Welded beam cost optimization method based on improved frost ice optimization algorithm

Abstract

The invention discloses a welded beam cost optimization method based on an improved frost ice optimization algorithm, and belongs to the field of artificial intelligence and swarm intelligence. The method comprises the steps that the thickness of a welded joint of a welded beam, the length of the welded joint, the height of the beam and the width of the beam serve as preset targets, constraint conditions are set according to bending stress, shear stress, buckling critical load and tail end errors of the beam on the basis of the preset targets, and a welded beam total cost target function is constructed; constructing an improved frost ice optimization algorithm by fusing a replacement strategy and a crisscross strategy through Tent chaotic mapping; using an improved frost ice optimization algorithm to solve the welded beam total cost objective function; and verifying the improved frost ice optimization algorithm. According to the method, the diversity of an initial population is increased by introducing Tent chaotic mapping; a fusion replacement strategy is provided, and the later development capability is improved; a new individual generated by introducing the segmented crisscross strategy has better performance.

Description

Technical Field

[0001] The present invention relates to the fields of artificial intelligence and swarm intelligence technology, and in particular to a welding beam cost optimization method based on an improved frost and ice optimization algorithm. Background Art

[0002] Welded beam optimization algorithms face a tension between global search and local exploitation when solving complex problems. Overemphasizing global search can make it difficult for the algorithm to converge to the optimal solution, while overemphasizing local exploitation can easily lead to trapping in local optima. Finding a balance between these two goals—enabling full exploration of the solution space while rapidly converging to the global optimal solution—is a key challenge in optimization algorithm design.

[0003] Traditional welded beam optimization algorithms often have difficulty effectively handling this contradiction, resulting in problems such as slow convergence and low accuracy when solving high-dimensional complex problems. In addition, how to design a mechanism that can adaptively adjust the search strategy so that the algorithm can flexibly switch the focus of global exploration and local development at different stages is also an urgent problem to be solved. In practical applications, welded beam optimization algorithms also need to consider how to effectively handle different types of constraints and how to improve the robustness of the algorithm while ensuring convergence. Solving these problems is of great significance to improving the practicality of optimization algorithms in various complex engineering problems. Summary of the Invention

[0004] In order to solve the above technical problems, the present invention provides a welding beam cost optimization method based on an improved frost and ice optimization algorithm.

[0005] To implement the above technology, the specific steps are as follows:

[0006] S1. Taking the weld seam thickness, weld seam length, beam height, and beam width of the welded beam as preset targets, and setting constraints based on the preset targets using bending stress, shear stress, critical buckling load, and beam end error, the total cost objective function of the welded beam is constructed.

[0007] The build steps are as follows:

[0008] Taking the weld seam thickness, weld seam length, beam height and beam width of the welded beam as preset targets, the objective function is obtained, and the expression is as follows:

[0009]

[0010] Wherein, x1 represents the thickness of the weld seam of the welded beam h; x2 represents the length of the weld seam of the welded beam s; x3 represents the height of the welded beam a; x4 represents the height of the welded beam b;

[0011] Taking bending stress, shear stress, critical buckling load, and welded beam tail end error as constraint conditions, the constraint function is obtained, and the expression is as follows:

[0012]

[0013] Where, represents the shear stress of the welded beam Should be less than the maximum shear stress τ max , used to prevent weld failure; Indicates the end error of the welded beam Should be less than the maximum welding beam tail end error δ max , used to ensure structural rigidity; The critical buckling load P of the welded beam should be less than the critical buckling load of the welded beam. Used to prevent instability and collapse; Represents the bending stress of the welded beam Should be less than the maximum allowable stress σ of the welded beam max , used to prevent breakage;

[0014] The objective function meets the preset conditions, which include:

[0015]

[0016] The constraint function satisfies the preset conditions, which include:

[0017]

[0018] Where τ' represents the first-order derivative of shear stress, which is calculated as: τ" represents the second-order derivative of shear stress and is calculated as: R represents the first coefficient, which is calculated as: M represents the second coefficient, which is calculated as: Where L represents the length of the welded beam; J represents the third coefficient, which is calculated as follows:

[0019]

[0020] Where, E represents the applied force;

[0021]

[0022] S2. Construct an improved frost and ice optimization algorithm through tent chaos mapping, fusion replacement strategy and vertical and horizontal cross strategy; use the improved frost and ice optimization algorithm to solve the total cost objective function of the welded beam, obtain the optimization result, and complete the welded beam cost optimization;

[0023] The frost and ice optimization algorithm is as follows: the input population sequentially undergoes an initial stage, a soft frost search stage, and a hard frost development stage; in the present invention, the input population is the total cost objective function of the welded beam;

[0024] Improvements include:

[0025] In the initialization stage of the algorithm, the initial population is initialized by the Tent chaotic map to make the population distribution more uniform, and the first population is obtained;

[0026] In the hard frost development stage, the worst individual and the best individual in the first population are replaced in multiple dimensions by a fusion replacement strategy to enhance the algorithm's late development capability, thus obtaining the second population;

[0027] Introducing cross-cutting strategies, including:

[0028] Optimize the soft cream search phase using a horizontal crossover strategy to improve global search capabilities;

[0029] Use the vertical crossover strategy to optimize the hard frost development stage to improve development capacity, obtain the third population, and determine the global optimal solution through the fitness value of the third population;

[0030] The steps of applying the total cost objective function of the welded beam to the improved frost and ice optimization algorithm for solution include:

[0031] 1) Initialize the population: The Frost Ice Optimization Algorithm uses each agent as the algorithm's search agent, and the group of agents consisting of all agents is used as the algorithm's population;

[0032] Initialize the entire frost population. The frost population consists of frost agents and frost particles. Each frost agent consists of frost particles. The frost population R can be represented by a matrix. In this invention, frost agents represent a complete design solution, that is, a set of (x1, x2, x3, x4); frost particles represent a specific variable in the weld beam design (that is, a dimension of the frost agent).

