducing Chebyshev chaotic mapping to improve the key parameters of the position update strategy in the whale optimization algorithm Perform chaos to improve the algorithm's ability to explore feasible solution space;
[0048] Parameters in the bubble net predation strategy of the WOA algorithm Plays an important role, parameters Controlling the size of the bubble net during whale hunting affects the search range and efficiency of the whale population in the solution space. With a larger search range, whales can search for potential optimal solutions in the solution space faster, but it also increases the randomness of the search. On the contrary, a smaller search range makes the search more localized and refined, but it may also cause the algorithm to fall into a local optimal solution.
[0049] By adjusting the value of parameter l, the performance and convergence speed of the WOA algorithm can be adjusted to achieve better optimization results. Chebyshev chaotic mapping has strong randomness and nonlinear characteristics. Integrating Chebyshev chaotic mapping into the value process of parameter l can make the changes of parameter l in iteration more diversified and random, thereby improving the algorithm's exploration ability in the solution space.
[0050] The Chebyshev chaotic mapping formula is:
[0051]
[0052] Among them, the parameters The value of is usually 4, and the chaotic value range of the Chebyshev chaotic map is [-1,1].
[0053] Improvement strategy 4: Introduce reverse learning strategy to increase population diversity and avoid the algorithm falling into local optimality;
[0054] In the process of finding the optimal solution, especially in the later stage of algorithm iteration, the WOA algorithm may focus on the local optimal solution and ignore the search for the global optimal solution, causing the algorithm to fall into the local optimum. The present invention introduces a reverse learning strategy to avoid the algorithm from falling into the local optimum and increase the diversity of the population. Reverse learning generation formula:
[0055]
[0056] in, is the position of the i-th whale individual, is the corresponding individual position of the whale after reverse learning, ub is the upper bound of the solution space, and lb is the lower bound of the solution space.
[0057] After the optimization algorithm completes the position update in each iteration, the reverse learning strategy is used to perform reverse learning on each individual in the population to generate N corresponding individuals, namely the reverse learning population. The fitness values of the whale individuals in the two populations are calculated, and the individuals with the best fitness values are selected to enter the next iteration. The reverse learning strategy can increase the population diversity of the algorithm, jump out of the local optimum, and select the best individual to enter the next iteration process, which improves the population quality of each iteration of the algorithm, thereby improving the convergence speed and optimization effect of the optimization algorithm.
[0058] In step 3, the improved whale optimization algorithm is used to optimize the parameters of the mathematical model of the tension / compression spring design problem, and the specific implementation steps of obtaining the global optimal solution and optimal parameter combination of the tension / compression spring design problem are as follows:
[0059] Step 3.1: Set the parameters of the ICWOA (Improved Whale Optimization Based on Chaotic Multi-Strategy Fusion) algorithm, including: solution space upper bound ub, solution space lower bound lb, population size PopSize, solution space dimension dim, and maximum number of iterations Maxiter. The solution space upper bound ub is the upper bound of the parameters in the tension / compression spring design problem, the solution space lower bound lb is the lower bound of the parameters in the tension / compression spring design problem, and the solution space dimension dim is the number of parameters in the tension / compression spring design problem.
[0060] Step 3.2: Use the improved Sinusoidal chaotic mapping formula to generate an initial population, and according to the population size PopSize, generate PopSize parameter combinations of the tension / compression spring design problem as the initial population of the whale optimization algorithm;
[0061] Step 3.3: Calculate the fitness value of each whale individual in the initialized population, that is, the value of the objective function in the tension / compression spring design problem, save the parameter combination of the tension / compression spring design that minimizes the value of the objective function as the optimal whale position, and save the minimum value of the objective function, that is, the optimal fitness value, as the current optimal solution of the whale;
[0062] Step 3.4: Generate a random number P and update the parameters A, C, and a in the location update strategy;
[0063] Step 3.5: Update the position. When P>=0.5, use the bubble net predation strategy to update the position, where the parameters are The Chebyshev chaotic map is used for iterative update. When P<0.5, the value of |A| is used for judgment. If |A|>=1, the position is updated according to the random search strategy. If |A|<1, the position is updated according to the prey encirclement strategy.
