CN105512721A - Multi-target cell difference method - Google Patents
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Abstract
The invention discloses a multi-target difference method, and the method comprises the steps: 1, building a target function, and also building a constraint condition for a multi-target problem containing constraint; 2, carrying out the random initialization of a population; 3, selecting two more excellent individuals from the neighbors of each current individual through a two-element championship contest, enabling the excellent individuals and the current individual to serve as a male parent, carrying out differential variation and crossing operation to obtain a filial generation, and calculating the target function value of the filial generation; 4, selecting the poorest one from the neighbors according to the order and a congestion distance; 5, repeatedly carrying out the steps 3 and 4 till the evolution of the last individual is completed; 6, ordering the individuals of an external file set according to the order and the congestion distance after the evolution of each generation, wherein the individuals with the lower order is placed in front, the individual with the larger congestion distance is placed in front after the individuals with the same order are compared, and the individuals exceeding the population is eliminated; 7, enabling the individuals of the whole external file set to serve as the population of the next evolution, enabling the individuals to be randomly distributed in a two-dimensional grid, and carrying out the operation till the ending condition of evolution is met.
Description
Technical field
The present invention relates to a kind of optimization method of multiple goal cellular difference.
Background technology
A large amount of multi-objective optimization question is there is in Science and engineering field.The essential distinction of multi-objective optimization question and single-object problem is, the former solution is not unique, but the disaggregation of a compromise, i.e. Pareto disaggregation.Tradition multi-objective optimization algorithm comprises weighted method, leash law and Objective Programming etc.These methods are all that multi-objective problem is converted to single-objective problem, and its shortcoming needs sufficient priori, and be difficult to processing target noise, and robustness is poor.Because the objective function of multi-objective optimization question and constraint may be non-linear, non-differentiabilities or discontinuous, traditional mathematic programming methods often efficiency is lower, and they for weighted value or the given order of target more responsive.
Evolvement method is a kind of stochastic global optimization method of simulating nature evolutionary process, and it adopts the solution of the mode search problem of message exchange between collective search and individual in population.Compared with classic method, its advantage is: first, and evolutionary search process has randomness, is not easily absorbed in local optimum; Secondly, it has concurrency, can evolve simultaneously search out multiple solution, is applicable to multi-objective optimization question; 3rd, can process discontinuous, the problems such as non-differentiability and Pareto forward position non-convex, do not need too much priori.
At present, existing two classes are divided into substantially to the research that cellular automaton is applied to genetic algorithm: the evolution rule that first kind cellular automaton enriches carrys out some genetic manipulation in alternative traditional genetic algorithm; Equations of The Second Kind is then based on Complex System Theory, and the individuality in population is assigned in cellular space by it, and the neighbours' structure by cellular Automation Model realizes the evolution that excellent solution spreads and guides other to separate in population.By constantly performing the genetic manipulation of local, finally realize the evolution of whole population.In cellular genetic method, the diffusion of this solution can make solution can keep the well balanced of overall search and local optimal searching well in search procedure, thus effectively avoids being absorbed in local optimum.2007, Alba etc., in the MANET broadcast strategy of Optimizing City, proposed cMOGA.Nebro etc. improve cMOGA, add the feedback mechanism of outside population, propose MOCell.In order to solve three target problems better, MOCell and DE algorithm mixes by Durillo etc., proposes CellDE.Zhang Yi etc. adopt a kind of replacement method of new defect individual, propose DECell.
Existing multiple goal cellular difference method often adopts external archival collection to collect the solution of some non-dominant, then after per generation evolution terminates, is fed back the non-domination solution of some by external archival collection to population.This feedback mechanism is unfavorable for distributing homogeneity and the coverage of Pareto front end, because this strategy does not consider the distribution of Pareto front end from whole population angle.
Summary of the invention
The present invention will overcome the above-mentioned shortcoming of prior art, provides a kind of optimization method of multiple goal cellular difference.
Multiple goal cellular difference method of the present invention, comprises the steps:
1, create multiple objective function, for the multi-objective problem containing constraint, also need to create constraint condition.
2, random initializtion is carried out to population, namely to the decision variable random initializtion of each individuality.Calculate the target function value of each individuality, then by the individual stochastic distribution in population in two-dimensional annular grid, and by current population stored in external archival collection.