[0033] In the initialization stage of the algorithm, the initial population is initialized by the Tent chaotic map to make the population distribution more uniform, and the first population is obtained;

[0034] The mathematical expression of Tent chaos map is:

[0035]

[0036] Where, X n Indicates the current sequence value; X n+1 represents the next sequence value; μ represents the control parameter; f μ represents the control parameter function representation;

[0037] 2) Soft frost search stage: In a breeze, the growth of soft frost is highly random. Frost particles can freely cover most of the surface of the attached object, but grow slowly in the same direction.

[0038] A soft frost search strategy is proposed by taking advantage of the strong randomness and coverage of frost generation, that is, the position of frost particles is calculated according to the motion characteristics of frost particles.

[0039] Use the horizontal cross strategy in the vertical and horizontal cross strategy to optimize the soft frost search phase to improve the global search capability;

[0040] Horizontal crossover: This refers to the arithmetic crossover of two different frost particles in all dimensions, allowing them to learn from each other and generate new frost particles. These new frost particles incorporate the strengths of different individuals, thus helping to search the solution space over a wider range. The horizontal crossover strategy further ensures that the population does not quickly converge to a local optimum during the global search phase by recombining genes, avoiding loss of diversity in search directions. Before executing the horizontal crossover strategy, the two different frost particles are set as parent individuals and the horizontal crossover calculation is performed. The expression is as follows:

[0041]

[0042] Where q1 and q2 represent the first and second random numbers, q1 and q2∈[0,1]; c1 and c2 represent the third and fourth random numbers, c1 and c2∈[0,1]; Y(i,d), Y(j,d) represent the dth dimension of the first parent Y(i) and the second parent Y(j); Represents the first expression of the horizontal crossover strategy;

[0043] The second expression of the horizontal crossover strategy is represented;

[0044] 3) Hard frost puncture stage: Under strong wind conditions, the growth of hard frost is simpler and more regular than that of soft frost. It is only affected by strong winds and grows in the same direction to form punctures;

[0045] During the hard frost development phase, the algorithm's late development capability is enhanced by performing multi-dimensional replacement of poor individuals with the best individuals in the first population through a fusion replacement strategy to obtain a second population, including: obtaining a fitness value for each individual in the first population and determining a global optimal individual based on the fitness value; obtaining an adaptive weight factor for each individual in the first population, wherein the adaptive weight factor is calculated according to an exponential decay formula based on the number of iterations;

[0046] Fusion replacement strategy in the hard frost development phase: Based on the expressions for the soft frost search strategy and the adhesion coefficient, the soft and hard frost phases of the original RIME algorithm are controlled by the parameter E', which increases with the number of iterations. Therefore, the probability of entering the soft frost phase is very small in the early stages of the RIME algorithm, but as the number of iterations increases, the probability of searching for soft frost increases, which in turn reduces the probability of entering the hard frost phase. Furthermore, in the hard frost phase, when the fitness of a frost individual is poor, it can only be replaced by borrowing some dimensional information from the current global optimal solution.

[0047] Taking these two points into account, the original RIME algorithm's development capabilities significantly decline in the later stages of iteration. To further improve this capability, this paper proposes a fusion replacement strategy. The mutation mechanism in this fusion replacement strategy is a probabilistic event within the local search mechanism, and an adaptively adjusted weight factor is introduced, which decreases nonlinearly with the number of iterations. This adaptive adjustment helps the algorithm adopt the most appropriate search strategy at different stages, and can dynamically adjust search accuracy based on the current search results and solution quality. This meticulous exploration helps discover subtle structures in complex objective functions, further identifying high-quality solutions that were not identified in the initial stages, and ultimately improving the quality of dimensional replacement for poorly performing frost particles. The expression is as follows:

[0048]

[0049] Where α(t) represents the decrement factor used in each iteration, and its value range is [0.3, 0.1]; min Indicates that the final value of the decrement factor is 0.1; α max The initial value of the decrement factor is 0.3; k is a constant that controls the decrement rate and is set to 3; it and Max_iter represent the current and maximum iteration times; r is a fifth random number representing uniform distribution, with a value range of [-0.5, 0.5]; ρ mut The mutation probability is 0.3; rand() represents random value;

[0050] The fusion replacement strategy introduces mutation probability and local search capabilities after the original RIME algorithm uses normalized fitness values ​​to decide whether to replace the global optimal individual. To a certain extent, it corrects the original hard frost puncture mechanism's practice of simply replacing certain frost particle dimensions with the corresponding values ​​of the global optimal solution, reducing the possibility of the algorithm's local convergence being too fast, leading to premature convergence. The local search mechanism fine-tunes the current solution near the global optimum in an attempt to find a solution slightly better than the global optimum. This introduces a more fine-grained development capability to the algorithm, which can further optimize the accuracy of the solution and avoid falling into the local optimum. In terms of mutation probability, after parameter sensitivity analysis, the mutation probability is set to 0.3, which has better robustness. Mutation increases the diversity of the population through small-probability local adjustments, which can effectively escape the local optimal solution and help the algorithm find a better solution in the search space.

[0051] A vertical and horizontal crossover strategy is introduced. The horizontal crossover strategy is used to optimize the soft frost search stage to improve the global search capability, and the vertical crossover strategy is used to optimize the hard frost development stage to improve the development capability, thus obtaining the third population.

[0052] Segmented vertical and horizontal crossover strategy: Due to the influence of parameter E', the probability of the frost optimization algorithm entering the soft frost search phase is very small, resulting in weak early search capabilities. Based on the characteristic that frost is formed by the random condensation of multiple frost particles in a breeze environment, a vertical and horizontal crossover strategy is introduced.