[0064] Step 3.6: Perform boundary detection on the positions of individuals in the population, and use the improved boundary detection method to update the positions of individuals that are beyond the solution space;
[0065] Step 3.7: Use the reverse learning strategy to update the population and perform boundary detection;
[0066] Step 3.8: Calculate the fitness value of the individual whale after the position update, and compare it with the optimal solution of the previous position update process. If the fitness value of the current whale position is better than the previous optimal solution, the fitness value of the new individual whale is used to replace the previous optimal solution.
[0067] Step 3.9: Repeat steps 3.4 to 3.8 until the number of iterations reaches the maximum number of iterations Maxiter, and output the whale individual corresponding to the optimal fitness value, that is, the optimal parameter combination for the tension / compression spring design problem.
[0068] Specifically, the suitability of the setting of the maximum number of iterations Maxiter is adjusted according to the specific problem and the convergence speed of the algorithm. By trying different settings of the number of iterations and observing the performance of the algorithm on the training set and the validation set, the number of iterations that enables the algorithm to converge to a better solution within a reasonable time can be selected.
[0069] The beneficial effects of the present invention are as follows: by improving the multi-strategy of the whale optimization algorithm, the tension / compression spring design problem can be solved more efficiently and accurately. The present invention initializes the whale population by using the improved Sinusoidal chaotic mapping, increases the population diversity, improves the quality of the initial population, and avoids the algorithm from falling into the local optimum. The boundary detection method is improved so that individuals beyond the feasible solution space are close to the current optimal solution, thereby improving the algorithm convergence speed. The Chebyshev chaotic mapping is introduced to replace the random parameter l, thereby improving the algorithm's stability and exploration ability and improving the algorithm's optimization performance. The reverse learning strategy is used to increase the population diversity, avoid the algorithm from falling into the local optimum, and at the same time improve the algorithm convergence speed. BRIEF DESCRIPTION OF THE DRAWINGS
[0070] Figure 1 It is a schematic diagram of the structure of the present invention;
[0071] Figure 2 is a schematic diagram of the design of the tension / compression spring of the present invention;
[0072] Figure 3 It is the Sinusoidal chaotic mapping distribution diagram and histogram diagram of the present invention. Among them, Figure (a) is the Sinusoidal chaotic mapping distribution; Figure (b) is the Sinusoidal chaotic mapping histogram; Figure (c) is the I-Sinusoidal chaotic mapping distribution; Figure (d) is the I-Sinusoidal chaotic mapping histogram;
[0073] Figure 4 It is the Sinusoidal chaotic map Lyapunov and bifurcation diagram of the present invention. Among them, Figure (a) is the Sinusoidal chaotic map Lyapunov; Figure (b) is the Sinusoidal chaotic map bifurcation diagram; Figure (c) is the I-Sinusoidal chaotic map Lyapunov; Figure (d) is the I-Sinusoidal chaotic map bifurcation diagram.
[0074] Figure 5 It is a comparison diagram of the convergence curves of different algorithms of the present invention under the benchmark function. DETAILED DESCRIPTION
[0075] The present invention will be further described below in conjunction with the accompanying drawings and specific implementation methods.
[0076] Example 1: Figure 1 As shown, a stretch spring optimization method based on the optimization method of the improved whale algorithm by fusion of chaotic multi-strategy is shown, and the steps are as follows:
[0077] Step 1: Determine the parameters of the tension / compression spring design problem; In the tension / compression spring design problem, the main optimization goal is to minimize the weight of the spring by finding the best combination of three variable parameters (wire diameter d, average coil diameter D, and number of active coils L) under a series of constraints.
[0078] Step 2: Determine the objective function and constraints of the tension / compression spring design problem; the design diagram of the tension / compression spring is as follows Figure 2 As shown. Let the variables d, D, and L be The mathematical model of the tension / compression spring design problem is as follows:
[0079] The target variable, i.e. the individuals of the whale population, is: (1)
[0080] Objective function: (2)
[0081] Constraints: (3)
[0082] in, is the minimum deflection constraint; is the shear constraint; is the impact frequency constraint; The outer diameter constraint.
[0083] The variable value range is .
