CN117688966A - Improved dung beetle optimization algorithm - Google Patents

Abstract

The invention discloses an improved dung beetle optimization algorithm, which comprises the following steps: the position of the rolling ball dung beetle is updated by adopting an improved sine algorithm position updating formula at the position of all dung beetles updated by the dung beetle optimizing algorithm, global exploration and local development capabilities of the dung beetle MSA are given, the search range is enlarged, the global exploration capability is improved, and the possibility of sinking into local optimum is reduced.

Description

Improved dung beetle optimization algorithm

Technical Field

The invention belongs to the technical field of optimization algorithms, and relates to an improved dung beetle optimization algorithm.

Background

Optimization problems have long been the focus of research. By optimization problem is meant that under certain conditions, the optimal solution or parameter is found among a plurality of solution or parameter values to optimize one or more functions, the optimization problem being that the best solution is found in all possible cases. Optimization problems exist in a variety of real-world fields, including fault diagnosis, path planning, aerospace, and various military industries. With the continuous and deep development of science and technology, the difficulty and complexity of engineering optimization problems in various fields are increasing, for example, in practical engineering, many design applications need to obtain an optimal solution in a short time and under a highly complex constraint, which is a very challenging task, so that a more efficient solution algorithm is needed. Through observation of natural physical laws and biological habits, various meta-heuristics have been developed and used for various optimization problems, which can find optimal solutions for various challenging and complex optimization problems, and which have the ability to solve complex and multidimensional problems in a short time.

Various meta-heuristic algorithms have been studied so far to solve optimization problems in the real world, such as genetic algorithms (Genetic Algorithm, GA), particle swarm optimization (Particle Swarm Optimization, PSO), differential evolution (Differential evolution, DE), gray-wolf optimization algorithms (Grey Wolf Optimizer, GWO), whale optimization algorithms (Whale Optimization algorithm, WOA), butterfly optimization algorithms (butterfly optimization algorithm, BOA) [15], sparrow Search Algorithms (SSA), chimpanzee optimization algorithms (Chimp Optimization Algorithm, chOA), arithmetic optimization algorithms (Arithmetic Optimization Algorithm, AOA), artificial jellyfish search algorithms (artificial Jellyfish Search optimizer, JS). Compared with the traditional optimization technology, the algorithms show better performance in practical application, and the meta-heuristic optimization algorithm has the advantages of strong stability, easiness in implementation and the like, and also has better performance in solving complex optimization problems. However, to date, researchers have used only very limited naturally inspired features, and there is room for many algorithms to develop.

The dung beetle optimizer (Dung beetle optimizer, DBO) algorithm is a novel group intelligent optimization algorithm proposed by XUE J equal to 2023, and the algorithm inspiration is derived from social behaviors of the dung beetle group, namely rolling, dancing, foraging, stealing and propagation behaviors of the dung beetles. Compared with classical algorithms such as particle swarm optimization algorithm and genetic algorithm, the dung beetle optimization algorithm divides different survival tasks according to the division of the dung beetle population, wherein the population is divided into four types of dung beetles, namely rolling ball dung beetles, breeding dung beetles (breeding balls), small dung beetles and stealing dung beetles. To further illustrate the practical application potential of the DBO algorithm, XUE J et al successfully applied the DBO algorithm to a number of engineering design issues. The experimental result shows that the proposed DBO algorithm can effectively treat practical application problems. In general, the DBO algorithm combines global exploration and local development, and has the characteristics of high convergence rate and high solving precision.

However, no Free Luncheon (NFL) theorem has logically demonstrated that none of the meta-heuristic algorithms are suitable for solving all optimization problems. This motivates us to develop new meta-heuristics to solve different problems. The dung beetle optimizing algorithm has the characteristics of strong optimizing capability and high convergence rate. However, the method has the defects that the global exploration and local development capability are unbalanced, the local optimization is easy to fall into, and the global exploration capability is weak.

Disclosure of Invention

The invention aims to solve the technical problems that: an improved dung beetle optimization algorithm is provided to solve the technical problems in the prior art.

The technical scheme adopted by the invention is as follows: an improved dung beetle optimization algorithm, the method comprising the steps of:

1) Initializing a dung beetle group and dung beetle optimizing algorithm parameters;

2) Calculating the fitness value of all the dung beetles according to the objective function;

3) Updating the positions of all dung beetles: judging whether the fitness value ST in the step 2) is larger than a threshold delta or not, and if the fitness value ST is larger than the threshold delta, updating the position of the rolling ball dung beetle by adopting a position updating mathematical model of the rolling ball dung beetle; if ST is less than or equal to delta, adopting an improved sine algorithm position updating formula to update the position of the ball dung beetle;

4) Judging whether each dung beetle is out of the boundary;

5) Updating the current optimal solution and the fitness value thereof;

6) Repeating the steps 3) -5) until the iteration times t meet the termination criterion, and outputting the global optimal solution and the fitness value thereof.

Further, the modified sine algorithm location update formula in step 3) above is as follows:

x _i (t+1)＝ω _t x _i (t)+r ₁ ×sin(r ₂ )×[r ₃ p _i (t)-x _i (t)] (9)

wherein t is the current iteration number omega _t Is the inertial weight, x _i (t) is the ith position component, p, of individual X in the t-th iteration _i (t) the ith component of the optimal individual position variable in the t-th iteration, r ₁ As a nonlinear decreasing function, r ₂ Is interval [0,2 pi ]]Random number on r ₃ Is the interval [ -2,2]A random number on the table;

wherein r is ₁ Is set using a non-linear decreasing pattern and a cosine function between 0 and pi is used to determine r ₁ Value change:

wherein omega is _max And omega _min Represents ω _t Maximum and minimum of ω _max ＝0.8，ω _min =0.75678, T represents the current iteration number, T _max Representing a maximum number of iterations;

inertial weight omega _t An adaptive variable inertia weight strategy is adopted, wherein the inertia weight is linearly reduced along with the increase of the iteration times:

the sine and cosine guiding mechanism is introduced, the improved sine algorithm is used as a strategy for replacing tangential dancing of the dung beetles to embed the dung beetle optimizing algorithm, namely, the whole individual dung beetles are subjected to sinusoidal operation in the rolling stage to guide the individual dung beetles to be updated to a new dung beetle position, and the improved formula is as follows:

wherein ST.epsilon.is (0.5, 1).