3, from the surrounding neighbours of each current individual, select two more outstanding individualities by binary championship, using them and current individual jointly as male parent, then carry out differential variation, interlace operation obtains filial generation, and calculate the target function value of filial generation.
Neighbours' structure of cellular automaton has a lot, adopts Moore type neighbours structure here.In Moore type, adjacent eight cellulars in the upper and lower, left and right of a cellular, upper left, lower-left, upper right, bottom right are the neighbours of this cellular.
If population scale is N, d is the dimension of solution space.X
r1, x
r2, x
r3be three male parents.V
ifor the vector that makes a variation, u
ifor filial generation vector.
A () differential variation operates
v
i.j＝x
r1.j+F·(x
r2.j-x
r3.j)，i∈[1,N]，j∈[1,d]
F is the zoom factor between [0,1].
The interlace operation of (b) difference
Rand
i.jfor equally distributed random number between [0,1], CR is the intersection constant between [0,1], rand
j∈ [1,2 ..., d].
4, according to order and crowding distance, the poorest person in neighbours is selected.Whole evolutionary process is divided into two stages according to the difference of replacement policy.In the first stage, replacement policy I is adopted to evolve.In subordinate phase, replacement policy II is adopted to evolve.
Replacement policy I: if the new individuality produced is not arranged by the poorest individuality in current individual and neighbours, then the poorest person in current individual and neighbours is replaced, and the individuality newly produced is added external archival collection.
Replacement policy II: once the new individuality produced arranges the poorest individuality in current individual or neighbours, then the poorest person in current individual or neighbours is replaced, and current individual is added external archival collection.
5, the step of 3 and 4 is repeated, until complete the evolution of last individuality.
6, per generation evolve terminate after, according to order and crowding distance the individuality to external archival collection sort (order low come before; The individuality that order is identical compares crowding distance, before what crowding distance was large come), and reject the individuality exceeding population scale.
7, the individuality concentrated by whole external archival as the population of evolving next time, and by its stochastic distribution in two-dimensional grid, continues to evolve until meet the end condition of evolving.
Conventional cellular difference method have employed partial feedback mechanism, concrete operating process is such: use external archival collection to collect the non-domination solution in evolutionary process, then, after the present age, evolution terminated, the next individuality substituting random selecting from grid of non-domination solution (being generally 20% of population quantity) of some is chosen from external archival collection.
This patent proposes a kind of new feedback mechanism: before often for evolution, parent population is added external archival concentrate, external archival collection is used to again collect the non-domination solution in evolutionary process simultaneously, then after the present age, evolution terminated, the non-dominant exceeding population scale is rejected individual according to order and crowding distance, using remaining all individualities as follow-on male parent, and they are assigned randomly in two-dimensional annular reticulate texture.
The present invention proposes a kind of new multiple goal cellular difference method, its feedback mechanism and replacement policy have done essence change.New feedback mechanism is from the distribution angle of whole Pareto front end, it enhances the effect in the search of whole population, simultaneously by the individual stochastic distribution of whole population in two-dimensional annular reticulate texture, whole population can be avoided again to be well absorbed in local optimum.It is uncertain that replacement policy I to take into account in Moore type neighbours structure the quality of individuality in whole population of not arranging mutually, therefore takes the collected strategy of filial generation will do not arranged mutually with parent and the poorest neighbours.In order to improve the convergence of solution further, propose replacement policy II.Test shows: new feedback mechanism and new replacement policy improve the front end distributing homogeneity understood, and improve the coverage of front end.
Advantage of the present invention is: improve the front end distributing homogeneity understood, improve the coverage of front end.
Accompanying drawing explanation
Fig. 1 is NCellDE method implementing procedure.
Fig. 2 is Moore type neighbours structures.
Fig. 3 is the Pareto front end that two kinds of methods obtain when solving Osyczka2.
Fig. 4 is the Pareto front end that two kinds of methods obtain when solving DTLZ2.
Fig. 5 is reducer structure schematic diagram.
Fig. 6 is the Pareto front end obtained Gear Reducer Optimal.
Embodiment
Below in conjunction with accompanying drawing, further illustrate the present invention.
The multiple goal cellular difference method (NCellDE) of new feedback mechanism of the present invention and replacement policy.Its flow process as shown in Figure 1.
Concrete steps are as follows:
1, create multiple objective function, for the multi-objective problem containing constraint, also need to create constraint condition.