[0053] Compared with RIME, which only uses horizontal and vertical crossovers sequentially in the final stage of the algorithm to improve search capabilities, the present invention uses a horizontal crossover strategy to enhance the global search capability of the soft frost search stage based on the different optimization characteristics of the hard and soft frost stages, and uses a vertical crossover strategy to enhance the development capability of the hard frost penetration stage;

[0054] Vertical crossover: The original RIME is prone to falling into local optimality in the later stages of iteration. This is often caused by some frost particles falling into local optimality in a certain dimension as their development capabilities weaken during the population update process. Vertical crossover is an arithmetic crossover that operates on all individuals between two different dimensions. The parent population of the vertical crossover search comes from the population of the dominant solution of the horizontal crossover, which can better prevent the population from falling into local optimality. Each crossover only updates one dimension, while the other dimensions remain unchanged, providing stagnant dimensions with an opportunity to escape the local optimality without destroying another dimension that may be optimal. The d1th and d2th dimensions of the particles are randomly selected for vertical crossover, and the offspring individuals are obtained by the following formula:

[0055]

[0056] Where q represents the sixth random number on [0, 1]; Represents the offspring particles generated by the parent generation after vertical intersection of the d1th dimension and the d2th dimension;

[0057] The offspring particles generated by vertical crossover compete with the parent generation, and particles with better fitness are retained. Since the vertical and horizontal crossover strategy will increase the time complexity of the present invention (ILRIME), the present invention adopts the single-point crossover strategy, which takes advantage of the simplicity and high computational efficiency of single-point crossover.

[0058] Finally, the global optimal solution is determined based on the fitness value of the third population: for each individual in the third population, its fitness value is calculated; the individuals are sorted according to the fitness value; the individual with the best fitness value is obtained as a candidate solution; if the current number of iterations is less than the maximum number of iterations, the step of replacing the dimensions of the individuals in the first population using the fusion replacement strategy is returned; if the current number of iterations is equal to the maximum number of iterations, the candidate solution is output as the global optimal solution;

[0059] S3. Verify the improved frost and ice optimization algorithm;

[0060] Beneficial effects of the present invention:

[0061] (1) The present invention introduces the Tent chaotic map to initialize the population position and increase the diversity of the initial population.

[0062] (2) The present invention proposes a fusion replacement strategy, which improves the probability of discovering new and better frost ice particles through local small-probability mutation and adaptive dynamic search, and replaces the poor frost ice particle dimensional information to improve the later development capabilities.

[0063] (3) This paper introduces a segmented vertical and horizontal crossover strategy. Unlike existing studies, this paper applies a horizontal crossover strategy to the soft frost search phase, which is beneficial for combining the beneficial characteristics of different individuals, generating new individuals with potentially better performance, and enhancing the search capability of RIME. Applying a vertical crossover strategy to the hard frost development phase allows for a multi-dimensional crossover between poor frost individuals and the optimal individuals, resulting in new individuals with better performance. BRIEF DESCRIPTION OF THE DRAWINGS

[0064] Figure 1 is a flow chart of the steps of the present invention;

[0065] Figure 2 This is a convergence curve diagram in an embodiment of the present invention, where part (a) represents the result of a single-modal function; parts (b) and (c) represent the results of two multimodal functions under different fitness values; parts (d), (e) and (f) represent the results of three hybrid functions under different fitness values; and parts (g), (h) and (i) represent the results of three combined functions under different fitness values. DETAILED DESCRIPTION

[0066] The present invention is further described in detail below with reference to specific embodiments.

[0067] A method for optimizing the cost of welded beams based on an improved frost and ice optimization algorithm comprises the following steps:

[0068] Example 1

[0069] S1. Taking the weld seam thickness, weld seam length, beam height, and beam width of the welded beam as preset targets, and setting constraints based on the preset targets using bending stress, shear stress, critical buckling load, and beam end error, the total cost objective function of the welded beam is constructed.

[0070] The build steps are as follows:

[0071] Taking the weld seam thickness, weld seam length, beam height and beam width of the welded beam as preset targets, the objective function is obtained, and the expression is as follows:

[0072]

[0073] Wherein, x1 represents the thickness of the weld seam of the welded beam h; x2 represents the length of the weld seam of the welded beam s; x3 represents the height of the welded beam a; x4 represents the height of the welded beam b;

[0074] Taking bending stress, shear stress, critical buckling load, and welded beam tail end error as constraint conditions, the constraint function is obtained, and the expression is as follows:

[0075]

[0076] Where, represents the shear stress of the welded beam Should be less than the maximum shear stress τ max , used to prevent weld failure; Indicates the end error of the welded beam Should be less than the maximum welding beam tail end error δ max , used to ensure structural rigidity; The critical buckling load P of the welded beam should be less than the critical buckling load of the welded beam. Used to prevent instability and collapse; Represents the bending stress of the welded beam Should be less than the maximum allowable stress σ of the welded beam max , used to prevent breakage;

[0077] The objective function meets the preset conditions, which include:

[0078]

[0079] The constraint function satisfies the preset conditions, which include:

[0080]

[0081] Where τ' represents the first-order derivative of shear stress, which is calculated as: τ" represents the second-order derivative of shear stress and is calculated as: R represents the first coefficient, which is calculated as: M represents the second coefficient, which is calculated as: Where L represents the length of the welded beam; J represents the third coefficient, which is calculated as follows:

[0082]

[0083] Where, E represents the applied force;

[0084]

[0085] In this embodiment, P = 6000 lb (pounds); L = 14 in, (inch); E = 30 × 10 6 psi (pounds per square inch); δ max =0.25in(inch);τ max =13600 psi (pounds per square inch); σ max =3000psi (pounds per square inch).

[0086] S2. Construct an improved frost and ice optimization algorithm through tent chaos mapping, fusion replacement strategy and vertical and horizontal cross strategy; use the improved frost and ice optimization algorithm to solve the total cost objective function of the welded beam, obtain the optimization result, and complete the welded beam cost optimization;

[0087] The frost and ice optimization algorithm is as follows: the input population sequentially undergoes an initial stage, a soft frost search stage, and a hard frost development stage; in the present invention, the input population is the total cost objective function of the welded beam;

[0088] Improvements include:

[0089] In the initialization stage of the algorithm, the initial population is initialized by the Tent chaotic map to make the population distribution more uniform, and the first population is obtained;

[0090] In the hard frost development stage, the worst individual and the best individual in the first population are replaced in multiple dimensions by a fusion replacement strategy to enhance the algorithm's late development capability, thus obtaining the second population;

[0091] Introducing cross-cutting strategies, including:

[0092] Optimize the soft cream search phase using a horizontal crossover strategy to improve global search capabilities;

[0093] Use the vertical crossover strategy to optimize the hard frost development stage to improve development capacity, obtain the third population, and determine the global optimal solution through the fitness value of the third population;

[0094] The steps of applying the total cost objective function of the welded beam to the improved frost and ice optimization algorithm for solution include:

[0095] 1) Initialize the population: The Frost Ice Optimization Algorithm uses each agent as the algorithm's search agent, and the group of agents consisting of all agents is used as the algorithm's population;

[0096] Initialize the entire frost population. The frost population consists of frost agents and frost particles. Each frost agent consists of frost particles. The frost population R can be represented by a matrix. In this invention, frost agents represent a complete design solution, that is, a set of (x1, x2, x3, x4); frost particles represent a specific variable in the weld beam design (that is, a dimension of the frost agent).