Further, initializing the dung beetle group in the step 1) and initializing individual positions of the dung beetles by using Bernoulli mapping.

Further, in the step 5), the current optimal solution is perturbed by adopting an adaptive gaussian-cauchy hybrid variation perturbation strategy to generate a new solution, and whether to perform position update is determined according to a greedy rule.

Further, the adaptive gaussian-cauchy hybrid variation perturbation strategy is: the optimal individual is mutated, the positions before and after mutation are compared, and the optimal position is selected to be substituted into the next iteration, and the specific formula is as follows:

H ^b (t)＝X ^b (t)*(1+μ ₁ *Gauss(σ)+μ ₂ *cauchy(σ)) (13)

wherein mu is ₁ ＝t/T _max ，μ ₂ ＝1-t/T _max Is X ^b (t) optimal position of individual X in the t-th iteration, H ^b (t) is the optimal position X in the t-th iteration ^b (t) at a position after Gaussian-cauchy mixture disturbance, gauss (sigma) is a Gaussian mutation operator, cauchy (sigma) is a cauchy mutation operator;

and (3) introducing a greedy rule, determining whether to update the position by comparing the fitness values of the new position and the old position, wherein the greedy rule is shown in a formula (14), and f (x) represents the fitness value of the x position:

the invention has the beneficial effects that: compared with the prior art, the invention has the following effects:

1) The invention explores the global in MSA (modified sine algorithm)And the local development capability is given to the dung beetles, the sine function in the MSA is adopted for iterative optimization, the global exploration capability is stronger, and meanwhile, the self-adaptive variable inertia weight coefficient omega is introduced in the position updating process of the MSA _t The algorithm can fully search the local area, so that the global exploration and the local development capability are well balanced;

2) Introducing a sine and cosine guiding mechanism to further optimize the capacity of the DBO algorithm to coordinate global exploration and local development;

3) The introduction of the sine guiding mechanism is improved, on one hand, the blindness defect of a DBO algorithm position updating strategy can be greatly overcome, because sine operation is adopted, when the direction cannot be determined by the dung beetles, the dung beetles can conduct information communication with the current optimal individual, the quick transmission of information in a population is promoted, each dung beetle individual can fully absorb the position difference information between the dung beetles and the optimal individual so as to guide the general individual to gradually approach the optimal individual, and the defect that the original algorithm lacks inter-individual information communication is overcome; on the other hand, aiming at the problem that the original algorithm is easy to fall into local optimum, the MSA guiding mechanism enables the dung beetle individual to freely perform global exploration and local optimization within the area range given by the algorithm, so that the search space is enlarged to a certain extent, and the dung beetle individual gradually converges on the same optimum solution, namely the objective function value, thereby improving the global optimizing capability of the algorithm. Also as can be seen from formula (10), r ₁ The search distance and direction of the dung beetles are controlled, the optimizing mode of the DBO algorithm is optimized, and the self-adaptive coefficient omega can be seen from the formula (11) _t The search space is gradually reduced, and the inertia weight is reduced along with the increase of the iteration times. The larger inertia weight in the initial stage of algorithm iteration can ensure that the algorithm keeps better global exploration capacity, and the smaller inertia weight in the later stage of the iteration is beneficial to improving the local development capacity of the algorithm. Therefore, the convergence accuracy of the algorithm can be improved, and the convergence speed of the algorithm can be accelerated.

4) The initial stage of DBO uses chaotic mapping to generate a highly diversified initial population, the diversity of the initial solution is improved, the traversing uniformity and the convergence speed of Bernoulli mapping are suitable for being used as the initialization of the chaotic population, and the Bernoulli mapping can be used as a chaotic sequence for generating an optimization algorithm through strict mathematical reasoning;

5) The self-adaptive Gaussian-Kexie hybrid disturbance variation disturbance strategy provided by the invention has uncertainty on the result of variation operation, and if variation operation is carried out on all individuals, the calculated amount is necessarily increased, and the searching efficiency of an algorithm is reduced; the optimal individuals are subjected to mutation, the positions before and after the mutation are compared, the position with the higher priority is selected to be substituted into the next iteration, on one hand, the information of the optimal individuals is fully utilized, on the other hand, the population diversity is increased, and the search range is enlarged;

6) In the iterative process of the algorithm, the individual dung beetles are disturbed according to the formula (13), and in the early stage of iterative process of the algorithm, the individual position distribution gap is larger because of uneven population distribution, so the algorithm mainly carries out larger-amplitude disturbance on the individual through the Cauchy distribution function, thereby generating various individuals, fully utilizing the current position information, adding random interference information, facilitating the algorithm to jump out of local optimum, and enabling the algorithm to carry out global exploration; with the continuous progress of algorithm iteration, most individual positions of the dung beetles cannot change too much, at the moment, the algorithm mainly perturbs the population through Gaussian distribution function coefficients, so that the algorithm is helped to converge rapidly and develop deeply, and meanwhile, the inter-dimension interference problem under the high-dimension problem is solved. In summary, the adaptive Gaussian-Cauchy hybrid variation perturbation strategy fully utilizes the characteristics of Cauchy and Gaussian distribution functions to generate new individuals, enhances the diversity of the population, and helps the algorithm coordinate the ability of local development and global exploration.

7) After the variation disturbance update is carried out, a greedy rule is introduced, so that the problem that the disturbance strategy cannot determine the adaptability value of the new position obtained after disturbance variation is superior to that of the original position can be effectively solved.