2, random initializtion is carried out to population, namely to the decision variable random initializtion of each individuality.Calculate the target function value of each individuality, then by the individual stochastic distribution in population in two-dimensional annular grid, and by current population stored in external archival collection.
3, from the surrounding neighbours of each current individual, select two more outstanding individualities by binary championship, using them and current individual jointly as male parent, then carry out differential variation, interlace operation obtains filial generation, and calculate the target function value of filial generation.
Neighbours' structure of cellular automaton has a lot, adopts Moore type neighbours' structure (Fig. 2) here.In Moore type, adjacent eight cellulars in the upper and lower, left and right of a cellular, upper left, lower-left, upper right, bottom right are the neighbours of this cellular.
If population scale is N, d is the dimension of solution space.X
r1, x
r2, x
r3be three male parents.V
ifor the vector that makes a variation, u
ifor filial generation vector.
A () differential variation operates
v
i.j＝x
r1.j+F·(x
r2.j-x
r3.j)，i∈[1,N]，j∈[1,d]
F is the zoom factor between [0,1].
The interlace operation of (b) difference
Rand
i.jfor equally distributed random number between [0,1], CR is the intersection constant between [0,1], rand
j∈ [1,2 ..., d].
4, according to order and crowding distance, the poorest person in neighbours is selected.Whole evolutionary process is divided into two stages according to the difference of replacement policy.In the first stage, replacement policy I is adopted to evolve.In subordinate phase, replacement policy II is adopted to evolve.
Replacement policy I: if the new individuality produced is not arranged by the poorest individuality in current individual and neighbours, then the poorest person in current individual and neighbours is replaced, and the individuality newly produced is added external archival collection.
Replacement policy II: once the new individuality produced arranges the poorest individuality in current individual or neighbours, then the poorest person in current individual or neighbours is replaced, and current individual is added external archival collection.
5, the step of 3 and 4 is repeated, until complete the evolution of last individuality.
6, per generation evolve terminate after, according to order and crowding distance the individuality to external archival collection sort (order low come before; The individuality that order is identical compares crowding distance, before what crowding distance was large come), and reject the individuality exceeding population scale.
7, the individuality concentrated by whole external archival as the population of evolving next time, and by its stochastic distribution in two-dimensional grid, continues to evolve until meet the end condition of evolving.
Performance test
In order to the performance of evaluation algorithms, adopt here and apart from (GD), distribution index (△) and hypervolume index (HV), algorithm is evaluated from generation to generation.GD is less, shows that the convergence of separating is better.△ is less, shows that the distribution of Pareto front end is more even.HV is larger, and illustrate that the diversity of the Pareto front end obtained, convergence are better, namely front end spreadability is better.In order to verify its performance, NCellDE and CellDE is made test comparison.3 unconfined multiple goal trial functions (ZDT2, ZDT3, DTLZ2) and 1 is selected to test these two kinds of methods containing constrained multiple goal trial function (Osyczka2).In testing, if population scale is 100, maximum evolutionary generation 5000 generation, intersection constant CR=0.5, zoom factor F=0.6.In last 1000 generations of NCellDE, adopt strategy II to continue to evolve.The Pareto front end that two kinds of methods obtain when solving Osyczka2, DTLZ2 problem as shown in Figure 3, Figure 4, is shown 1-table 3 simultaneously and be sets forth the statistics of two kinds of algorithms about △, GD, HV tri-kinds of indexs.
The front end that the Pareto front end that Fig. 3 and Fig. 4 reflects NCellDE acquisition intuitively obtains than CellDE is more even.
Table 1 distributing homogeneity index △
Table 2 spreadability index HV
Table 3 convergence index GD
From table 1, table 2, distributing homogeneity, the spreadability index of the Pareto front end that NCellDE obtains are better than CellDE.As shown in Table 3, in ZDT2 and Osyczka2 test, generation distance (GD) of NCellDE is more bigger than CellDE, and in ZDT3 and DTLZ2, NCellDE is better than CellDE.In sum, the solution that this patent is tried to achieve, its combination property is better.