[0084] Step 3: Build an improved whale optimization algorithm using multiple strategies;
[0085] Improvement strategy 1. Chaotic mapping is a nonlinear dynamic system with the characteristics of nonlinearity, randomness, and ergodicity. Sinusoidal chaotic mapping combines the characteristics of sine function and nonlinear dynamic system, and has periodicity and complexity. The chaotic value distribution of Sinusoidal chaotic mapping is mainly concentrated between [0.5, 0.9]. Therefore, it is necessary to improve Sinusoidal chaotic mapping to make its distribution more uniform. The improved distribution effect is as follows: Figure 3 The improved Sinusoidal chaotic mapping expression is:
[0086] (4)
[0087] Figure 4 It shows that the improved Sinusoidal chaotic mapping has better chaotic characteristics and helps to increase the diversity of the population. It enhances the exploration ability of individual whales in the population in the feasible solution space, effectively avoids the algorithm from falling into the local optimal solution, and improves the global search ability of the algorithm.
[0088] Improvement strategy 2: In swarm intelligence optimization algorithms, boundary detection is a common processing method used to ensure that the individual positions generated by the optimization algorithm are always within the solution space. This invention will improve the traditional boundary detection method so that individuals beyond the solution space can more effectively approach the optimal solution of the problem, thereby improving the convergence speed of the optimization algorithm. Specific improved boundary detection formula:
[0089] (5)
[0090] in, is the current optimal position, is the current individual position, is the corrected individual position, ub is the upper bound of the solution space, lb is the lower bound of the solution space, and r is a random number in [0,1]. Individuals that are beyond the scope of the solution space are randomly adjusted near the current optimal solution to help individuals get closer to the optimal solution.
[0091] Improvement strategy three, by adjusting the value of parameter l, adjust the performance and convergence speed of the WOA algorithm to achieve better optimization results. Chebyshev chaotic mapping has strong randomness and nonlinear characteristics. Integrating Chebyshev chaotic mapping into the value process of parameter l can make the change of parameter l in iteration more diversified and random, thereby improving the algorithm's exploration ability in the solution space.
[0092] The Chebyshev chaotic mapping formula is:
[0093] (6)
[0094] Among them, the parameters The value of is usually 4, and the chaotic value range of the Chebyshev chaotic map is [-1,1].
[0095] Improvement strategy 4: The present invention introduces a reverse learning strategy, and the reverse learning generation formula is:
[0096] (7)
[0097] in, is the position of the i-th whale individual, is the corresponding individual position of the whale after reverse learning, ub is the upper bound of the solution space, and lb is the lower bound of the solution space.
[0098] After the optimization algorithm completes the position update in each iteration, the reverse learning strategy is used to perform reverse learning on each individual in the population to generate N corresponding individuals, namely the reverse learning population. The fitness values of the whale individuals in the two populations are calculated, and the individuals with the best fitness values are selected to enter the next iteration. The reverse learning strategy can increase the population diversity of the algorithm, jump out of the local optimum, and select the best individual to enter the next iteration process, which improves the population quality of each iteration of the algorithm, thereby improving the convergence speed and optimization effect of the optimization algorithm.
[0099] Step 4: Use the improved whale optimization algorithm to optimize the parameters of the mathematical model of the tension / compression spring design problem, so as to obtain the global optimal solution and optimal parameter combination of the tension / compression spring design problem. The specific process of solving is:
[0100] Step 4.1: Set the parameters of an optimization method based on the improved whale algorithm based on chaotic multi-strategy fusion: population size PopSize, solution space dimension dim (the number of parameters in the tension / compression spring design problem), solution space upper bound ub (the upper bound of each parameter in the tension / compression spring design problem), solution space lower bound lb (the lower bound of each parameter in the tension / compression spring design problem) and maximum number of iterations Maxiter.
[0101] Step 4.2: In each iteration, the first individual in each population is randomly generated, and then the initial population of whales is iteratively generated according to the improved Sinusoidal chaotic map initialization formula (4). Each whale individual represents a possible combination of spring design parameters. In a population size of PopSize and a search space of dim dimension, the population is denoted by X, where X is a matrix of PopSize rows and dim columns, each row represents a whale individual, the first column of X represents parameter d, the second column represents parameter D, and the third column represents parameter L. The position of the i-th whale in the solution space can be represented by .
[0102] Step 4.3: Substitute the data of each individual in the population into formula (2) to obtain the objective function value of the corresponding individual, and store the objective function value of each individual in the population in fitness. Determine the optimal objective function value in fitness, record it as GbestScore, and take the individual corresponding to the optimal objective function value as the optimal individual, record it as GbestPositon.