In summary, the algorithm provided by the invention can accelerate the convergence speed, improve the convergence precision, and find global optimum instead of local optimum.

Drawings

FIG. 1 is a conceptual model diagram of a boundary selection strategy;

FIG. 2 is a graph of the scale distribution of each search agent in the DBO algorithm;

FIG. 3 is a Bernoulli mapping chaotic sequence diagram;

FIG. 4 is a MSADBO flow chart;

FIG. 5 is a schematic diagram of a pressure vessel design problem.

Detailed Description

The invention will be further described with reference to specific examples.

The concept of the dung beetle optimization algorithm is derived from the inspiration of rolling balls, dancing, foraging, breeding and stealing behavior of dung beetles, and five different updating rules are designed to help find out that a high-quality solution seeker has higher energy reserve. Each dung beetle group consists of four different proxy dung beetles, namely rolling ball dung beetles, breeding dung beetles (breeding balls), small dung beetles and stealing dung beetles.

1. Rolling ball dung beetle

The dung beetles have an interesting habit, the faeces are formed into balls, then the balls are rolled to an ideal position, and in the rolling process, the dung beetles need to keep the faeces balls rolling on a straight line through celestial bodies. To simulate rolling ball behavior, the dung beetles need to move in a given direction throughout the search space. During the rolling process, the position of the dung beetle of the rolling ball is updated, and the rolling mathematical model can be expressed as:

wherein t represents the current iteration number, x _i (t) represents the position information of the ith dung beetle at the t iteration, k E (0,0.2 represents a constant representing a deflection coefficient, b represents a constant belonging to, alpha is a natural coefficient, and is assigned to be-1 or 1 (see algorithm 1), X) ^w Representing the global worst position, deltax is used to simulate the change in light intensity.

When the dung beetle encounters an obstacle and cannot advance, the direction of the dung beetle needs to be adjusted through dancing to obtain a new route, and a tangential function is used for obtaining a new rolling direction in order to simulate the dance behavior of the dung beetle. Once the dung beetle has successfully determined a new direction, it should continue to roll the ball backwards. Thus, the position of the ball dung beetle is updated and defined as follows:

x _i (t+1)＝x _i (t)+tan(θ)|x _i (t)-x _i (t-1)| (2)

if θ is equal to 0, pi/2 or pi, the position of the dung beetle will not be updated.

2. Catharsii breeding (Breeding ball)

In nature, the dung balls are rolled to a safe place and hidden by the dung beetles. In order to provide a safe environment for their offspring, the selection of a suitable spawning location is critical to the dung beetles. Inspired by the discussion above, a boundary selection strategy is provided to simulate the oviposition area of female dung beetles, which is:

wherein X is ^* Indicating the current local best position, lb ^* And Ub ^* Respectively represent the lower and upper bounds of the spawning area, where r=1-T/T _max ，T _max Representing the maximum number of iterations, lb and Ub represent the lower and upper bounds, respectively, of the optimization problem.

Once the spawning area is determined, female dung beetles select the breeding ball of the area for spawning. It should be mentioned that for the DBO algorithm, only one egg is produced per female dung beetle in each iteration. Furthermore, as is clear from formula (3), the boundary range of the spawning area is dynamically varied, which is mainly determined by the R value. Thus, the position of the breeding balls is also dynamic in an iterative process, expressed as:

B _i (t+1)＝X ^* +b ₁ ×(B _i (t)-Lb ^* )+b ₂ ×(B _i (t)-Ub ^* ) (4)

wherein B is _i (t) is the position information of the ith breeding ball at the t-th iteration, b ₁ And b ₂ Representing two independent random vectors of size 1 x D, D representing the dimension of the optimization problem. Note that the position of the breeding balls is strictly limited to a certain range, i.e., the spawning area. (see algorithm 2)

As shown in FIG. 1, the blue dot represents the position of the ball dung beetle, the current local optimum position X ^* Represented by using a brown large circle, X ^* The black small circles around represent the breeding balls. Note that each breeding ball has one dung beetle egg. In addition, small circles of red color represent the upper and lower limits of the boundary.

3. Small dung beetle

Some adult dung beetles climb out of the ground to find food, which is called small dung beetles. In addition, an optimal foraging area is required to be established to guide foraging of the dung beetles, and the foraging process of the dung beetles in the nature is simulated. Specifically, the boundaries of the optimal foraging area are defined as follows:

wherein X is ^b Indicating global optimum position, lb ^b And Ub ^b The lower and upper limits of the best foraging area are respectively represented, and other parameters are defined in formula (3). Therefore, the position of the small dung beetles is updated as follows:

x _i (t+1)＝x _i (t)+C ₁ ×(x _i (t)-Lb ^b )+C ₂ ×(x _i (t)-Ub ^b ) (6)

wherein x is _i (t) represents the position information of the ith small dung beetle in the t-th iteration, C ₁ Representing random numbers subject to normal distribution, C ₂ Representing the random vector belonging to (0, 1).

4. Theft dung beetle

On the other hand, some dung beetles called thieves steal dung balls from other dung beetles. Furthermore, as can be seen from formula (5), X ^b Is the best food source. Thus, it can be assumed that X ^b The vicinity is the best place to compete for food. In the iterative process, the position information of the stealing dung beetles is updated, and can be described as follows:

x _i (t+1)＝X ^b +S×g×(|x _i (t)-X ^* |+|x _i (t)-X ^b |) (7)

wherein x is _i (t) the position information g of the ith thief at the t iteration is a random vector with the size of 1 x D which obeys normal distribution, and S represents a constant.

