CN104036332A - Average gradient value and improved multi-objective particle swarm optimization based robust optimization system - Google Patents
Average gradient value and improved multi-objective particle swarm optimization based robust optimization system Download PDFInfo
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Abstract
The invention discloses an average gradient value and improved multi-objective particle swarm optimization based robust optimization system. The average gradient value and improved multi-objective particle swarm optimization based robust optimization system comprises a parameter receiving module, an objective function obtaining module, a robustness calculation module, a function integration module and a function solving module; the parameter receiving module is used for receiving process parameters to be optimized according to the engineering actual situation and an optimization objective; the objective function obtaining module is used for obtaining an objective function model of the optimization objective and the process parameters to be optimized; the robustness calculation module is used for setting an uncertain domain and calculating the robustness through an average gradient value of uncertain domain samples; the function integration module is used for enabling an objective function and the robustness to form a double-objective optimization objective function; the function solving module is used for performing double-objective optimum value solution with the objective function and the robustness as the optimization objective. According to the average gradient value and improved multi-objective particle swarm optimization based robust optimization system, the applicable engineering problem range is broad and the evaluation of the robustness and the calculation of the original objective function serve as two optimization sub-objectives simultaneously to enable a designer to obtain a reasonable compromise solution between the performance index and the variation amplitude generated by the uncertain domain according to actual needs.
Description
Technical field
The present invention relates to field of computer data processing, relate in particular to a kind of robust optimization system based on improving multi-objective particle swarm optimization.
Background technology
Robust optimization is a kind of new optimization method solving under inner structure (as parameter) and external environment condition (as disturbance) uncertain environment, when optimizing beginning, just to consider the uncertainty of Optimized model, by the method for optimizing, make the unification that result is insensitive to uncertain factor and performance index are optimum of optimizing.Classical robust Optimal methods is mainly used in operational research, research be the protruding problems such as linear programming problem, quadratic programming problem and Semidefinite Programming with multi-form data uncertainty.Yet the engineering problem of real world often right and wrong is protruding, even there is no mathematic(al) representation, the classical way in operational research is also improper in engineering design.In existing engineering problem, the robust optimization solution how solving without analytical expression, nonlinearity, the more high a series of realistic problems of decision space dimension becomes robust optimization field urgent problem.
Summary of the invention
The technical problem to be solved in the present invention is to solve for engineering problem in prior art the difficulty running into, a kind of robust optimization system based on improving multi-objective particle swarm optimization is provided, this system can, using the calculating of the evaluation of robustness and former objective function simultaneously as the sub-goal of two optimizations, make deviser between performance index and the amplitude of variation of uncertain territory generation, select rational compromise solution according to actual needs.
The technical solution adopted for the present invention to solve the technical problems is:
A robust optimization system based on average gradient value and improvement multi-objective particle swarm optimization, comprising:
Parameter receiver module, for receiving according to engineering actual conditions and the definite technological parameter to be optimized of optimization aim;
Objective function acquisition module, for obtaining the objective function model of optimization aim and technological parameter to be optimized;
Function Synthesis module, for forming complex optimum objective function by objective function and robustness;
Function solves module, for to objective function and robustness as complex optimum target, carry out optimal value and solve; Function solves module and comprises:
Initialization unit, for initial setting up population scale, the maximum algebraically of iteration, stochastic variable R1, R2 between maximum Inertia Weight and minimum Inertia Weight and [0,1];
Initialization of population unit, for initialization population, by the position of CVT method initialization particle and the speed of initialization particle;
Fitness function computing unit, for calculating the fitness function of each particle; Specifically using the position of particle as decision variable, according to the objective function model calculating target function value of optimization aim and technological parameter to be optimized; Utilize the robustness of the average gradient value calculating particle of sample point in the uncertain territory of particle;
Position storage unit, for according to Pareto domination principle, is stored in the position of the non-domination particle in population in outside files;
Wherein non-domination particle is defined as follows: establishing p and q is two different individualities in population, when p domination q, must meet following two conditions: (1), to all sub-goals, p is poor unlike q; (2) at least there is a sub-goal, make p better than q; Q is called domination particle, and while there is no other particle dominations p in population, p is called non-domination particle;
Particle optimal location unit, for the memory files of each particle of initialization, records this particle optimal location up to the present by the memory files of particle;