The optimal design of application example-speed reduction unit
1, designing requirement
The optimal design of speed reduction unit is a classical multi-objective optimization question containing constraint.The optimization aim of this problem is the minimum (f of volume of speed reduction unit
1), the minimum (f of stress of axle 1
2).The power input of speed reduction unit is 10kW, and high speed shaft rotating speed is 800r/min, and rotating ratio is 3.The springform of gear material measures 200GPa, and form factor is 2.54, and pressure angle is 20 degree, and the relative facewidth of gear is between 5 ~ 12.The permissible bending stress of tooth root is 22.5MPa, the allowable contact stress 346.6MPa of the flank of tooth.Centre distance between gear is no more than 80cm.The deflection deformation of the first axle 1 and the second axle 2 is no more than 0.001, and the permissible stress of the first axle 1 is 1300MPa, and the permissible stress of the second axle 2 is 1100MPa.As shown in Figure 5, the design variable of this problem is respectively facewidth x
1, module x
2, pinion wheel number of teeth x
3, distance x between clutch shaft bearing 11
4, distance x between the second bearing 22
5, the first axle 1 diameter x
6, the second axle 2 diameter x
7.
The scope of design variable is:
2.6≤x
1≤3.6
0.7≤x
2≤0.8
17≤x
3≤28
7.3≤x
4≤8.3
7.3≤x
5≤8.3
2.9≤x
6≤3.9
5≤x
7≤5.5
2, according to designing requirement, the mathematical model of this problem is set up
s.t.
g
5＝x
2x
3-40≤0,
g
8＝1.9-x
4+1.5x
6≤0,g
9＝1.9-x
5+1.1x
7≤0,
g
10＝f
2(x)≤1300,
G
1: the bending stress constraint condition of gear; g
2: the contact stress constraint condition of gear; g
3, g
4: the amount of deflection constraint of axle; g
5~ g
7: the dimension constraint of gear; g
8, g
9: the physical dimension constraint of axle; g
10, g
11: the permissible stress constraint of axle.
3, NCellDE Optimization Steps
1) create multiple objective function (formula 1), create constraint condition (g
1~ g
11).
2) population scale is set to 100, in each variable range, to the design variable random initializtion of each individuality.Here, the violation degree of constraint condition is converted into a target function value, namely increases a target again.Calculate the target function value of each individuality, then by the individual stochastic distribution in population in two-dimensional annular grid, and by current population stored in external archival collection.
3) from the surrounding neighbours of each current individual, select two more outstanding individualities by binary championship, using them and current individual jointly as male parent, then carry out differential variation, interlace operation obtains filial generation, and calculate the target function value of filial generation.
A () differential variation operates
v
i.j＝x
r1.j+0.6·(x
r2.j-x
r3.j)，i∈[1,100]，j∈[1,7]
Zoom factor F is set to 0.6.
The interlace operation of (b) difference
The constant CR that intersects is set to 0.5.
4) according to order and crowding distance, the poorest person in neighbours is selected.Whole evolutionary process is divided into two stages according to the difference of replacement policy.In the first stage, replacement policy I is adopted to evolve for 2400 generations.In subordinate phase, replacement policy II is adopted to evolve for 800 generations.
Replacement policy in NCellDE is as follows:
Replacement policy I: if the new individuality produced is not arranged by the poorest individuality in current individual and neighbours, then the poorest person in current individual and neighbours is replaced, and the individuality newly produced is added external archival collection.
Replacement policy II: once the new individuality produced arranges the poorest individuality in current individual or neighbours, then the poorest person in current individual or neighbours is replaced, and current individual is added external archival collection.
5) step of 3 and 4 is repeated, until complete the evolution of last individuality.
6) per generation evolve terminate after, according to order and crowding distance the individuality to external archival collection sort (order low come before; The individuality that order is identical compares crowding distance, before what crowding distance was large come), and reject the individuality exceeding population scale, population scale is controlled 100.
7) individuality concentrated by whole external archival as the population of evolving next time, and by its stochastic distribution in two-dimensional grid, continues to evolve until evolved to for 3200 generations.
The Pareto front end that Fig. 6 obtains when giving this problem of Algorithm for Solving.
When method optimizing, by module x
2, pinion wheel number of teeth x
3as continuous variable process.Consider practical application scene, also will choose from the solution obtained that the number of teeth is integer, modulus is the solution of standard module.The occurrence of the design variable obtained is in table 4.
Table 4 design parameter and target function value (having 100 groups)
4, according to the result of trying to achieve optimization solution, in conjunction with the design object of deviser, corresponding optimization solution is selected.