[0097] The expression of frost population R is as follows:

[0098]

[0099] Where x ij Represents frost particles; i represents the ordinal number of the frost, i.e. R i =[x1,x2,x3,x4]; j represents the ordinal number of the frost particle, i.e. R i,1 =x1、R i,2 =x2、R i,3 =x3 and R i,4 =x4;

[0100] In the initialization stage of the algorithm, the initial population is initialized by the Tent chaotic map to make the population distribution more uniform, and the first population is obtained;

[0101] Tent mapping initializes the population: Population initialization is the first step in the metaheuristic algorithm and has a significant impact on the algorithm's performance and results. The initialization distribution of the population in the original frost ice algorithm (RIME) uses the Pure Random Search (PRS) method, which uses uniform random sampling within the entire solution space. The characteristic of uniform distribution is that each region in the search space has an equal probability of selection. From the characteristics of uniform distribution, it can be seen that it relies on the number of samples to ensure full coverage of the entire search space. If the number of samples is too small, the possibility of uniform exploration of the search space will be directly reduced. The expectation and variance of the uniform distribution are analyzed by the following formula to illustrate the degree to which it is affected by the sample size:

[0102] For a uniform distribution U(a', b'), the expectation is: The variance is: When the uniform distribution U(a', b') = U(0,1), 30 samples and 1000 samples are drawn for comparison. When the sample size is 30, the mean fluctuates between 0.45-0.55 and the variance fluctuates between 0.07-0.1. When the sample size is 1000, the mean and variance are close to the theoretical values ​​of 0.5 and 0.0833;

[0103] It can be concluded that when the sample size is small, the mean and variance fluctuate greatly, resulting in the inability to guarantee sufficient coverage of the uniform distribution. As the sample size increases, the mean gradually approaches the theoretical expectation, and the variance becomes more stable. This shows that the uniform distribution is more susceptible to randomness when the sample size is small, resulting in insufficient coverage of the search space. Therefore, when the population size is small, the uniform distribution initialization method will weaken the algorithm's exploration ability, thereby affecting the optimization effect. To improve these limitations, the Tent mapping is introduced to replace the original random initialization. The mathematical expression of the Tent mapping is:

[0104]

[0105] Where, X n Indicates the current sequence value; X n+1 represents the next sequence value; μ represents the control parameter; f μ represents the control parameter function representation;

[0106] 2) Soft frost search stage: In a breeze, the growth of soft frost is highly random. Frost particles can freely cover most of the surface of the attached object, but grow slowly in the same direction.

[0107] A soft frost search strategy is proposed by taking advantage of the strong randomness and coverage of frost generation. The position of frost particles is calculated according to the motion characteristics of frost particles. The expression is as follows:

[0108]

[0109] Where, Indicates the new position of the updated particle; R best,j represents the jth frost particle of the best frost agent in the frost population R; r1 represents the first parameter, r1∈(-1,1), which is used to control the direction of particle movement; the angle θ changes with the number of iterations; β represents the environmental factor, which changes with the number of iterations to simulate the influence of the external environment and is used to ensure the convergence of the algorithm; Ub ij and Lb ij represents the upper and lower bounds of the escape space; h' represents the adhesion; r2 represents the second parameter, r2∈(0,1); E' represents the adhesion coefficient, which affects the condensation probability of the cream and increases with the number of iterations;

[0110] The expression for how the angle θ changes with the number of iterations is as follows:

[0111]

[0112] Where t is the current iteration number and T is the maximum iteration number of the algorithm;

[0113] β represents the environmental factor, which changes with the number of iterations to simulate the influence of the external environment and is used to ensure the convergence of the algorithm. The expression is as follows:

[0114]

[0115] Where [·] indicates rounding; W represents the environmental factor parameter, which is 5 and is used to control the number of steps of the step function;

[0116] E' represents the adhesion coefficient, which affects the condensation probability of the cream and increases with the number of iterations. The expression is as follows:

[0117]

[0118] Use the horizontal cross strategy in the vertical and horizontal cross strategy to optimize the soft frost search phase to improve the global search capability;

[0119] Horizontal crossover: This refers to the arithmetic crossover of two different frost particles in all dimensions, allowing them to learn from each other and generate new frost particles. These new frost particles incorporate the strengths of different individuals, thus helping to search the solution space over a wider range. The horizontal crossover strategy further ensures that the population does not quickly converge to a local optimum during the global search phase by recombining genes, avoiding loss of diversity in search directions. Before executing the horizontal crossover strategy, the two different frost particles are set as parent individuals and the horizontal crossover calculation is performed. The expression is as follows:

[0120]

[0121] Where q1 and q2 represent the first and second random numbers, q1 and q2∈[0,1]; c1 and c2 represent the third and fourth random numbers, c1 and c2∈[0,1]; Y(i,d), Y(j,d) represent the dth dimension of the first parent Y(i) and the second parent Y(j); Represents the first expression of the horizontal crossover strategy;

[0122] The second expression of the horizontal crossover strategy is represented;

[0123] 3) Hard frost puncture stage: Under strong wind conditions, the growth of hard frost is simpler and more regular than that of soft frost. It is only affected by strong winds and grows in the same direction to form punctures;

[0124] Based on this characteristic, the frost penetration mechanism is proposed, and the displacement formula between particles is expressed as:

[0125]

[0126] Where r3 represents the third parameter, r3∈(-1,1); F normr (S i ) represents the normalized value of the current frost fitness value, indicating the probability of selecting the i-th frost agent;

[0127] In the hard frost development stage, the second population is obtained by performing multi-dimensional replacement of the poor individuals with the best individuals in the first population through the fusion replacement strategy to enhance the algorithm's late development ability;

[0128] Fusion replacement strategy in the hard frost development phase: Based on the expressions for the soft frost search strategy and the adhesion coefficient, the soft and hard frost phases of the original RIME algorithm are controlled by the parameter E', which increases with the number of iterations. Therefore, the probability of entering the soft frost phase is very small in the early stages of the RIME algorithm, but as the number of iterations increases, the probability of searching for soft frost increases, which in turn reduces the probability of entering the hard frost phase. Furthermore, in the hard frost phase, when the fitness of a frost individual is poor, it can only be replaced by borrowing some dimensional information from the current global optimal solution.