[0103] Step 4.4: Update the position of the whale according to the specified iteration strategy. Let t be the current iteration number, and t is increased by 1 after each iteration. The specific iteration process is:
[0104] Step 4.4.1: Update the parameters a, A, C using formulas (8), (9), (10), and use the Chebyshev chaotic mapping formula (6) to update the value of parameter l.
[0105] (8)
[0106] (9)
[0107] (10)
[0108] Step 4.4.2: Randomly generate a random number P between [0,1]. When P>=0.5, select the bubble net predation strategy (12) to update the position. When P<0.5, make a judgment based on the value of |A|. If |A|>=1, select the random search strategy (13) to update the position. If |A|<1, use the encirclement strategy (11) to update the position.
[0109] Surrounding prey strategy: (11)
[0110] Bubble net predation strategy: (12)
[0111] Random search strategy: (13)
[0112] Step 4.4.3: After the position is updated, use formula (5) to perform boundary detection on the whale individuals in the population. Then use the reverse learning strategy formula (7) to generate a reverse learning population. Calculate the fitness values of the whale individuals in the generated reverse learning population and the original population, and select the individuals with the best fitness values of the first PopSize to enter the next iteration.
[0113] Calculate the fitness value of individual whales in the whale population, determine the optimal objective function value, update GbestScore, and take the individual corresponding to the optimal objective function value as the optimal individual, and update GbestPositon.
[0114] Step 4.5: Stop the iteration when the set number of iterations is reached. At this time, the value of GbestPositon is the global optimal solution, that is, the solved tension / compression spring design parameters.
[0115] The tension / compression spring design solution parameters of the present invention and the traditional whale optimization algorithm were used respectively, and the parameters were independently run 30 times, the number of parameter populations was 50, and the maximum number of iterations was 500. The solution data is shown in Table 1:
[0116] Table 1 Statistical results of tension / compression springs
[0117]
[0118] The experimental results in Table 1 show that the optimization performance of the ICWOA algorithm in the tension / compression spring design problem is better than that of the original WOA algorithm and is more stable.
[0119] Table 2 Solution results of tension / compression spring
[0120]
[0121] From the analysis of the data in Table 2, it can be seen that both methods can meet the parameter range and the four constraints involved in the tension / compression spring. However, the objective function value 0.01272302 obtained by the present invention is less than the objective function value 0.01286709 obtained by using the existing whale optimization algorithm. It can be seen that the improved whale optimization algorithm parameter solution accuracy for the tension / compression spring of the present invention is higher than that of the existing whale optimization algorithm.
[0122] In order to verify the performance of the IWOA algorithm, 15 benchmark functions were selected for optimization. First, the IWOA algorithm was compared with a variety of swarm intelligence optimization algorithms to verify the superiority of the IWOA algorithm. Then, it was compared with different improvement strategies to verify the effectiveness of the improvements in the IWOA algorithm.
[0123] Three different types of benchmark functions were selected, among which F1~F5 are unimodal benchmark functions (Table 3), F6~F10 are multimodal benchmark functions (Table 4), and F11~F15 are fixed-dimensional multimodal benchmark functions (Table 5). The optimization performance of the ICWOA algorithm was verified by different function types.
[0124] Table 3 Unimodal benchmark functions
[0125]
[0126] Table 4 Multi-peak benchmark function
[0127]
[0128] Table 5 Fixed-dimensional multimodal benchmark functions
[0129]
[0130] The improved ICWOA algorithm is compared with the whale optimization algorithm (WOA), particle swarm optimization algorithm (PSO), cuckoo search algorithm (CS), genetic algorithm (GA), sine cosine optimization algorithm (SCA) and differential evolution algorithm (DE). In order to ensure the fairness and effectiveness of the experimental results, the common parameters of the algorithms are first unified, the population size (PopSize=50) is set to 50, and the maximum number of iterations (Maxiter = 500) is set to 500. Secondly, each algorithm is run independently for 30 times, and the results of 30 operations are recorded and the optimal value, average value and standard deviation are statistically used as the evaluation indicators of the performance of each algorithm. The parameter settings of each selected swarm intelligence optimization algorithm are shown in Table 6.