If deviser wishes that the volume of speed reduction unit is minimum, be not be concerned about very much to the stress of axle, then can choose the solution that sequence number is 75; If deviser wishes that the volume of speed reduction unit is as far as possible little, wish that again the stress of axle is also as far as possible little simultaneously, then can choose the solution that sequence number is 8.
Innovative point
Propose a kind of new multiple goal cellular difference method, its feedback mechanism and replacement policy have done essence change.New feedback mechanism is from the distribution angle of whole Pareto front end, it enhances the effect in the search of whole population, simultaneously by the individual stochastic distribution of whole population in two-dimensional annular reticulate texture, whole population can be avoided again to be well absorbed in local optimum.It is uncertain that replacement policy I to take into account in Moore type neighbours structure the quality of individuality in whole population of not arranging mutually, therefore takes the collected strategy of filial generation will do not arranged mutually with parent and the poorest neighbours.In order to improve the convergence of solution further, propose replacement policy II.Test shows: new feedback mechanism and new replacement policy improve the front end distributing homogeneity understood, and improve the coverage of front end.
Claims (2)
1. a multiple goal cellular difference method, comprises the steps:
Step 1, establishment multiple objective function, for the multi-objective problem containing constraint, also need to create constraint condition;
Step 2, random initializtion is carried out to population, namely to the decision variable random initializtion of each individuality; Calculate the target function value of each individuality, then by the individual stochastic distribution in population in two-dimensional annular grid, and by current population stored in external archival collection;
Step 3, from the surrounding neighbours of each current individual, select two more outstanding individualities by binary championship, using them and current individual jointly as male parent, then carry out differential variation, interlace operation obtains filial generation, and calculate the target function value of filial generation;
Neighbours' structure of cellular automaton has a lot, adopts Moore type neighbours structure here; In Moore type, adjacent eight cellulars in the upper and lower, left and right of a cellular, upper left, lower-left, upper right, bottom right are the neighbours of this cellular;
If population scale is N, d is the dimension of solution space; x
r1, x
r2, x
r3be three male parent vectors; v
ifor the vector that makes a variation, u
ifor filial generation vector;
A () differential variation operates
v
i.j＝x
r1.j+F·(x
r2.j-x
r3.j)，i∈[1,N]，j∈[1,d]
F is the zoom factor between [0,1], v
i.jfor variation vector v
iin a jth element, x
r1.jfor male parent vector x
r1in a jth element, other are analogized.
The interlace operation of (b) difference
Rand
i.jfor equally distributed random number between [0,1], CR is the intersection constant between [0,1], rand
j∈ [1,2 ..., d], u
i.jfor filial generation vector u
iin a jth element.
Step 4, according to order and crowding distance, select the poorest person in neighbours; Whole evolutionary process is divided into two stages according to the difference of replacement policy; In the first stage, replacement policy I is adopted to evolve; In subordinate phase, replacement policy II is adopted to evolve;
Replacement policy I: if the new individuality produced is not arranged by the poorest individuality in current individual and neighbours, then the poorest person in current individual and neighbours is replaced, and the individuality newly produced is added external archival collection;
Replacement policy II: once the new individuality produced arranges the poorest individuality in current individual or neighbours, then the poorest person in current individual or neighbours is replaced, and current individual is added external archival collection;
The step of step 5, repetition 3 and 4, until complete the evolution of last individuality;
Step 6, after every generation evolves and terminates, according to order and crowding distance, the individuality to external archival collection sorts, before what order was low come; The individuality that order is identical compares crowding distance, before what crowding distance was large come, and rejects the individuality exceeding population scale;
Step 7, the individuality concentrated by whole external archival as the population of evolving next time, and by its stochastic distribution in two-dimensional grid, continue to evolve until meet the end condition of evolving.