[0129] Taking these two points into account, the original RIME algorithm's development capabilities significantly decline in the later stages of iteration. To further improve this capability, this paper proposes a fusion replacement strategy. The mutation mechanism in this fusion replacement strategy is a probabilistic event within the local search mechanism, and an adaptively adjusted weight factor is introduced, which decreases nonlinearly with the number of iterations. This adaptive adjustment helps the algorithm adopt the most appropriate search strategy at different stages, and can dynamically adjust search accuracy based on the current search results and solution quality. This meticulous exploration helps discover subtle structures in complex objective functions, further identifying high-quality solutions that were not identified in the initial stages, and ultimately improving the quality of dimensional replacement for poorly performing frost particles. The expression is as follows:

[0130]

[0131] Where α(t) represents the decrement factor used in each iteration, and its value range is [0.3, 0.1]; min Indicates that the final value of the decrement factor is 0.1; α max The initial value of the decrement factor is 0.3; k is a constant that controls the decrement rate and is set to 3; it and Max_iter represent the current and maximum iteration times; r is a fifth random number representing uniform distribution, with a value range of [-0.5, 0.5]; ρ mutThe mutation probability is 0.3; rand() represents random value;

[0132] The fusion replacement strategy introduces mutation probability and local search capabilities after the original RIME algorithm uses normalized fitness values ​​to decide whether to replace the global optimal individual. To a certain extent, it corrects the original hard frost puncture mechanism's practice of simply replacing certain frost particle dimensions with the corresponding values ​​of the global optimal solution, reducing the possibility of the algorithm's local convergence being too fast, leading to premature convergence. The local search mechanism fine-tunes the current solution near the global optimum in an attempt to find a solution slightly better than the global optimum. This introduces a more fine-grained development capability to the algorithm, which can further optimize the accuracy of the solution and avoid falling into the local optimum. In terms of mutation probability, after parameter sensitivity analysis, the mutation probability is set to 0.3, which has better robustness. Mutation increases the diversity of the population through small-probability local adjustments, which can effectively escape the local optimal solution and help the algorithm find a better solution in the search space.

[0133] A vertical and horizontal crossover strategy is introduced. The horizontal crossover strategy is used to optimize the soft frost search stage to improve the global search capability, and the vertical crossover strategy is used to optimize the hard frost development stage to improve the development capability, thus obtaining the third population.

[0134] Segmented vertical and horizontal crossover strategy: Due to the influence of parameter E', the probability of the frost optimization algorithm entering the soft frost search stage is very small, resulting in weak early search capabilities. The vertical and horizontal crossover strategy is introduced based on the characteristic that frost is formed by the random condensation of multiple frost particles in a breeze environment. Compared with RIME, which only uses horizontal and vertical crossovers sequentially in the final stage of the algorithm to improve search capabilities, the present invention uses a horizontal crossover strategy to enhance the global search capabilities of the soft frost search stage based on the different optimization characteristics of the hard and soft frost stages, and uses a vertical crossover strategy to enhance the development capabilities of the hard frost puncture stage. To reduce the damage to the discovered optimal solution in the later stage, the crossover probability will be adaptively adjusted as the iteration proceeds, gradually decreasing from 0.7 to 0.3.

[0135] Vertical crossover: The original RIME is prone to falling into local optimality in the later stages of iteration. This is often caused by some frost particles falling into local optimality in a certain dimension as their development capabilities weaken during the population update process. Vertical crossover is an arithmetic crossover that operates on all individuals between two different dimensions. The parent population of the vertical crossover search comes from the population of the dominant solution of the horizontal crossover, which can better prevent the population from falling into local optimality. Each crossover only updates one dimension, while the other dimensions remain unchanged, providing stagnant dimensions with an opportunity to escape the local optimality without destroying another dimension that may be optimal. The d1th and d2th dimensions of the particles are randomly selected for vertical crossover, and the offspring individuals are obtained by the following formula:

[0136]

[0137] Where q represents the sixth random number on [0, 1]; Represents the offspring particles generated by the parent generation after vertical intersection of the d1th dimension and the d2th dimension;

[0138] The offspring particles generated by vertical crossover compete with the parent generation, and particles with better fitness are retained. Since the vertical and horizontal crossover strategy will increase the time complexity of the present invention (ILRIME), the present invention adopts the single-point crossover strategy, which takes advantage of the simplicity and high computational efficiency of single-point crossover.

[0139] Finally, the global optimal solution is determined based on the fitness value of the third population: for each individual in the third population, its fitness value is calculated; the individuals are sorted according to the fitness value; the individual with the best fitness value is obtained as a candidate solution; if the current number of iterations is less than the maximum number of iterations, the step of replacing the dimensions of the individuals in the first population using the fusion replacement strategy is returned; if the current number of iterations is equal to the maximum number of iterations, the candidate solution is output as the global optimal solution;

[0140] In this example, in order to verify the present invention, the population size is set to 30, the number of iterations is 1000 and 100, and the results are compared with 8 other comparison algorithms, including highly cited algorithms such as the whale optimization algorithm (WOA), the sine-cosine algorithm (SCA), the Harris Hawk algorithm (HHO) and the Archimedes optimization algorithm (AOA), as well as new algorithms proposed in the past two years such as the crayfish optimization algorithm (COA), the primitive frost ice optimization algorithm (RIME) and the two latest and most cited improved frost ice optimization algorithms (CCRIME and IDRM).