Wherein, particle optimal location is up to the present defined as follows: if particle is paid out and mixed a generation when evolution, the optimal location of particle is for working as former generation; If former generation is worked as in the previous generation of particle domination, the optimal location of particle is previous generation; If not arranging mutually as former generation and previous generation of particle, the optimal location of particle is for choosing at random one in both;
Optimal location selected cell, for the optimal location from outside files selected population;
Be implemented as follows: decision space is divided into a plurality of hypercubes, the population that the fitness value of each hypercube comprises according to its inside determines, first according to this fitness value, by roulette method, select hypercube, then from selected hypercube, determine optimum individual randomly;
The speed computing unit of particle, for calculating the speed of each particle;
Be implemented as follows: the product, the particle optimal location that calculate respectively Inertia Weight and particle previous generation speed arrive the distance of current location and the product of R1 and population optimal location to the distance of particle current location and the product of R2, by speed three products and that be made as particle; Wherein Inertia Weight is along with the increase of evolutionary generation is from maximum Inertia Weight to minimum Inertia Weight linear decrease;
Updating block, when not reaching the maximum algebraically of iteration when iteration, upgrades population particle; Comprise
Particle position upgrades subelement, for the position of new particle more, the position of particle previous generation is added to the speed of this particle, and keeps particle in search volume;
Fitness function upgrades subelement, for the fitness function of new particle more;
Outside files upgrade subelement, for upgrading the particle representative in each hypercube of outside files and division; Current non-domination solution is inserted into outside files, and the solution of being arranged will be deleted from outside files; When outside files are full, preferentially preserve the solution in the region that in object space, particle is few;
Memory files upgrade subelement, for the memory files of new particle more;
According to Pareto domination mechanism, when the current position of particle is better than the position in memory files, the desired positions of this particle needs to upgrade; When the current position of particle is worse than memory files, the desired positions of this particle does not need to upgrade; If both do not arrange mutually, choose at random one.
Press such scheme, in described initialization of population unit, the position of initialization particle adopts following methods:
1.1) a given density function ρ (x), a positive integer q, constant α
1, α
2, β
1and β
2, wherein: α
2>0, β
2>0, α
1+ α
2=1, β
1+ β
2=1;
Point set
expression will initialized population, i=1 wherein ..., N; For i=1 ..., N, arranges j
iinitial value is 1; By Monte Carlo method, at decision space, select an initial point set
1.2) according to density function ρ (x), at decision space, select q point
1.3) for set
in each point, will gather
in from z
inearest point (is put z
ivoronoi region in point) conclude set w
iin, if set w
ifor sky, z
iremain unchanged; Otherwise, calculate w
ithe mean place u of point in set
i, and upgrade z according to following expression formula
i;
1.4) if the particle assembly after upgrading meets convergence criterion, renewal stops; Otherwise, forward step 1.2 to).
Press such scheme, in described fitness function computing unit, the robustness of calculating particle comprises the following steps:
2.1), establishing the dimension that current particle exists the decision variable of disturbance is c, the dimension in this uncertain territory of particle is also c, every one dimension in uncertain territory is evenly divided into t section, c and t represent respectively because of prime number and number of levels, according to select uniform designs table to carry out factor level data because of prime number, number of levels, arrange, in the uncertain territory of current particle, by uniform designs table, choose sample point set; The uncertain territory of described particle is artificial default in advance;
2.2) calculate the target function value of each sample point and the absolute difference of the target function value of current particle and the decision variable of each sample point to the Euclidean distance of the decision variable of current particle, the ratio of the absolute difference of record object function and the Euclidean distance of decision variable;
2.3) calculate the mean value of the ratio of the absolute difference of all sample point objective functions and the Euclidean distance of decision variable, be the robustness evaluation value of this particle.
The beneficial effect that the present invention produces is:
1. the applicable engineering problem wide scope of the present invention;
2. system of the present invention, using the calculating of the evaluation of robustness and former objective function simultaneously as the sub-goal of two optimizations, makes deviser between performance index and the amplitude of variation of uncertain territory generation, select rational compromise solution according to actual needs;
3. system of the present invention adopts the improvement multi-objective particle based on CVT initialization of population and Dynamic Inertia weights, make to find ability that robust optimizes disaggregation than based on random initializtion population and fixedly the multiple-objection optimization particle cluster algorithm of Inertia Weight greatly strengthen, algorithm convergence is faster;
4. system of the present invention has proposed a kind of method of evaluation robustness of novelty, and the sample point of choosing in the uncertain territory of evaluation point arrives the average gradient value of evaluation point as the robustness evaluation value of this evaluation point;
5. system of the present invention adopts uniform Design to avoid calculated amount to be exponent increase with the increase of decision space dimension;
6. system of the present invention is used relatively less and equally distributed sample point to calculate the object space of evaluation point in uncertain territory and the relative amplitude of variation of decision space, has embodied fully the concept of robustness.