On the basis of the foregoing discussion, the pseudo code of the proposed DBO algorithm is shown as algorithm 3. First, let T _max For the maximum number of iterations, N is the size of the population of particles. All agents of the DBO algorithm are then randomly initialized, with their allocation settings as shown in fig. 2. In this figure, the ratio of each agent in the dung beetle is represented by a fan ratio graph (the ratio of each agent can be adjusted according to the specific situation). Specifically, as shown in fig. 2, the ratio of the rolling ball dung beetles is 20%, the ratio of the breeding balls is 20%, the ratio of the small dung beetles is 25%, and the ratio of the stealing dung beetles is 35%. The species scale was assumed to be 30 in the experiment. It is worth mentioning that the blue, yellow, green and red rectangles represent the dung beetles, the breeding balls, the small dung beetles and the stealing dung beetles of the rolling balls respectively.

Then, according to the 2 nd-27 th steps of the algorithm 3, knowing the positions of the rolling ball dung beetles, the breeding balls, the small dung beetles and the stealing dung beetles in the optimization processCan be updated continuously, and finally, the optimal position X is output ^b And fitness values thereof.

In summary, for any optimization problem, the DBO algorithm is a new meta-heuristic optimization algorithm, which has six main steps, and can be summarized as follows:

a) Initializing parameters of a dung beetle group and a DBO algorithm;

b) Calculating the fitness value of all the dung beetles according to the objective function;

c) Updating the positions of all dung beetles;

d) Judging whether each dung beetle is out of the boundary;

e) Updating the current optimal solution and the fitness value thereof;

f) Repeating the steps until t meets the termination criterion, and outputting the global optimal solution and the fitness value thereof.

The DBO algorithm is superior to other algorithms in terms of optimization problems. However, it is still very challenging for DBO to obtain an optimal solution that is ideal. In addition, its ability to solve complex problems is also undesirable. The dung beetle optimizing algorithm has the characteristics of strong optimizing capability and high convergence rate. However, the method has the defects that the global exploration and local development capability are unbalanced, the local optimization is easy to fall into, and the global exploration capability is weak. Therefore, in order to improve the tracking performance of DBO, three strategies for reinforcing DBO are provided in the application, and the original DBO algorithm cannot well balance the two stages of exploration and development; the MSADBO algorithm ameliorates this imbalance by a bernoulli mapping strategy, an embedded modified sinusoidal algorithm strategy, and an adaptive gaussian-cauchy variant perturbation.

Example 1: an improved dung beetle optimization algorithm, the method comprising the steps of:

1) Initializing a dung beetle group and dung beetle optimizing algorithm parameters;

the dung beetle optimization algorithm randomly initializes the positions of the population in the search space, but the method has three main disadvantages: a) The position distribution of the dung beetles is uneven; b) Global exploration ability is weak; c) Population diversity is low and tends to fall into local optima. To increase the diversity of the initial solution, a chaotic map is used at the initial stage of DBO to generate a highly diverse initial population.

Chaotic mapping is a method combining certainty with randomness, and chaos has the characteristics of randomness, aperiodicity and the like. The chaotic variable replaces a random variable in the intelligent algorithm process in the initialization position updating process, so that the search range of the chaotic mapping strategy to the solution space is wider than that of a random search strategy. Thus, for a random initialization process, chaotic initialization can improve the search breadth of the optimization algorithm.

There are many different chaotic mappings at present, mainly logistic mapping, chebyshev mapping, bernoulli mapping, tent mapping, circle mapping, etc. Studies of SAITO A and the like show that the traversal uniformity and the convergence speed of Bernoulli mapping are suitable for being used as chaotic population initialization, and through strict mathematical reasoning, the Bernoulli mapping can be used as a chaotic sequence for generating an optimization algorithm. Therefore, bernoulli mapping is adopted to initialize individual positions of dung beetles, the Bernoulli mapping relation is firstly utilized to project the obtained value into a chaotic variable space, and then the generated chaotic value is mapped into an algorithm initial space through linear transformation, wherein the Bernoulli mapping has the following specific expression:

where β is the mapping parameter, β e (0, 1), set β=0.518, z ₀ =0.326 to achieve the best value effect.

2) Calculating the fitness value of all the dung beetles according to the objective function;

3) Updating the positions of all dung beetles: judging whether the fitness value ST in the step 2) is larger than a threshold delta or not, and if the fitness value ST is larger than the threshold delta, updating the position of the rolling ball dung beetle by adopting a position updating mathematical model of the rolling ball dung beetle; if ST is less than or equal to delta, adopting an improved sine algorithm position updating formula to update the position of the ball dung beetle;

the improved sine algorithm (MSA) strategy utilizes a sine function in mathematics to carry out iterative optimization, and has stronger global exploration capacity. While introducing adaptive variable inertia in the location update processWeight coefficient omega _t The algorithm can fully search the local area, so that the global exploration and the local development capability are well balanced.

The modified sine algorithm location update formula is as follows:

x _i (t+1)＝ω _t x _i (t)+r ₁ ×sin(r ₂ )×[r ₃ p _i (t)-x _i (t)] (9)

wherein t is the current iteration number omega _t Is the inertial weight, x _i (t) is the ith position component, p, of individual X in the t-th iteration _i (t) the ith component of the optimal individual position variable in the t-th iteration, r ₁ As a nonlinear decreasing function, r ₂ Is interval [0,2 pi ]]Random number on r ₃ Is the interval [ -2,2]A random number on the table;

wherein r is ₁ Is set using a non-linear decreasing pattern and a cosine function between 0 and pi is used to determine r ₁ Value change:

wherein omega is _max And omega _min Represents ω _t Maximum and minimum of ω _max ＝0.8，ω _min =0.75678, T represents the current iteration number, T _max Representing a maximum number of iterations;

inertial weight omega _t An adaptive variable inertia weight strategy is adopted, wherein the inertia weight is linearly reduced along with the increase of the iteration times:

wherein omega _max And omega _min Represents ω _t Maximum and minimum of ω _max ＝0.8，ω _min =0.75678, T represents the current iteration number, T _max Representing a maximum number of iterations;