[0141] The results are shown in Table 1 below;

[0142] Table 1: Welded beam design optimization results

[0143] algorithm <![CDATA[x1]]> <![CDATA[x2]]> <![CDATA[x3]]> <![CDATA[x4]]> f(x) COA 0.205 50 3.239 70 9.037 10 0.205 70 1.693 20 HHO 0.178 74 4.521 50 9.035 30 0.205 70 1.816 40 WOA 0.322 44 2.084 60 8.090 50 0.342 20 2.381 90 IDRM 0.204 86 3.325 90 9.074 30 0.213 20 1.758 70 CCR IME 0.201 56 2.953 60 9.956 30 0.201 50 1.769 40 RIME 0.244 23 2.955 90 8.105 70 0.255 70 1.886 00 SCA 0.195 12 3.359 00 10.000 00 0.201 70 1.825 90 AOA 0.341 96 2.599 60 6.543 00 0.415 80 2.508 50 ILR IME 0.206 47 3.242 70 9.091 50 0.205 30 1.691 10

[0144] The proposed method (ILRIME) was compared with eight algorithms, including the original RIME algorithm. The results are shown in Table 1. The ILRIME algorithm demonstrated the best results in terms of manufacturing cost for the welded beam design problem. When the values ​​of the key parameters were 0.20647, 3.2427, 9.0915, and 0.2053, respectively, the manufacturing cost of the welded beam was the lowest, at only 1.6911. These results demonstrate the superiority of the ILRIME algorithm in optimizing performance and reducing the welding cost of welded beams.

[0145] S3. Verify the improved frost and ice optimization algorithm;

[0146] In order to verify the effectiveness and robustness of ILRIME, this paper conducted experimental comparisons on the CEC2017 test set;

[0147] The experiment used Matlab 2021a and was conducted under the Windows 11 operating system. The experimental environment was equipped with 8GRAM and a 1.99GHz CPU;

[0148] The ILRIME algorithm was compared with eight other algorithms, including highly cited algorithms such as the Whale Optimization Algorithm (WOA), the Sin-Cosine Algorithm (SCA), the Harris Hawk Algorithm (HHO), and the Archimedes Optimization Algorithm (AOA), as well as new algorithms proposed in the past two years, such as the Crayfish Optimization Algorithm (COA), the Original Frost Ice Optimization Algorithm (RIME), and the two most recent and highly cited Improved Frost Ice Optimization Algorithms (CCRIME and IDRM). These algorithms have been widely verified and applied in past research and have strong optimization performance. Comparison with these algorithms further demonstrates the effectiveness of the improved algorithm proposed in this paper.

[0149] To ensure the fairness and objectivity of the experiment, the population size R is set to 30, the dimension Dim is set to 100 (the upper limit of the dimension of the CEC217 test set is 100), the maximum number of iterations T is set to 1000, and each algorithm is run independently on each test function 30 times, and the average value and standard deviation are recorded.

[0150] Test function setting: The present invention uses the CEC2017 test set to evaluate the performance of the algorithm. The functions in this test set are subjected to operations such as displacement and rotation, so that the global optimal solution of the function is not located at the origin, which can effectively avoid experimental bias. Compared with the standard test function CEC2017, it is more complex, and compared with the recent CEC2022 test function set, its test dimension upper limit is higher (CEC2022 is up to 20 dimensions, while CEC2017 is 100 dimensions). As the dimension increases, problem solving becomes more difficult, and the performance of the algorithm can be more comprehensively reflected. CEC2017 contains a total of 29 test functions. The detailed information is shown in Table 2, among which F1 and F3 are unimodal functions, F2 has been deleted because of instability, F4-F10 are multimodal functions, F11-F20 are mixed functions, and F21-F30 are combined functions. In order to more accurately test the performance of the algorithm in complex situations, this experiment uniformly sets the dimension to 100.

[0151] Table 2: CEC2017 test results

[0152]

[0153] Table 3 reports the mean and variance of the optimization results from 30 independent runs of each algorithm in 100 dimensions. Overall, the proposed algorithm, ILRIME, significantly outperforms the other eight compared algorithms in terms of accuracy and robustness on various types of functions. Its average fitness and variance win rates reach 86.20% and 55.17%, respectively, demonstrating its excellent global optimization capabilities and applicability to a wide range of objective functions. Specifically, for unimodal functions (F1), which focuses on testing the algorithm's development capabilities, ILRIME outperforms the other compared algorithms by 1 to 4 orders of magnitude in both average fitness and standard deviation on F1, demonstrating its excellent development capabilities. For multimodal functions (F4 to F10), which focus on testing the algorithm's exploration capabilities, the solution space exhibits multiple local optima. ILRIME achieves superiority on most multimodal functions, demonstrating its strong global search capabilities. Hybrid functions (F11 to F20), composed of a mixture of different benchmark functions, are similar to real-world optimization problems and are primarily used to evaluate the algorithm's adaptability. ILRIME achieved the best performance across most metrics on hybrid functions, demonstrating its potential for solving real-world optimization problems. Combination functions (F21 to F30), which are more complex than the other tested functions, achieved an even higher overall win rate of 80%, demonstrating its strong overall capabilities.

[0154] Table 3: Experimental results on the CEC2017 test set (100 dimensions)

[0155]

[0156] In order to intuitively show the convergence trend of ILRIME, Figure 2 The average convergence curves of different types of test functions are plotted, where 1 is a unimodal function, 4 and 10 are multimodal functions, 12, 15, and 18 are hybrid functions, and 22, 26, and 30 are combined functions. As can be seen from the figure, ILRIME converges faster and with higher accuracy on all types of functions, indicating that the proposed strategy can effectively balance the exploration and development of the algorithm and has a wider applicability than other algorithms. The ILRIME algorithm converges faster on unimodal functions, indicating that the proposed strategy can effectively improve the algorithm's development capabilities. At the same time, the ILRIME algorithm does not stagnate on multimodal functions, hybrid functions, and combined functions, indicating that the proposed strategy improves the algorithm's global search capability and its ability to escape local optimality.