[0131] Table 6 Algorithm parameter settings
[0132]
[0133] Table 7 shows the statistics of the ICWOA algorithm and the original WOA algorithm, PSO, CS, GA, SCA and DE algorithms running 30 times independently on 15 benchmark functions. From Table 7, it can be seen that for the single-peak benchmark functions (F1~F5), the optimization performance of the ICWOA algorithm is better than other optimization algorithms in terms of optimal value, average value or standard deviation, indicating that the ICWOA algorithm has a higher improvement in the optimization performance of single-peak benchmark functions compared with other swarm intelligence optimization algorithms. Although the optimization effect of the ICWOA algorithm on function F6 is slightly lower than that of the CS algorithm, the optimization effect of the ICWOA algorithm is still slightly better than that of other swarm intelligence optimization algorithms. Since the experimental data is only retained to three significant digits, the optimization performance of the ICWAO algorithm on function F11 does not show significant differences compared with the original WOA algorithm, PSO, CS, GA, SCA and DE algorithms. On functions F13 and F15, the theoretical optimal value of the function can be found accurately and stably. On functions F12 and F14, the average values show good optimization effects. For function F12, PSO and DE algorithms fell into local optimal solutions during multiple independent runs. Excluding PSO and DE algorithms, the minimum index of ICWOA algorithm is slightly lower than that of SCA algorithm, but the average value of ICWOA algorithm is higher than that of other optimization algorithms, and has a better standard deviation. In general, ICWAO algorithm has improved the optimization effect of multi-peak benchmark functions compared with other swarm intelligence optimization algorithms.
[0134] Table 7 Comparison of ICWOA algorithm and different optimization algorithms
[0135]
[0136] Figure 5The convergence curves of eight optimization algorithms on 15 functions are shown. By comparing the convergence curves of different algorithms on different functions, it can be seen that the IWOA algorithm has shown relatively strong convergence effects on most benchmark functions. However, the convergence speed of the IWOA algorithm on the three fixed-dimensional multi-peak benchmark functions of F11, F13 and F14 has not been significantly improved. This may be because the dimensions of the three functions F11, F13 and F14 are relatively low, resulting in the other algorithms selected in the experiment being able to converge quickly and achieve higher optimization effects.
[0137] In order to further study the effectiveness of each improvement strategy in the IWOA algorithm and the impact of each improvement on the IWOA algorithm, a single improvement strategy is selected to improve the original WOA algorithm and compared with the original WOA algorithm and the ICWOA algorithm on 15 benchmark functions. The WOA improved algorithms with a single improvement strategy are: WOA based on improved chaotic map initialization (WOA-I), WOA based on edge detection at the optimal position (WOA-B), WOA based on parameter chaos (WOA-L), and WOA based on reverse learning strategy to diversify the population (WOA-M). The algorithm parameters are consistent with the original WOA algorithm, the population size is 50, and the maximum number of iterations is 500. Under one function, each algorithm runs independently 30 times. Table 8 shows the statistical results of the five improved optimization algorithms and the original WOA algorithm running independently 30 times on 15 benchmark functions.
[0138] Table 8 Optimization results of different improvement strategies
[0139]
[0140] It can be seen from Table 8 that the improvement strategy (WOA-M) that introduces the reverse learning strategy plays a decisive role among the various improvement strategies, and the improvement method (WOA-L) that introduces the parameter chaos strategy is slightly inferior. At the same time, other improvement strategies also play an auxiliary role in algorithm improvement. However, a single improvement strategy has different limits on the improvement effect of the optimization performance of different benchmark functions, and cannot guarantee that the optimization performance under different functions can be effectively improved. The experimental results show that the ICWOA algorithm that introduces four improvement strategies at the same time shows a better optimization effect overall.
[0141] The specific implementation modes of the present invention are described in detail above in conjunction with the accompanying drawings, but the present invention is not limited to the above implementation modes, and various changes can be made within the knowledge scope of ordinary technicians in this field without departing from the purpose of the present invention.