in order to further optimize the capacity of a DBO algorithm for coordinating global exploration and local development, a sine and cosine guide mechanism is introduced, a sine algorithm is improved to be used as a strategy for replacing tangential dancing of the dung beetles to embed a dung beetle optimizing algorithm, namely, the whole dung beetle individual is subjected to sinusoidal operation guide in a rolling stage, the dung beetle individual is updated to a new dung beetle position, and the improved formula is as follows:

wherein ST epsilon (0.5, 1)]The method comprises the steps of carrying out a first treatment on the surface of the The introduction of the sine guiding mechanism is improved, on one hand, the blindness defect of the DBO algorithm position updating strategy can be greatly overcome, because of the sine operation, dung beetle individuals can exchange information with optimal individuals, the quick transmission of information in a population is promoted, each dung beetle individual can fully absorb the position difference information of the dung beetle individuals and the optimal individuals so as to guide the common individuals to gradually approach the optimal individuals, and the defect that the original algorithm lacks inter-individual information exchange is overcome. On the other hand, aiming at the problem that the original algorithm is easy to fall into local optimization, the MSA guiding mechanism enables the dung beetle individual to further perform global exploration and local optimization in different area ranges of the same space, the search space can be enlarged to a certain extent, and finally the dung beetle individual converges on the same optimal solution, namely the objective function value, so that the global optimization capacity of the algorithm is improved. Also as can be seen from formula (10), r ₁ The search distance and direction of the dung beetles are controlled, the optimizing mode of the DBO algorithm is optimized, and the self-adaptive coefficient omega can be seen from the formula (11) _t The search space is gradually reduced, and the inertia weight is reduced along with the increase of the iteration times. The larger inertia weight in the initial stage of algorithm iteration can ensure that the algorithm keeps better global exploration capacity, and the smaller inertia weight in the later stage of the iteration is beneficial to improving the local development capacity of the algorithm. Therefore, the convergence accuracy of the algorithm can be improved, and the convergence speed of the algorithm can be accelerated.

4) Judging whether each dung beetle is out of the boundary;

5) Updating the current optimal solution and the fitness value thereof;

in the later stage of basic dung beetle algorithm iteration, individual quick assimilation of dung beetles, the dung beetle population is gathered to the current position and near the optimal position rapidly, and the value of the dung beetle population is converged to the optimal solution, so that if the current optimal position is locally optimal, the dung beetle population can be concentrated in the locally optimal position for searching, the population diversity is reduced, and the situation that local optimal stagnation easily occurs due to difficulty in jumping out of the local optimal position is avoided. To solve this problem, mutation operations are generally used to interfere with individuals to increase population diversity and jump out of local optima. The diversity of the dung beetle population can be improved by executing the mutation disturbance operation, so that the algorithm can jump out of the local optimal solution, enter other areas of the solution space and continue searching until the global optimal solution is found finally.

The mutation operator is introduced into the intelligent optimization algorithm, so that the diversity of the population can be enhanced, and the algorithm can be prevented from being trapped into local minima. The cauchy mutation and the Gaussian mutation are two common mutation operators in the intelligent optimization algorithm, and the two mutation operators have respective advantages and disadvantages. The search range of the cauchy variation is larger than that of the Gaussian variation, but the excessive step length is easy to jump away from the optimal value to generate poorer offspring; whereas gaussian variation has good searching ability in a small range because it can produce smaller variation values with a larger probability.

The current optimal solution is perturbed by adopting a self-adaptive Gaussian-Kexil mixed variation perturbation strategy to generate a new solution, and whether the position is updated is determined according to a greedy rule.

The provided adaptive Gaussian-Kexie hybrid disturbance variation disturbance strategy has uncertainty of a variation operation result, if all individuals are subjected to variation operation, the calculated amount is increased, the searching efficiency of an algorithm is reduced, the optimal individuals are subjected to variation, the positions before and after the variation are compared, and the optimal positions are selected to be substituted into the next iteration, so that the information of the optimal individuals is fully utilized, the population diversity is increased, and the searching range is enlarged.

The adaptive Gaussian-Cauchy hybrid variation disturbance strategy is: the optimal individual is mutated, the positions before and after mutation are compared, and the optimal position is selected to be substituted into the next iteration, and the specific formula is as follows:

H ^b (t)＝X ^b (t)*(1+μ ₁ *Gauss(σ)+μ ₂ *cauchy(σ)) (13)

wherein mu is ₁ ＝t/T _max ，μ ₂ ＝1-t/T _max Is X ^b (t) optimal position of individual X in the t-th iteration, H ^b (t) is the optimal position X in the t-th iteration ^b (t) at a position after Gaussian-cauchy mixture disturbance, gauss (sigma) is a Gaussian mutation operator, cauchy (sigma) is a cauchy mutation operator;

in the iterative process of the algorithm, the individual dung beetles are disturbed according to the formula (13), and in the early stage of iterative process of the algorithm, the individual position distribution gap is larger because of uneven population distribution, so the algorithm mainly carries out larger-amplitude disturbance on the individual through the Cauchy distribution function, thereby generating various individuals, fully utilizing the current position information, adding random interference information, facilitating the algorithm to jump out of local optimum, and enabling the algorithm to carry out global exploration; with the continuous progress of algorithm iteration, most individual positions of the dung beetles cannot change too much, at the moment, the algorithm mainly perturbs the population through Gaussian distribution function coefficients, so that the algorithm is helped to converge rapidly and explore deeply, and meanwhile, the inter-dimensional interference problem under the high-dimensional problem is solved. In summary, the adaptive Gaussian-Cauchy hybrid variation perturbation strategy fully utilizes the characteristics of Cauchy and Gaussian distribution functions to generate new individuals, enhances the diversity of the population, and helps the algorithm coordinate the ability of local development and global exploration.