Accompanying drawing explanation
Below in conjunction with drawings and Examples, the invention will be further described, in accompanying drawing:
Fig. 1 is the system architecture schematic diagram of the embodiment of the present invention;
Fig. 2 is the function schematic diagram of the beneficial effect checking of the embodiment of the present invention;
Fig. 3 is the optimum results disaggregation schematic diagram of the beneficial effect checking of the embodiment of the present invention;
Fig. 4 is the employing random initializtion of the embodiment of the present invention and the fixing disaggregation schematic diagram of the multi-objective particle swarm algorithm of Inertia Weight;
Fig. 5 is the disaggregation schematic diagram of the use CVT method initialization of the embodiment of the present invention and the multi-objective particle swarm algorithm of Dynamic Inertia weights.
Embodiment
In order to make object of the present invention, technical scheme and advantage clearer, below in conjunction with embodiment, the present invention is further elaborated.Should be appreciated that specific embodiment described herein, only in order to explain the present invention, is not intended to limit the present invention.
As shown in Figure 1, a kind of robust optimization system based on average gradient value and improvement multi-objective particle swarm optimization, comprising:
Parameter receiver module, for receiving according to engineering actual conditions and the definite technological parameter to be optimized of optimization aim;
Objective function acquisition module, for obtaining the objective function model of optimization aim and technological parameter to be optimized;
Function Synthesis module, for forming two optimization aim functions by objective function and robustness;
Function solves module, for to objective function and robustness as two optimization aim, carry out optimal value and solve; Function solves module and comprises:
Initialization unit, for initial setting up population scale, the maximum algebraically of iteration, stochastic variable R1, R2 between maximum Inertia Weight and minimum Inertia Weight and [0,1];
Initialization of population unit, for initialization population, by the position of CVT method initialization particle and the speed of initialization particle;
Fitness function computing unit, for calculating the fitness function of each particle; Specifically using the position of particle as decision variable, according to the objective function model calculating target function value of optimization aim and technological parameter to be optimized; Utilize the robustness of the average gradient value calculating particle of sample point in the uncertain territory of particle;
Position storage unit, for according to Pareto domination principle, is stored in the position of the non-domination particle in population in outside files;
Wherein non-domination particle is defined as follows: establishing p and q is two different individualities in population, when p domination q, must meet following two conditions: (1), to all sub-goals, p is poor unlike q; (2) at least there is a sub-goal, make p better than q; Q is called domination particle, and while there is no other particle dominations p in population, p is called non-domination particle;
Particle optimal location unit, for the memory files of each particle of initialization, records this particle optimal location up to the present by the memory files of particle;
Wherein, particle optimal location is up to the present defined as follows: if particle is paid out and mixed a generation when evolution, the optimal location of particle is for working as former generation; If former generation is worked as in the previous generation of particle domination, the optimal location of particle is previous generation; If not arranging mutually as former generation and previous generation of particle, the optimal location of particle is for choosing at random one in both;
Optimal location selected cell, for the optimal location from outside files selected population;
Be implemented as follows: decision space is divided into a plurality of hypercubes, the population that the fitness value of each hypercube comprises according to its inside determines, first according to this fitness value, by roulette method, select hypercube, then from selected hypercube, determine optimum individual randomly;
The speed computing unit of particle, for calculating the speed of each particle;
Be implemented as follows: the product, the particle optimal location that calculate respectively Inertia Weight and particle previous generation speed arrive the distance of current location and the product of R1 and population optimal location to the distance of particle current location and the product of R2, by speed three products and that be made as particle; Wherein Inertia Weight is along with the increase of evolutionary generation is from maximum Inertia Weight to minimum Inertia Weight linear decrease;
Updating block, when not reaching the maximum algebraically of iteration when iteration, upgrades population particle; Comprise
Particle position upgrades subelement, for the position of new particle more, the position of particle previous generation is added to the speed of this particle, and keeps particle in search volume;
Fitness function upgrades subelement, for the fitness function of new particle more;
Outside files upgrade subelement, for upgrading the particle representative in each hypercube of outside files and division; Current non-domination solution is inserted into outside files, and the solution of being arranged will be deleted from outside files; When outside files are full, preferentially preserve the solution in the region that in object space, particle is few;
Memory files upgrade subelement, for the memory files of new particle more;
According to Pareto domination mechanism, when the current position of particle is better than the position in memory files, the desired positions of this particle needs to upgrade; When the current position of particle is worse than memory files, the desired positions of this particle does not need to upgrade; If both do not arrange mutually, choose at random one.
In initialization of population unit, the position of initialization particle adopts following methods:
1.1) a given density function ρ (x), a positive integer q, constant α
1, α
2, β
1and β
2, wherein: α
2>0, β
2>0, α
1+ α
2=1, β
1+ β
2=1;
Point set
expression will initialized population, i=1 wherein ..., N; For i=1 ..., N, arranges j
iinitial value is 1; By Monte Carlo method, at decision space, select an initial point set
1.2) according to density function ρ (x), at decision space, select q point
1.3) for set
in each point, will gather
in from z
inearest point (is put z
ivoronoi region in point) conclude set w
iin, if set w
ifor sky, z
iremain unchanged; Otherwise, calculate w
ithe mean place u of point in set
i, and upgrade z according to following expression formula
i;
1.4) if the particle assembly after upgrading meets convergence criterion, renewal stops; Otherwise, forward step 1.2 to).
In fitness function computing unit, the robustness of calculating particle comprises the following steps:
2.1), establishing the dimension that current particle exists the decision variable of disturbance is c, the dimension in this uncertain territory of particle is also c, every one dimension in uncertain territory is evenly divided into t section, c and t represent respectively because of prime number and number of levels, according to select suitable uniform designs table to carry out factor level data because of prime number, number of levels, arrange, in the uncertain territory of current particle, by uniform designs table, choose sample point set; The uncertain territory of described particle is artificial default;
2.2) calculate the target function value of each sample point and the absolute difference of the target function value of current particle and the decision variable of each sample point to the Euclidean distance of the decision variable of current particle, the ratio of the absolute difference of record object function and the Euclidean distance of decision variable;
2.3) calculate the mean value of the ratio of the absolute difference of all sample point objective functions and the Euclidean distance of decision variable, be the robustness evaluation value of this particle.
The checking of system beneficial effect of the present invention:
1. in system of the present invention, the evaluation of robustness is defined as follows: the mean value of the ratio of the absolute difference of sample point objective function and the Euclidean distance of decision variable in the employing uncertain territory of evaluation point.In the uncertain territory of evaluation point, sample, the absolute difference and the sample point that calculate the objective function of sample point and the objective function of evaluation point arrive the Euclidean distance of evaluation point at decision space, the gradient of this sample point is: the ratio of the absolute difference of objective function and decision space Euclidean distance, the robustness of evaluation point is the mean value of all sample point gradients in uncertain territory.
As Fig. 2, using robustness as objective function, be optimized and solve, uncertain territory is made as x ± 0.1, the solution that the robustness evaluation method of using the present invention to propose obtains is 10.0, corresponding robustness evaluation value is 0.004728, has reacted this and has been in neighborhood comparatively smoothly, and robustness is good, in figure, take robustness as the solution that objective function obtains is a c, confirmed the validity of the method.
2. the present invention adopts uniform Design sampling, the choosing of sample point in uncertain territory: uniform Design sampling.Exist every one dimension of disturbance to be all evenly divided into the equal segments that quantity is equal decision space, be designated as number of levels t, decision space dimension is designated as because of prime number c, according to select suitable uniform designs table to carry out factor level data because of prime number, number of levels, arrange, according to uniform designs table, at decision space, choose corresponding sample point.
Compare existing Grid Sampling method, fiery initial orbit design problem in combination, the decision space that has disturbance is 8 dimensions, in uncertain territory, every one dimension is divided into 50 sections, if adopt Grid Sampling, sampling number is 50
8, and uniform Design sampling only needs 50 times.
3. adopt the thought of multiple-objection optimization, the result of optimization is not single solution, but the disaggregation of an one-dimensional manifold is shown in Fig. 3.
4, the Inertia Weight of multi-objective particle: the Dynamic Inertia weights of linear decrease.Inertia Weight is by following expression formula linear change in population search procedure, and at the iteration initial stage, Inertia Weight is larger, and algorithm has stronger global optimizing ability; Along with the increase of number of iterations, population is restrained gradually, and Inertia Weight diminishes, and algorithm has stronger local optimal searching ability.
Expression formula:
Inertia Weight=maximum Inertia Weight-(maximum Inertia Weight-minimum Inertia Weight) * iteration algebraically ÷ greatest iteration algebraically
The performance comparison of random initializtion and fixedly Inertia Weight and the multi-objective particle swarm algorithm based on the initialization of CVT method and Dynamic Inertia weights.Fig. 4 is for adopting random initializtion and the fixing disaggregation of the multi-objective particle swarm algorithm of Inertia Weight, Fig. 5 is for being used the disaggregation of the multi-objective particle swarm algorithm of CVT method initialization and Dynamic Inertia weights, both operation algebraically was for 500 generations, can find out and adopt the algorithm optimization disaggregation of CVT method initialization and Dynamic Inertia weights to be more evenly distributed, the optimization solution finding is more, meanwhile, in the situation that iteration algebraically is identical, adopt the algorithm convergence of CVT method initialization and Dynamic Inertia weights faster.
Should be understood that, for those of ordinary skills, can be improved according to the above description or convert, and all these improvement and conversion all should belong to the protection domain of claims of the present invention.
Claims (4)
1. the robust optimization system based on average gradient value and improvement multi-objective particle swarm optimization, is characterized in that, comprising:
Parameter receiver module, for receiving according to engineering actual conditions and the definite technological parameter to be optimized of optimization aim;
Objective function acquisition module, for obtaining the objective function model of optimization aim and technological parameter to be optimized;
Robustness computing module, for setting uncertain territory, utilizes the average gradient value of uncertain territory sample, calculates robustness;
Function Synthesis module, for forming Bi-objective optimization aim function by objective function and robustness;
Function solves module, for to objective function and robustness as optimization aim, carry out Bi-objective Optimization Solution.
2. robust optimization system according to claim 1, is characterized in that, described function solves module and comprises:
Initialization unit, for initial setting up population scale, the maximum algebraically of iteration, stochastic variable R1, R2 between maximum Inertia Weight and minimum Inertia Weight and [0,1];
Initialization of population unit, for initialization population, by the position of CVT method initialization particle and the speed of initialization particle;
Fitness function computing unit, for calculating the fitness function of each particle; Specifically, using the position of particle as decision variable, described decision variable is for representing technological parameter to be optimized; According to the objective function model calculating target function value of optimization aim and technological parameter to be optimized; Utilize the robustness of the average gradient value calculating particle of sample point in the uncertain territory of particle;
Position storage unit, for according to Pareto domination principle, is stored in the position of the non-domination particle in population in outside files;
Wherein non-domination particle is defined as follows: establishing p and q is two different individualities in population, when p domination q, must meet following two conditions: (1), to all sub-goals, p is poor unlike q; (2) at least there is a sub-goal, make p better than q; Q is called domination particle, and while there is no other particle dominations p in population, p is called non-domination particle;
Particle optimal location unit, for the memory files of each particle of initialization, records this particle optimal location up to the present by the memory files of particle;
Wherein, particle optimal location is up to the present defined as follows: if particle is paid out and mixed a generation when evolution, the optimal location of particle is for working as former generation; If former generation is worked as in the previous generation of particle domination, the optimal location of particle is previous generation; If not arranging mutually as former generation and previous generation of particle, the optimal location of particle is for choosing at random one in both;
Optimal location selected cell, for the optimal location from outside files selected population;
The speed computing unit of particle, for calculating the speed of each particle;
Updating block, when not reaching the maximum algebraically of iteration when iteration, upgrades population particle;
Comprise:
Particle position upgrades subelement, for the position of new particle more, the position of particle previous generation is added to the speed of this particle, and keeps particle in search volume;
Fitness function upgrades subelement, for the fitness function of new particle more;
Outside files upgrade subelement, for upgrading the particle representative in each hypercube of outside files and division; Current non-domination solution is inserted into outside files, and the solution of being arranged will be deleted from outside files; When outside files are full, preferentially preserve the solution in the region that in object space, particle is few;
Memory files upgrade subelement, for the memory files of new particle more;
According to Pareto domination mechanism, when the current position of particle is better than the position in memory files, the desired positions of this particle needs to upgrade; When the current position of particle is worse than memory files, the desired positions of this particle does not need to upgrade; If both do not arrange mutually, choose at random one.
3. robust optimization system according to claim 1, is characterized in that, in described initialization of population unit, the position of initialization particle adopts following methods:
1.1) a given density function ρ (x), a positive integer q, constant α
1, α
2, β
1and β
2, wherein: α
2>0, β
2>0, α
1+ α
2=1, β
1+ β
2=1;
Point set
expression will initialized population, i=1 wherein ..., N; For i=1 ..., N, arranges j
iinitial value is 1; By Monte Carlo method, at decision space, select an initial point set
1.2) according to density function ρ (x), at decision space, select q point
1.3) for set
in each point, will gather
in from z
inearest point (is put z
ivoronoi region in point) conclude set w
iin, if set w
ifor sky, z
iremain unchanged; Otherwise, calculate w
ithe mean place u of point in set
i, and upgrade z according to following expression formula
i;
1.4) if the particle assembly after upgrading meets convergence criterion, renewal stops; Otherwise, forward step 1.2 to).
4. robust optimization system according to claim 1, is characterized in that, in described fitness function computing unit, the robustness of calculating particle comprises the following steps:
2.1), establishing the dimension that current particle exists the decision variable of disturbance is c, the dimension in this uncertain territory of particle is also c, every one dimension in uncertain territory is evenly divided into t section, c and t represent respectively because of prime number and number of levels, according to select uniform designs table to carry out factor level data because of prime number, number of levels, arrange, in the uncertain territory of current particle, by uniform designs table, choose sample point set; The uncertain territory of described particle is artificial default;
2.2) calculate the target function value of each sample point and the absolute difference of the target function value of current particle and the decision variable of each sample point to the Euclidean distance of the decision variable of current particle, the ratio of the absolute difference of record object function and the Euclidean distance of decision variable;
2.3) calculate the mean value of the ratio of the absolute difference of all sample point objective functions and the Euclidean distance of decision variable, be the robustness evaluation value of this particle.
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| CN104809517A (en) * | 2015-04-16 | 2015-07-29 | 四川大学 | Building construction site security double-layer dynamic optimizing method |
| CN111080020A (en) * | 2019-12-23 | 2020-04-28 | 中山大学 | Robustness evaluation method and device for drilling arrangement scheme |
| CN112327923A (en) * | 2020-11-19 | 2021-02-05 | 中国地质大学（武汉） | Multi-unmanned aerial vehicle collaborative path planning method |
| CN113627025A (en) * | 2021-08-16 | 2021-11-09 | 湘潭大学 | Optimized design method for sheet forming process |
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| CN102999678A (en) * | 2012-12-26 | 2013-03-27 | 北京航空航天大学 | Nonlinear multi-target range robust optimization based automobile noise reduction method |
| CN103606967A (en) * | 2013-11-26 | 2014-02-26 | 华中科技大学 | Dispatching method for achieving robust operation of electrical power system |
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| US6606529B1 (en) * | 2000-06-09 | 2003-08-12 | Frontier Technologies, Inc. | Complex scheduling method and device |
| CN102999678A (en) * | 2012-12-26 | 2013-03-27 | 北京航空航天大学 | Nonlinear multi-target range robust optimization based automobile noise reduction method |
| CN103606967A (en) * | 2013-11-26 | 2014-02-26 | 华中科技大学 | Dispatching method for achieving robust operation of electrical power system |
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| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN104809517A (en) * | 2015-04-16 | 2015-07-29 | 四川大学 | Building construction site security double-layer dynamic optimizing method |
| CN111080020A (en) * | 2019-12-23 | 2020-04-28 | 中山大学 | Robustness evaluation method and device for drilling arrangement scheme |
| CN111080020B (en) * | 2019-12-23 | 2023-03-31 | 中山大学 | Robustness evaluation method and device for drilling arrangement scheme |
| CN112327923A (en) * | 2020-11-19 | 2021-02-05 | 中国地质大学（武汉） | Multi-unmanned aerial vehicle collaborative path planning method |
| CN113627025A (en) * | 2021-08-16 | 2021-11-09 | 湘潭大学 | Optimized design method for sheet forming process |
| CN113627025B (en) * | 2021-08-16 | 2023-05-26 | 湘潭大学 | Optimal design method for sheet forming process |
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Application publication date: 20140910 |