Claims
1. An optimization method based on chaotic multi-strategy fusion to improve the whale algorithm, characterized by: Step 1: Construct a mathematical model of the tension / compression spring design problem; Step 2: Use a variety of improvement strategies to improve the traditional whale optimization algorithm; Step 3: Use the improved whale optimization algorithm to optimize the parameters of the mathematical model of the tension / compression spring design problem to obtain the global optimal solution and optimal parameter combination of the tension / compression spring design problem; The step 1 is specifically as follows: Step 1.1: Determine the variable parameters of the tension / compression spring design problem, which are the wire diameter d, the average coil diameter D, and the number of effective coils L; Step 1.2: Determine the objective function and constraints of the tension / compression spring design problem; The step 2 is specifically as follows: Improvement strategy 1: Introduce the improved Sinusoidal chaotic mapping to initialize the population, improve the population quality and population diversity. The improved Sinusoidal chaotic mapping expression is: ； in, is the value of the nth iteration, is the value of the next iteration, and a is a control parameter; Improvement strategy 2: Improve the boundary detection method and increase the convergence speed of the algorithm. The specific improved boundary detection formula is: ； in, is the current optimal position, is the current individual position, is the corrected individual position, ub is the upper bound of the solution space, lb is the lower bound of the solution space, and r is a random number in [0,1]; Improvement strategy 3: Introducing Chebyshev chaotic mapping to improve the key parameters of the position update strategy in the whale optimization algorithm Chaos is performed to improve the algorithm's exploration ability in the feasible solution space. The Chebyshev chaos mapping formula is: ； Among them, β is the order; Improvement strategy 4: Introduce reverse learning strategy to increase population diversity and avoid the algorithm falling into local optimality. The reverse learning generation formula is: ； in, is the position of the i-th whale individual, is the corresponding individual position of the whale after reverse learning, ub is the upper bound of the solution space, and lb is the lower bound of the solution space; The step 3 is specifically as follows: Step 3.1: Set the ICWOA algorithm parameters, including: solution space upper bound ub, solution space lower bound lb, population size PopSize, solution space dimension dim, and maximum number of iterations Maxiter. The solution space upper bound ub is the upper bound of the parameters in the tension / compression spring design problem, the solution space lower bound lb is the lower bound of the parameters in the tension / compression spring design problem, and the solution space dimension dim is the number of parameters in the tension / compression spring design problem. Step 3.2: Use the improved Sinusoidal chaotic mapping formula to generate an initial population, and according to the population size PopSize, generate PopSize parameter combinations of the tension / compression spring design problem as the initial population of the whale optimization algorithm; Step 3.3: Calculate the fitness value of each whale individual in the initialized population, that is, the value of the objective function in the tension / compression spring design problem, save the parameter combination of the tension / compression spring design that minimizes the value of the objective function as the optimal whale position, and save the minimum value of the objective function, that is, the optimal fitness value, as the current optimal solution of the whale; Step 3.4: Generate a random number P and update the parameters A, C, and a in the location update strategy; Step 3.5: Update the position. When P>=0.5, use the bubble net predation strategy to update the position, where the parameters are The Chebyshev chaotic map is used for iterative update. When P<0.5, the value of |A| is used for judgment. If |A|>=1, the position is updated according to the random search strategy. If |A|<1, the position is updated according to the prey encirclement strategy. Step 3.6: Perform boundary detection on the positions of individuals in the population, and use the improved boundary detection method to update the positions of individuals that are beyond the solution space; Step 3.7: Use the reverse learning strategy to update the population and perform boundary detection; Step 3.8: Calculate the fitness value of the individual whale after the position update, and compare it with the optimal solution of the previous position update process. If the fitness value of the current whale position is better than the previous optimal solution, the fitness value of the new individual whale is used to replace the previous optimal solution. Step 3.9: Repeat steps 3.4 to 3.8 until the number of iterations reaches the maximum number of iterations Maxiter, and output the whale individual corresponding to the optimal fitness value, that is, the optimal parameter combination for the tension / compression spring design problem.
2. According to claim 1, the optimization method based on chaotic multi-strategy fusion improved whale algorithm is characterized in that: The step 1.2 is specifically as follows: the variable parameters of the wire diameter d, the average coil diameter D and the number of effective coils L are respectively recorded as , under the constraints of minimum deflection, shear, impact frequency and outer diameter, minimizing the weight of the tension / compression spring is defined as the objective function, and the expression is: ； The constraints for the tension / compression spring design problem are: ； in, is the minimum deflection constraint, is the shear constraint, is the impact frequency constraint, The outer diameter constraint.