2. apply a reductor Optimization Design for multiple goal cellular difference method, comprise the steps:
Step 21. determines designing requirement;
The optimal design of speed reduction unit is a classical multi-objective optimization question containing constraint; The optimization aim of this problem is the minimum (f of volume of speed reduction unit
1), the minimum (f of stress of axle 1
2); The power input of speed reduction unit is 10kW, and high speed shaft rotating speed is 800r/min, and rotating ratio is 3; The springform of gear material measures 200GPa, and form factor is 2.54, and pressure angle is 20 degree, and the relative facewidth of gear is between 5 ~ 12; The permissible bending stress of tooth root is 22.5MPa, the allowable contact stress 346.6MPa of the flank of tooth; Centre distance between gear is no more than 80cm; The deflection deformation of the first axle (1) and the second axle (2) is no more than 0.001, and the permissible stress of the first axle (1) is 1300MPa, and the permissible stress of the second axle (2) is 1100MPa; The design variable of this problem is respectively facewidth x
1, module x
2, pinion wheel number of teeth x
3, distance x between clutch shaft bearing (11)
4, distance x between the second bearing (22)
5, the first axle (1) diameter x
6, the second axle (2) diameter x
7;
The scope of design variable is:
2.6≤x
1≤3.6
0.7≤x
2≤0.8
17≤x
3≤28
7.3≤x
4≤8.3
7.3≤x
5≤8.3
2.9≤x
6≤3.9
5≤x
7≤5.5
Step 22, the designing requirement determined according to step 1, set up the mathematical model of this problem;
F
volume, f
1x volume (being jointly made up of the volume of axle and the volume of gear) that () is reducer casing, f
stress, f
2x () is the stress of axle 1.
s.t.
G
1: the bending stress constraint condition of gear; g
2: the contact stress constraint condition of gear; g
3, g
4: the amount of deflection constraint of axle; g
5~ g
7: the dimension constraint of gear; g
8, g
9: the physical dimension constraint of axle; g
10, g
11: the permissible stress constraint of axle;
Step 23. implements NCellDE Optimization Steps;
1) create multiple objective function (formula 1), create constraint condition (g
1~ g
11);
2) population scale is set to 100, in each variable range, to the design variable random initializtion of each individuality; Here, the violation degree of constraint condition is converted into a target function value, namely increases a target again; Calculate the target function value of each individuality, then by the individual stochastic distribution in population in two-dimensional annular grid, and by current population stored in external archival collection;
3) from the surrounding neighbours of each current individual, select two more outstanding individualities by binary championship, using them and current individual jointly as male parent, then carry out differential variation, interlace operation obtains filial generation, and calculate the target function value of filial generation;
A () differential variation operates
v
i.j＝x
r1.j+0.6·(x
r2.j-x
r3.j)，i∈[1,100]，j∈[1,7]
Zoom factor F is set to 0.6;
The interlace operation of (b) difference
The constant CR that intersects is set to 0.5;
4) according to order and crowding distance, the poorest person in neighbours is selected; Whole evolutionary process is divided into two stages according to the difference of replacement policy; In the first stage, replacement policy I is adopted to evolve for 2400 generations; In subordinate phase, replacement policy II is adopted to evolve for 800 generations;
Replacement policy in NCellDE is as follows:
Replacement policy I: if the new individuality produced is not arranged by the poorest individuality in current individual and neighbours, then the poorest person in current individual and neighbours is replaced, and the individuality newly produced is added external archival collection;
Replacement policy II: once the new individuality produced arranges the poorest individuality in current individual or neighbours, then the poorest person in current individual or neighbours is replaced, and current individual is added external archival collection;
5) step of 3 and 4 is repeated, until complete the evolution of last individuality;
6) per generation evolve terminate after, according to order and crowding distance the individuality to external archival collection sort (order low come before; The individuality that order is identical compares crowding distance, before what crowding distance was large come), and reject the individuality exceeding population scale, population scale is controlled 100;
7) individuality concentrated by whole external archival as the population of evolving next time, and by its stochastic distribution in two-dimensional grid, continues to evolve until evolved to for 3200 generations;
When method optimizing, by module x
2, pinion wheel number of teeth x
3as continuous variable process; Consider practical application scene, also will choose from the solution obtained that the number of teeth is integer, modulus is the solution of standard module; The occurrence of the design variable obtained is in table 4;
Table 4 design parameter and target function value (having 100 groups)
Step 24, basis try to achieve the result of optimization solution, in conjunction with the design object of deviser, select corresponding optimization solution;
If deviser wishes that the volume of speed reduction unit is minimum, be not be concerned about very much to the stress of axle, then can choose the solution that sequence number is 75; If deviser wishes that the volume of speed reduction unit is as far as possible little, wish that again the stress of axle is also as far as possible little simultaneously, then can choose the solution that sequence number is 8.
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