Although the capability of the algorithm for jumping out the local space can be enhanced through the disturbance strategy, the adaptability value of the new position obtained after disturbance variation is better than that of the original position cannot be determined, so that a greedy rule is introduced after variation disturbance update, whether the position is to be updated is determined by comparing the adaptability values of the new position and the old position, the greedy rule is shown as a formula (14), and f (x) represents the adaptability value of the x position:

6) Repeating the steps 3) -5) until the iteration times t meet the termination criterion, and outputting the global optimal solution and the fitness value thereof.

MSADBO complexity analysis: the purpose of algorithm analysis is to select a suitable algorithm and to improve the algorithm, and the time complexity is one of the important indicators of the algorithm convergence speed. The invention selects a large O representation to calculate the time complexity.

The derivation method of the large O representation is as follows:

a) Replacing all additional constants at run time with constant 1;

b) In the modified run-time function, only the highest-order item is reserved;

c) If the highest order term exists and is not 1, the constant multiplied by that term is removed. The result is a large O-sequence.

According to the calculation method, a Dung Beetle Optimization (DBO) algorithm and a dung beetle optimization algorithm (MSADBO) guided by an improved sine algorithm are selected as comparison algorithms. The calculation result of DBO is O (NxD x T). Likewise, MSADBO does not introduce new loops into the calculation process, nor does it change its order. Thus, the result is still O (NxDxT).

To illustrate the feasibility and effect of the invention, the following simulation experiments and comparisons were performed.

MSADBO is based on 23 basis functions and compares popular algorithms with the other 10 basis functions. These functions have been validated in a number of documents. These reference functions are shown in tables 2-4, where the dimensions represent the dimensions of the set functions. The range is the boundary of the function search space, f _min The best fitness value for the function. It is notable that the statistical results are given, including the Best value (Best), average value (Average) and standard deviation (STD). The operating systems of the simulation experiments are Windows11 operating systems, and the CPU is 12th GenCore ^TM i7-12700H, the main frequency is 2.30GHz, the memory is 16GB (4800 MHz), the display card is NVIDIA GeForce RTX 3060 Laptop GPU, and the programming software is MATLAB R2020b. The number of search agents is set to 30 in a unified way, and the maximum iteration time isThe number is 500, and each algorithm runs independently 30 times.

Finding the minima of these functions is similar to the search agent finding the location of the prey minima. Our goal is to find the fitness function that is the smallest possible. First, the algorithm generates an initial population in the search space, searching in a set manner. In addition to selecting MSADBO for experimentation, a number of well-established optimization algorithms, including SSA, DBO, GWO, PSO, SCA, chOA, MSA and BOA as comparison algorithms, were selected and tested with the same benchmarks to demonstrate the superiority of the proposed MSADBO. These methods were chosen because of their wide range of performance characteristics in terms of exploration and development.

Table 1 summarizes the parameter settings of the corresponding algorithm. It should be noted that the values of these parameters are set according to empirical values, ω among the parameters of the MSADBO algorithm presented herein _max And omega _min Represents ω _t Maximum, minimum, k e (0,0.2)]Represents a constant representing the deflection coefficient, and b represents a constant belonging to (0, 1). The algorithm parameters can be adjusted in a reasonable interval according to different application environments, and ω is selected through a large amount of experimental comparison _max ＝0.9,ω _min =0.782, k=0.1, b=0.3 to achieve the most balanced effect.

TABLE 1 parameter variable settings

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Global exploration analysis:

table 2 shows 7 unimodal functions (F1-F7). Since the unimodal function has and has only a minimum, it can be used to verify the development performance of the algorithm. F1-F7 in tables 5-7 gives their performance in different dimensions. The results show that the proposed algorithm performs better on most functions than the popular algorithm chosen (best results are shown in bold).

Table 2 unimodal basis function

Local development analysis: table 3 shows 6 multimodal functions (F8-F13). Unlike unimodal functions, multimodal functions have multiple extrema and increase as the dimension increases. Thus, the multimodal function can be used to evaluate the exploratory capabilities of the algorithm. The properties in different dimensions are given in tables 5-7 for F8-F13. The results show that the proposed algorithm performs better on most functions than the popular algorithm chosen (best results are shown in bold).

TABLE 3 multimodal reference function

TABLE 4 fixed dimension multimodal reference function

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Experimental results of tables 5F1-F13 (dimension=30)

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Experimental results of tables 6F1-F13 (dimension=50)

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Experimental results of tables 7F1-F13 (dimension=100)

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F14-F23 are also multimodal functions, their functional expressions are shown in Table 4. But since their dimensions were fixed, their results are shown in table 8, respectively. Since there is no difference in experimental results, many algorithms can obtain the optimal solution, so the optimal value is not marked. Through comparison, the proposed algorithm is well balanced in the global exploration and local development phases.

