CN114065896A - Multi-target decomposition evolution algorithm based on neighborhood adjustment and angle selection strategy - Google Patents

Multi-target decomposition evolution algorithm based on neighborhood adjustment and angle selection strategy Download PDF

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CN114065896A
CN114065896A CN202111239833.XA CN202111239833A CN114065896A CN 114065896 A CN114065896 A CN 114065896A CN 202111239833 A CN202111239833 A CN 202111239833A CN 114065896 A CN114065896 A CN 114065896A
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李薇
苑俊清
樊瑶驰
罗浩男
雷洲
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Xian University of Technology
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Abstract

The invention provides a multi-objective decomposition evolution algorithm based on neighborhood adjustment and angle selection strategies, which comprises the following steps: initializing to obtain an initial population P (t), weight vectors associated with each parent individual of the initial population P (t), and all values S of history memory neighborhood archiverTInitializing to a fixed value; distributing the weight vector neighborhood size T associated with each parent individual in the initial population P (T) based on the adaptive neighborhood adjustment strategyi(ii) a Breeding to generate filial individuals to form a new population Q (t); based on an angle selection mechanism, traversing all the descendant individuals in the new population Q (t) to select excellent descendant individuals to replace parent individuals; updating STA value of ST,pos,t+1And judging whether population evolution is finished. Compared with the existing MOEA/D and the improved algorithm thereof, the multi-target decomposition evolution algorithm can effectively solve the continuous complex multi-target problem and has better convergence and accuracy.

Description

Multi-target decomposition evolution algorithm based on neighborhood adjustment and angle selection strategy
Technical Field
The invention belongs to the technical field of multi-objective optimization research in evolutionary computation, and particularly relates to a multi-objective decomposition evolutionary algorithm based on a neighborhood adjustment and angle selection strategy.
Background
In scientific research and engineering practice, it is often involved in simultaneously solving multiple conflicting objectives, which are collectively referred to in the field of optimization as Multi-Objective optimization Problems (Multi-Objective issues). Solving the multi-target problem has very important practical significance and scientific research value. The multi-objective optimization theory is urgently needed to guide the production and the life of people.
In general, all objective functions in a multi-objective optimization are conflicting, so there is no point in the decision space or objective space where all objective function values are optimal at the same time. Due to this special property, unlike the single objective optimization problem, the optimal solution of the multi-objective optimization problem is not a single solution but a set of compromise solutions. In practice, the trade-off between goals can be defined in pareto optima by giving the following definitions: 1. let u, n ∈ RmIf, if
Figure BDA0003319021050000011
Satisfy ui≤υiAnd is
Figure BDA0003319021050000012
uj＜vjIf u dominates v, it is noted that u < v. 2. If there is no solution x ∈ Ω, so that F (x) < F (x)*) Then the variable x is decided*Referred to as Pareto optimal solution. Pareto optimal solution set (PS) is a set of Pareto optimal solutions, namely:
Figure BDA0003319021050000013
Figure BDA0003319021050000014
pareto Frontier (PF) is the set of target vectors corresponding to the Pareto optimal solution, i.e.: PF ═ f (x) x ∈ PS }. Currently, commonly used multi-objective evolutionary algorithms can be divided into three categories: 1. guiding the population to search and evolve by utilizing the pareto advantage relation among population solutions based on the pareto advantage algorithm; 2. an index-based algorithm by buildingA performance index to guide the search and evolution of the population; 3. a multi-objective optimization problem is decomposed into a plurality of single-objective subproblems by using weight vectors based on a decomposition algorithm.
The multi-objective evolutionary algorithm solves each sub-problem to finally obtain the PF of the whole multi-objective optimization problem, for example: firstly, converting a multi-objective optimization problem into a series of single-objective optimization subproblems by utilizing a weight vector preset in a target space based on a decomposed multi-objective evolutionary algorithm-MOEA/D; then, a group of subproblems is collaboratively evolved through a neighborhood concept defined by the relation between the subproblems, and a solution which is wanted by us is finally found, a more complex calculation index is not introduced, only less calculation resources are needed in the iteration process of the population, and the time complexity is lower; the existence of the weight vector enables the whole algorithm to have a hidden diversity protection mechanism, which is beneficial to the diversity distribution of the population and can generate a uniformly distributed solution by using a smaller population; the neighborhood in the MOEA/D realizes information sharing among individuals, and an aggregation function is introduced into the algorithm to optimize corresponding targets without considering conflicts among the targets, so that the method has the advantages of being very beneficial to adaptability distribution, rapid convergence of populations and the like. However, the decomposition-based multi-objective evolutionary algorithm-MOEA/D also has the following disadvantages:
the neighborhood used in MOEA/D is completely fixed, which not only can not meet the demand of the species group on prospectiveness and development, but also can cause certain resource waste.
2. The weight vector in the decomposition-based algorithm implies a population diversity protection mechanism, but MOEA/D adopting a TCH aggregation method does not fully utilize the characteristic.
Therefore, there is a need for improvement of the existing multi-objective evolutionary algorithm, MOEA/D.
Disclosure of Invention
In order to solve the defects of the multi-objective evolutionary algorithm MOEA/D, the invention discloses a multi-objective decomposition evolutionary algorithm based on neighborhood adjustment and angle selection strategies, and the adaptive neighborhood adjustment strategy is adopted to distribute the neighborhood size of the weight vector associated with each parent individual, so that the weight vectors of the parent individuals/child individuals have neighborhoods with different sizes in different evolution stages, and the requirements of different conditions on exploration and development are met. Meanwhile, the multi-objective decomposition evolution algorithm of the invention also improves the environment selection process, and selects excellent offspring individuals close to the weight vector as much as possible according to an angle selection mechanism by fully utilizing the distribution information of the individuals in the population so as to increase the diversity of the population and make the finally obtained solution distribution more uniform.
The technical scheme for realizing the purpose of the invention is as follows: a multi-objective decomposition evolution algorithm based on neighborhood adjustment and angle selection strategies comprises the following steps:
s1, acquiring a population in the problem search space, and initializing to obtain an initial population P (t) containing N parent individuals; carrying out grid design on a target space in a search space, and obtaining weight vectors associated with each grid and each parent individual of an initial population P (t) through WS-conversion; establishing a history memory neighborhood archiver with H entries, and storing all values S of the history memory neighborhood archiverTInitializing to a fixed value, and initializing an ideal point according to a population objective function value;
s2, distributing neighborhood size T to weight vector associated with each parent individual in initial population P (T) based on adaptive neighborhood adjustment strategyi；
S3, reproducing each parent individual in the initial population P (t) to generate an offspring individual, and forming a new population Q (t);
s4, traversing all the descendant individuals in the new population Q (t) based on an angle selection mechanism, and selecting excellent descendant individuals to replace parent individuals;
s5, S for updating history memory neighborhood archiverTA value of ST,pos,t+1；
And S6, judging whether the population evolution is finished or not, outputting the population after the evolution if the population evolution is finished, and repeating the steps S2 to S5 until the evolution is finished if the population evolution is not finished.
Further, in step S2, a neighborhood size T is assigned to the weight vector associated with each parent individual in the initial population p (T)iThe method comprises：
S201, randomly selecting an index number k in a history memory neighborhood archive device, wherein the k belongs to [1, H ];
s202, obtaining a numerical value S corresponding to the index number k according to the index number kT，kAnd according to the formula Ti＝randci(ST，k0.1) calculating the neighborhood size TiWherein randc (μ, σ) represents a Cauchy distributed random value with μ as a mean and σ as a variance.
Furthermore, in step S202, if T is detectediIf the set threshold range is exceeded, the T is again determinedi＝randci(ST，k0.1) calculating until T satisfying the threshold range is generatediThe value is obtained.
Further, in the step S4, based on the angle selection mechanism, the method of traversing all the offspring individuals in the new population q (t) and selecting the superior offspring individual to replace the parent individual comprises:
s401, calculating an aggregation function value of each offspring individual in the new population Q (t), and calculating an angle theta between each weight vector in a neighborhood where the offspring individual is located and the associated parent individualiCalculating the included angle theta between the child individual and all the weight vectors in the neighborhoodi；
S402, updating the offspring individuals in the neighborhood, and selecting excellent offspring individuals to replace parent individuals;
s4021, when t is less than or equal to 0.8 × maxgen, maxgen is the maximum evolution algebra, and for a certain weight vector K in the neighborhood of a descendant individual, K belongs to [1, i ];
if the aggregation function value of the offspring is superior to that of the parent
Figure BDA0003319021050000041
The child individual is an excellent child individual, and the parent individual associated with the weight vector K is replaced;
if the aggregation function value of the offspring individual is superior to that of the parent individual, but
Figure BDA0003319021050000042
Then by the formula
Figure BDA0003319021050000043
Determining the probability that the descendant replaces the parent associated with the weight vector K,
Figure BDA0003319021050000044
an improved value of an aggregation function for the offspring individual relative to the parent individual;
s4022, when 0.8 × maxgen is less than or equal to t, only comparing the aggregation function value of the filial generation individual with that of the parent generation individual, and if the filial generation individual has the aggregation function value
Figure BDA0003319021050000051
If the number of the child individuals is smaller than that of the parent individuals, the child individuals replace the parent individuals associated with the weight vectors K for excellent child individuals;
s403, adding the excellent offspring individuals replacing the parent individuals into the GoodT matrix, and recording the excellent offspring individuals
Figure BDA0003319021050000052
Into dif _ val.
Further, in the above step S5,
Figure BDA0003319021050000053
wherein G represents an algebraic generation of evolution; k is an element of [1, H ]]Representing the index value in the archive, and determining the updated position in the history memory neighborhood archiver; mean is a measure ofWL(GoodT) A weighted lehmer average representing the size of the neighborhood.
Further, the mean mentioned aboveWL(GoodT) The calculation formula of (2) is as follows:
Figure BDA0003319021050000054
wherein
Figure BDA0003319021050000055
Is the improved value of the aggregation function for the offspring individual relative to the parent individual.
Compared with the prior art, the invention has the beneficial effects that:
1. after the neighborhood self-adaptive strategy based on historical performance is adopted, the weight vectors associated with the parent individuals/the child individuals can have neighborhoods with different sizes in different evolution stages, on one hand, the waste of computing resources can be reduced, on the other hand, the balance between population development and exploration is favorably maintained, the global search capability and the local search capability of the algorithm are balanced, and the population is prevented from falling into local optimum.
2. The new selection mechanism (angle selection mechanism) can effectively select the child individuals which are close to the weight vector and have better distribution, improves the diversity of the population, enables the obtained solution to be more uniformly distributed, and optimizes the performance of the algorithm.
3. Compared with the existing MOEA/D and partial improved algorithm thereof, the multi-target decomposition evolution algorithm (MOEA/D-ADN) provided by the invention can effectively solve the continuous complex multi-target problem.
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In order to more clearly illustrate the technical solution of the embodiment of the present invention, the drawings used in the description of the embodiment will be briefly introduced below. It should be apparent that the drawings in the following description are only for illustrating the embodiments of the present invention or technical solutions in the prior art more clearly, and that other drawings can be obtained by those skilled in the art without any inventive work.
FIG. 1 is a block diagram of a flow of a multi-objective decomposition-evolution algorithm based on neighborhood adjustment and angle selection strategies according to the present invention;
FIG. 2 is a flowchart of an embodiment of an algorithm of a multi-objective decomposition evolution algorithm based on a neighborhood adjustment and angle selection strategy;
FIG. 3 is a solution set obtained by running the multi-objective decomposition evolution algorithm based on the neighborhood adjustment and angle selection strategy on the test function WFG8 according to the present invention;
FIG. 4 is a solution set obtained by running the prior art algorithm MOEA/D on the test function WFG 8;
FIG. 5 is a solution set resulting from the prior art algorithm MOEA/D-CMA running on test function WFG 8.
FIG. 6 is a solution set resulting from the prior art algorithm MOEA/D-DE running on test function WFG 8.
FIG. 7 is a solution set resulting from running the prior art algorithm MOEA/D-M2M on test function WFG 8.
FIG. 8 is a solution set resulting from the prior art algorithm MOEA/D-Pas running on test function WFG 8.
Fig. 9 is a standard Pareto front of test function WFG 8.
Detailed Description
The invention will be further described with reference to specific embodiments, and the advantages and features of the invention will become apparent as the description proceeds. These examples are illustrative only and do not limit the scope of the present invention in any way. It will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention, and that such changes and modifications may be made without departing from the spirit and scope of the invention.
The specific embodiment provides a multi-objective decomposition evolution algorithm based on neighborhood adjustment and angle selection strategies, as shown in fig. 1 and 2, comprising the following steps:
step 1: the initialization process of the population, the weight vector and the history memory neighborhood archiver comprises the steps of S1:
s1, acquiring a population in the problem search space, and initializing to obtain an initial population P (t) containing N parent individuals; carrying out grid design on a target space in a search space, and obtaining weight vectors associated with each grid and each parent individual of an initial population P (t) through WS-conversion; establishing a history memory neighborhood archiver with H entries, and storing all values S of the history memory neighborhood archiverTInitializing to a fixed value, and initializing an ideal point according to a population objective function value.
Specifically, a simplex mesh design method may be selected in the target space to divide the target space into N meshes, and WS-transform initialization is performed on the N meshes to obtain N weight vectors, where a more specific WS-transform method is as follows:
Figure BDA0003319021050000071
specifically, the size H of the neighborhood archive in the history memory neighborhood archiver is set to be a fixed value, and the value S in the neighborhood archive is initializedT,i(i is more than or equal to 1 and less than or equal to H) is a constant value, and the filing update position pos is initialized to 1;
specifically, the population is initialized to an initial population p (t) containing N parent individuals, where t is an evolution algebra and an initial value thereof is 0, and the population is initialized in the following manner: first, a continuous, uniformly distributed decision variable is generated in the decision space, i.e.
Figure BDA0003319021050000081
i is the dimension of the decision variable and satisfies 1 ≦ i ≦ N, k is the population size and satisfies 1 ≦ k ≦ N, xi_up、xi_lowRepresenting the upper and lower bounds of a certain dimension of the decision variables; secondly, calculating and generating an objective function value of the parent individual; thirdly, initializing the ideal point according to the target function value of the individual in P (t). It should be noted that the objective function value and the objective function value initialization ideal point are calculated by using the existing method, and are not described again.
Step 2: the evolutionary propagation process, namely selecting the parents to perform operations such as mutation, intersection and the like to generate the next generation until the population evolution is completed, comprises the following steps S2 to S3.
S2, distributing neighborhood size T to weight vector associated with each parent individual in initial population P (T) based on adaptive neighborhood adjustment strategyi；
S201, randomly selecting an index number k in a history memory neighborhood archive device, wherein the k belongs to [1, H ];
s202, obtaining a numerical value S corresponding to the index number k according to the index number kT，kAnd according to the formula Ti＝randci(ST，k0.1) calculating the neighborhood size TiWherein randc (μ, σ) represents a Cauchy distributed random value with μ as a mean and σ as a variance. Preferably, if TiIf the set threshold range is exceeded, the T is again determinedi＝randci(ST，k，01) calculating until T satisfying a threshold range is generatediThe value is obtained.
S3, breeding each parent in the initial population p (t) to generate an offspring individual, and forming a new population q (t), specifically, breeding each parent in the initial population p (t) generates an offspring individual by a variant crossing manner.
Step 3: the environment selecting process, which selects the excellent descendant to enter the next generation by traversing all descendant individuals in q (t), includes the steps S4:
s4, traversing all the descendant individuals in the new population Q (t) based on an angle selection mechanism, and selecting excellent descendant individuals to replace parent individuals;
s401, calculating an aggregation function value of each offspring individual in the new population Q (t), and calculating an angle theta between each weight vector in a neighborhood where the offspring individual is located and the associated parent individualiCalculating the included angle theta between the child individual and all the weight vectors in the neighborhoodi；
S402, updating the offspring individuals in the neighborhood, and selecting excellent offspring individuals to replace parent individuals;
s4021, when t is less than or equal to 0.8 × maxgen, maxgen is the maximum evolution algebra, and for a certain weight vector K in the neighborhood of a descendant individual, K belongs to [1, i ];
if the aggregation function value of the offspring is superior to that of the parent
Figure BDA0003319021050000091
The child individual is an excellent child individual, and the parent individual associated with the weight vector K is replaced;
if the aggregation function value of the offspring individual is superior to that of the parent individual, but
Figure BDA0003319021050000092
Then by the formula
Figure BDA0003319021050000093
Determining the probability that the descendant replaces the parent associated with the weight vector K,
Figure BDA0003319021050000094
an improved value of an aggregation function for the offspring individual relative to the parent individual;
s4022, when 0.8 × maxgen is less than or equal to t, only comparing the aggregation function value of the filial generation individual with that of the parent generation individual, and if the filial generation individual has the aggregation function value
Figure BDA0003319021050000095
If the number of the child individuals is smaller than that of the parent individuals, the child individuals replace the parent individuals associated with the weight vectors K for excellent child individuals;
s403, adding the excellent offspring individuals replacing the parent individuals into the GoodT matrix, and recording the excellent offspring individuals
Figure BDA0003319021050000096
In dif _ val, in this step, the superior offspring of the parent is successfully replaced, and the neighborhood size T of the weight vector associated with the parent that produces the superior offspring is considered as the neighborhood size T of the weight vector associated with the parentiIs excellent.
Step 4: updating the file, i.e. updating the S of the archive archiver in the history memory neighborhood after a generation of individuals in the population has evolvedTValue including step S5, S for updating history memory neighborhood archiverTA value of ST，pos，t+1；
In particular, the method comprises the following steps of,
Figure BDA0003319021050000101
wherein G represents an algebraic generation of evolution; k is an element of [1, H ]]Representing the index value in the archive, and determining the updated position in the history memory neighborhood archiver; mean is a measure ofWL(GoodT) A weighted lehmer average representing the size of the neighborhood. Mean aboveWL(GoodT) The calculation formula of (2) is as follows:
Figure BDA0003319021050000102
Figure BDA0003319021050000103
wherein
Figure BDA0003319021050000104
Is the improved value of the aggregation function for the offspring individual relative to the parent individual.
It should be noted that, at the beginning, the archive update position pos is initialized to 1, and S is updated every timeTThe value of pos is incremented by one, and if pos > H, the value of pos needs to be reset to 1.
Step 5: judging whether population evolution is finished or not and whether termination conditions are met or not, wherein the method comprises the following steps of S6:
and S6, judging whether the population evolution is finished or not, outputting the population after the evolution if the population evolution is finished, and repeating the steps S2 to S5 until the evolution is finished if the population evolution is not finished.
Specifically, if the function evaluation frequency FES is less than or equal to MAXFES, it indicates that the evolution is not completed, and the steps S2 to S5 need to be repeated before judgment; if FES > MAXFES, the population evolution is completed, and the current population P (t) is output, wherein MAXFES represents the maximum function evaluation times.
The effect of the multi-objective decomposition evolution algorithm based on the neighborhood adjustment and angle selection strategy is further illustrated by the following simulation experiment:
first, given test functions, the performance of the proposed algorithm was tested by selecting multi-mode multi-target test functions with various problem characteristics (e.g., spoofing, deflection, multi-mode) from the WFG test suite, WFG 1-9, the test set of which is specifically shown in Table 1 below.
Next, algorithm parameters are initialized, and in this experiment, the target number is set to m-3. The number of decision variables is set to n-2 (m-1) + 20. The population size N of the test function is 91, the maximum evaluation times MAXFES is 36400, and the test function is independently operated for 20 times. During initialization, H is set to N/10, ST,i(1. ltoreq. i.ltoreq.H) is set to N/10.
Table 1: WFG 1-9 test function set
Figure BDA0003319021050000111
Figure BDA0003319021050000121
In order to better explain the multi-objective decomposition evolution algorithm machine, the existing MOEA/D algorithm and the variant algorithms thereof MOEA/D-DE, MOEA/D-M2M, MOEA/D-CMA and MOEA/D-Pas are adopted to carry out experiments on the same test functions, and the performances of the multi-objective decomposition evolution algorithm machine are compared.
The selected test performance index is an Inverted Generated Distance (IGD), the IGD represents an average distance between a real Pareto front edge of the multi-target problem and an individual solution set obtained by the algorithm, the IGD index is adopted to evaluate the performance of the algorithm, and reliable information about diversity and convergence of the obtained solution can be provided.
Suppose P*For a set evenly distributed on the real Pareto frontier, P is the individual solution set obtained by the algorithm, IGD measures P*And P, the calculation formula is as follows:
Figure BDA0003319021050000131
Figure BDA0003319021050000132
wherein d (v, P) represents from P*The euclidean distance between the starting point v and the nearest member of P. As can be known from the definition of IGD, the IGD index can measure the diversity and convergence of the solution finally obtained by the algorithm at the same time, and the smaller the IGD value is, the more uniformly the obtained solution set is distributed in the target space, and the smaller the distance from the real PF is.
Comparing the operation result of the multi-objective decomposition evolution algorithm with IGD indexes obtained by the operation results of the algorithms such as MOEA/D, MOEA/D-DE, MOEA/D-M2M, MOEA/D-CMA, MOEA/D-Pas and the like: all tests were run 20 times independently and the mean and standard deviation of the metric values were recorded and the results were statistically analyzed using a Wilcoxon rank sum test with a significance level of 0.05, as shown in table 2 below, where the symbols "+", "-" and "═ respectively indicate that the current algorithm gave significantly better, significantly worse and statistically similar results to the MOEA/D.
TABLE 2 results of experiments with IGD index obtained by running five algorithms 20 times on WFG 1-WFG 9
Figure BDA0003319021050000133
Figure BDA0003319021050000141
The data in the table is divided into two rows, the first row being the mean of the indices found in 20 independent runs, the second row being the standard deviation, the bold data representing the best results for this example of the problem in this experiment. From the data in the table, one can derive: in 9 examples, the multi-objective decomposition evolution algorithm (MOEA/D-AND) of the invention is optimal on 6 example problems, AND has better convergence AND accuracy in 6 algorithms.
In order to better compare the solution sets obtained by the operation results of the multi-objective decomposition evolution algorithm (MOEA/D-ADN) of the invention with the solution sets obtained by the operation results of the algorithms such as MOEA/D, MOEA/D-DE, MOEA/D-CMA, MOEA/D-Pas and MOEA/D-M2M, the operation results of the algorithms on a three-objective test function WFG8 are compared, and the operation results are shown in FIGS. 3 to 9, wherein 9 represents a standard Pareto front edge of the three-objective test function WFG8, gray points in the graphs 3 to 9 represent the distribution of the solution sets finally obtained by the algorithms on a target space, and the distribution can be seen from the graphs: on the three-target WFG8 problem, the solution sets obtained by the two algorithms in FIG. 3 and FIG. 5 are distributed well, the coincidence rate with the standard Pareto front edge is the highest among several algorithms, and the rest algorithms are inferior to the two algorithms and have disordered solution distribution.
Finally, it should be noted that the above embodiments are only used for illustrating the technical solutions of the present invention, and are preferred embodiments of the present invention, not limiting the same; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can modify the technical solutions described in the foregoing embodiments, or make various changes and modifications to the present invention without departing from the spirit and scope of the present invention, and therefore, the present invention should be covered by the protection scope of the present invention.

Claims (6)

1. A multi-objective decomposition evolution algorithm based on neighborhood adjustment and angle selection strategies is characterized by comprising the following steps:
s1, acquiring a population in the problem search space, and initializing to obtain an initial population P (t) containing N parent individuals; carrying out grid design on a target space in a search space, and obtaining weight vectors associated with each grid and each parent individual of an initial population P (t) through WS-conversion; establishing a history memory neighborhood archiver with H entries, and storing all values S of the history memory neighborhood archiverTInitializing to a fixed value, and initializing an ideal point according to a population objective function value;
s2, distributing neighborhood size T to weight vector associated with each parent individual in initial population P (T) based on adaptive neighborhood adjustment strategyi；
S3, reproducing each parent individual in the initial population P (t) to generate an offspring individual, and forming a new population Q (t);
s4, traversing all the descendant individuals in the new population Q (t) based on an angle selection mechanism, and selecting excellent descendant individuals to replace parent individuals;
s5, S for updating history memory neighborhood archiverTA value of ST,pos,t+1；
And S6, judging whether the population evolution is finished or not, outputting the population after the evolution if the population evolution is finished, and repeating the steps S2 to S5 until the evolution is finished if the population evolution is not finished.
2. The multi-objective decomposition evolution algorithm based on neighborhood adjustment and angle selection strategy according to claim 1, wherein in step S2, a neighborhood size T is assigned to the weight vector associated with each parent individual in the initial population p (T)iThe method comprises the following steps:
s201, randomly selecting an index number k in a history memory neighborhood archive device, wherein the k belongs to [1, H ];
s202, obtaining a numerical value S corresponding to the index number k according to the index number kT，kAnd according to the formula Ti＝randci(ST,k0.1) calculating the neighborhood size TiWherein randc (μ, σ) represents a Cauchy distributed random value with μ as a mean and σ as a variance.
3. The multi-objective decomposition evolution algorithm based on neighborhood adjustment and angle selection strategy according to claim 2, characterized in that: in step S202, if T is foundiIf the set threshold range is exceeded, the T is again determinedi＝randci(ST,k0.1) calculating until T satisfying the threshold range is generatediThe value is obtained.
4. The multi-objective decomposition evolution algorithm based on neighborhood adjustment and angle selection strategy according to claim 1, wherein in step S4, based on the angle selection mechanism, all the descendant individuals in the new population q (t) are traversed, and the method for selecting the superior descendant individual to replace the parent individual is as follows:
s401, calculating an aggregation function value of each offspring individual in the new population Q (t), and calculating an angle theta between each weight vector in a neighborhood where the offspring individual is located and the associated parent individualiCalculating the included angle theta between the child individual and all the weight vectors in the neighborhoodi；
S402, updating the offspring individuals in the neighborhood, and selecting excellent offspring individuals to replace parent individuals;
s4021, when t is less than or equal to 0.8 × maxgen, maxgen is the maximum evolution algebra, and for a certain weight vector K in the neighborhood of a descendant individual, K belongs to [1, i ];
if the aggregation function value of the offspring individual is superior to that of the parent individual, and thetak≤θkIf so, the child individual is an excellent child individual, and replaces the parent individual associated with the weight vector K;
if the aggregation function value of the child individual is superior to that of the parent individual, thetak>θkThen by the formula
Figure FDA0003319021040000021
Determining the probability that the descendant replaces the parent associated with the weight vector K,
Figure FDA0003319021040000022
an improved value of an aggregation function for the offspring individual relative to the parent individual;
s4022, when 0.8 × maxgen is less than or equal to t, only comparing the aggregation function value of the filial generation individual with that of the parent generation individual, and if the filial generation individual has the aggregation function value
Figure FDA0003319021040000023
If the number of the child individuals is smaller than that of the parent individuals, the child individuals replace the parent individuals associated with the weight vectors K for excellent child individuals;
s403, adding the excellent offspring individuals replacing the parent individuals into the GoodT matrix, and recording the excellent offspring individuals
Figure FDA0003319021040000031
Into dif _ val.
5. The multi-objective decomposition evolution algorithm based on neighborhood adjustment and angle selection strategy according to claim 1, characterized in that: in the step S5, in the step S,
Figure FDA0003319021040000032
wherein G represents an algebraic generation of evolution; k is an element of [1, H ]]Representing the index value in the archive, and determining the updated position in the history memory neighborhood archiver; mean is a measure ofWL(GoodT) A weighted lehmer average representing the size of the neighborhood.
6. The multi-objective decomposition evolution algorithm based on neighborhood adjustment and angle selection strategy according to claim 5, characterized in that: mean is a measure ofWL(GoodT) The calculation formula of (2) is as follows:
Figure FDA0003319021040000033
wherein
Figure FDA0003319021040000034
Is the improved value of the aggregation function for the offspring individual relative to the parent individual.
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CN117077981A (en) * 2023-10-16 2023-11-17 四川大学 Method and device for distributing stand by fusing neighborhood search variation and differential evolution

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CN117035373A (en) * 2023-10-09 2023-11-10 中国电建集团山东电力管道工程有限公司 Intelligent management method and system for pipeline prefabrication production line
CN117035373B (en) * 2023-10-09 2024-01-23 中国电建集团山东电力管道工程有限公司 Intelligent management method and system for pipeline prefabrication production line
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