[0157] Ablation experiment: This section analyzes the effectiveness of the three improved strategies proposed in this paper on a total of 9 test functions, including F1, F3-F10, at CEC2017. The parameter settings are the same as those of the test functions mentioned above. The experimental results are shown in Table 4. Among them, TRIME represents the RIME algorithm that uses only Tent mapping; r1RIME represents the RIME algorithm that uses only the fusion replacement strategy; r2RIME represents the RIME algorithm that uses only the segmented vertical and horizontal cross strategy; Tr1RIME represents the RIME algorithm that uses Tent mapping and fusion replacement strategy; r2RIME represents the RIME algorithm that uses Tent mapping and segmented vertical and horizontal cross strategy; r3RIME represents the RIME algorithm that uses the fusion replacement strategy and segmented vertical and horizontal cross strategy.

[0158] The experimental results show that the performance of Tr1RIME and Tr2RIME, which combine a single strategy with Tent mapping, is better than that of r1RIME and r2RIME, which combine a single strategy. The comprehensive performance of r3RIME, which combines the two strategies, is second only to ILRIME. This is mainly attributed to the following reasons:

[0159] (1) Tent mapping makes the initialization of the frost population more uniform, avoiding the problem of uneven population distribution caused by random initialization. The overall performance of TRIME is slightly improved compared with the original RIME. In Tr1RIME and Tr2RIME, Tent mapping is combined with the fusion replacement strategy and the segmented vertical and horizontal cross strategy respectively. The two algorithms are similar to and slightly better than r1RIME and r2RIME in terms of convergence trend and convergence speed. This shows that Tent mapping can make the frost population more diverse, enhance the stability of optimization to a certain extent, and thus further strengthen the exploration ability of the algorithm.

[0160] (2) When the algorithm is not trapped in a local optimum, the fusion replacement strategy uses its own local search mechanism with the help of an adaptive weight factor to select the most appropriate search method for the current stage as the iteration proceeds. In the F4 function, r1RIME, Tr1RIME, r3RIME, and ILRIME all show stronger development capabilities, indicating that r1RIME can quickly develop the target area compared to original RIME, r2RIME, and Tr2RIME.

[0161] (3) The segmented vertical and horizontal crossover strategy performs segmented crossover operations based on the different binding characteristics of hard frost and soft frost particles. Through crossover operations, the information exchange between frost particles is enhanced, and the search potential of poor individuals is explored. Among them, the horizontal crossover operation acts on the soft frost search stage, allowing individuals to exchange information in all dimensions, thereby expanding the search range of individuals; the vertical crossover operation acts on the hard frost development stage, allowing all individuals between two different dimensions to perform arithmetic crossover, avoiding the stagnation of the entire algorithm caused by some frost particles falling into the local optimum due to a certain dimension, and helping the algorithm to jump out of the local optimum. Maintaining a balance between search and development while improving performance.

[0162] Table 4: Ablation experiment data statistics

[0163]

[0164] The experimental results show that the performance of Tr1RIME and Tr2RIME, which combine a single strategy with Tent mapping, is better than that of r1RIME and r2RIME, which combine a single strategy. The comprehensive performance of r3RIME, which combines the two strategies, is second only to ILRIME. This is mainly attributed to the following reasons:

[0165] (1) Tent mapping makes the initialization of the frost population more uniform, avoiding the problem of uneven population distribution caused by random initialization. The overall performance of TRIME is slightly improved compared with the original RIME. In Tr1RIME and Tr2RIME, Tent mapping is combined with the fusion replacement strategy and the segmented vertical and horizontal cross strategy respectively. The two algorithms are similar to and slightly better than r1RIME and r2RIME in terms of convergence trend and convergence speed. This shows that Tent mapping can make the frost population more diverse, enhance the stability of optimization to a certain extent, and thus further strengthen the exploration ability of the algorithm.

[0166] (2) When the algorithm is not trapped in a local optimum, the fusion replacement strategy uses its own local search mechanism with the help of an adaptive weight factor to select the most appropriate search method for the current stage as the iteration proceeds. In the F4 function, r1RIME, Tr1RIME, r3RIME, and ILRIME all show stronger development capabilities, indicating that r1RIME can quickly develop the target area compared to original RIME, r2RIME, and Tr2RIME.

[0167] (3) The segmented vertical and horizontal crossover strategy performs segmented crossover operations based on the different binding characteristics of hard frost and soft frost particles. Through crossover operations, the information exchange between frost particles is enhanced, and the search potential of poor individuals is explored. Among them, the horizontal crossover operation acts on the soft frost search stage, allowing individuals to exchange information in all dimensions, thereby expanding the search range of individuals; the vertical crossover operation acts on the hard frost development stage, allowing all individuals between two different dimensions to perform arithmetic crossover, avoiding the stagnation of the entire algorithm caused by some frost particles falling into the local optimum due to a certain dimension, and helping the algorithm to jump out of the local optimum. Maintaining a balance between search and development while improving performance.

[0168] Wilcoxon rank-sum test and Friedman test: To more comprehensively examine the performance of ILRIME, the nonparametric Wilcoxon rank-sum test and Friedman test were used to analyze the algorithms. The Wilcoxon rank-sum test is a nonparametric test used to compare the differences between algorithms. The test results are expressed as p-values. A p-value less than 0.05 indicates a significant difference in optimization results between the two algorithms. "+ / = / -" indicates the number of test functions for which ILRIME is superior, equivalent, or inferior to the comparison algorithm, respectively. The Friedman test evaluates the performance of multiple comparison algorithms and calculates the average rank, allowing for intuitive comparison of the differences between the algorithms. A smaller average rank indicates better performance. Table 5 summarizes the statistical results of the Wilcoxon rank-sum test and Friedman test. According to the Wilcoxon rank-sum test results, ILRIME significantly outperforms CCRIME, RIME, and IDRIME on most test functions and significantly surpasses HHO, WOA, COA, SCA, and AOA on all test functions. In addition, ILRIME achieved the highest ranking in the Friedman test. Combining the experimental results obtained from the two tests, it can be concluded that ILRIME shows superior performance compared with mainstream algorithms when solving complex problems. Therefore, ILRIME is more competitive than other algorithms.

[0169] Table 5: Wilconxon rank sum test and Friedman test results

[0170]

[0171] Example 2

[0172] In order to further verify the effect of the present invention, an experimental analysis of the node coverage problem of a wireless sensor network (WSN) was conducted using the improved frost optimization algorithm constructed in Example 1.

[0173] Two experiments were conducted to verify the performance of ILRIME on WSN node coverage. A three-dimensional space with a length, width, and height of 20 was set. The sensing radius, Rl, was set to 4 and 5, the number of sensors was set to 30, and each algorithm had a maximum iteration count of 100 and was run independently 30 times.

[0174] The experimental results are shown in Table 6.

[0175] Table 6 Coverage statistics

[0176]

[0177] As can be seen from Table 6, when the perception radius is 4 and 5, the optimal coverage and average coverage of ILRIME reach 75.11%, 72.24%, 97.15%, and 94.29% respectively, which all achieve the best results, demonstrating the effectiveness and potential of ILRIME in practical applications.

[0178] To address the problems of population diversity degradation, poor early exploration capability, poor late development capability, and poor optimization accuracy in the early iterations of the Rime Optimization Algorithm (RIME), this paper proposes a chaotic fusion replacement crossover Rime Optimization Algorithm, ILRIME. This algorithm applies tent mapping, fusion replacement strategy, and segmented vertical and horizontal crossover strategy to the Rime Optimization Algorithm to improve the algorithm's global search and escape local optimality capabilities, effectively solving the RIME algorithm's search and development imbalance problem for high-dimensional problems. The performance of ILRIME was evaluated on the CEC2017 function test set, and the effectiveness of the three improved strategies was verified through ablation experiments. Finally, the experimental results were subjected to non-parametric statistical analysis using the Wilcoxon rank sum test and the Friedman test. The results show that compared with other comparison algorithms, ILRIME exhibits better optimization accuracy, robustness, and statistical significance. In addition, the present invention also applies ILRIME to the design optimization of welded beam structures and WSN node coverage problems. The results show that compared with the comparison algorithm, the LRIME algorithm achieves the best results in both welded beam structure design optimization and WSN node coverage problems, proving the application value of the algorithm in practical applications.

[0179] It will be apparent to those skilled in the art that the present application is not limited to the details of the exemplary embodiments described above and that the present application can be implemented in other specific forms without departing from the spirit or essential characteristics of the present application. Therefore, the embodiments should be considered in all respects as illustrative and non-restrictive, and the scope of the present application is defined by the appended claims, not the foregoing description, and all variations within the meaning and range of equivalents of the claims are intended to be included therein. Any reference sign in a claim should not be construed as limiting the claim to which it relates.

Claims

1. A welding beam cost optimization method based on an improved frost and ice optimization algorithm, characterized in that: The following steps are involved: S1. Taking the weld seam thickness, weld seam length, beam height, and beam width of the welded beam as preset targets, and based on the preset targets, setting constraints with bending stress, shear stress, critical load for buckling, and end error of the beam, the total cost objective function of the welded beam is constructed. S2, construct an improved frost and ice optimization algorithm through tent chaos mapping, fusion replacement strategy and vertical and horizontal cross strategy; The improved frost and ice optimization algorithm is used to solve the total cost objective function of the welded beam, and the optimization result is obtained to complete the welded beam cost optimization; S3. Verify the improved frost and ice optimization algorithm.

2. The welding beam cost optimization method based on the improved frost and ice optimization algorithm according to claim 1, characterized in that: The method of constructing the improved frost ice optimization algorithm through the tent chaos map, fusion replacement strategy and vertical and horizontal cross strategy is as follows: In the initialization phase of the algorithm, the initial population is initialized by the Tent chaotic map to obtain the first population; In the hard frost development stage, the worst individual and the best individual in the first population are replaced in multiple dimensions through the fusion replacement strategy to obtain the second population; Introducing cross-cutting strategies, including: Optimize the soft cream search phase using a horizontal crossover strategy to improve global search capabilities; The third population was obtained by optimizing the hard frost development stage using a vertical crossover strategy to improve development capacity; A global optimal solution is determined according to the fitness value of the third population.

3. The welding beam cost optimization method based on the improved frost and ice optimization algorithm according to claim 1 or 2, characterized in that: The expression of the Tent chaos map is as follows: In the formula, in the formula, X n Indicates the current sequence value; X n+1 represents the next sequence value; μ represents the control parameter; f μ Represents the control parameter function representation.

4. The method for optimizing the cost of welded beams based on an improved frost and ice optimization algorithm according to claim 2, wherein: In the hard frost development stage, the worst individual and the best individual in the first population are subjected to multi-dimensional replacement by a fusion replacement strategy to enhance the late development capability of the algorithm, thereby obtaining a second population, including: Obtaining a fitness value of each individual in the first population, and determining a global optimal individual according to the fitness value; For each individual in the first population, an adaptive weight factor is obtained, wherein the adaptive weight factor is calculated according to an exponential decay formula based on the number of iterations, and the exponential decay formula is: Where α(t) represents the decrement factor used in each iteration; min represents the final value of the decrement factor; α max represents the initial value of the decrement factor; k represents the constant that controls the decrement rate; it and Max_iter represent the current number of iterations and the maximum number of iterations; r is the fifth random number representing uniform distribution.

5. The welding beam cost optimization method based on the improved frost and ice optimization algorithm according to claim 2, characterized in that: The expression of the horizontal cross strategy is: Where q1 and q2 represent the first and second random numbers, q1 and q2∈[0,1]; c1 and c2 represent the third and fourth random numbers, c1 and c2∈[0,1]; Y(i,d), Y(j,d) represent the dth dimension of the first parent Y(i) and the second parent Y(j); The expression of the vertical crossover strategy is: Where q represents the sixth random number on [0, 1]; Represents the child particles generated by the parent generation after vertical intersection of the d1th dimension and the d2th dimension.

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