TABLE 8 fixed dimension multimodal reference function experimental results

MSADBO

SSA

DBO

GWO

PSO

SCA

ChOA

MSA

BOA

F14

Best

9.98E-01

9.98E-01

9.98E-01

9.98E-01

9.98E-01

9.98E-01

9.98E-01

9.98E-01

9.98E-01

Average

1.13E+00

6.40E+00

1.49E+00

4.30E+00

9.98E-01

2.05E+00

9.98E-01

6.12E+00

1.16E+00

STD

5.03E-01

4.93E+00

8.14E-01

4.06E+00

1.58E-10

1.90E+00

2.36E-04

4.25E+00

4.58E-01

F15

Best

3.08E-04

3.07E-04

3.07E-04

3.07E-04

5.22E-04

3.35E-04

1.23E-03

3.46E-04

3.75E-04

Average

3.73E-04

3.92E-04

6.85E-04

5.77E-03

1.16E-02

1.04E-03

1.29E-03

4.85E-04

1.99E-03

STD

1.21E-04

2.68E-04

3.43E-04

8.95E-03

9.50E-03

3.84E-04

4.44E-05

1.12E-04

1.91E-03

F16

Best

-1.03E+00

-1.03E+00

-1.03E+00

-1.03E+00

-1.03E+00

-1.03E+00

-1.03E+00

-1.03E+00

-1.03E+00

Average

-1.03E+00

-1.03E+00

-1.03E+00

-1.03E+00

-1.03E+00

-1.03E+00

-1.03E+00

-1.03E+00

-1.04E+00

STD

3.81E-09

4.61E-16

5.83E-16

2.36E-08

9.70E-06

3.21E-05

8.57E-06

3.72E-05

8.55E-03

F17

Best

3.98E-01

3.98E-01

3.98E-01

3.98E-01

3.98E-01

3.98E-01

3.98E-01

3.98E-01

3.98E-01

Average

3.98E-01

3.98E-01

3.98E-01

3.98E-01

3.98E-01

3.99E-01

3.98E-01

4.00E-01

1.22E+00

STD

7.54E-08

0.00E+00

0.00E+00

7.90E-07

7.33E-08

8.60E-04

3.86E-04

1.71E-03

2.29E+00

F18

Best

3.00E+00

3.00E+00

3.00E+00

3.00E+00

3.00E+00

3.00E+00

3.00E+00

3.00E+00

3.00E+00

Average

3.00E+00

5.70E+00

3.90E+00

3.00E+00

3.00E+00

3.00E+00

3.00E+00

3.00E+00

4.41E+00

STD

1.38E-07

8.24E+00

4.93E+00

4.60E-05

8.60E-06

1.50E-04

1.23E-04

3.44E-05

5.07E+00

F19

Best

-3.86E+00

-3.86E+00

-3.86E+00

-3.86E+00

-3.86E+00

-3.86E+00

-3.86E+00

-3.86E+00

-2.94E+00

Average

-3.86E+00

-3.86E+00

-3.86E+00

-3.86E+00

-3.86E+00

-3.86E+00

-3.85E+00

-3.85E+00

-3.64E+00

STD

6.68E-07

2.11E-15

1.44E-03

2.38E-03

3.21E-03

3.22E-03

1.74E-03

4.05E-03

2.42E-01

F20

Best

-3.32E+00

-3.32E+00

-3.32E+00

-3.32E+00

-3.32E+00

-3.21E+00

-3.08E+00

-3.08E+00

-1.59E+00

Average

-3.32E+00

-3.28E+00

-3.22E+00

-3.25E+00

-3.05E+00

-2.72E+00

-2.64E+00

-2.91E+00

-2.52E+00

STD

6.05E-02

5.83E-02

1.14E-01

9.34E-02

1.97E-01

5.50E-01

5.15E-01

1.42E-01

3.53E-01

F21

Best

-1.02E+01

-1.02E+01

-1.02E+01

-1.02E+01

-1.02E+01

-5.00E+00

-4.99E+00

-4.93E+00

-1.77E+00

Average

-1.01E+01

-8.11E+00

-7.86E+00

-9.65E+00

-9.22E+00

-1.91E+00

-2.47E+00

-4.58E+00

-1.35E+00

STD

5.23E-02

2.54E+00

2.70E+00

1.54E+00

2.15E+00

1.70E+00

2.06E+00

2.17E-01

2.28E-01

F22

Best

-1.04E+01

-1.04E+01

-1.04E+01

-1.04E+01

-1.04E+01

-7.06E+00

-5.06E+00

-4.98E+00

-2.89E+00

Average

-1.04E+01

-8.81E+00

-8.09E+00

-9.96E+00

-9.50E+00

-3.85E+00

-3.69E+00

-4.49E+00

-1.69E+00

STD

7.12E-03

2.48E+00

2.69E+00

1.71E+00

2.38E+00

1.90E+00

1.92E+00

4.32E-01

5.39E-01

F23

Best

-1.05E+01

-1.05E+01

-1.05E+01

-1.05E+01

-1.05E+01

-8.57E+00

-5.12E+00

-6.99E+00

-3.18E+00

Average

-1.05E+01

-8.55E+00

-8.65E+00

-9.99E+00

-1.03E+01

-3.48E+00

-4.22E+00

-4.63E+00

-1.85E+00

STD

8.93E-03

2.65E+00

2.75E+00

2.06E+00

9.88E-01

1.92E+00

1.66E+00

6.11E-01

4.88E-01

Meanwhile, the convergence process of different algorithms in the iteration process can be obtained through the graphical method. Through verification and comparison, the MSADBO algorithm not only has better final convergence accuracy than other algorithms, but also has faster convergence speed than other algorithms. Statistical analysis rank sum test:

in the above statistical process, we calculated the mean and standard deviation of 30 experimental results for each algorithm to compare the superiority of the algorithm. To further compare the differences between the proposed algorithm and other algorithms, we performed a statistical test, the method used was called Wilcoxon rank sum non-parametric statistical test. We calculate the p-value to compare the differences between the algorithms. We determine the significance of the statistics by calculating the p-value. If the p value is <0.05, we can determine that there is a significant difference between the two algorithms. The calculation results are shown in Table 9. Wherein values greater than 0.05 are underlined. NaN indicated that the results of both algorithms were too similar to be significant. The total number of data with significant differences is counted in the last column.

Only a few NaN marks and underlined values are listed in table 9, indicating that the reference function-based MSADBO is less similar to competitor's search results. Therefore, the optimization performance of the MSADBO on 23 reference functions is obviously different from that of other meta-heuristic algorithms and basic DBO. In connection with the analysis herein we can see that the overall performance of MSADBO is most prominent among the many meta-heuristics.

TABLE 9 statistical analysis results

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The feasibility of the invention was verified using pressure vessel design problems, the main idea being to translate the actual optimization problem into a mathematical model, then find the optimal solution using various algorithms,is a fitness function in the original algorithm, +.>Representing search space, x ₁ ,x ₂ …x _n Representing different dimensions. The final goal is to minimize the use of materials based on satisfying various mechanical properties. Thus, they have some equal and unequal constraints. The algorithm should have a way to handle constraints in order to accommodate these engineering design issues. Thus, the simplest constraint processing method (penalty function) can be effectively used to process constraints in an algorithm. That is, if the search agent violates any constraints, it will be assigned a large objective function value. Thus, after the next iteration, it will be automatically replaced by a new search agent. Pressure vessel design Problem (PVD), as shown in fig. 5, minimizes the manufacturing cost of the vessel while meeting the use requirements. The four variables to be optimized are the shell thickness Ts, the head thickness Th, the inner radius R and the cylindrical section length L without considering the head, and the mathematical model is as follows:

setting:

objective function:

constraint conditions:

the value range is as follows:

x ₁ and x ₂ Is an integer multiple of 0.0625, x is more than or equal to 10 ₃ ,x ₄ ≤200

MSADBO was compared to SSA, DBO, GWO, PSO, SCA, chOA, MSA and BOA. From the results of table 10, it can be seen that MSADBO ranks first in addressing pressure vessel design issues, with minimal overall cost.

Table 10 comparison of PVD problem MSADBO

In summary, the invention provides an improved dung beetle optimizer, which mainly aims to enable rolling ball dung beetles to be embedded into an improved sine algorithm, so that the searching range is enlarged. Meanwhile, the dung beetle optimizing algorithm for balancing exploration and development strategies is changed, and the local development capability of the dung beetles and the global exploration capability of the improved sine algorithm are developed.

In the embodiment of the invention, 23 standard basis functions are adopted to test MSADBO performance. Experimental results show that the MSADBO algorithm has higher convergence speed and higher convergence accuracy than the multi-element heuristic optimization algorithm. In addition, the algorithm is also applied to practical engineering design problems such as pressure vessel design, and the MSADBO is obviously improved through various experimental results.

The foregoing is merely illustrative of the present invention, and the scope of the present invention is not limited thereto, and any person skilled in the art can easily think about variations or substitutions within the scope of the present invention, and therefore, the scope of the present invention shall be defined by the scope of the appended claims.

Claims (5)

1. An improved dung beetle optimization algorithm is characterized in that: the method comprises the following steps:

1) Initializing a dung beetle group and dung beetle optimizing algorithm parameters;

2) Calculating the fitness value of all the dung beetles according to the objective function;

3) Updating the positions of all dung beetles: judging whether the fitness value ST in the step 2) is larger than a threshold delta or not, and if the fitness value ST is larger than the threshold delta, updating the position of the rolling ball dung beetle by adopting a position updating mathematical model of the rolling ball dung beetle; if ST is less than or equal to delta, adopting an improved sine algorithm position updating formula to update the position of the ball dung beetle;

4) Judging whether each dung beetle is out of the boundary;

5) Updating the current optimal solution and the fitness value thereof;

6) Repeating the steps 3) to 5) until the iteration times t meet the termination criterion, and outputting the global optimal solution and the fitness value thereof.

2. An improved dung beetle optimization algorithm in accordance with claim 1 wherein: the modified sine algorithm location update formula in step 3) is as follows:

x _i (t+1)＝ω _t x _i (t)+r ₁ ×sin(r ₂ )×[r ₃ p _i (t)-x _i (t)] (9)

wherein t is the current iteration number omega _t Is the inertial weight, x _i (t) is the ith position component, p, of individual X in the t-th iteration _i (t) isThe ith component of the best individual position variable in the t-th iteration, r ₁ As a nonlinear decreasing function, r ₂ Is interval [0,2 pi ]]Random number on r ₃ Is the interval [ -2,2]A random number on the table;

wherein r is ₁ Is set using a non-linear decreasing pattern and a cosine function between 0 and pi is used to determine r ₁ Value change:

wherein omega is _max And omega _min Represents ω _t T represents the current iteration number, T _max Representing a maximum number of iterations;

inertial weight omega _t An adaptive variable inertia weight strategy is adopted, wherein the inertia weight is linearly reduced along with the increase of the iteration times:

the sine and cosine guiding mechanism is introduced, the improved sine algorithm is used as a strategy for replacing tangential dancing of the dung beetles to embed the dung beetle optimizing algorithm, namely, the whole individual dung beetles are subjected to sinusoidal operation in the rolling stage to guide the individual dung beetles to be updated to a new dung beetle position, and the improved formula is as follows:

wherein ST.epsilon.is (0.5, 1).

3. An improved dung beetle optimization algorithm in accordance with claim 1 wherein: initializing individual positions of the dung beetles by using Bernoulli mapping in the step 1).

4. An improved dung beetle optimization algorithm in accordance with claim 1 wherein: and 5) perturbing the current optimal solution by adopting a self-adaptive Gaussian-cauchy mixed variation perturbation strategy to generate a new solution, and determining whether to update the position according to a greedy rule.

5. An improved dung beetle optimization algorithm in accordance with claim 4 wherein: the adaptive Gaussian-Cauchy hybrid variation disturbance strategy is: the optimal individual is mutated, the positions before and after mutation are compared, and the optimal position is selected to be substituted into the next iteration, and the specific formula is as follows:

H ^b (t)＝X ^b (t)*(1+μ ₁ *Gauss(σ)+μ ₂ *cauchy(σ)) (13)

wherein mu is ₁ ＝t/T _max ，μ ₂ ＝1-t/T _max Is X ^b (t) optimal position of individual X in the t-th iteration, H ^b (t) is the optimal position X in the t-th iteration ^b (t) at a position after Gaussian-cauchy mixture disturbance, gauss (sigma) is a Gaussian mutation operator, cauchy (sigma) is a cauchy mutation operator;

after mixing disturbance, a greedy rule is introduced, whether the position is to be updated is determined by comparing the fitness values of the new position and the old position, the greedy rule is shown as a formula (14), and f (x) represents the fitness value of the x position: